We study upper and lower bounds on the expectile risk measure of risky portfolios when the joint distribution of the risky components is not fully specified. First, we summarize methods for obtaining bounds when only the marginal distributions of the components are known, but not their...

Dybvig ( 1988a , 1988b ) solves in a complete market setting the problem of finding a payoff that is cheapest possible in reaching a given target distribution (“cost-efficient payoff”). In the presence of ambiguity, the distribution of a payoff is, however, no longer known with certainty. We...

Actuarial fairness pertains to the situation in which the price of an insurance contract is equal to its expected outcome. We show that actuarial fairness leads to "unfairness" in that annuitants with higher survival rates can choose a better payoff in the sense of second-order stochastic...