# Arbitrage Pricing Theory

The CAPM seems to have the following limitations:

1. It is based on highly restrictive assumptions
2. There are serious doubts about its testability
3. The market factor is not the sole factor influencing stock returns

The arbitrage pricing theory (APT), originally developed by Stephen A Ross seeks to overcome the shortcomings of the CAPM. It is a novel and different approach to determining asset prices.

Return Generating Process:

The CAPM assumes that investors utilize a mean variance framework. The APT is much more general in that asset prices can be influenced by factors beyond means and variances. The APT assumes that the return on any stock is linearly related to a set of factors also referred to as systematic factors or risk factors as given in Eq 5

Ri = ai + bi1I1+ bi2 + … + bij IJ + ei —————- Eq 5

where Ri = return on stock i
ai = expected return on stock i if all factors have a value of zero
Ij = value of jth factor which influences the return on stock i( j=1,…J)
ei = random error term which has a mean of zero and variance of σ2ei

This model rests on the following assumptions:

E (ei ej) = 0 for all i and j where i = j

What are the factors I1,… Ij, that impact on stock returns? The APT does not specify these factors. It merely says that stock returns are related in a linear manner to a limited number of factors (or systematic influences).

Equilibrium Risk return Relationship:

Given the return generating process reflected in Eq 5, the APT established an equilibrium risk return relationship. The key idea which guides the development of the equilibrium relationship is the law of one price which says that two identical things cannot sell at different process. Applied to portfolios, it means that two portfolios that have the same risk cannot offer different expected returns if it were so, arbitrageurs will step in and their actions will ensure that the law of one price is satisfied.

The difference between the CAPM and the APT is vividly reflected in the following situation. A private airplane is lost in the clouds. The pilot calls the air controller and asks his whereabouts. Employing a uni-dimensional model like the CAPM, the air controller responds 150 miles from New Delhi. This hardly helps the pilot. Instead, if the air controller uses a multidimensional model like the APT, he would provide information on latitude, longitude and altitude. This definitely helps the pilot.

Empirical Evidence:

The APT has been empirically tested using two different approaches. In the first approach, the technique of factor analysis
(a statistical technique) is applied to stock returns to discover the basic factors. These are then examined to see whether they correspond to some economic or behavioral variables. Empirical studies done so far suggest that there is hardly any consistency in terms of (1) the number of basic factors, (2) the interpretation that may be put on these factors (typically the factors identified are artificial constructs representing several economic variables) and the stability of these factors from test to test.

In the second approach, factors are specified a priority, rather than extracted by analyzing stock returns. The classic work of Roll and Ross typifies this approach. They employ four factors: industrial production, inflation rate, term structure of interest rates, and default risk premium. Sensitivity to unanticipated changes in these factors provides explanation for differences in expected returns among stocks in their study.