| Summary: | We study a continuous-time mean-risk portfolio choice problem in which an agent, with or without the bankruptcy constraint, chooses among the portfolios that achieve an exogenously given expected terminal wealth target with the objective of minimizing the risk of his portfolio. The risk is measured either by a so-called weighted value-at-risk risk measure, which is a generalization of value-at-risk and conditional value-at-risk, or by a law-invariant coherent risk measure. For the WVaR case, we obtain sufficient and necessary conditions for the well-posedness of the problem and the existence of the optimal solution. We find that the optimal value does not depend on the expected terminal wealth target, implying a vertical efficient frontier. In addition, the asymptotically optimal strategy for the agent is to invest little in an extremely risky but highly rewarded lottery and invest the rest in the risk-free asset in the absence of the bankruptcy constraint or in an asset with medium risk in the presence of the bankruptcy constraint. For the case of coherent risk measures, we obtain similarly that the optimal value is independent of the expected terminal wealth target.
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