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).
• Table 1. Observable policy characteristics used as covariates
• Table 2. Percentage and number of claims by count and year
• Table 3. Average severity (AvgSev) by claim count and calendar year
• Figure 1. Frequency and average severity by calendar year
• Figure 2. Graphical relationship of frequency and average severity, per policyholder
• Table 4. Goodness-of-fit test for the frequency component
• Figure 3. log-QQ plots of fitting gamma to average severity for each calendar year
• Table 5. Regression estimates of the negative binomial model for frequency
• Table 6. Regression estimates of the gamma model for average severity
• Table 7. Regression estimates for the aggregate loss models based on Tweedie
• Table 8. Validation measures for the five models
• Figure 4. The Lorenz curve and the Gini index values for the five models

## Abstract

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