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Econ 134A Fall 2012
Test 2 solution sketches
Average: 41.68 points
What counts as 100%: 54.55 points
(2 students with 55 points; this also counts as 100%)
Joe Izu takes out a car loan of
$50,000 today

Joe Izu takes out a car loan of $50,000 today.
He makes 72 monthly payments of $1,000
each, starting one month from today. He also
makes 2 additional payments in order to fully
pay off the loan. One of these payments will be
2 months from today and one will be 84
months from today. The payment 84 months
from today will be twice the amount of the
payment 2 months from today. How much will
the final payment be if the stated annual
discount rate is 18%, compounded monthly?
Joe Izu takes out a car loan of
$50,000 today


Monthly rate is 0.18/12 = 0.015 = 1.5%
PV of $1,000 payments


PV of other 2 payments


(1,000/0.015)(1 – 1/1.01572) = $43,844.67
$50,000 - $43,844.67 = $6,155.33
Let Y be the payment 84 months from
now


6,155.33 = (Y/2)/1.0152 + Y/1.01584
Solve for Y to get $7,976.83
Solve each of the following

(a) An investment portfolio has annual
returns of 10%, –60%, 45%, 14%, 9%,
2%, and 30% over each of seven years.
What is the geometric average return over
this seven-year period?


Take the seventh root of
(1.1)(0.4)(1.45)(1.14)(1.09)(1.02)(1.3) to get
1.00716208
The geometric average is 1.00716208 – 1, or
0.00716208
Solve each of the following

(b) Stella will receive 4 payments. Each
payment will be $1,000 every six
months, starting six months from today.
The effective annual discount rate is
13%, and interest is compounded
continuously. What is the total present
value of the 4 payments?
Solve each of the following

(b) Stella will
receive 4
payments. Each
payment will be
$1,000 every
six months,
starting six
months from
today. The
effective
annual discount
rate is 13%,
and interest is
compounded
continuously.
What is the
total present
value of the 4
payments?



Two ways to find the rate every 6 months
 Since the effective rate is 13% annually, we can just
take the square root of 1.13 – 1, or 6.30146%
 Find the stated rate, which is ln(1.13), or 12.2218%;
then take exp(0.5*.0122218) – 1, or 6.30146%
PV = 1000/(1.0630146) + 1000/(1.0630146)2 +
1000/(1.0630146)3 + 1000/(1.0630146)4 = $3,441.33
OR
PV = 1000/(1.13)1/2 + 1000/1.13 + 1000/(1.13)3/2 +
1000/(1.13)2 = $3,441.33
Solve each of the following

(c) A perpetuity pays $5,000 every
three years, starting one year from
today. What is the present value of this
perpetuity if the effective annual
discount rate is 16%?


Effective rate every 3 years is (1.16)3 – 1 =
56.0896%
PV = (5,000/0.560896) * (1.16)2 =
$11,995.09
Solve each of the following

(d) There are three states of the world, each
with one-third probability of occurring: High,
Medium, and Low. When times are High, Stock
X has a rate of return of 35%, and stock Y has
a rate of return of 3%. When times are
Medium, Stock X has a rate of return of 24%
and stock Y has a rate of return of 12%. When
times are Low, Stock X has a rate of return of
7% and stock Y has a rate of return of 11%.
What is the correlation of Stock X and Stock Y?
Solve each of the following

(d) There are three
states of the world,
each with one-third
probability of
occurring: High,
Medium, and Low.
When times are
High, Stock X has a
rate of return of
35%, and stock Y
has a rate of return
of 3%. When times
are Medium, Stock
X has a rate of
return of 24% and
stock Y has a rate
of return of 12%.
When times are
Low, Stock X has a
rate of return of 7%
and stock Y has a
rate of return of
11%. What is the
correlation of Stock
X and Stock Y?


Find the arithmetic average of each stock:
22% for Stock X and 8.67% for Stock Y
Find the covariance of X and Y




(1/3) [(.35-.22)(.03-.0867) + (.24-.22)(.12-.0867)
+ (.07-.22)(.11-.0867), or –0.0034
σX2 = (1/3)[(.35-.22)2 + (.24-.22)2 +
(.07-.22)2]  σX = 0.11518
σY2 = (1/3)[(.03-.0867)2 + (.12-.0867)2 +
(.11-.0867)2]  σY2 = 0.040277
Corr(X,Y) = –0.0034/σXσY = – 0.7329
Solve each of the following

(e) Suppose that the daily price for each share of
Alominyo, Inc., stock is a random walk with each day’s
movement in price independent of the previous day’s.
Every day, the stock can either go up with probability
60% or down by $1 with probability 40%. However, over
the past five days, the stock has gone up by $1 every day.
What is the probability that the stock will be the same
price two days from today?




Two possibilities: (up, down) or (down, up)
P(up, down) = .4 * .6 = .24
P(down, up) = .6 * .4 = .24
Total probability is .24 + .24, or .48
Solve each of the following

(f) The ZipDoodle machine can be
purchased today for $5,500, and lasts 6
years. Maintenance costs of $700 have to
be incurred three times. The first
maintenance cost occurs 18 months from
today, the second 3 years from today, and
the third 54 months from today. If the
effective annual discount rate is 21%,
what is the equivalent annual cost of the
machine? (Note: All costs are in real
dollars.)
Solve each of the following

(f) The ZipDoodle machine can be purchased today for $5,500, and lasts 6
years. Maintenance costs of $700 have to be incurred three times. The first
maintenance cost occurs 18 months from today, the second 3 years from
today, and the third 54 months from today. If the effective annual discount
rate is 21%, what is the equivalent annual cost of the machine? (Note: All
costs are in real dollars.)


Total cost = 5500 + 700/1.213/2 +
700/1.213 + 700/1.219/2 = $6,717.92
EAC  6717.92 = (C/.21)[1 – 1/1.216] 
C = $2,070.48
Leo’s Batons, Inc.

Leo’s Batons, Inc., has the following
characteristics: The beta for the company
is 1.6; the annual dividend of $6 will be
paid later today; the annual dividend will
go up by 4% each year. You may also find
the following information useful in solving
this problem: Dividends for this stock will
be paid forever; the rate of return for riskfree assets is 3%; the rate of return to the
market is 8%. What is the present value of
a share of Leo’s Batons stock?
Leo’s Batons, Inc.



Leo’s Batons, Inc., has the following characteristics: The beta
for the company is 1.6; the annual dividend of $6 will be paid
later today; the annual dividend will go up by 4% each year.
You may also find the following information useful in solving
this problem: Dividends for this stock will be paid forever; the
rate of return for risk-free assets is 3%; the rate of return to
the market is 8%. What is the present value of a share of
Leo’s Batons stock?
Return = risk-free rate + beta * market premium
= 3% + 1.6(8% – 3%) = 11%
PV = 6 + 6(1.04)/(0.11 – 0.04) = $95.14
Stock Q and Stock K

Suppose that Stock Q and Stock K have a
correlation value of ρ = –1. Stock Q has
an expected return of 5% and standard
deviation 10%. Stock K also has an
expected return of 5% and standard
deviation 10%. Today, each stock is
valued at $150 per share. Over the next
year, Stock K will go up by $5. How much
will Stock Q go up by next year? (Please
completely justify your answer to get full
credit.)
Stock Q and Stock K





Suppose that Stock Q and Stock K have a correlation value of ρ = –1. Stock Q has an expected
return of 5% and standard deviation 10%. Stock K also has an expected return of 5% and
standard deviation 10%. Today, each stock is valued at $150 per share. Over the next year, Stock
K will go up by $5. How much will Stock Q go up by next year? (Please completely justify your
answer to get full credit.)
If someone invests $150 in each stock, then XQ = XK = 0.5
σQK = Corr(Q,K) * s.d.(Q) * s.d.(K) = –1 * 0.1 * 0.1 = –0.01
Variance of a portfolio with $150 invested in each stock is
2
2
2 2
 XQ σQ + 2XQXK σQK + XK σK =
.52 * .12 + 2 * .5 * .5 * (–.01) + .52 * .12 = 0
Since the variance of the portfolio is 0, then investing in one
share of each stock guarantees a return of 5%
 5% of $300 is $15
 If Stock K has a return of $5, then the return of Stock Q must
be $10
Level of difficulty

Easy (34 points)





Joe Izu
Geometric average
A perpetuity pays $5K
every 3 years…
ZipDoodle
Leo’s Batons

Easy-medium (5
points)


Hard (8 points)



…3 states of the
world…
Stella
Alominyo, Inc.
Very hard (8 points)

Stock Q/Stock K