Duality in mean-variance frontiers with conditioning information

Portfolio and stochastic discount factor mean-variance frontiers are usually regarded as dual objects. However, the Hansen and Richard (1987) and Gallant, Hansen and Tauchen (1990) unconditional frontiers are not dual unless some strong conditions hold. We characterise the objects that are always dual to those frontiers, which are not generally proper SDFs or returns. We avoid the common practice of parametrically specifying conditional moments of returns, estimating instead the frontiers with easily implementable sieve methods, which have a managed portfolio interpretation. We empirically assess the validity of SDFs with constant risk prices and the relevance of predictability for portfolio choice.