Dynamic Portfolio Choice When Risk Is Measured by Weighted VaR

We seek to characterize the trading behavior of an agent, in the context of a continuous-time portfolio choice model, if she measures the risk by a so called weighted value-at-risk (VaR), which is a generalization of both VaR and conditional VaR. We show that when bankruptcy is allowed the agent displays extreme risk-taking behaviors, unless the downside risk is significantly penalized, in which case an asymptotically optimal strategy is to invest a very small amount of money in an extremely risky but highly rewarding lottery, and save the rest in the risk-free asset. When bankruptcy is prohibited, extreme risk-taking behaviors are prevented in most cases in which the asymptotically optimal strategy is to spend a very small amount of money in an extremely risky but highly rewarding lottery and put the rest in an asset with moderate risk. Finally, we show that the trading behaviors remain qualitatively the same if the weighted VaR is replaced by a law-invariant coherent risk measure.