Abstract
In insurance loss reserving, a large portion of zeros are expected at the later development periods of an incremental loss triangle. Negative losses occur frequently in the incremental loss triangle due to actuarial practices such as subrogation and salvation. The nature of the distributions assumed by most stochastic models, such as the lognormal and over-dispersed Poisson distributions, brings restrictions on the zeros and negatives appearing in the loss triangle. In this paper, we propose a Bayesian mixture model for stochastic reserving under the situation where there are both zeros and negatives in the incremental loss triangle. A multinomial regression model will be applied to model the sign of the loss data, while the lognormal distribution is assumed for the loss magnitudes of negatives and positives. Bayesian generalized linear models will be fitted for both the mixture and magnitude models. The model will be implemented using the Markov chain Monte Carlo (MCMC) techniques in BUGS. Our model provides a realistic tool for stochastic reserving in the cases of zeros and negatives.
Highlights
Determining an appropriate amount of loss reserve is very important for the financial stability of an insurance company
A large portion of zeros are expected at the later development periods of an incremental loss triangle
The nature of the distributions assumed by most stochastic models, such as the lognormal and over-dispersed Poisson distributions, brings restrictions on the zeros and negatives appearing in the loss triangle
Summary
Determining an appropriate amount of loss reserve is very important for the financial stability of an insurance company. For stochastic reserving (England and Verrall, 2002), specific distributions such as the lognormal (Kremer, 1982), over-dispersed Poisson (Renshaw and Verrall, 1998; England, Verrall and Wuthrich, 2012), negative binomial (Verrall, 2000) and gamma (de Alba and Nieto-Barajas, 2008) are assumed for the loss reserving data. For these models, classical generalized linear model (Nelder and Wedderburn, 1972) structures can be fitted to the mean or other parameters of the reserve distribution. We will propose a Bayesian mixture model for handling the situation when there are both zeros and negatives in the loss reserving data.
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