passing through B and C shows all the portfolios that can be formed from those two stocks.

Now observe point E on the AB curve and point F on the BC curve. These points represent two

188

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189

6 Efficient Diversification

35

30

C

F

Expected return (%)

25

20

B

15

E

10 A

5

0

0 10 20 30 40

Standard deviation (%)

F I G U R E 6.9

Portfolios constructed with three stocks (A, B, and C)

portfolios chosen from the set of AB combinations and BC combinations. The curve that

passes through E and F in turn represents all the portfolios that can be constructed from port-

folios E and F. Since E and F are themselves constructed from A, B, and C, this curve also may

be viewed as depicting some of the portfolios that can be constructed from these three securi-

ties. Notice that curve EF extends the investment opportunity set to the northwest, which is

the desired direction.

Now we can continue to take other points (each representing portfolios) from these three

curves and further combine them into new portfolios, thus shifting the opportunity set even

farther to the northwest. You can see that this process would work even better with more

stocks. Moreover, the efficient frontier, the boundary or “envelope” of all the curves thus de-

veloped, will lie quite away from the individual stocks in the northwesterly direction, as

shown in Figure 6.10.

The analytical technique to derive the efficient frontier of risky assets was developed by

Harry Markowitz at the University of Chicago in 1951 and ultimately earned him the Nobel

Prize in economics. We will sketch his approach here.

First, we determine the risk-return opportunity set. The aim is to construct the efficient frontier

northwestern-most portfolios in terms of expected return and standard deviation from the uni-

Graph representing a

verse of securities. The inputs are the expected returns and standard deviations of each asset

set of portfolios that

in the universe, along with the correlation coefficients between each pair of assets. These data maximizes expected

come from security analysis, to be discussed in Part Four. The graph that connects all the return at each level

northwestern-most portfolios is called the efficient frontier of risky assets. It represents of portfolio risk.

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Fifth Edition

190 Part TWO Portfolio Theory

F I G U R E 6.10 Portfolio

expected return

The efficient frontier of

risky assets and E(rP)

Efficient

individual assets

frontier of

risky assets

Minimum Individual

variance assets

portfolio

σP

Portfolio standard deviation

the set of portfolios that offers the highest possible expected rate of return for each level of

portfolio standard deviation. These portfolios may be viewed as efficiently diversified. One

such frontier is shown in Figure 6.10.

Expected return-standard deviation combinations for any individual asset end up inside

the efficient frontier, because single-asset portfolios are inefficient”they are not efficiently

diversified.

When we choose among portfolios on the efficient frontier, we can immediately discard

portfolios below the minimum-variance portfolio. These are dominated by portfolios on the

upper half of the frontier with equal risk but higher expected returns. Therefore, the real

choice is among portfolios on the efficient frontier above the minimum-variance portfolio.

Various constraints may preclude a particular investor from choosing portfolios on the

efficient frontier, however. If an institution is prohibited by law from taking short positions

in any asset, for example, the portfolio manager must add constraints to the computer-

optimization program that rule out negative (short) positions.

Short sale restrictions are only one possible constraint. Some clients may want to assure a

minimum level of expected dividend yield. In this case, data input must include a set of ex-

pected dividend yields. The optimization program is made to include a constraint to ensure

that the expected portfolio dividend yield will equal or exceed the desired level. Another com-

mon constraint forbids investments in companies engaged in “undesirable social activity.”

In principle, portfolio managers can tailor an efficient frontier to meet any particular ob-

jective. Of course, satisfying constraints carries a price tag. An efficient frontier subject to a

number of constraints will offer a lower reward-to-variability ratio than a less constrained one.

Clients should be aware of this cost and may want to think twice about constraints that are not

mandated by law.

Deriving the efficient frontier may be quite difficult conceptually, but computing

and graphing it with any number of assets and any set of constraints is quite straightforward.

For a small number of assets, and in the absence of constraints beyond the obvious one that

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

Essentials of Investments, Companies, 2003

Fifth Edition

EXCE L Applications www.mhhe.com

> Efficient Frontier for Many Stocks

Excel spreadsheets can be used to construct an efficient frontier for a group of individual securi-

ties or a group of portfolios of securities. The Excel model “Efficient Portfolio” is built using a sam-

ple of actual returns on stocks that make up a part of the Dow Jones Industrial Average Index. The

efficient frontier is graphed, similar to Figure 6.10, using various possible target returns. The

model is built for eight securities and can be easily modified for any group of eight assets.

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 G H I

1

2

3 TKR SYM Return S.D.

4 C 46.6 34.8

5 GE 37.3 25.0

6 HD 41.8 31.4

7 INTC 46.0 45.9

8 JNJ 24.6 26.2

9 MRK 32.6 31.0

10 SBC 19.0 28.1

11 WMT 41.2 31.4

12

13 Correlation Matrix

14

C GE HD INTC JNJ MRK SBC WMT

15

16 C 1.00 0.54 0.26 0.26 0.35 0.29 0.25 0.40

17 GE 0.54 1.00 0.58 0.26 0.29 0.20 0.34 0.52

18 HD 0.26 0.58 1.00 -0.09 -0.02 -0.12 0.15 0.58

19 INTC 0.26 0.26 -0.09 1.00 0.09 0.11 -0.05 -0.02

20 JNJ 0.35 0.29 -0.02 0.09 1.00 0.58 0.28 0.28

21 MRK 0.29 0.20 -0.12 0.11 0.58 1.00 0.37 0.12

22 SBC 0.25 0.34 0.15 -0.05 0.28 0.37 1.00 0.16

23 WMT 0.40 0.52 0.58 -0.02 0.28 0.12 0.16 1.00

24

25 Covariance Matrix

C GE HD INTC JNJ MRK SBC WMT

26

27 C 1211.55 468.81 282.30 419.81 320.52 308.52 239.86 440.95

28 GE 468.81 627.47 451.99 299.86 189.64 158.28 240.96 409.29

29 HD 282.30 451.99 983.39 -133.54 -17.19 -117.25 133.28 566.72

30 INTC 419.81 299.86 -133.54 2106.34 113.73 151.78 -63.77 -34.46

31 JNJ 320.52 189.64 -17.19 113.73 686.88 473.15 203.37 229.77

32 MRK 308.52 158.28 -117.25 151.78 473.15 961.63 324.53 119.16

33 SBC 239.86 240.96 133.28 -63.77 203.37 324.53 790.22 140.90

34 WMT 440.95 409.29 566.72 -34.46 229.77 119.16 140.90 987.13

portfolio proportions must sum to 1.0, the efficient frontier can be computed and graphed with

a spreadsheet program.

Choosing the Optimal Risky Portfolio

The second step of the optimization plan involves the risk-free asset. Using the current risk-

free rate, we search for the capital allocation line with the highest reward-to-variability ratio

(the steepest slope), as shown in Figures 6.5 and 6.6.

191

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192 Part TWO Portfolio Theory

The CAL formed from the optimal risky portfolio (O) will be tangent to the efficient fron-

tier of risky assets discussed above. This CAL dominates all alternative feasible lines (the

dashed lines that are drawn through the frontier). Portfolio O, therefore, is the optimal risky

portfolio.

The Preferred Complete Portfolio and the Separation Property

Finally, in the third step, the investor chooses the appropriate mix between the optimal risky

portfolio (O) and T-bills, exactly as in Figure 6.7.

A portfolio manager will offer the same risky portfolio (O) to all clients, no matter what

their degrees of risk aversion. Risk aversion comes into play only when clients select their de-

sired point on the CAL. More risk-averse clients will invest more in the risk-free asset and less

in the optimal risky portfolio O than less risk-averse clients, but both will use portfolio O as

the optimal risky investment vehicle.

separation This result is called a separation property, introduced by James Tobin (1958), the 1983

property Nobel Laureate for economics: It implies that portfolio choice can be separated into two in-

dependent tasks. The first task, which includes steps one and two, determination of the opti-

The property that

mal risky portfolio (O), is purely technical. Given the particular input data, the best risky

implies portfolio

portfolio is the same for all clients regardless of risk aversion. The second task, construction

choice can be

separated into two of the complete portfolio from bills and portfolio O, however, depends on personal preference.

independent tasks:

Here the client is the decision maker.

(1) determination

Of course, the optimal risky portfolio for different clients may vary because of portfolio

of the optimal risky

constraints such as dividend yield requirements, tax considerations, or other client prefer-

portfolio, which is

ences. Our analysis, though, suggests that a few portfolios may be sufficient to serve the de-

a purely technical

problem, and (2) the mands of a wide range of investors. We see here the theoretical basis of the mutual fund

personal choice of

industry.

the best mix of the

If the optimal portfolio is the same for all clients, professional management is more effi-

risky portfolio and

cient and less costly. One management firm can serve a number of clients with relatively small

the risk-free asset.

incremental administrative costs.

The (computerized) optimization technique is the easiest part of portfolio construction. If

different managers use different input data to develop different efficient frontiers, they will of-