Implications for Optimal Investment

In addition to the implementable relation between risk and expected return, described

by equation (5.5), the CAPM also implies a rule for optimal investment:

Result 5.7

Under the assumptions of the CAPM, if a risk-free asset exists, every investor should opti-mally hold a combination of the market portfolio and a risk-free asset.

According to the CAPM, the major difference between the portfolios of Jack and Jill

derives entirely from their differing weights on the risk-free asset. This is demonstrated

in Example 5.8.

Example 5.8:Portfolio Weights That Include the Risk-Free Asset

Consider one-month U.S.Treasury bills as the risk-free asset.Ten million T-bills are issued

for $9,900 each.Jack holds 4 million T-bills, and Jill holds 6 million.If Jack has $200 billion

in wealth, what are the portfolio weights of Jack and Jill given the data in the previous exam-

ple, which indicated that the aggregate wealth invested in risky assets is $268 billion?

Answer:The total wealth in the world is the value of the risky assets, $268 billion, plus

the value of the T-bills, $99 billion, which sum to a total of $367 billion.Thus, if Jack has

$200 billion, Jill has $167 billion.Jack spends $39.6 billion on T-bills, which makes his port-

folio weight on T-bills $39.6 billion/$200 billion .20.Jill spends $59.4 billion on T-bills, mak-

ing her T-bill portfolio weight approximately .36.Thus, Jack owns $160.4 billion/$268 billion

of the shares of the three risky assets and Jill owns $107.6 billion/$268 billion.After some

calculation, the four portfolio weights (respectively, the risk-free asset, HP, IBM, and CPQ)

for Jack are approximately (.20, .20, .50, .10) and the weights for Jill are (.36, .16, .40, .08).

Note, from Example 5.8, that the last three weights in Jack’s and Jill’s portfolios—

that is, weights of .20, .50, and .10 (Jack) and .16, .40, and .08 (Jill) on HP, IBM, and

CPQ, respectively—are the market portfolio’s weights if they are rescaled to sum to 1.

Obviously, this result follows from both Jack’s and Jill’s portfolios being combinations

of the tangency (market) portfolio and the risk-free asset.

To understand the importance of Result 5.7, think again about the inputs needed

to find the tangency portfolio. With thousands of securities to choose from, an investor

would need to calculate not only thousands of mean returns, but also millions of covari-

ances. Such a daunting task would surely require a professional portfolio manager.

However, the CAPM suggests that none of this is necessary; investors can do just as

well by investing in the market portfolio.

The 1970s, 1980s, and 1990s witnessed tremendous growth in the use of passively

managed index portfolios as vehicles for investment in the pension fund, mutual fund,

and life insurance industries. These portfolios attempted to mimic the return behavior of

value-weighted portfolios like the S&P500. One of the major reasons behind this trend

was the popularization of the CAPM, a theory which suggested that the mean-standard

deviation trade-off from investing in the market portfolio cannot be improved upon.

Grinblatt329Titman: Financial	II. Valuing Financial Assets	5. Mean329Variance Analysis	© The McGraw329Hill
Markets and Corporate		and the Capital Asset	Companies, 2002
Strategy, Second Edition		Pricing Model

Chapter 5

Mean-Variance Analysis and the Capital Asset Pricing Model

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