Linear conditional expectation, return distributions, and capital asset pricing theories

We show that E[X(g(Y₁, …, Y_n)] (where E[.] is the expectation operator) can be decomposed into a product of two expected values plus a sum of n comovement terms, if X, Y₁, …, Y_n follow a distribution that admits linear conditional expectation (LCE). We then apply this relation to show that if each asset return is LCE distributed with the market and/or the factors, many capital asset pricing models and the mutual fund separation theorem can be obtained. A well-known example of a class of distributions that admits LCE is the elliptical distributions, of which the normal is a special case. A larger family, not mentioned in the existing literature, that admits LCE is the Pearson system. As a result, the distribution assumption to derive the capital asset pricing theories can be relaxed to the wider LCE family. We also present the relation of the LCE family to Ross's (1978) separating distribution family.