Portfolio optimization under the Value-at-Risk constraint

Abstract

In this paper we analyse the effects arising from imposing a Value-at-Risk constraint in an agent's portfolio selection problem. The financial market is incomplete and consists of multiple risky assets (stocks) plus a risk-free asset. The stocks are modelled as exponential Brownian motions with random drift and volatility. The risk of the trading portfolio is re-evaluated dynamically, hence the agent must satisfy the Value-at-Risk constraint continuously. We derive the optimal consumption and portfolio allocation policy in closed form for the case of logarithmic utility. The non-logarithmic CRRA utilities are considered as well, when the randomness of market coefficients is independent of the Brownian motion driving the stocks. The portfolio selection, a stochastic control problem, is reduced, in this context, to a deterministic control one, which is analysed, and a numerical treatment is proposed.

Keywords:

• Value-at-Risk (VaR)
• Utility functions
• Portfolio optimization
• Portfolio theory
• Portfolio management

Acknowledgements

The author would like to thank Ulrich Haussmann, Philip Loewen and two anonymous referees for helpful discussions and comments. This work was supported by NSERC under research grant 88051 and NCE grant 30354 (MITACS).