The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 436 Wednesday, February 4, 2004 1:08 PM

436 The Mathematics of Financial Modeling and Investment Management

E [] r = λ 0 ιιιι+ Bλλλλ

where λλλλare risk premia. This relationship says that each asset’s return is

equal to the risk-free rate λ 0 plus a linear combination of factors.

In the original formulation, the above linear relationship holds only

approximately in the limit of an infinite economy. Any finite number of

assets can be mispriced, that is, violate the above relationship. The APT

relationship can be made rigorous with additional restrictions on agent

behavior.

T esting APT

The original formulation of APT does not identify factors. Subsequently

a number of researchers tried to tackle the problem. As we will see in

Chapter 18, factors can be either exogenously given factors or abstract

factors formed by particular portfolios. A number of studies have tried

to identify macroeconomic factors responsible for stock returns.^3 Statis-

tical techniques such as factor analysis or principal components analysis

have also been used.

The approximate nature of APT makes it difficult to test it. In fact,

the APT holds only in the limit of an infinite economy while any finite

number of securities can be arbitrarily priced without affecting the arbi-

trage principle. For this reason it has been suggested that APT cannot be

tested at all.^4 Based on a given selection of factors APT has been tested

with the techniques that we will explain in the following sections.

Testing APT when Factors are Portfolios

Suppose that factors are given portfolios and that there is a risk-free

asset. This means that it is known (or at least assumed) that the model

in excess returns takes the form

zt = aBf+ t + εεεεt

(^3) See, for example, Chen, Nai-Fu, Richard R. Roll, and Stephen A. Ross, “Economic
Forces and the Stock Market,” Journal of Business 59, no. 3 (1986), pp. 383–404
and Michael A. Berry, Edwin Burmeister, and Marjorie B. McElroy, “Sorting out
Risk Using Known APT Factors,” Financial Analysts Journal 44, no. 2 (1988), pp.
29–42.
(^4) Phoebus J. Dhrymes, Irwin Friend, and N. Bulent Gultekin, “A Critical Re-Exami-
nation of the Empirical Evidence on the Arbitrage Pricing Theory,” Journal of Fi-
nance 39, no. 2 (1988), pp. 323–346.