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Actuarial Science and Insurance Series: insurance-and-pensions

Archimedean copulas derived from utility functions

Author(s): Dr Jaap Spreeuw - Cass Business School;

Archimedean copulas are an important subclass of dependency models, while risk attitude of individuals is usually modelled by means of a utility function that is increasing and convex in terms of wealth. This publication explores the relationship between Archimedean copulas and utility functions, thus enhancing the interpretation of Archimedean copulas and therefore informing the risk modeller choosing the most appropriate copula.

For instance, we show that these copulas can be interpreted as certainty equivalents. (To clarify: if an individual is indifferent between participating in a lottery and receiving a monetary amount with certainty, then that amount is defined as the certainty equivalent). Also, the coefficient of relative risk aversion, a well known concept originating from Arrow and Pratt in the 1960’s, gives information about the strength of dependence featured by the corresponding Archimedean copula. It also informs us how the dependence between the risks involved, e.g. the dependence between times of death of a couple, evolves over time

We also show that in this way new Archimedean copula families can be constructed with potentially desirable properties. One of these families, being rich and flexible, is calibrated to a portfolio of annuities on two lives extracted from a large North American insurer.

The full working paper, which was accepted for publication, is available to download below. A definitive version was subsequently published in Insurance: Mathematics and Economics, Volume 59, November 2014,

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