Capital allocation for a sum of dependent compound mixed Poisson variables: a recursive algorithm approach. (English) Zbl 1417.62300

Summary: The sum of independent compound Poisson random variables is a widely used stochastic model in many economic applications, including non-life insurance, credit and operational risk management, and environmental sciences. In this article we generalize this model by introducing dependence among Poisson frequency variables through a latent random variable in a linear fashion, which can be translated as a common underlying risk factors affecting the frequencies of individual compound Poisson variables. Despite its natural interpretation, this generalization leads to a highly complicated model with no closed-form distribution function. For this dependent compound mixed Poisson sum with an arbitrary severity distribution, we obtain the Laplace transform and further develop a new recursive algorithm to efficiently compute the probability mass function, extending the well-known Panjer recursion. Furthermore, based on this recursion, we derive another recursive scheme to determine the capital allocation associated with the conditional tail expectation, a popular risk management exercise. A numerical example is presented for the illustration of our findings.


62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
62E15 Exact distribution theory in statistics


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