## Abstract

In Finance and Actuarial Science, the multivariate elliptical family of distributions is a famous and well-used model for continuous risks. However, it has an essential shortcoming: all its univariate marginal distributions are the same, up to location and scale transformations. For example, all marginals of the multivariate Student's t-distribution, an important member of the elliptical family, have the same number of degrees of freedom. We introduce a new approach to generate a multivariate distribution whose marginals are elliptical random variables, while in general, each of the risks has different elliptical distribution, which is important when dealing with insurance and financial data. The proposal is an alternative to the elliptical copula distribution where, in many cases, it is very difficult to calculate its risk measures and risk capital allocation. We study the main characteristics of the proposed model: characteristic and density functions, expectations, covariance matrices and expectation of the linear regression vector. We calculate important risk measures for the introduced distributions, such as the value at risk and tail value at risk, and the risk capital allocation of the aggregated risks.

Original language | English |
---|---|

Pages (from-to) | 6-24 |

Number of pages | 19 |

Journal | Annals of Actuarial Science |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - 22 Mar 2022 |

### Bibliographical note

Funding Information:This research was supported by the Israel Science Foundation (Grant N 1686/17). The authors are grateful to the anonymous reviewer for his/her useful comments.

Publisher Copyright:

© 2022 Cambridge University Press. All rights reserved.

## Keywords

- Elliptical distributions
- Multi-spherical distributions
- Risk measures
- Spherical distributions
- Tail value at risk
- Value at risk

## ASJC Scopus subject areas

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