Transform approach for operational risk modeling: value-at-risk and tail conditional expectation

ABSTRACT To quantify the aggregate losses from operational risk, we employ an actuarial risk model, ie, we consider the compound Cox model of operational risk to deal with the stochastic nature of its frequency rate in real situations. A shot noise process is used for this purpose. A compound Poisson model is also considered as its counterpart for the case where the operational loss frequency rate is deterministic. As the loss amounts arising due to mismanagement of operational risks are extremes in practice, we assume the loss sizes are log gamma, Fréchet and truncated Gumbel. We also use an exponential distribution for the case of non-extreme losses. Employing a loss distribution approach, we derive the analytical/explicit forms of the Laplace transform of the distribution of aggregate operational losses. The value-at-risk (VaR) and tail conditional expectation (TCE, also known as TailVaR) are used to evaluate the operational risk capital charge. Fast Fourier transform is used to approximate VaR and TCE numerically and the figures of the distributions of aggregate operational losses are provided. Numerical comparisons of VaR and TCE obtained using two compound processes are also made.