Conditional value-at-risk bounds for compound Poisson risks and a normal approximation

A considerable number of equivalent formulas defining conditional value-at-risk and expected shortfall are gathered together. Then we present a simplemethod to bound the conditional value-at-risk of compound Poisson loss distributions under incomplete information about its severity distribution, which is assumed to have a known finite range, mean, and variance. This important class of nonnormal loss distributions finds applications in actuarial science, where it is able to model the aggregate claims of an insurance-risk business.