CHAPTER 06 - EFFICIENT DIVERSIFICATION
6.
a. Without doing any math, the severe recession is worse and the boom is better.
Thus, there appears to be a higher variance, yet the mean is probably the same
since the spread is equally larger on both the high and low side. The mean return,
however, should be higher since there is higher probability given to the higher
returns.
b. Calculation of mean return and variance for the stock fund:
(A)
(B)
(D)
(E)
(F)
Deviation
Col. B
from

Rate of
Expected Squared
Col. C
Scenario
Probability Return
Return Deviation
Severe recession
0.05
-40
-2
-51.2 2621.44
Mild recession
0.25
-14
-3.5
-25.2
635.04
6.8
5.8
33.64
Normal growth
0.4
17
9.9
21.8
475.24
Boom
0.3
33
11.2
Expected Return =
Variance =
Standard Deviation =
c. Calculation of covariance:
(A)
(B)
Scenario
Severe recession
Mild recession
Normal growth
Boom
Probability
0.05
0.25
0.4
0.3
(C)
(C)
(D)
Deviation from
Mean Return
Stock
Fund
-51.2
-25.2
5.8
21.8
Bond
Fund
-14
10
3
-10
(E)
Col. C

Col. D
716.8
-252
17.4
-218
Covariance =
(G)
Col. B

Col. F
131.07
158.76
13.46
142.57
445.86
21.12
(F)
Col. B

Col. E
35.84
-63
6.96
-65.4
-85.6
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.
7.
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
Severe recession
Probability
0.1
Rate of
Return
-40
(D)
Col. B

Col. C
(E)
(F)
Deviation
from
Expected Squared
Return Deviation
-2
-51.2
2621.44
(G)
Col. B

Col. F
131.07
Mild recession
0.2
-14
-3.5
-25.2
635.04
158.76
Normal growth
0.35
17
6.8
5.8
33.64
13.46
Boom
0.35
33
9.9
21.8
475.24
142.57
Expected Return =
11.2
Variance =
445.86
Standard Deviation =
21.12
c. Calculation of covariance
(A)
Scenario
Severe recession
Mild recession
Normal growth
Boom
(B)
Probability
0.1
0.2
0.35
0.35
(C)
(D)
Deviation from
Mean Return
Stock
Fund
-51.2
-25.2
5.8
21.8
Bond
Fund
-14
10
3
-10
(E)
Col. C

Col. D
716.8
-252
17.4
-218
Covariance =
(F)
Col. B

Col. E
71.68
-50.4
6.09
-76.3
-48.93
Covariance has decreased because the probabilities of the more extreme returns in
the recession and boom periods are now higher. This gives more weight to the
extremes in the mean calculation, thus making their deviation from the mean less
pronounced.
8. The parameters of the opportunity set are:
E(rS) = 15%, E(rB) = 9%, S = 32%, B = 23%,  = 0.15, rf = 5.5%
From the standard deviations and the correlation coefficient we generate the covariance
matrix [note that Cov(rS, rB) = SB]:
Bonds Stocks
Bonds
529.0
110.4
Stocks
110.4 1024.0
The minimum-variance portfolio proportions are:
w Min (S) 
 2B  Cov(rS , rB )
 S2   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%

 Min  w S2 S2  w 2B  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
Exp. return
100.00
80.00
68.58
60.00
40.00
29.25
20.00
00.00
9.00
10.20
10.89
11.40
12.60
13.25
13.80
15.00
Std dev.
23.00
20.37
19.94
20.18
22.50
24.57
26.68
32.00
Minimum variance
Tangency portfolio
Investment Opportunity Set
Expected Return (%)
20
15
10
5
0
0
10
20
30
Standard Deviation (%)
9.
The graph approximates the points:
Minimum Variance Portfolio
Tangency Portfolio
E(r)
10.89%
13.25%

19.94%
24.57%
40
10. The reward-to-variability ratio of the optimal CAL is:
E(rp )  rf 13.25  5.5

 0.3154
p
24.57
11.
a. The equation for the CAL is:
E(rC )  rf 
E(rp )  rf
p
 C  5.5  0.3154 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:
c.
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
12. 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:
P = [(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 standard deviation by 45 basis points.