Dependence uncertainty bounds for the expectile of a portfolio

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 interdependence (unconstrained bounds). In particular, we provide the best-possible upper bound and the best-possible lower bound (under some conditions), as well as numerical procedures to compute them. We also derive simple analytic bounds that appear adequate in various situations of interest. Second, we study bounds when some information on interdependence is available (constrained bounds). When the variance of the portfolio is known, a simple-to-compute upper bound is provided, and we illustrate that it may significantly improve the unconstrained upper bound. We also show that the unconstrained lower bound cannot be readily improved using variance information. Next, we derive improved bounds when the bivariate distributions of each of the risky components and a risk factor are known. When the factor induces a positive dependence among the components, it is typically possible to improve the unconstrained lower bound. Finally, the unconstrained dependence uncertainty spreads of expected shortfall, value-at-risk and the expectile are compared.