  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.
