Measuring Tail Dependence for Aggregate Collateral Losses Using Bivariate Compound Cox Process with Shot Noise Intensity
39 Pages Posted: 19 Jan 2010
Date Written: Janaury 18, 2010
Abstract
A catastrophic event such as flood, storm, hail, bushfire and earthquake brings about damages in properties, motors and interruption of businesses collaterally. Also a couple of losses incurred collaterally from the World Trade Centre (WTC) catastrophe, Hurricane Katrina and Victorian Bushfire. However it has not been developed a suitable model for insurance companies either to measure tail dependence between these collateral losses or relevant risk measures that can be used as insurance risk premiums. The first aim of this paper is to measure tail dependence between collateral losses as insurance industry is more concerned with dependence between extreme losses. The second is to calculate conditional probabilities and conditional expectations as relevant risk measures. To achieve these aims, we use bivariate compound process where a Cox process with shot noise intensity is used to count collateral losses from catastrophic events. Homogeneous Poisson process is also examined as its counterpart for the case where the catastrophic loss frequency rate is deterministic. Using a member of Farlie-Gumbel-Morgenstern copula with exponential margins, we derive explicit expressions of joint Laplace transforms of aggregate collateral losses. Fast Fourier transform is used to obtain the joint distributions of aggregate collateral losses, with which we calculate relevant risk measures. The figures of the joint distributions of collateral losses, their contours and numerical calculations of risk measures are provided.
Keywords: Aggregate collateral losses, bivariate compopund Cox/Poisson process, shot noise process, Farlie-Gumbel-Morgenstern copula, tail dependence, Fast Fourier transform
JEL Classification: G13, G22
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