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

The arbitrage pricing theory (APT) attributes differences in expected returns to exposure to systematic risk factors. Two aspects of the APT are considered. Firstly, the factors in the statistical asset pricing model are related to a theoretically consistent set of factors defined by their conditional covariation with the stochastic discount factor (SDF) used to price securities within inter-temporal asset pricing models. It is shown that risk premia arise from non-zero correlation of observed factors with SDF and the pricing errors arise from the correlation of the errors in the statistical model with SDF. Secondly, the estimates of factor risk premia using portfolios are compared to those obtained using individual securities. It is shown that in the presence of pricing errors consistent estimation of risk premia requires a large number of not fully diversified portfolios. Also, in general, it is not possible to rank estimators using individual securities and portfolios in terms of their small sample bias.