Confidence bounds for discounted loss reserves

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Abstract

In this paper we give some methods to set up confidence bounds for the discounted IBNR reserve. We start with a loglinear regression model and estimate the parameters by maximum likelihood such as given, for example, by Doray [Insur.: Math. Econ. 18 (1996) 43]. The knowledge of the distribution function of the discounted IBNR reserve (S) will help us to determine the initial reserve, for example, through the 95th percentile F−1S(0.95). The results are based on convex order techniques, such that our approximations for the distribution function of S are larger or smaller, in convex order sense, than the true distribution function of S.

Introduction

An important problem in insurance is to determine the provision for claims already incurred but not yet reported (hence IBNR), or not fully paid. The past data used to construct estimates for the future payments consist of a triangle of incremental claims Yij, as depicted in Fig. 1. This is the simplest shape of data that can be obtained and it avoids having to introduce complicated notation to cope with all possible situations.

The random variables Yij with i,j=1,2,…,t denote the claim figures for year of origin i and development year j, meaning that the claims were paid in calendar year i+j−1. Year of origin, year of development and calendar year act as explanatory variables for the observation Yij. For (i,j) combinations with i+jt+1, Yij has already been observed, otherwise it is a future observation. Next to claims actually paid, these figures can also be used to denote quantities such as loss ratios. To a large extent, it is irrelevant whether incremental or cumulative data are used when considering claims reserving in a stochastic context.

The purpose is to complete this run-off triangle to a square, and even to a rectangle if estimates are required pertaining to development years of which no data are recorded in the run-off triangle at hand. To this end, the actuary can make use of a variety of techniques. The inherent uncertainty is described by the distribution of possible outcomes, and one needs to arrive at the best estimate of the reserve. Loss reserving deals with the determination of the uncertain present value of an unknown amount of future payments. Since this amount is very important for an insurance company and its policyholders, these inherent uncertainties are no excuse for providing anything less than a rigorous scientific analysis. In order for the reserve estimate truly to represent the actuary’s “best estimate” of the needed reserve, both the determination of the expected value of unpaid losses and the appropriate discount should reflect the actuary’s best estimates (i.e. should not be dictated by others or regulatory requirements). Since the reserve is a provision for the future payment of unpaid losses, we believe the estimate loss reserve should reflect the time value of money. In many situations this discounted reserve is useful, for example, dynamic financial analysis, assessing profitability and pricing, identifying risk based capital needs, loss portfolio transfers, etc. Ideally the discounted loss reserve would also be acceptable for regulatory reporting. However, many current regulations do not permit it. Undiscounted loss reserves include in fact a certain risk margin depending on the level of the interest rate. In this paper we consider the discounted IBNR reserve and impose an explicit margin based on a risk measure (for example, VaR) from the distribution of the total discounted reserve.

As a first attempt to analyze the discounted IBNR reserve, we consider here a simple loglinear statistical model to describe the past and future payments. So, the total IBNR reserve will be a sum of lognormal random variables which implies that its exact distribution function (d.f.) cannot be determined analytically. Considering the discounted IBNR reserve (S), we have to incorporate a certain dependence structure. This will be explained in detail in the next section. In general, it is hard or even impossible to determine the quantiles of S analytically, because in any realistic model for the return process the random variable S will be a sum of strongly dependent random variables. The “true” multivariate distribution function of the lower triangle cannot be determined in most cases, because the mutual dependencies are not known, or are difficult to cope with. We suggest to solve this problem by calculating upper and lower bounds for this sum of dependent random variables making efficient use of the available information. These bounds are based on a general technique for deriving lower and upper bounds for stop-loss premiums of sums of dependent random variables, as explained in Kaas et al. (2000). The first approximation we will consider for the d.f. of the discounted IBNR reserve is derived by approximating the dependence structure between the random variables involved by a comonotonic dependence structure. The second approximation, which is derived by considering conditional expectations, takes part of the dependence structure into account. We will include a numerical comparison of our approximations with a simulation study. The second approximation turns out to perform quite well. For details of this technique we refer to Dhaene et al., 2002a, Dhaene et al., 2002b and the references therein.

The choice of an appropriate statistical model is an important matter. Furthermore within a stochastic framework, there is considerable flexibility in the choice of predictor structures. In England and Verrall (2002) the reader finds an excellent review of possible stochastic models. An appropriate model will enable the calculation of the distribution of the reserve that reflects the process variability producing the future payments, and accounts for the estimation error and statistical uncertainty (in the sense given in Taylor and Ashe (1983)). It is necessary to be able to estimate the variability of claims reserves, and ideally to be able to estimate a full distribution of possible outcomes so that percentiles (or other risk measures of this distribution) can be obtained. Next, recognizing the estimation error involved with the parameter estimates, confidence intervals for these measures constitute another desirable part of the output. Here, putting the emphasis on the discounting aspect of the reserve, we consider simple loglinear models. Doray (1996) studied these models extensively, taking into account the estimation error on the parameters and the statistical prediction error in the model. This class of models have some significant disadvantages. We need to impose that each incremental value should be greater than zero. Moreover predictions from this model can yield unusable results. In the future the authors intend to deal with other statistical models as well.

This paper is set out as follows. Section 2 gives a summary of results on loglinear models in claims reserving. In Section 3 we state stochastic bounds for the scalar product of two independent random vectors, where the marginal distribution functions of each vector are given, but the dependence structures are unknown. We will describe how these results can be used for discounted IBNR evaluations. Finally, we will calculate the cdf’s of these bounds. Some numerical illustrations for a simulated data set are provided in Section 4, together with a discussion of the estimation error using a bootstrap approach. We also graphically illustrate the obtained bounds. Most of the proofs are deferred to the appendix.

Section snippets

Loglinear models

We consider the following loglinear regression model: Zi=lnYi=Xiβ+ϵi,Yi>0,where

  • Yi is the ith element of the data vector Y, of dimension t(t+1)/2;

  • X is the regression matrix of dimension [t(t+1)/2]p, the ith row is denoted by Xi, and element (i,j) is denoted Xij;

  • β is the vector (of dimension p) of unknown parameters;

  • ϵi are the independent normal random errors with mean 0 and variance σ2.

In matrix notation this linear model can be represented as Z=lnY=Xβ+ϵ,ϵ∼N(0,σ2I).The normal

Methodology

Because the discounted IBNR reserve is a sum of dependent lognormal random variables, its distribution function cannot be determined analytically. Therefore, instead of calculating the exact distribution, we will look for bounds, in the sense of “more favorable/less dangerous” and “less favorable/more dangerous”, with a simpler structure. This technique is common practice in the actuarial literature. When lower and upper bounds are close to each other, together they can provide reliable

Numerical illustrations

In this section we illustrate the effectiveness of the bounds derived for the discounted IBNR reserve S. We investigate the accuracy of the proposed bounds, by comparing their cdf to the empirical cdf obtained with Monte Carlo simulation, which serves as a close approximation to the exact distribution of S. To analyze the precision of the derived bounds (given the choice of the stochastic model), we built a non-cumulative run-off triangle ourselves based on the chain-ladder model (3). So, the

Conclusions and possibilities for future research

In this paper, we considered the problem of deriving the distribution function of the present value of a triangle of claim payments that are discounted using some given stochastic return process. Because an explicit expression for the distribution function is hard to obtain, even when starting from a classical loglinear regression model, we presented three approximations for this distribution function, in the sense that these approximations are larger or smaller in convex order sense than the

Acknowledgements

The authors acknowledge financial support by the Onderzoeksfonds K.U. Leuven (GOA/02: Actuariële, financiële en statistische aspecten van afhankelijkheden in Verzekerings- en financiële portefeuilles). The authors like to thank the referee for the valuable comments, which helped to improve the presentation significantly.

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