Arbitrage Pricing Theory (APT)

The APT, developed by Stephen Ross (1976), is the theoretical foundation for multi-factor asset pricing. Unlike the CAPM, which derives a single risk factor from mean-variance optimization, the APT derives a multi-factor pricing relation from no-arbitrage conditions alone.

Model specification

Assume returns follow a k-factor generating process:

${\tilde{R}}_i=E_i+b_{i1}{\tilde{f}}_1+\cdots +b_{ik}{\tilde{f}}_k+{ϵ}_i$

where ${\tilde{f}}_j$ are mean-zero common factors and ${ϵ}_i$ is idiosyncratic noise. In a large, well-diversified economy, no-arbitrage requires:

$E_i\approx {\lambda }_0+b_{i1}{\lambda }_1+\cdots +b_{ik}{\lambda }_k$

where ${\lambda }_0$ is the risk-free rate and ${\lambda }_j$ are factor risk premia. The pricing relation holds approximately for all assets and exactly in the limit as the number of assets grows.

Key assumptions

• Returns generated by a linear factor model
• Enough assets to diversify away idiosyncratic risk
• No arbitrage opportunities persist

Notably absent: assumptions about utility functions, normality of returns, or the efficiency of the market portfolio.

Theoretical contribution

The APT solved a fundamental problem with the CAPM. The CAPM requires (a) quadratic utility or normal returns, and (b) that the market portfolio is mean-variance efficient. Ross showed that neither assumption is needed. The pricing relation emerges from a much weaker condition: in a large economy, well-diversified portfolios with zero net investment and zero systematic risk must earn zero expected return.

The theory does not specify what the factors are or how many exist. This agnosticism is both a strength (generality) and a weakness (lack of testable restrictions without specifying factors). Empirical implementations must identify the factors, leading to the statistical factor models and characteristic-based models that followed.

Relationship to other models

The capm is a special case of the APT with a single factor (the market return). The fama-french-three-factor, fama-french-five-factor, and q-factor-model can all be interpreted as empirical APT models with specific factor choices.

The APT also provides the theoretical justification for commercial risk models like msci-barra-gem and mac3-global-equity-risk-model, which use multiple factors to model the covariance structure of returns.

Limitations

• The approximate pricing relation has an error term that vanishes asymptotically but may matter in finite samples
• Factor identification is not specified by the theory
• Testability debates (Shanken 1982; Dybvig and Ross 1985) parallel the Roll (1977) critique of the CAPM

Sources

• The Arbitrage Theory of Capital Asset Pricing (File, DOI)