Time consistency of the mean-risk problem

Kováčová, Gabriela and Rudloff, Birgit ORCID: https://orcid.org/0000-0003-1675-5451 (2021) Time consistency of the mean-risk problem. Operations Research, 69 (4). pp. 1100-1117. ISSN 1526-5463

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Abstract

Choosing a portfolio of risky assets over time that maximizes the expected return at the same time as it minimizes portfolio risk is a classical problem in mathematical finance and is referred to a sthe dynamic Markowitz problem (when the risk is measured by variance)or, more generally, the dynamic mean-risk problem. In most of the literature, the mean-risk problem 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 not applicable. For the purpose of this paper we focus on the discrete time setup, and we will use a time-consistent dynamic convex risk measure to evaluate the risk of a portfolio. We will show that, when we do not scalarize the problem but leave it in its original form as a vector optimization problem, the upper images, whose boundaries contain the efficient frontiers, recurse backward in time under very mild assumptions. Thus, the dynamic mean-risk problem does satisfy a Bellman’s principle, but a more general one, that seems more appropriate for a vector optimization problem: a set-valued Bellman’s principle. We will present conditions under which this recursion can be exploited directly to compute a solution in the spirit of dynamic programming. Numerical examples illustrate the proposed method. The obtained results open the door for a new branch in mathematics: dynamic multivariate programming.

Item Type: Article
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Keywords: mean-risk problem, portfolio selection problem, vector optimization, dynamic programming, Bellman’s principle, algorithms
Divisions: Departments > Finance, Accounting and Statistics > Statistics and Mathematics
Version of the Document: Published
Depositing User: Gertraud Novotny
Date Deposited: 23 Mar 2021 14:31
Last Modified: 16 Nov 2021 16:18
Related URLs:
FIDES Link: https://bach.wu.ac.at/d/research/results/86431/
URI: https://epub.wu.ac.at/id/eprint/8053

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