Ross_9e_FCF_sml

Solutions ManualFundamentals of Corporate Finance 9 th edition Ross, Westerfield, and Jordan Updated 12-20-2008 CHAPTER 1 INTRODUCTION TO CORPORATE FINANCE Answers to Concepts Review and Critical Thinking Questions 1. Capital budgeting (deciding hether to e!pand a "anufacturing plant#, capital structure (deciding hether to issue ne e$uit% and use the proceeds to retire outstanding debt#, and or&ing capital "anage"ent ("odif%ing the fir"'s credit collection polic% ith its custo"ers#( 2. )isad*antages+ unli"ited liabilit%, li"ited life, difficult% in transferring onership, hard to raise capital funds( ,o"e ad*antages+ si"pler, less regulation, the oners are also the "anagers, so"eti"es personal ta! rates are better than corporate ta! rates( 3. -he pri"ar% disad*antage of the corporate for" is the double ta!ation to shareholders of distributed earnings and di*idends( ,o"e ad*antages include+ li"ited liabilit%, ease of transferabilit%, abilit% to raise capital, and unli"ited life( 4. .n response to ,arbanes-/!le%, s"all fir"s ha*e elected to go dar& because of the costs of co"pliance( -he costs to co"pl% ith ,arbo! can be se*eral "illion dollars, hich can be a large percentage of a s"all fir"s profits( 0 "a1or cost of going dar& is less access to capital( ,ince the fir" is no longer publicl% traded, it can no longer raise "one% in the public "ar&et( 0lthough the co"pan% ill still ha*e access to ban& loans and the pri*ate e$uit% "ar&et, the costs associated ith raising funds in these "ar&ets are usuall% higher than the costs of raising funds in the public "ar&et( . -he treasurer's office and the controller's office are the to pri"ar% organi2ational groups that report directl% to the chief financial officer( -he controller's office handles cost and financial accounting, ta! "anage"ent, and "anage"ent infor"ation s%ste"s, hile the treasurer's office is responsible for cash and credit "anage"ent, capital budgeting, and financial planning( -herefore, the stud% of corporate finance is concentrated ithin the treasur% group's functions( !. -o "a!i"i2e the current "ar&et *alue (share price# of the e$uit% of the fir" (hether it's publicl%- traded or not#( ". .n the corporate for" of onership, the shareholders are the oners of the fir"( -he shareholders elect the directors of the corporation, ho in turn appoint the fir"'s "anage"ent( -his separation of onership fro" control in the corporate for" of organi2ation is hat causes agenc% proble"s to e!ist( 3anage"ent "a% act in its on or so"eone else's best interests, rather than those of the shareholders( .f such e*ents occur, the% "a% contradict the goal of "a!i"i2ing the share price of the e$uit% of the fir"( #. 0 pri"ar% "ar&et transaction( B-2 SOLUTIONS $. .n auction "ar&ets li&e the 45,6, bro&ers and agents "eet at a ph%sical location (the e!change# to "atch bu%ers and sellers of assets( )ealer "ar&ets li&e 40,)07 consist of dealers operating at dispersed locales ho bu% and sell assets the"sel*es, co""unicating ith other dealers either electronicall% or literall% o*er-the-counter( 1%. ,uch organi2ations fre$uentl% pursue social or political "issions, so "an% different goals are concei*able( /ne goal that is often cited is re*enue "ini"i2ation8 i(e(, pro*ide hate*er goods and ser*ices are offered at the loest possible cost to societ%( 0 better approach "ight be to obser*e that e*en a not-for-profit business has e$uit%( -hus, one anser is that the appropriate goal is to "a!i"i2e the *alue of the e$uit%( 11. 9resu"abl%, the current stoc& *alue reflects the ris&, ti"ing, and "agnitude of all future cash flos, both short-ter" and long-ter"( .f this is correct, then the state"ent is false( 12. 0n argu"ent can be "ade either a%( 0t the one e!tre"e, e could argue that in a "ar&et econo"%, all of these things are priced( -here is thus an opti"al le*el of, for e!a"ple, ethical and:or illegal beha*ior, and the fra"eor& of stoc& *aluation e!plicitl% includes these( 0t the other e!tre"e, e could argue that these are non-econo"ic pheno"ena and are best handled through the political process( 0 classic (and highl% rele*ant# thought $uestion that illustrates this debate goes so"ething li&e this+ ;0 fir" has esti"ated that the cost of i"pro*ing the safet% of one of its products is <=0 "illion( >oe*er, the fir" belie*es that i"pro*ing the safet% of the product ill onl% sa*e <20 "illion in product liabilit% clai"s( What should the fir" do?@ 13. -he goal ill be the sa"e, but the best course of action toard that goal "a% be different because of differing social, political, and econo"ic institutions( 14. -he goal of "anage"ent should be to "a!i"i2e the share price for the current shareholders( .f "anage"ent belie*es that it can i"pro*e the profitabilit% of the fir" so that the share price ill e!ceed <=A, then the% should fight the offer fro" the outside co"pan%( .f "anage"ent belie*es that this bidder or other unidentified bidders ill actuall% pa% "ore than <=A per share to ac$uire the co"pan%, then the% should still fight the offer( >oe*er, if the current "anage"ent cannot increase the *alue of the fir" be%ond the bid price, and no other higher bids co"e in, then "anage"ent is not acting in the interests of the shareholders b% fighting the offer( ,ince current "anagers often lose their 1obs hen the corporation is ac$uired, poorl% "onitored "anagers ha*e an incenti*e to fight corporate ta&eo*ers in situations such as this( 1. We ould e!pect agenc% proble"s to be less se*ere in countries ith a relati*el% s"all percentage of indi*idual onership( Beer indi*idual oners should reduce the nu"ber of di*erse opinions concerning corporate goals( -he high percentage of institutional onership "ight lead to a higher degree of agree"ent beteen oners and "anagers on decisions concerning ris&% pro1ects( .n addition, institutions "a% be better able to i"ple"ent effecti*e "onitoring "echanis"s on "anagers than can indi*idual oners, based on the institutions' deeper resources and e!periences ith their on "anage"ent( -he increase in institutional onership of stoc& in the United ,tates and the groing acti*is" of these large shareholder groups "a% lead to a reduction in agenc% proble"s for U(,( corporations and a "ore efficient "ar&et for corporate control( CHAPTER 1 B-3 1!. >o "uch is too "uch? Who is orth "ore, Ra% .rani or -iger Woods? -he si"plest anser is that there is a "ar&et for e!ecuti*es 1ust as there is for all t%pes of labor( 6!ecuti*e co"pensation is the price that clears the "ar&et( -he sa"e is true for athletes and perfor"ers( >a*ing said that, one aspect of e!ecuti*e co"pensation deser*es co""ent( 0 pri"ar% reason e!ecuti*e co"pensation has gron so dra"aticall% is that co"panies ha*e increasingl% "o*ed to stoc&-based co"pensation( ,uch "o*e"ent is ob*iousl% consistent ith the atte"pt to better align stoc&holder and "anage"ent interests( .n recent %ears, stoc& prices ha*e soared, so "anage"ent has cleaned up( .t is so"eti"es argued that "uch of this reard is si"pl% due to rising stoc& prices in general, not "anagerial perfor"ance( 9erhaps in the future, e!ecuti*e co"pensation ill be designed to reard onl% differential perfor"ance, i(e(, stoc& price increases in e!cess of general "ar&et increases( CHAPTER 2 FINANCIAL STATEMENTS, TAXES AND CASH FLOW Answers to Concepts Review and Critical Thinking Questions 1. Ci$uidit% "easures ho $uic&l% and easil% an asset can be con*erted to cash ithout significant loss in *alue( .t's desirable for fir"s to ha*e high li$uidit% so that the% ha*e a large factor of safet% in "eeting short-ter" creditor de"ands( >oe*er, since li$uidit% also has an opportunit% cost associated ith itDna"el% that higher returns can generall% be found b% in*esting the cash into producti*e assetsDlo li$uidit% le*els are also desirable to the fir"( .t's up to the fir"'s financial "anage"ent staff to find a reasonable co"pro"ise beteen these opposing needs( 2. -he recognition and "atching principles in financial accounting call for re*enues, and the costs associated ith producing those re*enues, to be ;boo&ed@ hen the re*enue process is essentiall% co"plete, not necessaril% hen the cash is collected or bills are paid( 4ote that this a% is not necessaril% correct8 it's the a% accountants ha*e chosen to do it( 3. >istorical costs can be ob1ecti*el% and precisel% "easured hereas "ar&et *alues can be difficult to esti"ate, and different anal%sts ould co"e up ith different nu"bers( -hus, there is a tradeoff beteen rele*ance ("ar&et *alues# and ob1ecti*it% (boo& *alues#( 4. )epreciation is a non-cash deduction that reflects ad1ust"ents "ade in asset boo& *alues in accordance ith the "atching principle in financial accounting( .nterest e!pense is a cash outla%, but it's a financing cost, not an operating cost( . 3ar&et *alues can ne*er be negati*e( ."agine a share of stoc& selling for E<20( -his ould "ean that if %ou placed an order for 100 shares, %ou ould get the stoc& along ith a chec& for <2,000( >o "an% shares do %ou ant to bu%? 3ore generall%, because of corporate and indi*idual ban&ruptc% las, net orth for a person or a corporation cannot be negati*e, i"pl%ing that liabilities cannot e!ceed assets in "ar&et *alue( !. Bor a successful co"pan% that is rapidl% e!panding, for e!a"ple, capital outla%s ill be large, possibl% leading to negati*e cash flo fro" assets( .n general, hat "atters is hether the "one% is spent isel%, not hether cash flo fro" assets is positi*e or negati*e( ". .t's probabl% not a good sign for an established co"pan%, but it ould be fairl% ordinar% for a start- up, so it depends( #. Bor e!a"ple, if a co"pan% ere to beco"e "ore efficient in in*entor% "anage"ent, the a"ount of in*entor% needed ould decline( -he sa"e "ight be true if it beco"es better at collecting its recei*ables( .n general, an%thing that leads to a decline in ending 4WC relati*e to beginning ould CHAPTER 2 B-5 ha*e this effect( 4egati*e net capital spending ould "ean "ore long-li*ed assets ere li$uidated than purchased( B-6 SOLUTIONS $. .f a co"pan% raises "ore "one% fro" selling stoc& than it pa%s in di*idends in a particular period, its cash flo to stoc&holders ill be negati*e( .f a co"pan% borros "ore than it pa%s in interest, its cash flo to creditors ill be negati*e( 1%. -he ad1ust"ents discussed ere purel% accounting changes8 the% had no cash flo or "ar&et *alue conse$uences unless the ne accounting infor"ation caused stoc&holders to re*alue the deri*ati*es( 11. 6nterprise *alue is the theoretical ta&eo*er price( .n the e*ent of a ta&eo*er, an ac$uirer ould ha*e to ta&e on the co"pan%Fs debt, but ould poc&et its cash( 6nterprise *alue differs significantl% fro" si"ple "ar&et capitali2ation in se*eral a%s, and it "a% be a "ore accurate representation of a fir"Fs *alue( .n a ta&eo*er, the *alue of a fir"Fs debt ould need to be paid b% the bu%er hen ta&ing o*er a co"pan%( -his enterprise *alue pro*ides a "uch "ore accurate ta&eo*er *aluation because it includes debt in its *alue calculation( 12. .n general, it appears that in*estors prefer co"panies that ha*e a stead% earnings strea"( .f true, this encourages co"panies to "anage earnings( Under G009, there are nu"erous choices for the a% a co"pan% reports its financial state"ents( 0lthough not the reason for the choices under G009, one outco"e is the abilit% of a co"pan% to "anage earnings, hich is not an ethical decision( 6*en though earnings and cash flo are often related, earnings "anage"ent should ha*e little effect on cash flo (e!cept for ta! i"plications#( .f the "ar&et is ;fooled@ and prefers stead% earnings, shareholder ealth can be increased, at least te"poraril%( >oe*er, gi*en the $uestionable ethics of this practice, the co"pan% (and shareholders# ill lose *alue if the practice is disco*ered( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -o find oner's e$uit%, e "ust construct a balance sheet as follos+ Halance ,heet C0 <A,100 CC <I,=00 4B0 2=,800 C-) J,I00 /6 ?? -0 <28,900 -C K /6 <28,900 We &no that total liabilities and oner's e$uit% (-C K /6# "ust e$ual total assets of <28,900( We also &no that -C K /6 is e$ual to current liabilities plus long-ter" debt plus oner's e$uit%, so oner's e$uit% is+ /6 L <28,900 E J,I00 E I,=00 L <1J,200 4WC L C0 E CC L <A,100 E I,=00 L <800 CHAPTER 2 B-7 2. -he inco"e state"ent for the co"pan% is+ .nco"e ,tate"ent ,ales <A8M,000 Costs 2IJ,000 )epreciation I=,000 6H.- <29M,000 .nterest =2,000 6H- <2MI,000 -a!es(=AN# 92,I00 4et inco"e <1J1,M00 3. /ne e$uation for net inco"e is+ 4et inco"e L )i*idends O 0ddition to retained earnings Rearranging, e get+ 0ddition to retained earnings L 4et inco"e E )i*idends L <1J1,M00 E J=,000 L <98,M00 4. 69, L 4et inco"e : ,hares L <1J1,M00 : 8A,000 L <2(02 per share )9, L )i*idends : ,hares L <J=,000 : 8A,000 L <0(8M per share . -o find the boo& *alue of current assets, e use+ 4WC L C0 E CC( Rearranging to sol*e for current assets, e get+ C0 L 4WC O CC L <=80,000 O 1,I00,000 L <1,I80,000 -he "ar&et *alue of current assets and fi!ed assets is gi*en, so+ Hoo& *alue C0 L <1,I80,000 3ar&et *alue C0 L <1,M00,000 Hoo& *alue 4B0 L <=,J00,000 3ar&et *alue 4B0 L <I,900,000 Hoo& *alue assets L <A,180,000 3ar&et *alue assets L <M,A00,000 !. -a!es L 0(1A(<A0P# O 0(2A(<2AP# O 0(=I(<2AP# O 0(=9(<2=MP E 100P# L <JA,290 ". -he a*erage ta! rate is the total ta! paid di*ided b% net inco"e, so+ 0*erage ta! rate L <JA,290 : <2=M,000 L =1(90N -he "arginal ta! rate is the ta! rate on the ne!t <1 of earnings, so the "arginal ta! rate L =9N( B-8 SOLUTIONS #. -o calculate /CB, e first need the inco"e state"ent+ .nco"e ,tate"ent ,ales <2J,A00 Costs 1=,280 )epreciation 2,=00 6H.- <11,920 .nterest 1,10A -a!able inco"e <10,81A -a!es (=AN# =,J8A 4et inco"e < J,0=0 /CB L 6H.- O )epreciation E -a!es L <11,920 O 2,=00 E =,J8A L <10,I=A $. 4et capital spending L 4B0end E 4B0beg O )epreciation 4et capital spending L <I,200,000 E =,I00,000 O =8A,000 4et capital spending L <1,18A,000 1%. Change in 4WC L 4WCend E 4WCbeg Change in 4WC L (C0end E CCend# E (C0beg E CCbeg# Change in 4WC L (<2,2A0 E 1,J10# E (<2,100 E 1,=80# Change in 4WC L <AI0 E J20 L E<180 11. Cash flo to creditors L .nterest paid E 4et ne borroing Cash flo to creditors L .nterest paid E (C-)end E C-)beg# Cash flo to creditors L <1J0,000 E (<2,900,000 E 2,M00,000# Cash flo to creditors L E<1=0,000 12. Cash flo to stoc&holders L )i*idends paid E 4et ne e$uit% Cash flo to stoc&holders L )i*idends paid E Q(Co""onend O 09.,end# E (Co""onbeg O 09.,beg#R Cash flo to stoc&holders L <I90,000 E Q(<81A,000 O A,A00,000# E (<JI0,000 O A,200,000#R Cash flo to stoc&holders L <11A,000 4ote, 09., is the additional paid-in surplus( 13. Cash flo fro" assets L Cash flo to creditors O Cash flo to stoc&holders L E<1=0,000 O 11A,000 L E<1A,000 Cash flo fro" assets L E<1A,000 L /CB E Change in 4WC E 4et capital spending L E<1A,000 L /CB E (E<8A,000# E 9I0,000 /perating cash flo L E<1A,000 E 8A,000 O 9I0,000 /perating cash flo L <8I0,000 CHAPTER 2 B-9 &ntermediate 14. -o find the /CB, e first calculate net inco"e( .nco"e ,tate"ent ,ales <19M,000 Costs 10I,000 /ther e!penses M,800 )epreciation 9,100 6H.- <JM,100 .nterest 1I,800 -a!able inco"e <M1,=00 -a!es 21,IAA 4et inco"e <=9,8IA )i*idends <10,I00 0dditions to R6 <29,IIA a. /CB L 6H.- O )epreciation E -a!es L <JM,100 O 9,100 E 21,IAA L <M=,JIA b. CBC L .nterest E 4et ne C-) L <1I,800 E (EJ,=00# L <22,100 4ote that the net ne long-ter" debt is negati*e because the co"pan% repaid part of its long- ter" debt( c. CB, L )i*idends E 4et ne e$uit% L <10,I00 E A,J00 L <I,J00 d. We &no that CB0 L CBC O CB,, so+ CB0 L <22,100 O I,J00 L <2M,800 CB0 is also e$ual to /CB E 4et capital spending E Change in 4WC( We alread% &no /CB( 4et capital spending is e$ual to+ 4et capital spending L .ncrease in 4B0 O )epreciation L <2J,000 O 9,100 L <=M,100 4o e can use+ CB0 L /CB E 4et capital spending E Change in 4WC <2M,800 L <M=,JIA E =M,100 E Change in 4WC ,ol*ing for the change in 4WC gi*es <8IA, "eaning the co"pan% increased its 4WC b% <8IA( 1. -he solution to this $uestion or&s the inco"e state"ent bac&ards( ,tarting at the botto"+ 4et inco"e L )i*idends O 0ddition to ret( earnings L <1,A00 O A,100 L <M,M00 B-10 SOLUTIONS 4o, loo&ing at the inco"e state"ent+ 6H- E 6H- S -a! rate L 4et inco"e Recogni2e that 6H- S -a! rate is si"pl% the calculation for ta!es( ,ol*ing this for 6H- %ields+ 6H- L 4. : (1E ta! rate# L <M,M00 : (1 E 0(=A# L <10,1AI 4o %ou can calculate+ 6H.- L 6H- O .nterest L <10,1AI O I,A00 L <1I,MAI -he last step is to use+ 6H.- L ,ales E Costs E )epreciation <1I,MAI L <I1,000 E 19,A00 E )epreciation ,ol*ing for depreciation, e find that depreciation L <M,8IM 1!. -he balance sheet for the co"pan% loo&s li&e this+ Halance ,heet Cash <19A,000 0ccounts pa%able <I0A,000 0ccounts recei*able 1=J,000 4otes pa%able 1M0,000 .n*entor% 2MI,000 Current liabilities <AMA,000 Current assets <A9M,000 Cong-ter" debt 1,19A,=00 -otal liabilities <1,JM0,=00 -angible net fi!ed assets 2,800,000 .ntangible net fi!ed assets J80,000 Co""on stoc& ?? 0ccu"ulated ret( earnings 1,9=I,000 -otal assets <I,1JM,000 -otal liab( K oners' e$uit% <I,1JM,000 -otal liabilities and oners' e$uit% is+ -C K /6 L CC O C-) O Co""on stoc& O Retained earnings ,ol*ing for this e$uation for e$uit% gi*es us+ Co""on stoc& L <I,1JM,000 E 1,9=I,000 E 1,JM0,=00 L <I81,J00 1". -he "ar&et *alue of shareholders' e$uit% cannot be negati*e( 0 negati*e "ar&et *alue in this case ould i"pl% that the co"pan% ould pa% %ou to on the stoc&( -he "ar&et *alue of shareholders' e$uit% can be stated as+ ,hareholders' e$uit% L 3a! Q(-0 E -C#, 0R( ,o, if -0 is <8,I00, e$uit% is e$ual to <1,100, and if -0 is <M,J00, e$uit% is e$ual to <0( We should note here that the boo& *alue of shareholders' e$uit% can be negati*e( CHAPTER 2 B-11 1#. a. -a!es Groth L 0(1A(<A0,000# O 0(2A(<2A,000# O 0(=I(<1=,000# L <18,1J0 -a!es .nco"e L 0(1A(<A0,000# O 0(2A(<2A,000# O 0(=I(<2A,000# O 0(=9(<2=A,000# O 0(=I(<8,IMA,000# L <2,992,000 b. 6ach fir" has a "arginal ta! rate of =IN on the ne!t <10,000 of ta!able inco"e, despite their different a*erage ta! rates, so both fir"s ill pa% an additional <=,I00 in ta!es( 1$. .nco"e ,tate"ent ,ales <J=0,000 C/G, A80,000 0K, e!penses 10A,000 )epreciation 1=A,000 6H.- E<90,000 .nterest JA,000 -a!able inco"e E<1MA,000 -a!es (=AN# 0 a. 4et inco"e E<1MA,000 b. /CB L 6H.- O )epreciation E -a!es L E<90,000 O 1=A,000 E 0 L <IA,000 c. 4et inco"e as negati*e because of the ta! deductibilit% of depreciation and interest e!pense( >oe*er, the actual cash flo fro" operations as positi*e because depreciation is a non-cash e!pense and interest is a financing e!pense, not an operating e!pense( 2%. 0 fir" can still pa% out di*idends if net inco"e is negati*e8 it 1ust has to be sure there is sufficient cash flo to "a&e the di*idend pa%"ents( Change in 4WC L 4et capital spending L 4et ne e$uit% L 0( (Gi*en# Cash flo fro" assets L /CB E Change in 4WC E 4et capital spending Cash flo fro" assets L <IA,000 E 0 E 0 L <IA,000 Cash flo to stoc&holders L )i*idends E 4et ne e$uit% L <2A,000 E 0 L <2A,000 Cash flo to creditors L Cash flo fro" assets E Cash flo to stoc&holders Cash flo to creditors L <IA,000 E 2A,000 L <20,000 Cash flo to creditors L .nterest E 4et ne C-) 4et ne C-) L .nterest E Cash flo to creditors L <JA,000 E 20,000 L <AA,000 21. a. .nco"e ,tate"ent ,ales <22,800 Cost of goods sold 1M,0A0 )epreciation I,0A0 6H.- < 2,J00 .nterest 1,8=0 -a!able inco"e < 8J0 -a!es (=IN# 29M 4et inco"e < AJI b. /CB L 6H.- O )epreciation E -a!es L <2,J00 O I,0A0 E 29M L <M,IAI B-12 SOLUTIONS c. Change in 4WC L 4WCend E 4WCbeg L (C0end E CCend# E (C0beg E CCbeg# L (<A,9=0 E =,1A0# E (<I,800 E 2,J00# L <2,J80 E 2,100 L <M80 4et capital spending L 4B0end E 4B0beg O )epreciation L <1M,800 E 1=,MA0 O I,0A0 L <J,200 CB0 L /CB E Change in 4WC E 4et capital spending L <M,IAI E M80 E J,200 L E<1,I2M -he cash flo fro" assets can be positi*e or negati*e, since it represents hether the fir" raised funds or distributed funds on a net basis( .n this proble", e*en though net inco"e and /CB are positi*e, the fir" in*ested hea*il% in both fi!ed assets and net or&ing capital8 it had to raise a net <1,I2M in funds fro" its stoc&holders and creditors to "a&e these in*est"ents( d. Cash flo to creditors L .nterest E 4et ne C-) L <1,8=0 E 0 L <1,8=0 Cash flo to stoc&holders L Cash flo fro" assets E Cash flo to creditors L E<1,I2M E 1,8=0 L E<=,2AM We can also calculate the cash flo to stoc&holders as+ Cash flo to stoc&holders L )i*idends E 4et ne e$uit% ,ol*ing for net ne e$uit%, e get+ 4et ne e$uit% L <1,=00 E (E=,2AM# L <I,AAM -he fir" had positi*e earnings in an accounting sense (4. T 0# and had positi*e cash flo fro" operations( -he fir" in*ested <M80 in ne net or&ing capital and <J,200 in ne fi!ed assets( -he fir" had to raise <1,I2M fro" its sta&eholders to support this ne in*est"ent( .t acco"plished this b% raising <I,AAM in the for" of ne e$uit%( 0fter pa%ing out <1,=00 of this in the for" of di*idends to shareholders and <1,8=0 in the for" of interest to creditors, <1,I2M as left to "eet the fir"'s cash flo needs for in*est"ent( 22. a. -otal assets 2008 L <MA= O 2,M91 L <=,=II -otal liabilities 2008 L <2M1 O 1,I22 L <1,M8= /ners' e$uit% 2008 L <=,=II E 1,M8= L <1,MM1 -otal assets 2009 L <J0J O =,2I0 L <=,9IJ -otal liabilities 2009 L <29= O 1,A12 L <1,80A /ners' e$uit% 2009 L <=,9IJ E 1,80A L <2,1I2 b. 4WC 2008 L C008 E CC08 L <MA= E 2M1 L <=92 4WC 2009 L C009 E CC09 L <J0J E 29= L <I1I Change in 4WC L 4WC09 E 4WC08 L <I1I E =92 L <22 CHAPTER 2 B-13 c. We can calculate net capital spending as+ 4et capital spending L 4et fi!ed assets 2009 E 4et fi!ed assets 2008 O )epreciation 4et capital spending L <=,2I0 E 2,M91 O J=8 L <1,28J ,o, the co"pan% had a net capital spending cash flo of <1,28J( We also &no that net capital spending is+ 4et capital spending L Bi!ed assets bought E Bi!ed assets sold <1,28J L <1,=A0 E Bi!ed assets sold Bi!ed assets sold L <1,=A0 E 1,28J L <M= -o calculate the cash flo fro" assets, e "ust first calculate the operating cash flo( -he inco"e state"ent is+ &ncome 'tatement ,ales < 8,280(00 Costs =,8M1(00 )epreciation e!pense J=8 (00 6H.- <=,M81(00 .nterest e!pense 211 (00 6H- <=,IJ0(00 -a!es (=AN# 1,21A(A0 4et inco"e <2,2AM(A0 ,o, the operating cash flo is+ /CB L 6H.- O )epreciation E -a!es L <=,M81 O J=8 E 1,21I(A0 L <=,20I(A0 0nd the cash flo fro" assets is+ Cash flo fro" assets L /CB E Change in 4WC E 4et capital spending( L <=,20I(A0 E 22 E 1,28J L <1,89A(A0 d. 4et ne borroing L C-)09 E C-)08 L <1,A12 E 1,I22 L <90 Cash flo to creditors L .nterest E 4et ne C-) L <211 E 90 L <121 4et ne borroing L <90 L )ebt issued E )ebt retired )ebt retired L <2J0 E 90 L <180 Challenge 23. 4et capital spending L 4B0end E 4B0beg O )epreciation L (4B0end E 4B0beg# O ()epreciation O 0)beg# E 0)beg L (4B0end E 4B0beg#O 0)end E 0)beg L (4B0end O 0)end# E (4B0beg O 0)beg# L B0end E B0beg B-14 SOLUTIONS 24. a. -he ta! bubble causes a*erage ta! rates to catch up to "arginal ta! rates, thus eli"inating the ta! ad*antage of lo "arginal rates for high inco"e corporations( b. -a!es L 0(1A(<A0,000# O 0(2A(<2A,000# O 0(=I(<2A,000# O 0(=9(<2=A,000# L <11=,900 0*erage ta! rate L <11=,900 : <==A,000 L =IN -he "arginal ta! rate on the ne!t dollar of inco"e is =I percent( Bor corporate ta!able inco"e le*els of <==A,000 to <10 "illion, a*erage ta! rates are e$ual to "arginal ta! rates( -a!es L 0(=I(<10,000,000# O 0(=A(<A,000,000# O 0(=8(<=,===,===#L <M,I1M,MMJ 0*erage ta! rate L <M,I1M,MMJ : <18,===,==I L =AN -he "arginal ta! rate on the ne!t dollar of inco"e is =A percent( Bor corporate ta!able inco"e le*els o*er <18,===,==I, a*erage ta! rates are again e$ual to "arginal ta! rates( c. -a!es L 0(=I(<200,000# L <M8,000 <M8,000 L 0(1A(<A0,000# O 0(2A(<2A,000# O 0(=I(<2A,000# O U(<100,000#8 U(<100,000# L <M8,000 E 22,2A0 U L <IA,JA0 : <100,000 U L IA(JAN 2. Halance sheet as of )ec( =1, 2008 Cash <=,J92 0ccounts pa%able <=,98I 0ccounts recei*able A,021 4otes pa%able J=2 .n*entor% 8,92J Current liabilities <I,J1M Current assets <1J,JI0 Cong-ter" debt <12,J00 4et fi!ed assets <=1,80A /nersF e$uit% =2,129 -otal assets <I9,AIA -otal liab( K e$uit% <I9,AIA Halance sheet as of )ec( =1, 2009 Cash <I,0I1 0ccounts pa%able <I,02A 0ccounts recei*able A,892 4otes pa%able J1J .n*entor% 9,AAA Current liabilities <I,JI2 Current assets <19,I88 Cong-ter" debt <1A,I=A 4et fi!ed assets <==,921 /nersF e$uit% ==,2=2 -otal assets <A=,I09 -otal liab( K e$uit% <A=,I09 CHAPTER 2 B-15 2008 .nco"e ,tate"ent 2009 .nco"e ,tate"ent ,ales <J,2==(00 ,ales <8,08A(00 C/G, 2,I8J(00 C/G, 2,9I2(00 /ther e!penses A91(00 /ther e!penses A1A(00 )epreciation 1,0=8(00 )epreciation 1,08A(00 6H.- <=,11J(00 6H.- <=,AI=(00 .nterest I8A(00 .nterest AJ9(00 6H- <2,M=2(00 6H- <2,9MI(00 -a!es (=IN# 89I(88 -a!es (=IN# 1,00J(JM 4et inco"e <1,J=J(12 4et inco"e <1,9AM(2I )i*idends <882(00 )i*idends <1,011(00 0dditions to R6 8AA(12 0dditions to R6 9IA(2I 2!. /CB L 6H.- O )epreciation E -a!es L <=,AI= O 1,08A E 1,00J(JM L <=,M20(2I Change in 4WC L 4WCend E 4WCbeg L (C0 E CC# end E (C0 E CC# beg L (<19,I88 E I,JI2# E (<1J,JI0 E I,J1M# L <1,J22 4et capital spending L 4B0end E 4B0beg O )epreciation L <==,921 E =1,80A O 1,08A L <=,201 Cash flo fro" assets L /CB E Change in 4WC E 4et capital spending L <=,M20(2I E 1,J22 E =,201 L E<1,=02(JM Cash flo to creditors L .nterest E 4et ne C-) 4et ne C-) L C-)end E C-)beg Cash flo to creditors L <AJ9 E (<1A,I=A E 12,J00# L E<2,1AM 4et ne e$uit% L Co""on stoc&end E Co""on stoc&beg Co""on stoc& O Retained earnings L -otal oners' e$uit% 4et ne e$uit% L (/6 E R6# end E (/6 E R6# beg L /6end E /6beg O R6beg E R6end R6end L R6beg O 0dditions to R608 ∴ 4et ne e$uit% L /6end E /6beg O R6beg E (R6beg O 0dditions to R608# L /6end E /6beg E 0dditions to R6 4et ne e$uit% L <==,2=2 E =2,129 E 9IA(2I L <1AJ(JM CB, L )i*idends E 4et ne e$uit% CB, L <1,011 E 1AJ(JM L <8A=(2I 0s a chec&, cash flo fro" assets is E<1,=02(JM( CB0 L Cash flo fro" creditors O Cash flo to stoc&holders CB0 L E<2,1AM O 8A=(2I L E<1,=02(JM CHAPTER 3 WORKING WITH FINANCIAL STATEMENTS Answers to Concepts Review and Critical Thinking Questions 1. a. .f in*entor% is purchased ith cash, then there is no change in the current ratio( .f in*entor% is purchased on credit, then there is a decrease in the current ratio if it as initiall% greater than 1(0( b. Reducing accounts pa%able ith cash increases the current ratio if it as initiall% greater than 1(0( c. Reducing short-ter" debt ith cash increases the current ratio if it as initiall% greater than 1(0( d. 0s long-ter" debt approaches "aturit%, the principal repa%"ent and the re"aining interest e!pense beco"e current liabilities( -hus, if debt is paid off ith cash, the current ratio increases if it as initiall% greater than 1(0( .f the debt has not %et beco"e a current liabilit%, then pa%ing it off ill reduce the current ratio since current liabilities are not affected( e. Reduction of accounts recei*ables and an increase in cash lea*es the current ratio unchanged( f. .n*entor% sold at cost reduces in*entor% and raises cash, so the current ratio is unchanged( g. .n*entor% sold for a profit raises cash in e!cess of the in*entor% recorded at cost, so the current ratio increases( 2. -he fir" has increased in*entor% relati*e to other current assets8 therefore, assu"ing current liabilit% le*els re"ain unchanged, li$uidit% has potentiall% decreased( 3. 0 current ratio of 0(A0 "eans that the fir" has tice as "uch in current liabilities as it does in current assets8 the fir" potentiall% has poor li$uidit%( .f pressed b% its short-ter" creditors and suppliers for i""ediate pa%"ent, the fir" "ight ha*e a difficult ti"e "eeting its obligations( 0 current ratio of 1(A0 "eans the fir" has A0N "ore current assets than it does current liabilities( -his probabl% represents an i"pro*e"ent in li$uidit%8 short-ter" obligations can generall% be "et co"- pletel% ith a safet% factor built in( 0 current ratio of 1A(0, hoe*er, "ight be e!cessi*e( 0n% e!cess funds sitting in current assets generall% earn little or no return( -hese e!cess funds "ight be put to better use b% in*esting in producti*e long-ter" assets or distributing the funds to shareholders( 4. a. 7uic& ratio pro*ides a "easure of the short-ter" li$uidit% of the fir", after re"o*ing the effects of in*entor%, generall% the least li$uid of the fir"'s current assets( b. Cash ratio represents the abilit% of the fir" to co"pletel% pa% off its current liabilities ith its "ost li$uid asset (cash#( c. -otal asset turno*er "easures ho "uch in sales is generated b% each dollar of fir" assets( d. 6$uit% "ultiplier represents the degree of le*erage for an e$uit% in*estor of the fir"8 it "easures the dollar orth of fir" assets each e$uit% dollar has a clai" to( e. Cong-ter" debt ratio "easures the percentage of total fir" capitali2ation funded b% long-ter" debt( CHAPTER 3 B-17 f. -i"es interest earned ratio pro*ides a relati*e "easure of ho ell the fir"'s operating earnings can co*er current interest obligations( g. 9rofit "argin is the accounting "easure of botto"-line profit per dollar of sales( h. Return on assets is a "easure of botto"-line profit per dollar of total assets( i. Return on e$uit% is a "easure of botto"-line profit per dollar of e$uit%( (. 9rice-earnings ratio reflects ho "uch *alue per share the "ar&et places on a dollar of accounting earnings for a fir"( . Co""on si2e financial state"ents e!press all balance sheet accounts as a percentage of total assets and all inco"e state"ent accounts as a percentage of total sales( Using these percentage *alues rather than no"inal dollar *alues facilitates co"parisons beteen fir"s of different si2e or business t%pe( Co""on-base %ear financial state"ents e!press each account as a ratio beteen their current %ear no"inal dollar *alue and so"e reference %ear no"inal dollar *alue( Using these ratios allos the total groth trend in the accounts to be "easured( !. 9eer group anal%sis in*ol*es co"paring the financial ratios and operating perfor"ance of a particular fir" to a set of peer group fir"s in the sa"e industr% or line of business( Co"paring a fir" to its peers allos the financial "anager to e*aluate hether so"e aspects of the fir"'s operations, finances, or in*est"ent acti*ities are out of line ith the nor", thereb% pro*iding so"e guidance on appropriate actions to ta&e to ad1ust these ratios if appropriate( 0n aspirant group ould be a set of fir"s hose perfor"ance the co"pan% in $uestion ould li&e to e"ulate( -he financial "anager often uses the financial ratios of aspirant groups as the target ratios for his or her fir"8 so"e "anagers are e*aluated b% ho ell the% "atch the perfor"ance of an identified aspirant group( ". Return on e$uit% is probabl% the "ost i"portant accounting ratio that "easures the botto"-line perfor"ance of the fir" ith respect to the e$uit% shareholders( -he )u 9ont identit% e"phasi2es the role of a fir"'s profitabilit%, asset utili2ation efficienc%, and financial le*erage in achie*ing an R/6 figure( Bor e!a"ple, a fir" ith R/6 of 20N ould see" to be doing ell, but this figure "a% be "isleading if it ere "arginall% profitable (lo profit "argin# and highl% le*ered (high e$uit% "ultiplier#( .f the fir"'s "argins ere to erode slightl%, the R/6 ould be hea*il% i"pacted( #. -he boo&-to-bill ratio is intended to "easure hether de"and is groing or falling( .t is closel% folloed because it is a baro"eter for the entire high-tech industr% here le*els of re*enues and earnings ha*e been relati*el% *olatile( $. .f a co"pan% is groing b% opening ne stores, then presu"abl% total re*enues ould be rising( Co"paring total sales at to different points in ti"e "ight be "isleading( ,a"e-store sales control for this b% onl% loo&ing at re*enues of stores open ithin a specific period( 1%. a. Bor an electric utilit% such as Con 6d, e!pressing costs on a per &iloatt hour basis ould be a a% to co"pare costs ith other utilities of different si2es( b. Bor a retailer such as ,ears, e!pressing sales on a per s$uare foot basis ould be useful in co"paring re*enue production against other retailers( c. Bor an airline such as ,outhest, e!pressing costs on a per passenger "ile basis allos for co"parisons ith other airlines b% e!a"ining ho "uch it costs to fl% one passenger one "ile( B-18 SOLUTIONS d. Bor an on-line ser*ice pro*ider such as 0/C, using a per call basis for costs ould allo for co"parisons ith s"aller ser*ices( 0 per subscriber basis ould also "a&e sense( e. Bor a hospital such as >ol% Cross, re*enues and costs e!pressed on a per bed basis ould be useful( f. Bor a college te!tboo& publisher such as 3cGra->ill:.rin, the leading publisher of finance te!tboo&s for the college "ar&et, the ob*ious standardi2ation ould be per boo& sold( 11. Reporting the sale of -reasur% securities as cash flo fro" operations is an accounting ;tric&@, and as such, should constitute a possible red flag about the co"panies accounting practices( Bor "ost co"panies, the gain fro" a sale of securities should be placed in the financing section( .ncluding the sale of securities in the cash flo fro" operations ould be acceptable for a financial co"pan%, such as an in*est"ent or co""ercial ban&( 12. .ncreasing the pa%ables period increases the cash flo fro" operations( -his could be beneficial for the co"pan% as it "a% be a cheap for" of financing, but it is basicall% a one ti"e change( -he pa%ables period cannot be increased indefinitel% as it ill negati*el% affect the co"pan%'s credit rating if the pa%ables period beco"es too long( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. Using the for"ula for 4WC, e get+ 4WC L C0 E CC C0 L CC O 4WC L <=,J20 O 1,=J0 L <A,090 ,o, the current ratio is+ Current ratio L C0 : CC L <A,090:<=,J20 L 1(=J ti"es 0nd the $uic& ratio is+ 7uic& ratio L (C0 E .n*entor%# : CC L (<A,090 E 1,9A0# : <=,J20 L 0(8I ti"es 2. We need to find net inco"e first( ,o+ 9rofit "argin L 4et inco"e : ,ales 4et inco"e L ,ales(9rofit "argin# 4et inco"e L (<29,000,000#(0(08# L <2,=20,000 R/0 L 4et inco"e : -0 L <2,=20,000 : <1J,A00,000 L (1=2M or 1=(2MN CHAPTER 3 B-19 -o find R/6, e need to find total e$uit%( -C K /6 L -) O -6 -6 L -C K /6 E -) -6 L <1J,A00,000 E M,=00,000 L <11,200,000 R/6 L 4et inco"e : -6 L 2,=20,000 : <11,200,000 L (20J1 or 20(J1N 3. Recei*ables turno*er L ,ales : Recei*ables Recei*ables turno*er L <=,9I=,J09 : <I=1,28J L 9(1I ti"es )a%s' sales in recei*ables L =MA da%s : Recei*ables turno*er L =MA : 9(1I L =9(92 da%s -he a*erage collection period for an outstanding accounts recei*able balance as =9(92 da%s( 4. .n*entor% turno*er L C/G, : .n*entor% .n*entor% turno*er L <I,10A,M12 : <I0J,A=I L 10(0J ti"es )a%s' sales in in*entor% L =MA da%s : .n*entor% turno*er L =MA : 10(0J L =M(2= da%s /n a*erage, a unit of in*entor% sat on the shelf =M(2= da%s before it as sold( . -otal debt ratio L 0(M= L -) : -0 ,ubstituting total debt plus total e$uit% for total assets, e get+ 0(M= L -) : (-) O -6# ,ol*ing this e$uation %ields+ 0(M=(-6# L 0(=J(-)# )ebt:e$uit% ratio L -) : -6 L 0(M= : 0(=J L 1(J0 6$uit% "ultiplier L 1 O ):6 L 2(J0 !. 4et inco"e L 0ddition to R6 O )i*idends L <I=0,000 O 1JA,000 L <M0A,000 6arnings per share L 4. : ,hares L <M0A,000 : 210,000 L <2(88 per share )i*idends per share L )i*idends : ,hares L <1JA,000 : 210,000 L <0(8= per share Hoo& *alue per share L -6 : ,hares L <A,=00,000 : 210,000 L <2A(2I per share 3ar&et-to-boo& ratio L ,hare price : HV9, L <M= : <2A(2I L 2(A0 ti"es 9:6 ratio L ,hare price : 69, L <M= : <2(88 L 21(8J ti"es ,ales per share L ,ales : ,hares L <I,A00,000 : 210,000 L <21(I= 9:, ratio L ,hare price : ,ales per share L <M= : <21(I= L 2(9I ti"es B-20 SOLUTIONS ". R/6 L (93#(-0-#(63# R/6 L ((0AA#(1(1A#(2(80# L (1JJ1 or 1J(J1N #. -his $uestion gi*es all of the necessar% ratios for the )u9ont .dentit% e!cept the e$uit% "ultiplier, so, using the )u9ont .dentit%+ R/6 L (93#(-0-#(63# R/6 L (182J L ((0M8#(1(9A#(63# 63 L (182J : ((0M8#(1(9A# L 1(=8 ):6 L 63 E 1 L 1(=8 E 1 L 0(=8 $. )ecrease in in*entor% is a source of cash )ecrease in accounts pa%able is a use of cash .ncrease in notes pa%able is a source of cash .ncrease in accounts recei*able is a use of cash Changes in cash L sources E uses L <=JA E 190 O 210 E 10A L <290 Cash increased b% <290 1%. 9a%ables turno*er L C/G, : 0ccounts pa%able 9a%ables turno*er L <28,=8I : <M,10A L I(MA ti"es )a%s' sales in pa%ables L =MA da%s : 9a%ables turno*er )a%s' sales in pa%ables L =MA : I(MA L J8(A1 da%s -he co"pan% left its bills to suppliers outstanding for J8(A1 da%s on a*erage( 0 large *alue for this ratio could i"pl% that either (1# the co"pan% is ha*ing li$uidit% proble"s, "a&ing it difficult to pa% off its short-ter" obligations, or (2# that the co"pan% has successfull% negotiated lenient credit ter"s fro" its suppliers( 11. 4e in*est"ent in fi!ed assets is found b%+ 4et in*est"ent in B0 L (4B0end E 4B0beg# O )epreciation 4et in*est"ent in B0 L <8=A O 1I8 L <98= -he co"pan% bought <98= in ne fi!ed assets8 this is a use of cash( 12. -he e$uit% "ultiplier is+ 63 L 1 O ):6 63 L 1 O 0(MA L 1(MA /ne for"ula to calculate return on e$uit% is+ R/6 L (R/0#(63# R/6 L (08A(1(MA# L (1I0= or 1I(0=N CHAPTER 3 B-21 R/6 can also be calculated as+ R/6 L 4. : -6 ,o, net inco"e is+ 4. L R/6(-6# 4. L ((1I0=#(<AI0,000# L <JA,J=A 13. through 1+ 2%%# )13 2%%$ )13 )14 )1 0ssets Current assets Cash <8,I=M 2(8MN <10,1AJ =(1=N 1(20I0 1(09M1 0ccounts recei*able 21,A=0 J(29N 2=,I0M J(21N 1(08J1 0(989J .n*entor% =8,JM0 1=(12N I2,MA0 1=(1IN 1(100I 1(001J -otal <M8,J2M 2=(2MN <JM,21= 2=(I8N 1(1089 1(009A Bi!ed assets 4et plant and e$uip"ent 22M,J0M JM(JIN 2I8,=0M JM(A2N 1(09A= 0(99J1 -otal assets <29A,I=2 100N <=2I,A19 100N 1(098A 1(0000 Ciabilities and /ners' 6$uit% Current liabilities 0ccounts pa%able <I=,0A0 1I(AJN <IM,821 1I(I=N 1(08JM 0(9901 4otes pa%able 18,=8I M(22N 1J,=82 A(=MN 0(9IAA 0(8M08 -otal <M1,I=I 20(J9N <MI,20= 19(J8N 1(0IA1 0(9A1I Cong-ter" debt 2A,000 8(IMN =2,000 9(8MN 1(2800 1(1MA= /nersF e$uit% Co""on stoc& and paid-in surplus <I0,000 1=(AIN <I0,000 12(==N 1(0000 0(910I 0ccu"ulated retained earnings 1M8,998 AJ(20N 188,=1M A8(0=N 1(11I= 1(01II -otal <208,998 J0(JIN <228,=1M J0(=MN 1(092I 0(99IA -otal liabilities and onersF e$uit% <29A,I=2 100N <=2I,A19 100N 1(098A 1(0000 -he co""on-si2e balance sheet ansers are found b% di*iding each categor% b% total assets( Bor e!a"ple, the cash percentage for 2008 is+ <8,I=M : <29A,I=2 L (028M or 2(8MN -his "eans that cash is 2(8MN of total assets( B-22 SOLUTIONS -he co""on-base %ear ansers for 7uestion 1I are found b% di*iding each categor% *alue for 2009 b% the sa"e categor% *alue for 2008( Bor e!a"ple, the cash co""on-base %ear nu"ber is found b%+ <10,1AJ : <8,I=M L 1(20I0 -his "eans the cash balance in 2009 is 1(20I0 ti"es as large as the cash balance in 2008( -he co""on-si2e, co""on-base %ear ansers for 7uestion 1A are found b% di*iding the co""on- si2e percentage for 2009 b% the co""on-si2e percentage for 2008( Bor e!a"ple, the cash calculation is found b%+ =(1=N : 2(8MN L 1(09M1 -his tells us that cash, as a percentage of assets, increased b% 9(M1N( 1!. 2008 ,ources:Us es 2008 0ssets Current assets Cash <8,I=M <1,J21 U <10,1AJ 0ccounts recei*able 21,A=0 1,8JM U 2=,I0M .n*entor% =8,JM0 =,890 U I2,MA0 -otal <M8,J2M <J,I8J U <JM,21= Bi!ed assets 4et plant and e$uip"ent <22M,J0M <21,M00 U <2I8,=0M -otal assets <29A,I=2 <29,08J U <=2I,A19 Ciabilities and /ners' 6$uit% Current liabilities 0ccounts pa%able <I=,0A0 =,JJ1 , <IM,821 4otes pa%able 18,=8I E1,002 U 1J,=82 -otal <M1,I=I 2,JM9 , <MI,20= Cong-ter" debt 2A,000 <J,000 , =2,000 /nersF e$uit% Co""on stoc& and paid-in surplus <I0,000 <0 <I0,000 0ccu"ulated retained earnings 1M8,998 19,=18 , 188,=1M -otal <208,998 <19,=18 , <228,=1M -otal liabilities and onersF e$uit% <29A,I=2 <29,08J , <=2I,A19 -he fir" used <29,08J in cash to ac$uire ne assets( .t raised this a"ount of cash b% increasing liabilities and oners' e$uit% b% <29,08J( .n particular, the needed funds ere raised b% internal financing (on a net basis#, out of the additions to retained earnings, an increase in current liabilities, and b% an issue of long-ter" debt( CHAPTER 3 B-23 1". a. Current ratio L Current assets : Current liabilities Current ratio 2008 L <M8,J2M : <M1,I=I L 1(12 ti"es Current ratio 2009 L <JM,21= : <MI,20= L 1(19 ti"es b. 7uic& ratio L (Current assets E .n*entor%# : Current liabilities 7uic& ratio 2008 L (<MJ,J2M E =8,JM0# : <M1,I=I L 0(I9 ti"es 7uic& ratio 2009 L (<JM,21= E I2,MA0# : <MI,20= L 0(A2 ti"es c. Cash ratio L Cash : Current liabilities Cash ratio 2008 L <8,I=M : <M1,I=I L 0(1I ti"es Cash ratio 2009 L <10,1AJ : <MI,20= L 0(1M ti"es d. 4WC ratio L 4WC : -otal assets 4WC ratio 2008 L (<M8,J2M E M1,I=I# : <29A,I=2 L 2(IJN 4WC ratio 2009 L (<JM,21= E MI,20=# : <=2I,A19 L =(J0N e. )ebt-e$uit% ratio L -otal debt : -otal e$uit% )ebt-e$uit% ratio 2008 L (<M1,I=I O 2A,000# : <208,998 L 0(I1 ti"es )ebt-e$uit% ratio 2009 L (<MI,20M O =2,000# : <228,=1M L 0(I2 ti"es 6$uit% "ultiplier L 1 O ):6 6$uit% "ultiplier 2008 L 1 O 0(I1 L 1(I1 6$uit% "ultiplier 2009 L 1 O 0(I2 L 1(I2 f. -otal debt ratio L (-otal assets E -otal e$uit%# : -otal assets -otal debt ratio 2008 L (<29A,I=2 E 208,998# : <29A,I=2 L 0(29 -otal debt ratio 2009 L (<=2I,A19 E 228,=1M# : <=2I,A19 L 0(=0 Cong-ter" debt ratio L Cong-ter" debt : (Cong-ter" debt O -otal e$uit%# Cong-ter" debt ratio 2008 L <2A,000 : (<2A,000 O 208,998# L 0(11 Cong-ter" debt ratio 2009 L <=2,000 : (<=2,000 O 228,=1M# L 0(12 &ntermediate 1#. -his is a "ulti-step proble" in*ol*ing se*eral ratios( -he ratios gi*en are all part of the )u9ont .dentit%( -he onl% )u9ont .dentit% ratio not gi*en is the profit "argin( .f e &no the profit "argin, e can find the net inco"e since sales are gi*en( ,o, e begin ith the )u9ont .dentit%+ R/6 L 0(1A L (93#(-0-#(63# L (93#(, : -0#(1 O ):6# ,ol*ing the )u9ont .dentit% for profit "argin, e get+ 93 L Q(R/6#(-0#R : Q(1 O ):6#(,#R 93 L Q(0(1A#(<=,10A#R : Q(1 O 1(I#( <A,J2M#R L (0==9 4o that e ha*e the profit "argin, e can use this nu"ber and the gi*en sales figure to sol*e for net inco"e+ 93 L (0==9 L 4. : , 4. L (0==9(<A,J2M# L <19I(0M B-24 SOLUTIONS 1$. -his is a "ulti-step proble" in*ol*ing se*eral ratios( .t is often easier to loo& bac&ard to deter"ine here to start( We need recei*ables turno*er to find da%s' sales in recei*ables( -o calculate recei*ables turno*er, e need credit sales, and to find credit sales, e need total sales( ,ince e are gi*en the profit "argin and net inco"e, e can use these to calculate total sales as+ 93 L 0(08J L 4. : ,ales L <218,000 : ,ales8 ,ales L <2,A0A,JIJ Credit sales are J0 percent of total sales, so+ Credit sales L <2,A1A,JIJ(0(J0# L <1,JAI,02= 4o e can find recei*ables turno*er b%+ Recei*ables turno*er L Credit sales : 0ccounts recei*able L <1,JAI,02= : <1=2,8A0 L 1=(20 ti"es )a%s' sales in recei*ables L =MA da%s : Recei*ables turno*er L =MA : 1=(20 L 2J(MA da%s 2%. -he solution to this proble" re$uires a nu"ber of steps( Birst, re"e"ber that C0 O 4B0 L -0( ,o, if e find the C0 and the -0, e can sol*e for 4B0( Using the nu"bers gi*en for the current ratio and the current liabilities, e sol*e for C0+ CR L C0 : CC C0 L CR(CC# L 1(2A(<8JA# L <1,09=(JA -o find the total assets, e "ust first find the total debt and e$uit% fro" the infor"ation gi*en( ,o, e find the sales using the profit "argin+ 93 L 4. : ,ales 4. L 93(,ales# L (09A(<A,8J0# L <AI9(10 We no use the net inco"e figure as an input into R/6 to find the total e$uit%+ R/6 L 4. : -6 -6 L 4. : R/6 L <AI9(10 : (18A L <2,9M8(11 4e!t, e need to find the long-ter" debt( -he long-ter" debt ratio is+ Cong-ter" debt ratio L 0(IA L C-) : (C-) O -6# .n*erting both sides gi*es+ 1 : 0(IA L (C-) O -6# : C-) L 1 O (-6 : C-)# ,ubstituting the total e$uit% into the e$uation and sol*ing for long-ter" debt gi*es the folloing+ 2(222 L 1 O (<2,9M8(11 : C-)# C-) L <2,9M8(11 : 1(222 L <2,I28(IA CHAPTER 3 B-25 4o, e can find the total debt of the co"pan%+ -) L CC O C-) L <8JA O 2,I28(IA L <=,=0=(IA 0nd, ith the total debt, e can find the -)K6, hich is e$ual to -0+ -0 L -) O -6 L <=,=0=(IA O 2,9M8(11 L <M,2J1(AM 0nd finall%, e are read% to sol*e the balance sheet identit% as+ 4B0 L -0 E C0 L <M,2J1(AM E 1,09=(JA L <A,1JJ(81 21. Child+ 9rofit "argin L 4. : , L <=(00 : <A0 L (0M or MN ,tore+ 9rofit "argin L 4. : , L <22,A00,000 : <JA0,000,000 L (0= or =N -he ad*ertise"ent is referring to the store's profit "argin, but a "ore appropriate earnings "easure for the fir"'s oners is the return on e$uit%( R/6 L 4. : -6 L 4. : (-0 E -)# R/6 L <22,A00,000 : (<I20,000,000 E 280,000,000# L (1M0J or 1M(0JN 22. -he solution re$uires substituting to ratios into a third ratio( Rearranging ):-0+ Bir" 0 Bir" H ) : -0 L (=A ) : -0 L (=0 (-0 E 6# : -0 L (=A (-0 E 6# : -0 L (=0 (-0 : -0# E (6 : -0# L (=A (-0 : -0# E (6 : -0# L (=0 1 E (6 : -0# L (=A 1 E (6 : -0# L (=0 6 : -0 L (MA 6 : -0 L (=0 6 L (MA(-0# 6 L (J0 (-0# Rearranging R/0, e find+ 4. : -0 L (12 4. : -0 L (11 4. L (12(-0# 4. L (11(-0# ,ince R/6 L 4. : 6, e can substitute the abo*e e$uations into the R/6 for"ula, hich %ields+ R/6 L (12(-0# : (MA(-0# L (12 : (MA L 18(IMN R/6 L (11(-0# : (J0 (-0# L (11 : (J0 L 1A(J1N 23. -his proble" re$uires %ou to or& bac&ard through the inco"e state"ent( Birst, recogni2e that 4et inco"e L (1 E t#6H-( 9lugging in the nu"bers gi*en and sol*ing for 6H-, e get+ 6H- L <1=,1M8 : (1 E 0(=I# L <19,9A1(A2 4o, e can add interest to 6H- to get 6H.- as follos+ 6H.- L 6H- O .nterest paid L <19,9A1(A2 O =,M0A L <2=,AAM(A2 B-26 SOLUTIONS -o get 6H.-) (earnings before interest, ta!es, and depreciation#, the nu"erator in the cash co*erage ratio, add depreciation to 6H.-+ 6H.-) L 6H.- O )epreciation L <2=,AAM(A2 O 2,=82 L <2A,9=8(A2 4o, si"pl% plug the nu"bers into the cash co*erage ratio and calculate+ Cash co*erage ratio L 6H.-) : .nterest L <2A,9=8(A2 : <=,M0A L J(20 ti"es 24. -he onl% ratio gi*en hich includes cost of goods sold is the in*entor% turno*er ratio, so it is the last ratio used( ,ince current liabilities is gi*en, e start ith the current ratio+ Current ratio L 1(I0 L C0 : CC L C0 : <=MA,000 C0 L <A11,000 Using the $uic& ratio, e sol*e for in*entor%+ 7uic& ratio L 0(8A L (C0 E .n*entor%# : CC L (<A11,000 E .n*entor%# : <=MA,000 .n*entor% L C0 E (7uic& ratio S CC# .n*entor% L <A11,000 E (0(8A S <=MA,000# .n*entor% L <200,JA0 .n*entor% turno*er L A(82 L C/G, : .n*entor% L C/G, : <200,JA0 C/G, L <1,1MI,=A0 2. 93 L 4. : , L EW1=,I82,000 : W1=8,J9= L E0(09J1 or E9(J1N 0s long as both net inco"e and sales are "easured in the sa"e currenc%, there is no proble"8 in fact, e!cept for so"e "ar&et *alue ratios li&e 69, and HV9,, none of the financial ratios discussed in the te!t are "easured in ter"s of currenc%( -his is one reason h% financial ratio anal%sis is idel% used in international finance to co"pare the business operations of fir"s and:or di*isions across national econo"ic borders( -he net inco"e in dollars is+ 4. L 93 S ,ales 4. L E0(09J1(<2JI,21=,000# L E<2M,M=M,=AA 2!. 'hort)term solvenc ratios: Current ratio L Current assets : Current liabilities Current ratio 2008 L <AM,2M0 : <=8,9M= L 1(II ti"es Current ratio 2009 L <M0,AA0 : <I=,2=A L 1(I0 ti"es 7uic& ratio L (Current assets E .n*entor%# : Current liabilities 7uic& ratio 2008 L (<AM,2M0 E 2=,08I# : <=8,9M= L 0(8A ti"es 7uic& ratio 2009 L (<M0,AA0 E 2I,MA0# : <I=,2=A L 0(8= ti"es Cash ratio L Cash : Current liabilities Cash ratio 2008 L <21,8M0 : <=8,9M= L 0(AM ti"es Cash ratio 2009 L <22,0A0 : <I=,2=A L 0(A1 ti"es CHAPTER 3 B-27 Asset utili*ation ratios: -otal asset turno*er L ,ales : -otal assets -otal asset turno*er L <=0A,8=0 : <=21,0JA L 0(9A ti"es .n*entor% turno*er L Cost of goods sold : .n*entor% .n*entor% turno*er L <210,9=A : <2I,MA0 L 8(AM ti"es Recei*ables turno*er L ,ales : 0ccounts recei*able Recei*ables turno*er L <=0A,8=0 : <1=,8A0 L 22(08 ti"es +ong)term solvenc ratios: -otal debt ratio L (-otal assets E -otal e$uit%# : -otal assets -otal debt ratio 2008 L (<290,=28 E 1JM,=MA# : <290,=28 L 0(=9 -otal debt ratio 2009 L (<=21,0JA E 192,8I0# : <=21,0JA L 0(I0 )ebt-e$uit% ratio L -otal debt : -otal e$uit% )ebt-e$uit% ratio 2008 L (<=8,9M= O JA,000# : <1JM,=MA L 0(MA )ebt-e$uit% ratio 2009 L (<I=,2=A O 8A,000# : <192,8I0 L 0(MM 6$uit% "ultiplier L 1 O ):6 6$uit% "ultiplier 2008 L 1 O 0(MA L 1(MA 6$uit% "ultiplier 2009 L 1 O 0(MM L 1(MM -i"es interest earned L 6H.- : .nterest -i"es interest earned L <M8,0IA : <11,9=0 L A(J0 ti"es Cash co*erage ratio L (6H.- O )epreciation# : .nterest Cash co*erage ratio L (<M8,0IA O 2M,8A0# : <11,9=0 L J(9A ti"es ,rofitabilit ratios: 9rofit "argin L 4et inco"e : ,ales 9rofit "argin L <=M,IJA : <=0A,8=0 L 0(119= or 11(9=N Return on assets L 4et inco"e : -otal assets Return on assets L <=M,IJA : <=21,0JA L 0(11=M or 11(=MN Return on e$uit% L 4et inco"e : -otal e$uit% Return on e$uit% L <=M,IJA : <192,8I0 L 0(1891 or 18(91N 2". -he )u9ont identit% is+ R/6 L (93#(-0-#(63# R/6 L (0(119=#(0(9A#(1(MM# L 0(1891 or 18(91N B-28 SOLUTIONS 2#. ,3/C.R0 G/CB C/R9( ,tate"ent of Cash Blos Bor 2009 Cash* 'eginning o+ the ,ear < 21,8M0 Operating activities 4et inco"e < =M,IJA 9lus+ )epreciation < 2M,8A0 .ncrease in accounts pa%able =,A=0 .ncrease in other current liabilities 1,JI2 Cess+ .ncrease in accounts recei*able < (2,A=I# .ncrease in in*entor% (1,AMM# Net cash from operating activities < MI,I9J &nvestment activities Bi!ed asset ac$uisition <(A=,=0J# Net cash from investment activities <(A=,=0J# Financing activities .ncrease in notes pa%able < (1,000# )i*idends paid (20,000# .ncrease in long-ter" debt 10,000 Net cash from financing activities <(11,000# Net increase in cash < 190 Cash* end o+ ,ear < 22,0A0 2$. 6arnings per share L 4et inco"e : ,hares 6arnings per share L <=M,IJA : 2A,000 L <1(IM per share 9:6 ratio L ,hares price : 6arnings per share 9:6 ratio L <I= : <1(IM L 29(IJ ti"es )i*idends per share L )i*idends : ,hares )i*idends per share L <20,000 : 2A,000 L <0(80 per share Hoo& *alue per share L -otal e$uit% : ,hares Hoo& *alue per share L <192,8I0 : 2A,000 shares L <J(J1 per share CHAPTER 3 B-29 3ar&et-to-boo& ratio L ,hare price : Hoo& *alue per share 3ar&et-to-boo& ratio L <I= : <J(J1 L A(AJ ti"es 96G ratio L 9:6 ratio : Groth rate 96G ratio L 29(IJ : 9 L =(2J ti"es 3%. Birst, e ill find the "ar&et *alue of the co"pan%'s e$uit%, hich is+ 3ar&et *alue of e$uit% L ,hares S ,hare price 3ar&et *alue of e$uit% L 2A,000(<I=# L <1,0JA,000 -he total boo& *alue of the co"pan%'s debt is+ -otal debt L Current liabilities O Cong-ter" debt -otal debt L <I=,2=A O 8A,000 L <128,2=A 4o e can calculate -obin's 7, hich is+ -obin's 7 L (3ar&et *alue of e$uit% O Hoo& *alue of debt# : Hoo& *alue of assets -obin's 7 L (<1,0JA,000 O 128,2=A# : <=21,0JA -obin's 7 L =(JA Using the boo& *alue of debt i"plicitl% assu"es that the boo& *alue of debt is e$ual to the "ar&et *alue of debt( -his ill be discussed in "ore detail in later chapters, but this assu"ption is generall% true( Using the boo& *alue of assets assu"es that the assets can be replaced at the current *alue on the balance sheet( -here are se*eral reasons this assu"ption could be flaed( Birst, inflation during the life of the assets can cause the boo& *alue of the assets to understate the "ar&et *alue of the assets( ,ince assets are recorded at cost hen purchased, inflation "eans that it is "ore e!pensi*e to replace the assets( ,econd, i"pro*e"ents in technolog% could "ean that the assets could be replaced ith "ore producti*e, and possibl% cheaper, assets( .f this is true, the boo& *alue can o*erstate the "ar&et *alue of the assets( Binall%, the boo& *alue of assets "a% not accuratel% represent the "ar&et *alue of the assets because of depreciation( )epreciation is done according to so"e schedule, generall% straight-line or 30CR,( -hus, the boo& *alue and "ar&et *alue can often di*erge( CHAPTER 4 LONG-TERM FINANCIAL PLANNING AND GROWTH Answers to Concepts Review and Critical Thinking Questions 1. -he reason is that, ulti"atel%, sales are the dri*ing force behind a business( 0 fir"'s assets, e"plo%ees, and, in fact, 1ust about e*er% aspect of its operations and financing e!ist to directl% or indirectl% support sales( 9ut differentl%, a fir"'s future need for things li&e capital assets, e"plo%ees, in*entor%, and financing are deter"ined b% its future sales le*el( 2. -o assu"ptions of the sustainable groth for"ula are that the co"pan% does not ant to sell ne e$uit%, and that financial polic% is fi!ed( .f the co"pan% raises outside e$uit%, or increases its debt- e$uit% ratio it can gro at a higher rate than the sustainable groth rate( /f course the co"pan% could also gro faster than its profit "argin increases, if it changes its di*idend polic% b% increasing the retention ratio, or its total asset turno*er increases( 3. -he internal groth rate is greater than 1AN, because at a 1AN groth rate the negati*e 6B4 indicates that there is e!cess internal financing( .f the internal groth rate is greater than 1AN, then the sustainable groth rate is certainl% greater than 1AN, because there is additional debt financing used in that case (assu"ing the fir" is not 100N e$uit%-financed#( 0s the retention ratio is increased, the fir" has "ore internal sources of funding, so the 6B4 ill decline( Con*ersel%, as the retention ratio is decreased, the 6B4 ill rise( .f the fir" pa%s out all its earnings in the for" of di*idends, then the fir" has no internal sources of funding (ignoring the effects of accounts pa%able#8 the internal groth rate is 2ero in this case and the 6B4 ill rise to the change in total assets( 4. -he sustainable groth rate is greater than 20N, because at a 20N groth rate the negati*e 6B4 indicates that there is e!cess financing still a*ailable( .f the fir" is 100N e$uit% financed, then the sustainable and internal groth rates are e$ual and the internal groth rate ould be greater than 20N( >oe*er, hen the fir" has so"e debt, the internal groth rate is ala%s less than the sustainable groth rate, so it is a"biguous hether the internal groth rate ould be greater than or less than 20N( .f the retention ratio is increased, the fir" ill ha*e "ore internal funding sources a*ailable, and it ill ha*e to ta&e on "ore debt to &eep the debt:e$uit% ratio constant, so the 6B4 ill decline( Con*ersel%, if the retention ratio is decreased, the 6B4 ill rise( .f the retention rate is 2ero, both the internal and sustainable groth rates are 2ero, and the 6B4 ill rise to the change in total assets( . 9resu"abl% not, but, of course, if the product had been much less popular, then a si"ilar fate ould ha*e aaited due to lac& of sales( !. ,ince custo"ers did not pa% until ship"ent, recei*ables rose( -he fir"'s 4WC, but not its cash, increased( 0t the sa"e ti"e, costs ere rising faster than cash re*enues, so operating cash flo CHAPTER 4 B-31 declined( -he fir"'s capital spending as also rising( -hus, all three co"ponents of cash flo fro" assets ere negati*el% i"pacted( B-32 SOLUTIONS ". 0pparentl% notX .n hindsight, the fir" "a% ha*e underesti"ated costs and also underesti"ated the e!tra de"and fro" the loer price( #. Binancing possibl% could ha*e been arranged if the co"pan% had ta&en $uic& enough action( ,o"eti"es it beco"es apparent that help is needed onl% hen it is too late, again e"phasi2ing the need for planning( $. 0ll three ere i"portant, but the lac& of cash or, "ore generall%, financial resources ulti"atel% spelled doo"( 0n inade$uate cash resource is usuall% cited as the "ost co""on cause of s"all business failure( 1%. )e"anding cash up front, increasing prices, subcontracting production, and i"pro*ing financial resources *ia ne oners or ne sources of credit are so"e of the options( When orders e!ceed capacit%, price increases "a% be especiall% beneficial( Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. .t is i"portant to re"e"ber that e$uit% ill not increase b% the sa"e percentage as the other assets( .f e*er% other ite" on the inco"e state"ent and balance sheet increases b% 1A percent, the pro for"a inco"e state"ent and balance sheet ill loo& li&e this+ 9ro for"a inco"e state"ent 9ro for"a balance sheet ,ales < 2M,IA0 0ssets <18,1J0 )ebt < A,980 Costs 19,20A 6$uit% 12,190 4et inco"e < J,2IA -otal < 18,1J0 -otal < 18,1J0 .n order for the balance sheet to balance, e$uit% "ust be+ 6$uit% L -otal liabilities and e$uit% E )ebt 6$uit% L <18,1J0 E A,980 6$uit% L <12,190 6$uit% increased b%+ 6$uit% increase L <12,190 E 10,M00 6$uit% increase L <1,A90 CHAPTER 4 B-33 4et inco"e is <J,2IA but e$uit% onl% increased b% <1,A908 therefore, a di*idend of+ )i*idend L <J,2IA E 1,A90 )i*idend L <A,MAA "ust ha*e been paid( )i*idends paid is the plug *ariable( 2. >ere e are gi*en the di*idend a"ount, so di*idends paid is not a plug *ariable( .f the co"pan% pa%s out one-half of its net inco"e as di*idends, the pro for"a inco"e state"ent and balance sheet ill loo& li&e this+ 9ro for"a inco"e state"ent 9ro for"a balance sheet ,ales <2M,IA0(00 0ssets <18,1J0(0 0 )ebt < A,980(00 Costs 19,20A(00 6$uit% 1I,222(A0 4et inco"e < J,2IA(00 -otal <18,1J0(0 0 -otal <19,I22(A0 )i*idends <=,M22(A0 0dd( to R6 <=,M22(A0 4ote that the balance sheet does not balance( -his is due to 6B4( -he 6B4 for this co"pan% is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <18,1J0 E 19,I22(A0 6B4 L E<1,2A2(A0 3. 0n increase of sales to <J,I2I is an increase of+ ,ales increase L (<J,I2I E M,=00# : <M,=00 ,ales increase L (18 or 18N 0ssu"ing costs and assets increase proportionall%, the pro for"a financial state"ents ill loo& li&e this+ 9ro for"a inco"e state"ent 9ro for"a balance sheet ,ales < J,I=I 0ssets < 21,A9I )ebt < 12,I00 Costs I,A90 6$uit% 8,JII 4et inco"e < 2,8II -otal < 21,A9I -otal < 21,1II .f no di*idends are paid, the e$uit% account ill increase b% the net inco"e, so+ 6$uit% L <A,900 O 2,8II 6$uit% L <8,JII ,o the 6B4 is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <21,A9I E 21,1II L <IA0 B-34 SOLUTIONS CHAPTER 4 B-35 4. 0n increase of sales to <21,8I0 is an increase of+ ,ales increase L (<21,8I0 E 19,A00# : <19,A00 ,ales increase L (12 or 12N 0ssu"ing costs and assets increase proportionall%, the pro for"a financial state"ents ill loo& li&e this+ 9ro for"a inco"e state"ent 9ro for"a balance sheet ,ales < 21,8I0 0ssets <109,JM0 )ebt <A2,A00 Costs 1M,800 6$uit% J9,208 6H.- A,0I0 -otal <109,JM0 -otal <99,IAM -a!es (I0N# 2,01M 4et inco"e < =,02I -he pa%out ratio is constant, so the di*idends paid this %ear is the pa%out ratio fro" last %ear ti"es net inco"e, or+ )i*idends L (<1,I00 : <2,J00#(<=,02I# )i*idends L <1,AM8 -he addition to retained earnings is+ 0ddition to retained earnings L <=,02I E 1,AM8 0ddition to retained earnings L <1,IAM 0nd the ne e$uit% balance is+ 6$uit% L <IA,A00 O 1,IAM 6$uit% L <IM,9AM ,o the 6B4 is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <109,JM0 E 99,IAM 6B4 L <10,=0I . 0ssu"ing costs and assets increase proportionall%, the pro for"a financial state"ents ill loo& li&e this+ 9ro for"a inco"e state"ent 9ro for"a balance sheet ,ales <I,8=0(00 C0 <I,1I0(00 CC <2,1IA(00 Costs =,J9A(00 B0 9,08A(00 C-) =,MA0(00 -a!able inco"e <1,0=A(00 6$uit% M,1A9(8M -a!es (=IN# =A1(90 -0 <1=,22A(00 -otal )K6 <12,22I(8M 4et inco"e < M8=(10 B-36 SOLUTIONS -he pa%out ratio is I0 percent, so di*idends ill be+ )i*idends L 0(I0(<M8=(10# )i*idends L <2J=(2I -he addition to retained earnings is+ 0ddition to retained earnings L <M8=(10 E 2J=(2I 0ddition to retained earnings L <I09(8M ,o the 6B4 is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <1=,22A E 12,22I(8M 6B4 L <1,000(1I !. -o calculate the internal groth rate, e first need to calculate the R/0, hich is+ R/0 L 4. : -0 R/0 L <2,2M2 : <=9,1A0 R/0 L (0AJ8 or A(J8N -he plobac& ratio, b, is one "inus the pa%out ratio, so+ b L 1 E (=0 b L (J0 4o e can use the internal groth rate e$uation to get+ .nternal groth rate L (R/0 S b# : Q1 E (R/0 S b#R .nternal groth rate L Q0(0AJ8((J0#R : Q1 E 0(0AJ8((J0#R .nternal groth rate L (0I21 or I(21N ". -o calculate the sustainable groth rate, e first need to calculate the R/6, hich is+ R/6 L 4. : -6 R/6 L <2,2M2 : <21,MA0 R/6 L (10IA or 10(IAN -he plobac& ratio, b, is one "inus the pa%out ratio, so+ b L 1 E (=0 b L (J0 4o e can use the sustainable groth rate e$uation to get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L Q0(10IA((J0#R : Q1 E 0(10IA((J0#R ,ustainable groth rate L (0J89 or J(89N CHAPTER 4 B-37 #. -he "a!i"u" percentage sales increase is the sustainable groth rate( -o calculate the sustainable groth rate, e first need to calculate the R/6, hich is+ R/6 L 4. : -6 R/6 L <8,910 : <AM,000 R/6 L (1A91 or 1A(91N -he plobac& ratio, b, is one "inus the pa%out ratio, so+ b L 1 E (=0 b L (J0 4o e can use the sustainable groth rate e$uation to get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L Q(1A91((J0#R : Q1 E (1A91((J0#R ,ustainable groth rate L (12A= or 12(A=N ,o, the "a!i"u" dollar increase in sales is+ 3a!i"u" increase in sales L <I2,000((12A=# 3a!i"u" increase in sales L <A,2MI(0= $. 0ssu"ing costs *ar% ith sales and a 20 percent increase in sales, the pro for"a inco"e state"ent ill loo& li&e this+ >6.R J/R)04 C/R9/R0-./4 9ro Bor"a .nco"e ,tate"ent ,ales <IA,M00(00 Costs 22,080(00 -a!able inco"e <2=,A20(00 -a!es (=IN# J,99M(80 4et inco"e < 1A,A2=(20 -he pa%out ratio is constant, so the di*idends paid this %ear is the pa%out ratio fro" last %ear ti"es net inco"e, or+ )i*idends L (<A,200:<12,9=M#(<1A,A2=(20# )i*idends L <M,2I0(00 0nd the addition to retained earnings ill be+ 0ddition to retained earnings L <1A,A2=(20 E M,2I0 0ddition to retained earnings L <9,28=(20 B-38 SOLUTIONS 1%. Helo is the balance sheet ith the percentage of sales for each account on the balance sheet( 4otes pa%able, total current liabilities, long-ter" debt, and all e$uit% accounts do not *ar% directl% ith sales( >6.R J/R)04 C/R9/R0-./4 Halance ,heet (<# (N# (<# (N# 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =,0A0 8(0= 0ccounts pa%able < 1,=00 =(I2 0ccounts recei*able M,900 18(1M 4otes pa%able M,800 n:a .n*entor% J,M00 20(00 -otal < 8,100 n:a -otal < 1J,AA0 IM(18 Cong-ter" debt 2A,000 n:a Bi!ed assets /ners' e$uit% 4et plant and Co""on stoc& and e$uip"ent =I,A00 90(J9 paid-in surplus <1A,000 n:a Retained earnings =,9A0 n:a -otal < 18,9A0 n:a -otal liabilities and oners' -otal assets < A2,0A0 1=M(9J e$uit% < A2,0A0 n:a 11. 0ssu"ing costs *ar% ith sales and a 1A percent increase in sales, the pro for"a inco"e state"ent ill loo& li&e this+ >6.R J/R)04 C/R9/R0-./4 9ro Bor"a .nco"e ,tate"ent ,ales <I=,J00(00 Costs 21,1M0(00 -a!able inco"e <22,AI0(00 -a!es (=IN# J,MM=(M0 4et inco"e < 1I,8JM(I0 -he pa%out ratio is constant, so the di*idends paid this %ear is the pa%out ratio fro" last %ear ti"es net inco"e, or+ )i*idends L (<A,200:<12,9=M#(<1I,8JM(I0# )i*idends L <A,980(00 0nd the addition to retained earnings ill be+ 0ddition to retained earnings L <1I,8JM(I0 E A,980 0ddition to retained earnings L <8,89M(I0 -he ne accu"ulated retained earnings on the pro for"a balance sheet ill be+ 4e accu"ulated retained earnings L <=,9A0 O 8,89M(I0 4e accu"ulated retained earnings L <12,8IM(I0 CHAPTER 4 B-39 -he pro for"a balance sheet ill loo& li&e this+ >6.R J/R)04 C/R9/R0-./4 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =,A0J(A0 0ccounts pa%able < 1,I9A(00 0ccounts recei*able J,9=A(00 4otes pa%able M,800(00 .n*entor% 8,JI0(00 -otal < 8,29A(00 -otal <20,182(A0 Cong-ter" debt 2A,000(00 Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent =9(MJA(00 Co""on stoc& and paid-in surplus < 1A,000(00 Retained earnings 12,8IM(I0 -otal < 2J,8IM(I0 -otal liabilities and oners' -otal assets < A9,8AJ(A0 e$uit% < M1,1I1(I0 ,o the 6B4 is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <A9,8AJ(A0 E M1,1I1(I0 6B4 L E<1,28=(90 12. We need to calculate the retention ratio to calculate the internal groth rate( -he retention ratio is+ b L 1 E (20 b L (80 4o e can use the internal groth rate e$uation to get+ .nternal groth rate L (R/0 S b# : Q1 E (R/0 S b#R .nternal groth rate L Q(08((80#R : Q1 E (08((80#R .nternal groth rate L (0M8I or M(8IN 13. We need to calculate the retention ratio to calculate the sustainable groth rate( -he retention ratio is+ b L 1 E (2A b L (JA 4o e can use the sustainable groth rate e$uation to get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L Q(1A((JA#R : Q1 E (1A((JA#R ,ustainable groth rate L (12M8 or 12(M8N B-40 SOLUTIONS 14. We first "ust calculate the R/6 to calculate the sustainable groth rate( -o do this e "ust reali2e to other relationships( -he total asset turno*er is the in*erse of the capital intensit% ratio, and the e$uit% "ultiplier is 1 O ):6( Using these relationships, e get+ R/6 L (93#(-0-#(63# R/6 L ((082#(1:(JA#(1 O (I0# R/6 L (1A=1 or 1A(=1N -he plobac& ratio is one "inus the di*idend pa%out ratio, so+ b L 1 E (<12,000 : <I=,000# b L (J209 4o e can use the sustainable groth rate e$uation to get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L Q(1A=1((J209#R : Q1 E (1A=1((J209#R ,ustainable groth rate L (12I0 or 12(I0N 1. We "ust first calculate the R/6 using the )u9ont ratio to calculate the sustainable groth rate( -he R/6 is+ R/6 L (93#(-0-#(63# R/6 L ((0J8#(2(A0#(1(80# R/6 L (=A10 or =A(10N -he plobac& ratio is one "inus the di*idend pa%out ratio, so+ b L 1 E (M0 b L (I0 4o e can use the sustainable groth rate e$uation to get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L Q(=A10((I0#R : Q1 E (=A10((I0#R ,ustainable groth rate L (1M== or 1M(==N &ntermediate 1!. -o deter"ine full capacit% sales, e di*ide the current sales b% the capacit% the co"pan% is currentl% using, so+ Bull capacit% sales L <AA0,000 : (9A Bull capacit% sales L <AJ8,9IJ -he "a!i"u" sales groth is the full capacit% sales di*ided b% the current sales, so+ 3a!i"u" sales groth L (<AJ8,9IJ : <AA0,000# E 1 3a!i"u" sales groth L (0A2M or A(2MN CHAPTER 4 B-41 1". -o find the ne le*el of fi!ed assets, e need to find the current percentage of fi!ed assets to full capacit% sales( )oing so, e find+ Bi!ed assets : Bull capacit% sales L <II0,000 : <AJ8,9IJ Bi!ed assets : Bull capacit% sales L (JM 4e!t, e calculate the total dollar a"ount of fi!ed assets needed at the ne sales figure( -otal fi!ed assets L (JM(<M=0,000# -otal fi!ed assets L <IJ8,800 -he ne fi!ed assets necessar% is the total fi!ed assets at the ne sales figure "inus the current le*el of fi!ed assts( 4e fi!ed assets L <IJ8,800 E II0,000 4e fi!ed assets L <=8,800 1#. We ha*e all the *ariables to calculate R/6 using the )u9ont identit% e!cept the profit "argin( .f e find R/6, e can sol*e the )u9ont identit% for profit "argin( We can calculate R/6 fro" the sustainable groth rate e$uation( Bor this e$uation e need the retention ratio, so+ b L 1 E (=0 b L (J0 Using the sustainable groth rate e$uation and sol*ing for R/6, e get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R (12 L QR/6((J0#R : Q1 E R/6((J0#R R/6 L (1A=1 or 1A(=1N 4o e can use the )u9ont identit% to find the profit "argin as+ R/6 L 93(-0-#(63# (1A=1 L 93(1 : 0(JA#(1 O 1(20# 93 L ((1A=1# : Q(1 : 0(JA#(2(20#R 93 L (0A22 or A(22N 1$. We ha*e all the *ariables to calculate R/6 using the )u9ont identit% e!cept the e$uit% "ultiplier( Re"e"ber that the e$uit% "ultiplier is one plus the debt-e$uit% ratio( .f e find R/6, e can sol*e the )u9ont identit% for e$uit% "ultiplier, then the debt-e$uit% ratio( We can calculate R/6 fro" the sustainable groth rate e$uation( Bor this e$uation e need the retention ratio, so+ b L 1 E (=0 b L (J0 Using the sustainable groth rate e$uation and sol*ing for R/6, e get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R (11A L QR/6((J0#R : Q1 E R/6((J0#R R/6 L (1IJ= or 1I(J=N B-42 SOLUTIONS 4o e can use the )u9ont identit% to find the e$uit% "ultiplier as+ R/6 L 93(-0-#(63# (1IJ= L ((0M2#(1 : (M0#63 63 L ((1IJ=#((M0# : (0M2 63 L 1(I= ,o, the ):6 ratio is+ ):6 L 63 E 1 ):6 L 1(I= E 1 ):6 L 0(I= 2%. We are gi*en the profit "argin( Re"e"ber that+ R/0 L 93(-0-# We can calculate the R/0 fro" the internal groth rate for"ula, and then use the R/0 in this e$uation to find the total asset turno*er( -he retention ratio is+ b L 1 E (2A b L (JA Using the internal groth rate e$uation to find the R/0, e get+ .nternal groth rate L (R/0 S b# : Q1 E (R/0 S b#R (0J L QR/0((JA#R : Q1 E R/0((JA#R R/0 L (08J2 or 8(J2N 9lugging R/0 and 93 into the e$uation e began ith and sol*ing for -0-, e get+ R/0 L (93#(-0-# (08J2 L (0A(93# -0- L (08J2 : (0A -0- L 1(JI ti"es 21. We should begin b% calculating the ):6 ratio( We calculate the ):6 ratio as follos+ -otal debt ratio L (MA L -) : -0 .n*erting both sides e get+ 1 : (MA L -0 : -) 4e!t, e need to recogni2e that -0 : -) L 1 O -6 : -) ,ubstituting this into the pre*ious e$uation, e get+ 1 : (MA L 1 O -6 :-) CHAPTER 4 B-43 ,ubtract 1 (one# fro" both sides and in*erting again, e get+ ):6 L 1 : Q(1 : (MA# E 1R ):6 L 1(8M With the ):6 ratio, e can calculate the 63 and sol*e for R/6 using the )u9ont identit%+ R/6 L (93#(-0-#(63# R/6 L ((0I8#(1(2A#(1 O 1(8M# R/6 L (1J1I or 1J(1IN 4o e can calculate the retention ratio as+ b L 1 E (=0 b L (J0 Binall%, putting all the nu"bers e ha*e calculated into the sustainable groth rate e$uation, e get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L Q(1J1I((J0#R : Q1 E (1J1I((J0#R ,ustainable groth rate L (1=MI or 1=(MIN 22. -o calculate the sustainable groth rate, e first "ust calculate the retention ratio and R/6( -he retention ratio is+ b L 1 E <9,=00 : <1J,A00 b L (IM8M 0nd the R/6 is+ R/6 L <1J,A00 : <A8,000 R/6 L (=01J or =0(1JN ,o, the sustainable groth rate is+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L Q(=01J((IM8M#R : Q1 E (=01J((IM8M#R ,ustainable groth rate L (1MIJ or 1M(IJN .f the co"pan% gros at the sustainable groth rate, the ne le*el of total assets is+ 4e -0 L 1(1MIJ(<8M,000 O A8,000# L <1MJ,J10(8I -o find the ne le*el of debt in the co"pan%'s balance sheet, e ta&e the percentage of debt in the capital structure ti"es the ne le*el of total assets( -he additional borroing ill be the ne le*el of debt "inus the current le*el of debt( ,o+ 4e -) L Q) : () O 6#R(-0# 4e -) L Q<8M,000 : (<8M,000 O A8,000#R(<1MJ,J10(8I# 4e -) L <100,1M0(MI B-44 SOLUTIONS 0nd the additional borroing ill be+ 0dditional borroing L <100,1M0(0I E 8M,000 0dditional borroing L <1I,1M0(MI -he groth rate that can be supported ith no outside financing is the internal groth rate( -o calculate the internal groth rate, e first need the R/0, hich is+ R/0 L <1J,A00 : (<8M,000 O A8,000# R/0 L (121A or 12(1AN -his "eans the internal groth rate is+ .nternal groth rate L (R/0 S b# : Q1 E (R/0 S b#R .nternal groth rate L Q(121A((IM8M#R : Q1 E (121A((IM8M#R .nternal groth rate L (0M0I or M(0IN 23. ,ince the co"pan% issued no ne e$uit%, shareholders' e$uit% increased b% retained earnings( Retained earnings for the %ear ere+ Retained earnings L 4. E )i*idends Retained earnings L <19,000 E 2,A00 Retained earnings L <1M,A00 ,o, the e$uit% at the end of the %ear as+ 6nding e$uit% L <1=A,000 O 1M,A00 6nding e$uit% L <1A1,A00 -he R/6 based on the end of period e$uit% is+ R/6 L <19,000 : <1A1,A00 R/6 L (12AI or 12(AIN -he plobac& ratio is+ 9lobac& ratio L 0ddition to retained earnings:4. 9lobac& ratio L <1M,A00 : <19,000 9lobac& ratio L (8M8I or 8M(8IN Using the e$uation presented in the te!t for the sustainable groth rate, e get+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L Q(12AI((8M8I#R : Q1 E (12AI((8M8I#R ,ustainable groth rate L (1222 or 12(22N -he R/6 based on the beginning of period e$uit% is R/6 L <1M,A00 : <1=A,000 R/6 L (1I0J or 1I(0JN CHAPTER 4 B-45 Using the shortened e$uation for the sustainable groth rate and the beginning of period R/6, e get+ ,ustainable groth rate L R/6 S b ,ustainable groth rate L (1I0J S (8M8I ,ustainable groth rate L (1222 or 12(22N Using the shortened e$uation for the sustainable groth rate and the end of period R/6, e get+ ,ustainable groth rate L R/6 S b ,ustainable groth rate L (12AI S (8M8I ,ustainable groth rate L (1089 or 10(89N Using the end of period R/6 in the shortened sustainable groth rate results in a groth rate that is too lo( -his ill ala%s occur hene*er the e$uit% increases( .f e$uit% increases, the R/6 based on end of period e$uit% is loer than the R/6 based on the beginning of period e$uit%( -he R/6 (and sustainable groth rate# in the abbre*iated e$uation is based on e$uit% that did not e!ist hen the net inco"e as earned( 24. -he R/0 using end of period assets is+ R/0 L <19,000 : <2A0,000 R/0 L (0JM0 or J(M0N -he beginning of period assets had to ha*e been the ending assets "inus the addition to retained earnings, so+ Heginning assets L 6nding assets E 0ddition to retained earnings Heginning assets L <2A0,000 E 1M,A00 Heginning assets L <2==,A00 0nd the R/0 using beginning of period assets is+ R/0 L <19,000 : <2==,A00 R/0 L (081I or 8(1IN Using the internal groth rate e$uation presented in the te!t, e get+ .nternal groth rate L (R/0 S b# : Q1 E (R/0 S b#R .nternal groth rate L Q(081I((8M8I#R : Q1 E (081I((8M8I#R .nternal groth rate L (0J0J or J(0JN Using the for"ula R/0 S b, and end of period assets+ .nternal groth rate L (0JM0 S (8M8I .nternal groth rate L (0MM0 or M(M0N Using the for"ula R/0 S b, and beginning of period assets+ .nternal groth rate L (081I S (8M8I .nternal groth rate L (0J0J or J(0JN B-46 SOLUTIONS 2. 0ssu"ing costs *ar% ith sales and a 20 percent increase in sales, the pro for"a inco"e state"ent ill loo& li&e this+ 3//,6 -/UR, .4C( 9ro Bor"a .nco"e ,tate"ent ,ales < 1,11I,800 Costs 8MJ,M00 /ther e!penses 22,800 6H.- < 22I,I00 .nterest 1I,000 -a!able inco"e < 210,I00 -a!es(=AN# J=,MI0 4et inco"e < 1=M,JM0 -he pa%out ratio is constant, so the di*idends paid this %ear is the pa%out ratio fro" last %ear ti"es net inco"e, or+ )i*idends L (<==,J=A:<112,IA0#(<1=M,JM0# )i*idends L <I1,028 0nd the addition to retained earnings ill be+ 0ddition to retained earnings L <1=M,JM0 E I1,028 0ddition to retained earnings L <9A,J=2 -he ne retained earnings on the pro for"a balance sheet ill be+ 4e retained earnings L <182,900 O 9A,J=2 4e retained earnings L <2J8,M=2 -he pro for"a balance sheet ill loo& li&e this+ 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =0,=M0 0ccounts pa%able < 81,M00 0ccounts recei*able I8,8I0 4otes pa%able 1J,000 .n*entor% 10I,280 -otal < 98,M00 -otal < 18=,I80 Cong-ter" debt 1A8,000 Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent I9A,M00 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 2J8,M=2 -otal < I18,M=2 -otal liabilities and oners' -otal assets < MJ9,080 e$uit% < MJA,2=2 CHAPTER 4 B-47 ,o the 6B4 is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <MJ9,080 E MJA,2=2 6B4 L <=,8I8 2!. Birst, e need to calculate full capacit% sales, hich is+ Bull capacit% sales L <929,000 : (80 Bull capacit% sales L <1,1M1,2A0 -he capital intensit% ratio at full capacit% sales is+ Capital intensit% ratio L Bi!ed assets : Bull capacit% sales Capital intensit% ratio L <I1=,000 : <1,1M1,2A0 Capital intensit% ratio L (=AAMA -he fi!ed assets re$uired at full capacit% sales is the capital intensit% ratio ti"es the pro1ected sales le*el+ -otal fi!ed assets L (=AAMA(<1,1M1,2A0# L <=9M,I80 ,o, 6B4 is+ 6B4 L (<18=,I80 O =9M,I80# E <M1=,80M L E<9A,2J2 4ote that this solution assu"es that fi!ed assets are decreased (sold# so the co"pan% has a 100 percent fi!ed asset utili2ation( .f e assu"e fi!ed assets are not sold, the anser beco"es+ 6B4 L (<18=,I80 O I1=,000# E <M1=,80M L E<1MM,1AI 2". -he ):6 ratio of the co"pan% is+ ):6 L (<8A,000 O 1A8,000# : <=22,900 ):6 L (JA2M ,o the ne total debt a"ount ill be+ 4e total debt L (JA2M(<I18,M=2# 4e total debt L <=1A,0II -his is the ne total debt for the co"pan%( Gi*en that our calculation for 6B4 is the a"ount that "ust be raised e!ternall% and does not increase spontaneousl% ith sales, e need to subtract the spontaneous increase in accounts pa%able( -he ne le*el of accounts pa%able ill be, hich is the current accounts pa%able ti"es the sales groth, or+ ,pontaneous increase in accounts pa%able L <M8,000((20# ,pontaneous increase in accounts pa%able L <1=,M00 B-48 SOLUTIONS -his "eans that <1=,M00 of the ne total debt is not raised e!ternall%( ,o, the debt raised e!ternall%, hich ill be the 6B4 is+ 6B4 L 4e total debt E (Heginning C-) O Heginning CC O ,pontaneous increase in 09# 6B4 L <=1A,0II E (<1A8,000 O M8,000 O 1J,000 O 1=,M00# L <A8,III -he pro for"a balance sheet ith the ne long-ter" debt ill be+ 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =0,=M0 0ccounts pa%able < 81,M00 0ccounts recei*able II,I00 4otes pa%able 1J,000 .n*entor% 10I,280 -otal < 98,M00 -otal < 18=,I80 Cong-ter" debt 21M,III Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent I9A,M00 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 2J8,M=2 -otal < I18,M=2 -otal liabilities and oners' -otal assets < M9J,080 e$uit% < J==,MJM -he funds raised b% the debt issue can be put into an e!cess cash account to "a&e the balance sheet balance( -he e!cess debt ill be+ 6!cess debt L <J==,MJM E M9J,080 L <AI,A9M -o "a&e the balance sheet balance, the co"pan% ill ha*e to increase its assets( We ill put this a"ount in an account called e!cess cash, hich ill gi*e us the folloing balance sheet+ CHAPTER 4 B-49 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =0,=M0 0ccounts pa%able < 81,M00 6!cess cash AI,A9M 0ccounts recei*able II,I00 4otes pa%able 1J,000 .n*entor% 10I,280 -otal < 98,M00 -otal < 2=8,0JM Cong-ter" debt 21M,III Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent I9A,M00 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 2J8,M=2 -otal < I18,M=2 -otal liabilities and oners' -otal assets < J==,MJM e$uit% < J==,MJM -he e!cess cash has an opportunit% cost that e discussed earlier( .ncreasing fi!ed assets ould also not be a good idea since the co"pan% alread% has enough fi!ed assets( 0 li&el% scenario ould be the repurchase of debt and e$uit% in its current capital structure eights( -he co"pan%'s debt-assets and e$uit% assets are+ )ebt-assets L (JA2M : (1 O (JA2M# L (I= 6$uit%-assets L 1 : (1 O (JA2M# L (AJ ,o, the a"ount of debt and e$uit% needed ill be+ -otal debt needed L (I=(<M9J,080# L <291,M00 6$uit% needed L (AJ(<M9J,080# L <=8J,I80 ,o, the repurchases of debt and e$uit% ill be+ )ebt repurchase L (<98,M00 O 21M,III# E 291,M00 L <2=,III 6$uit% repurchase L <I18,M=2 E =8J,I80 L <=1,1A2 0ssu"ing all of the debt repurchase is fro" long-ter" debt, and the e$uit% repurchase is entirel% fro" the retained earnings, the final pro for"a balance sheet ill be+ B-50 SOLUTIONS 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =0,=M0 0ccounts pa%able < 81,M00 0ccounts recei*able II,I00 4otes pa%able 1J,000 .n*entor% 10I,280 -otal < 98,M00 -otal < 18=,I80 Cong-ter" debt 19=,000 Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent I9A,M00 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 2IJ,I80 -otal < =8J,I80 -otal liabilities and oners' -otal assets < M9J,080 e$uit% < M9J,080 Challenge 2#. -he pro for"a inco"e state"ents for all three groth rates ill be+ 3//,6 -/UR, .4C( 9ro Bor"a .nco"e ,tate"ent -. / 'ales 0rowth 12/ 'ales 0rowth 1./ 'ales 0rowth ,ales <1,0M8,=A0 <1,11I,800 <1,1M1,2A0 Costs 8=1,IA0 8MJ,M00 90=,JA0 /ther e!penses 21,8A0 22,800 2=,JA0 6H.- <21A,0A0 <22I,I00 <2==,JA0 .nterest 1I,000 1I,000 1I,000 -a!able inco"e <201,0A0 <210,I00 <219,JA0 -a!es (=AN# J0,=M8 J=,MI0 JM,91= 4et inco"e <1=0,M8= <1=M,JM0 <1I2,8=8 )i*idends <=9,20A <I1,028 <I2,8A1 0dd to R6 91,IJ8 9A,J=2 99,98M We ill calculate the 6B4 for the 1A percent groth rate first( 0ssu"ing the pa%out ratio is constant, the di*idends paid ill be+ )i*idends L (<==,J=A:<112,IA0#(<1=0,M8=# )i*idends L <=9,20A 0nd the addition to retained earnings ill be+ CHAPTER 4 B-51 0ddition to retained earnings L <1=0,M8= E =9,20A 0ddition to retained earnings L <91,IJ8 -he ne retained earnings on the pro for"a balance sheet ill be+ 4e retained earnings L <182,900 O 91,IJ8 4e retained earnings L <2JI,=J8 -he pro for"a balance sheet ill loo& li&e this+ -./ 'ales 0rowth+ 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < 29,09A 0ccounts pa%able < J8,200 0ccounts recei*able IM,80A 4otes pa%able 1J,000 .n*entor% 99,9=A -otal < 9A,200 -otal < 1JA,8=A Cong-ter" debt < 1A8,000 Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent IJI,9A0 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 2JI,=J8 -otal < I1I,=J8 -otal liabilities and oners' -otal assets < MA0,J8A e$uit% < MMJ,AJ8 ,o the 6B4 is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <MA0,J8A E MMJ,AJ8 6B4 L E<1M,J9= 0t a 20 percent groth rate, and assu"ing the pa%out ratio is constant, the di*idends paid ill be+ )i*idends L (<==,J=A:<112,IA0#(<1=M,JM0# )i*idends L <I1,028 0nd the addition to retained earnings ill be+ 0ddition to retained earnings L <1=M,JM0 E I1,028 0ddition to retained earnings L <9A,J=2 -he ne retained earnings on the pro for"a balance sheet ill be+ 4e retained earnings L <182,900 O 9A,J=2 4e retained earnings L <2J8,M=2 B-52 SOLUTIONS -he pro for"a balance sheet ill loo& li&e this+ 12/ 'ales 0rowth+ 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =0,=M0 0ccounts pa%able < 81,M00 0ccounts recei*able I8,8I0 4otes pa%able 1J,000 .n*entor% 10I,280 -otal < 98,M00 -otal < 18=,I80 Cong-ter" debt < 1A8,000 Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent I9A,M00 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 2J8,M=2 -otal < I18,M=2 -otal liabilities and oners' -otal assets < MJ9,080 e$uit% < MJA,2=2 ,o the 6B4 is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <MJ9,080 E MJA,2=2 6B4 L <=,8I8 0t a 2A percent groth rate, and assu"ing the pa%out ratio is constant, the di*idends paid ill be+ )i*idends L (<==,J=A:<112,IA0#(<1I2,8=8# )i*idends L <I2,8A1 0nd the addition to retained earnings ill be+ 0ddition to retained earnings L <1I2,8=8 E I2,8A1 0ddition to retained earnings L <99,98M -he ne retained earnings on the pro for"a balance sheet ill be+ 4e retained earnings L <182,900 O 99,98M 4e retained earnings L <282,88M -he pro for"a balance sheet ill loo& li&e this+ CHAPTER 4 B-53 1./ 'ales 0rowth+ 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =1,M2A 0ccounts pa%able < 8A,000 0ccounts recei*able A0,8JA 4otes pa%able 1J,000 .n*entor% 108,M2A -otal < 102,000 -otal < 191,12A Cong-ter" debt < 1A8,000 Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent A1M,2A0 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 282,88M -otal < I22,88M -otal liabilities and oners' -otal assets < J0J,=JA e$uit% < M82,88M ,o the 6B4 is+ 6B4 L -otal assets E -otal liabilities and e$uit% 6B4 L <J0J,=JA E M82,88M 6B4 L <2I,889 2$. -he pro for"a inco"e state"ents for all three groth rates ill be+ 3//,6 -/UR, .4C( 9ro Bor"a .nco"e ,tate"ent 12/ 'ales 0rowth 32/ 'ales 0rowth 3./ 'ales 0rowth ,ales <1,11I,800 <1,20J,J00 <1,2AI,1A0 Costs 8MJ,M00 9=9,900 9JM,0A0 /ther e!penses 22,800 2I,J00 2A,MA0 6H.- <22I,I00 <2I=,100 <2A2,IA0 .nterest 1I,000 1I,000 1I,000 -a!able inco"e <210,I00 <229,100 <2=8,IA0 -a!es (=AN# J=,MI0 80,18A 8=,IA8 4et inco"e <1=M,JM0 <1I8,91A <1AI,99= )i*idends <I1,028 <II,MJA <IM,I98 0dd to R6 9A,J=2 10I,2I1 108,I9A 0t a =0 percent groth rate, and assu"ing the pa%out ratio is constant, the di*idends paid ill be+ )i*idends L (<=0,810:<102,J00#(<1=A,9I8# )i*idends L <I0,J8I 0nd the addition to retained earnings ill be+ B-54 SOLUTIONS 0ddition to retained earnings L <1=A,9I8 E I0,J8I 0ddition to retained earnings L <10I,2I1 -he ne addition to retained earnings on the pro for"a balance sheet ill be+ 4e addition to retained earnings L <182,900 O 10I,2I1 4e addition to retained earnings L <28J,1I1 -he ne total debt ill be+ 4e total debt L (JAAM(<I2J,1I1# 4e total debt L <=21,IIJ ,o, the ne long-ter" debt ill be the ne total debt "inus the ne short-ter" debt, or+ 4e long-ter" debt L <=21,IIJ E 10A,I00 4e long-ter" debt L <A8,0IJ -he pro for"a balance sheet ill loo& li&e this+ 'ales growth rate 4 32/ and debt5e!uit ratio 4 .6.17: 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =2,890 0ccounts pa%able < 88,I00 0ccounts recei*able A2,910 4otes pa%able 1J,000 .n*entor% 112,9J0 -otal < 10A,I00 -otal < 198,JJ0 Cong-ter" debt 21M,0IJ Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent A=M,900 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 28J,1I1 -otal < I2J,1I1 -otal liabilities and oners' -otal assets < J=A,MJ0 e$uit% < JI8,A8J ,o the e!cess debt raised is+ 6!cess debt L <JI8,A8J E J=A,MJ0 6!cess debt L <12,91J 0t a =A percent groth rate, and assu"ing the pa%out ratio is constant, the di*idends paid ill be+ )i*idends L (<=0,810:<102,J00#(<1AI,99=# )i*idends L <IM,I98 CHAPTER 4 B-55 0nd the addition to retained earnings ill be+ 0ddition to retained earnings L <1AI,99= E IM,I98 0ddition to retained earnings L <108,I9A -he ne retained earnings on the pro for"a balance sheet ill be+ 4e retained earnings L <182,900 O 108,I9A 4e retained earnings L <291,=9A -he ne total debt ill be+ 4e total debt L (JA2AA(<I=1,=9A# 4e total debt L <=2I,MI8 ,o, the ne long-ter" debt ill be the ne total debt "inus the ne short-ter" debt, or+ 4e long-ter" debt L <=2I,MI8 E 108,800 4e long-ter" debt L <21A,8I8 B-56 SOLUTIONS 'ales growth rate 4 3./ and debt5e!uit ratio 4 .6.1..: 3//,6 -/UR, .4C( 9ro Bor"a Halance ,heet 0ssets Ciabilities and /ners' 6$uit% Current assets Current liabilities Cash < =I,1AA 0ccounts pa%able < 91,800 0ccounts recei*able AI,9IA 4otes pa%able 1J,000 .n*entor% 11J,=1A -otal < 108,800 -otal < 20M,I1A Cong-ter" debt < 21A,8I8 Bi!ed assets 4et plant and /ners' e$uit% e$uip"ent AAJ,AA0 Co""on stoc& and paid-in surplus < 1I0,000 Retained earnings 291,=9A -otal < I=1,=9A -otal liabilities and oners' -otal assets < JM=,9MA e$uit% < JAM,0I= ,o the e!cess debt raised is+ 6!cess debt L <JAM,0I= E JM=,9MA 6!cess debt L E<J,922 0t a =A percent groth rate, the fir" ill need funds in the a"ount of <J,922 in addition to the e!ternal debt alread% raised( ,o, the 6B4 ill be+ 6B4 L <AJ,8I8 O J,922 6B4 L <MA,JJ0 3%. We "ust need the R/6 to calculate the sustainable groth rate( -he R/6 is+ R/6 L (93#(-0-#(63# R/6 L ((0MJ#(1 : 1(=A#(1 O 0(=0# R/6 L (0MIA or M(IAN 4o e can use the sustainable groth rate e$uation to find the retention ratio as+ ,ustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R ,ustainable groth rate L (12 L Q(0MIA(b#R : Q1 E (0MIA(b# b L 1(MM -his i"plies the pa%out ratio is+ 9a%out ratio L 1 E b 9a%out ratio L 1 E 1(MM 9a%out ratio L E0(MM CHAPTER 4 B-57 -his is a negati*e di*idend pa%out ratio of MM percent, hich is i"possible( -he groth rate is not consistent ith the other constraints( -he loest possible pa%out rate is 0, hich corresponds to retention ratio of 1, or total earnings retention( -he "a!i"u" sustainable groth rate for this co"pan% is+ 3a!i"u" sustainable groth rate L (R/6 S b# : Q1 E (R/6 S b#R 3a!i"u" sustainable groth rate L Q(0MIA(1#R : Q1 E (0MIA(1#R 3a!i"u" sustainable groth rate L (0M90 or M(90N 31. We &no that 6B4 is+ 6B4 L .ncrease in assets E 0ddition to retained earnings -he increase in assets is the beginning assets ti"es the groth rate, so+ .ncrease in assets L 0 × g -he addition to retained earnings ne!t %ear is the current net inco"e ti"es the retention ratio, ti"es one plus the groth rate, so+ 0ddition to retained earnings L (4. × b#(1 O g# 0nd rearranging the profit "argin to sol*e for net inco"e, e get+ 4. L 93(,# ,ubstituting the last three e$uations into the 6B4 e$uation e started ith and rearranging, e get+ 6B4 L 0(g# E 93(,#b(1 O g# 6B4 L 0(g# E 93(,#b E Q93(,#bRg 6B4 L E 93(,#b O Q0 E 93(,#bRg 32. We start ith the 6B4 e$uation e deri*ed in 9roble" =1 and set it e$ual to 2ero+ 6B4 L 0 L E 93(,#b O Q0 E 93(,#bRg ,ubstituting the rearranged profit "argin e$uation into the internal groth rate e$uation, e ha*e+ .nternal groth rate L Q93(,#b R : Q0 E 93(,#bR ,ince+ R/0 L 4. : 0 R/0 L 93(,# : 0 We can substitute this into the internal groth rate e$uation and di*ide both the nu"erator and deno"inator b% 0( -his gi*es+ .nternal groth rate L YQ93(,#bR : 0Z : YQ0 E 93(,#bR : 0Z B-58 SOLUTIONS .nternal groth rate L b(R/0# : Q1 E b(R/0#R CHAPTER 4 B-59 -o deri*e the sustainable groth rate, e "ust reali2e that to "aintain a constant ):6 ratio ith no e!ternal e$uit% financing, 6B4 "ust e$ual the addition to retained earnings ti"es the ):6 ratio+ 6B4 L ():6#Q93(,#b(1 O g#R 6B4 L 0(g# E 93(,#b(1 O g# ,ol*ing for g and then di*iding nu"erator and deno"inator b% 0+ ,ustainable groth rate L 93(,#b(1 O ):6# : Q0 E 93(,#b(1 O ):6 #R ,ustainable groth rate L QR/0(1 O ):6 #bR : Q1 E R/0(1 O ):6 #bR ,ustainable groth rate L b(R/6# : Q1 E b(R/6#R 33. .n the folloing deri*ations, the subscript ;6@ refers to end of period nu"bers, and the subscript ;H@ refers to beginning of period nu"bers( -6 is total e$uit% and -0 is total assets( Bor the sustainable groth rate: ,ustainable groth rate L (R/66 S b# : (1 E R/66 S b# ,ustainable groth rate L (4.:-66 S b# : (1 E 4.:-66 S b# We "ultipl% this e$uation b%+ (-66 : -66# ,ustainable groth rate L (4. : -66 S b# : (1 E 4. : -66 S b# S (-66 : -66# ,ustainable groth rate L (4. S b# : (-66 E 4. S b# Recogni2e that the nu"erator is e$ual to beginning of period e$uit%, that is+ (-66 E 4. S b# L -6H ,ubstituting this into the pre*ious e$uation, e get+ ,ustainable rate L (4. S b# : -6H Which is e$ui*alent to+ ,ustainable rate L (4. : -6H# S b ,ince R/6H L 4. : -6H -he sustainable groth rate e$uation is+ ,ustainable groth rate L R/6H S b Bor the internal groth rate+ .nternal groth rate L (R/06 S b# : (1 E R/06 S b# .nternal groth rate L (4. : -06 S b# : (1 E 4. : -06 S b# B-60 SOLUTIONS We "ultipl% this e$uation b%+ (-06 : -06# .nternal groth rate L (4. : -06 S b# : (1 E 4. : -06 S b# S (-06 : -06# .nternal groth rate L (4. S b# : (-06 E 4. S b# Recogni2e that the nu"erator is e$ual to beginning of period assets, that is+ (-06 E 4. S b# L -0H ,ubstituting this into the pre*ious e$uation, e get+ .nternal groth rate L (4. S b# : -0H Which is e$ui*alent to+ .nternal groth rate L (4. : -0H# S b ,ince R/0H L 4. : -0H -he internal groth rate e$uation is+ .nternal groth rate L R/0H S b CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. -he four parts are the present *alue (9V#, the future *alue (BV#, the discount rate ( r#, and the life of the in*est"ent (t#( 2. Co"pounding refers to the groth of a dollar a"ount through ti"e *ia rein*est"ent of interest earned( .t is also the process of deter"ining the future *alue of an in*est"ent( )iscounting is the process of deter"ining the *alue toda% of an a"ount to be recei*ed in the future( 3. Buture *alues gro (assu"ing a positi*e rate of return#8 present *alues shrin&( 4. -he future *alue rises (assu"ing it's positi*e#8 the present *alue falls( . .t ould appear to be both decepti*e and unethical to run such an ad ithout a disclai"er or e!planation( !. .t's a reflection of the ti"e *alue of "one%( -3CC gets to use the <2I,099( .f -3CC uses it isel%, it ill be orth "ore than <100,000 in thirt% %ears( ". -his ill probabl% "a&e the securit% less desirable( -3CC ill onl% repurchase the securit% prior to "aturit% if it is to its ad*antage, i(e( interest rates decline( Gi*en the drop in interest rates needed to "a&e this *iable for -3CC, it is unli&el% the co"pan% ill repurchase the securit%( -his is an e!a"ple of a ;call@ feature( ,uch features are discussed at length in a later chapter( #. -he &e% considerations ould be+ (1# .s the rate of return i"plicit in the offer attracti*e relati*e to other, si"ilar ris& in*est"ents? and (2# >o ris&% is the in*est"ent8 i(e(, ho certain are e that e ill actuall% get the <100,000? -hus, our anser does depend on ho is "a&ing the pro"ise to repa%( $. -he -reasur% securit% ould ha*e a so"ehat higher price because the -reasur% is the strongest of all borroers( 1%. -he price ould be higher because, as ti"e passes, the price of the securit% ill tend to rise toard <100,000( -his rise is 1ust a reflection of the ti"e *alue of "one%( 0s ti"e passes, the ti"e until receipt of the <100,000 gros shorter, and the present *alue rises( .n 2019, the price ill probabl% be higher for the sa"e reason( We cannot be sure, hoe*er, because interest rates could be "uch higher, or -3CC's financial position could deteriorate( 6ither e*ent ould tend to depress the securit%'s price( B-62 SOLUTIONS Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he si"ple interest per %ear is+ <A,000 S (08 L <I00 ,o after 10 %ears %ou ill ha*e+ <I00 S 10 L <I,000 in interest( -he total balance ill be <A,000 O I,000 L <9,000 With co"pound interest e use the future *alue for"ula+ BV L 9V(1 Or# t BV L <A,000(1(08# 10 L <10,J9I(M2 -he difference is+ <10,J9I(M2 E 9,000 L <1,J9I(M2 2. -o find the BV of a lu"p su", e use+ BV L 9V(1 O r# t BV L <2,2A0(1(10# 11 L < M,I19(A1 BV L <8,JA2(1(08# J L < 1I,999(=9 BV L <JM,=AA(1(1J# 1I L <M8J,JMI(1J BV L <18=,J9M(1(0J# 8 L <=1A,J9A(JA 3. -o find the 9V of a lu"p su", e use+ 9V L BV : (1 O r8 t 9V L <1A,IA1 : (1(0J# M L < 10,29A(MA 9V L <A1,AAJ : (1(1=# J L < 21,91I(8A 9V L <88M,0J= : (1(1I# 2= L < I=,A1M(90 9V L <AA0,1MI : (1(09# 18 L <11M,M=1(=2 CHAPTER 5 B-63 4. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for r, e get+ r L (BV : 9V# 1 : t E 1 BV L <29J L <2I0(1 O r# 2 8 r L (<29J : <2I0# 1:2 E 1 L 11(2IN BV L <1,080 L <=M0(1 O r# 10 8 r L (<1,080 : <=M0# 1:10 E 1 L 11(M1N BV L <18A,=82 L <=9,000(1 O r# 1A 8 r L (<18A,=82 : <=9,000# 1:1A E 1 L 10(9AN BV L <A=1,M18 L <=8,2M1(1 O r# =0 8 r L (<A=1,M18 : <=8,2M1# 1:=0 E 1 L 9(1JN . -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for t, e get+ t L ln(BV : 9V# : ln(1 O r# BV L <1,28I L <AM0(1(09# t 8 t L ln(<1,28I: <AM0# : ln 1(09 L 9(M= %ears BV L <I,=I1 L <810(1(10# t 8 t L ln(<I,=I1: <810# : ln 1(10 L 1J(M1 %ears BV L <=MI,A18 L <18,I00(1(1J# t 8 t L ln(<=MI,A18 : <18,I00# : ln 1(1J L 19(02 %ears BV L <1J=,I=9 L <21,A00(1(1A# t 8 t L ln(<1J=,I=9 : <21,A00# : ln 1(1A L 1I(9I %ears !. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for r, e get+ r L (BV : 9V# 1 : t E 1 r L (<290,000 : <AA,000# 1:18 E 1 L (09M8 or 9(M8N B-64 SOLUTIONS ". -o find the length of ti"e for "one% to double, triple, etc(, the present *alue and future *alue are irrele*ant as long as the future *alue is tice the present *alue for doubling, three ti"es as large for tripling, etc( -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for t, e get+ t L ln(BV : 9V# : ln(1 O r# -he length of ti"e to double %our "one% is+ BV L <2 L <1(1(0J# t t L ln 2 : ln 1(0J L 10(2I %ears -he length of ti"e to $uadruple %our "one% is+ BV L <I L <1(1(0J# t t L ln I : ln 1(0J L 20(I9 %ears 4otice that the length of ti"e to $uadruple %our "one% is tice as long as the ti"e needed to double %our "one% (the difference in these ansers is due to rounding#( -his is an i"portant concept of ti"e *alue of "one%( #. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for r, e get+ r L (BV : 9V# 1 : t E 1 r L (<=1I,M00 : <200,=00# 1:J E 1 L (0MMM or M(MMN $. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for t, e get+ t L ln(BV : 9V# : ln(1 O r# t L ln (<1J0,000 : <I0,000# : ln 1(0A= L 28(02 %ears 1%. -o find the 9V of a lu"p su", e use+ 9V L BV : (1 O r8 t 9V L <MA0,000,000 : (1(0JI# 20 L <1AA,89=,I00(1= CHAPTER 5 B-65 11. -o find the 9V of a lu"p su", e use+ 9V L BV : (1 O r8 t 9V L <1,000,000 : (1(10# 80 L <I88(19 12. -o find the BV of a lu"p su", e use+ BV L 9V(1 O r# t BV L <A0(1(0IA# 10A L <A,08=(J1 13. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for r, e get+ r L (BV : 9V# 1 : t E 1 r L (<1,2M0,000 : <1A0# 1:112 E 1 L (08I0 or 8(I0N -o find the BV of the first pri2e, e use+ BV L 9V(1 O r# t BV L <1,2M0,000(1(08I0# == L <18,0AM,I09(9I 14. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for r, e get+ r L (BV : 9V# 1 : t E 1 r L (<I=,12A : <1# 1:11= E 1 L (0990 or 9(90N 1. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for r, e get+ r L (BV : 9V# 1 : t E 1 r L (<10,=11,A00 : <12,=JJ,A00# 1:I E 1 L E I(IMN 4otice that the interest rate is negati*e( -his occurs hen the BV is less than the 9V( B-66 SOLUTIONS &ntermediate 1!. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for r, e get+ r L (BV : 9V# 1 : t E 1 a. 9V L <100,000 : (1 O r# =0 L <2I,099 r L (<100,000 : <2I,099# 1:=0 E 1 L (0I8M or I(8MN b. 9V L <=8,2M0 : (1 O r# 12 L <2I,099 r L (<=8,2M0 : <2I,099# 1:12 E 1 L (0=9= or =(9=N c. 9V L <100,000 : (1 O r# 18 L <=8,2M0 r L (<100,000 : <=8,2M0# 1:18 E 1 L (0AI8 or A(I8N 1". -o find the 9V of a lu"p su", e use+ 9V L BV : (1 O r8 t 9V L <1J0,000 : (1(12# 9 L <M1,=0=(J0 1#. -o find the BV of a lu"p su", e use+ BV L 9V(1 O r# t BV L <I,000(1(11# IA L <I=8,120(9J BV L <I,000(1(11# =A L <1AI,299(I0 Hetter start earl%X 1$. We need to find the BV of a lu"p su"( >oe*er, the "one% ill onl% be in*ested for si! %ears, so the nu"ber of periods is si!( BV L 9V(1 O r# t BV L <20,000(1(08I# M L <=2,II9(== CHAPTER 5 B-67 2%. -o anser this $uestion, e can use either the BV or the 9V for"ula( Hoth ill gi*e the sa"e anser since the% are the in*erse of each other( We ill use the BV for"ula, that is+ BV L 9V(1 O r# t ,ol*ing for t, e get+ t L ln(BV : 9V# : ln(1 O r# t L ln(<JA,000 : <10,000# : ln(1(11# L 19(=1 ,o, the "one% "ust be in*ested for 19(=1 %ears( >oe*er, %ou ill not recei*e the "one% for another to %ears( Bro" no, %ou'll ait+ 2 %ears O 19(=1 %ears L 21(=1 %ears Calculator Solutions 1. 6nter 10 8N <A,000 - ./0 &1 &MT 21 ,ol*e for <10,J9I(M2 <10,J9I(M2 E 9,000 L <1,J9I(M2 2. 6nter 11 10N <2,2A0 - ./0 &1 &MT 21 ,ol*e for <M,I19(A1 6nter J 8N <8,JA2 - ./0 &1 &MT 21 ,ol*e for <1I,999(=9 6nter 1I 1JN <JM,=AA - ./0 &1 &MT 21 ,ol*e for <M8J,JMI(1J 6nter 8 JN <18=,J9M - ./0 &1 &MT 21 ,ol*e for <=1A,J9A(JA 3. 6nter M JN <1A,IA1 - ./0 &1 &MT 21 ,ol*e for <10,29A(MA B-68 SOLUTIONS 6nter J 1=N <A1,AAJ - ./0 &1 &MT 21 ,ol*e for <21,91I(8A 6nter 2= 1IN <88M,0J= - ./0 &1 &MT 21 ,ol*e for <I=,A1M(90 6nter 18 9N <AA0,1MI - ./0 &1 &MT 21 ,ol*e for <11M,M=1(=2 4. 6nter 2 <2I0 t<29J - ./0 &1 &MT 21 ,ol*e for 11(2IN 6nter 10 <=M0 t<1,080 - ./0 &1 &MT 21 ,ol*e for 11(M1N 6nter 1A <=9,000 t<18A,=82 - ./0 &1 &MT 21 ,ol*e for 10(9AN 6nter =0 <=8,2M1 t<A=1,M18 - ./0 &1 &MT 21 ,ol*e for 9(1JN . 6nter 9N <AM0 t<1,28I - ./0 &1 &MT 21 ,ol*e for 9(M= 6nter 10N <810 t<I,=I1 - ./0 &1 &MT 21 ,ol*e for 1J(M1 6nter 1JN <18,I00 t<=MI,A18 - ./0 &1 &MT 21 ,ol*e for 19(02 CHAPTER 5 B-69 6nter 1AN <21,A00 t<1J=,I=9 - ./0 &1 &MT 21 ,ol*e for 1I(9I !. 6nter 18 <AA,000 t<290,000 - ./0 &1 &MT 21 ,ol*e for 9(M8N ". 6nter JN <1 t<2 - ./0 &1 &MT 21 ,ol*e for 10(2I 6nter JN <1 t<I - ./0 &1 &MT 21 ,ol*e for 20(I9 #. 6nter J <200,=00 t<=1I,M00 - ./0 &1 &MT 21 ,ol*e for M(MMN $. 6nter A(=0N <I0,000 t<1J0,000 - ./0 &1 &MT 21 ,ol*e for 28(02 1%. 6nter 20 J(IN <MA0,000,000 - ./0 &1 &MT 21 ,ol*e for <1AA,89=,I00(1= 11. 6nter 80 10N <1,000,000 - ./0 &1 &MT 21 ,ol*e for <I88(19 12. 6nter 10A I(A0N <A0 - ./0 &1 &MT 21 ,ol*e for <A,08=(J1 B-70 SOLUTIONS 13. 6nter 112 t<1A0 <1,2M0,000 - ./0 &1 &MT 21 ,ol*e for 8(I0N 6nter == 8(I0N <1,2M0,000 - ./0 &1 &MT 21 ,ol*e for <18,0AM,I0I(9I 14. 6nter 11= <1 [<I=,12A - ./0 &1 &MT 21 ,ol*e for 9(90N 1. 6nter I t<12,=JJ,A00 <10,=11,A00 - ./0 &1 &MT 21 ,ol*e for EI(IMN 1!. a. 6nter =0 t<2I,099 <100,000 - ./0 &1 &MT 21 ,ol*e for I(8MN 1!. b. 6nter 12 t<2I,099 <=8,2M0 - ./0 &1 &MT 21 ,ol*e for =(9=N 1!. c. 6nter 18 t<=8,2M0 <100,000 - ./0 &1 &MT 21 ,ol*e for A(I8N 1". 6nter 9 12N <1J0,000 - ./0 &1 &MT 21 ,ol*e for <M1,=0=(J0 1#. 6nter IA 11N <I,000 - ./0 &1 &MT 21 ,ol*e for <I=8,120(9J 6nter =A 11N <I,000 - ./0 &1 &MT 21 ,ol*e for <1AI,299(I0 CHAPTER 5 B-71 1$. 6nter M 8(I0N <20,000 - ./0 &1 &MT 21 ,ol*e for <=2,II9(== 2%. 6nter 11N t<10,000 <JA,000 - ./0 &1 &MT 21 ,ol*e for 19(=1 Bro" no, %ou'll ait 2 O 19(=1 L 21(=1 %ears CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. -he four pieces are the present *alue (9V#, the periodic cash flo (C#, the discount rate (r#, and the nu"ber of pa%"ents, or the life of the annuit%, t( 2. 0ssu"ing positi*e cash flos, both the present and the future *alues ill rise( 3. 0ssu"ing positi*e cash flos, the present *alue ill fall and the future *alue ill rise( 4. .t's decepti*e, but *er% co""on( -he basic concept of ti"e *alue of "one% is that a dollar toda% is not orth the sa"e as a dollar to"orro( -he deception is particularl% irritating gi*en that such lotteries are usuall% go*ern"ent sponsoredX . .f the total "one% is fi!ed, %ou ant as "uch as possible as soon as possible( -he tea" (or, "ore accuratel%, the tea" oner# ants 1ust the opposite( !. -he better deal is the one ith e$ual install"ents( ". 5es, the% should( 09Rs generall% don't pro*ide the rele*ant rate( -he onl% ad*antage is that the% are easier to co"pute, but, ith "odern co"puting e$uip"ent, that ad*antage is not *er% i"portant( #. 0 fresh"an does( -he reason is that the fresh"an gets to use the "one% for "uch longer before interest starts to accrue( -he subsid% is the present *alue (on the da% the loan is "ade# of the interest that ould ha*e accrued up until the ti"e it actuall% begins to accrue( $. -he proble" is that the subsid% "a&es it easier to repa% the loan, not obtain it( >oe*er, abilit% to repa% the loan depends on future e"plo%"ent, not current need( Bor e!a"ple, consider a student ho is currentl% need%, but is preparing for a career in a high-pa%ing area (such as corporate financeX#( ,hould this student recei*e the subsid%? >o about a student ho is currentl% not need%, but is preparing for a relati*el% lo- pa%ing 1ob (such as beco"ing a college professor#? CHAPTER 6 B-73 1%. .n general, *iatical settle"ents are ethical( .n the case of a *iatical settle"ent, it is si"pl% an e!change of cash toda% for pa%"ent in the future, although the pa%"ent depends on the death of the seller( -he purchaser of the life insurance polic% is bearing the ris& that the insured indi*idual ill li*e longer than e!pected( 0lthough *iatical settle"ents are ethical, the% "a% not be the best choice for an indi*idual( .n a %usiness 9ee: article (/ctober =1, 200A#, options ere e!a"ined for a J2 %ear old "ale ith a life e!pectanc% of 8 %ears and a <1 "illion dollar life insurance polic% ith an annual pre"iu" of <=J,000( -he four options ere+ 1# Cash the polic% toda% for <100,000( 2# ,ell the polic% in a *iatical settle"ent for <2JA,000( =# Reduce the death benefit to <=JA,000, hich ould &eep the polic% in force for 12 %ears ithout pre"iu" pa%"ents( I# ,top pa%ing pre"iu"s and don't reduce the death benefit( -his ill run the cash *alue of the polic% to 2ero in A %ears, but the *iatical settle"ent ould be orth <IJA,000 at that ti"e( .f he died ithin A %ears, the beneficiaries ould recei*e <1 "illion( Ulti"atel%, the decision rests on the indi*idual on hat the% percei*e as best for the"sel*es( -he *alues that ill affect the *alue of the *iatical settle"ent are the discount rate, the face *alue of the polic%, and the health of the indi*idual selling the polic%( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -o sol*e this proble", e "ust find the 9V of each cash flo and add the"( -o find the 9V of a lu"p su", e use+ 9V L BV : (1 O r8 t 9V\10N L <9A0 : 1(10 O <1,0I0 : 1(10 2 O <1,1=0 : 1(10 = O <1,0JA : 1(10 I L <=,=0M(=J 9V\18N L <9A0 : 1(18 O <1,0I0 : 1(18 2 O <1,1=0 : 1(18 = O <1,0JA : 1(18 I L <2,J9I(22 9V\2IN L <9A0 : 1(2I O <1,0I0 : 1(2I 2 O <1,1=0 : 1(2I = O <1,0JA : 1(2I I L <2,I89(88 2. -o find the 9V0, e use the e$uation+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r # 0t a A percent interest rate+ U\AN+ 9V0 L <M,000YQ1 E (1:1(0A# 9 R : (0A Z L <I2,MIM(9= 5\AN+ 9V0 L <8,000YQ1 E (1:1(0A# M R : (0A Z L <I0,M0A(AI B-74 SOLUTIONS 0nd at a 1A percent interest rate+ U\1AN+ 9V0 L <M,000YQ1 E (1:1(1A# 9 R : (1A Z L <28,M29(A0 5\1AN+ 9V0 L <8,000YQ1 E (1:1(1A# M R : (1A Z L <=0,2JA(8M 4otice that the 9V of cash flo U has a greater 9V at a A percent interest rate, but a loer 9V at a 1A percent interest rate( -he reason is that U has greater total cash flos( 0t a loer interest rate, the total cash flo is "ore i"portant since the cost of aiting (the interest rate# is not as great( 0t a higher interest rate, 5 is "ore *aluable since it has larger cash flos( 0t the higher interest rate, these bigger cash flos earl% are "ore i"portant since the cost of aiting (the interest rate# is so "uch greater( 3. -o sol*e this proble", e "ust find the BV of each cash flo and add the"( -o find the BV of a lu"p su", e use+ BV L 9V(1 O r8 t BV\8N L <9I0(1(08# = O <1,090(1(08# 2 O <1,=I0(1(08# O <1,I0A L <A,=0J(J1 BV\11N L <9I0(1(11# = O <1,090(1(11# 2 O <1,=I0(1(11# O <1,I0A L <A,A20(9M BV\2IN L <9I0(1(2I# = O <1,090(1(2I# 2 O <1,=I0(1(2I# O <1,I0A L <M,A=I(81 4otice e are finding the *alue at 5ear I, the cash flo at 5ear I is si"pl% added to the BV of the other cash flos( .n other ords, e do not need to co"pound this cash flo( 4. -o find the 9V0, e use the e$uation+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r # 9V0\1A %rs+ 9V0 L <A,=00YQ1 E (1:1(0J# 1A R : (0JZ L <I8,2J1(9I 9V0\I0 %rs+ 9V0 L <A,=00YQ1 E (1:1(0J# I0 R : (0JZ L <J0,MA8(0M 9V0\JA %rs+ 9V0 L <A,=00YQ1 E (1:1(0J# JA R : (0JZ L <JA,2I0(J0 -o find the 9V of a perpetuit%, e use the e$uation+ 9V L C : r 9V L <A,=00 : (0J L <JA,J1I(29 4otice that as the length of the annuit% pa%"ents increases, the present *alue of the annuit% approaches the present *alue of the perpetuit%( -he present *alue of the JA %ear annuit% and the present *alue of the perpetuit% i"pl% that the *alue toda% of all perpetuit% pa%"ents be%ond JA %ears is onl% <IJ=(A9( CHAPTER 6 B-75 . >ere e ha*e the 9V0, the length of the annuit%, and the interest rate( We ant to calculate the annuit% pa%"ent( Using the 9V0 e$uation+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r # 9V0 L <=I,000 L <CYQ1 E (1:1(0JMA# 1A R : (0JMAZ We can no sol*e this e$uation for the annuit% pa%"ent( )oing so, e get+ C L <=I,000 : 8(JIAI8 L <=,88J(J2 !. -o find the 9V0, e use the e$uation+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r # 9V0 L <J=,000YQ1 E (1:1(08A# 8 R : (08AZ L <I11,MM0(=M ". >ere e need to find the BV0( -he e$uation to find the BV0 is+ BV0 L CYQ(1 O r8 t E 1R : rZ BV0 for 20 %ears L <I,000Q(1(112 20 E 1# : (112R L <2M2,J81(1M BV0 for I0 %ears L <I,000Q(1(112 I0 E 1# : (112R L <2,IA9,0J2(M= 4otice that because of e!ponential groth, doubling the nu"ber of periods does not "erel% double the BV0( #. >ere e ha*e the BV0, the length of the annuit%, and the interest rate( We ant to calculate the annuit% pa%"ent( Using the BV0 e$uation+ BV0 L CYQ(1 O r# t E 1R : rZ <90,000 L <CQ(1(0M8 10 E 1# : (0M8R We can no sol*e this e$uation for the annuit% pa%"ent( )oing so, e get+ C L <90,000 : 1=(M8MM2 L <M,AJA(JJ $. >ere e ha*e the 9V0, the length of the annuit%, and the interest rate( We ant to calculate the annuit% pa%"ent( Using the 9V0 e$uation+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r# <A0,000 L CYQ1 E (1:1(0JA# J R : (0JAZ We can no sol*e this e$uation for the annuit% pa%"ent( )oing so, e get+ C L <A0,000 : A(29MM0 L <9,II0(02 1%. -his cash flo is a perpetuit%( -o find the 9V of a perpetuit%, e use the e$uation+ 9V L C : r 9V L <2A,000 : (0J2 L <=IJ,222(22 B-76 SOLUTIONS 11. >ere e need to find the interest rate that e$uates the perpetuit% cash flos ith the 9V of the cash flos( Using the 9V of a perpetuit% e$uation+ 9V L C : r <=JA,000 L <2A,000 : r We can no sol*e for the interest rate as follos+ r L <2A,000 : <=JA,000 L (0MMJ or M(MJN 12. Bor discrete co"pounding, to find the 60R, e use the e$uation+ 60R L Q1 O (09R : m#R m E 1 60R L Q1 O ((08 : I#R I E 1 L (082I or 8(2IN 60R L Q1 O ((1M : 12#R 12 E 1 L (1J2= or 1J(2=N 60R L Q1 O ((12 : =MA#R =MA E 1 L (12JA or 12(JAN -o find the 60R ith continuous co"pounding, e use the e$uation+ 60R L e $ E 1 60R L e (1A E 1 L (1M18 or 1M(18N 13. >ere e are gi*en the 60R and need to find the 09R( Using the e$uation for discrete co"pounding+ 60R L Q1 O (09R : m#R m E 1 We can no sol*e for the 09R( )oing so, e get+ 09R L mQ(1 O 60R# 1:m E 1R 60R L (08M0 L Q1 O (09R : 2#R 2 E 1 09R L 2Q(1(08M0# 1:2 E 1R L (08I2 or 8(I2N 60R L (1980 L Q1 O (09R : 12#R 12 E 1 09R L 12Q(1(1980# 1:12 E 1R L (1820 or 18(20N 60R L (09I0 L Q1 O (09R : A2#R A2 E 1 09R L A2Q(1(09I0# 1:A2 E 1R L (0899 or 8(99N ,ol*ing the continuous co"pounding 60R e$uation+ 60R L e $ E 1 We get+ 09R L ln(1 O 60R# 09R L ln(1 O (1MA0# 09R L (1A2J or 1A(2JN CHAPTER 6 B-77 14. Bor discrete co"pounding, to find the 60R, e use the e$uation+ 60R L Q1 O (09R : m#R m E 1 ,o, for each ban&, the 60R is+ Birst 4ational+ 60R L Q1 O ((1I20 : 12#R 12 E 1 L (1A1M or 1A(1MN Birst United+ 60R L Q1 O ((1IA0 : 2#R 2 E 1 L (1A0= or 1A(0=N 4otice that the higher 09R does not necessaril% "ean the higher 60R( -he nu"ber of co"pounding periods ithin a %ear ill also affect the 60R( 1. -he reported rate is the 09R, so e need to con*ert the 60R to an 09R as follos+ 60R L Q1 O (09R : m#R m E 1 09R L mQ(1 O 60R# 1:m E 1R 09R L =MAQ(1(1M# 1:=MA E 1R L (1I8A or 1I(8AN -his is decepti*e because the borroer is actuall% pa%ing annuali2ed interest of 1MN per %ear, not the 1I(8AN reported on the loan contract( 1!. Bor this proble", e si"pl% need to find the BV of a lu"p su" using the e$uation+ BV L 9V(1 O r8 t .t is i"portant to note that co"pounding occurs se"iannuall%( -o account for this, e ill di*ide the interest rate b% to (the nu"ber of co"pounding periods in a %ear#, and "ultipl% the nu"ber of periods b% to( )oing so, e get+ BV L <2,100Q1 O ((08I:2#R =I L <8,A0A(9= 1". Bor this proble", e si"pl% need to find the BV of a lu"p su" using the e$uation+ BV L 9V(1 O r8 t .t is i"portant to note that co"pounding occurs dail%( -o account for this, e ill di*ide the interest rate b% =MA (the nu"ber of da%s in a %ear, ignoring leap %ear#, and "ultipl% the nu"ber of periods b% =MA( )oing so, e get+ BV in A %ears L <I,A00Q1 O ((09=:=MA#R A(=MA# L <J,1M=(MI BV in 10 %ears L <I,A00Q1 O ((09=:=MA#R 10(=MA# L <11,I0=(9I BV in 20 %ears L <I,A00Q1 O ((09=:=MA#R 20(=MA# L <28,899(9J B-78 SOLUTIONS 1#. Bor this proble", e si"pl% need to find the 9V of a lu"p su" using the e$uation+ 9V L BV : (1 O r8 t .t is i"portant to note that co"pounding occurs dail%( -o account for this, e ill di*ide the interest rate b% =MA (the nu"ber of da%s in a %ear, ignoring leap %ear#, and "ultipl% the nu"ber of periods b% =MA( )oing so, e get+ 9V L <A8,000 : Q(1 O (10:=MA# J(=MA# R L <28,80I(J1 1$. -he 09R is si"pl% the interest rate per period ti"es the nu"ber of periods in a %ear( .n this case, the interest rate is =0 percent per "onth, and there are 12 "onths in a %ear, so e get+ 09R L 12(=0N# L =M0N -o find the 60R, e use the 60R for"ula+ 60R L Q1 O (09R : m#R m E 1 60R L (1 O (=0# 12 E 1 L 2,229(81N 4otice that e didn't need to di*ide the 09R b% the nu"ber of co"pounding periods per %ear( We do this di*ision to get the interest rate per period, but in this proble" e are alread% gi*en the interest rate per period( 2%. We first need to find the annuit% pa%"ent( We ha*e the 9V0, the length of the annuit%, and the interest rate( Using the 9V0 e$uation+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r# <M8,A00 L <CQ1 E Y1 : Q1 O ((0M9:12#R M0 Z : ((0M9:12#R ,ol*ing for the pa%"ent, e get+ C L <M8,A00 : A0(M222A2 L <1,=A=(1A -o find the 60R, e use the 60R e$uation+ 60R L Q1 O (09R : m#R m E 1 60R L Q1 O ((0M9 : 12#R 12 E 1 L (0J12 or J(12N 21. >ere e need to find the length of an annuit%( We &no the interest rate, the 9V, and the pa%"ents( Using the 9V0 e$uation+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r# <18,000 L <A00YQ1 E (1:1(01=# t R : (01=Z CHAPTER 6 B-79 4o e sol*e for t+ 1:1(01= t L 1 E YQ(<18,000#:(<A00#R((01=#Z 1:1(01= t L 0(A=2 1(01= t L 1:(0(A=2# L 1(8J9J t L ln 1(8J9J : ln 1(01= L I8(8M "onths 22. >ere e are tr%ing to find the interest rate hen e &no the 9V and BV( Using the BV e$uation+ BV L 9V(1 O r# <I L <=(1 O r# r L I:= E 1 L ==(==N per ee& -he interest rate is ==(==N per ee&( -o find the 09R, e "ultipl% this rate b% the nu"ber of ee&s in a %ear, so+ 09R L (A2#==(==N L 1,J==(==N 0nd using the e$uation to find the 60R+ 60R L Q1 O (09R : m#R m E 1 60R L Q1 O (====R A2 E 1 L =1=,91M,A1A(M9N 23. >ere e need to find the interest rate that e$uates the perpetuit% cash flos ith the 9V of the cash flos( Using the 9V of a perpetuit% e$uation+ 9V L C : r <9A,000 L <1,800 : r We can no sol*e for the interest rate as follos+ r L <1,800 : <9A,000 L (0189 or 1(89N per "onth -he interest rate is 1(89N per "onth( -o find the 09R, e "ultipl% this rate b% the nu"ber of "onths in a %ear, so+ 09R L (12#1(89N L 22(JIN 0nd using the e$uation to find an 60R+ 60R L Q1 O (09R : m#R m E 1 60R L Q1 O (0189R 12 E 1 L 2A(2MN 24. -his proble" re$uires us to find the BV0( -he e$uation to find the BV0 is+ BV0 L CYQ(1 O r8 t E 1R : rZ BV0 L <=00QYQ1 O ((10:12# R =M0 E 1Z : ((10:12#R L <MJ8,1IM(=8 B-80 SOLUTIONS 2. .n the pre*ious proble", the cash flos are "onthl% and the co"pounding period is "onthl%( -his assu"ption still holds( ,ince the cash flos are annual, e need to use the 60R to calculate the future *alue of annual cash flos( .t is i"portant to re"e"ber that %ou ha*e to "a&e sure the co"pounding periods of the interest rate is the sa"e as the ti"ing of the cash flos( .n this case, e ha*e annual cash flos, so e need the 60R since it is the true annual interest rate %ou ill earn( ,o, finding the 60R+ 60R L Q1 O (09R : m#R m E 1 60R L Q1 O ((10:12#R 12 E 1 L (10IJ or 10(IJN Using the BV0 e$uation, e get+ BV0 L CYQ(1 O r8 t E 1R : rZ BV0 L <=,M00Q(1(10IJ =0 E 1# : (10IJR L <MIJ,M2=(IA 2!. -he cash flos are si"pl% an annuit% ith four pa%"ents per %ear for four %ears, or 1M pa%"ents( We can use the 9V0 e$uation+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r# 9V0 L <2,=00YQ1 E (1:1(00MA# 1M R : (00MAZ L <=I,8I=(J1 2". -he cash flos are annual and the co"pounding period is $uarterl%, so e need to calculate the 60R to "a&e the interest rate co"parable ith the ti"ing of the cash flos( Using the e$uation for the 60R, e get+ 60R L Q1 O (09R : m#R m E 1 60R L Q1 O ((11:I#R I E 1 L (11IM or 11(IMN 0nd no e use the 60R to find the 9V of each cash flo as a lu"p su" and add the" together+ 9V L <J2A : 1(11IM O <980 : 1(11IM 2 O <1,=M0 : 1(11IM I L <2,=20(=M 2#. >ere the cash flos are annual and the gi*en interest rate is annual, so e can use the interest rate gi*en( We si"pl% find the 9V of each cash flo and add the" together( 9V L <1,MA0 : 1(08IA O <I,200 : 1(08IA = O <2,I=0 : 1(08IA I L <M,AJ0(8M &ntermediate 2$. -he total interest paid b% Birst ,i"ple Han& is the interest rate per period ti"es the nu"ber of periods( .n other ords, the interest b% Birst ,i"ple Han& paid o*er 10 %ears ill be+ (0J(10# L (J Birst Co"ple! Han& pa%s co"pound interest, so the interest paid b% this ban& ill be the BV factor of <1, or+ (1 O r# 10 CHAPTER 6 B-81 ,etting the to e$ual, e get+ ((0J#(10# L (1 O r# 10 E 1 r L 1(J 1:10 E 1 L (0AIA or A(IAN 3%. >ere e need to con*ert an 60R into interest rates for different co"pounding periods( Using the e$uation for the 60R, e get+ 60R L Q1 O (09R : m#R m E 1 60R L (1J L (1 O r# 2 E 18 r L (1(1J# 1:2 E 1 L (081J or 8(1JN per si! "onths 60R L (1J L (1 O r# I E 18 r L (1(1J# 1:I E 1 L (0I00 or I(00N per $uarter 60R L (1J L (1 O r# 12 E 18 r L (1(1J# 1:12 E 1 L (01=2 or 1(=2N per "onth 4otice that the effecti*e si! "onth rate is not tice the effecti*e $uarterl% rate because of the effect of co"pounding( 31. >ere e need to find the BV of a lu"p su", ith a changing interest rate( We "ust do this proble" in to parts( 0fter the first si! "onths, the balance ill be+ BV L <A,000 Q1 O ((01A:12#R M L <A,0=J(M2 -his is the balance in si! "onths( -he BV in another si! "onths ill be+ BV L <A,0=J(M2Q1 O ((18:12#R M L <A,A08(=A -he proble" as&s for the interest accrued, so, to find the interest, e subtract the beginning balance fro" the BV( -he interest accrued is+ .nterest L <A,A08(=A E A,000(00 L <A08(=A 32. We need to find the annuit% pa%"ent in retire"ent( /ur retire"ent sa*ings ends and the retire"ent ithdraals begin, so the 9V of the retire"ent ithdraals ill be the BV of the retire"ent sa*ings( ,o, e find the BV of the stoc& account and the BV of the bond account and add the to BVs( ,toc& account+ BV0 L <J00QYQ1 O ((11:12# R =M0 E 1Z : ((11:12#R L <1,9M=,1M=(82 Hond account+ BV0 L <=00QYQ1 O ((0M:12# R =M0 E 1Z : ((0M:12#R L <=01,=AI(A1 ,o, the total a"ount sa*ed at retire"ent is+ <1,9M=,1M=(82 O =01,=AI(A1 L <2,2MI,A18(== ,ol*ing for the ithdraal a"ount in retire"ent using the 9V0 e$uation gi*es us+ 9V0 L <2,2MI,A18(== L <CQ1 E Y1 : Q1 O ((09:12#R =00 Z : ((09:12#R C L <2,2MI,A18(== : 119(1M1M L <19,00=(JM= ithdraal per "onth B-82 SOLUTIONS 33. We need to find the BV of a lu"p su" in one %ear and to %ears( .t is i"portant that e use the nu"ber of "onths in co"pounding since interest is co"pounded "onthl% in this case( ,o+ BV in one %ear L <1(1(011J# 12 L <1(1A BV in to %ears L <1(1(011J# 2I L <1(=2 -here is also another co""on alternati*e solution( We could find the 60R, and use the nu"ber of %ears as our co"pounding periods( ,o e ill find the 60R first+ 60R L (1 O (011J# 12 E 1 L (1I98 or 1I(98N Using the 60R and the nu"ber of %ears to find the BV, e get+ BV in one %ear L <1(1(1I98# 1 L <1(1A BV in to %ears L <1(1(1I98# 2 L <1(=2 6ither "ethod is correct and acceptable( We ha*e si"pl% "ade sure that the interest co"pounding period is the sa"e as the nu"ber of periods e use to calculate the BV( 34. >ere e are finding the annuit% pa%"ent necessar% to achie*e the sa"e BV( -he interest rate gi*en is a 12 percent 09R, ith "onthl% deposits( We "ust "a&e sure to use the nu"ber of "onths in the e$uation( ,o, using the BV0 e$uation+ ,tarting toda%+ BV0 L CQYQ1 O ((12:12# R I80 E 1Z : ((12:12#R C L <1,000,000 : 11,JMI(JJ L <8A(00 ,tarting in 10 %ears+ BV0 L CQYQ1 O ((12:12# R =M0 E 1Z : ((12:12#R C L <1,000,000 : =,I9I(9M L <28M(1= ,tarting in 20 %ears+ BV0 L CQYQ1 O ((12:12# R 2I0 E 1Z : ((12:12#R C L <1,000,000 : 989(2AA L <1,010(8M 4otice that a deposit for half the length of ti"e, i(e( 20 %ears *ersus I0 %ears, does not "ean that the annuit% pa%"ent is doubled( .n this e!a"ple, b% reducing the sa*ings period b% one-half, the deposit necessar% to achie*e the sa"e ending *alue is about tel*e ti"es as large( 3. ,ince e are loo&ing to $uadruple our "one%, the 9V and BV are irrele*ant as long as the BV is three ti"es as large as the 9V( -he nu"ber of periods is four, the nu"ber of $uarters per %ear( ,o+ BV L <= L <1(1 O r# (12:=# r L (=1M1 or =1(M1N CHAPTER 6 B-83 3!. ,ince e ha*e an 09R co"pounded "onthl% and an annual pa%"ent, e "ust first con*ert the interest rate to an 60R so that the co"pounding period is the sa"e as the cash flos( 60R L Q1 O ((10 : 12#R 12 E 1 L (10IJ1= or 10(IJ1=N 9V01 L <9A,000 YQ1 E (1 : 1(10IJ1=# 2 R : (10IJ1=Z L <1M=,8=9(09 9V02 L <IA,000 O <J0,000YQ1 E (1:1(10IJ1=# 2 R : (10IJ1=Z L <1MA,J2=(AI 5ou ould choose the second option since it has a higher 9V( 3". We can use the present *alue of a groing perpetuit% e$uation to find the *alue of %our deposits toda%( )oing so, e find+ 9V L C YQ1:(r E g#R E Q1:(r E g#R S Q(1 O g#:(1 O r#R t Z 9V L <1,000,000YQ1:((08 E (0A#R E Q1:((08 E (0A#R S Q(1 O (0A#:(1 O (08#R =0 Z 9V L <19,01M,AM=(18 3#. ,ince %our salar% gros at I percent per %ear, %our salar% ne!t %ear ill be+ 4e!t %ear's salar% L <A0,000 (1 O (0I# 4e!t %ear's salar% L <A2,000 -his "eans %our deposit ne!t %ear ill be+ 4e!t %ear's deposit L <A2,000((0A# 4e!t %ear's deposit L <2,M00 ,ince %our salar% gros at I percent, %ou deposit ill also gro at I percent( We can use the present *alue of a groing perpetuit% e$uation to find the *alue of %our deposits toda%( )oing so, e find+ 9V L C YQ1:(r E g#R E Q1:(r E g#R S Q(1 O g#:(1 O r#R t Z 9V L <2,M00YQ1:((11 E (0I#R E Q1:((11 E (0I#R S Q(1 O (0I#:(1 O (11#R I0 Z 9V L <=I,=99(IA 4o, e can find the future *alue of this lu"p su" in I0 %ears( We find+ BV L 9V(1 O r# t BV L <=I,=MM(IA(1 O (11# I0 BV L <2,2=A,99I(=1 -his is the *alue of %our sa*ings in I0 %ears( B-84 SOLUTIONS 3$. -he relationship beteen the 9V0 and the interest rate is+ 9V0 falls as r increases, and 9V0 rises as r decreases BV0 rises as r increases, and BV0 falls as r decreases -he present *alues of <9,000 per %ear for 10 %ears at the *arious interest rates gi*en are+ 9V0\10N L <9,000YQ1 E (1:1(10# 1A R : (10Z L <M8,IAI(J2 9V0\AN L <9,000YQ1 E (1:1(0A# 1A R : (0AZ L <9=,I1M(92 9V0\1AN L <9,000YQ1 E (1:1(1A# 1A R : (1AZ L <A2,M2M(== 4%. >ere e are gi*en the BV0, the interest rate, and the a"ount of the annuit%( We need to sol*e for the nu"ber of pa%"ents( Using the BV0 e$uation+ BV0 L <20,000 L <=I0QYQ1 O ((0M:12#R t E 1 Z : ((0M:12#R ,ol*ing for t, e get+ 1(00A t L 1 O Q(<20,000#:(<=I0#R((0M:12# t L ln 1(29I118 : ln 1(00A L A1(M9 pa%"ents 41. >ere e are gi*en the 9V0, nu"ber of periods, and the a"ount of the annuit%( We need to sol*e for the interest rate( Using the 9V0 e$uation+ 9V0 L <J=,000 L <1,IA0QY1 E Q1 : (1 O r#R M0 Z: rR -o find the interest rate, e need to sol*e this e$uation on a financial calculator, using a spreadsheet, or b% trial and error( .f %ou use trial and error, re"e"ber that increasing the interest rate loers the 9V0, and decreasing the interest rate increases the 9V0( Using a spreadsheet, e find+ r L 0(A9IN -he 09R is the periodic interest rate ti"es the nu"ber of periods in the %ear, so+ 09R L 12(0(A9IN# L J(1=N CHAPTER 6 B-85 42. -he a"ount of principal paid on the loan is the 9V of the "onthl% pa%"ents %ou "a&e( ,o, the present *alue of the <1,1A0 "onthl% pa%"ents is+ 9V0 L <1,1A0Q(1 E Y1 : Q1 O ((0M=A:12#RZ =M0 # : ((0M=A:12#R L <18I,81J(I2 -he "onthl% pa%"ents of <1,1A0 ill a"ount to a principal pa%"ent of <18I,81J(I2( -he a"ount of principal %ou ill still oe is+ <2I0,000 E 18I,81J(I2 L <AA,182(A8 -his re"aining principal a"ount ill increase at the interest rate on the loan until the end of the loan period( ,o the balloon pa%"ent in =0 %ears, hich is the BV of the re"aining principal ill be+ Halloon pa%"ent L <AA,182(A8Q1 O ((0M=A:12#R =M0 L <=M8,9=M(AI 43. We are gi*en the total 9V of all four cash flos( .f e find the 9V of the three cash flos e &no, and subtract the" fro" the total 9V, the a"ount left o*er "ust be the 9V of the "issing cash flo( ,o, the 9V of the cash flos e &no are+ 9V of 5ear 1 CB+ <1,J00 : 1(10 L <1,AIA(IA 9V of 5ear = CB+ <2,100 : 1(10 = L <1,AJJ(JM 9V of 5ear I CB+ <2,800 : 1(10 I L <1,912(II ,o, the 9V of the "issing CB is+ <M,AA0 E 1,AIA(IA E 1,AJJ(JM E 1,912(II L <1,A1I(=A -he $uestion as&s for the *alue of the cash flo in 5ear 2, so e "ust find the future *alue of this a"ount( -he *alue of the "issing CB is+ <1,A1I(=A(1(10# 2 L <1,8=2(=M 44. -o sol*e this proble", e si"pl% need to find the 9V of each lu"p su" and add the" together( .t is i"portant to note that the first cash flo of <1 "illion occurs toda%, so e do not need to discount that cash flo( -he 9V of the lotter% innings is+ 9V 3 <1,000,000 O <1,A00,000:1(09 O <2,000,000:1(09 2 O <2,A00,000:1(09 = O <=,000,000:1(09 I O <=,A00,000:1(09 A O <I,000,000:1(09 M O <I,A00,000:1(09 J O <A,000,000:1(09 8 O <A,A00,000:1(09 9 O <M,000,000:1(09 10 9V L <22,812,8J=(I0 4. >ere e are finding interest rate for an annuit% cash flo( We are gi*en the 9V0, nu"ber of periods, and the a"ount of the annuit%( We should also note that the 9V of the annuit% is not the a"ount borroed since e are "a&ing a don pa%"ent on the arehouse( -he a"ount borroed is+ 0"ount borroed L 0(80(<2,900,000# L <2,=20,000 B-86 SOLUTIONS Using the 9V0 e$uation+ 9V0 L <2,=20,000 L <1A,000QY1 E Q1 : (1 O r#R =M0 Z: rR Unfortunatel% this e$uation cannot be sol*ed to find the interest rate using algebra( -o find the interest rate, e need to sol*e this e$uation on a financial calculator, using a spreadsheet, or b% trial and error( .f %ou use trial and error, re"e"ber that increasing the interest rate loers the 9V0, and decreasing the interest rate increases the 9V0( Using a spreadsheet, e find+ r L 0(AM0N -he 09R is the "onthl% interest rate ti"es the nu"ber of "onths in the %ear, so+ 09R L 12(0(AM0N# L M(J2N 0nd the 60R is+ 60R L (1 O (00AM0# 12 E 1 L (0M9= or M(9=N 4!. -he profit the fir" earns is 1ust the 9V of the sales price "inus the cost to produce the asset( We find the 9V of the sales price as the 9V of a lu"p su"+ 9V L <1MA,000 : 1(1= I L <101,19J(A9 0nd the fir"'s profit is+ 9rofit L <101,19J(A9 E 9I,000(00 L <J,19J(A9 -o find the interest rate at hich the fir" ill brea& e*en, e need to find the interest rate using the 9V (or BV# of a lu"p su"( Using the 9V e$uation for a lu"p su", e get+ <9I,000 L <1MA,000 : ( 1 O r# I r L (<1MA,000 : <9I,000# 1:I E 1 L (1A10 or 1A(10N 4". We ant to find the *alue of the cash flos toda%, so e ill find the 9V of the annuit%, and then bring the lu"p su" 9V bac& to toda%( -he annuit% has 18 pa%"ents, so the 9V of the annuit% is+ 9V0 L <I,000YQ1 E (1:1(10# 18 R : (10Z L <=2,80A(MA ,ince this is an ordinar% annuit% e$uation, this is the 9V one period before the first pa%"ent, so it is the 9V at t L J( -o find the *alue toda%, e find the 9V of this lu"p su"( -he *alue toda% is+ 9V L <=2,80A(MA : 1(10 J L <1M,8=I(I8 4#. -his $uestion is as&ing for the present *alue of an annuit%, but the interest rate changes during the life of the annuit%( We need to find the present *alue of the cash flos for the last eight %ears first( -he 9V of these cash flos is+ 9V02 L <1,A00 QY1 E 1 : Q1 O ((0J:12#R 9M Z : ((0J:12#R L <110,021(=A CHAPTER 6 B-87 4ote that this is the 9V of this annuit% e!actl% se*en %ears fro" toda%( 4o e can discount this lu"p su" to toda%( -he *alue of this cash flo toda% is+ 9V L <110,021(=A : Q1 O ((11:12#R 8I L <A1,120(== 4o e need to find the 9V of the annuit% for the first se*en %ears( -he *alue of these cash flos toda% is+ 9V01 L <1,A00 QY1 E 1 : Q1 O ((11:12#R 8I Z : ((11:12#R L <8J,M0I(=M -he *alue of the cash flos toda% is the su" of these to cash flos, so+ 9V L <A1,120(== O 8J,M0I(=M L <1=8,J2I(M8 4$. >ere e are tr%ing to find the dollar a"ount in*ested toda% that ill e$ual the BV0 ith a &non interest rate, and pa%"ents( Birst e need to deter"ine ho "uch e ould ha*e in the annuit% account( Binding the BV of the annuit%, e get+ BV0 L <1,200 QYQ 1 O ((08A:12#R 180 E 1Z : ((08A:12#R L <I=I,1I=(M2 4o e need to find the 9V of a lu"p su" that ill gi*e us the sa"e BV( ,o, using the BV of a lu"p su" ith continuous co"pounding, e get+ BV L <I=I,1I=(M2 L 9Ve (08(1A# 9V L <I=I,1I=(M2e ;1(20 L <1=0,JM1(AA %. -o find the *alue of the perpetuit% at t L J, e first need to use the 9V of a perpetuit% e$uation( Using this e$uation e find+ 9V L <=,A00 : (0M2 L <AM,IA1(M1 Re"e"ber that the 9V of a perpetuit% (and annuit%# e$uations gi*e the 9V one period before the first pa%"ent, so, this is the *alue of the perpetuit% at t L 1I( -o find the *alue at t L J, e find the 9V of this lu"p su" as+ 9V L <AM,IA1(M1 : 1(0M2 J L <=J,0A1(I1 1. -o find the 09R and 60R, e need to use the actual cash flos of the loan( .n other ords, the interest rate $uoted in the proble" is onl% rele*ant to deter"ine the total interest under the ter"s gi*en( -he interest rate for the cash flos of the loan is+ 9V0 L <2A,000 L <2,I1M(MJY(1 E Q1 : (1 O r#R 12 # : r Z 0gain, e cannot sol*e this e$uation for r, so e need to sol*e this e$uation on a financial calculator, using a spreadsheet, or b% trial and error( Using a spreadsheet, e find+ r L 2(=M1N per "onth B-88 SOLUTIONS ,o the 09R is+ 09R L 12(2(=M1N# L 28(==N 0nd the 60R is+ 60R L (1(02=M1# 12 E 1 L (=2=1 or =2(=1N 2. -he cash flos in this proble" are se"iannual, so e need the effecti*e se"iannual rate( -he interest rate gi*en is the 09R, so the "onthl% interest rate is+ 3onthl% rate L (10 : 12 L (008== -o get the se"iannual interest rate, e can use the 60R e$uation, but instead of using 12 "onths as the e!ponent, e ill use M "onths( -he effecti*e se"iannual rate is+ ,e"iannual rate L (1(008==# M E 1 L (0A11 or A(11N We can no use this rate to find the 9V of the annuit%( -he 9V of the annuit% is+ 9V0 \ %ear 8+ <J,000YQ1 E (1 : 1(0A11# 10 R : (0A11Z L <A=,JJM(J2 4ote, this is the *alue one period (si! "onths# before the first pa%"ent, so it is the *alue at %ear 8( ,o, the *alue at the *arious ti"es the $uestions as&ed for uses this *alue 8 %ears fro" no( 9V \ %ear A+ <A=,JJM(J2 : 1(0A11 M L <=9,888(== 4ote, %ou can also calculate this present *alue (as ell as the re"aining present *alues# using the nu"ber of %ears( -o do this, %ou need the 60R( -he 60R is+ 60R L (1 O (008=# 12 E 1 L (10IJ or 10(IJN ,o, e can find the 9V at %ear A using the folloing "ethod as ell+ 9V \ %ear A+ <A=,JJM(J2 : 1(10IJ = L <=9,888(== -he *alue of the annuit% at the other ti"es in the proble" is+ 9V \ %ear =+ <A=,JJM(J2 : 1(0A11 10 L <=2,M8I(88 9V \ %ear =+ <A=,JJM(J2 : 1(10IJ A L <=2,M8I(88 9V \ %ear 0+ <A=,JJM(J2 : 1(0A11 1M L <2I,2I=(MJ 9V \ %ear 0+ <A=,JJM(J2 : 1(10IJ 8 L <2I,2I=(MJ 3. a. .f the pa%"ents are in the for" of an ordinar% annuit%, the present *alue ill be+ 9V0 L C(Y1 E Q1:(1 O r# t RZ : r #8 9V0 L <10,000QY1 E Q1 : (1 O (11#R A Z: (11R 9V0 L <=M,9A8(9J CHAPTER 6 B-89 .f the pa%"ents are an annuit% due, the present *alue ill be+ 9V0due L (1 O r# 9V0 9V0due L (1 O (11#<=M,9A8(9J 9V0due L <I1,02I(IM b. We can find the future *alue of the ordinar% annuit% as+ BV0 L CYQ(1 O r# t E 1R : rZ BV0 L <10,000YQ(1 O (11# A E 1R : (11Z BV0 L <M2,2J8(01 .f the pa%"ents are an annuit% due, the future *alue ill be+ BV0due L (1 O r# BV0 BV0due L (1 O (11#<M2,2J8(01 BV0due L <M9,128(M0 c. 0ssu"ing a positi*e interest rate, the present *alue of an annuit% due ill ala%s be larger than the present *alue of an ordinar% annuit%( 6ach cash flo in an annuit% due is recei*ed one period earlier, hich "eans there is one period less to discount each cash flo( 0ssu"ing a positi*e interest rate, the future *alue of an ordinar% due ill ala%s higher than the future *alue of an ordinar% annuit%( ,ince each cash flo is "ade one period sooner, each cash flo recei*es one e!tra period of co"pounding( 4. We need to use the 9V0 due e$uation, that is+ 9V0due L (1 O r# 9V0 Using this e$uation+ 9V0due L <M8,000 L Q1 O ((0J8A:12#R S CQY1 E 1 : Q1 O ((0J8A:12#R M0 Z : ((0J8A:12# <MJ,AA8(0M L <CY1 E Q1 : (1 O (0J8A:12# M0 RZ : ((0J8A:12# C L <1,=MI(99 4otice, hen e find the pa%"ent for the 9V0 due, e si"pl% discount the 9V of the annuit% due bac& one period( We then use this *alue as the 9V of an ordinar% annuit%( . -he pa%"ent for a loan repaid ith e$ual pa%"ents is the annuit% pa%"ent ith the loan *alue as the 9V of the annuit%( ,o, the loan pa%"ent ill be+ 9V0 L <I2,000 L C YQ1 E 1 : (1 O (08# A R : (08Z C L <10,A19(1J -he interest pa%"ent is the beginning balance ti"es the interest rate for the period, and the principal pa%"ent is the total pa%"ent "inus the interest pa%"ent( -he ending balance is the beginning balance "inus the principal pa%"ent( -he ending balance for a period is the beginning balance for the ne!t period( -he a"orti2ation table for an e$ual pa%"ent is+ B-90 SOLUTIONS 5ear Heginning Halance -otal 9a%"ent .nterest 9a%"ent 9rincipal 9a%"ent 6nding Halance 1 <I2,000(00 <10,A19(1J <=,=M0(00 <J,1A9(1J <=I,8I0(8= 2 =I,8I0(8= 10,A19(1J 2,J8J(2J J,J=1(90 2J,108(92 = 2J,108(92 10,A19(1J 2,1M8(J1 8,=A0(IM 18,JA8(IJ I 18,JA8(IJ 10,A19(1J 1,A00(M8 9,018(I9 9,J=9(9J A 9,J=9(9J 10,A19(1J JJ9(20 9,J=9(9J 0(00 .n the third %ear, <2,1M8(J1 of interest is paid( -otal interest o*er life of the loan L <=,=M0 O 2,J8J(2J O 2,1M8(J1 O 1,A00(M8 O JJ9(20 -otal interest o*er life of the loan L <10,A9A(8M !. -his a"orti2ation table calls for e$ual principal pa%"ents of <8,I00 per %ear( -he interest pa%"ent is the beginning balance ti"es the interest rate for the period, and the total pa%"ent is the principal pa%"ent plus the interest pa%"ent( -he ending balance for a period is the beginning balance for the ne!t period( -he a"orti2ation table for an e$ual principal reduction is+ 5ear Heginning Halance -otal 9a%"ent .nterest 9a%"ent 9rincipal 9a%"ent 6nding Halance 1 <I2,000(00 <11,JM0(00 <=,=M0(00 <8,I00(00 <==,M00(00 2 ==,M00(00 11,088(00 2,M88(00 8,I00(00 2A,200(00 = 2A,200(00 10,I1M(00 2,01M(00 8,I00(00 1M,800(00 I 1M,800(00 9,JII(00 1,=II(00 8,I00(00 8,I00(00 A 8,I00(00 9,0J2(00 MJ2(00 8,I00(00 0(00 .n the third %ear, <2,01M of interest is paid( -otal interest o*er life of the loan L <=,=M0 O 2,M88 O 2,01M O 1,=II O MJ2 L <10,080 4otice that the total pa%"ents for the e$ual principal reduction loan are loer( -his is because "ore principal is repaid earl% in the loan, hich reduces the total interest e!pense o*er the life of the loan( Challenge ". -he cash flos for this proble" occur "onthl%, and the interest rate gi*en is the 60R( ,ince the cash flos occur "onthl%, e "ust get the effecti*e "onthl% rate( /ne a% to do this is to find the 09R based on "onthl% co"pounding, and then di*ide b% 12( ,o, the pre-retire"ent 09R is+ 60R L (10 L Q1 O (09R : 12#R 12 E 18 09R L 12Q(1(10# 1:12 E 1R L (09AJ or 9(AJN 0nd the post-retire"ent 09R is+ 60R L (0J L Q1 O (09R : 12#R 12 E 18 09R L 12Q(1(0J# 1:12 E 1R L (0MJ8 or M(J8N CHAPTER 6 B-91 Birst, e ill calculate ho "uch he needs at retire"ent( -he a"ount needed at retire"ent is the 9V of the "onthl% spending plus the 9V of the inheritance( -he 9V of these to cash flos is+ 9V0 L <20,000Y1 E Q1 : (1 O (0MJ8:12# 12(2A# RZ : ((0MJ8:12# L <2,88A,I9M(IA 9V L <900,000 : Q1 O ((0MJ8:12#R =00 L <1MA,82I(2M ,o, at retire"ent, he needs+ <2,88A,I9M(IA O 1MA,82I(2M L <=,0A1,=20(J1 >e ill be sa*ing <2,A00 per "onth for the ne!t 10 %ears until he purchases the cabin( -he *alue of his sa*ings after 10 %ears ill be+ BV0 L <2,A00QYQ 1 O ((09AJ:12#R 12(10# E 1Z : ((09AJ:12#R L <I99,MA9(MI 0fter he purchases the cabin, the a"ount he ill ha*e left is+ <I99,MA9(MI E =80,000 L <119,MA9(MI >e still has 20 %ears until retire"ent( When he is read% to retire, this a"ount ill ha*e gron to+ BV L <119,MA9(MIQ1 O ((09AJ:12#R 12(20# L <80A,010(2= ,o, hen he is read% to retire, based on his current sa*ings, he ill be short+ <=,0A1,=20(J1 E 80A,010(2= L <2,2IM,=10(I8 -his a"ount is the BV of the "onthl% sa*ings he "ust "a&e beteen %ears 10 and =0( ,o, finding the annuit% pa%"ent using the BV0 e$uation, e find his "onthl% sa*ings ill need to be+ BV0 L <2,2IM,=10(I8 L CQYQ 1 O ((10I8:12#R 12(20# E 1Z : ((10I8:12#R C L <=,12J(II #. -o anser this $uestion, e should find the 9V of both options, and co"pare the"( ,ince e are purchasing the car, the loest 9V is the best option( -he 9V of the leasing is si"pl% the 9V of the lease pa%"ents, plus the <99( -he interest rate e ould use for the leasing option is the sa"e as the interest rate of the loan( -he 9V of leasing is+ 9V L <99 O <IA0Y1 E Q1 : (1 O (0J:12# 12(=# RZ : ((0J:12# L <1I,MJ2(91 -he 9V of purchasing the car is the current price of the car "inus the 9V of the resale price( -he 9V of the resale price is+ 9V L <2=,000 : Q1 O ((0J:12#R 12(=# L <18,MAI(82 -he 9V of the decision to purchase is+ <=2,000 E 18,MAI(82 L <1=,=IA(18 B-92 SOLUTIONS .n this case, it is cheaper to bu% the car than leasing it since the 9V of the purchase cash flos is loer( -o find the brea&e*en resale price, e need to find the resale price that "a&es the 9V of the to options the sa"e( .n other ords, the 9V of the decision to bu% should be+ <=2,000 E 9V of resale price L <1I,MJ2(91 9V of resale price L <1J,=2J(09 -he resale price that ould "a&e the 9V of the lease *ersus bu% decision is the BV of this *alue, so+ Hrea&e*en resale price L <1J,=2J(09Q1 O ((0J:12#R 12(=# L <21,=M=(01 $. -o find the $uarterl% salar% for the pla%er, e first need to find the 9V of the current contract( -he cash flos for the contract are annual, and e are gi*en a dail% interest rate( We need to find the 60R so the interest co"pounding is the sa"e as the ti"ing of the cash flos( -he 60R is+ 60R L Q1 O ((0AA:=MA#R =MA E 1 L A(MAN -he 9V of the current contract offer is the su" of the 9V of the cash flos( ,o, the 9V is+ 9V L <J,000,000 O <I,A00,000:1(0AMA O <A,000,000:1(0AMA 2 O <M,000,000:1(0AMA = O <M,800,000:1(0AMA I O <J,900,000:1(0AMA A O <8,800,000:1(0AMA M 9V L <=8,M10,I82(AJ -he pla%er ants the contract increased in *alue b% <1,I00,000, so the 9V of the ne contract ill be+ 9V L <=8,M10,I82(AJ O 1,I00,000 L <I0,010,I82(AJ -he pla%er has also re$uested a signing bonus pa%able toda% in the a"ount of <9 "illion( We can si"pl% subtract this a"ount fro" the 9V of the ne contract( -he re"aining a"ount ill be the 9V of the future $uarterl% pa%chec&s( <I0,010,I82(AJ E 9,000,000 L <=1,010,I82(AJ -o find the $uarterl% pa%"ents, first reali2e that the interest rate e need is the effecti*e $uarterl% rate( Using the dail% interest rate, e can find the $uarterl% interest rate using the 60R e$uation, ith the nu"ber of da%s being 91(2A, the nu"ber of da%s in a $uarter (=MA : I#( -he effecti*e $uarterl% rate is+ 6ffecti*e $uarterl% rate L Q1 O ((0AA:=MA#R 91(2A E 1 L (01=8I or 1(=8IN 4o e ha*e the interest rate, the length of the annuit%, and the 9V( Using the 9V0 e$uation and sol*ing for the pa%"ent, e get+ 9V0 L <=1,010,I82(AJ L CYQ1 E (1:1(01=8I# 2I R : (01=8IZ C L <1,A2J,IM=(JM CHAPTER 6 B-93 !%. -o find the 09R and 60R, e need to use the actual cash flos of the loan( .n other ords, the interest rate $uoted in the proble" is onl% rele*ant to deter"ine the total interest under the ter"s gi*en( -he cash flos of the loan are the <2A,000 %ou "ust repa% in one %ear, and the <21,2A0 %ou borro toda%( -he interest rate of the loan is+ <2A,000 L <21,2A0(1 O r# r L (<2A,000 : 21,2A0# E 1 L (1JMA or 1J(MAN Hecause of the discount, %ou onl% get the use of <21,2A0, and the interest %ou pa% on that a"ount is 1J(MAN, not 1AN( !1. >ere e ha*e cash flos that ould ha*e occurred in the past and cash flos that ould occur in the future( We need to bring both cash flos to toda%( Hefore e calculate the *alue of the cash flos toda%, e "ust ad1ust the interest rate so e ha*e the effecti*e "onthl% interest rate( Binding the 09R ith "onthl% co"pounding and di*iding b% 12 ill gi*e us the effecti*e "onthl% rate( -he 09R ith "onthl% co"pounding is+ 09R L 12Q(1(08# 1:12 E 1R L (0JJ2 or J(J2N -o find the *alue toda% of the bac& pa% fro" to %ears ago, e ill find the BV of the annuit%, and then find the BV of the lu"p su"( )oing so gi*es us+ BV0 L (<IJ,000:12# QYQ 1 O ((0JJ2:12#R 12 E 1Z : ((0JJ2:12#R L <I8,M99(=9 BV L <I8,M99(=9(1(08# L <A2,A9A(=I 4otice e found the BV of the annuit% ith the effecti*e "onthl% rate, and then found the BV of the lu"p su" ith the 60R( 0lternati*el%, e could ha*e found the BV of the lu"p su" ith the effecti*e "onthl% rate as long as e used 12 periods( -he anser ould be the sa"e either a%( 4o, e need to find the *alue toda% of last %ear's bac& pa%+ BV0 L (<A0,000:12# QYQ 1 O ((0JJ2:12#R 12 E 1Z : ((0JJ2:12#R L <A1,80J(8M 4e!t, e find the *alue toda% of the fi*e %ear's future salar%+ 9V0 L (<AA,000:12#YQY1 E Y1 : Q1 O ((0JJ2:12#R 12(A# ZR : ((0JJ2:12#ZL <22J,A=9(1I -he *alue toda% of the 1ur% aard is the su" of salaries, plus the co"pensation for pain and suffering, and court costs( -he aard should be for the a"ount of+ 0ard L <A2,A9A(=I O A1,80J(8M O 22J,A=9(1I O 100,000 O 20,000 L <IA1,9I2(=I 0s the plaintiff, %ou ould prefer a loer interest rate( .n this proble", e are calculating both the 9V and BV of annuities( 0 loer interest rate ill decrease the BV0, but increase the 9V0( ,o, b% a loer interest rate, e are loering the *alue of the bac& pa%( Hut, e are also increasing the 9V of the future salar%( ,ince the future salar% is larger and has a longer ti"e, this is the "ore i"portant cash flo to the plaintiff( B-94 SOLUTIONS !2. 0gain, to find the interest rate of a loan, e need to loo& at the cash flos of the loan( ,ince this loan is in the for" of a lu"p su", the a"ount %ou ill repa% is the BV of the principal a"ount, hich ill be+ Coan repa%"ent a"ount L <10,000(1(08# L <10,800 -he a"ount %ou ill recei*e toda% is the principal a"ount of the loan ti"es one "inus the points( 0"ount recei*ed L <10,000(1 E (0=# L <9,J00 4o, e si"pl% find the interest rate for this 9V and BV( <10,800 L <9,J00(1 O r# r L (<10,800 : <9,J00# E 1 L (11=I or 11(=IN !3. -his is the sa"e $uestion as before, ith different *alues( ,o+ Coan repa%"ent a"ount L <10,000(1(11# L <11,100 0"ount recei*ed L <10,000(1 E (02# L <9,800 <11,100 L <9,800(1 O r# r L (<11,100 : <9,800# E 1 L (1=2J or 1=(2JN -he effecti*e rate is not affected b% the loan a"ount since it drops out hen sol*ing for r( !4. Birst e ill find the 09R and 60R for the loan ith the refundable fee( Re"e"ber, e need to use the actual cash flos of the loan to find the interest rate( With the <2,=00 application fee, %ou ill need to borro <2I2,=00 to ha*e <2I0,000 after deducting the fee( ,ol*ing for the pa%"ent under these circu"stances, e get+ 9V0 L <2I2,=00 L C YQ1 E 1:(1(00AMMJ# =M0 R:(00AMMJZ here (00AMMJ L (0M8:12 C L <1,AJ9(M1 We can no use this a"ount in the 9V0 e$uation ith the original a"ount e ished to borro, <2I0,000( ,ol*ing for r, e find+ 9V0 L <2I0,000 L <1,AJ9(M1QY1 E Q1 : (1 O r#R =M0 Z: rR ,ol*ing for r ith a spreadsheet, on a financial calculator, or b% trial and error, gi*es+ r L 0(AJIAN per "onth 09R L 12(0(AJIAN# L M(89N 60R L (1 O (00AJIA# 12 E 1 L J(12N CHAPTER 6 B-95 With the nonrefundable fee, the 09R of the loan is si"pl% the $uoted 09R since the fee is not considered part of the loan( ,o+ 09R L M(80N 60R L Q1 O ((0M8:12#R 12 E 1 L J(02N !. He careful of interest rate $uotations( -he actual interest rate of a loan is deter"ined b% the cash flos( >ere, e are told that the 9V of the loan is <1,000, and the pa%"ents are <I1(1A per "onth for three %ears, so the interest rate on the loan is+ 9V0 L <1,000 L <I1(1AQY1 E Q1 : (1 O r#R =M Z : r R ,ol*ing for r ith a spreadsheet, on a financial calculator, or b% trial and error, gi*es+ r L 2(=0N per "onth 09R L 12(2(=0N# L 2J(M1N 60R L (1 O (02=0# 12 E 1 L =1(=9N .t's called add-on interest because the interest a"ount of the loan is added to the principal a"ount of the loan before the loan pa%"ents are calculated( !!. >ere e are sol*ing a to-step ti"e *alue of "one% proble"( 6ach $uestion as&s for a different possible cash flo to fund the sa"e retire"ent plan( 6ach sa*ings possibilit% has the sa"e BV, that is, the 9V of the retire"ent spending hen %our friend is read% to retire( -he a"ount needed hen %our friend is read% to retire is+ 9V0 L <10A,000YQ1 E (1:1(0J# 20 R : (0JZ L <1,112,=J1(A0 -his a"ount is the sa"e for all three parts of this $uestion( a. .f %our friend "a&es e$ual annual deposits into the account, this is an annuit% ith the BV0 e$ual to the a"ount needed in retire"ent( -he re$uired sa*ings each %ear ill be+ BV0 L <1,112,=J1(A0 L CQ(1(0J =0 E 1# : (0JR C L <11,JJM(01 b. >ere e need to find a lu"p su" sa*ings a"ount( Using the BV for a lu"p su" e$uation, e get+ BV L <1,112,=J1(A0 L 9V(1(0J# =0 9V L <1IM,129(0I B-96 SOLUTIONS c. .n this proble", e ha*e a lu"p su" sa*ings in addition to an annual deposit( ,ince e alread% &no the *alue needed at retire"ent, e can subtract the *alue of the lu"p su" sa*ings at retire"ent to find out ho "uch %our friend is short( )oing so gi*es us+ BV of trust fund deposit L <1A0,000(1(0J# 10 L <29A,0J2(J0 ,o, the a"ount %our friend still needs at retire"ent is+ BV L <1,112,=J1(A0 E 29A,0J2(J0 L <81J,298(80 Using the BV0 e$uation, and sol*ing for the pa%"ent, e get+ <81J,298(80 L CQ(1(0J =0 E 1# : (0JR C L <8,MA2(2A -his is the total annual contribution, but %our friend's e"plo%er ill contribute <1,A00 per %ear, so %our friend "ust contribute+ BriendFs contribution L <8,MA2(2A E 1,A00 L <J,1A2(2A !". We ill calculate the nu"ber of periods necessar% to repa% the balance ith no fee first( We si"pl% need to use the 9V0 e$uation and sol*e for the nu"ber of pa%"ents( Without fee and annual rate L 19(80N+ 9V0 L <10,000 L <200YQ1 E (1:1(01MA# t R : (01MA Z here (01MA L (198:12 ,ol*ing for t, e get+ 1:1(01MA t L 1 E (<10,000:<200#((01MA# 1:1(01MA t L (1JA t L ln (1:(1JA# : ln 1(01MA t L 10M(A0 "onths Without fee and annual rate L M(20N+ 9V0 L <10,000 L <200YQ1 E (1:1(00A1MJ# t R : (00A1MJ Z here (00A1MJ L (0M2:12 ,ol*ing for t, e get+ 1:1(00A1MJ t L 1 E (<10,000:<200#((00A1MJ# 1:1(00A1MJ t L (JI1J t L ln (1:(JI1J# : ln 1(00A1MJ t L AJ(99 "onths 4ote that e do not need to calculate the ti"e necessar% to repa% %our current credit card ith a fee since no fee ill be incurred( -he ti"e to repa% the ne card ith a transfer fee is+ CHAPTER 6 B-97 With fee and annual rate L M(20N+ 9V0 L <10,200 L <200Y Q1 E (1:1(00A1MJ# t R : (00A1MJ Z here (00A1MJ L (082:12 ,ol*ing for t, e get+ 1:1(00A1MJ t L 1 E (<10,200:<200#((00A1MJ# 1:1(00A1MJ t L (J=MA t L ln (1:(J=MA# : ln 1(00A1MJ t L A9(=A "onths !#. We need to find the BV of the pre"iu"s to co"pare ith the cash pa%"ent pro"ised at age MA( We ha*e to find the *alue of the pre"iu"s at %ear M first since the interest rate changes at that ti"e( ,o+ BV1 L <900(1(12# A L <1,A8M(11 BV2 L <900(1(12# I L <1,I1M(1J BV= L <1,000(1(12# = L <1,I0I(9= BVI L <1,000(1(12# 2 L <1,2AI(I0 BVA L <1,100(1(12# 1 L <1,2=2(00 Value at %ear si! L <1,A8M(11 O 1,I1M(1J O 1,I0I(9= O 1,2AI(I0 O 1,2=2(00 O 1,100 Value at %ear si! L <J,99=(M0 Binding the BV of this lu"p su" at the child's MA th birthda%+ BV L <J,99=(M0(1(08# A9 L <JI9,IA2(AM -he polic% is not orth bu%ing8 the future *alue of the deposits is <JI9,IA2(AM, but the polic% contract ill pa% off <A00,000( -he pre"iu"s are orth <2I9,IA2(AM "ore than the polic% pa%off( 4ote, e could also co"pare the 9V of the to cash flos( -he 9V of the pre"iu"s is+ 9V L <900:1(12 O <900:1(12 2 O <1,000:1(12 = O <1,000:1(12 I O <1,100:1(12 A O <1,100:1(12 M 9V L <I,0I9(81 0nd the *alue toda% of the <A00,000 at age MA is+ 9V L <A00,000:1(08 A9 L <A,==2(9M 9V L <A,==2(9M:1(12 M L <2,J01(8I -he pre"iu"s still ha*e the higher cash flo( 0t ti"e 2ero, the difference is <1,=IJ(9J( Whene*er %ou are co"paring to or "ore cash flo strea"s, the cash flo ith the highest *alue at one ti"e ill ha*e the highest *alue at an% other ti"e( >ere is a $uestion for %ou+ ,uppose %ou in*est <1,=IJ(9J, the difference in the cash flos at ti"e 2ero, for si! %ears at a 12 percent interest rate, and then for A9 %ears at an 8 percent interest rate( >o "uch ill it B-98 SOLUTIONS be orth? Without doing calculations, %ou &no it ill be orth <2I9,IA2(AM, the difference in the cash flos at ti"e MAX !$. -he "onthl% pa%"ents ith a balloon pa%"ent loan are calculated assu"ing a longer a"orti2ation schedule, in this case, =0 %ears( -he pa%"ents based on a =0-%ear repa%"ent schedule ould be+ 9V0 L <JA0,000 L C(Y1 E Q1 : (1 O (081:12#R =M0 Z : ((081:12## C L <A,AAA(M1 4o, at ti"e L 8, e need to find the 9V of the pa%"ents hich ha*e not been "ade( -he balloon pa%"ent ill be+ 9V0 L <A,AAA(M1(Y1 E Q1 : (1 O (081:12#R 12(22# Z : ((081:12## 9V0 L <M8=,J00(=2 "%. >ere e need to find the interest rate that "a&es the 9V0, the college costs, e$ual to the BV0, the sa*ings( -he 9V of the college costs are+ 9V0 L <20,000QY1 E Q1 : (1 O r# I RZ : r R 0nd the BV of the sa*ings is+ BV0 L <9,000YQ(1 O r# M E 1 R : r Z ,etting these to e$uations e$ual to each other, e get+ <20,000QY1 E Q1 : (1 O r#R I Z : r R L <9,000YQ (1 O r# M E 1 R : r Z Reducing the e$uation gi*es us+ (1 O r# M E 11,000(1 O r# I O 29,000 L 0 4o e need to find the roots of this e$uation( We can sol*e using trial and error, a root-sol*ing calculator routine, or a spreadsheet( Using a spreadsheet, e find+ r L 8(0JN "1. >ere e need to find the interest rate that "a&es us indifferent beteen an annuit% and a perpetuit%( -o sol*e this proble", e need to find the 9V of the to options and set the" e$ual to each other( -he 9V of the perpetuit% is+ 9V L <20,000 : r 0nd the 9V of the annuit% is+ 9V0 L <28,000QY1 E Q1 : (1 O r#R 20 Z : r R CHAPTER 6 B-99 ,etting the" e$ual and sol*ing for r, e get+ <20,000 : r 4 <28,000Q Y1 E Q1 : (1 O r#R 20 Z : r R <20,000 : <28,000 L 1 E Q1 : (1 O r#R 20 (J1I= 1:20 L 1 : (1 O r# r L (0MIM or M(IMN "2. -he cash flos in this proble" occur e*er% to %ears, so e need to find the effecti*e to %ear rate( /ne a% to find the effecti*e to %ear rate is to use an e$uation si"ilar to the 60R, e!cept use the nu"ber of da%s in to %ears as the e!ponent( (We use the nu"ber of da%s in to %ears since it is dail% co"pounding8 if "onthl% co"pounding as assu"ed, e ould use the nu"ber of "onths in to %ears(# ,o, the effecti*e to-%ear interest rate is+ 6ffecti*e 2-%ear rate L Q1 O ((10:=MA#R =MA(2# E 1 L (221I or 22(1IN We can use this interest rate to find the 9V of the perpetuit%( )oing so, e find+ 9V L <1A,000 :(221I L <MJ,JM0(0J -his is an i"portant point+ Re"e"ber that the 9V e$uation for a perpetuit% (and an ordinar% annuit%# tells %ou the 9V one period before the first cash flo( .n this proble", since the cash flos are to %ears apart, e ha*e found the *alue of the perpetuit% one period (to %ears# before the first pa%"ent, hich is one %ear ago( We need to co"pound this *alue for one %ear to find the *alue toda%( -he *alue of the cash flos toda% is+ 9V L <MJ,JM0(0J(1 O (10:=MA# =MA L <JI,88A(II -he second part of the $uestion assu"es the perpetuit% cash flos begin in four %ears( .n this case, hen e use the 9V of a perpetuit% e$uation, e find the *alue of the perpetuit% to %ears fro" toda%( ,o, the *alue of these cash flos toda% is+ 9V L <MJ,JM0(0J : (1 O (221I# L <AA,IJ8(J8 "3. -o sol*e for the 9V0 due+ 9V0 L # (1 (((( # (1 # (1 2 t r C r C r C + + + + + + 9V0due L # (1 (((( # (1 1 - t r C r C C + + + + + 9V0due L # (1 (((( # (1 # (1 # (1 2 , _ ¸ ¸ + + + + + + + t r C r C r C r 9V0due L (1 O r# 9V0 0nd the BV0 due is+ BV0 L C O C(1 O r# O C(1 O r# 2 O ]( O C(1 O r# t E 1 BV0due L C(1 O r# O C(1 O r# 2 O ]( O C(1 O r# t BV0due L (1 O r#QC O C(1 O r# O ]( O C(1 O r# t E 1 R BV0due L (1 O r#BV0 B-100 SOLUTIONS "4. We need to find the lu"p su" pa%"ent into the retire"ent account( -he present *alue of the desired a"ount at retire"ent is+ 9V L BV:(1 O r# t 9V L <2,000,000:(1 O (11# I0 9V L <=0,JM8(82 -his is the *alue toda%( ,ince the sa*ings are in the for" of a groing annuit%, e can use the groing annuit% e$uation and sol*e for the pa%"ent( )oing so, e get+ 9V L C YQ1 E ((1 O g#:(1 O r## t R : (r E g#Z <=0,JM8(82 L CYQ1 E ((1 O (0=#:(1 O (11## I0 R : ((11 E (0=#Z C L <2,A91(AM -his is the a"ount %ou need to sa*e ne!t %ear( ,o, the percentage of %our salar% is+ 9ercentage of salar% L <2,A91(AM:<I0,000 9ercentage of salar% L (0MI8 or M(I8N 4ote that this is the percentage of %our salar% %ou "ust sa*e each %ear( ,ince %our salar% is increasing at = percent, and the sa*ings are increasing at = percent, the percentage of salar% ill re"ain constant( ". a. -he 09R is the interest rate per ee& ti"es A2 ee&s in a %ear, so+ 09R L A2(JN# L =MIN 60R L (1 O (0J# A2 E 1 L =2(J2A= or =,2J=(A=N b. .n a discount loan, the a"ount %ou recei*e is loered b% the discount, and %ou repa% the full principal( With a J percent discount, %ou ould recei*e <9(=0 for e*er% <10 in principal, so the ee&l% interest rate ould be+ <10 L <9(=0(1 O r# r L (<10 : <9(=0# E 1 L (0JA= or J(A=N 4ote the dollar a"ount e use is irrele*ant( .n other ords, e could use <0(9= and <1, <9= and <100, or an% other co"bination and e ould get the sa"e interest rate( 4o e can find the 09R and the 60R+ 09R L A2(J(A=N# L =91(I0N 60R L (1 O (0JA=# A2 E 1 L I2(A=98 or I,2A=(98N CHAPTER 6 B-101 c. Using the cash flos fro" the loan, e ha*e the 9V0 and the annuit% pa%"ents and need to find the interest rate, so+ 9V0 L <M8(92 L <2AQY1 E Q1 : (1 O r#R I Z: r R Using a spreadsheet, trial and error, or a financial calculator, e find+ r L 1M(JAN per ee& 09R L A2(1M(JAN# L 8J0(99N 60R L 1(1MJA A2 E 1 L =1I1(JIJ2 or =1I,1JI(J2N "!. -o anser this, e need to diagra" the perpetuit% cash flos, hich are+ (4ote, the subscripts are onl% to differentiate hen the cash flos begin( -he cash flos are all the sa"e a"ount(# ](( C= C2 C2 C1 C1 C1 -hus, each of the increased cash flos is a perpetuit% in itself( ,o, e can rite the cash flos strea" as+ C1:R C2:R C=:R CI:R ]( ,o, e can rite the cash flos as the present *alue of a perpetuit%, and a perpetuit% of+ C2:R C=:R CI:R ]( -he present *alue of this perpetuit% is+ 9V L (C:R# : R L C:R 2 ,o, the present *alue e$uation of a perpetuit% that increases b% C each period is+ 9V L C:R O C:R 2 B-102 SOLUTIONS "". We are onl% concerned ith the ti"e it ta&es "one% to double, so the dollar a"ounts are irrele*ant( ,o, e can rite the future *alue of a lu"p su" as+ BV L 9V(1 O R# t <2 L <1(1 O R# t ,ol*ing for t, e find+ ln(2# L tQln(1 O R#R t L ln(2# : ln(1 O R# ,ince R is e!pressed as a percentage in this case, e can rite the e!pression as+ t L ln(2# : ln(1 O R:100# -o si"plif% the e$uation, e can "a&e use of a -a%lor ,eries e!pansion+ ln(1 O R# L R E R 2 :2 O R = := E ((( ,ince R is s"all, e can truncate the series after the first ter"+ ln(1 O R# L R Co"bine this ith the solution for the doubling e!pression+ t L ln(2# : (R:100# t L 100ln(2# : R t L M9(=1IJ : R -his is the e!act (appro!i"ate# e!pression, ,ince M9(=1IJ is not easil% di*isible, and e are onl% concerned ith an appro!i"ation, J2 is substituted( "#. We are onl% concerned ith the ti"e it ta&es "one% to double, so the dollar a"ounts are irrele*ant( ,o, e can rite the future *alue of a lu"p su" ith continuousl% co"pounded interest as+ <2 L <1e Rt 2 L e Rt Rt L ln(2# Rt L (M9=1IJ t L (M9=1IJ : R ,ince e are using interest rates hile the e$uation uses deci"al for", to "a&e the e$uation correct ith percentages, e can "ultipl% b% 100+ t L M9(=1IJ : R CHAPTER 6 B-103 Calculator Solutions 1. C2o <0 C2o <0 C2o <0 C%1 <9A0 C%1 <9A0 C%1 <9A0 2%1 1 2%1 1 2%1 1 C%2 <1,0I0 C%2 <1,0I0 C%2 <1,0I0 2%2 1 2%2 1 2%2 1 C%3 <1,1=0 C%3 <1,1=0 C%3 <1,1=0 2%3 1 2%3 1 2%3 1 C%4 <1,0JA C%4 <1,0JA C%4 <1,0JA 2%4 1 2%4 1 2%4 1 . L 10 . L 18 . L 2I 49V C9- 49V C9- 49V C9- <=,=0M(=J <2,J9I(22 <2,I89(88 2. 6nter 9 AN <M,000 - ./0 &1 &MT 21 ,ol*e for <I2,MIM(9= 6nter M AN <8,000 - ./0 &1 &MT 21 ,ol*e for <I0,M0A(AI 6nter 9 1AN <M,000 - ./0 &1 &MT 21 ,ol*e for <28,M29(A0 6nter A 1AN <8,000 - ./0 &1 &MT 21 ,ol*e for <=0,2JA(8M 3. 6nter = 8N <9I0 - ./0 &1 &MT 21 ,ol*e for <1,18I(1= 6nter 2 8N <1,090 - ./0 &1 &MT 21 ,ol*e for <1,2J1(=8 6nter 1 8N <1,=I0 - ./0 &1 &MT 21 ,ol*e for <1,IIJ(20 BV L <1,18I(1= O 1,2J1(=8 O 1,IIJ(20 O 1,I0A L <A,=0J(J1 B-104 SOLUTIONS 6nter = 11N <9I0 - ./0 &1 &MT 21 ,ol*e for <1,28A(AJ 6nter 2 11N <1,090 - ./0 &1 &MT 21 ,ol*e for <1,=I2(99 6nter 1 11N <1,=I0 - ./0 &1 &MT 21 ,ol*e for <1,I8J(I0 BV L <1,28A(AJ O 1,=I2(99 O 1,I8J(I0 O 1,I0A L <A,A20(9M 6nter = 2IN <9I0 - ./0 &1 &MT 21 ,ol*e for <1,J92(2= 6nter 2 2IN <1,090 - ./0 &1 &MT 21 ,ol*e for <1,MJA(98 6nter 1 2IN <1,=I0 - ./0 &1 &MT 21 ,ol*e for <1,MM1(M0 BV L <1,J92(2= O 1,MJA(98 O 1,MM1(M0 O 1,I0A L <M,A=I(81 4. 6nter 1A JN <A,=00 - ./0 &1 &MT 21 ,ol*e for <I8,2J1(9I 6nter I0 JN <A,=00 - ./0 &1 &MT 21 ,ol*e for <J0,MA8(0M 6nter JA JN <A,=00 - ./0 &1 &MT 21 ,ol*e for <JA,2I0(J0 CHAPTER 6 B-105 . 6nter 1A J(MAN <=I,000 - ./0 &1 &MT 21 ,ol*e for <=,88J(J2 !. 6nter 8 8(AN <J=,000 - ./0 &1 &MT 21 ,ol*e for <I11,MM0(=M ". 6nter 20 11(2N <I,000 - ./0 &1 &MT 21 ,ol*e for <2M2,J81(1M 6nter I0 11(2N <I,000 - ./0 &1 &MT 21 ,ol*e for <2,IA9,0J2(M= #. 6nter 10 M(8N <90,000 - ./0 &1 &MT 21 ,ol*e for <M,AJA(JJ $. 6nter J J(AN <J0,000 - ./0 &1 &MT 21 ,ol*e for <9,II0(02 12. 6nter 8N I -4M 522 C/0 ,ol*e for 8(2IN 6nter 1MN 12 -4M 522 C/0 ,ol*e for 1J(2=N 6nter 12N =MA -4M 522 C/0 ,ol*e for 12(JAN 13. 6nter 8(MN 2 -4M 522 C/0 ,ol*e for 8(2IN B-106 SOLUTIONS 6nter 19(8N 12 -4M 522 C/0 ,ol*e for 18(20N 6nter 9(I0N A2 -4M 522 C/0 ,ol*e for 8(99N 14. 6nter 1I(2N 12 -4M 522 C/0 ,ol*e for 1A(1MN 6nter 1I(AN 2 -4M 522 C/0 ,ol*e for 1A(0=N 1. 6nter 1MN =MA -4M 522 C/0 ,ol*e for 1I(8AN 1!. 6nter 1J S 2 8(IN:2 <2,100 - ./0 &1 &MT 21 ,ol*e for <8,A0A(9= 1". 6nter A × =MA 9(=N : =MA <I,A00 - ./0 &1 &MT 21 ,ol*e for <J,1M=(MI 6nter 10 × =MA 9(=N : =MA <I,A00 - ./0 &1 &MT 21 ,ol*e for <11,I0=(9I 6nter 20 × =MA 9(=N : =MA <I,A00 - ./0 &1 &MT 21 ,ol*e for <28,899(9J 1#. 6nter J × =MA 10N : =MA <A8,000 - ./0 &1 &MT 21 ,ol*e for <28,80I(J1 CHAPTER 6 B-107 1$. 6nter =M0N 12 -4M 522 C/0 ,ol*e for 2,229(81N 2%. 6nter M0 M(9N : 12 <M8,A00 - ./0 &1 &MT 21 ,ol*e for <1,=A=(1A 6nter M(9N 12 -4M 522 C/0 ,ol*e for J(12N 21. 6nter 1(=N <18,000 t<A00 - ./0 &1 &MT 21 ,ol*e for I8(8M 22. 6nter 1,J==(==N A2 -4M 522 C/0 ,ol*e for =1=,91M,A1A(M9N 23. 6nter 22(JIN 12 -4M 522 C/0 ,ol*e for 2A(2MN 24. 6nter =0 × 12 10N : 12 <=00 - ./0 &1 &MT 21 ,ol*e for <MJ8,1IM(=8 2. 6nter 10(00N 12 -4M 522 C/0 ,ol*e for 10(IJN 6nter =0 10(IJN <=,M00 - ./0 &1 &MT 21 ,ol*e for <MIJ,M2=(IA 2!. 6nter I × I 0(MAN <2,=00 - ./0 &1 &MT 21 ,ol*e for <=I,8I=(J1 B-108 SOLUTIONS CHAPTER 6 B-109 2". 6nter 11(00N I -4M 522 C/0 ,ol*e for 11(IMN C2o <0 C%1 <J2A 2%1 1 C%2 <980 2%2 1 C%3 <0 2%3 1 C%4 <1,=M0 2%4 1 . L 11(IMN 49V C9- <2,=20(=M 2#. C2o <0 C%1 <1,MA0 2%1 1 C%2 <0 2%2 1 C%3 <I,200 2%3 1 C%4 <2,I=0 2%4 1 . L 8(IAN 49V C9- <M,AJ0(8M 3%. 6nter 1JN 2 -4M 522 C/0 ,ol*e for 1M(==N 1M(==N : 2 L 8(1JN 6nter 1JN I -4M 522 C/0 ,ol*e for 1M(01N 1M(01N : I L I(00N 6nter 1JN 12 -4M 522 C/0 ,ol*e for 1A(80N 1A(80N : 12 L 1(=2N B-110 SOLUTIONS 31. 6nter M 1(A0N : 12 <A,000 - ./0 &1 &MT 21 ,ol*e for <A,0=J(M2 6nter M 18N : 12 <A,0=J(M2 - ./0 &1 &MT 21 ,ol*e for <A,A08(=A <A,A08(=A E A,000 L <A08(=A 32. ,toc& account+ 6nter =M0 11N : 12 <J00 - ./0 &1 &MT 21 ,ol*e for <1,9M=,1M=(82 Hond account+ 6nter =M0 MN : 12 <=00 - ./0 &1 &MT 21 ,ol*e for <=01,=AI(A1 ,a*ings at retire"ent L <1,9M=,1M=(82 O =01,=AI(A1 L <2,2MI,A18(== 6nter =00 9N : 12 <2,2MI,A18(== - ./0 &1 &MT 21 ,ol*e for <19,00=(JM 33. 6nter 12 1(1JN <1 - ./0 &1 &MT 21 ,ol*e for <1(1A 6nter 2I 1(1JN <1 - ./0 &1 &MT 21 ,ol*e for <1(=2 34. 6nter I80 12N : 12 <1,000,000 - ./0 &1 &MT 21 ,ol*e for <8A(00 6nter =M0 12N : 12 <1,000,000 - ./0 &1 &MT 21 ,ol*e for <28M(1= CHAPTER 6 B-111 6nter 2I0 12N : 12 <1,000,000 - ./0 &1 &MT 21 ,ol*e for <1,010(8M 3. 6nter 12 : = t<1 <= - ./0 &1 &MT 21 ,ol*e for =1(M1N 3!. 6nter 10(00N 12 -4M 522 C/0 ,ol*e for 10(IJN 6nter 2 10(IJN <9A,000 - ./0 &1 &MT 21 ,ol*e for <1M=,8=9(09 C2o <IA,000 C%1 <JA,000 2%1 2 . L 10(IJN 49V C9- <1MA,J2=(9I 3$. 6nter 1A 10N <9,000 - ./0 &1 &MT 21 ,ol*e for <M8,IAI(J2 6nter 1A AN <9,000 - ./0 &1 &MT 21 ,ol*e for <9=,I2M(92 6nter 1A 1AN <9,000 - ./0 &1 &MT 21 ,ol*e for <A2,M2M(== 4%. 6nter MN : 12 t<=I0 <20,000 - ./0 &1 &MT 21 ,ol*e for A1(M9 B-112 SOLUTIONS 41. 6nter M0 <J=,000 t<1,IA0 - ./0 &1 &MT 21 ,ol*e for 0(A9IN 0(A9IN × 12 L J(1=N 42. 6nter =M0 M(=AN : 12 <1,1A0 - ./0 &1 &MT 21 ,ol*e for <18I,81J(I2 <2I0,000 E 18I,81J(I2 L <AA,182(A8 6nter =M0 M(=AN : 12 <AA,182(A8 - ./0 &1 &MT 21 ,ol*e for <=M8,9=M(AI 43. C2o <0 C%1 <1,J00 2%1 1 C%2 <0 2%2 1 C%3 <2,100 2%3 1 C%4 <2,800 2%4 1 . L 10N 49V C9- <A,0=A(MA 9V of "issing CB L <M,AA0 E A,0=A(MA L <1,A1I(=A Value of "issing CB+ 6nter 2 10N <1,A1I(=A - ./0 &1 &MT 21 ,ol*e for <1,8=2(=M CHAPTER 6 B-113 44. C2o <1,000,000 C%1 <1,A00,000 2%1 1 C%2 <2,A00,000 2%2 1 C%3 <2,800,000 2%3 1 C%4 <=,000,000 2%4 1 C% <=,A00,000 2% 1 C%! <I,000,000 2%! 1 C%" <I,A00,000 2%" 1 C%# <A,000,000 2%# 1 C%$ <A,A00,000 2%$ 1 C%1% <M,000,000 . L 9N 49V C9- <22,812,8J= 4. 6nter =M0 (80(<2,900,000# t<1A,000 - ./0 &1 &MT 21 ,ol*e for 0(AM0N 09R L 0(AM0N × 12 L M(J2N 6nter M(J2N 12 -4M 522 C/0 ,ol*e for M(9=N 4!. 6nter I 1=N <1MA,000 - ./0 &1 &MT 21 ,ol*e for <101,19J(A9 9rofit L <101,19J(A9 E 9I,000 L <J,19J(A9 6nter I t<9I,000 <1MA,000 - ./0 &1 &MT 21 ,ol*e for 1A(10N B-114 SOLUTIONS 4". 6nter 18 10N <I,000 - ./0 &1 &MT 21 ,ol*e for <=2,80A(MA 6nter J 10N <=2,80A(MA - ./0 &1 &MT 21 ,ol*e for <1M,8=I(I8 4#. 6nter 8I JN : 12 <1,A00 - ./0 &1 &MT 21 ,ol*e for <8J,M0I(=M 6nter 9M 11N : 12 <1,A00 - ./0 &1 &MT 21 ,ol*e for <110,021(=A 6nter 8I JN : 12 <110,021(=A - ./0 &1 &MT 21 ,ol*e for <A1,120(== <8J,M0I(=M O A1,120(== L <1=8,J2I(M8 4$. 6nter 1A S 12 8(AN:12 <1,200 - ./0 &1 &MT 21 ,ol*e for <I=I,1I=(M2 BV L <I=I,1I=(M2 L 9V e (08(1A# 8 9V L <I=I,1I=(M2e ;1(20 L <1=0,JM1(AA %. 9V\ t L 1I+ <=,A00 : 0(0M2 L <AM,IA1(M1 6nter J M(2N <AM,IA1(M1 - ./0 &1 &MT 21 ,ol*e for <=J,0A1(I1 1. 6nter 12 <2A,000 t<2,I1M(MJ - ./0 &1 &MT 21 ,ol*e for 2(=M1N 09R L 2(=M1N × 12 L 28(==N 6nter 28(==N 12 -4M 522 C/0 CHAPTER 6 B-115 ,ol*e for =2(=1N B-116 SOLUTIONS 2. 3onthl% rate L (10 : 12 L (008=8 se"iannual rate L (1(008=# M E 1 L A(11N 6nter 10 A(11N <J,000 - ./0 &1 &MT 21 ,ol*e for <A=,JJM(J2 6nter M A(11N <A=,JJM(J2 - ./0 &1 &MT 21 ,ol*e for <=9,888(== 6nter 10 A(11N <A=,JJM(J2 - ./0 &1 &MT 21 ,ol*e for <=2,M8I(88 6nter 1M A(11N <A=,JJM(J2 - ./0 &1 &MT 21 ,ol*e for <2I,2I=(MJ 3. a. 6nter A 11N t<10,000 - ./0 &1 &MT 21 ,ol*e for <=M,9A8(9J 2 nd HG4 2 nd ,6- 6nter A 11N t<10,000 - ./0 &1 &MT 21 ,ol*e for <I1,02I(IM b. 6nter A 11N t<10,000 - ./0 &1 &MT 21 ,ol*e for <M2,2J8(01 2 nd HG4 2 nd ,6- 6nter A 11N t<10,000 - ./0 &1 &MT 21 ,ol*e for <M9,128(M0 CHAPTER 6 B-117 4. 2 nd 67- 2 nd S5T 6nter M0 J(8AN : 12 <M8,000 - ./0 &1 &MT 21 ,ol*e for <1,=MI(99 ". 9re-retire"ent 09R+ 6nter 10N 12 -4M 522 C/0 ,ol*e for 9(AJN 9ost-retire"ent 09R+ 6nter JN 12 -4M 522 C/0 ,ol*e for M(J8N 0t retire"ent, he needs+ 6nter =00 M(J8N : 12 <20,000 <900,000 - ./0 &1 &MT 21 ,ol*e for <=,0A1,=20(J1 .n 10 %ears, his sa*ings ill be orth+ 6nter 120 J(J2N : 12 <2,A00 - ./0 &1 &MT 21 ,ol*e for <I99,MA9(MI 0fter purchasing the cabin, he ill ha*e+ <I99,MA9(MI E =80,000 L <119,MA9(MI 6ach "onth beteen %ears 10 and =0, he needs to sa*e+ 6nter 2I0 9(AJN : 12 <119,MA9(MI <=,0A1,=20(J1[ - ./0 &1 &MT 21 ,ol*e for <=,12J(II #. 9V of purchase+ 6nter =M JN : 12 <2=,000 - ./0 &1 &MT 21 ,ol*e for <18,MAI(82 <=2,000 E 18,MAI(82 L <1=,=IA(18 B-118 SOLUTIONS 9V of lease+ 6nter =M JN : 12 <IA0 - ./0 &1 &MT 21 ,ol*e for <1I,AJ=(99 <1I,AJ=(91 O 99 L <1I,MJ2(91 Hu% the car( 5ou ould be indifferent hen the 9V of the to cash flos are e$ual( -he present *alue of the purchase decision "ust be <1I,MJ2(91( ,ince the difference in the to cash flos is <=2,000 E 1I,MJ2(91 L <1J,=2J(09, this "ust be the present *alue of the future resale price of the car( -he brea&- e*en resale price of the car is+ 6nter =M JN : 12 <1J,=2J(09 - ./0 &1 &MT 21 ,ol*e for <21,=M=(01 $. 6nter A(A0N =MA -4M 522 C/0 ,ol*e for A(MAN C2o <J,000,000 C%1 <I,A00,000 2%1 1 C%2 <A,000,000 2%2 1 C%3 <M,000,000 2%3 1 C%4 <M,800,000 2%4 1 C% <J,900,000 2% 1 C%! <8,800,000 2%! 1 . L A(MAN 49V C9- <=8,M10,I82(AJ 4e contract *alue L <=8,M10,I82(AJ O 1,I00,000 L <I0,010,I82(AJ 9V of pa%"ents L <I0,010,I82(AJ E 9,000,000 L <=1,010,I82(AJ 6ffecti*e $uarterl% rate L Q1 O ((0AA:=MA#R 91(2A E 1 L (01=8I or 1(=8IN 6nter 2I 1(=8IN <=1,010,I82(AJ - ./0 &1 &MT 21 ,ol*e for <1,A2J,IM=(JM CHAPTER 6 B-119 !%. 6nter 1 <21,2A0 t<2A,000 - ./0 &1 &MT 21 ,ol*e for 1J(MAN !1. 6nter 8N 12 -4M 522 C/0 ,ol*e for J(J2N 6nter 12 J(J2N : 12 <IJ,000 : 12 - ./0 &1 &MT 21 ,ol*e for <I8,M99(=9 6nter 1 8N <I8,M99(=9 - ./0 &1 &MT 21 ,ol*e for <A2,A9A(=I 6nter 12 J(J2N : 12 <A0,000 : 12 - ./0 &1 &MT 21 ,ol*e for <A1,80J(8M 6nter M0 J(J2N : 12 <AA,000 : 12 - ./0 &1 &MT 21 ,ol*e for <22J,A=9(1I 0ard L <A2,A9A(=I O A1,80J(8M O 22J,A=9(1I O 100,000 O 20,000 L <IA1,9I2(=I !2. 6nter 1 <9,J00 t<10,800 - ./0 &1 &MT 21 ,ol*e for 11(=IN !3. 6nter 1 <9,800 t<11,200 - ./0 &1 &MT 21 ,ol*e for 1I(29N !4. Refundable fee+ With the <2,=00 application fee, %ou ill need to borro <2I2,=00 to ha*e <2I0,000 after deducting the fee( ,ol*e for the pa%"ent under these circu"stances( 6nter =0 × 12 M(80N : 12 <2I2,=00 - ./0 &1 &MT 21 ,ol*e for <1,AJ9(M1 B-120 SOLUTIONS 6nter =0 × 12 <2I0,000 t<1,AJ9(M1 - ./0 &1 &MT 21 ,ol*e for 0(AJIAN 09R L 0(AJIAN × 12 L M(89N 6nter M(89N 12 -4M 522 C/0 ,ol*e for J(12N Without refundable fee+ 09R L M(80N 6nter M(80N 12 -4M 522 C/0 ,ol*e for J(02N !. 6nter =M <1,000 t<I1(1A - ./0 &1 &MT 21 ,ol*e for 2(=0N 09R L 2(=0N × 12 L 2J(M1N 6nter 2J(M1N 12 -4M 522 C/0 ,ol*e for =1(=9N !!. What she needs at age MA+ 6nter 20 JN <10A,000 - ./0 &1 &MT 21 ,ol*e for <1,112,=J1(A0 a. 6nter =0 JN <1,112,=J1(A0 - ./0 &1 &MT 21 ,ol*e for <11,JJM(01 b. 6nter =0 JN <1,112,=J1(A0 - ./0 &1 &MT 21 ,ol*e for <1IM,129(0I c. 6nter 10 JN <1A0,000 - ./0 &1 &MT 21 ,ol*e for <29A,0J2(J0 CHAPTER 6 B-121 0t MA, she is short+ <1,112,=J1(A0 E 29A,0J2(A0 L <81J,298(80 6nter =0 JN [<81J,298(80 - ./0 &1 &MT 21 ,ol*e for <8,MA2(2A >er e"plo%er ill contribute <1,A00 per %ear, so she "ust contribute+ <8,MA2(2A E 1,A00 L <J,1A2(2A per %ear !". Without fee+ 6nter 19(8N : 12 <10,000 t<200 - ./0 &1 &MT 21 ,ol*e for 10M(A0 6nter M(8N : 12 <10,000 t<200 - ./0 &1 &MT 21 ,ol*e for AJ(99 With fee+ 6nter M(8N : 12 <10,200 t<200 - ./0 &1 &MT 21 ,ol*e for A9(=A !#. Value at 5ear M+ 6nter A 12N <900 - ./0 &1 &MT 21 ,ol*e for <1,A8M(11 6nter I 12N <900 - ./0 &1 &MT 21 ,ol*e for <1,I1M(1J 6nter = 12N <1,000 - ./0 &1 &MT 21 ,ol*e for <1,I0I(9= 6nter 2 12N <1,000 - ./0 &1 &MT 21 ,ol*e for <1,2AI(I0 B-122 SOLUTIONS 6nter 1 12N <1,100 - ./0 &1 &MT 21 ,ol*e for <1,2=2 ,o, at 5ear A, the *alue is+ <1,A8M(11 O 1,I1M(1J O 1,I0I(9= O 1,2AI(I0 O 1,2=2 O 1,100 L <J,99=(M0 0t 5ear MA, the *alue is+ 6nter A9 8N <J,99=(M0 - ./0 &1 &MT 21 ,ol*e for <JI9,IA2(AM -he polic% is not orth bu%ing8 the future *alue of the deposits is <JI9,IA2(AM but the polic% contract ill pa% off <A00,000( !$. 6nter =0 × 12 8(1N : 12 <JA0,000 - ./0 &1 &MT 21 ,ol*e for <A,AAA(M1 6nter 22 × 12 8(1N : 12 <A,AAA(M1 - ./0 &1 &MT 21 ,ol*e for <M8=,J00(=2 "%. C2o t<9,000 C%1 t<9,000 2%1 A C%2 <20,000 2%2 I .RR C9- 8(0JN ". a. A&R 3 "8 × 2 3 3!48 6nter =MIN A2 -4M 522 C/0 ,ol*e for =,2J2(A=N b. 6nter 1 <9(=0 t<10(00 - ./0 &1 &MT 21 ,ol*e for J(A=N CHAPTER 6 B-123 09R L J(A=N × A2 L =91(I0N 6nter =91(I0N A2 -4M 522 C/0 ,ol*e for I,2A=(98N c. 6nter I <M8(92 t<2A - ./0 &1 &MT 21 ,ol*e for 1M(JAN 09R L 1M(JAN × A2 L 8J0(99N 6nter 8J0(99N A2 -4M 522 C/0 ,ol*e for =1I,1JI(J2N CHAPTER 7 INTEREST RATES AND BOND VALUATION Answers to Concepts Review and Critical Thinking Questions 1. 4o( 0s interest rates fluctuate, the *alue of a -reasur% securit% ill fluctuate( Cong-ter" -reasur% securities ha*e substantial interest rate ris&( 2. 0ll else the sa"e, the -reasur% securit% ill ha*e loer coupons because of its loer default ris&, so it ill ha*e greater interest rate ris&( 3. 4o( .f the bid price ere higher than the as& price, the i"plication ould be that a dealer as illing to sell a bond and i""ediatel% bu% it bac& at a higher price( >o "an% such transactions ould %ou li&e to do? 4. 9rices and %ields "o*e in opposite directions( ,ince the bid price "ust be loer, the bid %ield "ust be higher( . -here are to benefits( Birst, the co"pan% can ta&e ad*antage of interest rate declines b% calling in an issue and replacing it ith a loer coupon issue( ,econd, a co"pan% "ight ish to eli"inate a co*enant for so"e reason( Calling the issue does this( -he cost to the co"pan% is a higher coupon( 0 put pro*ision is desirable fro" an in*estor's standpoint, so it helps the co"pan% b% reducing the coupon rate on the bond( -he cost to the co"pan% is that it "a% ha*e to bu% bac& the bond at an unattracti*e price( !. Hond issuers loo& at outstanding bonds of si"ilar "aturit% and ris&( -he %ields on such bonds are used to establish the coupon rate necessar% for a particular issue to initiall% sell for par *alue( Hond issuers also si"pl% as& potential purchasers hat coupon rate ould be necessar% to attract the"( -he coupon rate is fi!ed and si"pl% deter"ines hat the bond's coupon pa%"ents ill be( -he re$uired return is hat in*estors actuall% de"and on the issue, and it ill fluctuate through ti"e( -he coupon rate and re$uired return are e$ual onl% if the bond sells for e!actl% at par( ". 5es( ,o"e in*estors ha*e obligations that are deno"inated in dollars8 i(e(, the% are no"inal( -heir pri"ar% concern is that an in*est"ent pro*ide the needed no"inal dollar a"ounts( 9ension funds, for e!a"ple, often "ust plan for pension pa%"ents "an% %ears in the future( .f those pa%"ents are fi!ed in dollar ter"s, then it is the no"inal return on an in*est"ent that is i"portant( #. Co"panies pa% to ha*e their bonds rated si"pl% because unrated bonds can be difficult to sell8 "an% large in*estors are prohibited fro" in*esting in unrated issues( CHAPTER 7 B-125 $. -reasur% bonds ha*e no credit ris& since it is bac&ed b% the U(,( go*ern"ent, so a rating is not necessar%( Jun& bonds often are not rated because there ould be no point in an issuer pa%ing a rating agenc% to assign its bonds a lo rating (it's li&e pa%ing so"eone to &ic& %ouX#( B-126 SOLUTIONS 1%. -he ter" structure is based on pure discount bonds( -he %ield cur*e is based on coupon-bearing issues( 11. Hond ratings ha*e a sub1ecti*e factor to the"( ,plit ratings reflect a difference of opinion a"ong credit agencies( 12. 0s a general constitutional principle, the federal go*ern"ent cannot ta! the states ithout their consent if doing so ould interfere ith state go*ern"ent functions( 0t one ti"e, this principle as thought to pro*ide for the ta!-e!e"pt status of "unicipal interest pa%"ents( >oe*er, "odern court rulings "a&e it clear that Congress can re*o&e the "unicipal e!e"ption, so the onl% basis no appears to be historical precedent( -he fact that the states and the federal go*ern"ent do not ta! each other's securities is referred to as ;reciprocal i""unit%(@ 13. Cac& of transparenc% "eans that a bu%er or seller can't see recent transactions, so it is "uch harder to deter"ine hat the best bid and as& prices are at an% point in ti"e( 14. Co"panies charge that bond rating agencies are pressuring the" to pa% for bond ratings( When a co"pan% pa%s for a rating, it has the opportunit% to "a&e its case for a particular rating( With an unsolicited rating, the co"pan% has no input( 1. 0 100-%ear bond loo&s li&e a share of preferred stoc&( .n particular, it is a loan ith a life that al"ost certainl% e!ceeds the life of the lender, assu"ing that the lender is an indi*idual( With a 1un& bond, the credit ris& can be so high that the borroer is al"ost certain to default, "eaning that the creditors are *er% li&el% to end up as part oners of the business( .n both cases, the ;e$uit% in disguise@ has a significant ta! ad*antage( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he %ield to "aturit% is the re$uired rate of return on a bond e!pressed as a no"inal annual interest rate( Bor noncallable bonds, the %ield to "aturit% and re$uired rate of return are interchangeable ter"s( Unli&e 5-3 and re$uired return, the coupon rate is not a return used as the interest rate in bond cash flo *aluation, but is a fi!ed percentage of par o*er the life of the bond used to set the coupon pa%"ent a"ount( Bor the e!a"ple gi*en, the coupon rate on the bond is still 10 percent, and the 5-3 is 8 percent( 2. 9rice and %ield "o*e in opposite directions8 if interest rates rise, the price of the bond ill fall( -his is because the fi!ed coupon pa%"ents deter"ined b% the fi!ed coupon rate are not as *aluable hen interest rates riseDhence, the price of the bond decreases( CHAPTER 7 B-127 NOTE: Most problems do not e<plicitl list a par value for bonds. Even though a bond can have an par value# in general# corporate bonds in the =nited 'tates will have a par value of >-#222. 9e will use this par value in all problems unless a different par value is e<plicitl stated. 3. -he price of an% bond is the 9V of the interest pa%"ent, plus the 9V of the par *alue( 4otice this proble" assu"es an annual coupon( -he price of the bond ill be+ 9 L <JA(Y1 E Q1:(1 O (08JA#R 10 Z : (08JA# O <1,000Q1 : (1 O (08JA# 10 R L <918(89 We ould li&e to introduce shorthand notation here( Rather than rite (or t%pe, as the case "a% be# the entire e$uation for the 9V of a lu"p su", or the 9V0 e$uation, it is co""on to abbre*iate the e$uations as+ 9V.B?#t L 1 : (1 O r8 t hich stands for 9resent Value .nterest Bactor 9V.B0?#t L (Y1 E Q1:(1 O r8R t Z : r # hich stands for 9resent Value .nterest Bactor of an 0nnuit% -hese abbre*iations are short hand notation for the e$uations in hich the interest rate and the nu"ber of periods are substituted into the e$uation and sol*ed( We ill use this shorthand notation in re"ainder of the solutions &e%( 4. >ere e need to find the 5-3 of a bond( -he e$uation for the bond price is+ 9 L <9=I L <90(9V.B0?/,9# O <1,000(9V.B?N,9# 4otice the e$uation cannot be sol*ed directl% for ?( Using a spreadsheet, a financial calculator, or trial and error, e find+ ? L 5-3 L 10(1AN .f %ou are using trial and error to find the 5-3 of the bond, %ou "ight be ondering ho to pic& an interest rate to start the process( Birst, e &no the 5-3 has to be higher than the coupon rate since the bond is a discount bond( -hat still lea*es a lot of interest rates to chec&( /ne a% to get a starting point is to use the folloing e$uation, hich ill gi*e %ou an appro!i"ation of the 5-3+ 0ppro!i"ate 5-3 L Q0nnual interest pa%"ent O (9rice difference fro" par : 5ears to "aturit%#R : Q(9rice O 9ar *alue# : 2R ,ol*ing for this proble", e get+ 0ppro!i"ate 5-3 L Q<90 O (<MI : 9R : Q(<9=I O 1,000# : 2R L 10(0IN -his is not the e!act 5-3, but it is close, and it ill gi*e %ou a place to start( B-128 SOLUTIONS . >ere e need to find the coupon rate of the bond( 0ll e need to do is to set up the bond pricing e$uation and sol*e for the coupon pa%"ent as follos+ 9 L <1,0IA L C(9V.B0J(AN,1=# O <1,000(9V.BJ(AN,=M# ,ol*ing for the coupon pa%"ent, e get+ C L <80(AI -he coupon pa%"ent is the coupon rate ti"es par *alue( Using this relationship, e get+ Coupon rate L <80(AI : <1,000 L (080A or 8(0AN !. -o find the price of this bond, e need to reali2e that the "aturit% of the bond is 10 %ears( -he bond as issued one %ear ago, ith 11 %ears to "aturit%, so there are 10 %ears left on the bond( 0lso, the coupons are se"iannual, so e need to use the se"iannual interest rate and the nu"ber of se"iannual periods( -he price of the bond is+ 9 L <=I(A0(9V.B0=(JN,20# O <1,000(9V.B=(JN,20# L <9MA(10 ". >ere e are finding the 5-3 of a se"iannual coupon bond( -he bond price e$uation is+ 9 L <1,0A0 L <I2(9V.B0?/,20# O <1,000(9V.B?/,20# ,ince e cannot sol*e the e$uation directl% for ?, using a spreadsheet, a financial calculator, or trial and error, e find+ ? L =(8=JN ,ince the coupon pa%"ents are se"iannual, this is the se"iannual interest rate( -he 5-3 is the 09R of the bond, so+ 5-3 L 2 × =(8=JN L J(MJN #. >ere e need to find the coupon rate of the bond( 0ll e need to do is to set up the bond pricing e$uation and sol*e for the coupon pa%"ent as follos+ 9 L <92I L C(9V.B0=(IN,29# O <1,000(9V.B=(IN,29# ,ol*ing for the coupon pa%"ent, e get+ C L <29(8I ,ince this is the se"iannual pa%"ent, the annual coupon pa%"ent is+ 2 S <29(8I L <A9(M8 0nd the coupon rate is the annual coupon pa%"ent di*ided b% par *alue, so+ Coupon rate L <A9(M8 : <1,000 Coupon rate L (0A9J or A(9JN CHAPTER 7 B-129 $. -he appro!i"ate relationship beteen no"inal interest rates (?#, real interest rates (r#, and inflation (h# is+ ? L r O h 0ppro!i"ate r L (0J E (0=8 L(0=2 or =(20N -he Bisher e$uation, hich shos the e!act relationship beteen no"inal interest rates, real interest rates, and inflation is+ (1 O ?# L (1 O r#(1 O h# (1 O (0J# L (1 O r#(1 O (0=8# 6!act r L Q(1 O (0J# : (1 O (0=8#R E 1 L (0=08 or =(08N 1%. -he Bisher e$uation, hich shos the e!act relationship beteen no"inal interest rates, real interest rates, and inflation is+ (1 O ?# L (1 O r#(1 O h# ? L (1 O (0IJ#(1 O (0=# E 1 L (0J8I or J(8IN 11. -he Bisher e$uation, hich shos the e!act relationship beteen no"inal interest rates, real interest rates, and inflation is+ (1 O ?# L (1 O r#(1 O h# h L Q(1 O (1I# : (1 O (09#R E 1 L (0IA9 or I(A9N 12. -he Bisher e$uation, hich shos the e!act relationship beteen no"inal interest rates, real interest rates, and inflation is+ (1 O ?# L (1 O r#(1 O h# r L Q(1 O (11I# : (1(0I8#R E 1 L (0M=0 or M(=0N 13. -his is a bond since the "aturit% is greater than 10 %ears( -he coupon rate, located in the first colu"n of the $uote is M(12AN( -he bid price is+ Hid price L 120+0J L 120 J:=2 L 120(218JAN× <1,000 L <1,202(18JA -he pre*ious da%'s as& price is found b%+ 9re*ious da%'s as&ed price L -oda%'s as&ed price E Change L 120 8:=2 E (A:=2# L 120 =:=2 -he pre*ious da%'s price in dollars as+ 9re*ious da%'s dollar price L 120(I0MN× <1,000 L <1,20I(0M B-130 SOLUTIONS 14. -his is a pre"iu" bond because it sells for "ore than 100N of face *alue( -he current %ield is+ Current %ield L 0nnual coupon pa%"ent : 9rice L <JA:<1,=A1(AM2A L A(9J8N -he 5-3 is located under the ;0s&ed 5ield@ colu"n, so the 5-3 is I(IJN( -he bid-as& spread is the difference beteen the bid price and the as& price, so+ Hid-0s& spread L 1=A+0M E 1=A+0A L 1:=2 &ntermediate 1. >ere e are finding the 5-3 of se"iannual coupon bonds for *arious "aturit% lengths( -he bond price e$uation is+ P = C(PVIFAR%,t) + $1,000(PVIFR%,t) U+ 90 L <80(9V.B0MN,1=# O <1,000(9V.BMN,1=# L <1,1JJ(0A 91 L <80(9V.B0MN,12# O <1,000(9V.BMN,12# L <1,1MJ(M8 9= L <80(9V.B0MN,10# O <1,000(9V.BMN,10# L <1,1IJ(20 98 L <80(9V.B0MN,A# O <1,000(9V.BMN,A# L <1,08I(2A 912 L <80(9V.B0MN,1# O <1,000(9V.BMN,1# L <1,018(8J 91= L <1,000 5+ 90 L <M0(9V.B08N,1=# O <1,000(9V.B8N,1=# L <8I1(92 91 L <M0(9V.B08N,12# O <1,000(9V.B8N,12# L <8I9(28 9= L <M0(9V.B08N,10# O <1,000(9V.B8N,10# L <8MA(80 98 L <M0(9V.B08N,A# O <1,000(9V.B8N,A# L <920(1A 912 L <M0(9V.B08N,1# O <1,000(9V.B8N,1# L <981(I8 91= L <1,000 0ll else held e$ual, the pre"iu" o*er par *alue for a pre"iu" bond declines as "aturit% approaches, and the discount fro" par *alue for a discount bond declines as "aturit% approaches( -his is called ;pull to par(@ .n both cases, the largest percentage price changes occur at the shortest "aturit% lengths( 0lso, notice that the price of each bond hen no ti"e is left to "aturit% is the par *alue, e*en though the purchaser ould recei*e the par *alue plus the coupon pa%"ent i""ediatel%( -his is because e calculate the clean price of the bond( CHAPTER 7 B-131 1!. 0n% bond that sells at par has a 5-3 e$ual to the coupon rate( Hoth bonds sell at par, so the initial 5-3 on both bonds is the coupon rate, 9 percent( .f the 5-3 suddenl% rises to 11 percent+ 9,a" L <IA(9V.B0A(AN,M# O <1,000(9V.BA(AN,M# L <9A0(0I 9)a*e L <IA(9V.B0A(AN,I0# O <1,000(9V.BA(AN,I0# L <8=9(AI -he percentage change in price is calculated as+ 9ercentage change in price L (4e price E /riginal price# : /riginal price ∆9,a"N L (<9A0(0I E 1,000# : <1,000 L E A(00N ∆9)a*eN L (<8=9(AI E 1,000# : <1,000 L E 1M(0AN .f the 5-3 suddenl% falls to J percent+ 9,a" L <IA(9V.B0=(AN,M# O <1,000(9V.B=(AN,M# L <1,0A=(29 9)a*e L <IA(9V.B0=(AN,I0# O <1,000(9V.B=(AN,I0# L <1,21=(AA ∆9,a"N L (<1,0A=(29 E 1,000# : <1,000 L O A(==N ∆9)a*eN L (<1,21=(AA E 1,000# : <1,000 L O 21(=MN 0ll else the sa"e, the longer the "aturit% of a bond, the greater is its price sensiti*it% to changes in interest rates( 1". .nitiall%, at a 5-3 of 8 percent, the prices of the to bonds are+ 9J L <20(9V.B0IN,18# O <1,000(9V.BIN,18# L <JIM(81 9P L <M0(9V.B0IN,18# O <1,000(9V.BIN,18# L <1,2A=(19 .f the 5-3 rises fro" 8 percent to 10 percent+ 9J L <20(9V.B0AN,18# O <1,000(9V.BAN,18# L <MI9(=1 9P L <M0(9V.B0AN,18# O <1,000(9V.BAN,18# L <1,11M(90 -he percentage change in price is calculated as+ 9ercentage change in price L (4e price E /riginal price# : /riginal price ∆9JN L (<MI9(=1 E JIM(81# : <JIM(81 L E 1=(0MN ∆9PN L (<1,11M(90 E 1,2A=(19# : <1,2A=(19 L E 10(88N B-132 SOLUTIONS .f the 5-3 declines fro" 8 percent to M percent+ 9J L <20(9V.B0=N,18# O <1,000(9V.B=N,18# L <8M2(IM 9P L <M0(9V.B0=N,18# O <1,000(9V.B=N,18# L <1,I12(M1 ∆9JN L (<8M2(IM E JIM(81# : <JIM(81 L O 1A(I9N ∆9PN L (<1,I12(M1 E 1,2A=(19# : <1,2A=(19 L O 12(J2N 0ll else the sa"e, the loer the coupon rate on a bond, the greater is its price sensiti*it% to changes in interest rates( 1#. -he bond price e$uation for this bond is+ 90 L <1,0M8 L <IM(9V.B0?/,18# O <1,000(9V.B?/,18# Using a spreadsheet, financial calculator, or trial and error e find+ ? L I(0MN -his is the se"iannual interest rate, so the 5-3 is+ 5-3 L 2 × I(0MN L 8(12N -he current %ield is+ Current %ield L 0nnual coupon pa%"ent : 9rice L <92 : <1,0M8 L (08M1 or 8(M1N -he effecti*e annual %ield is the sa"e as the 60R, so using the 60R e$uation fro" the pre*ious chapter+ 6ffecti*e annual %ield L (1 O 0(0I0M# 2 E 1 L (0829 or 8(29N 1$. -he co"pan% should set the coupon rate on its ne bonds e$ual to the re$uired return( -he re$uired return can be obser*ed in the "ar&et b% finding the 5-3 on outstanding bonds of the co"pan%( ,o, the 5-3 on the bonds currentl% sold in the "ar&et is+ 9 L <9=0 L <I0(9V.B0?/,I0# O <1,000(9V.B?/,I0# Using a spreadsheet, financial calculator, or trial and error e find+ ? L I(=J=N -his is the se"iannual interest rate, so the 5-3 is+ 5-3 L 2 × I(=J=N L 8(JAN CHAPTER 7 B-133 2%. 0ccrued interest is the coupon pa%"ent for the period ti"es the fraction of the period that has passed since the last coupon pa%"ent( ,ince e ha*e a se"iannual coupon bond, the coupon pa%"ent per si! "onths is one-half of the annual coupon pa%"ent( -here are four "onths until the ne!t coupon pa%"ent, so to "onths ha*e passed since the last coupon pa%"ent( -he accrued interest for the bond is+ 0ccrued interest L <JI:2 S 2:M L <12(== 0nd e calculate the clean price as+ Clean price L )irt% price E 0ccrued interest L <9M8 E 12(== L <9AA(MJ 21. 0ccrued interest is the coupon pa%"ent for the period ti"es the fraction of the period that has passed since the last coupon pa%"ent( ,ince e ha*e a se"iannual coupon bond, the coupon pa%"ent per si! "onths is one-half of the annual coupon pa%"ent( -here are to "onths until the ne!t coupon pa%"ent, so four "onths ha*e passed since the last coupon pa%"ent( -he accrued interest for the bond is+ 0ccrued interest L <M8:2 S I:M L <22(MJ 0nd e calculate the dirt% price as+ )irt% price L Clean price O 0ccrued interest L <1,0J= O 22(MJ L <1,09A(MJ 22. -o find the nu"ber of %ears to "aturit% for the bond, e need to find the price of the bond( ,ince e alread% ha*e the coupon rate, e can use the bond price e$uation, and sol*e for the nu"ber of %ears to "aturit%( We are gi*en the current %ield of the bond, so e can calculate the price as+ Current %ield L (0JAA L <80:90 90 L <80:(0JAA L <1,0A9(M0 4o that e ha*e the price of the bond, the bond price e$uation is+ 9 L <1,0A9(M0 L <80Q(1 E (1:1(0J2# t # : (0J2 R O <1,000:1(0J2 t We can sol*e this e$uation for t as follos+ <1,0A9(M0(1(0J2# t L <1,111(11(1(0J2# t E 1,111(11 O 1,000 111(11 L A1(A1(1(0J2# t 2(1AJ0 L 1(0J2 t t L log 2(1AJ0 : log 1(0J2 L 11(0M ≈ 11 %ears -he bond has 11 %ears to "aturit%( B-134 SOLUTIONS 23. -he bond has 1I %ears to "aturit%, so the bond price e$uation is+ 9 L <1,089(M0 L <=M(9V.B0?/,28# O <1,000(9V.B?/,28# Using a spreadsheet, financial calculator, or trial and error e find+ ? L =(11MN -his is the se"iannual interest rate, so the 5-3 is+ 5-3 L 2 × =(11MN L M(2=N -he current %ield is the annual coupon pa%"ent di*ided b% the bond price, so+ Current %ield L <J2 : <1,089(M0 L (0MM1 or M(M1N 24. a( -he bond price is the present *alue of the cash flos fro" a bond( -he 5-3 is the interest rate used in *aluing the cash flos fro" a bond( b( .f the coupon rate is higher than the re$uired return on a bond, the bond ill sell at a pre"iu", since it pro*ides periodic inco"e in the for" of coupon pa%"ents in e!cess of that re$uired b% in*estors on other si"ilar bonds( .f the coupon rate is loer than the re$uired return on a bond, the bond ill sell at a discount since it pro*ides insufficient coupon pa%"ents co"pared to that re$uired b% in*estors on other si"ilar bonds( Bor pre"iu" bonds, the coupon rate e!ceeds the 5-38 for discount bonds, the 5-3 e!ceeds the coupon rate, and for bonds selling at par, the 5-3 is e$ual to the coupon rate( c( Current %ield is defined as the annual coupon pa%"ent di*ided b% the current bond price( Bor pre"iu" bonds, the current %ield e!ceeds the 5-3, for discount bonds the current %ield is less than the 5-3, and for bonds selling at par *alue, the current %ield is e$ual to the 5-3( .n all cases, the current %ield plus the e!pected one-period capital gains %ield of the bond "ust be e$ual to the re$uired return( 2. -he price of a 2ero coupon bond is the 9V of the par, so+ a( 90 L <1,000:1(0IA A0 L <110(J1 b( .n one %ear, the bond ill ha*e 2I %ears to "aturit%, so the price ill be+ 91 L <1,000:1(0IA I8 L <120(90 CHAPTER 7 B-135 -he interest deduction is the price of the bond at the end of the %ear, "inus the price at the beginning of the %ear, so+ 5ear 1 interest deduction L <120(90 E 110(J1 L <10(19 -he price of the bond hen it has one %ear left to "aturit% ill be+ 92I L <1,000:1(0IA 2 L <91A(J= 5ear 2I interest deduction L <1,000 E 91A(J= L <8I(2J c( 9re*ious .R, regulations re$uired a straight-line calculation of interest( -he total interest recei*ed b% the bondholder is+ -otal interest L <1,000 E 110(J1 L <889(29 -he annual interest deduction is si"pl% the total interest di*ided b% the "aturit% of the bond, so the straight-line deduction is+ 0nnual interest deduction L <889(29 : 2A L <=A(AJ d( -he co"pan% ill prefer straight-line "ethods hen alloed because the *aluable interest deductions occur earlier in the life of the bond( 2!. a( -he coupon bonds ha*e an 8N coupon hich "atches the 8N re$uired return, so the% ill sell at par( -he nu"ber of bonds that "ust be sold is the a"ount needed di*ided b% the bond price, so+ 4u"ber of coupon bonds to sell L <=0,000,000 : <1,000 L =0,000 -he nu"ber of 2ero coupon bonds to sell ould be+ 9rice of 2ero coupon bonds L <1,000:1(0I M0 L <9A(0M 4u"ber of 2ero coupon bonds to sell L <=0,000,000 : <9A(0M L =1A,A89 b( -he repa%"ent of the coupon bond ill be the par *alue plus the last coupon pa%"ent ti"es the nu"ber of bonds issued( ,o+ Coupon bonds repa%"ent L =0,000(<1,0I0# L <=2,I00,000 -he repa%"ent of the 2ero coupon bond ill be the par *alue ti"es the nu"ber of bonds issued, so+ ^eroes+ repa%"ent L =1A,A89(<1,000# L <=1A,A88,822 B-136 SOLUTIONS c( -he total coupon pa%"ent for the coupon bonds ill be the nu"ber bonds ti"es the coupon pa%"ent( Bor the cash flo of the coupon bonds, e need to account for the ta! deductibilit% of the interest pa%"ents( -o do this, e ill "ultipl% the total coupon pa%"ent ti"es one "inus the ta! rate( ,o+ Coupon bonds+ (=0,000#(<80#(1E(=A# L <1,AM0,000 cash outflo 4ote that this is cash outflo since the co"pan% is "a&ing the interest pa%"ent( Bor the 2ero coupon bonds, the first %ear interest pa%"ent is the difference in the price of the 2ero at the end of the %ear and the beginning of the %ear( -he price of the 2eroes in one %ear ill be+ 91 L <1,000:1(0I A8 L <102(82 -he %ear 1 interest deduction per bond ill be this price "inus the price at the beginning of the %ear, hich e found in part b, so+ 5ear 1 interest deduction per bond L <102(82 E 9A(0M L <J(JM -he total cash flo for the 2eroes ill be the interest deduction for the %ear ti"es the nu"ber of 2eroes sold, ti"es the ta! rate( -he cash flo for the 2eroes in %ear 1 ill be+ Cash flos for 2eroes in 5ear 1 L (=1A,A89#(<J(JM#((=A# L <8AM,800(00 4otice the cash flo for the 2eroes is a cash inflo( -his is because of the ta! deductibilit% of the i"puted interest e!pense( -hat is, the co"pan% gets to rite off the interest e!pense for the %ear e*en though the co"pan% did not ha*e a cash flo for the interest e!pense( -his reduces the co"pan%'s ta! liabilit%, hich is a cash inflo( )uring the life of the bond, the 2ero generates cash inflos to the fir" in the for" of the interest ta! shield of debt( We should note an i"portant point here+ .f %ou find the 9V of the cash flos fro" the coupon bond and the 2ero coupon bond, the% ill be the sa"e( -his is because of the "uch larger repa%"ent a"ount for the 2eroes( 2". We found the "aturit% of a bond in 9roble" 22( >oe*er, in this case, the "aturit% is indeter"inate( 0 bond selling at par can ha*e an% length of "aturit%( .n other ords, hen e sol*e the bond pricing e$uation as e did in 9roble" 22, the nu"ber of periods can be an% positi*e nu"ber( 2#. We first need to find the real interest rate on the sa*ings( Using the Bisher e$uation, the real interest rate is+ (1 O ?# L (1 O r#(1 O h# 1 O (11 L (1 O r#(1 O (0=8# r L (0M9I or M(9IN CHAPTER 7 B-137 4o e can use the future *alue of an annuit% e$uation to find the annual deposit( )oing so, e find+ BV0 L CYQ(1 O r# t E 1R : rZ <1,A00,000 L <CQ(1(0M9I I0 E 1# : (0M9IR C L <J,M=J(JM Challenge 2$. -o find the capital gains %ield and the current %ield, e need to find the price of the bond( -he current price of Hond 9 and the price of Hond 9 in one %ear is+ 9+ 90 L <120(9V.B0JN,A# O <1,000(9V.BJN,A# L <1,11M(M9 91 L <120(9V.B0JN,I# O <1,000(9V.BJN,I# L <1,09J(19 Current %ield L <120 : <1,11M(M9 L (10JA or 10(JAN -he capital gains %ield is+ Capital gains %ield L (4e price E /riginal price# : /riginal price Capital gains %ield L (<1,09J(19 E 1,111(M9# : <1,11M(M9 L E(01JA or E1(JAN -he current price of Hond ) and the price of Hond ) in one %ear is+ )+ 90 L <M0(9V.B0JN,A# O <1,000(9V.BJN,A# L <88=(=1 91 L <M0(9V.B0JN,I# O <1,000(9V.BJN,I# L <902(81 Current %ield L <M0 : <88=(81 L (0MJ9 or M(J9N Capital gains %ield L (<902(81 E 88=(=1# : <88=(=1 L O(0221 or O2(21N 0ll else held constant, pre"iu" bonds pa% high current inco"e hile ha*ing price depreciation as "aturit% nears8 discount bonds do not pa% high current inco"e but ha*e price appreciation as "aturit% nears( Bor either bond, the total return is still 9N, but this return is distributed differentl% beteen current inco"e and capital gains( 3%. a( -he rate of return %ou e!pect to earn if %ou purchase a bond and hold it until "aturit% is the 5-3( -he bond price e$uation for this bond is+ 90 L <1,0M0 L <J0(9V.B0?/,10# O <1,000(9V.B ?/,10# Using a spreadsheet, financial calculator, or trial and error e find+ ? L 5-3 L M(18N B-138 SOLUTIONS b( -o find our >95, e need to find the price of the bond in to %ears( -he price of the bond in to %ears, at the ne interest rate, ill be+ 92 L <J0(9V.B0A(18N,8# O <1,000(9V.BA(18N,8# L <1,11M(92 -o calculate the >95, e need to find the interest rate that e$uates the price e paid for the bond ith the cash flos e recei*ed( -he cash flos e recei*ed ere <J0 each %ear for to %ears, and the price of the bond hen e sold it( -he e$uation to find our >95 is+ 90 L <1,0M0 L <J0(9V.B0?/,2# O <1,11M(92(9V.B?/,2# ,ol*ing for ?, e get+ ? L >95 L 9(1JN -he reali2ed >95 is greater than the e!pected 5-3 hen the bond as bought because interest rates dropped b% 1 percent8 bond prices rise hen %ields fall( 31. -he price of an% bond (or financial instru"ent# is the 9V of the future cash flos( 6*en though Hond 3 "a&es different coupons pa%"ents, to find the price of the bond, e 1ust find the 9V of the cash flos( -he 9V of the cash flos for Hond 3 is+ 93 L <1,100(9V.B0=(AN,1M#(9V.B=(AN,12# O <1,I00(9V.B0=(AN,12#(9V.B=(AN,28# O <20,000(9V.B=(AN,I0# 93 L <19,018(J8 4otice that for the coupon pa%"ents of <1,I00, e found the 9V0 for the coupon pa%"ents, and then discounted the lu"p su" bac& to toda%( Hond 4 is a 2ero coupon bond ith a <20,000 par *alue, therefore, the price of the bond is the 9V of the par, or+ 94 L <20,000(9V.B=(AN,I0# L <A,0A1(IA 32. -o calculate this, e need to set up an e$uation ith the callable bond e$ual to a eighted a*erage of the noncallable bonds( We ill in*est U percent of our "one% in the first noncallable bond, hich "eans our in*est"ent in Hond = (the other noncallable bond# ill be (1 E U#( -he e$uation is+ C2 L C1 U O C=(1 E U# 8(2A L M(A0 U O 12(1 E U# 8(2A L M(A0 U O 12 E 12 U U L 0(M8181 ,o, e in*est about M8 percent of our "one% in Hond 1, and about =2 percent in Hond =( -his co"bination of bonds should ha*e the sa"e *alue as the callable bond, e!cluding the *alue of the call( ,o+ 92 L 0(M818191 O 0(=18199= 92 L 0(M8181(10M(=JA# O 0(=1819(1=I(9M8JA# 92 L 11A(IJ=0 CHAPTER 7 B-139 -he call *alue is the difference beteen this i"plied bond *alue and the actual bond price( ,o, the call *alue is+ Call *alue L 11A(IJ=0 E 10=(A0 L 11(9J=0 0ssu"ing <1,000 par *alue, the call *alue is <119(J=( 33. In general, this is not likely to happen, although it can (and did). The reason this bond has a negative YTM is that it is a callable U.S. Treasury bond. Market participants know this. Given the high coupon rate of the bond, it is extremely likely to be called, which means the bondholder will not receive all the cash flows promised. A better measure of the return on a callable bond is the yield to call (YTC). The YTC calculation is the basically the same as the YTM calculation, but the number of periods is the number of periods until the call date. If the YTC were calculated on this bond, it would be positive. 34. -o find the present *alue, e need to find the real ee&l% interest rate( -o find the real return, e need to use the effecti*e annual rates in the Bisher e$uation( ,o, e find the real 60R is+ (1 O ?# L (1 O r#(1 O h# 1 O (08I L (1 O r#(1 O (0=J# r L (0IA= or I(A=N Now, to find the weekly interest rate, we need to find the APR. Using the equation for discrete compounding: 60R L Q1 O (09R : m#R m E 1 We can sol*e for the 09R( )oing so, e get+ 09R L mQ(1 O 60R# 1:m E 1R 09R L A2Q(1 O (0IA=# 1:A2 E 1R 09R L (0II= or I(I=N ,o, the ee&l% interest rate is+ Wee&l% rate L 09R : A2 Wee&l% rate L (0II= : A2 Wee&l% rate L (0009 or 0(09N 4o e can find the present *alue of the cost of the roses( -he real cash flos are an ordinar% annuit%, discounted at the real interest rate( ,o, the present *alue of the cost of the roses is+ 9V0 L C(Y1 E Q1:(1 O r#R t Z : r# 9V0 L <A({1 – [1/(1 + .0009)] 30(52) } / .0009) PVA = $4,312.13 B-140 SOLUTIONS 3. -o anser this $uestion, e need to find the "onthl% interest rate, hich is the 09R di*ided b% 12( We also "ust be careful to use the real interest rate( -he Bisher e$uation uses the effecti*e annual rate, so, the real effecti*e annual interest rates, and the "onthl% interest rates for each account are+ ,toc& account+ (1 O ?# L (1 O r#(1 O h# 1 O (11 L (1 O r#(1 O (0I# r L (0MJ= or M(J=N 09R L mQ(1 O 60R# 1:m E 1R 09R L 12Q(1 O (0MJ=# 1:12 E 1R 09R L (0MA= or M(A=N 3onthl% rate L 09R : 12 3onthl% rate L (0MA= : 12 3onthl% rate L (00AI or 0(AIN Hond account+ (1 O ?# L (1 O r#(1 O h# 1 O (0J L (1 O r#(1 O (0I# r L (0288 or 2(88N 09R L mQ(1 O 60R# 1:m E 1R 09R L 12Q(1 O (0288# 1:12 E 1R 09R L (028A or 2(8AN 3onthl% rate L 09R : 12 3onthl% rate L (028A : 12 3onthl% rate L (002I or 0(2IN 4o e can find the future *alue of the retire"ent account in real ter"s( -he future *alue of each account ill be+ ,toc& account+ BV0 L C {(1 + r ) t – 1] / r} FVA = $900{[(1 + .0054) 360 – 1] / .0054]} FVA = $1,001,704.05 Bond account: BV0 L C {(1 + r ) t – 1] / r} FVA = $450{[(1 + .0024) 360 – 1] / .0024]} FVA = $255,475.17 The total future value of the retirement account will be the sum of the two accounts, or: Account value = $1,001,704.05 + 255,475.17 Account value = $1,257,179.22 CHAPTER 7 B-141 Now we need to find the monthly interest rate in retirement. We can use the same procedure that we used to find the monthly interest rates for the stock and bond accounts, so: (1 O ?# L (1 O r#(1 O h# 1 O (09 L (1 O r#(1 O (0I# r L (0I81 or I(81N 09R L mQ(1 O 60R# 1:m E 1R 09R L 12Q(1 O (0I81# 1:12 E 1R 09R L (0IJ0 or I(J0N 3onthl% rate L 09R : 12 3onthl% rate L (0IJ0 : 12 3onthl% rate L (00=9 or 0(=9N 4o e can find the real "onthl% ithdraal in retire"ent( Using the present *alue of an annuit% e$uation and sol*ing for the pa%"ent, e find+ 9V0 L C(Y1 E Q1:(1 O r8R t Z : r # <1,2AJ,1J9(22 L C(Y1 E Q1:(1 O (00=9#R =00 Z : (00=9# C L <J,1=I(82 -his is the real dollar a"ount of the "onthl% ithdraals( -he no"inal "onthl% ithdraals ill increase b% the inflation rate each "onth( -o find the no"inal dollar a"ount of the last ithdraal, e can increase the real dollar ithdraal b% the inflation rate( We can increase the real ithdraal b% the effecti*e annual inflation rate since e are onl% interested in the no"inal a"ount of the last ithdraal( ,o, the last ithdraal in no"inal ter"s ill be+ BV L 9V(1 O r# t BV L <J,1=I(82(1 O (0I# (=0 O 2A# BV L <M1,M90(29 Calculator Solutions 3. 6nter 10 8(JAN <JA <1,000 - ./0 &1 &MT 21 ,ol*e for <918(89 4. 6nter 9 [<9=I <90 <1,000 - ./0 &1 &MT 21 ,ol*e for 10(1AN . 6nter 1= J(AN [<1,0IA <1,000 - ./0 &1 &MT 21 ,ol*e for <80(AI Coupon rate L <80(AI : <1,000 L 8(0AN B-142 SOLUTIONS !. 6nter 20 =(J0N <=I(A0 <1,000 - ./0 &1 &MT 21 ,ol*e for <9MA(10 ". 6nter 20 [<1,0A0 <I2 <1,000 - ./0 &1 &MT 21 ,ol*e for =(8=JN =(8=JN × 2 L J(MJN #. 6nter 29 =(I0N [<92I <1,000 - ./0 &1 &MT 21 ,ol*e for <29(8I (<29(8I : <1,000#(2# L A(9JN 1. Hond U 90 6nter 1= MN <80 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,1JJ(0A 91 6nter 12 MN <80 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,1MJ(M8 9= 6nter 10 MN <80 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,1IJ(20 98 6nter A MN <80 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,08I(2A 912 6nter 1 MN <80 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,018(8J CHAPTER 7 B-143 Hond 5 90 6nter 1= 8N <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <8I1(92 91 6nter 12 8N <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <8I9(28 9= 6nter 10 8N <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <8MA(80 98 6nter A 8N <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <920(1A 912 6nter 1 8N <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <981(I8 1!. .f both bonds sell at par, the initial 5-3 on both bonds is the coupon rate, 9 percent( .f the 5-3 suddenl% rises to 11 percent+ 9,a" 6nter M A(AN <IA <1,000 - ./0 &1 &MT 21 ,ol*e for <9A0(0I ∆9,a"N L (<9A0(0I E 1,000# : <1,000 L E A(00N 9)a*e 6nter I0 A(AN <IA <1,000 - ./0 &1 &MT 21 ,ol*e for <8=9(AI ∆9)a*eN L (<8=9(AI E 1,000# : <1,000 L E 1M(0AN .f the 5-3 suddenl% falls to J percent+ 9,a" 6nter M =(AN <IA <1,000 - ./0 &1 &MT 21 ,ol*e for <1,0A=(29 ∆9,a"N L (<1,0A=(29 E 1,000# : <1,000 L O A(==N B-144 SOLUTIONS 9)a*e 6nter I0 =(AN <IA <1,000 - ./0 &1 &MT 21 ,ol*e for <1,121=(AA ∆9)a*eN L (<1,21=(AA E 1,000# : <1,000 L O 21(=MN 0ll else the sa"e, the longer the "aturit% of a bond, the greater is its price sensiti*it% to changes in interest rates( 1". .nitiall%, at a 5-3 of 8 percent, the prices of the to bonds are+ 9J 6nter 18 IN <20 <1,000 - ./0 &1 &MT 21 ,ol*e for <JIM(81 9P 6nter 18 IN <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,2A=(19 .f the 5-3 rises fro" 8 percent to 10 percent+ 9J 6nter 18 AN <20 <1,000 - ./0 &1 &MT 21 ,ol*e for <MI9(=1 ∆9JN L (<MI9(=1 E JIM(81# : <JIM(81 L E 1=(0MN 9P 6nter 18 AN <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,11M(90 ∆9PN L (<1,11M(90 E 1,2A=(19# : <1,2A=(19 L E 10(88N .f the 5-3 declines fro" 8 percent to M percent+ 9J 6nter 18 =N <20 <1,000 - ./0 &1 &MT 21 ,ol*e for <8M2(IM ∆9JN L (<8M2(IM E JIM(81# : <JIM(81 L O 1A(I9N 9P 6nter 18 =N <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,I12(M1 ∆9PN L (<1,I12(M1 E 1,2A=(19# : <1,2A=(19 L O 12(J2N 0ll else the sa"e, the loer the coupon rate on a bond, the greater is its price sensiti*it% to changes in interest rates( CHAPTER 7 B-145 1#. 6nter 18 [<1,0M8 <IM <1,000 - ./0 &1 &MT 21 ,ol*e for I(0MN I(0MN × 2 L 8(12N 6nter 8(12 N 2 -4M 522 C/0 ,ol*e for 8(29N 1$. -he co"pan% should set the coupon rate on its ne bonds e$ual to the re$uired return8 the re$uired return can be obser*ed in the "ar&et b% finding the 5-3 on outstanding bonds of the co"pan%( 6nter I0 [<9=0 <=A <1,000 - ./0 &1 &MT 21 ,ol*e for I(=J=N I(=J=N × 2 L 8(JAN 22. Current %ield L (0JAA L <90:90 8 90 L <90:(0JAA L <1,0A9(M0 6nter J(2N [<1,0A9(M0 <80 <1,000 - ./0 &1 &MT 21 ,ol*e for 11(0M 11(0M or ≈ 11 %ears 23. 6nter 28 [<1,089(M0 <=M <1,000 - ./0 &1 &MT 21 ,ol*e for =(11MN =(11MN S 2 L M(2=N 2. a. 9o 6nter A0 I(AN <1,000 - ./0 &1 &MT 21 ,ol*e for <110(J1 b. 91 6nter I8 I(AN <1,000 - ./0 &1 &MT 21 ,ol*e for <120(90 %ear 1 interest deduction L <120(90 E 110(J1 L <10(19 919 6nter 1 I(AN <1,000 - ./0 &1 &MT 21 ,ol*e for <91A(J= %ear 2A interest deduction L <1,000 E 91A(J= L <8I(2J B-146 SOLUTIONS c. Total interest = $1,000 – 110.71 = $889.29 0nnual interest deduction L <889(29 : 2A L <=A(AJ d( -he co"pan% ill prefer straight-line "ethod hen alloed because the *aluable interest deductions occur earlier in the life of the bond( 2!. a( -he coupon bonds ha*e an 8N coupon rate, hich "atches the 8N re$uired return, so the% ill sell at par8 _ of bonds L <=0,000,000:<1,000 L =0,000( Bor the 2eroes+ 6nter M0 IN <1,000 - ./0 &1 &MT 21 ,ol*e for <9A(0M <=0,000,000:<9A(0M L =1A,A89 ill be issued( b( Coupon bonds+ repa%"ent L =0,000(<1,080# L <=2,I00,000 ^eroes+ repa%"ent L =1A,A89(<1,000# L <=1A,A88,822 c( Coupon bonds+ (=0,000#(<80#(1 E(=A# L <1,AM0,000 cash outflo ^eroes+ 6nter A8 IN <1,000 - ./0 &1 &MT 21 ,ol*e for <102(82 %ear 1 interest deduction L <102(82 E 9A(0M L <J(JM (=1A,A89#(<J(JM#((=A# L <8AM,800(00 cash inflo )uring the life of the bond, the 2ero generates cash inflos to the fir" in the for" of the interest ta! shield of debt( 2$. Hond 9 90 6nter A JN <120 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,11M(M9 91 6nter I JN <120 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,09J(19 Current %ield L <120 : <1,11M(M9 L 10(JAN Capital gains %ield L (<1,09J(19 E 1,11M(M9# : <1,11M(M9 L E1(JAN Hond ) 90 6nter A JN <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <88=(=1 CHAPTER 7 B-147 91 6nter I JN <M0 <1,000 - ./0 &1 &MT 21 ,ol*e for <902(81 Current %ield L <M0 : <88=(=1 L M(J9N Capital gains %ield L (<902(81 E 88=(=1# : <88=(=1 L 2(21N All else held constant, premium bonds pay high current income while having price depreciation as maturity nears; discount bonds do not pay high current income but have price appreciation as maturity nears. For either bond, the total return is still 9%, but this return is distributed differently between current income and capital gains. 3%. a. 6nter 10 [<1,0M0 <J0 <1,000 - ./0 &1 &MT 21 ,ol*e for M(18N -his is the rate of return %ou e!pect to earn on %our in*est"ent hen %ou purchase the bond( b. 6nter 8 A(18N <J0 <1,000 - ./0 &1 &MT 21 ,ol*e for <1,11M(92 -he >95 is+ 6nter 2 [<1,0M0 <J0 <1,11M(92 - ./0 &1 &MT 21 ,ol*e for 9(1JN -he reali2ed >95 is greater than the e!pected 5-3 hen the bond as bought because interest rates dropped b% 1 percent8 bond prices rise hen %ields fall( 31. 93 C2o <0 C%1 <0 2%1 12 C%2 <1,100 2%2 1M C%3 <1,I00 2%3 11 C%4 <21,I00 2%4 1 . L =(AN 49V C9- <19,018(J8 94 6nter I0 =(AN <20,000 - ./0 &1 &MT 21 ,ol*e for <A,0A1(IA CHAPTER 8 STOCK VALUATION Answers to Concepts Review and Critical Thinking Questions 1. -he *alue of an% in*est"ent depends on the present *alue of its cash flos8 i(e(, hat in*estors ill actuall% recei*e( -he cash flos fro" a share of stoc& are the di*idends( 2. .n*estors belie*e the co"pan% ill e*entuall% start pa%ing di*idends (or be sold to another co"pan%#( 3. .n general, co"panies that need the cash ill often forgo di*idends since di*idends are a cash e!pense( 5oung, groing co"panies ith profitable in*est"ent opportunities are one e!a"ple8 another e!a"ple is a co"pan% in financial distress( -his $uestion is e!a"ined in depth in a later chapter( 4. -he general "ethod for *aluing a share of stoc& is to find the present *alue of all e!pected future di*idends( -he di*idend groth "odel presented in the te!t is onl% *alid (i# if di*idends are e!pected to occur fore*er, that is, the stoc& pro*ides di*idends in perpetuit%, and (ii# if a constant groth rate of di*idends occurs fore*er( 0 *iolation of the first assu"ption "ight be a co"pan% that is e!pected to cease operations and dissol*e itself so"e finite nu"ber of %ears fro" no( -he stoc& of such a co"pan% ould be *alued b% appl%ing the general "ethod of *aluation e!plained in this chapter( 0 *iolation of the second assu"ption "ight be a start-up fir" that isn't currentl% pa%ing an% di*idends, but is e!pected to e*entuall% start "a&ing di*idend pa%"ents so"e nu"ber of %ears fro" no( -his stoc& ould also be *alued b% the general di*idend *aluation "ethod e!plained in this chapter( . -he co""on stoc& probabl% has a higher price because the di*idend can gro, hereas it is fi!ed on the preferred( >oe*er, the preferred is less ris&% because of the di*idend and li$uidation preference, so it is possible the preferred could be orth "ore, depending on the circu"stances( !. -he to co"ponents are the di*idend %ield and the capital gains %ield( Bor "ost co"panies, the capital gains %ield is larger( -his is eas% to see for co"panies that pa% no di*idends( Bor co"panies that do pa% di*idends, the di*idend %ields are rarel% o*er fi*e percent and are often "uch less( ". 5es( .f the di*idend gros at a stead% rate, so does the stoc& price( .n other ords, the di*idend groth rate and the capital gains %ield are the sa"e( #. .n a corporate election, %ou can bu% *otes (b% bu%ing shares#, so "one% can be used to influence or e*en deter"ine the outco"e( 3an% ould argue the sa"e is true in political elections, but, in principle at least, no one has "ore than one *ote( $. .t ouldn't see" to be( .n*estors ho don't li&e the *oting features of a particular class of stoc& are under no obligation to bu% it( 1%. .n*estors bu% such stoc& because the% ant it, recogni2ing that the shares ha*e no *oting poer( 9resu"abl%, in*estors pa% a little less for such shares than the% ould otherise( CHAPTER 8 B-149 11. 9resu"abl%, the current stoc& *alue reflects the ris&, ti"ing and "agnitude of all future cash flos, both short-ter" and long-ter"( .f this is correct, then the state"ent is false( 12. .f this assu"ption is *iolated, the to-stage di*idend groth "odel is not *alid( .n other ords, the price calculated ill not be correct( )epending on the stoc&, it "a% be "ore reasonable to assu"e that the di*idends fall fro" the high groth rate to the lo perpetual groth rate o*er a period of %ears, rather than in one %ear( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he constant di*idend groth "odel is+ 9t L )t S (1 O g# : (? E g# ,o the price of the stoc& toda% is+ 90 L )0 (1 O g# : (? E g# L <1(9A (1(0M# : ((11 E (0M# L <I1(=I -he di*idend at %ear I is the di*idend toda% ti"es the BV.B for the groth rate in di*idends and four %ears, so+ 9= L )= (1 O g# : (? E g# L )0 (1 O g# I : (? E g# L <1(9A (1(0M# I : ((11 E (0M# L <I9(2I We can do the sa"e thing to find the di*idend in 5ear 1M, hich gi*es us the price in 5ear 1A, so+ 91A L )1A (1 O g# : (? E g# L )0 (1 O g# 1M : (? E g# L <1(9A (1(0M# 1M : ((11 E (0M# L <99(0J -here is another feature of the constant di*idend groth "odel+ -he stoc& price gros at the di*idend groth rate( ,o, if e &no the stoc& price toda%, e can find the future *alue for an% ti"e in the future e ant to calculate the stoc& price( .n this proble", e ant to &no the stoc& price in three %ears, and e ha*e alread% calculated the stoc& price toda%( -he stoc& price in three %ears ill be+ 9= L 90(1 O g# = L <I1(=I(1 O (0M# = L <I9(2I 0nd the stoc& price in 1A %ears ill be+ 91A L 90(1 O g# 1A L <I1(=I(1 O (0M# 1A L <99(0J 2. We need to find the re$uired return of the stoc&( Using the constant groth "odel, e can sol*e the e$uation for ?( )oing so, e find+ ? L ()1 : 90# O g L (<2(10 : <I8(00# O (0A L (09=8 or 9(=8N B-150 SOLUTIONS 3. -he di*idend %ield is the di*idend ne!t %ear di*ided b% the current price, so the di*idend %ield is+ )i*idend %ield L )1 : 90 L <2(10 : <I8(00 L (0I=8 or I(=8N -he capital gains %ield, or percentage increase in the stoc& price, is the sa"e as the di*idend groth rate, so+ Capital gains %ield L AN 4. Using the constant groth "odel, e find the price of the stoc& toda% is+ 90 L )1 : (? E g# L <=(0I : ((11 E (0=8# L <I2(22 . -he re$uired return of a stoc& is "ade up of to parts+ -he di*idend %ield and the capital gains %ield( ,o, the re$uired return of this stoc& is+ ? L )i*idend %ield O Capital gains %ield L (0M= O (0A2 L (11A0 or 11(A0N !. We &no the stoc& has a re$uired return of 11 percent, and the di*idend and capital gains %ield are e$ual, so+ )i*idend %ield L 1:2((11# L (0AA L Capital gains %ield 4o e &no both the di*idend %ield and capital gains %ield( -he di*idend is si"pl% the stoc& price ti"es the di*idend %ield, so+ )1 L (0AA(<IJ# L <2(A9 -his is the di*idend ne!t %ear( -he $uestion as&s for the di*idend this %ear( Using the relationship beteen the di*idend this %ear and the di*idend ne!t %ear+ )1 L )0(1 O g# We can sol*e for the di*idend that as 1ust paid+ <2(A9 L )0(1 O (0AA# )0 L <2(A9 : 1(0AA L <2(IA ". -he price of an% financial instru"ent is the 9V of the future cash flos( -he future di*idends of this stoc& are an annuit% for 11 %ears, so the price of the stoc& is the 9V0, hich ill be+ 90 L <9(JA(9V.B010N,11# L <M=(== #. -he price a share of preferred stoc& is the di*idend di*ided b% the re$uired return( -his is the sa"e e$uation as the constant groth "odel, ith a di*idend groth rate of 2ero percent( Re"e"ber, "ost preferred stoc& pa%s a fi!ed di*idend, so the groth rate is 2ero( Using this e$uation, e find the price per share of the preferred stoc& is+ ? L ):90 L <A(A0:<108 L (0A09 or A(09N CHAPTER 8 B-151 $. We can use the constant di*idend groth "odel, hich is+ 9t L )t S (1 O g# : (? E g# ,o the price of each co"pan%'s stoc& toda% is+ Red stoc& price L <2(=A : ((08 E (0A# L <J8(== 5ello stoc& price L <2(=A : ((11 E (0A# L <=9(1J Hlue stoc& price L <2(=A : ((1I E (0A# L <2M(11 0s the re$uired return increases, the stoc& price decreases( -his is a function of the ti"e *alue of "one%+ 0 higher discount rate decreases the present *alue of cash flos( .t is also i"portant to note that relati*el% s"all changes in the re$uired return can ha*e a dra"atic i"pact on the stoc& price( &ntermediate 1%. -his stoc& has a constant groth rate of di*idends, but the re$uired return changes tice( -o find the *alue of the stoc& toda%, e ill begin b% finding the price of the stoc& at 5ear M, hen both the di*idend groth rate and the re$uired return are stable fore*er( -he price of the stoc& in 5ear M ill be the di*idend in 5ear J, di*ided b% the re$uired return "inus the groth rate in di*idends( ,o+ 9M L )M (1 O g# : (? E g# L )0 (1 O g# J : (? E g# L <=(A0 (1(0A# J : ((10 E (0A# L <98(A0 4o e can find the price of the stoc& in 5ear =( We need to find the price here since the re$uired return changes at that ti"e( -he price of the stoc& in 5ear = is the 9V of the di*idends in 5ears I, A, and M, plus the 9V of the stoc& price in 5ear M( -he price of the stoc& in 5ear = is+ 9= L <=(A0(1(0A# I : 1(12 O <=(A0(1(0A# A : 1(12 2 O <=(A0(1(0A# M : 1(12 = O <98(A0 : 1(12 = 9= L <80(81 Binall%, e can find the price of the stoc& toda%( -he price toda% ill be the 9V of the di*idends in 5ears 1, 2, and =, plus the 9V of the stoc& in 5ear =( -he price of the stoc& toda% is+ 90 L <=(A0(1(0A# : 1(1I O <=(A0(1(0A# 2 : (1(1I# 2 O <=(A0(1(0A# = : (1(1I# = O <80(81 : (1(1I# = 90 L <M=(IJ 11. >ere e ha*e a stoc& that pa%s no di*idends for 10 %ears( /nce the stoc& begins pa%ing di*idends, it ill ha*e a constant groth rate of di*idends( We can use the constant groth "odel at that point( .t is i"portant to re"e"ber that general constant di*idend groth for"ula is+ 9t L Q)t S (1 O g#R : (? E g# -his "eans that since e ill use the di*idend in 5ear 10, e ill be finding the stoc& price in 5ear 9( -he di*idend groth "odel is si"ilar to the 9V0 and the 9V of a perpetuit%+ -he e$uation gi*es %ou the 9V one period before the first pa%"ent( ,o, the price of the stoc& in 5ear 9 ill be+ 99 L )10 : (? E g# L <10(00 : ((1I E (0A# L <111(11 B-152 SOLUTIONS -he price of the stoc& toda% is si"pl% the 9V of the stoc& price in the future( We si"pl% discount the future stoc& price at the re$uired return( -he price of the stoc& toda% ill be+ 90 L <111(11 : 1(1I 9 L <=I(1J 12. -he price of a stoc& is the 9V of the future di*idends( -his stoc& is pa%ing four di*idends, so the price of the stoc& is the 9V of these di*idends using the re$uired return( -he price of the stoc& is+ 90 L <10 : 1(11 O <1I : 1(11 2 O <18 : 1(11 = O <22 : 1(11 I O <2M : 1(11 A L <M=(IA 13. With supernor"al di*idends, e find the price of the stoc& hen the di*idends le*el off at a constant groth rate, and then find the 9V of the future stoc& price, plus the 9V of all di*idends during the supernor"al groth period( -he stoc& begins constant groth in 5ear I, so e can find the price of the stoc& in 5ear I, at the beginning of the constant di*idend groth, as+ 9I L )I (1 O g# : (? E g# L <2(00(1(0A# : ((12 E (0A# L <=0(00 -he price of the stoc& toda% is the 9V of the first four di*idends, plus the 9V of the 5ear = stoc& price( ,o, the price of the stoc& toda% ill be+ 90 L <11(00 : 1(11 O <8(00 : 1(11 2 O <A(00 : 1(11 = O <2(00 : 1(11 I O <=0(00 : 1(11 I L <I0(09 14. With supernor"al di*idends, e find the price of the stoc& hen the di*idends le*el off at a constant groth rate, and then find the 9V of the futures stoc& price, plus the 9V of all di*idends during the supernor"al groth period( -he stoc& begins constant groth in 5ear I, so e can find the price of the stoc& in 5ear =, one %ear before the constant di*idend groth begins as+ 9= L )= (1 O g# : (? E g# L )0 (1 O g-# = (1 O g1# : (? E g# 9= L <1(80(1(=0# = (1(0M# : ((1= E (0M# 9= L <A9(88 -he price of the stoc& toda% is the 9V of the first three di*idends, plus the 9V of the 5ear = stoc& price( -he price of the stoc& toda% ill be+ 90 L <1(80(1(=0# : 1(1= O <1(80(1(=0# 2 : 1(1= 2 O <1(80(1(=0# = : 1(1= = O <A9(88 : 1(1= = 90 L <I8(J0 We could also use the to-stage di*idend groth "odel for this proble", hich is+ 90 L Q)0(1 O g1#:(R E g1#RY1 E Q(1 O g1#:(1 O R#R - ZO Q(1 O g1#:(1 O R#R - Q)0(1 O g1#:(R E g1#R 90 3 Q<1(80(1(=0#:((1= E (=0#RQ1 E (1(=0:1(1=# = R O Q(1 O (=0#:(1 O (1=#R = Q<1(80(1(0M#:((1= E (0M#R 90 3 <I8(J0 1. >ere e need to find the di*idend ne!t %ear for a stoc& e!periencing supernor"al groth( We &no the stoc& price, the di*idend groth rates, and the re$uired return, but not the di*idend( Birst, e need to reali2e that the di*idend in 5ear = is the current di*idend ti"es the BV.B( -he di*idend in 5ear = ill be+ )= L )0 (1(2A# = CHAPTER 8 B-153 0nd the di*idend in 5ear I ill be the di*idend in 5ear = ti"es one plus the groth rate, or+ )I L )0 (1(2A# = (1(1A# -he stoc& begins constant groth in 5ear I, so e can find the price of the stoc& in 5ear I as the di*idend in 5ear A, di*ided b% the re$uired return "inus the groth rate( -he e$uation for the price of the stoc& in 5ear I is+ 9I L )I (1 O g# : (? E g# 4o e can substitute the pre*ious di*idend in 5ear I into this e$uation as follos+ 9I L )0 (1 O g-# = (1 O g1# (1 O g3# : (? E g# 9I L )0 (1(2A# = (1(1A# (1(08# : ((1= E (08# L I8(A2)0 When e sol*e this e$uation, e find that the stoc& price in 5ear I is I8(A2 ti"es as large as the di*idend toda%( 4o e need to find the e$uation for the stoc& price toda%( -he stoc& price toda% is the 9V of the di*idends in 5ears 1, 2, =, and I, plus the 9V of the 5ear I price( ,o+ 90 L )0(1(2A#:1(1= O )0(1(2A# 2 :1(1= 2 O )0(1(2A# = :1(1= = O )0(1(2A# = (1(1A#:1(1= I O I8(A2)0:1(1= I We can factor out )0 in the e$uation, and co"bine the last to ter"s( )oing so, e get+ 90 L <JM L )0Y1(2A:1(1= O 1(2A 2 :1(1= 2 O 1(2A = :1(1= = O Q(1(2A# = (1(1A# O I8(A2R : 1(1= I Z Reducing the e$uation e*en further b% sol*ing all of the ter"s in the braces, e get+ <JM L <=I(J9)0 )0 L <JM : <=I(J9 )0 L <2(18 -his is the di*idend toda%, so the pro1ected di*idend for the ne!t %ear ill be+ )1 L <2(18(1(2A# )1 L <2(J= 1!. -he constant groth "odel can be applied e*en if the di*idends are declining b% a constant percentage, 1ust "a&e sure to recogni2e the negati*e groth( ,o, the price of the stoc& toda% ill be+ 90 L )0 (1 O g# : (? E g# 90 L <10(IM(1 E (0I# : Q((11A E (E(0I#R 90 L <MI(J8 1". We are gi*en the stoc& price, the di*idend groth rate, and the re$uired return, and are as&ed to find the di*idend( Using the constant di*idend groth "odel, e get+ 90 L <MI L )0 (1 O g# : (? E g# B-154 SOLUTIONS ,ol*ing this e$uation for the di*idend gi*es us+ )0 L <MI((10 E (0IA# : (1(0IA# )0 L <=(=J 1#. -he price of a share of preferred stoc& is the di*idend pa%"ent di*ided b% the re$uired return( We &no the di*idend pa%"ent in 5ear 20, so e can find the price of the stoc& in 5ear 19, one %ear before the first di*idend pa%"ent( )oing so, e get+ 919 L <20(00 : (0MI 919 L <=12(A0 -he price of the stoc& toda% is the 9V of the stoc& price in the future, so the price toda% ill be+ 90 L <=12(A0 : (1(0MI# 19 90 L <9M(1A 1$. -he annual di*idend paid to stoc&holders is <1(I8, and the di*idend %ield is 2(1 percent( Using the e$uation for the di*idend %ield+ )i*idend %ield L )i*idend : ,toc& price We can plug the nu"bers in and sol*e for the stoc& price+ (021 L <1(I8 : 90 90 L <1(I8:(021 L <J0(I8 -he ;4et Chg@ of the stoc& shos the stoc& decreased b% <0(2= on this da%, so the closing stoc& price %esterda% as+ 5esterda%'s closing price L <J0(I8 O 0(2= L <J0(J1 -o find the net inco"e, e need to find the 69,( -he stoc& $uote tells us the 9:6 ratio for the stoc& is 19( ,ince e &no the stoc& price as ell, e can use the 9:6 ratio to sol*e for 69, as follos+ 9:6 L 19 L ,toc& price : 69, L <J0(I8 : 69, 69, L <J0(I8 : 19 L <=(J1 We &no that 69, is 1ust the total net inco"e di*ided b% the nu"ber of shares outstanding, so+ 69, L 4. : ,hares L <=(J1 L 4. : 2A,000,000 4. L <=(J1(2A,000,000# L <92,J=1,8=0 2%. We can use the to-stage di*idend groth "odel for this proble", hich is+ 90 L Q)0(1 O g1#:(R E g1#RY1 E Q(1 O g1#:(1 O R#R - ZO Q(1 O g1#:(1 O R#R - Q)0(1 O g2#:(R E g2#R 90 3 Q<1(2A(1(28#:((1= E (28#RQ1 E (1(28:1(1=# 8 R O Q(1(28#:(1(1=#R 8 Q<1(2A(1(0M#:((1= E (0M#R 90 3 <M9(AA CHAPTER 8 B-155 21. We can use the to-stage di*idend groth "odel for this proble", hich is+ 90 L Q)0(1 O g1#:(R E g1#RY1 E Q(1 O g1#:(1 O R#R - ZO Q(1 O g1#:(1 O R#R - Q)0(1 O g2#:(R E g2#R 90 3 Q<1(JI(1(2A#:((12 E (2A#RQ1 E (1(2A:1(12# 11 R O Q(1(2A#:(1(12#R 11 Q<1(JI(1(0M#:((12 E (0M#R 90 3 <1I2(1I Challenge 22. We are as&ed to find the di*idend %ield and capital gains %ield for each of the stoc&s( 0ll of the stoc&s ha*e a 1A percent re$uired return, hich is the su" of the di*idend %ield and the capital gains %ield( -o find the co"ponents of the total return, e need to find the stoc& price for each stoc&( Using this stoc& price and the di*idend, e can calculate the di*idend %ield( -he capital gains %ield for the stoc& ill be the total return (re$uired return# "inus the di*idend %ield( W+ 90 L )0(1 O g# : (? E g# L <I(A0(1(10#:((19 E (10# L <AA(00 )i*idend %ield L )1:90 L <I(A0(1(10#:<AA(00 L (09 or 9N Capital gains %ield L (19 E (09 L (10 or 10N U+ 90 L )0(1 O g# : (? E g# L <I(A0:((19 E 0# L <2=(M8 )i*idend %ield L )1:90 L <I(A0:<2=(M8 L (19 or 19N Capital gains %ield L (19 E (19 L 0N 5+ 90 L )0(1 O g# : (? E g# L <I(A0(1 E (0A#:((19 O (0A# L <1J(81 )i*idend %ield L )1:90 L <I(A0(0(9A#:<1J(81 L (2I or 2IN Capital gains %ield L (19 E (2I L E(0A or EAN ^+ 92 L )2(1 O g# : (? E g# L )0(1 O g-# 2 (1 O g1#:(? E g2# L <I(A0(1(20# 2 (1(12#:((19 E (12# L <10=(M8 90 L <I(A0 (1(20# : (1(19# O <I(A0 (1(20# 2 : (1(19# 2 O <10=(M8 : (1(19# 2 L <82(== )i*idend %ield L )1:90 L <I(A0(1(20#:<9M(10 L (0MM or M(MN Capital gains %ield L (19 E (0MM L (12I or 12(IN .n all cases, the re$uired return is 19N, but the return is distributed differentl% beteen current inco"e and capital gains( >igh groth stoc&s ha*e an appreciable capital gains co"ponent but a relati*el% s"all current inco"e %ield8 con*ersel%, "ature, negati*e-groth stoc&s pro*ide a high current inco"e but also price depreciation o*er ti"e( 23. a( Using the constant groth "odel, the price of the stoc& pa%ing annual di*idends ill be+ 90 L )0(1 O g# : (? E g# L <=(20(1(0M#:((12 E (0M# L <AM(A= B-156 SOLUTIONS b( .f the co"pan% pa%s $uarterl% di*idends instead of annual di*idends, the $uarterl% di*idend ill be one-fourth of annual di*idend, or+ 7uarterl% di*idend+ <=(20(1(0M#:I L <0(8I8 -o find the e$ui*alent annual di*idend, e "ust assu"e that the $uarterl% di*idends are rein*ested at the re$uired return( We can then use this interest rate to find the e$ui*alent annual di*idend( .n other ords, hen e recei*e the $uarterl% di*idend, e rein*est it at the re$uired return on the stoc&( ,o, the effecti*e $uarterl% rate is+ 6ffecti*e $uarterl% rate+ 1(12 (2A E 1 L (028J -he effecti*e annual di*idend ill be the BV0 of the $uarterl% di*idend pa%"ents at the effecti*e $uarterl% re$uired return( .n this case, the effecti*e annual di*idend ill be+ 6ffecti*e )1 L <0(8I8(BV.B02(8JN,I# L <=(AI 4o, e can use the constant groth "odel to find the current stoc& price as+ 90 L <=(AI:((12 E (0M# L <A9(02 4ote that e cannot si"pl% find the $uarterl% effecti*e re$uired return and groth rate to find the *alue of the stoc&( -his ould assu"e the di*idends increased each $uarter, not each %ear( 24. >ere e ha*e a stoc& ith supernor"al groth, but the di*idend groth changes e*er% %ear for the first four %ears( We can find the price of the stoc& in 5ear = since the di*idend groth rate is constant after the third di*idend( -he price of the stoc& in 5ear = ill be the di*idend in 5ear I, di*ided b% the re$uired return "inus the constant di*idend groth rate( ,o, the price in 5ear = ill be+ 9= L <2(IA(1(20#(1(1A#(1(10#(1(0A# : ((11 E (0A# L <MA(08 -he price of the stoc& toda% ill be the 9V of the first three di*idends, plus the 9V of the stoc& price in 5ear =, so+ 90 L <2(IA(1(20#:(1(11# O <2(IA(1(20#(1(1A#:1(11 2 O <2(IA(1(20#(1(1A#(1(10#:1(11 = O <MA(08:1(11 = 90 L <AA(J0 2. >ere e ant to find the re$uired return that "a&es the 9V of the di*idends e$ual to the current stoc& price( -he e$uation for the stoc& price is+ 9 L <2(IA(1(20#:(1 O ?# O <2(IA(1(20#(1(1A#:(1 O ?# 2 O <2(IA(1(20#(1(1A#(1(10#:(1 O ?# = O Q<2(IA(1(20#(1(1A#(1(10#(1(0A#:(? E (0A#R:(1 O ?# = L <M=(82 We need to find the roots of this e$uation( Using spreadsheet, trial and error, or a calculator ith a root sol*ing function, e find that+ ? L 10(2IN CHAPTER 8 B-157 2!. 6*en though the $uestion concerns a stoc& ith a constant groth rate, e need to begin ith the e$uation for to-stage groth gi*en in the chapter, hich is+ 90 L 1 1 0 1 ? ) g 8 g @ " + 1 1 ] 1 ¸ , _ ¸ ¸ + + t ? g 1 1 - 1 1 O t t ?8 @ , + 1 We can e!pand the e$uation (see 9roble" 2J for "ore detail# to the folloing+ 90 L 1 1 0 1 ? ) g 8 g @ " + 1 1 ] 1 ¸ , _ ¸ ¸ + + t ? g 1 1 - 1 1 O t ? g , _ ¸ ¸ + + 1 1 1 2 2 1 ? ) g 8 g "@ + ,ince the groth rate is constant, g1 L g2 , so+ 90 L ? ) g g8 @ " + 1 0 1 1 ] 1 ¸ , _ ¸ ¸ + + t ? g ) 1 1 1 O t ? g , _ ¸ ¸ + + 1 1 ? ) g g8 "@ + 1 ,ince e ant the first t di*idends to constitute one-half of the stoc& price, e can set the to ter"s on the right hand side of the e$uation e$ual to each other, hich gi*es us+ g - R g# (1 ) 0 + 1 1 ] 1 ¸ , _ ¸ ¸ + + t ? g ) 1 1 1 L t ? g , _ ¸ ¸ + + 1 1 ? ) g g8 "@ + 1 ,ince g - R g# (1 ) 0 + appears on both sides of the e$uation, e can eli"inate this, hich lea*es+ 1 E t , _ ¸ ¸ + + R 1 g 1 L t ? g , _ ¸ ¸ + + 1 1 ,ol*ing this e$uation, e get+ 1 L t , _ ¸ ¸ + + R 1 g 1 O t ? g , _ ¸ ¸ + + 1 1 1 L 2 t , _ ¸ ¸ + + R 1 g 1 1:2 L t ? g , _ ¸ ¸ + + 1 1 B-158 SOLUTIONS t ln , _ ¸ ¸ + + R 1 g 1 L ln(0(A# t L , _ ¸ ¸ + + ? g 1 1 ln ln(0(A# -his e!pression ill tell %ou the nu"ber of di*idends that constitute one-half of the current stoc& price( 2". -o find the *alue of the stoc& ith to-stage di*idend groth, consider that the present *alue of the first t di*idends is the present *alue of a groing annuit%( 0dditionall%, to find the price of the stoc&, e need to add the present *alue of the stoc& price at ti"e t( ,o, the stoc& price toda% is+ 90 L 9V of t di*idends O 9V(9t# Using g1 to represent the first groth rate and substituting the e$uation for the present *alue of a groing annuit%, e get+ 90 L )1 1 1 1 1 1 ] 1 ¸ , _ ¸ ¸ + + 1 1 1 1 1 ? ) g ? g ) t O 9V(9t# ,ince the di*idend in one %ear ill increase at g1, e can re-rite the e!pression as+ 90 L )0(1 O g1# 1 1 1 1 1 ] 1 ¸ , _ ¸ ¸ + + 1 1 1 1 1 ? ) g ? g ) t O 9V(9t# 4o e can re-rite the e$uation again as+ 90 L 1 1 0 1 ? ) g 8 g @ " + 1 1 ] 1 ¸ , _ ¸ ¸ + + t 1 R 1 g 1 - 1 O 9V(9t# -o find the price of the stoc& at ti"e t, e can use the constant di*idend groth "odel, or+ 9t L 2 1 t g - R ) + -he di*idend at t A - ill ha*e gron at g1 for t periods, and at g2 for one period, so+ )t O 1 L )0(1 O g1# t (1 O g2# CHAPTER 8 B-159 ,o, e can re-rite the e$uation as+ 9t L 2 2 1 1 1 ? ) g 8 g @ 8 g "@ t + + 4e!t, e can find *alue toda% of the future stoc& price as+ 9V(9t# L 2 2 1 1 1 ? ) g 8 g @ 8 g "@ t + + S t ?8 @ + 1 1 hich can be ritten as+ 9V(9t# L t ? g , _ ¸ ¸ + + 1 1 1 S 2 2 1 ? ) g 8 g "@ + ,ubstituting this into the stoc& price e$uation, e get+ 90 L 1 1 0 1 ? ) g 8 g @ " + 1 1 ] 1 ¸ , _ ¸ ¸ + + t ? g ) 1 1 1 1 O t ? g , _ ¸ ¸ + + 1 1 1 S 2 2 1 ? ) g 8 g "@ + .n this e$uation, the first ter" on the right hand side is the present *alue of the first t di*idends, and the second ter" is the present *alue of the stoc& price hen constant di*idend groth fore*er begins( 2#. -o find the e!pression hen the groth rate for the first stage is e!actl% e$ual to the re$uired return, consider e can find the present *alue of the di*idends in the first stage as+ 9V L 1 1 1 0 # 1 ( # (1 ? g " + + O 2 2 1 0 # 1 ( # (1 ? g " + + O = = 1 0 # 1 ( # (1 ? g " + + O ] ,ince g1 is e$ual to ?, each of the terns reduces to+ 9V L "0 O "0 O "0 O ]( 9V L t S "0 ,o, the e!pression for the price of a stoc& hen the first groth rate is e!actl% e$ual to the re$uired return is+ 9t L t S "0 O ( ) ( ) 2 2 1 0 1 1 g ? g g " t − + × + × CHAPTER 9 -5T &R5S5-T 1A9:5 A-; 4T<5R .-15STM5-T CR.T5R.A Answers to Concepts Review and Critical Thinking Questions 1. 0 pa%bac& period less than the pro1ect's life "eans that the 49V is positi*e for a 2ero discount rate, but nothing "ore definiti*e can be said( Bor discount rates greater than 2ero, the pa%bac& period ill still be less than the pro1ect's life, but the 49V "a% be positi*e, 2ero, or negati*e, depending on hether the discount rate is less than, e$ual to, or greater than the .RR( -he discounted pa%bac& includes the effect of the rele*ant discount rate( .f a pro1ect's discounted pa%bac& period is less than the pro1ect's life, it "ust be the case that 49V is positi*e( 2. .f a pro1ect has a positi*e 49V for a certain discount rate, then it ill also ha*e a positi*e 49V for a 2ero discount rate8 thus, the pa%bac& period "ust be less than the pro1ect life( ,ince discounted pa%bac& is calculated at the sa"e discount rate as is 49V, if 49V is positi*e, the discounted pa%bac& period "ust be less than the pro1ect's life( .f 49V is positi*e, then the present *alue of future cash inflos is greater than the initial in*est"ent cost8 thus 9. "ust be greater than 1( .f 49V is positi*e for a certain discount rate R, then it ill be 2ero for so"e larger discount rate R`8 thus the .RR "ust be greater than the re$uired return( 3. a. 9a%bac& period is si"pl% the accounting brea&-e*en point of a series of cash flos( -o actuall% co"pute the pa%bac& period, it is assu"ed that an% cash flo occurring during a gi*en period is reali2ed continuousl% throughout the period, and not at a single point in ti"e( -he pa%bac& is then the point in ti"e for the series of cash flos hen the initial cash outla%s are full% reco*ered( Gi*en so"e predeter"ined cutoff for the pa%bac& period, the decision rule is to accept pro1ects that pa%bac& before this cutoff, and re1ect pro1ects that ta&e longer to pa%bac&( b. -he orst proble" associated ith pa%bac& period is that it ignores the ti"e *alue of "one%( .n addition, the selection of a hurdle point for pa%bac& period is an arbitrar% e!ercise that lac&s an% steadfast rule or "ethod( -he pa%bac& period is biased toards short-ter" pro1ects8 it full% ignores an% cash flos that occur after the cutoff point( c. )espite its shortco"ings, pa%bac& is often used because (1# the anal%sis is straightforard and si"ple and (2# accounting nu"bers and esti"ates are readil% a*ailable( 3aterialit% considerations often arrant a pa%bac& anal%sis as sufficient8 "aintenance pro1ects are another e!a"ple here the detailed anal%sis of other "ethods is often not needed( ,ince pa%bac& is biased toards li$uidit%, it "a% be a useful and appropriate anal%sis "ethod for short-ter" pro1ects here cash "anage"ent is "ost i"portant( 4. a. -he discounted pa%bac& is calculated the sa"e as is regular pa%bac&, ith the e!ception that each cash flo in the series is first con*erted to its present *alue( -hus discounted pa%bac& pro*ides a "easure of financial:econo"ic brea&-e*en because of this discounting, 1ust as regular pa%bac& pro*ides a "easure of accounting brea&-e*en because it does not discount the cash flos( Gi*en so"e predeter"ined cutoff for the discounted pa%bac& period, the decision rule is to accept CHAPTER 9 B-161 pro1ects hose discounted cash flos pa%bac& before this cutoff period, and to re1ect all other pro1ects( B-162 SOLUTIONS b. -he pri"ar% disad*antage to using the discounted pa%bac& "ethod is that it ignores all cash flos that occur after the cutoff date, thus biasing this criterion toards short-ter" pro1ects( 0s a result, the "ethod "a% re1ect pro1ects that in fact ha*e positi*e 49Vs, or it "a% accept pro1ects ith large future cash outla%s resulting in negati*e 49Vs( .n addition, the selection of a cutoff point is again an arbitrar% e!ercise( c. )iscounted pa%bac& is an i"pro*e"ent on regular pa%bac& because it ta&es into account the ti"e *alue of "one%( Bor con*entional cash flos and strictl% positi*e discount rates, the discounted pa%bac& ill ala%s be greater than the regular pa%bac& period( . a. -he a*erage accounting return is interpreted as an a*erage "easure of the accounting perfor"ance of a pro1ect o*er ti"e, co"puted as so"e a*erage profit "easure attributable to the pro1ect di*ided b% so"e a*erage balance sheet *alue for the pro1ect( -his te!t co"putes 00R as a*erage net inco"e ith respect to a*erage (total# boo& *alue( Gi*en so"e predeter"ined cutoff for 00R, the decision rule is to accept pro1ects ith an 00R in e!cess of the target "easure, and re1ect all other pro1ects( b. 00R is not a "easure of cash flos and "ar&et *alue, but a "easure of financial state"ent accounts that often bear little rese"blance to the rele*ant *alue of a pro1ect( .n addition, the selection of a cutoff is arbitrar%, and the ti"e *alue of "one% is ignored( Bor a financial "anager, both the reliance on accounting nu"bers rather than rele*ant "ar&et data and the e!clusion of ti"e *alue of "one% considerations are troubling( )espite these proble"s, 00R continues to be used in practice because (1# the accounting infor"ation is usuall% a*ailable, (2# anal%sts often use accounting ratios to anal%2e fir" perfor"ance, and (=# "anagerial co"pensation is often tied to the attain"ent of certain target accounting ratio goals( !. a. 49V is si"pl% the present *alue of a pro1ect's cash flos( 49V specificall% "easures, after considering the ti"e *alue of "one%, the net increase or decrease in fir" ealth due to the pro1ect( -he decision rule is to accept pro1ects that ha*e a positi*e 49V, and re1ect pro1ects ith a negati*e 49V( b. 49V is superior to the other "ethods of anal%sis presented in the te!t because it has no serious flas( -he "ethod una"biguousl% ran&s "utuall% e!clusi*e pro1ects, and can differentiate beteen pro1ects of different scale and ti"e hori2on( -he onl% drabac& to 49V is that it relies on cash flo and discount rate *alues that are often esti"ates and not certain, but this is a proble" shared b% the other perfor"ance criteria as ell( 0 pro1ect ith 49V L <2,A00 i"plies that the total shareholder ealth of the fir" ill increase b% <2,A00 if the pro1ect is accepted( ". a. -he .RR is the discount rate that causes the 49V of a series of cash flos to be e!actl% 2ero( .RR can thus be interpreted as a financial brea&-e*en rate of return8 at the .RR, the net *alue of the pro1ect is 2ero( -he .RR decision rule is to accept pro1ects ith .RRs greater than the discount rate, and to re1ect pro1ects ith .RRs less than the discount rate( b. .RR is the interest rate that causes 49V for a series of cash flos to be 2ero( 49V is preferred in all situations to .RR8 .RR can lead to a"biguous results if there are non-con*entional cash flos, and it also a"biguousl% ran&s so"e "utuall% e!clusi*e pro1ects( >oe*er, for stand-alone pro1ects ith con*entional cash flos, .RR and 49V are interchangeable techni$ues( c. .RR is fre$uentl% used because it is easier for "an% financial "anagers and anal%sts to rate perfor"ance in relati*e ter"s, such as ;12N@, than in absolute ter"s, such as ;<IM,000(@ .RR "a% be a preferred "ethod to 49V in situations here an appropriate discount rate is un&non are uncertain8 in this situation, .RR ould pro*ide "ore infor"ation about the pro1ect than ould 49V( CHAPTER 9 B-163 #. a. -he profitabilit% inde! is the present *alue of cash inflos relati*e to the pro1ect cost( 0s such, it is a benefit:cost ratio, pro*iding a "easure of the relati*e profitabilit% of a pro1ect( -he profitabilit% inde! decision rule is to accept pro1ects ith a 9. greater than one, and to re1ect pro1ects ith a 9. less than one( b. 9. L (49V O cost#:cost L 1 O (49V:cost#( .f a fir" has a bas&et of positi*e 49V pro1ects and is sub1ect to capital rationing, 9. "a% pro*ide a good ran&ing "easure of the pro1ects, indicating the ;bang for the buc&@ of each particular pro1ect( $. Bor a pro1ect ith future cash flos that are an annuit%+ 9a%bac& L . : C 0nd the .RR is+ 0 L E . O C : .RR ,ol*ing the .RR e$uation for .RR, e get+ .RR L C : . 4otice this is 1ust the reciprocal of the pa%bac&( ,o+ .RR L 1 : 9H Bor long-li*ed pro1ects ith relati*el% constant cash flos, the sooner the pro1ect pa%s bac&, the greater is the .RR( 1%. -here are a nu"ber of reasons( -o of the "ost i"portant ha*e to do ith transportation costs and e!change rates( 3anufacturing in the U(,( places the finished product "uch closer to the point of sale, resulting in significant sa*ings in transportation costs( .t also reduces in*entories because goods spend less ti"e in transit( >igher labor costs tend to offset these sa*ings to so"e degree, at least co"pared to other possible "anufacturing locations( /f great i"portance is the fact that "anufacturing in the U(,( "eans that a "uch higher proportion of the costs are paid in dollars( ,ince sales are in dollars, the net effect is to i""uni2e profits to a large e!tent against fluctuations in e!change rates( -his issue is discussed in greater detail in the chapter on international finance( 11. -he single biggest difficult%, b% far, is co"ing up ith reliable cash flo esti"ates( )eter"ining an appropriate discount rate is also not a si"ple tas&( -hese issues are discussed in greater depth in the ne!t se*eral chapters( -he pa%bac& approach is probabl% the si"plest, folloed b% the 00R, but e*en these re$uire re*enue and cost pro1ections( -he discounted cash flo "easures (discounted pa%bac&, 49V, .RR, and profitabilit% inde!# are reall% onl% slightl% "ore difficult in practice( 12. 5es, the% are( ,uch entities generall% need to allocate a*ailable capital efficientl%, 1ust as for-profits do( >oe*er, it is fre$uentl% the case that the ;re*enues@ fro" not-for-profit *entures are not tangible( Bor e!a"ple, charitable gi*ing has real opportunit% costs, but the benefits are generall% hard to "easure( -o the e!tent that benefits are "easurable, the $uestion of an appropriate re$uired return re"ains( 9a%bac& rules are co""onl% used in such cases( Binall%, realistic cost:benefit anal%sis along the lines indicated should definitel% be used b% the U(,( go*ern"ent and ould go a long a% toard balancing the budgetX B-164 SOLUTIONS 13. -he 3.RR is calculated b% finding the present *alue of all cash outflos, the future *alue of all cash inflos to the end of the pro1ect, and then calculating the .RR of the to cash flos( 0s a result, the cash flos ha*e been discounted or co"pounded b% one interest rate (the re$uired return#, and then the interest rate beteen the to re"aining cash flos is calculated( 0s such, the 3.RR is not a true interest rate( .n contrast, consider the .RR( .f %ou ta&e the initial in*est"ent, and calculate the future *alue at the .RR, %ou can replicate the future cash flos of the pro1ect e!actl%( 14. -he state"ent is incorrect( .t is true that if %ou calculate the future *alue of all inter"ediate cash flos to the end of the pro1ect at the re$uired return, then calculate the 49V of this future *alue and the initial in*est"ent, %ou ill get the sa"e 49V( >oe*er, 49V sa%s nothing about rein*est"ent of inter"ediate cash flos( -he 49V is the present *alue of the pro1ect cash flos( What is actuall% done ith those cash flos once the% are generated is not rele*ant( 9ut differentl%, the *alue of a pro1ect depends on the cash flos generated b% the pro1ect, not on the future *alue of those cash flos( -he fact that the rein*est"ent ;or&s@ onl% if %ou use the re$uired return as the rein*est"ent rate is also irrele*ant si"pl% because rein*est"ent is not rele*ant in the first place to the *alue of the pro1ect( /ne ca*eat+ /ur discussion here assu"es that the cash flos are trul% a*ailable once the% are generated, "eaning that it is up to fir" "anage"ent to decide hat to do ith the cash flos( .n certain cases, there "a% be a re$uire"ent that the cash flos be rein*ested( Bor e!a"ple, in international in*esting, a co"pan% "a% be re$uired to rein*est the cash flos in the countr% in hich the% are generated and not ;repatriate@ the "one%( ,uch funds are said to be ;bloc&ed@ and rein*est"ent beco"es rele*ant because the cash flos are not trul% a*ailable( 1. -he state"ent is incorrect( .t is true that if %ou calculate the future *alue of all inter"ediate cash flos to the end of the pro1ect at the .RR, then calculate the .RR of this future *alue and the initial in*est"ent, %ou ill get the sa"e .RR( >oe*er, as in the pre*ious $uestion, hat is done ith the cash flos once the% are generated does not affect the .RR( Consider the folloing e!a"ple+ C0 C1 C2 .RR 9ro1ect 0 E<100 <10 <110 10N ,uppose this <100 is a deposit into a ban& account( -he .RR of the cash flos is 10 percent( )oes the .RR change if the 5ear 1 cash flo is rein*ested in the account, or if it is ithdran and spent on pi22a? 4o( Binall%, consider the %ield to "aturit% calculation on a bond( .f %ou thin& about it, the 5-3 is the .RR on the bond, but no "ention of a rein*est"ent assu"ption for the bond coupons is suggested( -he reason is that rein*est"ent is irrele*ant to the 5-3 calculation8 in the sa"e a%, rein*est"ent is irrele*ant in the .RR calculation( /ur ca*eat about bloc&ed funds applies here as ell( CHAPTER 9 B-165 Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -o calculate the pa%bac& period, e need to find the ti"e that the pro1ect has reco*ered its initial in*est"ent( 0fter three %ears, the pro1ect has created+ <1,M00 O 1,900 O 2,=00 L <A,800 in cash flos( -he pro1ect still needs to create another+ <M,I00 E A,800 L <M00 in cash flos( )uring the fourth %ear, the cash flos fro" the pro1ect ill be <1,I00( ,o, the pa%bac& period ill be = %ears, plus hat e still need to "a&e di*ided b% hat e ill "a&e during the fourth %ear( -he pa%bac& period is+ 9a%bac& L = O (<M00 : <1,I00# L =(I= %ears 2. -o calculate the pa%bac& period, e need to find the ti"e that the pro1ect has reco*ered its initial in*est"ent( -he cash flos in this proble" are an annuit%, so the calculation is si"pler( .f the initial cost is <2,I00, the pa%bac& period is+ 9a%bac& L = O (<10A : <JMA# L =(1I %ears -here is a shortcut to calculate the future cash flos are an annuit%( Just di*ide the initial cost b% the annual cash flo( Bor the <2,I00 cost, the pa%bac& period is+ 9a%bac& L <2,I00 : <JMA L =(1I %ears Bor an initial cost of <=,M00, the pa%bac& period is+ 9a%bac& L <=,M00 : <JMA L I(J1 %ears -he pa%bac& period for an initial cost of <M,A00 is a little tric&ier( 4otice that the total cash inflos after eight %ears ill be+ -otal cash inflos L 8(<JMA# L <M,120 .f the initial cost is <M,A00, the pro1ect ne*er pa%s bac&( 4otice that if %ou use the shortcut for annuit% cash flos, %ou get+ 9a%bac& L <M,A00 : <JMA L 8(A0 %ears -his anser does not "a&e sense since the cash flos stop after eight %ears, so again, e "ust conclude the pa%bac& period is ne*er( B-166 SOLUTIONS 3. 9ro1ect 0 has cash flos of <19,000 in 5ear 1, so the cash flos are short b% <21,000 of recapturing the initial in*est"ent, so the pa%bac& for 9ro1ect 0 is+ 9a%bac& L 1 O (<21,000 : <2A,000# L 1(8I %ears 9ro1ect H has cash flos of+ Cash flos L <1I,000 O 1J,000 O 2I,000 L <AA,000 during this first three %ears( -he cash flos are still short b% <A,000 of recapturing the initial in*est"ent, so the pa%bac& for 9ro1ect H is+ H+ 9a%bac& L = O (<A,000 : <2J0,000# L =(019 %ears Using the pa%bac& criterion and a cutoff of = %ears, accept pro1ect 0 and re1ect pro1ect H( 4. When e use discounted pa%bac&, e need to find the *alue of all cash flos toda%( -he *alue toda% of the pro1ect cash flos for the first four %ears is+ Value toda% of 5ear 1 cash flo L <I,200:1(1I L <=,M8I(21 Value toda% of 5ear 2 cash flo L <A,=00:1(1I 2 L <I,0J8(18 Value toda% of 5ear = cash flo L <M,100:1(1I = L <I,11J(== Value toda% of 5ear I cash flo L <J,I00:1(1I I L <I,=81(=9 -o find the discounted pa%bac&, e use these *alues to find the pa%bac& period( -he discounted first %ear cash flo is <=,M8I(21, so the discounted pa%bac& for a <J,000 initial cost is+ )iscounted pa%bac& L 1 O (<J,000 E =,M8I(21#:<I,0J8(18 L 1(81 %ears Bor an initial cost of <10,000, the discounted pa%bac& is+ )iscounted pa%bac& L 2 O (<10,000 E =,M8I(21 E I,0J8(18#:<I,11J(== L 2(AI %ears 4otice the calculation of discounted pa%bac&( We &no the pa%bac& period is beteen to and three %ears, so e subtract the discounted *alues of the 5ear 1 and 5ear 2 cash flos fro" the initial cost( -his is the nu"erator, hich is the discounted a"ount e still need to "a&e to reco*er our initial in*est"ent( We di*ide this a"ount b% the discounted a"ount e ill earn in 5ear = to get the fractional portion of the discounted pa%bac&( .f the initial cost is <1=,000, the discounted pa%bac& is+ )iscounted pa%bac& L = O (<1=,000 E =,M8I(21 E I,0J8(18 E I,11J(==# : <I,=81(=9 L =(2M %ears . R L 0N+ = O (<2,100 : <I,=00# L =(I9 %ears discounted pa%bac& L regular pa%bac& L =(I9 %ears R L AN+ <I,=00:1(0A O <I,=00:1(0A 2 O <I,=00:1(0A = L <11,J09(9J <I,=00:1(0A I L <=,A=J(M2 discounted pa%bac& L = O (<1A,000 E 11,J09(9J# : <=,A=J(M2 L =(9= %ears CHAPTER 9 B-167 R L 19N+ <I,=00(9V.B019N,M# L <1I,MM2(0I -he pro1ect ne*er pa%s bac&( !. /ur definition of 00R is the a*erage net inco"e di*ided b% the a*erage boo& *alue( -he a*erage net inco"e for this pro1ect is+ 0*erage net inco"e L (<1,9=8,200 O 2,201,M00 O 1,8JM,000 O 1,=29,A00# : I L <1,8=M,=2A 0nd the a*erage boo& *alue is+ 0*erage boo& *alue L (<1A,000,000 O 0# : 2 L <J,A00,000 ,o, the 00R for this pro1ect is+ 00R L 0*erage net inco"e : 0*erage boo& *alue L <1,8=M,=2A : <J,A00,000 L (2II8 or 2I(I8N ". -he .RR is the interest rate that "a&es the 49V of the pro1ect e$ual to 2ero( ,o, the e$uation that defines the .RR for this pro1ect is+ 0 L E <=I,000 O <1M,000:(1O.RR# O <18,000:(1O.RR# 2 O <1A,000:(1O.RR# = Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 20(9JN ,ince the .RR is greater than the re$uired return e ould accept the pro1ect( #. -he 49V of a pro1ect is the 9V of the outflos "inus the 9V of the inflos( -he e$uation for the 49V of this pro1ect at an 11 percent re$uired return is+ 49V L E <=I,000 O <1M,000:1(11 O <18,000:1(11 2 O <1A,000:1(11 = L <A,991(I9 0t an 11 percent re$uired return, the 49V is positi*e, so e ould accept the pro1ect( -he e$uation for the 49V of the pro1ect at a =0 percent re$uired return is+ 49V L E <=I,000 O <1M,000:1(=0 O <18,000:1(=0 2 O <1A,000:1(=0 = L E<I,21=(9= 0t a =0 percent re$uired return, the 49V is negati*e, so e ould re1ect the pro1ect( $. -he 49V of a pro1ect is the 9V of the outflos "inus the 9V of the inflos( ,ince the cash inflos are an annuit%, the e$uation for the 49V of this pro1ect at an 8 percent re$uired return is+ 49V L E<1=8,000 O <28,A00(9V.B08N, 9# L <I0,0=M(=1 0t an 8 percent re$uired return, the 49V is positi*e, so e ould accept the pro1ect( B-168 SOLUTIONS -he e$uation for the 49V of the pro1ect at a 20 percent re$uired return is+ 49V L E<1=8,000 O <28,A00(9V.B020N, 9# L E<2=,11J(IA 0t a 20 percent re$uired return, the 49V is negati*e, so e ould re1ect the pro1ect( We ould be indifferent to the pro1ect if the re$uired return as e$ual to the .RR of the pro1ect, since at that re$uired return the 49V is 2ero( -he .RR of the pro1ect is+ 0 L E<1=8,000 O <28,A00(9V.B0.RR, 9# .RR L 1I(A9N 1%. -he .RR is the interest rate that "a&es the 49V of the pro1ect e$ual to 2ero( ,o, the e$uation that defines the .RR for this pro1ect is+ 0 L E<19,A00 O <9,800:(1O.RR# O <10,=00:(1O.RR# 2 O <8,M00:(1O.RR# = Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 22(MIN 11. -he 49V of a pro1ect is the 9V of the outflos "inus the 9V of the inflos( 0t a 2ero discount rate (and onl% at a 2ero discount rate#, the cash flos can be added together across ti"e( ,o, the 49V of the pro1ect at a 2ero percent re$uired return is+ 49V L E<19,A00 O 9,800 O 10,=00 O 8,M00 L <9,200 -he 49V at a 10 percent re$uired return is+ 49V L E<19,A00 O <9,800:1(1 O <10,=00:1(1 2 O <8,M00:1(1 = L <I,=82(J9 -he 49V at a 20 percent re$uired return is+ 49V L E<19,A00 O <9,800:1(2 O <10,=00:1(2 2 O <8,M00:1(2 = L <J9M(=0 0nd the 49V at a =0 percent re$uired return is+ 49V L E<19,A00 O <9,800:1(= O <10,=00:1(= 2 O <8,M00:1(= = L E<1,9A2(II 4otice that as the re$uired return increases, the 49V of the pro1ect decreases( -his ill ala%s be true for pro1ects ith con*entional cash flos( Con*entional cash flos are negati*e at the beginning of the pro1ect and positi*e throughout the rest of the pro1ect( CHAPTER 9 B-169 12. a. -he .RR is the interest rate that "a&es the 49V of the pro1ect e$ual to 2ero( -he e$uation for the .RR of 9ro1ect 0 is+ 0 L E<I=,000 O <2=,000:(1O.RR# O <1J,900:(1O.RR# 2 O <12,I00:(1O.RR# = O <9,I00:(1O.RR# I Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 20(IIN -he e$uation for the .RR of 9ro1ect H is+ 0 L E<I=,000 O <J,000:(1O.RR# O <1=,800:(1O.RR# 2 O <2I,000:(1O.RR# = O <2M,000:(1O.RR# I Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 18(8IN 6!a"ining the .RRs of the pro1ects, e see that the .RR0 is greater than the .RRH, so .RR decision rule i"plies accepting pro1ect 0( -his "a% not be a correct decision8 hoe*er, because the .RR criterion has a ran&ing proble" for "utuall% e!clusi*e pro1ects( -o see if the .RR decision rule is correct or not, e need to e*aluate the pro1ect 49Vs( b. -he 49V of 9ro1ect 0 is+ 49V0 L E<I=,000 O <2=,000:1(11O <1J,900:1(11 2 O <12,I00:1(11 = O <9,I00:1(11 I 49V0 L <J,A0J(M1 0nd the 49V of 9ro1ect H is+ 49VH L E<I=,000 O <J,000:1(11 O <1=,800:1(11 2 O <2I,000:1(11 = O <2M,000:1(11 I 49VH L <9,182(29 -he 49VH is greater than the 49V0, so e should accept 9ro1ect H( c. -o find the crosso*er rate, e subtract the cash flos fro" one pro1ect fro" the cash flos of the other pro1ect( >ere, e ill subtract the cash flos for 9ro1ect H fro" the cash flos of 9ro1ect 0( /nce e find these differential cash flos, e find the .RR( -he e$uation for the crosso*er rate is+ Crosso*er rate+ 0 L <1M,000:(1OR# O <I,100:(1OR# 2 E <11,M00:(1OR# = E <1M,M00:(1OR# I Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ R L 1A(=0N 0t discount rates abo*e 1A(=0N choose pro1ect 08 for discount rates belo 1A(=0N choose pro1ect H8 indifferent beteen 0 and H at a discount rate of 1A(=0N( B-170 SOLUTIONS 13. -he .RR is the interest rate that "a&es the 49V of the pro1ect e$ual to 2ero( -he e$uation to calculate the .RR of 9ro1ect U is+ 0 L E<1A,000 O <8,1A0:(1O.RR# O <A,0A0:(1O.RR# 2 O <M,800:(1O.RR# = Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 1M(AJN Bor 9ro1ect 5, the e$uation to find the .RR is+ 0 L E<1A,000 O <J,J00:(1O.RR# O <A,1A0:(1O.RR# 2 O <J,2A0:(1O.RR# = Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 1M(IAN -o find the crosso*er rate, e subtract the cash flos fro" one pro1ect fro" the cash flos of the other pro1ect, and find the .RR of the differential cash flos( We ill subtract the cash flos fro" 9ro1ect 5 fro" the cash flos fro" 9ro1ect U( .t is irrele*ant hich cash flos e subtract fro" the other( ,ubtracting the cash flos, the e$uation to calculate the .RR for these differential cash flos is+ Crosso*er rate+ 0 L <IA0:(1OR# E <100:(1OR# 2 E <IA0:(1OR# = Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ R L 11(J=N -he table belo shos the 49V of each pro1ect for different re$uired returns( 4otice that 9ro1ect 5 ala%s has a higher 49V for discount rates belo 11(J= percent, and ala%s has a loer 49V for discount rates abo*e 11(J= percent( R <49VU <49V5 0N <A,000(00 <A,100(00 AN <=,21M(A0 <=,2MJ(=M 10N <1,M91(A9 <1,J0=(2= 1AN <=JM(A9 <=AM(J8 20N E<JMM(20 E<811(=I 2AN E<1,JMM(I0 E<1,8=2(00 14. a. -he e$uation for the 49V of the pro1ect is+ 49V L E<IA,000,000 O <J8,000,000:1(1 E <1I,000,000:1(1 2 L <1=,I82,1I2(8M -he 49V is greater than 0, so e ould accept the pro1ect( CHAPTER 9 B-171 b. -he e$uation for the .RR of the pro1ect is+ 0 L E<IA,000,000 O <J8,000,000:(1O.RR# E <1I,000,000:(1O.RR# 2 Bro" )escartes rule of signs, e &no there are potentiall% to .RRs since the cash flos change signs tice( Bro" trial and error, the to .RRs are+ .RR L A=(00N, EJ9(MJN When there are "ultiple .RRs, the .RR decision rule is a"biguous( Hoth .RRs are correct, that is, both interest rates "a&e the 49V of the pro1ect e$ual to 2ero( .f e are e*aluating hether or not to accept this pro1ect, e ould not ant to use the .RR to "a&e our decision( 1. -he profitabilit% inde! is defined as the 9V of the cash inflos di*ided b% the 9V of the cash outflos( -he e$uation for the profitabilit% inde! at a re$uired return of 10 percent is+ 9. L Q<J,=00:1(1 O <M,900:1(1 2 O <A,J00:1(1 = R : <1I,000 L 1(18J -he e$uation for the profitabilit% inde! at a re$uired return of 1A percent is+ 9. L Q<J,=00:1(1A O <M,900:1(1A 2 O <A,J00:1(1A = R : <1I,000 L 1(09I -he e$uation for the profitabilit% inde! at a re$uired return of 22 percent is+ 9. L Q<J,=00:1(22 O <M,900:1(22 2 O <A,J00:1(22 = R : <1I,000 L 0(98= We ould accept the pro1ect if the re$uired return ere 10 percent or 1A percent since the 9. is greater than one( We ould re1ect the pro1ect if the re$uired return ere 22 percent since the 9. is less than one( 1!. a. -he profitabilit% inde! is the 9V of the future cash flos di*ided b% the initial in*est"ent( -he cash flos for both pro1ects are an annuit%, so+ 9.. L <2J,000(9V.B010N,= # : <A=,000 L 1(2MJ 9... L <9,100(9V.B010N,=# : <1M,000 L 1(I1I -he profitabilit% inde! decision rule i"plies that e accept pro1ect .., since 9... is greater than the 9..( b. -he 49V of each pro1ect is+ 49V. L E<A=,000 O <2J,000(9V.B010N,=# L <1I,1IA(00 49V.. L E<1M,000 O <9,100(9V.B010N,=# L <M,M=0(=A -he 49V decision rule i"plies accepting 9ro1ect ., since the 49V. is greater than the 49V..( B-172 SOLUTIONS c. Using the profitabilit% inde! to co"pare "utuall% e!clusi*e pro1ects can be a"biguous hen the "agnitude of the cash flos for the to pro1ects are of different scale( .n this proble", pro1ect . is roughl% = ti"es as large as pro1ect .. and produces a larger 49V, %et the profitabilit% inde! criterion i"plies that pro1ect .. is "ore acceptable( 1". a( -he pa%bac& period for each pro1ect is+ 0+ = O (<180,000:<=90,000# L =(IM %ears H+ 2 O (<9,000:<18,000# L 2(A0 %ears -he pa%bac& criterion i"plies accepting pro1ect H, because it pa%s bac& sooner than pro1ect 0( b. -he discounted pa%bac& for each pro1ect is+ 0+ <20,000:1(1A O <A0,000:1(1A 2 O <A0,000:1(1A = L <88,0JI(=0 <=90,000:1(1A I L <222,98=(JJ )iscounted pa%bac& L = O (<=90,000 E 88,0JI(=0#:<222,98=(JJ L =(9A %ears H+ <19,000:1(1A O <12,000:1(1A 2 O <18,000:1(1A = L <=J,I=0(JM <10,A00:1(1A I L <M,00=(I1 )iscounted pa%bac& L = O (<I0,000 E =J,I=0(JM#:<M,00=(I1 L =(I= %ears -he discounted pa%bac& criterion i"plies accepting pro1ect H because it pa%s bac& sooner than 0( c( -he 49V for each pro1ect is+ 0+ 49V L E<=00,000 O <20,000:1(1A O <A0,000:1(1A 2 O <A0,000:1(1A = O <=90,000:1(1A I 49V L <11,0A8(0J H+ 49V L E<I0,000 O <19,000:1(1A O <12,000:1(1A 2 O <18,000:1(1A = O <10,A00:1(1A I 49V L <=,I=I(1M 49V criterion i"plies e accept pro1ect 0 because pro1ect 0 has a higher 49V than pro1ect H( d. -he .RR for each pro1ect is+ 0+ <=00,000 L <20,000:(1O.RR# O <A0,000:(1O.RR# 2 O <A0,000:(1O.RR# = O <=90,000:(1O.RR# I Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 1M(20N CHAPTER 9 B-173 H+ <I0,000 L <19,000:(1O.RR# O <12,000:(1O.RR# 2 O <18,000:(1O.RR# = O <10,A00:(1O.RR# I Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 19(A0N .RR decision rule i"plies e accept pro1ect H because .RR for H is greater than .RR for 0( e. -he profitabilit% inde! for each pro1ect is+ 0+ 9. L (<20,000:1(1A O <A0,000:1(1A 2 O <A0,000:1(1A = O <=90,000:1(1A I # : <=00,000 L 1(0=J H+ 9. L (<19,000:1(1A O <12,000:1(1A 2 O <18,000:1(1A = O <10,A00:1(1A I # : <I0,000 L 1(08M 9rofitabilit% inde! criterion i"plies accept pro1ect H because its 9. is greater than pro1ect 0's( f. .n this instance, the 49V criteria i"plies that %ou should accept pro1ect 0, hile profitabilit% inde!, pa%bac& period, discounted pa%bac&, and .RR i"pl% that %ou should accept pro1ect H( -he final decision should be based on the 49V since it does not ha*e the ran&ing proble" associated ith the other capital budgeting techni$ues( -herefore, %ou should accept pro1ect 0( 1#. 0t a 2ero discount rate (and onl% at a 2ero discount rate#, the cash flos can be added together across ti"e( ,o, the 49V of the pro1ect at a 2ero percent re$uired return is+ 49V L E<M8I,M80 O 2M=,2J9 O 29I,0M0 O 22J,M0I O 1JI,=AM L <2JI,M19 .f the re$uired return is infinite, future cash flos ha*e no *alue( 6*en if the cash flo in one %ear is <1 trillion, at an infinite rate of interest, the *alue of this cash flo toda% is 2ero( ,o, if the future cash flos ha*e no *alue toda%, the 49V of the pro1ect is si"pl% the cash flo toda%, so at an infinite interest rate+ 49V L E<M8I,M80 -he interest rate that "a&es the 49V of a pro1ect e$ual to 2ero is the .RR( -he e$uation for the .RR of this pro1ect is+ 0 L E<M8I,M80 O <2M=,2J9:(1O.RR# O <29I,0M0:(1O.RR# 2 O <22J,M0I:(1O.RR# = O 1JI,=AM:(1O.RR# I Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 1M(2=N B-174 SOLUTIONS 1$. -he 3.RR for the pro1ect ith all three approaches is+ "iscounting approach: .n the discounting approach, e find the *alue of all cash outflos to ti"e 0, hile an% cash inflos re"ain at the ti"e at hich the% occur( ,o, the discounting the cash outflos to ti"e 0, e find+ -i"e 0 cash flo L E<1M,000 E <A,100 : 1(10 A -i"e 0 cash flo L E<19,1MM(J0 ,o, the 3.RR using the discounting approach is+ 0 L E<19,1MM(J0 O <M,100:(1O3.RR# O <J,800:(1O3.RR# 2 O <8,I00:(1O3.RR# = O M,A00:(1O3.RR# I Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ 3.RR L 18(18N ?einvestment approach: .n the rein*est"ent approach, e find the future *alue of all cash e!cept the initial cash flo at the end of the pro1ect( ,o, rein*esting the cash flos to ti"e A, e find+ -i"e A cash flo L <M,100(1(10 I # O <J,800(1(10 = # O <8,I00(1(10 2 # O <M,A00(1(10# E <A,100 -i"e A cash flo L <=1,A2M(81 ,o, the 3.RR using the discounting approach is+ 0 L E<1M,000 O <=1,A2M(81:(1O3.RR# A <=1,A2M(81 : <1M,000 L (1O3.RR# A 3.RR L (<=1,A2M(81 : <1M,000# 1:A E 1 3.RR L (1IA= or 1I(A=N Combination approach: .n the co"bination approach, e find the *alue of all cash outflos at ti"e 0, and the *alue of all cash inflos at the end of the pro1ect( ,o, the *alue of the cash flos is+ -i"e 0 cash flo L E<1M,000 E <A,100 : 1(10 A -i"e 0 cash flo L E<19,1MM(J0 -i"e A cash flo L <M,100(1(10 I # O <J,800(1(10 = # O <8,I00(1(10 2 # O <M,A00(1(10# -i"e A cash flo L <=M,M2M(81 ,o, the 3.RR using the discounting approach is+ 0 L E<19,1MM(J0 O <=M,M2M(81:(1O3.RR# A <=M,M2M(81 : <19,1MM(J0 L (1O3.RR# A 3.RR L (<=M,M2M(81 : <19,1MM(J0# 1:A E 1 3.RR L (1=8= or 1=(8=N CHAPTER 9 B-175 &ntermediate 2%. With different discounting and rein*est"ent rates, e need to "a&e sure to use the appropriate interest rate( -he 3.RR for the pro1ect ith all three approaches is+ "iscounting approach: .n the discounting approach, e find the *alue of all cash outflos to ti"e 0 at the discount rate, hile an% cash inflos re"ain at the ti"e at hich the% occur( ,o, the discounting the cash outflos to ti"e 0, e find+ -i"e 0 cash flo L E<1M,000 E <A,100 : 1(11 A -i"e 0 cash flo L E<19,02M(M0 ,o, the 3.RR using the discounting approach is+ 0 L E<19,02M(M0 O <M,100:(1O3.RR# O <J,800:(1O3.RR# 2 O <8,I00:(1O3.RR# = O M,A00:(1O3.RR# I Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ 3.RR L 18(AAN ?einvestment approach: .n the rein*est"ent approach, e find the future *alue of all cash e!cept the initial cash flo at the end of the pro1ect using the rein*est"ent rate( ,o, the rein*esting the cash flos to ti"e A, e find+ -i"e A cash flo L <M,100(1(08 I # O <J,800(1(08 = # O <8,I00(1(08 2 # O <M,A00(1(08# E <A,100 -i"e A cash flo L <29,8I2(A0 ,o, the 3.RR using the discounting approach is+ 0 L E<1M,000 O <29,8I2(A0:(1O3.RR# A <29,8I2(A0 : <1M,000 L (1O3.RR# A 3.RR L (<29,8I2(A0 : <1M,000# 1:A E 1 3.RR L (1=28 or 1=(28N Combination approach: .n the co"bination approach, e find the *alue of all cash outflos at ti"e 0 using the discount rate, and the *alue of all cash inflos at the end of the pro1ect using the rein*est"ent rate( ,o, the *alue of the cash flos is+ -i"e 0 cash flo L E<1M,000 E <A,100 : 1(11 A -i"e 0 cash flo L E<19,02M(M0 -i"e A cash flo L <M,100(1(08 I # O <J,800(1(08 = # O <8,I00(1(08 2 # O <M,A00(1(08# -i"e A cash flo L <=I,9I2(A0 B-176 SOLUTIONS ,o, the 3.RR using the discounting approach is+ 0 L E<19,02M(M0 O <=I,9I2(A0:(1O3.RR# A <=I,9I2(A0 : <19,02M(M0 L (1O3.RR# A 3.RR L (<=I,9I2(A0 : <19,02M(M0# 1:A E 1 3.RR L (129= or 12(9=N 21. ,ince the 49V inde! has the cost subtracted in the nu"erator, 49V inde! L 9. E 1( 22. a. -o ha*e a pa%bac& e$ual to the pro1ect's life, gi*en C is a constant cash flo for 4 %ears+ C L .:4 b. -o ha*e a positi*e 49V, . a C (9V.B0?N, N#( -hus, C T . : (9V.B0?N, N#( c. Henefits L C (9V.B0?/# N# L 2 S costs L 2. C L 2. : (9V.B0?/# N# Challenge 23. Gi*en the se*en %ear pa%bac&, the orst case is that the pa%bac& occurs at the end of the se*enth %ear( -hus, the orst-case+ 49V L E<J2I,000 O <J2I,000:1(12 J L E<=9M,I99(1J -he best case has infinite cash flos be%ond the pa%bac& point( -hus, the best-case 49V is infinite( 24. -he e$uation for the .RR of the pro1ect is+ 0 L E<1,A12 O <8,A8M:(1 O .RR# E <18,210:(1 O .RR# 2 O <1J,100:(1 O .RR# = E <M,000:(1 O .RR# I Using )escartes rule of signs, fro" loo&ing at the cash flos e &no there are four .RRs for this pro1ect( 6*en ith "ost co"puter spreadsheets, e ha*e to do so"e trial and error( Bro" trial and error, .RRs of 2AN, ==(==N, I2(8MN, and MM(MJN are found( We ould accept the pro1ect hen the 49V is greater than 2ero( ,ee for %ourself if that 49V is greater than 2ero for re$uired returns beteen 2AN and ==(==N or beteen I2(8MN and MM(MJN( 2. a. >ere the cash inflos of the pro1ect go on fore*er, hich is a perpetuit%( Unli&e ordinar% perpetuit% cash flos, the cash flos here gro at a constant rate fore*er, hich is a groing perpetuit%( .f %ou re"e"ber bac& to the chapter on stoc& *aluation, e presented a for"ula for *aluing a stoc& ith constant groth in di*idends( -his for"ula is actuall% the for"ula for a groing perpetuit%, so e can use it here( -he 9V of the future cash flos fro" the pro1ect is+ 9V of cash inflos L C-:(? E g# 9V of cash inflos L <8A,000:((1= E (0M# L <1,21I,28A(J1 CHAPTER 9 B-177 49V is the 9V of the outflos "inus the 9V of the inflos, so the 49V is+ 49V of the pro1ect L E<1,I00,000 O 1,21I,28A(J1 L E<18A,J1I(29 -he 49V is negati*e, so e ould re1ect the pro1ect( b. >ere e ant to &no the "ini"u" groth rate in cash flos necessar% to accept the pro1ect( -he "ini"u" groth rate is the groth rate at hich e ould ha*e a 2ero 49V( -he e$uation for a 2ero 49V, using the e$uation for the 9V of a groing perpetuit% is+ 0 L E<1,I00,000 O <8A,000:((1= E g# ,ol*ing for g, e get+ g L (0M9= or M(9=N 2!. -he .RR of the pro1ect is+ <A8,000 L <=I,000:(1O.RR# O <IA,000:(1O.RR# 2 Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 22(1IN 0t an interest rate of 12 percent, the 49V is+ 49V L <A8,000 E <=I,000:1(12 E <IA,000:1(12 2 49V L E<8,2=0(8J 0t an interest rate of 2ero percent, e can add cash flos, so the 49V is+ 49V L <A8,000 E <=I,000 E <IA,000 49V L E<21,000(00 0nd at an interest rate of 2I percent, the 49V is+ 49V L <A8,000 E <=I,000:1(2I E <IA,000:1(2I 2 49V L O<1,=1I(2M -he cash flos for the pro1ect are uncon*entional( ,ince the initial cash flo is positi*e and the re"aining cash flos are negati*e, the decision rule for .RR in in*alid in this case( -he 49V profile is upard sloping, indicating that the pro1ect is "ore *aluable hen the interest rate increases( B-178 SOLUTIONS 2". -he .RR is the interest rate that "a&es the 49V of the pro1ect e$ual to 2ero( ,o, the .RR of the pro1ect is+ 0 L <20,000 E <2M,000 : (1 O .RR# O <1=,000 : (1 O .RR# 2 6*en though it appears there are to .RRs, a spreadsheet, financial calculator, or trial and error ill not gi*e an anser( -he reason is that there is no real .RR for this set of cash flos( .f %ou e!a"ine the .RR e$uation, hat e are reall% doing is sol*ing for the roots of the e$uation( Going bac& to high school algebra, in this proble" e are sol*ing a $uadratic e$uation( .n case %ou don't re"e"ber, the $uadratic e$uation is+ ! L a ac b b 2 I 2 − t − .n this case, the e$uation is+ ! L # 000 2M ( 2 # 000 1= #( 000 20 ( I # 000 2M ( # 000 2M ( 2 # # # # # − − t − − -he s$uare root ter" or&s out to be+ MJM,000,000 E 1,0I0,000,000 L E=MI,000,000 -he s$uare root of a negati*e nu"ber is a co"ple! nu"ber, so there is no real nu"ber solution, "eaning the pro1ect has no real .RR( 2#. Birst, e need to find the future *alue of the cash flos for the one %ear in hich the% are bloc&ed b% the go*ern"ent( ,o, rein*esting each cash inflo for one %ear, e find+ 5ear 2 cash flo L <20A,000(1(0I# L <21=,200 5ear = cash flo L <2MA,000(1(0I# L <2JA,M00 5ear I cash flo L <=IM,000(1(0I# L <=A9,8I0 5ear A cash flo L <220,000(1(0I# L <228,800 ,o, the 49V of the pro1ect is+ 49V L E<IA0,000 O <21=,200:1(11 2 O <2JA,M00:1(11 = O <=A9,8I0:1(11 I O <228,800:1(11 A 49V L E<2,M2M(== 0nd the .RR of the pro1ect is+ 0 L E<IA0,000 O <21=,200:(1 O .RR# 2 O <2JA,M00:(1 O .RR# = O <=A9,8I0:(1 O .RR# I O <228,800:(1 O .RR# A Using a spreadsheet, financial calculator, or trial and error to find the root of the e$uation, e find that+ .RR L 10(89N While this "a% loo& li&e a 3.RR calculation, it is not an 3.RR, rather it is a standard .RR calculation( ,ince the cash inflos are bloc&ed b% the go*ern"ent, the% are not a*ailable to the co"pan% for a CHAPTER 9 B-179 period of one %ear( -hus, all e are doing is calculating the .RR based on hen the cash flos actuall% occur for the co"pan%( Calculator Solutions ". C2o E<=I,000 C%1 <1M,000 2%1 1 C%2 <18,000 2%2 1 C%3 <1A,000 2%3 1 .RR C9- 20(9JN #. C2o E<=I,000 C2o E<=I,000 C%1 <1M,000 C%1 <1M,000 2%1 1 2%1 1 C%2 <18,000 C%2 <18,000 2%2 1 2%2 1 C%3 <1A,000 C%3 <1A,000 2%3 1 2%3 1 . L 11N . L =0N 49V C9- 49V C9- <A,991(I9 E<I,21=(9= $. C2o E<1=8,000 C2o E<1=8,000 C2o E<1=8,000 C%1 <28,A00 C%1 <28,A00 C%1 <28,A00 2%1 9 2%1 9 2%1 9 . L 8N . L 20N .RR C9- 49V C9- 49V C9- 1I(A9N <I0,0=M(=1 E<2=,11J(IA B-180 SOLUTIONS 1%. C2o E<19,A00 C%1 <9,800 2%1 1 C%2 <10,=00 2%2 1 C%3 <8,M00 2%3 1 .RR C9- 22(MIN 11. C2o E<19,A00 C2o E<19,A00 C%1 <9,800 C%1 <9,800 2%1 1 2%1 1 C%2 <10,=00 C%2 <10,=00 2%2 1 2%2 1 C%3 <8,M00 C%3 <8,M00 2%3 1 2%3 1 . L 0N . L 10N 49V C9- 49V C9- <9,200 <I(=82(J9 C2o E<19,A00 C2o E<19,A00 C%1 <9,800 C%1 <9,800 2%1 1 2%1 1 C%2 <10,=00 C%2 <10,=00 2%2 1 2%2 1 C%3 <8,M00 C%3 <8,M00 2%3 1 2%3 1 . L 20N . L =0N 49V C9- 49V C9- <J9M(=0 E<1,9A2(II 12. ,ro(ect A C2o E<I=,000 C2o E<I=,000 C%1 <2=,000 C%1 <2=,000 2%1 1 2%1 1 C%2 <1J,900 C%2 <1J,900 2%2 1 2%2 1 C%3 <12,I00 C%3 <12,I00 2%3 1 2%3 1 C%4 <9,I00 C%4 <9,I00 2%4 1 2%4 1 .RR C9- . L 11N 20(IIN 49V C9- <J,A0J(M1 CHAPTER 9 B-181 ,ro(ect % C2o E<I=,000 C2o E<I=,000 C%1 <J,000 C%1 <J,000 2%1 1 2%1 1 C%2 <1=,800 C%2 <1=,800 2%2 1 2%2 1 C%3 <2I,000 C%3 <2I,000 2%3 1 2%3 1 C%4 <2M,000 C%4 <2M,000 2%4 1 2%4 1 .RR C9- . L 11N 18(8IN 49V C9- <9,182(29 Crossover rate C2o <0 C%1 <1M,000 2%1 1 C%2 <I,100 2%2 1 C%3 E<11,M00 2%3 1 C%4 E<1M,M00 2%4 1 .RR C9- 1A(=0N 13. ,ro(ect B C2o E<1A,000 C2o E<1A,000 C2o E<1A,000 C%1 <8,1A0 C%1 <8,1A0 C%1 <8,1A0 2%1 1 2%1 1 2%1 1 C%2 <A,0A0 C%2 <A,0A0 C%2 <A,0A0 2%2 1 2%2 1 2%2 1 C%3 <M,800 C%3 <M,800 C%3 <M,800 2%3 1 2%3 1 2%3 1 . L 0N . L 1AN . L 2AN 49V C9- 49V C9- 49V C9- <A,000(00 <=JM(A9 E<1,JMM(I0 B-182 SOLUTIONS ,ro(ect C C2o E<1A,000 C2o E<1A,000 C2o E<1A,000 C%1 <J,J00 C%1 <J,J00 C%1 <J,J00 2%1 1 2%1 1 2%1 1 C%2 <A,1A0 C%2 <A,1A0 C%2 <A,1A0 2%2 1 2%2 1 2%2 1 C%3 <J,2A0 C%3 <J,2A0 C%3 <J,2A0 2%3 1 2%3 1 2%3 1 . L 0N . L 1AN . L 2AN 49V C9- 49V C9- 49V C9- <A,100(00 <=AM(J8 E<1,8=2(00 Crossover rate C2o <0 C%1 <IA0 2%1 1 C%2 E<100 2%2 1 C%3 E<IA0 2%3 1 .RR C9- 11(J=N 14. C2o E<IA,000,000 C2o E<IA,000,000 C%1 <J8,000,000 C%1 <J8,000,000 2%1 1 2%1 1 C%2 E<1I,000,000 C%2 E<1I,000,000 2%2 1 2%2 1 . L 10N .RR C9- 49V C9- A=(00N <1=,I82,1I2(8M Financial calculators will only give you one IRR, even if there are multiple IRRs. Using trial and error, or a root solving calculator, the other IRR is –79.67%. CHAPTER 9 B-183 1. C2o <0 C2o <0 C2o <0 C%1 <J,=00 C%1 <J,=00 C%1 <J,=00 2%1 1 2%1 1 2%1 1 C%2 <M,900 C%2 <M,900 C%2 <M,900 2%2 1 2%2 1 2%2 1 C%3 <A,J00 C%3 <A,J00 C%3 <A,J00 2%3 1 2%3 1 2%3 1 . L 10N . L 1AN . L 22N 49V C9- 49V C9- 49V C9- <1M,M21(=I <1A,=1=(0M <1=,JA8(I9 \10N+ 9. L <1M,M21(=I : <1I,000 L 1(18J \1AN+ 9. L <1A,=1=(0M : <1I,000 L 1(09I \22N+ 9. L <1=,JA8(I9 : <1I,000 L 0(98= 1!. ,ro(ect & C2o <0 C2o E<A=,000 C%1 <2J,000 C%1 <2J,000 2%1 = 2%1 = . L 10N . L 10N 49V C9- 49V C9- <MJ,1IA(00 <1I,1IA(00 9. L <MJ,1IA(00 : <A=,000 L 1(2MJ ,ro(ect && C2o <0 C2o E<1M,000 C%1 <9,100 C%1 <9,100 2%1 = 2%1 = . L 10N . L 10N 49V C9- 49V C9- <22,M=0(=A <M,M=0(=A 9. L <22,M=0(=A : <1M,000 L 1(I1I 1". CF@A8 c. d. e. C+o E<=00,000 C2o E<=00,000 C2o <0 C%1 <20,000 C%1 <20,000 C%1 <20,000 2%1 1 2%1 1 2%1 1 C%2 <A0,000 C%2 <A0,000 C%2 <A0,000 2%2 2 2%2 2 2%2 2 C%3 <=90,000 C%3 <=90,000 C%3 <=90,000 2%3 1 2%3 1 2%3 1 . L 1AN .RR C9- . L 1AN 49V C9- 1M(20N 49V C9- <11,0A8(0J <=11,0A8(0J 9. L <=11,0A8(0J : <=00,000 L 1(0=J B-184 SOLUTIONS CF@%8 c. d. e. C2o E<I0,000 C2o E<I0,000 C2o <0 C%1 <19,000 C%1 <19,000 C%1 <19,000 2%1 1 2%1 1 2%1 1 C%2 <12,000 C%2 <12,000 C%2 <12,000 2%2 1 2%2 1 2%2 1 C%3 <18,000 C%3 <18,000 C%3 <18,000 2%3 1 2%3 1 2%3 1 C%4 <10,A00 C%4 <10,A00 C%4 <10,A00 2%4 1 2%4 1 2%4 1 . L 1AN .RR C9- . L 1AN 49V C9- 19(A0N 49V C9- <=,I=I(1M <I=,I=I(1M 9. L <I=,I=I(1M : <I0,000 L 1(08M f. .n this instance, the 49V criteria i"plies that %ou should accept pro1ect 0, hile pa%bac& period, discounted pa%bac&, profitabilit% inde!, and .RR i"pl% that %ou should accept pro1ect H( -he final decision should be based on the 49V since it does not ha*e the ran&ing proble" associated ith the other capital budgeting techni$ues( -herefore, %ou should accept pro1ect 0( 1#. C2o E<M8I,M80 C2o E<M8I,M80 C%1 <2M=,2J9 C%1 <2M=,2J9 2%1 1 2%1 1 C%2 <29I,0M0 C%2 <29I,0M0 2%2 1 2%2 1 C%3 <22J,M0I C%3 <22J,M0I 2%3 1 2%3 1 C%4 <1JI,=AM C%4 <1JI,=AM 2%4 1 2%4 1 . L 0N .RR C9- 49V C9- 1M(2=N <2JI,M19 CHAPTER 10 MA=.-7 CA&.TA9 .-15STM5-T ;5C.S.4-S Answers to Concepts Review and Critical Thinking Questions 1. .n this conte!t, an opportunit% cost refers to the *alue of an asset or other input that ill be used in a pro1ect( -he rele*ant cost is hat the asset or input is actuall% orth toda%, not, for e!a"ple, hat it cost to ac$uire( 2. Bor ta! purposes, a fir" ould choose 30CR, because it pro*ides for larger depreciation deductions earlier( -hese larger deductions reduce ta!es, but ha*e no other cash conse$uences( 4otice that the choice beteen 30CR, and straight-line is purel% a ti"e *alue issue8 the total depreciation is the sa"e, onl% the ti"ing differs( 3. .t's probabl% onl% a "ild o*er-si"plification( Current liabilities ill all be paid, presu"abl%( -he cash portion of current assets ill be retrie*ed( ,o"e recei*ables on't be collected, and so"e in*entor% ill not be sold, of course( Counterbalancing these losses is the fact that in*entor% sold abo*e cost (and not replaced at the end of the pro1ect's life# acts to increase or&ing capital( -hese effects tend to offset one another( 4. 3anage"ent's discretion to set the fir"'s capital structure is applicable at the fir" le*el( ,ince an% one particular pro1ect could be financed entirel% ith e$uit%, another pro1ect could be financed ith debt, and the fir"'s o*erall capital structure re"ains unchanged, financing costs are not rele*ant in the anal%sis of a pro1ect's incre"ental cash flos according to the stand-alone principle( . -he 60C approach is appropriate hen co"paring "utuall% e!clusi*e pro1ects ith different li*es that ill be replaced hen the% ear out( -his t%pe of anal%sis is necessar% so that the pro1ects ha*e a co""on life span o*er hich the% can be co"pared8 in effect, each pro1ect is assu"ed to e!ist o*er an infinite hori2on of 4-%ear repeating pro1ects( 0ssu"ing that this t%pe of anal%sis is *alid i"plies that the pro1ect cash flos re"ain the sa"e fore*er, thus ignoring the possible effects of, a"ong other things+ (1# inflation, (2# changing econo"ic conditions, (=# the increasing unreliabilit% of cash flo esti"ates that occur far into the future, and (I# the possible effects of future technolog% i"pro*e"ent that could alter the pro1ect cash flos( !. )epreciation is a non-cash e!pense, but it is ta!-deductible on the inco"e state"ent( -hus depreciation causes ta!es paid, an actual cash outflo, to be reduced b% an a"ount e$ual to the depreciation ta! shield t c )( 0 reduction in ta!es that ould otherise be paid is the sa"e thing as a cash inflo, so the effects of the depreciation ta! shield "ust be added in to get the total incre"ental afterta! cash flos( ". -here are to particularl% i"portant considerations( -he first is erosion( Will the essentiali2ed boo& si"pl% displace copies of the e!isting boo& that ould ha*e otherise been sold? -his is of special concern gi*en the loer price( -he second consideration is co"petition( Will other publishers step in B-186 SOLUTIONS and produce such a product? .f so, then an% erosion is "uch less rele*ant( 0 particular concern to boo& publishers (and producers of a *ariet% of other product t%pes# is that the publisher onl% "a&es "one% fro" the sale of ne boo&s( -hus, it is i"portant to e!a"ine hether the ne boo& ould displace sales of used boo&s (good fro" the publisher's perspecti*e# or ne boo&s (not good#( -he concern arises an% ti"e there is an acti*e "ar&et for used product( #. )efinitel%( -he da"age to 9orsche's reputation is definitel% a factor the co"pan% needed to consider( .f the reputation as da"aged, the co"pan% ould ha*e lost sales of its e!isting car lines( $. /ne co"pan% "a% be able to produce at loer incre"ental cost or "ar&et better( 0lso, of course, one of the to "a% ha*e "ade a "ista&eX 1%. 9orsche ould recogni2e that the outsi2ed profits ould dindle as "ore product co"es to "ar&et and co"petition beco"es "ore intense( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he <M "illion ac$uisition cost of the land si! %ears ago is a sun& cost( -he <M(I "illion current afterta! *alue of the land is an opportunit% cost if the land is used rather than sold off( -he <1I(2 "illion cash outla% and <890,000 grading e!penses are the initial fi!ed asset in*est"ents needed to get the pro1ect going( -herefore, the proper %ear 2ero cash flo to use in e*aluating this pro1ect is <M,I00,000 O 1I,200,000 O 890,000 L <21,I90,000 2. ,ales due solel% to the ne product line are+ 19,000(<1=,000# L <2IJ,000,000 .ncreased sales of the "otor ho"e line occur because of the ne product line introduction8 thus+ I,A00(<A=,000# L <2=8,A00,000 in ne sales is rele*ant( 6rosion of lu!ur% "otor coach sales is also due to the ne "id-si2e ca"pers8 thus+ 900(<91,000# L <81,900,000 loss in sales is rele*ant( -he net sales figure to use in e*aluating the ne line is thus+ <2IJ,000,000 O 2=8,A00,000 E 81,900,000 L <I0=,M00,000 CHAPTER 10 B-187 3. We need to construct a basic inco"e state"ent( -he inco"e state"ent is+ ,ales < 8=0,000 Variable costs I98,000 Bi!ed costs 181,000 )epreciation JJ,000 6H- < JI,000 -a!es\=AN 2A,900 4et inco"e < I8,100 4. -o find the /CB, e need to co"plete the inco"e state"ent as follos+ ,ales < 82I,A00 Costs A=8,900 )epreciation 12M,A00 6H- < 1A9,100 -a!es\=IN AI,09I 4et inco"e < 10A,00M -he /CB for the co"pan% is+ /CB L 6H.- O )epreciation E -a!es /CB L <1A9,100 O 12M,A00 E AI,09I /CB L <2=1,A0M -he depreciation ta! shield is the depreciation ti"es the ta! rate, so+ )epreciation ta! shield L tc)epreciation )epreciation ta! shield L (=I(<12M,A00# )epreciation ta! shield L <I=,010 -he depreciation ta! shield shos us the increase in /CB b% being able to e!pense depreciation( . -o calculate the /CB, e first need to calculate net inco"e( -he inco"e state"ent is+ ,ales < 108,000 Variable costs A1,000 )epreciation M,800 6H- < A0,200 -a!es\=AN 1J,AJ0 4et inco"e < =2,M=0 Using the "ost co""on financial calculation for /CB, e get+ /CB L 6H.- O )epreciation E -a!es /CB L <A0,200 O M,800 E 1J,AJ0 /CB L <=9,I=0 B-188 SOLUTIONS -he top-don approach to calculating /CB %ields+ /CB L ,ales E Costs E -a!es /CB L <108,000 E A1,000 E 1J,AJ0 /CB L <=9,I=0 -he ta!-shield approach is+ /CB L (,ales E Costs#(1 E tC# O tC)epreciation /CB L (<108,000 E A1,000#(1 E (=A# O (=A(M,800# /CB L <=9,I=0 0nd the botto"-up approach is+ OCF = Net income + Depreciation OCF = $32,630 + 6,800 OCF = $39,430 All four methods of calculating OCF should always give the same answer. !. -he 30CR, depreciation schedule is shon in -able 10(J( -he ending boo& *alue for an% %ear is the beginning boo& *alue "inus the depreciation for the %ear( Re"e"ber, to find the a"ount of depreciation for an% %ear, %ou "ultipl% the purchase price of the asset ti"es the 30CR, percentage for the %ear( -he depreciation schedule for this asset is+ 5ear Heginning Hoo& Value 30CR, )epreciation 6nding Hoo& *alue 1 <1,080,000(00 0(1I29 <1AI,==2(00 <92A,MM8(00 2 92A,MM8(00 0(2II9 2MI,I92(00 MM1,1JM(00 = MM1,1JM(00 0(1JI9 188,892(00 IJ2,28I(00 I IJ2,28I(00 0(12I9 1=I,892(00 ==J,=92(00 A ==J,=92(00 0(089= 9M,III(00 2I0,9I8(00 M 2I0,9I8(00 0(0892 9M,==M(00 1II,M12(00 J 1II,M12(00 0(089= 9M,III(00 I8,1M8(00 8 I8,1M8(00 0(0IIM I8,1M8(00 0 ". -he asset has an 8 %ear useful life and e ant to find the HV of the asset after A %ears( With straight- line depreciation, the depreciation each %ear ill be+ 0nnual depreciation L <AI8,000 : 8 0nnual depreciation L <M8,A00 ,o, after fi*e %ears, the accu"ulated depreciation ill be+ 0ccu"ulated depreciation L A(<M8,A00# 0ccu"ulated depreciation L <=I2,A00 CHAPTER 10 B-189 -he boo& *alue at the end of %ear fi*e is thus+ HVA L <AI8,000 E =I2,A00 HVA L <20A,A00 -he asset is sold at a loss to boo& *alue, so the depreciation ta! shield of the loss is recaptured( 0fterta! sal*age *alue L <10A,000 O (<20A,A00 E 10A,000#(0(=A# 0fterta! sal*age *alue L <1I0,1JA -o find the ta!es on sal*age *alue, re"e"ber to use the e$uation+ -a!es on sal*age *alue L (HV E 3V#tc -his e$uation ill ala%s gi*e the correct sign for a ta! inflo (refund# or outflo (pa%"ent#( #. -o find the HV at the end of four %ears, e need to find the accu"ulated depreciation for the first four %ears( We could calculate a table as in 9roble" M, but an easier a% is to add the 30CR, depreciation a"ounts for each of the first four %ears and "ultipl% this percentage ti"es the cost of the asset( We can then subtract this fro" the asset cost( )oing so, e get+ HVI L <J,900,000 E J,900,000(0(2000 O 0(=200 O 0(1920 O 0(11A2# HVI L <1,=MA,120 -he asset is sold at a gain to boo& *alue, so this gain is ta!able( 0fterta! sal*age *alue L <1,I00,000 O (<1,=MA,120 E 1,I00,000#((=A# 0fterta! sal*age *alue L <1,=8J,J92 $. Using the ta! shield approach to calculating /CB (Re"e"ber the approach is irrele*ant8 the final anser ill be the sa"e no "atter hich of the four "ethods %ou use(#, e get+ /CB L (,ales E Costs#(1 E tC# O tC)epreciation /CB L (<2,MA0,000 E 8I0,000#(1 E 0(=A# O 0(=A(<=,900,000:=# /CB L <1,M=1,A00 1%. ,ince e ha*e the /CB, e can find the 49V as the initial cash outla% plus the 9V of the /CBs, hich are an annuit%, so the 49V is+ 49V L E<=,900,000 O <1,M=1,A00(9V.B012N,=# 49V L <18,A8J(J1 B-190 SOLUTIONS 11. -he cash outflo at the beginning of the pro1ect ill increase because of the spending on 4WC( 0t the end of the pro1ect, the co"pan% ill reco*er the 4WC, so it ill be a cash inflo( -he sale of the e$uip"ent ill result in a cash inflo, but e also "ust account for the ta!es hich ill be paid on this sale( ,o, the cash flos for each %ear of the pro1ect ill be+ 5ear Cash Blo 0 E<I,200,000 L E<=,900,000 E =00,000 1 1,M=1,A00 2 1,M=1,A00 = 2,0M8,000 L <1,M=1,A00 O =00,000 O 210,000 O (0 E 210,000#((=A# 0nd the 49V of the pro1ect is+ 49V L E<I,200,000 O <1,M=1,A00(9V.B012N,2# O (<2,0M8,000 : 1(12 = # 49V L <29,2J9(J9 12. Birst e ill calculate the annual depreciation for the e$uip"ent necessar% for the pro1ect( -he depreciation a"ount each %ear ill be+ 5ear 1 depreciation L <=,900,000(0(====# L <1,299,8J0 5ear 2 depreciation L <=,900,000(0(IIIA# L <1,J==,AA0 5ear = depreciation L <=,900,000(0(1I81# L <AJJ,A90 ,o, the boo& *alue of the e$uip"ent at the end of three %ears, hich ill be the initial in*est"ent "inus the accu"ulated depreciation, is+ Hoo& *alue in = %ears L <=,900,000 E (<1,299,8J0 O 1,J==,AA0 O AJJ,A90# Hoo& *alue in = %ears L <288,990 -he asset is sold at a loss to boo& *alue, so this loss is ta!able deductible( 0fterta! sal*age *alue L <210,000 O (<288,990 E 210,000#(0(=A# 0fterta! sal*age *alue L <2=J,MIJ -o calculate the /CB, e ill use the ta! shield approach, so the cash flo each %ear is+ /CB L (,ales E Costs#(1 E tC# O tC)epreciation 5ear Cash Blo 0 E<I,200,000 L E<=,900,000 E =00,000 1 1,M=1,IAI(A0 L (<1,810,000#((MA# O 0(=A(<1,299,8J0# 2 1,J8=,2I2(A0 L (<1,810,000#((MA# O 0(=A(<1,J==,AA0# = 1,91M,=0=(00 L (<1,810,000#((MA# O 0(=A(<AJJ,A90# O <2=J,MIJ O =00,000 Re"e"ber to include the 4WC cost in 5ear 0, and the reco*er% of the 4WC at the end of the pro1ect( -he 49V of the pro1ect ith these assu"ptions is+ 49V L E<I,200,000 O (<1,M=1,IAI(A0:1(12# O (<1,J8=,2I2(A0:1(12 2 # O (<1,91M,=0=(00:1(12 = # 49V L <I2,2=2(I= CHAPTER 10 B-191 13. Birst e ill calculate the annual depreciation of the ne e$uip"ent( .t ill be+ 0nnual depreciation L <AM0,000:A 0nnual depreciation L <112,000 4o, e calculate the afterta! sal*age *alue( -he afterta! sal*age *alue is the "ar&et price "inus (or plus# the ta!es on the sale of the e$uip"ent, so+ 0fterta! sal*age *alue L 3V O (HV E 3V#tc Ver% often the boo& *alue of the e$uip"ent is 2ero as it is in this case( .f the boo& *alue is 2ero, the e$uation for the afterta! sal*age *alue beco"es+ 0fterta! sal*age *alue L 3V O (0 E 3V#tc 0fterta! sal*age *alue L 3V(1 E tc# We ill use this e$uation to find the afterta! sal*age *alue since e &no the boo& *alue is 2ero( ,o, the afterta! sal*age *alue is+ 0fterta! sal*age *alue L <8A,000(1 E 0(=I# 0fterta! sal*age *alue L <AM,100 Using the ta! shield approach, e find the /CB for the pro1ect is+ /CB L <1MA,000(1 E 0(=I# O 0(=I(<112,000# /CB L <1IM,980 4o e can find the pro1ect 49V( 4otice e include the 4WC in the initial cash outla%( -he reco*er% of the 4WC occurs in 5ear A, along ith the afterta! sal*age *alue( 49V L E<AM0,000 E 29,000 O <1IM,980(9V.B010N,A# O Q(<AM,100 O 29,000# : 1(10 A R 49V L <21,010(2I 14. Birst e ill calculate the annual depreciation of the ne e$uip"ent( .t ill be+ 0nnual depreciation charge L <J20,000:A 0nnual depreciation charge L <1II,000 -he afterta! sal*age *alue of the e$uip"ent is+ 0fterta! sal*age *alue L <JA,000(1 E 0(=A# 0fterta! sal*age *alue L <I8,JA0 Using the ta! shield approach, the /CB is+ /CB L <2M0,000(1 E 0(=A# O 0(=A(<1II,000# /CB L <219,I00 B-192 SOLUTIONS 4o e can find the pro1ect .RR( -here is an unusual feature that is a part of this pro1ect( 0ccepting this pro1ect "eans that e ill reduce 4WC( -his reduction in 4WC is a cash inflo at 5ear 0( -his reduction in 4WC i"plies that hen the pro1ect ends, e ill ha*e to increase 4WC( ,o, at the end of the pro1ect, e ill ha*e a cash outflo to restore the 4WC to its le*el before the pro1ect( We also "ust include the afterta! sal*age *alue at the end of the pro1ect( -he .RR of the pro1ect is+ 49V L 0 L E<J20,000 O 110,000 O <219,I00(9V.B0.RRN,A# O Q(<I8,JA0 E 110,000# : (1O.RR# A R .RR L 21(MAN 1. -o e*aluate the pro1ect ith a <=00,000 cost sa*ings, e need the /CB to co"pute the 49V( Using the ta! shield approach, the /CB is+ /CB L <=00,000(1 E 0(=A# O 0(=A(<1II,000# L <2IA,I00 49V L E<J20,000 O 110,000 O <2IA,I00(9V.B020N,A# O Q(<I8,JA0 E 110,000# : (1(20# A R 49V L <99,281(22 -he 49V ith a <2I0,000 cost sa*ings is+ /CB L <2I0,000(1 E 0(=A# O 0(=A(<1II,000# /CB L <20M,I00 49V L E<J20,000 O 110,000 O <20M,I00(9V.B020N,A# O Q(<I8,JA0 E 110,000# : (1(20# A R 49V L E<1J,=A2(MM We ould accept the pro1ect if cost sa*ings ere <=00,000, and re1ect the pro1ect if the cost sa*ings ere <2I0,000( -he re$uired preta! cost sa*ings that ould "a&e us indifferent about the pro1ect is the cost sa*ings that results in a 2ero 49V( -he 49V of the pro1ect is+ 49V L 0 L E<J20,000 O <110,000 O /CB(9V.B020N,A# O Q(<I8,JA0 E 110,000# : (1(20# A R ,ol*ing for the /CB, e find the necessar% /CB for 2ero 49V is+ /CB L <212,202(=8 Using the ta! shield approach to calculating /CB, e get+ /CB L <212,202(=8 L (, E C#(1 E 0(=A# O 0(=A(<1II,000# (, E C# L <2I8,92M(J= -he cost sa*ings that ill "a&e us indifferent is <2I8,92M(J=( CHAPTER 10 B-193 1!. -o calculate the 60C of the pro1ect, e first need the 49V of the pro1ect( 4otice that e include the 4WC e!penditure at the beginning of the pro1ect, and reco*er the 4WC at the end of the pro1ect( -he 49V of the pro1ect is+ 49V L E<2J0,000 E 2A,000 E <I2,000(9V.B011N,A# O <2A,000:1(11 A L E<I=A,=91(=9 4o e can find the 60C of the pro1ect( -he 60C is+ 60C L E<I=A,=91(=9 : (9V.B011N,A# L E<11J,80=(98 1". We ill need the afterta! sal*age *alue of the e$uip"ent to co"pute the 60C( 6*en though the e$uip"ent for each product has a different initial cost, both ha*e the sa"e sal*age *alue( -he afterta! sal*age *alue for both is+ Hoth cases+ afterta! sal*age *alue L <I0,000(1 E 0(=A# L <2M,000 -o calculate the 60C, e first need the /CB and 49V of each option( -he /CB and 49V for -echron . is+ /CB L E<MJ,000(1 E 0(=A# O 0(=A(<290,000:=# L E9,J1M(MJ 49V L E<290,000 E <9,J1M(MJ(9V.B010N,=# O (<2M,000:1(10 = # L E<29I,M29(J= 60C L E<29I,M29(J= : (9V.B010N,=# L E<118,IJI(9J 0nd the /CB and 49V for -echron .. is+ /CB L E<=A,000(1 E 0(=A# O 0(=A(<A10,000:A# L <12,9A0 49V L E<A10,000 O <12,9A0(9V.B010N,A# O (<2M,000:1(10 A # L E<III,JMA(=M 60C L E<III,JMA(=M : (9V.B010N,A# L E<11J,=2J(98 -he to "illing "achines ha*e une$ual li*es, so the% can onl% be co"pared b% e!pressing both on an e$ui*alent annual basis, hich is hat the 60C "ethod does( -hus, %ou prefer the -echron .. because it has the loer (less negati*e# annual cost( 1#. -o find the bid price, e need to calculate all other cash flos for the pro1ect, and then sol*e for the bid price( -he afterta! sal*age *alue of the e$uip"ent is+ 0fterta! sal*age *alue L <J0,000(1 E 0(=A# L <IA,A00 4o e can sol*e for the necessar% /CB that ill gi*e the pro1ect a 2ero 49V( -he e$uation for the 49V of the pro1ect is+ 49V L 0 L E<9I0,000 E JA,000 O /CB(9V.B012N,A# O Q(<JA,000 O IA,A00# : 1(12 A R B-194 SOLUTIONS ,ol*ing for the /CB, e find the /CB that "a&es the pro1ect 49V e$ual to 2ero is+ /CB L <9IM,M2A(0M : 9V.B012N,A L <2M2,M0=(01 -he easiest a% to calculate the bid price is the ta! shield approach, so+ /CB L <2M2,M0=(01 L Q(9 E *#7 E BC R(1 E t c # O t c ) <2M2,M0=(01 L Q(9 E <9(2A#(18A,000# E <=0A,000 R(1 E 0(=A# O 0(=A(<9I0,000:A# 9 L <12(AI &ntermediate 1$. Birst, e ill calculate the depreciation each %ear, hich ill be+ )1 L <AM0,000(0(2000# L <112,000 )2 L <AM0,000(0(=200# L <1J9,200 )= L <AM0,000(0(1920# L <10J,A20 )I L <AM0,000(0(11A2# L <MI,A12 -he boo& *alue of the e$uip"ent at the end of the pro1ect is+ HVI L <AM0,000 E (<112,000 O 1J9,200 O 10J,A20 O MI,A12# L <9M,JM8 -he asset is sold at a loss to boo& *alue, so this creates a ta! refund( 0fter-ta! sal*age *alue L <80,000 O (<9M,JM8 E 80,000#(0(=A# L <8A,8M8(80 ,o, the /CB for each %ear ill be+ /CB1 L <210,000(1 E 0(=A# O 0(=A(<112,000# L <1J2,J00 /CB2 L <210,000(1 E 0(=A# O 0(=A(<1J9,200# L <19M,220 /CB= L <210,000(1 E 0(=A# O 0(=A(<10J,A20# L <1J1,1=2 /CBI L <210,000(1 E 0(=A# O 0(=A(<MI,A12# L <1A9,0J9(20 4o e ha*e all the necessar% infor"ation to calculate the pro1ect 49V( We need to be careful ith the 4WC in this pro1ect( 4otice the pro1ect re$uires <20,000 of 4WC at the beginning, and <=,000 "ore in 4WC each successi*e %ear( We ill subtract the <20,000 fro" the initial cash flo, and subtract <=,000 each %ear fro" the /CB to account for this spending( .n 5ear I, e ill add bac& the total spent on 4WC, hich is <29,000( -he <=,000 spent on 4WC capital during 5ear I is irrele*ant( Wh%? Well, during this %ear the pro1ect re$uired an additional <=,000, but e ould get the "one% bac& i""ediatel%( ,o, the net cash flo for additional 4WC ould be 2ero( With all this, the e$uation for the 49V of the pro1ect is+ 49V L E <AM0,000 E 20,000 O (<1J2,J00 E =,000#:1(09 O (<19M,220 E =,000#:1(09 2 O (<1J1,1=2 E =,000#:1(09 = O (<1A9,0J9(20 O 29,000 O 8A,8M8(80#:1(09 I 49V L <M9,811(J9 CHAPTER 10 B-195 2%. .f e are tr%ing to decide beteen to pro1ects that ill not be replaced hen the% ear out, the proper capital budgeting "ethod to use is 49V( Hoth pro1ects onl% ha*e costs associated ith the", not sales, so e ill use these to calculate the 49V of each pro1ect( Using the ta! shield approach to calculate the /CB, the 49V of ,%ste" 0 is+ /CB0 L E<110,000(1 E 0(=I# O 0(=I(<I=0,000:I# /CB0 L E<=M,0A0 49V0 L E<I=0,000 E <=M,0A0(9V.B011N,I# 49V0 L E<AI1,8I=(1J 0nd the 49V of ,%ste" H is+ /CBH L E<98,000(1 E 0(=I# O 0(=I(<AJ0,000:M# /CBH L E<=2,=80 49VH L E<AJ0,000 E <=2,=80(9V.B011N,M# 49VH L E<J0M,98I(82 .f the s%ste" ill not be replaced hen it ears out, then ,%ste" 0 should be chosen, because it has the "ore positi*e 49V( 21. .f the e$uip"ent ill be replaced at the end of its useful life, the correct capital budgeting techni$ue is 60C( Using the 49Vs e calculated in the pre*ious proble", the 60C for each s%ste" is+ 60C0 L E<AI1,8I=(1J : (9V.B011N,I# 60C0 L E<1JI,MA0(== 60CH L E <J0M,98I(82 : (9V.B011N,M# 60CH L E<1MJ,11I(MI .f the con*e%or belt s%ste" ill be continuall% replaced, e should choose ,%ste" H since it has the "ore positi*e 60C( 22. -o find the bid price, e need to calculate all other cash flos for the pro1ect, and then sol*e for the bid price( -he afterta! sal*age *alue of the e$uip"ent is+ 0fter-ta! sal*age *alue L <AI0,000(1 E 0(=I# 0fter-ta! sal*age *alue L <=AM,I00 4o e can sol*e for the necessar% /CB that ill gi*e the pro1ect a 2ero 49V( -he current afterta! *alue of the land is an opportunit% cost, but e also need to include the afterta! *alue of the land in fi*e %ears since e can sell the land at that ti"e( -he e$uation for the 49V of the pro1ect is+ 49V L 0 L E<I,100,000 E 2,J00,000 E M00,000 O /CB(9V.B012N,A# E <A0,000(9V.B012N,I# O Y(<=AM,I00 O M00,000 O I(A0,000# O =,200,000R : 1(12 A Z B-196 SOLUTIONS ,ol*ing for the /CB, e find the /CB that "a&es the pro1ect 49V e$ual to 2ero is+ /CB L <A,0J9,929(11 : 9V.B012N,A /CB L <1,I09,221(JJ -he easiest a% to calculate the bid price is the ta! shield approach, so+ /CB L <1,I09,221(JJ L Q(9 E *#7 E BC R(1 E tC# O tc) <1,I09,221(JJ L Q(9 E <0(00A#(100,000,000# E <9A0,000R(1 E 0(=I# O 0(=I(<I,100,000:A# 9 L <0(0=1M= 23. 0t a gi*en price, ta&ing accelerated depreciation co"pared to straight-line depreciation causes the 49V to be higher8 si"ilarl%, at a gi*en price, loer net or&ing capital in*est"ent re$uire"ents ill cause the 49V to be higher( -hus, 49V ould be 2ero at a loer price in this situation( .n the case of a bid price, %ou could sub"it a loer price and still brea&-e*en, or sub"it the higher price and "a&e a positi*e 49V( 24. ,ince e need to calculate the 60C for each "achine, sales are irrele*ant( 60C onl% uses the costs of operating the e$uip"ent, not the sales( Using the botto" up approach, or net inco"e plus depreciation, "ethod to calculate /CB, e get+ 3achine 0 3achine H Variable costs E<=,A00,000 E<=,000,000 Bi!ed costs E1J0,000 E1=0,000 )epreciation EI8=,=== EAMM,MMJ 6H- E<I,1A=,=== E<=,M9M,MMJ -a! 1,IA=,MMJ 1,29=,8== 4et inco"e E<2,M99,MMJ E<2,I02,8== O )epreciation I8=,=== AMM,MMJ /CB E<2,21M,=== E<1,8=M,1MJ -he 49V and 60C for 3achine 0 is+ 49V0 L E<2,900,000 E <2,21M,===(9V.B010N,M# 49V0 L E<12,AA2,J09(IM 60C0 L E <12,AA2(J09(IM : (9V.B010N,M# 60C0 L E<2,882,19I(JI 0nd the 49V and 60C for 3achine H is+ 49VH L E<A,100,000 E 1,8=M,1MJ(9V.B010N,9# 49VH L E<1A,MJI,A2J(AM 60CH L E <1A,MJI,A2J(AM : (9V.B010N,9# 60CH L E<2,J21,J==(I2 5ou should choose 3achine H since it has a "ore positi*e 60C( CHAPTER 10 B-197 2. 0 &iloatt hour is 1,000 atts for 1 hour( 0 M0-att bulb burning for A00 hours per %ear uses =0,000 att hours, or =0 &iloatt hours( ,ince the cost of a &iloatt hour is <0(101, the cost per %ear is+ Cost per %ear L =0(<0(101# Cost per %ear L <=(0= -he M0-att bulb ill last for 1,000 hours, hich is 2 %ears of use at A00 hours per %ear( ,o, the 49V of the M0-att bulb is+ 49V L E<0(A0 E <=(0=(9V.B010N,2# 49V L E<A(JM 0nd the 60C is+ 60C L E<A(8= : (9V.B010N,2# 60C L E<=(=2 4o e can find the 60C for the 1A-att CBC( 0 1A-att bulb burning for A00 hours per %ear uses J,A00 atts, or J(A &iloatts( 0nd, since the cost of a &iloatt hour is <0(101, the cost per %ear is+ Cost per %ear L J(A(<0(101# Cost per %ear L <0(JAJA -he 1A-att CBC ill last for 12,000 hours, hich is 2I %ears of use at A00 hours per %ear( ,o, the 49V of the CBC is+ 49V L E<=(A0 E <0(JAJA(9V.B010N,2I# 49V L E<10(=1 0nd the 60C is+ 60C L E<10(8A : (9V.B010N,2I# 60C L E<1(1A -hus, the CBC is "uch cheaper( Hut see our ne!t to $uestions( 2!. -o sol*e the 60C algebraicall% for each bulb, e can set up the *ariables as follos+ W L light bulb attage C L cost per &iloatt hour > L hours burned per %ear 9 L price the light bulb -he nu"ber of atts use b% the bulb per hour is+ W9> L W : 1,000 0nd the &iloatt hours used per %ear is+ P95 L W9> S > B-198 SOLUTIONS -he electricit% cost per %ear is therefore+ 6C5 L P95 S C -he 49V of the decision to but the light bulb is+ 49V L E 9 E 6C5(9V.B0RN,t# 0nd the 60C is+ 60C L 49V : (9V.B0RN,t# ,ubstituting, e get+ 60C L QE9 E (W : 1,000 S > S C#9V.B0RN,tR : 9B.V0RN,t We need to set the 60C of the to light bulbs e$ual to each other and sol*e for C, the cost per &iloatt hour( )oing so, e find+ QE<0(A0 E (M0 : 1,000 S A00 S C#9V.B010N,2R : 9V.B010N,2 L QE<=(A0 E (1A : 1,000 S A00 S C#9V.B010N,2IR : 9V.B010N,2I C L <0(00IA09 ,o, unless the cost per &iloatt hour is e!tre"el% lo, it "a&es sense to use the CBC( Hut hen should %ou replace the incandescent bulb? ,ee the ne!t $uestion( 2". We are again sol*ing for the brea&e*en &iloatt hour cost, but no the incandescent bulb has onl% A00 hours of useful life( .n this case, the incandescent bulb has onl% one %ear of life left( -he brea&e*en electricit% cost under these circu"stances is+ QE<0(A0 E (M0 : 1,000 S A00 S C#9V.B010N,1R : 9V.B010N,1 L QE<=(A0 E (1A : 1,000 S A00 S C#9V.B010N,2IR : 9V.B010N,2I C L E<0(00J1=1 Unless the electricit% cost is negati*e (4ot *er% li&el%X#, it does not "a&e financial sense to replace the incandescent bulb until it burns out( 2#. -he debate beteen incandescent bulbs and CBCs is not 1ust a financial debate, but an en*iron"ental one as ell( -he nu"bers belo correspond to the nu"bered ite"s in the $uestion+ 1( -he e!tra heat generated b% an incandescent bulb is aste, but not necessaril% in a heated structure, especiall% in northern cli"ates( 2( ,ince CBCs last so long, fro" a financial *iepoint, it "ight "a&e sense to ait if prices are declining( =( Hecause of the nontri*ial health and disposal issues, CBCs are not as attracti*e as our pre*ious anal%sis suggests( CHAPTER 10 B-199 I( Bro" a co"pan%'s perspecti*e, the cost of replacing or&ing incandescent bulbs "a% outeigh the financial benefit( >oe*er, since CBCs last longer, the cost of replacing the bulbs ill be loer in the long run( A( Hecause incandescent bulbs use "ore poer, "ore coal has to be burned, hich generates "ore "ercur% in the en*iron"ent, potentiall% offsetting the "ercur% concern ith CBCs( M( 0s in the pre*ious $uestion, if C/2 production is an en*iron"ental concern, the the loer poer consu"ption fro" CBCs is a benefit( J( CBCs re$uire "ore energ% to "a&e, potentiall% offsetting (at least partiall%# the energ% sa*ings fro" their use( Wor&er safet% and site conta"ination are also negati*es for CBCs( 8( -his fact fa*ors the incandescent bulb because the purchasers ill onl% recei*e part of the benefit fro" the CBC( 9( -his fact fa*ors aiting for ne technolog%( 10( -his fact also fa*ors aiting for ne technolog%( While there is ala%s a ;best@ anser, this $uestion shos that the anal%sis of the ;best@ anser is not ala%s eas% and "a% not be possible because of inco"plete data( 0s for ho to better legislate the use of CBCs, our anal%sis suggests that re$uiring the" in ne construction "ight "a&e sense( Rental properties in general should probabl% be re$uired to use CBCs (h% rentals?#( 0nother piece of legislation that "a&es sense is re$uiring the producers of CBCs to suppl% a disposal &it and proper disposal instructions ith each one sold( Binall%, e need "uch better research on the ha2ards associated ith bro&en bulbs in the ho"e and or&place and proper procedures for dealing ith bro&en bulbs( 2$. ,urpriseX 5ou should definitel% upgrade the truc&( >ere's h%( 0t 10 "pg, the truc& burns 12,000 : 10 L 1,200 gallons of gas per %ear( -he ne truc& ill burn 12,000 : 12(A L 9M0 gallons of gas per %ear, a sa*ings of 2I0 gallons per %ear( -he car burns 12,000 : 2A L I80 gallons of gas per %ear, hile the ne car ill burn 12,000 : I0 L =00 gallons of gas per %ear, a sa*ings of 180 gallons per %ear, so it's not e*en close( -his anser "a% stri&e %ou as counterintuiti*e, so let's consider an e!tre"e case( ,uppose the car gets M,000 "pg, and %ou could upgrade to 12,000 "pg( ,hould %ou upgrade? 9robabl% not since %ou ould onl% sa*e one gallon of gas per %ear( ,o, the reason %ou should upgrade the truc& is that it uses so "uch "ore gas in the first place( 4otice that the anser doesn't depend on the cost of gasoline, "eaning that if %ou upgrade, %ou should ala%s upgrade the truc&( .n fact, it doesn't depend on the "iles dri*en, as long as the "iles dri*en are the sa"e( B-200 SOLUTIONS 3%. ,urpriseX 5ou should definitel% upgrade the truc&( >ere's h%( 0t 10 "pg, the truc& burns 12,000 : 10 L 1,200 Challenge 31. We ill begin b% calculating the afterta! sal*age *alue of the e$uip"ent at the end of the pro1ect's life( -he afterta! sal*age *alue is the "ar&et *alue of the e$uip"ent "inus an% ta!es paid (or refunded#, so the afterta! sal*age *alue in four %ears ill be+ -a!es on sal*age *alue L (HV E 3V#tC -a!es on sal*age *alue L (<0 E I00,000#((=8# -a!es on sal*age *alue L E<1A2,000 3ar&et price <I00,000 -a! on sale E1A2,000 0fterta! sal*age *alue <2I8,000 4o e need to calculate the operating cash flo each %ear( Using the botto" up approach to calculating operating cash flo, e find+ 5ear 0 5ear 1 5ear 2 5ear = 5ear I Re*enues <2,I9M,000 <=,=AI,000 <=,0I2,000 <2,18I,000 Bi!ed costs I2A,000 I2A,000 I2A,000 I2A,000 Variable costs =JI,I00 A0=,100 IAM,=00 =2J,M00 )epreciation 1,=99,8M0 1,8MM,900 M22,020 =11,220 6H- <29M,JI0 <AA9,000 <1,A=8,M80 <1,120,180 -a!es 112,JM1 212,I20 A8I,M98 I2A,MM8 4et inco"e <18=,9J9 <=IM,A80 <9A=,982 <M9I,A12 /CB <1,A8=,8=9 <2,21=,I80 <1,AJM,002 <1,00A,J=2 Capital spending E<I,200,000 <2I8,000 Cand E1,A00,000 1,M00,000 4WC E12A,000 12A,000 -otal cash flo E<A,82A,000 <1,A8=,8=9 <2,21=,I80 <1,AJM,002 <2,9J8,J=2 4otice the calculation of the cash flo at ti"e 0( -he capital spending on e$uip"ent and in*est"ent in net or&ing capital are cash outflos are both cash outflos( -he afterta! selling price of the land is also a cash outflo( 6*en though no cash is actuall% spent on the land because the co"pan% alread% ons it, the afterta! cash flo fro" selling the land is an opportunit% cost, so e need to include it in the anal%sis( -he co"pan% can sell the land at the end of the pro1ect, so e need to include that *alue as ell( With all the pro1ect cash flos, e can calculate the 49V, hich is+ 49V L E<A,82A,000 O <1,A8=,8=9 : 1(1= O <2,21=,I80 : 1(1= 2 O <1,AJM,002 : 1(1= = O <2,9J8,J=2 : 1(1= I 49V L <229,2MM(82 CHAPTER 10 B-201 -he co"pan% should accept the ne product line( 32. -his is an in-depth capital budgeting proble"( 9robabl% the easiest /CB calculation for this proble" is the botto" up approach, so e ill construct an inco"e state"ent for each %ear( Heginning ith the initial cash flo at ti"e 2ero, the pro1ect ill re$uire an in*est"ent in e$uip"ent( -he pro1ect ill also re$uire an in*est"ent in 4WC( -he initial 4WC in*est"ent is gi*en, and the subse$uent 4WC in*est"ent ill be 1A percent of the ne!t %ear's sales( .n this case, it ill be 5ear 1 sales( Reali2ing e need 5ear 1 sales to calculate the re$uired 4WC capital at ti"e 0, e find that 5ear 1 sales ill be <=A,=I0,000( ,o, the cash flo re$uired for the pro1ect toda% ill be+ Capital spending E<2I,000,000 .nitial 4WC E1,800,000 -otal cash flo E<2A,800,000 4o e can begin the re"aining calculations( ,ales figures are gi*en for each %ear, along ith the price per unit( -he *ariable costs per unit are used to calculate total *ariable costs, and fi!ed costs are gi*en at <1,200,000 per %ear( -o calculate depreciation each %ear, e use the initial e$uip"ent cost of <2I "illion, ti"es the appropriate 30CR, depreciation each %ear( -he re"ainder of each inco"e state"ent is calculated belo( 4otice at the botto" of the inco"e state"ent e added bac& depreciation to get the /CB for each %ear( -he section labeled ;4et cash flos@ ill be discussed belo+ 5ear 1 2 = I A 6nding boo& *alue <20,AJ0,I00 <1I,M92,800 <10,I9A,200 <J,I9J,M00 <A,=AI,I00 ,ales <=A,=I0,000 <=9,900,000 <I8,MI0,000 <A0,920,000 <==,0M0,000 Variable costs 2I,MIA,000 2J,82A,000 ==,920,000 =A,A10,000 2=,0AA,000 Bi!ed costs 1,200,000 1,200,000 1,200,000 1,200,000 1,200,000 )epreciation =,I29,M00 A,8JJ,M00 I,19J,M00 2,99J,M00 2,1I=,200 6H.- <M,0MA,I00 <I,99J,I00 <9,=22,I00 <11,212,I00 <M,MM1,800 -a!es 2,122,890 1,JI9,090 =,2M2,8I0 =,92I,=I0 2,==1,M=0 4et inco"e <=,9I2,A10 <=,2I8,=10 <M,0A9,AM0 <J,288,0M0 <I,==0,1J0 )epreciation =,I29,M00 A,8JJ,M00 I,19J,M00 2,99J,M00 2,1I=,200 /perating cash flo <J,=J2,110 <9,12A,910 <10,2AJ,1M0 <10,28A,MM0 <M,IJ=,=J0 Net cash flows /perating cash flo <J,=J2,110 <9,12A,910 <10,2AJ,1M0 <10,28A,MM0 <M,IJ=,=J0 Change in 4WC EM8I,000 E1,=11,000 E=I2,000 2,MJ9,000 1,IA8,000 Capital spending 0 0 0 0 I,99I,0I0 -otal cash flo <M,M88,110 <J,81I,910 <9,91A,1M0 <12,9MI,MM0 <12,92A,I10 0fter e calculate the /CB for each %ear, e need to account for an% other cash flos( -he other cash flos in this case are 4WC cash flos and capital spending, hich is the afterta! sal*age of the e$uip"ent( -he re$uired 4WC capital is 1A percent of the increase in sales in the ne!t %ear( We ill B-202 SOLUTIONS or& through the 4WC cash flo for 5ear 1( -he total 4WC in 5ear 1 ill be 1A percent of sales increase fro" 5ear 1 to 5ear 2, or+ .ncrease in 4WC for 5ear 1 L (1A(<=9,900,000 E =A,=I0,000# .ncrease in 4WC for 5ear 1 L <M8I,000 4otice that the 4WC cash flo is negati*e( ,ince the sales are increasing, e ill ha*e to spend "ore "one% to increase 4WC( .n 5ear I, the 4WC cash flo is positi*e since sales are declining( 0nd, in 5ear A, the 4WC cash flo is the reco*er% of all 4WC the co"pan% still has in the pro1ect( -o calculate the afterta! sal*age *alue, e first need the boo& *alue of the e$uip"ent( -he boo& *alue at the end of the fi*e %ears ill be the purchase price, "inus the total depreciation( ,o, the ending boo& *alue is+ 6nding boo& *alue L <2I,000,000 E (<=,I29,M00 O A,8JJ,M00 O I,19J,M00 O 2,99J,M00 O 2,1I=,200# 6nding boo& *alue L <A,=AI,I00 -he "ar&et *alue of the used e$uip"ent is 20 percent of the purchase price, or <I(8 "illion, so the afterta! sal*age *alue ill be+ 0fterta! sal*age *alue L <I,800,000 O (<A,=AI,I00 E I,800,000#((=A# 0fterta! sal*age *alue L <I,99I,0I0 -he afterta! sal*age *alue is included in the total cash flos are capital spending( 4o e ha*e all of the cash flos for the pro1ect( -he 49V of the pro1ect is+ 49V L E<2A,800,000 O <M,M88,110:1(18 O <J,81I,910:1(18 2 O <9,91A,1M0:1(18 = O <12,9MI,MM0:1(18 I O <12,92A,I10:1(18 A 49V L <=,8A1,9A2(2= 0nd the .RR is+ 49V L 0 L E<2A,800,000 O <M,M88,110:(1 O .RR# O <J,81I,910:(1 O .RR# 2 O <9,91A,1M0:(1 O .RR# = O <12,9MI,MM0:(1 O .RR# I O <12,92A,I10:(1 O .RR# A .RR L 2=(M2N We should accept the pro1ect( 33. -o find the initial preta! cost sa*ings necessar% to bu% the ne "achine, e should use the ta! shield approach to find the /CB( We begin b% calculating the depreciation each %ear using the 30CR, depreciation schedule( -he depreciation each %ear is+ )1 L <M10,000(0(====# L <20=,=1= )2 L <M10,000(0(IIII# L <2J1,1IA )= L <M10,000(0(1I82# L <90,=I1 )I L <M10,000(0(0JI1# L <IA,201 Using the ta! shield approach, the /CB each %ear is+ /CB1 L (, E C#(1 E 0(=A# O 0(=A(<20=,=1=# CHAPTER 10 B-203 /CB2 L (, E C#(1 E 0(=A# O 0(=A(<2J1,1IA# /CB= L (, E C#(1 E 0(=A# O 0(=A(<90,=I1# /CBI L (, E C#(1 E 0(=A# O 0(=A(<IA,201# /CBA L (, E C#(1 E 0(=A# 4o e need the afterta! sal*age *alue of the e$uip"ent( -he afterta! sal*age *alue is+ 0fter-ta! sal*age *alue L <I0,000(1 E 0(=A# L <2M,000 -o find the necessar% cost reduction, e "ust reali2e that e can split the cash flos each %ear( -he /CB in an% gi*en %ear is the cost reduction (, E C# ti"es one "inus the ta! rate, hich is an annuit% for the pro1ect life, and the depreciation ta! shield( -o calculate the necessar% cost reduction, e ould re$uire a 2ero 49V( -he e$uation for the 49V of the pro1ect is+ 49V L 0 L E<M10,000 E AA,000 O (, E C#(0(MA#(9V.B012N,A# O 0(=A(<20=,=1=:1(12 O <2J1,1IA:1(12 2 O <90,=I1:1(12 = O <IA,201:1(12 I # O (<AA,000 O 2M,000#:1(12 A ,ol*ing this e$uation for the sales "inus costs, e get+ (, E C#(0(MA#(9V.B012N,A# L <IIJ,288(MJ (, E C# L <190,89A(JI 34. a. -his proble" is basicall% the sa"e as 9roble" 18, e!cept e are gi*en a sales price( -he cash flo at -i"e 0 for all three parts of this $uestion ill be+ Capital spending E<9I0,000 Change in 4WC EJA,000 -otal cash flo E<1,01A,000 We ill use the initial cash flo and the sal*age *alue e alread% found in that proble"( Using the botto" up approach to calculating the /CB, e get+ Assume price per unit 4 >-3 and units5ear 4 -D.#222 5ear 1 2 = I A ,ales <2,I0A,000 <2,I0A,000 <2,I0A,000 <2,I0A,000 <2,I0A,000 Variable costs 1,J11,2A0 1,J11,2A0 1,J11,2A0 1,J11,2A0 1,J11,2A0 Bi!ed costs =0A,000 =0A,000 =0A,000 =0A,000 =0A,000 )epreciation 188,000 188,000 188,000 188,000 188,000 6H.- 200,JA0 200,JA0 200,JA0 200,JA0 200,JA0 -a!es (=AN# J0,2M= J0,2M= J0,2M= J0,2M= J0,2M= 4et .nco"e 1=0,I88 1=0,I88 1=0,I88 1=0,I88 1=0,I88 )epreciation 188,000 188,000 188,000 188,000 188,000 /perating CB <=18,I88 <=18,I88 <=18,I88 <=18,I88 <=18,I88 5ear 1 2 = I A /perating CB <=18,I88 <=18,I88 <=18,I88 <=18,I88 <=18,I88 Change in 4WC 0 0 0 0 JA,000 Capital spending 0 0 0 0 IA,A00 -otal CB <=18,I88 <=18,I88 <=18,I88 <=18,I88 <I=8,988 B-204 SOLUTIONS With these cash flos, the 49V of the pro1ect is+ 49V L E<9I0,000 E JA,000 O <=18,I88(9V.B012N,A# O Q(<JA,000 O IA,A00# : 1(12 A R 49V L <201,IA1(10 .f the actual price is abo*e the bid price that results in a 2ero 49V, the pro1ect ill ha*e a positi*e 49V( 0s for the cartons sold, if the nu"ber of cartons sold increases, the 49V ill increase, and if the costs increase, the 49V ill decrease( b( -o find the "ini"u" nu"ber of cartons sold to still brea&e*en, e need to use the ta! shield approach to calculating /CB, and sol*e the proble" si"ilar to finding a bid price( Using the initial cash flo and sal*age *alue e alread% calculated, the e$uation for a 2ero 49V of the pro1ect is+ 49V L 0 L E<9I0,000 E JA,000 O /CB(9V.B012N,A# O Q(<JA,000 O IA,A00# : 1(12 A R ,o, the necessar% /CB for a 2ero 49V is+ /CB L <9IM,M2A(0M : 9V.B012N,A L <2M2,M0=(01 4o e can use the ta! shield approach to sol*e for the "ini"u" $uantit% as follos+ /CB L <2M2,M0=(01 L Q(9 E *#7 E BC R(1 E t c # O t c ) $262,603.01 L Q(<1=(00 E 9(2A#7 E =0A,000 R(1 E 0(=A# O 0(=A(<9I0,000:A# 7 L 1M2,0J= 0s a chec&, e can calculate the 49V of the pro1ect ith this $uantit%( -he calculations are+ 5ear 1 2 = I A ,ales <2,10M,9I9 <2,10M,9I9 <2,10M,9I9 <2,10M,9I9 <2,10M,9I9 Variable costs 1,I99,1JM 1,I99,1JM 1,I99,1JM 1,I99,1JM 1,I99,1JM Bi!ed costs =0A,000 =0A,000 =0A,000 =0A,000 =0A,000 )epreciation 188,000 188,000 188,000 188,000 188,000 6H.- 11I,JJI 11I,JJI 11I,JJI 11I,JJI 11I,JJI -a!es (=AN# I0,1J1 I0,1J1 I0,1J1 I0,1J1 I0,1J1 4et .nco"e JI,M0= JI,M0= JI,M0= JI,M0= JI,M0= )epreciation 188,000 188,000 188,000 188,000 188,000 /perating CB <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= 5ear 1 2 = I A /perating CB <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= Change in 4WC 0 0 0 0 JA,000 Capital spending 0 0 0 0 IA,A00 -otal CB <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= <=8=,10= 49V L E<9I0,000 E JA,000 O <2M2,M0=(9V.B012N,A# O Q(<JA,000 O IA,A00# : 1(12 A R ≈ <0 4ote, the 49V is not e!actl% e$ual to 2ero because e had to round the nu"ber of cartons sold8 %ou cannot sell one-half of a carton( CHAPTER 10 B-205 c( -o find the highest le*el of fi!ed costs and still brea&e*en, e need to use the ta! shield approach to calculating /CB, and sol*e the proble" si"ilar to finding a bid price( Using the initial cash flo and sal*age *alue e alread% calculated, the e$uation for a 2ero 49V of the pro1ect is+ 49V L 0 L E<9I0,000 E JA,000 O /CB(9V.B012N,A# O Q(<JA,000 O IA,A00# : 1(12 A R /CB L <9IM,M2A(0M : 9V.B012N,A L <2M2,M0=(01 4otice this is the sa"e /CB e calculated in part b( 4o e can use the ta! shield approach to sol*e for the "a!i"u" le*el of fi!ed costs as follos+ /CB L <2M2,M0=(01 L Q(9E*#7 E BC R(1 E tC# O tC) <2M2,M0=(01 L Q(<1=(00 E 9(2A#(18A,000# E BCR(1 E 0(=A# O 0(=A(<9I0,000:A# BC L <=90,9JM(1A 0s a chec&, e can calculate the 49V of the pro1ect ith this le*el of fi!ed costs( -he calculations are+ 5ear 1 2 = I A ,ales <2,I0A,000 <2,I0A,000 <2,I0A,000 <2,I0A,000 <2,I0A,000 Variable costs 1,J11,2A0 1,J11,2A0 1,J11,2A0 1,J11,2A0 1,J11,2A0 Bi!ed costs =90,9JM =90,9JM =90,9JM =90,9JM =90,9JM )epreciation 188,000 188,000 188,000 188,000 188,000 6H.- 11I,JJI 11I,JJI 11I,JJI 11I,JJI 11I,JJI -a!es (=AN# I0,1J1 I0,1J1 I0,1J1 I0,1J1 I0,1J1 4et .nco"e JI,M0= JI,M0= JI,M0= JI,M0= JI,M0= )epreciation 188,000 188,000 188,000 188,000 188,000 /perating CB <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= 5ear 1 2 = I A /perating CB <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= Change in 4WC 0 0 0 0 JA,000 Capital spending 0 0 0 0 IA,A00 -otal CB <2M2,M0= <2M2,M0= <2M2,M0= <2M2,M0= <=8=,10= 49V L E<9I0,000 E JA,000 O <2M2,M0=(9V.B012N,A# O Q(<JA,000 O IA,A00# : 1(12 A R ≈ <0 3. We need to find the bid price for a pro1ect, but the pro1ect has e!tra cash flos( ,ince e don't alread% produce the &e%board, the sales of the &e%board outside the contract are rele*ant cash flos( ,ince e &no the e!tra sales nu"ber and price, e can calculate the cash flos generated b% these sales( -he cash flo generated fro" the sale of the &e%board outside the contract is+ 1 2 = I ,ales <8AA,000 <1,J10,000 <2,280,000 <1,I2A,000 Variable costs A2A,000 1,0A0,000 1,I00,000 8JA,000 6H- <==0,000 <MM0,000 <880,000 <AA0,000 -a! 1=2,000 2MI,000 =A2,000 220,000 4et inco"e (and /CB# <198,000 <=9M,000 <A28,000 <==0,000 B-206 SOLUTIONS ,o, the addition to 49V of these "ar&et sales is+ 49V of "ar&et sales L <198,000:1(1= O <=9M,000:1(1= 2 O <A28,000:1(1= = O <==0,000:1(1= I 49V of "ar&et sales L <1,0A=,MJ2(99 5ou "a% ha*e noticed that e did not include the initial cash outla%, depreciation or fi!ed costs in the calculation of cash flos fro" the "ar&et sales( -he reason is that it is irrele*ant hether or not e include these here( Re"e"ber, e are not onl% tr%ing to deter"ine the bid price, but e are also deter"ining hether or not the pro1ect is feasible( .n other ords, e are tr%ing to calculate the 49V of the pro1ect, not 1ust the 49V of the bid price( We ill include these cash flos in the bid price calculation( -he reason e stated earlier that hether e included these costs in this initial calculation as irrele*ant is that %ou ill co"e up ith the sa"e bid price if %ou include these costs in this calculation, or if %ou include the" in the bid price calculation( 4e!t, e need to calculate the afterta! sal*age *alue, hich is+ 0fterta! sal*age *alue L <2JA,000(1 E (I0# L <1MA,000 .nstead of sol*ing for a 2ero 49V as is usual in setting a bid price, the co"pan% president re$uires an 49V of <100,000, so e ill sol*e for a 49V of that a"ount( -he 49V e$uation for this pro1ect is (re"e"ber to include the 4WC cash flo at the beginning of the pro1ect, and the 4WC reco*er% at the end#+ 49V L <100,000 L E<=,I00,000 E 9A,000 O 1,0A=,MJ2(99 O /CB (9V.B01=N,I# O Q(<1MA,000 O 9A,000# : 1(1= I R ,ol*ing for the /CB, e get+ /CB L <2,=81,8MI(1I : 9V.B01=N,I L <800,JM8(90 4o e can sol*e for the bid price as follos+ /CB L <800,JM8(90 L Q(9 E *#7 E BC R(1 E tC# O tC) <800,JM8(90 L Q(9 E <1JA#(1J,A00# E <M00,000R(1 E 0(I0# O 0(I0(<=,I00,000:I# 9 L <2A=(1J 3!. a. ,ince the to co"puters ha*e une$ual li*es, the correct "ethod to anal%2e the decision is the 60C( We ill begin ith the 60C of the ne co"puter( Using the depreciation ta! shield approach, the /CB for the ne co"puter s%ste" is+ /CB L (<1IA,000#(1 E (=8# O (<J80,000 : A#((=8# L <1I9,180 4otice that the costs are positi*e, hich represents a cash inflo( -he costs are positi*e in this case since the ne co"puter ill generate a cost sa*ings( -he onl% initial cash flo for the ne co"puter is cost of <J80,000( We ne!t need to calculate the afterta! sal*age *alue, hich is+ 0fterta! sal*age *alue L <1A0,000(1 E (=8# L <9=,000 4o e can calculate the 49V of the ne co"puter as+ 49V L E<J80,000 O <1I9,180(9V.B012N,A# O <9=,000 : 1(12 A CHAPTER 10 B-207 49V L E<189,IM8(J9 0nd the 60C of the ne co"puter is+ 60C L E<189,IM8(J9 : (9V.B012N,A# L E<A2,AM0(I9 0nal%2ing the old co"puter, the onl% /CB is the depreciation ta! shield, so+ /CB L <1=0,000((=8# L <I9,I00 -he initial cost of the old co"puter is a little tric&ier( 5ou "ight assu"e that since e alread% on the old co"puter there is no initial cost, but e can sell the old co"puter, so there is an opportunit% cost( We need to account for this opportunit% cost( -o do so, e ill calculate the afterta! sal*age *alue of the old co"puter toda%( We need the boo& *alue of the old co"puter to do so( -he boo& *alue is not gi*en directl%, but e are told that the old co"puter has depreciation of <1=0,000 per %ear for the ne!t three %ears, so e can assu"e the boo& *alue is the total a"ount of depreciation o*er the re"aining life of the s%ste", or <=90,000( ,o, the afterta! sal*age *alue of the old co"puter is+ 0fterta! sal*age *alue L <210,000 O (<=90,000 E 210,000#((=8# L <=JJ,200 -his is the initial cost of the old co"puter s%ste" toda% because e are forgoing the opportunit% to sell it toda%( We ne!t need to calculate the afterta! sal*age *alue of the co"puter s%ste" in to %ears since e are ;bu%ing@ it toda%( -he afterta! sal*age *alue in to %ears is+ 0fterta! sal*age *alue L <M0,000 O (<1=0,000 E M0,000#((=8# L <8M,M00 4o e can calculate the 49V of the old co"puter as+ 49V L E<=JJ,200 O <I9,I00(9V.B011N,2# O 1=M,000 : 1(12 2 49V L E<22I,MIJ(I9 0nd the 60C of the old co"puter is+ 60C L E<22I,MJI(I9 : (9V.B012N,2# L E<1=2,9=9(IJ 6*en if e are going to replace the s%ste" in to %ears no "atter hat our decision toda%, e should replace it toda% since the 60C is "ore positi*e( B-208 SOLUTIONS b. .f e are onl% concerned ith hether or not to replace the "achine no, and are not orr%ing about hat ill happen in to %ears, the correct anal%sis is 49V( -o calculate the 49V of the decision on the co"puter s%ste" no, e need the difference in the total cash flos of the old co"puter s%ste" and the ne co"puter s%ste"( Bro" our pre*ious calculations, e can sa% the cash flos for each co"puter s%ste" are+ t 4e co"puter /ld co"puter )ifference 0 E<J80,000 E<=JJ,200 E<I02,800 1 1I9,180 I9,I00 99,J80 2 1I9,180 1=M,000 1=,180 = 1I9,180 0 1I9,180 I 1I9,180 0 1I9,180 A 2I2,180 0 2I2,180 ,ince e are onl% concerned ith "arginal cash flos, the cash flos of the decision to replace the old co"puter s%ste" ith the ne co"puter s%ste" are the differential cash flos( -he 49V of the decision to replace, ignoring hat ill happen in to %ears is+ 49V L E<I02,800 O <99,J80:1(12 O <1=,180:1(12 2 O <1I9,180:1(1I = O <1I9,180:1(1I I O <2I2,180:1(1I A 49V L <=A,20A(J0 .f e are not concerned ith hat ill happen in to %ears, e should replace the old co"puter s%ste"( CHAPTER 11 PROJECT ANALYSIS AND EVALUATION Answers to Concepts Review and Critical Thinking Questions 1. Borecasting ris& is the ris& that a poor decision is "ade because of errors in pro1ected cash flos( -he danger is greatest ith a ne product because the cash flos are probabl% harder to predict( 2. With a sensiti*it% anal%sis, one *ariable is e!a"ined o*er a broad range of *alues( With a scenario anal%sis, all *ariables are e!a"ined for a li"ited range of *alues( 3. .t is true that if a*erage re*enue is less than a*erage cost, the fir" is losing "one%( -his "uch of the state"ent is therefore correct( 0t the "argin, hoe*er, accepting a pro1ect ith "arginal re*enue in e!cess of its "arginal cost clearl% acts to increase operating cash flo( 4. .t "a&es ages and salaries a fi!ed cost, dri*ing up operating le*erage( . Bi!ed costs are relati*el% high because airlines are relati*el% capital intensi*e (and airplanes are e!pensi*e#( ,&illed e"plo%ees such as pilots and "echanics "ean relati*el% high ages hich, because of union agree"ents, are relati*el% fi!ed( 3aintenance e!penses are significant and relati*el% fi!ed as ell( !. Bro" the shareholder perspecti*e, the financial brea&-e*en point is the "ost i"portant( 0 pro1ect can e!ceed the accounting and cash brea&-e*en points but still be belo the financial brea&-e*en point( -his causes a reduction in shareholder (%our# ealth( ". -he pro1ect ill reach the cash brea&-e*en first, the accounting brea&-e*en ne!t and finall% the financial brea&-e*en( Bor a pro1ect ith an initial in*est"ent and sales after, this ordering ill ala%s appl%( -he cash brea&-e*en is achie*ed first since it e!cludes depreciation( -he accounting brea&-e*en is ne!t since it includes depreciation( Binall%, the financial brea&-e*en, hich includes the ti"e *alue of "one%, is achie*ed( #. ,oft capital rationing i"plies that the fir" as a hole isn't short of capital, but the di*ision or pro1ect does not ha*e the necessar% capital( -he i"plication is that the fir" is passing up positi*e 49V pro1ects( With hard capital rationing the fir" is unable to raise capital for a pro1ect under an% circu"stances( 9robabl% the "ost co""on reason for hard capital rationing is financial distress, "eaning ban&ruptc% is a possibilit%( $. -he i"plication is that the% ill face hard capital rationing( B-210 SOLUTIONS Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. a. -he total *ariable cost per unit is the su" of the to *ariable costs, so+ -otal *ariable costs per unit L <A(I= O =(1= -otal *ariable costs per unit L <8(AM b. -he total costs include all *ariable costs and fi!ed costs( We need to "a&e sure e are including all *ariable costs for the nu"ber of units produced, so+ -otal costs L Variable costs O Bi!ed costs -otal costs L <8(AM(280,000# O <J20,000 -otal costs L <=,11M,800 c. -he cash brea&e*en, that is the point here cash flo is 2ero, is+ 7C L <J20,000 : (<19(99 E 8(AM# 7C L M2,992(1= units 0nd the accounting brea&e*en is+ 70 L (<J20,000 O 220,000# : (<19(99 E 8(AM# 70 L 82,2=9(J2 units 2. -he total costs include all *ariable costs and fi!ed costs( We need to "a&e sure e are including all *ariable costs for the nu"ber of units produced, so+ -otal costs L (<2I(8M O 1I(08#(120,000# O <1,AA0,000 -otal costs L <M,222,800 -he "arginal cost, or cost of producing one "ore unit, is the total *ariable cost per unit, so+ 3arginal cost L <2I(8M O 1I(08 3arginal cost L <=8(9I CHAPTER 11 B-211 -he a*erage cost per unit is the total cost of production, di*ided b% the $uantit% produced, so+ 0*erage cost L -otal cost : -otal $uantit% 0*erage cost L <M,222,800:120,000 0*erage cost L <A1(8M 3ini"u" acceptable total re*enue L A,000(<=8(9I# 3ini"u" acceptable total re*enue L <19I,J00 0dditional units should be produced onl% if the cost of producing those units can be reco*ered( 3. -he base-case, best-case, and orst-case *alues are shon belo( Re"e"ber that in the best-case, sales and price increase, hile costs decrease( .n the orst-case, sales and price decrease, and costs increase( Unit ,cenario Unit ,ales Unit 9rice Variable Cost Bi!ed Costs Hase 9A,000 <1,900(00 <2I0(00 <I,800,000 Hest 109,2A0 <2,18A(00 <20I(00 <I,080,000 Worst 80,JA0 <1,M1A(00 <2JM(00 <A,A20,000 4. 0n esti"ate for the i"pact of changes in price on the profitabilit% of the pro1ect can be found fro" the sensiti*it% of 49V ith respect to price+ ∆49V:∆9( -his "easure can be calculated b% finding the 49V at an% to different price le*els and for"ing the ratio of the changes in these para"eters( Whene*er a sensiti*it% anal%sis is perfor"ed, all other *ariables are held constant at their base-case *alues( . a( -o calculate the accounting brea&e*en, e first need to find the depreciation for each %ear( -he depreciation is+ )epreciation L <J2I,000:8 )epreciation L <90,A00 per %ear 0nd the accounting brea&e*en is+ 70 L (<J80,000 O 90,A00#:(<I= E 29# 70 L M2,1J9 units -o calculate the accounting brea&e*en, e "ust reali2e at this point (and onl% this point#, the /CB is e$ual to depreciation( ,o, the )/C at the accounting brea&e*en is+ )/C L 1 O BC:/CB L 1 O BC:) )/C L 1 O Q<J80,000:<90,A00R )/C L 9(919 b. We ill use the ta! shield approach to calculate the /CB( -he /CB is+ /CBbase L Q(9 E *#7 E BCR(1 E tc# O tc) /CBbase L Q(<I= E 29#(90,000# E <J80,000R(0(MA# O 0(=A(<90,A00# /CBbase L <=I=,MJA B-212 SOLUTIONS 4o e can calculate the 49V using our base-case pro1ections( -here is no sal*age *alue or 4WC, so the 49V is+ 49Vbase L E<J2I,000 O <=I=,MJA(9V.B01AN,8# 49Vbase L <818,180(22 -o calculate the sensiti*it% of the 49V to changes in the $uantit% sold, e ill calculate the 49V at a different $uantit%( We ill use sales of 9A,000 units( -he 49V at this sales le*el is+ /CBne L Q(<I= E 29#(9A,000# E <J80,000R(0(MA# O 0(=A(<90,A00# /CBne L <=89,1JA 0nd the 49V is+ 49Vne L E<J2I,000 O <=89,1JA(9V.B01AN,8# 49Vne L <1,022,=A=(=A ,o, the change in 49V for e*er% unit change in sales is+ ∆49V:∆, L (<1,022,=A=(=A E 818,180(22#:(9A,000 E 90,000# ∆49V:∆, L O<I0(8=A .f sales ere to drop b% A00 units, then 49V ould drop b%+ 49V drop L <I0(8=A(A00# L <20,I1J(=1 5ou "a% onder h% e chose 9A,000 units( Hecause it doesn't "atterX Whate*er sales nu"ber e use, hen e calculate the change in 49V per unit sold, the ratio ill be the sa"e( c. -o find out ho sensiti*e /CB is to a change in *ariable costs, e ill co"pute the /CB at a *ariable cost of <=0( 0gain, the nu"ber e choose to use here is irrele*ant+ We ill get the sa"e ratio of /CB to a one dollar change in *ariable cost no "atter hat *ariable cost e use( ,o, using the ta! shield approach, the /CB at a *ariable cost of <=0 is+ /CBne L Q(<I= E =0#(90,000# E J80,000R(0(MA# O 0(=A(<90,A00# /CBne L <28A,1JA ,o, the change in /CB for a <1 change in *ariable costs is+ ∆/CB:∆* L (<28A,1JA,IA0 E =I=,MJA#:(<=0 E 29# ∆/CB:∆* L E<A8,A00 .f *ariable costs decrease b% <1 then, /CB ould increase b% <A8,A00 CHAPTER 11 B-213 !. We ill use the ta! shield approach to calculate the /CB for the best- and orst-case scenarios( Bor the best-case scenario, the price and $uantit% increase b% 10 percent, so e ill "ultipl% the base case nu"bers b% 1(1, a 10 percent increase( -he *ariable and fi!ed costs both decrease b% 10 percent, so e ill "ultipl% the base case nu"bers b% (9, a 10 percent decrease( )oing so, e get+ /CBbest L YQ(<I=#(1(1# E (<29#(0(9#R(90,000#(1(1# E <J80,000(0(9#Z(0(MA# O 0(=A(<90,A00# /CBbest L <9=9,A9A -he best-case 49V is+ 49Vbest L E<J2I,000 O <9=9,A9A(9V.B01AN,8# 49Vbest L <=,I92,2MI(8A Bor the orst-case scenario, the price and $uantit% decrease b% 10 percent, so e ill "ultipl% the base case nu"bers b% (9, a 10 percent decrease( -he *ariable and fi!ed costs both increase b% 10 percent, so e ill "ultipl% the base case nu"bers b% 1(1, a 10 percent increase( )oing so, e get+ /CBorst L YQ(<I=#(0(9# E (<29#(1(1#R(90,000#(0(9# E <J80,000(1(1#Z(0(MA# O 0(=A(<90,A00# /CBorst L E<1M8,00A -he orst-case 49V is+ 49Vorst L E<J2I,000 E <1M8,00A(9V.B01AN,8# 49Vorst L E<1,IJJ,892(IA ". -he cash brea&e*en e$uation is+ 7C L BC:(9 E *# 0nd the accounting brea&e*en e$uation is+ 70 L (BC O )#:(9 E *# Using these e$uations, e find the folloing cash and accounting brea&e*en points+ (1#+ 7C L <1I3:(<=,020 E 2,2JA# 70 L (<1I3 O M(A3#:(<=,020 E 2,2JA# 7C L 18,J92 70 L 2J,A1J (2#+ 7C L <J=,000:(<=8 E 2J# 70 L (<J=,000 O 1A0,000#:(<=8 E 2J# 7C L M,M=M 70 L 20,2J= (=#+ 7C L <1,200:(<11 E I# 70 L (<1,200 O 8I0#:(<11 E I# 7C L 1J1 70 L 291 B-214 SOLUTIONS #. We can use the accounting brea&e*en e$uation+ 70 L (BC O )#:(9 E *# to sol*e for the un&non *ariable in each case( )oing so, e find+ (1#+ 70 L 112,800 L (<820,000 O )#:(<I1 E =0# ) L <I20,800 (2#+ 70 L 1MA,000 L (<=(23 O 1(1A3#:(9 E <I=# 9 L <M9(=M (=#+ 70 L I,=8A L (<1M0,000 O 10A,000#:(<98 E *# * L <=J(AJ $. -he accounting brea&e*en for the pro1ect is+ 70 L Q<M,000 O (<18,000:I#R:(<AJ E =2# 70 L AI0 0nd the cash brea&e*en is+ 7C L <9,000:(<AJ E =2# 7C L =M0 0t the financial brea&e*en, the pro1ect ill ha*e a 2ero 49V( ,ince this is true, the initial cost of the pro1ect "ust be e$ual to the 9V of the cash flos of the pro1ect( Using this relationship, e can find the /CB of the pro1ect "ust be+ 49V L 0 i"plies <18,000 L /CB(9V.B012N,I# /CB L <A,92M(22 Using this /CB, e can find the financial brea&e*en is+ 7B L (<9,000 O <A,92M(22#:(<AJ E =2# L A9J 0nd the )/C of the pro1ect is+ )/C L 1 O (<9,000:<A,92M(22# L 2(A19 1%. .n order to calculate the financial brea&e*en, e need the /CB of the pro1ect( We can use the cash and accounting brea&e*en points to find this( Birst, e ill use the cash brea&e*en to find the price of the product as follos+ 7C L BC:(9 E *# 1=,200 L <1I0,000:(9 E <2I# 9 L <=I(M1 CHAPTER 11 B-215 4o that e &no the product price, e can use the accounting brea&e*en e$uation to find the depreciation( )oing so, e find the annual depreciation "ust be+ 70 L (BC O )#:(9 E *# 1A,A00 L (<1I0,000 O )#:(<=I(M1 E 2I# )epreciation L <2I,=9I We no &no the annual depreciation a"ount( 0ssu"ing straight-line depreciation is used, the initial in*est"ent in e$uip"ent "ust be fi*e ti"es the annual depreciation, or+ .nitial in*est"ent L A(<2I,=9I# L <121,9J0 -he 9V of the /CB "ust be e$ual to this *alue at the financial brea&e*en since the 49V is 2ero, so+ <121,9J0 L /CB(9V.B01MN,A# /CB L <=J,2A0(M9 We can no use this /CB in the financial brea&e*en e$uation to find the financial brea&e*en sales $uantit% is+ 7B L (<1I0,000 O =J,2A0(M9#:(<=I(M1 E 2I# 7B L 1M,J12 11. We &no that the )/C is the percentage change in /CB di*ided b% the percentage change in $uantit% sold( ,ince e ha*e the original and ne $uantit% sold, e can use the )/C e$uation to find the percentage change in /CB( )oing so, e find+ )/C L N∆/CB : N∆7 ,ol*ing for the percentage change in /CB, e get+ N∆/CB L ()/C#(N∆7# N∆/CB L =(I0Q(J0,000 E MA,000#:MA,000R N∆/CB L (2M1A or 2M(1AN -he ne le*el of operating le*erage is loer since BC:/CB is s"aller( 12. Using the )/C e$uation, e find+ )/C L 1 O BC : /CB =(I0 L 1 O <1=0,000:/CB /CB L <AI,1MJ -he percentage change in $uantit% sold at A8,000 units is+ Nb7 L (A8,000 E MA,000# : MA,000 Nb7 L E(10JJ or E10(JJN B-216 SOLUTIONS ,o, using the sa"e e$uation as in the pre*ious proble", e find+ Nb/CB L =(I0(E10(JJN# Nb7 L E=M(M2N ,o, the ne /CB le*el ill be+ 4e /CB L (1 E (=MM2#(<AI,1MJ# 4e /CB L <=I,=== 0nd the ne )/C ill be+ 4e )/C L 1 O (<1=0,000:<=I,===# 4e )/C L I(J8M 13. -he )/C of the pro1ect is+ )/C L 1 O (<J=,000:<8J,A00# )/C L 1(8=I= .f the $uantit% sold changes to 8,A00 units, the percentage change in $uantit% sold is+ N∆7 L (8,A00 E 8,000#:8,000 Nb7 L (0M2A or M(2AN ,o, the /CB at 8,A00 units sold is+ N∆/CB L )/C(N∆7# Nb/CB L 1(8=I=((0M2A# Nb/CB L (11IM or 11(IMN -his "a&es the ne /CB+ 4e /CB L <8J,A00(1(11IM# 4e /CB L <9J,A=1 0nd the )/C at 8,A00 units is+ )/C L 1 O (<J=,000:<9J,A=1# )/C L 1(JI8A 14. We can use the e$uation for )/C to calculate fi!ed costs( -he fi!ed cost "ust be+ )/C L 2(=A L 1 O BC:/CB BC L (2(=A E 1#<I1,000 BC L <A8,080 .f the output rises to 11,000 units, the percentage change in $uantit% sold is+ N∆7 L (11,000 E 10,000#:10,000 CHAPTER 11 B-217 Nb7 L (10 or 10(00N B-218 SOLUTIONS -he percentage change in /CB is+ N∆/CB L 2(=A((10# Nb/CB L (2=A0 or 2=(A0N ,o, the operating cash flo at this le*el of sales ill be+ /CB L <I=,000(1(2=A# /CB L <A=,10A .f the output falls to 9,000 units, the percentage change in $uantit% sold is+ N∆7 L (9,000 E 10,000#:10,000 Nb7 L E(10 or E10(00N -he percentage change in /CB is+ N∆/CB L 2(=A(E(10# Nb/CB L E(2=A0 or E2=(A0N ,o, the operating cash flo at this le*el of sales ill be+ /CB L <I=,000(1 E (2=A# /CB L <=2,89J 1. Using the e$uation for )/C, e get+ )/C L 1 O BC:/CB 0t 11,000 units )/C L 1 O <A8,0A0:<A=,10A )/C L 2(09=1 0t 9,000 units )/C L 1 O <A8,0A0:<=2,89A )/C L 2(JMIJ &ntermediate 1!. a( 0t the accounting brea&e*en, the .RR is 2ero percent since the pro1ect reco*ers the initial in*est"ent( -he pa%bac& period is 4 %ears, the length of the pro1ect since the initial in*est"ent is e!actl% reco*ered o*er the pro1ect life( -he 49V at the accounting brea&e*en is+ 49V L . Q(1:4#(9V.B0RN,4# E 1R b( 0t the cash brea&e*en le*el, the .RR is E100 percent, the pa%bac& period is negati*e, and the 49V is negati*e and e$ual to the initial cash outla%( CHAPTER 11 B-219 c( -he definition of the financial brea&e*en is here the 49V of the pro1ect is 2ero( .f this is true, then the .RR of the pro1ect is e$ual to the re$uired return( .t is i"possible to state the pa%bac& period, e!cept to sa% that the pa%bac& period "ust be less than the length of the pro1ect( ,ince the discounted cash flos are e$ual to the initial in*est"ent, the undiscounted cash flos are greater than the initial in*est"ent, so the pa%bac& "ust be less than the pro1ect life( 1". Using the ta! shield approach, the /CB at 110,000 units ill be+ /CB L Q(9 E *#7 E BCR(1 E tC# O tC()# /CB L Q(<=2 E 19#(110,000# E 210,000R(0(MM# O 0(=I(<I90,000:I# /CB L <8IM,8A0 We ill calculate the /CB at 111,000 units( -he choice of the second le*el of $uantit% sold is arbitrar% and irrele*ant( 4o "atter hat le*el of units sold e choose, e ill still get the sa"e sensiti*it%( ,o, the /CB at this le*el of sales is+ /CB L Q(<=2 E 19#(111,000# E 210,000R(0(MM# O 0(=I(<I90,000:I# /CB L <8AA,I=0 -he sensiti*it% of the /CB to changes in the $uantit% sold is+ ,ensiti*it% L ∆/CB:∆7 L (<8IM,8A0 E 8AA,I=0#:(110,000 E 111,000# ∆/CB:∆7 L O<8(A8 /CB ill increase b% <A(28 for e*er% additional unit sold( 1#. 0t 110,000 units, the )/C is+ )/C L 1 O BC:/CB )/C L 1 O (<210,000:<8IM,8A0# )/C L 1(2I80 -he accounting brea&e*en is+ 70 L (BC O )#:(9 E *# 7 0 L Q<210,000 O (<I90,000:I#R:(<=2 E 19# 70 L 2A,AJM 0nd, at the accounting brea&e*en le*el, the )/C is+ )/C L 1 O Q<210,000:(<I90,000:I#R )/C L 2(J1I= B-220 SOLUTIONS 1$. a( -he base-case, best-case, and orst-case *alues are shon belo( Re"e"ber that in the best-case, sales and price increase, hile costs decrease( .n the orst-case, sales and price decrease, and costs increase( ,cenario Unit sales Variable cost Bi!ed costs Hase 190 <11,200 <I10,000 Hest 209 <10,080 <=M9,000 Worst 1J1 <12,=20 <IA1,000 Using the ta! shield approach, the /CB and 49V for the base case esti"ate is+ /CBbase L Q(<18,000 E 11,200#(190# E <I10,000R(0(MA# O 0(=A(<1,J00,000:I# /CBbase L <J22,0A0 49Vbase L E<1,J00,000 O <J22,0A0(9V.B012N,I# 49Vbase L <I9=,118(10 -he /CB and 49V for the orst case esti"ate are+ /CBorst L Q(<18,000 E 12,=20#(1J1# E <IA1,000R(0(MA# O 0(=A(<1,J00,000:I# /CBorst L <I8M,9=2 49Vorst L E<1,J00,000 O <I8M,9=2(9V.B012N,I# 49Vorst L E<221,01J(I1 0nd the /CB and 49V for the best case esti"ate are+ /CBbest L Q(<18,000 E 10,080#(209# E <=M9,000R(0(MA# O 0(=A(<1,J00,000:I# /CBbest L <98I,8=2 49Vbest L E<1,J00,000 O <98I,8=2(9V.B012N,I# 49Vbest L <1,291,2J8(8= b( -o calculate the sensiti*it% of the 49V to changes in fi!ed costs e choose another le*el of fi!ed costs( We ill use fi!ed costs of <I20,000( -he /CB using this le*el of fi!ed costs and the other base case *alues ith the ta! shield approach, e get+ /CB L Q(<18,000 E 11,200#(190# E <I10,000R(0(MA# O 0(=A(<1,J00,000:I# /CB L <J1A,AA0 0nd the 49V is+ 49V L E<1,J00,000 O <J1A,AA0(9V.B012N,I# 49V L <IJ=,=JA(=2 -he sensiti*it% of 49V to changes in fi!ed costs is+ ∆49V:∆BC L (<I9=,118(10 E IJ=,=JA(=2#:(<I10,000 E I20,000# ∆49V:∆BC L E<1(9JI CHAPTER 11 B-221 Bor e*er% dollar BC increase, 49V falls b% <1(9JI( B-222 SOLUTIONS c( -he cash brea&e*en is+ 7C L BC:(9 E *# 7 C L <I10,000:(<18,000 E 11,200# 7C L M0 d( -he accounting brea&e*en is+ 70 L (BC O )#:(9 E *# 7 0 L Q<I10,000 O (<1,J00,000:I#R:(<18,000 E 11,200# 70 L 12= 0t the accounting brea&e*en, the )/C is+ )/C L 1 O BC:/CB )/C L 1 O (<I10,000:<I2A,000# L 1(9MIJ Bor each 1N increase in unit sales, /CB ill increase b% 1(9MIJN( 2%. -he "ar&eting stud% and the research and de*elop"ent are both sun& costs and should be ignored( We ill calculate the sales and *ariable costs first( ,ince e ill lose sales of the e!pensi*e clubs and gain sales of the cheap clubs, these "ust be accounted for as erosion( -he total sales for the ne pro1ect ill be+ ,ales 4e clubs <JA0 × A1,000 L <=8,2A0,000 6!p( clubs <1,200 × (E11,000# L E1=,200,000 Cheap clubs <I20 × 9,A00 L =,990,000 <29,0I0,000 Bor the *ariable costs, e "ust include the units gained or lost fro" the e!isting clubs( 4ote that the *ariable costs of the e!pensi*e clubs are an inflo( .f e are not producing the sets an%"ore, e ill sa*e these *ariable costs, hich is an inflo( ,o+ Var( costs 4e clubs E<==0 × A1,000 L E<1M,8=0,000 6!p( clubs E<MA0 × (E11,000# L J,1A0,000 Cheap clubs E<190 × 9,A00 L E1,80A,000 E<11,I8A,000 -he pro for"a inco"e state"ent ill be+ ,ales <29,0I0,000 Variable costs 11,I8A,000 Costs 8,100,000 )epreciation =,200,000 6H- <M,2AA,000 -a!es 2,A02,000 4et inco"e <=,JA=,000 CHAPTER 11 B-223 Using the botto" up /CB calculation, e get+ /CB L 4. O )epreciation L <=,JA=,000 O =,200,000 /CB L <M,9A=,000 ,o, the pa%bac& period is+ 9a%bac& period L = O <2,J91,000:<M,9A=,000 9a%bac& period L =(I01 %ears -he 49V is+ 49V L E<22,I00,000 E 1,2A0,000 O <M,9A=,000(9V.B010N,J# O <1,2A0,000:1(10 J 49V L <10,8I1,AM=(M9 0nd the .RR is+ .RR L E<22,I00,000 E 1,2A0,000 O <M,9A=,000(9V.B0.RRN,J# O <1,2A0,000:.RR J .RR L 22(MIN 21. -he best case and orst cases for the *ariables are+ Hase Case Hest Case Worst Case Unit sales (ne# A1,000 AM,100 IA,900 9rice (ne# <JA0 <82A <MJA VC (ne# <==0 <29J <=M= Bi!ed costs <8,100,000 <J,290,000 <8,910,000 ,ales lost (e!pensi*e# 11,000 9,900 12,100 ,ales gained (cheap# 9,A00 10,IA0 8,AA0 Hest-case We ill calculate the sales and *ariable costs first( ,ince e ill lose sales of the e!pensi*e clubs and gain sales of the cheap clubs, these "ust be accounted for as erosion( -he total sales for the ne pro1ect ill be+ ,ales 4e clubs <JA0 × AM,100 L <IM,282,A00 6!p( clubs <1,200 × (E9,900# L E 11,880,000 Cheap clubs <I20 × 10,IA0 L I,=89,000 <=8,J91,A00 Bor the *ariable costs, e "ust include the units gained or lost fro" the e!isting clubs( 4ote that the *ariable costs of the e!pensi*e clubs are an inflo( .f e are not producing the sets an%"ore, e ill sa*e these *ariable costs, hich is an inflo( ,o+ Var( costs 4e clubs E<29J × AM,100 L E<1M,MM1,J00 6!p( clubs E<MA0 × (E9,900# L M,I=A,000 Cheap clubs E<190 × 10,IA0 L E 1,98A,A00 E<12,212,200 B-224 SOLUTIONS -he pro for"a inco"e state"ent ill be+ ,ales <=8,J91,A00 Variable costs 12,212,200 Costs J,290,000 )epreciation =,200,000 6H- 1M,089,=00 -a!es M,I=A,J20 4et inco"e <9,MA=,A80 Using the botto" up /CB calculation, e get+ /CB L 4et inco"e O )epreciation L <9,MA=,A80 O =,200,000 /CB L <12,8A=,A80 0nd the best-case 49V is+ 49V L E<22,I00,000 E 1,2A0,000 O <12,8A=,A80(9V.B010N,J# O 1,2A0,000:1(10 J 49V L <=9,AM8,0A8(=9 Worst-case We ill calculate the sales and *ariable costs first( ,ince e ill lose sales of the e!pensi*e clubs and gain sales of the cheap clubs, these "ust be accounted for as erosion( -he total sales for the ne pro1ect ill be+ ,ales 4e clubs <MJA × IA,900 L <=0,982,A00 6!p( clubs <1,200 × (E 12,100# L E 1I,A20,000 Cheap clubs <I20 × 8,AA0 L =,A91,000 <20,0A=,A00 Bor the *ariable costs, e "ust include the units gained or lost fro" the e!isting clubs( 4ote that the *ariable costs of the e!pensi*e clubs are an inflo( .f e are not producing the sets an%"ore, e ill sa*e these *ariable costs, hich is an inflo( ,o+ Var( costs 4e clubs E<=M= × IA,900 L E<1M,MM1,J00 6!p( clubs E<MA0 × (E 12,100# L J,8MA,000 Cheap clubs E<190 × 8,AA0 L E 1,M2I,A00 E<10,I21,200 -he pro for"a inco"e state"ent ill be+ ,ales <20,0A=,A00 Variable costs 10,I21,200 Costs 8,910,000 )epreciation =,200,000 6H- E 2,IJJ,J00 -a!es 991,080 `assu"es a ta! credit 4et inco"e E<1,I8M,M20 CHAPTER 11 B-225 Using the botto" up /CB calculation, e get+ /CB L 4. O )epreciation L E<1,I8M,M20 O =,200,000 /CB L <1,J1=,=80 0nd the orst-case 49V is+ 49V L E<22,I00,000 E 1,2A0,000 O <1,J1=,=80(9V.B010N,J# O 1,2A0,000:1(10 J 49V L E<1I,MMJ,100(92 22. -o calculate the sensiti*it% of the 49V to changes in the price of the ne club, e si"pl% need to change the price of the ne club( We ill choose <800, but the choice is irrele*ant as the sensiti*it% ill be the sa"e no "atter hat price e choose( We ill calculate the sales and *ariable costs first( ,ince e ill lose sales of the e!pensi*e clubs and gain sales of the cheap clubs, these "ust be accounted for as erosion( -he total sales for the ne pro1ect ill be+ ,ales 4e clubs <800 × A1,000 L <I0,800,000 6!p( clubs <1,200 × (E11,000# L E1=,200,000 Cheap clubs <I20 × 9,A00 L =,990,000 <=1,A90,000 Bor the *ariable costs, e "ust include the units gained or lost fro" the e!isting clubs( 4ote that the *ariable costs of the e!pensi*e clubs are an inflo( .f e are not producing the sets an%"ore, e ill sa*e these *ariable costs, hich is an inflo( ,o+ Var( costs 4e clubs E<==0 × A1,000 L E<1M,8=0,000 6!p( clubs E<MA0 × (E11,000# L J,1A0,000 Cheap clubs E<190 × 9,A00 L E1,80A,000 E<11,I8A,000 -he pro for"a inco"e state"ent ill be+ ,ales <=1,A90,000 Variable costs 11,I8A,000 Costs 8,100,000 )epreciation =,200,000 6H- 8,80A,000 -a!es =,A22,000 4et inco"e < A,28=,000 Using the botto" up /CB calculation, e get+ /CB L 4. O )epreciation L <A,28=,000 O =,200,000 /CB L <8,I8=,000 B-226 SOLUTIONS 0nd the 49V is+ 49V L E<22,I00,000 E 1,2A0,000 O <8,I8=,000(9V.B010N,J# O 1,2A0,000:1(10 J 49V L <18,290,2II(I8 ,o, the sensiti*it% of the 49V to changes in the price of the ne club is+ ∆49V:∆9 L (<10,8I1,AM=(M9 E 18,290,2II(I8#:(<JA0 E 800# ∆49V:∆9 L <1I8,9J=(M2 Bor e*er% dollar increase (decrease# in the price of the clubs, the 49V increases (decreases# b% <1I8,9J=(M2( -o calculate the sensiti*it% of the 49V to changes in the $uantit% sold of the ne club, e si"pl% need to change the $uantit% sold( We ill choose A2,000 units, but the choice is irrele*ant as the sensiti*it% ill be the sa"e no "atter hat $uantit% e choose( We ill calculate the sales and *ariable costs first( ,ince e ill lose sales of the e!pensi*e clubs and gain sales of the cheap clubs, these "ust be accounted for as erosion( -he total sales for the ne pro1ect ill be+ ,ales 4e clubs <JA0 × A2,000 L <=9,000,000 6!p( clubs <1,200 × (E11,000# L E1=,200,000 Cheap clubs <I20 × 9,A00 L =,990,000 <29,J90,000 Bor the *ariable costs, e "ust include the units gained or lost fro" the e!isting clubs( 4ote that the *ariable costs of the e!pensi*e clubs are an inflo( .f e are not producing the sets an%"ore, e ill sa*e these *ariable costs, hich is an inflo( ,o+ Var( costs 4e clubs E<==0 × A2,000 L E<1J,1M0,000 6!p( clubs E<MA0 × (E11,000# L J,1A0,000 Cheap clubs E<190 × 9,A00 L E1,80A,000 E<11,81A,000 -he pro for"a inco"e state"ent ill be+ ,ales <29,J90,000 Variable costs 11,81A,000 Costs 8,100,000 )epreciation =,200,000 6H- M,MJA,000 -a!es 2,MJ0,000 4et inco"e < I,00A,000 CHAPTER 11 B-227 Using the botto" up /CB calculation, e get+ /CB L 4. O )epreciation L <I,00A,000 O =,200,000 /CB L <J,20A,000 -he 49V at this $uantit% is+ 49V L E<22,I00,000 E <1,2A0,000 O <J,20A,000(9V.B010N,J# O <1,2A0,000:1(10 J 49V L <12,0M8,I0A(2= ,o, the sensiti*it% of the 49V to changes in the $uantit% sold is+ ∆49V:∆7 L (<10,8I1,AM=(M9 E 12,0M8,I0A(2=#:(A1,000 E A2,000# ∆49V:∆7 L <1,22M(8I Bor an increase (decrease# of one set of clubs sold per %ear, the 49V increases (decreases# b% <1,22M(8I( 23. a. Birst e need to deter"ine the total additional cost of the h%brid( -he h%brid costs "ore to purchase and "ore each %ear, so the total additional cost is+ -otal additional cost L <A,IA0 O M(<I00# -otal additional cost L <J,8A0 4e!t, e need to deter"ine the cost per "ile for each *ehicle( -he cost per "ile is the cost per gallon of gasoline di*ided b% the "iles per gallon, or+ Cost per "ile for traditional L <=(M0:2= Cost per "ile for traditional L <0(1AMA22 Cost per "ile for h%brid L <=(M0:2A Cost per "ile for h%brid L <0(1II000 ,o, the sa*ings per "ile dri*en for the h%brid ill be+ ,a*ings per "ile L <0(1AMA22 E 0(1II000 ,a*ings per "ile L <0(012A22 We can no deter"ine the brea&e*en point b% di*iding the total additional cost b% the sa*ings per "ile, hich is+ -otal brea&e*en "iles L <J,8A0 : <0(012A22 -otal brea&e*en "iles L M2M,910 ,o, the "iles %ou ould need to dri*e per %ear is the total brea&e*en "iles di*ided b% the nu"ber of %ears of onership, or+ 3iles per %ear L M2M,910 "iles : M %ears 3iles per %ear L 10I,I8A "iles:%ear B-228 SOLUTIONS b. Birst, e need to deter"ine the total "iles dri*en o*er the life of either *ehicle, hich ill be+ -otal "iles dri*en L M(1A,000# -otal "iles dri*en L 90,000 ,ince e &no the total additional cost of the h%brid fro" part a, e can deter"ine the necessar% sa*ings per "ile to "a&e the h%brid financiall% attracti*e( -he necessar% cost sa*ings per "ile ill be+ Cost sa*ings needed per "ile L <J,8A0 : 90,000 Cost sa*ings needed per "ile L <0(08J22 4o e can find the price per gallon for the "iles dri*en( .f e let 9 be the price per gallon, the necessar% price per gallon ill be+ 9:2= E 9:2A L <0(08J22 9(1:2= E 1:2A# L <0(08J22 9 L <2A(08 c. -o find the nu"ber of "iles it is necessar% to dri*e, e need the present *alue of the costs and sa*ings to be e$ual to 2ero( .f e let 3)95 e$ual the "iles dri*en per %ear, the brea&e*en e$uation for the h%brid car as+ Cost L 0 L E<A,IA0 E <I00(9V.B010N,M# O <0(012A22(3)95#(9V.B010N,M# -he sa*ings per "ile dri*en, <0(012A22, is the sa"e as e calculated in part a( ,ol*ing this e$uation for the nu"ber of "iles dri*en per %ear, e find+ <0(012A22(3)95#(9V.B010N,M# L <J,192(10 3)95(9V.B010N,M# L AJI,=M9(II 3iles dri*en per %ear L 1=1,8J9 -o find the cost per gallon of gasoline necessar% to "a&e the h%brid brea& e*en in a financial sense, if e let C,9G e$ual the cost sa*ings per gallon of gas, the cost e$uation is+ Cost L 0 L E<A,IA0 E <I00(9V.B010N,M# O C,9G(1A,000#(9V.B010N,M# ,ol*ing this e$uation for the cost sa*ings per gallon of gas necessar% for the h%brid to brea&e*en fro" a financial sense, e find+ C,9G(1A,000#(9V.B010N,M# L <J,192(10 C,9G(9V.B010N,M# L <0(IJ9IJ Cost sa*ings per gallon of gas L <0(110091 4o e can find the price per gallon for the "iles dri*en( .f e let 9 be the price per gallon, the necessar% price per gallon ill be+ 9:2= E 9:2A L <0(110091 9(1:2= E 1:2A# L <0(110091 9 L <=1(MA CHAPTER 11 B-229 d. -he i"plicit assu"ption in the pre*ious anal%sis is that each car depreciates b% the sa"e dollar a"ount( 24. a. -he cash flo per plane is the initial cost di*ided b% the brea&e*en nu"ber of planes, or+ Cash flo per plane L <1=,000,000,000 : 2I9 Cash flo per plane L <A2,208,8=A b. .n this case the cash flos are a perpetuit%( ,ince e &no the cash flo per plane, e need to deter"ine the annual cash flo necessar% to deli*er a 20 percent return( Using the perpetuit% e$uation, e find+ 9V L C :R <1=,000,000,000 L C : (20 C L <2,M00,000,000 -his is the total cash flo, so the nu"ber of planes that "ust be sold is the total cash flo di*ided b% the cash flo per plane, or+ 4u"ber of planes L <2,M00,000,000 : <A2,208,8=A 4u"ber of planes L I9(80 or about A0 planes per %ear c. .n this case the cash flos are an annuit%( ,ince e &no the cash flo per plane, e need to deter"ine the annual cash flo necessar% to deli*er a 20 percent return( Using the present *alue of an annuit% e$uation, e find+ 9V L C(9V.B020N,10# <1=,000,000,000 L C(9V.B020N,10# C L <=,100,J9A,8=9 -his is the total cash flo, so the nu"ber of planes that "ust be sold is the total cash flo di*ided b% the cash flo per plane, or+ 4u"ber of planes L <=,100,J9A,8=9 : <A2,208,8=A 4u"ber of planes L A9(=9 or about M0 planes per %ear Challenge 2. a. -he ta! shield definition of /CB is+ /CB L Q(9 E *#7 E BC R(1 E tC# O tC) Rearranging and sol*ing for 7, e find+ (/CB E tC)#:(1 E tC# L (9 E *#7 E BC 7 L YBC O Q(/CB E tC)#:(1 E tC#RZ:(9 E *# B-230 SOLUTIONS b. -he cash brea&e*en is+ 7C L <A00,000:(<I0,000 E 20,000# 7C L 2A 0nd the accounting brea&e*en is+ 70 L Y<A00,000 O Q(<J00,000 E <J00,000(0(=8##:0(M2RZ:(<I0,000 E 20,000# 70 L M0 -he financial brea&e*en is the point at hich the 49V is 2ero, so+ /CBB L <=,A00,000:9V.B020N,A /CBB L <1,1J0,=28(9M ,o+ 7B L QBC O (/CB E tC S )#:(1 E tC#R:(9 E *# 7B L Y<A00,000 O Q<1,1J0,=28(9M E (=8(<J00,000#R:(1 E (=8#Z:(<I0,000 E 20,000# 7B L 9J(9= ≈ 98 c. 0t the accounting brea&-e*en point, the net inco"e is 2ero( -his using the botto" up definition of /CB+ /CB L 4. O ) We can see that /CB "ust be e$ual to depreciation( ,o, the accounting brea&e*en is+ 70 L YBC O Q() E tC)#:(1 E t#RZ:(9 E *# 70 L (BC O )#:(9 E *# 70 L (BC O /CB#:(9 E *# -he ta! rate has cancelled out in this case( 2!. -he )/C is e!pressed as+ )/C L N∆/CB : N∆7 )/C L YQ(/CB1 E /CB0#:/CB0R : Q(71 E 70#:70RZ -he /CB for the initial period and the first period is+ /CB1 L Q(9 E *#71 E BCR(1 E tC# O tC) /CB0 L Q(9 E *#70 E BCR(1 E tC# O tC) -he difference beteen these to cash flos is+ /CB1 E /CB0 L (9 E *#(1 E tC#(71 E 70# CHAPTER 11 B-231 )i*iding both sides b% the initial /CB e get+ (/CB1 E /CB0#:/CB0 L (9 E *#( 1E tC#(71 E 70# : /CB0 Rearranging e get+ Q(/CB1 E /CB0#:/CB 0 RQ(71 E 70#:70R L Q(9 E *#(1 E tC#70R:/CB0 L Q/CB0 E tC) O BC(1 E t#R:/CB0 )/C L 1 O QBC(1 E t# E tC)R:/CB0 2". a( Using the ta! shield approach, the /CB is+ /CB L Q(<2=0 E 18A#(=A,000# E <IA0,000R(0(M2# O 0(=8(<=,200,000:A# /CB L <9I0,J00 0nd the 49V is+ 49V L E<=,200,000 E =M0,000 O <9I0,J00(9V.B01=N,A# O Q<=M0,000 O <A00,000(1 E (=8#R:1(1= A 49V L <112,=08(M0 b( .n the orst-case, the /CB is+ /CBorst L YQ(<2=0#(0(9# E 18AR(=A,000# E <IA0,000Z(0(M2# O 0(=8(<=,M80,000:A# /CBorst L <IJ8,080 0nd the orst-case 49V is+ 49Vorst L E<=,M80,000 E <=M0,000(1(0A# O <IJ8,080(9V.B01=N,A# O Q<=M0,000(1(0A# O <A00,000(0(8A#(1 E (=8#R:1(1= A 49Vorst L E<2,028,=01(A8 -he best-case /CB is+ /CBbest L YQ<2=0(1(1# E 18AR(=A,000# E <IA0,000Z(0(M2# O 0(=8(<2,J20,000:A# /CBbest L <1,I0=,=20 0nd the best-case 49V is+ 49Vbest L E <2,J20,000 E <=M0,000(0(9A# O <1,I0=,=20(9V.B01=N,A# O Q<=M0,000(0(9A# O <A00,000(1(1A#(1 E (=8#R:1(1= A 49Vbest L <2,2A2,918(J9 2#. -o calculate the sensiti*it% to changes in $uantit% sold, e ill choose a $uantit% of =M,000( -he /CB at this le*el of sale is+ /CB L Q(<2=0 E 18A#(=M,000# E <IA0,000R(0(M2# O 0(=8(<=,200,000:A# /CB L <9M8,M00 B-232 SOLUTIONS -he sensiti*it% of changes in the /CB to $uantit% sold is+ ∆/CB:∆7 L (<9M8,M00 E 9I0,J00#:(=M,000 E =A,000# ∆/CB:∆7 L O<2J(90 -he 49V at this le*el of sales is+ 49V L E<=,200,000 E <=M0,000 O <9M8,M00(9V.B01=N,A# O Q<=M0,000 O <A00,000(1 E (=8#R:1(1= A 49V L <210,I=9(=M 0nd the sensiti*it% of 49V to changes in the $uantit% sold is+ ∆49V:∆7 L (<210,I=9(=M E 112,=08(M0##:(=M,000 E =A,000# ∆49V:∆7 L O<98(1= 5ou ouldn't ant the $uantit% to fall belo the point here the 49V is 2ero( We &no the 49V changes <98(1= for e*er% unit sale, so e can di*ide the 49V for =A,000 units b% the sensiti*it% to get a change in $uantit%( )oing so, e get+ <112,=08(M0 L <98(1=(∆7# ∆7 L 1,1II Bor a 2ero 49V, e need to decrease sales b% 1,1II units, so the "ini"u" $uantit% is+ 73in L =A,000 E 1,1II 73in L ==,8AM 2$. 0t the cash brea&e*en, the /CB is 2ero( ,etting the ta! shield e$uation e$ual to 2ero and sol*ing for the $uantit%, e get+ /CB L 0 L Q(<2=0 E 18A#7C E <IA0,000R(0(M2# O 0(=8(<=,200,000:A# 7C L 1,28= -he accounting brea&e*en is+ 70 L Q<IA0,000 O (<=,200,000:A#R:(<2=0 E 18A# 70 L 2I,222 Bro" 9roble" 28, e &no the financial brea&e*en is ==,8AM units( CHAPTER 11 B-233 3%. Using the ta! shield approach to calculate the /CB, the )/C is+ )/C L 1 O Q<IA0,000(1 E 0(=8# E 0(=8(<=,200,000:A#R: <9I0,J00 )/C L 1(0=80M -hus a 1N rise leads to a 1(0=80MN rise in /CB( .f 7 rises to =M,000, then -he percentage change in $uantit% is+ ∆7 L (=M,000 E =A,000#:=A,000 L (028AJ or 2(8AJN ,o, the percentage change in /CB is+ N∆/CB L 2(8AJN(1(0=80M# N∆/CB L 2(9MA9N Bro" 9roble" 2M+ ∆/CB:/CB L (<9M8,M00 E 9I0,J00#:<9I0,J00 ∆/CB:/CB L 0(029MA9 .n general, if 7 rises b% 1,000 units, /CB rises b% 2(9MA9N( CHAPTER 12 S4M5 95SS4-S 2R4M CA&.TA9 MAR=5T <.ST4R0 Answers to Concepts Review and Critical Thinking Questions 1. -he% all ish the% hadX ,ince the% didn't, it "ust ha*e been the case that the stellar perfor"ance as not foreseeable, at least not b% "ost( 2. 0s in the pre*ious $uestion, it's eas% to see after the fact that the in*est"ent as terrible, but it probabl% asn't so eas% ahead of ti"e( 3. 4o, stoc&s are ris&ier( ,o"e in*estors are highl% ris& a*erse, and the e!tra possible return doesn't attract the" relati*e to the e!tra ris&( 4. /n a*erage, the onl% return that is earned is the re$uired returnDin*estors bu% assets ith returns in e!cess of the re$uired return (positi*e 49V#, bidding up the price and thus causing the return to fall to the re$uired return (2ero 49V#8 in*estors sell assets ith returns less than the re$uired return (negati*e 49V#, dri*ing the price loer and thus causing the return to rise to the re$uired return (2ero 49V#( . -he "ar&et is not ea& for" efficient( !. 5es, historical infor"ation is also public infor"ation8 ea& for" efficienc% is a subset of se"i-strong for" efficienc%( ". .gnoring trading costs, on a*erage, such in*estors "erel% earn hat the "ar&et offers8 stoc& in*est"ents all ha*e a 2ero 49V( .f trading costs e!ist, then these in*estors lose b% the a"ount of the costs( #. Unli&e ga"bling, the stoc& "ar&et is a positi*e su" ga"e8 e*er%bod% can in( 0lso, speculators pro*ide li$uidit% to "ar&ets and thus help to pro"ote efficienc%( $. -he 63> onl% sa%s, ithin the bounds of increasingl% strong assu"ptions about the infor"ation processing of in*estors, that assets are fairl% priced( 0n i"plication of this is that, on a*erage, the t%pical "ar&et participant cannot earn e!cessi*e profits fro" a particular trading strateg%( >oe*er, that does not "ean that a fe particular in*estors cannot outperfor" the "ar&et o*er a particular in*est"ent hori2on( Certain in*estors ho do ell for a period of ti"e get a lot of attention fro" the financial press, but the scores of in*estors ho do not do ell o*er the sa"e period of ti"e generall% get considerabl% less attention fro" the financial press( 1%. a. .f the "ar&et is not ea& for" efficient, then this infor"ation could be acted on and a profit earned fro" folloing the price trend( Under (2#, (=#, and (I#, this infor"ation is full% i"pounded in the current price and no abnor"al profit opportunit% e!ists( CHAPTER 12 B-235 b( Under (2#, if the "ar&et is not se"i-strong for" efficient, then this infor"ation could be used to bu% the stoc& ;cheap@ before the rest of the "ar&et disco*ers the financial state"ent ano"al%( ,ince (2# is stronger than (1#, both i"pl% that a profit opportunit% e!ists8 under (=# and (I#, this infor"ation is full% i"pounded in the current price and no profit opportunit% e!ists( c. Under (=#, if the "ar&et is not strong for" efficient, then this infor"ation could be used as a profitable trading strateg%, b% noting the bu%ing acti*it% of the insiders as a signal that the stoc& is underpriced or that good nes is i""inent( ,ince (1# and (2# are ea&er than (=#, all three i"pl% that a profit opportunit% e!ists( 4ote that this assu"es the indi*idual ho sees the insider trading is the onl% one ho sees the trading( .f the infor"ation about the trades "ade b% co"pan% "anage"ent is public infor"ation, it ill be discounted in the stoc& price and no profit opportunit% e!ists( Under (I#, this infor"ation does not signal an% profit opportunit% for traders8 an% pertinent infor"ation the "anager-insiders "a% ha*e is full% reflected in the current share price( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he return of an% asset is the increase in price, plus an% di*idends or cash flos, all di*ided b% the initial price( -he return of this stoc& is+ R L Q(<102 E 91# O 2(I0R : <91 L (1IJ= or 1I(J=N 2. -he di*idend %ield is the di*idend di*ided b% price at the beginning of the period price, so+ )i*idend %ield L <2(I0 : <91 L (02MI or 2(MIN 0nd the capital gains %ield is the increase in price di*ided b% the initial price, so+ Capital gains %ield L (<102 E 91# : <91 L (1209 or 12(09N 3. Using the e$uation for total return, e find+ R L Q(<8= E 91# O 2(I0R : <91 L E(0M1A or EM(1AN 0nd the di*idend %ield and capital gains %ield are+ )i*idend %ield L <2(I0 : <91 L (02MI or 2(MIN Capital gains %ield L (<8= E 91# : <91 L E(08J9 or E8(J9N >ere's a $uestion for %ou+ Can the di*idend %ield e*er be negati*e? 4o, that ould "ean %ou ere pa%ing the co"pan% for the pri*ilege of oning the stoc&( .t has happened on bonds( B-236 SOLUTIONS 4. -he total dollar return is the increase in price plus the coupon pa%"ent, so+ -otal dollar return L <1,0J0 E 1,0I0 O J0 L <100 -he total percentage return of the bond is+ R L Q(<1,0J0 E 1,0I0# O J0R : <1,0I0 L (09M2 or 9(M2N 4otice here that e could ha*e si"pl% used the total dollar return of <100 in the nu"erator of this e$uation( Using the Bisher e$uation, the real return as+ (1 O R# L (1 O r#(1 O h# r L (1(09M2 : 1(0I# E 1 L (0AI0 or A(I0N . -he no"inal return is the stated return, hich is 12(=0 percent( Using the Bisher e$uation, the real return as+ (1 O R# L (1 O r#(1 O h# r L (1(12=#:(1(0=1# E 1 L (0892 or 8(92N !. Using the Bisher e$uation, the real returns for long-ter" go*ern"ent and corporate bonds ere+ (1 O R# L (1 O r#(1 O h# rG L 1(0A8:1(0=1 E 1 L (02M2 or 2(M2N rC L 1(0M2:1(0=1 E 1 L (0=01 or =(01N ". -he a*erage return is the su" of the returns, di*ided b% the nu"ber of returns( -he a*erage return for each stoc& as+ [ ] N J(80 or (0J80 A 09 ( 1M ( 1J ( 21 ( 08 ( 1 · + − + + · 1 ] 1 ¸ · ∑ · N < B N i i CHAPTER 12 B-237 [ ] N M0 ( 1I or (1IM0 A 2M ( 21 ( 1I ( =8 ( 1M ( 1 · + − + + · 1 ] 1 ¸ · ∑ · N C N i i Re"e"bering bac& to ;sadistics,@ e calculate the *ariance of each stoc& as+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } 0I8M80 ( 1IM ( 2M ( 1IM ( 21 ( 1IM ( 1I ( 1IM ( =8 ( 1IM ( 1M ( 1 A 1 020MJ0 ( 0J8 ( 09 ( 0J8 ( 1M ( 0J8 ( 1J ( 0J8 ( 21 ( 0J8 ( 08 ( 1 A 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 · − + − − + − + − + − − · · − + − − + − + − + − − · − 1 ] 1 ¸ − · ∑ · C B N i i B N < < σ σ σ -he standard de*iation is the s$uare root of the *ariance, so the standard de*iation of each stoc& is+ σB L ((020MJ0# 1:2 L (1I=8 or 1I(=8N σC L ((0I8M80# 1:2 L (220M or 22(0MN #. We ill calculate the su" of the returns for each asset and the obser*ed ris& pre"iu" first( )oing so, e get+ 5ear Carge co( stoc& return --bill return Ris& pre"iu" 19J0 =(9IN M(A0N −2(AMN 19J1 1I(=0 I(=M 9(9I 19J2 18(99 I(2= 1I(JM 19J= E1I(M9 J(29 E21(98 19JI E2M(IJ J(99 E=I(IM 19JA =J(2= A(8J =1(=M ==(=0 =M(2I E2(9I a( -he a*erage return for large co"pan% stoc&s o*er this period as+ Carge co"pan% stoc&s a*erage return L ==(=0N : M L A(AAN 0nd the a*erage return for --bills o*er this period as+ - --bills a*erage return L =M(2IN : M L M(0IN U B-238 SOLUTIONS b( Using the e$uation for *ariance, e find the *ariance for large co"pan% stoc&s o*er this period as+ Variance L 1:AQ((0=9I E (0AAA# 2 O ((1I=0 E (0AAA# 2 O ((1899 E (0AAA# 2 O (E(1IM9 E (0AAA# 2 O (E(2MIJ E (0AAA# 2 O ((=J2= E (0AAA# 2 R Variance L 0(0A=9MJ 0nd the standard de*iation for large co"pan% stoc&s o*er this period as+ ,tandard de*iation L (0(0A=9MJ# 1:2 L 0(2=2= or 2=(2=N Using the e$uation for *ariance, e find the *ariance for --bills o*er this period as+ Variance L 1:AQ((0MA0 E (0M0I# 2 O ((0I=M E (0M0I# 2 O ((0I2= E (0M0I# 2 O ((0J29 E (0M0I# 2 O ((0J99 E (0M0I# 2 O ((0A8J E (0M0I# 2 R Variance L 0(0002=I 0nd the standard de*iation for --bills o*er this period as+ ,tandard de*iation L (0(0002=I# 1:2 L 0(01A= or 1(A=N c( -he a*erage obser*ed ris& pre"iu" o*er this period as+ 0*erage obser*ed ris& pre"iu" L E2(9IN : M L E0(I9N -he *ariance of the obser*ed ris& pre"iu" as+ Variance L 1:AQ(E(02AM E (E(00I9## 2 O ((099I E (E(00I9## 2 O ((1IJM E (E(00I9### 2 O (E(2198 E (E(00I9## 2 O (E(=IIM E (E(00I9## 2 O ((=1=M E (E(00I9## 2 R Variance L 0(0A9A1J 0nd the standard de*iation of the obser*ed ris& pre"iu" as+ ,tandard de*iation L (0(0A9A1J# 1:2 L 0(2II0 or 2I(I0N d. Hefore the fact, for "ost assets the ris& pre"iu" ill be positi*e8 in*estors de"and co"pensation o*er and abo*e the ris&-free return to in*est their "one% in the ris&% asset( 0fter the fact, the obser*ed ris& pre"iu" can be negati*e if the asset's no"inal return is une!pectedl% lo, the ris&- free return is une!pectedl% high, or if so"e co"bination of these to e*ents occurs( $. a( -o find the a*erage return, e su" all the returns and di*ide b% the nu"ber of returns, so+ 0*erage return L ((0J E(12 O(11 O(=8 O(1I#:A L (11M0 or 11(M0N CHAPTER 12 B-239 b( Using the e$uation to calculate *ariance, e find+ Variance L 1:IQ((0J E (11M# 2 O (E(12 E (11M# 2 O ((11 E (11M# 2 O ((=8 E (11M# 2 O ((1I E (11M# 2 R Variance L 0(0=20=0 ,o, the standard de*iation is+ ,tandard de*iation L (0(0=2=0# 1:2 L 0(1J90 or 1J(90N 1%. a( -o calculate the a*erage real return, e can use the a*erage return of the asset, and the a*erage inflation in the Bisher e$uation( )oing so, e find+ (1 O R# L (1 O r#(1 O h# r L (1(1M0:1(0=A# E 1 L (0J8= or J(8=N b( -he a*erage ris& pre"iu" is si"pl% the a*erage return of the asset, "inus the a*erage ris&-free rate, so, the a*erage ris& pre"iu" for this asset ould be+ R R9 · E f R L (11M0 E (0I2 L (0JI0 or J(I0N 11. We can find the a*erage real ris&-free rate using the Bisher e$uation( -he a*erage real ris&-free rate as+ (1 O R# L (1 O r#(1 O h# f r L (1(0I2:1(0=A# E 1 L (00M8 or 0(M8N 0nd to calculate the a*erage real ris& pre"iu", e can subtract the a*erage ris&-free rate fro" the a*erage real return( ,o, the a*erage real ris& pre"iu" as+ r rp · E f r L J(8=N E 0(M8N L J(1AN 12. --bill rates ere highest in the earl% eighties( -his as during a period of high inflation and is consistent ith the Bisher effect( B-240 SOLUTIONS &ntermediate 13. -o find the real return, e first need to find the no"inal return, hich "eans e need the current price of the bond( Going bac& to the chapter on pricing bonds, e find the current price is+ 91 L <80(9V.B0JN,M# O <1,000(9V.BJN,M# L <1,0IJ(MJ ,o the no"inal return is+ R L Q(<1,0IJ(MJ E 1,0=0# O 80R:<1,0=0 L (09I8 or 9(I8N 0nd, using the Bisher e$uation, e find the real return is+ 1 O R L (1 O r#(1 O h# r L (1(09I8:1(0I2# E 1 L (0A0J or A(0JN 14. >ere e &no the a*erage stoc& return, and four of the fi*e returns used to co"pute the a*erage return( We can or& the a*erage return e$uation bac&ard to find the "issing return( -he a*erage return is calculated as+ (A2A L (0J E (12 O (18 O (19 O R R L (20A or 20(AN -he "issing return has to be 20(A percent( 4o e can use the e$uation for the *ariance to find+ Variance L 1:IQ((0J E (10A# 2 O (E(12 E (10A# 2 O ((18 E (10A# 2 O ((19 E (10A# 2 O ((20A E (10A# 2 R Variance L 0(018MJA 0nd the standard de*iation is+ ,tandard de*iation L (0(018MJA# 1:2 L 0(1=MJ or 1=(MJN 1. -he arith"etic a*erage return is the su" of the &non returns di*ided b% the nu"ber of returns, so+ 0rith"etic a*erage return L ((0= O (=8 O (21 E (1A O (29 E (1=# : M 0rith"etic a*erage return L (10A0 or 10(A0N Using the e$uation for the geo"etric return, e find+ Geo"etric a*erage return L Q(1 O R1# S (1 O R2# S ] S (1 O RT#R 1:T E 1 Geo"etric a*erage return L Q(1 O (0=#(1 O (=8#(1 O (21#(1 E (1A#(1 O (29#(1 E (1=#R (1:M# E 1 Geo"etric a*erage return L (08M0 or 8(M0N Re"e"ber, the geo"etric a*erage return ill ala%s be less than the arith"etic a*erage return if the returns ha*e an% *ariation( CHAPTER 12 B-241 1!. -o calculate the arith"etic and geo"etric a*erage returns, e "ust first calculate the return for each %ear( -he return for each %ear is+ R1 L (<J=(MM E M0(18 O 0(M0# : <M0(18 L (2=I0 or 2=(I0N R2 L (<9I(18 E J=(MM O 0(MI# : <J=(MM L (28J= or 28(J=N R= L (<89(=A E 9I(18 O 0(J2# : <9I(18 L E(0I=M or EI(=MN RI L (<J8(I9 E 89(=A O 0(80#: <89(=A L E(112M or 11(2MN RA L (<9A(0A E J8(I9 O 1(20# : <J8(I9 L (22M= or 12(M=N -he arith"etic a*erage return as+ R0 L (0(2=I0 O 0(28J= E 0(0I=M E 0(112M O 0(22M=#:A L 0(118= or 11(8=N 0nd the geo"etric a*erage return as+ RG L Q(1 O (2=I0#(1 O (28J=#(1 E (0I=M#(1 E(112M#(1 O (22M=#R 1:A E 1 L 0(10A8 or 10(A8N 1". Coo&ing at the long-ter" corporate bond return histor% in Bigure 12(10, e see that the "ean return as M(2 percent, ith a standard de*iation of 8(I percent( .n the nor"al probabilit% distribution, appro!i"atel% 2:= of the obser*ations are ithin one standard de*iation of the "ean( -his "eans that 1:= of the obser*ations are outside one standard de*iation aa% fro" the "ean( /r+ 9r(Ra E2(2 or RT1I(M# ≈ 1 : = Hut e are onl% interested in one tail here, that is, returns less than E2(2 percent, so+ 9r(Ra E2(2# ≈ 1 : M 5ou can use the 2-statistic and the cu"ulati*e nor"al distribution table to find the anser as ell( )oing so, e find+ 2 L (U E c#:σ 2 L (E2(2N E M(2#:8(IN L E1(00 Coo&ing at the 2-table, this gi*es a probabilit% of 1A(8JN, or+ 9r(Ra E2(2# ≈ (1A8J or 1A(8JN -he range of returns %ou ould e!pect to see 9A percent of the ti"e is the "ean plus or "inus 2 standard de*iations, or+ 9AN le*el+ R∈ µ d[ 2σ L M(2N [ 2(8(IN# L E10(M0N to 2=(00N B-242 SOLUTIONS -he range of returns %ou ould e!pect to see 99 percent of the ti"e is the "ean plus or "inus = standard de*iations, or+ 99N le*el+ R∈ µ d[ =σ L M(2N [ =(8(IN# L E19(00N to =1(I0N 1#. -he "ean return for s"all co"pan% stoc&s as 1J(1 percent, ith a standard de*iation of =2(M percent( )oubling %our "one% is a 100N return, so if the return distribution is nor"al, e can use the 2-statistic( ,o+ 2 L (U E c#:σ 2 L (100N E 1J(1#:=2(MN L 2(AI= standard de*iations abo*e the "ean -his corresponds to a probabilit% of ≈ 0(AAN, or once e*er% 200 %ears( -ripling %our "one% ould be+ 2 L (200N E 1J(1#:=2(MN L A(M10 standard de*iations abo*e the "ean( -his corresponds to a probabilit% of about (000001N, or about once e*er% 1 "illion %ears( 1$. .t is i"possible to lose "ore than 100 percent of %our in*est"ent( -herefore, return distributions are truncated on the loer tail at E100 percent( 2%. -o find the best forecast, e appl% Hlu"e's for"ula as follos+ R(A# L =9 1 - A S 11(9N O =9 A - I0 S 1A(=N L 1I(9AN R(10# L =9 1 - 10 S 11(9N O =9 10 - I0 S 1A(=N L 1I(A2N R(20# L =9 1 - 20 S 11(9N O =9 20 - I0 S 1A(=N L 1=(MIN 21. -he best forecast for a one %ear return is the arith"etic a*erage, hich is 12(= percent( -he geo"etric a*erage, found in -able 12(I is 10(I percent( -o find the best forecast for other periods, e appl% Hlu"e's for"ula as follos+ R(A# L 1 - 82 1 - A S 10(IN O 1 - 82 A - 82 S 12(=N L 12(21N R(20) = 1 - 82 1 - 20 × 10.4% + 1 - 82 20 - 82 × 12.3% = 11.85% R(30) = 1 - 82 1 - =0 × 10.4% + 1 - 82 =0 - 82 × 12.3% = 11.62% CHAPTER 12 B-243 22. -o find the real return e need to use the Bisher e$uation( Re-riting the Bisher e$uation to sol*e for the real return, e get+ r L Q(1 O R#:(1 O h#R E 1 ,o, the real return each %ear as+ 5ear --bill return .nflation Real return 19J= 0(0J29 0(08J1 E0(01=1 19JI 0(0J99 0(12=I E0(0=8J 19JA 0(0A8J 0(0M9I E0(0100 19JM 0(0A0J 0(0I8M 0(0020 19JJ 0(0AIA 0(0MJ0 E0(011J 19J8 0(0JMI 0(0902 E0(012J 19J9 0(10AM 0(1=29 E0(02I1 1980 0(1210 0(12A2 E0(00=J 0(M19J 0(JI=8 E0(1120 a. -he a*erage return for --bills o*er this period as+ 0*erage return L 0(M19 : 8 0*erage return L (0JJA or J(JAN 0nd the a*erage inflation rate as+ V 0*erage inflation L 0(JI=8 : 8 W 0*erage inflation L (09=0 or 9(=0N U b. Using the e$uation for *ariance, e find the *ariance for --bills o*er this period as+ Variance L 1:JQ((0J29 E (0JJA# 2 O ((0J99 E (0JJA# 2 O ((0A8J E (0JJA# 2 O ((0A0J E (0JJA# 2 O ((0AIA E (0JJA# 2 O ((0JMI E (0JJA# 2 O ((10AM E (0JJA# 2 O ((1210 − (0JJA# 2 R Variance L 0(000M1M 0nd the standard de*iation for --bills as+ ,tandard de*iation L (0(000M1M# 1:2 ,tandard de*iation L 0(02I8 or 2(I8N -he *ariance of inflation o*er this period as+ Variance L 1:JQ((08J1 E (09=0# 2 O ((12=I E (09=0# 2 O ((0M9I E (09=0# 2 O ((0I8M E (09=0# 2 O ((0MJ0 E (09=0# 2 O ((0902 E (09=0# 2 O ((1=29 E (09=0# 2 O ((12A2 − (09=0# 2 R Variance L 0(0009J1 0nd the standard de*iation of inflation as+ ,tandard de*iation L (0(0009J1# 1:2 ,tandard de*iation L 0(0=12 or =(12N B-244 SOLUTIONS c( -he a*erage obser*ed real return o*er this period as+ 0*erage obser*ed real return L E(1122 : 8 0*erage obser*ed real return L E(01I0 or E1(I0N d( -he state"ent that --bills ha*e no ris& refers to the fact that there is onl% an e!tre"el% s"all chance of the go*ern"ent defaulting, so there is little default ris&( ,ince --bills are short ter", there is also *er% li"ited interest rate ris&( >oe*er, as this e!a"ple shos, there is inflation ris&, i(e( the purchasing poer of the in*est"ent can actuall% decline o*er ti"e e*en if the in*estor is earning a positi*e return( Challenge 23. Using the 2-statistic, e find+ 2 L (U E c#:σ 2 L (0N E 12(=#:20(0N L E0(M1A 9r(R≤0# ≈ 2M(9=N 24. Bor each of the $uestions as&ed here, e need to use the 2-statistic, hich is+ 2 L (U E c#:σ a( 2 1 L (10N E M(2#:8(IN L 0(IA2I -his 2-statistic gi*es us the probabilit% that the return is less than 10 percent, but e are loo&ing for the probabilit% the return is greater than 10 percent( Gi*en that the total probabilit% is 100 percent (or 1#, the probabilit% of a return greater than 10 percent is 1 "inus the probabilit% of a return less than 10 percent( Using the cu"ulati*e nor"al distribution table, e get+ 9r(R≥10N# L 1 E 9r(R≤10N# L 1 E (MJIA ≈ =2(AAN Bor a return greater than 0 percent+ 2 2 L (0N E M(2#:8(IN L E0(J=81 9r(R≥10N# L 1 E 9r(R≤10N# L 1 E (JM98 ≈ 2=(02N b. -he probabilit% that --bill returns ill be greater than 10 percent is+ 2 = L (10N E =(8#:=(1N L 2 9r(R≥10N# L 1 E 9r(R≤10N# L 1 E (9JJ2 ≈ 2(28N CHAPTER 12 B-245 0nd the probabilit% that --bill returns ill be less than 0 percent is+ 2 I L (0N E =(8#:=(1N L E1(22A8 9r(R≤0# ≈ 11(01N c( -he probabilit% that the return on long-ter" corporate bonds ill be less than EI(18 percent is+ 2 A L (EI(18N E M(2#:8(IN L E1(2=AJ 9r(R≤EI(18N# ≈ 10(8=N 0nd the probabilit% that --bill returns ill be greater than 10(AM percent is+ 2 M L (10(AMN E =(8#:=(1N L 2(180M 9r(R≥10(AMN# L 1 E 9r(R≤10(AMN# L 1 E (982= ≈ 1(IMN CHAPTER 13 RISK, RETURN, AND THE SECURITY MARKET LINE Answers to Concepts Review and Critical Thinking Questions 1. ,o"e of the ris& in holding an% asset is uni$ue to the asset in $uestion( H% in*esting in a *ariet% of assets, this uni$ue portion of the total ris& can be eli"inated at little cost( /n the other hand, there are so"e ris&s that affect all in*est"ents( -his portion of the total ris& of an asset cannot be costlessl% eli"inated( .n other ords, s%ste"atic ris& can be controlled, but onl% b% a costl% reduction in e!pected returns( 2. .f the "ar&et e!pected the groth rate in the co"ing %ear to be 2 percent, then there ould be no change in securit% prices if this e!pectation had been full% anticipated and priced( >oe*er, if the "ar&et had been e!pecting a groth rate other than 2 percent and the e!pectation as incorporated into securit% prices, then the go*ern"ent's announce"ent ould "ost li&el% cause securit% prices in general to change8 prices ould drop if the anticipated groth rate had been "ore than 2 percent, and prices ould rise if the anticipated groth rate had been less than 2 percent( 3. a. s%ste"atic b. uns%ste"atic c. both8 probabl% "ostl% s%ste"atic d. uns%ste"atic e. uns%ste"atic f. s%ste"atic 4. a. a change in s%ste"atic ris& has occurred8 "ar&et prices in general ill "ost li&el% decline( b. no change in uns%ste"atic ris&8 co"pan% price ill "ost li&el% sta% constant( c. no change in s%ste"atic ris&8 "ar&et prices in general ill "ost li&el% sta% constant( d. a change in uns%ste"atic ris& has occurred8 co"pan% price ill "ost li&el% decline( e. no change in s%ste"atic ris&8 "ar&et prices in general ill "ost li&el% sta% constant( . 4o to both $uestions( -he portfolio e!pected return is a eighted a*erage of the asset returns, so it "ust be less than the largest asset return and greater than the s"allest asset return( !. Balse( -he *ariance of the indi*idual assets is a "easure of the total ris&( -he *ariance on a ell- di*ersified portfolio is a function of s%ste"atic ris& onl%( ". 5es, the standard de*iation can be less than that of e*er% asset in the portfolio( >oe*er, βp cannot be less than the s"allest beta because βp is a eighted a*erage of the indi*idual asset betas( #. 5es( .t is possible, in theor%, to construct a 2ero beta portfolio of ris&% assets hose return ould be e$ual to the ris&-free rate( .t is also possible to ha*e a negati*e beta8 the return ould be less than the CHAPTER 13 B-247 ris&-free rate( 0 negati*e beta asset ould carr% a negati*e ris& pre"iu" because of its *alue as a di*ersification instru"ent( $. ,uch la%offs generall% occur in the conte!t of corporate restructurings( -o the e!tent that the "ar&et *ies a restructuring as *alue-creating, stoc& prices ill rise( ,o, it's not la%offs per se that are being cheered on( 4onetheless, Wall ,treet does encourage corporations to ta&es actions to create *alue, e*en if such actions in*ol*e la%offs( 1%. 6arnings contain infor"ation about recent sales and costs( -his infor"ation is useful for pro1ecting future groth rates and cash flos( -hus, une!pectedl% lo earnings often lead "ar&et participants to reduce esti"ates of future groth rates and cash flos8 price drops are the result( -he re*erse is often true for une!pectedl% high earnings( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he portfolio eight of an asset is total in*est"ent in that asset di*ided b% the total portfolio *alue( Birst, e ill find the portfolio *alue, hich is+ -otal *alue L 180(<IA# O 1I0(<2J# L <11,880 -he portfolio eight for each stoc& is+ Weight0 L 180(<IA#:<11,880 L (M818 WeightH L 1I0(<2J#:<11,880 L (=182 2. -he e!pected return of a portfolio is the su" of the eight of each asset ti"es the e!pected return of each asset( -he total *alue of the portfolio is+ -otal *alue L <2,9A0 O =,J00 L <M,MA0 ,o, the e!pected return of this portfolio is+ 6(Rp# L (<2,9A0:<M,MA0#(0(11# O (<=,J00:<M,MA0#(0(1A# L (1=2= or 1=(2=N 3. -he e!pected return of a portfolio is the su" of the eight of each asset ti"es the e!pected return of each asset( ,o, the e!pected return of the portfolio is+ 6(Rp# L (M0((09# O (2A((1J# O (1A((1=# L (11M0 or 11(M0N B-248 SOLUTIONS 4. >ere e are gi*en the e!pected return of the portfolio and the e!pected return of each asset in the portfolio, and are as&ed to find the eight of each asset( We can use the e$uation for the e!pected return of a portfolio to sol*e this proble"( ,ince the total eight of a portfolio "ust e$ual 1 (100N#, the eight of ,toc& 5 "ust be one "inus the eight of ,toc& U( 3athe"aticall% spea&ing, this "eans+ 6(Rp# L (12I L (1IU O (10A(1 E U# We can no sol*e this e$uation for the eight of ,toc& U as+ (12I L (1IU O (10A E (10AU (019 L (0=AU U L 0(AI28AJ ,o, the dollar a"ount in*ested in ,toc& U is the eight of ,toc& U ti"es the total portfolio *alue, or+ .n*est"ent in U L 0(AI28AJ(<10,000# L <A,I28(AJ 0nd the dollar a"ount in*ested in ,toc& 5 is+ .n*est"ent in 5 L (1 E 0(AI28AJ#(<10,000# L <I,AJI(I= . -he e!pected return of an asset is the su" of the probabilit% of each return occurring ti"es the probabilit% of that return occurring( ,o, the e!pected return of the asset is+ 6(R# L (2A(E(08# O (JA((21# L (1=JA or 1=(JAN !. -he e!pected return of an asset is the su" of the probabilit% of each return occurring ti"es the probabilit% of that return occurring( ,o, the e!pected return of the asset is+ 6(R# L (20(E(0A# O (A0((12# O (=0((2A# L (12A0 or 12(A0N ". -he e!pected return of an asset is the su" of the probabilit% of each return occurring ti"es the probabilit% of that return occurring( ,o, the e!pected return of each stoc& asset is+ 6(R0# L (1A((0A# O (MA((08# O (20((1=# L (08AA or 8(AAN 6(RH# L (1A(E(1J# O (MA((12# O (20((29# L (110A or 11(0AN -o calculate the standard de*iation, e first need to calculate the *ariance( -o find the *ariance, e find the s$uared de*iations fro" the e!pected return( We then "ultipl% each possible s$uared de*iation b% its probabilit%, then add all of these up( -he result is the *ariance( ,o, the *ariance and standard de*iation of each stoc& is+ σ0 2 L(1A((0A E (08AA# 2 O (MA((08 E (08AA# 2 O (20((1= E (08AA# 2 L (000M0 σ0 L ((000M0# 1:2 L (02IM or 2(IMN CHAPTER 13 B-249 σH 2 L(1A(E(1J E (110A# 2 O (MA((12 E (110A# 2 O (20((29 E (110A# 2 L (018=0 σH L ((018=0# 1:2 L (1=A= or 1=(A=N #. -he e!pected return of a portfolio is the su" of the eight of each asset ti"es the e!pected return of each asset( ,o, the e!pected return of the portfolio is+ 6(Rp# L (2A((08# O (AA((1A# O (20((2I# L (1A0A or 1A(0AN .f e on this portfolio, e ould e!pect to get a return of 1A(0A percent( $. a. -o find the e!pected return of the portfolio, e need to find the return of the portfolio in each state of the econo"%( -his portfolio is a special case since all three assets ha*e the sa"e eight( -o find the e!pected return in an e$uall% eighted portfolio, e can su" the returns of each asset and di*ide b% the nu"ber of assets, so the e!pected return of the portfolio in each state of the econo"% is+ Hoo"+ 6(Rp# L ((0J O (1A O (==#:= L (18== or 18(==N Hust+ 6(Rp# L ((1= O (0= −(0M#:= L (0=== or =(==N -o find the e!pected return of the portfolio, e "ultipl% the return in each state of the econo"% b% the probabilit% of that state occurring, and then su"( )oing this, e find+ 6(Rp# L (=A((18==# O (MA((0===# L (08A8 or 8(A8N b. -his portfolio does not ha*e an e$ual eight in each asset( We still need to find the return of the portfolio in each state of the econo"%( -o do this, e ill "ultipl% the return of each asset b% its portfolio eight and then su" the products to get the portfolio return in each state of the econo"%( )oing so, e get+ Hoo"+ 6(Rp# L (20((0J# O(20((1A# O (M0((==# L(2I20 or 2I(20N Hust+ 6(Rp# L (20((1=# O(20((0=# O (M0(−(0M# L E(00I0 or E0(I0N 0nd the e!pected return of the portfolio is+ 6(Rp# L (=A((2I20# O (MA(−(00I# L (0821 or 8(21N -o find the *ariance, e find the s$uared de*iations fro" the e!pected return( We then "ultipl% each possible s$uared de*iation b% its probabilit%, than add all of these up( -he result is the *ariance( ,o, the *ariance and standard de*iation of the portfolio is+ σp 2 L (=A((2I20 E (0821# 2 O (MA(−(00I0 E (0821# 2 L (01=JMJ B-250 SOLUTIONS 1%. a. -his portfolio does not ha*e an e$ual eight in each asset( We first need to find the return of the portfolio in each state of the econo"%( -o do this, e ill "ultipl% the return of each asset b% its portfolio eight and then su" the products to get the portfolio return in each state of the econo"%( )oing so, e get+ Hoo"+ 6(R p # L (=0((=# O (I0((IA# O (=0((==# L (=M90 or =M(90N Good+ 6(R p # L (=0((12# O (I0((10# O (=0((1A# L (1210 or 12(10N 9oor+ 6(R p # L (=0((01# O (I0(E(1A# O (=0(E(0A# L E(0J20 or EJ(20N Hust+ 6(R p # L (=0(E(0M# O (I0(E(=0# O (=0(E(09# L E(1MA0 or E1M(A0N 0nd the e!pected return of the portfolio is+ 6(R p # L (1A((=M90# O (IA((1210# O (=A(E(0J20# O (0A(E(1MA0# L (0JMI or J(MIN b. -o calculate the standard de*iation, e first need to calculate the *ariance( -o find the *ariance, e find the s$uared de*iations fro" the e!pected return( We then "ultipl% each possible s$uared de*iation b% its probabilit%, than add all of these up( -he result is the *ariance( ,o, the *ariance and standard de*iation of the portfolio is+ σp 2 L (1A((=M90 E (0JMI# 2 O (IA((1210 E (0JMI# 2 O (=A(E(0J20 E (0JMI# 2 O (0A(E(1MA0 E (0JMI# 2 σp 2 L (02I=M σp L ((02I=M# 1:2 L (1AM1 or 1A(M1N 11. -he beta of a portfolio is the su" of the eight of each asset ti"es the beta of each asset( ,o, the beta of the portfolio is+ βp L (2A((8I# O (20(1(1J# O (1A(1(11# O (I0(1(=M# L 1(1A 12. -he beta of a portfolio is the su" of the eight of each asset ti"es the beta of each asset( .f the portfolio is as ris&% as the "ar&et it "ust ha*e the sa"e beta as the "ar&et( ,ince the beta of the "ar&et is one, e &no the beta of our portfolio is one( We also need to re"e"ber that the beta of the ris&-free asset is 2ero( .t has to be 2ero since the asset has no ris&( ,etting up the e$uation for the beta of our portfolio, e get+ βp L 1(0 L 1 : = (0# O 1 : = (1(=8# O 1 : = (βU# ,ol*ing for the beta of ,toc& U, e get+ βU L 1(M2 CHAPTER 13 B-251 13. C093 states the relationship beteen the ris& of an asset and its e!pected return( C093 is+ 6(Ri# L Rf O Q6(R3# E RfR S βi ,ubstituting the *alues e are gi*en, e find+ 6(Ri# L (0A2 O ((11 E (0A2#(1(0A# L (1129 or 11(29N 14. We are gi*en the *alues for the C093 e!cept for the β of the stoc&( We need to substitute these *alues into the C093, and sol*e for the β of the stoc&( /ne i"portant thing e need to reali2e is that e are gi*en the "ar&et ris& pre"iu"( -he "ar&et ris& pre"iu" is the e!pected return of the "ar&et "inus the ris&-free rate( We "ust be careful not to use this *alue as the e!pected return of the "ar&et( Using the C093, e find+ 6(Ri# L (102 L (0IAO (08Aβi βi L 0(MJ 1. >ere e need to find the e!pected return of the "ar&et using the C093( ,ubstituting the *alues gi*en, and sol*ing for the e!pected return of the "ar&et, e find+ 6(Ri# L (1=A L (0AA O Q6(R3# E (0AAR(1(1J# 6(R3# L (12=I or 12(=IN 1!. >ere e need to find the ris&-free rate using the C093( ,ubstituting the *alues gi*en, and sol*ing for the ris&-free rate, e find+ 6(Ri# L (1I L Rf O ((11A E Rf#(1(IA# (1I L Rf O (1MMJA E 1(IARf Rf L (0A9I or A(9IN 1". a. 0gain e ha*e a special case here the portfolio is e$uall% eighted, so e can su" the returns of each asset and di*ide b% the nu"ber of assets( -he e!pected return of the portfolio is+ 6(Rp# L ((1M O (0I8#:2 L (10I0 or 10(I0N B-252 SOLUTIONS b. We need to find the portfolio eights that result in a portfolio ith a β of 0(9A( We &no the β of the ris&-free asset is 2ero( We also &no the eight of the ris&-free asset is one "inus the eight of the stoc& since the portfolio eights "ust su" to one, or 100 percent( ,o+ βp L 0(9A L ,(1(=A# O (1 E ,#(0# 0(9A L 1(=A, O 0 E 0, , L 0(9A:1(=A , L (J0=J 0nd, the eight of the ris&-free asset is+ Rf L 1 E (J0=J L (29M= c. We need to find the portfolio eights that result in a portfolio ith an e!pected return of 8 percent( We also &no the eight of the ris&-free asset is one "inus the eight of the stoc& since the portfolio eights "ust su" to one, or 100 percent( ,o+ 6(Rp# L (08 L (1M, O (0I8(1 E ,# (08 L (1M, O (0I8 E (0I8, (0=2 L (112, , L (28AJ ,o, the β of the portfolio ill be+ βp L (28AJ(1(=A# O (1 E (28AJ#(0# L 0(=8M d. ,ol*ing for the β of the portfolio as e did in part a, e find+ βp L 2(J0 L ,(1(=A# O (1 E ,#(0# , L 2(J0:1(=A L 2 Rf L 1 E 2 L E1 -he portfolio is in*ested 200N in the stoc& and E100N in the ris&-free asset( -his represents borroing at the ris&-free rate to bu% "ore of the stoc&( 1#. Birst, e need to find the β of the portfolio( -he β of the ris&-free asset is 2ero, and the eight of the ris&-free asset is one "inus the eight of the stoc&, the β of the portfolio is+ ep L W(1(2A# O (1 E W#(0# L 1(2AW ,o, to find the β of the portfolio for an% eight of the stoc&, e si"pl% "ultipl% the eight of the stoc& ti"es its β( CHAPTER 13 B-253 6*en though e are sol*ing for the β and e!pected return of a portfolio of one stoc& and the ris&-free asset for different portfolio eights, e are reall% sol*ing for the ,3C( 0n% co"bination of this stoc&, and the ris&-free asset ill fall on the ,3C( Bor that "atter, a portfolio of an% stoc& and the ris&-free asset, or an% portfolio of stoc&s, ill fall on the ,3C( We &no the slope of the ,3C line is the "ar&et ris& pre"iu", so using the C093 and the infor"ation concerning this stoc&, the "ar&et ris& pre"iu" is+ 6(RW# L (1A2 L (0A= O 3R9(1(2A# 3R9 L (099:1(2A L (0J92 or J(92N ,o, no e &no the C093 e$uation for an% stoc& is+ 6(Rp# L (0A= O (0J9=βp -he slope of the ,3C is e$ual to the "ar&et ris& pre"iu", hich is 0(0J92( Using these e$uations to fill in the table, e get the folloing results+ W 6(Rp# ep 0(00N A(=0N 0(000 2A(00N J(J8N 0(=1= A0(00N 10(2AN 0(M2A JA(00N 12(J=N 0(9=8 100(00N 1A(20N 1(2A0 12A(00N 1J(M8N 1(AM= 1A0(00N 20(1AN 1(8JA 1$. -here are to a%s to correctl% anser this $uestion( We ill or& through both( Birst, e can use the C093( ,ubstituting in the *alue e are gi*en for each stoc&, e find+ 6(R5# L (08 O (0JA(1(=0# L (1JJA or 1J(JAN .t is gi*en in the proble" that the e!pected return of ,toc& 5 is 18(A percent, but according to the C093, the return of the stoc& based on its le*el of ris&, the e!pected return should be 1J(JA percent( -his "eans the stoc& return is too high, gi*en its le*el of ris&( ,toc& 5 plots abo*e the ,3C and is under*alued( .n other ords, its price "ust increase to reduce the e!pected return to 1J(JA percent( Bor ,toc& ^, e find+ 6(R^# L (08 O (0JA(0(J0# L (1=2A or 1=(2AN -he return gi*en for ,toc& ^ is 12(1 percent, but according to the C093 the e!pected return of the stoc& should be 1=(2A percent based on its le*el of ris&( ,toc& ^ plots belo the ,3C and is o*er*alued( .n other ords, its price "ust decrease to increase the e!pected return to 1=(2A percent( B-254 SOLUTIONS We can also anser this $uestion using the reard-to-ris& ratio( 0ll assets "ust ha*e the sa"e reard- to-ris& ratio( -he reard-to-ris& ratio is the ris& pre"iu" of the asset di*ided b% its β( We are gi*en the "ar&et ris& pre"iu", and e &no the β of the "ar&et is one, so the reard-to-ris& ratio for the "ar&et is 0(0JA, or J(A percent( Calculating the reard-to-ris& ratio for ,toc& 5, e find+ Reard-to-ris& ratio 5 L ((18A E (08# : 1(=0 L (0808 -he reard-to-ris& ratio for ,toc& 5 is too high, hich "eans the stoc& plots abo*e the ,3C, and the stoc& is under*alued( .ts price "ust increase until its reard-to-ris& ratio is e$ual to the "ar&et reard- to-ris& ratio( Bor ,toc& ^, e find+ Reard-to-ris& ratio ^ L ((121 E (08# : (J0 L (0A8M -he reard-to-ris& ratio for ,toc& ^ is too lo, hich "eans the stoc& plots belo the ,3C, and the stoc& is o*er*alued( .ts price "ust decrease until its reard-to-ris& ratio is e$ual to the "ar&et reard- to-ris& ratio( 2%. We need to set the reard-to-ris& ratios of the to assets e$ual to each other, hich is+ ((18A E Rf#:1(=0 L ((121 E Rf#:0(J0 We can cross "ultipl% to get+ 0(J0((18A E Rf# L 1(=0((121 E Rf# ,ol*ing for the ris&-free rate, e find+ 0(129A E 0(J0Rf L 0(1AJ= E 1(=0Rf Rf L (0IM= or I(M=N &ntermediate 21. Bor a portfolio that is e$uall% in*ested in large-co"pan% stoc&s and long-ter" bonds+ Return L (12(=0N O A(80N#:2 L 9(0AN Bor a portfolio that is e$uall% in*ested in s"all stoc&s and -reasur% bills+ Return L (1J(10N O =(80N#:2 L 10(IAN CHAPTER 13 B-255 22. We &no that the reard-to-ris& ratios for all assets "ust be e$ual( -his can be e!pressed as+ Q6(R0# E RfR:β0 L Q6(RH# E RfR:eH -he nu"erator of each e$uation is the ris& pre"iu" of the asset, so+ R90:β0 L R9H:βH We can rearrange this e$uation to get+ βH:β0 L R9H:R90 .f the reard-to-ris& ratios are the sa"e, the ratio of the betas of the assets is e$ual to the ratio of the ris& pre"iu"s of the assets( 23. a. We need to find the return of the portfolio in each state of the econo"%( -o do this, e ill "ultipl% the return of each asset b% its portfolio eight and then su" the products to get the portfolio return in each state of the econo"%( )oing so, e get+ Hoo"+ 6(Rp# L (I((2I# O (I((=M# O (2((AA# L (=A00 or =A(00N 4or"al+ 6(Rp# L (I((1J# O (I((1=# O (2((09# L (1=80 or 1=(80N Hust+ 6(Rp# L (I((00# O (I(E(28# O (2(E(IA# L E(2020 or E20(20N 0nd the e!pected return of the portfolio is+ 6(Rp# L (=A((=A# O (A0((1=8# O (1A(E(202# L (1M12 or 1M(12N -o calculate the standard de*iation, e first need to calculate the *ariance( -o find the *ariance, e find the s$uared de*iations fro" the e!pected return( We then "ultipl% each possible s$uared de*iation b% its probabilit%, than add all of these up( -he result is the *ariance( ,o, the *ariance and standard de*iation of the portfolio is+ σ 2 p L (=A((=A E (1M12# 2 O (A0((1=8 E (1M12# 2 O (1A(E(202 E (1M12# 2 σ 2 p L (0=2A= σp L ((0=2A=# 1:2 L (180I or 18(0IN b. -he ris& pre"iu" is the return of a ris&% asset, "inus the ris&-free rate( --bills are often used as the ris&-free rate, so+ R9i L 6(Rp# E Rf L (1M12 E (0=80 L (12=2 or 12(=2N B-256 SOLUTIONS c. -he appro!i"ate e!pected real return is the e!pected no"inal return "inus the inflation rate, so+ 0ppro!i"ate e!pected real return L (1M12 E (0=A L (12M2 or 12(M2N -o find the e!act real return, e ill use the Bisher e$uation( )oing so, e get+ 1 O 6(Ri# L (1 O h#Q1 O e(ri#R 1(1M12 L (1(0=A0#Q1 O e(ri#R e(ri# L (1(1M12:1(0=A# E 1 L (1219 or 12(19N -he appro!i"ate real ris& pre"iu" is the e!pected return "inus the ris&-free rate, so+ 0ppro!i"ate e!pected real ris& pre"iu" L (1M12 E (0=8 L (12=2 or 12(=2N -he e!act e!pected real ris& pre"iu" is the appro!i"ate e!pected real ris& pre"iu", di*ided b% one plus the inflation rate, so+ 6!act e!pected real ris& pre"iu" L (11M8:1(0=A L (1190 or 11(90N 24. ,ince the portfolio is as ris&% as the "ar&et, the β of the portfolio "ust be e$ual to one( We also &no the β of the ris&-free asset is 2ero( We can use the e$uation for the β of a portfolio to find the eight of the third stoc&( )oing so, e find+ βp L 1(0 L 0((8A# O H(1(20# O C(1(=A# O Rf(0# ,ol*ing for the eight of ,toc& C, e find+ C L (=2I0JI ,o, the dollar in*est"ent in ,toc& C "ust be+ .n*est in ,toc& C L (=2I0JI(<1,000,000# L <=2I,0JI(0J We &no the total portfolio *alue and the in*est"ent of to stoc&s in the portfolio, so e can find the eight of these to stoc&s( -he eights of ,toc& 0 and ,toc& H are+ 0 L <210,000 : <1,000,000 L (210 H L <=20,000:<1,000,000 L (=20 CHAPTER 13 B-257 We also &no the total portfolio eight "ust be one, so the eight of the ris&-free asset "ust be one "inus the asset eight e &no, or+ 1 L 0 O H O C O Rf L 1 E (210 E (=20 E (=2I0JI E Rf Rf L (1IA92M ,o, the dollar in*est"ent in the ris&-free asset "ust be+ .n*est in ris&-free asset L (1IA92M(<1,000,000# L <1IA,92A(9= Challenge 2. We are gi*en the e!pected return of the assets in the portfolio( We also &no the su" of the eights of each asset "ust be e$ual to one( Using this relationship, e can e!press the e!pected return of the portfolio as+ 6(Rp# L (18A L U((1J2# O 5((1=M# (18A L U((1J2# O (1 E U#((1=M# (18A L (1J2U O (1=M E (1=MU (0I9 L (0=MU U L 1(=M111 0nd the eight of ,toc& 5 is+ 5 L 1 E 1(=M111 5 L E(=M111 -he a"ount to in*est in ,toc& 5 is+ .n*est"ent in ,toc& 5 L E(=M111(<100,000# .n*est"ent in ,toc& 5 L E<=M,111(11 0 negati*e portfolio eight "eans that %ou short sell the stoc&( .f %ou are not fa"iliar ith short selling, it "eans %ou borro a stoc& toda% and sell it( 5ou "ust then purchase the stoc& at a later date to repa% the borroed stoc&( .f %ou short sell a stoc&, %ou "a&e a profit if the stoc& decreases in *alue( -o find the beta of the portfolio, e can "ultipl% the portfolio eight of each asset ti"es its beta and su"( ,o, the beta of the portfolio is+ β9 L 1(=M111(1(I0# O (E(=M111#(0(9A# β9 L 1(AM ( B-258 SOLUTIONS 2!. -he a"ount of s%ste"atic ris& is "easured b% the β of an asset( ,ince e &no the "ar&et ris& pre"iu" and the ris&-free rate, if e &no the e!pected return of the asset e can use the C093 to sol*e for the β of the asset( -he e!pected return of ,toc& . is+ 6(R.# L (2A((11# O (A0((29# O (2A((1=# L (20A0 or 20(A0N Using the C093 to find the β of ,toc& ., e find+ (20A0 L (0I O (08β. β. L 2(0M -he total ris& of the asset is "easured b% its standard de*iation, so e need to calculate the standard de*iation of ,toc& .( Heginning ith the calculation of the stoc&'s *ariance, e find+ σ. 2 L (2A((11 E (20A0# 2 O (A0((29 E (20A0# 2 O (2A((1= E (20A0# 2 σ. 2 L (00J28 σ. L ((00J28# 1:2 L (08A= or 8(A=N Using the sa"e procedure for ,toc& .., e find the e!pected return to be+ 6(R..# L (2A(E(I0# O (A0((10# O (2A((AM# L (0900 Using the C093 to find the β of ,toc& .., e find+ (0900 L (0I O (08β.. β.. L 0(M= 0nd the standard de*iation of ,toc& .. is+ σ.. 2 L (2A(E(I0 E (0900# 2 O (A0((10 E (0900# 2 O (2A((AM E (0900# 2 σ.. 2 L (11A=0 σ.. L ((11A=0# 1:2 L (==9M or ==(9MN 0lthough ,toc& .. has "ore total ris& than ., it has "uch less s%ste"atic ris&, since its beta is "uch s"aller than .'s( -hus, . has "ore s%ste"atic ris&, and .. has "ore uns%ste"atic and "ore total ris&( ,ince uns%ste"atic ris& can be di*ersified aa%, . is actuall% the ;ris&ier@ stoc& despite the lac& of *olatilit% in its returns( ,toc& . ill ha*e a higher ris& pre"iu" and a greater e!pected return( CHAPTER 13 B-259 2". >ere e ha*e the e!pected return and beta for to assets( We can e!press the returns of the to assets using C093( .f the C093 is true, then the securit% "ar&et line holds as ell, hich "eans all assets ha*e the sa"e ris& pre"iu"( ,etting the ris& pre"iu"s of the assets e$ual to each other and sol*ing for the ris&-free rate, e find+ ((1=2 E Rf#:1(=A L ((101 E Rf#:(80 (80((1=2 E Rf# L 1(=A((101 E Rf# (10AM E (8Rf L (1=M=A E 1(=ARf (AARf L (0=0JA Rf L (0AA9 or A(A9N 4o using C093 to find the e!pected return on the "ar&et ith both stoc&s, e find+ (1=2 L (0AA9 O 1(=A(R3 E (0AA9# (101 L (0AA9 O (80(R3 E (0AA9# R3 L (112= or 11(2=N R3 L (112= or 11(2=N 2#. a. -he e!pected return of an asset is the su" of the probabilit% of each return occurring ti"es the probabilit% of that return occurring( ,o, the e!pected return of each stoc& is+ 6(R0# L (1A(E(08# O (J0((1=# O (1A((I8# L (1A10 or 1A(10N 6(RH# L (1A(E(0A# O (J0((1I# O (1A((29# L (1=I0 or 1=(I0N b. We can use the e!pected returns e calculated to find the slope of the ,ecurit% 3ar&et Cine( We &no that the beta of ,toc& 0 is (2A greater than the beta of ,toc& H( -herefore, as beta increases b% (2A, the e!pected return on a securit% increases b% (01J (L (1A10 E (1=I0#( -he slope of the securit% "ar&et line (,3C# e$uals+ ,lope,3C L Rise : Run ,lope,3C L .ncrease in e!pected return : .ncrease in beta ,lope,3C L ((1A10 E (1=I0# : (2A ,lope,3C L (0M80 or M(80N B-260 SOLUTIONS ,ince the "ar&et's beta is 1 and the ris&-free rate has a beta of 2ero, the slope of the ,ecurit% 3ar&et Cine e$uals the e!pected "ar&et ris& pre"iu"( ,o, the e!pected "ar&et ris& pre"iu" "ust be M(8 percent( We could also sol*e this proble" using C093( -he e$uations for the e!pected returns of the to stoc&s are+ 6(R0# L (1A1 L Rf O (βH O (2A#(3R9# 6(RH# L (1=I L Rf O βH(3R9# We can rerite the C093 e$uation for ,toc& 0 as+ (1A1 L Rf O βH(3R9# O (2A(3R9# ,ubtracting the C093 e$uation for ,toc& H fro" this e$uation %ields+ (01J L (2A3R9 3R9 L (0M8 or M(8N hich is the sa"e anser as our pre*ious result( CHAPTER 14 COST OF CAPITAL Answers to Concepts Review and Critical Thinking Questions 1. .t is the "ini"u" rate of return the fir" "ust earn o*erall on its e!isting assets( .f it earns "ore than this, *alue is created( 2. Hoo& *alues for debt are li&el% to be "uch closer to "ar&et *alues than are e$uit% boo& *alues( 3. 4o( -he cost of capital depends on the ris& of the pro1ect, not the source of the "one%( 4. .nterest e!pense is ta!-deductible( -here is no difference beteen preta! and afterta! e$uit% costs( . -he pri"ar% ad*antage of the )CB "odel is its si"plicit%( -he "ethod is disad*antaged in that (1# the "odel is applicable onl% to fir"s that actuall% pa% di*idends8 "an% do not8 (2# e*en if a fir" does pa% di*idends, the )CB "odel re$uires a constant di*idend groth rate fore*er8 (=# the esti"ated cost of e$uit% fro" this "ethod is *er% sensiti*e to changes in g, hich is a *er% uncertain para"eter8 and (I# the "odel does not e!plicitl% consider ris&, although ris& is i"plicitl% considered to the e!tent that the "ar&et has i"pounded the rele*ant ris& of the stoc& into its "ar&et price( While the share price and "ost recent di*idend can be obser*ed in the "ar&et, the di*idend groth rate "ust be esti"ated( -o co""on "ethods of esti"ating g are to use anal%sts' earnings and pa%out forecasts or to deter"ine so"e appropriate a*erage historical g fro" the fir"'s a*ailable data( !. -o pri"ar% ad*antages of the ,3C approach are that the "odel e!plicitl% incorporates the rele*ant ris& of the stoc& and the "ethod is "ore idel% applicable than is the di*idend discount "odel "odel, since the ,3C doesn't "a&e an% assu"ptions about the fir"'s di*idends( -he pri"ar% disad*antages of the ,3C "ethod are (1# three para"eters (the ris&-free rate, the e!pected return on the "ar&et, and beta# "ust be esti"ated, and (2# the "ethod essentiall% uses historical infor"ation to esti"ate these para"eters( -he ris&-free rate is usuall% esti"ated to be the %ield on *er% short "aturit% --bills and is, hence, obser*able8 the "ar&et ris& pre"iu" is usuall% esti"ated fro" historical ris& pre"iu"s and, hence, is not obser*able( -he stoc& beta, hich is unobser*able, is usuall% esti"ated either b% deter"ining so"e a*erage historical beta fro" the fir" and the "ar&et's return data, or b% using beta esti"ates pro*ided b% anal%sts and in*est"ent fir"s( ". -he appropriate afterta! cost of debt to the co"pan% is the interest rate it ould ha*e to pa% if it ere to issue ne debt toda%( >ence, if the 5-3 on outstanding bonds of the co"pan% is obser*ed, the co"pan% has an accurate esti"ate of its cost of debt( .f the debt is pri*atel%-placed, the fir" could still esti"ate its cost of debt b% (1# loo&ing at the cost of debt for si"ilar fir"s in si"ilar ris& classes, (2# loo&ing at the a*erage debt cost for fir"s ith the sa"e credit rating (assu"ing the fir"'s pri*ate debt is rated#, or (=# consulting anal%sts and in*est"ent ban&ers( 6*en if the debt is publicl% traded, an additional co"plication is hen the fir" has "ore than one issue outstanding8 these issues rarel% ha*e the sa"e %ield because no to issues are e*er co"pletel% ho"ogeneous( B-262 SOLUTIONS #. a. -his onl% considers the di*idend %ield co"ponent of the re$uired return on e$uit%( b. -his is the current %ield onl%, not the pro"ised %ield to "aturit%( .n addition, it is based on the boo& *alue of the liabilit%, and it ignores ta!es( c. 6$uit% is inherentl% "ore ris&% than debt (e!cept, perhaps, in the unusual case here a fir"'s assets ha*e a negati*e beta#( Bor this reason, the cost of e$uit% e!ceeds the cost of debt( .f ta!es are considered in this case, it can be seen that at reasonable ta! rates, the cost of e$uit% does e!ceed the cost of debt( $. R,up L (12 O (JA((08# L (1800 or 18(00N Hoth should proceed( -he appropriate discount rate does not depend on hich co"pan% is in*esting8 it depends on the ris& of the pro1ect( ,ince ,uperior is in the business, it is closer to a pure pla%( -herefore, its cost of capital should be used( With an 18N cost of capital, the pro1ect has an 49V of <1 "illion regardless of ho ta&es it( 1%. .f the different operating di*isions ere in "uch different ris& classes, then separate cost of capital figures should be used for the different di*isions8 the use of a single, o*erall cost of capital ould be inappropriate( .f the single hurdle rate ere used, ris&ier di*isions ould tend to recei*e "ore funds for in*est"ent pro1ects, since their return ould e!ceed the hurdle rate despite the fact that the% "a% actuall% plot belo the ,3C and, hence, be unprofitable pro1ects on a ris&-ad1usted basis( -he t%pical proble" encountered in esti"ating the cost of capital for a di*ision is that it rarel% has its on securities traded on the "ar&et, so it is difficult to obser*e the "ar&et's *aluation of the ris& of the di*ision( -o t%pical a%s around this are to use a pure pla% pro!% for the di*ision, or to use sub1ecti*e ad1ust"ents of the o*erall fir" hurdle rate based on the percei*ed ris& of the di*ision( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. With the infor"ation gi*en, e can find the cost of e$uit% using the di*idend groth "odel( Using this "odel, the cost of e$uit% is+ R6 L Q<2(I0(1(0AA#:<A2R O (0AA L (10=J or 10(=JN 2. >ere e ha*e infor"ation to calculate the cost of e$uit% using the C093( -he cost of e$uit% is+ R6 L (0A= O 1(0A((12 E (0A=# L (12=I or 12(=IN 3. We ha*e the infor"ation a*ailable to calculate the cost of e$uit% using the C093 and the di*idend groth "odel( Using the C093, e find+ R6 L (0A O 0(8A((08# L (1180 or 11(80N CHAPTER 14 B-263 0nd using the di*idend groth "odel, the cost of e$uit% is R6 L Q<1(M0(1(0M#:<=JR O (0M L (10A8 or 10(A8N Hoth esti"ates of the cost of e$uit% see" reasonable( .f e re"e"ber the historical return on large capitali2ation stoc&s, the esti"ate fro" the C093 "odel is about to percent higher than a*erage, and the esti"ate fro" the di*idend groth "odel is about one percent higher than the historical a*erage, so e cannot definiti*el% sa% one of the esti"ates is incorrect( Gi*en this, e ill use the a*erage of the to, so+ R6 L ((1180 O (10A8#:2 L (1119 or 11(19N 4. -o use the di*idend groth "odel, e first need to find the groth rate in di*idends( ,o, the increase in di*idends each %ear as+ g1 L (<1(12 E 1(0A#:<1(0A L (0MMJ or M(MJN g2 L (<1(19 E 1(12#:<1(12 L (0M2A or M(2AN g= L (<1(=0 E 1(19#:<1(19 L (092I or 9(2IN gI L (<1(I= E 1(=0#:<1(=0 L (1000 or 10(00N ,o, the a*erage arith"etic groth rate in di*idends as+ g L ((0MMJ O (0M2A O (092I O (1000#:I L (080I or 8(0IN Using this groth rate in the di*idend groth "odel, e find the cost of e$uit% is+ R6 L Q<1(I=(1(080I#:<IA(00R O (080I L (11IJ or 11(IJN Calculating the geo"etric groth rate in di*idends, e find+ <1(I= L <1(0A(1 O g# I g L (080= or 8(0=N -he cost of e$uit% using the geo"etric di*idend groth rate is+ R6 L Q<1(I=(1(080=#:<IA(00R O (080= L (11IM or 11(IMN . -he cost of preferred stoc& is the di*idend pa%"ent di*ided b% the price, so+ R9 L <M:<9M L (0M2A or M(2AN !. -he preta! cost of debt is the 5-3 of the co"pan%'s bonds, so+ 90 L <1,0J0 L <=A(9V.B0RN,=0# O <1,000(9V.BRN,=0# R L =(1=JN 5-3 L 2 S =(1=JN L M(2JN 0nd the afterta! cost of debt is+ R) L (0M2J(1 E (=A# L (0I08 or I(08N B-264 SOLUTIONS ". a. -he preta! cost of debt is the 5-3 of the co"pan%'s bonds, so+ 90 L <9A0 L <I0(9V.B0RN,IM# O <1,000(9V.BRN,IM# R L I(2I9N 5-3 L 2 S I(2I9N L 8(A0N b. -he afterta! cost of debt is+ R) L (08A0(1 E (=A# L (0AA2 or A(A2N c. -he after-ta! rate is "ore rele*ant because that is the actual cost to the co"pan%( #. -he boo& *alue of debt is the total par *alue of all outstanding debt, so+ HV) L <80,000,000 O =A,000,000 L <11A,000,000 -o find the "ar&et *alue of debt, e find the price of the bonds and "ultipl% b% the nu"ber of bonds( 0lternati*el%, e can "ultipl% the price $uote of the bond ti"es the par *alue of the bonds( )oing so, e find+ 3V) L (9A(<80,000,000# O (M1(<=A,000,000# 3V) L <JM,000,000 O 21,=A0,000 3V) L <9J,=A0,000 -he 5-3 of the 2ero coupon bonds is+ 9^ L <M10 L <1,000(9V.BRN,1I# R L =(A9IN 5-3 L 2 S =(A9IN L J(19N ,o, the afterta! cost of the 2ero coupon bonds is+ R^ L (0J19(1 E (=A# L (0IMJ or I(MJN -he afterta! cost of debt for the co"pan% is the eighted a*erage of the afterta! cost of debt for all outstanding bond issues( We need to use the "ar&et *alue eights of the bonds( -he total afterta! cost of debt for the co"pan% is+ R) L (0AA2(<JM:<9J(=A# O (0IMJ(<21(=A:<9J(=A# L (0A=I or A(=IN $. a. Using the e$uation to calculate the W0CC, e find+ W0CC L (M0((1I# O (0A((0M# O (=A((08#(1 E (=A# L (10A2 or 10(A2N b. ,ince interest is ta! deductible and di*idends are not, e "ust loo& at the after-ta! cost of debt, hich is+ (08(1 E (=A# L (0A20 or A(20N >ence, on an after-ta! basis, debt is cheaper than the preferred stoc&( CHAPTER 14 B-265 1%. >ere e need to use the debt-e$uit% ratio to calculate the W0CC( )oing so, e find+ W0CC L (1A(1:1(MA# O (09((MA:1(MA#(1 E (=A# L (11I0 or 11(I0N 11. >ere e ha*e the W0CC and need to find the debt-e$uit% ratio of the co"pan%( ,etting up the W0CC e$uation, e find+ W0CC L (0890 L (12(6:V# O (0J9():V#(1 E (=A# Rearranging the e$uation, e find+ (0890(V:6# L (12 O (0J9((MA#():6# 4o e "ust reali2e that the V:6 is 1ust the e$uit% "ultiplier, hich is e$ual to+ V:6 L 1 O ):6 (0890():6 O 1# L (12 O (0A1=A():6# 4o e can sol*e for ):6 as+ (0MJMA():6# L (0=1 ):6 L (82=I 12. a. -he boo& *alue of e$uit% is the boo& *alue per share ti"es the nu"ber of shares, and the boo& *alue of debt is the face *alue of the co"pan%'s debt, so+ HV6 L 11,000,000(<M# L <MM,000,000 HV) L <J0,000,000 O AA,000,000 L <12A,000,000 ,o, the total *alue of the co"pan% is+ V L <MM,000,000 O 12A,000,000 L <191,000,000 0nd the boo& *alue eights of e$uit% and debt are+ 6:V L <MM,000,000:<191,000,000 L (=IAA ):V L 1 E 6:V L (MAIA b. -he "ar&et *alue of e$uit% is the share price ti"es the nu"ber of shares, so+ 3V6 L 11,000,000(<M8# L <JI8,000,000 Using the relationship that the total "ar&et *alue of debt is the price $uote ti"es the par *alue of the bond, e find the "ar&et *alue of debt is+ 3V) L (9=(<J0,000,000# O 1(0I(<AA,000,000# L <122,=00,000 B-266 SOLUTIONS -his "a&es the total "ar&et *alue of the co"pan%+ V L <JI8,000,000 O 122,=00,000 L <8J0,=00,000 0nd the "ar&et *alue eights of e$uit% and debt are+ 6:V L <JI8,000,000:<8J0,=00,000 L (8A9A ):V L 1 E 6:V L (1I0A c. -he "ar&et *alue eights are "ore rele*ant( 13. Birst, e ill find the cost of e$uit% for the co"pan%( -he infor"ation pro*ided allos us to sol*e for the cost of e$uit% using the di*idend groth "odel, so+ R6 L Q<I(10(1(0M#:<M8R O (0M L (12=9 or 12(=9N 4e!t, e need to find the 5-3 on both bond issues( )oing so, e find+ 91 L <9=0 L <=A(9V.B0RN,I2# O <1,000(9V.BRN,I2# R L =(8=8N 5-3 L =(8=8N S 2 L J(M8N 92 L <1,0I0 L <I0(9V.B0RN,12# O <1,000(9V.BRN,12# R L =(A8IN 5-3 L =(A8IN S 2 L J(1JN -o find the eighted a*erage afterta! cost of debt, e need the eight of each bond as a percentage of the total debt( We find+ )1 L (9=(<J0,000,000#:<122,=00,000 L (A=2= )2 L 1(0I(<AA,000,000#:<122,=00,000 L (IMJJ 4o e can "ultipl% the eighted a*erage cost of debt ti"es one "inus the ta! rate to find the eighted a*erage afterta! cost of debt( -his gi*es us+ R) L (1 E (=A#Q((A=2=#((0JM8# O ((IMJJ#((0J1J#R L (0I8I or I(8IN Using these costs e ha*e found and the eight of debt e calculated earlier, the W0CC is+ W0CC L (8A9A((12=9# O (1I0A((0I8I# L (11== or 11(==N 14. a. Using the e$uation to calculate W0CC, e find+ W0CC L (09I L (1:2(0A#((1I# O (1(0A:2(0A#(1 E (=A#R) R) L (0JJ2 or J(J2N CHAPTER 14 B-267 b. Using the e$uation to calculate W0CC, e find+ W0CC L (09I L (1:2(0A#R6 O (1(0A:2(0A#((0M8# R6 L (121= or 12(1=N 15. We will begin by finding the market value of each type of financing. We find: MVD = 8,000($1,000)(0.92) = $7,360,000 MV E = 250,000($57) = $14,250,000 3V9 L 1A,000(<9=# L <1,=9A,000 0nd the total "ar&et *alue of the fir" is+ V L <J,=M0,000 O 1I,2A0,000 O 1,=9A,000 L <2=,00A,000 4o, e can find the cost of e$uit% using the C093( -he cost of e$uit% is+ R6 L (0IA O 1(0A((08# L (1290 or 12(90N -he cost of debt is the 5-3 of the bonds, so+ 90 L <920 L <=2(A0(9V.B0RN,I0# O <1,000(9V.BRN,I0# R L =(M=2N 5-3 L =(M=2N S 2 L J(2MN 0nd the afterta! cost of debt is+ R) L (1 E (=A#((0J2M# L (0IJ2 or I(J2N -he cost of preferred stoc& is+ R9 L <A:<9= L (0A=8 or A(=8N 4o e ha*e all of the co"ponents to calculate the W0CC( -he W0CC is+ W0CC L (0IJ2(J(=M:2=(00A# O (1290(1I(2A:2=(00A# O (0A=8(1(=9A:2=(00A# L (098= or 9(8=N 4otice that e didn't include the (1 E tC# ter" in the W0CC e$uation( We used the afterta! cost of debt in the e$uation, so the ter" is not needed here( 1!. a. We will begin by finding the market value of each type of financing. We find: 3V) L 10A,000(<1,000#(0(9=# L <9J,MA0,000 3V6 L 9,000,000(<=I# L <=0M,000,000 3V9 L 2A0,000(<91# L <22,JA0,000 0nd the total "ar&et *alue of the fir" is+ V L <9J,MA0,000 O =0M,000,000 O 22,JA0,000 L <I2M,I00,000 B-268 SOLUTIONS ,o, the "ar&et *alue eights of the co"pan%'s financing is+ ):V L <9J,MA0,000:<I2M,I00,000 L (2290 9:V L <22,JA0,000:<I2M,I00,000 L (0A=I 6:V L <=0M,000,000:<I2M,I00,000 L (J1JM b. Bor pro1ects e$uall% as ris&% as the fir" itself, the W0CC should be used as the discount rate( Birst e can find the cost of e$uit% using the C093( -he cost of e$uit% is+ R6 L (0A O 1(2A((08A# L (1AM= or 1A(M=N -he cost of debt is the 5-3 of the bonds, so+ 90 L <9=0 L <=J(A(9V.B0RN,=0# O <1,000(9V.BRN,=0# R L I(1M=N 5-3 L I(1M=N S 2 L 8(==N 0nd the afterta! cost of debt is+ R) L (1 E (=A#((08==# L (0AI1 or A(I1N -he cost of preferred stoc& is+ R9 L <M:<91 L (0MA9 or M(A9N 4o e can calculate the W0CC as+ W0CC L (0AI1((2290# O (1AM=((J1JM# O (0MA9((0A=I# L (1280 or 12(80N 1". a. 9ro1ects U, 5 and ^( b. Using the C093 to consider the pro1ects, e need to calculate the e!pected return of the pro1ect gi*en its le*el of ris&( -his e!pected return should then be co"pared to the e!pected return of the pro1ect( .f the return calculated using the C093 is loer than the pro1ect e!pected return, e should accept the pro1ect, if not, e re1ect the pro1ect( 0fter considering ris& *ia the C093+ 6QWR L (0A O (80((11 E (0A# L (0980 a (10, so accept W 6QUR L (0A O (90((11 E (0A# L (10I0 a (12, so accept U 6Q5R L (0A O 1(IA((11 E (0A# L (1=J0 T (1=, so re1ect 5 6Q^R L (0A O 1(M0((11 E (0A# L (1IM0 a (1A, so accept ^ c. 9ro1ect W ould be incorrectl% re1ected8 9ro1ect 5 ould be incorrectl% accepted( 1#. a. >e should loo& at the eighted a*erage flotation cost, not 1ust the debt cost( CHAPTER 14 B-269 b. -he eighted a*erage floatation cost is the eighted a*erage of the floatation costs for debt and e$uit%, so+ f- L (0A((JA:1(JA# O (08(1:1(JA# L (0MJ1 or M(J1N c. -he total cost of the e$uip"ent including floatation costs is+ 0"ount raised(1 E (0MJ1# L <20,000,000 0"ount raised L <20,000,000:(1 E (0MJ1# L <21,I=9,A10 6*en if the specific funds are actuall% being raised co"pletel% fro" debt, the flotation costs, and hence true in*est"ent cost, should be *alued as if the fir"'s target capital structure is used( 1$. We first need to find the eighted a*erage floatation cost( )oing so, e find+ f- L (MA((09# O (0A((0M# O (=0((0=# L (0J1 or J(1N 0nd the total cost of the e$uip"ent including floatation costs is+ 0"ount raised(1 E (0J1# L <IA,000,000 0"ount raised L <IA,000,000:(1 E (0J1# L <I8,I1=,12A &ntermediate 2%. Using the debt-e$uit% ratio to calculate the W0CC, e find+ W0CC L ((90:1(90#((0I8# O (1:1(90#((1=# L (0912 or 9(12N ,ince the pro1ect is ris&ier than the co"pan%, e need to ad1ust the pro1ect discount rate for the additional ris&( Using the sub1ecti*e ris& factor gi*en, e find+ 9ro1ect discount rate L 9(12N O 2(00N L 11(12N We ould accept the pro1ect if the 49V is positi*e( -he 49V is the 9V of the cash outflos plus the 9V of the cash inflos( ,ince e ha*e the costs, e 1ust need to find the 9V of inflos( -he cash inflos are a groing perpetuit%( .f %ou re"e"ber, the e$uation for the 9V of a groing perpetuit% is the sa"e as the di*idend groth e$uation, so+ 9V of future CB L <2,J00,000:((1112 E (0I# L <=J,9I=,J8J -he pro1ect should onl% be underta&en if its cost is less than <=J,9I=,J8J since costs less than this a"ount ill result in a positi*e 49V( 21. -he total cost of the e$uip"ent including floatation costs as+ -otal costs L <1A,000,000 O 8A0,000 L <1A,8A0,000 B-270 SOLUTIONS Using the e$uation to calculate the total cost including floatation costs, e get+ 0"ount raised(1 E f-# L 0"ount needed after floatation costs <1A,8A0,000(1 E f-# L <1A,000,000 f- L (0A=M or A(=MN 4o, e &no the eighted a*erage floatation cost( -he e$uation to calculate the percentage floatation costs is+ f- L (0A=M L (0J(6:V# O (0=():V# We can sol*e this e$uation to find the debt-e$uit% ratio as follos+ (0A=M(V:6# L (0J O (0=():6# We "ust recogni2e that the V:6 ter" is the e$uit% "ultiplier, hich is (1 O ):6#, so+ (0A=M():6 O 1# L (08 O (0=():6# ):6 L 0(M929 22. -o find the afterta! cost of debt for the co"pan%, e need to find the eighted a*erage of the four debt issues( We ill begin b% calculating the "ar&et *alue of each debt issue, hich is+ 3V1 L 1(0=(<I0,000,000# 3V1 L <I1,200,000 3V2 L 1(08(<=A,000,000# 3V2 L <=J,800,000 3V= L 0(9J(<AA,000,000# 3V= L <A=,A00,000 3VI L 1(11(<I0,000,000# 3VI L <AA,A00,000 ,o, the total "ar&et *alue of the co"pan%'s debt is+ 3V) L <I1,200,000 O =J,800,000 O A=,=A0,000 O AA,A00,000 3V) L <18J,8A0,000 -he eight of each debt issue is+ 1 L <I1,200,000:<18J,8A0,000 1 L (219= or 21(9=N 2 L <=J,800,000:<18J,8A0,000 2 L (2012 or 20(12N = L <A=,A00,000:<18J,8A0,000 = L (28I0 or 28(I0N CHAPTER 14 B-271 I L <AA,A00,000:<18J,8A0,000 I L (29AI or 29(AIN 4e!t, e need to find the 5-3 for each bond issue( -he 5-3 for each issue is+ 91 L <1,0=0 L <=A(9V.B0RN,10# O <1,000(9V.BRN,10# R1 L 2(JM8N 5-31 L =(1IMN S 2 5-31 L M(29N 92 L <1,080 L <I2(A0(9V.B0RN,1M# O <1,000(9V.BRN,1M# R2 L =(A8IN 5-32 L =(A8IN S 2 5-32 L J(1JN 9= L <9J0 L <I1(9V.B0RN,=1# O <1,000(9V.BRN,=1# R= L =(MAIN 5-3= L I(2JMN S 2 5-3= L 8(AIN 9I L <1,110 L <I9(9V.B0RN,A0# O <1,000(9V.BRN,A0# RI L I(=AMN 5-3I L I(=AMN S 2 5-3I L 8(J1N -he eighted a*erage 5-3 of the co"pan%'s debt is thus+ 5-3 L (219=((0M29# O (2012 ((0J1J# O (28I0((08AI# O (29AI((08J1# 5-3 L (0J82 or J(82N 0nd the afterta! cost of debt is+ R) L (0J82(1 E (0=I# R) L (0A1M or A(1MN 23. a. Using the di*idend discount "odel, the cost of e$uit% is+ R6 L Q(0(80#(1(0A#:<M1R O (0A R6 L (0M=8 or M(=8N b. Using the C093, the cost of e$uit% is+ R6 L (0AA O 1(A0((1200 E (0AA0# R6 L (1A2A or 1A(2AN c. When using the di*idend groth "odel or the C093, %ou "ust re"e"ber that both are esti"ates for the cost of e$uit%( 0dditionall%, and perhaps "ore i"portantl%, each "ethod of esti"ating the cost of e$uit% depends upon different assu"ptions( B-272 SOLUTIONS Challenge 24. We can use the debt-e$uit% ratio to calculate the eights of e$uit% and debt( -he debt of the co"pan% has a eight for long-ter" debt and a eight for accounts pa%able( We can use the eight gi*en for accounts pa%able to calculate the eight of accounts pa%able and the eight of long-ter" debt( -he eight of each ill be+ 0ccounts pa%able eight L (20:1(20 L (1J Cong-ter" debt eight L 1:1(20 L (8= ,ince the accounts pa%able has the sa"e cost as the o*erall W0CC, e can rite the e$uation for the W0CC as+ W0CC L (1:1(J#((1I# O (0(J:1(J#Q((20:1(2#W0CC O (1:1(2#((08#(1 E (=A#R ,ol*ing for W0CC, e find+ W0CC L (082I O (I118Q((20:1(2#W0CC O (0I==R W0CC L (082I O ((0M8M#W0CC O (01J8 ((9=1I#W0CC L (1002 W0CC L (10JM or 10(JMN We ill use basicall% the sa"e e$uation to calculate the eighted a*erage floatation cost, e!cept e ill use the floatation cost for each for" of financing( )oing so, e get+ Blotation costs L (1:1(J#((08# O (0(J:1(J#Q((20:1(2#(0# O (1:1(2#((0I#R L (0M08 or M(08N -he total a"ount e need to raise to fund the ne e$uip"ent ill be+ 0"ount raised cost L <IA,000,000:(1 E (0M08# 0"ount raised L <IJ,912,=1J ,ince the cash flos go to perpetuit%, e can calculate the present *alue using the e$uation for the 9V of a perpetuit%( -he 49V is+ 49V L E<IJ,912,=1J O (<M,200,000:(10JM# 49V L <9,J19,JJJ 2. We can use the debt-e$uit% ratio to calculate the eights of e$uit% and debt( -he eight of debt in the capital structure is+ ) L 1(20 : 2(20 L (AIAA or AI(AAN 0nd the eight of e$uit% is+ 6 L 1 E (AIAA L (IAIA or IA(IAN CHAPTER 14 B-273 4o e can calculate the eighted a*erage floatation costs for the *arious percentages of internall% raised e$uit%( -o find the portion of e$uit% floatation costs, e can "ultipl% the e$uit% costs b% the percentage of e$uit% raised e!ternall%, hich is one "inus the percentage raised internall%( ,o, if the co"pan% raises all e$uit% e!ternall%, the floatation costs are+ f- L (0(AIAA#((08#(1 E 0# O (0(IAIA#((0=A# f- L (0AAA or A(AAN -he initial cash outflo for the pro1ect needs to be ad1usted for the floatation costs( -o account for the floatation costs+ 0"ount raised(1 E (0AAA# L <1IA,000,000 0"ount raised L <1IA,000,000:(1 E (0AAA# 0"ount raised L <1A=,A12,99= .f the co"pan% uses M0 percent internall% generated e$uit%, the floatation cost is+ f- L (0(AIAA#((08#(1 E 0(M0# O (0(IAIA#((0=A# f- L (0==M or =(=MN 0nd the initial cash flo ill be+ 0"ount raised(1 E (0==M# L <1IA,000,000 0"ount raised L <1IA,000,000:(1 E (0==M# 0"ount raised L <1A0,0IJ,0=J .f the co"pan% uses 100 percent internall% generated e$uit%, the floatation cost is+ f- L (0(AIAA#((08#(1 E 1# O (0(IAIA#((0=A# f- L (0191 or 1(91N 0nd the initial cash flo ill be+ 0"ount raised(1 E (0191# L <1IA,000,000 0"ount raised L <1IA,000,000:(1 E (0191# 0"ount raised L <1IJ,822,0AJ 2!. -he <I "illion cost of the land = %ears ago is a sun& cost and irrele*ant8 the <A(1 "illion appraised *alue of the land is an opportunit% cost and is rele*ant( -he <M "illion land *alue in A %ears is a rele*ant cash flo as ell( -he fact that the co"pan% is &eeping the land rather than selling it is uni"portant( -he land is an opportunit% cost in A %ears and is a rele*ant cash flo for this pro1ect( -he "ar&et *alue capitali2ation eights are+ 3V) L 2I0,000(<1,000#(0(9I# L <22A,M00,000 3V6 L 9,000,000(<J1# L <M=9,000,000 3V9 L I00,000(<81# L <=2,I00,000 -he total "ar&et *alue of the co"pan% is+ V L <22A,M00,000 O M=9,000,000 O =2,I00,000 L <89J,000,000 B-274 SOLUTIONS 4e!t e need to find the cost of funds( We ha*e the infor"ation a*ailable to calculate the cost of e$uit% using the C093, so+ R6 L (0A O 1(20((08# L (1IM0 or 1I(M0N -he cost of debt is the 5-3 of the co"pan%'s outstanding bonds, so+ 90 L <9I0 L <=J(A0(9V.B0RN,I0# O <1,000(9V.BRN,I0# R L I(0AMN 5-3 L I(0AMN S 2 L 8(11N 0nd the afterta! cost of debt is+ R) L (1 E (=A#((0811# L (0A2J or A(2JN -he cost of preferred stoc& is+ R9 L <A(A0:<81 L (0MJ9 or M(J9N a. -he eighted a*erage floatation cost is the su" of the eight of each source of funds in the capital structure of the co"pan% ti"es the floatation costs, so+ f- L (<M=9:<89J#((08# O (<=2(I:<89J#((0M# O (<22A(M:<89J#((0I# L (0M92 or M(92N -he initial cash outflo for the pro1ect needs to be ad1usted for the floatation costs( -o account for the floatation costs+ 0"ount raised(1 E (0M92# L <=A,000,000 0"ount raised L <=A,000,000:(1 E (0M92# L <=J,M02,JMA ,o the cash flo at ti"e 2ero ill be+ CB0 L E<A,100,000 E =J,M02,JMA E 1,=000,000 L E<II,002,JMA -here is an i"portant ca*eat to this solution( -his solution assu"es that the increase in net or&ing capital does not re$uire the co"pan% to raise outside funds8 therefore the floatation costs are not included( >oe*er, this is an assu"ption and the co"pan% could need to raise outside funds for the 4WC( .f this is true, the initial cash outla% includes these floatation costs, so+ -otal cost of 4WC including floatation costs+ <1,=00,000:(1 E (0M92# L <1,=9M,MJI -his ould "a&e the total initial cash flo+ CB0 L E<A,100,000 E =J,M02,JMA E 1,=9M,MJI L E<II,099,I=9 CHAPTER 14 B-275 b. -o find the re$uired return on this pro1ect, e first need to calculate the W0CC for the co"pan%( -he co"pan%'s W0CC is+ W0CC L Q(<M=9:<89J#((1IM0# O (<=2(I:<89J#((0MJ9# O (<22A(M:<89J#((0A2J#R L (119J -he co"pan% ants to use the sub1ecti*e approach to this pro1ect because it is located o*erseas( -he ad1ust"ent factor is 2 percent, so the re$uired return on this pro1ect is+ 9ro1ect re$uired return L (119J O (02 L (1=9J c. -he annual depreciation for the e$uip"ent ill be+ <=A,000,000:8 L <I,=JA,000 ,o, the boo& *alue of the e$uip"ent at the end of fi*e %ears ill be+ HVA L <=A,000,000 E A(<I,=JA,000# L <1=,12A,000 ,o, the afterta! sal*age *alue ill be+ 0fterta! sal*age *alue L <M,000,000 O (=A(<1=,12A,000 E M,000,000# L <8,I9=,JA0 d. Using the ta! shield approach, the /CB for this pro1ect is+ /CB L Q(9 E *#7 E BCR(1 E t# O tC) /CB L Q(<10,900 E 9,I00#(18,000# E J,000,000R(1 E (=A# O (=A(<=A,000,000:8# L <1I,A=1,2A0 e. -he accounting brea&e*en sales figure for this pro1ect is+ 70 L (BC O )#:(9 E *# L (<J,000,000 O I,=JA,000#:(<10,900 E 9,I00# L J,A8= units f. We ha*e calculated all cash flos of the pro1ect( We 1ust need to "a&e sure that in 5ear A e add bac& the afterta! sal*age *alue and the reco*er% of the initial 4WC( -he cash flos for the pro1ect are+ Cear Flow Cash 0 E<II,002,JMA 1 1I,A=1,2A0 2 1I,A=1,2A0 = 1I,A=1,2A0 I 1I,A=1,2A0 A =0,=2A,000 Using the re$uired return of 1=(9J percent, the 49V of the pro1ect is+ 49V L E<II,002,JMA O <1I,A=1,2A0(9V.B01=(9JN,I# O <=0,=2A,000:1(1=9J A 49V L <1I,1=0,J1=(81 B-276 SOLUTIONS 0nd the .RR is+ 49V L 0 L E<II,002,JMA O <1I,A=1,2A0(9V.B0.RRN,I# O <=0,=2A,000:(1 O .RR# A .RR L 2A(2AN .f the initial 4WC is assu"ed to be financed fro" outside sources, the cash flos are+ Cear Flow Cash 0 E<II,099,I=9 1 1I,A=1,2A0 2 1I,A=1,2A0 = 1I,A=1,2A0 I 1I,A=1,2A0 A =0,=2A,000 With this assu"ption, and the re$uired return of 1=(9J percent, the 49V of the pro1ect is+ 49V L E<II,099,I=9 O <1I,A=1,2A0(9V.B01=(9JN,I# O <=0,=2A,000:1(1=9J A 49V L <1I,0=I,0=9(MJ 0nd the .RR is+ 49V L 0 L E<II,099,I=9 O <1I,A=1,2A0(9V.B0.RRN,I# O <=0,=2A,000:(1 O .RR# A .RR L 2A(1AN CHAPTER 15 RAISING CAPITAL Answers to Concepts Review and Critical Thinking Questions 1. 0 co"pan%'s internall% generated cash flo pro*ides a source of e$uit% financing( Bor a profitable co"pan%, outside e$uit% "a% ne*er be needed( )ebt issues are larger because large co"panies ha*e the greatest access to public debt "ar&ets (s"all co"panies tend to borro "ore fro" pri*ate lenders#( 6$uit% issuers are fre$uentl% s"all co"panies going public8 such issues are often $uite s"all( 2. Bro" the pre*ious $uestion, econo"ies of scale are part of the anser( He%ond this, debt issues are si"pl% easier and less ris&% to sell fro" an in*est"ent ban&'s perspecti*e( -he to "ain reasons are that *er% large a"ounts of debt securities can be sold to a relati*el% s"all nu"ber of bu%ers, particularl% large institutional bu%ers such as pension funds and insurance co"panies, and debt securities are "uch easier to price( 3. -he% are ris&ier and harder to "ar&et fro" an in*est"ent ban&'s perspecti*e( 4. 5ields on co"parable bonds can usuall% be readil% obser*ed, so pricing a bond issue accuratel% is "uch less difficult( . .t is clear that the stoc& as sold too cheapl%, so 6%etech had reason to be unhapp%( !. 4o, but, in fairness, pricing the stoc& in such a situation is e!tre"el% difficult( ". .t's an i"portant factor( /nl% M(A "illion of the shares ere underpriced( -he other =2 "illion ere, in effect, priced co"pletel% correctl%( #. -he e*idence suggests that a non-underritten rights offering "ight be substantiall% cheaper than a cash offer( >oe*er, such offerings are rare, and there "a% be hidden costs or other factors not %et identified or ell understood b% researchers( $. >e could ha*e done orse since his access to the o*ersubscribed and, presu"abl%, underpriced issues as restricted hile the bul& of his funds ere allocated to stoc&s fro" the undersubscribed and, $uite possibl%, o*erpriced issues( 1%. a. -he price ill probabl% go up because .9/s are generall% underpriced( -his is especiall% true for s"aller issues such as this one( b. .t is probabl% safe to assu"e that the% are ha*ing trouble "o*ing the issue, and it is li&el% that the issue is not substantiall% underpriced( B-278 SOLUTIONS Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. a. -he ne "ar&et *alue ill be the current shares outstanding ti"es the stoc& price plus the rights offered ti"es the rights price, so+ 4e "ar&et *alue L A00,000(<81# O M0,000(<J0# L <II,J00,000 b( -he nu"ber of rights associated ith the old shares is the nu"ber of shares outstanding di*ided b% the rights offered, so+ 4u"ber of rights needed L A00,000 old shares:M0,000 ne shares L 8(== rights per ne share c( -he ne price of the stoc& ill be the ne "ar&et *alue of the co"pan% di*ided b% the total nu"ber of shares outstanding after the rights offer, hich ill be+ 9U L <II,J00,000:(A00,000 O M0,000# L <J9(82 d. -he *alue of the right Value of a right L <81(00 E J9(82 L <1(18 e( 0 rights offering usuall% costs less, it protects the proportionate interests of e!isting share-holders and also protects against underpricing( 2. a. -he "a!i"u" subscription price is the current stoc& price, or <A=( -he "ini"u" price is an%thing greater than <0( b. -he nu"ber of ne shares ill be the a"ount raised di*ided b% the subscription price, so+ 4u"ber of ne shares L <I0,000,000:<I8 L 8==,=== shares 0nd the nu"ber of rights needed to bu% one share ill be the current shares outstanding di*ided b% the nu"ber of ne shares offered, so+ 4u"ber of rights needed L I,100,000 shares outstanding:8==,=== ne shares L I(92 CHAPTER 15 B-279 c( 0 shareholder can bu% I(92 rights on shares for+ I(92(<A=# L <2M0(JM -he shareholder can e!ercise these rights for <I8, at a total cost of+ <2M0(JM O I8 L <=08(JM -he in*estor ill then ha*e+ 6!-rights shares L 1 O I(92 6!-rights shares L A(92 -he e!-rights price per share is+ 9U L QI(92(<A=# O <I8R:A(92 L <A2(1M ,o, the *alue of a right is+ Value of a right L <A= E A2(1M L <0(8I d( Hefore the offer, a shareholder ill ha*e the shares oned at the current "ar&et price, or+ 9ortfolio *alue L (1,000 shares#(<A=# L <A=,000 0fter the rights offer, the share price ill fall, but the shareholder ill also hold the rights, so+ 9ortfolio *alue L (1,000 shares#(<A2(1M# O (1,000 rights#(<0(8I# L <A=,000 3. Using the e$uation e deri*ed in 9roble" 2, part c to calculate the price of the stoc& e!-rights, e can find the nu"ber of shares a shareholder ill ha*e e!-rights, hich is+ 9U L <JI(80 L Q4(<81# O <I0R:(4 O 1# 4 L A(M1= -he nu"ber of ne shares is the a"ount raised di*ided b% the per-share subscription price, so+ 4u"ber of ne shares L <20,000,000:<I0 L A00,000 0nd the nu"ber of old shares is the nu"ber of ne shares ti"es the nu"ber of shares e!-rights, so+ 4u"ber of old shares L A(M1=(A00,000# L 2,80M,IA2 B-280 SOLUTIONS 4. .f %ou recei*e 1,000 shares of each, the profit is+ 9rofit L 1,000(<J# E 1,000(<A# L <2,000 ,ince %ou ill onl% recei*e one-half of the shares of the o*ersubscribed issue, %our profit ill be+ 6!pected profit L A00(<J# E 1,000(<A# L E<1,A00 -his is an e!a"ple of the inner's curse( . Using U to stand for the re$uired sale proceeds, the e$uation to calculate the total sale proceeds, including floatation costs is+ U(1 E (09# L <M0,000,000 U L <MA,9=I,0MM re$uired total proceeds fro" sale( ,o the nu"ber of shares offered is the total a"ount raised di*ided b% the offer price, hich is+ 4u"ber of shares offered L <MA,9=I,0MM:<21 L =,1=9,J1J !. -his is basicall% the sa"e as the pre*ious proble", e!cept e need to include the <900,000 of e!penses in the a"ount the co"pan% needs to raise, so+ U(1 E (09# L (<M0,000,000 O 900,00# U L <MM,92=,0JJ re$uired total proceeds fro" sale( 4u"ber of shares offered L <MM,92=,0JJ:<21 L =,18M,81= ". We need to calculate the net a"ount raised and the costs associated ith the offer( -he net a"ount raised is the nu"ber of shares offered ti"es the price recei*ed b% the co"pan%, "inus the costs associated ith the offer, so+ 4et a"ount raised L (10,000,000 shares#(<18(20# E 900,000 E =20,000 L <180,J80,000 -he co"pan% recei*ed <180,J80,000 fro" the stoc& offering( 4o e can calculate the direct costs( 9art of the direct costs are gi*en in the proble", but the co"pan% also had to pa% the underriters( -he stoc& as offered at <20 per share, and the co"pan% recei*ed <18(20 per share( -he difference, hich is the underriters spread, is also a direct cost( -he total direct costs ere+ -otal direct costs L <900,000 O (<20 E 18(20#(10,000,000 shares# L <18,900,000 We are gi*en part of the indirect costs in the proble"( 0nother indirect cost is the i""ediate price appreciation( -he total indirect costs ere+ -otal indirect costs L <=20,000 O (<2A(M0 E 20#(10,000,000 shares# L <AM,=20,000 CHAPTER 15 B-281 -his "a&es the total costs+ -otal costs L <18,900,000 O AM,=20,000 L <JA,220,000 -he floatation costs as a percentage of the a"ount raised is the total cost di*ided b% the a"ount raised, so+ Blotation cost percentage L <JA,220,000:<180,J80,000 L (I1M1 or I1(M1N #. -he nu"ber of rights needed per ne share is+ 4u"ber of rights needed L 120,000 old shares:2A,000 ne shares L I(8 rights per ne share( Using 9R/ as the rights-on price, and 9, as the subscription price, e can e!press the price per share of the stoc& e!-rights as+ 9U L Q49R/ O 9,R:(4 O 1# a. 9U L QI(8(<9I# O <9IR:(I(80 O 1# L <9I(008 4o change( b( 9U L QI(8(<9I# O <90R:(I(80 O 1# L <9=(=18 9rice drops b% <0(M9 per share( c( 9U L QI(8(<9I# O <8AR:(I(80 O 1# L <92(IA8 9rice drops b% <1(AA per share( &ntermediate $. a. -he nu"ber of shares outstanding after the stoc& offer ill be the current shares outstanding, plus the a"ount raised di*ided b% the current stoc& price, assu"ing the stoc& price doesn't change( ,o+ 4u"ber of shares after offering L 8,000,000 O <=A,000,000:<A0 L 8,J00,000 ,ince the boo& *alue per share is <18, the old boo& *alue of the shares is the current nu"ber of shares outstanding ti"es 18( Bro" the pre*ious solution, e can see the co"pan% ill sell J00,000 shares, and these ill ha*e a boo& *alue of <A0 per share( -he su" of these to *alues ill gi*e us the total boo& *alue of the co"pan%( .f e di*ide this b% the ne nu"ber of shares outstanding( )oing so, e find the ne boo& *alue per share ill be+ 4e boo& *alue per share L Q8,000,000(<18# O J00,000(<A0#R:8,J00,000 L <2=(A= -he current 69, for the co"pan% is+ 69,0 L 4.0:,hares0 L <1J,000,000:8,000,000 shares L <2(1= per share 0nd the current 9:6 is+ (9:6#0 L <A0:<2(1= L 2=(A= B-282 SOLUTIONS .f the net inco"e increases b% <1,100,000, the ne 69, ill be+ 69,1 L 4.1:shares1 L <18,100,000:8,J00,000 shares L <2(08 per share 0ssu"ing the 9:6 re"ains constant, the ne share price ill be+ 91 L (9:6#0(69, 1 # L 2=(A=(<2(08# L <I8(9A -he current "ar&et-to-boo& ratio is+ Current "ar&et-to-boo& L <A0:<18 L 2(JJ8 Using the ne share price and boo& *alue per share, the ne "ar&et-to-boo& ratio ill be+ 4e "ar&et-to-boo& L <I8(9A:<20(AJ L 2(=J9 0ccounting dilution has occurred because ne shares ere issued hen the "ar&et-to-boo& ratio as less than one8 "ar&et *alue dilution has occurred because the fir" financed a negati*e 49V pro1ect( -he cost of the pro1ect is gi*en at <=A "illion( -he 49V of the pro1ect is the cost of the ne pro1ect plus the ne "ar&et *alue of the fir" "inus the current "ar&et *alue of the fir", or+ 49V L E<=A,000,000 O Q8,J00,000(<I8(9A# E 8,000,000(<A0#R L E<9,11J,MIJ b( Bor the price to re"ain unchanged hen the 9:6 ratio is constant, 69, "ust re"ain constant( -he ne net inco"e "ust be the ne nu"ber of shares outstanding ti"es the current 69,, hich gi*es+ 4.1 L (8,J00,000 shares#(<2(1= per share# L <18,I8J,A00 1%. -he total e$uit% of the co"pan% is total assets "inus total liabilities, or+ 6$uit% L <8,000,000 E =,I00,000 6$uit% L <I,M00,000 ,o, the current R/6 of the co"pan% is+ R/60 L 4.0:-60 L <900,000:<I,M00,000 L (19AJ or 19(AJN -he ne net inco"e ill be the R/6 ti"es the ne total e$uit%, or+ 4.1 L (R/60#(-61# L (19AJ(<I,M00,000 O 8A0,000# L <1,0MM,=0I -he co"pan%'s current earnings per share are+ 69,0 L 4.0:,hares outstanding0 L <900,000:=0,000 shares L <=0(00 -he nu"ber of shares the co"pan% ill offer is the cost of the in*est"ent di*ided b% the current share price, so+ 4u"ber of ne shares L <8A0,000:<8I L 10,119 CHAPTER 15 B-283 -he earnings per share after the stoc& offer ill be+ 69,1 L <1,0MM,=0I:I0,119 shares L <2M(A8 -he current 9:6 ratio is+ (9:6#0 L <8I:<2M(A8 L 2(800 0ssu"ing the 9:6 re"ains constant, the ne stoc& price ill be+ 91 L 2(800(<2M(A8# L <JI(I2 -he current boo& *alue per share and the ne boo& *alue per share are+ HV9,0 L -60:shares0 L <I,M00,000:=0,000 shares L <1A=(== per share HV9,1 L -61:shares1 L (<I,M00,000 O 8A0,000#:I0,119 shares L <1=A(8A per share ,o the current and ne "ar&et-to-boo& ratios are+ 3ar&et-to-boo&0 L <8I:<1A=(== L 0(AIJ8 3ar&et-to-boo&1 L <JI(I2:<1=A(8A L 0(AIJ8 -he 49V of the pro1ect is the cost of the pro1ect plus the ne "ar&et *alue of the fir" "inus the current "ar&et *alue of the fir", or+ 49V L E<8A0,000 O Q<JI(I2(I0,119# E <8I(=0,000#R L E<=8I,=I8 0ccounting dilution ta&es place here because the "ar&et-to-boo& ratio is less than one( 3ar&et *alue dilution has occurred since the fir" is in*esting in a negati*e 49V pro1ect( 11. Using the 9:6 ratio to find the necessar% 69, after the stoc& issue, e get+ 91 L <8I L 2(800(69,1# 69,1 L <=0(00 -he additional net inco"e le*el "ust be the 69, ti"es the ne shares outstanding, so+ 4. L <=0(10,119 shares# L <=0=,AJ1 0nd the ne R/6 is+ R/61 L <=0=,AJ1:<8A0,000 L (=AJ1 or =A(J1N B-284 SOLUTIONS 4e!t, e need to find the 49V of the pro1ect( -he 49V of the pro1ect is the cost of the pro1ect plus the ne "ar&et *alue of the fir" "inus the current "ar&et *alue of the fir", or+ 49V L E<8A0,000 O Q<8I(I0,119# E <8I(=0,000#R L <0 0ccounting dilution still ta&es place, as HV9, still falls fro" <1A=(== to <1=A(8A, but no "ar&et dilution ta&es place because the fir" is in*esting in a 2ero 49V pro1ect( 12. -he nu"ber of ne shares is the a"ount raised di*ided b% the subscription price, so+ 4u"ber of ne shares L <M0,000,000:<9, 0nd the e!-rights nu"ber of shares (4# is e$ual to+ 4 L /ld shares outstanding:4e shares outstanding 4 L 19,000,000:(<M0,000,000:<9,# 4 L 0(0=1MJ9, We &no the e$uation for the e!-rights stoc& price is+ 9U L Q49R/ O 9,R:(4 O 1# We can substitute in the nu"bers e are gi*en, and then substitute the to pre*ious results( )oing so, and sol*ing for the subscription price, e get+ 9U L <J1 L Q4(<JM# O <9,R:(4 O 1# <J1 L Q<JM(0(0=1MJ9,# O 9,R:(0(0=1MJ9, O 1# <J1 L <2I(0MMJ9,:(1 O 0(0=1MJ9,# 9, L <2J(I8 13. Using 9R/ as the rights-on price, and 9, as the subscription price, e can e!press the price per share of the stoc& e!-rights as+ 9U L Q49R/ O 9,R:(4 O 1# 0nd the e$uation for the *alue of a right is+ Value of a right L 9R/ E 9U ,ubstituting the e!-rights price e$uation into the e$uation for the *alue of a right and rearranging, e get+ Value of a right L 9R/ E YQ49R/ O 9,R:(4 O 1#Z Value of a right L Q(4 O 1#9R/ E 49R/ E 9,R:(4O1# Value of a right L Q9R/ E 9,R:(4 O 1# CHAPTER 15 B-285 14. -he net proceeds to the co"pan% on a per share basis is the subscription price ti"es one "inus the underriter spread, so+ 4et proceeds to the co"pan% L <2=(1 E (0M# L <21(M2 per share ,o, to raise the re$uired funds, the co"pan% "ust sell+ 4e shares offered L <A,M00,000:<21(M2 L 2A9,019 -he nu"ber of rights needed per share is the current nu"ber of shares outstanding di*ided b% the ne shares offered, or+ 4u"ber of rights needed L MA0,000 old shares:2A9,019 ne shares 4u"ber of rights needed L 2(A1 rights per share -he e!-rights stoc& price ill be+ 9U L Q49R/ O 9,R:(4 O 1# 9U L Q2(A1(<A0# O 2=R:(2(A1 O 1# L <I2(=1 ,o, the *alue of a right is+ Value of a right L <A0 E I2(=1 L <J(M9 0nd %our proceeds fro" selling %our rights ill be+ 9roceeds fro" selling rights L A,000(<J(M9# L <=8,IMJ(I1 1. Using the e$uation for *aluing a stoc& e!-rights, e find+ 9U L Q49R/ O 9,R:(4 O 1# 9U L QI(<M0# O <=AR:(I O 1# L <AA -he stoc& is correctl% priced( Calculating the *alue of a right, e find+ Value of a right L 9R/ E 9U Value of a right L <M0 E A= L <J ,o, the rights are underpriced( 5ou can create an i""ediate profit on the e!-rights da% if the stoc& is selling for <A= and the rights are selling for <= b% e!ecuting the folloing transactions+ Hu% I rights in the "ar&et for I(<=# L <12( Use these rights to purchase a ne share at the subscription price of <=A( .""ediatel% sell this share in the "ar&et for <A=, creating an instant <M profit( CHAPTER 16 FINANCIAL LEVERAGE AND CAPITAL STRUCTURE POLICY Answers to Concepts Review and Critical Thinking Questions 1. Husiness ris& is the e$uit% ris& arising fro" the nature of the fir"'s operating acti*it%, and is directl% related to the s%ste"atic ris& of the fir"'s assets( Binancial ris& is the e$uit% ris& that is due entirel% to the fir"'s chosen capital structure( 0s financial le*erage, or the use of debt financing, increases, so does financial ris& and, hence, the o*erall ris& of the e$uit%( -hus, Bir" H could ha*e a higher cost of e$uit% if it uses greater le*erage( 2. 4o, it doesn't follo( While it is true that the e$uit% and debt costs are rising, the &e% thing to re"e"ber is that the cost of debt is still less than the cost of e$uit%( ,ince e are using "ore and "ore debt, the W0CC does not necessaril% rise( 3. Hecause "an% rele*ant factors such as ban&ruptc% costs, ta! as%""etries, and agenc% costs cannot easil% be identified or $uantified, it's practicall% i"possible to deter"ine the precise debt-e$uit% ratio that "a!i"i2es the *alue of the fir"( >oe*er, if the fir"'s cost of ne debt suddenl% beco"es "uch "ore e!pensi*e, it's probabl% true that the fir" is too highl% le*eraged( 4. -he "ore capital intensi*e industries, such as airlines, cable tele*ision, and electric utilities, tend to use greater financial le*erage( 0lso, industries ith less predictable future earnings, such as co"puters or drugs, tend to use less financial le*erage( ,uch industries also ha*e a higher concentration of groth and startup fir"s( /*erall, the general tendenc% is for fir"s ith identifiable, tangible assets and relati*el% "ore predictable future earnings to use "ore debt financing( -hese are t%picall% the fir"s ith the greatest need for e!ternal financing and the greatest li&elihood of benefiting fro" the interest ta! shelter( . .t's called le*erage (or ;gearing@ in the UP# because it "agnifies gains or losses( !. >o"e"ade le*erage refers to the use of borroing on the personal le*el as opposed to the corporate le*el( ". /ne anser is that the right to file for ban&ruptc% is a *aluable asset, and the financial "anager acts in shareholders' best interest b% "anaging this asset in a%s that "a!i"i2e its *alue( -o the e!tent that a ban&ruptc% filing pre*ents ;a race to the courthouse steps,@ it ould see" to be a reasonable use of the process( #. 0s in the pre*ious $uestion, it could be argued that using ban&ruptc% las as a sord "a% si"pl% be the best use of the asset( Creditors are aare at the ti"e a loan is "ade of the possibilit% of ban&ruptc%, and the interest charged incorporates it( CHAPTER 16 B-287 $. /ne side is that Continental as going to go ban&rupt because its costs "ade it unco"petiti*e( -he ban&ruptc% filing enabled Continental to restructure and &eep fl%ing( -he other side is that Continental abused the ban&ruptc% code( Rather than renegotiate labor agree"ents, Continental si"pl% abrogated the" to the detri"ent of its e"plo%ees( .n this, and the last se*eral, $uestions, an i"portant thing to &eep in "ind is that the ban&ruptc% code is a creation of la, not econo"ics( 0 strong argu"ent can ala%s be "ade that "a&ing the best use of the ban&ruptc% code is no different fro", for e!a"ple, "ini"i2ing ta!es b% "a&ing best use of the ta! code( .ndeed, a strong case can be "ade that it is the financial "anager's dut% to do so( 0s the case of Continental illustrates, the code can be changed if sociall% undesirable outco"es are a proble"( 1%. -he basic goal is to "ini"i2e the *alue of non-"ar&eted clai"s( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. a. 0 table outlining the inco"e state"ent for the three possible states of the econo"% is shon belo( -he 69, is the net inco"e di*ided b% the A,000 shares outstanding( -he last ro shos the percentage change in 69, the co"pan% ill e!perience in a recession or an e!pansion econo"%( Recession 4or"al 6!pansion 6H.- <1I,000 <28,000 <=M,I00 .nterest 0 0 0 4. <1I,000 <28,000 <=M,I00 69, < 2(80 < A(M0 < J(28 N∆69, EA0 EEE O=0 b. .f the co"pan% undergoes the proposed recapitali2ation, it ill repurchase+ ,hare price L 6$uit% : ,hares outstanding ,hare price L <2A0,000:A,000 ,hare price L <A0 ,hares repurchased L )ebt issued : ,hare price ,hares repurchased L<90,000:<A0 ,hares repurchased L 1,800 B-288 SOLUTIONS -he interest pa%"ent each %ear under all three scenarios ill be+ .nterest pa%"ent L <90,000((0J# L <M,=00 -he last ro shos the percentage change in 69, the co"pan% ill e!perience in a recession or an e!pansion econo"% under the proposed recapitali2ation( Recession 4or"al 6!pansion 6H.- <1I,000 <28,000 <=M,I00 .nterest M,=00 M,=00 M,=00 4. <J,J00 <21,J00 <=0,100 69, <2(I1 < M(J8 <9(I1 N∆69, EMI(A2 EEE O=8(J1 2. a. 0 table outlining the inco"e state"ent ith ta!es for the three possible states of the econo"% is shon belo( -he share price is still <A0, and there are still A,000 shares outstanding( -he last ro shos the percentage change in 69, the co"pan% ill e!perience in a recession or an e!pansion econo"%( Recession 4or"al 6!pansion 6H.- <1I,000 <28,000 <=M,I00 .nterest 0 0 0 -a!es I,900 9,8A0 12,JI0 4. <9,100 <18,200 <2=,M00 69, <1(82 <=(MI <I(J= N∆69, EA0 EEE O=0 b. 0 table outlining the inco"e state"ent ith ta!es for the three possible states of the econo"% and assu"ing the co"pan% underta&es the proposed capitali2ation is shon belo( -he interest pa%"ent and shares repurchased are the sa"e as in part b of 9roble" 1( Recession 4or"al 6!pansion 6H.- <1I,000 <28,000 <=M,I00 .nterest M,=00 M,=00 M,=00 -a!es 2,M9A J,A9A 10,A=A 4. <A,00A <1I,10A <19,AMA 69, <1(AM <I(I1 <M(11 N∆69, EMI(A2 EEE O=8(J1 4otice that the percentage change in 69, is the sa"e both ith and ithout ta!es( CHAPTER 16 B-289 3. a. ,ince the co"pan% has a "ar&et-to-boo& ratio of 1(0, the total e$uit% of the fir" is e$ual to the "ar&et *alue of e$uit%( Using the e$uation for R/6+ R/6 L 4.:<2A0,000 -he R/6 for each state of the econo"% under the current capital structure and no ta!es is+ Recession 4or"al 6!pansion R/6 (0AM0 (1120 (1IAM N∆R/6 EA0 EEE O=0 -he second ro shos the percentage change in R/6 fro" the nor"al econo"%( b. .f the co"pan% underta&es the proposed recapitali2ation, the ne e$uit% *alue ill be+ 6$uit% L <2A0,000 E 90,000 6$uit% L <1M0,000 ,o, the R/6 for each state of the econo"% is+ R/6 L 4.:<1M0,000 Recession 4or"al 6!pansion R/6 (0I81 (1=AM (1881 N∆R/6 EMI(A2 EEE O=8(J1 c. .f there are corporate ta!es and the co"pan% "aintains its current capital structure, the R/6 is+ R/6 (0=MI (0J28 (09IM N∆R/6 EA0 EEE O=0 .f the co"pan% underta&es the proposed recapitali2ation, and there are corporate ta!es, the R/6 for each state of the econo"% is+ R/6 (0=1= (0882 (122= N∆R/6 EMI(A2 EEE O=8(J1 4otice that the percentage change in R/6 is the sa"e as the percentage change in 69,( -he percentage change in R/6 is also the sa"e ith or ithout ta!es( 4. a. Under 9lan ., the unle*ered co"pan%, net inco"e is the sa"e as 6H.- ith no corporate ta!( -he 69, under this capitali2ation ill be+ 69, L <=A0,000:1M0,000 shares 69, L <2(19 B-290 SOLUTIONS Under 9lan .., the le*ered co"pan%, 6H.- ill be reduced b% the interest pa%"ent( -he interest pa%"ent is the a"ount of debt ti"es the interest rate, so+ 4. L <A00,000 E (08(<2,800,000# 4. L <12M,000 0nd the 69, ill be+ 69, L <12M,000:80,000 shares 69, L <1(A8 9lan . has the higher 69, hen 6H.- is <=A0,000( b. Under 9lan ., the net inco"e is <A00,000 and the 69, is+ 69, L <A00,000:1M0,000 shares 69, L <=(1= Under 9lan .., the net inco"e is+ 4. L <A00,000 E (08(<2,800,000# 4. L <2JM,000 0nd the 69, is+ 69, L <2JM,000:80,000 shares 69, L <=(IA 9lan .. has the higher 69, hen 6H.- is <A00,000( c. -o find the brea&e*en 6H.- for to different capital structures, e si"pl% set the e$uations for 69, e$ual to each other and sol*e for 6H.-( -he brea&e*en 6H.- is+ 6H.-:1M0,000 L Q6H.- E (08(<2,800,000#R:80,000 6H.- L <II8,000 . We can find the price per share b% di*iding the a"ount of debt used to repurchase shares b% the nu"ber of shares repurchased( )oing so, e find the share price is+ ,hare price L <2,800,000:(1M0,000 E 80,000# ,hare price L <=A(00 per share -he *alue of the co"pan% under the all-e$uit% plan is+ V L <=A(00(1M0,000 shares# L <A,M00,000 0nd the *alue of the co"pan% under the le*ered plan is+ V L <=A(00(80,000 shares# O <2,800,000 debt L <A,M00,000 CHAPTER 16 B-291 !. a. -he inco"e state"ent for each capitali2ation plan is+ & && All)e!uit 6H.- <=9,000 <=9,000 <=9,000 .nterest 1M,000 2I,000 0 4. <2=,000 <1A,000 <=9,000 69, < =(29 < =(00 < =(AA -he all-e$uit% plan8 9lan .. has the loest 69,( b. -he brea&e*en le*el of 6H.- occurs hen the capitali2ation plans result in the sa"e 69,( -he 69, is calculated as+ 69, L (6H.- E R))#:,hares outstanding -his e$uation calculates the interest pa%"ent (R))# and subtracts it fro" the 6H.-, hich results in the net inco"e( )i*iding b% the shares outstanding gi*es us the 69,( Bor the all-e$uit% capital structure, the interest ter" is 2ero( -o find the brea&e*en 6H.- for to different capital structures, e si"pl% set the e$uations e$ual to each other and sol*e for 6H.-( -he brea&e*en 6H.- beteen the all-e$uit% capital structure and 9lan . is+ 6H.-:11,000 L Q6H.- E (10(<1M0,000#R:J,000 6H.- L <II,000 0nd the brea&e*en 6H.- beteen the all-e$uit% capital structure and 9lan .. is+ 6H.-:11,000 L Q6H.- E (10(<2I0,000#R:A,000 6H.- L <II,000 -he brea&-e*en le*els of 6H.- are the sa"e because of 3K3 9roposition .( c. ,etting the e$uations for 69, fro" 9lan . and 9lan .. e$ual to each other and sol*ing for 6H.-, e get+ Q6H.- E (10(<1M0,000#R:J,000 L Q6H.- E (10(<2I0,000#R:A,000 6H.- L <II,000 -his brea&-e*en le*el of 6H.- is the sa"e as in part b again because of 3K3 9roposition .( B-292 SOLUTIONS d. -he inco"e state"ent for each capitali2ation plan ith corporate inco"e ta!es is+ I && All)e!uit 6H.- <=9,000 <=9,000 <=9,000 .nterest 1M,000 2I,000 0 -a!es 9,200 M,000 1A,M00 4. < 1=,800 < 9,000 < 2=,I00 69, < 1(9J < 1(80 < 2(1= -he all-e$uit% plan still has the highest 69,8 9lan .. still has the loest 69,( We can calculate the 69, as+ 69, L Q(6H.- E R))#(1 E tC#R:,hares outstanding -his is si"ilar to the e$uation e used before, e!cept no e need to account for ta!es( 0gain, the interest e!pense ter" is 2ero in the all-e$uit% capital structure( ,o, the brea&e*en 6H.- beteen the all-e$uit% plan and 9lan . is+ 6H.-(1 E (I0#:11,000 L Q6H.- E (10(<1M0,000#R(1 E (I0#:J,000 6H.- L <II,000 -he brea&e*en 6H.- beteen the all-e$uit% plan and 9lan .. is+ 6H.-(1 E (I0#:11,000 L Q6H.- E (10(<2I0,000#R(1 E (I0#:A,000 6H.- L <II,000 0nd the brea&e*en beteen 9lan . and 9lan .. is+ Q6H.- E (10(<1M0,000#R(1 E (I0#:J,000 L Q6H.- E (10(<2I0,000#R(1 E (I0#:A,000 6H.- L <II,000 -he brea&-e*en le*els of 6H.- do not change because the addition of ta!es reduces the inco"e of all three plans b% the sa"e percentage8 therefore, the% do not change relati*e to one another( ". -o find the *alue per share of the stoc& under each capitali2ation plan, e can calculate the price as the *alue of shares repurchased di*ided b% the nu"ber of shares repurchased( ,o, under 9lan ., the *alue per share is+ 9 L <1M0,000:(11,000 E J,000 shares# 9 L <I0 per share 0nd under 9lan .., the *alue per share is+ 9 L <2I0,000:(11,000 E A,000 shares# 9 L <I0 per share -his shos that hen there are no corporate ta!es, the stoc&holder does not care about the capital structure decision of the fir"( -his is 3K3 9roposition . ithout ta!es( CHAPTER 16 B-293 #. a. -he earnings per share are+ 69, L <=2,000:8,000 shares 69, L <I(00 ,o, the cash flo for the co"pan% is+ Cash flo L <I(00(100 shares# Cash flo L <I00 b. -o deter"ine the cash flo to the shareholder, e need to deter"ine the 69, of the fir" under the proposed capital structure( -he "ar&et *alue of the fir" is+ V L <AA(8,000# V L <II0,000 Under the proposed capital structure, the fir" ill raise ne debt in the a"ount of+ ) L 0(=A(<II0,000# ) L <1AI,000 in debt( -his "eans the nu"ber of shares repurchased ill be+ ,hares repurchased L <1AI,000:<AA ,hares repurchased L 2,800 Under the ne capital structure, the co"pan% ill ha*e to "a&e an interest pa%"ent on the ne debt( -he net inco"e ith the interest pa%"ent ill be+ 4. L <=2,000 E (08(<1AI,000# 4. L <19,M80 -his "eans the 69, under the ne capital structure ill be+ 69, L <19,M80:(8,000 E 2,800# shares 69, L <=(J8IM ,ince all earnings are paid as di*idends, the shareholder ill recei*e+ ,hareholder cash flo L <=(J8IM(100 shares# ,hareholder cash flo L <=J8(IM c. -o replicate the proposed capital structure, the shareholder should sell =A percent of their shares, or =A shares, and lend the proceeds at 8 percent( -he shareholder ill ha*e an interest cash flo of+ .nterest cash flo L =A(<AA#((08# .nterest cash flo L <1AI B-294 SOLUTIONS -he shareholder ill recei*e di*idend pa%"ents on the re"aining MA shares, so the di*idends recei*ed ill be+ )i*idends recei*ed L <=(J8IM(MA shares# )i*idends recei*ed L <2IM -he total cash flo for the shareholder under these assu"ptions ill be+ -otal cash flo L <1AI O 2IM -otal cash flo L <I00 -his is the sa"e cash flo e calculated in part a( d. -he capital structure is irrele*ant because shareholders can create their on le*erage or unle*er the stoc& to create the pa%off the% desire, regardless of the capital structure the fir" actuall% chooses( $. a. -he rate of return earned ill be the di*idend %ield( -he co"pan% has debt, so it "ust "a&e an interest pa%"ent( -he net inco"e for the co"pan% is+ 4. L <80,000 E (08(<=00,000# 4. L <AM,000 -he in*estor ill recei*e di*idends in proportion to the percentage of the co"pan%'s share the% on( -he total di*idends recei*ed b% the shareholder ill be+ )i*idends recei*ed L <AM,000(<=0,000:<=00,000# )i*idends recei*ed L <A,M00 ,o the return the shareholder e!pects is+ R L <A,M00:<=0,000 R L (18MJ or 18(MJN b. -o generate e!actl% the sa"e cash flos in the other co"pan%, the shareholder needs to "atch the capital structure of 0HC( -he shareholder should sell all shares in U5^( -his ill net <=0,000( -he shareholder should then borro <=0,000( -his ill create an interest cash flo of+ .nterest cash flo L (08(E<=0,000# .nterest cash flo L E<2,I00 -he in*estor should then use the proceeds of the stoc& sale and the loan to bu% shares in 0HC( -he in*estor ill recei*e di*idends in proportion to the percentage of the co"pan%'s share the% on( -he total di*idends recei*ed b% the shareholder ill be+ )i*idends recei*ed L <80,000(<M0,000:<M00,000# )i*idends recei*ed L <8,000 CHAPTER 16 B-295 -he total cash flo for the shareholder ill be+ -otal cash flo L <8,000 E 2,I00 -otal cash flo L <A,M00 -he shareholders return in this case ill be+ R L <A,M00:<=0,000 R L (18MJ or 18(MJN c. 0HC is an all e$uit% co"pan%, so+ R6 L R0 L <80,000:<M00,000 R6 L (1=== or 1=(==N -o find the cost of e$uit% for U5^ e need to use 3K3 9roposition .., so+ R6 L R0 O (R0 E R)#():6#(1 E tC# R6 L (1=== O ((1=== E (08#(1#(1# R6 L (18MJ or 18(MJN d. -o find the W0CC for each co"pan% e need to use the W0CC e$uation+ W0CC L (6:V#R6 O ():V#R)(1 E tC# ,o, for 0HC, the W0CC is+ W0CC L (1#((1===# O (0#((08# W0CC L (1=== or 1=(==N 0nd for U5^, the W0CC is+ W0CC L (1:2#((18MJ# O (1:2#((08# W0CC L (1=== or 1=(==N When there are no corporate ta!es, the cost of capital for the fir" is unaffected b% the capital structure8 this is 3K3 9roposition .. ithout ta!es( 1%. With no ta!es, the *alue of an unle*ered fir" is the interest rate di*ided b% the unle*ered cost of e$uit%, so+ V L 6H.-:W0CC <2=,000,000 L 6H.-:(09 6H.- L (09(<2=,000,000# 6H.- L <2,0J0,000 B-296 SOLUTIONS 11. .f there are corporate ta!es, the *alue of an unle*ered fir" is+ VU L 6H.-(1 E tC#:RU Using this relationship, e can find 6H.- as+ <2=,000,000 L 6H.-(1 E (=A#:(09 6H.- L <=,18I,M1A(=8 -he W0CC re"ains at 9 percent( )ue to ta!es, 6H.- for an all-e$uit% fir" ould ha*e to be higher for the fir" to still be orth <2= "illion( 12. a. With the infor"ation pro*ided, e can use the e$uation for calculating W0CC to find the cost of e$uit%( -he e$uation for W0CC is+ W0CC L (6:V#R6 O ():V#R)(1 E tC# -he co"pan% has a debt-e$uit% ratio of 1(A, hich i"plies the eight of debt is 1(A:2(A, and the eight of e$uit% is 1:2(A, so W0CC L (10 L (1:2(A#R6 O (1(A:2(A#((0J#(1 E (=A# R6 L (1818 or 18(18N b. -o find the unle*ered cost of e$uit% e need to use 3K3 9roposition .. ith ta!es, so+ R6 L RU O (RU E R)#():6#(1 E tC# (1818 L RU O (RU E (0J#(1(A#(1 E (=A# RU L (12MM or 12(MMN c. -o find the cost of e$uit% under different capital structures, e can again use 3K3 9roposition .. ith ta!es( With a debt-e$uit% ratio of 2, the cost of e$uit% is+ R6 L RU O (RU E R)#():6#(1 E tC# R6 L (12MM O ((12MM E (0J#(2#(1 E (=A# R6 L (2001 or 20(01N With a debt-e$uit% ratio of 1(0, the cost of e$uit% is+ R6 L (12MM O ((12MM E (0J#(1#(1 E (=A# R6 L (1M=I or 1M(=IN 0nd ith a debt-e$uit% ratio of 0, the cost of e$uit% is+ R6 L (12MM O ((12MM E (0J#(0#(1 E (=A# R6 L RU L (12MM or 12(MMN CHAPTER 16 B-297 13. a( Bor an all-e$uit% financed co"pan%+ W0CC L RU L R6 L (11 or 11N b. -o find the cost of e$uit% for the co"pan% ith le*erage e need to use 3K3 9roposition .. ith ta!es, so+ R6 L RU O (RU E R)#():6#(1 E tC# R6 L (11 O ((11 E (082#((2A:(JA#((MA# R6 L (11M1 or 11(M1N c. Using 3K3 9roposition .. ith ta!es again, e get+ R6 L RU O (RU E R)#():6#(1 E tC# R6 L (11 O ((11 E (082#((A0:(A0#(1 E (=A# R6 L (1282 or 12(82N d. -he W0CC ith 2A percent debt is+ W0CC L (6:V#R6 O ():V#R)(1 E tC# W0CC L (JA((11M1# O (2A((082#(1 E (=A# W0CC L (100I or 10(0IN 0nd the W0CC ith A0 percent debt is+ W0CC L (6:V#R6 O ():V#R)(1 E tC# W0CC L (A0((1282# O (A0((082#(1 E (=A# W0CC L (0908 or 9(08N 14. a. -he *alue of the unle*ered fir" is+ VU L 6H.-(1 E tC#:RU VU L <92,000(1 E (=A#:(1A VU L <=98,MMM(MJ b. -he *alue of the le*ered fir" is+ VU L VU O t C ) VU L <=98,MMM(MJ O (=A(<M0,000# VU L <I19,MMM(MJ B-298 SOLUTIONS 1. We can find the cost of e$uit% using 3K3 9roposition .. ith ta!es( )oing so, e find+ R6 L RU O (RU E R)#():6#(1 E t# R6 L (1A O ((1A E (09#(<M0,000:<=98,MMJ#(1 E (=A# R6 L (1AMA or 1A(MAN Using this cost of e$uit%, the W0CC for the fir" after recapitali2ation is+ W0CC L (6:V#R6 O ():V#R)(1 E tC# W0CC L (1AMA(<=98,MMJ:<I19,MMJ# O (09(1 E (=A#(<M0,000:<I19,MMJ# W0CC L (1I2A or 1I(2AN When there are corporate ta!es, the o*erall cost of capital for the fir" declines the "ore highl% le*eraged is the fir"'s capital structure( -his is 3K3 9roposition . ith ta!es( &ntermediate 1!. -o find the *alue of the le*ered fir" e first need to find the *alue of an unle*ered fir"( ,o, the *alue of the unle*ered fir" is+ VU L 6H.-(1 E tC#:RU VU L (<MI,000#(1 E (=A#:(1A VU L <2JJ,===(== 4o e can find the *alue of the le*ered fir" as+ VC L VU O tC) VC L <2JJ,===(== O (=A(<9A,000# VC L <=10,A8=(== 0ppl%ing 3K3 9roposition . ith ta!es, the fir" has increased its *alue b% issuing debt( 0s long as 3K3 9roposition . holds, that is, there are no ban&ruptc% costs and so forth, then the co"pan% should continue to increase its debt:e$uit% ratio to "a!i"i2e the *alue of the fir"( 1". With no debt, e are finding the *alue of an unle*ered fir", so+ VU L 6H.-(1 E tC#:RU VU L <1I,000(1 E (=A#:(1M VU L <AM,8JA With debt, e si"pl% need to use the e$uation for the *alue of a le*ered fir"( With A0 percent debt, one-half of the fir" *alue is debt, so the *alue of the le*ered fir" is+ VC L VU O tC():V#VU VC L <AM,8JA O (=A((A0#(<AM,8JA# VC L <MM,828(1= CHAPTER 16 B-299 0nd ith 100 percent debt, the *alue of the fir" is+ VC L VU O tC():V#VU VC L <AM,8JA O (=A(1(0#(<AM,8JA# VC L <JM,J81(2A 1#. a. -o purchase A percent of Pnight's e$uit%, the in*estor ould need+ Pnight in*est"ent L (0A(<1,M=2,000# L <81,M00 0nd to purchase A percent of Veblen ithout borroing ould re$uire+ Veblen in*est"ent L (0A(<2,A00,000# L <12A,000 .n order to co"pare dollar returns, the initial net cost of both positions should be the sa"e( -herefore, the in*estor ill need to borro the difference beteen the to a"ounts, or+ 0"ount to borro L <12A,000 E 81,M00 L <I=,I00 0n in*estor ho ons A percent of Pnight's e$uit% ill be entitled to A percent of the fir"'s earnings a*ailable to co""on stoc& holders at the end of each %ear( While Pnight's e!pected operating inco"e is <I00,000, it "ust pa% <J2,000 to debt holders before distributing an% of its earnings to stoc&holders( ,o, the a"ount a*ailable to this shareholder ill be+ Cash flo fro" Pnight to shareholder L (0A(<I00,000 E J2,000# L <1M,I00 Veblen ill distribute all of its earnings to shareholders, so the shareholder ill recei*e+ Cash flo fro" Veblen to shareholder L (0A(<I00,000# L <20,000 >oe*er, to ha*e the sa"e initial cost, the in*estor has borroed <I=,I00 to in*est in Veblen, so interest "ust be paid on the borroings( -he net cash flo fro" the in*est"ent in Veblen ill be+ 4et cash flo fro" Veblen in*est"ent L <20,000 E (0M(<I=,I00# L <1J,=9M Bor the sa"e initial cost, the in*est"ent in Veblen produces a higher dollar return( b. Hoth of the to strategies ha*e the sa"e initial cost( ,ince the dollar return to the in*est"ent in Veblen is higher, all in*estors ill choose to in*est in Veblen o*er Pnight( -he process of in*estors purchasing Veblen's e$uit% rather than Pnight's ill cause the "ar&et *alue of Veblen's e$uit% to rise and the "ar&et *alue of Pnight's e$uit% to fall( 0n% differences in the dollar returns to the to strategies ill be eli"inated, and the process ill cease hen the total "ar&et *alues of the to fir"s are e$ual( B-300 SOLUTIONS Challenge 1$. 3K3 9roposition .. states+ R6 L RU O (RU E R)#():6#(1 E tC# 0nd the e$uation for W0CC is+ W0CC L (6:V#R6 O ():V#R)(1 E tC# ,ubstituting the 3K3 9roposition .. e$uation into the e$uation for W0CC, e get+ W0CC L (6:V#QRU O (RU E R)#():6#(1 E tC#R O ():V#R)(1 E tC# Rearranging and reducing the e$uation, e get+ W0CC L RUQ(6:V# O (6:V#():6#(1 E tC#R O R)(1 E tC#Q():V# E (6:V#():6#R W0CC L RUQ(6:V# O ():V#(1 E tC#R W0CC L RUQY(6O)#:VZ E tC():V#R W0CC L RUQ1 E tC():V#R 2%. -he return on e$uit% is net inco"e di*ided b% e$uit%( 4et inco"e can be e!pressed as+ 4. L (6H.- E R))#(1 E tC# ,o, R/6 is+ R6 L (6H.- E R))#(1 E tC#:6 4o e can rearrange and substitute as follos to arri*e at 3K3 9roposition .. ith ta!es+ R6 L Q6H.-(1 E tC#:6R E QR)():6#(1 E tC#R R6 L RUVU:6 E QR)():6#(1 E tC#R R6 L RU(VC E tC)#:6 E QR)():6#(1 E tC#R R6 L RU(6 O ) E tC)#:6 E QR)():6#(1 E tC#R R6 L RU O (RU E R)#():6#(1 E tC# CHAPTER 16 B-301 21. 3K3 9roposition .., ith no ta!es is+ R6 L RU O (RU E Rf#():6# 4ote that e use the ris&-free rate as the return on debt( -his is an i"portant assu"ption of 3K3 9roposition ..( -he C093 to calculate the cost of e$uit% is e!pressed as+ R6 L β6(R3 E Rf# O Rf We can rerite the C093 to e!press the return on an unle*ered co"pan% as+ R0 L β0(R3 E Rf# O Rf We can no substitute the C093 for an unle*ered co"pan% into 3K3 9roposition ..( )oing so and rearranging the ter"s e get+ R6 L β0(R3 E Rf# O Rf O Qβ0(R3 E Rf# O Rf E RfR():6# R6 L β0(R3 E Rf# O Rf O Qβ0(R3 E Rf#R():6# R6 L (1 O ):6#β0(R3 E Rf# O Rf 4o e set this e$uation e$ual to the C093 e$uation to calculate the cost of e$uit% and reduce+ β6(R3 E Rf# O Rf L (1 O ):6#β0(R3 E Rf# O Rf β6(R3 E Rf# L (1 O ):6#β0(R3 E Rf# β6 L β0(1 O ):6# 22. Using the e$uation e deri*ed in 9roble" 21+ β6 L β0(1 O ):6# -he e$uit% beta for the respecti*e asset betas is+ )ebt-e$uit% ratio 6$uit% beta 0 1(1 O 0# L 1 1 1(1 O 1# L 2 A 1(1 O A# L M 20 1(1 O 20# L 21 -he e$uit% ris& to the shareholder is co"posed of both business and financial ris&( 6*en if the assets of the fir" are not *er% ris&%, the ris& to the shareholder can still be large if the financial le*erage is high( -hese higher le*els of ris& ill be reflected in the shareholder's re$uired rate of return R6, hich ill increase ith higher debt:e$uit% ratios( CHAPTER 17 DIVIDENDS AND DIVIDEND POLICY Answers to Concepts Review and Critical Thinking Questions 1. )i*idend polic% deals ith the ti"ing of di*idend pa%"ents, not the a"ounts ulti"atel% paid( )i*idend polic% is irrele*ant hen the ti"ing of di*idend pa%"ents doesn't affect the present *alue of all future di*idends( 2. 0 stoc& repurchase reduces e$uit% hile lea*ing debt unchanged( -he debt ratio rises( 0 fir" could, if desired, use e!cess cash to reduce debt instead( -his is a capital structure decision( 3. Brida%, )ece"ber 29 is the e!-di*idend da%( Re"e"ber not to count Januar% 1 because it is a holida%, and the e!changes are closed( 0n%one ho bu%s the stoc& before )ece"ber 29 is entitled to the di*idend, assu"ing the% do not sell it again before )ece"ber 29( 4. 4o, because the "one% could be better in*ested in stoc&s that pa% di*idends in cash hich benefit the fundholders directl%( . -he change in price is due to the change in di*idends, not due to the change in di*idend polic( )i*idend polic% can still be irrele*ant ithout a contradiction( !. -he stoc& price dropped because of an e!pected drop in future di*idends( ,ince the stoc& price is the present *alue of all future di*idend pa%"ents, if the e!pected future di*idend pa%"ents decrease, then the stoc& price ill decline( ". -he plan ill probabl% ha*e little effect on shareholder ealth( -he shareholders can rein*est on their on, and the shareholders "ust pa% the ta!es on the di*idends either a%( >oe*er, the shareholders ho ta&e the option "a% benefit at the e!pense of the ones ho don't (because of the discount#( 0lso as a result of the plan, the fir" ill be able to raise e$uit% b% pa%ing a 10N flotation cost (the discount#, hich "a% be a s"aller discount than the "ar&et flotation costs of a ne issue for so"e co"panies( #. .f these fir"s 1ust ent public, the% probabl% did so because the% ere groing and needed the additional capital( Groth fir"s t%picall% pa% *er% s"all cash di*idends, if the% pa% a di*idend at all( -his is because the% ha*e nu"erous pro1ects a*ailable, and the% rein*est the earnings in the fir" instead of pa%ing cash di*idends( $. -he stoc& price drop on the e!-di*idend date should be loer( With ta!es, stoc& prices should drop b% the a"ount of the di*idend, less the ta!es in*estors "ust pa% on the di*idends( 0 loer ta! rate loers the in*estors' ta! liabilit%( 1%. With a high ta! on di*idends and a lo ta! on capital gains, in*estors, in general, ill prefer capital gains( .f the di*idend ta! rate declines, the attracti*eness of di*idends increases( CHAPTER 17 B-303 Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he afterta! di*idend is the preta! di*idend ti"es one "inus the ta! rate, so+ 0fterta! di*idend L <I(M0(1 E (1A# L <=(91 -he stoc& price should drop b% the afterta! di*idend a"ount, or+ 6!-di*idend price L <80(=J E =(91 L <JM(IM 2. a. -he shares outstanding increases b% 10 percent, so+ 4e shares outstanding L =0,000(1(10# L ==,000 4e shares issued L =,000 ,ince the par *alue of the ne shares is <1, the capital surplus per share is <29( -he total capital surplus is therefore+ Capital surplus on ne shares L =,000(<29# L <8J,000 Co""on stoc& (<1 par *alue# < ==,000 Capital surplus =J2,000 Retained earnings AA9,180 <9MI,180 b. -he shares outstanding increases b% 2A percent, so+ 4e shares outstanding L =0,000(1(2A# L =J,A00 4e shares issued L J,A00 ,ince the par *alue of the ne shares is <1, the capital surplus per share is <29( -he total capital surplus is therefore+ Capital surplus on ne shares L J,A00(<29# L <21J,A00 Co""on stoc& (<1 par *alue# < =J,A00 Capital surplus A02,A00 Retained earnings I2I,180 <9MI,180 B-304 SOLUTIONS 3. a. -o find the ne shares outstanding, e "ultipl% the current shares outstanding ti"es the ratio of ne shares to old shares, so+ 4e shares outstanding L =0,000(I:1# L 120,000 -he e$uit% accounts are unchanged e!cept the par *alue of the stoc& is changed b% the ratio of ne shares to old shares, so the ne par *alue is+ 4e par *alue L <1(1:I# L <0(2A per share( b. -o find the ne shares outstanding, e "ultipl% the current shares outstanding ti"es the ratio of ne shares to old shares, so+ 4e shares outstanding L =0,000(1:A# L M,000( -he e$uit% accounts are unchanged e!cept the par *alue of the stoc& is changed b% the ratio of ne shares to old shares, so the ne par *alue is+ 4e par *alue L <1(A:1# L <A(00 per share( 4. -o find the ne stoc& price, e "ultipl% the current stoc& price b% the ratio of old shares to ne shares, so+ a. <90(=:A# L <AI(00 b. <90(1:1(1A# L <J8(2M c. <90(1:1(I2A# L <M=(1M d. <90(J:I# L <1AJ(A0 e. -o find the ne shares outstanding, e "ultipl% the current shares outstanding ti"es the ratio of ne shares to old shares, so+ a: =A0,000(A:=# L A8=,=== b: =A0,000(1(1A# L I02,A00 c: =A0,000(1(I2A# L I98,JA0 d: =A0,000(I:J# L 200,000 . -he stoc& price is the total "ar&et *alue of e$uit% di*ided b% the shares outstanding, so+ 90 L <=08,A00 e$uit%:8,000 shares L <=8(AM per share .gnoring ta! effects, the stoc& price ill drop b% the a"ount of the di*idend, so+ 9U L <=8(AM E 1(=0 L <=J(2M CHAPTER 17 B-305 -he total di*idends paid ill be+ <1(=0 per share(8,000 shares# L <10,I00 -he e$uit% and cash accounts ill both decline b% <10,I00( !. Repurchasing the shares ill reduce cash and shareholders' e$uit% b% <10,I00( -he shares repurchased ill be the total purchase a"ount di*ided b% the stoc& price, so+ ,hares bought L <10,I00:<=8(AM L 2M9(M9 0nd the ne shares outstanding ill be+ 4e shares outstanding L 8,000 E 2M9(M9 L J,J=0(=1 0fter repurchase, the ne stoc& price is+ ,hare price L (<=08,A00 E 10,I00#:J,J=0(=1 shares L <=8(AM -he repurchase is effecti*el% the sa"e as the cash di*idend because %ou either hold a share orth <=8(AM, or a share orth <=J(2M and <1(=0 in cash( -herefore, %ou participate in the repurchase according to the di*idend pa%out percentage8 %ou are unaffected( ". -he stoc& price is the total "ar&et *alue of e$uit% di*ided b% the shares outstanding, so+ 90 L <IJ1,000 e$uit%:12,000 shares L <=9(2A per share -he shares outstanding ill increase b% 2A percent, so+ 4e shares outstanding L 12,000(1(2A# L 1A,000 -he ne stoc& price is the "ar&et *alue of e$uit% di*ided b% the ne shares outstanding, so+ 9U L <IJ1,000:1A,000 shares L <=1(I0 #. With a stoc& di*idend, the shares outstanding ill increase b% one plus the di*idend a"ount, so+ 4e shares outstanding L I0M,000(1(1A# L IMM,900 -he capital surplus is the capital paid in e!cess of par *alue, hich is <1, so+ Capital surplus for ne shares L M0,900(<=I# L <2,0J0,M00 -he ne capital surplus ill be the old capital surplus plus the additional capital surplus for the ne shares, so+ Capital surplus L <1,=I0,000 O 2,0J0,M00 L <=,I10,M00 B-306 SOLUTIONS -he ne e$uit% portion of the balance sheet ill loo& li&e this+ Co""on stoc& (<1 par *alue# < IMM,900 Capital surplus =,I10,M00 Retained earnings 1,29A,A00 <A,1J=,000 $. -he onl% e$uit% account that ill be affected is the par *alue of the stoc&( -he par *alue ill change b% the ratio of old shares to ne shares, so+ 4e par *alue L <1(1:I# L <0(2A per share( -he total di*idends paid this %ear ill be the di*idend a"ount ti"es the nu"ber of shares outstanding( -he co"pan% had I0M,000 shares outstanding before the split( We "ust re"e"ber to ad1ust the shares outstanding for the stoc& split, so+ -otal di*idends paid this %ear L <0(8A(I0M,000 shares#(I:1 split# L <1,=80,I00 -he di*idends increased b% 10 percent, so the total di*idends paid last %ear ere+ Cast %ear's di*idends L <1,=80,I00:1(10 L <1,2AI,909(09 0nd to find the di*idends per share, e si"pl% di*ide this a"ount b% the shares outstanding last %ear( )oing so, e get+ )i*idends per share last %ear L <1,2AI,909(09:I0M,000 shares L <=(09 &ntermediate 1%. -he price of the stoc& toda% is the 9V of the di*idends, so+ 90 L <2(=0:1(1A O <A=:1(1A 2 L <I2(08 -o find the e$ual to %ear di*idends ith the sa"e present *alue as the price of the stoc&, e set up the folloing e$uation and sol*e for the di*idend (4ote+ -he di*idend is a to %ear annuit%, so e could sol*e ith the annuit% factor as ell#+ <I2(08 L ):1(1A O ):1(1A 2 ) L <2A(88 We no &no the cash flo per share e ant each of the ne!t to %ears( We can find the price of stoc& in one %ear, hich ill be+ 91 L <A=:1(1A L <IM(09 ,ince %ou on 1,000 shares, in one %ear %ou ant+ Cash flo in 5ear one L 1,000(<2A(88# L <2A,881(I0 CHAPTER 17 B-307 Hut %ou'll onl% get+ )i*idends recei*ed in one %ear L 1,000(<2(=0# L <2,=00 -hus, in one %ear %ou ill need to sell additional shares in order to increase %our cash flo( -he nu"ber of shares to sell in %ear one is+ ,hares to sell at ti"e one L (<2A,881(I0 E 2,=00#:<IM(09 L A11(MJ shares 0t 5ear 2, %ou cash flo ill be the di*idend pa%"ent ti"es the nu"ber of shares %ou still on, so the 5ear 2 cash flo is+ 5ear 2 cash flo L <A=(1,000 E A11(MJ# L <2A,881(I0 11. .f %ou onl% ant <JA0 in 5ear 1, %ou ill bu%+ (<2,=00 E JA0#:<IM(09 L ==(M= shares at ti"e 1( 5our di*idend pa%"ent in 5ear 2 ill be+ 5ear 2 di*idend L (1,000 O ==(M=#(<A=# L <AI,J82(A0 4ote, the present *alue of each cash flo strea" is the sa"e( Helo e sho this b% finding the present *alues as+ 9V L <JA0:1(1A O <AI,J82(A0:1(1A 2 L <I2,0JA(M1 9V L 1,000(<2(=0#:1(1A O 1,000(<A=#:1(1A 2 L <I2,0JA(M1 12. a. .f the co"pan% "a&es a di*idend pa%"ent, e can calculate the ealth of a shareholder as+ )i*idend per share L <9,000:1,000 shares L <9(00 -he stoc& price after the di*idend pa%"ent ill be+ 9U L <MI E 9 L <AA per share -he shareholder ill ha*e a stoc& orth <AA and a <9 di*idend for a total ealth of <MI( .f the co"pan% "a&es a repurchase, the co"pan% ill repurchase+ ,hares repurchased L <9,000:<MI L 1I0(M= shares .f the shareholder lets their shares be repurchased, the% ill ha*e <MI in cash( .f the shareholder &eeps their shares, the%'re still orth <MI( B-308 SOLUTIONS b. .f the co"pan% pa%s di*idends, the current 69, is <1(=0, and the 9:6 ratio is+ 9:6 L <AA:<1(=0 L I2(=1 .f the co"pan% repurchases stoc&, the nu"ber of shares ill decrease( -he total net inco"e is the 69, ti"es the current nu"ber of shares outstanding( )i*iding net inco"e b% the ne nu"ber of shares outstanding, e find the 69, under the repurchase is+ 69, L <1(=0(1,000#:(1,000 − 1I0(M=# L <1(A1 -he stoc& price ill re"ain at <MI per share, so the 9:6 ratio is+ 9:6 L <MI:<1(A1 L I2(=1 c. 0 share repurchase ould see" to be the preferred course of action( /nl% those shareholders ho ish to sell ill do so, gi*ing the shareholder a ta! ti"ing option that he or she doesn't get ith a di*idend pa%"ent( Challenge 13. 0ssu"ing no capital gains ta!, the afterta! return for the Gordon Co"pan% is the capital gains groth rate, plus the di*idend %ield ti"es one "inus the ta! rate( Using the constant groth di*idend "odel, e get+ 0fterta! return L g O )(1 E t# L (1A ,ol*ing for g, e get+ (1A L g O (0A(1 E (=A# g L (11JA -he e$ui*alent preta! return for Gordon Co"pan%, hich pa%s no di*idend, is+ 9reta! return L g O ) L (11JA O (0A L (1MJA or 1M(JAN 14. Using the e$uation for the decline in the stoc& price e!-di*idend for each of the ta! rate policies, e get+ (90 E 9U#:) L (1 E -9#:(1 E -G# a. 90 E 9U L )(1 E 0#:(1 E 0# 90 E 9U L ) b. 90 E 9U L )(1 E (1A#:(1 E 0# 90 E 9U L (8A) c. 90 E 9U L )(1 E (1A#:(1 E (=0# 90 E 9U L 1(21I=) CHAPTER 17 B-309 d. With this ta! polic%, e si"pl% need to "ultipl% the personal ta! rate ti"es one "inus the di*idend e!e"ption percentage, so+ 90 E 9U L )Q1 E ((=A#((=0#R:(1 E (=A# 90 E 9U L 1(=JJ) e. ,ince different in*estors ha*e idel% *ar%ing ta! rates on ordinar% inco"e and capital gains, di*idend pa%"ents ha*e different after-ta! i"plications for different in*estors( -his differential ta!ation a"ong in*estors is one aspect of hat e ha*e called the clientele effect( 1. ,ince the <2,000,000 cash is after corporate ta!, the full a"ount ill be in*ested( ,o, the *alue of each alternati*e is+ Alternative - : -he fir" in*ests in --bills or in preferred stoc&, and then pa%s out as special di*idend in = %ears &f the firm invests in T)%ills+ .f the fir" in*ests in --bills, the afterta! %ield of the --bills ill be+ 0fterta! corporate %ield L (0A(1 E (=A# 0fterta! corporate %ield L (0=2A or =(2AN ,o, the future *alue of the corporate in*est"ent in --bills ill be+ BV of in*est"ent in --bills L <2,000,000(1 O (0=2A# = BV of in*est"ent in --bills L <2,201,I0M(1M ,ince the future *alue ill be paid to shareholders as a di*idend, the afterta! cash flo ill be+ 0fterta! cash flo to shareholders L <2,201,I0M(1M(1 E (1A# 0fterta! cash flo to shareholders L <1,8J1,19A(2= &f the firm invests in preferred stoc:: .f the fir" in*ests in preferred stoc&, the assu"ption ould be that the di*idends recei*ed ill be rein*ested in the sa"e preferred stoc&( -he preferred stoc& ill pa% a di*idend of+ 9referred di*idend L (08(<2,000,000# 9referred di*idend L <1M0,000 ,ince J0 percent of the di*idends are e!cluded fro" ta!+ -a!able preferred di*idends L (1 E (J0#(<220,000# -a!able preferred di*idends L <I8,000 0nd the ta!es the co"pan% "ust pa% on the preferred di*idends ill be+ -a!es on preferred di*idends L (=A(<I8,000# -a!es on preferred di*idends L <1M,800 B-310 SOLUTIONS ,o, the afterta! di*idend for the corporation ill be+ 0fterta! corporate di*idend L <1M0,000 E 1M,800 0fterta! corporate di*idend L <1I=,200 -his "eans the afterta! corporate di*idend %ield is+ 0fterta! corporate di*idend %ield L <1I=,200 : <2,000,000 0fterta! corporate di*idend %ield L (0J1M or J(1MN -he future *alue of the co"pan%'s in*est"ent in preferred stoc& ill be+ BV of in*est"ent in preferred stoc& L <2,000,000(1 O (0J1M# = BV of in*est"ent in preferred stoc& L <2,IM1,09=(I8 ,ince the future *alue ill be paid to shareholders as a di*idend, the afterta! cash flo ill be+ 0fterta! cash flo to shareholders L <2,IM1,09=(I8(1 E (1A# 0fterta! cash flo to shareholders L <2,091,92M(IM Alternative 1: -he fir" pa%s out di*idend no, and indi*iduals in*est on their on( -he afterta! cash recei*ed b% shareholders no ill be+ 0fterta! cash recei*ed toda% L <2,000,000(1 E (1A# 0fterta! cash recei*ed toda% L <1,J00,000 The individuals invest in Treasur bills: .f the shareholders in*est the current afterta! di*idends in -reasur% bills, the afterta! indi*idual %ield ill be+ 0fterta! indi*idual %ield on --bills L (0A(1 E (=1# 0fterta! indi*idual %ield on --bills L (0=IA or =(IAN ,o, the future *alue of the indi*idual in*est"ent in -reasur% bills ill be+ BV of in*est"ent in --bills L <1,J00,000(1 O (0=IA# = BV of in*est"ent in --bills L <1,882,090(08 The individuals invest in preferred stoc:: .f the indi*idual in*ests in preferred stoc&, the assu"ption ould be that the di*idends recei*ed ill be rein*ested in the sa"e preferred stoc&( -he preferred stoc& ill pa% a di*idend of+ 9referred di*idend L (08(<1,J00,000# 9referred di*idend L <1=M,000 CHAPTER 17 B-311 0nd the ta!es on the preferred di*idends ill be+ -a!es on preferred di*idends L (=1(<1=M,000# -a!es on preferred di*idends L <I2,1M0 ,o, the afterta! preferred di*idend ill be+ 0fterta! preferred di*idend L <1=M,000 E I2,1M0 0fterta! preferred di*idend L <9=,8I0 -his "eans the afterta! indi*idual di*idend %ield is+ 0fterta! corporate di*idend %ield L <9=,8I0 : <1,J00,000 0fterta! corporate di*idend %ield L (0AA2 or A(A2N -he future *alue of the indi*idual in*est"ent in preferred stoc& ill be+ BV of in*est"ent in preferred stoc& L <1,J00,000(1 O (0AA2# = BV of in*est"ent in preferred stoc& L <1,99J,=IA(8I -he afterta! cash flo for the shareholders is "a!i"i2ed hen the fir" in*ests the cash in the preferred stoc&s and pa%s a special di*idend later( 1!. a. Cet < be the ordinar% inco"e ta! rate( -he indi*idual recei*es an after-ta! di*idend of+ 0fterta! di*idend L <1,000(1 E <# hich she in*ests in -reasur% bonds( -he -reasur% bond ill generate afterta! cash flos to the in*estor of+ 0fterta! cash flo fro" -reasur% bonds L <1,000(1 E <#Q1 O (0M(1 E <#R .f the fir" in*ests the "one%, its proceeds are+ Bir" proceeds L <1,000Q1 O (0M(1 E (=A#R 0nd the proceeds to the in*estor hen the fir" pa%s a di*idend ill be+ 9roceeds if fir" in*ests first L (1 E <#Y<1,000Q1 O (0M(1 E (=A#RZ -o be indifferent, the in*estor's proceeds "ust be the sa"e hether she in*ests the afterta! di*idend or recei*es the proceeds fro" the fir"'s in*est"ent and pa%s ta!es on that a"ount( -o find the rate at hich the in*estor ould be indifferent, e can set the to e$uations e$ual, and sol*e for <( )oing so, e find+ <1,000(1 E <#Q1 O (0M(1 E <#R L (1 E <#Y<1,000Q1 O (0M(1 E (=A#RZ 1 O (0M(1 E <# L 1 O (0M(1 E (=A# < L (=A or =AN 4ote that this argu"ent does not depend upon the length of ti"e the in*est"ent is held( B-312 SOLUTIONS b. 5es, this is a reasonable anser( ,he is onl% indifferent if the afterta! proceeds fro" the <1,000 in*est"ent in identical securities are identical( -hat occurs onl% hen the ta! rates are identical( c. ,ince both in*estors ill recei*e the sa"e pre-ta! return, %ou ould e!pect the sa"e anser as in part a( 5et, because Carlson en1o%s a ta! benefit fro" in*esting in stoc& (J0 percent of inco"e fro" stoc& is e!e"pt fro" corporate ta!es#, the ta! rate on ordinar% inco"e hich induces indifference, is "uch loer( 0gain, set the to e$uations e$ual and sol*e for <+ <1,000(1 E <#Q1 O (09(1 E <#R L (1 E <#(<1,000Y1 O (09Q(J0 O (1 E (J0#(1 E (=A#RZ# 1 O (09(1 E <# L 1 O (09Q(J0 O (1 E (J0#(1 E (=A#R < L (10A0 or 10(A0N d. .t is a co"pelling argu"ent, but there are legal constraints, hich deter fir"s fro" in*esting large su"s in stoc& of other co"panies( CHAPTER 18 SHORT-TERM FINANCE AND PLANNING Answers to Concepts Review and Critical Thinking Questions 1. -hese are fir"s ith relati*el% long in*entor% periods and:or relati*el% long recei*ables periods( -hus, such fir"s tend to &eep in*entor% on hand, and the% allo custo"ers to purchase on credit and ta&e a relati*el% long ti"e to pa%( 2. -hese are fir"s that ha*e a relati*el% long ti"e beteen the ti"e purchased in*entor% is paid for and the ti"e that in*entor% is sold and pa%"ent recei*ed( -hus, these are fir"s that ha*e relati*el% short pa%ables periods and:or relati*el% long recei*able c%cles( 3. a. Use+ -he cash balance declined b% <200 to pa% the di*idend( b. ,ource+ -he cash balance increased b% <A00, assu"ing the goods bought on pa%ables credit ere sold for cash( c. Use+ -he cash balance declined b% <900 to pa% for the fi!ed assets( d. Use+ -he cash balance declined b% <M2A to pa% for the higher le*el of in*entor%( e. Use+ -he cash balance declined b% <1,200 to pa% for the rede"ption of debt( 4. Carr%ing costs ill decrease because the% are not holding goods in in*entor%( ,hortage costs ill probabl% increase depending on ho close the suppliers are and ho ell the% can esti"ate need( -he operating c%cle ill decrease because the in*entor% period is decreased( . ,ince the cash c%cle e$uals the operating c%cle "inus the accounts pa%able period, it is not possible for the cash c%cle to be longer than the operating c%cle if the accounts pa%able period is positi*e( 3oreo*er, it is unli&el% that the accounts pa%able period ould e*er be negati*e since that i"plies the fir" pa%s its bills before the% are incurred( !. .t lengthened its pa%ables period, thereb% shortening its cash c%cle( .t ill ha*e no effect on the operating c%cle( ". -he supplier's recei*ables period ill increase, thereb% increasing their operating and cash c%cles( #. .t is so"eti"es argued that large fir"s ta&e ad*antage of s"aller fir"s b% threatening to ta&e their business elsehere( >oe*er, considering a "o*e to another supplier to get better ter"s is the nature of co"petiti*e free enterprise( $. -he% ould li&e toX -he pa%ables period is a sub1ect of "uch negotiation, and it is one aspect of the price a fir" pa%s its suppliers( 0 fir" ill generall% negotiate the best possible co"bination of pa%ables B-314 SOLUTIONS period and price( -%picall%, suppliers pro*ide strong financial incenti*es for rapid pa%"ent( -his issue is discussed in detail in a later chapter on credit polic%( CHAPTER 18 B-315 1%. Hlue,&% ill need less financing because it is essentiall% borroing "ore fro" its suppliers( 0"ong other things, Hlue,&% ill li&el% need less short-ter" borroing fro" other sources, so it ill sa*e on interest e!pense( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. a. 4o change( 0 di*idend paid for b% the sale of debt ill not change cash since the cash raised fro" the debt offer goes i""ediatel% to shareholders( b. 4o change( -he real estate is paid for b% the cash raised fro" the debt, so this ill not change the cash balance( c. 4o change( .n*entor% and accounts pa%able ill increase, but neither ill i"pact the cash account( d. )ecrease( -he short-ter" ban& loan is repaid ith cash, hich ill reduce the cash balance( e( )ecrease( -he pa%"ent of ta!es is a cash transaction( f( )ecrease( -he preferred stoc& ill be repurchased ith cash( g. 4o change( 0ccounts recei*able ill increase, but cash ill not increase until the sales are paid off( h( )ecrease( -he interest is paid ith cash, hich ill reduce the cash balance( i( .ncrease( When pa%"ents for pre*ious sales, or accounts recei*able, are paid off, the cash balance increases since the pa%"ent "ust be "ade in cash( (. )ecrease( -he accounts pa%able are reduced through cash pa%"ents to suppliers( :. )ecrease( >ere the di*idend pa%"ents are "ade ith cash, hich is generall% the case( -his is different fro" part a here debt as raised to "a&e the di*idend pa%"ent( l. 4o change( -he short-ter" note ill not change the cash balance( m. )ecrease( -he utilit% bills "ust be paid in cash( n. )ecrease( 0 cash pa%"ent ill reduce cash( o. .ncrease( .f "ar&etable securities are sold, the co"pan% ill recei*e cash fro" the sale( B-316 SOLUTIONS 2. The total liabilities and equity of the company are the net book worth, or market value of equity, plus current liabilities and long-term debt, so: -otal liabilities and e$uit% L <10,=80 O 1,IA0 O J,A00 -otal liabilities and e$uit% L <19,==0 -his is also e$ual to the total assets of the co"pan%( ,ince total assets are the su" of all assets, and cash is an asset, the cash account "ust be e$ual to total assets "inus all other assets, so+ Cash L <19,==0 E 1A,190 E 2,10A Cash L <2,0=A We ha*e 4WC other than cash, so the total 4WC is+ 4WC L <2,10A O 2,0=A 4WC L <I,1I0 We can find total current assets b% using the 4WC e$uation( 4WC is e$ual to+ 4WC L C0 E CC <I,1I0 L C0 E <1,IA0 C0 L <A,A90 3. a. .ncrease( .f recei*ables go up, the ti"e to collect the recei*ables ould increase, hich increases the operating c%cle( b. .ncrease( .f credit repa%"ent ti"es are increased, custo"ers ill ta&e longer to pa% their bills, hich ill lead to an increase in the operating c%cle( c. )ecrease( .f the in*entor% turno*er increases, the in*entor% period decreases( d. 4o change( -he accounts pa%able period is part of the cash c%cle, not the operating c%cle( e. )ecrease( .f the recei*ables turno*er increases, the recei*ables period decreases( f. 4o change( 9a%"ents to suppliers affects the accounts pa%able period, hich is part of the cash c%cle, not the operating c%cle( 4. a. .ncrease8 .ncrease( .f the ter"s of the cash discount are "ade less fa*orable to custo"ers, the accounts recei*able period ill lengthen( -his ill increase both the cash c%cle and the operating c%cle( b. .ncrease8 4o change( -his ill shorten the accounts pa%able period, hich ill increase the cash c%cle( .t ill ha*e no effect on the operating c%cle since the accounts pa%able period is not part of the operating c%cle( CHAPTER 18 B-317 c. )ecrease8 )ecrease( .f "ore custo"ers pa% in cash, the accounts recei*able period ill decrease( -his ill decrease both the cash c%cle and the operating c%cle( d. )ecrease8 )ecrease( 0ssu"e the accounts pa%able period does not change( Beer ra "aterials purchased ill reduce the in*entor% period, hich ill decrease both the cash c%cle and the operating c%cle( e. )ecrease8 4o change( .f "ore ra "aterials are purchased on credit, the accounts pa%able period ill tend to increase, hich ould decrease the cash c%cle( We should sa% that this "a% not be the case( -he accounts pa%able period is a decision "ade b% the co"pan%'s "anage"ent( -he co"pan% could increase the accounts pa%able account and still "a&e the pa%"ents in the sa"e nu"ber of da%s( -his ould lea*e the accounts pa%able period unchanged, hich ould lea*e the cash c%cle unchanged( -he change in credit purchases "ade on credit ill not affect the in*entor% period or the accounts pa%able period, so the operating c%cle ill not change( f. .ncrease8 .ncrease( .f "ore goods are produced for in*entor%, the in*entor% period ill increase( -his ill increase both the cash c%cle and operating c%cle( . a. 0 IA-da% collection period i"plies all recei*ables outstanding fro" the pre*ious $uarter are collected in the current $uarter, and+ (90 E IA#:90 L 1:2 of current sales are collected( ,o+ E- E1 E3 EF Heginning recei*ables <=M0(00 <=9A(00 <=J0(00 <I=A(00 ,ales J90(00 JI0(00 8J0(00 9A0(00 Cash collections (JAA(00# (JMA(00# (80A(00# (910(00# 6nding recei*ables <=9A(00 <=J0(00 <I=A(00 <IJA(00 b. 0 M0-da% collection period i"plies all recei*ables outstanding fro" pre*ious $uarter are collected in the current $uarter, and+ (90-M0#:90 L 1:= of current sales are collected( ,o+ E- E1 E3 EF Heginning recei*ables <=M0(00 <A2M(MJ <I9=(== <A80(00 ,ales J90(00 JI0(00 8J0(00 9A0(00 Cash collections (M2=(==# (JJ=(==# (J8=(==# (89M(MJ# 6nding recei*ables <A2M(MJ <I9=(== <A80(00 <M==(== B-318 SOLUTIONS c. 0 =0-da% collection period i"plies all recei*ables outstanding fro" pre*ious $uarter are collected in the current $uarter, and+ (90-=0#:90 L 2:= of current sales are collected( ,o+ E- E1 E3 EF Heginning recei*ables <=M0(00 <2M=(== <2IM(MJ <290(00 ,ales J90(00 JI0(00 8J0(00 9A0(00 Cash collections (88M(MJ# (JAM(MJ# (82M(MJ# (92=(==# 6nding recei*ables <2M=(== <2IM(MJ <290(00 <=1M(MJ !. -he operating c%cle is the in*entor% period plus the recei*ables period( -he in*entor% turno*er and in*entor% period are+ .n*entor% turno*er L C/G,:0*erage in*entor% .n*entor% turno*er L <AM,=8I:YQ<9,J80 O 11,=80R:2Z .n*entor% turno*er L A(=29= ti"es .n*entor% period L =MA da%s:.n*entor% turno*er .n*entor% period L =MA da%s:A(=29= .n*entor% period L M8(I9 da%s 0nd the recei*ables turno*er and recei*ables period are+ Recei*ables turno*er L Credit sales:0*erage recei*ables Recei*ables turno*er L <89,80I:YQ<I,108 O I,9=8R:2Z Recei*ables turno*er L 19(8AA0 ti"es Recei*ables period L =MA da%s:Recei*ables turno*er Recei*ables period L =MA da%s:19(8AA0 Recei*ables period L 18(=8 da%s ,o, the operating c%cle is+ /perating c%cle L M8(I9 da%s O 18(=8 da%s /perating c%cle L 8M(8J da%s -he cash c%cle is the operating c%cle "inus the pa%ables period( -he pa%ables turno*er and pa%ables period are+ 9a%ables turno*er L C/G,:0*erage pa%ables 9a%ables turno*er L <AM,=8I:YQ<J,M=M O J,92JR:2Z 9a%ables turno*er L J(2IA9 ti"es 9a%ables period L =MA da%s:9a%ables turno*er 9a%ables period L =MA da%s:J(2IA9 9a%ables period L A0(=J da%s CHAPTER 18 B-319 ,o, the cash c%cle is+ Cash c%cle L 8M(8J da%s E A0(=J da%s Cash c%cle L =M(A0 da%s -he fir" is recei*ing cash on a*erage =M(A0 da%s after it pa%s its bills( ". .f e factor i""ediatel%, e recei*e cash on an a*erage of =2 da%s sooner( -he nu"ber of periods in a %ear are+ 4u"ber of periods L =MA:=2 4u"ber of periods L 11(I0M= -he 60R of this arrange"ent is+ 60R L (1 O 9eriodic rate# " E 1 60R L (1 O 1(A:98(A# 11(I0M= E 1 60R L (1881 or 18(81N #. a. -he pa%ables period is 2ero since the co"pan% pa%s i""ediatel%( -he pa%"ent in each period is =0 percent of ne!t period's sales, so+ E- E1 E3 EF 9a%"ent of accounts <2A8(00 <2J9(00 <29J(00 <282(90 b. ,ince the pa%ables period is 90 da%s, the pa%"ent in each period is =0 percent of the last period's sales, so+ E- E1 E3 EF 9a%"ent of accounts <2IM(00 <2A8(00 <2J9(00 <29J(00 c. ,ince the pa%ables period is M0 da%s, the pa%"ent in each period is 2:= of last $uarter's orders, plus 1:= of this $uarter's orders, or+ 7uarterl% pa%"ents L 2:=((=0# ti"es current sales O 1:=((=0# ne!t period sales( E- E1 E3 EF 9a%"ent of accounts <2A0(00 <2MA(00 <28A(00 <292(=0 B-320 SOLUTIONS $. ,ince the pa%ables period is M0 da%s, the pa%ables in each period ill be+ 9a%ables each period L 2:= of last $uarter's orders O 1:= of this $uarter's orders 9a%ables each period L 2:=((JA# ti"es current sales O 1:=((JA# ne!t period sales E- E1 E3 EF 9a%"ent of accounts <J22(A0 <J=2(A0 <8IJ(A0 <89J(A0 Wages, ta!es, other e!penses 19M(00 18M(00 21I(00 2A0(00 Cong-ter" financing e!penses 90(00 90(00 90(00 90(00 -otal <1,008(A 0 <1,008(A 0 <1,1A1(A 0 <1,2=J(A 0 1%. a. -he 4o*e"ber sales "ust ha*e been the total uncollected sales "inus the uncollected sales fro" )ece"ber, di*ided b% the collection rate to "onths after the sale, so+ 4o*e"ber sales L (<1J=,000 E 1=M,000#:0(1A 4o*e"ber sales L <2IM,MMM(MJ b. -he )ece"ber sales are the uncollected sales fro" )ece"ber di*ided b% the collection rate of the pre*ious "onths' sales, so+ )ece"ber sales L <1=M,000:0(=A )ece"ber sales L <=88,AJ1(I= c. -he collections each "onth for this co"pan% are+ Collections L (1A(,ales fro" 2 "onths ago# O (20(Cast "onth's sales# O (MA (Current sales# Januar% collections L (1A(<2IM,MMM(MJ# O (20(<=88,AJ1(I=# O (MA(<2=A,000# Januar% collections L <2MJ,IMI(29 Bebruar% collections L (1A(<=88,AJ1(I=# O (20(<2=A,000# O (MA(<2M0,000# Bebruar% collections L <2JI,28A(J1 3arch collections L (1A(<2=A,000# O (20(<2M0,000# O (MA(<29A,000# 3arch collections L <2J9,000(00 CHAPTER 18 B-321 11. -he sales collections each "onth ill be+ ,ales collections L (=A(current "onth sales# O (M0(pre*ious "onth sales# Gi*en this collection, the cash budget ill be+ April May June Heginning cash balance <1I0,000 <101,M00 <10I,100 Cash receipts Cash collections fro" credit sales 28=,A00 =M1,I00 =J1,J00 -otal cash a*ailable <I2=,A00 IM=,000 IJA,800 Cash disburse"ents 9urchases <1M8,000 <1IJ,800 <1JM,=00 Wages, ta!es, and e!penses A=,800 A1,000 J8,=00 .nterest 1=,100 1=,100 1=,100 6$uip"ent purchases 8J,000 1IJ,000 0 -otal cash disburse"ents <=21,900 <=A8,900 <2MJ,J00 6nding cash balance <101,M00 <10I,100 <208,100 12. .te" Source/Use Amount Cash ,ource <1,100 0ccounts recei*able Use E<I,=00 .n*entories Use E<=,MJ0 9ropert%, plant, and e$uip"ent Use E<12,J20 0ccounts pa%able ,ource <2,M00 0ccrued e!penses Use E<810 Cong-ter" debt ,ource <=,000 Co""on stoc& ,ource <A,000 0ccu"ulated retained earnings ,ource <1,M10 &ntermediate 13. a. .f %ou borro <A0,000,000 for one "onth, %ou ill pa% interest of+ .nterest L <A0,000,000((00MI# .nterest L <=20,000 >oe*er, ith the co"pensating balance, %ou ill onl% get the use of+ 0"ount recei*ed L <A0,000,000 E A0,000,000((0A# 0"ount recei*ed L <IJ,A00,000 -his "eans the periodic interest rate is+ 9eriodic interest L <=20,000:<IJ,A00,000 9eriodic interest L (00MJ=J or 0(MJIN B-322 SOLUTIONS ,o, the 60R is+ 60R L Q1 O (<=20,000:<IJ,A00,000#R 12 E 1 60R L (08=9 or 8(=9N b. -o end up ith <1A,000,000, %ou "ust borro+ 0"ount to borro L <1A,000,000:(1 E (0A# 0"ount to borro L <1A,J89,IJ=(M8 -he total interest %ou ill pa% on the loan is+ -otal interest paid L <1A,J89,IJ=(M8(1(00MI# M E 1A,J89,IJ=(M8 -otal interest paid L <M1M,100(02 14. a. -he 60R of %our in*est"ent account is+ 60R L 1(012 I E 1 60R L (0I89 or I(89N b. -o calculate the 60R of the loan, e can di*ide the interest on the loan b% the a"ount of the loan( -he interest on the loan includes the opportunit% cost of the co"pensating balance( -he opportunit% cost is the a"ount of the co"pensating balance ti"es the potential interest rate %ou could ha*e earned( -he co"pensating balance is onl% on the unused portion of the credit line, so+ /pportunit% cost L (0I(<J0,000,000 E IA,000,000#(1(012# I E (0I(<J0,000,000 E IA,000,000# /pportunit% cost L <I8,8J0(9= 0nd the interest %ou ill pa% to the ban& on the loan is+ .nterest cost L <IA,000,000(1(02=# I E IA,000,000 .nterest cost L <I,28A,0=2(MA ,o, the 60R of the loan in the a"ount of <IA "illion is+ 60R L (<I,28A,0=2(MA O I8,8J0(9=#:<IA,000,000 60R L (09M= or 9(M=N c. -he co"pensating balance is onl% applied to the unused portion of the credit line, so the 60R of a loan on the full credit line is+ 60R L 1(02= I E 1 60R L (09A2 or 9(A2N CHAPTER 18 B-323 1. a. 0 IA-da% collection period "eans sales collections each $uarter are+ Collections L 1:2 current sales O 1:2 old sales 0 =M-da% pa%ables period "eans pa%ables each $uarter are+ 9a%ables L =:A current orders O 2:A old orders ,o, the cash inflos and disburse"ents each $uarter are+ E- E1 E3 EF Heginning recei*ables <M8(00 <10A(00 <90(00 <122(A0 ,ales 210(00 180(00 2IA(00 280(00 Collection of accounts 1J=(00 19A(00 212(A0 2M2(A0 6nding recei*ables <10A(00 <90(00 <122(A0 <1I0(00 9a%"ent of accounts <8M(I0 <98(AA <119(J0 <11A(20 Wages, ta!es, and e!penses A2(A0 IA(00 M1(2A J0(00 Capital e!penditures 80(00 .nterest K di*idends 12(00 12(00 12(00 12(00 -otal cash disburse"ents <1A0(90 <2=A(AA <192(9A <19J(20 -otal cash collections <1J=(00 <19A(00 <212(A0 <2M2(A0 -otal cash disburse"ents 1A0(90 2=A(AA 192(9A 19J(20 4et cash inflo <22(10 <(I0(AA# <19(AA <MA(=0 -he co"pan%'s cash budget ill be+ W.C)C0-, .4C( Cash Hudget (in "illions# Q1 Q2 Q3 Q4 Heginning cash balance <MI(00 <8M(10 <IA(AA <MA(10 4et cash inflo 22(10 EI0(AA 19(AA MA(=0 6nding cash balance <8M(10 <IA(AA <MA(10 <1=0(I0 3ini"u" cash balance E=0(00 E=0(00 E=0(00 E=0(00 Cu"ulati*e surplus (deficit# <AM(10 <1A(AA <=A(10 <100(I0 B-324 SOLUTIONS With a <=0 "illion "ini"u" cash balance, the short-ter" financial plan ill be+ W.C)C0-, .4C( ,hort--er" Binancial 9lan (in "illions# b. Q1 Q2 Q3 Q4 Heginning cash balance <=0(00 <=0(00 <=0(00 <=0(00 4et cash inflo 22(10 EI0(AA 19(AA MA(=0 4e short-ter" in*est"ents E22(J8 0 E19(90 EMA(J0 .nco"e on short-ter" in*est"ents 0(M8 1(1I 0(=A 0(I0 ,hort-ter" in*est"ents sold 0 =9(I1 0 0 4e short-ter" borroing 0 0 0 0 .nterest on short-ter" borroing 0 0 0 0 ,hort-ter" borroing repaid 0 0 0 0 6nding cash balance <=0(00 <=0(00 <=0(00 <=0(00 3ini"u" cash balance E=0(00 E=0(00 E=0(00 E=0(00 Cu"ulati*e surplus (deficit# <0 <0 <0 <0 Heginning short-ter" in*est"ents <=I(00 <AM(J8 <0 <1J(=J 6nding short-ter" in*est"ents <AM(J8 <1J(=J <19(90 <8=(IM Heginning short-ter" debt <0 <0 <0 <0 6nding short-ter" debt <0 <0 <0 <0 Helo %ou ill find the interest paid (or recei*ed# for each $uarter+ 71+ e!cess funds at start of $uarter of <=I in*ested for 1 $uarter earns (02(<=I# L <0(M8 inco"e 72+ e!cess funds of <AM(J8 in*ested for 1 $uarter earns (02(<AM(J8# L <1(1I in inco"e 7=+ e!cess funds of <1J(=J in*ested for 1 $uarter earns (02(<1J(=J# L <0(=A in inco"e 7I+ e!cess funds of <19(90 in*ested for 1 $uarter earns (02(<19(90# L <0(I0 in inco"e 4et cash cost L <0(M8 O 1(1I O 0(=A O 0(I0 L <2(AM CHAPTER 18 B-325 1!. a. With a "ini"u" cash balance of <A0 "illion, the short-ter" financial plan ill be+ W.C)C0-, .4C( ,hort--er" Binancial 9lan (in "illions# Q1 Q2 Q3 Q4 Heginning cash balance <A0(00 <A0(00 <A0(00 <A0(00 4et cash inflo 22(10 EI0(AA 19(AA MA(=0 4e short-ter" in*est"ents E22(=8 0 E1M(00 EMA(M2 .nco"e on short-ter" in*est"ents 0(28 0(J= 0 0(=2 ,hort-ter" in*est"ents sold 0 =M(=8 0 0 4e short-ter" borroing 0 =(II 0 0 .nterest on short-ter" borroing 0 0 E0(10 0 ,hort-ter" borroing repaid 0 0 E=(II 0 6nding cash balance <A0(00 <A0(00 <A0(00 <A0(00 3ini"u" cash balance EA0(00 EA0(00 EA0(00 EA0(00 Cu"ulati*e surplus (deficit# <0 <0 <0 <0 Heginning short-ter" in*est"ents <1I(00 <=M(=8 <0 <0 6nding short-ter" in*est"ents <=M(=8 <0 <1M(00 <MA(9I Heginning short-ter" debt <0 <0 <=(II <0 6nding short-ter" debt <0 <=(II <0 <0 Helo %ou ill find the interest paid (or recei*ed# for each $uarter+ 71+ e!cess funds at start of $uarter of <1I in*ested for 1 $uarter earns (02(<1I# L <0(28 inco"e 72+ e!cess funds of <=M(=8 in*ested for 1 $uarter earns (02(<=M(=8# L <0(J= in inco"e 7=+ shortage of funds of <=(II borroed for 1 $uarter costs (0=(<=(II# L <0(10 in interest 7I+ e!cess funds of <1M(00 in*ested for 1 $uarter earns (02(<1M(00# L <0(=2 in inco"e 4et cash cost L <0(28 O 0(J= E 0(10 O 0(=2 L <1(22 B-326 SOLUTIONS b. 0nd ith a "ini"u" cash balance of <10 "illion, the short-ter" financial plan ill be+ W.C)C0-, .4C( ,hort--er" Binancial 9lan (in "illions# Q1 Q2 Q3 Q4 Heginning cash balance <10(00 <10(00 <10(00 <10(00 4et cash inflo 22(10 EI0(AA 19(AA MA(=0 4e short-ter" in*est"ents E2=(18 0 E20(=1 EMA(J1 .nco"e on short-ter" in*est"ents 1(08 1(AI 0(JM 0(I1 ,hort-ter" in*est"ents sold 0 =9(01 0 0 4e short-ter" borroing 0 0 0 0 .nterest on short-ter" borroing 0 0 0 0 ,hort-ter" borroing repaid 0 0 0 0 6nding cash balance <10(00 <10(00 <10(00 <10(00 3ini"u" cash balance E10(00 E10(00 E10(00 E10(00 Cu"ulati*e surplus (deficit# 0 0 0 0 Heginning short-ter" in*est"ents <AI(00 <JJ(18 0 <=8(1J 6nding short-ter" in*est"ents <JJ(18 <=8(1J <20(=1 <10I(29 Heginning short-ter" debt 0 0 0 0 6nding short-ter" debt 0 0 0 0 Helo %ou ill find the interest paid (or recei*ed# for each $uarter+ 71+ e!cess funds at start of $uarter of <AI in*ested for 1 $uarter earns (02(<AI# L <1(08 inco"e 72+ e!cess funds of <JJ(18 in*ested for 1 $uarter earns (02(<JJ(18# L <1(AI in inco"e 7=+ e!cess funds of <=8(1J in*ested for 1 $uarter earns (02(<=8(1J# L <0(JM in interest 7I+ e!cess funds of <20(=1 in*ested for 1 $uarter earns (02(<20(=1# L <0(I1 in inco"e 4et cash cost L <1(08 O 1(AI O 0(JM O 0(I1 L <=(J9 ,ince cash has an opportunit% cost, the fir" can boost its profit if it &eeps its "ini"u" cash balance lo and in*ests the cash instead( >oe*er, the tradeoff is that in the e*ent of unforeseen circu"stances, the fir" "a% not be able to "eet its short-run obligations if enough cash is not a*ailable( CHAPTER 18 B-327 Challenge 1". a. Bor e*er% dollar borroed, %ou pa% interest of+ .nterest L <1((019# L <0(019 5ou also "ust "aintain a co"pensating balance of I percent of the funds borroed, so for each dollar borroed, %ou ill onl% recei*e+ 0"ount recei*ed L <1(1 E (0I# L <0(9M We can ad1ust the 60R e$uation e ha*e been using to account for the co"pensating balance b% di*iding the 60R b% one "inus the co"pensating balance, so+ 60R L Q(1(019# I E 1R:(1 E (0I# 60R L (081IA or 8(1AN 0nother a% to calculate the 60R is using the BV.B (or 9V.B#( Bor each dollar borroed, e "ust repa%+ 0"ount oed L <1(1(019# I 0"ount oed L <1(0J82 0t the end of the %ear the co"pensating ill be returned, so %our net cash flo at the end of the %ear ill be+ 6nd of %ear cash flo L <1(0J82 E (0I 6nd of %ear cash flo L <1(0=82 -he present *alue of the end of %ear cash flo is the a"ount %ou recei*e at the beginning of the %ear, so the 60R is+ BV L 9V(1 O R# <1(0=82 L <0(9M(1 O R# R L <1(0=82:<0(9M E 1 60R L (081IA or 8(1AN b. -he 60R is the a"ount of interest paid on the loan di*ided b% the a"ount recei*ed hen the loan is originated( -he a"ount of interest %ou ill pa% on the loan is the a"ount of the loan ti"es the effecti*e annual interest rate, so+ .nterest L <1=0,000,000Q(1(019# I E 1R .nterest L <10,1MA,1M=(M2 Bor hate*er loan a"ount %ou ta&e, %ou ill onl% recei*e 9M percent of that a"ount since %ou "ust "aintain a I percent co"pensating balance on the portion of the credit line used( -he credit line also has a fee of (1I0 percent, so %ou ill onl% get to use+ 0"ount recei*ed L (9M(<1=0,000,000# E (001I0(<I00,000,000# 0"ount recei*ed L <12I,2I0,000 B-328 SOLUTIONS ,o, the 60R of the loan is+ 60R L <10,1MA,1M=(M2:<12I,2I0,000 60R L (08182 or 8(18N 1#. 5ou ill pa% interest of+ .nterest L <2A,000,000((10# L <2,A00,000 0dditionall%, the co"pensating balance on the loan is+ Co"pensating balance L <2A,000,000((0A# L <1,2A0,000 ,ince this is a discount loan, %ou ill recei*e the loan a"ount "inus the interest pa%"ent( 5ou ill also not get to use the co"pensating balance( ,o, the a"ount of "one% %ou ill actuall% recei*e on a <2A "illion loan is+ Cash recei*ed L <2A,000,000 E 2,A00,000 E 1,2A0,000 L <21,2A0,000 -he 60R is the interest a"ount di*ided b% the loan a"ount, so+ 60R L <2,A00,000:<21,2A0,000 60R L (11JM or 11(JMN We can also use the BV.B (or 9V.B# here to calculate the 60R( 5our cash flo at the beginning of the %ear is <21,2A0,000( 0t the end of the %ear, %our cash flo loan repa%"ent, but %ou ill also recei*e %our co"pensating balance bac&, so+ 6nd of %ear cash flo L <2A,000,000 E 1,2A0,000 6nd of %ear cash flo L <2=,JA0,000 ,o, using the ti"e *alue of "one%, the 60R is+ <2=,JA0,000 L <21,2A0,000(1 O R# R L <2=,JA0,000:<21,2A0,000 E 1 60R L (11JM or 11(JMN CHAPTER 19 CAS< A-; 9.Q:.;.T0 MA-A75M5-T Answers to Concepts Review and Critical Thinking Questions 1. 5es( /nce a fir" has "ore cash than it needs for operations and planned e!penditures, the e!cess cash has an opportunit% cost( .t could be in*ested (b% shareholders# in potentiall% "ore profitable a%s( 7uestion 10 discusses another reason( 2. .f it has too "uch cash it can si"pl% pa% a di*idend, or, "ore li&el% in the current financial en*iron"ent, bu% bac& stoc&( .t can also reduce debt( .f it has insufficient cash, then it "ust either borro, sell stoc&, or i"pro*e profitabilit%( 3. 9robabl% not( Creditors ould probabl% ant substantiall% "ore( 4. .n the case of Bord, the co"pan% generall% argued that it held cash to guard against future econo"ic donturns( Bor Gold"an ,achs, in*est"ent ban&s traditionall% hold large cash positions, although the a"ount of cash in earl% 2008 as an all-ti"e high( . Cash "anage"ent is associated "ore ith the collection and disburse"ent of cash( Ci$uidit% "anage"ent is broader and concerns the opti"al le*el of li$uid assets needed b% a fir"( -hus, for e!a"ple, a co"pan%'s stoc&piling of cash is li$uidit% "anage"ent8 hereas, e*aluating a loc&bo! s%ste" is cash "anage"ent( !. ,uch instru"ents go b% a *ariet% of na"es, but the &e% feature is that the di*idend ad1usts, &eeping the price relati*el% stable( -his price stabilit%, along ith the di*idend ta! e!e"ption, "a&es so-called ad1ustable rate preferred stoc& *er% attracti*e relati*e to interest-bearing instru"ents( ". 4et disburse"ent float is "ore desirable because the ban& thin&s the fir" has "ore "one% than it actuall% does, and the fir" is, therefore, recei*ing interest on funds it has alread% spent( #. -he fir" has a net disburse"ent float of <A00,000( .f this is an ongoing situation, the fir" "a% be te"pted to rite chec&s for "ore than it actuall% has in its account( $. a. 0bout the onl% disad*antage to holding --bills are the generall% loer %ields co"pared to alternati*e "one% "ar&et in*est"ents( b. ,o"e ordinar% preferred stoc& issues pose both credit and price ris&s that are not consistent ith "ost short-ter" cash "anage"ent plans( c. -he pri"ar% disad*antage of 4C)s is the nor"all% large transactions si2es, hich "a% not be feasible for the short-ter" in*est"ent plans of "an% s"aller to "ediu"-si2ed corporations( B-330 SOLUTIONS d. -he pri"ar% disad*antages of the co""ercial paper "ar&et are the higher default ris& characteristics of the securit% and the lac& of an acti*e secondar% "ar&et hich "a% e!cessi*el% restrict the fle!ibilit% of corporations to "eet their li$uidit% ad1ust"ent needs( CHAPTER 19 B-331 e. -he pri"ar% disad*antages of R04s is that so"e possess non-tri*ial le*els of default ris&, and also, corporations are so"ehat restricted in the t%pe and a"ount of these ta!-e!e"pts that the% can hold in their portfolios( f. -he pri"ar% disad*antage of the repo "ar&et is the generall% *er% short "aturities a*ailable( 1%. -he concern is that e!cess cash on hand can lead to poorl% thought-out "anage"ent decisions( -he thought is that &eeping cash le*els relati*el% lo forces "anage"ent to pa% careful attention to cash flo and capital spending( 11. 0 potential ad*antage is that the $uic&er pa%"ent often "eans a better price( -he disad*antage is that doing so increases the fir"'s cash c%cle( 12. -his is reall% a capital structure decision( .f the fir" has an opti"al capital structure, pa%ing off debt "o*es it to an under-le*eraged position( >oe*er, a co"bination of debt reduction and stoc& bu%-bac&s could be structured to lea*e capital structure unchanged( 13. .t is unethical because %ou ha*e essentiall% tric&ed the grocer% store into "a&ing %ou an interest-free loan, and the grocer% store is har"ed because it could ha*e earned interest on the "one% instead of loaning it to %ou( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he a*erage dail% float is the a*erage a"ount of chec&s recei*ed per da% ti"es the a*erage nu"ber of da%s dela%, di*ided b% the nu"ber of da%s in a "onth( 0ssu"ing =0 da%s in a "onth, the a*erage dail% float is+ 0*erage dail% float L I(<1AM,000#:=0 0*erage dail% float L <20,800 2. a. -he disburse"ent float is the a*erage "onthl% chec&s ritten ti"es the a*erage nu"ber of da%s for the chec&s to clear, so+ )isburse"ent float L I(<1I,000# )isburse"ent float L <AM,000 -he collection float is the a*erage "onthl% chec&s recei*ed ti"es the a*erage nu"ber of da%s for the chec&s to clear, so+ Collection float L 2(E<2M,000# Collection float L E<A2,000 B-332 SOLUTIONS -he net float is the disburse"ent float plus the collection float, so+ 4et float L <AM,000 E A2,000 4et float L <I,000 b. -he ne collection float ill be+ Collection float L 1(E<2M,000# Collection float L E<2M,000 0nd the ne net float ill be+ 4et float L <AM,000 E 2M,000 4et float L <=0,000 3. a. -he collection float is the a*erage dail% chec&s recei*ed ti"es the a*erage nu"ber of da%s for the chec&s to clear, so+ Collection float L =(<19,000# Collection float L <AJ,000 b. -he fir" should pa% no "ore than the a"ount of the float, or <AJ,000, to eli"inate the float( c. -he "a!i"u" dail% charge the fir" should be illing to pa% is the collection float ti"es the dail% interest rate, so+ 3a!i"u" dail% charge L <AJ,000((00019# Maximum daily charge = $10.83 4. a. -otal float L I(<1J,000# O A(<M,000# -otal float L <98,000 b. -he a*erage dail% float is the total float di*ided b% the nu"ber of da%s in a "onth( 0ssu"ing =0 da%s in a "onth, the a*erage dail% float is+ 0*erage dail% float L <98,000:=0 0*erage dail% float L <=,2MM(MJ c. -he a*erage dail% receipts are the a*erage dail% chec&s recei*ed di*ided b% the nu"ber of da%s in a "onth( 0ssu"ing a =0 da% "onth+ 0*erage dail% receipts L (<1J,000 O M,000#:=0 0*erage dail% receipts L <JMM(MJ -he eighted a*erage dela% is the su" of the da%s to clear a chec&, ti"es the a"ount of the chec& di*ided b% the a*erage dail% receipts, so+ Weighted a*erage dela% L I(<1M,000:<2=,000# O A(<M,000:<2=,000# Weighted a*erage dela% L I(2M da%s CHAPTER 19 B-333 . -he a*erage dail% collections are the nu"ber of chec&s recei*ed ti"es the a*erage *alue of a chec&, so+ 0*erage dail% collections L <108(8,A00# 0*erage dail% collections L <918,000 -he present *alue of the loc&bo! ser*ice is the a*erage dail% receipts ti"es the nu"ber of da%s the collection is reduced, so+ 9V L (2 da% reduction#(<918,000# 9V L <1,8=M,000 -he dail% cost is a perpetuit%( -he present *alue of the cost is the dail% cost di*ided b% the dail% interest rate( ,o+ 9V of cost L <22A:(0001M 9V of cost L <1,I0M,2A0 -he fir" should ta&e the loc&bo! ser*ice( -he 49V of the loc&bo! is the cost plus the present *alue of the reduction in collection ti"e, so+ 49V L E<1,I0M,2A0 O 1,8=M,000 49V L <I29,JA0 -he annual sa*ings e!cluding the cost ould be the future *alue of the sa*ings "inus the costs, so+ 0nnual sa*ings L <1,8=M,000(1(0001M# =MA E 1,8=M,000 0nnual sa*ings L <110,I0M(0A 0nd the annual cost ould be the future *alue of the dail% cost, hich is an annuit%, so+ 0nnual cost L <22A(BV.B0=MA,(01MN# 0nnual cost L <8I,AM=(IM ,o, the annual net sa*ings ould be+ 0nnual net sa*ings L <110,I0M(0A E 8I,AM=(IM 0nnual net sa*ings L <2A,8I2(A9 !. a. -he a*erage dail% float is the su" of the percentage each chec& a"ount is of the total chec&s recei*ed ti"es the nu"ber of chec&s recei*ed ti"es the a"ount of the chec& ti"es the nu"ber of da%s until the chec& clears, di*ided b% the nu"ber of da%s in a "onth( 0ssu"ing a =0 da% "onth, e get+ 0*erage dail% float L Q(M0(A,=00#(<AA#(2# O (I0(A,=00#(<80#(=#R:=0 0*erage dail% float L <28,M20 /n a*erage, there is <28,M20 that is uncollected and not a*ailable to the fir"( B-334 SOLUTIONS b. -he total collections are the su" of the percentage of each chec& a"ount recei*ed ti"es the total chec&s recei*ed ti"es the a"ount of the chec&, so+ -otal collections L (M0(A,=00#(<AA# O (I0(A,=00#(<80# -otal collections L <=II,A00 -he eighted a*erage dela% is the su" of the a*erage nu"ber of da%s a chec& of a specific a"ount is dela%ed, ti"es the percentage that chec& a"ount "a&es up of the total chec&s recei*ed, so+ Weighted a*erage dela% L 2Q(M0(A,=00#(<AA#:<=II,A00R O =Q(I0(A,=00#(<80# :<=II,A00R Weighted a*erage dela% L 2(I9 da%s -he a*erage dail% float is the eighted a*erage dela% ti"es the a*erage chec&s recei*ed per da%( 0ssu"ing a =0 da% "onth, e get+ 0*erage dail% float L 2(I9(<=II,A00:=0 da%s# 0*erage dail% float L <28,M20 c. -he "ost the fir" should pa% is the total a"ount of the a*erage float, or <28,M20( d. -he a*erage dail% interest rate is+ 1(0J L (1 O R# =MA R L (018AIN per da% -he dail% cost of float is the a*erage dail% float ti"es the dail% interest rate, so+ )ail% cost of the float L <28,M20((00018AI# )ail% cost of the float L <A(=1 e. -he "ost the fir" should pa% is still the a*erage dail% float( Under the reduced collection ti"e assu"ption, e get+ 4e a*erage dail% float L 1(A(<=II,A00:=0# 4e a*erage dail% float L <1J,22A ". a. -he present *alue of adopting the s%ste" is the nu"ber of da%s collections are reduced ti"es the a*erage dail% collections, so+ 9V L =(=8A#(<1,10A# 9V L <1,2JM,2JA b. -he 49V of adopting the s%ste" is the present *alue of the sa*ings "inus the cost of adopting the s%ste"( -he cost of adopting the s%ste" is the present *alue of the fee per transaction ti"es the nu"ber of transactions( -his is a perpetuit%, so+ 49V L <1,2JM,2JA E Q<0(A0(=8A#:(0002R 49V L <=1=,JJA CHAPTER 19 B-335 c. -he net cash flos is the present *alue of the a*erage dail% collections ti"es the dail% interest rate, "inus the transaction cost per da%, so+ 4et cash flo per da% L <1,2JM,2JA((0002# E <0(A0(=8A# 4et cash flo per da% L <M2(JM -he net cash flo per chec& is the net cash flo per da% di*ided b% the nu"ber of chec&s recei*ed per da%, or+ 4et cash flo per chec& L <M2(JM:=8A 4et cash flo per chec& L <0(1M 0lternati*el%, e could find the net cash flo per chec& as the nu"ber of da%s the s%ste" reduces collection ti"e ti"es the a*erage chec& a"ount ti"es the dail% interest rate, "inus the transaction cost per chec&( )oing so, e confir" our pre*ious anser as+ 4et cash flo per chec& L =(<1,10A#((0002# E <0(A0 4et cash flo per chec& L <0(1M per chec& #. a. -he reduction in cash balance fro" adopting the loc&bo! is the nu"ber of da%s the s%ste" reduces collection ti"e ti"es the a*erage dail% collections, so+ Cash balance reduction L =(<1IA,000# Cash balance reduction L <I=A,000 b. -he dollar return that can be earned is the a*erage dail% interest rate ti"es the cash balance reduction( -he a*erage dail% interest rate is+ 0*erage dail% rate L 1(09 1:=MA E 1 0*erage dail% rate L (02=MN per da% -he dail% dollar return that can be earned fro" the reduction in da%s to clear the chec&s is+ )ail% dollar return L <I=A,000((0002=M# )ail% dollar return L <102(J2 c. .f the co"pan% ta&es the loc&bo!, it ill recei*e three pa%"ents earl%, ith the first pa%"ent occurring toda%( We can use the dail% interest rate fro" part b, so the sa*ings are+ ,a*ings L <1IA,000 O <1IA,000(9V.B0(02=MN,2# ,a*ings L <I=I,89J(=2 .f the loc&bo! pa%"ents occur at the end of the "onth, e need the effecti*e "onthl% interest rate, hich is+ 3onthl% interest rate L 1(09 1:12 E 1 3onthl% interest rate L 0(J20JN B-336 SOLUTIONS 0ssu"ing the loc&bo! pa%"ents occur at the end of the "onth, the loc&bo! pa%"ents, hich are a perpetuit%, ill be+ 9V L C:R <I=I,89J(=2 L C : (00J20J C L <=,1=I(=A .t could also be assu"ed that the loc&bo! pa%"ents occur at the beginning of the "onth( .f so, e ould need to use the 9V of a perpetuit% due, hich is+ 9V L C O C : R ,ol*ing for C+ C L (9V S R# : (1 O R# C L (I=I,89J(=2 S (00J20J# : (1 O (00J20J# C L <=,112(02 $. -he interest that the co"pan% could earn ill be the a"ount of the chec&s ti"es the nu"ber of da%s it ill dela% pa%"ent ti"es the nu"ber of ee&s that chec&s ill be disbursed ti"es the dail% interest rate, so+ .nterest L <9=,000(J#(A2:2#((0001A# .nterest L <2,A=8(90 1%. -he benefit of the ne arrange"ent is the <I "illion in accelerated collections since the ne s%ste" ill speed up collections b% one da%( -he cost is the ne co"pensating balance, but the co"pan% ill reco*er the e!isting co"pensating balance, so+ 49V L <I,000,000 E (<I00,000 E A00,000# 49V L <=,900,000 -he co"pan% should proceed ith the ne s%ste"( -he sa*ings are the 49V ti"es the annual interest rate, so+ 4et sa*ings L <=,900,000((0A# 4et sa*ings L <19A,000 &ntermediate 11. -o find the 49V of ta&ing the loc&bo!, e first need to calculate the present *alue of the sa*ings( -he present *alue of the sa*ings ill be the reduction in collection ti"e ti"es the a*erage dail% collections, so+ 9V L 2(JA0#(<980# 9V L <1,IJ0,000 0nd the dail% interest rate is+ )ail% interest rate L 1(0J0 1:=MA E 1 )ail% interest rate L (00019 or (019N per da% CHAPTER 19 B-337 -he transaction costs are a perpetuit%( -he cost per da% is the cost per transaction ti"es the nu"ber of transactions per da%, so the 49V of ta&ing the loc&bo! is+ 49V L <1,IJ0,000 E Q<0(=A(980#:(00019R 49V L <AI,01A(1J Without the fee, the loc&bo! s%ste" should be accepted( -o calculate the 49V of the loc&bo! ith the annual fee, e can si"pl% use the 49V of the loc&bo! ithout the annual fee and subtract the addition cost( -he annual fee is a perpetuit%, so, ith the fee, the 49V of ta&ing the loc&bo! is+ 49V L <AI,01A(1J E Q<A,000:(0JR 49V L E<1J,I1=(I0 With the fee, the loc&bo! s%ste" should not be accepted( 12. -o find the "ini"u" nu"ber of pa%"ents per da% needed to "a&e the loc&bo! s%ste" feasible is the nu"ber of chec&s that "a&es the 49V of the decision e$ual to 2ero( -he a*erage dail% interest rate is+ )ail% interest rate L 1(0A 1:=MA E 1 )ail% interest rate L (01=IN per da% -he present *alue of the sa*ings is the a*erage pa%"ent a"ount ti"es the da%s the collection period is reduced ti"es the nu"ber of custo"ers( -he costs are the transaction fee and the annual fee( Hoth are perpetuities( -he total transaction costs are the transaction costs per chec& ti"es the nu"ber of chec&s( -he e$uation for the 49V of the pro1ect, here 4 is the nu"ber of chec&s transacted per da%, is+ 49V L 0 L (<A,=00#(1#4 E <0(10(4#:(0001=I E <20,000:(0A <I00,000 L <A,=004 E <JI8(0A4 <I,AA1(9A4 L <I00,000 4 L 8J(8J ≈ 88 custo"ers per da% B-338 SOLUTIONS APPENDIX 19A 1. a. )ecrease( -his ill loer the trading costs, hich ill cause a decrease in the target cash balance( b. )ecrease( -his ill increase the holding cost, hich ill cause a decrease in the target cash balance( c. .ncrease( -his ill increase the a"ount of cash that the fir" has to hold in non-interest bearing accounts, so the% ill ha*e to raise the target cash balance to "eet this re$uire"ent( d. )ecrease( .f the credit rating i"pro*es, then the fir" can borro "ore easil%, alloing it to loer the target cash balance and borro if a cash shortfall occurs( e. .ncrease( .f the cost of borroing increases, the fir" ill need to hold "ore cash to protect against cash shortfalls as its borroing costs beco"e "ore prohibiti*e( f. 6ither( -his depends so"ehat on hat the fees appl% to, but if direct fees are established, then the co"pensating balance "a% be loered, thus loering the target cash balance( .f, on the other hand, fees are charged on the nu"ber of transactions, then the fir" "a% ish to hold a higher cash balance so the% are not transferring "one% into the account as often( 2. -he target cash balance using the H0- "odel is+ C ` L Q(2- S B#:RR 1:2 C ` L Q2(<8,A00#(<2A#:(0MR 1:2 C ` L <2,MM1(IA -he initial balance should be <2,MM1(IA, and hene*er the balance drops to <0, another <2,MM1(IA should be transferred in( 3. -he holding cost is the a*erage dail% cash balance ti"es the interest rate, so+ >olding cost L (<1,=00#((0A# >olding cost L <MA(00 -he trading costs are the total cash needed ti"es the replenishing costs, di*ided b% the a*erage dail% balance ti"es to, so+ -rading cost L Q(<I=,000#(<8#R:Q(<1,=00#(2#R -rading cost L <1=2(=1 -he total cost is the su" of the holding cost and the trading cost, so+ -otal cost L <MA(00 O 1=2(=1 -otal cost L <19J(=1 CHAPTER 19 B-339 -he target cash balance using the H0- "odel is+ C ` L Q(2- S B#:RR 1:2 C ` L Q2(<I=,000#(<8#:(0AR 1:2 C ` L <=,J09(IA -he% should increase their a*erage dail% cash balance to+ 4e a*erage cash balance L <=,J09(IA:2 4e a*erage cash balance L <1,8AI(J2 -his ould "ini"i2e the costs( -he ne total cost ould be+ 4e total cost L (<1,8IA(J2#((0A# O Q(<I=,000#(<8#R:Q2(<1,8AI(J2#R 4e total cost L <18A(IJ 4. a. -he opportunit% costs are the a"ount transferred ti"es the interest rate, di*ided b% to, so+ /pportunit% cost L (<1,A00#((0A#:2 /pportunit% cost L <=J(A0 -he trading costs are the total cash balance ti"es the trading cost per transaction, di*ided b% the a"ount transferred, so+ -rading cost L (<1M,000#(<2A#:<1,A00 -rading cost L <2MM(MJ -he fir" &eeps too little in cash because the trading costs are "uch higher than the opportunit% costs( b. -he target cash balance using the H0- "odel is+ C ` L Q(2- S B#:RR 1:2 C ` L Q2(<1M,000#(<2A#:(0AR 1:2 C ` L <I,000 . -he total cash needed is the cash shortage per "onth ti"es tel*e "onths, so+ -otal cash L 12(<1I0,000# -otal cash L <1,M80,000 -he target cash balance using the H0- "odel is+ C ` L Q(2- S B#:RR 1:2 C ` L Q2(<1,M80,000#(<A00#:(0AJR 1:2 C ` L <1J1,MJ9(02 B-340 SOLUTIONS -he co"pan% should in*est+ .n*est L <M90,000 E 1J1,MJ9(02 .n*est L <A18,=20(98 of its current cash holdings in "ar&etable securities to bring the cash balance don to the opti"al le*el( /*er the rest of the %ear, sell securities+ ,ell securities L <1,M80,000:<1J1,MJ9(02 ,ell securities L 9(J9 ≈ 10 ti"es( !. -he loer li"it is the "ini"u" balance alloed in the account, and the upper li"it is the "a!i"u" balance alloed in the account( When the account balance drops to the loer li"it+ ,ecurities sold L <80,000 E I=,000 ,ecurities sold L <=J,000 in "ar&etable securities ill be sold, and the proceeds deposited in the account( -his "o*es the account balance bac& to the target cash le*el( When the account balance rises to the upper li"it, then+ ,ecurities purchased L <12A,000 E 80,000 ,ecurities purchased L <IA,000 of "ar&etable securities ill be purchased( -his e!penditure brings the cash le*el bac& don to the target balance of <80,000( ". -he target cash balance using the 3iller-/rr "odel is+ C ` L C O (=:I S B S σ 2 : RR 1:= C ` L <1,A00 O Q=:I(<I0#(<J0# 2 :(00021R 1:= C ` L <2,=8J(90 -he upper li"it is+ U ` L = S C ` E 2 S C U ` L =(<2,=8J(90# E 2(<1,A00# U ` L <I,1M=(J1 CHAPTER 19 B-341 When the balance in the cash account drops to <1,A00, the fir" sells+ ,ell L <2,=8J(90 E 1,A00 ,ell L <88J(90 of "ar&etable securities( -he proceeds fro" the sale are used to replenish the account bac& to the opti"al target le*el of C ` ( Con*ersel%, hen the upper li"it is reached, the fir" bu%s+ Hu% L <I,1M=(J1 E 2,=8J(90 Hu% L <1,JJA(81 of "ar&etable securities( -his e!penditure loers the cash le*el bac& don to the opti"al le*el of <2,=8J(90( #. 0s *ariance increases, the upper li"it and the spread ill increase, hile the loer li"it re"ains unchanged( -he loer li"it does not change because it is an e!ogenous *ariable set b% "anage"ent( 0s the *ariance increases, hoe*er, the a"ount of uncertaint% increases( When this happens, the target cash balance, and therefore the upper li"it and the spread, ill need to be higher( .f the *ariance drops to 2ero, then the loer li"it, the target balance, and the upper li"it ill all be the sa"e( $. -he a*erage dail% interest rate is+ )ail% rate L 1(0J 1:=MA E 1 )ail% rate L (00018A or (018AN per da% -he target cash balance using the 3iller-/rr "odel is+ C ` L C O (=:I S B S σ 2 : RR 1:= C ` L <1M0,000 O Q=:I(<=00#(<890,000#:(00018AR 1:= C ` L <1J0,2M0(IJ -he upper li"it is+ U ` L = S C ` E 2 S C U ` L =(<1J0,2M0(IJ# E 2(<1M0,000# U ` L <190,J81(I1 1%. Using the H0- "odel and sol*ing for R, e get+ C ` L Q(2- S B#:RR 1:2 <2,J00 L Q2(<28,000#(<10#:RR 1:2 R L Q2(<28,000#(<10#R:<2,J00 2 R L (0JM8 or J(M8N CHAPTER 20 CR5;.T A-; .-15-T4R0 MA-A75M5-T Answers to Concepts Review and Critical Thinking Questions 1. a. 0 sight draft is a co""ercial draft that is pa%able i""ediatel%( b. 0 ti"e draft is a co""ercial draft that does not re$uire i""ediate pa%"ent( c. 0 ban&ers acceptance is hen a ban& guarantees the future pa%"ent of a co""ercial draft( d. 0 pro"issor% note is an ./U that the custo"er signs( e. 0 trade acceptance is hen the bu%er accepts the co""ercial draft and pro"ises to pa% it in the future( 2. -rade credit is usuall% granted on open account( -he in*oice is the credit instru"ent( 3. Credit costs+ cost of debt, probabilit% of default, and the cash discount 4o-credit costs+ lost sales -he su" of these are the carr%ing costs( 4. -. Character+ deter"ines if a custo"er is illing to pa% his or her debts( 1. Capacit%+ deter"ines if a custo"er is able to pa% debts out of operating cash flo( 3. Capital+ deter"ines the custo"er's financial reser*es in case proble"s occur ith operating cash flo( F. Collateral+ assets that can be li$uidated to pa% off the loan in case of default( .. Conditions+ custo"er's abilit% to eather an econo"ic donturn and hether such a donturn is li&el%( . -. 9erishabilit% and collateral *alue 1. Consu"er de"and 3. Cost, profitabilit%, and standardi2ation F. Credit ris& .. -he si2e of the account 7. Co"petition 6. Custo"er t%pe .f the credit period e!ceeds a custo"er's operating c%cle, then the fir" is financing the recei*ables and other aspects of the custo"er's business that go be%ond the purchase of the selling fir"'s "erchandise( !. a. H+ 0 is li&el% to sell for cash onl%, unless the product reall% or&s( .f it does, then the% "ight grant longer credit periods to entice bu%ers( b. 0+ Candlords ha*e significantl% greater collateral, and that collateral is not "obile( c. 0+ ,ince 0's custo"ers turn o*er in*entor% less fre$uentl%, the% ha*e a longer in*entor% period, and thus, ill "ost li&el% ha*e a longer credit period as ell( d. H+ ,ince 0's "erchandise is perishable and H's is not, H ill probabl% ha*e a longer credit period( CHAPTER 20 B-343 e. 0+ Rugs are fairl% standardi2ed and the% are transportable, hile carpets are custo" fit and are not particularl% transportable( ". -he three "ain categories of in*entor% are+ ra "aterial (initial inputs to the fir"'s production process#, or&-in-progress (partiall% co"pleted products#, and finished goods (products read% for sale#( Bro" the fir"'s perspecti*e, the de"and for finished goods is independent fro" the de"and for the other t%pes of in*entor%( -he de"and for ra "aterial and or&-in-progress is deri*ed fro", or dependent on, the fir"'s needs for these in*entor% t%pes in order to achie*e the desired le*els of finished goods( #. J.- s%ste"s reduce in*entor% a"ounts( 0ssu"ing no ad*erse effects on sales, in*entor% turno*er ill increase( ,ince assets ill decrease, total asset turno*er ill also increase( Recalling the )u9ont e$uation, an increase in total asset turno*er, all else being e$ual, has a positi*e effect on R/6( $. Carr%ing costs should be e$ual to order costs( ,ince the carr%ing costs are lo relati*e to the order costs, the fir" should increase the in*entor% le*el( 1%. ,ince the price of co"ponents can decline $uic&l%, )ell does not ha*e in*entor% hich is purchased and then declines $uic&l% in *alue before it is sold( .f this happens, the in*entor% "a% be sold at a loss( While this approach is *aluable, it is difficult to i"ple"ent( Bor e!a"ple, )ell "anufacturing plants ill often ha*e areas set aside that are for the suppliers( When parts are needed, it is a "atter of going across the floor to get ne parts( .n fact, "0st co"puter "anufacturers are tr%ing to i"ple"ent si"ilar in*entor% s%ste"s( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. a. -here are =0 da%s until account is o*erdue( .f %ou ta&e the full period, %ou "ust re"it+ Re"ittance L I00(<12A# Re"ittance L <A0,000 b. -here is a 1 percent discount offered, ith a 10 da% discount period( .f %ou ta&e the discount, %ou ill onl% ha*e to re"it+ Re"ittance L (1 E (01#(<A0,000# Re"ittance L <I9,A00 c. -he i"plicit interest is the difference beteen the to re"ittance a"ounts, or+ ."plicit interest L <A0,000 E I9,A00 ."plicit interest L <A00 B-344 SOLUTIONS -he nu"ber of da%s' credit offered is+ )a%s' credit L =0 E 10 )a%s' credit L 20 da%s 2. -he recei*ables turno*er is+ Recei*ables turno*er L =MA:0*erage collection period Recei*ables turno*er L =MA:=M Recei*ables turno*er L 10(1=9 ti"es 0nd the a*erage recei*ables are+ 0*erage recei*ables L ,ales:Recei*ables period 0*erage recei*ables L <IJ,000,000 : 10(1=9 0*erage recei*ables L <I,M=A,M1M 3. a. -he a*erage collection period is the percentage of accounts ta&ing the discount ti"es the discount period, plus the percentage of accounts not ta&ing the discount ti"es the da%s' until full pa%"ent is re$uired, so+ 0*erage collection period L (MA(10 da%s# O (=A(=0 da%s# 0*erage collection period L 1J da%s b. 0nd the a*erage dail% balance is+ 0*erage balance L 1,=00(<1,J00#(1J#(12:=MA# 0*erage balance L <1,2=A,1J8(08 4. The daily sales are: Daily sales = $19,400 / 7 Daily sales = $2,771.43 ,ince the a*erage collection period is =I da%s, the a*erage accounts recei*able is+ 0*erage accounts recei*able L <2,JJ1(I=(=I# 0*erage accounts recei*able L <9I,228(AJ . -he interest rate for the ter" of the discount is+ .nterest rate L (01:(99 .nterest rate L (0101 or 1(01N 0nd the interest is for+ =A E 10 L 2A da%s CHAPTER 20 B-345 ,o, using the 60R e$uation, the effecti*e annual interest rate is+ 60R L (1 O 9eriodic rate# " E 1 60R L (1(0101# =MA:2A E 1 60R L (1A80 or 1A(80N a. -he periodic interest rate is+ .nterest rate L (02:(98 .nterest rate L (020I or 2(0IN 0nd the 60R is+ 60R L (1(020I# =MA:2A E 1 60R L (=I=1 or =I(=1N b. -he 60R is+ 60R L (1(0101# =MA:A0 E 1 60R L (0JM1 or L J(M1N c. -he 60R is+ 60R L (1(0101# =MA:20 E 1 60R L (201= or 20(1=N !. -he recei*ables turno*er is+ Recei*ables turno*er L =MA:0*erage collection period Recei*ables turno*er L =MA:=9 Recei*ables turno*er L 9(=A90 ti"es 0nd the annual credit sales are+ 0nnual credit sales L Recei*ables turno*er S 0*erage dail% recei*ables 0nnual credit sales L 9(=A90(<IJ,A00# 0nnual credit sales L <III,AA1(28 ". -he total sales of the fir" are e$ual to the total credit sales since all sales are on credit, so+ -otal credit sales L A,M00(<I2A# -otal credit sales L <2,=80,000 -he a*erage collection period is the percentage of accounts ta&ing the discount ti"es the discount period, plus the percentage of accounts not ta&ing the discount ti"es the da%s' until full pa%"ent is re$uired, so+ 0*erage collection period L (M0(10# O (I0(I0# 0*erage collection period L 22 da%s B-346 SOLUTIONS -he recei*ables turno*er is =MA di*ided b% the a*erage collection period, so+ Recei*ables turno*er L =MA:22 Recei*ables turno*er L 1M(A91 ti"es 0nd the a*erage recei*ables are the credit sales di*ided b% the recei*ables turno*er so+ 0*erage recei*ables L <2,=80,000:1M(A91 0*erage recei*ables L <1I=,IA2(0A .f the fir" increases the cash discount, "ore people ill pa% sooner, thus loering the a*erage collection period( .f the 0C9 declines, the recei*ables turno*er increases, hich ill lead to a decrease in the a*erage recei*ables( #. -he a*erage collection period is the net credit ter"s plus the da%s o*erdue, so+ 0*erage collection period L =0 O 8 0*erage collection period L =8 da%s -he recei*ables turno*er is =MA di*ided b% the a*erage collection period, so+ Recei*ables turno*er L =MA:=8 Recei*ables turno*er L 9(M0A= ti"es 0nd the a*erage recei*ables are the credit sales di*ided b% the recei*ables turno*er so+ 0*erage recei*ables L <8,I00,000 : 9(M0A= 0*erage recei*ables L <8JI,A20(AA $. a. -he cash outla% for the credit decision is the *ariable cost of the engine( .f this is a one-ti"e order, the cash inflo is the present *alue of the sales price of the engine ti"es one "inus the default probabilit%( ,o, the 49V per unit is+ 49V L E<1,M00,000 O (1 E (00A#(<1,8J0,000#:1(029 49V L <208,211(8M per unit -he co"pan% should fill the order( b. -o find the brea&e*en probabilit% of default, π, e si"pl% use the 49V e$uation fro" part a, set it e$ual to 2ero, and sol*e for π( )oing so, e get+ 49V L 0 L E<1,M00,000 O (1 E π#(<1,8J0,000#:1(029 π L (119M or 11(9MN We ould not accept the order if the default probabilit% as higher than 11(9M percent( CHAPTER 20 B-347 c. .f the custo"er ill beco"e a repeat custo"er, the cash inflo changes( -he cash inflo is no one "inus the default probabilit%, ti"es the sales price "inus the *ariable cost( We need to use the sales price "inus the *ariable cost since e ill ha*e to build another engine for the custo"er in one period( 0dditionall%, this cash inflo is no a perpetuit%, so the 49V under these assu"ptions is+ 49V L E<1,M00,000 O (1 E (00A#(<1,8J0,000 E 1,M00,000#:(029 49V L <J,MM=,J9=(10 per unit -he co"pan% should fill the order( -he brea&e*en default probabilit% under these assu"ptions is+ 49V L 0 L E<1,M00,000 O (1 E π#(<1,8J0,000 E 1,M00,000#:(029 π L (8281 or 82(81N We ould not accept the order if the default probabilit% as higher than 82(81 percent( -his default probabilit% is "uch higher than in part b because the custo"er "a% beco"e a repeat custo"er( d. .t is assu"ed that if a person has paid his or her bills in the past, the% ill pa% their bills in the future( -his i"plies that if so"eone doesn't default hen credit is first granted, then the% ill be a good custo"er far into the future, and the possible gains fro" the future business outeigh the possible losses fro" granting credit the first ti"e( 1%. -he cost of sitching is the lost sales fro" the e!isting polic% plus the incre"ental *ariable costs under the ne polic%, so+ Cost of sitching L <J20(1,=0A# O <I9A(1,=80 E 1,=0A# Cost of sitching L <9JM,J2A -he benefit of sitching is the ne sales price "inus the *ariable costs per unit, ti"es the incre"ental units sold, so+ Henefit of sitching L (<J20 E I9A#(1,=80 E 1,=0A# Henefit of sitching L <1M,8JA -he benefit of sitching is a perpetuit%, so the 49V of the decision to sitch is+ 49V L E<9JM,2JA O <1M,8JA:(01A 49V L <1I8,2JA(00 -he fir" ill ha*e to bear the cost of sales for one "onth before the% recei*e an% re*enue fro" credit sales, hich is h% the initial cost is for one "onth( Recei*ables ill gro o*er the one "onth credit period and ill then re"ain stable ith pa%"ents and ne sales offsetting one another( 11. -he carr%ing costs are the a*erage in*entor% ti"es the cost of carr%ing an indi*idual unit, so+ Carr%ing costs L (2,A00:2#(<9# L <11,2A0 B-348 SOLUTIONS -he order costs are the nu"ber of orders ti"es the cost of an order, so+ /rder costs L (A2#(<1,J00# L <88,I00 -he econo"ic order $uantit% is+ 6/7 L Q(2- S B#:CCR 1:2 6/7 L Q2(A2#(2,A00#(<1,J00#:<9R 1:2 6/7 L J,00J(9= -he fir"'s polic% is not opti"al, since the carr%ing costs and the order costs are not e$ual( -he co"pan% should increase the order si2e and decrease the nu"ber of orders( 12. -he carr%ing costs are the a*erage in*entor% ti"es the cost of carr%ing an indi*idual unit, so+ Carr%ing costs L (=00:2#(<I1# L <M,1A0 -he order costs are the nu"ber of orders ti"es the cost of an order, so+ Restoc&ing costs L A2(<9A# L <I,9I0 -he econo"ic order $uantit% is+ 6/7 L Q(2- S B#:CCR 1:2 6/7 L Q2(A2#(=00#(<9A#:<I1R 1:2 6/7 L 2M8(8J -he nu"ber of orders per %ear ill be the total units sold per %ear di*ided b% the 6/7, so+ 4u"ber of orders per %ear L A2(=00#:2M8(8J 4u"ber of orders per %ear L A8(02 -he fir"'s polic% is not opti"al, since the carr%ing costs and the order costs are not e$ual( -he co"pan% should decrease the order si2e and increase the nu"ber of orders( &ntermediate 13. -he total carr%ing costs are+ Carr%ing costs L (7:2# × CC here CC is the carr%ing cost per unit( -he restoc&ing costs are+ Restoc&ing costs L B × (-:7# ,etting these e$uations e$ual to each other and sol*ing for 7, e find+ CC × (7:2# L B × (-:7# 7 2 L 2 × B × - :CC 7 L Q2B × - :CCR 1:2 L 6/7 CHAPTER 20 B-349 14. -he cash flo fro" either polic% is+ Cash flo L (9 E *#7 ,o, the cash flos fro" the old polic% are+ Cash flo fro" old polic% L (<91 E IJ#(=,8A0# Cash flo fro" old polic% L <1M9,I00 0nd the cash flo fro" the ne polic% ould be+ Cash flo fro" ne polic% L (<9I E IJ#(=,9I0# Cash flo fro" ne polic%L <18A,180 ,o, the incre"ental cash flo ould be+ .ncre"ental cash flo L <18A,180 E 1M9,I00 .ncre"ental cash flo L <1A,J80 -he incre"ental cash flo is a perpetuit%( -he cost of initiating the ne polic% is+ Cost of ne polic% L EQ97 O *(7′ E 7#R ,o, the 49V of the decision to change credit policies is+ 49V L EQ(<9I#(=,8A0# O (<IJ#(=,9I0 E =,8A0#R O <1A,J80:(02A 49V L <2JM,M20 1. -he cash flo fro" the old polic% is+ Cash flo fro" old polic% L (<290 E 2=0#(1,10A# Cash flo fro" old polic% L <MM,=00 0nd the cash flo fro" the ne polic% ill be+ Cash flo fro" ne polic% L (<29A E 2=I#(1,12A# Cash flo fro" ne polic%L <M8,M2A -he incre"ental cash flo, hich is a perpetuit%, is the difference beteen the old polic% cash flos and the ne polic% cash flos, so+ .ncre"ental cash flo L <MM,=00 E M8,M2A .ncre"ental cash flo L <2,=2A B-350 SOLUTIONS -he cost of sitching credit policies is+ Cost of ne polic% L EQ97 O 7(*′ E *# O *′(7′ E 7#R .n this cost e$uation, e need to account for the increased *ariable cost for all units produced( -his includes the units e alread% sell, plus the increased *ariable costs for the incre"ental units( ,o, the 49V of sitching credit policies is+ 49V L EQ(<290#(1,10A# O (1,10A#(<2=I E 2=0# O (<2=I#(1,12A E 1,120#R O (<2,=2A:(009A# 49V L E<8I,81=(1M 1!. .f the cost of subscribing to the credit agenc% is less than the sa*ings fro" collection of the bad debts, the co"pan% should subscribe( -he cost of the subscription is+ Cost of the subscription L <I90 O <A(A00# Cost of the subscription L <2,9A0 0nd the sa*ings fro" ha*ing no bad debts ill be+ ,a*ings fro" not selling to bad credit ris&s L (<I90#(A00#(0(0I# ,a*ings fro" not selling to bad credit ris&s L <9,800 ,o, the co"pan%'s net sa*ings ill be+ 4et sa*ings L <9,800 E 2,9A0 4et sa*ings L <M,8A0 -he co"pan% should subscribe to the credit agenc%( Challenge 1". -he cost of sitching credit policies is+ Cost of ne polic% L EQ97 O 7(*′ E *# O *′(7′ E 7#R 0nd the cash flo fro" sitching, hich is a perpetuit%, is+ Cash flo fro" ne polic% L Q7′(9′ E *# E 7(9 E *#R -o find the brea&e*en $uantit% sold for sitching credit policies, e set the 49V e$ual to 2ero and sol*e for 7′( )oing so, e find+ 49V L 0 L EQ(<91#(=,8A0# O (<IJ#(7′ E =,8A0#R O Q(7′#(<9I E IJ# E (=,8A0#(<91 E IJ#R:(02A 0 L E<=A0,=A0 E <IJ7′ O <180,9A0 O <1,8807′ E <M,JJM,000 <1,8==7′ L <M,9IA,I00 7′ L =,J89(09 CHAPTER 20 B-351 1#. We can use the e$uation for the 49V e constructed in 9roble" 1J( Using the sales figure of I,100 units and sol*ing for 9′, e get+ 49V L 0 L QE(<91#(=,8A0# E (<IJ#(I,100 E =,8A0#R O Q(9′ E IJ#(I,100# E (<91 E IJ#(=,8A0#R:(02A 0 L E<=A0,=A0 E 11,JA0 O <1MI,0009′ E J,J08,000 E M,JJM,000 <1MI,0009′ L <1I,8IM,100 9′ L <90(A= 1$. Bro" 9roble" 1A, the incre"ental cash flo fro" the ne credit polic% ill be+ .ncre"ental cash flo L 7′(9′ E *′# E 7(9 E *# 0nd the cost of the ne polic% is+ Cost of ne polic% L EQ97 O 7(*′ E *# O *′(7′ E 7#R ,etting the 49V e$ual to 2ero and sol*ing for 9′, e get+ 49V L 0 L EQ(<290#(1,10A# O (<2=I E 2=0#(1,10A# O (<2=I#(1,12A E 1,10A#R O Q(1,12A#(9′ E 2=I# E (1,10A#(<290 E 2=0#R:(009A 0 L EQ<=20,IA0 O I,I20 O I,M80R O <118,I21(0A9′ E 2J,J10,A2M(=2 E M,9J8,9IJ(=J <118,I21(0A9′ L <=A,019,02=(M8 9′ L <29A(J2 2%. ,ince the co"pan% sells J00 suits per ee&, and there are A2 ee&s per %ear, the total nu"ber of suits sold is+ -otal suits sold L J00 S A2 L =M,I00 0nd, the 6/7 is A00 suits, so the nu"ber of orders per %ear is+ /rders per %ear L =M,I00 : A00 L J2(80 -o deter"ine the da% hen the ne!t order is placed, e need to deter"ine hen the last order as placed( ,ince the suits arri*ed on 3onda% and there is a = da% dela% fro" the ti"e the order as placed until the suits arri*e, the last order as placed Brida%( ,ince there are fi*e da%s beteen the orders, the ne!t order ill be placed on Wednesda% 0lternati*el%, e could consider that the store sells 100 suits per da% (J00 per ee& : J da%s#( -his i"plies that the store ill be at the safet% stoc& of 100 suits on ,aturda% hen it opens( ,ince the suits "ust arri*e before the store opens on ,aturda%, the% should be ordered = da%s prior to account for the deli*er% ti"e, hich again "eans the suits should be ordered in Wednesda%( B-352 SOLUTIONS APPENDIX 20A 1. -he cash flo fro" the old polic% is the $uantit% sold ti"es the price, so+ Cash flo fro" old polic% L I0,000(<A10# Cash flo fro" old polic% L <20,I00,000 -he cash flo fro" the ne polic% is the $uantit% sold ti"es the ne price, all ti"es one "inus the default rate, so+ Cash flo fro" ne polic% L I0,000(<A=J#(1 E (0=# Cash flo fro" ne polic% L <20,8=A,M00 -he incre"ental cash flo is the difference in the to cash flos, so+ .ncre"ental cash flo L <20,8=A,M00 E 20,I00,000 .ncre"ental cash flo L <I=A,M00 -he cash flos fro" the ne polic% are a perpetuit%( -he cost is the old cash flo, so the 49V of the decision to sitch is+ 49V L E<20,I00,000 O <I=A,M00:(02A 49V L E<2,9JM,000 2. a. -he old price as a percentage of the ne price is+ <90:<91(8I L (98 ,o the discount is+ )iscount L 1 E (98 L (02 or 2N -he credit ter"s ill be+ Credit ter"s+ 2:1A, net =0 b. We are unable to deter"ine for certain since no infor"ation is gi*en concerning the percentage of custo"ers ho ill ta&e the discount( >oe*er, the "a!i"u" recei*ables ould occur if all custo"ers too& the credit, so+ Recei*ables L =,=00(<90# Recei*ables L <29J,000 (at a "a!i"u"# c. ,ince the $uantit% sold does not change, *ariable cost is the sa"e under either plan( CHAPTER 20 B-353 d. 4o, because+ d E π L (02 E (11 d E π L E(09 or E9N -herefore the 49V ill be negati*e( -he 49V is+ 49V L E=,=00(<90# O (=,=00#(<91(8I#((02 E (11#:((01# 49V L E<=,02=,A92 -he brea&e*en credit price is+ 9(1 O r#:(1 E π# L <90(1(01#:((89# 9 L <102(1= -his i"plies that the brea&e*en discount is+ Hrea&e*en discount L 1 E (<90:<102(1=# Hrea&e*en discount L (1188 or 11(88N -he 49V at this discount rate is+ 49V L E=,=00(<90# O (=,=00#(<102(1=#((1188 E (11#:((01# 49V ≈ 0 3. a. -he cost of the credit polic% sitch is the $uantit% sold ti"es the *ariable cost( -he cash inflo is the price ti"es the $uantit% sold, ti"es one "inus the default rate( -his is a one-ti"e, lu"p su", so e need to discount this *alue one period( )oing so, e find the 49V is+ 49V L E1A(<JM0# O (1 E (2#(1A#(<1,1I0#:1(02 49V L <2,011(JM -he order should be ta&en since the 49V is positi*e( b. -o find the brea&e*en default rate, π, e 1ust need to set the 49V e$ual to 2ero and sol*e for the brea&e*en default rate( )oing so, e get+ 49V L 0 L E1A(<JM0# O (1 E π#(12#(<1,1I0#:1(02 π L (=200 or =2(00N c. 6ffecti*el%, the cash discount is+ Cash discount L (<1,1I0 E 1,090#:<1,1I0 Cash discount L (0I=9 or I(=9N ,ince the discount rate is less than the default rate, credit should not be granted( -he fir" ould be better off ta&ing the <1,090 up-front than ta&ing an 80N chance of "a&ing <1,1I0( B-354 SOLUTIONS 4. a. -he cash discount is+ Cash discount L (<JA E J1#:<JA Cash discount L (0A== or A(==N -he default probabilit% is one "inus the probabilit% of pa%"ent, or+ )efault probabilit% L 1 E (90 )efault probabilit% L (10 ,ince the default probabilit% is greater than the cash discount, credit should not be granted8 the 49V of doing so is negati*e( b. )ue to the increase in both $uantit% sold and credit price hen credit is granted, an additional incre"ental cost is incurred of+ 0dditional cost L (M,200#(<== E =2# O (M,900 E M,200#(<==# 0dditional cost L <29,=00 -he brea&e*en price under these assu"ptions is+ 49V L 0 L E<29,=00 E (M,200#(<J1# O YM,900Q(1 E (10#9′ E <==R E M,200(<J1 E =2#Z:(1(00JA = E 1# 49V L E<=I,100 E II0,200 O 2J=,9I0(=19′ E 10,0II,IJ8(08 E 10,MMM,IM8(1M <21,18A,2IM(2I L <2J=,9I0(=19′ 9′ L <JJ(=2 c. -he credit report is an additional cost, so e ha*e to include it in our anal%sis( -he 49V hen using the credit reports is+ 49V L M,200(=2# E (90(M,900#== E M,200(J1# E M,900(<1(A0# O YM,900Q0(90(JA E ==# E 1(A0R E M,200(J1 E =2#Z:(1(00JA = E 1# 49V L <198,I00 E 20I,9=0 E II0,200 E 10,=A0 O =8I,IAJ(J= 49V L E<JI,M22(2J -he reports should not be purchased and credit should not be granted( CHAPTER 20 B-355 . We can e!press the old cash flo as+ /ld cash flo L (9 E *#7 0nd the ne cash flo ill be+ 4e cash flo L (9 E *#(1 E α#7′ O α7′ Q(1 E π#9′ E *R ,o, the incre"ental cash flo is .ncre"ental cash flo L E(9 E *#7 O (9 E *#(1 E α#7′ O α7′ Q(1 E π#9′ E *R .ncre"ental cash floL (9 E *#(7′ E 7# O α7′ Q(1 E π#9′ E 9R -hus+ 49V L (9 E *#(7′ E 7# E α97′ O 1 ] 1 ¸ ′ ′ + ′ R 9Z - 9 # - Y(1 7 7# - 7 *#( - (9 π α CHAPTER 21 INTERNATIONAL CORPORATE FINANCE Answers to Concepts Review and Critical Thinking Questions 1. a. -he dollar is selling at a pre"iu" because it is "ore e!pensi*e in the forard "ar&et than in the spot "ar&et (,Br 1(A= *ersus ,Br 1(A0#( b. -he franc is e!pected to depreciate relati*e to the dollar because it ill ta&e "ore francs to bu% one dollar in the future than it does toda%( c. .nflation in ,it2erland is higher than in the United ,tates, as are interest rates( 2. -he e!change rate ill increase, as it ill ta&e progressi*el% "ore pesos to purchase a dollar( -his is the relati*e 999 relationship( 3. a. -he 0ustralian dollar is e!pected to ea&en relati*e to the dollar, because it ill ta&e "ore 0< in the future to bu% one dollar than it does toda%( b. -he inflation rate in 0ustralia is higher( c. 4o"inal interest rates in 0ustralia are higher8 relati*e real rates in the to countries are the sa"e( 4. 0 5an&ee bond is "ost accuratel% described b% d( . .t depends( Bor e!a"ple, if a countr%'s currenc% strengthens, i"ports beco"e cheaper (good#, but its e!ports beco"e "ore e!pensi*e for others to bu% (bad#( -he re*erse is true for currenc% depreciation( !. 0dditional ad*antages include being closer to the final consu"er and, thereb%, sa*ing on transportation, significantl% loer ages, and less e!posure to e!change rate ris&( )isad*antages include political ris& and costs of super*ising distant operations( ". /ne &e% thing to re"e"ber is that di*idend pa%"ents are "ade in the ho"e currenc%( 3ore generall%, it "a% be that the oners of the "ultinational are pri"aril% do"estic and are ulti"atel% concerned about their ealth deno"inated in their ho"e currenc% because, unli&e a "ultinational, the% are not internationall% di*ersified( #. a. Balse( .f prices are rising faster in Great Hritain, it ill ta&e "ore pounds to bu% the sa"e a"ount of goods that one dollar can bu%8 the pound ill depreciate relati*e to the dollar( b. Balse( -he forard "ar&et ould alread% reflect the pro1ected deterioration of the euro relati*e to the dollar( /nl% if %ou feel that there "ight be additional, unanticipated ea&ening of the euro that isn't reflected in forard rates toda% ill the forard hedge protect %ou against additional declines( CHAPTER 21 B-357 c. -rue( -he "ar&et ould onl% be correct on a*erage, hile %ou ould be correct all the ti"e( $. a. 0"erican e!porters+ their situation in general i"pro*es because a sale of the e!ported goods for a fi!ed nu"ber of euros ill be orth "ore dollars( 0"erican i"porters+ their situation in general orsens because the purchase of the i"ported goods for a fi!ed nu"ber of euros ill cost "ore in dollars( b. 0"erican e!porters+ the% ould generall% be better off if the Hritish go*ern"ent's intentions result in a strengthened pound( 0"erican i"porters+ the% ould generall% be orse off if the pound strengthens( c. 0"erican e!porters+ ould generall% be "uch orse off, because an e!tre"e case of fiscal e!pansion li&e this one ill "a&e 0"erican goods prohibiti*el% e!pensi*e to bu%, or else Hra2ilian sales, if fi!ed in reais, ould beco"e orth an unacceptabl% lo nu"ber of dollars( 0"erican i"porters+ ould generall% be "uch better off, because Hra2ilian goods ill beco"e "uch cheaper to purchase in dollars( 1%. .R9 is the "ost li&el% to hold because it presents the easiest and least costl% "eans to e!ploit an% arbitrage opportunities( Relati*e 999 is least li&el% to hold since it depends on the absence of "ar&et i"perfections and frictions in order to hold strictl%( Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. Using the $uotes fro" the table, e get+ a. <100(f0(MA09:<1# L fMA(09 b. <1(A=M= c. fA3(<1(A=M=:f# L <J,M81,MJ2 d. 4e ^ealand dollar e. 3e!ican peso f. (910(=A8I:<1#(<1(A=M=:f1# L 91A(91I0:f -his is a cross rate( g. 3ost *aluable+ Puait dinar L <=(JA9A Ceast *aluable+ Vietna" dong L <0(0000M020 B-358 SOLUTIONS 2. a. 5ou ould prefer W100, since+ (W100#(W1:<0(A1=A# L <19I(JI b. 5ou ould still prefer W100( Using the <:W e!change rate and the ,B:W e!change rate to find the a"ount of ,iss francs W100 ill bu%, e get+ (W100#(<1(9IJI:W1#(<:,B 0(9A=1# L ,B 2I0(=22J c. Using the $uotes in the boo& to find the ,B:W cross rate, e find+ (,B 0(9A=1:<#(<1(9IJI:W1# L ,B 2(0I=2:W1 -he W:,B e!change rate is the in*erse of the ,B:W e!change rate, so+ W1:,B 2(0I=2 L W0(I89I:,B 1 3. a. B180 L g10M(9M (per <#( -he %en is selling at a pre"iu" because it is "ore e!pensi*e in the forard "ar&et than in the spot "ar&et (<0(0092I1 *ersus <0(009=I9=#( b. B90 L <0(9J0M:C<1( -he dollar is selling at a discount because it is less e!pensi*e in the forard "ar&et than in the spot "ar&et (<0(9J1M *ersus <0(9J0M#( c. -he *alue of the dollar ill fall relati*e to the %en, since it ta&es "ore dollars to bu% one %en in the future than it does toda%( -he *alue of the dollar ill rise relati*e to the Canadian dollar, because it ill ta&e feer dollars to bu% one Canadian dollar in the future than it does toda%( 4. a. -he U(,( dollar, since one Canadian dollar ill bu%+ (Can<1#:(Can<1(0M:<1# L <0(9I=I b. -he cost in U(,( dollars is+ (Can<2(A0#:(Can<1(0M:<1# L <2(=M 0"ong the reasons that absolute 999 doesn't hold are tariffs and other barriers to trade, transactions costs, ta!es, and different tastes( c. -he U(,( dollar is selling at a pre"iu", because it is "ore e!pensi*e in the forard "ar&et than in the spot "ar&et (Can<1(11 *ersus Can<1(0M#( d. -he Canadian dollar is e!pected to depreciate in *alue relati*e to the dollar, because it ta&es "ore Canadian dollars to bu% one U(,( dollar in the future than it does toda%( e. .nterest rates in the United ,tates are probabl% loer than the% are in Canada( . a. -he cross rate in g:W ter"s is+ (g112:<1#(<1(9=:W1# L g21M(1M:W1 CHAPTER 21 B-359 b. -he %en is $uoted too lo relati*e to the pound( -a&e out a loan for <1 and bu% g112( Use the g112 to purchase pounds at the cross-rate, hich ill gi*e %ou+ g112(W1:g209# L W0(A=A89 Use the pounds to bu% bac& dollars and repa% the loan( -he cost to repa% the loan ill be+ W0(A=A89(<1(9=:W1# L <1(0=I= 5ou arbitrage profit is <0(0=I= per dollar used( !. We can rearrange the interest rate parit% condition to anser this $uestion( -he e$uation e ill use is+ RBC L (Bt E ,0#:,0 O RU, Using this relationship, e find+ Great Hritain+ RBC L (W0(A20I E W0(A1=A#:W0(A1=A O (022 L (0=AI or =(AIN Japan+ RBC L (g10M(9M E g108(21#:g108(21 O (022 L (010I or 1(0IN ,it2erland+ RBC L (,Br 1(0IJ8 E ,Br 1(0I92#:,Br 1(0I92 O (022 L (020J or 2(0JN ". .f e in*est in the U(,( for the ne!t three "onths, e ill ha*e+ <=0,000,000(1(00=J# = L <=0,==I,2==(M2 .f e in*est in Great Hritain, e "ust e!change the dollars toda% for pounds, and e!change the pounds for dollars in three "onths( 0fter "a&ing these transactions, the dollar a"ount e ould ha*e in three "onths ould be+ (<=0,000,000#(W0(AA:<1#(1(00A1# = :(W0(AM:<1# L <29,91J,=92(29 -he co"pan% should in*est in the U(,( #. Using the relati*e purchasing poer parit% e$uation+ Bt L ,0 S Q1 O (hBC E hU,#R t We find+ ^2(2M L ^2(1JQ1 O (hBC E hU,#R = hBC E hU, L (^2(2M:^2(1J# 1:= E 1 hBC E hU, L (01=M or 1(=MN .nflation in 9oland is e!pected to e!ceed that in the U(,( b% 1(=MN o*er this period( B-360 SOLUTIONS $. -he profit ill be the $uantit% sold, ti"es the sales price "inus the cost of production( -he production cost is in ,ingapore dollars, so e "ust con*ert this to U(,( dollars( )oing so, e find that if the e!change rates sta% the sa"e, the profit ill be+ 9rofit L =0,000Q<1A0 E Y(,<20I(J0#:(,<1(=80=:<1#ZR 9rofit L <A0,9MJ(18 .f the e!change rate rises, e "ust ad1ust the cost b% the increased e!change rate, so+ 9rofit L =0,000Q<1A0 E Y(,<20I(J0#:(1(1(,<1(=80=:<1##ZR 9rofit L <IAA,I2I(J1 .f the e!change rate falls, e "ust ad1ust the cost b% the decreased e!change rate, so+ 9rofit L =0,000Q<1A0 E Y(,<20I(J0#:(0(9(,<1(=80=:<1##ZR 9rofit L E<II=,=M9(80 -o calculate the brea&e*en change in the e!change rate, e need to find the e!change rate that "a&e the cost in ,ingapore dollars e$ual to the selling price in U(,( dollars, so+ <1A0 L ,<20I(J0:,- ,- L ,<1(=MIJ:<1 ,- L E0(011= or E1(1=N decline 1%. a. .f .R9 holds, then+ B180 L (Pr A(1A#Q1 O ((0AJ E (0=8#R 1:2 B180 L Pr A(198J ,ince gi*en B180 is PrA(22, an arbitrage opportunit% e!ists8 the forard pre"iu" is too high( Horro Pr1 toda% at A(JN interest( 0gree to a 180-da% forard contract at Pr A(22( Con*ert the loan proceeds into dollars+ Pr 1 (<1:Pr A(1A# L <0(19I1J .n*est these dollars at =(8N, ending up ith <0(19JJ8( Con*ert the dollars bac& into &rone as <0(19JJ8(Pr A(22:<1# L Pr 1(0=2I1 Repa% the Pr 1 loan, ending ith a profit of+ Pr1(0=2I1 E Pr1(02JJ1 L Pr 0(0IM9 b. -o find the forard rate that eli"inates arbitrage, e use the interest rate parit% condition, so+ B180 L (Pr A(1A#Q1 O ((0AJ E (0=8#R 1:2 B180 L Pr A(198J CHAPTER 21 B-361 11. -he international Bisher effect states that the real interest rate across countries is e$ual( We can rearrange the international Bisher effect as follos to anser this $uestion+ RU, E hU, L RBC E hBC hBC L RBC O hU, E RU, a. h0U, L (0I O (0=9 E (0A8 h0U, L (021 or 2(1N b. hC04 L (0J O (0=9 E (0A8 hC04 L (0A1 or A(1N c. h-0. L (09 O (0=9 E (0A8 h-0. L (0J1 or J(1N 12. a. -he %en is e!pected to get ea&er, since it ill ta&e "ore %en to bu% one dollar in the future than it does toda%( b. hU, E hJ09 ≈ (g11M(0= E g11I(=2#:g11I(=2 hU, E hJ09 L 0(01A0 or 1(A0N (1 O (01A0# I E 1 L 0(0M12 or M(12N -he appro!i"ate inflation differential beteen the U(,( and Japan is M(12N annuall%( 13. We need to find the change in the e!change rate o*er ti"e so e need to use the interest rate parit% relationship+ Bt L ,0 S Q1 O (RBC E RU,#R t Using this relationship, e find the e!change rate in one %ear should be+ B1 L 1A2(9=Q1 O ((08M E (0I9#R 1 B1 L >UB 1A8(A9 -he e!change rate in to %ears should be+ B2 L 1A2(9=Q1 O ((08M E (0I9#R 2 B2 L >UB 1MI(IM 0nd the e!change rate in fi*e %ears should be+ BA L 1A2(9=Q1 O ((08M E (0I9#R A BA L >UB 18=(=9 B-362 SOLUTIONS &ntermediate 14. Birst, e need to forecast the future spot rate for each of the ne!t three %ears( Bro" interest rate and purchasing poer parit%, the e!pected e!change rate is+ 6(,-# L Q(1 O RU,# : (1 O RBC#R - ,0 ,o+ 6(,1# L (1(0I80 : 1(0I10# 1 <1(28:f L <1(288M:f 6(,2# L (1(0I80 : 1(0I10# 2 <1(28:f L <1(29J=:f 6(,=# L (1(0I80 : 1(0I10# = <1(28:f L <1(=0M0:f 4o e can use these future spot rates to find the dollar cash flos( -he dollar cash flo each %ear ill be+ 5ear 0 cash flo L Ef<1I,000,000(<1(28:f# L E<1J,920,000(00 5ear 1 cash flo L f<2,100,000(<1(288M:f# L <2,J0M,0JI(9= 5ear 2 cash flo L f<=,I00,000(<1(29J=:f# L <I,I10,J2A(12 5ear = cash flo L (fI,=00,000 O 9,M00,000#(<1(=0M0:f# L <18,1A=,==A(=0 0nd the 49V of the pro1ect ill be+ 49V L E<1J,920,000 O <2,J0M,0JI(9=:1(1= O <I,I10,J2A(12:1(1= 2 O <18,1A=,==A(=0:1(1= = 49V L <A10,1J=(=0 1. a. ."plicitl%, it is assu"ed that interest rates on't change o*er the life of the pro1ect, but the e!change rate is pro1ected to decline because the 6urosiss rate is loer than the 6urodollar rate( b. We can use relati*e purchasing poer parit% to calculate the dollar cash flos at each ti"e( -he e$uation is+ 6Q,tR L (,Br 1(09#Q1 O ((0J E (08#R t 6Q,tR L 1(09((99# t ,o, the cash flos each %ear in U(,( dollar ter"s ill be+ t ,Br 6Q, t R U,< 0 E2I(03 1(0900 E<22,018,=I8(M= 1 OM(M3 1(0J91 <M,11M,20J(9A 2 OM(M3 1(0M8= <M,1JJ,98J(8= = OM(M3 1(0AJM <M,2I0,=91(JA I OM(M3 1(0IJ0 <M,=0=,I2M(01 A OM(M3 1(0=MM <M,=MJ,09M(98 CHAPTER 21 B-363 0nd the 49V is+ 49V L E<22,018,=I8(M2 O <M,11M,20J(9A:1(12 O <M,1JJ,98J(8=:1(12 2 O <M,2I0,=91(JA :1(12 = O <M,=0=,I2M(01:1(12 I O <M,=MJ,09M(98:1(12 A 49V L <I28,19A(99 c. Rearranging the relati*e purchasing poer parit% e$uation to find the re$uired return in ,iss francs, e get+ R,Br L 1(12Q1 O ((0J E (08#R E 1 R,Br L 10(88N ,o the 49V in ,iss francs is+ 49V L E,Br 2I(03 O ,Br M(M3(9V.B010(88N,A# 49V L ,Br IMM,J==(M= Con*erting the 49V to dollars at the spot rate, e get the 49V in U(,( dollars as+ 49V L (,Br IMM,J==(M=#(<1:,Br 1(09# 49V L <I28,19A(99 1!. a. -o construct the balance sheet in dollars, e need to con*ert the account balances to dollars( 0t the current e!change rate, e get+ 0ssets L solaris 2=,000 : (< : solaris 1(20# L <19,1MM(MJ )ebt L solaris 9,000 : (< : solaris 1(20# L <J,A00(00 6$uit% L solaris 1I,000 : (< : solaris 1(20# L <19,1MM(MJ b. .n one %ear, if the e!change rate is solaris 1(I0:<, the accounts ill be+ 0ssets L solaris 2=,000 : (< : solaris 1(I0# L <1M,I28(AJ )ebt L solaris 9,000 : (< : solaris 1(I0# L <M,I28(AJ 6$uit% L solaris 1I,000 : (< : solaris 1(I0# L <10,000(00 b. .f the e!change rate is solaris 1(12:<, the accounts ill be+ 0ssets L solaris 2=,000 : (< : solaris 1(12# L <20,A=A(J1 )ebt L solaris 9,000 : (< : solaris 1(12# L <8,0=A(J1 6$uit% L solaris 1I,000 : (< : solaris 1(12# L <12,A00(00 B-364 SOLUTIONS Challenge 1". Birst, e need to construct the end of %ear balance sheet in solaris( ,ince the co"pan% has retained earnings, the e$uit% account ill increase, hich necessaril% i"plies the assets ill also increase b% the sa"e a"ount( ,o, the balance sheet at the end of the %ear in solaris ill be+ Halance ,heet (solaris# Ciabilities <9,000(00 6$uit% 1A,2A0(00 0ssets <2I,2A0(00 -otal liabilities K e$uit% <2I,2A0(00 4o e need to con*ert the balance sheet accounts to dollars, hich gi*es us+ 0ssets L solaris 2I,2A0 : (< : solaris 1(2I# L <19,AAM(IA )ebt L solaris 9,000 : (< : solaris 1(2I# L <J,2A8(0M 6$uit% L solaris 1A,2A0 : (< : solaris 1(2I# L <12,298(=9 1#. a. -he do"estic Bisher effect is+ 1 O ?=' L (1 O r='#(1 O h='# 1 O r=' L (1 O ?='#:(1 O h='# -his relationship "ust hold for an% countr%, that is+ 1 O rFC L (1 O ?FC#:(1 O hFC# -he international Bisher effect states that real rates are e$ual across countries, so+ 1 O r=' L (1 O ?='#:(1 O h='# L (1 O ?FC#:(1 O hFC# L 1 O rFC b. -he e!act for" of unbiased interest rate parit% is+ 6Q'tR L Ft L ,0 Q(1 O ?FC#:(1 O ?='8R t c. -he e!act for" for relati*e 999 is+ 6Q'tR L '0 Q(1 O hFC#:(1 O h='#R t CHAPTER 21 B-365 d. Bor the ho"e currenc% approach, e calculate the e!pected currenc% spot rate at ti"e t as+ 6Q'tR L (f0(A#Q1(0J:1(0AR t L (f0(A#(1(019# t We then con*ert the euro cash flos using this e$uation at e*er% ti"e, and find the present *alue( )oing so, e find+ 49V L E Qf23:(f0(A#R O Yf0(93:Q1(019(f0(A#RZ:1(1 O Yf0(93:Q1(019 2 (f0(A#RZ:1(1 2 O Yf0(93:Q1(019 = (f0(A:<1#RZ:1(1 = 49V L <=1M,2=0(J2 Bor the foreign currenc% approach e first find the return in the euros as+ ? FC L 1(10(1(0J:1(0A# E 1 L 0(121 4e!t, e find the 49V in euros as+ 49V L E f23 O (f0(93#:1(121 O (f0(93#:1(121 2 O (f0(93#:1(121 = L f1A8,11A(=M 0nd finall%, e con*ert the euros to dollars at the current e!change rate, hich is+ 49V (<# L f1A8,11A(=M :(f0(A:<1# L <=1M,2=0(J2 CHAPTER 22 BEHAVIORAL FINANCE: IMPLICATIONS FOR FINANCIAL MANAGEMENT Answers to Concepts Review and Critical Thinking Questions 1. -he least li&el% li"it to arbitrage is fir"-specific ris&( Bor e!a"ple, in the =Co":9al" case, the stoc&s are perfect substitutes after accounting for the e!change ratio( 0n in*estor could in*est in a ris& neutral portfolio b% purchasing the underpriced asset and selling the o*erpriced asset( When the prices of the assets re*ert to an e$uilibriu", the positions could be closed( 2. /*erconfidence is the belief that one's abilities are greater than the% are( 0n o*erconfident financial "anager could belie*e that the% are correct in the face of e*idence to the contrar%( Bor e!a"ple, the financial "anager could belie*e that a ne product ill be a great success (or failure# e*en though "ar&et research points to the contrar%( -his could "ean that the co"pan% in*ests or in*ests too "uch in the ne product or "isses out on the ne in*est"ent, an opportunit% cost( .n each case, shareholder *alue is not "a!i"i2ed( 3. Bra"e dependence is the argu"ent that an in*estor's choice is dependent on the a% the $uestion is posed( 0n in*estor can fra"e a decision proble" in broad ter"s (li&e ealth# or in narro ter"s (li&e gains and losses#( Hroad and narro fra"es often lead the in*estor to "a&e different choices( While it is hu"an nature to use a narro fra"e (li&e gains and losses#, doing so can lead to irrational decisions( Using broad fra"es, li&e o*erall ealth, results in better in*est"ent decisions( 4. 0 noise trader is so"eone hose trades are not based on infor"ation or financiall% "eaningful anal%sis( 4oise traders could, in principle, act together to orsen a "ispricing in the short-run( 4oise trader ris& is i"portant because the orsening of a "ispricing could force the arbitrageur to li$uidate earl% and sustain steep losses( . As long as it is a fair coin the probability in both cases is 50 percent as coins have no memory. Although many believe the probability of flipping a tail would be greater given the long run of heads, this is an example of the gambler’s fallacy. !. Taken at face value, this fact suggests that markets have become more efficient. The increasing ease with which information is available over the Internet lends strength to this conclusion. On the other hand, during this particular period, large-capitalization growth stocks were the top performers. Value-weighted indexes such as the S&P 500 are naturally concentrated in such stocks, thus making them especially hard to beat during this period. So, it may be that the dismal record compiled by the pros is just a matter of bad luck or benchmark error. CHAPTER 22 B-367 ". The statement is false because every investor has a different risk preference. Although the expected return from every well-diversified portfolio is the same after adjusting for risk, investors still need to choose funds that are consistent with their particular risk level. 9. Behavioral finance attempts to explain both the 1987 stock market crash and the Internet bubble by changes in investor sentiment and psychology. These changes can lead to non-random price behavior. $. Heha*ioral finance states that the "ar&et is not efficient( 0dherents argue that+ 1# .n*estors are not rational( 2# )e*iations fro" rationalit% are si"ilar across in*estors( =# 0rbitrage, being costl%, ill not eli"inate inefficiencies. 1%. Bra"e dependence "eans that the decision "ade is affected b% the a% in hich the $uestion is as&ed( .n this e!a"ple, consider that the <J8 is a sun& cost( 5ou ill not get this "one% bac& hether or not %ou accept the deal( .n this case, the *alues fro" the deal are a gain of <J8 ith 20 percent probabilit% or a loss of <22 ith an 80 percent probabilit%( -he e!pected *alue of the deal is <J8((20# E <22((80# L E<2( 4otice this is the sa"e as the difference beteen the loss of <J8 and the e!pected loss of <80 hich e calculated using no net loss and a loss of <100( CHAPTER 23 RISK MANAGEMENT: AN INTRODUCTION TO FINANCIAL ENGINEERING Answers to Concepts Review and Critical Thinking Questions 1. ,ince the fir" is selling futures, it ants to be able to deli*er the lu"ber8 therefore, it is a supplier( ,ince a decline in lu"ber prices ould reduce the inco"e of a lu"ber supplier, it has hedged its price ris& b% selling lu"ber futures( Cosses in the spot "ar&et due to a fall in lu"ber prices are offset b% gains on the short position in lu"ber futures( 2. Hu%ing call options gi*es the fir" the right to purchase por& bellies8 therefore, it "ust be a consu"er of por& bellies( While a rise in por& bell% prices is bad for the consu"er, this ris& is offset b% the gain on the call options8 if por& bell% prices actuall% decline, the consu"er en1o%s loer costs, hile the call option e!pires orthless( 3. Borard contracts are usuall% designed b% the parties in*ol*ed for their specific needs and are rarel% sold in the secondar% "ar&et8 forards are so"ehat custo"i2ed financial contracts( 0ll gains and losses on the forard position are settled at the "aturit% date( Butures contracts are standardi2ed to facilitate their li$uidit% and to allo the" to be effecti*el% traded on organi2ed futures e!changes( Gains and losses on futures are "ar&ed-to-"ar&et dail%( -he default ris& is greatl% reduced ith futures, since the e!change acts as an inter"ediar% beteen the to parties, guaranteeing perfor"ance8 default ris& is also reduced because the dail% settle"ent procedure &eeps large loss positions fro" accu"ulating( 5ou "ight prefer to use forards instead of futures if %our hedging needs ere different fro" the standard contract si2e and "aturit% dates offered b% the futures contract( 4. -he fir" is hurt b% declining oil prices, so it should sell oil futures contracts( -he fir" "a% not be able to create a perfect hedge because the $uantit% of oil it needs to hedge doesn't "atch the standard contract si2e on crude oil futures, or perhaps the e!act settle"ent date the co"pan% re$uires isn't a*ailable on these futures (e!posing the fir" to basis ris&#, or "a%be the fir" produces a different grade of crude oil than that specified for deli*er% in the futures contract( . -he fir" is directl% e!posed to fluctuations in the price of natural gas, since it is a natural gas user( .n addition, the fir" is indirectl% e!posed to fluctuations in the price of oil( .f oil beco"es less e!pensi*e relati*e to natural gas, its co"petitors ill en1o% a cost ad*antage relati*e to the fir"( !. Hu%ing the call options is a for" of insurance polic% for the fir"( .f cotton prices rise, the fir" is protected b% the call, hile if prices actuall% decline, the% can 1ust allo the call to e!pire orthless( >oe*er, options hedges are costl% because of the initial pre"iu" that "ust be paid( -he futures contract can be entered into at no initial cost, ith the disad*antage that the fir" is loc&ing in one price for cotton8 it can't profit fro" cotton price declines( CHAPTER 23 B-369 ". -he put option on the bond gi*es the oner the right to sell the bond at the option's stri&e price( .f bond prices decline, the oner of the put option profits( >oe*er, since bond prices and interest rates "o*e in opposite directions, if the put oner profits fro" a decline in bond prices, he ould also profit fro" a rise in interest rates( >ence, a call option on interest rates is conceptuall% the sa"e thing as a put option on bond prices( #. -he co"pan% ould li&e to loc& in the current lo rates, or at least be protected fro" a rise in rates, alloing for the possibilit% of benefit if rates actuall% fall( -he for"er hedge could be i"ple"ented b% selling bond futures8 the latter could be i"ple"ented b% bu%ing put options on bond prices or bu%ing call options on interest rates( $. 0 sap contract is an agree"ent beteen parties to e!change assets o*er se*eral ti"e inter*als in the future( -he sap contract is usuall% an e!change of cash flos, but not necessaril% so( ,ince a forard contract is also an agree"ent beteen parties to e!change assets in the future, but at a single point in ti"e, a sap can be *ieed as a series of forard contracts ith different settle"ent dates( -he fir" participating in the sap agree"ent is e!posed to the default ris& of the dealer, in that the dealer "a% not "a&e the cash flo pa%"ents called for in the contract( -he dealer faces the sa"e ris& fro" the contracting part%, but can "ore easil% hedge its default ris& b% entering into an offsetting sap agree"ent ith another part%( 1%. -he fir" ill borro at a fi!ed rate of interest, recei*e fi!ed rate pa%"ents fro" the dealer as part of the sap agree"ent, and "a&e floating rate pa%"ents bac& to the dealer8 the net position of the fir" is that it has effecti*el% borroed at floating rates( 11. -ransactions e!posure is the short-ter" e!posure due to uncertain prices in the near future( 6cono"ic e!posure is the long-ter" e!posure due to changes in o*erall econo"ic conditions( -here are a *ariet% of instru"ents a*ailable to hedge transaction e!posure, but *er% fe long-ter" hedging instru"ents e!ist( .t is "uch "ore difficult to hedge against econo"ic e!posure, since funda"ental changes in the business generall% "ust be "ade to offset long-run changes in the econo"ic en*iron"ent( 12. -he ris& is that the dollar ill strengthen relati*e to the %en, since the fi!ed %en pa%"ents in the future ill be orth feer dollars( ,ince this i"plies a decline in the <:g e!change rate, the fir" should sell %en futures( 13. a. Hu% oil and natural gas futures contracts, since these are probabl% %our pri"ar% resource costs( .f it is a coal-fired plant, a cross-hedge "ight be i"ple"ented b% selling natural gas futures, since coal and natural gas prices are so"ehat negati*el% related in the "ar&et8 coal and natural gas are so"ehat substitutable( b. Hu% sugar and cocoa futures, since these are probabl% %our pri"ar% co""odit% inputs( c. ,ell corn futures, since a record har*est i"plies lo corn prices( d. Hu% sil*er and platinu" futures, since these are pri"ar% co""odit% inputs re$uired in the "anufacture of photographic fil"( e. ,ell natural gas futures, since e!cess suppl% in the "ar&et i"plies lo prices( f. 0ssu"ing the ban& doesn't resell its "ortgage portfolio in the secondar% "ar&et, bu% bond futures( g. ,ell stoc& inde! futures, using an inde! "ost closel% associated ith the stoc&s in %our fund, such as the ,K9 100 or the 3a1or 3ar&et .nde! for large blue-chip stoc&s( h. Hu% ,iss franc futures, since the ris& is that the dollar ill ea&en relati*e to the franc o*er the ne!t si! "onth, hich i"plies a rise in the <:,Br e!change rate( i. ,ell euro futures, since the ris& is that the dollar ill strengthen relati*e to the euro o*er the ne!t three "onths, hich i"plies a decline in the <:f e!change rate( B-370 SOLUTIONS 14. ,%sco "ust ha*e felt that the co"bination of fi!ed plus sap ould result in an o*erall better rate( .n other ords, *ariable rate a*ailable *ia a sap "a% ha*e been "ore attracti*e than the rate a*ailable fro" issuing a floating-rate bond( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. -he initial price is <=,122 per "etric ton and each contract is for 10 "etric tons, so the initial contract *alue is+ .nitial contract *alue L (<=,122 per ton#(10 tons per contract# L <=1,220 0nd the final contract *alue is+ Binal contract *alue L (<=,081 per ton#(10 tons per contract# L <=0,810 5ou ill ha*e a loss on this futures position of+ Coss on futures contract L <=1,220 E =0,810 L <I10 2. -he price $uote is <1M(M0J per ounce and each contract is for A,000 ounces, so the initial contract *alue is+ .nitial contract *alue L (<1M(M0J per o2(#(A,000 o2( per contract# L <8=,0=A 0t a final price of <1M(81 per ounce, the *alue of the position is+ Binal contract *alue L (<1M(81 per o2(#(A,000 o2( per contract# L <8I,0A0 ,ince this is a short position, there is a net loss of+ <8I,0A0 E 8=,0=A L <1,01A per contract ,ince %ou sold fi*e contracts, the net loss is+ 4et loss L A(<1,01A# L <A,0JA CHAPTER 23 B-371 0t a final price of <1M(=2 per ounce, the *alue of the position is+ Binal contract *alue L (<1M(=2 per o2(#(A,000 o2( per contract# L <81,M00 ,ince this is a short position, there is a net gain of <8=,0=A E 81,M00 L <1,I=A ,ince %ou sold fi*e contracts, the net gain is+ 4et gain L A(<1,I=A# L <J,1JA With a short position, %ou "a&e a profit hen the price falls, and incur a loss hen the price rises( 3. -he price $uote is <0(08=A per pound and each contract is for 1A,000 pounds, so the cost per contract is+ Cost L (<0(08=A per pound#(1A,000 pounds per contract# L <1,2A2(A0 .f the price of orange 1uice at e!piration is <1(29 per pound, the call is out of the "one% since the stri&e price is abo*e the spot price( -he contracts ill e!pire orthless, so %our loss ill be the initial in*est"ent of <1,2A=( .f orange 1uice prices at contract e!piration are <1(MJ per pound, the call is in the "one% since the price per pound is abo*e the stri&e price( -he pa%off on %our position is the current price "inus the stri&e price, ti"es the 1A,000 pounds per contract, or+ 9a%off L (<1(MJ E 1(I0#(1A,000# L <I,0A0 0nd the profit is the pa%off "inus the initial cost of the contract, or+ 9rofit L <I,0A0 E 1,2A2(A0 L <2,J9J(A0 4. -he call options gi*e the "anager the right to purchase oil futures contracts at a futures price of <1I0 per barrel( -he "anager ill e!ercise the option if the price rises abo*e <1I0( ,elling put options obligates the "anager to bu% oil futures contracts at a futures price of <1I0 per barrel( -he put holder ill e!ercise the option if the price falls belo <1I0( -he pa%offs per barrel are+ /il futures price+ <1=A <1=J <1I0 <1I= <1IA Value of call option position+ 0 0 0 = A Value of put option position+ EA E= 0 0 0 -otal *alue+ E<A E<= <0 <= <A -he pa%off profile is identical to that of a forard contract ith a <1I0 stri&e price( . -he price $uote is <0(1880 per pound and each contract is for 1A,000 pounds, so the cost per contract is+ Cost L (<0(1880 per pound#(1A,000 pounds per contract# L <2,820(00 B-372 SOLUTIONS .f the price of orange 1uice at e!piration is <1(1I per pound, the put is in the "one% since the stri&e price is greater than the spot price( -he pa%off on %our position is the stri&e price "inus the current price, ti"es the 1A,000 pounds per contract, or+ 9a%off L (<1(=A E 1(1I#(1A,000# L <=,1A0 0nd the profit is the pa%off "inus the initial cost of the contract, or+ 9rofit L <=,1A0 E 2,820 L <==0 .f the price of orange 1uice at e!piration is <1(IJ per pound, the put is out of the "one% since the stri&e price is less than the spot price( -he contracts ill e!pire orthless, so %our loss ill be the initial in*est"ent of <2,820( &ntermediate !. a. 5ou're concerned about a rise in corn prices, so %ou ould bu% )ece"ber contracts( ,ince each contract is for A,000 bushels, the nu"ber of contracts %ou ould need to bu% is+ 4u"ber of contracts to bu% L 10A,000:A,000 L 21 H% doing so, %ou're effecti*el% loc&ing in the settle price in )ece"ber, 2008 of <J(MA per bushel of corn, or+ -otal price for 10A,000 bushels L 21(<J(MA#(A,000# L <80=,2A0 b. .f the price of corn at e!piration is <J(I1 per bushel, the *alue of %ou futures position is+ Value of future position L 21(<J(I1#(A,000# L <JJ8,0A0 .gnoring an% transaction costs, %our loss on the futures position ill be+ Coss L <80=,2A0 E JJ8,0A0 L <2A,200 While the price of the corn %our fir" needs has beco"e <2A,200 less e!pensi*e since June, %our loss fro" the futures position has netted out this loer cost( ". a( U5^ has a co"parati*e ad*antage relati*e to 0HC in borroing at fi!ed interest rates, hile 0HC has a co"parati*e ad*antage relati*e to U5^ in borroing at floating interest rates( ,ince the spread beteen 0HC and U5^'s fi!ed rate costs is onl% 1N, hile their differential is 2N in floating rate "ar&ets, there is an opportunit% for a =N total gain b% entering into a fi!ed for floating rate sap agree"ent( CHAPTER 23 B-373 b. .f the sap dealer "ust capture 2N of the a*ailable gain, there is 1N left for 0HC and U5^( 0n% di*ision of that gain is feasible8 in an actual sap deal, the di*isions ould probabl% be negotiated b% the dealer( /ne possible co"bination is ½N for 0HC and ½N for U5^+ 0HC 0HC 0HC 0HC 0HC )ealer C.H/R O1N 10(AN U5^ C.H/R O2(AN O2(AN 10(0N )ebt 3ar&et C.H/R O1N )ebt 3ar&et 10N Challenge #. -he financial engineer can replicate the pa%offs of oning a put option b% selling a forard contract and bu%ing a call( Bor e!a"ple, suppose the forard contract has a settle price of <A0 and the e!ercise price of the call is also <A0( -he pa%offs belo sho that the position is the sa"e as oning a put ith an e!ercise price of <A0+ 9rice of coal+ <I0 <IA <A0 <AA <M0 Value of call option position+ 0 0 0 A 10 Value of forard position+ 10 A 0 EA E10 -otal *alue+ <10 <A <0 <0 <0 Value of put position+ <10 <A <0 <0 <0 -he pa%offs for the co"bined position are e!actl% the sa"e as those of oning a put( -his "eans that, in general, the relationship beteen puts, calls, and forards "ust be such that the cost of the to strategies ill be the sa"e, or an arbitrage opportunit% e!ists( .n general, gi*en an% to of the instru"ents, the third can be s%nthesi2ed( CHAPTER 24 OPTIONS AND CORPORATE FINANCE Answers to Concepts Review and Critical Thinking Questions 1. 0 call option confers the right, ithout the obligation, to bu% an asset at a gi*en price on or before a gi*en date( 0 put option confers the right, ithout the obligation, to sell an asset at a gi*en price on or before a gi*en date( 5ou ould bu% a call option if %ou e!pect the price of the asset to increase( 5ou ould bu% a put option if %ou e!pect the price of the asset to decrease( 0 call option has unli"ited potential profit, hile a put option has li"ited potential profit8 the underl%ing asset's price cannot be less than 2ero( 2. a. -he bu%er of a call option pa%s "one% for the right to bu%(((( b. -he bu%er of a put option pa%s "one% for the right to sell(((( c. -he seller of a call option recei*es "one% for the obligation to sell(((( d. -he seller of a put option recei*es "one% for the obligation to bu%(((( 3. -he intrinsic *alue of a call option is 3a! Q, E 6,0R( .t is the *alue of the option at e!piration( 4. -he *alue of a put option at e!piration is 3a!Q6 E ,,0R( H% definition, the intrinsic *alue of an option is its *alue at e!piration, so 3a!Q6 E ,,0R is the intrinsic *alue of a put option( . -he call is selling for less than its intrinsic *alue8 an arbitrage opportunit% e!ists( Hu% the call for <10, e!ercise the call b% pa%ing <=A in return for a share of stoc&, and sell the stoc& for <A0( 5ou'*e "ade a ris&less <A profit( !. -he prices of both the call and the put option should increase( -he higher le*el of donside ris& still results in an option price of 2ero, but the upside potential is greater since there is a higher probabilit% that the asset ill finish in the "one%( ". Balse( -he *alue of a call option depends on the total *ariance of the underl%ing asset, not 1ust the s%ste"atic *ariance( #. -he call option ill sell for "ore since it pro*ides an unli"ited profit opportunit%, hile the potential profit fro" the put is li"ited (the stoc& price cannot fall belo 2ero#( $. -he *alue of a call option ill increase, and the *alue of a put option ill decrease( 1%. -he reason the% don't sho up is that the U(,( go*ern"ent uses cash accounting8 i(e(, onl% actual cash inflos and outflos are counted, not contingent cash flos( Bro" a political perspecti*e, debt guarantees ould "a&e the deficit larger, so that is another reason not to count the"X Whether the% should be included depends on hether e feel cash accounting is appropriate or not, but these contingent liabilities should be "easured and reported( -he% currentl% are not, at least not in a s%ste"atic fashion( CHAPTER 24 B-375 11. -he option to abandon reflects our abilit% to shut don a pro1ect if it is losing "one%( ,ince this option acts to li"it losses, e ill underesti"ate 49V if e ignore it( B-376 SOLUTIONS 12. -he option to e!pand reflects our abilit% to increase production if the ne product sells "ore than e initiall% e!pected( ,ince this option increases the potential future cash flos be%ond our initial esti"ate, e ill underesti"ate 49V if e ignore it( 13. -his is a good e!a"ple of the option to e!pand( 14. With oil, for e!a"ple, e can si"pl% stop pu"ping if prices drop too far, and e can do so $uic&l%( -he oil itself is not affected8 it 1ust sits in the ground until prices rise to a point here pu"ping is profitable( Gi*en the *olatilit% of natural resource prices, the option to suspend output is *er% *aluable( 1. -here are to possible benefits( Birst, aarding e"plo%ee stoc& options "a% better align the interests of the e"plo%ees ith the interests of the stoc&holders, loering agenc% costs( ,econdl%, if the co"pan% has little cash a*ailable to pa% top e"plo%ees, e"plo%ee stoc& options "a% help attract $ualified e"plo%ees for less pa%( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. a. -he *alue of the call is the stoc& price "inus the present *alue of the e!ercise price, so+ C0 L <MM E Q<AA:1(0AMR L <1=(92 -he intrinsic *alue is the a"ount b% hich the stoc& price e!ceeds the e!ercise price of the call, so the intrinsic *alue is <11( b. -he *alue of the call is the stoc& price "inus the present *alue of the e!ercise price, so+ C0 L <MM E Q<IA:1(0AMR L <2=(=9 -he intrinsic *alue is the a"ount b% hich the stoc& price e!ceeds the e!ercise price of the call, so the intrinsic *alue is <21( c. -he *alue of the put option is <0 since there is no possibilit% that the put ill finish in the "one%( -he intrinsic *alue is also <0( 2. a. -he calls are in the "one%( -he intrinsic *alue of the calls is <A( b. -he puts are out of the "one%( -he intrinsic *alue of the puts is <0( CHAPTER 24 B-377 c. -he 3ar call and the /ct put are "ispriced( -he call is "ispriced because it is selling for less than its intrinsic *alue( .f the option e!pired toda%, the arbitrage strateg% ould be to bu% the call for <2(80, e!ercise it and pa% <80 for a share of stoc&, and sell the stoc& for <8A( 0 ris&less profit of <2(20 results( -he /ctober put is "ispriced because it sells for less than the Jul% put( -o ta&e ad*antage of this, sell the Jul% put for <=(90 and bu% the /ctober put for <=(MA, for a cash inflo of <0(2A( -he e!posure of the short position is co"pletel% co*ered b% the long position in the /ctober put, ith a positi*e cash inflo toda%( 3. a. 6ach contract is for 100 shares, so the total cost is+ Cost L 10(100 shares:contract#(<=(2=# Cost L <=,2=0 b. .f the stoc& price at e!piration is <1=I, the pa%off is+ 9a%off L 10(100#(<1=I E 120# 9a%off L <1I,000 .f the stoc& price at e!piration is <12M, the pa%off is+ 9a%off L 10(100#(<12M E 120# 9a%off L <M,000 c. Re"e"bering that each contract is for 100 shares of stoc&, the cost is+ Cost L 10(100#(<9(10# Cost L <9,100 -he "a!i"u" gain on the put option ould occur if the stoc& price goes to <0( We also need to subtract the initial cost, so+ 3a!i"u" gain L 10(100#(<120# E <9,100 3a!i"u" gain L <110,900 .f the stoc& price at e!piration is <109, the position ill be orth+ 9osition *alue L 10(100#(<120 E 109# 9osition *alue L <11,000 0nd %our profit ill be+ 9rofit L <11,000 E 9,100 9rofit L <1,900 d. 0t a stoc& price of <108 the put is in the "one%( 0s the riter %ou ill lose+ 4et gain(loss# L <9,100 E 10(100#(<120 E 108# 4et gain(loss# L E<2,900 B-378 SOLUTIONS 0t a stoc& price of <1=2 the put is out of the "one%, so the riter ill "a&e the initial cost+ 4et gain L <9,100 0t the brea&e*en, %ou ould reco*er the initial cost of <9,100, so+ <9,100 L 10(100#(<120 E ,-# ,- L <110(90 Bor ter"inal stoc& prices abo*e <110(90, the riter of the put option "a&es a net profit (ignoring transaction costs and the effects of the ti"e *alue of "one%#( 4. a. -he *alue of the call is the stoc& price "inus the present *alue of the e!ercise price, so+ C0 L <J0 E MA:1(0A C0 L <8(10 b. Using the e$uation presented in the te!t to pre*ent arbitrage, e find the *alue of the call is+ <J0 L Q(<8M E M2#:(<8M E JA#RC0 O <M2:1(0A C0 L <A(02 . a. -he *alue of the call is the stoc& price "inus the present *alue of the e!ercise price, so+ C0 L <8A E <MA:1(0M C0 L <2=(M8 b. Using the e$uation presented in the te!t to pre*ent arbitrage, e find the *alue of the call is+ <8A L Q(<9A E JA#:(<9A E J0#RC0 O <JA:1(0M C0 L <1J(81 !. 6ach option contract is for 100 shares of stoc&, so the price of a call on one share is+ C0 L <1,=00:100 shares per contract C0 L <1= Using the no arbitrage "odel, e find that the price of the stoc& is+ ,0 L <1=Q(<MJ E I8#:(<MJ E M0#R O <I8:1(08 ,0 L <J8(J= ". a. -he e$uit% can be *alued as a call option on the fir" ith an e!ercise price e$ual to the *alue of the debt, so+ 60 L <1,0A0 E Q<1,000:1(0JR 60 L <11A(I2 CHAPTER 24 B-379 b. -he current *alue of debt is the *alue of the fir"'s assets "inus the *alue of the e$uit%, so+ )0 L <1,0A0 E 11A(I2 )0 L <9=I(A8 We can use the face *alue of the debt and the current "ar&et *alue of the debt to find the interest rate, so+ .nterest rate L Q<1,000:<9=I(A8R E 1 .nterest rate L (0J or JN c. -he *alue of the e$uit% ill increase( -he debt then re$uires a higher return8 therefore the present *alue of the debt is less hile the *alue of the fir" does not change( #. a. Using the no arbitrage *aluation "odel, e can use the current "ar&et *alue of the fir" as the stoc& price, and the par *alue of the bond as the stri&e price to *alue the e$uit%( )oing so, e get+ <1,1I0 L Q(<1,I=0 E 920#:(<1,I=0 E 1,000#R60 O Q<920:1(0MR 60 L <229(I0 -he current *alue of the debt is the *alue of the fir"'s assets "inus the *alue of the e$uit%, so+ )0 L <1,1I0 E 229(I0 )0 L <910(M0 b. Using the no arbitrage "odel as in part a, e get+ <1,1I0 L Q(<1,M00 E 800#:(<1,M00 E 1,000#R60 O Q<800:1(0MR 60 L <288(9M -he stoc&holders ill prefer the ne asset structure because their potential gain increases hile their "a!i"u" potential loss re"ains unchanged( $. -he con*ersion ratio is the par *alue di*ided b% the con*ersion price, so+ Con*ersion ratio L <1,000:<=A Con*ersion ratio L 28(AJ -he con*ersion *alue is the con*ersion ratio ti"es the stoc& price, so+ Con*ersion *alue L 28(AJ(<IM# Con*ersion *alue L <1,=1I(29 B-380 SOLUTIONS 1%. a. -he "ini"u" bond price is the greater of the straight bond *alue or the con*ersion price( -he straight bond *alue is+ ,traight bond *alue L <2M(9V.B0=(AN,M0# O <1,000:1(0=A M0 ,traight bond *alue L <JJA(A0 -he con*ersion ratio is the par *alue di*ided b% the con*ersion price, so+ Con*ersion ratio L <1,000:<AA Con*ersion ratio L 18(18 -he con*ersion *alue is the con*ersion ratio ti"es the stoc& price, so+ Con*ersion *alue L 18(18(<I1# Con*ersion *alue L <JIA(IA -he "ini"u" *alue for this bond is the straight bond *alue of <JJA(A0( b. -he option e"bedded in the bond adds the e!tra *alue( 11. a. -he "ini"u" bond price is the greater of the straight bond *alue or the con*ersion *alue( -he straight bond *alue is+ ,traight bond *alue L <=A(9V.B0I(AN,M0# O <1,000:1(0IA M0 ,traight bond *alue L <J9=(M2 -he con*ersion ratio is the par *alue di*ided b% the con*ersion price, so+ Con*ersion ratio L <1,000:<IA Con*ersion ratio L 22(22 -he con*ersion price is the con*ersion ratio ti"es the stoc& price, so+ Con*ersion *alue L 22(22(<=9# Con*ersion *alue L <8MM(MJ -he "ini"u" *alue for this bond is the con*ertible floor *alue of <8MM(MJ( b. -he con*ersion pre"iu" is the difference beteen the current stoc& price and con*ersion price, di*ided b% the current stoc& price, so+ Con*ersion pre"iu" L (<IA E =9#:<=9 L (1A=8 or 1A(=8N CHAPTER 24 B-381 12. -he *alue of the bond ithout arrants is+ ,traight bond *alue L <IA(9V.B0JN,1A# O <1,000:1(0J 1A ,traight bond *alue L <JJ2(=0 -he *alue of the arrants is the selling price of the bond "inus the *alue of the bond ithout arrants, so+ -otal arrant *alue L <1,000 E JJ2(=0 -otal arrant *alue L <22J(J0 ,ince the bond has 2A arrants attached, the price of each arrant is+ 9rice of one arrant L <22J(J0:2A 9rice of one arrant L <9(11 13. .f e purchase the "achine toda%, the 49V is the cost plus the present *alue of the increased cash flos, so+ 49V0 L E<1,800,000 O <=20,000(9V.B012N,10# 49V0 L <8,0J1(=J We should not necessaril% purchase the "achine toda%, but rather e ould ant to purchase the "achine hen the 49V is the highest( ,o, e need to calculate the 49V each %ear( -he 49V each %ear ill be the cost plus the present *alue of the increased cash sa*ings( We "ust be careful hoe*er( .n order to "a&e the correct decision, the 49V for each %ear "ust be ta&en to a co""on date( We ill discount all of the 49Vs to toda%( )oing so, e get+ 5ear 1+ 49V1 L QE<1,M80,000 O <=20,000(9V.B012N,9#R : 1(12 49V1 L <22,=AJ(08 5ear 2+ 49V2 L QE<1,AM0,000 O <=20,000(9V.B012N,8#R : 1(12 2 49V2 L <2=,M=2(A9 5ear =+ 49V= L QE<1,II0,000 O <=20,000(9V.B012N,J#R : 1(12 = 49V= L <1I,A21(81 5ear I+ 49VI L QE<1,=20,000 O <=20,000(9V.B012N,M#R : 1(12 I 49VI L E<2,JMI(29 5ear A+ 49VA L QE<1,200,000 O <=20,000(9V.B012N,A#R : 1(12 A 49VA L E<2M,=M9(2I 5ear M+ 49VM L QE<1,200,000 O <=20,000(9V.B012N,I#R : 1(12 M 49VM L E<11A,A=M(=2 -he co"pan% should purchase the "achine 2 %ears fro" no hen the 49V is the highest( B-382 SOLUTIONS &ntermediate 14. a( -he base-case 49V is+ 49V L E<2,=00,000 O <A10,000(9V.B01IN,10# 49V L <=M0,218(98 b. We ould abandon the pro1ect if the cash flo fro" selling the e$uip"ent is greater than the present *alue of the future cash flos( We need to find the sale $uantit% here the to are e$ual, so+ <1,A00,000 L (<M8#7(9V.B01IN,9# 7 L <1,A00,000:Q<M8(I(9IMI#R 7 L I,IM0 0bandon the pro1ect if 7 a I,IM0 units, because the 49V of abandoning the pro1ect is greater than the 49V of the future cash flos( c. -he <1,A00,000 is the "ar&et *alue of the pro1ect( .f %ou continue ith the pro1ect in one %ear, %ou forego the <1,A00,000 that could ha*e been used for so"ething else( 1. a. .f the pro1ect is a success, present *alue of the future cash flos ill be+ 9V future CBs L <M8(9,A00#(9V.B01IN,9# 9V future CBs L <=,19A,=AM(21 Bro" the pre*ious $uestion, if the $uantit% sold is I,000, e ould abandon the pro1ect, and the cash flo ould be <1,A00,000( ,ince the pro1ect has an e$ual li&elihood of success or failure in one %ear, the e!pected *alue of the pro1ect in one %ear is the a*erage of the success and failure cash flos, plus the cash flo in one %ear, so+ 6!pected *alue of pro1ect at %ear 1 L Q(<=,19A,=AM(21 O <1,A00,000#:2R O <A10,000 6!pected *alue of pro1ect at %ear 1 L <2,8AJ,MJ8(10 -he 49V is the present *alue of the e!pected *alue in one %ear plus the cost of the e$uip"ent, so+ 49V L E<2,=00,000 O (<2,8AJ,MJ8(10#:1(1I 49V L <20M,J=A(18 b( .f e couldn't abandon the pro1ect, the present *alue of the future cash flos hen the $uantit% is I,000 ill be+ 9V future CBs L <M8(I,000#(9V.B01IN,9# 9V future CBs L <1,=IA,I1=(1I -he gain fro" the option to abandon is the abandon"ent *alue "inus the present *alue of the cash flos if e cannot abandon the pro1ect, so+ Gain fro" option to abandon L <1,A00,000 E 1,=IA,I1=(1I Gain fro" option to abandon L <1AI,A8M(8M CHAPTER 24 B-383 We need to find the *alue of the option to abandon ti"es the li&elihood of abandon"ent( ,o, the *alue of the option to abandon toda% is+ /ption *alue L ((A0#(<1AI,A8M(8M#:1(1I /ption *alue L <MJ,801(2A 1!. .f the pro1ect is a success, present *alue of the future cash flos ill be+ 9V future CBs L <M8(19,000#(9V.B01IN,9# 9V future CBs L <M,=90,J12(I1 .f the sales are onl% I,000 units, fro" 9roble" _1I, e &no e ill abandon the pro1ect, ith a *alue of <1,A00,000( ,ince the pro1ect has an e$ual li&elihood of success or failure in one %ear, the e!pected *alue of the pro1ect in one %ear is the a*erage of the success and failure cash flos, plus the cash flo in one %ear, so+ 6!pected *alue of pro1ect at %ear 1 L Q(<M,=90,J12(I1 O <1,A00,000#:2R O <A10,000 6!pected *alue of pro1ect at %ear 1 L <I,IAA,=AM(21 -he 49V is the present *alue of the e!pected *alue in one %ear plus the cost of the e$uip"ent, so+ 49V L E<2,=00,000 O <I,IAA,=AM(21:1(1I 49V L <1,M08,20J(20 -he gain fro" the option to e!pand is the present *alue of the cash flos fro" the additional units sold, so+ Gain fro" option to e!pand L <M8(9,A00#(9V.B01IN,9# Gain fro" option to e!pand L <=,19A,=AM(21 We need to find the *alue of the option to e!pand ti"es the li&elihood of e!pansion( We also need to find the *alue of the option to e!pand toda%, so+ /ption *alue L ((A0#(<=,19A,=AM(21#:1(1I /ption *alue L <1,I01,IJ2(02 1". a. -he *alue of the call is the "a!i"u" of the stoc& price "inus the present *alue of the e!ercise price, or 2ero, so+ C0 L 3a!Q<MA E (<JA:1(0A#,0R C0 L <0 -he option isn't orth an%thing( b. -he stoc& price is too lo for the option to finish in the "one%( -he "ini"u" return on the stoc& re$uired to get the option in the "one% is+ 3ini"u" stoc& return L (<JA E MA#:<MA 3ini"u" stoc& return L (1A=8 or 1A(=8N hich is "uch higher than the ris&-free rate of interest( B-384 SOLUTIONS 1#. H is the "ore t%pical case8 0 presents an arbitrage opportunit%( 5ou could bu% the bond for <800 and i""ediatel% con*ert it into stoc& that can be sold for <1,000( 0 ris&less <200 profit results( 1$. a. -he con*ersion ratio is gi*en at 22( -he con*ersion price is the par *alue di*ided b% the con*ersion ratio, so+ Con*ersion price L <1,000:22 Con*ersion price L <IA(IA -he con*ersion pre"iu" is the percent increase in stoc& price that results in no profit hen the bond is con*erted, so+ Con*ersion pre"iu" L (<IA(IA E =A#:<=A Con*ersion pre"iu" L (298J or 29(8JN b. -he straight bond *alue is+ ,traight bond *alue L <=0(9V.B0I(AN,I0# O <1,000:1(0IA I0 ,traight bond *alue L <J2=(98 0nd the con*ersion *alue is the con*ersion ratio ti"es the stoc& price, so+ Con*ersion *alue L 22(<=A# Con*ersion *alue L <JJ0(00 c. We si"pl% need to set the straight bond *alue e$ual to the con*ersion ratio ti"es the stoc& price, and sol*e for the stoc& price, so+ <J2=(98 L 22, , L <=2(91 d. -here are actuall% to option *alues to consider ith a con*ertible bond( -he con*ersion option *alue, defined as the "ar&et *alue less the floor *alue, and the speculati*e option *alue, defined as the floor *alue less the straight bond *alue( When the con*ersion *alue is less than the straight- bond *alue, the speculati*e option is orth 2ero( Con*ersion option *alue L <9M0 E JJ0 L <190 ,peculati*e option *alue L <JJ0 E J2=(98 L <IM(02 -otal option *alue L <190(00 O IM(02 L <2=M(02 2%. a. -he 49V of the pro1ect is the su" of the present *alue of the cash flos generated b% the pro1ect( -he cash flos fro" this pro1ect are an annuit%, so the 49V is+ 49V L E<JA,000,000 O <18,000,000(9V.B01IN,8# 49V L <8,I99,AA0(09 CHAPTER 24 B-385 b. -he co"pan% should abandon the pro1ect if the 9V of the re*ised cash flos for the ne!t J %ears is less than the pro1ect's afterta! sal*age *alue( ,ince the option to abandon the pro1ect occurs in %ear 1, discount the re*ised cash flos to %ear 1 as ell( -o deter"ine the le*el of e!pected cash flos belo hich the co"pan% should abandon the pro1ect, calculate the e$ui*alent annual cash flos the pro1ect "ust earn to e$ual the afterta! sal*age *alue( We ill sol*e for C2, the re*ised cash flo beginning in %ear 2( ,o, the re*ised annual cash flo belo hich it "a&es sense to abandon the pro1ect is+ 0fterta! sal*age *alue L C2(9V.B01IN,J# <=0,000,000 L C2(9V.B01IN,9# C2 L <=0,000,000 : 9V.B01IN,J C2 L <M,99A,JJ1(=2 Challenge 21. -he straight bond *alue toda% is+ ,traight bond *alue L <AI(9V.B09N,2A# O <1,000:1(09 2A ,traight bond *alue L <MIM(=9 0nd the con*ersion *alue of the bond toda% is+ Con*ersion *alue L <I1(I0(<1,000:<1A0# Con*ersion *alue L <2JM(00 We e!pect the bond to be called hen the con*ersion *alue increases to <1,=00, so e need to find the nu"ber of periods it ill ta&e for the current con*ersion *alue to reach the e!pected *alue at hich the bond ill be con*erted( )oing so, e find+ <2JM(00(1(11# t L <1,=00 t L 1I(8A %ears -he bond ill be called in 1I(8A %ears( -he bond *alue is the present *alue of the e!pected cash flos( -he cash flos ill be the annual coupon pa%"ents plus the con*ersion price( -he present *alue of these cash flos is+ Hond *alue L <AI(9V.B09N,1I(8A# O <1,=00:1(09 1I(8A L <J9I(M8 B-386 SOLUTIONS 22. We ill use the botto" up approach to calculate the operating cash flo( 0ssu"ing e operate the pro1ect for all four %ears, the cash flos are+ 5ear 0 1 2 = I ,ales <9,100,000 <9,100,000 <9,100,000 <9,100,000 /perating costs =,J00,000 =,J00,000 =,J00,000 =,J00,000 )epreciation =,000,000 =,000,000 =,000,000 =,000,000 6H- <2,I00,000 <2,I00,000 <2,I00,000 <2,I00,000 -a! 912,000 912,000 912,000 912,000 4et inco"e <1,I88,000 <1,I88,000 <1,I88,000 <1,I88,000 O)epreciation =,000,000 =,000,000 =,000,000 =,000,000 /perating CB <I,I88,000 <I,I88,000 <I,I88,000 <I,I88,000 Change in 4WC E<900,000 0 0 0 <900,000 Capital spending E<12,000,000 0 0 0 0 -otal cash flo E<12,900,000 <I,I88,000 <I,I88,000 <I,I88,000 <A,=88,000 -here is no sal*age *alue for the e$uip"ent( -he 49V is+ 49V L E<12,900,000 O <I,I88,000(9V.B01=N,=# O <A,=88,000:1(1= I 49V L <1,001,I1I(1M b. -he cash flos if e abandon the pro1ect after one %ear are+ 5ear 0 1 ,ales <9,100,000 /perating costs =,J00,000 )epreciation =,000,000 6H- <2,I00,000 -a! 912,000 4et inco"e <1,I88,000 O)epreciation =,000,000 /perating CB <I,I88,000 Change in 4WC E<900,000 <900,000 Capital spending E<12,000,000 <8,A0I,000 -otal cash flo E<12,900,000 <1=,892,000 -he boo& *alue of the e$uip"ent is+ Hoo& *alue L <12,000,000 E (1#(<12,000,000:I# Hoo& *alue L <9,000,000 CHAPTER 24 B-387 ,o, the ta!es on the sal*age *alue ill be+ -a!es L (<9,000,000 E 8,200,000#((=8# -a!es L <=0I,000 -his "a&es the afterta! sal*age *alue+ 0fterta! sal*age *alue L <8,200,000 O =0I,000 0fterta! sal*age *alue L <8,A0I,000 -he 49V if e abandon the pro1ect after one %ear is+ 49V L E<12,900,000 O <1=,892,000:1(1= 49V L E<M0M,19I(M9 .f e abandon the pro1ect after to %ears, the cash flos are+ 5ear 0 1 2 ,ales <9,100,000 <9,100,000 /perating costs =,J00,000 =,J00,000 )epreciation =,000,000 =,000,000 6H- <2,I00,000 <2,I00,000 -a! 912,000 912,000 4et inco"e <1,I88,000 <1,I88,000 O)epreciation =,000,000 =,000,000 /perating CB <I,I88,000 <I,I88,000 Change in 4WC E<900,000 0 <900,000 Capital spending E<12,000,000 0 <M,0M2,000 -otal cash flo E<12,900,000 <I,I88,000 <11,IA0,000 -he boo& *alue of the e$uip"ent is+ Hoo& *alue L <12,000,000 E (2#(<12,000,000:I# Hoo& *alue L <M,000,000 ,o the ta!es on the sal*age *alue ill be+ -a!es L (<M,000,000 E M,100,000#((=8# -a!es L E<=8,000 -his "a&es the afterta! sal*age *alue+ 0fterta! sal*age *alue L <M,100,000 E =8,000 0fterta! sal*age *alue L <M,0M2,000 B-388 SOLUTIONS -he 49V if e abandon the pro1ect after to %ears is+ 49V L E<12,900,000 O <I,I88,000:1(1= O <11,IA0,000:1(1= 2 49V L <=8,J10(9I .f e abandon the pro1ect after three %ears, the cash flos are+ 5ear 0 1 2 = ,ales <9,100,000 <9,100,000 <9,100,000 /perating costs =,J00,000 =,J00,000 =,J00,000 )epreciation =,000,000 =,000,000 =,000,000 6H- <2,I00,000 <2,I00,000 <2,I00,000 -a! 912,000 912,000 912,000 4et inco"e <1,I88,000 <1,I88,000 <1,I88,000 O)epreciation =,000,000 =,000,000 =,000,000 /perating CB <I,I88,000 <I,I88,000 <I,I88,000 Change in 4WC E<900,000 0 0 <900,000 Capital spending E<12,000,000 0 0 <I,0AI,000 -otal cash flo E<12,900,000 <I,I88,000 <I,I88,000 <9,II2,000 -he boo& *alue of the e$uip"ent is+ Hoo& *alue L <12,000,000 E (=#(<12,000,000:I# Hoo& *alue L <=,000,000 ,o the ta!es on the sal*age *alue ill be+ -a!es L (<=,000,000 E I,J00,000#((=8# -a!es L E<MIM,000 -his "a&es the afterta! sal*age *alue+ 0fterta! sal*age *alue L <I,J00,000 E MIM,000 0fterta! sal*age *alue L <I,0AI,000 -he 49V if e abandon the pro1ect after three %ears is+ 49V L E<12,900,000 O <I,I88,000(9V.B01=N,2# O <9,II2,000:1(1= = 49V L <1,1=0,22=(=M We should abandon the e$uip"ent after three %ears since the 49V of abandoning the pro1ect after three %ears has the highest 49V( CHAPTER 25 OPTION VALUATION Answers to Concepts Review and Critical Thinking Questions 1. .ncreasing the ti"e to e!piration increases the *alue of an option( -he reason is that the option gi*es the holder the right to bu% or sell( -he longer the holder has that right, the "ore ti"e there is for the option to increase (or decrease in the case of a put# in *alue( Bor e!a"ple, i"agine an out-of-the-"one% option that is about to e!pire( Hecause the option is essentiall% orthless, increasing the ti"e to e!piration ould ob*iousl% increase its *alue( 2. 0n increase in *olatilit% acts to increase both call and put *alues because the greater *olatilit% increases the possibilit% of fa*orable in-the-"one% pa%offs( 3. .nterest rate increases are good for calls and bad for puts( -he reason is that if a call is e!ercised in the future, e ha*e to pa% a fi!ed a"ount at that ti"e( -he higher the interest rate, the loer the present *alue of that fi!ed a"ount( -he re*erse is true for puts in that e recei*e a fi!ed a"ount( 4. .f %ou bu% a put option on a stoc& that %ou alread% on %ou guarantee that %ou can sell the stoc& for the e!ercise price of the put( -hus, %ou ha*e effecti*el% insured %ourself against a stoc& price decline belo this point( -his is the protecti*e put strateg%( . -he intrinsic *alue of a call is 3a!Q, E 6, 0R( -he intrinsic *alue of a put is 3a!Q6 E ,, 0R( -he intrinsic *alue of an option is the *alue at e!piration( !. -he ti"e *alue of both a call option and a put option is the difference beteen the price of the option and the intrinsic *alue( Bor both t%pes of options, as "aturit% increases, the ti"e *alue increases since %ou ha*e a longer ti"e to reali2e a price increase (decrease#( 0 call option is "ore sensiti*e to the "aturit% of the contract( ". ,ince %ou ha*e a large nu"ber of stoc& options in the co"pan%, %ou ha*e an incenti*e to accept the second pro1ect, hich ill increase the o*erall ris& of the co"pan% and reduce the *alue of the fir"'s debt( >oe*er, accepting the ris&% pro1ect ill increase %our ealth, as the options are "ore *aluable hen the ris& of the fir" increases( #. Rearranging the put-call parit% for"ula, e get+ , E 9V(6# L C E 9( ,ince e &no that the stoc& price and e!ercise price are the sa"e, assu"ing a positi*e interest rate, the left hand side of the e$uation "ust be greater than 2ero( -his i"plies the price of the call "ust be higher than the price of the put in this situation( $. Rearranging the put-call parit% for"ula, e get+ , E 9V(6# L C E 9( .f the call and the put ha*e the sa"e price, e &no C E 9 L 0( -his "ust "ean the stoc& price is e$ual to the present *alue of the e!ercise price, so the put is in-the-"one%( B-390 SOLUTIONS 1%. 0 stoc& can be replicated using a long call (to capture the upside gains#, a short put (to reflect the donside losses# and a --bill (to reflect the ti"e *alue co"ponent E the ;ait@ factor#( CHAPTER 25 B-391 Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. With continuous co"pounding, the BV is+ BV L <1,000 × e (11(9# L <2,M91(2= 2. With continuous co"pounding, the 9V is+ 9V L <1A,000 × e E(09(8# L <J,=01(28 3. Using put-call parit% and sol*ing for the put price, e get+ <M2 O 9 L <M0e E((02M#((2A# O <I(10 9 L <1(J1 4. Using put-call parit% and sol*ing for the call price e get+ <IJ O <A(08 L <A0e E((0I8#((A# O C C L <=(2J . Using put-call parit% and sol*ing for the stoc& price e get+ , O <=(10 L <J0e E((0I8#(=:12# O <I(=A , L <J0(I2 !. Using put-call parit%, e can sol*e for the ris&-free rate as follos+ <M9(=8 O <1(0A L <MAe ER(I:12# O <M(2J <MI(1M L <MAe ER(I:12# 0(98J1 L e ER(I:12# ln(0(98J1# L ln(e ER(I:12# # E0(01=0 L ER(I:12# Rf L (0=90 or =(90N ". Using put-call parit%, e can sol*e for the ris&-free rate as follos+ <MM(81 O <8(10 L <J0e ER(A:12# O <M(12 <M8(J9 L <J0e ER(A:12# 0(982J L e ER(A:12# ln(0(982J# L ln(e ER(A:12# # E0(01JI L ER(A:12# Rf L (0I18 or I(18N B-392 SOLUTIONS #. Using the Hlac&-,choles option pricing "odel to find the price of the call option, e find+ d1 L Qln(<M9:<J0# O ((0M O (I1 2 :2# × (=:12#R : ((I1 × 12 : = # L (10AA d2 L (10AA E ((I1 × 12 : = # L E(099A 4(d1# L (AI20 4(d2# L (IM0I 9utting these *alues into the Hlac&-,choles "odel, e find the call price is+ C L <M9((AI20# E (<J0e E(0M((2A# #((IM0I# L <A(MA Using put-call parit%, the put price is+ 9ut L <J0e E(0M((2A# O A(MA E M9 L <A(M1 $. Using the Hlac&-,choles option pricing "odel to find the price of the call option, e find+ d1 L Qln(<8M:<90# O ((0AA O (M2 2 :2# × (I:12#R : ((M2 × 12 : I # L (10=2 d2 L (10=2 E ((M2 × 12 : I # L E(2AI8 4(d1# L (AI11 4(d2# L (=99A 9utting these *alues into the Hlac&-,choles "odel, e find the call price is+ C L <8M((AI11# E (<90e E(0AA(I:12# #((=9AA# L <11(2I Using put-call parit%, the put price is+ 9ut L <90e E(0AA(I:12# O 11(2I E 8M L <1=(M0 1%. -he delta of a call option is 4(d1#, so+ d1 L Qln(<89:<8A# O ((0A O (=9 2 :2# × (JAR : ((=9 × JA ( # L (I1M1 4(d1# L (MM1= Bor a call option the delta is (MM1=( Bor a put option, the delta is+ 9ut delta L (MM1= E 1 L E(==8J -he delta tells us the change in the price of an option for a <1 change in the price of the underl%ing asset( CHAPTER 25 B-393 11. Using the Hlac&-,choles option pricing "odel, ith a hstoc&' price is <1,900,000 and an e!ercise price is <2,0A0,000, the price %ou should recei*e is+ d1 L Qln(<1,900,000:<2,0A0,000# O ((0A O (20 2 :2# × (12:12#R : ((20 × 12 : 12 # L E(0299 d2 L E(0299 E ((20 × 12 : 12 # L E(2299 4(d1# L (I881 4(d2# L (I091 9utting these *alues into the Hlac&-,choles "odel, e find the call price is+ C L <1,900,000((I881# E (<2,0A0,000e E(0A(1# #((I091# L <129,M1A(91 12. Using the call price e found in the pre*ious proble" and put-call parit%, %ou ould need to pa%+ 9ut L <2,0A0,000e E(0A(1# O 129,M1A(91 E 1,900,000 L <1J9,M=M(2= 5ou ould ha*e to pa% <1J9,M=M(2= in order to guarantee the right to sell the land for <2,0A0,000( 13. Using the Hlac&-,choles option pricing "odel to find the price of the call option, e find+ d1 L Qln(<8I:<80# O ((0M O (A= 2 :2# × (M:12#R : ((A= × # 12 : M ( # L (=9JM d2 L (=9JM E ((A= × 12 : M # L (0229 4(d1# L (MAIA 4(d2# L (A091 9utting these *alues into the Hlac&-,choles "odel, e find the call price is+ C L <8I((MAIA# E (<80e E(0M((A0# #((A091# L <1A(IM Using put-call parit%, e find the put price is+ 9ut L <80e E(0M((A0# O 1A(IM E 8I L <9(09 a. -he intrinsic *alue of each option is+ Call intrinsic *alue L 3a!Q, E 6, 0R L <I 9ut intrinsic *alue L 3a!Q6 E ,, 0R L <0 B-394 SOLUTIONS b. /ption *alue consists of ti"e *alue and intrinsic *alue, so+ Call option *alue L .ntrinsic *alue O -i"e *alue <1A(IM L <I O -V -V L <11(IM 9ut option *alue L .ntrinsic *alue O -i"e *alue <9(09 L <0 O -V -V L <9(09 c. -he ti"e pre"iu" (theta# is "ore i"portant for a call option than a put option, therefore, the ti"e pre"iu" is, in general, larger for a call option( 14. Using put-call parit%, the price of the put option is+ <2J(0A O 9 L <=0e E(0A(1:=# O <=(81 9 L <M(2M &ntermediate 1. .f the e!ercise price is e$ual to 2ero, the call price ill e$ual the stoc& price, hich is <8A( 1!. .f the standard de*iation is 2ero, d1 and d2 go to O∞, so 4(d1# and 4(d2# go to 1( -his is the no ris& call option for"ula e discussed in an earlier chapter, so+ C L , E 6e Ert C L <JI E <J0e E(0A(M:12# L <A(J= 1". .f the standard de*iation is infinite, d1 goes to positi*e infinit% so 4(d1# goes to 1, and d2 goes to negati*e infinit% so 4(d2# goes to 0( .n this case, the call price is e$ual to the stoc& price, hich is <I2( 1#. We can use the Hlac&-,choles "odel to *alue the e$uit% of a fir"( Using the asset *alue of <1M,100 as the stoc& price, and the face *alue of debt of <1A,000 as the e!ercise price, the *alue of the fir"'s e$uit% is+ d1 L Qln(<1M,100:<1A,000# O ((0M O (=2 2 :2# × 1R : ((=2 × 1 # L (AM8J d2 L (AM8J E ((=2 × 1 # L (2I8J 4(d1# L (J1A2 4(d2# L (A982 9utting these *alues into the Hlac&-,choles "odel, e find the e$uit% *alue is+ 6$uit% L <1M,100((J1A2# E (<1A,000e E(0M(1# #((A982# L <=,0MI(AI CHAPTER 25 B-395 -he *alue of the debt is the fir" *alue "inus the *alue of the e$uit%, so+ ) L <1M,100 E =,0MI(AI L <1=,0=A(IM 1$. a. We can use the Hlac&-,choles "odel to *alue the e$uit% of a fir"( Using the asset *alue of <1J,M00 as the stoc& price, and the face *alue of debt of <1A,000 as the e!ercise price, the *alue of the fir" if it accepts pro1ect 0 is+ d1 L Qln(<1J,M00:<1A,000# O ((0M O (IM 2 :2# × 1R : ((IM × 1 # L (J0J9 d2 L (J0J9 E ((IM × 1 # L (2IJ9 4(d1# L (JM0A 4(d2# L (A9J9 9utting these *alues into the Hlac&-,choles "odel, e find the e$uit% *alue is+ 60 L <1J,M00((JM0A# E (<1A,000e E(0M(1# #((A9J9# L <I,9=8(M0 -he *alue of the debt is the fir" *alue "inus the *alue of the e$uit%, so+ )0 L <1J,M00 E I,9=8(M0 L <12,MM1(I0 0nd the *alue of the fir" if it accepts 9ro1ect H is+ d1 L Qln(<18,200:<1A,000# O ((0M O (2I 2 :2# × 1R : ((2I × 1 # L 1(1JAJ d2 L 1(1JAJ E ((2I × 1 # L (9=AJ 4(d1# L (8801 4(d2# L (82A= 9utting these *alues into the Hlac&-,choles "odel, e find the e$uit% *alue is+ 6H L <18,200((8801# E (<1A,000e E(0M(1# #((82A=# L <I,=M0(22 -he *alue of the debt is the fir" *alue "inus the *alue of the e$uit%, so+ )H L <18,200 E I,=M0(22 L <1=,8=9(J8 b. 0lthough the 49V of pro1ect H is higher, the e$uit% *alue ith pro1ect 0 is higher( While 49V represents the increase in the *alue of the assets of the fir", in this case, the increase in the *alue of the fir"'s assets resulting fro" pro1ect H is "ostl% allocated to the debtholders, resulting in a s"aller increase in the *alue of the e$uit%( ,toc&holders ould, therefore, prefer pro1ect 0 e*en though it has a loer 49V( B-396 SOLUTIONS c. 5es( .f the sa"e group of in*estors ha*e e$ual sta&es in the fir" as bondholders and stoc&holders, then total fir" *alue "atters and pro1ect H should be chosen, since it increases the *alue of the fir" to <18,200 instead of <1J,M00( d. ,toc&holders "a% ha*e an incenti*e to ta&e on "ore ris&%, less profitable pro1ects if the fir" is le*eraged8 the higher the fir"'s debt load, all else the sa"e, the greater is this incenti*e( 2%. We can use the Hlac&-,choles "odel to *alue the e$uit% of a fir"( Using the asset *alue of <2J,=00 as the stoc& price, and the face *alue of debt of <2A,000 as the e!ercise price, the *alue of the fir"'s e$uit% is+ d1 L Qln(<2J,=00:<2A,000# O ((0M O (I= 2 :2# × 1R : ((I= × 1 # L (AA92 d2 L (AA92 E ((I= × 1 # L (1292 4(d1# L (J120 4(d2# L (AA1I 9utting these *alues into the Hlac&-,choles "odel, e find the e$uit% *alue is+ 6$uit% L <2J,=00((J120# E (<2A,000e E(0M(1# #((AA1I# L <M,IAA(02 -he *alue of the debt is the fir" *alue "inus the *alue of the e$uit%, so+ ) L <2J,=00 E M,IAA(02 L <20,8II(98 -he return on the co"pan%'s debt is+ <20,8II(98 L <2A,000e ER(1# (8==8 L e ER R) L Eln((8==8# L (1818 or 18(18N 21. a. -he co"bined *alue of e$uit% and debt of the to fir"s is+ 6$uit% L <=,0MI(AI O M,IAA(02 L <9,A19(AM )ebt L <1=,0=A(IM O 20,8II(98 L <==,880(II b. Bor the ne fir", the co"bined "ar&et *alue of assets is <I=,I00, and the co"bined face *alue of debt is <I0,000( Using Hlac&-,choles to find the *alue of e$uit% for the ne fir", e find+ d1 L Qln(<I=,I00:<I0,000# O ((0M O (19 2 :2# × 1R : ((19 × 1 # L (8I02 d2 L (8I02 E ((19 × 1 # L (MA02 4(d1# L (J99M 4(d2# L (JI22 CHAPTER 25 B-397 9utting these *alues into the Hlac&-,choles "odel, e find the e$uit% *alue is+ 6 L <I=,I00((J99M# E (<I0,000e E(0M(1# #((JI22# L <M,JI2(92 -he *alue of the debt is the fir" *alue "inus the *alue of the e$uit%, so+ ) L <I=,I00 E M,JI2(92 L <=M,MAJ(08 c. -he change in the *alue of the fir"'s e$uit% is+ 6$uit% *alue change L <M,JI2(92 E 9,A19(AM L E<2,JJM(MA -he change in the *alue of the fir"'s debt is+ )ebt L <=M,MAJ(08 E ==,880(II L <2,JJM(MA d. .n a purel% financial "erger, hen the standard de*iation of the assets declines, the *alue of the e$uit% declines as ell( -he shareholders ill lose e!actl% the a"ount the bondholders gain( -he bondholders gain as a result of the coinsurance effect( -hat is, it is less li&el% that the ne co"pan% ill default on the debt( 22. a. Using Hlac&-,choles "odel to *alue the e$uit%, e get+ d1 L Qln(<1M,000,000:<2A,000,000# O ((0M O (I1 2 :2# × 10R : ((I1 × 0 1 # L (JMM8 d2 L (JMM8 E ((I1 × 0 1 # L E(A29J 4(d1# L (JJ8I 4(d2# L (2982 9utting these *alues into Hlac&-,choles+ 6 L <1M,000,000((JJ8I# E (<2A,000,000e E(0M(10# #((2982# L <8,=M=,J1A(92 b. -he *alue of the debt is the fir" *alue "inus the *alue of the e$uit%, so+ ) L <1M,000,000 E 8,=M=,J1A(92 L <J,M=M,28I(08 c. Using the e$uation for the 9V of a continuousl% co"pounded lu"p su", e get+ <J,M=M,28I(08 L <2A,000,000e ER(10# (=0AA L e ER(10# R) L E(1:10#ln((=0AA# L (118M or 11(8MN B-398 SOLUTIONS d. Using Hlac&-,choles "odel to *alue the e$uit%, e get+ d1 L Qln(<1M,JA0,000:<2A,000,000# O ((0M O (I1 2 :2# × 10R : ((I1 × 0 1 # L (8022 d2 L (8022 E ((== × 0 1 # L E(I9II 4(d1# L (J888 4(d2# L (=10A 9utting these *alues into Hlac&-,choles+ 6 L <1M,JA0,000((J888# E (<2A,000,000e E(0M(10# #((=10A# L <8,9A1,IAI(== e( -he *alue of the debt is the fir" *alue "inus the *alue of the e$uit%, so+ ) L <1M,JA0,000 E 8,9A1,IAI(== L <J,J98,AIA(MJ Using the e$uation for the 9V of a continuousl% co"pounded lu"p su", e get+ <J,J98,AIA(MJ L <2A,000,000e ER(10# (=119 L e ER10 R) L E(1:10#ln((=119# L (11MA or 11(MAN When the fir" accepts the ne pro1ect, part of the 49V accrues to bondholders( -his increases the present *alue of the bond, thus reducing the return on the bond( 0dditionall%, the ne pro1ect "a&es the fir" safer in the sense it increases the *alue of assets, thus increasing the probabilit% the call ill end in-the-"one% and the bondholders ill recei*e their pa%"ent( Challenge 23. a. Using the e$uation for the 9V of a continuousl% co"pounded lu"p su", e get+ 9V L <I0,000 × e E(0J(2# L <=I,JJI(== b. Using Hlac&-,choles "odel to *alue the e$uit%, e get+ d1 L Qln(<1J,000:<I0,000# O ((0J O (M0 2 :2# × 2R : ((M0 × 2 # L E(I192 d2 L E(I192 E ((M0 × 2 # L E1(2MJJ 4(d1# L (==JM 4(d2# L (102A 9utting these *alues into Hlac&-,choles+ 6 L <1J,000((==JM# E (<I0,000e E(0J(2# #((102A# L <2,1JA(AA CHAPTER 25 B-399 0nd using put-call parit%, the price of the put option is+ 9ut L <I0,000e E(0J(10# O 2,1JA(AA E 1J,000 L <19,9I9(88 c. -he *alue of a ris&% bond is the *alue of a ris&-free bond "inus the *alue of a put option on the fir"'s e$uit%, so+ Value of ris&% bond L <=I,JJI(== E 19,9I9(88 L <1I,82I(IA Using the e$uation for the 9V of a continuousl% co"pounded lu"p su" to find the return on debt, e get+ <1I,82I(IA L <I0,000e ER(2# (=J0M L e ER2 R) L E(1:2#ln((=J0M# L (I9M= or I9(M=N d. -he *alue of the debt ith fi*e %ears to "aturit% at the ris&-free rate is+ 9V L <I0,000 × e E(0J(A# L <28,18J(A2 Using Hlac&-,choles "odel to *alue the e$uit%, e get+ d1 L Qln(<1J,000:<I0,000# O ((0J O (M0 2 :2# × AR : ((M0 × A # L (29=9 d2 L (29=9 E ((M0 × A # L E1(0IJJ 4(d1# L (M1AM 4(d2# L (1IJI 9utting these *alues into Hlac&-,choles+ 6 L <1J,000((M1AM# E (<I0,000e E(0J(A# #((1IJI# L <M,=10(MM 0nd using put-call parit%, the price of the put option is+ 9ut L <I0,000e E(0J(A# O M,=10(MM E 1J,000 L <1J,I98(18 -he *alue of a ris&% bond is the *alue of a ris&-free bond "inus the *alue of a put option on the fir"'s e$uit%, so+ Value of ris&% bond L <28,18J(A2 E 1J,I98(18 L <10,M89(=I Using the e$uation for the 9V of a continuousl% co"pounded lu"p su" to find the return on debt, e get+ <10,M89(=I L <I0,000e ER(A# (2MJ2 L e ERA R) L E(1:A#ln((2MJ2# L (2M=9 or 2M(=9N B-400 SOLUTIONS -he *alue of the debt declines because of the ti"e *alue of "one%, i(e(, it ill be longer until shareholders recei*e their pa%"ent( >oe*er, the re$uired return on the debt declines( Under the current situation, it is not li&el% the co"pan% ill ha*e the assets to pa% off bondholders( Under the ne plan here the co"pan% operates for fi*e "ore %ears, the probabilit% of increasing the *alue of assets to "eet or e!ceed the face *alue of debt is higher than if the co"pan% onl% operates for to "ore %ears( 24. a. Using the e$uation for the 9V of a continuousl% co"pounded lu"p su", e get+ 9V L <A0,000 × e E(0J(A# L <=A,2=I(I0 b. Using Hlac&-,choles "odel to *alue the e$uit%, e get+ d1 L Qln(<IM,000:<A0,000# O ((0J O (II 2 :2# × AR : ((II × A # L (JM29 d2 L (JM29 E ((II × A # L E(2209 4(d1# L (JJJ2 4(d2# L (I12M 9utting these *alues into Hlac&-,choles+ 6 L <IM,000((JJJ2# E (<A0,000e E(0J(A# #((I12M# L <21,21M(JI 0nd using put-call parit%, the price of the put option is+ 9ut L <A0,000e E(0J(A# O 21,21M(JI E IM,000 L <10,IA1(1I c. -he *alue of a ris&% bond is the *alue of a ris&-free bond "inus the *alue of a put option on the fir"'s e$uit%, so+ Value of ris&% bond L <=A,2=I(I0 E 10,IA1(1I L <2I,J8=(2M Using the e$uation for the 9V of a continuousl% co"pounded lu"p su" to find the return on debt, e get+ <2I,J8=(2M L <A0,000e ER(A# (I9AJ L e ER(A# R) L E(1:A#ln((I9AJ# L (1I0I or 1I(0IN CHAPTER 25 B-401 d. Using the e$uation for the 9V of a continuousl% co"pounded lu"p su", e get+ 9V L <A0,000 × e E(0J(A# L <=A,2=I(I0 Using Hlac&-,choles "odel to *alue the e$uit%, e get+ d1 L Qln(<IM,000:<A0,000# O ((0J O (AA 2 :2# × AR : ((AA × A # L (8=1J d2 L (8=1J E ((AA × A # L E(=981 4(d1# L (J9J2 4(d2# L (=IA= 9utting these *alues into Hlac&-,choles+ 6 L <IM,000((J9J2# E (<A0,000e E(0J(A# #((=IA=# L <2I,A0M(A1 0nd using put-call parit%, the price of the put option is+ 9ut L <A0,000e E(0J(A# O 2I,A0M(A1 E IM,000 L <1=,JI0(92 -he *alue of a ris&% bond is the *alue of a ris&-free bond "inus the *alue of a put option on the fir"'s e$uit%, so+ Value of ris&% bond L <=A,2=I(I0 E 1=,JI0(92 L <21,I9=(I9 Using the e$uation for the 9V of a continuousl% co"pounded lu"p su" to find the return on debt, e get+ <21,I9=(I9 L <A0,000e ER(A# (I299 L e ER(A# R) L E(1:A#ln((I299# L (1M89 or 1M(89N -he *alue of the debt declines( ,ince the standard de*iation of the co"pan%'s assets increases, the *alue of the put option on the face *alue of the bond increases hich decreases the bond's current *alue( e. Bro" c and d, bondholders lose+ <21,I9=(I9 E =0,1=2(1A L E<=,289(JJ Bro" c and d, stoc&holders gain+ <2I,A0M(A1 E 21,21M(JI L <=,289(JJ -his is an agenc% proble" for bondholders( 3anage"ent, acting to increase shareholder ealth in this "anner, ill reduce bondholder ealth b% the e!act a"ount that shareholder ealth is increased( B-402 SOLUTIONS 2. a. Going bac& to the chapter on di*idends, the price of the stoc& ill decline b% the a"ount of the di*idend (less an% ta! effects#( -herefore, e ould e!pect the price of the stoc& to drop hen a di*idend is paid, reducing the upside potential of the call b% the a"ount of the di*idend( -he price of a call option ill decrease hen the di*idend %ield increases( b. Using the Hlac&-,choles "odel ith di*idends, e get+ d1 L Qln(<11I:<10A# O ((0A E (0= O (A0 2 :2# × (AR : ((A0 × A ( # L (I=JJ d2 L (I=JJ E ((A0 × A ( # L (08I1 4(d1# L (MM92 4(d2# L (A==A C L <11Ie E((0=#((A# ((MM92# E (<10Ae E(0A((A# #((A==A# L <20(A2 2!. a. Going bac& to the chapter on di*idends, the price of the stoc& ill decline b% the a"ount of the di*idend (less an% ta! effects#( -herefore, e ould e!pect the price of the stoc& to drop hen a di*idend is paid( -he price of put option ill increase hen the di*idend %ield increases( b. Using put-call parit% to find the price of the put option, e get+ <11Ie E(0=((A# O 9 L <10Ae E(0A((A# O 20(A2 9 L <10(M2 2". 4(d1# is the probabilit% that ;*@ is less than or e$ual to 4(d1#, so 1 E 4(d1# is the probabilit% that ;*@ is greater than 4(d1#( Hecause of the s%""etr% of the nor"al distribution, this is the sa"e thing as the probabilit% that ;*@ is less than 4(Ed1#( ,o+ 4(d1# E 1 L 4(Ed1#( 2#. Bro" put-call parit%+ , L E S e -?t A C ; ' ,ubstituting the Hlac&-,choles call option for"ula for C and using the result in the pre*ious $uestion produces the put option for"ula+ , L E S e -?t A C ; ' , 4 E S e -?t A ' G4(d1# E E S e -?t G4(d2# E ' , 4 ' G(4(d1# E 1# O E S e -?t G(1 E 4(d2## , 4 E S e -?t G4(Ed2# E ' G 4(Ed1# CHAPTER 25 B-403 2$. Hased on Hlac&-,choles, the call option is orth <A0X -he reason is that present *alue of the e!ercise price is 2ero, so the second ter" disappears( 0lso, d1 is infinite, so 4(d1# is e$ual to one( -he proble" is that the call option is 6uropean ith an infinite e!piration, so h% ould %ou pa% an%thing for it since %ou can never e!ercise it? -he parado! can be resol*ed b% e!a"ining the price of the stoc&( Re"e"ber that the call option for"ula onl% applies to a non-di*idend pa%ing stoc&( .f the stoc& ill ne*er pa% a di*idend, it (and a call option to bu% it at an% price# "ust be orthless( 3%. -he delta of the call option is 4(d1# and the delta of the put option is 4(d1# E 1( ,ince %ou are selling a put option, the delta of the portfolio is 4(d1# E Q4(d1# E 1R( -his lea*es the o*erall delta of %our position as 1( -his position ill change dollar for dollar in *alue ith the underl%ing asset( -his position replicates the dollar ;action@ on the underl%ing asset( CHAPTER 26 M5R75RS A-; ACQ:.S.T.4-S Answers to Concepts Review and Critical Thinking Questions 1. .n the purchase "ethod, assets are recorded at "ar&et *alue, and goodill is created to account for the e!cess of the purchase price o*er this recorded *alue( .n the pooling of interests "ethod, the balance sheets of the to fir"s are si"pl% co"bined8 no goodill is created( -he choice of accounting "ethod has no direct i"pact on the cash flos of the fir"s( 69, ill probabl% be loer under the purchase "ethod because reported inco"e is usuall% loer due to the re$uired a"orti2ation of the goodill created in the purchase( 2. a. Green"ail refers to the practice of pa%ing unanted suitors ho hold an e$uit% sta&e in the fir" a pre"iu" o*er the "ar&et *alue of their shares to eli"inate the potential ta&eo*er threat( b. 0 hite &night refers to an outside bidder that a target fir" brings in to ac$uire it, rescuing the fir" fro" a ta&eo*er b% so"e other unanted hostile bidder( c. 0 golden parachute refers to lucrati*e co"pensation and ter"ination pac&ages granted to "anage"ent in the e*ent the fir" is ac$uired( d. -he cron 1eels usuall% refer to the "ost *aluable or prestigious assets of the fir", hich in the e*ent of a hostile ta&eo*er atte"pt, the target so"eti"es threatens to sell( e. ,har& repellent generall% refers to an% defensi*e tactic e"plo%ed b% the fir" to resist hostile ta&eo*er atte"pts( f. 0 corporate raider usuall% refers to a person or fir" that speciali2es in the hostile ta&eo*er of other fir"s( g. 0 poison pill is an a"end"ent to the corporate charter granting the shareholders the right to purchase shares at little or no cost in the e*ent of a hostile ta&eo*er, thus "a&ing the ac$uisition prohibiti*el% e!pensi*e for the hostile bidder( h. 0 tender offer is the legal "echanis" re$uired b% the ,6C hen a bidding fir" goes directl% to the shareholders of the target fir" in an effort to purchase their shares( i. 0 le*eraged bu%out refers to the purchase of the shares of a publicl%-held co"pan% and its subse$uent con*ersion into a pri*atel%-held co"pan%, financed pri"aril% ith debt( 3. )i*ersification doesn't create *alue in and of itself because di*ersification reduces uns%ste"atic, not s%ste"atic, ris&( 0s discussed in the chapter on options, there is a "ore subtle issue as ell( Reducing uns%ste"atic ris& benefits bondholders b% "a&ing default less li&el%( >oe*er, if a "erger is done purel% to di*ersif% (i(e(, no operating s%nerg%#, then the 49V of the "erger is 2ero( .f the 49V is 2ero, and the bondholders are better off, then stoc&holders "ust be orse off( 4. 0 fir" "ight choose to split up because the neer, s"aller fir"s "a% be better able to focus on their particular "ar&ets( -hus, re*erse s%nerg% is a possibilit%( 0n added ad*antage is that perfor"ance e*aluation beco"es "uch easier once the split is "ade because the ne fir"'s financial results (and stoc& prices# are no longer co""ingled( CHAPTER 26 B-405 . .t depends on ho the% are used( .f the% are used to protect "anage"ent, then the% are not good for stoc&holders( .f the% are used b% "anage"ent to negotiate the best possible ter"s of a "erger, then the% are good for stoc&holders( !. /ne of the pri"ar% ad*antages of a ta!able "erger is the rite-up in the basis of the target fir"'s assets, hile one of the pri"ar% disad*antages is the capital gains ta! that is pa%able( -he situation is the re*erse for a ta!-free "erger( -he basic deter"inant of ta! status is hether or not the old stoc&holders ill continue to participate in the ne co"pan%, hich is usuall% deter"ined b% hether the% get an% shares in the bidding fir"( 0n CH/ is usuall% ta!able because the ac$uiring group pa%s off the current stoc&holders in full, usuall% in cash( ". 6cono"ies of scale occur hen a*erage cost declines as output le*els increase( 0 "erger in this particular case "ight "a&e sense because 6astern and Western "a% need less total capital in*est"ent to handle the pea& poer needs, thereb% reducing a*erage generation costs( #. 0"ong the defensi*e tactics often e"plo%ed b% "anage"ent are see&ing hite &nights, threatening to sell the cron 1eels, appealing to regulator% agencies and the courts (if possible#, and targeted share repurchases( Bre$uentl%, antita&eo*er charter a"end"ents are a*ailable as ell, such as poison pills, poison puts, golden parachutes, loc&up agree"ents, and super"a1orit% a"end"ents, but these re$uire shareholder appro*al, so the% can't be i""ediatel% used if ti"e is short( While target fir" shareholders "a% benefit fro" "anage"ent acti*el% fighting ac$uisition bids, in that it encourages higher bidding and "a% solicit bids fro" other parties as ell, there is also the danger that such defensi*e tactics ill discourage potential bidders fro" see&ing the fir" in the first place, hich har"s the shareholders( $. .n a cash offer, it al"ost surel% does not "a&e sense( .n a stoc& offer, "anage"ent "a% feel that one suitor is a better long-run in*est"ent than the other, but this is onl% *alid if the "ar&et is not efficient( .n general, the highest offer is the best one( 1%. Various reasons include+ (1# 0nticipated gains "a% be s"aller than thought8 (2# Hidding fir"s are t%picall% "uch larger, so an% gains are spread thinl% across shares8 (=# 3anage"ent "a% not be acting in the shareholders' best interest ith "an% ac$uisitions8 (I# Co"petition in the "ar&et for ta&eo*ers "a% force prices for target fir"s up to the 2ero 49V le*el8 and (A# 3ar&et participants "a% ha*e alread% discounted the gains fro" the "erger before it is announced( Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. Bor the "erger to "a&e econo"ic sense, the ac$uirer "ust feel the ac$uisition ill increase *alue b% at least the a"ount of the pre"iu" o*er the "ar&et *alue, so+ B-406 SOLUTIONS 3ini"u" econo"ic *alue L <A=8,000,000 E I9A,000,000 L <I=,000,000 CHAPTER 26 B-407 2. a8 ,ince neither co"pan% has an% debt, using the pooling "ethod, the asset *alue of the co"bined "ust e$ual the *alue of the e$uit%, so+ 0ssets L 6$uit% L =0,000(<1J# O 20,000(<M# L <M=0,000 b8 With the purchase "ethod, the assets of the co"bined fir" ill be the boo& *alue of Bir" U, the ac$uiring co"pan%, plus the "ar&et *alue of Bir" 5, the target co"pan%, so+ 0ssets fro" U L =0,000(<1J# L <A10,000 (boo& *alue# 0ssets fro" 5 L 20,000(<1M# L <=20,000 ("ar&et *alue# -he purchase price of Bir" 5 is the nu"ber of shares outstanding ti"es the su" of the current stoc& price per share plus the pre"iu" per share, so+ 9urchase price of 5 L 20,000(<1M O A# L <I20,000 -he goodill created ill be+ Goodill L <I20,000 E =20,000 L <100,000 0nd the total asset of the co"bined co"pan% ill be+ -otal assets U5 L -otal e$uit% U5 L <A10,000 O =20,000 O 100,000 L <9=0,000 3. .n the pooling "ethod, all accounts of both co"panies are added together to total the accounts in the ne co"pan%, so the post-"erger balance sheet ill be+ Meat Co.# post)merger Current assets <12,M00 Current liabilities < J,M00 Bi!ed assets =8,A00 Cong-ter" debt 11,I00 6$uit% =2,100 -otal <A1,100 <A1,100 4. ,ince the ac$uisition is funded b% long-ter" debt, the post-"erger balance sheet ill ha*e long-ter" debt e$ual to the original long-ter" debt of 3eat's balance sheet, plus the original long-ter" debt on Coaf's balance sheet, plus the ne long-ter" debt issue, so+ 9ost-"erger long-ter" debt L <9,=00 O 2,100 O 19,000 L <=0,I00 Goodill ill be created since the ac$uisition price is greater than the boo& *alue( -he goodill a"ount is e$ual to the purchase price "inus the "ar&et *alue of assets, plus the "ar&et *alue of the ac$uired co"pan%'s debt( Generall%, the "ar&et *alue of current assets is e$ual to the boo& *alue, so+ Goodill L <19,000 E (<12,A00 "ar&et *alue B0# E (<=,M00 "ar&et *alue C0# O (<1,800 O 2,100# Goodill L <M,800 B-408 SOLUTIONS 6$uit% ill re"ain the sa"e as the pre-"erger balance sheet of the ac$uiring fir"( Current assets and debt accounts ill be the su" of the to fir"'s pre-"erger balance sheet accounts, and the fi!ed assets ill be the su" of the pre-"erger fi!ed assets of the ac$uirer and the "ar&et *alue of fi!ed assets of the target fir"( -he post-"erger balance sheet ill be+ Meat Co.# post)merger Current assets <12,M00 Current liabilities < J,M00 4et fi!ed assets II,A00 Cong-ter" debt =0,I00 Goodill M,800 6$uit% 2A,900 -otal <M=,900 <M=,900 . .n the pooling "ethod, all accounts of both co"panies are added together to total the accounts in the ne co"pan%, so the post-"erger balance sheet ill be+ 'ilver Enterprises# post)merger Current assets < I,M00 Current liabilities < =,I00 /ther assets 1,=80 Cong-ter" debt M,A00 4et fi!ed assets 18,I00 6$uit% 1I,I80 -otal <2I,=80 <2I,=80 !. ,ince the ac$uisition is funded b% long-ter" debt, the post-"erger balance sheet ill ha*e long-ter" debt e$ual to the original long-ter" debt of ,il*er's balance sheet plus the ne long-ter" debt issue, so+ 9ost-"erger long-ter" debt L <M,A00 O 11,000 L <1J,A00 6$uit% ill re"ain the sa"e as the pre-"erger balance sheet of the ac$uiring fir"( Current assets, current liabilities, and other assets ill be the su" of the to fir"'s pre-"erger balance sheet accounts, and the fi!ed assets ill be the su" of the pre-"erger fi!ed assets of the ac$uirer and the "ar&et *alue of fi!ed assets of the target fir"( We can calculate the goodill as the plug *ariable hich "a&es the balance sheet balance( -he post-"erger balance sheet ill be+ 'ilver Enterprises# post)merger Current assets < I,M00 Current liabilities < =,I00 /ther assets 1,=80 Cong-ter" debt 1J,A00 4et fi!ed assets 20,000 6$uit% J,M00 Goodill 2,A20 -otal <28,A00 <28,A00 CHAPTER 26 B-409 ". a. -he cash cost is the a"ount of cash offered, so the cash cost is <J= "illion( -o calculate the cost of the stoc& offer, e first need to calculate the *alue of the target to the ac$uirer( -he *alue of the target fir" to the ac$uiring fir" ill be the "ar&et *alue of the target plus the 9V of the incre"ental cash flos generated b% the target fir"( -he cash flos are a perpetuit%, so V ` L <A8,000,000 O <2,I00,000:(10 L <82,000,000 -he cost of the stoc& offer is the percentage of the ac$uiring fir" gi*en up ti"es the su" of the "ar&et *alue of the ac$uiring fir" and the *alue of the target fir" to the ac$uiring fir"( ,o, the e$uit% cost ill be+ 6$uit% cost L (I0(<10J,000,000 O 82,000,000# L <JA,M00,000 b. -he 49V of each offer is the *alue of the target fir" to the ac$uiring fir" "inus the cost of ac$uisition, so+ 49V cash L <82,000,000 E J=,000,000 L <9,000,000 49V stoc& L <82,000,000 E JA,M00,000 L <M,I00,000 c. ,ince the 49V is greater ith the cash offer the ac$uisition should be in cash( #. a( -he 69, of the co"bined co"pan% ill be the su" of the earnings of both co"panies di*ided b% the shares in the co"bined co"pan%( ,ince the stoc& offer is one share of the ac$uiring fir" for three shares of the target fir", ne shares in the ac$uiring fir" ill increase b% one-third( ,o, the ne 69, ill be+ 69, L (<2IA,000 O J=0,000#:Q19I,000 O (1:=#(92,==J#R L <I(==8 -he "ar&et price of 9itt ill re"ain unchanged if it is a 2ero 49V ac$uisition( Using the 96 ratio, e find the current "ar&et price of 9itt stoc&, hich is+ 9 L 22(<J=0,000#:19I,000 L <82(J8 .f the ac$uisition has a 2ero 49V, the stoc& price should re"ain unchanged( -herefore, the ne 96 ill be+ 9:6 L <82(J8:<I(==8 L 19(09 b. -he *alue of Jolie to 9itt "ust be the "ar&et *alue of the co"pan% since the 49V of the ac$uisition is 2ero( -herefore, the *alue is+ V ` L <2IA,000(10(I# L <2,AI8,000 -he cost of the ac$uisition is the nu"ber of shares offered ti"es the share price, so the cost is+ Cost L (1:=#(92,==J#(<82(J8# L <2,AI8,000 B-410 SOLUTIONS ,o, the 49V of the ac$uisition is+ 49V L 0 L V ` O ∆V E Cost L <2,AI8,000 O ∆V E 2,AI8,000 ∆V L <0 0lthough there is no econo"ic *alue to the ta&eo*er, it is possible that 9itt is "oti*ated to purchase Jolie for other than financial reasons( $. a. -he 49V of the "erger is the "ar&et *alue of the target fir", plus the *alue of the s%nerg%, "inus the ac$uisition costs, so+ 49V L 1,A00(<18# O <M,000 E 1,A00(<20(A0# L <2,2A0 b. ,ince the 49V goes directl% to stoc&holders, the share price of the "erged fir" ill be the "ar&et *alue of the ac$uiring fir" plus the 49V of the ac$uisition, di*ided b% the nu"ber of shares outstanding, so+ ,hare price L Q=,I00(<I=# O <2,2A0R:=,I00 L <I=(MM c. -he "erger pre"iu" is the pre"iu" per share ti"es the nu"ber of shares of the target fir" outstanding, so the "erger pre"iu" is+ 3erger pre"iu" L 1,A00(<20(A0 E 18# L <=,JA0 d. -he nu"ber of ne shares ill be the nu"ber of shares of the target ti"es the e!change ratio, so+ 4e shares created L 1,A00(1:2# L JA0 ne shares -he *alue of the "erged fir" ill be the "ar&et *alue of the ac$uirer plus the "ar&et *alue of the target plus the s%nerg% benefits, so+ VH- L =,I00(<I=# O 1,A00(<18# O M,000 L <1J9,200 -he price per share of the "erged fir" ill be the *alue of the "erged fir" di*ided b% the total shares of the ne fir", hich is+ 9 L <1J9,200:(=,I00 O JA0# L <I=(18 e. -he 49V of the ac$uisition using a share e!change is the "ar&et *alue of the target fir" plus s%nerg% benefits, "inus the cost( -he cost is the *alue per share of the "erged fir" ti"es the nu"ber of shares offered to the target fir" shareholders, so+ NPV = 1,500($18) + $6,000 – 750($43.18) = $614.46 CHAPTER 26 B-411 &ntermediate 1%. -he cash offer is better for the target fir" shareholders since the% recei*e <20(A0 per share( .n the share offer, the target fir"'s shareholders ill recei*e+ 6$uit% offer *alue L (1:2#(<18(00# L <9(00 per share Bro" 9roble" 9, e &no the *alue of the "erged fir"'s assets ill be <1J9,200( -he nu"ber of shares in the ne fir" ill be+ ,hares in ne fir" L =,I00 O 1,A00! that is, the nu"ber of shares outstanding in the bidding fir", plus the nu"ber of shares outstanding in the target fir", ti"es the e!change ratio( -his "eans the post "erger share price ill be+ 9 L <1J9,200:(=,I00 O 1,A00!# -o "a&e the target fir"'s shareholders indifferent, the% "ust recei*e the sa"e ealth, so+ 1,A00(!#9 L 1,A00(<20(A0# -his e$uation shos that the ne offer is the shares outstanding in the target co"pan% ti"es the e!change ratio ti"es the ne stoc& price( -he *alue under the cash offer is the shares outstanding ti"es the cash offer price( ,ol*ing this e$uation for 9, e find+ 9 L <20(A0 : ! Co"bining the to e$uations, e find+ <1J9,200:(=,I00 O 1,A00!# L <20(A0 : ! ! L 0(IM9A -here is a si"pler solution that re$uires an econo"ic understanding of the "erger ter"s( .f the target fir"'s shareholders are indifferent, the bidding fir"'s shareholders are indifferent as ell( -hat is, the offer is a 2ero su" ga"e( Using the ne stoc& price produced b% the cash deal, e find+ 6!change ratio L <20(A0:<I=(MM L 0(IM9A 11. -he cost of the ac$uisition is+ Cost L 200(<11# L <2,200 ,ince the stoc& price of the ac$uiring fir" is <=I, the fir" ill ha*e to gi*e up+ ,hares offered L <2,200:<=I L MI(J1 shares a. -he 69, of the "erged fir" ill be the co"bined 69, of the e!isting fir"s di*ided b% the ne shares outstanding, so+ 69, L (<1,I00 O A00#:(A00 O MI(J1# L <=(=M B-412 SOLUTIONS b. -he 96 of the ac$uiring fir" is+ /riginal 9:6 L <=I:(<1,I00:A00# L 12(1I ti"es 0ssu"ing the 96 ratio does not change, the ne stoc& price ill be+ 4e 9 L <=(=M(12(1I# L <I0(8M c. .f the "ar&et correctl% anal%2es the earnings, the stoc& price ill re"ain unchanged since this is a 2ero 49V ac$uisition, so+ 4e 9:6 L <=I:<=(=M L 10(11 ti"es d. -he ne share price ill be the co"bined "ar&et *alue of the to e!isting co"panies di*ided b% the nu"ber of shares outstanding in the "erged co"pan%( ,o+ 9 L Q(A00#(<=I# O 200(<8#R:(A00 O MI(J1# L <=2(9I 0nd the 96 ratio of the "erged co"pan% ill be+ 9:6 L <=2(9I:<=(=M L 9(J9 ti"es 0t the proposed bid price, this is a negati*e 49V ac$uisition for 0 since the share price declines( -he% should re*ise their bid donard until the 49V is 2ero( 12. Heginning ith the fact that the 49V of a "erger is the *alue of the target "inus the cost, e get+ 49V L VH ` E Cost 49V L ∆V O VH E Cost 49V L ∆V E (Cost E VH# 49V L ∆V E 3erger pre"iu" 13. a. -he s%nerg% ill be the present *alue of the incre"ental cash flos of the proposed purchase( ,ince the cash flos are perpetual, the s%nerg% *alue is+ ,%nerg% *alue L <IA0,000 : (08 ,%nerg% *alue L <A,M2A,000 b. -he *alue of Blash-in-the-9an to Bl%-b%-4ight is the s%nerg% plus the current "ar&et *alue of Blash-in-the-9an, hich is+ Value L <A,M2A,000 O 1I,000,000 Value L <19,M2A,000 c. -he *alue of the cash option is the a"ount of cash paid, or <18(A "illion( -he *alue of the stoc& ac$uisition is the percentage of onership in the "erged co"pan%, ti"es the *alue of the "erged co"pan%, so+ ,toc& ac$uisition *alue L (=A(<19,M2A,000 O =1,000,000# ,toc& ac$uisition *alue L <1J,J18,JA0 CHAPTER 26 B-413 B-414 SOLUTIONS d. -he 49V is the *alue of the ac$uisition "inus the cost, so the 49V of each alternati*e is+ 49V of cash offer L <19,M2A,000 E 18,A00,000 49V of cash offer L <1,12A,000 49V of stoc& offer L <19,M2A,000 E 1J,J18,J80 49V of stoc& offer L <1,90M,2A0 e. -he ac$uirer should "a&e the stoc& offer since its 49V is greater( 14. a. -he nu"ber of shares after the ac$uisition ill be the current nu"ber of shares outstanding for the ac$uiring fir", plus the nu"ber of ne shares created for the ac$uisition, hich is+ 4u"ber of shares after ac$uisition L 9,000,000 O 2,A00,000 4u"ber of shares after ac$uisition L 11,A00,000 0nd the share price ill be the *alue of the co"bined co"pan% di*ided b% the shares outstanding, hich ill be+ 4e stoc& price L W=00,000,000 : 11,A00,000 4e stoc& price L W2M(09 b. Cet α e$ual the fraction of onership for the target shareholders in the ne fir"( We can set the percentage of onership in the ne fir" e$ual to the *alue of the cash offer, so+ α(W=00,000,000# L W90,000,000 α L (=0 or =0N ,o, the shareholders of the target fir" ould be e$uall% as ell off if the% recei*ed =0 percent of the stoc& in the ne co"pan% as if the% recei*ed the cash offer( -he onership percentage of the target fir" shareholders in the ne fir" can be e!pressed as+ /nership L 4e shares issued : (4e shares issued O Current shares of ac$uiring fir"# (=0 L 4e shares issued : (4e shares issued O 9,000,000# 4e shares issued L =,8AJ,1I= -o find the e!change ratio, e di*ide the ne shares issued to the shareholders of the target fir" b% the e!isting nu"ber of shares in the target fir", so+ 6!change ratio L 4e shares : 6!isting shares in target fir" 6!change ratio L =,8AJ,1I= : 90,000,000 6!change ratio L 0(I821 0n e!change ratio of 0(I821 shares of the "erged co"pan% for each share of the target co"pan% oned ould "a&e the *alue of the stoc& offer e$ui*alent to the *alue of the cash offer( CHAPTER 26 B-415 Challenge 1. a. -o find the *alue of the target to the ac$uirer, e need to find the share price ith the ne groth rate( We begin b% finding the re$uired return for shareholders of the target fir"( -he earnings per share of the target are+ 69,9 L <M80,000:A00,000 L <1(=M per share -he price per share is+ 99 L 11(A(<1(=M# L <1A(MI 0nd the di*idends per share are+ )9,9 L <=10,000:A00,000 L <0(M2 -he current re$uired return for 9ulit2er shareholders, hich incorporates the ris& of the co"pan% is+ R6 L Q<0(M2(1(0A#:<1A(MIR O (0A L (091M -he price per share of 9ulit2er ith the ne groth rate is+ 99 L <0(M2(1(0J#:((091M E (0J# L <=0(M8 -he *alue of the target fir" to the ac$uiring fir" is the nu"ber of shares outstanding ti"es the price per share under the ne groth rate assu"ptions, so+ V- ` L A00,000(<=0(M8# L <1A,==9,I08(M= b. -he gain to the ac$uiring fir" ill be the *alue of the target fir" to the ac$uiring fir" "inus the "ar&et *alue of the target, so+ Gain L <1A,==9,I08(M= E A00,000(<1A(MI# L <J,A19,I08(M= c. -he 49V of the ac$uisition is the *alue of the target fir" to the ac$uiring fir" "inus the cost of the ac$uisition, so+ 49V L <1A,==9,I08(M= E A00,000(<18# L <M,==9,I08(M= d. -he "ost the ac$uiring fir" should be illing to pa% per share is the offer price per share plus the 49V per share, so+ 3a!i"u" bid price L <18 O (<M,==9,I08(M=:A00,000# L <=0(M8 4otice, this is the sa"e *alue e calculated earlier in part a as the *alue of the target to the ac$uirer( B-416 SOLUTIONS e. -he price of the stoc& in the "erged fir" ould be the "ar&et *alue of the ac$uiring fir" plus the *alue of the target to the ac$uirer, di*ided b% the nu"ber of shares in the "erged fir", so+ 9B9 L (<AA,800,000 O 1A,==9,I08(M=#:(1,200,000 O 200,000# L <A0(81 -he 49V of the stoc& offer is the *alue of the target to the ac$uirer "inus the *alue offered to the target shareholders( -he *alue offered to the target shareholders is the stoc& price of the "erged fir" ti"es the nu"ber of shares offered, so+ 49V L <1A,==9,I08(M= E 200,000(<A0(81# L <A,1JM,M=A(9J f. 5es, the ac$uisition should go forard, and Bo!% should offer cash since the 49V is higher( g. Using the ne groth rate in the di*idend groth "odel, along ith the di*idend and re$uired return e calculated earlier, the price of the target under these assu"ptions is+ 99 L <0(M2(1(0M#:((091M E (0M# L <20(J8 0nd the *alue of the target fir" to the ac$uiring fir" is+ V9 ` L A00,000(<20(J8# L <10,=90,828(9A -he gain to the ac$uiring fir" ill be+ Gain L <10,=90,828(9A E A00,000(<1A(MI# L <2,AJ0,828(9A -he 49V of the cash offer is no+ 49V cash L <10,=90,828(9A E A00,000(<18# L <1,=90,828(9A 0nd the ne price per share of the "erged fir" ill be+ 9B9 L Q<AA,800,000 O 10,=90,828(9AR:(1,200,000 O 200,000# L <IJ(28 0nd the 49V of the stoc& offer under the ne assu"ption ill be+ 49V stoc& L <10,=90,828(9A E 200,000(<IJ(28# L <9=I,99M(2A 6*en ith the loer pro1ected groth rate, the both offers still ha*e a positi*e 49V, although both are significantl% loer( Bo!% should purchase 9ulit2er ith the cash offer( CHAPTER 27 95AS.-7 Answers to Concepts Review and Critical Thinking Questions 1. ,o"e &e% differences are+ (1# Cease pa%"ents are full% ta!-deductible, but onl% the interest portion of the loan is8 (2# -he lessee does not on the asset and cannot depreciate it for ta! purposes8 (=# .n the e*ent of a default, the lessor cannot force ban&ruptc%8 and (I# -he lessee does not obtain title to the asset at the end of the lease (absent so"e additional arrange"ent#( 2. -he less profitable one because leasing pro*ides, a"ong other things, a "echanis" for transferring ta! benefits fro" entities that *alue the" less to entities that *alue the" "ore( 3. 9otential proble"s include+ (1# Care "ust be ta&en in interpreting the .RR (a high or lo .RR is preferred depending on the setup of the anal%sis#8 and (2# Care "ust be ta&en to ensure the .RR under e!a"ination is not the i"plicit interest rate 1ust based on the lease pa%"ents( 4. a. Ceasing is a for" of secured borroing( .t reduces a fir"'s cost of capital onl% if it is cheaper than other for"s of secured borroing( -he reduction of uncertaint% is not particularl% rele*ant8 hat "atters is the 40C( b. -he state"ent is not ala%s true( Bor e!a"ple, a lease often re$uires an ad*ance lease pa%"ent or securit% deposit and "a% be i"plicitl% secured b% other assets of the fir"( c. Ceasing ould probabl% not disappear, since it does reduce the uncertaint% about sal*age *alue and the transactions costs of transferring onership( >oe*er, the use of leasing ould be greatl% reduced( . 0 lease "ust be disclosed on the balance sheet if one of the folloing criteria is "et+ -. -he lease transfers onership of the asset b% the end of the lease( .n this case, the fir" essentiall% ons the asset and ill ha*e access to its residual *alue( 1. -he lessee can purchase the asset at a price belo its fair "ar&et *alue (bargain purchase option# hen the lease ends( -he fir" essentiall% ons the asset and ill ha*e access to "ost of its residual *alue( 3. -he lease ter" is for JAN or "ore of the esti"ated econo"ic life of the asset( -he fir" basicall% has access to the "a1orit% of the benefits of the asset, ithout an% responsibilit% for the conse$uences of its disposal( F. -he present *alue of the lease pa%"ents is 90N or "ore of the fair "ar&et *alue of the asset at the start of the lease( -he fir" is essentiall% purchasing the asset on an install"ent basis( !. -he lease "ust "eet the folloing .R, standards for the lease pa%"ents to be ta! deductible+ -. -he lease ter" "ust be less than 80N of the econo"ic life of the asset( .f the ter" is longer, the lease is considered to be a conditional sale( 1. -he lease should not contain a bargain purchase option, hich the .R, interprets as an e$uit% interest in the asset( B-418 SOLUTIONS 3. -he lease pa%"ent schedule should not pro*ide for *er% high pa%"ents earl% and *er% lo pa%"ents late in the life of the lease( -his ould indicate that the lease is being used si"pl% to a*oid ta!es( F. Reneal options should be reasonable and based on the fair "ar&et *alue of the asset at reneal ti"e( -his indicates that the lease is for legiti"ate business purposes, not ta! a*oidance( ". 0s the ter" i"plies, off-balance sheet financing in*ol*es financing arrange"ents that are not re$uired to be reported on the fir"'s balance sheet( ,uch acti*ities, if reported at all, appear onl% in the footnotes to the state"ents( /perating leases (those that do not "eet the criteria in proble" A# pro*ide off-balance sheet financing( Bor accounting purposes, total assets ill be loer and so"e financial ratios "a% be artificiall% high( Binancial anal%sts are generall% not fooled b% such practices( -here are no econo"ic conse$uences, since the cash flos of the fir" are not affected b% ho the lease is treated for accounting purposes( #. -he lessee "a% not be able to ta&e ad*antage of the depreciation ta! shield and "a% not be able to obtain fa*orable lease arrange"ents for ;passing on@ the ta! shield benefits( -he lessee "ight also need the cash flo fro" the sale to "eet i""ediate needs, but ill be able to "eet the lease obligation cash flos in the future( $. ,ince the rele*ant cash flos are all afterta!, the afterta! discount rate is appropriate( 1%. Japan .nternational's financial position as such that the pac&age of leasing and bu%ing probabl% resulted in the o*erall best afterta! cost( .n particular, Japan .nternational "a% not ha*e been in a position to use all of the ta! credits and also "a% not ha*e had the credit strength to borro and bu% the plane ithout facing a credit dongrade and:or substantiall% higher rates( 11. -here is the ta! "oti*e, but, be%ond this, Genesis Cease Ci"ited &nos that, in the e*ent of a default, Japan .nternational ould relin$uish the plane, hich ould then be re-leased( Bungible assets, such as planes, hich can be readil% reclai"ed and redeplo%ed are good candidates for leasing( 12. -he plane ill be re-leased to Japan .nternational or another air transportation fir", used b% Genesis Cease Ci"ited, or it ill si"pl% be sold( -here is an acti*e "ar&et for used aircraft( Solutions to Questions and &ro'le(s NOTE: All end of chapter problems were solved using a spreadsheet. Man problems re!uire multiple steps. "ue to space and readabilit constraints# when these intermediate steps are included in this solutions manual# rounding ma appear to have occurred. $owever# the final answer for each problem is found without rounding during an step in the problem. %asic 1. We ill calculate cash flos fro" the depreciation ta! shield first( -he depreciation ta! shield is+ )epreciation ta! shield L (<A,800,000:I#((=A# L <A0J,A00 -he afterta! cost of the lease pa%"ents ill be+ 0fterta! lease pa%"ent L (<1,IA0,000#(1 E (=A# L <1,1=1,000 CHAPTER 27 B-419 ,o, the total cash flos fro" leasing are+ /CB L <A0J,A00 O 1,1=1,000 L <1,M=8,A00 -he afterta! cost of debt is+ 0fterta! debt cost L (08(1 E (=A# L (0A2 Using all of this infor"ation, e can calculate the 40C as+ 40C L <A,800,000 E <1,M=8,A00(9V.B0A(20N,I# L <1M,8A1(2A -he 40C is positi*e so %ou should lease( 2. .f e assu"e the lessor has the sa"e cost of debt and the sa"e ta! rate, the 40C to the lessor is the negati*e of our co"pan%'s 40C, so+ 40C L E <1M,8A1(2A 3. -o find the "a!i"u" lease pa%"ent that ould satisf% both the lessor and the lessee, e need to find the pa%"ent that "a&es the 40C e$ual to 2ero( Using the 40C e$uation and sol*ing for the /CB, e find+ 40C L 0 L <A,800,000 E /CB(9V.B0A(20N,I# /CB L <1,MI=,2JI(=A -he /CB for this lease is co"posed of the depreciation ta! shield cash flo, as ell as the afterta! lease pa%"ent( ,ubtracting out the depreciation ta! shield cash flo e calculated earlier, e find+ 0fterta! lease pa%"ent L <1,MI=,2JI(=A E A0J,A00 L <1,1=A,JJI(=A ,ince this is the afterta! lease pa%"ent, e can no calculate the brea&e*en preta! lease pa%"ent as+ Hrea&e*en lease pa%"ent L <1,1=A,JJI(=A:(1 E (=A# L <1,JIJ,=IA(1A 4. .f the ta! rate is 2ero, there is no depreciation ta! shield foregone( 0lso, the afterta! lease pa%"ent is the sa"e as the preta! pa%"ent, and the afterta! cost of debt is the sa"e as the preta! cost( ,o+ Cost of debt L (08 0nnual cost of leasing L leasing pa%"ent L <1,IA0,000 -he 40C to leasing ith these assu"ptions is+ 40C L <A,800,000 E <1,IA0,000(9V.B08N,I# L <=M,899(=0 B-420 SOLUTIONS . We alread% calculated the brea&e*en lease pa%"ent for the lessor in 9roble" =( ,ince the assu"ption about the lessor concerning the ta! rate ha*e not changed( ,o, the lessor brea&s e*en ith a pa%"ent of <1,JIJ,=IA(1A( Bor the lessee, e need to calculate the brea&e*en lease pa%"ent hich results in a 2ero 40C( Using the assu"ptions in 9roble" I, e find+ 40C L 0 L <A,800,000 E 93-(9V.B08N,I# 93- L <1,JA1,1I0(MJ ,o, the range of lease pa%"ents that ould satisf% both the lessee and the lessor are+ -otal pa%"ent range L <1,JIJ,=IA(1A to <1,JA1,1I0(MJ !. -he appropriate depreciation percentages for a =-%ear 30CR, class asset can be found in Chapter 10( -he depreciation percentages are (====, (IIIA, (1I81, and (0JI1( -he cash flos fro" leasing are+ 5ear 1+ (<A,800,000#((====#((=A# O <1,1=1,000 L <1,80J,A99 5ear 2+ (<A,800,000#((IIIA#((=A# O <1,1=1,000 L <2,0==,==A 5ear =+ (<A,800,000#((1I81#((=A# O <1,1=1,000 L <1,I=1,MI= 5ear I+ (<A,800,000#((0JI1#((=A# O <1,1=1,000 L <1,281,I2= 40C L <A,800,000 E <1,80J,A99:1(0A2 E <2,0==,==A:1(0A2 2 E <1,I=1,MI=:1(0A2 = E <1,281,I2=:1(0A2 I 40C L −<=1,II1(MM -he "achine should not be leased( -his is because of the accelerated ta! benefits due to depreciation, hich represents a cost in the decision to lease co"pared to an ad*antage of the decision to purchase( &ntermediate ". -he preta! cost sa*ings are not rele*ant to the lease *ersus bu% decision, since the fir" ill definitel% use the e$uip"ent and reali2e the sa*ings regardless of the financing choice "ade( -he depreciation ta! shield is+ )epreciation ta! shield lost L (<8,000,000:A#((=I# L <AII,000 0nd the afterta! lease pa%"ent is+ 0fterta! lease pa%"ent L <1,900,000(1 E (=I# L <1,2AI,000 -he afterta! cost of debt is+ 0fterta! debt cost L (09(1 E (=I# L (0A9I or A(9IN With these cash flos, the 40C is+ 40C L <8,000,000 E 1,2AI,000 E <1,2AI,000(9V.B0A(9IN,I# E <AII,000(9V.B0A(9IN,A# L <99,A09(8M -he e$uip"ent should be leased( CHAPTER 27 B-421 -o find the "a!i"u" pa%"ent, e find here the 40C is e$ual to 2ero, and sol*e for the pa%"ent( Using U to represent the "a!i"u" pa%"ent+ 40C L 0 L <8,000,000 E U(1(0A9I#(9V.B0A(9IN,A# E <AII,000(9V.B0A(9IN,A# U L <1,2JM,2M2(=A ,o the "a!i"u" preta! lease pa%"ent is+ 9reta! lease pa%"ent L <1,2JM,2M2(=A:(1 E (=I# L <1,9==,J=0(8I #. -he afterta! residual *alue of the asset is an opportunit% cost to the leasing decision, occurring at the end of the pro1ect life (%ear A#( 0lso, the residual *alue is not reall% a debt-li&e cash flo, since there is uncertaint% associated ith it at %ear 0( 4e*ertheless, although a higher discount rate "a% be appropriate, e'll use the afterta! cost of debt to discount the residual *alue as is co""on in practice( ,etting the 40C e$ual to 2ero+ 40C L 0 L <8,000,000 E U(1(0A9I#(9V.B0A(9IN,A# E AII,000(9V.B0A(9IN,A# E A00,000:1(0A9I A U L <1,192,I=J(0M ,o, the "a!i"u" preta! lease pa%"ent is+ 9reta! lease pa%"ent L <1,192,I=J(0M:(1 E (=I# L <1,80M,J22(81 $. -he securit% deposit is a cash outflo at the beginning of the lease and a cash inflo at the end of the lease hen it is returned( -he 40C ith these assu"ptions is+ 40C L <8,000,000 E 200,000 E 1,2AI,000 E <1,2AI,000(9V.B0A(9IN,I# E <AII,000(9V.B0A(9IN,A# O <200,000:1(0A9I A 40C L <I9,=8A(19 With the securit% deposit, the fir" should still lease the e$uip"ent rather than bu% it, because the 40C is greater than 2ero( We could also sol*e this proble" another a%( Bro" 9roble" J, e &no that the 40C ithout the securit% deposit is <99,A09(8M, so, if e find the present *alue of the securit% deposit, e can si"pl% add this to <99,A09(8M( -he present *alue of the securit% deposit is+ 9V of securit% deposit L E<200,000 O <200,000:1(0A9I A L E<A0,12I(MJ ,o, the 40C ith the securit% deposit is+ 40C L <99,A09(8M E A0,12I(MJ L <I9,=8A(19 1%. a. -he different borroing rates are irrele*ant( 0 basic tenant of capital budgeting is that the return of a pro1ect depends on the ris& of the pro1ect( ,ince the lease pa%"ents are affected b% the ris&iness of the lessee, the lessee's cost of debt is the appropriate interest rate for the anal%sis b% both co"panies( B-422 SOLUTIONS b. ,ince the both co"panies ha*e the sa"e ta! rate, there is onl% one lease pa%"ent that ill result in a 2ero 40C for each co"pan%( We ill calculate cash flos fro" the depreciation ta! shield first( -he depreciation ta! shield is+ )epreciation ta! shield L (<==0,000:=#((=I# L <=J,I00 -he afterta! cost of debt is the lessee's cost of debt, hich is+ 0fterta! debt cost L (09(1 E (=I# L (0A9I Using all of this infor"ation, e can calculate the lease pa%"ent as+ 40C L 0 L <==0,000 E 93-(1 E (=I#(9V.B0A(9IN,=# O <=J,I00(9V.B0A(9IN,=# 93- L <1=0,180(M= c. ,ince the lessor's ta! brac&et is unchanged, the 2ero 40C lease pa%"ent is the sa"e as e found in part b( -he lessee ill not reali2e the depreciation ta! shield, and the afterta! cost of debt ill be the sa"e as the preta! cost of debt( ,o, the lessee's "a!i"u" lease pa%"ent ill be+ 40C L 0 L E<==0,000 O 93-(9V.B09N,=# 93- L <1=0,=M8(0J Hoth parties ha*e positi*e 40C for lease pa%"ents beteen <1=0,180(M= and <1=0,=M8(0J( 11. -he 09R of the loan is the lease factor ti"es 2,I00, so+ 09R L 0(0029A(2,I00# L J(08N -o calculate the lease pa%"ent e first need the net capitali2ation cost, hich is the base capitali2ed cost plus an% other costs, "inus and don pa%"ent or rebates( ,o, the net capitali2ed cost is+ 4et capitali2ed cost L <=A,000 O IA0 E 2,000 4et capitali2ed cost L <==,IA0 -he depreciation charge is the net capitali2ed cost "inus the residual *alue, di*ided b% the ter" of the lease, hich is+ )epreciation charge L (<==,IA0 E 21,A00# : =M )epreciation charge L <==1(9I 4e!t, e can calculate the finance charge, hich is the net capitali2ed cost plus the residual *alue, ti"es the lease factor, or+ Binance charge L (<==,IA0 O 21,A00#(0(0029A# Binance charge L <1M2(10 0nd the ta!es on each "onthl% pa%"ent ill be+ -a!es L (<==1(9I O 1M2(10#(0(0J# -a!es L <=I(A8 CHAPTER 27 B-423 -he "onthl% lease pa%"ent is the su" of the depreciation charge, the finance charge, and ta!es, hich ill be+ Cease pa%"ent L <==1(9I O 1M2(10 O =I(A8 Cease pa%"ent L <A28(M= Challenge 12. With a four-%ear loan, the annual loan pa%"ent ill be <A,800,000 L 93-(9V.B08N,I# 93- L <1,JA1,1I0(MJ -he afterta! loan pa%"ent is found b%+ 0fterta! pa%"ent L 9reta! pa%"ent E .nterest ta! shield ,o, e need to find the interest ta! shield( -o find this, e need a loan a"orti2ation table since the interest pa%"ent each %ear is the beginning balance ti"es the loan interest rate of 8 percent( -he interest ta! shield is the interest pa%"ent ti"es the ta! rate( -he a"orti2ation table for this loan is+ 5ear Heginning balance -otal pa%"ent .nterest pa%"ent 9rincipal pa%"ent 6nding balance 1 <A,800,000(00 <1,JA1,1I0(MJ <IMI,000(00 <1,28J,1I0(MJ <I,A12,8A9(== 2 I,A12,8A9(== 1,JA1,1I0(MJ =M1,028(JA 1,=90,111(92 =,122,JIJ(I2 = =,122,JIJ(I2 1,JA1,1I0(MJ 2I9,819(J9 1,A01,=20(8J 1,M21,I2M(AI I 1,M21,I2M(AI 1,JA1,1I0(MJ 129,J1I(12 1,M21,I2M(AI 0(00 ,o, the total cash flos each %ear are+ 5ear Heginning balance 0fterta! loan pa%"ent /CB -otal cash flo 1 <1,JA1,1I0(MJ E <A,800,000((08#((=A# <1,A88,JI0(MJ E<1,M=8,A00 L E<I9,JA9(== 2 <1,JA1,1I0(MJ E <I,A12,8A9(== ((08#((=A# <1,M2I,J80(M0 E<1,M=8,A00 L E<1=,J19(I0 = <1,JA1,1I0(MJ E <=,122,JIJ(I2 ((08#((=A# <1,MM=,J0=(JI E<1,M=8,A00 L <2A,20=(JI I <1,JA1,1I0(MJ E <1,M21,I2M(AI((08#((=A# <1,J0A,JI0(J2 E<1,M=8,A00 L <MJ,2I0(J2 ,o, the 40C ith the loan pa%"ents is+ 40C L 0 E <I9,JA9(==:1(0A2 E <1=,J19(I0:1(0A2 2 O <2A,20=(JI:1(0A2 = O <MJ,2I0(J2:1(0A2 I 40C L <1M,8A1(2A -he 40C is the sa"e because the present *alue of the afterta! loan pa%"ents, discounted at the afterta! cost of capital (hich is the afterta! cost of debt# e$uals <A,800,000( B-424 SOLUTIONS