Loading [MathJax]/jax/output/SVG/fonts/TeX/fontdata.js
Jeong, Himchan, Emiliano A. Valdez, Jae Youn Ahn, and Sojung Carol Park. 2021. “Generalized Linear Mixed Models for Dependent Compound Risk Models.” Variance 14 (1).
Download all (4)
  • Figure 1. Frequency and average severity by calendar year
  • Figure 2. Graphical relationship of frequency and average severity, per policyholder
  • Figure 3. log-QQ plots of fitting gamma to average severity for each calendar year
  • Figure 4. The Lorenz curve and the Gini index values for the five models


In ratemaking, calculation of a pure premium has traditionally been based on modeling frequency and severity in an aggregated claims model. For simplicity, it has been a standard practice to assume the independence of loss frequency and loss severity. In recent years, there has been sporadic interest in the actuarial literature exploring models that depart from this independence. In this paper, the authors extend the work of Garrido, Genest, and Schulz (2016), which uses generalized linear models (GLMs) that account for dependence between frequency and severity and simultaneously incorporate rating factors to capture policyholder heterogeneity. In addition, they quantify and explain the contribution of the variability of claims among policyholders through the use of random effects using generalized linear mixed models (GLMMs). The authors calibrated their model using a portfolio of auto insurance contracts from a Singapore insurer where they observed claim counts and amounts from policyholders for a period of six years. They compared their results with the dependent GLM considered by Garrido, Genest, and Schulz; Tweedie models; and the case of independence. The dependent GLMM shows statistical evidence of positive dependence between frequency and severity. Using validation procedures, the authors find that the results demonstrate a superior model when random effects are considered within a GLMM framework.

Himchan Jeong and Emiliano A. Valdez were supported by the CAE Research Grant on Applying Data Mining Techniques in Actuarial Science funded by the Society of Actuaries (SOA).

Sojung Carol Park acknowledges support from the Institute of Management Research at Seoul National University.

Accepted: May 21, 2018 EDT