The Chairman (Mr A. J. Newman, F.I.A.): A very warm welcome to you all for the discussion on the link between classical reserving and granular reserving through double chain ladder (DCL) and its extensions.
The presenter is Professor Jens Nielsen. Professor Nielsen has worked as an appointed actuary and led various product development departments specialising in research and development. He became research director of RSA, with responsibilities in life as well as non-life, in 1999. From 2006 until 2012 he worked as an entrepreneur. He is still co-owner and board member of the Copenhagen-based ScienceFirst. He is co-author of over 100 scientific papers in journals on actuarial science, economics, econometrics and statistics. He is the author of a book on quantitative operational risk modelling and is currently professor of actuarial science at Cass Business School at City University.
Prof J. P. Nielsen (affiliate member): We are going to talk about reserving, which is probably one of the most important responsibilities of actuaries.
We will talk about continuous and aggregated claims reserving. We are trying to make the point that continuous, also called granular, reserving is not so different from aggregated reserving as some might think. There have been quite a few actuarial meetings lately where some have advocated that we adopt continuous granular approaches towards modelling. Aggregating discrete data does not work because you lose information. Continuous, however, will still be doing many of the same types of thing. We are fooling ourselves if we think we should start completely from afresh. We have to remember what we know. This is why I am trying to make the link today between granular and aggregated methods.
The baby in Figure 1 represents the actuaries actually doing claims reserving. Those actuaries have a lot of knowledge, which is not terribly well articulated. It is very hard to go to practical actuaries and elicit their wisdom in a short conversation. What they possess and practice could be called tacit knowledge. One has to work with practitioner actuaries for a while to learn their knowledge.
Scene 1
You should not behave like the evil man you can see in Figure 2. You should not start developing all kinds of methods ignoring the knowledge that is already there, even if it is unarticulated. If you do so, you are going to throw the baby out with the bathwater, and risk developing methods unrelated to the real issues.
Scene 2
A major issue is whether we have to make a major change from aggregate to continuous methods.
I have plotted standard reserving data on a triangle in Figure 3. It is not clear whether this is aggregate or continuous. This is paid data. It looks like a histogram. In a certain part of the literature, a histogram is a continuous non-parametric estimator of some shape.
Aggregate or continuous?
However, we can also call it aggregated data because it is obviously discrete. Actually, it is both aggregated and continuous depending on how you look at it. It turns out that the chain ladder (CL) is a projection of this aggregated data down on a multiplicative sub-space. This sounds complicated but it just means that when you take the closest multiplicative model to your paid data that is the CL. You project your data and you obtain a multiplicative model, which gives exactly the normal CL.
Perhaps, we can use the standard approaches with CLs in the granular data space? The CL is a structured histogram. We know from statistics that a histogram is a non-parametric estimate of something but a very inefficient estimate. You can do better than histograms.
The simplest and most natural extension is kernel smoothing. If you look at the graphs in Figure 4, to the left you have a local linear unstructured kernel density of something which is still on the well-known triangle. On the right, in the first graph, you have a projection down of the multiplicative space. We are doing exactly the same as the CL but now it is continuous. In the second graph we have used multiplicative bias correction. One of my first papers from 1994 introduced multiplicative bias correction to mathematical statistics. We should probably add some bias correction as we know that the worst shapes you could ever think of without bias correction are the shapes that are found in reserving.
What should we do? MBC, multiplicative bias correction
So you can do the various estimators shown on the left and then you can project down and we are actually very close to the CL. Everything we are doing can be related back to the classical CL.
On the right of Figure 5 are three different projections. The dots are the familiar CL. The two other projections are versions of non-parametric kernel smoothers that have been projected down. They are of course very similar. One could also have smoothed the dots, which is called the method of Sieves. The method of Sieves is not an efficient method here, so we do not adopt that approach.
What is chain ladder really modelling? CLM, chain ladder method; MBC, multiplicative bias correction
The CL is really modelling time effects. It is modelling effects in both the underwriting and the development direction. We have to be careful to remember those time effects when we use granular approaches. Many of the studies that I have seen on granular data forget what we knew from the CL. That is not what we really want. We want to add to what we already know, not just do something completely different. Therefore, remember about the continuous CL when going granular.
It is possible to show that the CL works in very general conditions. However, if you build very complex systems, then you get bias for every single step involved. That is an underlying law of mathematical statistics. One has to be careful when building very big systems that one is not just adding systematic error and being more wrong than even the simplest models. Clearly, when we have all this data, this big data revolution, and continuous data, we should make sure we do better and not worse than before.
This introduction is comparing the continuous and the aggregate. Sometimes, when I have given talks about the DCL, people came to me and said: “I think granular data is more important”. So, I have had to make this introduction just to make sure that it is understood that the insight we obtain from the aggregated data is also important when you go granular.
I will now tell you about the DCL, the Bornhuetter–Ferguson and what I think is the important thing: the DCL reproduces the CL results. Professor Richard Verrall, Dr Maria Dolores Martínez-Miranda and I had to do three Astin Bulletin papers before we confirmed this position. Finally, in 2012, we were able to obtain a vigorous statistical model exactly reproducing the numbers that we obtain from the CL from a system of two triangles. Then we had responses like “Why would you do that? Why would you spend so much of your own time, and that of other people, just to reproduce what we already have?”. The answer is that a proper mathematical model of the CL allows us to understand what is behind the CL so we know how to change it. The problem with the CL, as we have it today, is that there is so much going on underneath it is not understood by people. Therefore, they cannot change it as confidently as they would like.
The big goal I see for the near future is to understand how the different CLs interact and how to generalise them, particularly in granular situations. We can do it incrementally from where we are today into the new world and not the other way round: starting with the new world and then hoping we return to where we are today.
When you work with incurred data, I call it prior knowledge. In my view, given it is somehow created by people in the office it is not data in the strict sense, but it is really expert or prior knowledge. One cannot take incurred data and say: “Here we have some data” and then put it in a model, process it and obtain some numbers. Instead one has to take into account what incurred data really represents. It is an estimate, made by people, of the future. It is not data as such. So, the big issue is how do you make paid data, that is, real data, work together with incurred data. When I was working as a practicing actuary there were interesting discussions when paid data and incurred data gave a very different number. The difference was very important for the financial well-being of the company. Actually, the discussion that we had was not very intelligent. It was something on the lines of one person saying “I believe in incurred data” and someone else responding “I believe in paid data”. I am sure probably most people in the room have been part of such a discussion.
The challenge is to use incurred data and paid data in the same model, on the same scale, so they can be compared in a valid way with statistically recorded validation. We have been working on this aspect and we have a solution to it in this Sessional paper.
We have worked for many years with RSA, and every year we did a paper. The work was both practical and academic at the same time. Every year we hired three interns. We put them into RSA and they tried out the latest version of the method and came back and told us it was not working. We changed it and wrote another paper. All the papers about the CL approaches actually came about after initiatives from practitioners. It was not the other way round. We followed this approach for many years. Now we have made a lot of progress and are publishing in actuarial journals, operational research journals and mathematical statistical journals.
All of you know the claims-reserving exercise. Accident reports come in and you try to estimate claims incurred but not reported (IBNR) and claims reported but not settled (RBNS). We want to understand the size of the claims and their timing and distribution. We also want to understand the cash flows.
I am not going to go into the formulae for the DCL today. There are formulae in the papers and there also were presentations at GIRO, which included the formulae. I am going to give you the highlights.
In broad terms, the DCL is a statistical model that breaks down the classical CL into its components. You have two triangles. You have one for the counts and one for the paid claims. You run the CL twice and then, via some mathematics, you obtain the whole system. In a sense practicing actuaries can do twice what they always did and obtain a full mathematical statistical system.
When you have a full mathematical system, everything becomes easy. You can easily do RBNS and IBNR. You do not become confused about bootstrapping. People have complained about some problems with the bootstrap we are using at the moment, but if you have a full mathematical system of the underlying methodology, you do not have problems with the bootstrap.
In Figure 6 you see the paid and counts data. One of the reasons we were working with only two triangles is that data collection should be really easy. I was not trying to say “Come on, you should give us all our data”. It is very hard even to collect counts data. Academic actuaries are talking about big data and all kinds of complicated things like Monte Carlo methods and Bayesian algorithms. I am talking about collecting counts data, which is a very simple extension of what we normally do. Even this, however, is non-trivial in practice. It often goes wrong, the percentage of times it goes wrong is in double digits. It is important that we are careful when collecting our data. When that is said, this is the kind of data that we need. And it should be within reach.
The modelled data: two run-off triangles
In Figure 7 you see the system. You have counts with some parameters, some αs and some βs, and then you have paid with some αs and βs. Then you have delays. You have a severity mean, which some people would call an average cost per claim. The γ is the severity inflation in the underwriting year direction. It turns out, when you look at the normal CL method in detail, almost all of the difficulties come from changes in the severity inflation in the underwriting year direction. This is also the reason for using Bornhuetter–Ferguson and similar methods to improve and smooth estimates.
The double chain ladder model (a)
It is not all the γ that cause us problems. It is just the most recent ones. Those last few are the critical ones. We can identify this via the DCL method because we can identify all the parameters.
It turns out that the CL method still produces good reserves under much more complex systems that we cannot identify and that we cannot estimate using only counts data and paid data. We can show the CL method makes sense under very general conditions. Almost the only thing that you cannot do with the CL method is crossing. As long as factors including all kinds of dependencies and correlations are going in only one direction everything is fine. But they cannot cross. That is why, if you go to a practical actuarial workshop, it is always something that is crossing that the actuaries take up as their problem, such as calendar effects.
This generality is hard to replicate if you build the model from scratch. The CL is quite clever in some sense.
What you see in Figure 8 is the severity inflation. The paper presents different methods having different severity inflations. It turns out that the last few severity inflations are really the ones which are key. The payment triangle is based on the last few numbers. I think most of you who have tried reserving will know that in this particular data set we could do a CL and 85% of the results are based on the last few numbers: it is only a few numbers that are critical.
The double chain ladder model (b)
The major drawback of the classical CL method is that the last two or three severity inflation numbers are volatile. It is quite some insight into a CL model. It is because of this effect that we use the Bornhuetter–Ferguson method.
In DCL and Bornhuetter–Ferguson we are mixing not two triangles but three triangles, as in Figure 9. We are also using incurred data. We estimate the DCL using the incurred triangle instead of the paid triangle. Our theoretical definition of incurred data is that it is unbiased when producing case estimates of the future. Under that assumption, incurred data is estimating exactly the severity inflation via the DCL as the DCL itself was estimating. By running DCL on incurred instead of on paid, we obtain another version of the severity inflation, but this time from incurred data which is much more stable. Our assumption is that the claims estimates are exactly unbiased, conditional on what we know. We all know that is not always true.
Double chain ladder and Bornhuetter–Ferguson and reported but not settled preserving double chain ladder
The graph of the two different estimates of severity inflation looks more or less the same except for the last three numbers, where the estimate based on the incurred data is much lower. It turns out that the reserve using severity inflation based on incurred data is half as big as when we are doing paid CL basing the severity inflation on the paid data. So, the difference in the final reserve from changing the last three severity inflations is very large.
Some people have said you cannot call the approach DCL and Bornhuetter–Ferguson because this is not what Bornhuetter–Ferguson did. I have read the Bornhuetter–Ferguson paper and it is not clear exactly what they meant. We are thinking in the spirit of Bornhuetter–Ferguson in that what they wanted to do in their original paper was to use prior knowledge in the system. This is also what we are doing here. Our label of the method is a tribute to Bornhuetter–Ferguson, we are trying to transfer their ideas into a modern setting.
You can also do other things to use prior knowledge in the system. If you look at the incurred triangle and try to define what it is, you can also find the RBNSs. It is rather strange that when we do reserving, we tend not to look too much at the RBNSs. Sometimes we do not look at them at all. It is a big decision to change the RBNSs when reserving. Often RBNSs might be 85% of the reserve. If they are changed it should be done consciously, for clear reasons and by a documented amount. There will be a claim department setting these RBNSs professionally, if the reserving department thinks the numbers done by the claims department are say 30% wrong, then the departments should at least have a conversation. If appropriate, the reserving department should say that they have changed the RBNSs because they do not have confidence in what the claims department is doing.
You can do many things with the DCL. One can do an RBNS preserving version of it, where the end result relating to the RBNSs is the one with which you started. The way you do this is to put the RBNSs into the model then forward them into the future as a cash flow. Subsequently, you reach them again and you obtain something looking like a DCL, but the end result gives you exactly the RBNSs with which you started. You obtain a different IBNR because you have different parameters. This is probably the approach you should take if you trust your RBNSs. If you do not trust your RBNSs, then it probably should be articulated. That is what we call the preserving double chain ladder (PDCL). This approach is not fully developed and we do not think that we have yet found its full potential.
That is one more method that I will talk about today: the incurred double chain ladder (IDCL). This is all about putting prior knowledge, such as incurred data and RBNS, into the system. IDCL runs the classical incurred CL and rephases the severity inflation into exactly the one reproducing the incurred reserve. This gives a model of the paid CL and a model of the incurred CL on the same scale. We also have a model of an RBNS preserving CL.
We also created a new model compared with Bornhuetter–Ferguson. So, we have four different versions of the CL. They are all defined as proper statistical models within the same framework. This allows us to compare them and validate them against each other. Until now it has not been possible to validate paid versus incurred. They can also be validated against BDCL and PDCL. All these methods can be compared because they are defined on the same scale, so you can just cut-off diagonals and validate. It is the same way as you do validation today, but we have solved the problem of making sure that the numbers are comparable.
I want to highlight a new software package with which we are happy. The package is called Res-timator and it carries out all the DCL methods. You can obtain the tables, the numbers, the αs, the βs and the delay. Then you can look at the different methods with different severity inflations.
What you see in Figure 10 are IDCL, BDCL and DCL. The only thing that makes the three methods different is the severity inflation. You can see the blue one, which is the BDCL. The paid one is the green one that is jumping up at the end because it is too volatile. The red one is quite volatile because it is the man-made incurred. We tried quite a few data sets with RSA, maybe 40 data sets over the last few years, and it turns out that sometimes incurred does incredibly well, much better than paid, because paid is too volatile. Almost as often, paid does much better than incurred. BDCL, the blue one, is almost always sensible. So it seems that BDCL stabilises things and is a good compromise.
The kernel: incurred double chain ladder (IDCL) parameter estimation in Res-timator. DCL, double chain ladder; BDCL, double chain ladder and Bornhuetter–Ferguson
In Figure 11 there is an illustration that you can use DCL to split the CL into RBNS and IBNR so that you can discuss those numbers. All the models related to the DCL can do things like that. You can validate, split and bootstrap them. It is trivial because you can just run the existing model.
The best estimate: reported but not settled (RBNS)/incurred but not reported (IBNR) split using double chain ladder (DCL)
Figure 12 shows an example of validation. It is not clear how you should do validation because you could do calendar time, payments depending on development and underwriting year, or cut-off the total reserve of, say, the last 3 years. It depends on your objectives. Different measures of prediction sometimes give different answers.
Validation in Res-timator (1). DCL, double chain ladder; BDCL, double chain ladder and Bornhuetter–Ferguson
Figure 13 shows an example of the validation where you have the DCL, BDCL and IDCL.
Validation in Res-timator (2). DCL, double chain ladder; BDCL, double chain ladder and Bornhuetter–Ferguson; IDCL, incurred double chain ladder
The Chairman: We are looking forward to an interesting and stimulating debate and I would like to encourage as many members as possible to contribute.
Mr B. R. Raj, F.I.A.: Thank you for a very interesting discussion, you have brought new insight into the reserving methods with which we regularly work. I note, however, that the methods under discussion today utilise claims data only, in particular counts, paid and incurred data. Have you considered if the method can be extended to also incorporate exposure volumes, for example, premium volumes? The claims experience is often affected by experience in the exposure. For example, an increasing exposure volume could indicate that more policies are being written, which could in turn lead to more claims.
Prof Nielsen: I think that that is a very important and valid point. I am pretty sure we could do more or less the same thing inside the DCL, which would allow you to validate it, bootstrap it and things like that. We would need to discuss how you incorporate these volumes into the system at the moment and try to translate those ideas into a DCL vocabulary.
The Chairman: That seems like an avenue for research, which is what we always want.
Ms L. K. Showler, F.I.A.: When I do a lot of my reserving, I use Bornhuetter–Ferguson. With that there comes a prior estimate of the ultimates that I use. I suppose the way that you could introduce that is as a prior estimate of claim count, but I would not know what that is, and I do not think that many people would.
Prof Nielsen: Neither BDCL, IDCL nor PDCL use prior estimates of claims counts. BDCL and PDCL are, on the contrary, trying to extract the meaningful prior information from well-known incurred data.
Ms Showler: Say we are relatively new on a diagonal and we have only had three claims to date and the projection is showing another 14 to ultimate. But, in the past, you have seen 50 claims or 60 claims come through. Would you take 50 or 60 or would you take the 14 which is going to come through using the pure CL on the claim count?
Prof Nielsen: You are telling me you are doing real, man-made Bornhuetter–Ferguson types of projections because you do not yet have much real data. I guess, in those circumstances, that is what you have to do. New versions of DCL will include models of the calendar effect. In that case you would not need to cut-off and both the recent claims and the past 50–60 claims will be taken properly into account.
Mr J. P. Law (student): You are introducing slightly more parameters than the normal CL. Do you have any concerns about over-parameterisation, the amounts of data that you need or the size of the triangle to make it work?
Prof Nielsen: The DCL method reproduces the CL method with the same level of parameterisation. The extensions of DCL to include prior knowledge do, of course, stabilise the position. When it comes to the available amount of data, then the less data you have, the more you need to smooth. Remember, CL is a histogram. Smoothing in classical CL is typically expressed using months, quarters or years. With our structured histogram interpretation, the proper amount of smoothing can be chosen via validation. However, with very little data it would be better to use continuous CL, which has more optimal properties. So, with very little data, I would work with the continuous DCL method.
Mr Raj: You mentioned earlier that you had a large number of data sets with which to work in this analysis. Can you give us an understanding as to which classes of business you considered in this analysis and for which classes the DCL method worked particularly well? Also, are there any particular lines of business where you believe that this approach may not work so well?
Prof Nielsen: I do not have that data as it is all proprietary. A lot of the work was done in RSA. They did deal with all kinds of business. For example, they considered motor business with short developments. There was a bias towards doing personal accident, workers’ compensation and other tricky areas where they were particular interested in the answers. There was also a bias towards doing problematic data sets where there had been some calendar events in the data. Generally, they tended to give me the more challenging areas, although we also dealt with some simple ones.
Often we obtained similar results to those which they had already obtained. Our methods are, at the end of the day, not too different from what they are doing already. However, we obtained deeper insight into the results and, maybe, better ways of communicating them and talking about changing them.
We discovered one thing that was very interesting when we were running so many data sets: if there was a tendency that the paid was jumping up due to severity inflation then the Bornhuetter–Ferguson was very stable.
This led us into a new stream of research on what actuaries call operational time. That is the kind of learning that you obtain from running many data sets. If you see the same pattern across data sets then you get an idea of how to improve the method. Every time we run a data set we learn something. Then we run it again and then we learn again. It is probably a never-ending process.
But to return to your question, DCL and its extensions work in the same type of situations as CL. But BDCL and PDCL might work on data sets where CL is not working. However, DCL goes beyond the methods introduced here. It is easy to adjust DCL to different challenges, so if your data set is not fitting any of the methods given here, it might still be an easy exercise to adjust DCL to work also for that data set.
The Chairman: You mentioned that this is not just a non-life insurance problem and you are finding other avenues to which it can apply. Can you tell us a bit more?
Prof Nielsen: My own background was that I took my PhD in biostatistics at Berkeley in 1988. When you worked in San Francisco in 1988 and did biostatistics specialising in survival analysis, then you worked on the AIDS epidemic. If you look at the AIDS epidemic, then it is quite clear that the structure of the problem is the same as the structure of reserving.
Actually, there are many forecasting problems that are the same in biostatistics as in reserving. Maybe actuaries have something to offer to the biostatistics literature here because they have done a lot of papers on similar methods. However, the biostatisticians are driven via a classical academic process where they do not have the tacit knowledge that actuaries have gained from many years of working in the office with reserving.
When I talk about projecting down on a multiplicative space to obtain the continuous CL, then actuaries think it sounds technical. To high-level academic mathematical statisticians it is trivial. They almost would not write about that because it is so easy. To motivate them I said there is something very special about this reserving problem. It is that when you do a forecast in reserving everything parametric you estimate it from in-sample data. So, when you are forecasting, you are not extrapolating parameters or generating a time series as, for example, in the Lee–Carter model.
The CL approach almost lies between forecasting and goodness of fit. It is a method that is much more stable and reliable than forecasting using extrapolation and time series. A number of longevity models can run crazily in all kinds of directions because you are doing many things at the same time. That is real out-of-sample forecasting and is difficult and that can go anywhere.
On the other hand, we cannot only do goodness of fit because that relates to the past. The reserving problem is so special because it is a particular structure where you can forecast without extrapolation. For some reason this insight has never been published in statistics.
I went to the mathematical statisticians and I said we call this in-sample forecasting and we published it in the Annals of Statistics, which is the most technical and prestigious mathematical statistical journal in the world. The statisticians learned from the continuous CL and developed a new methodology in statistics, which is still known as in-sample forecasting.
I have big expectations that such co-operation with people such as top academics in statistics can give something back to reserving. I hope the interaction goes both ways with both parties learning from one another.
Mr M. G. White, F.I.A.: I would like to make a few observations from practical experience of looking at different portfolios. Your initial comments about outstanding claims, and therefore incurred data, not really being data do make great sense to me. What I always do when looking at a liability portfolio is to grasp the cumulative paids and then to compare the relative shape of those developments from one underwriting year to another. Then I do the same thing with the incurreds. Then I compare the two pictures and see how stable the development patterns are and which looks more plausible, the paid or the incurred, from underwriting year to underwriting year. What I tend to see is that companies lean more towards believing the story that the incurred data is telling than believing the story that the paid data is telling.
Whereas paid data, in terms of consistency of development shapes, tends, I find, to look much more consistent and much more persuasive. I have spent some time wondering why this is. My feeling as to the likely explanation is that the paid data responds more quickly to changes in ultimate claim costs. I think the case-reserving process, even assuming it is done in as unbiased a way as possible, is a lagging indicator. A good example there is the recent experience of the UK motor market.
Quite separately, I would also support the idea that you mentioned just now. The kind of work that we do in non-life reserving can be useful elsewhere. For example, I have been talking to a company that sells a service. But that sales process has many elements to it that unfold over time. First, you have to persuade the customer to install your product, and doing that involves overcoming all sorts of hurdles. Then once the product is installed, you make money each time the product is used. It is a simple enough problem and the actuarial approach is to divide the thing into stages, measure them and try to make sense of them. It is quite simple once you have explained it, but it comes more naturally to us.
Prof Nielsen: I agree that paid has this ability to pick up things immediately when they happen. It is very interesting that it is even more stable. I would like to understand that better. I hope you will be able to discuss it with me.
Mr White: I am happy to explain it in general terms. What I have described is dramatic and sometimes really persuasive.
Prof Nielsen: It is very interesting. I hope it can be incorporated in the DCL method as well, because the approach is still lacking visual methods, which bring better understanding to the whole approach.
Mr Law: Another method I have come across, which attempts to combine or make use of both paid and incurred data, is the Munich CL. Do you have an opinion on this method?
Prof Nielsen: I have studied the Munich CL but I do not really understand it. My first ever published paper was on something similar. It was in the early days of making predictions about AIDS. What people did in the very early days was regression analysis giving current status on various blood measures. They were running linear regressions into the future giving them fake events of onset of AIDS, because they did not have any data. We showed, in a paper published in Biometrika in 1993, that there exists no probability space where this makes sense. It was my first paper. You should not make a career out of writing papers like that because it is not going to win you any friends. When I read about the Munich CL I had the same feeling, that is, that they were throwing some linear regression around. No one really checked whether it made sense.
Prof A. D. Wilkie, F.F.A., F.I.A.: I am not an expert on reserving for anything, but I have had some experience in analysing different sorts of data. You mentioned the forecasting of mortality rates. That is obviously a triangulation sort of problem. For the underwriting year starting 100 years ago you have a lot of data. For the underwriting year starting last year you have very little data.
If you think of year of birth instead of underwriting year, you are just counting numbers, not amounts and you have a very good development, and year after year is very similar. The CL structure is used rather differently for mortality rates, but you are essentially doing the same thing. You are smoothing by age from underwriting year and then trying to project the change in the CL structure over the future. That, as you say, is very difficult when you are trying to forecast. It is a very different sort of problem. The numbers are usually sufficiently large with populations. Even with assured life or pensioners’ data, the numbers are big. So there are no problems, and the amount of data does not come into it.
The other area where I have had a little experience is in the very large claims in shipping liability insurance, where the numbers are tiny in comparison and the amounts are huge. The distribution of amounts becomes critical. Whether one is doing it by paid amounts or by estimated amounts, the numbers of cases are not sufficient to try to estimate the future terribly well from smoothing the paids and so on, because each case is very different. To mention cases, the estimated costs of Costa Concordia (a ship which ran aground) clearly depended on quite a lot of different elements. There were some injuries, there were some deaths, there was some damage and there were some absolutely enormous salvage costs. The only people that could do some estimate of the salvage costs were the individual estimators, engineers and shipping people. No actuary or statistical method was going to help there.
There is another ship, the Rena, that was wrecked on a coral reef off New Zealand, which is sacred to the Maoris. The problem there is whether you need to do something about it. The costs of salvage would be enormous, but how much is it also necessary to do to preserve the coral and to satisfy the peoples’ religious feelings? Similar issues come into the Costa Concordia case. It was not possible to take the wreck out to sea and dump it because there were bodies in it. They had to be recovered. In a country like Italy, dumping the bodies would not have been acceptable.
Very often problems are binary in nature. One example is tobacco liability. Which way is the court going to go, one way or another?
With very big claims I would think you need to rely more on the individual estimates from experts and recognise where there is a court case or something similar involved, it could go either way.
I always feel that the CL system works nicely when you have very large numbers of cases, as in motor claims. The amounts of individual motor claims are not running into hundreds of millions of pounds. Shipping claims are of that magnitude as are those for earthquakes and other similar cases.
Prof Nielsen: I agree with both points. Clearly, the longevity problem is the same problem. The issue is that there are two different academic cultures and histories involved. When you look at the reserving literature, it looks completely different from the longevity literature. It is not because the problems are different, it just so happened that different schools and different networks with different traditions studied the problem.
Triangles are not used in longevity but parallelograms. We actually did one such parallelogram in our recent forecasting paper coming out in IME (Mammen et al., Reference Mammen, Martínez-Miranda and Nielsen2015).
What I would like to see in longevity studies is the in-sample forecasting middle step. I am not against doing forecasts with a time series, for example, a Lee–Carter kind of time series forecasting. I would, however, like to see the forecast without the time series first, because the time series confuses me as it has so many numbers and is so volatile. I would like to do the investigation stepwise. Do the numbers first before having all these time series and extrapolations, because they do not really do much for my intuition. It is important to know what is going on with the long-term trend. One could, perhaps, investigate the long-term trend first and then add the volatility afterwards as we do in reserving. I think longevity could become more understandable, more intuitive, from such an approach.
I agree with the other point raised. In situations like those mentioned we need to take more dramatic action to solve the problem. I have worked on a book on quantitative operational risk estimation. Most of this book was about adding prior knowledge into statistics. You need to combine statistical models with a lot of insight. It is not really statistics, it is a combination of statistics and something else. This is the challenge, as I see it.
The Chairman: May I ask about validation? It is a big theme in actuarial work at the moment, with Solvency II coming along. Everything has to be formally validated in a manner which was not necessary in the past. This applies to both internal models and technical provisions, technical provisions obviously relating to this reserving question. All the validations that we have spoken about so far have been done by the practitioner who is coming up with the numbers. He or she takes off the last diagonal or some other set of points and sees what happens. How would you envisage validation being done by, and/or directed by, someone external to the process?
Prof Nielsen: Without access to the data?
The Chairman: Yes, that would be more conventional.
Prof Nielsen: I will say the validation presented here is what you can do with the kind of data we have at the moment. Obviously, if you have the original granular data, you can do more. You can split the data in two and validate one against the other and do all kinds of things.
Doing an external validation without access to data is a difficult issue. I will think about it.
Mr White: This is a response to what Professor Wilkie was discussing where you have a claims situation which is not the product of lots of little things happening in a pattern.
When I came to the London market, I was privileged to secure about half a day of David Craighead’sFootnote 1 time. He showed me through the numbers of some of his clients. He was reserving for things like reinsurance and retrocession accounts where there had been a catastrophe. He was looking at individual years and finding that they all developed differently from each other. He had ways of fitting formulae, which seemed to make sense and more or less worked. The key point was that years were not different. You had to have some insight as to how losses would flow round the market to make a guess as to how big they ultimately would become.
The Chairman: It remains for me to express my own thanks and the thanks of all of us to the speaker, Professor Nielsen, and also to all those who have participated in the evening’s discussion.
