Money & Capital Markets
Fall 2007
Assignment #3
Due: September 13th Selected Answers
1. (20 points) In your own words and without any math, write an intuitive description of
the results and practice of mean-variance analysis.
No answer provided. I want you to really think about how to state mean-variance
rather than memorizing a statement. In other words, how would you describe this
to a friend?
2. (20 points) In your own words and without any math, write an intuitive description of
the separation theorem.
No answer provided.
3. (10 points) A three-asset portfolio has the following characteristics:
Assets
X
Y
Z
E(r)
15%
10%
6%
Standard Deviation
22%
8%
3%
Weight
.50
.40
.10
What is the expected return on this three-asset portfolio?
Recall the expected rate of return on a portfolio is merely the weighted average of
the individual returns (the weights being the proportion of the individual stock in
the portfolio).
E(r) = (.5)(15) + (.4)(10) + (.1)(6) = 12.1%
The standard deviations would be used if we wanted to calculate the variance of the
portfolio.
4. (10 points) A pension fund manager is considering two mutual funds. The first is a
stock fund. The second is a long-term government and corporate bond fund. The
information for each fund is the following:
Fund
Expected Return
Standard Deviation
Stock
20%
30%
Bond
10%
20%
The correlation between the funds is .20 (which happens to be a good historical
approximation to the actual correlation between stocks and bonds by the way). What is
the expected return and standard deviation of a portfolio composed of:
a. Only the stock fund
b.
c.
d.
e.
f.
25% stock fund and 75% bond fund
50% stock fund and 50% bond fund
75% stock fund and 25% bond fund
Only the bond fund
Graph your results with the standard deviation on the horizontal axis and expected
return on the vertical axis.
We use two equations for this problem. First, the expected rate of return on the
portfolio is merely the weighted average of the expected rates of return for the stock
and bond:
E (rP )  Ws E (rs )  WB E (rB )
where the subscript P is for the Portfolio, S for Stock, and B for Bond. In a two
asset case, such as this one, the weights could be written as
WS  1  WB
at times this simplifies things. Second, we need the equation for the variance (and,
standard deviation) for the portfolio. We know that this is generally NOT merely
the weighted average of the individual variances – I know, I know, I keep saying this
in class but it is important. The formula is
 P2  (WS  S ) 2  (WB B ) 2  2(WS  S )(WB B ) 
where the Greek letter rho (at the very end of the formula) is the correlation
coefficient. Now, to solve this problem we simply substitute in the given rates of
returns, standard deviations, correlation coefficient, and weights. The weights are
the only input that we change going from parts (a) to (d). We did the 75% in stock/
25% in bond case in class. The following are the answers.
Allocation
Stocks
Bonds
Portfolio
Return
Portfolio
Risk
1
0
20
30.00
0.95
0.9
0.85
0.8
0.05
0.1
0.15
0.2
19.5
19
18.5
18
28.72
27.47
26.26
25.11
0.75
0.25
17.5
24.01
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.3
0.35
0.4
0.45
0.5
0.55
0.6
17
16.5
16
15.5
15
14.5
14
22.97
22.00
21.11
20.31
19.62
19.04
18.59
0.35
0.3
0.65
0.7
13.5
13
18.27
18.09
0.25
0.75
12.5
18.06
0.2
0.15
0.1
0.05
0.8
0.85
0.9
0.95
12
11.5
11
10.5
18.18
18.44
18.83
19.36
0
1
10
20.00
Efficient Frontier: Correlation = .20
21
19
Return
17
15
13
11
9
7
5
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
Risk
5. (10 points) Draw the familiar efficiency frontier with many possible assets. Now,
considering your personality, preferences, etc. – demonstrate with an indifference curve
the portfolio that you might choose. Explain your choice and the shape of your
indifference curve. Ask someone else how they view the trade-off between risk and
reward, attempt to discover the shape of their indifference curve and draw it on your
graph. Explain their portfolio choice.
No answer provided. The point is for you to think where you would choose to be on
the efficient frontier. By doing so, I hope you’ll begin to consider how this decision
is made (e.g., ‘I’m young, so I have plenty of time to wait out downturns and need
not be so concerned with a bad year here and there. Thus, I choose a portfolio on
the frontier with high expected returns and willing to accept the high risk associated
with it.’).
6. (10 points) Suppose a retired person is currently holding 100% of her wealth in the
stock of a very safe and mature corporation. The stock of this company (call it Stock 1)
has an expected rate of return of 10% with a standard deviation of 10%. An investment
manager advises our retired person to hold 80% of her wealth in stock 1 and 20% in a
stock 2. This stock 2 has been issued by a new, high-tech corporation. Stock 2 has an
expected rate of return of 30% with a standard deviation 50%. The correlation
coefficient between the two stocks is -.5.
a. If the retired person follows the investment manager’s advice (80% in stock 1 and
20% in stock 2), what will be the expected rate of return and standard deviation of the
resulting portfolio?
b. Illustrate the probable resulting graph – with the standard deviation on the horizontal
axis and expected return on the vertical axis - as the retired persons varies the percentages
of each stock held in her portfolio. Make sure to indicate where stock 1, stock 2, and the
suggested portfolio lie on the schedule in your graph.
c. Briefly evaluate whether the investment manager’s advice was good or bad.
a. A portfolio composed of 80% stock 1 and 20% stock 2 will have the following
expected rate of return and standard deviation.
E (rP )  (.8)(.10)  (.2)(.30)  .14  14%
 P2  (.8 * .10) 2  (.2 * .50) 2  2(.8)(.2)(.10)(.50)( .5)  84
 P  84  9.17%
Just for reference, portfolios with varying weights are:
Allocation
Stock 1
Stock 2
Portfolio
Return
Portfolio
Risk
100%
0%
10
10.00
95%
5%
11
8.53
90%
10%
12
7.81
85%
15%
13
8.05
80%
20%
14
9.17
75%
25%
15
10.90
70%
30%
16
13.00
65%
35%
17
15.32
60%
40%
18
17.78
55%
45%
19
20.32
50%
50%
20
22.91
45%
55%
21
25.55
40%
60%
22
28.21
35%
65%
23
30.90
30%
70%
24
33.60
25%
75%
25
36.31
20%
80%
26
39.04
15%
85%
27
41.77
10%
90%
28
44.51
5%
95%
29
47.25
0%
100%
30
50.00
b. The efficient frontier looks like the following.
Efficient Frontier
30
Return
25
20
15
10
5
5.00
15.00
25.00
35.00
45.00
Risk
The 80%/20% portfolio is on the upper portion of the efficient frontier, just after
the curve and right before it hits the 15% return.
c. I’d say it was very good advice. The investment manager was able to increase the
expected rate of return while decreasing risk – moving the client nearly straight
upwards from where they began.
7. (10 points) An investment manager has formed the efficient frontier by using the
mean-variance technique. The manager has two individuals in his office seeking advice
on investing. The first individual, Ms. Betty, has recently taken her first job as a lawyer
in a leading law firm and looks forward to many productive years in her career. The
second individual, Mr. Alan, is a few years away from retirement after a long career as a
partner in the law firm.
a. Using mean-variance analysis demonstrate graphically and explain the reasoning
behind the type of portfolio the investment manager would likely suggest to each
individual.
b. Later, after returning from a week long training session on the Separation Theorem,
the investment manager realizes that he has given the wrong advice. Assuming a riskfree asset exists, what will be the likely new advice of the investment manager to the two
individuals? Illustrate the advice with a new graph.
c. Even later, the risk-free rate declines. Demonstrate graphically the likely impact on
the two individual’s suggested portfolio’s of the decline in the risk-free rate.
a. Alan would be on the efficient frontier with lower risk and lower return than
Betty.
Return
Betty
Alan
Risk
b. With the introduction of the risk-free rate, both would hold the portfolio Z for
the risky part of their complete portfolio. Alan would put only a portion of his
entire wealth or investment in portfolio Z (e.g., 40%) and the remainder in the riskfree asset. Betty would put more than her entire wealth or investment in portfolio
Z. Hence, she borrows at the risk-free rate in order to buy more of portfolio Z. In
each case, the investors are able to do better than the efficient frontier.
Return
CAL
Betty
Z
Alan
rf
Risk
c. A decrease in the risk-free rate will move the vertical intercept down and make
the CAL steeper.
8. (10 points) A security analyst has been studying a particular corporation’s stock in
order to decide whether to issue a “buy” or “sell” order to his clients. The analyst has
been able to (1) estimate that next periods dividend will be $5, (2) the corporation paysout a constant 50% of its earnings in dividends, (3) the earnings and dividends should
grow at a 5% rate, and (4) 15% would be a proper discount rate for this stock. The
security analyst plans to use this information to apply the Gordon model (Dividend
Discount Model) to determine the proper price of the stock.
a. If the current market price of the stock is $55, should the analyst advise clients to buy
or sell this stock.
b. At the current market price of $55, what is the rate of return on the stock?
c. According to the Gordon model, what should the price-earnings ratio be for this stock?
a. The value of the stock according to the Gordon model is $50. Hence, the analyst
should not advise the client to buy at $55.
V 
$5
 $50
15%  5%
b. The rate of return is only 14%. Again, thinking about the discount rate as the
return we require from this stock, then we again advise not to buy.
r
D
$5
g
 5%  9%  5%  14%
P
$55
c. Here, we simply rearrange the Gordon model a bit.
P
D
bE


kg kg
P
b
50%
50%



5
E k  g 15%  5% 10%
The point is to observe what the price-earnings ratio depends upon (i.e., dividend
payout ratio, discount rate, and growth rate).