Final exam solution sketches

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Final exam solution sketches. 1:00 Lecture, Version A Note for multiple-choice questions: Choose the closest answer. Value of an Option.

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### Final exam solution sketches

1:00 Lecture, Version A

Note for multiple-choice questions: Choose the closest answer

Value of an Option
• Road End Records is currently valued at \$50 per share. The stock will go up by \$5 or down by \$1 every year, starting one year from today. Each year, the stock will go up in value each year with 50% probability and down with 50% probability, and each change in value is independent of other changes in value. You currently own a European call option with a \$60 exercise price, with an expiration date 50 months from today.
Value of an Option
• What is the present value of this option if the appropriate effective annual discount rate for the option is 15%?
• 4 price changes, at months 12, 24, 36, and 48
• Outcomes that have positive value (i.e. final price>\$50): UUUU, UUUD, UUDU, UDUU, DUUU
• 4 ups: Prob=.54=.0625; Share price=50+4*2=\$70
• 3 ups, 1 down: Prob=4*.54=.25; Price=50+3*5-1=\$64
• PV = .0625 * (70-60)/(1.1550/12) + .25 * (64-60)/(1.1550/12)
• PV = \$0.9077
Growing Dividends
• Goodbye Charlie stock will pay dividends annually, starting 5 months from today. The next dividend will be \$0.50, and will increase by 7% every subsequent year forever. What is the present value of this stock if the effective annual interest rate is 10%?
Growing Dividends
• PV if next payment in 1 year:
• \$0.50/(.1 - .07) = \$16.67
• Add 7 months interest to each payment:
• \$16.67 * (1.1)7/12 = \$17.62
Stock Returns
• Dya Glob stock is currently selling for \$78 per share. Over the last 5 years, the value of the stock went up by 6%, down by 10%, up by 18%, up by 2%, and up by 24%.
Stock Returns: Standard Deviation
• (a) What is the standard deviation of the sample of the changes in value over the last 5 years?
• Mean = 1/5*(.06 - .1 + .18 + .02 + .24)= .08
• Var = 1/4 * [(.06-.08)2 + (-.1-.08)2 + (.18-.08)2 + (.02-.08)2 + (.24-.08)2]Var = 1/4 * [.0004 + .0324 + .01 + .0036 + .0256] = 1/4 * 0.072 = 0.018
• S.D. = (0.018)1/2 = 0.1342 = 13.42%
Stock Returns: Calculating Past Prices
• (b) How much was Dya Glob stock worth 5 years ago?
• (1.06)(0.9)(1.18)(1.02)(1.24) * X = \$78
• 1.42381 * X = 78
• X = \$54.78
Stock Returns:Confidence Intervals
• (c) What is the 95.4% confidence interval for the rate of return of this stock, based on the changes over the last 5 years? (Hint: To find the 95.4% confidence interval, you must be within 2 standard errors.)
• S.E.=s.d./(n1/2)= 13.42% / 51/2 = 6%
• C.I. = mean ± 2 * S.E. = 8% ± 2 * 6%
• C.I. = (-4%, 20%)
Loan Amortization
• Amy will borrow \$800,000 today to start her own television channel, the Smile Channel International. The money will be amortized with equal payments made every 2 years for the next 40 years. (Note that the first payment will be made 2 years from today.) If the effective annual interest rate is 14% for the loan, how much will each payment be to completely repay the loan over the next 40 years?
Loan Amortization
• Note: 20 payments every 2 years
• Rate every 2 years = 1.142 – 1 = .2996
• 800,000 = C/.2996 * [1 – 1/1.299620]
• 800,000 = 3.3201 * C
• C = \$240,956
Bond Yields
• Robert is quoted a price for a bond of \$1,000. This bond has a face value of \$1,300. Three coupons of 5% each will be paid. The first coupon will be paid later today, the second coupon will be paid 6 months from today, and the third will be paid 1 year from today. If the bond matures 1 year from today, what is the yield on this bond (expressed as an effective annual discount rate)?
Bond Yields
• Let X = 1+r, where r is a 6-month rate
• Coupon payments = .05 * 1300 = \$65
• 1000 = 65 + 65/X + 1365/X2935X2 – 65X – 1365 = 0187X – 13X – 273 = 0
• X = =
• X = 1.2435, -1.1740 (rate<0 not reasonable)
• 6-month rate r = .2435
• EAR = (1.2435)2 – 1 = 0.5463
Graphing Option Value
• Charlotte buys one put option with an exercise price of \$80 (per share) today, one call option with an exercise price of \$60 (per share), and one share of stock currently valued at \$70. The expiration date of all of these options is six months from now. Each option is for buying or selling one share. For simplicity in this problem, you can assume that the discount rate is 0%.
Graphing Option Value
• Draw a well-labeled graph that shows the value of a combination of the two options and one share of stock, as a function of the value of the stock at expiration. The vertical intercept should have the value of the combination of the assets. The horizontal intercept should have the value of the stock on the expiration date.
Graphing Option Value
• Let X = stock price on the expiration date
• If X<60:
• Call value = 0
• Put value = 80 – X
• Stock value = X
• Total = 80
• If 60<X<80:
• Call value = X – 60
• Put value = 80 – X
• Stock value = X
• Total = X + 20
• If X>80:
• Call value = X – 60
• Put value = 0
• Stock value = X
• Total = 2X – 60
Graphing Option Value

Combined Value of Assets

Slope = 2

Slope = 1

Stock Value at Expiration

Solving for Beta
• (a) The expected return on a security is 20%. The risk-free rate is assumed to be 10%. The expected return on the market is 13%. Use the CAPM model derived in class to determine the beta of this security.
• 20% = 10% + β* (13% - 10%)
• 20% = 10% + 3% * β
• 10% = 3% * β
• β = 3.333
Converting Discount Rates
• (b) If the stated annual discount rate is 8%, compounded every three months, what is the effective discount rate every 7 months?
• 3-month rate = 2%
• EAIR = (1.02)4 – 1 = 0.082432
• 7-month rate = (1.082432)7/12 – 1 = 4.7281%
Future Stock Prices
• (c) Streaky Striker Stock has just paid out its annual dividend of \$5 earlier today. The dividend will go up by 10% every year forever. What will the price of the stock be 1 year from today if the effective annual discount rate is 15%? (Note: Provide the price AFTER the dividend has been paid.)
• FV1 = 5(1.1)2 / (.15 – .1) = \$121