Time consistency of the mean-risk problem

Choosing a portfolio of risky assets over time that maximizes the expected returnat the same time as it minimizes portfolio risk is a classical problem in mathematicalfinanceandisreferredtoasthedynamicMarkowitzproblem(whentheriskismeasuredbyvariance)or, more generally, the dynamic mean-risk problem. In most of the literature, the mean-riskproblem is scalarized, and it is well known that this scalarized problem does not satisfythe (scalar) Bellman’s principle. Thus, the classical dynamic programming methods are notapplicable. For the purpose of this paper we focusonthediscretetimesetup,andwewilluseatime-consistent dynamic convex risk measure to evaluate the risk of a portfolio. We will showthat, when we do not scalarize the problem but leave it in its original form as a vector op-timization problem, the upper images, whose boundaries contain the efficient frontiers, recursebackward in time under very mild assumptions.Thus, the dynamic mean-risk problem doessatisfy a Bellman’s principle, but a more general one, that seems more appropriate for a vectoroptimization problem: a set-valued Bellman’s principle. We will present conditions underwhichthisrecursioncanbeexploited directly to compute a solution in the spirit of dynamicprogramming. Numerical examples illustrate the proposed method. The obtained results openthe door for a new branch in mathematics: dynamic multivariate programming.