## Abstract

Consider a portfolio containing heterogeneous risks. The premiums of the policyholders might not cover the amount of the payments which an insurance company pays the policyholders. When setting the premium, this risk has to be taken into consideration. On the other hand the premium that the insured pays has to be fair. This fairness is measured by a function of the difference between the risk and the premium paid-we call this function a distance function. For a given small probability of insolvency, we find the premium for each class, such that the distance function is minimized. Next we formulate and solve the dual problem, which is minimizing the insolvency probability under the constraint that the distance function does not exceed a given level. This paper generalizes a previous paper [Zaks, Y., Frostig, E., Levikson, B., 2006. Optimal pricing of a heterogeneous portfolio for a given risk level. Astin Bull. 36 (1), 161-185] where only a square distance function was considered.

Original language | English |
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Pages (from-to) | 459-467 |

Number of pages | 9 |

Journal | Insurance: Mathematics and Economics |

Volume | 40 |

Issue number | 3 |

DOIs | |

State | Published - May 2007 |

Externally published | Yes |

### Bibliographical note

Funding Information:The authors thank Professor Zinoviy Landsman for enlightening us on the basic properties of the elliptical distributions. We are grateful to the referee for her/his careful reading and comments that contributed greatly to the paper’s presentation. Yaniv Zaks acknowledges the financial support of the Erhard Center for Higher Studies and Research in Insurance at the Faculty of Management, Tel-Aviv University, Israel.

## Keywords

- Heterogeneous portfolio
- Majorization
- Schur convex functions