Discussion Paper: “The Mathematics of Excess Losses”

By Liang Hong

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My congratulations to Mr. Leigh J. Halliwell on this paper that clearly presents the mathematics of excess losses with an interesting example. I agree with him that the mathematics of excess losses is beautiful and powerful. However, the mathematics of excess losses also contains several subtle points that are not mentioned in the paper. This discussion note complements the article by clarifying some of these points. To be clear, it is not my intention to be critical of Mr. Halliwell. The purpose of this note is two-fold:

1. To clarify some important hidden points in the mathematics of excess losses;

2. To give references to some uncredited results.

For those ambitious actuaries who want to dig deeper for a full understanding of the rigorous mathematics of excess losses, this note also provides some directions for further studies.

For the convenience of readers, we will adopt the notations in Halliwell (2013). Throughout this note, X will denote a nonnegative random variable. F and G will denote the cumulative distribution function (CDF) and survival function of X, respectively.

Keywords: Excess Loss; Loss Distributions


Hong, Liang, "Discussion Paper: “The Mathematics of Excess Losses”," Variance 9:1, 2015, pp. 11-19.

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Variance (ISSN 1940-6452) is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Significant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submissions by non-CAS members is encouraged.