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Papush, Dmitry E., Aleksey S. Popelyukhin, and Jasmine G. Zhang. 2021. “Approximating the Aggregate Loss Distribution.” Variance 14 (2).
• Table 1. Distributions used for the approximation of aggregate loss
• Table 2a. Distributions used for casualty products
• Table 2b. Distributions used for property, non-catastrophe
• Table 3. Higher moments (skewness and excess kurtosis) in terms of CV
• Figure 2. Skewness/CV as a function of primary limit

## Abstract

Aggregate loss distributions have extensive applications in actuarial practice. Several approaches have been suggested to estimate the aggregate loss distribution, including the Heckman-Meyers method, the Panjer algorithm, and fast Fourier transformation, to name a few. All of these methods rely on separate assumptions about frequency and severity components of the aggregate losses. Quite often, however, obtaining frequency and severity expectations independently is not practical, and only aggregate information is available for analysis. In that case, the a priori assumption about the shape of the aggregate loss distribution becomes critical, especially for assessing the probability of very high aggregate loss values, in the tail.

In this work we seek to determine which statistical two-parameter distribution, out of several, serves best to approximate aggregate loss distributions for property and casualty products. We focus on ground-up losses limited by a per occurrence limit. These results are relevant for quota share agreements. In addition, we consider layer losses, the results of which are important for umbrella quota share transactions.

We simulate samples of aggregate loss, fit statistical distributions to the samples, and then use goodness-of-fit tests to determine the best-fitting distribution. In all realistic scenarios with limited losses, we find that the gamma distribution uniformly provides the most reasonable approximation to the aggregate loss.

Accepted: March 12, 2019 EDT

# Appendix X is aggregate loss
X^ is loss in an aggregate layer, namely max(min(X-x_1, 0), x_2-x_1) X is aggregate loss
X^ is loss in an aggregate layer, namely max(min(X-x_1, 0), x_2-x_1) X is aggregate loss
X^ is loss in an aggregate layer, namely max(min(X-x_1, 0), x_2-x_1) X is aggregate loss
X^ is loss in an aggregate layer, namely max(min(X-x_1, 0), x_2-x_1) X is aggregate loss
X^ is loss in an aggregate layer, namely max(min(X-x_1, 0), x_2-x_1) X is aggregate loss
X^ is loss in an aggregate layer, namely max(min(X-x_1, 0), x_2-x_1)

1. Indeed, the sum of identical exponentials has the characteristic function ${(1 - \frac{{it}}{\lambda})}^{- n}$ which is $Gamma(n,\lambda).$

2. According to Pentikäinen (1987), as long as the severity distribution is restricted to a limited interval, aggregate loss distributions with several matching moments approximate each other acceptably well.

3. Indeed, for Poisson distributed number of claims with mean $\lambda$: