# Obtaining Predictive Distributions for Reserves Which Incorporate Expert Opinion. R. J. Verall

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1 Obtaining Predictive Distributions for Reserves Which Incorporate Expert Opinion R. J. Verall 283

2 Obtaining Predictive Distributions for Reserves Which Incorporate Expert Opinion 1LJ.Verrall Faculty of Actuarial Science and Statistics Cass Business School City University Abstract This paper shows how expert opinion can be inserted into a stochastic framework for claims reserving. The reserving methods used are the chain-ladder technique and Bornhuetter-Ferguson, and the stochastic framework follows England and Verrall (2062). Although stochastic models have been studied, there are 2 main obstacles to their more frequent use in practice: ease of implementation and adaptability to user needs. This paper attempts to address these obstacles by utilising Bayesian methods, and describing in some detail the implementation, using freely available software and programs supplied in the Appendix. Keywords Bayesian Statistics, Bornhuetter-Ferguson, Chain-ladder, Claims Reserving, Risk Address for correspondence Prof. R.J.Verrall, Faculty of Actuarial Science and Statistics, Cass Business School, City University, Northampton Square, London. EC 1V 0HB 284

5 investigate the use of these models himself/herself, using the programs given in the Appendix. In section 7 we state some conclusions. 2. Notation and basic methods To begin with, we define the notation used in this paper, and in doing so briefly summarize the chain-ladder technique and the Bomhuetter-Ferguson method. Although the methods can also be applied to other shapes of data, in order that the notation should not get too complicated we make the assumption that the data is in the shape of a triangle. Thus, without loss of generality, we assume that the data consist of a triangle of incremental claims: c,, G,... c,. c~,... c~... C., This can be also written as {Co: j = 1... n - i + 1;i = 1... n }, where n is the number of accident years. C o is used to denote incremental claims, and Dv is used to denote the cumulative claims, defined by: J v.=zc. k~l One of the methods considered in this paper is the chain-ladder technique, and the development factors {2::j = 2... n}. The usual estimates of the development factors from the standard chain-ladder technique are n-j+l ZD. ~ j _ i=l E Di.j-i Note that we only consider forecasting claims up to the latest development year (n) so far observed, and no tail factors are applied. It would be possible to extend this to allow a tail factor, using the same methods, but no specific modelling is carried out in this paper of the shape of the run-offbeyond the latest development year. Thus, we n refer to cumulative claims up to development year n, Di, = ~ C a, as "ultimate k~l 287

7 There are a number of different approaches that can be taken to the chain-ladder technique, with various positivity constraints, all of which give the same reserve estimates as the chain-ladder technique. The connections between the chain-ladder technique and various stochastic models have been explored in a number of previous papers. For example, Mack (1993) takes a non-parametric approach and specifies only the first 2 moments for the cumulative claims. In Mack's model the mean and variance of D O I D~.j_j,A, cr 2 are,~.jdi,j_ I and o'~ Di.j_l, respectively. Estimates of all the parameters are derived, and the properties of the model are examined. As was stated in the introduction, one of the advantages of this approach is that the parameter estimates and prediction errors can be obtained just using a spreadsheet, without having recourse to a statistical package or any complex programming. The consequence of not specifying a distribution for the data is that there is no predictive distribution. Also, there are separate parameters in the variance that must also be estimated, separately from the estimation of the development factors. As a separate stream of research, models in the form of generalized linear models have also been considered. Renshaw and Verrall (1998) used an approach based on generalized linear models (McCullagh and Nelder, 1989) and examined the overdispersed Poisson model for incremental claims: C U I c, a,fl,~o ~ independent over-dispersed Poisson, with mean, m0, where log(mo)=c+ai+flj and oq=fll=o. The term "over-dispersed" requires some explanation. It is used here in connection with the Poisson distribution, and it means that if X - Poisson (/.t), then Y = ~px follows the over-dispersed Poisson distribution with E(Y) = q~/.t and V(Y) = q~2e(x) = q~2/.t. ~o is usually greater than 1 - hence the term "over- dispersed" - but this is not a necessity. It can also be used for other distributions, and we make use of it for the negative binomial distribution. As with the Poisson distribution, the over-dispersed negative binomial distribution is defined such that if X ~ negative binomial then Y = ~ox follows the over-dispersed negative binomial distribution. Furthermore, a quasi-likelihood approach is taken so that the claims data are not restricted to the positive integers. It can be seen that this formulation has some similarities with the model of Kremer (1982), but it has a number of advantages. It does not necessarily break down if there are negative incremental claims values, it gives the same reserve estimates as the chain-ladder technique, and it has been found to be more stable than the log-normal model of Kremer. For these reasons, we concentrate on it in this paper. There are a number of ways of writing this model, which are useful in different context (note that the reserve estimates are unaffected by the way the model is written). Another way of writing the over-dispersed Poisson model for the chain-ladder technique is as follows: n C U I x,y,q~ - independent over-dispersed Poisson, with mean xiyj, and Zyk =

8 Here x = {xl,x 2... x,} and y = {Y,,Yz... y,} are parameter vectors relating to the rows (accident years) and colunms (development years), respectively, of the run-off triangle. The parameter x i = E[Dm ], and so represents expected ultimate cumulative claims (up to the latest development year so far observed, n) for the ith accident year. The column parameters, Yi, can be interpreted as the proportions of ultimate claims which emerge in each development year. Although the over-dispersed Poisson models give the same reserve estimates as the chain-ladder technique (as long as the row and column sums of incremental claims are positive), the connection with the chain-ladder technique is not immediately apparent from this formulation of the model. For this reason, the negative binomial model was developed by Verrall (2000), building on the over-dispersed Poisson model. Verrall (2000) showed that the same predictive distribution can be obtained from a negative binomial model (also with the inclusion of an over-dispersion parameter). In this recursive approach, the incremental claims have an over-dispersed negative binomial distribution, with mean and variance (2j - 1)Di.:,_, and ~.j (2 s - 1)Dia-_,, respectively. Again, the reserve estimates are the same as the chain-ladder technique, and the same positivity constraints apply as for the over-dispersed Poisson model. It is clear from this that the colunm sums must be positive since a negative sum would result in a development factor less than 1 ( 2j < 1 ), causing the variance to be negative. It is important to note that exactly the same predictive distribution can be obtained from either the Poisson or negative binomial models. Verrall (2000) also argued that the model could be specified either for incremental or cumulative claims, with no difference in the results. The negative binomial model has the advantage that the form of the mean is exactly the same as that which naturally arises from the chain-ladder technique. In fact, by adding the previous cumulative claims, an equivalent model ford 0 I D~.j_~, 2, 4 has an over-dispersed negative binomial distribution, with mean and variance 2.jDij_, and ~j(3.j-1)d,.i_,, respectively. Here the connection with the chain-ladder technique is immediately apparent because of the format of the mean. A further model, which is not considered further in this paper, is closely connected with Mack's model, and deals with the problem of negative incremental claims. This model replaces the negative binomial by a Normal distribution, whose mean is unchanged, but whose variance is altered to accommodate the case when 2j < 1. Preserving as much of,~l,j(~j - 1)DI,j_~ as possible, the variance is still proportional to D~a_ ~, with the constant of proportionality depending on j, but a Normal approximation is used for the distribution of incremental claims. Thus, C o. I Dia-t,.,qJ is approximately Normally distributed, with mean and variance 290

9 Dij_,(2S-1 ) and ~jd,4_,, respectively, or Du I D~u_~,2,~b is approximately Normally distributed, with mean and variance )tjdiu_ ~ and OjDi.j_~, respectively. As for Mack's model, there is now another set of parameters in the variance that needs to be estimated. For each of these models, the mean square error of prediction can be obtained, allowing the construction of prediction intervals, for example. Claims reserving is a predictive process: given the data, we try to predict future claims. These models apply to all the data, both observed and future observations. The estimation is based on the observed data, and we require predictive distributions for the future observation. We use the expected value of the distribution of future claims as the prediction. When considering variability, attention is focused on the root mean squared error of prediction (RMSEP), also known as the prediction error. To explain what this is, we consider, for simplicity, a random variable, y, and a predicted value ). The mean squared error of prediction (MSEP) is the expected square difference between the actual outcome and the predicted value, EI(y-.~)21' and can be written as follows: El(y- ~)2 ]= E[((y - E[yD- (.~ - E[yD) 21. In order to obtain an estimate of this, it is necessary to plug-in ) instead ofy in the final expectation. Then the MSEP can be expanded as follows: El(Y- ~)~ ]~ El(Y- ELY]) 2 ]- 2E[(y - E[y])(~ - EL~])] + E[(~- ED3]) 2 ], Assuming future observations are independent of past observations, the middle term is zero, and E[(y - ~)2 ]~ E[(y - ELY])21+ El03 - ED3])2 ]. In words, this is prediction variance = process variance + estimation variance. It is important to understand the difference between the prediction error and the standard error. Strictly, the standard error is the square root of the estimation variance. The prediction error is concerned with the variability of a forecast, taking account of uncertainty in parameter estimation and also of the inherent variability in the data being forecast. Further details of this can be found in England and Verrall (2002). Using non-bayesian methods, these two components - the process variance and the estimation variance - are estimated separately, and section 7 of England and Verrall (2002) goes into a lot of detail about this. The direct calculation of these quantities 291

10 can be a tricky process, and this is one of the reasons for the popularity of the bootstrap. The bootstrap uses a fairly simple simulation approach to obtain simulated estimates of the prediction variance in a spreadsheet. Fortunately, the same advantages apply to the Bayesian methods: the full predictive distribution can be found using simulation methods, and the RMSEP can be obtained directly by calculating its standard deviation. In addition, it is preferable to have the full predictive distribution, rather than just the first 2 moments, which is another advantage of Bayesian methods. The purpose of this paper is to show how expert opinion, from sources other than the specific data set under consideration, can be incorporated into the predictive distributions of the reserves. We use the approach of generalized linear models outlined in this section, concentrating on the over-dispersed Poisson and negative binomial models. We begin with considering how it is possible to intervene in the development factors for the chain-ladder technique in section 4, and then consider the Bomhuetter-Ferguson method in section Incorporating expert opinion about the development factors In this section, a Bayesian model is specified which allows the practitioner to intervene in the estimation of the development factors for the chain-ladder technique. There are a number of ways in which this could be used, and we describe some possibilities in this section. It is expected that a practitioner would be able to extend these to cover situations which, although not specifically covered here, would also be useful. The cases considered here are the intervention in a development factor in a particular row particular, and the choice of how many years of data to use in the estimation. The reasons for intervening in these ways could be that there is information that the settlement pattern has changed, making it inappropriate to use the same development factor for each row. For the first case, what may happen in practice is that a development factor in a particular row is simply changed. Thus, although the same development parameters (and hence run-offpattern) is usually applied for all accident years, if there is some exogenous information that indicates that this is not appropriate, the practitioner may decide to apply a different development factor (or set of factors) in some, or all, rows. In the second case, it is common to look at, say, 5-year volume weighted averages in calculating the development factors, rather than using all the available data in the triangle. The Bayesian methods make this particularly easy to do, and are flexible enough to allow many possibilities. We use the negative binomial model described in section 3, with different development factors in each row. This is the model for the data, and we then specify prior distributions for the development factors. In this way, we can choose prior distributions that reproduce the chain-ladder results, or we can intervene and use prior distributions based on external knowledge. The model for incremental claims, C,71Di.j-~, 2, 4, is an over-dispersed negative binomial distribution, with mean and variance 292

11 (2i, j -1)Dij_, and ~,~.j (21j -1)D;j_,, respectively. We next need to define prior distributions for the development factors, 2lj. It is possible to set some of these equal to each other (within each column) in order to revert!o the standard chain-ladder model. This is done by setting 2;j=37 for i=l,2,...,n-i+l;j=l,2,...n and defining vague prior distributions for 2j (j = 1,2,...n). This was the approach taken in section 8.4 of England and Verrall (2002) and is very similar to that taken by de Alba (2002). This can provide a very straightforward method to obtain prediction errors and predictive distributions for the chain-ladder technique. However, we really want to move away from the basic chain-ladder technique, and construct Bayesian prior distributions that encompass the expert opinion about the development parameters. Suppose, for example, that we have a triangle. We consider the 2 possibilities for incorporating expert knowledge described above. To illustrate the first case, suppose that there is information that implies that the second development factor (from column 2 to column 3) should be given the value 2, for rows 8, 9,and 10, and that there is no indication that the other parameters should be treated differently from the standard chain-ladder technique. An appropriate way to treat this would be to specify )tij =2j for i=1,2... n-i+l;j=l,3,4,...,n 2;.~ =~ for i=1, &.2 = ~,2 = &0.2 The means and variances of the prior distributions of the parameters are chosen to reflect the expert opinion: As. 2 has a prior distribution with mean 2 and variance W, where Wis set to reflect the strength of the prior information 2j have prior distributions with a large variances. For the second case, we divide the data into two parts using the prior distributions. To do this, we set '~i.j = ~'j for i = n-i-3,n -i-2,n-i-l,n-i,n-i+l Aia =3.) for i=1,2... n-i-4 293

12 and give both 2y and 2) prior distributions with large variances so that they are estimated from the data. Adjustments to the specification are made in the later development years, where there are less than 5 rows. For these columns there is just one development parameter, 2.i. The specific form of the prior distribution (gamma, log-normal, etc) is usually chosen so that the numerical procedures in winbugs work as well as possible. These models are used as illustrations of the possibilities for incorporating expert knowledge about the development pattern, but it is (of course) possible to specify many other prior distributions. In Appendix 1, the winbugs code is supplied, which can be cut and pasted directly in order to examine these methods. Section 6 contains a number of examples including the ones described in this section. 5. A Bayesian model for the Bornhuetter-Ferguson method. In this section, we show how the Bomhuetter-Ferguson method can be considered in a Bayesian context, using the approach of Verrall (2004). For further background on the Bornhuetter-Ferguson method, see Mack (2000). In section 3, the over-dispersed Poisson model was defined as follows. n C o. I x, y, cp ~ independent over-dispersed Poisson, with mean xiyj, and ~Yk = 1, k=l and in the Bayesian context, we also require prior distributions for the parameters. The Bornhuetter-Ferguson method assumes that there is expert opinion about the level of each row, and we therefore concentrate first on the specification of prior distributions for these. The most convenient form to use is gamma distributions: x i I ctr,flr ~ independent r(ar,p,). There is a wide range of possible choices for the parameters of these prior distributions, a i and fir. It is easiest to consider the mean and variance of the gamma ai ai Mr distribution, y and --fl/~, respectively. These can be written as M r and -~--r' from which it can be seen that, for a given choice of M r, the variance can be altered by changing the value of/3,. To consider a simple example, suppose it has been decided that M r = The table below shows how the value of fl~ affects the variance of the prior distribution, while M~ is kept constant. al ~i Mi Mi P, lo

14 us to take into account the fact that the column parameters have been estimated when calculating the prediction errors, predictive distribution, etc. It is not required to include any information about the column parameters, and hence we use improper gamma distributions for the column parameters, and derive the posterior distributions of these using a standard Bayesian prior-posterior analysis. The result of this is a distribution which looks similar to the negative binomial model for the chain-ladder technique, but which is recursive in i instead of j: C o. ICe.j, C2.j... C~_~j, x, ~0 ~ over-dispersed negative binomial, with mean i-i (r,-1) y m=l -I - Comparing this to the mean of the chain-ladder model, (2: -l)di.,_, : (2j -1)~4 Cj.,, m=l it can be seen that they are identical in form, with the recursion either being across the rows, or down the columns. In the context of the Bornhuetter-Ferguson method, we now have the stochastic version of this model. The Bornhuetter-Ferguson method inserts values for the expected ultimate claims in each row, x/, in the form of the values, M~. In the Bayesian context, prior distributions will be defined for the parameters x/, as discussed above. However, the model has been reparameterised, with a new set of parameters, yg. Hence, it is necessary to define the relationship between the new parameters, Yi, and the original parameters, x~. This is given in the equation below, which can be used to find values of y; from the values of xi given in the prior distributions. k=n-i+2 )'i = CI",_, + 1 U r, I The Bornhuetter-Ferguson technique can be reproduced by using strong prior information for the row parameters, x, and the chain-ladder technique can be reproduced by using improper priors for the row parameters. In other words, the Bornhuetter-Ferguson technique assumes that we are completely sure about the values of the row parameters, and their prior distributions have very small variances, while the chain-ladder technique assumes there is no information and has very large variances. This has now defined a stochastic version of the Borrthuetter-Ferguson technique. Since the column parameters (the development factors) are dealt with first, using improper prior distributions, their estimates will be those implied by the chain-ladder 296

15 technique. Prior information can be defined in terms of distributions for the parameters xi, which can then be converted into values for the parameters y~, and this is implemented in section Implementation This section explains how the Bayesian models can be implemented, using the software package winbugs (Spiegelhalter et al, 1996) which is available from uk/bugs. The programs used in these illustrations are contained in the Appendix. The data set we consider in this section is taken from Taylor and Ashe (1983), and has also been used in a number of previous papers on stochastic reserving. The incremental claims data is given in table 1, together with the chain-ladder results for comparison purposes. Table 1. Data from Taylor and Ashe (1983) with the chain-ladder estimates 357, , , , , , , , ,894 1, , , ,507 1,001, ,016, , , , , ,189 1,562, , , , , , , , , , , , , , ,631 1,131, , ,061,648 1,443, , , , , , , ,286 67,948 Chain-ladder development factors: Chain-ladder reserve estimates: 2 94, , ,419, , ,920, , ,625,811 Overall 18,680,

18 Table 2. Chain-ladder results. The Prediction Error is equal to the Bayesian Standard Deviation Chainladder Bayesian Bayesian Prediction Reserve Mean Standard Error Deviation (%) Year 2 94,634 94, , % Year3 469, , ,400 47% Year 4 709, , ,600 37% Year 5 984, , ,100 31% Year6 1,419,459 1,424, ,700 26% Year 7 2,177,641 2,186, ,200 23% Year 8 3,920,301 3,935, ,000 20% Year9 4,278,972 4,315,000 1,068,000 25% Year 10 4,625,811 4,671,000 2,013,000 43% Overall 18,680,856 18,800,000 2,975,000 16% However, the results certainly confirm that we can reproduce the chain-ladder results, and produce the prediction errors. It is also possible to obtain other information about the model from winbugs. For example, it is possible to produce full predictive distributions, using "density" in the Sample Monitor Tool window. We have now described one implementation of a Bayesian model using winbugs. In the rest of this section, we consider the Bayesian models described in sections 4 and 5, in order to consider how expert opinion can be incorporated into the predictive distribution of reserves. In each case, the programs are available in the Appendix, and the results can be reproduced using steps 3 to 13, above. It should be noted that this is a simulation-based program, so that the results obtained may not match exactly the results given below. However, there should be no significant differences, with the differences that there are being due to simulation error. 6.2 Intervention in the chain-ladder technique We now consider using a prior distribution to intervene in some of the parameters of the chain-ladder model, instead of using prior distributions with large variances which just reproduce the chain-ladder estimates. The implementation is set up in section (ii) of the Appendix, and the program can be cut and pasted into winbugs and run following steps 3 onwards, above. We consider 2 cases, as discussed in section 4. For the first ease, we assume that there is information that implies that the second development factor (from column 2 to column 3) should be given the value 1.5, for rows 7, 8, 9,and 10, and that there is no indication that the other parameters should be treated differently from the standard chain-ladder technique. In order to implement this, the parameter for the second development factor for rows 7-10 is given a prior distribution with mean 1.5. We then look at two different choices for the prior variance for this parameter. Using a large variance means that the parameter is estimated separately from the other rows, but using the data without letting the prior mean influence it too greatly. We then use a 300

19 standard deviation of 0.1 for the prior distribution, so that the prior mean has a greater influence. We consider first the estimate of the second development factor. The chain ladder estimate is and the individual development factors for the triangle are shown in table 3. The rows for the second development factor that are modelled separately are shown in italics. The estimate using the Bayesian models is 1.68 for rows 1-6. When a large variance is used for the prior distribution of the development factor for rows 7-10, the estimate using the Bayesian model is With the smaller variance for this prior distribution, the estimate is 1.673, and has been drawn down towards the prior mean of 1.5. This clearly shows how the prior distributions can be used to influence the parameter estimates. Table 3. Individual development factors The effect on the reserve estimates is shown in table 4, which compares the reserves and prediction errors for the two cases outlined above with the results for the chainladder model (which could be produced using the program in 6.1 on this set of data). The chain-ladder figures are slightly different from those given in table 2 because this is a simulation method. Table 4. Reserves and prediction errors for the chain-ladder and Bayesian models Chain-ladder Large variance Small variance Reserve Prediction Reserve Prediction Reserve Prediction Error (%) Error (%) Error (%) Year2 97, % 95, % 95, % Year 3 471,200 46% 475,700 46% 470,500 47% Year4 711,100 38% 721,700 37% 714,400 37% Year 5 989,200 31% 996,800 31% 994,700 31% Year 6 1,424,000 27% 1,429,000 26% 1,428,000 27% Year 7 2,187,000 23% 2,196,000 23% 2,185,000 23% Year 8 3,930,000 20% 3,937,000 20% 3,932,000 20% Year 9 4,307,000 24% 4,998,000 27% 4,044,000 25% Year 10 4,674,000 43% 5,337,000 44% 4,496,000 43% Overall 18,790,000 16% 20,190,000 17% 18,360,000 16% 301

20 It is interesting to note that, in this case, the intervention has not had a marked effect on the prediction errors (in percentage terms). However, the prediction errors themselves have changed considerably, and this indicates that it is important to think of the prediction error as a percentage of the prediction. Other prior distributions could have a greater effect on the percentage prediction error. The second case we consider is when we use only the most recent data for the estimation of each development factor. For the last 3 development factors, all the data is used because there is no more than 3 years for each. For the other development factors, only the 3 most recent years are used. The estimates of the development factors are shown in table 5. The estimates of the first development factor are not affected by the change in the model (the small differences could be due to simulation error or the changes elsewhere). For the other development factors, the estimates can be seen to be affected by the model assumptions. Table 5. Development factors using 3 most recent years data separately Earlier rows Recent rows All rows The effect of using only the latest 3 years in the estimation of the development factors in the forecasting of outstanding claims can be seen in table 6. Table 6 Reserve estimates using 3 most recent years data Chain-ladder Bayesian Model.Reserve Prediction Reserve Prediction Error (%) Error (%) Year 2 97, % 94, % Year 3 471,200 46% 469,300 46% Year 4 711,100 38% 712,900 37% Year 5 989,200 31% 1,042,000 30% Year 6 1,424,000 27% 1,393,000 27% Year 7 2,187,000 23% 2,058,000 24% Year 8 3,930,000 20% 3,468,000 22% Year 9 4,307,000 24% 4,230,000 27% Year 10 4,674,000 43% 4,711,000 47% Overall 18,790,000 16% 18,180,000 18% 302

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