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  • 1.
    Lindholm, Mathias
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lindskog, Filip
    Stockholm University, Faculty of Science, Department of Mathematics.
    Wahl, Felix
    Stockholm University, Faculty of Science, Department of Mathematics.
    Estimation of conditional mean squared error of prediction for claims reserving2019In: Annals of Actuarial Science, ISSN 1748-4995, E-ISSN 1748-5002Article in journal (Refereed)
    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.

  • 2.
    Lindholm, Mathias
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lindskog, Filip
    Stockholm University, Faculty of Science, Department of Mathematics.
    Wahl, Felix
    Stockholm University, Faculty of Science, Department of Mathematics.
    Valuation of Non-Life Liabilities from Claims Triangles2017In: Risks, ISSN 1670-0139, E-ISSN 2227-9091, Vol. 53, no 3, article id 39Article in journal (Refereed)
    Abstract [en]

    This paper provides a complete program for the valuation of aggregate non-life insurance liability cash flows based on claims triangle data. The valuation is fully consistent with the principle of valuation by considering the costs associated with a transfer of the liability to a so-called reference undertaking subject to capital requirements throughout the runoff of the liability cash flow. The valuation program includes complete details on parameter estimation, bias correction and conservative estimation of the value of the liability under partial information. The latter is based on a new approach to the estimation of mean squared error of claims reserve prediction.

  • 3.
    Lindholm, Mathias
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Wahl, Felix
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the variance parameter estimator in general linear modelsManuscript (preprint) (Other academic)
    Abstract [en]

    In the present note we consider general linear models where the covariates may be both random and non-random, and where the only restrictions on the error terms are that they are independent and have finite fourth moments. For this class of models we analyse the variance parameter estimator. In particular we obtain finite sample size bounds for the variance of the variance parameter estimator which are independent of covariate information regardless of whether the covariates are random or not. For the case with random covariates this immediately yields bounds on the unconditional variance of the variance estimator  a situation which in general is analytically intractable. The situation with random covariates is illustrated in an example where a certain vector autoregressive model which appears naturally within the area of insurance mathematics is analysed. Further, the obtained bounds are sharp in the sense that both the lower and upper bound will converge to the same asymptotic limit when scaled with the sample size. By using the derived bounds it is simple to show convergence in mean square of the variance parameter estimator for both random and non-random covariates. Moreover, the derivation of the bounds for the above general linear model is based on a lemma which applies in greater generality. This is illustrated by applying the used techniques to a class of mixed effects models.

  • 4. Remes, Kari
    et al.
    Anttila, Pekka
    Silvennoinen, Raija
    Putkonen, Mervi
    Ollikainen, Hanna
    Terävä, Venla
    Sinisalo, Marjatta
    Kananen, Kristiina
    Schain, Frida
    Castren-Kortegangas, Päivi
    Järvinen, Tiina M.
    Pisini, Marta
    Wahl, Felix
    Stockholm University, Faculty of Science, Department of Mathematics.
    Dixon, Tricia
    Leval, Amy
    Real-world treatment outcomes in multiple myeloma: Multicenter registry results from Finland 2009-20132018In: PLoS ONE, ISSN 1932-6203, E-ISSN 1932-6203, Vol. 13, no 12, article id e0208507Article in journal (Refereed)
    Abstract [en]

    Outcomes for patients with multiple myeloma (MM) have improved with the advent of novel therapies, however, real-world evidence of outcomes in clinical practice is scarce. We conducted a multi-center registry study to build a reliable picture of treatment and patient outcomes in Finland. The aim of this study was also to understand any methodological challenges in assessing treatment outcomes using disease registry data. Methods: We carried out a retrospective, observational study using data from the national Finnish Hematology Registry (FHR) to provide real-world evidence of outcomes for all adult patients diagnosed with and treated for MM between 2009-2013 at one of the six regional hospitals, with at least six months of recorded follow-up. Patients were identified within the FHR by applying eligibility criteria of a diagnosis of MM and verifiable records of medical treatment and lines of treatment during the study period. Patients receiving allogenic stem cell transplantation were excluded from the cohort, as were individuals who only had monoclonal gammopathy of undetermined significance diagnosis and patients who had not initiated treatment during this period. Kaplan Meier curves were used to calculate overall survival and time to next treatment. Stratification was carried out by drug status (conventional/novel) and by autologous stem cell transplant (ASCT) status. Results: A total of 321 patients met the inclusion criteria and were included in this study. Overall survival (OS) was longest in patients who received first-line novel therapy and ASCT (median not reached during 60-month follow-up) versus 46.2 months for novel first-line therapy without ASCT and 25.6 months for first-line conventional therapy without ASCT. Similarly, median time to next treatment were 33.9 months, 12.6 months and 7.8 months, respectively. Conclusions: The adoption of novel treatments in MM in Finland has had substantial impact on patient outcomes. Given the reality of complex treatment combinations for MM and relatively low patient numbers, assessing individual treatment effectiveness will require substantial cohort sizes and advanced, collaborative analytics on an international scale.

  • 5.
    Wahl, Felix
    Stockholm University, Faculty of Science, Department of Mathematics.
    Explicit moments for a class of micro-models in non-life insurance2019In: Insurance, Mathematics & Economics, ISSN 0167-6687, E-ISSN 1873-5959, Vol. 89, p. 140-156Article in journal (Refereed)
    Abstract [en]

    This paper considers properties of the micro-model analysed in Antonio and Plat (2014). The main results are analytical expressions for the moments of the outstanding claims payments subdivided into IBNR claims and individual RBNS claims. These moments are possible to compute explicitly using the discretisation scheme for estimation and simulation used in Antonio and Plat (2014) since the expressions then do not involve any integrals that, typically, would require numerical solutions. Other aspects of the model that are investigated are properties of the maximum likelihood estimators of the model parameters, such as bias and consistency, and a way of computing prediction uncertainty in terms of the mean squared error of prediction that does not require simulations. Moreover, a brief discussion is given on how to compute moments or risk-measures of the claims development result (CDR) using simulations, which based on the results of the present paper can be done without any nested simulations. Based on this it is straightforward to compute the one-year Solvency Capital Requirement, which corresponds to the 99.5% Value-at-Risk of the one-year CDR. A brief numerical illustration is used to show the theoretical performance of the maximum likelihood estimators of the parameters in the claims development process under this model using a realistic set-up based on the case-study of Antonio and Plat (2014). Additionally, the paper ends with a short numerical illustration discussing the model's robustness under violations of an independence assumption.

  • 6.
    Wahl, Felix
    Stockholm University, Faculty of Science, Department of Mathematics.
    Micro-level claims reserving in non-life insurance2019Doctoral 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.

  • 7.
    Wahl, Felix
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lindholm, Mathias
    Stockholm University, Faculty of Science, Department of Mathematics.
    Verrall, Richard
    The collective reserving model2019In: Insurance, Mathematics & Economics, ISSN 0167-6687, E-ISSN 1873-5959, Vol. 87, p. 34-50Article in journal (Refereed)
    Abstract [en]

    This paper sets out a model for analysing claims development data, which we call the collective reserving model (CRM). The model is defined on the individual claim level and it produces separate IBNR and RBNS reserve estimators at the collective level without using any approximations. The CRM is based on ideas from a paper by Verrall, Nielsen and Jessen (VNJ) from 2010 in which a model is proposed that relies on a claim giving rise to a single payment. This is generalised by the CRM to the case of multiple payments per claim. All predictors of outstanding claims payments for the VNJ model are shown to hold for this new model. Moreover, the quasi-Poisson GLM estimation framework will be applicable as well, but without using an approximation. Furthermore, analytical expressions for the variance of the total outstanding claims payments are given, with a subdivision on IBNR and RBNS claims. To quantify the effect of allowing only one payment per claim, the model is related and compared to the VNJ model, in particular by looking at variance inequalities. The double chain ladder (DCL) method is discussed as an estimation method for this new model and it is shown that both the GLM- and DCL-based estimators are consistent in terms of an exposure measure. Lastly, both of these methods are shown to asymptotically reproduce the regular chain ladder reserve estimator when restricting predictions to the lower right triangle without the tail, motivating the chain ladder technique as a large-exposure approximation of this model. 

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