1. Final exam solution sketches Winter 2014, Version A Note for multiple-choice questions: Choose the closest answer

2. Profitability Index If the effective annual discount rate is 10%, then what is the profitability index if someone invests $900 today in a project that pays out $1250 three years from today? PV cash flows = 1250/(1.1) 3 = 939.14 P.I. = 939.14/900 = 1.0435

3. Confidence Interval 95.44% of the probability distribution is within 2 standard deviations of the mean of a normal distribution. Assume the historical equity risk premium is 12.5%, and the standard deviation of the equity risk premium is 18.0%. 256 years of data were used to make these estimates.

4. Confidence Interval Find the LOWER BOUND of the 95.44% confidence interval of the historical equity risk premium. Lower bound of 95% C.I.: = 12.5% - 2 * 18%/(256) 1/2 = 12.5% - 2 * 18%/16 = 12.5% - 2 * 1.125% = 10.25%

5. Perpetuity An asset promises to pay $6 per year forever, starting six months from today. The stated annual discount rate for this asset is 18%, compounded twice per year. What is the present value of this stream of payments? EAR = (1.09) 2 – 1 = 18.81% PV = 6/.1881 * 1.09 = $34.77 1 st payment is in 6 months, not 1 year

6. CAPM If the market return is 20%, the risk- free rate is 10%, and the beta of Stock X is 5, what is the expected annual rate of return for Stock X? Risk premium = 20% - 10% Expected return = 10% + 5*(20% - 10%) Expected return = 60%

7. Random Walk Use the following information to answer the next three questions: Suppose that the daily price of each share of Wibby Pig stock is a random walk with each day’s movement in price independent of the previous day’s price change. Every day, the stock can either go up or down by $3, each with 50% probability. The stock is currently valued at $60.

8. Random Walk Probability What is the probability that the value of the stock two days from now will be $60? The stock price two days from now will be $60 if the price path is either (up, down) or (down, up) Pr(up, down) = Pr(down, up) = 25% Pr(price = $60) = 2 * 25% = 50%

9. Call Option and Random Walk What is the present value of a European call option with an expiration date two days from now if the exercise price of the option is $62? Assume a daily discount rate of 0.05%, with daily discounting. Pr(value ≤ $62) = 3/4 Pr(value > $62) = 1/4 (Only up, up) PV = 1/4 * (66 - 62)/(1.0005) 2 = $0.99900

10. Put Option and Random Walk A put option has an exercise price of $53, and this option expires three days from today. What is the probability that this option will have positive value on the expiration date? Only down, down, down will lead to a price < $53 Pr(down, down, down) = (1/2) 3 = 1/8 Pr(down, down, down) = 12.5%

11. Cost of Equity and WACC Trackety’s Trains currently has $300,000 of stock issued, with no bonds. The current cost of equity is 10%. If the company sells $100,000 of bonds and uses this money to buy back $100,000 worth of stock, what is the new cost of equity? Assume that the cost of debt is 1% and that there are no other securities issued by Trackety’s Trains. You can also assume that the weighted average cost of capital is constant.

12. Cost of Equity and WACC R S = R 0 + B/S * (R 0 – R B ) R S = 10% + 1/2 * (10% – 1%) R S = 10% + 1/2 * (9%) R S = 14.5%

13. Cost of Equity and WACC Alternative Method: Before bond sale/stock purchase: R WACC = 0 * R B + 300,000/300,000 * R S R WACC = 10% After bond sale/stock purchase: S = 300,000 – 100,000 = 200,000 B = 100,000 10% = 1/3 * 1% + 2/3 * R S R S = 14.5%

14. Dividends Harptonia is a company that sells drinks with harps on the front label. Harptonia’s dividends are paid as follows: Dividends are paid every 4 months, with the next dividend to be paid 4 months from now. The next 3 dividend payments will be $1 per share. Each subsequent dividend payment will be 15% higher than the dividend payment made one year before.

15. Dividends If we assume that this company will pay dividends forever, what is the present value of this stock if the stated annual discount rate is 20%, compounded every 4 months? 4-month rate = 20%/3 = 6.66667% EAR = (1.06667) 3 – 1 = 21.36296% Year 1: PV = 1/1.06667 + 1/(1.06667) 2 + 1/(1.06667) 3 = $2.6404 PV = 2.6404 + 2.6404*1.15/(.21363-.15) PV = $50.36 Annual equivalent of 3 payments

16. Dividends Alternate Method: 3 annuities with annual payments that grow by 8%, but whose start dates are 4 months, 8 months, and 12 months Annuity with 1 st payment in 4 months: PV = 1/(.21363-.15) * (1.06667) 2 Annuity with 1 st payment in 8 months: PV = 1/(.21363-.15) * (1.06667) Annuity with 1 st payment in 12 months: PV = 1/(.21363-.15) Total PV = $50.36

17. Portfolio Standard Deviation Stock 1 has an 8% annual rate of return if state A occurs, 11% if state B occurs, and 20% if state C occurs. Stock 2 has a 15% annual rate of return if state A occurs, 8% if state B occurs, and 7% if state C occurs. Assume all 3 states occur with equal probability. What is the standard deviation of a portfolio that has 50% of money invested in stock 1 and 50% invested in stock 2?

18. Portfolio Standard Deviation Expected return Stock1 = (.08+.11+.2)/3 =.13 Expected return Stock2 = (.15+.08+.07)/3 =.1 Var Stock1 = 1/3 * [(.08-.13) 2 + (.11-.13) 2 + (.2-.13) 2 ] = 1/3 * [.0078] =.0026 Var Stock2 = 1/3 * [(.15-.1) 2 + (.08-.1) 2 + (.07-.1) 2 ] = 1/3 * [.0038] =.0012667 Cov 1,2 = 1/3 * [(.08-.13)(.15-.1) + (.11-.13)(.08-.1) + (.2-.13)(.07-.1)] Cov 1,2 = 1/3 * [-.0042] = -.0014

19. Portfolio Standard Deviation Variance of a portfolio: Var = (1/2) 2 (.0026) + 2(1/2)(1/2)(-.0014) + (1/2) 2 (.00126667) =.00065 –.0007 +.00031667 Var =.0002667 s.d. of portfolio = (.0002667) 1/2 = 1.6330%

20. Call Option Itty Bitty Ball Bell stock could have value of $50, $55, $60, or $65 two years from today. Each outcome occurs with equal probability. If a European call option with an exercise price of $58 and expiration date two years from today has a present value of $1.80, what is the effective annual discount rate of this option?

21. Call Option Exercise call option if price at expiration is $60 or $65 (prob of each is 1/4) $1.80 = 1/4 * (60-58)/(1+r) 2 + 1/4 * (65-58)/(1+r) 2 $1.80 = 1/4 * 1/(1+r) 2 * (2 + 7) (1+r) 2 = 9/4 * 1/1.8 = 1.25 1+r = 1.11803 r = 11.803%

22. College Savings Suppose that you are advising a couple with one child about how much they need to save for college. The child is currently 8 years old, and will start college at age 18. The first payment for college will be $50,000, to be paid 10 years from today. Subsequent annual payments of $50,000 each will be made until the child is 21 years old. The effective annual interest rate is 12%.

23. College Savings If the couple made a deposit of $X today into the account, this will be exactly enough to cover all of the child’s college expenses. Find X. PV CollegeCosts = 50,000/(1.12) 10 + 50,000/(1.12) 11 + 50,000/(1.12) 12 + 50,000/(1.12) 13 PV CollegeCosts = 16,098.66 + 14,373.81 + 12,833.75 + 11,458.71 PV CollegeCosts = $54,764.93

24. Growing & Constant Dividends A stock will pay a dividend of $1 later today. Over the next 10 years, the annual dividend will go up by 8% each year. After that, the dividend will remain constant forever. What is the present value of this stock if the effective annual discount rate is 10%?

25. Growing & Constant Dividends Div’d, year 0 = $1 Div’d, year 10 = 1 * (1.08) 10 = $2.1589 Years 0-9: 10 payment growing annuity (shifted 1 year earlier because 1 st payment in year 0) Years 10+: perpetuity with payment of $2.1589, discounted by 9 years because 1 st payment in year 10

26. Growing & Constant Dividends PV = 1/(.10-.08) * [1 – (1.08/1.10) 10 ] * 1.10 + 1(1.08) 10 /.10 * 1/(1.10) 9 PV = 50 * (1 -.832359) * 1.10 + 21.5892 * 1/2.35795 PV = 9.22025 + 9.15595 = 18.3762 PV = $18.38

27. Growing & Constant Dividends Alternate Method: Years 1-10: 10 payment growing annuity Years 11+: perpetuity with payment of $2.1589, discounted by 10 years because 1 st payment in year 11

28. Growing & Constant Dividends PV = 1 + 1.08/(.10-.08) * [1 – (1.08/1.10) 10 ] + 1(1.08) 10 /.10 * 1/(1.10) 10 PV = 1 + 9.0526 + 8.32359 = 18.3762 PV = $18.38