Generalized Mack Chain-Ladder Model of Reserving with Robust Estimation

Transcription

1 Generalized Mac Chain-Ladder Model of Reserving with Robust Estimation Przemyslaw Sloma Abstract n the present paper we consider the problem of stochastic claims reserving in the framewor of Development Factor Models DFM. More precisely, we provide the Generalized Mac Chain-Ladder GMCL Model that expands the approaches of Mac 1993, 1994, 1999, Saito 009 and Murphy et al. 01. Our general flexible tool of reserving provides the solution to the one of the major challenges of day-to-day actuarial practise which is quantifying the variability of reserves in the case where different methods of selecting Loss Developments Factors LDFs are applied. We develop the theoretical bacground to estimate the conditional mean square error of prediction MSEP of claims reserves that is consistent with actuarial practice in selecting the LDFs. Moreover, we present an example of GMCL s application in which we indicate how to bridge the estimation of parameters in the Chain-Ladder framewor with the robust estimation techniques. Finally, we show how our approach can be used in validation of the reserve ris evaluation in the Solvency context. Keywords: Non-life insurance, Stochastic claims reserving, Mac Chain-Ladder, Robust Estimation, Solvency. 1. ntroduction and motivation The provision for outstanding claims is one of the main components of technical provisions of insurance company s liabilities. Measuring the deviation of the true amount of reserves from its estimation is one of the major actuarial challenge. Senior managers, shareholders, rating agencies, and insurance regulators all have an interest in nowing the magnitude of these potential variations reserve uncertainty since companies with large potential deviations need more capital or reinsurance. One of the most nown method of reserving used in practice is the approach called Chain-Ladder. This method belongs to the family of Development Factor Models DFM. The first stochastic approach based on Chain - Ladder technique was proposed by Mac 1993, n this study author proposed the estimation of the mean square error of prediction MSEP of claims reserves based upon all-year volume-weighted average of loss development factors also called: lin ratios, age-to-age factors, report-to-report factors. The variance structure was supposed to be proportional to the development period s initial loss. This assumption is sufficient for the weighted average development factors to have optimal statistical properties BLUE-best linear unbiased estimate. Some authors pointed out that the estimation of Chain-Ladder factors is connected with the estimation in the framewor of linear model by weighted least squares WLS regression approach. They also observed that by modifying the original variance assumption from Mac 1993 the corresponding estimators of Chain-Ladder development factors eep their BLUE property see Murphy 1996, Barnett and Zehnwirth 000 and Saito 009. t is worth underlying that the modification of variance assumption leads to different point estimators for development factors arithmetic average, slope of regression etc., see Remar 3.1 for more details. One of the major challenge in every day actuarial practice is selecting the loss development factors. The adjustments to mae data more homogeneous are often justified for number of reasons: unstable run-off triangles, outliers, inaccurate and incomplete data, etc.,. Most actuaries use picing up rules of thumb and helpful approaches in to selecting the loss development factors LDFs. The interesting study by Blumsohn and Laufer 009 describes this address: przemyslaw.sloma@yahoo.fr Przemyslaw Sloma Preprint submitted to Variance May, 016

2 topic in great detail. n this project, a group of actuaries were ased to select LDFs for an incurred run-off triangle. The important number of ways of LDFs selection was provided by the participants. The approaches proposed to evaluate the estimation of expected value of reserves varied widely and the additional information about the error of prediction of this estimation could be helpful in decision maing. That is why, it is extremely important from a practical point of view to have a method that provides the estimation of conditional MSEP of ultimate claims or reserves in the context of LDFs selection by actuaries. n the present study we provide such a tool embedded in the theoretical framewor to quantify the standard error of prediction of the claims reserves in the case where some factors has been excluded from the estimation of model parameters. However, we do not judge whether these ad hoc approaches of selecting factors are correct or wrong. We rather assume that the expert judgment taen by an actuary could always be justified by his specific nowledge of considered business. Measuring the variability of the reserves in this context is poorly developed in the literature. That is why in practice actuaries and reserving software developers often use the proxy methods based on formula for MSEP derived in Mac The approximations mainly consist on replacing the main parameters by theirs estimators computed by the other approach without changing the main formula. This procedure is incorrect because in the Chain-Lader framewor the formula for MSEP depends among others on the standard error of Chain-Ladder factors and it is not accurate to simply plug-in the new estimators in the old formula. The other proxy method often used in practice consists on applying the coefficients of variation of ultimate loss from Mac 1993 ratio of square root of MSEP of ultimate loss over ultimate loss in order to derive the MSEP estimators of a new approach. t turns out that in general, these approximations are highly inappropriate see example in Section 6.3. We thin that, in some simple cases no curve fitting for LDFs for example the approximations mentioned above are the consequences of bad understanding of the main formula for estimation of MSEP in Mac Moreover the approximations used by actuaries and actuarial software developers could be avoided by using the more appropriate existing models. One of such model was proposed by Mac To our nowledge, this was the first study that showed how to measure the uncertainty of reserves in the situation when an actuary selects the LDFs. n our opinion, the important results obtained in this paper are not always used in practice because author, instead of explicit formula which can be found in Section 3.4, gave the recursive equation for estimation of MSEP of ultimate loss. The study of Mac 1999 is an important article that allows the fully understanding of the MSEP formula and avoid the inappropriate approximation when it is not necessary. We recall the details of this method in Section 3. One of the major limitations of this method is the underestimation the MSEP of ultimate loss in the case where the number of excluding data is important. We discussed this topic in details in Section 3.5. One of possible solution to overcome this difficulty is to extent the existing approach proposed by Mac n this study, we propose a general approach for stochastic claims reserving in the framewor of Chain-Ladder Model extending the model proposed by Mac 1999 and Murphy et al. 01. This extension is three fold. First, our general tool has a educational role and gives also the possibility to validate the results from other approaches. More precisely, our general formula for estimation of MSEP of outstanding loss liabilities can be used to fully understand the Mac 1993, 1994, 1999 model. Furthermore, under new solvency requirements-solvency, the insurance companies use the bootstrap-type stochastic reserving methods to determine the economic capital corresponding to the reserve ris. The bootstrap method allows estimation of a whole claims reserves distribution via resampling techniques and Monte Carlo simulations. t seems to be crucial for non-life insurance companies to be able to valid the results given by the industrial software where we do not have access to the code and when the number of shortcuts may be applied. Our approach can be used to validate the estimation of the first two moments of the loss distribution in the case where selection of development factors was employed and the different weights in estimation of Chain-Ladder factors and volatility parameters were used see Section 6.3 for more details. Second, our General Mac Chain-Ladder GMCL model can be used to construct the proxy solutions to overcome the limits of Mac 1999 approach which is the use of the same weights for parameters estimation see discussion in Section 3.6. This means that, for the methods where we eliminate the considerable number of observations we reduce as well the data for variability of the reserves. This mechanically impact the estimation of MSEP of loss liabilities which in such cases is, in general, underestimated. We propose then the possible solution to overcome this ind of difficulties see Section 6.1 for more details. Finally, the third and really important application from a practical point of view, consists on bridging the point estimation of Chain-Ladder parameters with the theory robust statistics. As mentioned above, the point estimators of chain-ladder factors can be obtain in the linear regression framewor by applying the weight least squares procedure.

3 t is well nown that the OLS estimators are fragile to the outliers. That is why we propose using the robust techniques of estimation such as: M-estimators, L p - estimators, etc see Section 6.. The reminder of this paper is organized as follows. n Section we present our notations and definitions. We recall in Section 3 the MCL and its main limitations. n Section 4, we present the GMCL and the main results are derived in Section 5. Finally, the Section 6 introduce the numerical applications of GMCL. All proofs are provided in the Appendix. The related topics such as: tail factor, curve fitting, diagnostics and validation of the main model hypothesis, are out of scope of this paper and will be treated elsewhere.. Notations and definitions.1. Run-off triangle Let C i, j denote the random variables cumulative payments, inccured, reported claims numbers, etc. for accident year i {1,...,} until development year j {1,...,J}, where the accident year is referred to as the year in which an event triggering insurance claims occurs. We assume that C i, j are random variables observable for calendar years i + j + 1 and non-observable to be predicted for calendar years i + j > + 1. The observable C i, j are represented by the so-called run-off trapezoids > J or run-off triangles = J. Table 1 gives an example of a typical run-off triangle. n order to simplify our notation, we assume that = J run-off triangle. However, all the results we present here can be easily extended to the case when the last accident year for which data is available is greater than the last development year, i.e., > J run-off trapezoid. Accident Development Year j Year i j... J 1 3 C i, j observations j 1 C i, j to be predicted Table 1: Run-off triangle = J.. Outstanding Reserves Let R i et R denote the outstanding claims liabilities for accident year i {1,..,}, and the total outstanding loss liabilities for all accident years, R i = C i, C i, i+1,.1 R = R i.. We use the term claims reserves to describe the prediction of the outstanding loss liabilities. Hence, let R i and R denote the claims reserves for accident year i, R i = Ĉ i, C i, i+1, i {1,...,}, and the total claims reserves for aggregated accident years, R = R i, respectively, where Ĉ i, is a predictor for C i,. 3

4 .3. Conditional Mean Square Error of Prediction MSEP As already stated above, finding suitable prediction of ultimate loss, is rather the beginning of the process of reserving and the insurers companies need to assess the variability of these amounts. We are interested then in the quantification of the prediction uncertainty of the ultimate loss, i.e., Ĉ i, and Ĉi,, or equivalently of claims reserves, i.e., R i and R = R i. For that, we have to choose an appropriate ris measure which determines a conception of measuring the "distance" between the prediction and the actual outcomes. n the present paper, following the actuarial literature, we quantify the prediction uncertainty using the most popular such measure, the so-called mean-square error of prediction MSEP. [ ] msepĉi D C i = E Ĉi C i D,.3 where msep Ĉ i D denote the claims data available at time t =. 3. Mac Chain-LadderMCL Model C i = E Ĉ i C i D,.4 D = {C i, j : i + j + 1},.5 A major every-day challenge of actuarial wor is selecting loss development factors for number of reasons outliers in triangle, inaccurate data, incompleteness, etc. Most actuaries use picing up rules of thumb and helpful approaches into selecting the loss ratios. n Blumsohn and Laufer 009 a group of actuaries were ased to select age-to age factors for 1-years triangle of umbrella business. The important number of ways of selecting loss ratios was provided by the participants. t is important then from practical point of view to have a method that provides the estimation of conditional MSEP of ultimate claims in the context of factor selection of actuaries. To the best of our nowledge, the article of Mac 1999, is one of the first study dealing with factors selection and variability of reserves estimation in the framewor of Chain-Ladder method. This paper is an extension of Mac n the remaining part of this section we recall the assumptions of the Mac Chain-Ladder MCL Model from Mac Afterwards, we present the numerical example illustrating the limits of this approach. Finally, we indicate the possible expansion of MCL method and its potential applications Model assumptions of MCL method Let define the individual development factors, for 1 i 1 and 1 1, Following Mac 1999, we assume MCL.1 There exist constants f > 0 such that F i, = C i,+1 /C i,. 3.1 EF i, C i,1,..,c i, = f. The parameters f are often called loss development factors LDF, lin ratios or age-to-age factors. MCL. There exist constants σ > 0 such that for all 1 i and 1 1 we have σ VarF i, C i,1,..,c i, = w i, Ci, α, with w i, [0,1]. 3. The parameters σ are referred here as variance parameters LDF. MCL.3 The accident years C i,1,..,c i, 1 i are independent 4

5 3.. Estimation of parameters in the MCL model Given the information D and for 1 1, the factors f are estimated by f = w i,ci, α F i, γ, α {0,1,}. 3.3 i, Given the information D and for 1, the variance parameters σ are estimated by σ = 1 1 w i, C α i, F i, f, α {0,1,}, 3.4 where represents the number of weights w i, different from 0, namely, := card{i : w i, 0}. Formula 3.4 does not yield an estimator for σ because it is not possible to estimate this parameter from the single observation C, /C,. Following Mac 1993, 1994, 1999, if f = 1 and if the claims development is believed to be finished after 1 years we can put σ = 0. f not, the simple formula of extrapolation can be applied by requiring σ 3 / σ = σ / σ. This leads to the following definition σ := min σ 4 / σ 3,min σ 3, σ. 3.5 Remar 3.1. The parameter α determines the different ways of estimation of f. For sae of simplicity, let us assume that w i, j = 1 for all i, j. We present below the possible choices of α and their interpretation. 1. f α = 1 we get the classical Chain Ladder estimate of f f = C i,f i, C i, = C i,+1 C i,, for f α = 0 we get the model for which the estimators of the age-to-age factors f are the straightforward average of the observed individual development factors F i, j defined via 3.1, i.e., f = 1 F i,, for f α = we get the model for which the estimators of the age-to-age factors f are the results of an ordinary regression of {C i,+1 } i {1,..., 1} against {C i, } i {1,..., } with intercept 0, i.e., f = C i, F i, C i, 3.3. Properties of estimators from MCL model Proposition 3.1. = C i,c i,+1 1 i=0 C i,, for 0 1. i The estimators f given in 3.3 are unbiased and uncorrelated. ii The estimators f of f have the minimal variance among all unbiased estimators of f which are the weighted average of the observed development factors F i,. iii The estimator σ, given in 3.4 is the unbiased estimator of the parameter σ. 5

6 iv Under the model assumptions MCL.1 and MCL.3 we have EC i, D = C i,+1 i f i,+1 i... f. This implies, together with the fact that f are uncorrelated, that Ĉ i, is unbiased estimator of EC i, D. v The expected values of the estimator Ĉ i, = C i,+1 i =+1 i for the ultimate claims amount and of the true ultimate claims amount C i, are equal, i.e., EĈ i, = EC i,, i. The proof is postponed to Appendix A.3. f, 3.4. Estimators of conditional MSEP in MCL model Single accident years Under assumptions of MCL Model we have the following estimator for the conditional estimation error of a single accident year i {,...,}: msepĉi, D C i, = Ĉi, = i+1 σ f 1 ŵ i, Ĉ α i, + 1 j=1 w j,c α j, where, for i + > + 1, we define ŵ i, := 1 and f j and σ j are given in 3.3 and respectively Aggregated accident years msep Ĉ i, D C i, = i= msepĉi, D C i, + where f j and σ j are given in 3.3 and respectively Numerical application of MCL method i= Ĉ i, j=i+1 Ĉ j, = i+1, 3.6 σ / f l=1 w l,cl, α, 3.7 As mentioned above, the factors selection methods are an integral part of every-day actuarial practice. We choose here from Blumsohn and Laufer 009 several such methods where estimates are computed as different averages using varying weights and varying number of accident years: all/3/5-years weighted average and all excluding higher and lower AEHL factor average. We consider as well other popular methods in actuarial practice based on sample median. More precisely, for RAA run-off triangle see Appendix B, Section B.8, we apply the MCL model from Mac 1999 with the following parameters. For all 5 methods described below we choose α = 0 f arithmetic averages of F i, and we compute the estimators f and σ according to formula 3.3 and respectively. This allows us to compare the results with sample Median method which is rather consistent with straight-forward average of development factors see method number 5 below 1 ALL AV: f are computed as arithmetic average of all individual lin ratios F i, j. More precisely, we define the weights in the following way: w i, j = 1 for all i, j. AEHL: f are computed as arithmetic average of all individual lin ratios excluding the highest and the lowest values of F i, j. More precisely, we define the weights in the following way: for fixed j, w i, j = 0 for i such that F i, j = F j, j and F i, j = F 1, j, where F, j for = 1,..., j denotes the order statistics of F i, j. For remaining indices i, for fixed j, we tae w i, j = 1 6

7 3 5 Years AV: f are computed as an arithmetic average of individual lin ratios F i, j from 5 latest accidents years. More precisely, we define the weights in the following way: w i, j = 1 for i = j,..., j 4. For remaining indices i, for fixed j, we tae w i, j = Years AV: f are computed as an arithmetic average of individual lin ratios F i, j from 3 latest accidents years. More precisely, we define the weights in the following way: w i, j = 1 for i = j,..., j. For remaining indices i, for fixed j, we tae w i, j = 0 5 Median: f are computed as an arithmetic average of individual lin ratios F i, j in the way to obtain the sample Median. More precisely, we put w i, j = 1 or w i, j = 0 in the way that the estimators of the age-to-age factors f are given by f = median{f i, : i {1,..., }}. The median denotes the sample median that for the sample X 1,...,X n is computed by X n+1 if n is odd median{x i : i {1,...,n}} := X n +X n +1 otherwise where X denotes the th order statistics of the sample X 1,...,X n. Remar 3.. n the case where there is only one observation in estimation of parameter σ odd number of data in sample median computation we choose the additional F i, factor in order to have observations and be able to apply the formula 3.4. n the Table below, we present the estimation of total amount of claims reserves R as well as the value of estimators of aggregated MSEP R. Recall that R := R i, where R i := Ĉ i, C i, i+1. We observe that to obtain R it is enough to have the estimators C i, of ultimate claims Ĉ i, for all accident year i. n consequence MSEP R = MSEP Ĉi, and we use the formula 3.7 to estimate this quantity. We compute as well the coefficient of variation of R, given by CV R = R/MSEP R 1/. The last lines of Table indicate the relative proportion of R and MSEP R 1/, for each of 5 methods considered, in comparison to the ALL AV method which is the reference method in our example. alpha=0 item/method ALL AV AEHL 5 Years AV 3 Years AV Median R MSEP R 1/ CV R 99% 3% 36% 43% 7% 1/1 /1 3/1 4/1 5/1 R % 100% 70% 81% 73% 58% MSEP R 1/ % 100% 3% 30% 3% 16% Table : Estimation of total amount of outstanding loss liabilities R, value of estimator of aggregated MSEP R 1/ and coefficient of variation CV R, for 5 methods 3.6. Limits of MCL method As can be seen in Table the 4 last methods columns -5 reduce significantly the estimation of MSEP R 1/ comparing to the first method ALL AV. For the methods 3, 4 and 5, this is mainly due to the elimination of relatively significant number of development factors from estimation especially for the first development years which correspond to the columns of the run-off triangle. This phenomena is especially seen in the case of sample Median method in which, for each development factor we eep at most of lin ratios F i, j in estimation of f. From statistical 7

8 point of view, this is clearly not enough to perform the robust estimation. As a consequence, this ind of methods reduce unnaturally the variability of reserves. This could be dangerous for example in terms of evaluation of the economical capital for reserve ris required by the new Solvency regime. Beyond of the limits stated above, there are some incoherences with application of weights w i, for the AEHL and sample Median methods. ndeed, the weights w i, should be C i, measurable random variables in order to be able to derive the main results of MCL approach see for example Proposition A.. Although for the method 5 Year AV and 3 Years AV we can fix the weights without nowing the information D nowing all observation in the run-off triangle, see.5, this is not a case for the AEHL and sample Median methods. The reason is that we need to now the observation F i, in order to specify the corresponding weights for those methods. That is why the weights w i, are not C i, measurable but rather D -measurable. This means that the formula for the expectation of f and Var f B are not correct. Regarding to sample Median method, the derivation of Var f B require the computation of the moments of order statistics see Jeng 010 and those are strongly related to the distribution of F i, j. To overcome these difficulties we propose solutions: the simple Proxy method see Section 6.1 and the more complex one based on a robust estimation see details in Section 6.. First approach is programmed to avoid artificial volatility increase and it is based on all lin ratios in estimation of volatility parameters σ scale parameters in linear regression. Second method consist on developing an approach that allows using any robust estimators of f location and σ scale parameters. 4. General Mac Chain-Ladder Model 4.1. Model Assumptions Before stating the main assumptions of our general approach, let assume, that functions f j : [0, [0, are Borel measurable. Let δ i, j be the non-negative random variables defined by, δ i, j := f j C i, j. Our model is formalized by the following assumptions: GMCL.1 There exist constants f > 0 such that EF i, C i,1,..,c i, = f. GMCL. There exist constants σ > 0 such that for all 1 i and 1 1 we have VarF i, C i,1,..,c i, = { σ δ i, if δ i, 0 a.s., if δ i, = 0 a.s., 4.1 where a.s. means almost surely. GMCL.3 The accident years C i,1,..,c i,j 1 i are independent We observe that from the above assumptions the main difference between MCL and GMCL lies in the variance assumption. This modification allows us to introduce different weights in estimation of the parameters f and σ. 4.. Model Estimators Suppose that functions g j : [0, [0, are Borel measurable. Let γ i, j be the non-negative random variables defined by, γ i, j := g j C i, j. 8

9 Given the information D, the factors f are estimated by f = γ i,f i, γ, for i, t becomes obvious from assumption GMCL. that in order to compute correctly the variance of f see Proposition A. in Appendix we have to assume that {if δ i, j = 0 then γ i, j = 0}. 4.3 Given the information D, the variance parameters σ are estimated by σ = 1 1 δ i, F i, f, for 1, 4.4 where represents the number of weights δ i, different from 0, namely, := card{i : δ i, 0}. n the analogue way to 3.4 we define Proposition 4.1. i The estimators f given in 4. are unbiased and uncorrelated. σ = min σ 4 / σ 3,min σ 3, σ. 4.5 ii For = 1,.., 1, if δ i, = γ i, for all i, then the estimators f of f have the minimal variance among all unbiased estimators of f which are the weighted average of the observed development factors F i,. For = 1,.., 1, if δ i, γ i,, for some i, then the relative efficiency of s.e. f γ δ B with respect to s.e. f γ=δ B, i.e., the ratio s.e. f γ=δ B s.e. f γ δ B := Var f γ δ B 1/ Var f γ=δ B = 1/ γ j, j=1 δ 1 j, {δ j, 0} j=1 γ j, 1 {δ j, 0} iii For = 1,.., 1, if δ i, j = γ i, j for all i, then the estimator σ, given in 4.4 is the unbiased estimator of the parameter σ. For = 1,.., 1, if δ i, γ i,,, for some i, then the bias of the estimator σ is given by the following formula E[ σ σ ] = σ 1 E δ i, γ j, j=1 δ 1 j, {δ j, 0} j=1 γ j, 1. iv Under the model assumptions GMCL.1 and GMCL.3 we have EC i, D = C i,+1 i f i,+1 i... f. This implies, together with the fact that f are uncorrelated, that Ĉ i, is unbiased estimator of EC i, D. v The expected values of the estimator Ĉ i, = C i,+1 i =+1 i for the ultimate claims amount and of the true ultimate claims amount C i, are equal, i.e., EĈ i, = EC i,, i. 9 f,

10 The proof of this Proposition is postponed to the Appendix A.4. Remar 4.1. f we set γ i, j = δ i, j = w i, j Ci, α j, for α {0,1,}, in 4. and 4.4 we get the assumptions of MCL model from Mac 1999 see also Mac 1993, 1994 and Saito 009 f we put γ i, j = δ i, j = w i, j C α j i, j, for α j R, in 4. and 4.4 we get the Stochastic Chain-Ladder Model from Murphy et al Main Results 5.1. Single Accident Years Result 5.1 Conditional MSEP estimator for a single accident year. where i, = msepĉi, D C i, = Ĉi, Γ i, + i,, 5.1 J 1 = i+1 Γ i, = J 1 = i+1 σ / f j=1 γ j, and f j and σ j are given in 4. and respectively. 5.. Aggregation over Prior Accident Year σ / f δ i,, 5. j=1 Result 5. Conditional MSEP estimator for aggregated years. msep Ĉ i D C i, = i= msepĉi D C i + i= Ĉ i, j=i+1 γ j, δ j, 1 {δ j, 0}, 5.3 Ĉ j, where f j and σ j are defined in 4. and , respectively. J 1 = i+1 σ / f l=1 γ l, l=1 γl, 1 {δ j, 0}, δ l, Applications of GMCL Model n our numerical example in Section 3.5 we have seen that the assumption about the same weights in estimation of parameters σ and f yields for some methods to an artificial reduction of variability of reserves amounts refer to Table. To overcome this difficulty we introduced the different weights γ i, j and δ i, j in computation of f and σ respectively. n the following application we indicate how one can possibly estimate the weights γ i, j and δ i, j and we point out some other interesting applications. 10

11 6.1. Method Proxy for Factors Selection n this section we tae in our general framewor f j C i, j := w γ i, j Cα i, j and g j := w δ i, j Cβ i, j which means that γ i, j := w γ i, j Cα i, j and δ i, j := w δ i, j Cβ i, j. n this so called Proxy method we impose using all lin ratios F i, j in estimation of parameters σ w δ i, j = 1, for all i, j. For all 5 methods presented below we tae α = β = 0. We turn bac to our numerical example from Section 3.5 and we evaluate the same estimators for already presented 5 methods with the only difference in weights of σ estimation. More precisely, 1 ALL AV: The σ are estimated with wδ i, j = 1 for all i, j. Parameters f are computed as a arithmetic average of all individual lin ratios F i, j. More precisely, we define the weights in the following way: w γ i, j = 1 for all i, j. AEHL: The σ are estimated with wδ i, j = 1 for all i, j. Parameters f are computed as a arithmetic average of all individual lin ratios excluding the highest and the lowest values of F i, j. More precisely, we define the weights in the following way: for fixed j, w γ i, j = 0 for i such that F i, j = F j, j and F i, j = F 1, j, where F, j for = 1,..., j denotes the order statistics of F i, j. For remaining indices i, for fixed j, we tae w γ i, j = Years AV: The σ are estimated with wδ i, j = 1 for all i, j. Parameters f are computed as an arithmetic average of individual lin ratios F i, j from 5 latest accidents years. More precisely, we define the weights in the following way: w γ i, j = 1 for i = j,..., j 4. For remaining indices i, for fixed j, we tae wγ i, j = Years AV: The σ are estimated with wδ i, j = 1 for all i, j. Parameters f are computed as an arithmetic average of individual lin ratios F i, j from 3 latest accidents years. More precisely, we define the weights in the following way: w γ i, j = 1 for i = j,..., j. For remaining indices i, for fixed j, we tae wγ i, j = 0 5 Median: The σ are estimated with wδ i, j = 1 for all i, j. Parameters f are computed as an arithmetic average of individual lin ratios F i, j in the way to obtain the sample Median. More precisely, we put w γ i, j = 1 or wγ i, j = 0 in the way that the estimators of the age-to-age factors f are given by f = median{f i, : i {1,..., }}, were median denotes the sample median which, for the sample X 1,...,X n, is computed by X n+1 if n is odd median{x i : i {1,...,n}} := X n +X n +1 otherwise and X denotes the th order statistics of the sample X 1,...,X n. n Table 3 we present the estimation of R and MSEP R using 5 methods described above. n terms of MSEP we see that, in generally, we have the values greater then our reference method ALL AV from column 1 which stays unchanged comparing to Table. This is not surprising because by selecting of the development factors we deacresed the estimated values of f and by using all observations F i, j in σ computation we mechanically increased the dispersion around the values of f. n view of our results from Table and Table 3, the proxy method overestimate in general the real MSEP, and can be then treated as its upper bound. However, it can be useful as a tool to perform the sensitivity analysis for testing the impact on the reserve volatility of excluding the specific set of lin ratios. Finally, it can be seen as a measure of relative prudence of other approach of measuring the variability of reserves by means of MSEP estimators. 6.. Robust Estimation in GMCL model From previous two numerical examples MCL vs. GMCL results, we observe that, in generally, the first approach underestimate and second overestimate the MSEP of claims reserves see Table and Table 3. n this section we present an intermediate solution for our general problem that allows to evaluate the estimation of MSEP of reserves 11

12 alpha=0 item/method ALL AV AEHL 5 Years AV 3 Years AV Median R MSEP R 1/ CV R 99% 134% 134% 166% 196% 1/1 /1 3/1 4/1 5/1 R % 100% 70% 81% 73% 58% MSEP R 1/ % 100% 95% 110% 13% 114% Table 3: Estimators of R, MSEP R and CV R. in case of development factor selection. This go-between solution is based on the robust statistics in estimation of model parameters f and σ. The term of robust statistics is meant in the sense of Huber and Ronchetti 009. As already mentioned, the assumption GMCL. about the conditional variance of F i, j allows to estimate the factors f in the framewor of linear regression obtained by the means of weighted least squares procedure see Murphy et al. 01 and the references therein. Although these estimators are easy to compute and have excellent theoretical properties see Proposition 4.1, they rely on quite strict assumptions, and their violation may lead to useless results. One of possible solutions to overcome this difficulty is to use the robust estimation techniques. The idea of robust statistics is to account for certain deviations from idealized model assumptions. Typically, robust methods reduce the influence of outlying observation on the estimator. We tae the following assumption in our general framewor of GMCL model: f j t := t α j and g j t := t β j, with α j,β j R to be estimated. This means that γ i, j := C α j i, j and δ i, j := C β j i, j. The following algorithm shows how one can estimate the parameters α j for j = 1,..., 1 and β for = 1,...,. As can be seen, the presented method is based on similar principle as the well nown moment estimation method from point estimation theory Algorithm for fitting α and β parameters Step 1: We select the robust estimators for f and its variance. We denote these estimators by f and Ṽ ar f respectively. These two quantities can be derived by numerous techniques vastly described in the literature such as: M-estimation, L p estimation, etc., see Huber and Ronchetti 009 or Trimmed Mean see Jeng 010 Step : For every = 1,..., 1, we find α by solving the following equation f = f, where f are given in equation 4., namely Cα i, F i, Cα i, = f. 6.1 The procedure how to select the consistent α together with the problem of existence of solution of equation 6.1 is treated in Murphy et al. 01 see Lemma 1 and the comments that follow it. Step 3: For every = 1,...,, we find β by solving following equation: V ar f = Ṽ ar f, where V ar f is given in A.8, namely 1 1 i= C β i, F i, f 1 j=1 γ j, C β i, j=1 γ j, = Ṽ ar f 6.

13 The parameters f are given in equation 4. with α estimated in Step. For = 1, since only one observation is available in our data, β need to be estimated by other approaches. The limits for the values of Ṽ ar f for which the solution of equation 6. exists are presented in Appendix E Numerical example n the present example we consider only the Median method already presented in previous numerical applications. We concentrate ourselves on that particular method because our goal is not to present the extensive case study but rather to illustrate the general principle and the main steps of this application. Observe that the method AEHL can be treated by the theory of robust estimation by means of trimmed mean estimators see Jeng 010. To apply the above fitting algorithm for sample Median method, we use the LAD least absolute deviation estimation procedure. The theoretical framewor of LAD is presented in Appendix D. The values of f are given by computing the sample median from F i, as described in Section 3.5. The standard errors of f are obtained via bootstrap techniques. n Table 4 we present the numerical values of f, s.e f := Ṽ ar f 1/ and CV f := s.e f / f. Note that the last value of s.e f can not be estimated from the data for the reasons discussed in the case of σ estimation see Section 3. only one observation available. This why we do not fit the parameter β via equation 6., but we put β := α f 4,597 1,599 1,1635 1,1657 1,1318 1,0335 1,0333 1,0180 1,009 s.e f 1,7974 0,1686 0,1411 0,070 0,047 0,051 0,0065 0,019 - CV f 4,% 10,5% 1,1%,3% 4,%,4% 0,6% 1,3% - Table 4: Estimation of f and s.e f using LAD technique corresponding to sample Median method The standard deviations of f from Table 4 were obtained by using the rq function integrated in free R software. Note that the value of f given by this function are slightly different from those presented in Section 3.. This is probably due to the optimisation algorithm that is used in R. Given that these differences are insignificant we decided to present in Table 4 the same numerical values of the estimators f as given in Section 3.. The corresponding R code is available on the request from the author. As mentioned in the algorithm and in Appendix E, the solution of 6.1 and 6. are not always available. For instance, in our example, there is no solution of equation 6.1 for = 3,4 and no solution of equation 6. for = 8. This means that for = 3, = 4, and = 8 the parameters α and β need to be specified in different way. This could be done using any other approach that is being judged appropriate by the actuary performing estimation. n our case we put α 3 = β 3 = α 6 = β 6 = α 8 = β 8 = 0 to have from one hand the optimal properties see Proposition 4.1 but also to be consistent with our choice of α = 0 in our two previous numerical applications see Section 3.5 and Section 6.1. The estimation of parameters α j and β j are stated in Table 5. The values of α j and β j for which we arbitrary put 0 are indicated with bold font characters. j α j 0,504 1, ,585-0, ,00 0 0,0000 β j 0,6605 0, ,707 1, , Table 5: Estimation of parameters α j and β j The MSEP and claims reserves amount estimators are stated in Tabbe 6. Observe that the robust estimation is a good compromise between the method with the same weights see Section 3.5 and the method where we use all lin ratios in σ estimation see Proxy method in Section

14 Median item/method MCL Robust Proxy R MSEP R 1/ CV R 7% 64% 196% Table 6: Median method with Robust estimation 6.3. Validation of results from reserving softwares The next very interesting and extremely important application of our GMCL model is the possibility of validation the results from industry reserving softwares. The stochastic Chain-Ladder type methods are used to evaluate the economic ris capital required by Solvency for so called reserve ris. n fact, this capital requirement for reserve ris is computed as the 99.5 th percentile value at ris of run-off result distribution profit/loss on reserves over one year. This means that Solvency defines the reserve ris in one-year time horizon which is different from the standard approach considering the distribution of the ultimate cost of claims. However, one of the methods to derive the one-year reserve ris is based on simple scaling of ultimate view. This technique is called emergence pattern approach and is calibrated using the results of Merz and Wüthrich 008, which is currently a very popular methodology throughout the maret and taen from the latest technical literature on this topic. The empirical loss distribution in ultimate view is often derived by using the bootstrap techniques and Monte Carlo simulations. The first technique is used to evaluate the estimation error and the second to approximate the process variance. This ind of bootstrap approaches is also available in ResQ software, which is used worldwidely within the P&C insurance maret. The usual question that can be ased is following: how one can validate the results from bootstrap method provided by reserving tools such as ResQ. One of the possible solution is to compare the estimation of the first moments of loss distribution from bootstrapping based on simulations, with the estimators of reserves and MSEP of reserves obtained by the explicit formulas. For the sae of simplicity, we assume that there is no factors selection. We use the RAA run-off triangle and we present the numerical results in Table 7. For all bootstrapping results we used simulations. We begin our analysis with the classical Chain-Ladder method in which the estimators of f are the all volume weighted average and are consistent with Mac More precisely, with the hypothesis of MCL method with α = 1, we compare the estimate of MSEP obtained by these two techniques: bootstrap from ResQ and explicit formula given in MCL approach. The corresponding numerical values are respectively: see ResQBoot 3 in Table 7 and see ResQMCL 4 in Table 7. We observe a good convergence for bootstrap the relative error is less then 1%. We consider now the different estimator of f computed as a simple arithmetic average of individual lin ratios F i, j. This is equivalent to tae α = 0 in MCL framewor. n that case, we observe that the estimates of MSEP for both methods become divergent: see ResQMac in Table 7 and see ResQBoot 1 in Table 7. This is due to the fact that the ResQMac method is obtained by approximation based on MCL formula with α = 1. n fact, according to the technical documentation, the ResQ estimates of parameters f and σ in the bootstrap approach are of the form up to multiplicative constant for bias reduction: f = 1 F i, and σ = 1 1 C i,f i, f. t is easily seen that these estimators are consistent with our general approach with α = 0 and β = 1 see 4. and 4.4 in Section 4. The MSEP estimator is equal to see GMCL 5 in Table 7. This shows that GMCL method allows to validate the results and detect the incoherences. Effectively, the choice of estimators in ResQ for the case α = 0 is not optimal in sense of Proposition 4.1. t remains unnown wether this deliberate or rather this is just a proxy approach that was judged correct. n regards to the approximation ResQMac, this shows that in construction of the proxy methods we can not just tae the MSEP formula for α = 1 as a starting point. ndeed, the MSEP formula changes if we modify the estimates of f because the variance of f is not the same, so it is not enough to plug in the new estimators of C i,, f and σ in the MSEP formula 5.4 with α = β = 1. This lac of understanding of this principle could be a reason of taing the no optimal hypothesis in bootstrap ResQBoot1 method. Finally, we observe that the results of ResQBoot 1 method validate our explicit formula for estimation of MSEP 14

15 of claims reserves in the framewor of our GMCL Model. alpha=0 alpha=1 alpha=0, beta=1 item/method ResQBoot ResQMac ResQBoot MCL GMCL R MSEP R 1/ Table 7: Comparison of ResQ estimators of R and MSEP R with MCL and GMCL Models 7. Conclusion n this paper we presented a general flexible tool for stochastic loss reserving and its variability. We developed our GMCL model to quantify the variability of reserves in the context of selecting development factors in the framewor of stochastic Chain Ladder method. We provided the theoretical and flexible bacground which covers some practises of actuaries and industrial providers of reserving softwares. Finally, we showed the way of bridging the Chain Ladder model and the robust estimation techniques. Our results can be applied in other approaches based on Chain-Ladder framewor lie: Multivariate Chain-Ladder, Univariate and Multivariate Bayesian Chain-Ladder, etc. One can derive the similar results in the context of one-year reserve ris for Solvency purposes see Time-series Chain-Ladder. This topic will be treated in our forthcoming paper. Some partial results can be found in Sloma 014 and Sloma Acnowledgments The author want to than the reviewers for their helpful comments and suggestions. 15

16 A. Mathematical Proofs We present here the proofs of our main results. Most of them are derived by simple rewriting the techniques applied in Mac 1993, A.1. Proof of Result 5.1 Due to the general rule EX c = VarX + EX c for any scalar c we have msepĉi D C i = E [ Ĉi C i To estimate VarC i, D we use the following D ] = VarC i D + EC i D Ĉ i. A.1 Lemma A.1. For i =,...,, we have, VarC i, D = l=+1 i E [ ] C i,l D δ i,l σ l =l+1 f. A. The proof of Lemma A.1 is postponed to Appendix A.5. [ ] C Note that the estimation of E i,l δ D i,l from equation A. is a crucial part of this proof. We choose to estimate this [ ] term by Ĉ i,l. This is due to the obvious observation that C i,l C is an unbiased estimate of E i,l δ δ and from the basic i,l i,l property of conditional expectation, namely: E [ E δ i,l [ C i,l δ i,l D ]] = E [ C i,l δ i,l ]. t is worth noting here that, in the case where δ i,l := Ci,l α with α R, in Saito 009, the author used the same technique of estimation without giving any justification or reason for that see proof of Lemma 4 and Estimate 8. Similarly, in the case where δ i,l := w i,l Ci,l α with α {0,1,} and w i,l [0,1], we find the same estimator in Mac More precisely, the author claims without proving that l=+1 i VarC i,l+1 D =l+1 f can be estimated via the quantity Ĉ i, l=+1 i s.e.f i,l / f l, where s.e.f i,l is an estimate of VarF i,l C i,1,..,c i,l. ndeed, this is [ C i,l Ĉ i,l ] achieved if we estimate E w i,l Ci,l α D by. w i,l Ĉi,l α However in Murphy et al. 01, the author used different approach based on normal approximation. Note that in the Section 6.3 we obtained that the above estimator of VarC i,l+1 D is consistent with that provided by bootstrap technique from ResQ software. This is shown for the particular run-off triangle and the assumption that C i, j are gamma-distributed random variables. t would be interesting to perform the extensive simulation study in order to examine the exactitude of this estimate with other data and probability distributions. We apply now Lemma A.1 with Ê [ C i,l δ i,l D ] = Ĉ i,l δ i,l and by replacing the unnown parameters f et σ with their estimators f and σ. Together with the equality Ĉ i,l = C +1 i l 1 =+1 i f see Proposition 4.1 v we conclude VarC i, D = = l=+1 i l=+1 i E = C i,+1 i σ l [ ] C i,l D δ i,l δ i,l C i,+1 i =+1 i f σ l =l+1 l 1 =+1 i l=+1 i f f = =l+1 σ l / f l δ i,l l=+1 i Ĉi,l σ l δ i,l f = C i,+1 i = Ĉ i, l=+1 i =l+1 f l=+1 i σ l / f l. δ i,l σ l / f l δ i,l f =+1 i A.3 16

17 We now turn to the second summand of the expression A.1. Because of Proposition 4.1 iv and v we have, EC i, D Ĉ i, = C i,+1 i f +1 i... f f +1 i... f. As can be easily seen, this expression cannot be estimated by replacing f with f. n order to estimate the right hand side of A.4 we use the same approach as in Mac 1993, Saito 009 followed the same technique of estimation. However in Murphy et al. 01 we can find different approach which was also presented in Buchwalder et al. 006a. t is worth noting that in the paper of Mac et al. 006 the authors criticised the approaches of Buchwalder et al. 006a and showed that the estimate of estimation error form Mac 1993 is hard to be improved see also Buchwalder et al. 006b. As the answer for the criticism of Mac on article of Buchwalder et al. 006a, the authors provided the bounds for estimation error and claimed that the Mac estimator, in some particular cases, is closed to these bounds see Wüthrich et al This should be confirmed by performing the extensive simulation study to quantify the different approaches of error estimation in stochastic Chain-Ladder framewor. We define, A.4 F = f +1 i... f f +1 i... f = S +1 i S, A.5 with S = f +1 i... f 1 f f f f +1 i... f 1 f f f = f +1 i... f 1 f f f f. A.6 This yields We estimate F using the following F = S +1 i S = =+1 i S + =+1 i j< S j S. A.7 Proposition A.1 Estimate of F. Let define, for 1, the set of observed C i, j up to development year, namely Then, we can estimate F by B = {C i, j : i + j + 1, j} D. F = l=+1 i f l =+1 i Var f B. f The proof of Proposition A.1 is postponed to Appendix A.5. t remains to determine the estimate of Var f B. We use the following Proposition A.. We assume 4.3. We have Var f B = σ γ j, j=1 δ 1 j, {δ j, 0}. A.8 j=1 γ j, 17

18 The proof of Proposition A. is postponed to Appendix A.5. Finally, using A.4 and Proposition A. we estimate EC i, D Ĉ i, by C i,+1 i f +1 i... f This completes the proof of Result 5.1. =+1 i σ f γ j, j=1 δ 1 j, {δ j, 0} = Ĉi, j=1 γ j, =+1 i σ f γ j, j=1 δ 1 j, {δ j, 0}. j=1 γ j, A.. Proof of Result 5. Overall standard error Following the definition in.4, we have msep Ĉ i, D C i, = E The independence of accident years yields Ĉ i = Var C i, D + Var C i, D = C i D E C i, D Var C i, D. where each term of the sum has already been calculated in the proof of the Result 5.1. Furthermore Taing together E C i, D Ĉ i, = E C i, D Ĉ i, = i, j Ĉ i, E C i, D Ĉ i, E C j, D Ĉ j, with where msep Ĉ i D C i, = i= msepĉi D C i + i< j F i = f +1 i... f f +1 i... f = C i,+1 i C j,+1 j F i F j, =+1 i S i, S i = f +1 i... f 1 f f f f. We can determine the estimator of F i F j in the analogous way as for F. Proposition A.3. We have F i F j = =+1 i Var f B f f +1 i... f f +1 j... f. 18

19 We finally conclude, from Proposition A.3 i< j A.3. Proof of Proposition 3.1 C i,+1 i C j,+1 j F i F j = i See Theorem p. 15 in Mac Ĉ i, Ĉ j, i= j=i+1 = i+1 σ / f l=1 γ l, ii See discussion on p.11, Corollary on p.141 and Appendix B on p.140 in Mac iii See Appendix E on p.151 in Mac l=1 γ l, 1 {δ j, 0} δ l,. iv See Theorem 1 p. 15 and discussion after the proof of Theorem on page 16 in Mac v see Appendix C p.14 in Mac 1994: A.4. Proof of Proposition 4.1 i, iv and v, see proofs of i,iv and v respectively in Proposition 3.1 ii. The first part of the statement regarding to the minimal variance of parameters f can be easily derived from the proof of ii in Proposition 3.1. The rest of the proof is easily seen from the Proposition A.. iii. Without loss of generality and to avoid the complexity of notation we present the proof for = for each, all weights δ i, are different from 0. We have, for 1, 1 σ = Since δ i, are σc i, measurable, we have δ i, F i, f = δ i, F i, δ i, F i, f + δ i, f. E 1 σ B = δ i, EF i, B δ i, EF i, f B + δ i, E f B. n the following derivation we use σc i, -measurability of γ i, and definition of f from 4.. Furthermore, the assumption GMCL.3 implies that F i, and F j, are independent for i j. From assumption GMCL.1 and GMCL. we easily see that EFi, B = σ δ + f i,. Taing together, EF i, f B = = = 1 l=1 γ l, 1 l=1 γ l, 1 l=1 γ l, j=1 γ j, EF i, F j, B = γ i, σ + f δ + i, γ j, f j i γ i, σ δ i, + f j=1 γ j, = σ 1 l=1 γ l, γ i, δ i, l=1 γ + f. l, γ i, EF i, B + j i γ j, EF i, B EF j, B A.9 19