EAST MIDDLE TECHNICAL UNIVERSITY
DEPARTMENT OF BUSINESS ADMINISTRATION
SPRING 2010
BA 5802 FINANCIAL MANAGEMENT
Dr. Danolu
Problem Set #1
Due on Wednesday, March 31, 2010
1. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an
expected return of 10% with a standard deviation of 30%.
a. In light of the apparent inferiority of gold with respect to both mean return and volatility, would
anyone hold gold? If so, demonstrate graphically why one would do so.
Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold
as a part of a portfolio. If the correlation between gold and stocks is sufficiently low, gold will be held
as a component in a portfolio, specifically, the optimal tangency portfolio.
25.00
Efficient
OptimalCAL
frontier
20.00
P
Stocks
15.00
10.00
Gold
5.00
0.00
0
10
20
30
40
StandardDeviation(%)
b. Given the data above, re-answer part (a) with the additional assumption that the correlation
coefficient between gold and stocks equals 1.0. Draw a graph illustrating why one would or would
not hold gold in ones portfolio. Could this set of assumptions for expected returns, standard
deviations, and correlation coefficients represent an equilibrium for the financial markets?
If the correlation between gold and stocks equals +1, then no one would hold gold. The optimal
portfolio would be comprised of bills and stocks only. Since the set of risk/return combinations of
stocks and gold would plot as a straight line with a negative slope (see the following graph), these
combinations would be dominated by the stock portfolio. Of course, this situation could not persist. If
no one desired gold, its price would fall and its expected rate of return would increase until it became
sufficiently attractive to include in a portfolio.
25
20
18
Stocks
15
10
Gold
5
r
f
0
0.00
10.00
20.00
30.00
40.00
Standard Deviation(%)
2.
Fora two - asset portfolio :
[
22
P = w11 + w22 + 2w1w2r1,212
22
]
1/2
Assume that the following two investment classes are available in the market:
Asset
Debt
Equity
E(ri)
0.08
0.13
i
0.12
0.20
Determine the portfolio standard deviation and expected return if you invest the following weights
in the two investment classes and when r D,E = +1.00, or rD,E = +0.50, or rD,E = 0.00, or rD,E = 0.50,
or rD,E = 1.00:
Case
A
B
C
D
E
F
G
H
I
J
K
w1
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
w2
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Tabulate your results and graph the (expected return, standard deviation) pairs with expected
returns on the vertical axis and standard deviations on the horizontal axis. In the graph, also show
the location of the individual assets. Comment on the graph.
Debt
Equity
E(ri)
0.08
0.13
0.12
0.20
Corr(D.E) = +1.00
Portfoli
o
A
B
C
D
E
F
G
H
I
J
K
wD
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
wE
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
E(rP)
0.130
0.125
0.120
0.115
0.110
0.105
0.100
0.095
0.090
0.085
0.080
P
0.20
0.19
0.18
0.18
0.17
0.16
0.15
0.14
0.14
0.13
0.12
Corr(D.E) = +0.50
Portfoli
o
A
B
C
D
E
F
G
H
I
J
K
wD
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
wE
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
E(rP)
0.130
0.125
0.120
0.115
0.110
0.105
0.100
0.095
0.090
0.085
0.080
P
0.20
0.19
0.17
0.16
0.15
0.14
0.13
0.13
0.12
0.12
0.12
Corr(D.E) = 0.00
Portfoli
o
A
B
C
D
E
F
G
H
I
J
K
wD
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
wE
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
E(rP)
0.130
0.125
0.120
0.115
0.110
0.105
0.100
0.095
0.090
0.085
0.080
P
0.20
0.18
0.16
0.14
0.13
0.12
0.11
0.10
0.10
0.11
0.12
Corr(D.E) = -0.50
Portfoli
o
A
B
C
D
E
F
G
H
I
J
K
wD
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
wE
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
E(rP)
0.130
0.125
0.120
0.115
0.110
0.105
0.100
0.095
0.090
0.085
0.080
P
0.20
0.17
0.15
0.13
0.10
0.09
0.08
0.07
0.08
0.10
0.12
Corr(D.E) = -1.00
Portfoli
o
A
B
C
D
E
F
G
H
I
J
K
wD
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
wE
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
E(rP)
0.130
0.125
0.120
0.115
0.110
0.105
0.100
0.095
0.090
0.085
0.080
P
0.20
0.17
0.14
0.10
0.07
0.04
0.01
0.02
0.06
0.09
0.12
3. The correlation coefficients between pairs of stocks are as follows:
Corr(A,B) = 0.85
Corr(A,C) = 0.60
Corr(A,D) = 0.45
Each stock has an expected return of 8% and a standard deviation of 20%.
a. If your entire portfolio is comprised of stock A and you can add some of only one other stock to
your portfolio, which stock, B, C, or D, would you choose? Explain your choice.
The correct is choice Stock C. Intuitively, we note that since all stocks have the same expected
rate of return and standard deviation, we choose the stock that will result in lowest risk. This is the
stock that has the lowest correlation with Stock A.
b. Would your answer to part (b) change for more risk-averse or more risk-tolerant investors?
Explain.
No, the answer to part (b) would not change, at least as long as investors are not risk lovers. Risk
neutral investors would not care which portfolio they held since all portfolios have an expected
return of 8%.
c. Suppose that in addition to investing in one more stock, you can invest in T-bills as well. Would
you change your answer to parts (a) and (b) if the T-bill rate is 8%? Explain.
No, the answers to parts (a) and (b) would not change. The efficient frontier of risky assets is
horizontal at 8% since each stock has the same expected return but different standard deviations.
Therefore, the new efficient frontier runs from the risk-free rate through this linear frontier. The
best portfolio is again, the one with the lowest variance. The optimal complete portfolio (the one
that is formed by investing in both the risk-free asset and the best risky portfolio) depends on the
individual investors risk aversion.
4. Suppose that the market can be described by the following three sources of systematic risk with
associated risk premiums:
Factor
Industrial production (I)
Interest rates (R)
Consumer confidence
(C)
Risk Premium (%)
1.24
0.91
1.03
Factor Beta
1.2
0.5
0.3
Calculate the equilibrium rate of return on this stock using the APT. The current market estimate
for the stocks expected return is 10% and the T-bill rate is 6%. Is the stock under- or over-priced?
Explain.
APT required return = r = 6 + (1.24 x 1.2) + (0.91 x 0.5) + (1.03 x 0.3) = 8.252 %
Since the stocks expected return, 10%, is higher than its required return, 8.252%, this stock is
currently underpriced. The demand for the stock will increase, pushing its price upward and its
expected return downward, until the expected return is just equal to the required return.
5. After graduation, you start working for an investment bank and your banks current portfolio of
$2 million is invested as follows:
Percent of
Total
Expected
Annual
Return
Annual
Standard
Deviation
$200,000
10
4.6%
1.6%
600,000
30
12.4%
19.5%
1,200,000
60
16.0%
29.9%
$2,000,000
100
13.8%
23.1%
Value
Short-term bonds
Domestic large-cap
equities
Domestic small-cap
equities
Total Portfolio
Your bank soon expects to receive an additional $2 million and plans to invest the entire amount in
an index fund that best complements the current portfolio. You are evaluating the four index funds
shown in the following table for their ability to produce a portfolio that will meet two criteria relative
to the current portfolio: (1) maintain or enhance expected return, and, (2) maintain or reduce
volatility.
Each fund is invested in an asset class that is not substantially represented in the current portfolio.
Index Fund Characteristics
Expected Annual
Standard
Deviation
(%)
25
Correlation of
Returns with Current
Portfolio
Fund A
Expected
Annual
Return
(%)
15
Fund B
11
22
+0.60
Fund C
16
25
+0.90
Fund D
14
22
+0.65
Index Fund
+0.80
State which fund you should recommend to your bank. Justify your choice by describing how
your chosen fund best meets both of your banks criteria. (No calculations required)
Fund D represents the single best addition to complement your bank's current portfolio, given the
selection criteria. First, Fund Ds expected return (14.0 percent) has the potential to increase the
portfolios return somewhat. Second, Fund Ds relatively low correlation with the current portfolio
(+0.65) indicates that Fund D will provide greater diversification benefits than any of the other
alternatives except Fund B. The result of adding Fund D should be a portfolio with approximately
the same expected return and somewhat lower volatility compared to the original portfolio.
The other three funds have shortcomings in terms of either expected return enhancement or
volatility reduction through diversification benefits. Fund A offers the potential for increasing the
portfolios return, but is too highly correlated to provide substantial volatility reduction benefits
through diversification. Fund B provides substantial volatility reduction through diversification
benefits, but is expected to generate a return well below the current portfolios return. Fund C has
the greatest potential to increase the portfolios return, but is too highly correlated with the current
portfolio to provide substantial volatility reduction benefits through diversification.