What is Arbitrage Pricing Theory?

The capital asset pricing model (CAPM) asserts that only a single number – a security’s beta against the market – is required to measure risk. At the core of arbitrage pricing theory (APT) is the recognition that several systematic factors affect security return.

The returns on an individual stock will depend upon a variety of anticipated and unanticipated events. Anticipated events will be incorporated by investors into their expectations of returns on individual stocks and thus will be incorporated into market prices. Generally, however, most of the return ultimately realized will result from unanticipated events. Of course, change itself is anticipated, and investors know that the most unlikely occurrence of all would be the exact realization of the most probable future scenario. But even though we realize that some unforeseen events will occur, we do not know their direction or their magnitude. What we can know is the sensitivity of returns to these events.

Systematic factors are the major sources of risk in portfolio returns. Actual portfolio returns depend upon the same set of common factors, but this does not mean that all portfolios perform identically. Different portfolios have different sensitivities to these factors.

Because the systematic factors are primary sources of risk, it follows that they are the principal determiEmpirical testing of APT is still in its infancy, and concrete results proving the APT or disproving the CAPM do not exist. For these reasons, it is useful to regard CAPM and APT as different variants of the true equilibrium pricing model. Both are, therefore, useful in supplying intuition into the way security prices and equilibrium returns are established tenants of the expected, as well as the actual, returns on portfolios. It is possible to see that the actual return, R, on any security or portfolio may be broken down into three constituent parts, as follows:

(Z)     R = E + bf + e

where:

E = expected return on the security
b = security’s sensitivity to change in the systematic factor
f = the actual return on the systematic factor
e = returns on the unsystematic factors

Equation Z merely states that the actual return equals the expected return, plus factor sensitivity times factor movement, plus residual risk.

Empirical work suggests that a three-or-four-factor model adequately captures the influence of systematic factors on stock-market returns. Equation Z may thus be expanded to:

R = E + (b1) (f1) + (b2) (f2) + (b3) (f3) + (b4) (f4) + e

Each of the four middle terms in this equation is the product of the returns on a particular economic factor and the given stock’s sensitivity to that factor. Suppose f3 is associated with labor productivity. As labor productivity unexpectedly increases, f3 is positive, and firms with high b3 would find their returns very high. The subtler rationale and higher mathematics of APT are left for development elsewhere.

What are these factors? They are the underlying economic forces that are the primary influences on the stock market. Research suggests that the most important factors are unanticipated inflation, changes in the expected level of industrial production, unanticipated shifts in risk premiums, and unanticipated movements in the shape of the term structure of interest rates.

The biggest problems in APT are factor identification and separating unanticipated from anticipated factor movements in the measurement of sensitivities. Anyone stock is so influenced by idiosyncratic forces that it is very difficult to determine the precise relationship between its return and a given factor. Far more critical is the measurement of the b’s. The b’s measure the sensitivity of returns to unanticipated movements in the factors. By just looking at how a given stock relates to, say, movements in the money supply, we would be including the influence of both anticipated and unanticipated changes, when only the latter is relevant.

Empirical testing of APT is still in its infancy, and concrete results proving the APT or disproving the CAPM do not exist. For these reasons, it is useful to regard CAPM and APT as different variants of the true equilibrium pricing model. Both are, therefore, useful in supplying intuition into the way security prices and equilibrium returns are established.

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