Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution and Dependence Introduced by a FGM Copula

In this chapter, with renewal argument, we derive higher simple moments of the Discounted Compound Delay Renewal Risk Process (DCDRRP) when introducing dependence between the inter-occurrence time and the subsequent claim size. To illustrate our results, we assume that the inter-occurrence time is following a delay-Poisson process and the claim amounts is following a mixture of Exponential distribution, we then provide numerical results for the first two moments. The dependence structure between the inter-occurrence time and the subsequent claim size is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we obtain a general formula for all the moments of the DCDRRP process

Distribution of multi-states models in health insurance under semi-Markovian assumptions

Abstract:In this paper we develop a system of integral and differential equations forth estate wise probability distributions of the present value of future payments on a multi-states life insurance policy under semi-Markov assumptions. We also provide formulas on the probability distribution through inversion techniques on the respective moment generating functions of the present value of future payments. The results are illustrated with applications to Value-at-Risk (VaR) and TailValue-at-Risk (TVaR) calculations

Moments of Phase-type aging modeling for health dependent costs

Abstract: In this paper, we use a discrete time Phase-type process to model the health care cost of an insurance contract by considering all possible critical health states of an individual with constant interest rate. From the moment generating function of the NPV, we derive a recursive formula of this Markov Reward Model (MRM)

Asymptotic tail probability for the discounted aggregate sums in a time dependent renewal risk model

This paper presents an extension of the classical compound Poisson risk model in which the inter-claim time arrivals and the claim amounts are structurally dependent. We derive the corresponding asymptotic tail probabilities for the discounted aggregate claims in a finite insurance contract under constant force of interest. The dependence assumption between the inter-claim times and the claim amounts is well suited for insurance contracts during extreme and catastrophic events. Based on the existing literature, we use heavytailed distributions for the discounted aggregate claims and derive the extreme value at risk (minimum capital requirement). Our results, based on a case study of ten million simulations, show that the independence assumption between the inter-claim times and the claim amounts lead to underestimating the minimum capital requirement proposed by the regulatory authorities

Gerber–Shiu function in a class of delayed and perturbed risk model with dependence

Abstract:This paper considers the risk model perturbed by a diffusion process with a time delay in the arrival of the first two claims and takes into account dependence between claim amounts and the claim inter-occurrence times. Assuming that the time arrival of the first claim follows a generalized mixed equilibrium distribution, we derive the integro-differential Equations of the Gerber–Shiu function and its defective renewal equations. For the situation where claim amounts follow exponential distribution, we provide an explicit expression of the Gerber–Shiu function. Numerical examples are provided to illustrate the ruin probability

Asymptotic tail probability of the discounted aggregate claims under homogeneous, non-homogeneous and mixed poisson risk model

Abstract: In this paper, we derive a closed-form expression of the tail probability of the aggregate discounted claims under homogeneous, non-homogeneous and mixed Poisson risk models with constant force of interest by using a general dependence structure between the inter-occurrence time and the claim sizes. This dependence structure is relevant since it is well known that under catastrophic or extreme events the inter-occurrence time and the claim severities are dependent

Reserves in the multi-state health insurance model with stochastic interest of diffusion type

In this paper, we consider the Markovian model for the actuarial modelling of health insurance policies modified by the inclusion of durational effects (the time elapsed since entering a given state) on the aggregate payment streams, where the force of interest is a diffusion process. We derive differential equations for the first moment of the present value of the aggregate amount of benefits. We also give two examples to illustrate our results.Keywords: Multi-state life insurance; semi-Markov model; counting process; first conditional moment; partial differential equations; Markov chai

Asymptotic Tail Probability of the Discounted Aggregate Claims under Homogeneous, Non-Homogeneous and Mixed Poisson Risk Model

In this paper, we derive a closed-form expression of the tail probability of the aggregate discounted claims under homogeneous, non-homogeneous and mixed Poisson risk models with constant force of interest by using a general dependence structure between the inter-occurrence time and the claim sizes. This dependence structure is relevant since it is well known that under catastrophic or extreme events the inter-occurrence time and the claim severities are dependent

Valuation of Equity-Linked Death Benefits on Two Lives with Dependence

The purpose of this paper is to investigate equity-linked death benefits for joint alive and last survivor individuals. Utilizing Farlie–Gumbel–Morgenstern (FGM) type dependency modeling framework, we first analyze the joint distribution of the couple (joint alive and last survival density) when marginal distributions follow mixed exponentials and weighted exponentials distributions. Then, we derive the price of the guaranteed minimum death benefit (GMDB) product. In addition, we provide closed analytical expressions of the price of some financial contingent claim contracts (classical and exotic options). Furthermore, we present some numerical results to support our theoretical results. We show in our numerical example that it is important to model the dependency between two lives (couple) since the price changes as the copula parameter changes