The Interplay between Utility and Risk in Portfolio Selection

In this talk we revisit the problem of portfolio selection, where an investor maximizes utility subject to a risk constraint. Our general framework accommodates a wide range of utility and risk functionals, including non-concave utilities such as those from prospect theory and non-convex risk measures such as Value at Risk. Our main contribution is a complete characterization of well-posedness for one-period utility-risk portfolio selection, accounting for the interplay between utility and risk objectives. Under mild regularity conditions, we show that well-posedness is characterized by a simple either-or criterion: either the utility or the risk functional must satisfy the axiom of sensitivity to large losses. We illustrate this criterion through several examples. For expected utility maximization without risk constraints, including non-concave utilities, we further show that well-posedness is fully characterized by the asymptotic loss-gain ratio, which captures the investor's relative weighting of large losses and gains.