Hansen and Jagannathan (HJ, 1991) provided bounds on the volatility of Stochastic Discount Factors (SDF) that proved extremely useful to diagnose and test asset pricing models. This nonparametric bound reflects a duality between the meanstandard deviation frontier for SDFs and the mean-variance frontier for portfolios of asset returns. We extend this fundamental contribution by proposing information bounds that minimize general convex functions of SDFs directly taking into account higher moments of returns. These Minimum Discrepancy bounds reflect a duality with finding the optimal portfolio of asset returns with a general HARA utility function. The maximum utility portfolio implies SDF estimators that are based on implied probabilities associated with the class of Generalized Empirical Likelihood estimators. We analyze the implications of these information bounds for the pricing of size portfolios and the performance evaluation of hedge funds
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