Arbitrage pricing theory, the stochastic discount factor and estimation of risk premia from portfolios

The arbitrage pricing theory (APT) attributes differences in expected returns to exposure to systematic risk factors, which are typically assumed to be strong. In this paper we consider two aspects of the APT. Firstly we relate the factors in the statistical factor model to a theoretically consistent set of factors defined by their conditional covariation with the stochastic discount factor (mt) used to price securities within inter-temporal asset pricing models. We show that risk premia arise from non-zero correlation of observed factors with mt; and the pricing errors arise from the correlation of the errors in the statistical factor model with mt: Secondly we compare estimates of factor risk premia using portfolios with the ones obtained using individual securities, and show that the identification conditions in terms of the strength of the factor are the same and that, in general, no clear cut ranking of the small sample bias of the two estimators is possible.