Present and Future ValueYou have just deposited $15,000 in an account earning 8% APR, compounded annually. If you leave the money in the account for 10 years, how much will you have? Show Solution Step 1) Time Line 0 | $15,000 Step 2) Formula FV = PV (1 + r ) t FV = $15,000(1 + .08)10 FV = $32,383.87 1 | 2 | 9 | 10 | ? … You have just inherited $25,000. You immediately place $5,000 in a mutual fund earning 12% APR, compounded quarterly. You plan to blow the other $20,000 on a trip to Las Vegas. Much to your chagrin, after 2 years in Vegas, you not only did not manage to blow your inheritance, you actually now have $50,000. You now take $10,000 of the $50,000 and place it in the same mutual fund. Interest rates remain constant. After 10 years and many sad tales, you return home penniless. However, that day you receive your statement from the mutual fund. You quickly tear it open and discover that your deposits are now worth…? Show Solution Step 1) Time Line 0 4 8 48 | | | … | | $5,000 $10,000 ??? Note that the time units of the time line are in quarters. Thus, you place the $10,000 in the account two years, or 8 quarters, from today. Finally, you examine your mutual fund statement 12 years or 48 quarters from the time you deposit the $5,000. Step 2) Formula Note that you will need to use the future value tool two times to answer this question. Also note, that APR = r * #pds/yr Thus, 12% = r * 4, è r = 3% Part a) FV of $5000. FV = PV (1 + r ) t FV = $5,000(1 + .03) 48 FV = $20,661.26 Part b)FV of $10,000 FV = PV (1 + r ) t FV = $10,000(1 + .03) 40 FV = $32,620.38 Total Value of portfolio = $20,661.26 + $32,620.38 = $53,281.64 Also note, you could solve this by finding the value of the portfolio at t = 8 after the $10,000 has been added to the FV of the $5000 after 2 years. Then find the future value of this amount after an addition 10 years (40 quarters). You will find that the answers are the exact same. You would like to save for the down payment for a new car. You have just invested $1,500 in shares of Dell. You believe that Dell will continue to earn its historical return of 15%, compounded monthly. The car you wish to purchase will have a required down payment of $2,200. Given this information, how many months must you wait before you can purchase the car? Show Solution Step 1) Time Line 0 | $1,500 ? | $2,200 | | … | Step 2) Formula Note that we are moving a single value through time, so the appropriate tool is the present/future value tool. However, since our time periods are in months, we need to determine the monthly rate: APR = r * #pds/yr 15% = r * 12 r = 1.25%/month Now we need to manipulate the present value formula to the form that we need it: FV = PV(1 + r ) t FV = (1 + r ) t PV FV t ln = ln(1 + r ) = t ln(1 + r ) PV FV ln PV = t ln(1 + r ) $2,200 ln $1,500 = 30.83months t= ln(1.0125) or about 2 ½ years. How much must you place in an account today in order to have accumulated $100,000 is 15 years? Assume that interest rates are 8%, compounded semi-annually. Show Solution Step 1) Time Line 0 1 2 29 30 | | | … | | ? $100,000 Note that the number of periods (t) is 30 (2 per year times 15 years). The rate/period is APR = r * #pds/yr 8% = r * 2 r = 4% (every 6 months) Step 2) Formula FV = PV(1 + r ) t FV PV = (1 + r ) t $100,000 PV = (1.04) 30 PV = $30,831.87 You uncle constantly brags about his outstanding performance in the stock market. Indeed, at the beginning of 1992, his portfolio of assets was worth a mere $25,000. By the end of 2000, his portfolio had appreciated in value to $750,000 (before he lost almost all of it when the market declined!). What was his annual rate of return on his portfolio? Show Solution Step 1) Time Line 1992 1993 1994 2000 2001 | | | … | | $25,000 $750,000 Note that since you started at the beginning of 1992 and ended at the end of 2000, the money was invested for 9 full years. Also note that we want to calculate the answer in years in order to have, well, an annual rate. Step 2) Formula FV = PV(1 + r ) t FV = (1 + r ) t PV FV PV FV PV 1/ t = (1 + r ) −1 = r 1/ 9 1/ t $750,000 $25,000 − 1 = 45.92%!! You have just deposited $1,000 in your mutual fund account and plan to keep it in the account for forty (40) years. The money earns 16% interest for the first 20 years and 8% for the last 20 years. Your sibling invests in a different account that earns 8% for the first 20 years and 16% for the next 20 years. Which one of you has the most money after 40 years? Show Solution Step 1) Time Line You: 0 | 16% $1,000 Sibling 20 | 40 | ??? 8% 0 | 8% $1,000 20 | 16% 40 | ??? Step 2) Formula Initially we will solve this in two steps. First, determine the value of your investment after 20 years (FV1) FV1 = PV(1+r)t FV1 = $1000(1.16)20 = $19,460.76 Next, determine the future value of $19,460.76 after 20 years (FV2) at 8%. FV2 = PV (1+r)t FV2 = $19,460.76(1.08)20 = $90,705.76 You could repeat this procedure for your sibling. However, note that there is a little short-cut. In the equation FV = $19,460.76(1+.08)20, the $19,460.76 is really just $1000(1.16)20. Thus, we could get to the value at time period 40 by conducting the following calculation: FV = $1000(1.16)20(1.08)20 = $90,705.76. So now we can calculate the amount that your sibling will have as: FV = $1000(1.08)20(1.16)20 = $19,705.76, the same amount. Although this may not be intuitive, recalling the A * B is the same as B * A, you can see that the two answers must be the same. Annuities You have just won the lottery, a record $500 million, paid in equal annual installments over the next 20 years. The first payment will be received immediately. If lottery officials wanted to pay you at the end of the year rather than at the beginning, how much must they offer you in order for you to be indifferent between the two? Assume the effective annual interest rate is 10%. Show Solution Note that since there are 20 equal payments summing to $500 million, then each payment is $25 million. Step 1) Time Line 0 1 2 19 20 | | | … | | Original $25M $25M $25M $25M $0 Proposed $0 PMT PMT PMT PMT Step 2) Formula Note that the best time to solve this question for is at t = 0. If we can determine the value of the original stream at t = 0, we can then treat this as the present value of the proposed stream of cash flows to determine the fair PMT. Recall the annuity tool: 1 1 PV = PMT − r r (1 + r ) t it is extremely important to note that this tool when applied to the original stream of cash flows will only tell you the value of the cash flows 1-19, NOT the value of the first $25M (recall the picture of how this tool works). So to the PV of the annuity, you will need to add $25M. 1 1 PV = $25M − = $209,123,002 .1 .1(1 + .1)19 So the total present value of the cash flows stream is $209,123,002 + $25,000,000 = $234,123,002 Next, we treat the $234,123,002 as the present value of the proposed payment stream. Now note, that the annuity tool will work if $234,123,002 is the present value, to determine the payment stream based on the next 20 periods. 1 1 $234,123,002 = PMT − .1 .1(1 + .1) 20 PMT = $27,500,000 What is the present value today of a constant stream of $500 payments to be received at the end of the year for the next 10 years. Interest rates are 8% APR, compounded annually. Show Solution Step 1) Time Line 0 | 1 | $500 2 | $500 9 | $500 10 | $500 … Note that there are two ways to solve this question. First, you could individually discount each of the 10 $500 payments back to t = 0 using the present value formula. However, a much shorter way is to simply use the annuity tool, which we will use. Step 2) Formula 1 1 PV = PMT − r r (1 + r ) t 1 1 PV = $500 − .08 .08(1.08)10 PV = $3,355.04 You are considering the purchase of a Titanic Bond, issued by the U.S. government. This bond promises the owner a payment of $50/year forever. If the price of this bond is $850, what will be your rate of return on this investment? Show Solution Step 1) Time Line | Step 2) Formula PMT r PMT r= PV $50 r= $850 r = 5.88% PV = | $50 | $50 | $50 | … You have just retired after a long career with a nest egg of $800,000. Your lifelong dream has been to purchase an RV and motor the roads of North and South America for the 20 expected years of your retirement. You estimate that you will require $60,000/year with which to live in retirement and you will withdraw the $60,000 at the beginning of each year. Thus, you will immediately withdraw $60,000 and will do so again for the next 19 years. Assume interest rates are 10% APR, compounded annually, how much can you afford to spend on an RV today? Show Solution Step 1) Time Line 0 | $60 RV Value 1 | $60 2 | $60 19 | $60 20 | $0 … To solve this, we will first need to subtract the present value of the 20 $60,000 payments from your retirement fund of $800,000. What is left over can be spent on the RV. Step 2) Formula Present value of twenty $60,000 payments, with the first to be received today. 1 1 PV = $60,000 + $60,000 − = $561,895.21 .1 .1(1.1)19 $800,000 − $561,895.21 = $238,104.79 Bonds You have just purchased a bond with an annual coupon of 7%, 8 years remaining until maturity, that is selling at a price such that the yield to maturity is 12%. Given this information, what is the price that you paid for the bond? Show Solution Time Line 0 | 1 | $70 2 | $70 7 | $70 8 | $1070 … 1 $1000 1 PB = $70 − = $751.62 + .12 .12(1.12) 8 (1.12) 8 Last year, you purchased a bond issued by IBM that offered 8% annual coupon payments, had 12 years to maturity, and had a yield to maturity of 8%. The first coupon of $80 has just been paid and current interest rates have changed such that the yield to maturity on the bond is now 4%. What was your rate of return and the current yield on this bond? Show Solution Step 1: First note that the price of the bond when you purchased it was $1000 because the coupon rate equaled the YTM. Current yield = Coupon/Pbond = $80/$1000 = 8% Coupon + ∆ Pr ice Investment Rate of Return = or coupon plus change in price divided by the initial investment. In order to determine this, we need to calculate the ∆ Price. After the first coupon is paid, the bond will be worth: 1 $1000 1 PB = $80 − = $1,350.42 + .04 .04(1.04)11 (1.04)11 Thus, ∆ Price = $1,350.42 - $1,000 = $350.42. $80 + $350.42 $430.42 = = 43.04% $1000 $1000 So Rate of Return = Der-Bond-Hauffe is a little-known firm that just issued a bond with a maturity of 6 years, a yield to maturity of 7%, and a price of $904.67. What must be the coupon rate of this bond? Show Solution Time Line 0 | $904.67 First, we need to determine how much of the current price is due to the payment of the face value. $1,000 6 PVface = (1 + .07) 1 | C 2 | C … 5 | C 6 | C+1000 = $666.34 Thus, $904.67 - $666.34 = $238.33 is the value of the coupon payments. Next, solve for what stream of 6 coupon payments has a present value of $238.33. 1 1 $238.33 = C − .07 .07(1.07) 6 C= $238.33 1 1 − .07 .07(1.07) 6 = $50 Stocks What is the price today of a stock that is expected to pay an annual dividend of $2.50 each year in perpetuity? The appropriate discount rate for a stock of this risk is 15%. Show Solution The promised payments are a simple perpetuity. Thus, the value is: PS = $2.50 = $16.67 .15 What is the price today of a stock that just paid a dividend of $1 and dividends are expected to grow at a constant rate of 4% per year, indefinitely. The required return on equity is 18%. Show Solution The payments associated with owning this stock are simply a growing perpetuity. We can use the Gordon Growth model to value this stream of payments. PS = DIV1 $1(1.04) = = $7.43 r − g (.18 − .04) You have just purchased a stock that currently pays no dividends. However, you expect that after 5 years, the company will begin to pay dividends on their stock of $2.00 per share, and that this amount will grow at a constant rate of 5% per year thereafter. Assuming that stocks with similar risks require a rate of return of 20%, what is a fair price to have paid for this security? Show Solution Time Line 0 1 2 3 4 5 6 7 DIV | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2.10 … … Once the dividends begin, this is a growing perpetuity. We can value the growing stream of dividends using the Gordon Growth model. Ps = DIV1 2.00 = = $13.33 r − g .20 − .05 Note that the stock will have a value of $13.33 at t = 5 (since we are treating the $2 as the beginning of the perpetuity). Thus, the present value (at t = 0) of this cash flow is: V0 = $13.33 1 .2 5 = $5.36 We could have achieved that same answer by treating the $2.10 payment as the first dividend, finding the value of the growing perpetuity at t = 6, adding $2 to this value, and discounting the entire amount back to t = 0. A firm has a return on equity of 20% and a plowback ratio of 40%. The risk of this firm is such that equityholders expect a 30% return. The firm just paid a dividend of $1. Given this information, what is the price of the stock? Show Solution Recall that growth = return on equity * plowback. Thus, g = .2 * .4 = .08. We now have sufficient information with which to value the stock. We know that the next dividend is expected to be $1(1.08) = $1.08, and that dividends are expected to grow by 8% each year thereafter. Thus, the price of the stock is: Ps = $1.08 $1.08 = = $4.91 .30 − .08 .22
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