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Risks 2014, 2(2), 195-210; doi:10.3390/risks2020195

Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims

Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney 2109, Australia
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Received: 1 November 2013 / Revised: 10 April 2014 / Accepted: 16 May 2014 / Published: 27 May 2014
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
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Abstract

We study the recursive moments of aggregate discounted claims, where the dependence between the inter-claim time and the subsequent claim size is considered. Using the general expression for the m-th order moment proposed by Léveillé and Garrido (Scand. Actuar. J. 2001, 2, 98–110), which takes the form of the Volterra integral equation (VIE), we used the method of successive approximation to derive the Neumann series of the recursive moments. We then compute the first two moments of aggregate discounted claims, i.e., its mean and variance, based on the Neumann series expression, where the dependence structure is captured by a Farlie–Gumbel–Morgenstern (FGM) copula, a Gaussian copula and a Gumbel copula with exponential marginal distributions. Insurance premium calculations with their figures are also illustrated.
Keywords: aggregate discounted claims; moments; copulas; Volterra integral equation; Neumann series; insurance premium aggregate discounted claims; moments; copulas; Volterra integral equation; Neumann series; insurance premium
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Mohd Ramli, S.N.; Jang, J. Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims. Risks 2014, 2, 195-210.

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