Chapter 07 – Optimal Risky Portfolios
CHAPTER 7: OPTIMAL RISKY PORTFOLIOS
PROBLEM SETS
1. (a) and (e).
2. (a) and (c). After real estate is added to the portfolio, there are four asset classes in the
portfolio: stocks, bonds, cash and real estate. Portfolio variance now includes a variance
4. The parameters of the opportunity set are:
From the standard deviations and the correlation coefficient we generate the covariance
matrix [note that Cov(rS, rB) = SB]:
Bonds
Stocks
Bonds
225
45
Stocks
45
900
The minimum-variance portfolio is computed as follows:
wMin(S) =
1739.0
)452(225900
45225
)r,r(Cov2
)r,r(Cov
BS
2
B
2
S
BS
2
B=
−+
−
=
−+
−
The minimum variance portfolio mean and standard deviation are:
E(rMin) = (0.1739 20) + (0.8261 12) = 13.39%
2/1
2
2
2
2
7-2
5.
Proportion
in stock fund
Expected
return
0.00%
12.00%
17.39%
13.39%
minimum variance
20.00%
13.60%
40.00%
15.20%
45.16%
15.61%
tangency portfolio
60.00%
16.80%
80.00%
18.40%
100.00%
20.00%
Graph shown below.
6.
Chapter 07 – Optimal Risky Portfolios
7-3
7. The proportion of the optimal risky portfolio invested in the stock fund is given by:
)r,r(Cov]r)r(Er)r(E[]r)r(E[]r)r(E[
)r,r(Cov]r)r(E[]r)r(E[
w
BSfBfS
2
SfB
2
BfS
BSfB
2
BfS
S−+−−−+−
−−−
=
4516.0
]45)812820[(]900)812[(]225)820[(
]45)812[(]225)820[( =
−+−−−+−
−−−
=
The mean and standard deviation of the optimal risky portfolio are:
E(rP) = (0.4516 20) + (0.5484 12) = 15.61%
8. The reward-to-volatility ratio of the optimal CAL is:
4601.0
54.16
861.15
r)r(E
p
fp =
−
=
−
9. a. If you require that your portfolio yield an expected return of 14%, then you can
find the corresponding standard deviation from the optimal CAL. The equation for
this CAL is:
CC
P
fp
fC 4601.08
r)r(E
r)r(E +=
−
+=
Setting E(rC) equal to 14%, we find that the standard deviation of the optimal
portfolio is 13.04%.
b. To find the proportion invested in the T-bill fund, remember that the mean of the
complete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal
combination of stocks and bonds (P). Let y be the proportion invested in the
portfolio P. The mean of any portfolio along the optimal CAL is:
To find the proportions invested in each of the funds, multiply 0.7884 times the
respective proportions of stocks and bonds in the optimal risky portfolio:
Chapter 07 – Optimal Risky Portfolios
10. Using only the stock and bond funds to achieve a portfolio expected return of 14%, we
must find the appropriate proportion in the stock fund (wS) and the appropriate
proportion in the bond fund (wB = 1 − wS) as follows:
So the proportions are 25% invested in the stock fund and 75% in the bond fund. The
standard deviation of this portfolio will be:
11. a.
Even though it seems that gold is dominated by stocks, gold might still be an
b. If the correlation between gold and stocks equals +1, then no one would hold gold.
The optimal CAL 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
7-5
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 be the risk-free
rate. To find the proportions of this portfolio [with the proportion 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 is:
The expected rate of return for this risk-free portfolio is:
13. False. If the borrowing and lending rates are not identical, then, depending on the tastes
Chapter 07 – Optimal Risky Portfolios
7-6
14. False. The portfolio standard deviation equals the weighted average of the component-
asset standard deviations only in the special case that all assets are perfectly positively
15. The probability distribution is:
Probability
Rate of Return
0.7
100%
0.3
−50%
17. The correct choice is 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.
More formally, we note that when all stocks have the same expected rate of return, the
)I(w1)J(w
MinMin
JI
J
I
−=
Chapter 07 – Optimal Risky Portfolios
7-7
Since all standard deviations are equal to 20%:
Cov(rI , rJ) = IJ = 400 and wMin(I) = wMin(J) = 0.5
18. No, the answer to Problem 17 would not change, at least as long as investors are not risk
19. No, the answers to Problems 17 and 18 would not change. The efficient frontier of risky
20. Rearranging the table (converting rows to columns), and computing serial correlation
results in the following table:
Nominal Rates
Small
company
stocks
Large
company
stocks
Long-term
government
bonds
Intermed-term
government
bonds
Treasury
bills
Inflation
1920s
-3.72
18.36
3.98
3.77
3.56
-1.00
1930s
7.28
-1.25
4.60
3.91
0.30
-2.04
1940s
20.63
9.11
3.59
1.70
0.37
5.36
1950s
19.01
19.41
0.25
1.11
1.87
2.22
1960s
13.72
7.84
1.14
3.41
3.89
2.52
1970s
8.75
5.90
6.63
6.11
6.29
7.36
1980s
12.46
17.60
11.50
12.01
9.00
5.10
1990s
13.84
18.20
8.60
7.74
5.02
2.93
Serial Correlation
0.46
-0.22
0.60
0.59
0.63
0.23
Chapter 07 – Optimal Risky Portfolios
7-8
For example: to compute serial correlation in decade nominal returns for large-company
stocks, we set up the following two columns in an Excel spreadsheet. Then, use the
Excel function “CORREL” to calculate the correlation for the data.
Decade
Previous
1930s
-1.25%
18.36%
1940s
9.11%
-1.25%
1950s
19.41%
9.11%
1960s
7.84%
19.41%
1970s
5.90%
7.84%
1980s
17.60%
5.90%
1990s
18.20%
17.60%
Note that each correlation is based on only seven observations, so we cannot arrive at
any statistically significant conclusions. Looking at the results, however, it appears that,
with the exception of large-company stocks, there is persistent serial correlation. (This
conclusion changes when we turn to real rates in the next problem.)
21. The table for real rates (using the approximation of subtracting a decade’s average
inflation from the decade’s average nominal return) is:
Real Rates
Small
company
stocks
Large
company
stocks
Long-term
government
bonds
Intermed-term
government
bonds
Treasury
bills
1920s
-2.72
19.36
4.98
4.77
4.56
1930s
9.32
0.79
6.64
5.95
2.34
1940s
15.27
3.75
-1.77
-3.66
-4.99
1950s
16.79
17.19
-1.97
-1.11
-0.35
1960s
11.20
5.32
-1.38
0.89
1.37
1970s
1.39
-1.46
-0.73
-1.25
-1.07
1980s
7.36
12.50
6.40
6.91
3.90
1990s
10.91
15.27
5.67
4.81
2.09
Serial Correlation
0.29
-0.27
0.38
0.11
0.00
While the serial correlation in decade nominal returns seems to be positive, it appears
that real rates are serially uncorrelated. The decade time series (although again too short
for any definitive conclusions) suggest that real rates of return are independent from
decade to decade.
Chapter 07 – Optimal Risky Portfolios
CFA PROBLEMS
1. a. Restricting the portfolio to 20 stocks, rather than 40 to 50 stocks, will increase the
risk of the portfolio, but it is possible that the increase in risk will be minimal.
Suppose that, for instance, the 50 stocks in a universe have the same standard
Pn
n
The effect of the reduction in n on the second term on the right-hand side would
be relatively small (since 49/50 is close to 19/20 and 2 is smaller than 2), but
the denominator of the first term would be 20 instead of 50. For example, if =
b. Hennessy could contain the increase in risk by making sure that he maintains
reasonable diversification among the 20 stocks that remain in his portfolio. This
entails maintaining a low correlation among the remaining stocks. For example, in
2. Risk reduction benefits from diversification are not a linear function of the number of
issues in the portfolio. Rather, the incremental benefits from additional diversification
are most important when you are least diversified. Restricting Hennesey to 10 instead of
3. The point is well taken because the committee should be concerned with the volatility of
the entire portfolio. Since Hennessy’s portfolio is only one of six well-diversified
7-11
12. a. Subscript OP refers to the original portfolio, ABC to the new stock, and NP to
the new portfolio.
= 2.2673% 2.27%
b. Subscript OP refers to the original portfolio, GS to government securities, and NP
to the new portfolio.
i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = (0.9 0.67) + (0.1 0.042) = 0.645%
c. Adding the risk-free government securities would result in a lower beta for the new
d. The comment is not correct. Although the respective standard deviations and expected
returns for the two securities under consideration are equal, the covariances between
each security and the original portfolio are unknown, making it impossible to draw the
e. i. Grace clearly expressed the sentiment that the risk of loss was more important to
her than the opportunity for return. Using variance (or standard deviation) as a
Chapter 07 – Optimal Risky Portfolios
7-12
ii. Two alternative risk measures that could be used instead of variance are:
Range of returns, which considers the highest and lowest expected returns in the
13. a. Systematic risk refers to fluctuations in asset prices caused by macroeconomic
factors that are common to all risky assets; hence systematic risk is often referred
b. Trudy should explain to the client that picking only the top five best ideas would
most likely result in the client holding a much more risky portfolio. The total risk
of a portfolio, or portfolio variance, is the combination of systematic risk and firm-
specific risk.
The systematic component depends on the sensitivity of the individual assets to
market movements as measured by beta. Assuming the portfolio is well
diversified, the number of assets will not affect the systematic risk component of