Various approximations of the total aggregate loss quantile function with application to operational risk

A compound Poisson distribution is the sum of independent and identically distributed random variables over a count variable that follows a Poisson distribution. Generally, this distribution is not tractable. However, it has many practical applications that require the estimation of the quantile function at a high percentile, eg, the 99.9th percentile. Without loss of generality, this paper focuses on the application to operational risk. We assume that the support of random variables is nonnegative, discrete and finite. We investigate the mechanics of the empirical aggregate loss bootstrap distribution and suggest different approximations of its quantile function. Furthermore, we study the impact of empirical moments and large losses on the empirical bootstrap capital at the 99.9% confidence level.