Test 2 solution sketches
Note for multiple-choice
questions: Choose the closest
answer
Average Equity Risk Premium


In 1985, the country of Urce had average bond
returns of 2.5%, followed by 3.6% in 1986 and
1.6% in 1987. The average return on stocks in
these years was 5.7%, -3.8%, and 10.5%.
What was the average equity risk premium in
Urce over this 3-year period? (Use the arithmetic
mean.)



Avg stock return=(5.7-3.8+10.5)/3 = 4.133%
Avg bond return=(2.5+3.6+1.6)/3 = 2.567%
Difference = 4.133 – 2.567 = 1.566%
PV of Stock with Dividends

Fire Top House Pizza stock will pay
dividends of $2 per share every year,
starting 6 months from today. What is
the PV of the stock if the effective
annual discount rate is 8%?

PV = 2/.08 * 1.08 = $25.98
PV of Stock with Growing
Dividend

Organ Power Chicken will not pay its first
dividend until 5 years from today. This
dividend will be $1, with each
subsequent payment increasing by
8.5%. What is the PV of the stock if the
effective annual discount rate is 15%?

PV = [1 / (1.15)4] * [1 / (.15-.085)] = $8.80
CAPM Expected Return

Treehouse of Antacid stock has a beta
value of 2.5 and an expected return of
10%. The risk-free rate of return is
currently 1.25%. What is the expected
return on the market?



10% = 1.25 + 2.5 * (X – 1.25)
8.75 = 2.5X – 3.125
X = 4.75%
Geometric Average Return

A portfolio of stock is worth $500 today.
It was worth $450 one year ago, $440
two years ago, and $435 three years
ago. What is the geometric average
rate of return over the last three years?

(500/435)1/3 – 1 = 4.75%
Lottery Payments

The Saw Mill Gum lottery will pay Elianna
$100,000 today. Each subsequent payment
will be $10,000 higher than the one before.
Her final payment will be $140,000. What is
the PV of these payments if her effective
annual discount rate is 14%?


PV = 100,000 + 110,000/1.14 + 120,000/(1.14)2
+ 130,000/(1.14)3 + 140,000/(1.14)4
PV = $459,464
Profitability Index


Bumble Bloop canned sardines buy a
machine that costs $50,000 today. The
machine leads to positive cash flows in the
future, starting at $5,000 in one year and
subsequent annual cash flows that increase
by 5% forever.
What is the profitability index of this machine
if the effective annual discount rate is 15%?


PV of cash flows = 5000/(.15-.05)=$50,000
P.I. = 50,000/50,000 = 1
Real Rate of Return with
Inflation

According to the CPI, a bundle that cost
$1,000 in 2011 would cost $1,020.69 in
2012. If an investment received a
nominal rate of return of 5% between
2011 and 2012, what is the real rate of
return?




Inflation = (1020.69–1000)/1000 = 2.069%
(1 + real)*(1 + inflation) = 1 + nominal
(1 + real)* 1.02069 = 1.05
Real = .028716 = 2.87%
Balloon Payment

Clayton is taking out an interest-only
home loan for $100,000 today. (His
monthly payments will only cover the
interest.) He will make 120 monthly
payments starting in one month, and a
final balloon payment 10 years from
today.
Balloon Payment

How much will this balloon payment be
if the stated annual interest rate is
12%, compounded monthly?


Balance at the beginning of each month is
$100,000
Balloon payment is $100,000
Growing Savings

Danica is trying to save up enough
money for her son’s racing lessons 10
years from now. The PV of the costs of
the racing lessons is $10,000. She will
save $X one year from today, and
increase this amount by 5% each year
for a total of 9 years. The total savings
will be exactly enough to cover the
racing lessons.
Growing Savings

Find X if the effective annual discount
rate is 8%.




$10,000=X * 1/(r - g) * [1–((1+g)/(1+r))9]
$10,000=X * 1/(.08-.05) * [1–(1.05/1.08)9]
$10,000=7.46499 * X
X = $1,339.59
PV of Growing Annuity

The current date is May 22, 2013.
Today, Benson will deposit $500 into a
bank account that earns 5% effective
annual interest. In the future, he will
make annual deposits on the same date
each year. The next deposit will be
$1,050, and each deposit growing by
5%.
PV of Growing Annuity

In what year will Benson’s account have
a PV of $8,500?

2013
2014
2015
…
2012


Growing annuity formula will not work (r=g)
Year
Year
Year
…
Year
0
1
2
8
PV
PV
PV
…
PV
= $500
= $1050/1.05 = $1000
= $1050*1.05/(1.05)2 = $1000
= $1000
PV = 500 + 8*1000 = $8,500
Answer is May 22, 2021
Stock Value

Almond Tar Fireplaces will pay quarterly
dividends of $5 every 3 months,
starting 2 months from now. What is
the PV of this stock if the effective
annual discount rate is 15%?


Quarterly rate = (1.15)1/4 – 1 = 3.55581%
Monthly rate = (1.15)1/12 – 1 = 1.17149%
Stock Value

If the first dividend were paid in 3
months:


PV = $5/.0355581 = $140.62
Since the first dividend is in 2 months:

PV = 140.62 * 1.0117149 = $142.26
Internal Rates of Return

There is a potentially-profitable gold
mine in the Purple Elephant Hills. The
company would have to pay $200
million today to open the mine. One
year from today, all of the gold
extracted will be sold for $450 million.
Two years from today, costs of $252
million must be paid to seal the mine.
There are no other costs or benefits.
IRR: Part (a)

Find all internal rates of return for the
mine if it is opened.



(In $millions)
0 = -200 + 450/(1+r) – 252/(1+r)2
Simplifies to: 0 = -100r2 + 25r – 1
−25± 252 −4(−100)(−1)
2(−100)

r=

r = .05 or .20
=
−25± 225
−200
IRR: Part (b)

For what discount rates should the mine
be opened? Show all work to justify
your answer.


Answer: 0.05 < r < 0.20
Option 1:
Equation for NPV (0=-100r2+25r-1) has
a=-100<0, which means it is a parabola
that opens down (positive NPV is between
the two roots).
IRR: Part (b)


Option 2:
Choose values in each range (r < .05,
.05 < r < .2, r > .2) and show what the
NPV is for those discount rates.
Example:



r=0  NPV = -$2 million
r=.1  NPV = $0.826 million
r=1  NPV = -$38 million