Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

185

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

11.68

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

17.97%

O

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

12

The optimal capital

11

allocation line with

Expected return (%)

Stocks

10 bonds, stocks,

E(ro) 8.68% and T-bills

9

O

8

7

6

Bonds

5

σo 17.97%

4

0 5 10 15 20 25 30 35

Standard deviation (%)

3

The proportion of portfolio O invested in bonds is:

2

[E(rB) rf] [E(rS) rf]

S B S BS

wB 2 2

[E(rB) rf] [E(rS) rf] [E(rB) rf E(rS) rf]

S B B S BS

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

CALo

8.68%

O, optimal risky

7.02% portfolio

5% C, complete

portfolio

σP

9.88% 17.97%

The CAL with our optimal portfolio has a slope of

8.68 5

SO .20

17.97

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%

C

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

187

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

18.14%

T-bills

45%

Stocks

36.86%

Portfolio O

55%

<

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:

CHECK

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

deviation.

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.

6.4 EFFICIENT DIVERSIFICATION

WITH MANY RISKY ASSETS

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.

A B C D E F

1

2

3

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%

10

11

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

25

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%

CAL(MV)

32

33

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