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
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Using discrete price distribution:
201 possible values $54.05 - $56.05 (inclusive)
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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
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OR using continuous uniform distribution:
Total area of any pdf must be 1
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Base of rectangle is 2, so height must equal 1/2
FV1 = 100 *
0.05 𝑥
𝑑𝑥
0
2
= 100 * [(0.05)2/4 –
(0)2/4] = 100 * .0025/4 = $0.0625
56.05 𝑥−56
𝑑𝑥)
56
2

(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 yintercept is 0.06 with a standard error of
0.005.
Interpreting SML Regressions:
Risk-Free Return
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(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)
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Risk premium = slope
R = 6% + 10%
R = 16%
Interpreting SML Regressions:
Confidence Interval

(c) What is the 95% confidence interval for
the y-intercept?

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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

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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

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
(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
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(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
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(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
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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

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If the effective annual interest rate for this
stock is 11%, what is the present value for
Growing Annuity
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 Discount by 6 years
Annuity
(constant
PV = 15 + 16.50(5.2851) + 83.67
dividends)
PV = $185.87