  This paper studies a mean-risk portfolio choice problem for log-returns in a
continuous-time, complete market. This is a growth-optimal problem with 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 up to the concave envelope of a
certain function, and obtain analytical expressions for the optimal wealth and
portfolio policy when the risk is measured by VaR or ES. In addition, we find
that 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. Our results advocate the use of mean-WVaR
criterion for log-returns instead of terminal wealth in dynamic portfolio
choice.
