## Abstract

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.

Original language | English |
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Pages (from-to) | 82-97 |

Number of pages | 16 |

Journal | North American Actuarial Journal |

Volume | 23 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2019 Jan 2 |

### Bibliographical note

Funding Information:This work is supported by the Basic Science Research Program of the National Research Foundation of Korea [NRF 2015R1A1A1A05027336].

Publisher Copyright:

© 2019, © 2019 Society of Actuaries.

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Economics and Econometrics
- Statistics, Probability and Uncertainty