# Simulation of the Annual Loss Distribution in Operational Risk Via Panjer Recursions and Volterra Integral Equations for Value at Risk and Expected Shortfall Estimation.

27 Pages Posted: 5 Jun 2017

See all articles by Gareth Peters

## Gareth Peters

Department of Actuarial Mathematics and Statistics, Heriot-Watt University; University College London - Department of Statistical Science; University of Oxford - Oxford-Man Institute of Quantitative Finance; London School of Economics & Political Science (LSE) - Systemic Risk Centre; University of New South Wales (UNSW) - Faculty of Science

University of Bristol - Department of Mathematics

## Arnaud Doucet

University of Cambridge - Department of Engineering

Date Written: June 4, 2017

### Abstract

Following the Loss Distributional Approach (LDA), this article develops two procedures for simulation of an annual loss distribution for modeling of Operational Risk. First, we provide an overview of the typical compound-process LDA used widely in Operational Risk modeling, before expanding upon the current literature on evaluation and simulation of annual loss distributions. We present two novel Monte Carlo simulation procedures. In doing so, we make use of Panjer recursions and the Volterra integral equation of the second kind to reformulate the problem of evaluation of the density of a random sum as the calculation of an expectation. We demonstrate the use of importance sampling and trans-dimensional Markov Chain Monte Carlo algorithms to efficiently evaluate this expectation. We further demonstrate their use in the calculation of Value at Risk and Expected Shortfall.

Keywords: Importance Sampling; Trans-dimensional Markov Chain Monte Carlo; Basel II Advanced Measurement Approach; Panjer Recursions; Volterra Integral Equations; Compound Processes; Loss Distributional Approach; Operational Risk; Value at Risk; Expected Shortfall

Suggested Citation

Peters, Gareth and Johansen, Adam M. and Doucet, Arnaud, Simulation of the Annual Loss Distribution in Operational Risk Via Panjer Recursions and Volterra Integral Equations for Value at Risk and Expected Shortfall Estimation. (June 4, 2017). Available at SSRN: https://ssrn.com/abstract=2980408 or http://dx.doi.org/10.2139/ssrn.2980408

## Paper statistics 