City, University of London Institutional Repository Citation: Haibu, M., Margraf, C., Miranda, M. D. M. and Nielsen, J. P. (2016). Cash flow generalisations of non-life insurance expert systems estimating outstanding liabilities. Expert Systems with Applications, 45, pp. 400-409. doi: 10.1016/j.eswa.2015.09.021 This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link: http://openaccess.city.ac.uk/12533/ Link to published version: http://dx.doi.org/10.1016/j.eswa.2015.09.021 Copyright and reuse: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. City Research Online: http://openaccess.city.ac.uk/ publications@city.ac.uk City Research Online http://openaccess.city.ac.uk/ mailto:publications@city.ac.uk Cash flow generalisations of non-life insurance expert systems estimating outstanding liabilities Munir Hiabu Munir.Hiabu.1@cass.city.ac.uk Carolin Margraf Carolin.Margraf.1@cass.city.ac.uk Maŕıa Dolores Mart́ınez Miranda mmiranda@ugr.es Jens Perch Nielsen Jens.Nielsen.1@city.ac.uk Cass Business School, City University London, U.K. September 17, 2015 Abstract For as long as anyone remembers non-life insurance companies have used the so called chain ladder method to reserve for outstanding liabilities. When historical payments of claims are used as observations then chain ladder can be understood as estimating a multiplicative model. In most non-life insur- ance companies a mixture of paid data and expert knowledge, incurred data, is used as observations instead of just payments. This paper considers re- cent statistical cash flow models for asset-liability hedging, capital allocation and other management decision tools, and develops two new such methods incorporating available incurred data expert knowledge into the outstanding liability cash flow model. These two new methods unbundle the incurred data to aggregates of estimates of the future cash flow. By a re-distribution to the right algorithm, the estimated future cash flow is incorporated in the overall estimation process and considered as data. A statistical validation technique is developed for these two new methods and they are compared to the other recent cash flow methods. The two methods show to have a very good per- formance on the real-life data set considered. Keywords: Stochastic Reserving; General Insurance; Chain Ladder; Claims In- flation; Incurred Data; Model Validation. 1 1 Introduction The non-life insurance business is an important part of the economy for most devel- oped countries with market revenues amounting to around five percent of GNP’s. The best estimate of outstanding liabilities - often called the reserve - is perhaps the single most important number on the balance sheet of most non-life insurance com- panies. Insufficient reserves is one common reason for non-life insurance companies to go broke. A mismanaged reserving process can also lead to spurious and volatile yearly results leading to uninformed management decisions. Finally, a smoother and more transparent reserving process leads to significant cost savings in almost any non-life insurance company. In this light it is perhaps surprising that statistical models for the most often used data set, the incurred run off triangle, are rarely considered in the literature. Incurred data is a mixture of historical payments of already settled claims and predicted severities of reported but not settled claims. The predicted severities are based on all available expert opinion in the company and are called case estimates. In that spirit, incurred data has added information about reported claims which are not available when only historical payments are considered. Actuaries often prefer working with chain ladder on the incurred triangle rather than on historical payments. But it is a bit unclear what this really means in terms of mathematical statistics. While incurred chain ladder probably makes good sense in a deterministic forecasting framework, the stochastic nature of the expert opinion in incurred data is not taken into account by this practice. There is a little literature acknowledging the added value of incurred data: the probably most famous Munich chain ladder approach by Quarg and Mack (2004), regression approaches by Halliwell (1997), Halliwell (2009), Venter (2008), and a paid-incurred chain reserving method by Posthuma et al. (2008), Merz and Wüthrich (2010), Happ et al. (2012) and Happ and Wüthrich (2013). These eight papers combine payment data and incurred data into one statistical model resulting in one reserve estimate. However, none of those papers take the common micro-structure of payment data and incurred data into account. Payment data and incurred data are both constructed via manipulations on the same underlying data; incurred data with the extra element of expert knowledge on expected future cash flows. To model this relationship one needs micro-structural assumptions about the underlying claim process. Pigeon et al. (2014) does exactly this. It is based on a recent trend to use so called granular data or micro data for reserving, see Antonio and Plat (2014) for one of the most interesting recent contributions in that area. See also Mart́ınez-Miranda et al. (2013a) for a continuous interpretation of the classical chain ladder methodology. While these approaches indeed seem to be favorable, 2 they are not well established yet and rely on data and granular information that is most often not available at hand. Double chain ladder, introduced in Mart́ınez-Miranda et al. (2012), builds on micro-structural assumptions but does not need granular data in the estimation procedure. It is based on the methods of Verrall et al. (2010) and Mart́ınez-Miranda et al. (2011) where the objective was to only rely on data that is already available in most reserving departments. It uses additional information of claim counts which is another triangle most often available in the data portfolio of a reserving department. The result is a full statistical model based on historical payments and counts capable of incorporating the information of the incurred data in a natural way. Agbeko et al. (2015) recently introduced a model reproducing the deterministic results of the earlier mentioned chain ladder method on incurred data by incorpo- rating the incurred data expert opinion into the well-defined full stochastic model of double chain ladder. One direct advantage of this approach is that the chain ladder model based on paid data can be validated against the chain ladder model based on incurred data. While the paid chain ladder and incurred chain ladder methods have been available for a long time as part of almost any non-life actuary’s tool kit, it has never before been possible to compare them against each other when only the typical aggregated data were available. Two other methods of incorporating expert knowledge of incurred data into these full cash flow models have been introduced in Mart́ınez-Miranda et al. (2013b) and Hiabu et al. (2016). The first of these two methods is extracting the inflation of the cost of a single claim from the incurred data and then incorporates that information in the double chain ladder model of Mart́ınez-Miranda et al. (2012). The second of these two methods suggests to incorporate a RBNS-preserving property. RBNS stands for Reported But Not yet Settled, and it can be estimated by the sum of all case estimates. This estimate is the best the claims department of an insurance company (with all the expert knowledge on the nature and severity of each claim available in such a department) is able to do. Hiabu et al. (2016) therefore produced a version of double chain ladder reproducing exactly the expert judgement of the RBNS reserves. All those mentioned double chain ladder extensions take advantage of the un- derlying structure of the incurred data, extract the relevant information from it and plug it into the original double chain ladder method. The advantage of this approach is that the simplicity and intuition of the simple chain ladder method is preserved and that the full statistical interpretation and stochastic cash flow formulation is inherited from double chain ladder. The two new stochastic cash flow methods developed in this paper build on the ideas and techniques of Hiabu et al. (2016). The first one treats the expert knowledge 3 of the incurred data as real data and incorporates it into the model, the second builds a second RBNS preserving cash flow model on top of this method. The approach is first to unbundle the incurred data to get back to the original aggregated RBNS numbers. These aggregated numbers are re-distributed according to the estimated delay such that the resulting algorithm takes both historical data and expert data into account in the final estimation. We therefore let the estimated future cash flow be incorporated in the overall estimation process by considering it as data. Note that both of these two new methods are cash flow models of the same nature as the models considered in Agbeko et al. (2015), and they can therefore be validated and compared to the models considered there. In the applied data example, this validation indicates, that the two new methods seem to take better advantage of the incurred expert data than previous methods did. Recent years have seen a growing interest in expert systems related to non- life insurance, see for example Belles-Sampera et al. (2014) and Abbasi and Guillén (2013), who consider ways of understanding risk in non-life insurance. Guelman and Guillén (2014) work with pricing of insurance claims and the customers sensitivity to that price; Guillen et al. (2012) and Kaishev et al. (2013) transfer knowledge from one business line to another to optimize cross-selling. Human judgement is important in all these insurance applications. When prices are set, there is a business intelligence department evaluating how much weight to put on the model at hand and how much weight to put on market prices as such. When risk is evaluated, human judgement calibrates the entering parameters. And when RBNS claims reserves are set, then there is an element of human judgement in the settlement of every single claim. It is also a human judgement when it is decided to use model based claims reserves for some subset of the claims, for example the smaller ones. We conclude this introduction by noting that Mart́ınez-Miranda et al. (2012) has two versions of double chain ladder; one version where the delay is not adjusted and another where it is adjusted. In this paper only the unadjusted version of double chain ladder is considered. One reason for the adjustment of the delay in Mart́ınez-Miranda et al. (2012) was to improve the performance of estimating the out-of-sample tail reserve. While this is a very important issue, it is beyond the scope of this paper to consider the out-of-sample tail reserve. The rest of the paper is structured as follows. Section 2 describes the data and the expert knowledge, introduces the notation and defines the model assump- tions. Section 3 discusses the outstanding loss liabilities point estimates. Section 4 describes four methods to estimate the parameters in the model: DCL, BDCL, PDCL, IDCL, EDCL and PEDCL. An application is considered in Section 5 and the validation of the six methods against each other is gone through in Section 6. Finally, Section 7 provides some concluding remarks. 4 2 Data and first moment assumptions This chapter introduces the data used in maybe every insurance reserving depart- ment to calculate their outstanding liabilities. Also the methods described in this paper rely on these data sets. They are often shortly called run-off triangles. These run-off triangles are the aggregated incurred counts (data), aggregated payments (data) and aggregated incurred payments (mixture of data and expert knowledge). All of those three objects have the same structural form, i.e., they live on the upper triangle I = {(i,j) : i = 1, . . . ,m,j = 0, . . . ,m− 1; i + j ≤ m}, m > 0, where m is the number of underwriting years observed. The parameter m also has another crucial role. If no tail-factors are considered, which will be assumed throughout this paper, then m−1 is the maximum delay, that is the time from the underwriting of the policy a claim is based on until its payment. This assumption is called being run-off, hence the name run-off triangles. Let’s first introduce the two data triangles. Aggregated incremental incurred counts: NI = {Nik : (i,k) ∈ I}, with Nik being the total number of claims of insurance incurred in year i which have been reported in year i + k, i.e. with k periods delay from year i. Aggregated incremental payments: XI = {Xij : (i,j) ∈I}, with Xij being the total payments from claims incurred in year i and paid with j periods delay from year i. A often confusing point is that the meaning of the second coordinate of the triangle I varies between the two different data. While in the counts triangle it represents the reporting delay, in the payments triangle it represents the development delay, that is reporting delay plus settlement delay. The definition of the incurred payments triangles is not that straight forward. To allow for a exact description, we first introduce micro structural variables of the claims process. We hereby follow the line of Mart́ınez-Miranda et al. (2012), since those variables will also play a key role in the underlying assumption of the double chain ladder model. By N paid ikl , we denote the number of the future payments originating from the Nik reported claims, which were finally paid with a delay of k+l, where l = 0, . . . ,m−1. Also, let X (h) ikl denote the individual settled payments which arise from N paid ikl , h = 1, . . . ,N paid ikl . Finally, we define Xikl = N paid ikl∑ h=0 X (h) ikl , (i,k) ∈I, l = 0, . . . ,m− 1, 5 i.e., those payments originating from underwriting year i, which are reported after a delay of k and paid with an overall delay of k + l. The aggregated incurred payments are then considered as unbiased estimators of∑m−1 l=0 Xikl. Technically, we model the expert knowledge as follows. Expert knowledge: Aggregated incurred payments: II = {Iik : (i,k) ∈I}, with Iik being Iik = k∑ s=0 m−1∑ l=0 E[Xisl| F(i+k)] − k−1∑ s=0 m−1∑ l=0 E[Xisl| F(i+k−1)], where Fh is an increasing filtration illustrating the expert knowledge at calen- dar period h. In this manuscript, we will only consider best estimates and therefore only need assumptions on the mean. We show that the classical CLM multiplicative structure holds under very general underlying dependencies on the mean. The first moment conditions of the DCL model are formulated below. For fixed i = 0, . . . ,m; k,l = 0, . . . ,m− 1, and h = 1, . . . ,Npaidikl , it holds that A1. The counts Nik are random variables with mean having a multiplicative parametriza- tion E[Nik] = αiβk, and identification ∑m−1 k=0 βk = 1. A2. The mean of the RBNS delay variables is E[N paid ikl |NI] = Nikπ̃l. A3. The mean of the individual payments size conditional on the number of pay- ments and the counts is given by E[X (h) ikl |N paid ikl ,NI] = µ̃lγi. Assumption A1 is the classical chain ladder assumption applied on the counts triangle, see also Mack (1991). The main point hereby is the multiplicativity be- tween underwriting year and reporting delay. Assumptions A2 and A3 are necessary to connect reporting delay, settlement delay and development delay - the main idea of DCL. See also Verrall et al. (2010), Mart́ınez-Miranda et al. (2011) and Mart́ınez- Miranda et al. (2012). Using A1 to A3, we have that E  N paid i,j−l∑ h=1 X (h) i,j−l,l|NI   = E  N paid i,j−l,l∑ h=1 E[X (h) i,j−l,l|NI,N paid i,j−l,l]|NI   = E[N paid i,j−l,lµ̃j−lγi|NI] = Ni,j−lπ̃lµ̃j−lγi. 6 Note that the observed aggregated payments can be written as Xij = j∑ l=0 Xi,j−l,l = j∑ l=0 N paid i,j−l,l∑ h=1 X (h) i,j−l,l. With the previous consideration, we derive E[Xij|NI] = γi k∑ l=0 Ni,j−lπ̃lµ̃j−l,l, and the unconditional mean is E[Xij] = αiγi j∑ l=0 βj−lµ̃j−l,lπ̃l. (1) Inspecting equation (1), we can reduce the amount of parameters by setting µ =∑m−1 l=0 π̃lµ̃l and πl = π̃lµ̃lµ −1, so that µπl = µ̃lπ̃l and therefore the unconditional mean of the payments becomes E[Xij] = αiγiµ j∑ l=0 βj−lπl. (2) Equation (2) is the key in deriving the outstanding loss liabilities. These are the values of (Xij) in the lower triangle. Consequently in the sequel, (α,β,π,γ,µ) = (α1, . . . ,αm,β0, . . . ,βm−1,π0, . . . ,πm−1,γ1, . . . ,γm,µ) are called the DCL parameters. In the next section, we will see that in a very natural way, we are able to distinguish between RBNS and IBNR claims. This is possible due to the separation of the development delay into the reporting delay β and the settlement delay π. 3 Forecast outstanding claims: the RBNS and IBNR reserves and predictive distributions In this section, we assume that the DCL parameters (α,β,π,γ,µ) are already derived and show how easily point forecasts of the RBNS and IBNR components of the reserve can be calculated. Note that when calculating the RBNS part, it is possible to replace the parameter (αi,βk) by the true value Nik, since the claims are already reported and thus Nik is observed. However, for the IBNR reserves, it is obviously necessary to use all DCL parameters, including the estimates of future numbers of 7 incurred claims αiβk. Using the notation of Verrall et al. (2010) and Mart́ınez-Miranda et al. (2011), we consider predictions over the triangle, J = J = {i = 2, . . . ,m; j = 0, . . . ,m − 1 with i + j ≥ m + 1}, illustrated in Figure 1. Figure 1: Index sets for aggregate claims data, assuming a maximum delay of m−1. We define the RBNS component as follows, where we consider two possibilities depending on whether the estimates of Nik are used or not. X̂ rbns(1) ij = j∑ l=i−m+j Ni,j−lπ̂lµ̂γ̂i, (i,j) ∈J . (3) and X̂ rbns(2) ij = j∑ l=i−m+j N̂i,j−lπ̂lµ̂γ̂i, (i,j) ∈J , (4) where N̂ij = α̂iβ̂j. In most cases, to shorten the notation, we will simply write X̂rbnsij for the RBNS estimates. However, whenever it is necessary, we will state which version is taken. The IBNR component always needs all DCL parameters: X̂ibnrij = i−m+j−1∑ l=0 N̂i,j−lπ̂lµ̂γ̂i, (i,j) ∈J . (5) The outstanding loss liabilities point estimates are then, X̂ij = X̂ rbns ij + X̂ ibnr ij (6) The outstanding liabilities per accident year are the row sums of IBNR and RBNS estimates. For a fixed i, we write J(i) = {j : (i,j) ∈ J}. Then the outstanding 8 liabilities per accident year i = 1, . . . ,m are R̂i = ∑ j∈J(i) X̂rbnsij + X̂ ibnr ij . (7) In the next section we describe several methods to derive the DCL parameters. 4 Estimation of the parameters in the Double Chain Ladder model To estimate the outstanding claims and thereby construct RBNS and IBNR reserves, we need to estimate the parameters involved in (2). In this section, we explore six different estimators. 4.1 The DCL method The DCL method is the original and maybe most simple method to derive the pa- rameters introduced in the previous section. It was introduced in Mart́ınez-Miranda et al. (2012). To estimate the DCL parameters in (2), assumptions on the pay- ments triangle XI are needed. DCL assumes the assumptions underlying the CLM method. B1 The payments Xij, with i = 1, . . . ,m, and j = 0, . . . ,m − 1, are random variables with mean having a multiplicative parametrization: E[Xij] = α̃iβ̃j, m−1∑ j=0 β̃j = 1. (8) Finally, merging (2) and (8), we conclude αiγiµ j∑ l=0 βj−lπl = α̃iβ̃k, and identify the parameters by αiµγi = α̃i, (9) j∑ l=0 βj−lπl = β̃j. (10) Again, many other micro-structure formulations might exist, thus the one spec- ified by (9) and (10) is only one of several possible. However, the above model can be considered as a detailed specification of the CLM. In Mart́ınez-Miranda et al. 9 (2013b) it is shown that if the RBNS component is calculated by (4), DCL com- pletely replicates the results of CLM. Now, the main idea to derive the DCL parameters is to estimate the chain ladder parameters (α̂, β̂) and (̂̃α, ̂̃β) (cf. A1, B1) by applying the classical chain ladder algorithm on the payments triangle XI and the counts triangle NI. Afterwards, the parameters left in (2) (this is (γ̂, µ̂, π̂)) can be calculated by simple algebra using (9) and (10). For illustration of the chain ladder algorithm, we assume an incremental triangle (Cij) (in our case this would be NI or XI), and that we want to estimate its chain ladder parameters (α̂, β̂). To apply the chain ladder algorithm, one has to transform the triangle (Cij) into a cumulative triangle (Dij): Dij = j∑ k=1 Cik. Then, the chain ladder algorithm can be applied on (Dij). It will produce estimates of development factors, λj, j = 1, 2, . . . ,m− 1, which can be described by λ̂j = ∑n−j+1 i=1 Dij∑n−j+1 i=1 Di,j−1 . These development factors can be converted into estimates of (α,β) using the fol- lowing identities which were derived in Verrall (1991). β̂0 = 1∏m−1 l=1 λ̂l β̂j = λ̂j − 1∏m−1 l=j λ̂l α̂i = m−i∑ j=0 Cij m−1∏ j=m−i+1 λ̂j Alternatively, analytical expressions for the estimators can also be derived directly (rather than using the chain ladder algorithm) and further details can be found in Kuang et al. (2009). Once the chain ladder parameters (α̂, β̂) and (̂̃α, ̂̃β) are derived, the settlement delay parameter π can be estimated just by solving the following linear system.   ̂̃ β0 ... ...̂̃ βm−1   =   β̂0 0 · · · 0 β̂1 β̂0 . . . 0 ... . . . . . . 0 β̂m−1 · · · β̂1 β̂0     π0 ... ... πm−1   . (11) 10 Let π̂ denote the solution of (11). Now we consider the estimation of the parameters involved in the means of indi- vidual payments. Of course, the model is technically over-parametrised since there are too many inflation parameters in (9). The simplest way to ensure identifiability is to set γ1 = 1, and then the estimate of µ, µ̂, can be obtained from µ̂ = ̂̃α1 α̂1 . (12) Using µ̂, the remaining estimates for γi, i = 2, . . . ,m, are directly derived from (9). The estimation procedure of double chain ladder is already programmed with the language R. We have used the R-package DCL (Mart́ınez-Miranda et al., 2013c) to derive Table 1, which shows the values of α̂, β̂, π̂ and γ̂ calculated from real data included in the DCL package. i k,l α̂(i) β̂(k) π̂(l) γ̂(i) 1 0 1078 0.7599 0.0592 1.0000 2 1 1890 0.2097 0.3098 1.1173 3 2 2066 0.0189 0.2032 1.4947 4 3 2353 0.0064 0.1996 1.7461 5 4 3015 0.0016 0.1388 2.1075 6 5 3727 0.0010 0.0440 2.0936 7 6 5057 0.0009 0.0227 2.2495 8 7 6483 0.0007 0.0095 2.1250 9 8 7727 0.0003 0.0018 1.9028 10 9 7134 0.0001 0.0029 2.0197 11 10 7319 0.0001 0.0002 2.0704 12 11 6152 0.0000 0.0026 2.2666 13 12 5242 0.0001 0.0019 2.3157 14 13 6150 0.0000 0.0032 2.4747 15 14 7028 0.0001 -0.0002 2.3829 16 15 6725 0.0000 0.0013 2.8391 17 16 5260 0.0000 -0.0004 3.1815 18 17 5869 0.0000 0.0000 4.1747 19 18 5953 0.0000 0.0000 6.7501 µ̂ = 2579 Table 1: DCL parameter estimates derived by the DCL method 11 4.2 The BDCL method The CLM and Bornhuetter-Ferguson (BF) methods are among the easiest claim reserving methods and, due to their simplicity, they are two of the most commonly used techniques in practice. Some recent papers on the BF method include Verrall (2004), Mack (2008), Schmidt and Zocher (2008), Alai et al. (2009) and Alai et al. (2010). The BF method introduced by Bornhuetter and Ferguson (1972) aims to ad- dress one of the well known weaknesses of CLM, which is the effect outliers can have on the estimates of outstanding claims. Especially the most recent underwriting years are the years with nearly no data and thus very sensitive to outliers. However, these recent underwriting years build the very major part of the outstanding claims. Hence, the CLM estimates of the outstanding liabilities might differ fatally from the true (unknown) values. Acknowledging this problem, the BF method incorporates prior knowledge from ex- perts and is therefore more robust than the CLM method, which relies completely on the data contained in the run-off triangle XI. In this section, we briefly summarize the Bornhutter-Ferguson double chain lad- der (BDCL) method introduced in Mart́ınez-Miranda et al. (2013b), which mimics BF in the framework of DCL. The BDCL method starts with identical steps as DCL but instead of using the estimate of the inflation parameters, γ and µ, from the tri- angle of paid claims XI, it deploys expert knowledge in the form of the incurred triangle II to adjust the estimation of the sensitive inflation parameter γ. This is done as follows. From assumptions A2,A3 and equation (9), we easily deduce that E[Iik] = αiµγiβk = α̃iβk. (13) Hence, the incurred triangle II has multiplicative mean and its underwriting year factor, α̃, is identical to the one of the payments triangle XI (cf. (8)). However, its estimation is less sensitive to outliers since it incorporates all incurred claims via expert knowledge. We conclude that we can replace the payments triangle by the incurred payments triangle when we calculate estimates of the inflation parameters, γ,µ, in (9). Note that the severity mean µ is going to remain the same since the first rows of XI and II are identical. Summarized, the BDCL-method can be carried out as follows. • Step 1: Parameter estimation. Estimate the DCL parameters (α,β,π,γ,µ) using the DCL method of Section 4.1 with the data in the triangles NI and XI and denote the parameter esti- mates by (α̂, β̂, π̂, γ̂, µ̂). 12 Repeat this estimation using the DCL method but replacing the triangle of paid claims XI by the triangle of incurred data II. Keep only the resulting estimated inflation parameters, denoted by γ̂BDCL. • Step 2: BF adjustment. Replace the inflation parameters γ̂ from the paid data by the estimate from the incurred triangle, γ̂BDCL. From Step 1 and Step 2, the final BDCL estimates of the DCL parameters are (α̂, β̂, π̂, γ̂BDCL, µ̂). ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 2 4 6 Underwriting period S e ve ri ty in fla tio n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ● DCL BDCL Figure 2: Plot of severity inflation estimates. DCL: γ̂i (green), BDCL: γ̂ BDCL i (blue). Using the R-package DCL (Mart́ınez-Miranda et al., 2013c), we derive Figure 2 which shows the severity inflation estimates derived by DCL and BDCL. BDCL, with the incorporated expert knowledge, seems to stabilize the severity inflation in the most recent underwriting years while keeping the values in the other years. The result is a more realistic estimate correcting the DCL parameter γ̂i exactly in its weakest point, that is in those years where the payments triangle XI has nearly no data. Again, those recent underwriting years contain the very major part of the outstanding liabilities. 4.3 The IDCL method This section gives a brief theoretical introduction to the IDCL method of Agbeko et al. (2015). In the BDCL definition, we incorporated an additional triangle of in- curred claims in order to produce a more stable estimate of the underwriting inflation parameter γi. The derived BDCL method becomes a variant of the Bornhuetter- Ferguson technique using prior knowledge contained in the incurred triangle. One natural question is whether the derived reserve estimate is the classical incurred 13 chain ladder. The answer is that this not the case; the IDCL method does not replicate the results obtained by applying the classical chain ladder method to the incurred triangle. Among practitioners, the incurred reserve seems to be more real- istic for many datasets compared to the classical paid chain ladder reserve. IDCL mimics the reserve estimate of chain ladder on the incurred triangles in the DCL framework. It is defined just by rescaling the underwriting inflation parameter es- timate from the DCL method. Specifically, we define a new scaled inflation factor estimate γ̂IDCL such that γ̂IDCLi = R̂∗i R̂i γ̂i, (14) where R∗i is the outstanding loss liabilities per accident year as predicted by applying the traditional CLM on incurred data, and (Ri,γ̂i) are the outstanding loss liabilities per accident year and the inflation parameter respectively, using the DCL method (cf. Section 4.1). The final IDCL estimates of the DCL parameters are then (α̂, β̂, π̂, γ̂IDCL, µ̂). With the new inflation parameter estimate γ̂IDCL, the outstanding liabilities derived by the DCL parameters completely replicate the CLM estimates on the incurred trian- gle. 4.4 The PDCL method This section gives a brief theoretical introduction to the PDCL method introduced in Hiabu et al. (2016). See also Nielsen (2016). In the last section, we have described a method which incorporates expert knowledge in form of the incurred triangle II. The values in II arise from case estimates for RBNS claims, developed in the case department of the insurance company, and claims which are already paid. Thus, if one subtracts these already paid claims (which are given via the payments trian- gle XI) from the incurred triangle, one can reconstruct the RBNS case estimates. However, as soon as this is done, it is obvious that these RBNS case estimates do not match with the RBNS estimates (3) and (4), using any DCL method (including BDCL). We conclude that the reserve department, using double chain ladder (and also chain ladder), calculates different RBNS estimates than those given by the case department. PDCL replaces the calculated RBNS estimates with the case estimates. The method, however, does not only preserve the RBNS case estimates by adding the IBNR esti- mates to conclude the reserve, it also takes the RBNS case estimates to correct the other DCL parameters. Therefore, also the total IBNR size will change. The first step is to construct a preliminary square (Sij), i = 1, . . . ,m, j = 14 0, . . . ,m− 1, which yields new estimators for the DCL parameters. The upper tri- angle of the square (i.e., (i,j) ∈ I) should have the same entries as the payments triangle (Xij). The lower triangle (i.e., (i,j) ∈J) should consist of preliminary esti- mates of the outstanding loss liabilities. The outstanding loss liabilites comprise an RBNS and an IBNR part (cf. (6)). However, only the IBNR part of these outstand- ing loss liabilities is estimated. For the RBNS component, the RBNS case estimates are taken. More precisely, one takes the DCL parameter estimates (α̂, β̂, π̂, γ̂, µ̂) and use these parameters to estimate the RBNS component (X̂rbnsij ) and IBNR compo- nent (X̂ibnrij ) using (4) and (5). As mentioned above, the RBNS estimate should be be equal to the RBNS case estimates, which can only be reconstructed per accident year. For i = 1, . . . ,m, they can be described as Xrbns.case.estimatei = m−i∑ j=0 Iij − m−i∑ j=0 Xij. (15) Hence, we define the RBNS preserving components X̂ rbns.pres ij = ∑m−i j=0 Iij − ∑m−i j=0 Xij∑ j∈J(i) X̂ rbns ij X̂rbnsij . (16) Note that ∑ j∈J(i) X̂ rbns.pres ij = X rbns.case.estimate i . Thus we define the preliminary square (Sij) as Sij =  Xij, if (i,j) ∈I,X̂rbns.presij + X̂ibnrij , if (i,j) ∈J . (17) One can easily see that payments square (Sij) has approximately multiplicative mean E[Sij] ≈ α̃iβ̃j. Therefore, we can use (Sij) to completely replace XI to estimate the DCL parameters (cf. (8)). Note that in the BDCL method we were only able to balance the estimator of the inflation parameter α̃i (cf. (13)). Again, while in the BDCL method, one uses the expert knowledge to only adjust the inflation parameters, here, one can take full advantage of the triangle II and also correct the delay parameters. Since (Sij) has a multiplicative structure, the CLM idea is used to estimate α̃i and β̃j. We define ̂̃αPDCLi = m−1∑ j=0 Sij, ̂̃ β PDCL j = ∑m i=1 Sij∑ (i,j)∈I∪J Sij . (18) 15 Exactly as in the previous sections, one can now apply (9) and (10) to derive the PDCL parameters (α̂i, β̂j, π̂ PDCL, γ̂PDCL ∗ , µ̂PDCL). Since this approach is still not RBNS preserving, γ̂PDCL ∗ is balanced by defining a new scaled inflation factor esti- mate γ̂PDCL such that γ̂PDCL = ∑m−i j=0 Iij − ∑m−i j=0 Xij X̂rbnsij , (19) where X̂rbnsij is calculated with the parameters (α̂, β̂, π̂ PDCL, γ̂PDCL ∗ , µ̂PDCL) using (4). 4.5 The EDCL method In this chapter we introduce a new method called expert double chain ladder (EDCL), indicating that expert knowledge in form of incurred data and RBNS’s are incorpo- rated into the system as pseudo data. The idea of the EDCL method is to replicate the basic steps of the previously introduced PDCL method (15)-(18), but without the adjustment of the severity inflation in (19). Instead those steps are iterated until convergence. The iteration forces a homogeneous solution which incorporates the incurred triangle, I, and the payment triangle, X, to one reserve. The discrep- ancy between estimated RBNS and RBNS provided by the case estimates can be explained by variation of the observations around their mean. Note that given the model assumptions, we are estimating the mean of the RBNS. The EDCL estimation can be described as follows. In the first step of the itera- tion, we start with the DCL parameter estimates (α̂, β̂, π̂, γ̂, µ̂), which we denote by (α̂, β̂, π̂EDCL,(0), γ̂EDCL,(0), µ̂EDCL,(0)). In the k−th step of the iteration, we take the EDCL parameter estimates we obtained in the (k−1)−th step (α̂, β̂, π̂EDCL,(k−1), γ̂EDCL,(k−1), µ̂EDCL,(k−1)) and use these parameters to estimate the RBNS component (X̂ rbns,(k−1) ij ), and IBNR com- ponent (X̂ ibnr,(k−1) ij ), using (4) and (5). Then, we calculate the k−th RBNS preserving components X̂ rbns.pres,(k) ij = ∑m−i j=0 Iij − ∑m−i j=0 Xij∑ j∈J(i) X̂ rbns,(k−1) ij X̂ rbns,(k−1) ij . Now, we are able to define the k−th preliminary square (S(k)ij ) as S (k) ij =  Xij, if (i,j) ∈I,X̂rbns.pres,(k)ij + X̂ibnr,(k−1)ij , if (i,j) ∈J . 16 Since (S (k) ij ) has an approximately multiplicative structure, we use the CLM idea to estimate α̃i and β̃j. We define ̂̃αEDCL,(k)i = m−1∑ j=0 S (k) ij , ̂̃ β EDCL,(k) j = ∑m i=1 S (k) ij∑ (i,j)∈I∪J S (k) ij . Exactly as in the previous sections, we can now apply (9) and (10) to derive the EDCL parameters (α̂, β̂, π̂EDCL,(k), γ̂EDCL,(k), µ̂EDCL,(k)). After iterating until convergence, we derive the final EDCL parameters denoted by (α̂, β̂, π̂EDCL, γ̂EDCL, µ̂EDCL). 4.6 The PEDCL method In this section, we introduce the RBNS-preserving expert double chain ladder (PEDCL). As the name suggests, this method builds on the PDCL method by using the EDCL parameters (α̂, β̂, π̂EDCL, γ̂EDCL, µ̂EDCL). As mentioned in the previous section, EDCL does not preserve the RBNS case estimates. The reason was mentioned in the beginning of the previous section and has a parallelism to the two different RBNS estimates in (3) and (4). If as in PDCL, one decides not to change the case estimates, one can thus replace the RBNS part by the RBNS derived from the case estimates. As this information is only available per underwriting year, it can be incorporated by changing the infla- tion parameter accordingly. More precisely, we define a new scaled inflation factor estimate as γ̂PEDCL = ∑m−i j=0 Iij − ∑m−i j=0 Xij X̂rbnsij , where X̂rbnsij is calculated with the parameters (α̂, β̂, π̂ EDCL, γ̂EDCL, µ̂EDCL) using (4). This new inflation parameter γ̂PEDCL is used to replace γ̂EDCL in the parameter set (α̂, β̂, π̂EDCL, γ̂EDCL, µ̂EDCL) when calculating only the RBNS part which pre- serves the case estimates. Therefore, in contrast to the previous methods, PEDCL possesses two inflation parameters. Firstly, γ̂PEDCL for the RBNS part, that is when calculating (4) and secondly γ̂EDCL for the IBNR part, that is when calculating (5). Note that since we are just using the EDCL parameters to calculate the IBNR part, the IBNR reserve of the PEDCL method is exactly the same as the IBNR reserve of the EDCL method. 5 Real Data Application In this section, we apply the methods to a data set obtained from a UK motor business. The data is also available via the DCL R package (Mart́ınez-Miranda 17 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 2 4 6 Underwriting period S e ve ri ty in fla tio n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● DCL BDCL IDCL PDCL EDCL PEDCL Figure 3: Plot of severity inflation estimates. et al., 2013c). Figure 3 shows a plot of the six severity inflation parameters derived by DCL, BDCL, IDCL, PDCL, EDCL and PEDCL. There were some rather rough corrections going on for IDCL,PDCL and PEDCL in the first five years. These five years represent less than 0.1% of the total loss liabilities estimates (cf. Table 2-4) and they are not really important. We have therefore taken the first five years out for these three estimation methods. Otherwise, these unimportant first five years would have dominated the graph and perhaps have confused the reader. Figure 3 shows that IDCL,PDCL and PEDCL still are quite volatile with a tendency to have lower severity inflations than DCL, BDCL and EDCL. This is because IDCL,PDCL and PEDCL adjust to high or low RBNS’s in certain years, while in particular BDCL and EDCL take a more balanced point of view. From a modeling perspective the sever- ity inflations resulting from BDCL and EDCL seem more attractive, because they are more stable and therefore seem more realistic. This graphs illustrate very well why the PEDCL method uses the EDCL parameters while estimating the IBNR re- serve. These parameters seem more realistic than the RBNS-preserving parameters. However, the RBNS-preserving parameters might be more realistic when estimating the somewhat realised RBNS reserve. If one believes the RBNS estimates are of good quality or are of the best possible quality one can do with the data, then one should of course use a method preserving these RBNS estimates such as PDCL and PEDCL. However, for the IBNR’s it is another matter. There is no expert knowl- edge available for the IBNR estimates and therefore one should use the methodology with the most credible parameters. We believe that EDCL is the method with the most credible parameters and PEDCL is therefore - in our opinion - the optimal methodology if one wishes to preserve the RBNS estimates. If one is looking for a more balanced view, where the RBNS’s are allowed to impact the parameters, but 18 where the observed data also should play a role in some sort of validation of the RBNS as data, then one should use the EDCL method. We believe that the DCL method, the EDCL method and the PEDCL method are the three best methods to consider for practising actuaries. To visualise these arguments, Table 2 shows the outstanding loss liabilities per underwriting year for all six methods mentioned in chapter 4 as well as the classical chain ladder method. Furthermore, the splitted values of RBNS and IBNR per underwriting year are illustrated in Table 3 and Table 4. i CLM DCL BDCL IDCL PDCL EDCL PEDCL 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2 0.0000 0.0005 0.0003 0.0005 0.0000 0.0000 0.0000 3 0.0000 0.0001 -0.0003 0.0001 0.0040 0.0001 0.0040 4 0.0000 0.0007 -0.0014 0.0007 -0.0095 0.0003 -0.0095 5 0.0173 0.0039 0.0151 0.0045 0.0365 0.0076 0.0365 6 0.0346 0.0309 0.0313 0.0122 0.0054 0.0135 0.0068 7 0.1381 0.1407 0.1408 0.0090 0.0014 0.0541 0.0040 8 0.2449 0.2485 0.2483 0.0927 0.0949 0.1102 0.0972 9 0.3522 0.3582 0.3563 0.0536 0.0591 0.1774 0.0643 10 0.3943 0.3818 0.3800 0.1507 0.1981 0.2148 0.2024 11 0.5524 0.5246 0.5206 0.1664 0.2515 0.3415 0.2595 12 0.6839 0.6309 0.6169 -0.1458 0.0015 0.3415 0.0300 13 1.0504 0.9764 0.9733 0.8648 1.2718 0.8896 1.2576 14 2.5361 2.5483 2.5164 2.0388 2.8302 2.3205 2.8153 15 5.7370 5.4483 5.2846 4.2095 6.2887 5.4225 6.2721 16 14.0889 15.4373 12.9824 6.8542 9.4207 11.9785 9.4735 17 21.0057 21.7407 17.0455 12.0924 13.4243 16.5608 13.5068 18 44.6877 44.4580 29.2840 23.0002 25.9634 30.6858 26.1851 19 98.9723 98.9722 41.8444 39.1522 42.1009 45.0575 42.8640 SUM 190.4957 191.9021 112.2385 88.5565 101.9427 114.3021 103.0696 Table 2: Outstanding loss liabilities per underwriting year in million 19 i CLM DCL BDCL IDCL PDCL EDCL PEDCL 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2 0.0000 0.0005 0.0003 0.0005 0.0000 0.0000 0.0000 3 0.0000 0.0001 -0.0003 0.0001 0.0040 0.0001 0.0040 4 0.0000 0.0007 -0.0014 0.0007 -0.0095 0.0003 -0.0095 5 0.0173 0.0039 0.0151 0.0045 0.0365 0.0076 0.0365 6 0.0329 0.0291 0.0295 0.0115 0.0050 0.0118 0.0050 7 0.1355 0.1382 0.1382 0.0088 0.0014 0.0516 0.0014 8 0.2399 0.2436 0.2434 0.0909 0.0923 0.1053 0.0923 9 0.3455 0.3515 0.3496 0.0526 0.0576 0.1707 0.0576 10 0.3822 0.3697 0.3680 0.1459 0.1903 0.2027 0.1903 11 0.5339 0.5062 0.5023 0.1606 0.2411 0.3231 0.2411 12 0.6550 0.6020 0.5887 -0.1392 0.0014 0.4387 0.0014 13 1.0034 0.9294 0.9265 0.8231 1.2101 0.8420 1.2101 14 2.4415 2.4537 2.4229 1.9631 2.7197 2.2249 2.7197 15 5.5906 5.3020 5.1427 4.0964 6.1235 5.2739 6.1235 16 13.8419 15.1902 12.7746 6.7444 9.2492 11.7542 9.2492 17 20.5131 21.2482 16.6593 11.8184 13.0995 16.1534 13.0995 18 42.7693 42.5397 28.0204 22.0078 24.8281 29.3288 24.8281 19 74.0936 74.0942 31.3260 29.3108 31.4544 33.6479 31.4544 SUM 162.5956 164.0027 99.5059 77.1009 89.3045 100.5369 89.3045 Table 3: RBNS per underwriting year in million 20 i CLM DCL BDCL IDCL PDCL EDCL PEDCL 1-5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 6 0.0018 0.0018 0.0018 0.0007 0.0004 0.0018 0.0018 7 0.0026 0.0026 0.0026 0.0002 0.0000 0.0026 0.0026 8 0.0049 0.0049 0.0049 0.0018 0.0026 0.0049 0.0049 9 0.0067 0.0067 0.0067 0.0010 0.0015 0.0067 0.0067 10 0.0121 0.0121 0.0120 0.0048 0.0078 0.0121 0.0121 11 0.0184 0.0184 0.0183 0.0058 0.0103 0.0184 0.0184 12 0.0289 0.0289 0.0282 -0.0067 0.0001 0.0285 0.0285 13 0.0471 0.0471 0.0469 0.0417 0.0617 0.0476 0.0476 14 0.0946 0.0946 0.0934 0.0757 0.1105 0.0957 0.0957 15 0.1464 0.1464 0.1419 0.1131 0.1652 0.1486 0.1486 16 0.2470 0.2471 0.2077 0.1097 0.1715 0.2243 0.2243 17 0.4926 0.4925 0.3862 0.2740 0.3248 0.4073 0.4073 18 1.9183 1.9183 1.2636 0.9924 1.1353 1.3570 1.3570 19 24.8787 24.8779 10.5185 9.8414 10.6465 11.4096 11.4096 SUM 27.9001 27.8993 99.5059 11.4556 12.6382 13.7651 13.7651 Table 4: IBNR per underwriting year in million 21 6 Model validation This section describes the validation process for the six methods DCL, BDCL, IDCL, PDCL, EDCL and IPDCL discussed in Section 4. The following validation was introduced in Agbeko et al. (2015). To our knowledge this is the only developed method which is able to validate reserving estimates based on incurred data with estimates based on paid data. That is to validate DCL against IDCL in our terminology. However, this validation methodology is sufficiently general to allow all these six procedures to be validated against each other. The validation process builds on back testing and is based on the fact that all introduced DCL methods provide reserve estimates by predicting into the same paid triangle, X. Therefore, all methods can be compared on the same scale so to speak. Below, we have omitted the most recent calendar year and the four most recent calendar years, respectively (in all three available triangles). Therefore, since our dataset consists of m = 19 years, there are 18 and 60 cells, respectively, to be compared with the true values. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● CLM DCL BDCL IDCL PDCL EDCL PEDCL− 1 0 0 0 0 0 0 0 5 0 0 0 0 0 1 5 0 0 0 0 0 1 periods cut P re d ic tio n m in u s o b se rv e d ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● CLM DCL BDCL IDCL PDCL EDCL PEDCL− 1 0 0 0 0 0 0 0 5 0 0 0 0 0 1 5 0 0 0 0 0 4 periods cut P re d ic tio n m in u s o b se rv e d Figure 4: Box plot of the cell errors Figure 4 shows two box plots of the respectively 18 and 60 errors calculated by taking the difference between estimated and true values. One conclusion is that DCL and CLM seem to be inferior to the five methods that take advantage of expert knowledge. Between these five methods, it seems that BDCL and EDCL have similar performance with a slight advantage to the new EDCL method. Also PDCL and PEDCL have similar performance, which was to be expected because they only differ in the IBNR reserves while having identical RBNS reserves. The omnipresent IDCL method does not provide convincing performance in this data example. Earlier studies based on a number of data sets have shown that IDCL sometimes have very good performance, but equally often fails badly. Of the three methods available at the time, DCL, BDCL and IDCL, only the BDCL method had 22 a stable performance. Sometimes DCL was winning convincingly, other times IDCL was the winner. In the long run, however, the eternal runner-up was the BDCL method, and most of the time it did almost as well - but not quite - as the best of the two other methods. We do, however, find EDCL a more convincing method than BDCL. While they have some similarities and while both have stable underwriting year severities as we saw in the application, EDCL seems more theoretically correct in its way of exploring the expert knowledge and we believe it will be replacing BDCL in the long run. Based on long-term considerations, we think DCL, EDCL and PEDCL should be sufficient in the practical actuaries tool box. BDCL, IDCL and PDCL are - in our view - less convincing methods. CLM DCL BDCL IDCL PDCL EDCL PEDCL 1 periods cut S u m o f a b so lu te e rr o rs 0 e + 0 0 1 e + 0 7 2 e + 0 7 3 e + 0 7 4 e + 0 7 CLM DCL BDCL IDCL PDCL EDCL PEDCL 4 periods cut S u m o f a b so lu te e rr o rs 0 e + 0 0 1 e + 0 7 2 e + 0 7 3 e + 0 7 4 e + 0 7 Figure 5: Bar plot for the sum of absolute cell errors CLM DCL BDCL IDCL PDCL EDCL PEDCL 1 periods cut R e la tiv e e rr o r 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 CLM DCL BDCL IDCL PDCL EDCL PEDCL 4 periods cut R e la tiv e e rr o r 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 Figure 6: Bar plot for the relative errors 23 In the top panels of Figure 5, we have plotted the sum of the absolute cell errors (`1 error). That is Sum of absolute cell errors = ∑ (i,j)∈B |X̂ij −Xij|, B = {(i,j)| i = 2, . . . ,m− c; j = 0, . . . ,m− c− 1; i + j = m− c + 1, . . . ,m}, where c is the number of recent calendar years omitted for back testing (here: 1 and 4). The relative errors, that is Sum of absolute cell errors Sum of absolute true values = ∑ (i,j)∈B |X̂ij −Xij|∑ (i,j)∈B |Xij| , is shown in the bottom panels of Figure 5. The conclusion Figure 5 is similar to the conclusion of Figure 4. 7 Conclusions This paper has developed two new methods combining classical chain ladder method- ology with expert knowledge via the double chain ladder methodology. The new EDCL introduces RBNS’s as pseudo data and uses an iterative procedure to im- prove the originally estimated DCL parameters that did not take RBNS information into account. It shows to have very good performance. The new PEDCL method preserves RBNS estimates also after the reserve has been estimated. Validation is introduced for these two new methods and they are compared to the previous meth- ods DCL, BDCL, IDCL and PDCL. Our conclusion is that while DCL always will be some kind of benchmark in the actuaries tool box, then EDCL and PEDCL seem to have sufficient quality to replace BDCL, IDCL and PDCL. That also means that EDCL and PEDCL seem to have sufficient quality to replace incurred chain ladder as the actuaries preferred method of incorporating incurred data expert knowledge. This is important, because most reserves in the actuarial practise use incurred chain ladder as the basis of estimation. This incurred chain ladder might be manually manipulated according to expert knowledge and the values of paid chain ladder. However, it is the most common basis of estimation in actuarial practise. Only the future will be able to show whether EDCL and PEDCL or other similar innovations will be able to replace actuaries habit of using the - in our view outdated - incurred chain ladder approach. 24 References Abbasi, B. and Guillén, M. (2013). Bootstrap control charts in monitoring value at risk in insurance. 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