On Some Layer-Based Risk Measures with Applications to Exponential Dispersion Models

Denote by X the set of actuarial risks, and let 0 ≤ X ∈ X be a random variable rv with cumulative distribution function cdf F x , decumulative distribution function ddf F x 1 − F x , and probability density function pdf f x . The functional H : X → 0,∞ is then referred to as a risk measure, and it is interpreted as the measure of risk inherent in X. Naturally, a quite significant number of risk measuring functionals have been proposed and studied, starting with the arguably oldest Value-at-Risk or VaR cf. 1 , and up to the distorted cf. 2–5 and weighted cf. 6, 7 classes of risk measures. More specifically, the Value-at-Risk risk measure is formulated, for every 0 < q < 1, as