On a risk model with claim investigation

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Abstract

In this paper, a queue-based claims investigation mechanism is considered to model an insurer’s claim processing practices. The resulting risk model may be viewed as a first step in developing models with more realistic claim investigation mechanisms. Related to claim investigations, claim settlement delays and time dependent payments have been studied in a ruin context by, e.g. Taylor (1979), Cai and Dickson (2002), and Trufin et al. (2011). However, little has been done on queue-based investigation mechanisms. We first demonstrate the impact of a particular claim investigation system on some common ruin-related quantities when claims arrive according to a compound Poisson process, and investigation times are of a combination of exponential form. Probabilistic interpretations for the defective renewal equation components are also provided. Finally, via numerical examples, we explore various risk management questions related to this problem such as how claim investigation strategies can help an insurer control its activities within its risk appetite.

Introduction

Contemporary insurer surplus models do not typically consider claim investigation practices in an explicit manner. Rather, features related to claim investigation have to be a priori embedded in the defined risk model of interest by adjusting the model’s parameters. In this paper, we directly model claim investigation practices by considering a particular queue-based claim investigation mechanism. Investigation practices and strategies are developed to determine the extent of liability and identify ineligible or inflated claims which are crucial components of a sound insurance practice. For example, Juri (2002) discusses a risk process where claims are sums of dependent random variables. Such processes allow for the modeling of (allocated) loss adjustment expenses generated by claim investigations. As a result of investigation practices, claim payments are often modified to reflect investigation findings, in addition to the natural delay accompanying the investigation process. Claim payments may also be delayed by queueing times which we discuss next.

An insurer’s investigation strategy is constrained by the number of investigators, time per investigation, and volume of claims, among others. Queues form as claims accumulate and claims are served according to some queueing discipline. The natural existence of queues in this context prompts the inclusion of a queue-based investigation mechanism in surplus modeling. Queueing mechanisms have been an intensive area of research for many decades. A seminal reference in the context of a single server is Cohen (1982). While there are well-known connections between ruin and queueing problems (e.g., Asmussen and Albrecher, 2010 or Sigman, 2006), there has been little study of queues in surplus models. Thus, the present model is a first step in a longer inquiry on the topic, furthering the strong ties between the two research disciplines.

A queue-based investigation mechanism will help to improve the realism of an insurer’s cash flow dynamics. Many analogous modifications to improve realism have been made in the ruin theory literature such as the inclusion of dividend payments (e.g., Lin et al., 2003) and tax payments (e.g., Albrecher and Hipp, 2007). More closely related to claim investigations, claim settlement delays and time dependent payments have also been considered in a ruin context, e.g., claims inflation (e.g., Taylor, 1979), interest rates (e.g., Cai and Dickson, 2002), and IBNR (e.g., Trufin et al., 2011 and references therein). Such features have also been discussed in an aggregate claim context under various assumptions for the number of claims process (e.g., Landriault et al., 2014 for the nonhomogeneous birth process case, as well as references therein). In this paper, we use a different approach to model claim settlement delay from the widely studied Chain-Ladder method and its variants (e.g., Hossack et al., 1999, Chapter 10) where a major concern is the time until payment. The present model is intended for modeling short-term claim liabilities and moreover, its aim is to involve some queueing features (such as congestion) in an insurer’s surplus analysis.

The present paper contains 4 sections. Section  2 provides a description of the proposed queue-based claim investigation mechanism, a mathematical definition of the surplus process and the definition of a generalized Gerber–Shiu function which will be the main subject matter of this paper. Section  3 derives a defective renewal equation (DRE) for this Gerber–Shiu function assuming the investigation time is of a combination of exponential form. Probabilistic interpretations for the DRE components are also provided. In Section  4, numerical examples are presented to illustrate the impact of claim investigation strategies on the ruin probability.

Section snippets

Claim investigation surplus process

For completeness, we first recall the definition of the Cramér–Lundberg surplus process U¯={Ut}t0, where Ut=u+ctSt. We note that u(u0) is the initial surplus level and c(c>0) is the level premium rate per unit time. The aggregate claim process S¯={St}t0 is assumed to be a compound Poisson process, i.e. has Poisson arrivals at rate λ>0 and positive jumps with density p. The time to ruin T is defined as T=inf{t0:Ut<0} with T= if Ut>0 for all t0. A comprehensive treatment of the

Gerber–Shiu analysis

In this section, we derive a defective renewal equation (DRE) satisfied by the Gerber–Shiu function (8). A discussion of the components of the DRE follows. Note that the existence of this DRE representation for m is not immediate given the superimposition of a queueing mechanism.

Numerical examples

We now measure the impact of variations in the claim investigation mechanism on the ruin probability ψ(u). Our objective is to confirm some intuitive risk management features of the model and more importantly, quantify their impact from a risk management standpoint.

As a baseline case, we assume claims arrive according to a Poisson process with rate λ=5, and claim sizes are distributed as a mixture of an Erlang-2 and an Erlang-5 distribution with LT p˜(s)=0.8(0.60.6+s)2+0.2(22+s)5,s0. A premium

Acknowledgments

M. Huynh acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Society of Actuaries Hickman scholarship. Support for D. Landriault and G.E. Willmot from grants (grant numbers 341316 and 36515, respectively) from NSERC is acknowledged. Support from the Munich Reinsurance Company is also acknowledged by G.E. Willmot, as is the support from the Canada Research Chair program by D. Landriault. Start-up Summer Research Support for T. Shi from the

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