# Arbitrage Pricing Theory (APT)

So far, so good, but if we consider the purpose for which the CAPM was originally intended, namely stock market investment, it has limitations. As we observed in Chapter Six, even the most actively managed, institutional portfolio funds periodically underperform relative to the market as a whole.

Leaving aside the questionable assumptions that investors are rational, markets are efficient and prices perform a "random walk" (dealt with in our Introduction and elsewhere) one early explanation of the variable performance of portfolios, institutional or otherwise, was provided by Roll's critique of the CAPM (1977).

According to Roll, it is not only impossible for the most discerning investor to establish the composition of the * true *market portfolio, but there is also no reason to assume that a security's expected return is only affected by systematic risk. In the same year, Firth (1977) also observed that if the stock market is so efficient at assimilating all relevant information into security prices, it is impossible to claim that it is either efficient or inefficient, since by definition there is no alternative measurement criterion.

Such criticisms are important, not because they invalidate the CAPM (most empirical tests support it). But because they gave credibility to an alternative approach to portfolio asset management and security price determination based on stock market efficiency presented by Ross (1976). This is termed * Arbitrage Pricing Theory *(APT).

Unlike the CAPM, which prices securities in relation to a * global market portfolio, *the APT possesses the advantage of pricing of securities

*The*

**in relation to each other.***(beta factor) CAPM focuses upon an assumed*

**single index***linear relationship between betas and expected returns (systemic risk plotted by the SML). The APT is a*

**specific***model that*

**general***systematic risk into smaller components, which need not be specified in advance. These define the*

**subdivides***(APP). Any macro-economic factors, including market sentiment, which impact upon investor returns may be incorporated into the APP (or ignored, if inconsequential.) For example, an unexpected change in the rate of inflation (purchasing power risk) might affect the price of securities generally. The advantage of the APT, however, is that it can be used to eliminate this risk specifically, such as a pension fund portfolio's requirement that it should be immune to inflation.*

**Arbitrage Pricing Plane**Statistical tests on the model, including those of Roll and Ross (1980), established that a * four factor *linear version of the APT is a more accurate predictor of security and portfolio returns than the

*(index) CAPM. Specifically, their APT states that the expected return is directly proportional to its sensitivity to the following:*

**single factor**1) Interest rates,

2) Inflation

3) Industrial productivity,

4) Investor risk attitudes.

The return equation for a four-factor APP conforms to the following * simple linear *relationship for the expected return on the

*th security in a portfolio:*

**j**Where:

* r *= expected rate of return on security j,

ri = expected return on factor i, (i = 1,2,3,4),

a = intercept,

bi = slope of ri.

The expected risk premium on the jth security is defined as the difference between its expected return (r) and the risk-free fate of interest (rf) associated with each factor's return (ri) and the security's sensitivity to each of these factors (bi). The four-factor equation is given by:

Like the * specific *CAPM, the

*APT is still a*

**general***model. Theoretically, it assumes that unsystematic (unique) risk can be eliminated in a well-diversified portfolio, leaving only the portfolio's sensitivity to unexpected changes in*

**linear***factors. Subsequent studies, such as Chen, Roll and Ross (1986) therefore focused upon identifying further significant factors and why the sensitivity of returns on a particular share to each factor will vary. However, the work of Dhrymes, Friend and Gultekin (1984) had already suggested that this line of research may be redundant. Their study concluded that as the number of portfolio constituents increases, a greater number of factors must be incorporated into the model. Thus, at the limit, the APT could be equivalent to the CAPM, which defines risk in terms of a*

**macro-economic***over-arching*

**single***relative to the return on the market portfolio.*

**micro-economic factor**For one of the first comprehensive reviews of the APT, which explains why even today it is not fully developed and its application has been less successful than the CAPM, you should read Elton, Gruber and Mei (1994). A more recent perspective on the APT is provided by Huberman and Wang (2005).