Dynamic Mean-VaR Portfolio Selection in Continuous Time

Abstract

In the existing literature, the value-at-risk (VaR) is one of the most representative downside risk measures due to its wide spectra of applications in practice. In this paper, we investigate the dynamic mean-VaR portfolio selection formulation, while the state-of-the-art has only witnessed static versions for mean-VaR portfolio selection. Our contributions are two-fold, 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 such that to derive the optimal portfolio policy. In particular, we have characterized the condition of 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: Dynamic portfolio selection, Value-at-risk, Quantile Method

JEL Classification: G11, C61