Moments of Compound Renewal Sums with Dependent Risks Using Mixing Exponential Models

Abstract

In this paper, we study the discounted renewal aggregate claims with a full dependence structure. Based on a mixing exponential model, the dependence among the inter-claim times, the claim sizes, as well as the dependence between the inter-claim times and the claim sizes are included. The main contribution of this paper is the derivation of the closed-form expressions for the higher moments of the discounted aggregate renewal claims. Then, explicit expressions of these moments are provided for specific copulas families and some numerical illustrations are given to analyze the impact of dependency on the moments of the discounted aggregate amount of claims.

Risk, Ruin and Survival

“…Furthermore, the authors have employed an analogous numerical method to tackle other Gerber-Shiu type functions, such as the Laplace transform of the time of ruin. Marri et al (2018):…”

Special Issue “Risk, Ruin and Survival: Decision Making in Insurance and Finance”

“…To illustrate, in non-life insurance, the same catastrophic event (e.g., a flood or an earthquake) may lead to frequent and high losses. Inspired by such observations, Marri et al (2018) study discounted renewal aggregate claims under full dependence structure: they allow dependence among the inter-claim times, dependence among the claim sizes, and also dependence between the inter-claim times and the claim sizes.…”

Risk aggregation and capital allocation using a new generalized Archimedean copula

“…Sarabia et al (2016) give explicit formulas for the probability density function of S n for some multivariate mixed exponential distributions for which the dependence structure is an Archimedean copula. Recently, Marri et al (2018) derive explicit expressions for the higher moments of the discounted aggregate renewal claims with dependence.…”