Heat Equation as a Tool for Outliers Mitigation in Run-Off Triangles for Valuing the Technical Provisions in Non-Life Insurance Business

Abstract

Estimating outstanding claims reserves in the non-life insurance business is often impaired by outlier-contaminated datasets. Widely used methods to eliminate outliers in non-life development triangles are either limiting the number of outliers by robust statistical methods or by change of development factors. However, the whole estimation process is likewise adversely affected so that: (i) the total sum of all triangle payments is not correct or (ii) the difference between the original triangle and its backward estimation via the bootstrap method is ineligible. In this paper, the properties of the heat equation are examined to obtain an outlier smoothing technique for development triangles. The heat equation in two dimensions is being applied on an outlier contaminated dataset where no individual data are available. As a result, we introduce a new methodology to (i) treat outliers in non-life development triangles, (ii) keep the total sum of all triangle payments, and (iii) provide acceptable differences between the original and the backward estimated triangle. Consequently, the outlying values are eliminated and the resulting development triangle could be used as an input for any claims reserving method without a need for further robustification or change of development factors. Additionally, the research on the application of heat equation in one dimension presented in this paper enables one to employ the bootstrap method using Pearson’s residuals in cases where the method was originally inapplicable due to development factors being lower than one.