We provide analytical results for a static portfolio optimization problem with two coherent risk measures. The use of two risk measures is motivated by joint decision-making for portfolio selection where the risk perception of the portfolio manager is of primary concern, hence, it appears in the objective function, and the risk perception of an external authority needs to be taken into account as well, which appears in the form of a risk constraint. The problem covers the risk minimization problem with an expected return constraint and the expected return maximization problem with a risk constraint, as special cases. For the general case of an arbitrary joint distribution for the asset returns, under certain conditions, we characterize the optimal portfolio as the optimal Lagrange multiplier associated to an equality-constrained dual problem. Then, we consider the special case of Gaussian returns for which it is possible to identify all cases where an optimal solution exists and to give an explicit formula for the optimal portfolio whenever it exists.

我们提供具有两个相关风险度量的静态投资组合优化问题的分析结果。在选择投资组合经理时，联合决策的动机是使用两种风险度量，因为投资组合经理的风险感知是首要考虑因素，因此，它出现在目标函数中，需要考虑外部机构的风险感知也考虑在内，这以风险约束的形式出现。作为特殊情况，该问题涵盖具有预期收益约束的风险最小化问题和具有风险约束的预期收益最大化问题。对于资产收益率的任意联合分布的一般情况，在某些条件下，我们将最优投资组合表征为与等式约束对偶问题相关的最优拉格朗日乘数。然后，