The Mathematics of Financial Modelingand Investment Management

17-Capital Asset Pricing Model Page 517 Wednesday, February 4, 2004 1:10 PM

Capital Asset Pricing Model 517

empirical analogue for the above equation is the following linear regres-

sion, called the characteristic line:

rit − rft = αi + βi [rMt − rft] + eit

where eit is the error term.

The beta term βi in the above regression is the estimate of the ratio

in the SML equation that is

cov(Ri, RM)

βi = -------------------------------

var(RM)

Substituting βi into the SML equation gives the beta-version of the

SML:

E(Ri) = Rf + βi [E(RM) − Rf]

This is the CAPM. It states that, given the assumptions of the

CAPM, the expected return on an individual asset is a positive linear

function of its index of systematic risk as measured by beta. The higher

the beta is, the higher the expected return.

An investor pursuing an active strategy will search for underpriced

securities to purchase and overpriced securities to avoid (sell if held in

the current portfolio, or sold short if permitted). If an investor believes

that the CAPM is the correct asset pricing model, then the SML can be

used to identify mispriced securities. A security where the market prices

it such that the expected return is less than the expected return as pre-

dicted by the SML is an undervalued security. In contrast, an overvalued

security is one where the market prices the security such that its

expected return is greater than that predicted by the 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 unsystematic risk that remains in individual securities that can be

diversified out of portfolios of securities. It follows that the only risk

investors will pay a premium to avoid is market risk. Hence, two assets

with the same amount of systematic risk will have the same expected

return. In equilibrium, only efficient portfolios will lie on both the CML

and the SML. This underscores the fact that the systematic risk measure,

beta, is most correctly considered as an index of the contribution of an

individual security to the systematic risk of a well-diversified portfolio

of securities.