### 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 |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*North American Actuarial Journal*,

*23*(1), 82-97. https://doi.org/10.1080/10920277.2018.1506705

}

*North American Actuarial Journal*, vol. 23, no. 1, pp. 82-97. https://doi.org/10.1080/10920277.2018.1506705

**Capital Allocation for a Sum of Dependent Compound Mixed Poisson Variables : A Recursive Algorithm Approach.** / Kim, Joseph H.T.; Jang, Jiwook; Pyun, Chaehyun.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Capital Allocation for a Sum of Dependent Compound Mixed Poisson Variables

T2 - A Recursive Algorithm Approach

AU - Kim, Joseph H.T.

AU - Jang, Jiwook

AU - Pyun, Chaehyun

PY - 2019/1/2

Y1 - 2019/1/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85061279857&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061279857&partnerID=8YFLogxK

U2 - 10.1080/10920277.2018.1506705

DO - 10.1080/10920277.2018.1506705

M3 - Article

AN - SCOPUS:85061279857

VL - 23

SP - 82

EP - 97

JO - North American Actuarial Journal

JF - North American Actuarial Journal

SN - 1092-0277

IS - 1

ER -