Expected present value of total dividends in a delayed claims risk model under stochastic interest rates

Abstract

In this paper, a compound binomial risk model with a constant dividend barrier under stochastic interest rates is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. A system of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is derived and solved. Explicit results are obtained when the claim sizes are $K_n$ distributed or the claim size distributions have finite support. Numerical results are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends.

Introduction

In reality, insurance claims may be delayed due to various reasons. Since the work by Waters and Papatriandafylou (1985), risk models with this special feature have been discussed by many authors in the literature. For example, Yuen and Guo (2001) studied a compound binomial model with delayed claims and obtained recursive formulas for the finite time ruin probabilities. Xiao and Guo (2007) obtained the recursive formula of the joint distribution of the surplus immediately prior to ruin and deficit at ruin in this model. Xie and Zou (2008) studied a risk model with delayed claims. Exact analytical expressions for the Laplace transforms of the ruin functions were obtained. Yuen et al. (2005) studied a risk model with delayed claims, in which the time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. A framework of delayed claims is built by introducing two kinds of individual claims, namely main claims and by-claims, and allowing possible delays of the occurrences of by-claims.

Dividend strategy for insurance risk models were first proposed by De Finetti (1957) to reflect more realistically the surplus cash flows in an insurance portfolio, and he found that the optimal strategy must be a barrier strategy. From then on, barrier strategies have been studied in a number of papers and books. For example, Claramunt et al. (2003) calculated the expected present value of dividends in a discrete time risk model with a barrier dividend strategy. Dickson and Waters (2004) showed how to use the compound binomial risk model to approximate the classical compound Poisson risk model in calculating the moments of discounted dividend payments. Other risk model involving dividend payments were studied by Zhou (2005), Gerber and Shiu (2004), Li and Garrido (2004), Wu and Li (2006), Frosting (2005), Bara et al. (2008) and the references therein.

All risk models described in the paragraph above relied on the assumption that the force of interest or the discount factor per period is a constant. Based on this assumption, it becomes evident that the discount factor per period embedded into the risk model fails to capture the uncertainty of the (future) risk-free rates of interest. In the compound binomial model with delayed claims and a dividend barrier proposed in this paper, discount factors are defined via the modelization of the one-period interest rates using a time-homogeneous Markov chain with a finite state space. The use of time-homogeneous Markov chains to model the interest rates is well documented in finance (see, e.g. Landriault (2008) in a discrete time framework). We derive the explicit expression for the expected present value of total dividends in our risk model.

The model proposed in this paper is a generalization of compound binomial risk model with paying dividends and classical risk model with delayed claims. It seems to be the first risk model with delayed claims and a constant dividend barrier in a financial market driven by a time-homogeneous Markov chain. We show that, the explicit expression for the expected present value of total dividends in this risk model can be obtained. The work of this paper can be seen as a complement to the work of Li (2008) that calculated the present value of total dividends in the compound binomial model under stochastic interest rates and extend the results of Li (2008) by introducing two kinds of individual claims, namely main claims and by-claims, and allowing possible delays of the occurrences of by-claims. The model considered in this paper is also related to the one considered by Wu and Li (2006). Although both models employ a discrete time risk model with dividends and delayed claims, our model differs from the one by Wu and Li (2006) as follows. Our model is more general than that of Wu and Li (2006) in that we assume the financial market is driven by a time-homogeneous Markov chain, while the discount factor per period is a constant in Wu and Li (2006).

It is obvious that the incorporation of the delayed claim and dividend payments makes the problem more interesting. It also complicates the evaluation of the expected present value of dividends. Because of the certainty of ruin for a risk model with a constant dividend barrier, the calculation of the expected discounted dividend payments is a major problem of interest (see, e.g. Wu and Li (2006)). Similar to method of Li (2008), we use the technique of generating functions to calculate the expected present value of total dividends for this risk model. Section 2 defines the model of interest, describes various payments, including the premiums, claims and dividends, and lists the notation. In Section 3, difference equations with certain boundary conditions are developed for the expected present value of total dividend payments prior to ruin. Then an explicit expression is derived, using the technique of generating functions. Moreover, closed-form solutions for the expected present value of dividends are obtained for two classes of claim size distributions in Section 4. Numerical examples are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends in Section 4.