SUGGESTED ANSWERS TO DISCUSSION QUESTIONS

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1 SUGGESTED ANSWERS TO DISCUSSION QUESTIONS Answers will vary according to student s selections, tastes, and preferences. SOLUTIONS TO PROBLEMS P4.1 The investor would earn $8.25 on a stock that paid $3.75 in current income and sold for $ Part of the total dollar return includes a $4.50 capital gain which is the difference between the proceeds of the sale and the original purchase price ($67.50 $63.00) of the stock. P4.3 a. Current income: = $2.70 b. Capital gain: $60 $50 = $10 c. Total return: 1) In dollars: $ $10.00 = $ ) As a percentage of the initial investment: $12.70 =.25 or 25% P4.5 a. Total return = Current income + Capital gains (or losses) where: Capital gains (or losses) = Ending price Beginning price (3) (4) (5) (1) (2) (1) (2) (3) + (4) Ending Beginning Capital Current Total Year Price Price = Gain + Income = Return 2001 $32.50 $30.00 $2.50 $1.00 $ b. Of course, there is no correct answer here, but one might forecast using the arithmetic average or the average one-year holding period return. (i) The arithmetic average: $ $3.70 $.70 + $ $6.75 = $4.37 5

2 (ii) The average holding period return (HPR): HPR = Ending price Beginning price + Current income = Total Return Beginning price Beginning Price (1) (2) (3) Total Beginning (1) (2) Year Return* Price HPR 2001 $3.50 $ % *From part (a) above. Average HPR = = 12.8% 5 b. (i) (ii) Forecasts Based on Based on for: Arithmetic Average Average HPR 2006 $4.37 ($45.00)*.128 = $ $4.37 ($49.00)**.128 = $6.27 * End of 2005 price gain in original data. ** For lack of information, we are assuming the 2006 return is $4.00 from capital gains and $1.76 from current income. c. Students should be made aware of the fact that many other forecasts are possible. Other factors may be relevant here: Will the pattern of two good years followed by a bad one continue? Do future prospects seem bright? (We will discuss forecasting returns on specific investment vehicles in later chapters.)

3 P4.7 The simple interest calculations for parts a. and b. can be presented in tabular form: b. a. Beginning Annual Ending Date Balance* Interest Balance** 1/1/05 $5,000 $5, = $300 $5,000 1/1/06 1,000 1, =60 1,000 1/1/07 3,000 3, =180 3,000 1/1/08 6,000 6, = 360 6,000 * Assuming all transactions occur at the beginning of the period. ** Since all interest earned is withdrawn, the ending account balance equals the beginning account balance. c. The true rate of interest is 6 percent, the same as the stated rate of interest since the simple interest method is being used. P4.9 Future Value of an Investment: FV n = Investment Amount x (FVIF k,n ) Investment Investment A FV 20 = PV FVIF 5%, 20 yrs. B FV 7 = PV FVIF 8%, 7 yrs. FV 20 = $ FV 7 = $4, FV 20 = $ FV 7 = $7,713 Calculator solution: $ Calculator solution: $7, C FV 10 = PV FVIF 9%, 10 yrs. D FV 12 = PV FVIF 10%, 12 yrs. FV 10 = $10, FV 12 = $25, FV 10 = $23,670 FV 12 = $78,450 Calculator solution: $23, Calculator solution: $78, E FV 5 = PV FVIF 11%, 5 yrs. FV 5 = $37, FV 5 = $62,345 Calculator solution: $62, P4.11 Future Value of an Annuity Investment: FVA k,n = Annual Deposit FVIFA k,n Investment A FVA 8%,10 yrs. = $2, = $36, Calc. Sol n. = $36,216.41

4 B FVA 12%,6 yrs. = $ = $4, Calc. Sol n. = $4, C FVA 20%,5 yrs. = $1, = $7,442 Calc. Sol n. = $7, D FVA 6%,8 yrs. = $12, = $118,764 Calc. Sol n. = $118, E FVA 14%,30 yrs. = $4, = $1,427,112 Calc. Sol n. = $1,427, (FVIFA from Appendix A, Table A.2) P4.13 The least you would accept for each investment is its future value at the end of six years: a. Future Value of an Investment: FV n = Investment Amount (FVIF k,n ) FV 9%,6 yrs. = $5, = $8,385 Calc. Sol n = $8, b. Future Value of an Annuity Investment: FVA k,n = Annual Deposit FVIFA k,n FVA 9%,6 yrs. = $2, = $15,046 Calc. Sol n. = $15, c. FV of $3,000 at 9% for 6 years + FVA of $1,000 deposit at 9% at end of each of the next five years: 1. FV 9%,6 yrs. = $3, (From App. A, Table A.1) = $5,031 Calc. Sol n = $5, FVA 9%,6 yrs. = $1, (From App. A, Table A.2) = $7,523 Calc. Sol n. = $7,523.33

5 3. Total = $5,03 + $7,523 = $12,554 d. Number End-of Year of Years Future Year Deposit to Compound FVIF,9% Value 1 $ $1, , Total FV= $3, P4.15 This problem uses present value to solve an investment problem. The amount at which the bond will sell today is the value today of its value at maturity (in eight years), given an interest rate of 6%:. PV = FV (PVIF k,n ) PV 6%, 8 yrs. = $1, = $627 (Calculator Solution = $627.41) P.4.17 This is a present value question. Calculate the present value of $10,000 using a PVIF 3%, 10 periods value. $10,000 x = $7,440. P4.19 a. Income End of Stream Year Income PVIF,15%Present Value A 1 $4, = $ 3, , = 2, , = 1, , = 572 $7,636 Calculator solution = $7, B 1 $1, = $ , = 1, , = 1, , = 2,288 $6,644 Calculator solution = $6, PVIF from Table A.3, Appendix A.

6 b. Income Stream A, with a present value of $7,636, is higher than Income Stream B s present value of $6.644 because the larger cash inflows occur in A in the early years when their present value is greater. The smaller cash flows are received further in the future. P4.21 Find the present value of the annuity and compare it to the lump sum payment. PVIVA 8%, 20 periods x $1,000,000 = = $9,818,000. Since this is less than $15 million, you should take the $15 million today. P4.23 The analysis of Terri s investment opportunities uses the formula: FV n = PV (FVIF k,n ) Investment Calculation Decision A $30,000 = $18,000 FVIF k,5 yrs. $30,000 = FVIF k,5 yrs. $18, = FVIF k,5 yrs. 10% < k < 11% Invest Investment Calculation Decision B $3,000 = $600 FVIF k, 20 yrs. $3,000 = FVIF k, 20 yrs. $600 5 = FVIF k, 20 8% < k < 9% Forgo C $10,000 = $3,500 FVIF k, 10 yrs. $10,000 = FVIF k, 10 yrs. $3, = FVIF k, 10 yrs. 11% < k < 12% Invest D $15,000 = $1,000 FVIF k, 40 yrs. $15,000 = FVIF k, 40 yrs. $1, = FVIF k, 40 yrs. 7% < k < 8% Forgo

7 An alternative approach accurately answering the same questions would be the calculation of the present value of cash inflows and comparing the results to the investment s cost. Investment A $30,000 x.621 = $18,630 (using PVIF 10%, 5 years ) Cost of Investment = $18,000 Return exceeds cost - Invest Investment B $3,000 x.149 = $447 (using PVIF 10%, 20 years ) Cost of Investment = $600 Cost exceeds return - forgo Investment C $10,000 x.386 = $3,860 (using PVIF 10%, 10 years ) Cost of Investment = $3,500 Return exceeds cost - Invest Investment D $15,000 x..022 = $330 (using PVIF 10%, 40 years ) Cost of Investment = $18,000 Cost exceeds return - forgo P4.25 $15,000 = PVIFA 1%, 50 payments $15,000 = $15,000 / = $ P4.27 a. Using the notation given in the chapter, the risk-free rate of interest for both Investments is: R = r* + IP = 3% + 5% = 8% b. The required returns for each investment are calculated as follows: r 1 = r* + IP + RP i or R F + RP i r A = 3% + 5% + 3% 8% + 3% = 11% r B = 3% + 5% + 5% 8% + 5% = 13%

8 P4.29 Holding period return (HPR) = Current income + Ending price Beginning price Beginning price HPR X = $ $ $0 + $ $29.00 $30.00 = $ 3.50 $30.00 $30.00 = 11.67% HPR Y = $0 + $0 + $0 + $ $56.00 $50.00 = $ $50.00 = 16.0% If the investments are held beyond a year, the capital gain (loss) component would not be realized and would likely change. Assuming they are of equal risk, Investment Y would be preferred since it offers the higher return (16.0% for Y versus 11.67% for X). P4.31 HPR = ($50 + ($1,000 - $950)) / $950 = $100/$950 = 10.5% P4.33 Using PFIV of 4%: $ 65 x = $ $ 70 x = $ $ 70 x = $ $7,965 x = $6, $6, P4.35 (1) (2) (3) ( 4) (5) Initial Future (1) (2) Approximate Investment Investment Value Years Discount Rate Yield* A $ 1,000 $ 1, % (.822) B 10,000 20, % (.513) C 400 2, % (.215) D 3,000 4, % (.747) E 5,500 25, % (.231) * From Table A.3, Appendix A.

9 P4.37 The yield for these investments is the discount rate that results in the stream of income equaling the initial investment. Investment A: Using the present value of an annuity formula: PVA = Annual deposit PVIFA k%,5 yrs. $8,500 = $2,500 PVIFA k%,5 yrs. $8,500 = PVIFA k%,5 yrs. $2, = PVIFA k%,5 yrs. Looking at Table A.4, Appendix A, the closest factor for five years occurs at 14% (3.433); therefore, this investment yields about 14%. Investment B: It is necessary to try several different discount rates to determine the yield for Investment B. One way to estimate a starting point is to use the average annual income in the formula used in Part A. and adjusting it based on whether the larger cash flows are received in the earlier or later years. The Internal Rate of Return (IRR) function on a business calculator makes the task easier. PVA = Annual deposit PVIFA k%,5 yrs. $9,500 = $3,000 PVIFA k%,5 yrs. $9,500 = PVIFA k%,5 yrs. $3, = PVIFA k%,5 yrs. The closest interest rate to in Table A.4, Appendix A, is 17%. Because the larger cash flows are received in the later years, 16% is a good starting point. (4) (5) (1) (2) (3) (1) (4) Year Income PVIF, 16% PV at 16% PVIF,15% PV at 15% 1 $2, $1, $1, , , , , , , , , , , , , PV of Income = $9, $9, Calculator Solution = $9, $9, The discount rate that results in a present value closest to $9,500 is 15%. Calculator solution for IRR = 15.36

10 P4.39 (4) (5) End of (1) (2) (3) (1) (2) (1) (4) Year Income 10% PVIF PV at 10% 11% PVIF PV at 11% $ 0 1 $ $ PV of Income = $ $ 5.58 The Yield is very close to 11% on this investment. Since the yield of 11% is greater than the minimum required return of 9.0%, the investment is recommended. This project would result in positive Net Present Value to the investor P = 7 years $1 x FVIF?%, 7 years = $2.21 FVIF = $2.21/$1 = 2.21 FVIF 12%, 7 years = 2.211, so the yield is about 12% P4.43 a. Investment A, with returns that vary widely from 1% to 26% appears to be more risky than Investment B, whose returns vary from 8% to 16%. b. n_ s = Σ ( r r) 2 i = 1 CV = Standard deviation average return Investment A: (1) (2) (3) (4) Year Return Average (1) - (2) (3) 2 r i Return, r r i r (r i - r) % 12% 7% 49%

11 S A = 434 = = 10.42% 5 1 CV = 10.42% = % Investment B: (1) (2) (3) (4) Year Return Average (1) - (2) (3) 2 r i Return, r r i r (r i - r) % 12% -4% 16% S A = 40 = 10 = 3.16% 5 1 CV = 3.16% = % c. Investment A, with a standard deviation of 10.42, is considerably more risky than Investment B, whose standard deviation is This confirms the conclusions reached in Part A. d. Because the real benefit of calculating the coefficient or variation is in comparing investments that have different average returns, the standard deviation is not improved upon.