Arbitrage pricing theory, or APT, is a multi-factor asset pricing model, developed primarily by Stephen Ross, that sets an asset’s expected return approximately equal to the risk-free rate plus, for each systematic risk factor, the asset’s sensitivity to that factor multiplied by that factor’s risk premium. The linear relation is a conclusion rather than an assumption, forced by the absence of arbitrage in a market carrying many assets whose returns share a common factor structure, and it is exact in the limit for well-diversified portfolios.
Ross set it out in 1976, in the Journal of Economic Theory article “The arbitrage theory of capital asset pricing” (Ross, 1976; Huberman and Wang, 2005).
The Formula
The model puts the required return on the left, and a risk-free base plus one term per factor on the right.
r = Rf + (B1 x G1) + (B2 x G2) + ... + (Bk x Gk)
r the expected return the asset's risk requires; where the asset is a
company's own equity it is the cost of equity capital
Rf the risk-free rate, the return on an asset treated as carrying no
default risk
Bk the factor loading, the sensitivity of the asset's return to the kth
economic factor. Written as a beta, one for each factor
Gk the risk premium on the kth beta, the extra expected return carried
by one unit of sensitivity to that factor
k the number of systematic factors, which the theory itself leaves openThat is the form used for multibeta models in a Federal Reserve study of the imputed cost of equity capital for priced services at the Reserve Banks (Green, Lopez and Wang, FRBNY Economic Policy Review, 2003). With a single factor the expression has the same shape as the capital asset pricing model, and APT was developed as a generalization of that model (Huberman and Wang, Federal Reserve Bank of New York Staff Reports, no. 216, 2005).
Where the Linear Relation Comes From
APT is a one-period model, and it is the preclusion of arbitrage over static portfolios of the assets that forces expected returns into a linear relation with the factor loadings, which are themselves proportional to the returns’ covariances with the factors (Huberman and Wang, 2005).
The no-arbitrage argument delivers that relation approximately rather than exactly, and exact arbitrage pricing, the equality written above, is proven in the limit for well-diversified portfolios (Huberman and Wang, 2005).
The word arbitrage is doing lighter work here than it looks. An investor who buys an asset plotting above the line and shorts a portfolio carrying the same systematic risk locks in a positive expected payoff, not a guaranteed one, so Yale’s investment theory course calls it an arbitrage in expectations rather than an arbitrage (Goetzmann, An Introduction to Investment Theory, Yale School of Management).
The Theory Names No Factors
Nothing in the theory says what the factors are. The main drawback of the multibeta family, APT included, is that economic theory does not specify the factors to be used in them, and the task of identifying them is left to empirical research (Green, Lopez and Wang, 2003). The macroeconomic lists that circulate alongside the model, inflation and industrial production and interest-rate spreads among them, come out of that empirical work rather than out of APT.
Because the theory is silent about the factors’ identities, any test of APT is a joint test of the pricing relation and of the correctness of the factors chosen (Huberman and Wang, 2005). A result that rejects the model may have rejected only the factor list.
A Worked Example
The inputs below are stated for the example rather than read from a market.
Rf risk-free rate 4.0% stated input
loading (B) premium (G) stated inputs
Factor 1 0.80 3.0%
Factor 2 -0.50 2.0%
Factor 3 0.30 1.5%
r = 4.0% + (0.80 x 3.0%) + (-0.50 x 2.0%) + (0.30 x 1.5%)
= 4.0% + 2.40% - 1.00% + 0.45%
= 5.85%A loading can run either way, so the negative loading on the second factor subtracts from the required return rather than adding to it.
What Is the Difference Between CAPM and Arbitrage Pricing Theory?
Both assert a linear relation between an asset’s expected return and its covariance with something else, but under CAPM that something is a single quantity, the return on the market portfolio, while APT prices covariance with each factor in the structure (Huberman and Wang, 2005). The derivations differ more than the formulas do, since APT is reached from arbitrage arguments while the intertemporal capital asset pricing model, the other main multi-factor approach, is reached from equilibrium arguments (Green, Lopez and Wang, 2003). APT also does not require that all investors behave alike (Goetzmann, Yale School of Management).
Beta, the risk-free rate, the equity risk premium and the split between systematic and diversifiable risk are worked through in CAPM equation.
Where the Model Is Used
APT lives in investment and academic asset-pricing work. The applications a Federal Reserve Bank of New York review examines critically are asset allocation, the computation of the cost of capital, and the performance evaluation of managed funds. That review concludes that applying the model to evaluate managed portfolios is contradictory to the no-arbitrage spirit of the model, because APT does not preclude arbitrage over dynamic portfolios (Huberman and Wang, 2005). What a cost-of-capital rate is once it reaches a valuation sits in cost of capital.
The theory’s silence about factors is one of the reasons it has stayed out of one place a reader might expect it. The private sector adjustment factor is the formula the Federal Reserve uses to impute a cost of capital into the prices of Federal Reserve Bank services, as the Monetary Control Act of 1980 requires. The review of that formula found multibeta models could be employed for the equity cost of capital, but that no consensus on the factors exists, so adopting any particular one would be open to criticism. It recommended against using them for a second reason too, that the academic literature shows multibeta models do not substantially improve the estimates, so the gain in accuracy would likely be too small to justify the burden of defending a deviation from the CAPM method (Green, Lopez and Wang, 2003).
