On the Arbitrage Pricing Theory

Christian Gilles and Stephen LeRoy

Economic Theory, 1991, vol. 1, issue 3, 213-29

Abstract: The Arbitrage Pricing Theory relates the expected rates of return on a sequence of primitive securities to their factor exposures, suggesting that factor risk is of critical importance in asset pricing. However, we show that if the sequence of primitive returns is replaced by a sequence of returns on portfolios formed from the primitive securities, then the factor subspace is arbitrary. The implication is that the theorems relating expected returns to factor risk require substantial reinterpretation. Our reinterpretation consists of a demonstration that exact and approximate factor pricing do not constitute substantive characterizations of asset pricing. Instead, they are implications of the characterization of the returns space as a Hilbert space (exact factor pricing corresponds to the Riesz representation theorem and approximate factor pricing is a consequence of Bessel's inequality).

Date: 1991
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