The empirical tests of the CAPM exposed two problems: the single-beta model misses systematic return variation captured by size, book-to-market, and other factors; and Roll's critique makes it logically impossible to confirm or deny the CAPM because the true market portfolio is unobservable. The Arbitrage Pricing Theory (APT), introduced by Ross (1976), resolves both problems. It derives multi-factor pricing from a weaker premise — no-arbitrage in large markets — without ever requiring the identification of a market portfolio.
The core logic: if returns are generated by a small number of common factors, then any well-diversified portfolio that has zero exposure to all factors carries no risk at all. A no-arbitrage argument forces its expected return to equal the riskless rate. From this, the equilibrium pricing equation follows: each asset's expected return is a linear function of its factor loadings, with factor risk premia as coefficients. The equation is structurally identical to the multi-index model — the APT's contribution is showing why that structure must arise from no-arbitrage, and what it implies for the risk premia.
Learning objectives. By the end of this note you should be able to:
- state the APT return-generating process and define factors, factor loadings, and factor risk premia;
- sketch the simple (geometric) proof and the algebraic no-arbitrage proof of the APT pricing equation;
- explain three approaches to estimating the APT (factor analysis, pre-specified macro variables, pre-specified portfolios);
- describe the Chen-Roll-Ross macro factors and the Fama-French portfolio factors;
- compare the APT with the CAPM on testability, flexibility, and guidance on factor signs.
In this lesson
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- The APT return-generating process
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- Simple proof: the geometric argument
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- Rigorous proof: the orthogonality argument
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- Estimating the APT: three approaches
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- APT vs CAPM: a comparative assessment
Before you start
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