A new methodology for financial and insurance operational risk capital estimation is proposed. It is based on using the finite time probability of (non)ruin as an operational risk measure, under a general ruin probability model, according to which operational losses may have any joint (dependent) discrete or continuous distribution, and the function, describing the accumulation of risk capital may be any nondecreasing, positive real function hHtL. The probability of nonruin is explicitly expressed using closed form expressions, derived by Kaishev (2000, 2004) and Ignatov Kaishev and Krachunov (2001) and by setting it to a high enough preassigned value, say 0.99, it is possible to obtain not just a value for the capital charge but a (dynamic) risk capital accumulation strategy hHtL.In view of its generality, the proposed methodology is capable of accommodating any (heavy tailed) distributions, such as the Generalized Pareto Distribution, the Lognormal distribution the g-and-h distribution and the GB2 distribution. Applying our methodology on numerical examples, we demonstrate that dependence in the loss severities may have a dramatic effect on the estimated risk capital. In addition we also show that one and the same high enough survival probability may be achieved by different risk capital accumulation strategies one of which may possibly be preferable to accumulating capital just linearly, as has been assumed by Embrechts et al. (2004). The proposed methodology takes into account also the effect of insurance on operational losses, in which case it is proposed to take the probability of joint survival of the financial institution and the insurance provider as a joint operational risk measure. The risk capital allocation strategy is then obtained in such a way that the probability of joint survival is equal to a preassigned high enough value, say 99.9 %.
An explicit formula for the probability of nonruin of an insurance company in a finite time interval is derived, assuming Poisson claim arrivals, any continuous joint distribution of the claim amounts and any nonnegative, increasing real function representing its premium income. The formula is compact and expresses the nonruin probability in terms of Appell polynomials. An example, illustrating its numerical convenience, is also given in the case of inverted Dirichlet-distributed claims and a linearly increasing premium-income function.
This is the accepted version of the paper.This version of the publication may differ from the final published version. A closed form expression, in terms of some functions which we call exponential Appell polynomials, for the probability of non-ruin of an insurance company, in a finite-time interval is derived, assuming independent, non-identically Erlang distributed claim inter-arrival times,
Permanent repository link. ., any continuous joint distribution of the claim amounts and any non-negative, non-decreasing real function, representing its premium income. In the special case when τ i ∼ Erlang (g i , λ) , i = 1, 2, . . . it is shown that our main result yields a formula for the probability of non-ruin expressed in terms of the classical Appell polynomials. We give another special case of our non-ruin probability formula for. ., i.e., when the inter-arrival times are non-identically exponentially distributed and also show that it coincides with the formula for Poisson claim arrivals, given in [18], when τ i ∼ Erlang(1, λ), i = 1, 2, . . .. The main result is extended further to a risk model in which inter-arrival times are dependent random variables, obtained by randomizing the Erlang shape or/and rate parameters. We give also some useful auxiliary results which characterize and express explicitly (and recurrently) the exponential Appell polynomials which appear in our finite time non-ruin probability formulae.
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