Final exam solution sketches
Winter 2014, Version A
Note 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?


PVcash flows = 1250/(1.1)3 = 939.14
P.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

Find the LOWER BOUND of the 95.44%
confidence interval of the historical
equity risk premium.


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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
1st payment is in 6 months, not 1 year
CAPM

If the market return is 20%, the riskfree 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 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.
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) = 3/4
Pr(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, down will lead to a
price < $53
Pr(down, down, down) = (1/2)3 = 1/8
Pr(down, down, down) = 12.5%
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.
Cost of Equity and WACC




RS
RS
RS
RS
=
=
=
=
R0 + B/S * (R0 – RB)
10% + 1/2 * (10% – 1%)
10% + 1/2 * (9%)
14.5%
Cost of Equity and WACC


Alternative Method:
Before bond sale/stock purchase:



After bond sale/stock purchase:



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RWACC = 0 * RB + 300,000/300,000 * RS
RWACC = 10%
S = 300,000 – 100,000 = 200,000
B = 100,000
10% = 1/3 * 1% + 2/3 * RS
RS = 14.5%
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.
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?



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
4-month rate = 20%/3 = 6.66667%
EAR = (1.06667)3 – 1 = 21.36296%
Year 1: PV = 1/1.06667 + 1/(1.06667)2 +
Annual equivalent
1/(1.06667)3 = $2.6404
of 3 payments
PV = 2.6404 + 2.6404*1.15/(.21363-.15)
PV = $50.36
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
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

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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
Portfolio Standard Deviation
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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%
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





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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%
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%.
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
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


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
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
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
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
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