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Standard deviation (%)
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

6 Efficient Diversification

Now consider the CAL that uses portfolio B instead of A. Portfolio B invests 80% in bonds
and 20% in stocks, providing an expected return of 6.80% with a standard deviation of
11.68%. Thus, the reward-to-variability ratio of any portfolio on the CAL of B is
6.80 5
SB .15
This is higher than the reward-to-variability ratio of the CAL of the variance-minimizing port-
folio A.
The difference in the reward-to-variability ratios is SB SA 0.02. This implies that port-
folio B provides 2 extra basis points (0.02%) of expected return for every percentage point in-
crease in standard deviation.
The higher reward-to-variability ratio of portfolio B means that its capital allocation line is
steeper than that of A. Therefore, CALB plots above CALA in Figure 6.5. In other words, com-
binations of portfolio B and the risk-free asset provide a higher expected return for any level
of risk (standard deviation) than combinations of portfolio A and the risk-free asset. Therefore,
all risk-averse investors would prefer to form their complete portfolio using the risk-free asset
with portfolio B rather than with portfolio A. In this sense, portfolio B dominates A.
But why stop at portfolio B? We can continue to ratchet the CAL upward until it reaches
the ultimate point of tangency with the investment opportunity set. This must yield the CAL
with the highest feasible reward-to-variability ratio. Therefore, the tangency portfolio (O) in
Figure 6.6 is the optimal risky portfolio to mix with T-bills, which may be defined as the optimal risky
risky portfolio resulting in the highest possible CAL. We can read the expected return and portfolio
standard deviation of portfolio O (for “optimal”) off the graph in Figure 6.6 as The best combination
of risky assets to be
E(rO) 8.68%
mixed with safe
assets to form the
complete portfolio.
which can be identified as the portfolio that invests 32.99% in bonds and 67.01% in stocks.3
We can obtain a numerical solution to this problem using a computer program.

F I G U R E 6.6
The optimal capital
allocation line with
Expected return (%)

10 bonds, stocks,
E(ro) 8.68% and T-bills
σo 17.97%
0 5 10 15 20 25 30 35
Standard deviation (%)

The proportion of portfolio O invested in bonds is:
[E(rB) rf] [E(rS) rf]
wB 2 2
[E(rB) rf] [E(rS) rf] [E(rB) rf E(rS) rf]
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

186 Part TWO Portfolio Theory

F I G U R E 6.7 E(rP)
The complete portfolio

O, optimal risky
7.02% portfolio
5% C, complete

9.88% 17.97%

The CAL with our optimal portfolio has a slope of
8.68 5
SO .20
which is the reward-to-variability ratio of portfolio O. This slope exceeds the slope of any
other feasible portfolio, as it must if it is to be the slope of the best feasible CAL.
In the last chapter we saw that the preferred complete portfolio formed from a risky port-
folio and a risk-free asset depends on the investor™s risk aversion. More risk-averse investors
will prefer low-risk portfolios despite the lower expected return, while more risk-tolerant in-
vestors will choose higher-risk, higher-return portfolios. Both investors, however, will choose
portfolio O as their risky portfolio since that portfolio results in the highest return per unit of
risk, that is, the steepest capital allocation line. Investors will differ only in their allocation of
investment funds between portfolio O and the risk-free asset.
Figure 6.7 shows one possible choice for the preferred complete portfolio, C. The investor
places 55% of wealth in portfolio O and 45% in Treasury bills. The rate of return and volatil-
ity of the portfolio are
E(rC) 5 0.55 (8.68 5) 7.02%
0.55 17.97 9.88%

In turn, we found above that portfolio O is formed by mixing the bond fund and stock fund
with weights of 32.99% and 67.01%. Therefore, the overall asset allocation of the complete
portfolio is as follows:

Weight in risk-free asset 45.00%
Weight in bond fund 0.3299 55% 18.14
Weight in stock fund 0.6701 55% 36.86
Total 100.00%

Figure 6.8 depicts the overall asset allocation. The allocation reflects considerations of both
efficient diversification (the construction of the optimal risky portfolio, O) and risk aversion
(the allocation of funds between the risk-free asset and the risky portfolio O to form the com-
plete portfolio, C).
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

6 Efficient Diversification

F I G U R E 6.8
The composition of
the complete
portfolio: The solution
to the asset allocation
Bonds problem

Portfolio O

4. A universe of securities includes a risky stock (X), a stock index fund (M), and Concept
T-bills. The data for the universe are:

Expected Return Standard Deviation
X 15% 50%
M 10 20
T-bills 5 0

The correlation coefficient between X and M is 0.2.
a. Draw the opportunity set of securities X and M.
b. Find the optimal risky portfolio (O) and its expected return and standard
c. Find the slope of the CAL generated by T-bills and portfolio O.
d. Suppose an investor places 2/9 (i.e., 22.22%) of the complete portfolio in the
risky portfolio O and the remainder in T-bills. Calculate the composition of the
complete portfolio.

We can extend the two-risky-assets portfolio construction methodology to cover the case of
many risky assets and a risk-free asset. First, we offer an overview. As in the two-risky-assets
example, the problem has three separate steps. To begin, we identify the best possible or most
efficient risk-return combinations available from the universe of risky assets. Next we deter-
mine the optimal portfolio of risky assets by finding the portfolio that supports the steepest
CAL. Finally, we choose an appropriate complete portfolio based on the investor™s risk aver-
sion by mixing the risk-free asset with the optimal risky portfolio.
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

E XC E L Applications www.mhhe.com

> Two-Security Portfolio

The Excel model “Two-Security Portfolio” is based on the asset allocation problem between stocks
and bonds that appears in this chapter. You can change correlations, mean returns, and standard
deviation of return for any two securities or, as it is used in the text example, any two portfolios. All
of the concepts that are covered in this section can be explored using the model.
You can learn more about this spreadsheet model by using the interactive version available on
our website at www.mhhe.com/bkm.

4 Asset Allocation Analysis: Risk and Return
5 Expected Standard Corr.
6 Return Deviation Coeff s,b Covariance
7 Bonds 6.00% 12.00% 0 0
8 Stocks 10.00% 25.00%
9 T-Bill 5.00% 0.00%
12 Weight Weight Expected Standard Reward to
13 Bonds Stocks Return Deviation Variability
14 1 0 6.0000% 12.0000% 0.08333
15 0.9 0.1 6.4000% 11.0856% 0.12629
16 0.8 0.2 6.8000% 10.8240% 0.16630
17 0.7 0.3 7.2000% 11.2610% 0.19536
18 0.6 0.4 7.6000% 12.3223% 0.21100
19 0.5 0.5 8.0000% 13.8654% 0.21637
20 0.4 0.6 8.4000% 15.7493% 0.21588
21 0.3 0.7 8.8000% 17.8664% 0.21269
22 0.2 0.8 9.2000% 20.1435% 0.20850
23 0.1 0.9 9.6000% 22.5320% 0.20415
24 0 1 10.0000% 25.0000% 0.20000
26 Minimum Variance Portfolio Short Sales No Short
27 Allowed Sales
28 Weight Bonds 0.81274 0.81274
29 Weight Stocks 0.18726 0.18726
30 Return 6.7490% 6.7490%
31 Risk 10.8183% 10.8183%

The Efficient Frontier of Risky Assets
To get a sense of how additional risky assets can improve the investor™s investment opportu-
nities, look at Figure 6.9. Points A, B, and C represent the expected returns and standard devi-
ations of three stocks. The curve passing through A and B shows the risk-return combinations


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