Final exam solution sketches

11:00 Lecture, Version A

Note for multiple-choice questions: Choose the closest

answer

Option Prices Manuela is considering buying a

European call option with an exercise price of $56 today. This option can be used to buy 100 shares of stock at the exercise price. The stock’s current value is $50, and the expiration date of the option is one year from today. From her estimate, the value of the stock will be some value from $54.05-$56.05 on the expiration date.

Option Prices She further estimates that each of these

values will occur with equal probability. (In other words, the probability of $54.05 occurring is the same as $54.06, $54.07, and any other value up to $56.05.) If Manuela is risk-neutral and her effective annual discount rate is 10%, what is the most that Manuela is willing to pay for this option, assuming her estimates are correct?

Option Prices Using discrete price distribution: 201 possible values $54.05 - $56.05

(inclusive) Prices with positive option value: $56.01 -

$56.05 PV = 1/201 * 1/1.1 * 100 * [(56.05-56) +

(56.04-56) + (56.03-56) + (56.02-56) + (56.01-56)]

PV = 1/201 * 1/1.1 * 100 * [.05+.04+.03+.02+.01]

PV = $0.06784

Option Prices OR using continuous uniform

distribution: Total area of any pdf must be 1

Base of rectangle is 2, so height must equal 1/2

FV1 = 100 * = 100 * [(0.05)2/4 – (0)2/4] = 100 * .0025/4 = $0.0625

(Can also find FV = 100 * ) PV = 0.0625/1.1 = $0.05682

Interpreting SML Regressions Betsy has run a regression of the

security market line, with beta and expected returns entered as decimals. (For example, an expected return of 5% is entered as 0.05 for the regression) Based on her sample size, she finds in her regression results that her point estimate for the slope is 0.1 with a standard error of 0.009. Her point estimate for the y-intercept is 0.06 with a standard error of 0.005.

Interpreting SML Regressions: Risk-Free Return (a) What is the estimated rate of return

for a risk-free asset? Explain your answer in 25 words or less.

The y-intercept is the estimated return for a risk-free asset

Rf = 0.06, or 6%

Interpreting SML Regressions: Return for Asset with β=1 (b) What is the estimated rate of return

for an asset with beta equal to 1? (Note: I am referring to the meaning of beta referred to in this class, not the beta meaning that is commonly referred to in econometrics classes.)

β = 1 R= Rf + 1 * (risk premium) Risk premium = slope

R = 6% + 10% R = 16%

Interpreting SML Regressions: Confidence Interval (c) What is the 95% confidence interval

for the y-intercept? If you do not have enough information for a

numeric answer, leave your answer in terms of as few variables as possible, and explicitly state what your variable(s) mean. If helpful, you can assume that the sample size is n.

C.I. = .06 ± t* (0.005) t* is the critical value for a t-distribution

with n-2 degrees of freedom We don’t know t* since we don’t know n

Synergy In 50 words or less, give an example of

a synergy. You need to explain why this example is a synergy within your 50-word limit.

Example: A supermarket finds a lower-cost way to produce jelly. This leads to increased sale of peanut butter, which is synergy.

Real Options Do all real options lead to increased NPV

for a company? Explain why or why not in 50 words or less.

No, some real options are involuntary. Recall the malpractice example from class.

Weighed Average Cost of Capital What is the weighted average cost of

capital for a company that has one-third of its value in stocks, two-thirds of its value in bonds, the rate of return of stocks is 7%, and the rate of return of bonds is 2%?

WACC = B/(B+S) * RB + S/(B+S) * RS WACC = 2/3 * .02 + 1/3 * .07 WACC = 0.3667 = 3.667%

PV with Payments Erick will receive $100 in 14 months.

What is the present value of this payment if Erick’s stated annual discount rate is 14%, compounded every 3 months?

PV = 100/(1.035)14/3 = $85.17

Returns with Different States of the World There are three known states of the

world, Seal, Walrus, and Whale. Each state occurs with one-third probability. When Seal occurs, Stock A has a rate of return of 6% and Stock B has a rate of return of 19%. When Walrus occurs, Stock A has a rate of return of 14% and Stock B has a rate of return of 2%. When Whale occurs, Stock A has a rate of return of 10% and Stock B has a rate of return of 15%.

Returns with Different States of the World (a) What is the average rate of return

for each stock. (Note: Both must be correct to receive credit.)

A: 1/3 * (6% + 14% + 10%) = 10% B: 1/3 * (19% + 2% + 15%) = 12%

Returns with Different States of the World (b) What is the standard deviation for

the rate of return of each stock? VarA = 1/3 * [(.06-.1)2 + (.14-.1)2 +

(.1-.1)2] = 0.001067 s.d.A = (0.001067)1/2 = 0.03266 =

3.266% VarB = 1/3 * [(.19-.12)2 + (.02-.12)2 +

(.15-.12)2] = 0.005267 s.d.B = (0.005267)1/2 = 0.07257 =

7.257%

Returns with Different States of the World (c) What is the correlation coefficient for

the rate of return for these two stocks? Cov = 1/3 * [(.06-.1)(.19-12) +

(.14-.1)(.02-.12) + (.1-.1)(.15-.12)] Cov = 1/3 * [-.0028 + (-.004) + 0] Cov = -0.002267 ρ = cov/(s.d.A*s.d.B) ρ = -0.002267/(0.03266*0.07257) ρ = -0.9563

Internal Rate of Return In the city of Corn Chowder, USA,

Maude has won the local lottery. The lottery promises to pay $1,000 every year forever, starting one year from today. Kleitos has offered to take all of the payments from the lottery in return for a single payment to Maude of $12,000 today. If Maude accepts the offer, what can you say about her effective annual discount rate?

Internal Rate of Return If Maude accepts the offer, then

$12,000 > $1,000/r r > 1,000/12,000 r > 0.0833 or r > 8.33%

PV of Shares of Stock Grubber Baby Food, Inc. is expected to

pay out a dividend of $15 per share later today, followed by 10% annual growth for each of the next 6 years. After that, they will pay constant dividends for each of the next 10 years. After that, Grubber will go out of business and pay nothing else to stockholders. (Note that the last dividend payment will be made 16 years from today.)

PV of Shares of Stock If the effective annual interest rate for

this stock is 11%, what is the present value for each share of stock?

PV = 15 + 15 * 1.1 * 1/(.11-.1) * [1 – (1.1/1.11)6] + 15(1.1)6/.11 * [1 – 1/(1.11)10] * 1/(1.11)6

PV = 15 + 16.50(5.2851) + 83.67 PV = $185.87

Growing Annuity

Annuity (constant dividends)

Discount by 6 years