  A risk measure that is consistent with the second-order stochastic dominance
and additive for sums of independent random variables can be represented as a
weighted entropic risk measure (WERM). The expected utility maximization
problem with risk controlled by WERM and a related risk minimization problem
are investigated in this paper. The latter is same to a problem of maximizing a
weighted average of constant-absolute-risk-aversion (CARA) certainty
equivalents. The solutions of all the optimization problems are explicitly
characterized and an iterative method of the solutions is provided.
