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# Stochastic reserving based upon mathematical statistics

## Final Report Summary - RESERVING (Stochastic reserving based upon mathematical statistics)

The claims reserve can be the single most important item on the balance sheet of non-life (or general) insurance companies. To set adequate reserves aside for liabilities that are not fully known and to account for the uncertainty of this item are of great importance when considering the risks of insolvency and the capital requirements for non-life insurers. The aim of this project has been to develop a coherent methodology based on classical mathematical statistics for the important area of reserving and thereby transform it into a more normal field based on state-of-the-art mathematical statistics. Existing methods are either based on simple aggregated triangular arrays of data, or methods so complicated that they are neither well understood nor operational in an insurance company. The key difference with this project is that it incorporates a new methodological approach based on modern smoothing theory, which allows the use of the easily available data as well as individual claims data.

The objectives of this project were defined as follows:
1) To develop models based on reported numbers of claims and aggregated claim payments, which provide better insights into the underlying drivers of the claims data
2) To investigate the use of external data to stabilise the predictions of claims liabilities
3) To incorporate models which allow for claims inflation
4) To develop new bootstrap methodologies in order to obtain predictive distributions
5) To transfer all the above results into a continuous data setting based on structured smoothing.
These objectives have been addressed in two research streams with a series of academic papers as the expected results.

The starting point in reserving is the simple classical chain-ladder method (CLM) that is very widely used by companies for all business lines and in all countries. While it could not be claimed that this method does not work, because it is being used every day in most non-life insurance companies all over the world, it is clear that the method has some significant weaknesses. One problem is that by modelling aggregate data, it becomes difficult to define a genuine mathematical statistical model of the data generating mechanism. This project is built on the base of the recently developed double chain ladder (DCL) method by Martínez-Miranda, Nielsen and Verrall, published at the beginning of this project in the actuarial journal Astin Bulletin. DCL translates classical CLM to a mathematical statistical framework. It consists of a well-defined model of frequencies with a delay function until payment of claims, and the mean of the severities of these payments are multiplicatively structured in a natural way. Such a model allows us to explain what classical reserving is, when it can be appropriate and when it can break down. We have been able to suggest improved statistical tools that are far away from the common ad hoc approaches sometimes used in the traditional actuarial practice. In this discrete framework of classical reserving, where the input data consists of classical run-off triangles plus, possibly, expert knowledge, we have developed the first research stream, taking the double chain ladder paper as a base. We have been working in this line during these two years and we have continued afterwards. We have addressed the problem of including expert knowledge in the statistical model in a coherent way, including calendar effects, and validating the model and the estimation methods. Besides this, using the same spirit of the reserving methodologies, we have been able to deal with the problem of asbestos mortality forecasting. All these problems have been analysed using real data provided by a major insurer and, as a result of this work, we have written a total of eight papers with two more are in progress. To the date of this report, six of these papers have been already published in peer-reviewed actuarial journals, one is under review and the other one has been published in a practitioner magazine. We have participated in the key actuarial conferences, given seminars by invitation and held discussions with actuaries to explain our developments. All of this made possible a good dissemination of the results among academics and practitioners, who also have benefit of having available free software implementing our developments in this stream. It is a contributed package for the open-source software R named as DCL, which is totally free and was published in October 2013.

The second research stream focussed on reserving with granular data. While continuous data has previously been used for reserving in academia as well as in practical reserving, generally speaking the results have somewhat disappointing. Our understanding for this somewhat disappointing destiny of continuous reserving so far is that the methods have been too far away from current reserving methods to catch on among actuaries. In this project we have developed a novel approach on continuous reserving data having a natural transition from classical chain ladder to the more sophisticated kernel smoothing methods. We have shown that the classical CLM is in fact only a structured histogram on a triangle. This point of view has allowed us to develop what we call “continuous chain ladder” as a natural extension of the original (discrete) CLM. Continuous chain ladder has been published in an interdisciplinary peer-reviewed journal in 2013. When the classical CLM groups the data and develops a multiplicative histogram model, we just decide to follow the same approach without grouping the data. We use kernel smoothers, which are the natural improvement on histograms. Continuous chain ladder is actually just the same estimation technique as classical CLM, but without grouping the data. In contrast to other complicated approaches using granular data, our approach has the advantage of taking the actuarial estimation methodology and translating it into the standard universe of modern structured smoothing models, opening the entire toolbox of methods developed in recent years to our disposal. After publishing this reformulation of classical reserving, we have been working to improve the methodology as well as extending the multiplicative model to include the calendar year effect. In this work we have involved statisticians with expertise in the theory and the practice of smoothing theory and structured models. This has been possible through collaborations established via the mobility agenda defined in the project. Currently expert academics from London and Oxford in the UK, Mannheim and Konstanz in Germany, Korea, Geneva and Spain are working and contributing to the reserving problem, which this project has formulated as a well-defined forecasting problem. Now reserving is a appealing problem open to the statistical community. Moreover, we are at a point in which our reserving work is formulated as fundamental papers of mathematical statistics aiming at the top outlets of mathematical statistics. They are more general in scope than just reserving. Longevity and other forecasting models are included. However, reserving plays the role as the driving source of inspiration and the leading empirical illustration. To the date of this report, we have written six papers in this research stream, three have been already published in peer-reviewed journals and the other three are now under revision. Three more papers of this type are now in progress.

As we anticipated, the research results in this project have an immediate application in the industry. This statement is supported by the fact that our developments have been introduced into the internal system of a major insurer and been validated during these years. The project has provided explicit solutions to the perhaps single most important methodological issue of Solvency II: monitoring the risk of outstanding liabilities. The new Solvency II framework is about promoting higher quality risk management, working with the grain of industry developments, and ensuring that the assessment of regulatory capital is integrated with firms’ wider capital management processes. This project has confronted this tough issue head on by developing advance research in the field while making the new techniques available to practitioners.

As well as this, this project has defined a genuine mobility plan which extends to different and significant research, academic and professional environments. Several international connections and networks have arisen from this project, which are a great contribution to European excellence and European competitiveness. In fact, some of the best researchers, who are capable of dealing properly with many appealing problems in stochastic reserving, in several key places in Europe, are now connected by this project. A new generation of researchers is also being trained at the moment. Two PhD students are now working in the topics of this project under the supervision of the Fellow and the scientists supervising the project. Several postgraduate students (MSc in Actuarial Science, MSc in Actuarial Management, MSc in Applied Statistics etc.) have written their master theses about the project developments. A specialized course in stochastic reserving is being taught at Cass Business School with the collaboration of the Fellow, and through several seminars it has been possible to introduce the undergraduate students to the reserving problem and the statistical perspective of it promoted by this project.