1. If the opportunity cost of capital is 14 percent, what is the net present value of the factory?

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1 MØA Fall 2011 PROBLEM SET: Hand in 1 Exercise 1. An investor buys a share for $100 and sells it five years later, at the end of the year, at the price of $ Each year the stock pays dividends of $12 per share. The annual average rate of inflation is 2%. What is the average effective real return on the investment? % % % % 5. I choose not to answer. Exercise 2. NPV (BM 3.4) [2] A factory costs $800,000. You reckon that it will produce an inflow after operating costs of $170,000 a year for 10 years. 1. If the opportunity cost of capital is 14 percent, what is the net present value of the factory? 2. What will the factory be worth at the end of 5 years? Exercise 3. House Sale (RWJ 4.7) [3] You are selling your house. The Smiths have offered you $115,000. They will pay you immediately. The Joneses have offered you $150,000, but they cannot pay you until three years from today. The prevailing interest rate is 10%. 1. Which offer should you choose? Exercise 4. Projects [3] A project costs 100 today. The project has positive cash flows of 100 in years one and two. At the end of the life of the project there are large environmental costs resulting in a negative cash flow in year 3 of 95. Determine the internal rate(s) of return for the project. Exercise 5. Machine [4] A company is considering its options for a machine to use in production. At a cost of 47 they can make some small repairs on their current machine which will make it last for 2 more years. At a higher cost of 90 they can make some more extensive repairs on their current machine which will make it last for 4 more years. A new machine costs 300 and will last for 8 years. The company is facing an interest rate of 10%. Determine the best action. Exercise 6. Q [2] Equity in the company Q has an expected return of 12%, a beta of 1.4 and a standard deviation of 20%. The current risk free interest is 10%. 1. What is the current expected market return? Exercise 7. You are considering paying 2 million NOK to buy a new house. You expect to sell your old house for 800 thousand, and need to finance the remainder by borrowing. The bank quotes you a 30 year loan with an 1

2 interest rate of 6% to be paid as a monthly annuity, i.e. in monthly installments of equal size. The first amount is due at the end of the first month. Interest is calculated using monthly compounding. What is the amount you will be paying each month? (a) NOK 6,982 (b) NOK 7,159 (c) NOK 7,195 (d) NOK 11,990 (e) I choose not to answer. Exercise 8. A corporation s Annual Report contains the following information: Sales: 2,000,000 kr. Variable costs: 850,000 kr. Overhead costs: 395,000 kr. Depreciation: 248,000 kr. Corporate tax rate: 34% Calculate the corporation s after-tax cash flows ,620 kr ,620 kr ,000 kr ,620 kr. 5. I choose not to answer. Exercise 9. Lucky Pierre [5] Hi-Tech industries is involved in genetic research. It has enjoyed the astounding dividend growth rate of 25% recently. Next period s dividend is expected to be $20 and the current share price is $ Your stockbroker, Lucky Pierre, believes that the stock is under-priced. He tells you that it has an expected return of 35% per year, but once Wall Street sees its mistake, the stock price will shoot up. What assumptions did Pierre make to reach his conclusion? 2. If Pierre is correct (and the expected return is 35%), what is the expected stock price next period? 3. You, having taken a course in Finance, do not really trust Pierre s work (remember, he gets paid on a commission basis.) You feel that the dividend growth will slow down after 1 more year of 25% to a modest 10% per year. What are the expected dividend payments for the firm for the next three years? 4. Suppose you expect the stock return to be 20% once the growth rate slows down to 10% (after next year). What will the expected stock price be in year 1? What is the current expected return on the stock this year? Exercise 10. [3] A bond promises the following sequence of payments: t = Cashflow X t =

3 The interest rates r t and prices d t of future risk free cash flows are as follows Interest rates are compounded annually. 1. Calculate the bond s price Exercise 11. Bond [4] t = r t = 5.3% 5.4% 5.6% 5.7% d t = The appropriate discount rate for cash flows received one year from now is 7.5%. The appropriate discount rate for cash flows received two years from now is 11%. The appropriate discount rate for cash flows received three years from now is 14%. Interest rates are compounded annually. 1. What is the price of a two year bond with a 6% (annual) coupon and a face value of 1000? 2. What is the yield to maturity of this bond? Exercise 12. [4] A 3 year bond with a face value of $100 makes annual coupon payments of 10%. The current interest rate (with annual compounding) is 9%. 1. Find the bond s current price. 2. Suppose the interest rate changes to 10%, determine the new price of the bond by direct calculation. 3. Instead of direct calculation, use duration to estimate the new price and compare it to the correct price. 4. Use convexity to improve on your estimation using duration. Exercise 13. Compounding [2] A bank quotes an interest rate of 10% with quarterly compounding. 1. What is the equivalent interest rate with continous compounding? Exercise 14. Doubling [2] How long does it take to double a $100 initial investment when investing at a 5% continously compounded interest rate? Exercise 15. Hilda Hornbill (BM 8.7) Hilda Hornbill has invested 60% of her money in share A and the remainder in share B. She assesses the prospects as follows. A B Expected return 15% 20% Standard deviation 20% 22% Correlation between returns What is the expected return and standard deviation of returns on her portfolio? 2. How would your answer be changed if the correlation coefficient were 0 or -0.5? 3. Is Ms. Hornbill s portfolio better or worse than one invested entirely in share A or is it not possible to say? 3

4 Empirical Solutions MØA Fall 2011 PROBLEM SET: Hand in 1 Exercise 1. The return r n on the five year investment in nominal terms: The IRR is: r n = 15%. 0 = In real terms, the exact calculation is It is an approximation only to estimate The correct answer is 3, 12.75% Exercise 2. NPV (BM 3.4) [2] 1. All numbers are in thousands: 12 (1 + r n ) + 12 (1 + r n ) (1 + r n ) (1 + r n ) 4 (1 + r n ) 2 r = 1 + r n 1 + i 1 = = = 12.75% 1.02 r r n i = = 0.13 PV of annuity = = NP V = = After 5 years, the value of is equal the PV of getting $170 per year for 5 years: Exercise 3. House Sale (RWJ 4.7) [3] = You need to compare the present values of the two offers. The PV of the Smiths offer is $115,000. The PV of the Joneses is: P V = = 112, 697 (1 + r) 3 Thus, the Smiths offer the best terms (highest PV), accept their offer. Exercise 4. Projects [3] The following picture shows the NPV as a function of the interest rate, and illustrates the fact that there are two solutions y to the problem of solving 0 = y (1 + y) (1 + y) 3 4

5 NPV Exercise 5. Machine [4] interest One way to go about this is to find the net present value of each alternative by repeating each to make it last 8 years. Small repairs Larger repairs NP V = 47 + The cheapest is the small repairs. Exercise 6. Q [2] Exercise 7. t = C t = ( ) ( ) ( ) 6 = t = C t = NP V = 90 + E[r m ] = E[r] r f β Annuity factor(r=0.005 n=360) = Amount = 1.2 mill / = (c) is correct Exercise ( ) 4 = E[r] = r f + (E[r m ] r f )β + r f = = 11.4% ( )(1 0.34) =

6 (Numbers in thousands) 1. is the correct answer Exercise 9. Lucky Pierre [5] 1. He has used the constant dividend growth assumption. P 0 = D 1 r g = 20 r 0.25 = 200 r 0.25 = = 0.10 r = P 0 = D 1 + P r P 1 = P 0 (1 + r) D 1 = = 250 D 1 = 20 (given) D 2 = = 22 D 3 = 20 (1.1) 2 = 24.2 D t = 20 (1.1) t 1 P 1 = D 2 r g = = 200 P 0 = P 1 + D = = r r 1 200(1 + r) = 240 r 1 = 40 = 0.20 = 20% 200 Exercise 10. [3] 1. 4 Bond Price = P t X t = = 115 t=1 Exercise 11. Bond [4] 1. Price P 0 = (1.11) 2 = Yield to maturity is found by calculating the IRR of the following cashflows The YTM is 10.89%. t = C t = IRR =

7 Exercise 12. [4] 1. The bond price: NP V = 0 + t = C t = ( ) ( ) ( ) 3 = If the interest rate increases to 10%, the bond will be selling at par, equal to 100, which can be confirmed with direct computation: NP V = Calculate the bond s duration: 10 ( ) ( ) ( ) 3 = 100 t C t P V (C t ) tp V (C t ) Sum Bondprice Duration 2.74 Modified duration: D = D 1 + r = = 2.51 Let us now calculate the change in the bond price P P = D y = = Which means theat the bond price changes to: ( ) P P + δp = P = = P 4. Calculate the bond s convexity t C t P V (C t ) P V (C t )(t 2 + t) Sum Bondprice Convexity 8.93 Recalculating the change in the bond price using convexity: P P = D y (Convexity) = (0.01)2 = = Use this to re-estimate the bond price: ( ( P P + δp = P ) ) P = ( ) =

8 Exercise 13. Compounding [2] 1. Equivalent annual rate Continous rate ( )4 1 = % ln(1 + r) = % Exercise 14. Doubling [2] It takes years Exercise 15. Hilda Hornbill (BM 8.7) e 0.05t = 2 t = ln 2.05 = A B ω E[ r] σ ρ AB E[ r p ] = ω A E[ r A ] + ω B E[ r B ] = 17% σ 2 ( r p ) = ωaσ 2 A 2 + ωbσ 2 B 2 +2ω A ω B σ A σ B ρ AB = ρ AB = ρ AB = σ p = = 18.08% 2. If ρ AB = 0, then σ 2 ( r p ) = ωaσ 2 A 2 + ωbσ 2 B 2 +2ω A ω B σ A σ B 0 = = σ( r p ) = 14.88%. If ρ AB =- 1 2, then σ 2 ( r p ) = = σ( r p ) = 10.76% 8

9 3. The portfolio is better, since it has higher expected return and lower variance. 9

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