# IPM_Quiz2_a__Solution Spring 2012

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```					            BUS 431: INVESTMENT AND PORTFOLIO MANAGEMENT
SPRING 2012, AUBG

Quiz 2(a)
Solution Guide

Problem 1 (5 points)
You are face with probability distribution of the HPR on the stock market index fund
given in Spreadsheet 5.1 in the textbook. Suppose that the risk-free interest rate is 6% per
year. You are contemplating investing \$107.55 in a 1-year CD and simultaneously buying
a call option on the stock market index fund with an exercise price of \$110 and
expiration of 1 year. What is the probability distribution of your dollar return at the end
of the year?

Solution:
The probability distribution of the dollar return on CD plus call option is:

Year-end                        Ending          Total
State of the                                        Ending
Probability     price of                        Value of     (combined)
Economy                                           Value of CD
stock                          Call           Value
Excellent             0.25         126.50          \$ 114.00       \$16.50         \$130.50
Good                  0.45         110.00          \$ 114.00       \$ 0.00         \$114.00
Poor                  0.25          89.75          \$ 114.00       \$ 0.00         \$114.00
Crash                 0.05          46.00          \$ 114.00       \$ 0.00         \$114.00

Remember that the cost of the index fund is \$100 per share, and the cost of the call option
is \$12.

Problem 2(5 points)
You manage a risky portfolio with expected return of 18% and standard deviation of
28%. The T-bill rate is 8%. Suppose that your risky portfolio includes the following
investment in the given proportions: Stock A – 25%; Stock B: 32%; and Stock C – 43%.
What are the investment proportions of your client’s overall portfolio, including the
proportions in T-bills? What is the reward-to-variability ratio (S) of your risky portfolio?

Solution:
Investment proportions:                  30.0% in T-bills

Risky portfolio           0.7 × 25% = 17.5% in Stock A
0.7 × 32% = 22.4% in Stock B
0.7 × 43% = 30.1% in Stock C

.18  .08
Your reward-to-volatility ratio: S               0.3571
.28

1
Problem 3 (10 points): Consider the two securities listed below:
 Risky security: E(R) = 10%, σ = 20%.
 Risk-free security: Rf = 5%.
You wish to form a portfolio combining the risky security and the risk-free security such
that you earn an expected return of 15%.
a. What weights would you need to place in the risky and risk-free securities to
earn a 15% expected return? What is the standard deviation of this portfolio?
b. Draw the capital allocation line (CAL). Label the points and the axes clearly.
What is the reward-to-variability ratio?
Now, suppose that instead of one risky security and one risk-free security, you can invest
in two risky securities (bond and stock mutual funds) as follows: Security 1: E(R) = 8%,
σ1 = 12%; Security 2: E(R) = 13%, σ2 = 20%, and the correlation coefficient between
them is 0.3.
c. Find the expected return and the standard deviation of the minimum-variance
portfolio (MVP) on the investment opportunity set. Draw a tangent from the
risk-free rate to the investment opportunity set.

Solution:
1. Let w = weight in the risky security and (1 - w) = weight in the risk-free security.
Then, w10%  (1  w)5%  15%  5w  10% . Therefore, w = 2.0, and (1 – w) =
−1.0. A negative value for (1 - w) indicates that the investor would borrow money at
the risk-less rate and invest that money in the risky asset.
2. The standard deviation  p  2.0  20 %  40 % . Therefore, reward-to-variability ratio
= (15 - 5)/40.0 = 0.25. This is just the slope of CAL. It can also be computed as (10 -
5)/20 = 0.25. Here I expect to see a graph presenting CAL (see the Figure 6.4 in the
text).
3. From the parameters of the opportunity set (standard deviation and correlation
coefficient) we generate the covariance matrix:

Security 1         Security 2
Security 1                  144                 72
Security 2                   72                400

The minimum-variance portfolio is constructed so that:

 2  Cov(rD, rE )              400  72
wD                                                         0.82
E

    2Cov(rD, rE )
2
D
2
E                      144  400  (2  72)

w E  1  w D  1  0.82  0.18 (D = bonds mutual fund, E = stock mutual fund)

(Security 1 = bond mutual fund; Security 2 = stock mutual fund)

Its expected return and standard deviation are:

E(rP )  0.82 (8%)  0.18(13 %)  8.90 %

2
 P  [(0.82) 2 (12%)2  (0.18) 2 (20%)2  2(0.82)(0.12)(0.30)(12%)(20%)]1 / 2 
 P  11.45%

Here I expect to see a graph for the investment opportunity set (see the Figure 7.7 in
the text)

3

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