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Stochastic claims reserving in non-life insurance: Bootstrap and smoothing models
Stockholm University, Faculty of Science, Department of Mathematics.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In practice there is a long tradition of actuaries calculating reserve estimates according to deterministic methods without explicit reference to a stochastic model. For instance, the chain-ladder was originally a deterministic reserving method. Moreover, the actuaries often make ad hoc adjustments of the methods, for example, smoothing of the chain-ladder development factors, in order to fit the data set under analysis.

However, stochastic models are needed in order to assess the variability of the claims reserve. The standard statistical approach would be to first specify a model, then find an estimate of the outstanding claims under that model, typically by maximum likelihood, and finally the model could be used to find the precision of the estimate. As a compromise between this approach and the actuary's way of working without reference to a model the object of the research area has often been to first construct a model and a method that produces the actuary's estimate and then use this model in order to assess the uncertainty of the estimate. A drawback of this approach is that the suggested models have been constructed to give a measure of the precision of the reserve estimate without the possibility of changing the estimate itself.

The starting point of this thesis is the inconsistency between the deterministic approaches used in practice and the stochastic ones suggested in the literature. On one hand, the purpose of Paper I is to develop a bootstrap technique which easily enables the actuary to use other development factor methods than the pure chain-ladder relying on as few model assumptions as possible. This bootstrap technique is then extended and applied to the separation method in Paper II. On the other hand, the purpose of Paper III is to create a stochastic framework which imitates the ad hoc deterministic smoothing of chain-ladder development factors which is frequently used in practice.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2011. , 48 p.
Keyword [en]
Bootstrap, Chain-ladder, Generalized linear model, Separation method, Smoothing, Stochastic claims reserving
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-55347ISBN: 978-91-7447-255-4OAI: oai:DiVA.org:su-55347DiVA: diva2:406884
Public defence
2011-06-10, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 10:00 (English)
Opponent
Supervisors
Available from: 2011-05-12 Created: 2011-03-10 Last updated: 2011-03-30Bibliographically approved
List of papers
1. Non-parametric and parametric bootstrap techniques for age-to-age development factor methods in stochastic claims reserving.
Open this publication in new window or tab >>Non-parametric and parametric bootstrap techniques for age-to-age development factor methods in stochastic claims reserving.
2009 (English)In: Scandinavian Actuarial Journal, ISSN 0346-1238, E-ISSN 1651-2030, no 4, 306-331 p.Article in journal (Refereed) Published
Abstract [en]

In the literature, one of the main objects of stochastic claims reserving is to find models underlying the chain-ladder method in order to analyze the variability of the outstanding claims, either analytically or by bootstrapping. In bootstrapping these models are used to find a full predictive distribution of the claims reserve, even though there is a long tradition of actuaries calculating the reserve estimate according to more complex algorithms than the chain-ladder, without explicit reference to an underlying model. In this paper we investigate existing bootstrap techniques and suggest two alternative bootstrap procedures, one non-parametric and one parametric, by which the predictive distribution of the claims reserve can be found for other age-to-age development factor methods than the chain-ladder, using some rather mild model assumptions. For illustration, the procedures are applied to three different development triangles.

Keyword
Bootstrap, Development factor method, Development triangle, Stochastic claims reserving
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-35100 (URN)10.1080/03461230903239738 (DOI)000283721900004 ()
Available from: 2010-01-14 Created: 2010-01-14 Last updated: 2011-03-29Bibliographically approved
2. Bootstrapping the separation method in claims reserving.
Open this publication in new window or tab >>Bootstrapping the separation method in claims reserving.
2010 (English)In: Astin Bulletin: Actuarial Studies in Non-Life Insurance, ISSN 0515-0361, Vol. 40, no 2, 845-869 p.Article in journal (Refereed) Published
Abstract [en]

The separation method was introduced by Verbeek (1972) in order to forecast numbers of excess claims and it was developed further by Taylor (1977) to be applicable to the average claim cost.The separation method differs from the chain-ladder in that when the chain-ladder only assumes claim proportionality between the development years, the separation method also separates the claim delay distribution from influences affecting the calendar years, e.g. inflation. Since the inflation contributes to the uncertainty in the estimate of the claims reserve it is important to consider its impact in the context of risk management, too.

In this paper we present a method for assessing the prediction error distribution of the separation method. To this end we introduce a parametric framework within the separation model which enables joint resampling of claim counts and claim amounts. As a result, the variability of Taylor's predicted reserves can be assessed by extending the parametric bootstrap techniques of Björkwall et al. (2009). The performance of the bootstrapped separation method and chain-ladder is compared for a real data set.

Keyword
Bootstrap, Chain-ladder, Development triangle, Inflation, Separation method, Stochastic claims reserving
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-35104 (URN)10.2143/AST.40.2.2061138 (DOI)
Available from: 2010-01-14 Created: 2010-01-14 Last updated: 2011-05-13Bibliographically approved
3. A generalized linear model with smoothing effects for claims reserving
Open this publication in new window or tab >>A generalized linear model with smoothing effects for claims reserving
2011 (English)In: Insurance, Mathematics & Economics, ISSN 0167-6687, E-ISSN 0617-6687, Vol. 49, no 1, 27-37 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we continue the development of the ideas introduced in England and Verrall (2001) by suggesting the use of a reparameterized version of the generalized linear model (GLM) which is frequently used in stochastic claims reserving. This model enables us to smooth the origin, development and calendar year parameters in a similar way as is often done in practice, but still keep the GLM structure. Specifically, we use this model structure in order to obtain reserve estimates and to systemize the model selection procedure that arises in the smoothing process. Moreover, we provide a bootstrap procedure to achieve a full predictive distribution.

Keyword
Bootstrap; Generalized linear model, Model selection, Smoothing, Stochastic claims reserving
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-52363 (URN)10.1016/j.insmatheco.2011.01.012 (DOI)000291838200004 ()
Available from: 2011-01-13 Created: 2011-01-13 Last updated: 2012-01-17Bibliographically approved

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