A Bayesian Approach to Excess of Loss Pure Premium Rating

By Jack Barnett

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This paper demonstrates a Bayesian approach for estimating loss costs associated with excess of loss reinsurance programs. The main features of this approach are that (1) prior severity dis-tributions are adjusted for historical emergence patterns under-lying the experience data, (2) maximum likelihood estimation is used to estimate a ground-up loss ratio for each prior severity distribution, (3) a posterior severity distribution is derived using a Bayesian approach, and (4) a posterior ground-up loss ratio is derived using a Bayesian approach. This paper illustrates a simple implementation of the approach and tests the model by simulating from known frequency and severity distributions and fitting the model to the simulated “data.”

Keywords Bayesian estimation, maximum likelihood estimation, emergence patterns, exposure curves


Barnett, Jack, "A Bayesian Approach to Excess of Loss Pure Premium Rating," Variance 13:1, 2020, pp. 54-79.

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Variance (ISSN 1940-6452) is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Significant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submissions by non-CAS members is encouraged.