## Ctm

The ratio Cov(rGM,rM)/CTM measures the contribution of GM stock to the variance of the market portfolio as a fraction of the total variance of the market portfolio. The ratio is called beta and is denoted by p. Using this measure, we can restate equation 9.6 as

8 For example, if 8 is 1% (.01 of wealth), then its square is .0001 of wealth, one-hundredth of the original value. The term SV|m will be smaller than 28^ by an order of magnitude.

CHAPTER 9 The Capital Asset Pricing Model 271

This expected return-beta relationship is the most familiar expression of the CAPM to practitioners. We will have a lot more to say about the expected return-beta relationship shortly.

We see now why the assumptions that made individuals act similarly are so useful. If everyone holds an identical risky portfolio, then everyone will find that the beta of each asset with the market portfolio equals the asset's beta with his or her own risky portfolio. Hence everyone will agree on the appropriate risk premium for each asset.

Does the fact that few real-life investors actually hold the market portfolio imply that the CAPM is of no practical importance? Not necessarily. Recall from Chapter 8 that reasonably well-diversified portfolios shed firm-specific risk and are left with mostly systematic or market risk. Even if one does not hold the precise market portfolio, a well-diversified portfolio will be so very highly correlated with the market that a stock's beta relative to the market will still be a useful risk measure.

In fact, several authors have shown that modified versions of the CAPM will hold true even if we consider differences among individuals leading them to hold different portfolios. For example, Brennan9 examined the impact of differences in investors' personal tax rates on market equilibrium, and Mayers10 looked at the impact of nontraded assets such as human capital (earning power). Both found that although the market portfolio is no longer each investor's optimal risky portfolio, the expected return-beta relationship should still hold in a somewhat modified form.

If the expected return-beta relationship holds for any individual asset, it must hold for any combination of assets. Suppose that some portfolio P has weight wk for stock k, where k takes on values 1, . . . , n. Writing out the CAPM equation 9.7 for each stock, and multiplying each equation by the weight of the stock in the portfolio, we obtain these equations, one for each stock:

+ wnE(rn) = wnrf + wn$n[E(rM) - rf] E(rP) = rf + $P[E(rM) - f

Summing each column shows that the CAPM holds for the overall portfolio because E(rP)

= 2 wkE(rk) is the expected return on the portfolio, and BP = 2 wkBk is the portfolio beta.

Incidentally, this result has to be true for the market portfolio itself,

E(M = rf + Pm[E(%) - rf] Indeed, this is a tautology because pM = 1, as we can verify by noting that p = Cov(rM, M

This also establishes 1 as the weighted average value of beta across all assets. If the market beta is 1, and the market is a portfolio of all assets in the economy, the weighted average

9 Michael J. Brennan, "Taxes, Market Valuation, and Corporate Finance Policy," National Tax Journal, December 1973.

10 David Mayers, "Nonmarketable Assets and Capital Market Equilibrium under Uncertainty," in Studies in the Theory of Capital Markets, ed. M. C. Jensen (New York: Praeger, 1972).

PART III Equilibrium in Capital Markets beta of all assets must be 1. Hence betas greater than 1 are considered aggressive in that investment in high-beta stocks entails above-average sensitivity to market swings. Betas below 1 can be described as defensive.

A word of caution: We are all accustomed to hearing that well-managed firms will provide high rates of return. We agree this is true if one measures the firm's return on investments in plant and equipment. The CAPM, however, predicts returns on investments in the securities of the firm.

Let us say that everyone knows a firm is well run. Its stock price will therefore be bid up, and consequently returns to stockholders who buy at those high prices will not be excessive. Security prices, in other words, already reflect public information about a firm's prospects; therefore only the risk of the company (as measured by beta in the context of the CAPM) should affect expected returns. In a rational market investors receive high expected returns only if they are willing to bear risk.

CONCEPT CHECK ^ QUESTION 3

Suppose that the risk premium on the market portfolio is estimated at 8% with a standard deviation of 22%. What is the risk premium on a portfolio invested 25% in GM and 75% in Ford, if they have betas of 1.10 and 1.25, respectively?

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