Final solution sketches
Note for multiple-choice
questions: Choose the closest
answer
Effective Annual Returns

Suppose a house was worth $540,000
on October 1, 2012. The same house
was worth $652,000 on May 1, 2013.
What was the effective annual rate of
return between these two dates?


Monthly rate = (652,000/540,000)1/7 – 1 =
2.729%
Yearly rate = (1.02729)12 – 1 = 38.1404%
Cash Flows

Dave is investing in a stuffed bug toy
company. He knows the following: if he
invests $1,000 today, he will receive $X
every year, forever, starting two years
from now. His effective annual discount
rate is 12%. The IRR for this project is
9%. Find X.
Cash Flows



1,000 = X/.09 * 1/1.09
X = 98.1
Note that EAIR is irrelevant here.
PV of Stock

A stock pays a $6 dividend every four
months, starting two months from
today. If the stated annual interest rate
is 15%, compounded monthly, what is
the PV of each share of stock?
PV of Stock



Rate every 4 months = (1.0125)4 – 1 =
5.0945%
Rate every 2 months = (1.0125)2 – 1 =
2.5156%
PV = 6/.050945 * 1.025156 = $120.74
Profitability Index

In order to produce new sunglasses,
Highway Jailhouse Sunglasses must pay
$10,000 today in costs. They will have a
net profit of $3,000 every year, starting
two years from today. What is the
profitability index of this project if the
effective annual discount rate is 15%?
Profitability Index


PVbenefits = 3000/.15 * 1/1.15 =
$17,391.30
PI = 17,391.30 / 10,000 = 1.73913
Option Values

For the next 3 questions: Angela has
purchased three put options with an
exercise price of $50, and four call
options with an exercise price of $45. All
options can be exercised for one share
of stock, and the expiration date for all
options will be one month from today.
The stated annual interest rate is 12%,
compounded monthly.
Option Values

On the expiration date, what is the total
value of the seven options if the stock
price will be $40?



Calls: worthless
Puts: 3 * (50 – 40) = $30
Total = $30
Option Values

On the expiration date, what is the total
value of the seven options if the stock
price will be $48?



Calls: 4 * (48 – 45) = $12
Puts: 3 * (50 – 48) = $6
Total = $18
Option Values

Angela predicts that the value of the
stock could be $46, $47, or $48, each
with 1/3 probability. What is the
expected value of the option if this
assessment is true?



FV = 1/3*[4*(46-45) + 3*(50-46)] +
1/3*[4*(47-45) + 3*(50-47)] +
1/3*[4*(48-45) + 3*(50-48)]
FV = 1/3 * (16 + 17 + 18) = $17
PV = 17 / 1.01 = $16.83
Standard Deviation of Stock
Returns

A sample of stocks has rates of return
of 5%, 10%, and 9%. Find the
standard deviation of this sample.

Arithmetic average = (.05 + .1 + .09) / 3
= .08
Standard Deviation of Stock
Returns




Variance= 1/2 * [(.05-.08)2 + (.1-.08)2 +
(.09-.08)2]
Variance= 1/2 * [.0009 + .0004 + .0001]
= .0007
Standard deviation = (.0007)1/2 = .02646
Standard deviation = 2.646%
Internal Rates of Return

If Jackson invests in a company, he will
pay $5,000 one year from today, and
he will receive $7,000 four years from
today. What is the internal rate of
return for this investment if Jackson’s
effective annual discount rate is 8%?



5000 / (1 + IRR) = 7000 / (1 + IRR)4
(1 + IRR)3 = 1.4
IRR = 11.869%
Continuously-compounded
Annuities

Heyward is set to receive $1,000 per
year for three years, starting one year
from today. His stated annual discount
rate is 5%, compounded continuously.
What is the PV of the three payments?

1000 / e.05 + 1000 / e.05*2 + 1000 / e.05*3 =
$2,716.77
Growing Dividends

Neon Yellow Paper, Inc. is expected to
pay a dividend of $2.50 per share later
today. The dividend is expected to go
up by 20% each of the next three
years, followed by a constant dividend
forever after. What is the PV of this
stock if the effective annual interest
rate for this stock is 14%?
Growing Dividends





PV0 = $2.50
PV1 = 2.50*1.2/1.14 = $2.6316
PV2 = 2.50*(1.22)/(1.142) = $2.7701
PV3 = 2.50*(1.23)/.14 * 1/(1.142) =
$23.744
Total PV = $31.645
Decreasing Dividends

Old Yucker stock is expected to pay its
next dividend of $1 per share one year
from today. The dividend will fall by 5%
each year forever. What is the PV of
this stock if the effective annual
discount rate is 6%?

1 / (.06-(-.05)) = 1/.11 = $9.0909
Correlated Returns

There are three known states of the
world, each with 1/3 probability: good,
okay, and bad. QWE stock has a 20%
return in the good state, 9% in okay,
and -2% in bad. ZXC stock has a 15%
return in good, 6% in okay, and 3% in
bad. What is the correlation of these
two stocks?
Correlated Returns


Note that there is a strong positive
correlation (better state  higher return)
AvgQWE = (20 + 9 – 2) / 3 = 9%


SDQWE = [1/3 * (.2-.09)2 + (.09-.09)2 +
(-.02-.09)2]1/2 = .0898146
AvgZXC = (15 + 6 + 3) / 3 = 8%

SDZXC = [1/3 * (.15-.08)2 + (.06-.08)2 +
(.03-.08)2]1/2 = .0509902
Correlated Returns


Cov(QWE, ZXC) = 1/3 *
[(.2-.09)(.15-.08) + (.09-.09)(.06-.08) +
(-.02-.09)(.03-.08)] = .0044
Corr = .0044 / (.0898146 * .0509902) =
0.960769
Equivalent Annual Cost

A new machine currently costs $500.
The first required maintenance cost will
be $25 one year from now. Each
subsequent year, the maintenance cost
doubles, until the last cost is paid in
four years. What is the equivalent
annual cost of the machine if the
effective annual discount rate is 10%?
Equivalent Annual Cost



PVcosts = 500 + 25/1.1 + 50/1.12 +
100/1.13 + 200/1.14 = $775.784
EAC: 775.784 = EAC/.1 * [1 – 1/(1.16)]
775.784 = 4.35526 * EAC
EAC = $178.126
Portfolio Expected Return/Risk

Two assets have zero correlation.
Suppose that you invested in a portfolio
of these assets that is NOT in the
efficient set. Which of the following is
possible if you reallocate the amount
of money you have invested in each
asset?
Portfolio Expected Return/Risk

Answer: Increased expected return at the
same level of risk AND increased expected
on the portfolio with decreased risk, NOT
same expected return with decreased risk
Opportunity Set
B
Yes
Yes
No
A
Efficient Set
Sharpe Ratio

In the Jacksonvilla stock market, the
average annual risk premium was 10%
in the 1970s, 15% in the 1980s, and
11% in the 1990s. The variance of the
risk premium over this 30-year period
was 0.04. What is the Sharpe ratio over
this three-decade span?
Sharpe Ratio



Avg risk premium = (.1 + .15 + .1) / 3
= .12
SD of risk premium = (.04)1/2 = 0.2
Sharpe ratio = .12 / .2 = 0.6
Profitability Index

Paulina Poundrock is considering a new
project that requires a $50,000
investment today. The next cash flow
will occur in two years, a positive
$10,000 cash flow. In three years, there
will be a positive $18,000 cash flow.
Each subsequent cash flow will be 8%
higher than the previous year’s.
Profitability Index

What is the profitability index of this
project if the effective annual discount
rate is 14%?


PVbenefits = 10,000/1.142 + 18,000/1.143
+ 18,000 * 1.08 / (.14-.08) * 1/1.143
= 7695 + 12,149 + 218,691
= $238,535
PI = 238,535 / 50,000 = 4.7707
Option Values

DPR stock sells for $100. Every month,
the stock value goes up by $5 with 60%
probability and down by $4 with 40%
probability. The last time the price
changed was yesterday. Laura is
considering buying a European call
option with exercise price of $108 and
expiration date in four months. Her
stated annual discount rate is 12%.
Option Values


What is the most that Laura is willing to
pay for this option?
Four possible ways for positive value:



UUUU ($120) with prob. = .64 = .1296
UUUD/UUDU/UDUU/DUUU ($111) with
prob. = 4 * .63 * .4 = .3456
PV of option = 1/1.014 * [.1296*(120-108)
+ .3456*(111-108)] = $2.4909
Random Walk Stock Value

For the next two questions: Arrowjones
stock is currently priced at $55. Over
the next year, the value of the stock
could go up by 20% (with probability
50%), up by 10% (with probability
25%), or go down by 5% (with
probability 25%).
Random Walk Stock Value

What is the standard deviation of the
rate of return over the next year?



Avg= .2(.5) + .1(.25) + (-.05)(.25)=.1125
Variance = 1/2*[(.2-.1125)2] + 1/4*[(.1.1125)2 + 1/4*[(-.05-.1125)2] = .01046875
SD = 0.102317 = 10.2317%
Random Walk Stock Value

Assume that the effective annual
discount rate is 10%. A European call
option with expiration date one year
from today has a current risk-neutral
value of $2. What is the exercise price
of the option?
Random Walk Stock Value

Possible stock values in 1 year:


Let X = exercise price




50% prob of $66, 25% prob of $60.50, 25% prob
of $52.25
.5 * (66 – X) / 1.10 = 2
66 – X = 4.40
X = $61.60
Note that an exercise price below $60.50 will
lead to more than $2 risk-neutral value
Calculating Dividends

RTT, Inc. will pay a dividend of $X later
today. The dividend will increase by
$0.50 each of the next 3 years. After
that, it will increase by 5% each year
forever. The effective annual discount
rate is 10%. What is X if the current
value of RTT is $50 per share?
Calculating Dividends




50 = X + (X+.5)/1.1 + (X+1)/1.12 +
(X+1.5)/1.13 + (X+1.5)(1.05)/(.1-.05) *
1/(1.13)
50 = 3.48685X + 2.40796 + (1.05X +
1.575)/.06655
50 = 19.2645X + 26.0744
X = 1.2420