Final exam solution sketches
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Transcript of Final exam solution sketches
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Final exam solution sketches
Winter 2014, Version A
Note for multiple-choice questions: Choose the closest
answer
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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
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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.
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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%
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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 = $34.771st payment is in 6 months, not 1
year
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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%
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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.
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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%
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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
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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%
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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.
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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%
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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%
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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.
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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
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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
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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?
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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
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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%
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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?
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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%
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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%.
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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
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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%?
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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
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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
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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
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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