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Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk by Pavel V. Shevchenko, Gareth W. Peters

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CHAPTERSEVEN

Single Risk Closed-Form Approximations of Asymptotic Tail Behaviour

In Chapter 6, a set of special sub-families of annual loss loss distribution approach (LDA) models were developed which admitted closed-form exact representations. That is, closed-form representations were obtained for the distribution of the annual loss Z = X1 + · · · + XN represented by a compound process model with LDA structure in which the frequency is N ~ Poisson(λ) and the severities are independent and identically distributed (i.i.d.) Xi(t) ~ FX(x), and N and Xi are independent. The exact distribution of the annual loss processes can be expressed analytically as a mixture distribution comprised convolved distributional components with Poisson mixing weights for N > 0,

images

Here, images = imagesr[X1 + X2 + · · · + Xnz] is the n-th convolution of the severity distribution FX(·) calculated recursively as

images

with images = 1 if z ≥ 0 and zero otherwise. Note, throughout this chapter we refer to F as the distribution for the severity, ...

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