Abstract
This paper studies a mean-risk portfolio choice problem for log-returns in a continuous-time, complete market. It is a growth-optimal portfolio choice problem under risk control. The risk of log-returns is measured by weighted Value-at-Risk (WVaR), which is a generalization of Value-at-Risk (VaR) and Expected Shortfall (ES). We characterize the optimal terminal wealth and obtain analytical expressions when risk is measured by VaR or ES. We demonstrate that using VaR increases losses while ES reduces losses during market downturns. Moreover, the efficient frontier is a concave curve that connects the minimum-risk portfolio with the growth optimal portfolio, as opposed to the vertical line when WVaR is used on terminal wealth, and thus allows for a meaningful characterization of the risk-return trade-off and aids investors in setting reasonable investment targets. We also apply our model to benchmarking and illustrate how investors with benchmarking may overperform/underperform the market depending on economic conditions.
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