Final solution sketches
Note for multiple-choice
questions: Choose the closest
answer
PV of Perpetuity

Jil is due to receive $50,000 every 2
years, forever, starting one year from
today. What is the PV of this perpetuity
if her effective annual discount rate is
7%?



Rate every 2 years = (1.07)2 – 1 = 14.49%
(In $1000’s): 50/.1449 * 1.07 = 369.22
PV = $369,220
Loan Payments



Remi is taking out a loan for a $20,000 car. He will
make a 20% down payment and borrow the rest,
and will pay off the loan with 40 monthly
payments. Each month, he will pay off a constant
amount of the principal. How much will the first
payment be if Remi’s stated annual interest rate is
12%, compounded monthly?
Loan = 0.8 * $20,000 = $16,000
1st payment:



Principal = 16,000/40 = $400
Interest = 16,000 * 0.01 = $160
1st payment = $400 + $160 = $560
Bonds

William’s bond will mature in 10 years.
The amount that he will receive in 10
years will be $8,000. Assume William’s
effective annual discount rate is 5%
over the next 3 years, and 9%
thereafter. What is the PV of this bond?

$8,000 / (1.05)3(1.09)7 = $3780.39
Discount Rates and IRRs

Liam is analyzing a project with cash
flows at multiple times. He knows that if
the discount rate is 14%, the NPV of the
project is positive. If the discount rate is
18%, the NPV is negative. Which
statement MUST be true?

NPV must hit zero somewhere between
14% and 18%, so an IRR must exist
between 14% and 18%
Leverage and Returns

Joel’s Textboks, Inc. is currently an unlevered
company. The current expected return on
equity is 12%. If the cost of debt is 7 an Joel’s
Textbooks decides to move 60% of the value of
the company to bonds, what will the new return
on equity be?




12% = 0.6 * 7% + 0.4 * X
12% = 4.2% + 0.4 * X
7.8% = 0.4 * X
X = 19.5%
Value of Options

Tlinches stock sells for $75/share today.
The value of the stock in one year will
come from a uniform distribution, with
lower bound $71 and upper bound $81.
What is the PV of a European put
option with exercise price $67 and
expiration date one year from today?

The stock will never be below $67 on the
expiration date, so PV = $0
Decreasing Dividends


Silly Spring Stock plans on paying out a dividend of
$1/share today. Each subsequent annual dividend
will be 1 cent less than the previous, until the
dividend paid is 1¢. After the 1¢ dividend is paid,
the company will go out of business. The effective
annual discount rate is 10%. What is the PV of this
stock?
Upper bound on PV is a dividend that goes on
forever and does not decrease


Upper bound: $1 / .1 = $10
All other answer choices are greater than $10, so it is
the closest answer
Calculating Dividends


Slimy Mushroom Music (SMM) stock
currently sells for $50/share. The beta
value for SMM is 1. The risk-free rate of
return is currently 4%, and the risk
premium is currently 7%. SMM will pay a
yearly dividend of $X every year forever,
starting one year from today. What is X?
Discount rate for SMM is .04 + .07 = .11


$50 = X / .11
X = $5.50
Peak Housing Prices

From lecture, in what year did median
housing prices peak in the United
States?


Answer: 2006
From lecture slides
Corporate Regulations

Which of the following was NOT listed
as a corporate regulation in the
textbook?


Answer: Prohibition of initial public
offerings (IPOs) on American stock
markets
We talked about recent IPOs in class
Geometric Average Return

A stock is worth $500/share today. One
year ago, the value was $450/share.
Two years ago, the value was
$400/share. Three years ago, the value
was $350/share. What has been the
geometric average rate of return over
the last three years?

(500/350)1/3 – 1 = 12.625%
Bonds with Changing Yields




A zero-coupon bond is purchased for $1,000 at
11 am today, with a face value of $1,250 to be
paid two years from today. Later today, at 1:30
pm, the yield to maturity (compounded on an
annual basis) changes to 10%. How much does
the value of the bond go up between 11 am
and 1:30 pm today?
Value at 11 am = $1,000
Value at 1:30 pm = 1,250/(1.1)2 = $1,033.06
Bond value went up by $33.06
Portfolio Standard Deviation

Stocks M and N are uncorrelated with
each other. Stock M has an expected
return of 5% and a standard deviation
of 6%. Stock N has an expected return
of 8% and a standard deviation of 10%.
What is the standard deviation of a
portfolio that has 30% of stock M and
70% of stock N?
Portfolio Standard Deviation

Variance = (.3)2(.06)2 + 0 + (.7)2(.1)2
(Since M and N are uncorrelated)

= .000324 + 0 + .0049
= .005224
Standard deviation = .072277
Option Value



U2B4 stock was valued at $60 three years
ago, and has had a geometric average rate
of return of 5% over the past three years.
Savannah bought an American call option
with an exercise price of $68 a few weeks
ago. The option expires today. What is the
current value of the option if Savannah’s
effective annual discount rate is 6%?
Current value = 60 * (1.05)3 = $69.46
Value of option = 69.46 – 68 = $1.46
Project Discount Rates



Janie’s Noodles currently has a companywide beta value of 2. A stock whose beta
is 1 has an expected return of 5%. The
risk-free rate is currently 3%. A new
project that Janie’s Noodles is considering
has a beta value of 1.5. What is the
relevant discount rate for this new project?
Risk premium = 5% - 3% = 2%
X = 3% + 1.5 * (2%) = 6%
Non-annual Discount Rates

If the effective annual discount rate is
10%, what is the effective discount rate
for 9 months?


Rate every 3 months = (1.1)1/4 – 1
= 2.41137%
Rate every 9 months = (1.0241137)3 – 1
= 7.40995%
Probability of Positive Option
Value

A stock is currently priced at $65. The
stock’s price will change by $2 per year
every year, starting 11 months from
today. The stock’s price will go up with
50% probability and down with 50%
probability. Each change in the stock’s
price is independent of all other price
changes.
Probability of Positive Option
Value

Sonia buys a European option with an
exercise price of $70 and an expiration
date in five years. What is the
probability that the option will have a
positive value on the expiration date?



Only times with positive value: UUUUU,
DUUUU, UDUUU, UUDUU, UUUDU, UUUUD
25 = 32 possible outcomes, equal chances
6/32 = 3/16 probability of positive value
Value of Bond Guarantees

Jenny’s Furniture Manufacturing (JFM)
is considering moving away from
California. In order to try to keep JFM in
the state, the government is offering to
guarantee any bond repayment for
bonds issued by JFM this year. JFM’s
normal cost of debt capital is 15%.
Value of Bond Guarantees

If JFM is able to issue $1,000,000 in bonds
at a coupon rate of 9%, what is the value of
this implicit subsidy?



NPV = 1,000,000 – [90,000/1.15 + 90,000/1.152
+ 90,000/1.153 + 90,000/1.154 + 90,000/1.155
+ 1,090,000/1.156]
NPV = 1,000,000 – [78,260.87 + 68,052.93 +
59,176.46 + 51,457.79 + 44,745.91 +
471,237.08
NPV = $227,068.96
Standard Deviation of Stock
Returns

Suppose the returns of a stock over a four-year
period are 15%, -6%, 19%, and 22%. Find the
standard deviation of this sample.





Arithmetic average = (15+(-6)+19+22) / 4 =
12.5%
Variance= 1/3 * [(.15-.125)2 + (-.06-.125)2 +
(.19-.125)2 + (.22-.125)2]
Variance= 1/3 * [.000625 + .034225 + .004225 +
.009025] = 1/3*[.0481] = .01603
Standard deviation = (.01603)1/2 = .126623
Standard deviation = 12.6623%
Annuity Payments





An annuity will pay $5 one year from today,
$(5+Y) two years from today, $(5+2Y) three
years from today, and $(5+3Y) four years from
today. The PV of this annuity is $15, and the
effective annual discount rate is 20%. Find Y.
5/1.2 + (5+Y)/1.22 + (5+2Y)/1.23 +
(5+3Y)/1.24 = 15
12.9437 + 3.29861*Y = 15
3.29861*Y = 2.0563
Y = 0.623384
Equivalent Annual Cost

The Jack Crock Griller costs $6,000 to
purchase today and lasts for 9 years.
Annual maintenance costs are required.
The first annual cost will be $100 one
year from today. Each subsequent
maintenance cost will be 6% higher
than the previous year’s cost. The final
cost will be 8 years from today.
Equivalent Annual Cost

What is the equivalent annual cost if the
stated annual discount rate is 12%,
compounded every six months?




EAIR = (1 + .12/2)2 – 1 = 12.36%
PV of costs = 6000 + 100*[1/(.1236-.06) –
1/(.1236-.06) * (1.06/1.1236)8] = $6,585.30
EAC: 6585.30 = X/.1236 * [1 – 1/(1.12369)]
6585.30 = 5.25612 * X
X = $1,252.88
Funding Future Withdrawals

Wendy has two financial objectives. She
would like to withdraw $500,000 every
year for 30 years, starting 5 years from
now. She would like to donate
$3,000,000 to a charity 30 years from
today. Wendy will make a single deposit
in 3 years to exactly fund these goals.
How much will she have to deposit if
the EAIR = 4%?
Funding Future Withdrawals

First objective:
FVYear 4 = 500,000/.04 * [1 – 1/(1.0430)]
= $8,646,016.65



FVYear 3 = 8,646,016.65/1.04 =
$8,313,477.55
Second objective:
FVYear 3 = 3,000,000/(1.0427) =
$1,040,449.71
Total in year 3 = $9,353,927.26