This paper is devoted to the perturbation theory for defective matrices. We consider the asymptotic expansions of the perturbed spectrum when a matrix $A$ is changed to $A+tE$, where $E\ne 0$ and $t>0$ is a small parameter. In particular, we analyse the rational exponents that may occur when the matrix $E$ varies over the sphere $\Vert E\Vert =\rho >0$. We partially characterize the leading exponents noting that the description of the set of all leading exponents remains an open problem.

A considerable number of equivalent formulas defining conditional value-at-risk and expected shortfall are gathered together. Then we present a simple method 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.

A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951).