Mean Square Error of Prediction in the Bornhuetter-Ferguson Claims Reserving Method

Transcription

1 Mean Square Error of Prediction in the Bornhuetter-Ferguson Caims Reserving Method Danie H. Aai 1 Michae Merz 2 Mario V. Wüthrich 3 Department of Mathematics, ETH Zurich, CH-8092 Zurich, Switzerand Apri 15, 2008 Abstract The prediction of adequate caims reserves is a maor subect in actuaria practice and science. Due to their simpicity, the Chain-adder CL and Bornhuetter-Ferguson BF methods are the most commony used caims reserving methods in practice. However, in contrast to the CL method, no estimator for the conditiona mean square error of prediction MSEP of the utimate caim has been derived in the BF method unti now, and as such, this paper aims to fi that gap. This wi be done in the framewor of generaized inear modes GLM using the overdispersed Poisson mode motivation for the use of CL factor estimates in the estimation of the caims deveopment pattern. Keywords: Caims Reserving, Bornhuetter-Ferguson, Overdispersed Poisson Distribution, Chain-adder Method, Generaized Linear Modes, Conditiona Mean Square Error of Prediction 1 danie.aai@math.ethz.ch 2 michae.merz@uni-tuebingen.de 3 mario.wuethrich@math.ethz.ch 1

2 1 Introduction Often in non-ife insurance, caims reserves are the argest item on the iabiity side of the baance sheet. Therefore, given the avaiabe information about the past, the prediction of an adequate amount to face the responsibiities assumed by the non-ife insurance company as we as the quantification of the uncertainties in these reserves are maor issues in actuaria practice and science e.g. Casuaty Actuaria Society [3] and Teuges-Sundt [21]. Due to their simpicity, the Chain-adder CL and Bornhuetter-Ferguson BF methods are among the easiest caims reserving methods and, therefore, the most commony used techniques in practice. In 1993, Mac [10] pubished a fundamenta artice on caims reserving regarding the estimation of the conditiona mean square error of prediction MSEP in the CL method. Unfortunatey, unti now, no estimator for the conditiona MSEP of the utimate caim has been derived in the BF method. The BF method goes bac to Bornhuetter-Ferguson [1]. Apart from its simpicity, the BF method is a very popuar caims reserving method since it is rather robust against outiers in the observations and aows for incorporating a priori nowedge from experts, premium cacuations or strategic business pans. Furthermore, in contrast to the CL method, the BF method has proven to be a very robust method, in particuar, against instabiity in the proportion of utimate caims paid in eary deveopment years. The BF method, as it was stated in the origina wor of Bornhuetter-Ferguson [1], was not formuated in a probabiistic way. The wor of Mac [11] and Verra [25] puts the BF method into a probabiistic framewor; we discuss this further beow. Before describing the method in detai we woud ie to mention that there are aso rather septic and critica opinions of the use of the BF method. The purpose of this paper is not to improve the method in view of this criticism but rather to expain from a probabiistic point of view, what is done when actuaries use the BF method at its current state and to derive anaytica estimators for the prediction uncertainty. The BF method is based on the simpe idea of stabiizing the BF estimate Ĉi,J BF using an initia estimate µ i of the utimate caim C i,j based on externa nowedge. Then it is standard practice to use the prior estimate µ i with the CL factor estimates f to predict the utimate caim. In this case, the CL method and the BF method ony 2

3 differ in the choice of the estimate for the utimate caim CL estimate versus prior estimate. Hence, in this regard the BF method is a variant of the CL method that uses externa information to obtain an initia estimate for the utimate caim. Mac [11] studied this from a probabiistic point of view. In his wor he anayzed the stochastic mode for given deterministic caims deveopment patterns, which are the anaogon to the CL factors, and random initia estimates µ i. Mac [11] then derived optima credibiity weighted averages between the CL and the BF method. However, in most practica appications the caims deveopment pattern and the CL factors are unnown and need to be estimated from the data. This adds an additiona source of uncertainty to the probem. Verra [25] has studied these uncertainties using a Bayesian approach to the BF method. If one uses an appropriate Bayesian approach with improper priors and an appropriate two stage procedure one then arrives at the BF method. We use a simiar procedure within generaized inear modes using maximum ieihood estimators MLE. This framewor aows for an anaytica estimate for the mean square error of prediction using asymptotic properties of MLE. Note that in a Bayesian framewor one can, in genera, ony give numerica answers using simuation techniques such as the Marov chain Monte Caro method. A criticism to the BF method as it is currenty used is that the use of the CL estimates f contradicts the basic idea of independence between the ast observed cumuative caims C i,i i and the estimated outstanding caims iabiities Ĉi,J BF Ci,I i, which was fundamenta to the BF method see Mac [12]. Therefore, Mac [12] proposed different estimators for the caims deveopment pattern. In this paper however, we do not foow this route. We rather use the we-own fact that the overdispersed Poisson mode eads to the same caims reserves and payout pattern as the CL mode. This means that we use the overdispersed Poisson mode motivation for the use of the CL factor estimates f. It is then straightforward to use generaized inear mode GLM methods for parameter estimation and to derive an estimator for the conditiona MSEP of the utimate caim in the BF method. Organization of the paper. In Section 2 we provide the notation and data structure. In Section 3 we give a short review of the CL and BF methods and compare these two techniques. Section 4 is dedicated to the overdispersed Poisson mode and its representation as a generaized inear mode. In Section 5 we give an estimation procedure for the conditiona MSEP in the BF method. Finay, in Section 6 we discuss 3

4 an exampe. 2 Notation and Data Structure Throughout, we assume the oss data for the run-off portfoio is given by a caims deveopment triange of observations. However, a caims reserving methods discussed in this paper can aso be appied to other shapes of oss data e.g. caims deveopment trapezoids. In this caims deveopment triange the indices i {0, 1,..., I} and {0, 1,..., J} with I J refer to accident years and deveopment years, respectivey. The incrementa caims i.e. incrementa payments, change of reported caim amount or number of newy reported caims for accident year i and deveopment year are denoted by X i, and cumuative caims i.e. cumuative payments, caims incurred or tota number of reported caims of accident year i up to deveopment year are given by C i, = X i,. 2.1 =0 We assume that the ast deveopment year is given by J, i.e. X i, 0 for a > J, and the ast accident year is given by I. Moreover, our assumption that we consider caims deveopment trianges impies I = J. accident deveopment year year i J 0 reaizations of. r.v. X i,, C i, I D I. predicted r.v. X i,, C i, I D c I Figure 1: Caims deveopment triange. Usuay, at time t = I i.e. caender year I, we have observations D I in the upper caims deveopment triange, defined as foows, D I = {X i, ; i + I}. 2.2 We need to predict the random variabes in its compement D c I = {X i, ; i + > I, i I}

5 Figure 1 shows the caims data structure for the caims deveopment triange described above. Furthermore, et R i and R denote the outstanding caims iabiities for accident year i at time I, R i = X i, = C i,j C i,i i for 1 i I, 2.4 =I i+1 and the tota outstanding caims iabiities for aggregated accident years, R = I R i, 2.5 i=1 respectivey. The prediction of the outstanding caims iabiities R i and R by socaed caims reserves or best estimates, as we as quantifying the uncertainty in this prediction, is the cassica actuaria caims reserving probem studied at every non-ife insurance company. 3 Bornhuetter-Ferguson and Chain-adder Methods In this section we give a short review of the CL and BF methods, which are the most commony used caims reserving methods in practice on account of their simpicity. Our review is simiar to the one given in Mac [11]. 3.1 Chain-adder Method The cassica actuaria iterature often expains the CL method as a pure computationa agorithm to estimate caim reserves. The first distribution-free stochastic mode was proposed by Mac [10]. Mode Assumptions 3.1 CL mode: There exists deterministic deveopment factors f 0,..., f J 1 > 0 such that for a 0 i I and a 1 J we have E[C i, C i,0,..., C i, 1 ] = E[C i, C i, 1 ] = f 1 C i, Caims C i, of different accident years i are independent. 5

6 An easy exercise in cacuating conditiona expectation eads to E[C i,j D I ] = f J 1 E[C i,j 1 C i,i i ] =... = C i,i i J 1 =I i f, 3.2 for a 1 i I, where the factors f are caed CL factors or deveopment factors. Given the observations D I and CL factors f, 3.2 gives a recursive agorithm for predicting the utimate caim C i,j. However, in most practica appications the CL factors f are not nown and have to be estimated from the data D I. It is we nown that the D I -measurabe estimators for the CL factors f, defined by f = I 1 C i,+1 i=0 I 1 i=0 C i,, 3.3 for a {0,..., J 1}, are unbiased and uncorreated see e.g. Mac [10]. However, they are not independent since the squares of two successive estimators f and f +1 are negativey correated see e.g. Mac et a. [13] and Wüthrich et a. [26]. The properties of the CL factor estimates f impy that, given C i,i i, the CL estimator of the utimate caim C i,j, defined by Ĉ i,j CL = Ci,I i J 1 =I i is an unbiased estimator for E[C i,j D I ]. f for a 1 i I, Bornhuetter-Ferguson Method The BF method goes bac to Bornhuetter-Ferguson [1]. Anaogousy to the CL method, the cassica actuaria iterature often expains the BF method as a pure computationa agorithm to estimate caim reserves athough there are severa stochastic modes that motivate the BF method. The foowing stochastic mode is consistent with the BF method. Mode Assumptions 3.2 BF mode: There exist parameters µ 0,..., µ I > 0 and a pattern β 0,..., β J > 0 with β J = 1 such that for a i {0,..., I}, {0,..., J 1} and {1,..., J } E[C i,0 ] = β 0 µ i, 3.5 E[C i,+ C i,0,..., C i, ] = C i, + β + β µ i

7 Caims C i, of different accident years i are independent. These assumptions impy E[C i, ] = β µ i and E[C i,j ] = µ i, 3.7 which is often used to expain the BF method see e.g. Radte-Schmidt [16]. The sequence β denotes the caims deveopment pattern and, if C i, are cumuative payments, β is the expected cumuative cashfow pattern aso caed payout pattern. Such a pattern is often used when one needs to buid maret-consistent/discounted reserves, where money vaues differ over time. Assumption 3.6 motivates the BF estimator for the utimate caim C i,j given by BF Ĉ i,j = Ci,I i + 1 I i µ i, 3.8 for a 1 i I, where I i is an appropriate estimate for β I i and µ i is a prior estimate for the expected utimate caim E[C i,j ]. In practice, µ i is an exogenousy determined estimate i.e. without the observations D I such as a pan vaue from a strategic business pan or the vaue used for premium cacuations. 3.3 Comparison of BF and CL Methods From the CL Assumptions 3.1 we obtain which impies that J 1 E[C i,j ] = E[C i, ] f, 3.9 E[C i, ] = J 1 = = f 1 E[C i,j], for a = 0,..., J If we compare this to the BF method see e.g.3.7 we find that J 1 = f 1 pays the roe of β. Therefore, these parameters are often viewed equay and if one nows the CL factors f one can construct a deveopment pattern β and vice versa. That is, in practice the β are usuay estimated by CL = J 1 = = 1 f,

8 where f are the CL factor estimates given in 3.3. Moreover, using the estimator CL for β in the BF method, we see that Ĉ i,j BF = C i,i i + Ĉ i,j CL = Ci,I i CL I i CL I i µ i, 3.12 CL Ĉi,J, 3.13 which means that the CL method and BF method ony differ in the choice of the estimator for the utimate caim C i,j prior estimate µ i versus CL estimate Ĉi,J CL. In other words, if we identify CL and J 1 = 1 f, the BF method is a variant of the CL method that uses externa information to obtain an initia estimate for the utimate caim. The main criticism of this approach is that the use of the CL factor estimates f contradicts the basic idea of independence between ast observed cumuative caims C i,i i and estimated outstanding caims iabiities Ĉi,J BF Ci,I i, which was fundamenta to the origin of the BF method see Mac [12]. Therefore, Mac [12] constructed different estimators for the caims deveopment pattern β. However, we do not foow this route here. We rather concentrate on the overdispersed Poisson mode motivation for the use of the CL factor estimates f and utiize the fact that the overdispersed Poisson mode is a GLM. It is then straightforward to use GLM methods for parameter estimation and to derive an estimator for the conditiona MSEP of the utimate caim in the BF method. 4 Overdispersed Poisson Mode and Generaized Linear Modes In this section we give a brief review of the overdispersed Poisson mode and its formuation in a GLM context. 4.1 Overdispersed Poisson Mode We define the overdispersed Poisson mode by first considering the exponentia dispersion famiy. Random variabe Y beongs to the exponentia dispersion famiy if its density or probabiity distribution function can be written as f Y y; θ, φ = exp {[yθ bθ]/aφ + cy, φ}. 4.1 The overdispersed Poisson mode is a member of this famiy with aφ = φ, bθ = e θ and cy, φ = ny. It differs from the Poisson mode in that the variance is not equa to the mean. This mode was introduced for caims reserving in a Bayesian context by 8

9 Verra [22, 23, 25] and Renshaw-Verra [18] and it is aso used in the GLM framewor see e.g. McCuagh-Neder [14] and Engand-Vera [4, 5]. It is we-nown in actuaria iterature that the overdispersed Poisson mode eads to the same caims reserves as the CL mode. This resut goes bac to Hachemeister-Stanard [7] and can be found e.g. in Mac [9] and Verra-Engand [24]. This means that athough the CL mode and the overdispersed Poisson mode are very different, they ead to the same reserve estimates, the difference in the two modes is reevant ony if we estimate higher moments. In the foowing, we wi utiize this correspondence as we do not motivate the use of the estimate CL I i by CL factor estimates f but rather by the maximum ieihood estimators MLEs in the overdispersed Poisson mode. Note that CL factor estimates are used to cacuate the MLEs. Mode Assumptions 4.1 Overdispersed Poisson Mode: The increments X i, are independent overdispersed Poisson distributed and there exist positive parameters γ 0,..., γ J, µ 0,..., µ I and φ > 0 such that E [X i, ] = m i, = µ i γ, 4.2 Var X i, = φ m i,, 4.3 with J =0 γ = 1. µ i are independent random variabes that are unbiased estimators of µ i = E [C i,j ] for a i. X i, and µ are independent for a i,,. From Assumptions 4.1 we obtain E[C i,+ C i,0,..., C i, ] = C i, + E[X i,+ ] = C i, + β + β µ i, 4.4 =1 where β = =0 γ. This means that the overdispersed Poisson mode satisfies Assumptions 3.2 and can aso be used to expain the BF method. Remars 4.2: The so-caed dispersion parameter φ does not depend on accident year i and deveopment year. The restriction in the overdispersed Poisson mode is that we require X i, to be non-negative. 9

10 The parameters γ define an expected incrementa reporting/cashfow pattern over the deveopment years. The exogenous estimator µ is a prior estimate for the expected utimate caim E[C,J ], it is soey based on externa data and expert opinion. Therefore, we assume that it is independent of the data X i, this is in the BF spirit as expained by Mac [12]. Moreover, in order to obtain a meaningfu mode, we assume that it is unbiased for the expected utimate caim. In this sense, we foow a pure BF method. There are different methods for estimating the parameters µ i and γ. In the foowing we use MLEs. The MLEs µ MLE i found by soving and γ MLE in the overdispersed Poisson mode are µ MLE i γ MLE I i =0 I i=0 γ MLE = µ MLE i = I i X i,, 4.5 =0 I X i,, 4.6 i=0 for a i {0,..., I} and {0,..., J} under the constraint that J =0 γmle = 1. Remars 4.3: Because of the mutipicative structure of the overdispersed Poisson mode see 4.2, the parameters µ i and γ can ony be determined up to a constant factor, i.e. µ i = cµ i and γ = γ /c woud give the same estimate for m i,. Therefore, we need to impose a side constraint. In our situation this becomes that the MLEs γ MLE form a deveopment pattern, i.e. that J =0 γmle = 1. If we now use these MLEs γ MLE for the estimation of the expected incrementa cashfow pattern γ we obtain the foowing BF estimator, BF I i Ĉ i,j = C i,i i + 1 γ MLE µ i 4.7 see e.g Moreover, under Mode Assumptions 4.1, it has been proved that =0 γ MLE = =0 J 1 CL = = 1 f,

11 see e.g. Mac [9] or Tayor [20] which impies that the BF estimator, Ĉ i,j BF = C i,i i + 1 CL I i µ i, 4.9 is perfecty motivated by the overdispersed Poisson mode and the use of the MLE γ MLE for γ. This is exacty the BF estimator as it is commony used in practice see e.g Henceforth, in this understanding we do not motivate the use of the estimate γ MLE CL I i by CL factor estimates f but rather by the MLEs in the overdispersed Poisson mode. This means that 4.8 provides the essentia steps for the use of CL in the BF method. Note that we use a simiar two stage procedure as described by Verra [25]. First we estimate the caims deveopment pattern γ and the exposures µ i using ML methods resuting in γ MLE and µ MLE i. Ony the simutaneous MLE of γ and µ i give the chain adder pattern given in 4.8. In the second step we then repace the MLE µ MLE i by an externa estimate µ i. It coud be argued that this is, in some sense, inconsistent, but it describes what practitioners do in the BF method to smoothen the caims reserves estimates. 4.2 Overdispersed Poisson Mode as a Generaized Linear Mode Renshaw [17] and Renshaw-Verra [18] were the first to impement the standard GLM techniques for the derivation of estimates for incrementa data in a caims reserving context. In this subsection we give an brief description of the overdispersed Poisson mode in the GLM framewor. For more detais on the overdispersed Poisson mode in the GLM context and on GLMs and their statistica bacground we refer to Engand- Verra [4] and McCuagh-Neder [14] or Fahrmeir-Tutz [6], respectivey. A specific GLM mode in a parametrization suitabe for caims reserving is fuy characterized by the foowing three components a the type of the Exponentia Dispersion Famiy for the random component X i, ; b the in function g reating the expectation of the random component X i, to the inear predictor η i, = Γ i, b, i.e. g E[X i, ] = η i, for a 0 i I and 0 i J; 11

12 c the design matrices Γ i, for a 0 i I and 0 i J. In the overdispersed Poisson mode the distribution of the random component is given by the overdispersed Poisson distribution, and for the mutipicative structure of Mode 4.1 it is straightforward to choose the og-in g = og as in function. Then we have η i, = g m i, = ogµ i + ogγ For generaized inear modes, it is easy to obtain MLEs of the parameters µ i and γ using standard GLM software. However, since the mutipicative structure of Mode 4.1 is overparameterized it becomes necessary to set constraints which coud tae a number of different forms. In the ast subsection we derived the MLEs γ MLE under the normaization assumption J =0 γmle = 1 in order to obtain a caims deveopment pattern see e.g. Remars 4.3. However, in the framewor of GLMs it is more natura and indeed convenient to choose the constraint µ 0 ogµ 0 = 0 and = 1, hence η 0, = ogγ for a, 4.11 see e.g This parametrization eads to the foowing vector of unnown parameters b = ogµ 1,..., ogµ I, ogγ 0,..., ogγ J, 4.12 and the 1 I + J + 1 design matrices Γ 0, = 0,..., 0, 0, 0,..., 0, e I++1, 0,..., 0, 4.13 Γ i, = 0,..., 0, e i, 0,..., 0, e I++1, 0,..., for 1 i I and 0 J, where the entries e i = 1 and e I++1 = 1 are on the i-th and the I th position, respectivey. We obtain the inear predictor η i, = Γ i, b Hence, we have now reduced the dimension from I +1 J +1 unnown parameters m i, to p = I + J + 1 unnown parameters ogµ i and ogγ. Using standard GLM software based on the Fisher scoring method, these parameters are then estimated with the MLE method. We obtain the MLEs b = ogµ1 GLM,..., ogµ I GLM, ogγ 0 GLM,..., ogγj GLM,

13 which impies a second payout pattern where for a = 0,..., J. The foowing reationships hod, γ GLM 0,..., γ GLM J, 4.17 γ GLM = exp og γ GLM 4.18 γ MLE = γglm =0 γ GLM for a = 0,..., J Remars 4.4: Note the superscripts MLE and GLM, which are used to differentiate between the two normaizations, one natura to maximum ieihood for caims reserving and the other more practica for GLM modeing purposes. In mutipicative modes ie the overdispersed Poisson mode it is natura to use the og-in g = og as in function since the systematic effects are additive on the scae given by the og-in function. Moreover, the og-in is the so-caed canonica in function for the overdispersed Poisson distribution that has convenient mathematica and statistica properties see e.g. McCuagh- Neder [14] or Fahrmeir-Tutz [6]. In the next section, reationship 4.19 wi be crucia to incorporate our resuts from GLM theory in the derivation of an estimate of the conditiona MSEP in the BF method. From GLM theory it is we-nown that the ML estimate 4.16 is asymptoticay mutivariate normay distributed with covariance matrix Cov b, b estimated by the inverse of the Fisher information matrix denoted by H 1 b, which is a standard output in a GLM software pacages see e.g. Paner [15] or Fahrmeir- Tutz [6]. 5 MSEP in the BF Method using GLM In this section we quantify the uncertainty in the estimation of the utimate caims C i,j and BF I i=1 C i,j by the estimators Ĉ i,j and BF I i=1 Ĉ i,j, respectivey, given the 13

14 observations D I. More precisey, our goa is to derive an estimate of the conditiona MSEP for singe accident years i {1,..., I} msep Ci,J D I Ĉ i,j BF = E [ C i,j Ĉ i,j = E =I i+1 BF 2 D I ] X i, 1 CL I i 5.1 µ i 2 D I, 5.2 as we as an estimate of the conditiona MSEP for aggregated accident years I BF I I 2 BF msep P I i=1 C i,j D I Ĉ i,j = E C i,j Ĉ i,j D I. 5.3 i=1 This is described in the next subsections. i=1 i=1 5.1 MSEP in the BF Method, Singe Accident Year We choose i {1,..., I}. Since the incrementa caims X i, conditiona MSEP 5.1 can be decouped in the foowing way E CL X i, 1 I i µ i 2 D I = =I i+1 =I i+1 + 2E VarX i, + E =I i+1 =I i+1 Xi, E[X i, ] E[X i, ] 1 =I i+1 Note that µ i is independent of X, for a,, that CL I i E[X i, ] 1 CL I i are independent the µ i 2 D I 5.4 CL I i µ i D I. is D I -measurabe and that E [ µ i ] = µ i see e.g and Mode Assumptions 4.1. Therefore, the ast term in the above equaity disappears and we get BF msep Ci,J D I Ĉ i,j = =I i+1 + µ 2 i Var X i, + 1 =I i+1 γ CL 2 I i Var µi γ MLE =I i Hence, the three terms on the right-hand side of 5.5 need to be estimated in order to get an appropriate estimate for the conditiona MSEP in the BF method. The first term as a conditiona process variance originates from the stochastic movement of X i,. The second and third term on the right-hand side of 5.5 constitute the conditiona estimation error which refects the uncertainty in the prior estimate µ i and the ML estimates γ MLE, respectivey. 14

15 5.1.1 Process Variance For the estimation of the conditiona process variance, Mode Assumptions 4.1 motivates the foowing estimator VarX i, = φ µ i =I i+1 = φ µ i 1 γ MLE CL I i, 5.6 where φ is an estimate of the dispersion parameter φ. Within the framewor of GLM we use different types of residuas Pearson, deviance, Anscombe, etc. to estimate φ see e.g. McCuagh-Neder [14] or Fahrmeir-Tutz [6]. In the foowing we wi use the Pearson residuas defined by where m i, is the GLM estimate of m i, given by R p i, = X i, m i, mi,, 5.7 m i, = µ GLM i γ GLM 5.8 for a 0 i + I. The estimate of the dispersion parameter is then given by 2 Rp i, 0 i+ I φ =, 5.9 N p where N = number of observations X i, in D I, i.e. N = D I 5.10 p = number of estimated parameters, i.e. p = I + J Estimation Error The conditiona estimation error is given by the second and third term on the righthand side of 5.5. This means that we need to quantify the voatiity of the prior estimates µ i and the MLEs γ MLE around the true parameters µ i and γ, respectivey. Prior estimate µ i : The second term 1 CL 2 I i Var µi 5.12 quantifies the uncertainty in the prior estimate µ i of the expected utimate caim E[C i,j ]. Since µ i is determined exogenousy see e.g. Remars 4.2 this can generay 15

16 ony be done using externa data ie maret statistics and expert opinion. The reguator, for exampe, provides an estimate for the coefficient of variation of µ i, denoted by Vco µ i, that quantifies how good the exogenous estimator µ i is. Statistica estimates based on impact studies for the determination of estimates Vco µ i exist, for exampe, in the context of the Swiss Sovency Test [19]. These studies suggest that 5% to 10% is a reasonabe range for Vco µ i. Hence the term 5.12 is estimated by 1 CL 2 I i Var µi = 1 CL 2 I i µ 2 i Vco µi Note that an appropriate choice for Vco µ i is crucia for a meaningfu anaysis. This choice is cosey reated to a Bayesian setup where one chooses an appropriate prior distribution for µ i see Mac [11]. Of course, the choice of this prior distribution and/or its coefficient of variation depends on the interna processes in the company. Ideay, this is determined using maret statistics as described above and simiary as used, for exampe, in the context of modeing operationa ris, see Lambrigger et a. [8]. Unfortunatey, in many cases there are no maret statistics avaiabe and one tries to adust the priors using interna data. However, this approach contradicts the BF method if we interpret it in the strict sense described by Mac [12], since the choice of the prior µ i shoud be independent from the observations X,. We woud ie to motivate further research into this direction, i.e. 1 finding appropriate priors and 2 describe the interna processes as they are used in practice. This coud ead to a new theory and method using Kaman fiters see Chapter 9 in Bühmann-Giser [2] and Chapter 10 in Tayor [20] to describe oss ratio prediction based on observations of past accident years. One then immediatey oses the independence assumptions and the conditiona MSEP no onger decoupes in a nice way. MLEs γ MLE : The estimation of the third term on the right-hand side of 5.5 requires more wor. We have to study the fuctuations of the MLEs γ MLE around the true parameters γ =I i+1 γ =I i+1 γ MLE

17 Negecting that MLEs have a possibe bias term we estimate 5.14 by Var γ MLE = Cov γ MLE, γ MLE =I i+1 = =,=I i+1,=i i+1,=i i+1 Cov γ GLM J =0 γglm Cov 1 1 +,, bγ GLM bγ GLM γ GLM J =0 γglm bγ GLM bγ GLM 5.15 see e.g Here, we restricted our probabiity space such that a soution to the above equations exist. We define = =0 γ GLM γ GLM and δ = E [ ] 5.16 for a = I i + 1,..., J. Hence we need to cacuate 1 1 Cov, for a, = I i + 1,..., J. We first do a Tayor approximation around δ. To this end we define the function fx = x with f 1 x = 1 + x and obtain for the first order Tayor approximation around δ fx fδ + f δ x δ = δ 1 + δ 2 x δ This impies that 1 1 Cov, = δ δ 2 Cov, 5.20 GLM γ Cov m γ GLM, γglm m γ GLM δ δ 2 It ony remains to cacuate the covariance terms on the right-hand side of We use the foowing inearization Tayor approximation for the exponentia function γ GLM GLM γ γ GLM = exp og γ GLM = γ GLM γ γ exp og γ γ GLM og γ γ GLM γ γ 1 + og og γ γ γ GLM 17

18 Using 5.21 we obtain for the covariance terms on the right-hand side of 5.20 GLM γ Cov γ GLM, γglm m γ GLM γ GLM γ m γ γ m GLM Cov og γ γ γ GLM, og γ GLM Now, we define the sighty modified design matrices Γ = 0,..., 0, e I++1, 0,..., 0 for = 0,..., J, 5.23 which impies that see 4.16 and 4.18 og = Γ b for = 0,..., J Hence Cov og GLM γ γ GLM γ GLM γ m GLM, og γ GLM = Γ Γ Cov b, b Γm Γ Using the inverse of the Fisher information matrix H b for the estimation of the covariance term Cov b, b cf. Remars 4.4 we obtain for 5.15 Var γ MLE δ δ =I i+1,=i i+1 m γ γ γ m γ Γ Γ H b 1 Γm Γ see e.g. Remars 4.4. Hence we define the estimator Var γ MLE 1 1 = 1 + δ δ =I i+1,=i i+1 bγ GLM bγ GLM m γ GLM γ m GLM Γ γ GLM γ GLM Γ H b 1 Γm Γ, where we set δ = J =0 see e.g This can be rewritten in matrix notation, we define the parameter c, = γ GLM γ GLM µ MLE 4 0 and γ = 0,..., 0, γ GLM 0,..., γ GLM J. Furthermore, we define [ Ψ, = c, γ H b 1 γ µ MLE 0 γ H b 1 Γ + Γ ] + µ MLE 2 0 H b 1,. Then we obtain Var γ MLE =I i+1 =,=I i Ψ, Putting the three estimates 5.6, 5.13 and 5.29 together we obtain the foowing estimator for the conditiona MSEP for a singe accident year: 18

19 Estimator 5.1 MSEP for the BF method, singe accident year Under Mode Assumptions 4.1 an estimator for the conditiona MSEP for a singe accident year i {I J + 1,..., I} is given by BF msep Ci,J D I Ĉ i,j = φ 1 CL I i µ i + 1 CL 2 I i µ 2 i Vco µi µ 2 i,=i i+1 Ψ,, see 5.6, 5.13 and MSEP in the BF Method, Aggregated Accident Years In this subsection we derive an estimate for the conditiona MSEP for aggregated accident years 5.3. We start by considering two different accident years i <, BF BF msep Ci,J +C,J D I Ĉ [ BF BF ] 2 i,j + Ĉ,J = E C i,j + C,J Ĉ i,j Ĉ,J D I. By the usua decomposition we find BF BF msep Ci,J +C,J D I Ĉ i,j + Ĉ,J 5.31 BF = msep Ci,J D I Ĉ BF i,j + msep C,J D I Ĉ,J µ i µ γ γ MLE γ. =I i+1 =I i+1 =I +1 =I +1 γ MLE That is, we need to give an estimate for the term on the right-hand side of Anaogousy to 5.14, we have to study the fuctuations of the MLEs γ MLE around the true parameters γ. Again, negecting the possibe bias of the MLEs, we estimate this term by Cov =I i+1 = γ MLE, =I i+1 =I +1 =I i+1 =I +1 γ MLE =I +1 Cov γ MLE, γ MLE δ δ n m γ n γ γ m γ Γn Γ H b 1 Γm Γ 19

20 see e.g Again, as with the singe accident year case, we restrict our probabiity space such that the above covariance exists. This motivates the foowing estimator for the covariance term 5.33 Υ i, = Ĉov γ MLE, =I i+1 γ MLE =I +1 = =I i+1 =I +1 Ψ, This eads to the foowing estimator for the conditiona MSEP for aggregated accident years: Estimator 5.2 MSEP for the BF method, aggregated accident years Under Mode Assumptions 4.1 an estimator for the conditiona MSEP for aggregated accident years is given by msep P I i=1 C i,j D I I BF Ĉ i,j = i=1 I BF msep Ci,J D I Ĉ i,j 5.35 i= i< I µ i µ Υ i,. The natura extension of this wor woud be to obtain some properties of Estimators 5.1 and 5.2, for exampe, asymptotic behaviour. However, we omit this presenty, because it woud go beyond the context of this wor. 6 Exampe and Simuation In this section we state an exampe and a simuation. 6.1 Exampe of MSEP in the BF Method Using the incrementa caims data provided in Tabe 1, which are scaed cumuative payments from property business, we cacuate the BF reserves and the estimators for the conditiona MSEP as derived for Estimator 5.1 singe accident year and Estimator 5.2 aggregated accident years. Furthermore, we compare these resuts with the estimators for conditiona MSEP of the overdispersed Poisson method/poisson GLM method, with reserves matching those of the CL method. However, before we can estimate the conditiona MSEP for the BF method, we must first specify the assumptions regarding utimate caim estimates µ i. We assume the prior estimates µ i for the expected utimate caims E[C i,j ] are given by Tabe 2. Furthermore, we have to specify the uncertainty in these estimates. To 20

21 this end we assume for the coefficient of variation a fat rate of 5% for a accident years, i.e. Vco µ i = 0.05 for 1 i I. 6.1 This assumption, as previousy stated, has been shown to be reasonabe in the Swiss Sovency Test see Swiss Sovency Test [19]. To view the effect of changing this input see Tabe 5 where we have presented the BF MSEP as a function of the coefficient of variation for the expected utimate estimates. For the dispersion parameter φ we obtain the estimate φ = 14, see e.g ,946,975 3,721, , , ,704 62,124 65,813 14,850 11,130 15, ,346,756 3,246, , ,797 67,824 36,603 52,752 11,186 11, ,269,090 2,976, , , ,703 65,444 53,545 8, ,863,015 2,683, , , ,976 88,340 4, ,778,885 2,745, , , , , ,184,793 2,828, , , , ,600,184 2,893, , , ,288,066 2,440, , ,290,793 2,357, ,675,568 Tabe 1: Observed incrementa caims X i,. Now we can cacuate the estimators for the conditiona MSEPs as given by Estimator 5.1 and 5.2 and obtain the vaues given in Tabe 3. If we compare these resuts to the resuts for the overdispersed Poisson GLM method in Tabe 4 we observe the foowing: a The caims reserve estimates are rather arge for the BF method, refecting the conservative prior estimates µ i for the expected utimate caims given by Tabe 2. b As a consequence of a the process standard deviations of the BF method are arger than the ones of the overdispersed Poisson GLM method. c The totas of prior and parameter uncertainty in the estimators µ i and CL are sighty higher than the estimation errors in the overdispersed Poisson GLM method. As a consequence, the conditiona MSEPs of the BF method are arger than the ones of the overdispersed Poisson GLM method. However, due to the conservative estimation of the caims reserves, the coefficients of variation are smaer than the ones of the overdispersed Poisson GLM mode. 21

22 i bµ i 0 11,653, ,367, ,962, ,616, ,044, ,480, ,413, ,126, ,986, ,618,437 Tabe 2: Prior estimates for the expected utimate caims. i BF process prior parameter prior and msep 1/2 Vco reserves std. dev. std. dev. β std. dev. parameter std. dev. 1 16,120 15, ,539 15,560 21, % 2 26,998 19,931 1,350 17,573 17,624 26, % 3 37,575 23,514 1,879 18,545 18,639 30, % 4 95,434 37,473 4,772 24,168 24,635 44, % 5 178,023 51,181 8,901 29,600 30,910 59, % 6 341,305 70,866 17,065 35,750 39,614 81, % 7 574,089 91,909 28,704 41,221 50, , % 8 1,318, ,294 65,932 53,175 84, , % 9 4,768, , ,419 75, , , % tota 7,356, , , , , , % Tabe 3: Estimates for BF reserves, process error 1/2, prior standard deviation, parameter standard deviation, tota of prior and parameter standard deviation, msep 1/2 and Vco for singe and aggregated accident years. i reserves process parameter msep 1/2 Vco std. dev. std. dev. 1 15,125 14,918 14,611 20, % 2 26,257 19,656 17,160 26, % 3 34,538 22,543 17,159 28, % 4 85,301 35,428 22,040 41, % 5 156,493 47,986 27,108 55, % 6 286,120 64,885 32,927 72, % 7 449,166 81,296 38,935 90, % 8 1,043, ,897 66, , % 9 3,950, , , , % tota 6,047, , , , % Tabe 4: Estimates for overdispersed Poisson reserves, process error 1/2, prior standard deviation, parameter standard deviation, msep 1/2 and Vco for singe and aggregated accident years. 22

23 dvcobµ i msep 1/ , , , , , , , , , , ,311 Tabe 5: The BF msep 1/2 as a function of Vco µ i. 6.2 Simuation In the derivation of Estimators 5.1 and 5.2 we used various approximations. We test the strength of our resuts by simuation. Assuming the incrementa caims X i, are overdispersed Poisson distributed with parameters γ MLE and µ MLE and dispersion parameter 6.2 see e.g. Mode Assumptions 4.1, we generate run-off trianges D I from which we cacuate for each simuation/run-off triange the estimated payout pattern CL. From the generated sampes we obtain the empirica distribution of the payout pattern, and hence cacuate its empirica standard deviation. β empirica estimated variance Var bβ CL dvar bβ CL with % 0.654% 0.653% % 0.486% 0.484% % 0.373% 0.370% % 0.317% 0.313% % 0.260% 0.258% % 0.220% 0.219% % 0.177% 0.175% % 0.162% 0.160% % 0.138% 0.137% % 0.000% 0.000% Tabe 6: Simuation resuts comparing empirica and estimated variance of the cumuative payout pattern. Tabe 6 provides the resuting empirica payout pattern, the empirica variance and the estimated variance for each deveopment year. From these resuts it is cear that the approximation of the variance of the cumuative payout pattern is very cose to the empirica vaue. This means that the approximation given in 5.26 performs very 23

24 we for a typica payout pattern in practice. Acnowedgements The authors woud ie to than Prof. Pau Embrechts for hepfu discussion reated to the paper. Furthermore, the first author is gratefu to ACE Limited for providing the financia support that made this research a reaity. References [1] Bornhuetter, R.L., Ferguson, R.E The actuary and IBNR. Proc CAS, Vo. LIX, [2] Bühmann, H., Giser, A A Course in Credibiity Theory and its Appications. Springer, Berin. [3] Casuaty Actuaria Society CAS Foundations of Casuaty Actuaria Science, fourth edition. [4] Engand, P.D., Verra, R.J Stochastic caims reserving in genera insurance. British Act. J. 8/3, [5] Engand, P.D., Verra, R.J Predictive distributions of outstanding iabiities in genera insurance. Annas of Actuaria Science 1, [6] Fahrmeir, L., Tutz, G Mutivariate Statistica Modeing Based on Generaized Linear Modes. 2nd edition, Springer-Verag, Berin. [7] Hachemeister, C., Stanard, J IBNR caims count estimation with static ag functions. Presented at 1975 ASTIN Cooquium, Portimao, Portuga. [8] Lambrigger, D.D., Shevcheno, P.V., Wüthrich, M.V The quantification of operationa ris using interna data, reevant externa data and expert opinion. J. Op. Ris 2/3, [9] Mac, T A simpe parametric mode for rating automobie insurance or estimating IBNR caims reserves. ASTIN Buetin 21/1, [10] Mac, T Distribution-free cacuation of the standard error chain adder reserves estimates. ASTIN Buetin 23/2, [11] Mac, T Credibe caims reserves: The Bentander method. ASTIN Buetin 30/2, [12] Mac, T Parameter estimation for Bornhuetter/Ferguson. CAS Forum, Fa 2006,

25 [13] Mac, T., Quarg, G., Braun, C The mean square error of prediction in the chain adder reserving method - a comment. ASTIN Buetin 36/2, [14] McCuagh, P., Neder, J.A Generaized Linear Modes. 2nd edition, Chapman and Ha, London. [15] Paner, H.H Operationa Riss: Modeing Anaytics. Wiey & Sons, New Yor. [16] Radte, M., Schmidt, K.D Handbuch zur Schadenreservierung. Verag Versicherungswirtschaft, Karsruhe. [17] Renshaw, A.E Caims reserving by oint modeing. Proceedings of the XXV ASTIN Cooquium Cannes. [18] Renshaw, A.E., Verra, R.J A stochastic mode underying the chain adder technique. British Act. J. 4/4, [19] Swiss Sovency Test BPV SST Technisches Doument, Version 2. Otober [20] Tayor, G Loss Reserving: An Actuaria Perspective. Kuwer Academic Pubishers, Boston. [21] Teuges, J.L., Sundt, B Encycopedia of Actuaria Science, Voume 1. Wiey & Sons Ltd, Chichester. [22] Verra, R.J Bayesian and empirica Bayes estimation for the chain adder mode. ASTIN Buetin 20/2, [23] Verra, R.J An investigation into stochastic caims reserving modes and the chain-adder technique. Insurance: Math. Econom. 26, [24] Verra, R.J., Engand, P.D Comments on: A comparison of stochastic modes that reproduce chain adder reserve estimates, by Mac and Venter. Insurance: Math. Econom. 26, [25] Verra, R.J A Bayesian generaized inear mode for the Bornhuetter-Ferguson method of caims reserving. North American Act. J. 8/3, [26] Wüthrich, M.V., Merz, M., Bühmann, H Bounds on the estimation error in the chain adder method. To appear in Scand. Act. J. 25