We introduce a portfolio framework that jointly integrates investors' aversion to downside risk and distributional ambiguity. We measure downside risk with the lower partial moment and capture ambiguity with a distributionally robust formulation using a Wasserstein ball around a reference distribution of asset returns. Our problem is general and allows for various downside risk measures, orders of the Wasserstein distance, and choices of the cost function. We provide a convex dual formulation of the distributionally robust problem and characterize the Slater condition analytically. When the cost function is the Mahalanobis distance and the reference distribution is elliptical, we derive an analytical solution for the optimal portfolio and show that it is located on the sample mean-variance efficient frontier. For other cost functions, the optimal portfolio can be located outside of the frontier. We also establish formal relations between our framework and two classical portfolio paradigms showing that the risk-aversion and ambiguity-aversion coefficients can be linked to a more intuitive downside return threshold. An empirical analysis on datasets of characteristic-sorted portfolios and individual stocks shows that our portfolio strategies deliver gains in terms of downside risk, risk-return tradeoff, and turnover relative to distributionally robust mean-variance strategies, non-robust strategies, and naive diversification.
