This was part of

**Dynamic Assessment Indices**## Time consistency and set-valued Bellman’s principle: evidence from mean-risk problem and acceptability maximization

**
Gabriela Kovacova, Vienna University of Economics and BusinessWednesday, May 11, 2022
**

**Abstract**: In this talk we consider time consistency properties of two problems of financial mathematics: Selecting a portfolio of risky assets which maximizes the expected terminal values at the same time as it minimizes portfolio risk is 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 satisfies an appropriate notion of time consistency, closely related to existence of a moving scalarization. Additionally, the mean-risk problem satisfies a Bellman's principle appropriate for a bi-objective optimization problem: a set-valued Bellman's principle. Acceptability maximization is the optimal investment problem, where a coherent acceptability index (CAI) is used to measure the portfolio performance. Using robust representations of dynamic CAIs in terms of a family of dynamic coherent risk measures (DCRMs), we establish an intriguing dichotomy: if the corresponding family of DCRMs and the market structure are recursive, then the acceptability maximization problem reduces to just a one-period problem and the maximal acceptability is constant across all states and times. On the other hand, if the family of DCRMs is not recursive, which is often the case, then the acceptability maximization problem ordinarily is a time-inconsistent stochastic control problem. For two particular dynamic CAIs - the dynamic risk-adjusted return on capital and the dynamic gain-to-loss ratio - we overcome this issue by considering related bi-objective problems and applying the set-valued Bellman principle.