Dynamic mean–VaR portfolio selection in continuous time

Abstract

The value-at-risk (VaR) is one of the most well-known downside risk measures due to its intuitive meaning and wide spectra of applications in practice. In this paper, we investigate the dynamic mean–VaR portfolio selection formulation in continuous time, while the majority of the current literature on mean–VaR portfolio selection mainly focuses on its static versions. Our contributions are twofold, in both building up a tractable formulation and deriving the corresponding optimal portfolio policy. By imposing a limit funding level on the terminal wealth, we conquer the ill-posedness exhibited in the original dynamic mean–VaR portfolio formulation. To overcome the difficulties arising from the VaR constraint and no bankruptcy constraint, we have combined the martingale approach with the quantile optimization technique in our solution framework to derive the optimal portfolio policy. In particular, we have characterized the condition for the existence of the Lagrange multiplier. When the opportunity set of the market setting is deterministic, the portfolio policy becomes analytical. Furthermore, the limit funding level not only enables us to solve the dynamic mean–VaR portfolio selection problem, but also offers a flexibility to tame the aggressiveness of the portfolio policy.

Keywords:

• Continuous time models
• Martingales
• Portfolio optimization
• Risk management
• Value at risk

AMS Subject Classifications:

• G11
• C61

Acknowledgements

The third author is also grateful to the support from Patrick Huen Wing Ming Chair Professorship of Systems Engineering & Engineering Management.

Notes

No potential conflict of interest was reported by the authors.

1 We denote the essential upper and lower bounds of z(T) as and , respectively. Generally speaking, we always have and . However, when is a stochastic process, it is possible that and . To simplify the notation, we only discuss the case and in this paper. Note that assumption 3.1 is stronger than the atomless assumption commonly used in the literature.

2 ‘a.s.’ stands for ‘almost surely’, which excludes events with zero occurrence probability. In the following discussion, we simply ignore such a term for the random variables that satisfy certain condition.

3 The PDE derived in Wachter (2002) is from the expected utility maximization with CRRA utility, which yields a smooth boundary at time T.

Additional information

Funding

This research work was partially supported by the Key Project of National Natural Science Foundation of China [grant number 71431008]; Natural Science Foundation of China [grant number 61573244], [grant number 71201102], [grant number 71671106], [grant number 71201094]; the State Key Program in the Major Research Plan of National Natural Science Foundation of China [grant number 91546202]; Program for Innovative Research Team of Shanghai University of Finance and Economicsand Hong Kong Research Grants Council [grant number CUHK 414513], [grant number 14204514].