Download Fin 3710 Investment Analysis

HW 4
Fin 3710 Investment Analysis
Professor Rui Yao
CHAPTER 6: EFFICIENT DIVERSIFICATION
1.
E(rP) = (0.5 × 15) + (0.4 × 10) + (0.10 × 6) = 12.1%
3.
a. The mean return should be equal to the value computed in the spreadsheet. The fund's
return is 3% lower in a recession, but 3% higher in a boom. However, the variance of
returns should be higher, reflecting the greater dispersion of outcomes in the three
scenarios.
b. Calculation of mean return and variance for the stock fund:
(A)
(B)
(C)
Scenario
Probability
Rate of
Return
Recession
Normal
Boom
c.
0.3
-14
0.4
13
0.3
30
Expected Return =
(D)
Col. B
×
Col. C
-4.2
5.2
9.0
10.0
(E)
Deviation
from
Expected
Return
-24.0
3.0
20.0
(F)
Squared
Deviation
(G)
Col. B
×
Col. G
576
9
400
Variance =
Standard Deviation =
172.8
3.6
120.0
296.4
17.22
Calculation of covariance:
(A)
Scenario
Recession
Normal
Boom
(B)
Probability
0.3
0.4
0.3
(C)
(D)
Deviation from
Mean Return
Stock
Fund
-24
3
20
Bond
Fund
10
0
-10
(E)
(F)
Col. C
Col. B
×
×
Col. D
Col. E
-240.0
-72
0.0
0
-200.0
-60
Covariance = -132
Covariance has increased because the stock returns are more extreme in the recession
and boom periods. This makes the tendency for stock returns to be poor when bond
returns are good (and vice versa) even more dramatic.
4.
a.
One would expect variance to increase because the probabilities of the extreme
outcomes are now higher.
b.
Calculation of mean return and variance for the stock fund:
(A)
(B)
(C)
Scenario
Probability
Rate of
Return
Recession
Normal
Boom
c.
0.4
-11
0.2
13
0.4
27
Expected Return =
(D)
Col. B
×
Col. C
-4.4
2.6
10.8
9.0
(E)
Deviation
from
Expected
Return
-20.0
4.0
18.0
(F)
Squared
Deviation
(G)
Col. B
×
Col. G
400
16
324
Variance =
Standard Deviation =
160.0
3.2
129.6
292.8
17.11
Calculation of covariance:
(A)
Scenario
Recession
Normal
Boom
(B)
(C)
(D)
Deviation from
Mean Return
Probability
0.4
0.2
0.4
Stock
Fund
-20
4
18
Bond
Fund
10
0
-10
(E)
(F)
Col. C
Col. B
×
×
Col. D
Col. E
-200.0
-80
0.0
0
-180.0
-72
Covariance = -152
Covariance has increased because the probabilities of the more extreme returns in the
recession and boom periods are now higher. This makes the tendency for stock
returns to be poor when bond returns are good (and vice versa) more dramatic.
6.
The parameters of the opportunity set are:
E(rS) = 15%, E(rB) = 9%, σS = 32%, σB = 23%, ρ = 0.15,rf = 5.5%
From the standard deviations and the correlation coefficient we generate the
covariance matrix [note that Cov(rS, rB) = ρσSσB]:
Bonds
Bonds
Stocks
529.0
110.4
Stocks
110.4
1024.0
The minimum-variance portfolio proportions are:
σ 2B − Cov(rS , rB )
w Min (S) = 2
σ S + σ 2B − 2Cov(rS , rB )
=
529 − 110.4
= 0.3142
1024 + 529 − (2 × 110.4)
wMin(B) = 0.6858
The mean and standard deviation of the minimum variance portfolio are:
E(rMin) = (0.3142 × 15%) + (0.6858 × 9%) = 10.89%
[
σ Min = w S2 σ S2 + w 2B σ 2B + 2w S w B Cov(rS , rB )
]
1
2
= [(0.31422 × 1024) + (0.68582 × 529) + (2 × 0.3142 × 0.6858 × 110.4)]1/2
= 19.94%
% in stocks
00.00
20.00
31.42
40.00
60.00
70.75
80.00
100.00
% in bonds
100.00
80.00
68.58
60.00
40.00
29.25
20.00
00.00
Exp. return
Std dev.
9.00
10.20
10.89
11.40
12.60
13.25
13.80
15.00
23.00
20.37
19.94
20.18
22.50
24.57
26.68
32.00
minimum variance
tangency portfolio
7.
Investment opportunity set
for stocks and bonds
18
CAL
16
S
14
12
10
B
min var
8
6
4
2
0
0
10
20
Standard Deviation (%)
30
40
The graph approximates the points:
E(r)
10.89%
13.25%
Minimum Variance Portfolio
Tangency Portfolio
8.
The reward-to-variability ratio of the optimal CAL is:
E (rp ) − rf
σp
9.
σ
19.94%
24.57%
a.
=
13.25 − 5.5
= 0.3154
24.57
The equation for the CAL is:
E (rC ) = rf +
E (rp ) − rf
σp
σ C = 5.5 + 0.3154σ C
Setting E(rC) equal to 12% yields a standard deviation of 20.61%.
b.
The mean of the complete portfolio as a function of the proportion invested in
the risky portfolio (y) is:
E(rC) = (l − y)rf + yE(rP) = rf + y[E(rP) − rf] = 5.5 + y(13.25− 5.5)
Setting E(rC) = 12% ⇒ y = 0.8387 (83.87% in the risky portfolio)
1 − y = 0.1613 (16.13% in T-bills)
From the composition of the optimal risky portfolio:
Proportion of stocks in complete portfolio = 0.8387 × 0.7075 = 0.5934
Proportion of bonds in complete portfolio = 0.8387 × 0.2925 = 0.2453
10.
Using only the stock and bond funds to achieve a mean of 12% we solve:
12 = 15wS + 9(1 − wS ) = 9 + 6wS ⇒ wS = 0.5
Investing 50% in stocks and 50% in bonds yields a mean of 12% and standard
deviation of:
σP = [(0.502 × 1024) + (0.502 × 529) + (2 × 0.50 × 0.50 × 110.4)] 1/2 = 21.06%
The efficient portfolio with a mean of 12% has a standard deviation of only
20.61%. Using the CAL reduces the SD by 45 basis points.
11.
a.
Although it appears that gold is dominated by stocks, gold can still be an
attractive diversification asset. If the correlation between gold and stocks is
sufficiently low, gold will be held as a component in the optimal portfolio.
b.
If gold had a perfectly positive correlation with stocks, gold would not be a
part of efficient portfolios. The set of risk/return combinations of stocks and
gold would plot as a straight line with a negative slope. (See the following
graph.) The graph shows that the stock-only portfolio dominates any portfolio
containing gold. This cannot be an equilibrium; the price of gold must fall
and its expected return must rise.
12
10
STOCKS
8
6
GOLD
4
2
0
0
10
20
30
Standard Deviation (%)
12.
Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio
can be created and the rate of return for this portfolio in equilibrium will always be
the risk-free rate. To find the proportions of this portfolio [with wA invested in
Stock A and wB = (1 – wA ) invested in Stock B], set the standard deviation equal to
zero. With perfect negative correlation, the portfolio standard deviation reduces to:
σP = Abs[wAσA − wBσB]
0 = 40 wA − 60(1 – wA) ⇒ wA = 0.60
The expected rate of return on this risk-free portfolio is:
E(r) = (0.60 × 8%) + (0.40 × 13%) = 10.0%
Therefore, the risk-free rate must also be 10.0%.