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Final exam solution sketches

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

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Final exam solution sketches

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  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? • PVcash 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 1st 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, downwill 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 • RS = R0 + B/S * (R0 – RB) • RS = 10% + 1/2 * (10% – 1%) • RS = 10% + 1/2 * (9%) • RS = 14.5%

  13. Cost of Equity and WACC • Alternative Method: • Before bond sale/stock purchase: • RWACC = 0 * RB + 300,000/300,000 * RS • RWACC = 10% • After bond sale/stock purchase: • S = 300,000 – 100,000 = 200,000 • B = 100,000 • 10% = 1/3 * 1% + 2/3 * RS • RS = 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 1st payment in 4 months: • PV = 1/(.21363-.15) * (1.06667)2 • Annuity with 1st payment in 8 months: • PV = 1/(.21363-.15) * (1.06667) • Annuity with 1st 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 returnStock1 = (.08+.11+.2)/3 = .13 • Expected returnStock2= (.15+.08+.07)/3 = .1 • VarStock1 = 1/3 * [(.08-.13)2 + (.11-.13)2 + (.2-.13)2] = 1/3 * [.0078] = .0026 • VarStock2= 1/3 * [(.15-.1)2+ (.08-.1)2+ (.07-.1)2] = 1/3 * [.0038] = .0012667 • Cov1,2 = 1/3 * [(.08-.13)(.15-.1) + (.11-.13)(.08-.1) + (.2-.13)(.07-.1)] • Cov1,2 = 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. • PVCollegeCosts = 50,000/(1.12)10 + 50,000/(1.12)11+ 50,000/(1.12)12 + 50,000/(1.12)13 • PVCollegeCosts= 16,098.66 + 14,373.81 + 12,833.75 + 11,458.71 • PVCollegeCosts= $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 1st payment in year 0) • Years 10+: perpetuity with payment of $2.1589, discounted by 9 years because 1st 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 1st 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

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