Abstract: Selecting a portfolio of risky assets which maximizes the expected terminal values at the same time as it minimizes portfolio risk is a classical problem in Mathematical Finance known as the mean-risk problem. The usual approach in the literature is to combine the mean and the risk to obtain a problem with a single objective. In a dynamic setting this scalarization, however, comes at the cost of time inconsistency.

We show that these difficulties disappear by considering the problem in its natural form, that is, as a bi-objective optimization problem. As such the mean-risk problem can be shown to satisfy an appropriate notion of time consistency, closely related to existence of a moving scalarization. Additionally, we show that the mean-risk problem satisfies a Bellman's principle appropriate for a bi-objective optimization problem: a set-valued Bellman's principle.

The talk is based on join work with Birgit Rudloff.