MODELING THE AGGREGATE LOSS DISTRIBUTION IN MOTOR THIRD-PARTY LIABILITY INSURANCE USING MONTE CARLO SIMULATION

Abstract

This study models the aggregate loss distribution for motor third-party liability insurance using Monte Carlo simulation. Aggregate loss estimation is essential because it depends on claim frequency and severity, which often exhibit overdispersion and heavy tails, making analytical solutions intractable and motivating simulation-based approaches for accurate tail-risk assessment. The objective of this study is to identify appropriate distributions for frequency and severity using French Motor Third-Party Liability (MTPL) insurance data and to construct the aggregate loss distribution through Monte Carlo simulation. The modeling procedure involves distribution selection, goodness-of-fit assessment using Chi-Square and Kolmogorov-Smirnov tests, graphical comparison, and model evaluation using the Akaike Information Criterion (AIC). The selected distributions are then combined to generate simulated aggregate losses, from which Value at Risk (VaR) and Tail Value at Risk (TVaR) are computed. The results show that the Zero-One-Two-Three Modified Negative Binomial (Z123M-NB) distribution provides the best fit for claim frequency, while the Burr XII distribution effectively represents claim severity. Monte Carlo simulation with 10 million iterations produces stable estimates of the aggregate loss mean and variance, and the estimated VaR at the 95%, 97.5%, and 99% confidence levels are 105.85, 1,506.61, and 3,629.14, with corresponding TVaR values of 4,122.93, 7,418.70, and 15,075.21, indicating substantial tail heaviness. The study is limited by the sensitivity of variance estimation under extreme severity values and the assumption of a continuous severity model. The novelty of this study lies in integrating the Z123M-NB frequency model with Burr XII severity within a Monte Carlo framework for real MTPL data, offering enhanced flexibility in modeling extreme aggregate losses.