The Mathematics of Financial Modelingand Investment Management

17-Capital Asset Pricing Model Page 516 Wednesday, February 4, 2004 1:10 PM

516 The Mathematics of Financial Modeling and Investment Management

The horizontal axis shows the number of holdings of different assets

(e.g., the number of common stock held of different issuers).

As can be seen, as the number of asset holdings increases, the level

of nonsystematic risk is almost completely eliminated (i.e., diversified

away). Studies of different asset classes support this. For example, for

common stock, several studies suggest that a portfolio size of about 20

randomly selected companies will completely eliminate nonsystematic

risk leaving only systematic risk.^2

SECURITY MARKET LINE

The CML represents an equilibrium condition in which the expected

return on a portfolio of assets is a linear function of the expected return

on the market portfolio. Individual assets do not fall on the CML.

Instead, it can be demonstrated that the following relationship holds for

individual assets:^3

ER()i – Rf

ER()i = Rf + ---------------------------cov(Ri,RM )

var(RM )

The above equation is called the security market line (SML).

In equilibrium, the expected return of individual securities will lie

on the SML and not on the CML. This is true because of the high degree

of nonsystematic risk that remains in individual assets that can be diver-

sified out of portfolios. In equilibrium, only efficient portfolios will lie

on both the CML and the SML.

The SML also can be expressed as

cov(Ri,RM )

ER()i = Rf + [ER()i – Rf]-------------------------------

var(RM )

The ratio cov(Ri,RM ) ⁄var(RM) can be estimated empirically using

return data for the market portfolio and the return on the asset. The

(^2) The first empirical study of this type was by Wayne H. Wagner and Sheila Lau,
“The Effect of Diversification on Risks,” Financial Analysts Journal (November–De-
cember 1971), p. 50.
(^3) For the proof, see William F. Sharpe, Portfolio Theory and Capital Markets (New
York, NY: McGraw Hill, 1970), pp. 86–91.