**15**

**ARBITRAGE PRICING THEORY AND FACTOR MODELS**

*I*n this chapter, we continue with our coverage of *asset pricing models* by explaining the arbitrage pricing theory (APT) formulated by Ross (1976). As indicated in the previous chapter, the APT is a general multifactor model for pricing assets. We begin by discussing a special case of a one-factor model for motivation, and then prove the general form of the APT. Regardless of the APT's validity, in practice, factor models are useful for estimating expected asset returns and their covariance matrix, which in turn are valuable for both portfolio management and risk analysis. Consequently, we provide a detailed treatment of factor models and their estimation.

## 15.1 APT AND CAPM

One of the fundamental problems in finance is to explain the cross-section differences in asset expected returns. Specifically, what factors can explain the observed differences? Since those factors that systematically affect the differences in expected returns are the risks investors are compensated for, the term *factors* is used interchangeably with the term *risk factors*.

The Capital Asset Pricing Model (CAPM) developed in the previous chapter asserts that expected returns are linearly related to a single systematic source, the risk of the market portfolio. Beta is a relative measure of market portfolio risk and we referred to this as beta risk. In contrast and also as mentioned at the end of the previous chapter, the APT asserts that the expected returns are linearly ...