Mean–Variance Portfolio Optimization with State‐Dependent Risk Aversion

24 Pages Posted: 13 Dec 2013

See all articles by Tomas Bjork

Tomas Bjork

Stockholm School of Economics - Swedish House of Finance

Agatha Murgoci

Aarhus University - School of Business and Social Sciences

Xun Yu Zhou

The Chinese University of Hong Kong (CUHK) - Department of Systems Engineering & Engineering Management

Date Written: January 2014

Abstract

The objective of this paper is to study the mean–variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time‐inconsistent control developed in Björk and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.

Keywords: mean–variance, time inconsistency, time‐inconsistent control, dynamic programming, stochastic control, Hamilton–Jacobi–Bellman equation

Suggested Citation

Bjork, Tomas and Murgoci, Agatha and Zhou, Xun Yu, Mean–Variance Portfolio Optimization with State‐Dependent Risk Aversion (January 2014). Mathematical Finance, Vol. 24, Issue 1, pp. 1-24, 2014. Available at SSRN: https://ssrn.com/abstract=2367076 or http://dx.doi.org/10.1111/j.1467-9965.2011.00515.x

Tomas Bjork (Contact Author)

Stockholm School of Economics - Swedish House of Finance ( email )

Drottninggatan 98
111 60 Stockholm
Sweden

Agatha Murgoci

Aarhus University - School of Business and Social Sciences ( email )

Nordre Ringgade 1
Aarhus C, DK-8000
Denmark

Xun Yu Zhou

The Chinese University of Hong Kong (CUHK) - Department of Systems Engineering & Engineering Management ( email )

Shatin, New Territories
Hong Kong
852 2609-8320 (Phone)
852 2603-5505 (Fax)

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