Model uncertainty assessment for symmetric and right-skewed distributions

In actuarial modeling, right-skewed distributions are of paramount importance. Typically, they have the property of becoming unimodal and symmetric after applying some increasing concave transformation $ \mathsf {T} $ T (e.g. log-transformation). In what follows, we refer to them as $ \mathsf {T} $ T-unimodal $ \mathsf {T} $ T-symmetric distributions. In this paper, we derive attainable bounds for the Value-at-Risk for such distributions when some partial information is available, such as information on the support, median, or certain moments. We also derive explicit upper and lower bounds for the Range Value-at-Risk under knowledge of unimodality, symmetry, mean, variance, and possibly bounded support. In passing, we provide a generalization of the Gauss inequality for symmetric distributions with known support. We show how these bounds improve on bounds available in the literature using a real-world automobile insurance claims dataset.