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## Test 1 solution sketches - econ.ucsb.edu

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Final exam solution sketches
Winter 2014,Version ANote for multiple-choice questions: Choose the closest answer
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?PVcashflows= 1250/(1.1)3= 939.14P.I. = 939.14/900 = 1.0435
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.
Confidence Interval
Findthe 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%
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
1stpayment is in 6 months, not 1 year
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%
Random Walk
Use the following informationto answer the nextthree questions:Suppose that the daily price of each share ofWibbyPig 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.
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%
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) = 3/4Pr(value > \$62) = 1/4 (Only up, up)PV = 1/4 * (66 - 62)/(1.0005)2= \$0.99900
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 aprice < \$53Pr(down, down, down) = (1/2)3= 1/8Pr(down, down, down) =12.5%
Cost of Equity and WACC
Trackety’sTrains 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 byTrackety’sTrains. You can also assume that the weighted average cost of capital is constant.
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%
Cost of Equity and WACC
Alternative Method:Before bond sale/stock purchase:RWACC= 0 * RB+ 300,000/300,000 * RSRWACC=10%After bond sale/stock purchase:S = 300,000 – 100,000 = 200,000B = 100,00010% = 1/3 * 1% + 2/3 * RSRS= 14.5%
Dividends
Harptoniais a company that sells drinks with harps on the front label.Harptonia’sdividends 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 madeone year before.
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=\$2.6404PV = 2.6404 + 2.6404*1.15/(.21363-.15)PV = \$50.36
Annual equivalent of 3 payments
Dividends
Alternate Method:3 annuities with annual payments that grow by 8%, but whose start dates are 4 months, 8 months, and 12 monthsAnnuity with 1stpayment in 4 months:PV = 1/(.21363-.15) * (1.06667)2Annuity with 1stpayment in8months:PV = 1/(.21363-.15) * (1.06667)Annuity with 1stpayment in12months:PV = 1/(.21363-.15)Total PV = \$50.36
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?
Portfolio Standard Deviation
Expected returnStock1= (.08+.11+.2)/3 = .13ExpectedreturnStock2=(.15+.08+.07)/3 = .1VarStock1= 1/3 * [(.08-.13)2+ (.11-.13)2+ (.2-.13)2] = 1/3 * [.0078] = .0026VarStock2= 1/3 *[(.15-.1)2+(.08-.1)2+(.07-.1)2] = 1/3 * [.0038] = .0012667Cov1,2= 1/3 * [(.08-.13)(.15-.1) + (.11-.13)(.08-.1) +(.2-.13)(.07-.1)]Cov1,2=1/3 * [-.0042] = -.0014
Portfolio Standard Deviation
Variance of a portfolio:Var= (1/2)2(.0026) + 2(1/2)(1/2)(-.0014) + (1/2)2(.00126667) = .00065– .0007 + .00031667Var= .0002667s.d.of portfolio = (.0002667)1/2= 1.6330%
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?
Call Option
Exercise call option if price at expiration is \$60 or \$65 (probof 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 = 1.251+r = 1.11803r = 11.803%
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. Subsequentannual paymentsof \$50,000 each will be made until the child is 21 years old. The effective annual interest rate is 12%.
College Savings
If the couple made a deposit of \$Xtoday into the account, this will be exactly enough to cover all of the child’s college expenses. FindX.PVCollegeCosts= 50,000/(1.12)10+50,000/(1.12)11+50,000/(1.12)12+ 50,000/(1.12)13PVCollegeCosts= 16,098.66 + 14,373.81 + 12,833.75 + 11,458.71PVCollegeCosts= \$54,764.93
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%?
Growing & Constant Dividends
Div’d, year 0 = \$1Div’d, year 10 = 1 * (1.08)10= \$2.1589Years 0-9: 10 payment growing annuity (shifted 1 year earlier because 1stpayment in year 0)Years 10+: perpetuity with payment of \$2.1589, discounted by 9 years because 1stpayment in year 10
Growing & Constant Dividends
PV = 1/(.10-.08) * [1 – (1.08/1.10)10]* 1.10 + 1(1.08)10/.10 * 1/(1.10)9PV = 50 * (1 - .832359) * 1.10 + 21.5892 * 1/2.35795PV = 9.22025 + 9.15595 = 18.3762PV = \$18.38
Growing & Constant Dividends
Alternate Method:Years 1-10: 10 payment growing annuityYears 11+: perpetuity with payment of \$2.1589, discounted by 10 years because 1stpayment in year 11
Growing & Constant Dividends
PV = 1 + 1.08/(.10-.08) * [1 – (1.08/1.10)10] + 1(1.08)10/.10 * 1/(1.10)10PV = 1+ 9.0526 + 8.32359 =18.3762PV = \$18.38

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