that is, if any security violates the expected return-beta relationship, then many investors (each relatively small) will tilt their portfolios so that their combined overall pressure on prices will restore an equilibrium that satisfies the relationship.

In contrast, the APT uses a single-factor security market assumption and arbitrage argu- ments to obtain the expected return-beta relationship for well-diversified portfolios. Be- cause it focuses on the no-arbitrage condition, without the further assumptions of the market or index model, the APT cannot rule out a violation of the expected return-beta re- lationship for any particular asset. For this, we need the CAPM assumptions and its domi- nance arguments.

11.5 A MULTIFACTOR APT

We have assumed so far that there is only one systematic factor affecting stock returns. This simplifying assumption is in fact too simplistic. It is easy to think of several factors driven by the business cycle that might affect stock returns: interest rate fluctuations, inflation rates, oil prices, and so on. Presumably, exposure to any of these factors will affect a stocks risk and hence its expected return. We can derive a multifactor version of the APT to ac- commodate these multiple sources of risk.

Suppose that we generalize the factor model expressed in equation 11.1 to a two-factor model:

ri E(ri) i1F1 i 2F2 ei (11.5) Factor 1 might be, for example, departures of GDP growth from expectations, and factor 2 might be unanticipated inflation. Each factor has a zero expected value because each mea- sures the surprise in the systematic variable rather than the level of the variable. Similarly, the firm-specific component of unexpected return, ei, also has zero expected value. Ex- tending such a two-factor model to any number of factors is straightforward.

Establishing a multifactor APT is similar to the one-factor case. But first we must intro- duce the concept of a factor portfolio, which is a well-diversified portfolio constructed to have a beta of 1 on one of the factors and a beta of 0 on any other factor. This is an easy re- striction to satisfy, because we have a large number of securities to choose from, and a rel- atively small number of factors. Factor portfolios will serve as the benchmark portfolios for a multifactor security market line.

Suppose that the two factor portfolios, called Portfolios 1 and 2, have expected returns E(r1) 10% and E(r2) 12%. Suppose further that the risk-free rate is 4%. The risk

III. Equilibrium In Capital Markets

11. Arbitrage Pricing Theory The McGraw−Hill Companies, 2001