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.
Highlights
As the occurrence of catastrophe events becomes more frequent, the assumption of independence between event occurrence and claim severity is no longer sufficient in insurance risk modeling, given its impact on pricing and reserving, capital allocation solvency, as well as regulatory systems
The first two moments of the aggregate discounted claims were obtained in [9] assuming the dependency between the claim sizes and the rates of claim occurrence affected by a Markovian environment, called the circumstance process
[11] adopted the same technique to derive the recursive moments of a Sparre Andersen risk process assuming a fairly general dependence structure between the inter-claim time and subsequent claim size variables, providing a simplified moments expression for assuming Erlang weights
Summary
As the occurrence of catastrophe events becomes more frequent, the assumption of independence between event occurrence and claim severity is no longer sufficient in insurance risk modeling, given its impact on pricing and reserving, capital allocation solvency, as well as regulatory systems. The asymptotical behaviour of a conditional tail probability dependence structure of claim sizes given the inter-claim arrival time was studied in [5,8]. [11] adopted the same technique to derive the recursive moments of a Sparre Andersen risk process assuming a fairly general dependence structure between the inter-claim time and subsequent claim size variables, providing a simplified moments expression for assuming Erlang weights.
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