Operational risk and insurance: a ruin-probabilistic reserving approach

Vladimir K. Kaishev, Dimitrina S. Dimitrova, Zvetan G. Ignatov

Abstract

ABSTRACT

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, within a general risk model. It allows for inhomogeneous operational loss frequency (dependent interarrival times) and dependent loss severities that may have any joint discrete or continuous distribution. Under the proposed methodology, operational risk capital assessment is viewed not as a one-off exercise, performed at some moment of time, but as dynamic reservation, following a certain risk capital accumulation function. The latter describes the accumulation of risk capital with time and may be any non-decreasing, positive real function h(t). Under these reasonably general assumptions, the probability of non-ruin is explicitly expressed using closed-form expressions, derived by Ignatov and Kaishev (2000, 2004, 2007) and Ignatov et al (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 also a (dynamic) risk capital accumulation strategy, h(t). In view of its generality, the proposed methodology is capable of accommodating any (heavy-tailed) distribution, such as the generalized Pareto distribution, the lognormal distribution, the g-and-h distribution and the generalized beta distribution of the second kind. Applying this 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 the same high enough survival probability may be achieved by different risk capital accumulation strategies, one of which may be preferable to accumulating capital just linearly, as has been assumed by Embrechts et al 2004). The proposed methodology also takes into account 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%.