Title: Computational tools for IBNR reserves calculation Author: Bc. Štěpán Gregor Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Helga Krafferová, UNIQA pojišťovna, a.s. Supervisor' s e-mail address: Helga.Krafferova@uniqa.cz Abstract: Technical provisions represent the liability of the insurance company to its clients. In this work we focus on the computational tools of incurred but not reported claims. The most popular calculation method is probably the chain-ladder method which will be in detail analyzed in this thesis. Finally, by applying the theory in this work we will demonstrate the calculation in the own developed software. Keywords: Insurance, IBNR reserve, chain-ladder

Title: Munich chain ladder method Author: Bc. Veronika Janáková Department: Department of probability and mathematical statistic Supervisor: Mgr. Martin Pleška Supervisor's e-mail address: martin.pleska@cz.pwc.com Abstract: The insurance company is obliged to create technical reserves to cover the risk from insurance activities. One of the main reserves in non-life insurance is the IBNR reserve - the reserve to cover the damages that already exist, but that have not been reported to the insurance company yet. The content and the general aim of this thesis is to provide a better orientation in the method used in the estimation of the reserve IBNR - the Munich Chain Ladder Method. Munich Chain Ladder Method was created as a modification of the standard Chain Ladder calculation - one of the most used methods in the estimation of the IBNR reserve. The thesis focuses on the reasons that led to the modification of standard Chain Ladder calculation and explains the calculation of the modified method. It presents the application ot the method on the real data as well as the comparison with the results of the classical and less complicated Chain Ladder method. Keywords: IBNR reserve, chain ladder, Munich chain ladder 1

In this thesis we provide a construction of a micro-level reserving model for an economic disability insurance portfolio. The model is based on the mathematical framework developed by Norberg (1993). The data considered is provided by Trygg-Hansa. The micro model tracks the development of each individual claim throughout its lifetime. The model setup is straightforward and in line with the insurance contract for economic disability, with levels of disability categorized by 50%, 75% and 100%. Model parameters are estimated with the reported claim development data, up to the valuation time Τ. Using the estimated model parameters the development of RBNS and IBNR claims are simulated. The results of the simulations are presented on several levels and compared with Mack Chain-Ladder estimates. The distributions of end states and times to settlement from the simulations follow patterns that are representative of the reported data. The estimated ultimate of the micro model is considerably lower than the Mack Chain-ladder estimate. The difference can partly be explained by lower claim occurrence intensity for recent accident years, which is a consequence of the decreasing number of reported claims in data. Furthermore, the standard error of the micro model is lower than the standard error produced by Mack Chain-Ladder. However, no conclusion regarding accuracy of the two reserving models can be drawn. Finally, it is concluded that the opportunities of micro modelling are promising however complemented by some concerns regarding data and parameter estimations. / I detta examensarbete ges ett förslag på uppbyggnaden av en mikro-modell för reservsättning. Modellen är baserad på det matematiska ramverket utvecklat av Norberg (1993). Data som används är tillhandahållen av Trygg-Hansa och berör försäkringar kopplade till ekonomisk invaliditet. Mikro-modellen följer utvecklingen av varje enskild skada, från skadetillfälle till stängning. Modellen har en enkel struktur som följer försäkringsvillkoren för den aktuella portföljen, med tillstånd för invaliditetsgrader om 50%, 75% respektive 100%. Modellparametrarna är estimerade utifrån den historiska utvecklingen på skador, fram till och med utvärderingstillfället Τ. Med hjälp av de estimerade parametrarna simuleras den framtida utvecklingen av RBNS- och IBNR-skador. Resultat av simuleringarna presenteras på era nivåer och jämförs med Mack Chain-Ladder estimatet. Den simulerade fördelningen av sluttillstånd och tid mellan rapportering och stängning, följer mönster som stöds av rapporterade data. Den estimerade slutkostnaden från mikro-modellen är betydlig lägre än motsvarande från Mack Chain-Ladder. Skillnaden kan delvis förklaras av en låg skadeintensitet för de senaste skadeåren, vilket är en konsekvens av färre rapporterade skador i data. Vidare så är standardfelet lägre för simuleringarna från mikro-modellen jämfört med standardfelet för Mack Chain-Ladder. Däremot kan inga slutsatser angående reservsättningsmetodernas precision dras. Slutligen, framförs möjligheterna för mikro-modellering som intressanta, kompletterat med några svårigheter gällande datautbud och parameterestimering.

Ke krytí rizik z pojištovací ?innosti je pojištovna povinna vytváret technické rezervy. V neživotním pojišt?ní je nejd?ležit?jší rezervou rezerva na pojistná pln?ní, zahrnující rezervy typu RBNS a IBNR. K odhadu výše t?chto rezerv se používají r?zné matematicko-statistické metody vycházející z historických pozorování pr?b?hu škod. Tato práce prezentuje zp?sob ur?ení pot?ebné výše rezervy v pojišt?ní odpov?dnosti z provozu motorových vozidel na základ? stochastického modelování odhadnutých o?ekávaných po?t? pojistných událostí a o?ekávané výše t?chto událostí.

This diploma thesis deals with technical reserves in non-life insurance, in particular with provisions for future claim payments for damages that have occurred, but has not yet been reported to the insurance company. This type of provision is known by the acronym IBNR. After the introductory section containing a general introduction to the issue of claims reserving in non-life insurance different approaches to modeling of IBNR reserves are briefly presented. Subsequently, full attention is given to Chain-ladder method, which is most frequently used in the actuarial practise for the purpose of claims reserving. This method is then presented progressively from its simplest form of a simple computing algorithm followed by Mack's stochastic model to the last theoretical part of this part describing extended form of Chain-ladder method with relations between different groups of insurance portfolio included. In the very last section, all the lessons are demonstrated on real data to give readers an idea of how the process of claims reserving works is in the common actuarial practice.

Orientadores: Ronaldo Dias, Nancy Lopes Garcia / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-06T03:32:13Z (GMT). No. of bitstreams: 1 Nascimento_FernandoFerrazdo_M.pdf: 973412 bytes, checksum: 97b1cf4137bd59eb99ae987305700439 (MD5) Previous issue date: 2006 / Resumo: Neste trabalho, comparamos diversas técnicas de previsão de IBNR (Incurred But Not Reported) para dados de um triângulo Run-OjJ, desde as mais simples, como por exemplo as técnicas Chain- Ladder e a técnica da Separação, até as técnicas mais sofisticadas, considerando modelos Log-Normais ou pela distribuição Poisson Composta. Além disso, nosso trabalho enfatiza a necessidade do uso de técnicas não-paramétricas, considerando um modelo de truncamento das variáveis. Foi possível mostrar que, mesmo não tendo nenhuma informação sobre a distribuição dos dados, é possível estimar o IBNR com menor erro e variabilidade do que as técnicas usuais conhecidas. Para fazer as comparações, foram realizadas simulações de sinistros ocorrendo através de um Processo de Poisson não homogêneo, e com dependência no tempo de relato e valor do sinistro. A medida de comparação utilizada foi o Erro Quadrático Médio (EQM) entre os valores simulados e os valores previstos por cada técnica. A abordagem paramétrica, quando os dados provém de uma distribuição Poisson Composta, apresentou o menor EQM dentre todas as técnicas. Entretanto, se não há informação sobre a distribuição dos dados, a técnica de Mista de truncamento foi a melhor entre as não-paramétricas / Abstract: We compare several forecast techniques for IBNR(Incurred But Not Reported) from a Run-Off triangle data, since the most simple techniques like Chain-Ladder and Separation Technique, to the more complex using Log-Normal models and Compound Poisson distribution. Therefore, exist the necessity of the use of Nonparametric techniques, using a model that consider variable Truncation. It was possible shown that, when we don't have any information about the data, it's possible estimate de IBNR forecasting with less mistake and variability than the usual techniques. For make the forecasting, we used claims simulations occurring by a nonhomogeneous Poisson process and with dependence entry the time to report and value paid for one claim. The measure of comparison used was the Mean Square Error (MSE) of simulated values and forecasting values for each technique. The parametric boarding when the data come from a Compound Poisson distribution, was the best MSE entry all techniques. However, when we don't have any information about the data, the Truncation Technique was the best of the nonparametric techniques / Mestrado / Mestre em Estatística

The main goal of this Diploma thesis is to describe an approach for modeling run-off triangles of nonlife insurance (calculation of IBNR reserve) based on state space models and apply the method to the selected run-off triangles. In difference from (Atherino a kol., 2010) the KFAS package in R software is used for modeling purposes in the numerical study at the end of the thesis. One provides a preview of various possibilities of data and model adjustment applied to the same run-off triangles in order to asses added value of these steps (logartihmic transformation of input data, interventions for outliers etc.). A special attention is devoted to lognormal modification of the basic state space model. An integral part of the numerical study in the thesis is a residual diagnostic of models and simulation approach to IBNR reserves. 1

La présente thèse vise à explorer différents types de modèles log-bilinéaires dans le domaine des sciences actuarielles. Le point de départ consiste en le modèle de Lee-Carter, utilisé pour les problèmes de projection de la mortalité. Différentes variantes sont développées, et notamment le modèle de Poisson log-bilinéaire. L'introduction de variables explicatives est également analysée. Enfin, une tentative de d'exportation de ces modèles au cas des triangles IBNR est effectuée.

La présente thèse vise à explorer différents types de modèles log-bilinéaires dans le domaine des sciences actuarielles. Le point de départ consiste en le modèle de Lee-Carter, utilisé pour les problèmes de projection de la mortalité. Différentes variantes sont développées, et notamment le modèle de Poisson log-bilinéaire. L'introduction de variables explicatives est également analysée. Enfin, une tentative de d'exportation de ces modèles au cas des triangles IBNR est effectuée.

This text describes wide variety of models for IBNR reserve estimation, among them we can find chain ladder, Munich chain ladder, Cape Cod or regression methods. Especially, it aims at identification of model assumptions, its verification, impacts of their non-fulfillment and resulting model modification. Moreover, this work deals with anomalies and specificity of historical data and tries to suggest solutions for this problems. The last part of the thesis is dedicated to real analysis of motor third party liability.