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The Covariance Between the Surplus Prior to and at Ruin in the Classical Risk Model

Published online by Cambridge University Press:  09 August 2013

Georgios Psarrakos  and
Konstadinos Politis
Georgios Psarrakos
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Demetriou Street, Piraeus 18534, Greece, E-mail: [email protected]
Konstadinos Politis
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Demetriou Street, Piraeus 18534, Greece, E-mail: [email protected]
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Abstract

For the classical model of risk theory, we consider the covariance between the surplus prior to and at ruin, given that ruin occurs. A general expression for this covariance is given when the initial surplus u is zero, and we show that the covariance (and hence the correlation coefficient) between these two variables is positive, zero or negative according to the equilibrium distribution of the claim size distribution having a coefficient of variation greater than, equal to, or less than one. For positive values of u, the formula for the covariance may not always lead to explicit results and we thus also study its asymptotic behaviour. Our results are illustrated by a number of examples.

Keywords

Ruin probabilityRenewal equationDeficit at ruinSurplus prior to ruinReliability classesCoefficient of variation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 2 , November 2012 , pp. 631 - 653
Copyright
Copyright © International Actuarial Association 2012

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