Chapter 6TIME VALUE OF MONEY 1. 2. Value five years hence of a deposit of Rs.1,000 at various interest rates is as follows: r = 8% FV5 = = 1000 x FVIF (8%, 5 years) 1000 x 1.469 = Rs.1469 r = 10% FV5 = = 1000 x FVIF (10%, 5 years) 1000 x 1.611 = Rs.1611 r = 12% FV5 = = 1000 x FVIF (12%, 5 years) 1000 x 1.762 = Rs.1762 r = 15% FV5 = = 1000 x FVIF (15%, 5 years) 1000 x 2.011 = Rs.2011 Rs.160,000 / Rs. 5,000 = 32 = 25 According to the Rule of 72 at 12 percent interest rate doubling takes place approximately in 72 / 12 = 6 years So Rs.5000 will grow to Rs.160,000 in approximately 5 x 6 years = 30 years 3. In 12 years Rs.1000 grows to Rs.8000 or 8 times. This is 2 3 times the initial deposit. Hence doubling takes place in 12 / 3 = 4 years. According to the Rule of 69, the doubling period is: 0.35 + 69 / Interest rate Equating this to 4 and solving for interest rate, we get Interest rate = 18.9%. 4. Saving Rs.2000 a year for 5 years and Rs.3000 a year for 10 years thereafter is equivalent to saving Rs.2000 a year for 15 years and Rs.1000 a year for the years 6 through 15. Hence the savings will cumulate to: 2000 x FVIFA (10%, 15 years) + 1000 x FVIFA (10%, 10 years) = 2000 x 31.772 + 1000 x 15.937 = Rs.79481. 5. Let A be the annual savings. 983.980 From the tables we find that FVIFA (20%.411) x 2% r = 16% + = 17.234 Using linear interpolation in the interval.404 = Rs.000 5.000 – 4.000 x 0.000 x PVIF (r = 15%. 9.000 x 0. the present value is: .3% (10.411) 8. A x FVIFA (12%.10.000.000 x PVIF (r = 12%. 10 years) FVIF (r. 10 years) = A x 17.327 = Rs.000 FVIFA (r. A = 1. The present value of Rs.6. we get: 20% + (10.930) 7. 1.549 = 1. 6 years) Using linear interpolation in the interval. 6 years) = 10.10 years) = = 5.4.930) r= x 4% = 20.000 1. 8 years) = 10. 8 years) = 10. 6 years) FVIFA (24%. 6 years) = 10.000 x FVIFA (r.040 r = 15% PV = 10. 10 years) = FVIF (18%.000. 8 years) = 10. 10 years) = 4.467 = Rs.000.000 So.549 = Rs.000 receivable after 8 years for various discount rates (r ) are: r = 10% PV = 10.000 / 1000 = 10 = = 9.411 5.670 r = 12% PV = 10.234 – 4.000 x 0.000 x PVIF(r = 10%.3.000 / 1000 = 5 From the tables we find that FVIF (16%.270 Assuming that it is an ordinary annuity.930 10. we get: (5.56.4.000 x FVIF (r. 1.000 – 9.000 / 17.4% (5.980 – 9. 4.22.000.000 x PVIFA (r.50.000 x PVIFA (15%. 5years) = 2. 15 years) = 10.000 / FVIF (10%.10 x PVIF (10%.847 = Rs.10.20.10 = Rs.000.797 = Rs. 2 years) = 1.494 .000 A = -----------------.000 beginning from the end of 15 years from now. To earn an annual income of Rs. Obviously.019 4. The amount that can be withdrawn annually is: 100. 30 years) 9.212 14. 8 years) x PVIF(12%.-----------.000 x 5.000 for 15 years when r = 15% is: 10. if the deposit earns 10% per year a sum of Rs.797 + 5.000 x 3.000 x 3.582 10.000 100.000 x PVIFA (10%.608 PVIFA (10%. Rs.10.000 x PVIFA (12%. The present value of the income stream is: 2.165 15.000 / 3. The present value of an annual pension of Rs.000 / 0. 2 years) + 5. Mr.58. 10 years) PVIFA (18%.621 = Rs.000 / Rs.Rs.470 The alternative is to receive a lumpsum of Rs.5.968 x 0.26.= Rs. 10 years) = = 5.00 From the tables we find that: PVIFA (15%.893 + 2.427 12.000 x 4.2.000 is required at the end of 14 years. The present value of the income stream is: 1.50.000 = 5. 1 year) + 2. 10 years) PVIFA (r. 13.20.10 years) = Rs. 5 years) + 3000/0.50.50.000 x 0. 11.4.10.7.797 = Rs.683.5.10 x 0.000 x PVIFA (10%.791 + 3000/0. 5 years) = 2. Jingo will be better off with the annual pension amount of Rs.13.000 x PVIF (12%.000 =.500 x 0. The amount that must be deposited to get this sum is: Rs.= ----------.791 = Rs. 14 years) = Rs.500 x PVIF (12%. 000 [1+( 0.5. 9 years) + Rs.1 17.600 x PVIF (12%.806 Rs.00 r = 15% + ---------------5.10.361 + Rs.21.2 PV (Stream C) = Rs.10.9.10. 7 years) + Rs.000 x 0.Using linear interpolation we get: 5. PV (Stream A) = Rs.700 x 0.019 – 4.400 x 0.200 x 0.191 Rs. 6 years) + Rs.851. 5 years) + Rs.000 x PVIF (12%.494 x 3% = 15.567 + Rs.1.797 + Rs. 10 years) = Rs.900 x 0.910 18.200 x PVIF (12%.5.100 x 0.030 19 A B C .600 x 0.5.452 + Rs.000 [1 + (0.16 / 4)]5x4 Rs. FV5 = = = = Rs.000 (1.000 x 2.893 + Rs.12/4)] 5x4 Rs.712 + Rs.800 x 0. PV (Stream B) = Rs.000 x 1.400 x PVIF (12%.500 x 0.404 + Rs. FV5 = = = = Rs.1% 16.1.100 x PVIF (12%.04)20 Rs.900 x PVIF (12%.9 Similarly. 3 years) + Rs.636 + Rs. 2 years) + Rs.2590.507 + Rs.019 – 5.2.800 x PVIF (12%.625. 8 years) + Rs.700 x PVIF (12%.300 x 0.3.500 x PVIF (12%.03)20 Rs. 4 years) + Rs. 1 year) + Rs.322 = Rs.300 x PVIF(12%.000 (1. 8 0.000 Rs. ∞ ) = Rs. 10 years) = 4.10 years) = Rs.12. the amount to be deposited now is: Rs.000 Rs.046 This means that the implied interest rate is nearly 15%.20.2 2.000 = = Rs.8 Investment required at the end of 8th year to yield an income of Rs.Stated rate (%) 12 Frequency of compounding 6 times 4 times 24 12 times (1 + 0.000 per year from the end of 9th year (beginning of 10th year) for ever: Rs.100.000 x FVIF (r.10 years) = = 4.12.100.12/6)6.000 To have a sum of Rs.5.100.000 From the tables we find that FVIF (15%.20.40. The interest rate implicit in the offer of Rs.000 after 10 years from now because I find a return of 15% quite acceptable.1 (1+0.12.6 2.24/4)4 –1 (1 + 0.000 after 10 years in lieu of Rs. I would choose Rs. .000 Rs.476 21.000 FVIF (r.000 now is: Rs.100.000 / 0.5.24/12)12-1 Effective rate (%) Difference between the effective rate and stated rate (%) 20. 24 = 12. 8 years) 2.000 at the end of 8 th year .20.6 = 26.000 x PVIFA(12%.5.12 = Rs.2 = 26.388 PVIF(12%.20. 2544 17.12) Rs. 30 years) = Rs.7.10.994 30 per cent of the pension amount is .335 A x 8.10. A constant deposit at the beginning of each year represents an annuity due. The discounted value of Rs.427 = Rs.000 = = Rs.100 evaluated at the end of 2020 is Rs.72.2. FV10 = Rs.000 x PVIFA (12%.283 23.000 at the end of 10 years the annual deposit should be A = Rs.335 If A is the amount deposited at the end of each year from 2015 to 2020 then A x FVIFA (12%.100 x PVIF (12%.424 = Rs. evaluated as at the beginning of 2024 (or end of 2023) is: Rs.10 / 2)]10x2 = Rs.335 A = Rs.530 If the inflation rate is 8% per year.72.854 x 0.51. 3 years) = Rs.10 years) = Rs.530 x 0. the value of Rs.50.530 x PVIF (8%.2000 receivable for 30 years.51. 6 years) = Rs.6326 25. 9 years) = Rs.72. PVIFA of an annuity due is equal to : PVIFA of an ordinary annuity x (1 + r) To provide a sum of Rs.000 x PVIFA (10%. 5 years) = Rs.549 x 1.20.000 x 9.50.72. evaluated as at the end of 9th year is: Rs.10.000 FVIFA(12%.854 x PVIF (10%.26.000 [1 + (0.000 x 2.51.463 = Rs.854 The present value of Rs.18.51. 10 years) x (1.20.335 / 8.000 x 3.18.653 = Rs.18.000 (1.115 = Rs. The discounted value of the annuity of Rs.26.100.100 x 0.605 = Rs.26.05)20 = Rs.115 = Rs.000 receivable at the beginning of each year from 2025 to 2029.26.2.12 24.20.12.854 is: Rs.22.50.18.712 = Rs. The discounted value of Rs.530 10 years from now. in terms of the current rupees is: Rs. 26. 149.149.01)60 = P x 1.= Rs.000 r = 1% + ---------------------21.10 million x PVIF (8%. The corresponding effective rate of interest per annum is [ (1. The discounted value of the debentures to be redeemed between 8 to 10 years evaluated at the end of the 5th year is: Rs. 24) = 21. the bank charges an interest rate of 1.735 + 0.01 (1. the discounted value of an annuity of Rs.149.53% Thus.3000 = 20 From the tables we find that: PVIFA(1%.0.1800 receivable at the end of each month for 180 months (15 years) is: Rs. 5 years) = Rs. 24 months) = Rs.0153)12 – 1 ] x 100 = 20% 28. 3 years) + Rs.1800 x ---------------.817 Rs.980 Rs.10 million (0.24) = PVIFA (2%.540 1.914 Using a linear interpolation 21.24) = Rs.P today on which the monthly interest rate is 1% P x (1.82. = Rs.817 = P 27.681) = Rs.244 – 20.01)180 .2.914 x 1% = 1.10 million x PVIF (8%.01)180 If Mr.6000 = Rs.149.980 -----------.30 x Rs.244 18.60.1 Rs.000 PVIFA (r.244 – 18.3000 x PVIFA(r.= Rs. Ramesh borrows Rs.980 Rs. 180) (1.60000 / Rs.1800 Assuming that the monthly interest rate corresponding to an annual interest rate of 12% is 1%.980 .53% per month.10 million x PVIF (8%.1800 x PVIFA (1%. 4 years) + Rs.794 + 0.21 million . 585 Loan Amortisation Schedule Year -----1 2 3 4 Beginning amount ------------500000 398415 282608 150588 Annual installment --------------171585 171585 171585 171585 Interest ----------70000 55778 39565 21082 Principal Remaining repaid balance ------------------------101585 398415 115807282608 132020 150588 150503 85* (*) rounding off error 31.868 Equated annual installment x 1 = 7.914 = Rs.100.000 PVIFA (10 %.867 = Rs.000 x PVIFA (10%. 200. Using a linear interpolation we get n=7+ 30.20. 5. Define n as the maturity period of the loan.4) = 500000 / 2.21 million A x 5. Rs.000 = 5.2.000 – 4.335 Thus n is between 7 and 8.867 = Rs.868 ----------------5.20.If A is the annual deposit to be made in the sinking fund for the years 1 to 5.20.2.000 x PVIFA(13%.000 From the tables we find that PVIFA (10%.000 / Rs.335 – 4. The value of n can be obtained from the equation. n) = = 1.867 = Rs.868 5.100.377 million 29.500 .21 million / 5.21 million A = Rs. then A x FVIFA (8%. n) PVIFA (13%.000 7. Let `n’ be the number of years for which a sum of Rs. 8 years) = 4. 5 years) = Rs.21 million A = 5.000 can be withdrawn annually. n) = Rs.3 years = 500000 / PVIFA(14%.2.500. n) = Rs.0.2. 7 years) = PVIFA (10%.171. 35. g = (1+g1)(1+g2) .000 x 6.10) = Rs.05) = Rs. Expected value of iron ore mined during year 1 = Rs.1 In this problem the growth rate in the value of oil produced.9725 / 0.000 (d) PV = 100.300 million x (0. then the growth rate of their product.------1+r PV = A(1+g) ----------------- .0.000 (a) PV = c/(r – g) = 12/[0.g = 12 x 0.614. 1 – (1 + g)n / (1 + i)n -----------------------i-g 1 – (1.15 = Rs.000.700.6yrs = 1.000 (c ) PV = 60.74135 / 0.16 – 0.10-0.600.80 million (b) 1+g n 1 .0.000PVIF10%.500 (e) PV = C/(r-g) = 35.000 Option e has the highest present value viz.500.000/(0.300 million Expected present value of the iron ore that can be mined over the next 15 years assuming a price escalation of 6% per annum in the price per tone of iron = Rs.564.03) . 32.03)] = Rs. of units and g2 the growth rate in price per unit.10yrs = 100.300 million x 33 34.06 = Rs.300 million x = Rs.1 = .06)15 / (1.000.000 (b) PV = 1.000 PVIFA10%.16)15 0.000 x 0. Rs.8 million It may be noted that if g1 is the growth rate in the no.564 = Rs.2224 million (a) PV = Rs.10 = Rs.12 – (-0.From the tables or otherwise it can be verified that PVIFA(13.30) = 7.------1+r PV = A(1+g) ----------------r. g = (1.000/r = 60.05)(1 +0.145 = Rs.000/0.500 Hence the maturity period of the loan is 30 years.0215 Present value of the well’s production = 1+g n 1 .700.77. ------1+r PV = A(1+g) ----------------r.328.434.1 = 1.0224 Present value of the well’s production = 1+g n 1 . the effective interest rate is 0.000 x 60) x ( 1-0.08 1 + 52 . 09)20 – ( 1.08)20 So the value of the savings at the end of 20 years = 100. The growth rate in the value of the oil production g = (1.0215)x 1 – (0.654.g = (80.633 36.0.9776 / 1.0. ( 1+ i )n – ( 1 + g)n FVIFGA = i-g (1.g = (50.0224)x 1 – (0.536 38 Assuming 52 weeks in an year. Future Value Interest Factor for Growing Annuity.09 – 0.1 = .0224 = $ 30.12)20 0.10)15 0.10 + 0.1 = 8.000 x 0.12 + 0.0215 = $ 16.06)(1 +0.9785 / 1.93 37.0832 .04) .000 x 50) x ( 1-0. 9.781.08 = Rs.r.32 percent 52 MINICASE--1 . 48.000. 15years) + 1.000 = Rs. How much money should Ramesh save each year for the next 15 years to be able to meet his investment objective? Ramesh’s current capital of Rs.506. How much money would Ramesh need when he reaches the age of 60 to meet his donation objective? 200. the annual savings must be : 1.487 x 0. 11yrs) = 200.4.Solution: 1.042.12) 2 1. 3yrs) x PVIF (10%.08 = 400.800 So.000 x 4.12 1– 1.600.535.606 + 1.200 This means that his savings in the next 15 years must grow to : 4.676 4.000 x 2.10)15 = 600.000 46 1 400.000 (1. What is the present value of Ramesh’s life time earnings? 400.800 1.177 = Rs 2. How much money would Ramesh need 15 years from now? 500.042. 15 years) = Rs.338 31.000 x PVIF (10%.800 = FVIFA (10%.535.000 x 239.317 = 157.535.000 x PVIFA (10% .506.000 400.000 2.000 will grow to : 600. 15years) = 500.12)14 15 15 .200 = Rs 1.000(1.000 x PVIFA (10%.000 – 2.000 x 0.000 x 7.000(1.239 = 3.000.772 3.803. each 1 Rupee deposited in the RD account becomes = FVIFA(0.12 = Rs. for a maturity value of Rs.634 For a RD maturity value of Rs.02)44 – 1} / 0.08/4) 4x20 = Rs.0402 which when compounded quarterly becomes at the end of 10 years = 3.08/12) = {(1.962 MINICASE--2 Solution: 1) Re.3.742 Similarly for a maturity value of Rs.61.02] = Rs.will be = 42.183.0.05) 10 = 32.3) = [{(1+0.32.00667) 3-1}/0.05) 11 = 42.634 = Rs.58 lakhs MBA expenses for year II at present = 25 lakhs.58.00667 x 1.0402 x [(1.00.0402 44 Rs.0402 Rs. 183.000 / (1+0.76 lakhs the monthly deposit needed .76 lakhs At the end of 3 months.0402 x [(1+0.000/183.3.08/4) 4x10 1]/ (0.0402 MBA expenses for year I at present = 20 lakhs.02 = Rs.76.3.300 lakhs Cumulative fixed deposit to be made now to get the above amount = 300.08/12)3 -1} / (0.236 2) Amount required for Jasleen’s marriage at the end of 20 years = Rs.0402 x {(1.1 deposit each at the end of month 0 1 becomes 2 3 4 Rs.53.634 if the deposit to be made is Rs.08/12.254. 20.000 / [3.08 – 0.00667 = Rs. After 11 years it would be = 25(1+0.3.02)40 – 1] / 0.08/12)] x (1+0.292 3) Year end 0 19 20 What deposit? 12L 1 2 3 4 5 6 7 8 Annuity Payments 9 10 11 12 12L 12L 12L Annuity Period 13 14 15 16 17 18 12L 12L 12L 12L 12L 12L . After 10 years it would be = 20(1+0.3.3.17.58 lakhs.0402 5 6 9 12 40 Rs.1. the monthly deposit to be made will be = 32.0402 Rs.0402 Rs.3.08/4) = 3.7.42. 12 lakhs With inflation at 5 percent.74.74.93.08/4)4x9 = Rs.05)] = Rs.Annuity needed per annum at the beginning of each year in real terms after 10 years = Rs.163 Amount to be deposited in cumulative fixed deposit now. this may be considered as a growing annuity for 10 years at a growth rate of 5 percent and discount rate of 10 percent.00.05)/(1+0.95.000 x (1+0. Present value of the annuity . to have a maturity value of Rs.93.10-0.432 .163 at the end of 9 years = 93.45.163/(1+0. as at the beginning of the 10th year from now = 12. in nominal terms.10)10 /(0.74.05)[ 1 –(1+0.