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Estimation of conditional mean squared error of prediction for claims reserving
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0002-0775-9680
Stockholm University, Faculty of Science, Department of Mathematics.
2019 (English)In: Annals of Actuarial Science, ISSN 1748-4995, E-ISSN 1748-5002Article in journal (Refereed) Epub ahead of print
Abstract [en]

This paper studies estimation of the conditional mean squared error of prediction, conditional on what is known at the time of prediction. The particular problem considered is the assessment of actuarial reserving methods given data in the form of run-off triangles (trapezoids), where the use of prediction assessment based on out-of-sample performance is not an option. The prediction assessment principle advocated here can be viewed as a generalisation of Akaike’s final prediction error. A direct application of this simple principle in the setting of a data-generating process given in terms of a sequence of general linear models yields an estimator of the conditional mean squared error of prediction that can be computed explicitly for a wide range of models within this model class. Mack’s distribution-free chain ladder model and the corresponding estimator of the prediction error for the ultimate claim amount are shown to be a special case. It is demonstrated that the prediction assessment principle easily applies to quite different data-generating processes and results in estimators that have been studied in the literature.

Place, publisher, year, edition, pages
2019.
Keywords [en]
Mean squared error of prediction, reserving methods, ultimate claim amount, claims development result, chain ladder method
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-174298DOI: 10.1017/S174849951900006XOAI: oai:DiVA.org:su-174298DiVA, id: diva2:1358028
Available from: 2019-10-06 Created: 2019-10-06 Last updated: 2019-12-04
In thesis
1. Micro-level claims reserving in non-life insurance
Open this publication in new window or tab >>Micro-level claims reserving in non-life insurance
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Actuarial reserving deals with the problem of predicting outstanding claims payments on policies issued up to today to find an appropriate amount of capital, the claims reserve or technical provisions, to set aside in order to be able to meet obligations to policyholders. Historically, and commonly still today, this has been approached using purely algorithmic and deterministic methods, not based in any statistical models. This thesis contains five individual papers, mainly concerned with statistical models for use in the area of reserving in non-life insurance.

Paper I sets out all the components needed for the valuation of aggregate non-life insurance liability cash flows based on data in the form of claims triangles. The paper contains all necessary ingredients for use in practice, including the estimation of model parameters and a bias correction of the plug-in estimator of the valuation formula. The valuation framework that the paper takes as its starting point is compatible with the view of the Solvency IIdirective on how to compute the value of the technical provisions, i.e. that the value should equal the amount which a so-called reference undertaking would demand in order to take over and handle the run-off of the liability cash flow.

Paper II deals with the problem of estimating the conditional mean squared error of prediction(MSEP), conditional on the observed data. The paper presents an approach that yields analytically computable estimators for a wide range of different models — otherwise readily computable using simple numerical methods — and, moreover, it shows that the approach reproduces the famous MSEP formula for the distribution-free chain ladder model given by Mack in 1993. The approach is particularly useful when considering run-off triangles since itis then not feasible to perform a prediction assessment based on out-of-sample performance.

Paper III is concerned with properties of the variance of the variance parameter estimator ina general linear model, mainly in the form of finite sample size bounds that are independent of the covariates and that are such that, asymptotically, the lower and upper bounds are the same. As opposed to the other papers of this thesis, this paper is purely theoretical without an immediate insurance context — except for a small example.

Paper IV introduces a discrete-time micro-model called the collective reserving model (CRM). The model is highly accessible since, even though it is a micro-model, it is modelled on the aggregate level using two triangles, one for the number of reported claims and one for the claims payments. The paper shows, among other things, how the model gives predictors of outstanding claims payments separately for incurred but not reported and reported but not settled claims, and, interestingly, shows that the chain ladder technique is a large exposure (e.g. the number of contracts) approximation of the CRM.

Paper V is chiefly concerned with deriving closed-form expressions for moments in a class of continuous-time micro-models. It is the first paper to accomplish this task, hopefully making continuous-time micro-models accessible to a broader audience.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2019. p. 36
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-174300 (URN)978-91-7797-865-7 (ISBN)978-91-7797-866-4 (ISBN)
Public defence
2019-12-04, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 5: Manuscript.

Available from: 2019-11-11 Created: 2019-10-13 Last updated: 2019-10-30Bibliographically approved

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