Abstract

High order stop-loss transforms provide a risk measure that enables some flexibility on the weight given to high or low values of the risk. We interpret stop-loss transforms as iterated distributions and prove a recursive representation for risks expressed as convolutions. We apply this to the case of gamma distributions with integer shape parameter, the Erlang distributions, proving that high order stop-loss transforms are equivalent to the tails of the exponential distribution. The latter result is also extended to general gamma distributions. Furthermore, we prove that this equivalence to exponential tails does not hold in general, by proving that the stop-loss transform for a Weilbull distribution degenerates, unless, of course, in the exponential case.

摘要

高阶止损转换提供了一种风险度量，可以在赋予高风险或低风险值的权重方面提供一定的灵活性。我们将止损转换解释为迭代分布，并证明了以卷积表示的风险的递归表示。我们将此应用于具有整数形状参数的伽马分布的情况，即 Erlang 分布，证明高阶止损变换等效于指数分布的尾部。后一个结果也扩展到一般的伽马分布。此外，我们通过证明 Weilbull 分布的止损变换退化，证明这种与指数尾部的等价性通常不成立，当然，除非在指数情况下。