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.

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

Three methods for claims reserving are compared on two data sets. The first two methods are the commonly used chain ladder method that uses aggregated payments and the relatively new method, double chain ladder, that apart from the payments data also uses the number of reported claims. The third method is more advanced, data on micro-level is needed such as the reporting delay and the number of payment periods for every single claim. The two data sets that are used consist of claims with typically shorter and longer settlement time, respectively. The questions considered are if you can gain anything from using a method that is more advanced than the chain ladder method and if the gain differs from the two data sets. The methods are compared by simulating the reserves distributions as well as comparing the point estimates of the reserve with the real out-of-sample reserve. The results show that there is no gain in using the micro-level method considered. The double chain lad- der method on the other hand performs better than the chain ladder method. The difference between the two data sets is that the reserve in the data set with longer settlement times is harder to estimate, but no difference can be seen when it comes to method choice. / Tre reservsättningsmetoder jämförs på två dataset. De första två metoderna är den välkända chain ladder-metoden som använder sig av aggregerade utbetalningar samt den relativt nya metoden double chain ladder som förutom utbetalningarna använder sig av antalet anmälda skador. Den tredje metoden baseras på mikro-nivå och kräver information om varje enskild skada, såsom anmälningstid och antalet utbetalningsperioder. De två dataseten som används är ett som innehåller skador med typiskt kortare avvecklingstider och ett som innehåller skador med typiskt längre avvecklingstider. Frågorna som behandlas är om man vinner något på att använda en mer avancerad metod än chain ladder och om det skiljer sig åt mellan dataseten. Metoderna jämförs genom simulering av reserven, men också genom att jämföra punktskattningar med den verkliga reserven. Resultaten visar att man I detta fall inte vinner något på att använda mikro-metoden. Double chain ladder å andra sidan presterar bättre än chain ladder. Skillnaden mellan de två dataseten är att det är svårare att estimera reserven när avvecklingstiden är längre, men ingen skillnad ses när det gäller val av metod

[pt] Este trabalho utiliza três metodologias para modelagem de sinistros IBNR apresentados no formato do triângulo de runoff com cauda, e verifica, por meio de quatro exercícios empíricos com dados reais, se existe uma abordagem estatisticamente mais eficaz. A primeira metodologia se baseia no método do chain ladder clássico, com uma extensão de cálculo de reserva para ano de calendário. A segunda metodologia baseia-se em modelos de regressão linear com heterocedasticidade, sob o arranjo usual do triângulo via duplo-índice. A terceira insere-se no arcabouço dos modelos de espaço de estado lineares e do filtro de Kalman, considerando, desta vez, a ordenação por linhas do triângulo de Atherino et al. (2010). Para todas as abordagens, efetivam-se derivações teóricas e implementações computacionais tanto dos cálculos de reservas IBNR totais e parciais, resultantes dos modelos estimados, quanto dos correspondentes erros médios quadráticos teóricos. Como conclusões desta Dissertação, citam-se: (i) apesar de superiores ao chain ladder, nenhuma das outras duas abordagens se destaca sistematicamente em relação à outra; (ii) a adoção do efeito cauda se mostrou computacional e tecnicamente viável; e (iii) há fatos estilizados nos dados, modelados sob as três abordagens, que possibilitariam a confecção de softwares de estimação de reserva. / [en] This work makes use of three methodologies for modeling IBNR data arranged in the runoff triangle with a tail effect, and evaluates their performances in four empirical examples. The first methodology is the traditional chain ladder, duly extended to calculate a reserve corresponding to the calendar year. The second methodology remains on linear regression models with heteroscedastic errors, under the well-established double index notation of the triangle. The third methodology uses the linear state space modeling and the theory of the Kalman filter, adopting, this time, the row-wise ordering proposed by Atherino et al. (2010). For each approach, theoretical results and numerical implementations are obtained, where both the punctual IBNR reserve estimators and their corresponding theoretical mean square errors are considered. The main conclusions from this Dissertation are: (i) even thought proving to be superior to the chain ladder, none of the remaining two approaches seems to outperform the other; (ii) the adding of a tail effect does not entail major theoretical and/or computational problems; and (iii) the approaches have uncovered stylized facts that would enable the planning of softwares for IBNR reserve estimation.

Point estimates of the Standard Chain Ladder method (CLM) and of the more complex Munich Chain Ladder method (MCL) are compared to real data on 38 different datasets in order to evaluate if MCL produces better predictions on average with a dataset from an arbitrary insurance portfolio. MCL is also examined to determine if the future paid and incurred claims converge as time progresses. A bootstrap model based on MCL (BMCL) is examined in order to evaluate its possibility to estimate the probability density function (PDF) of future claims and observable claim development results (OCDR). The results show that the paid and incurred predictions by MCL converge. The results also show that when considering all datasets MCL produce on average better estimations than CLM with paid data but no improvement can be seen with incurred data. Further the results show that by considering a subset of datasets which fulfil certain criteria, or by only considering accident years after 1999 the percentage of datasets in which MCL produce superior estimations increases. When examining BMCL one finds that it can produce estimated PDFs of ultimate reserves and OCDRs, however the mean of estimate of ultimate reserves does not converge to the MCL estimates nor do the mean of the OCDRs converge to zero. In order to get the right convergence the estimated OCDR PDFs are centered and the mean of the BMCL estimated ultimate reserve is set to the MCL estimate by multiplication. / Punktskattningar gjorda med Standard Chain Ladder (CLM) och den mer komplexa Munich Chain Ladder-metoden (MCL) jämförs med verklig data för 38 olika dataset för att evaluera om MCL ger bättre prediktioner i genomsnitt än CLM för en godtycklig försäkringsportfölj. MCLs prediktioner undersöks också för att se om de betalda och de kända skadekostnaderna konvergerar. En bootstrapmodell baserad på MCL (BMCL) undersöks för att utvärdera om möjligheterna att estimera täthetsfunktionen (probability density function, PDF) av framtida skadekostnader och av ”observable claim development results (OCDR)”. Resultaten visar att MCLs estimerade betalda och kända skadekostnader konvergerar. Resultaten visar även att när man evaluerar alla dataseten så ger MCL i genomsnitt bättre prediktioner än CLM med betald data, men ingen förbättring kan ses med CLM med känd skadekostnadsdata. Vidare visar resultaten även att genom att bara titta på dataset som uppfyller vissa krav, eller genom att bara använda olycksår efter 1999, så ökar andelen dataset där MCL ger bättre prediktioner än CLM.Vid evaluering av BMCL ser man att den kan producera estimerade PDF:er för ultimo-reserver och OCDR:er, men att medelvärdet av ultimo-reserv prediktionerna från BMCL inte konvergerar mot MCL-prediktionerna och att medelvärdet av OCDR:erna inte konvergerar mot noll. För att få rätt konvergens så centreras OCDR PDF:erna och ultimo-reservernas medelvärden sätts till motsvarande MCL-prediktionens värde genom multiplikation.

This thesis is dedicated to Solvency II, a regulatory framework for insurance and reinsurance companies effective in European Union. Firstly, it explains the notion solvency and also describes the principles of the regulation itself. Another part is focused on the calculation of solvency capital requirement and minimal capital requirement, using standard formula. The capital requirements are derived on the level of risk modules and their submodules. Furthermore, the topic of technical reserves is discussed and emphasis is placed on the derivation of mean square error of prediction. Described methods are Chain-Ladder and Bornhuetter-Ferguson. The last part of the thesis includes the calculation of capital requirements for real data. A program SolvencyII.xlsx that shows particular derivations is enclosed.

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

One of the main and crucial activities of an insurance company is to determine amount of technical reserves to be generated. If the insurance company performs in the non-life insurance branch, it focuses first of all on loss reserve which is generated to settle debts coming from insurance claims. To set the proper amount of this reserve, especially of the reserve on incurred but not reported losses (IBNR), mathematical and statistical methods are used. This thesis introduces one of the most used methods which is the chain ladder method. It presents the first chain ladder deterministic model then moves to its stochastic extension in a form of Mack's model and finally gets to the Munich chain ladder model, which takes into calculations not only data on losses paid but also losses incurred. In the theoretical part both these models (standard Mack's chain ladder and Munich chain ladder) are presented both separately and in a common context so that later in the analytical section they could be demonstrated on real data.

Dans la première partie de la thèse, nous nous intéressons à la convergence faible du processus empirique pondéré des copules. Nous fournissons la condition suffisante pour que cette convergence ait lieu vers un processus Gaussien limite. Nos résultats sont obtenus dans un espace de Banach L^p. Nous donnons des applications statistiques de ces résultats aux tests d'adéquation (tests of goodness of fit) pour les copules. Une attention spéciale est portée aux tests basées sur des statistiques de type Cramér-von Mises.Dans un second temps, nous étudions le problème de provisionnement stochastique pour une compagnie d'assurance non-vie. Les méthodes stochastiques sont utilisées afin d'évaluer la variabilité des réserves. Le point de départ pour cette thèse est une incohérence entre les méthodes utilisées en pratique et celles publiées dans la littérature. Pour remédier à cela, nous présentons un outil général de provisionnement stochastique à horizon ultime (Chapitre 3) et à un an (Chapitre 4), basé sur la méthode Chain Ladder. / The aim of this thesis is twofold. First, we concentrate on the study of weak convergence of weighted empirical copula processes. We provide sufficient conditions for this convergence to hold to a limiting Gaussian process. Our results are obtained in the framework of convergence in the Banach space $L^{p}$ ($1\leq p <\infty $). Statistical applications to goodness of fit (GOF) tests for copulas are given to illustrate these results. We pay special attention to GOF tests based on Cramér-von Mises type statistics. Second, we discuss the problem of stochastic claims reserving in general non-life insurance. Stochastic models are needed in order to assess the variability of the claims reserve. The starting point of this thesis is an observed inconsistency between the approaches used in practice and that suggested in the literature. To fill this gap, we present a general tool for measuring the uncertainty of reserves in the framework of ultimate (Chapter 3) and one-year time horizon (Chapter 4), based on the Chain-Ladder method.

[pt] A reserva IBNR é uma reserva de suma importância para as seguradoras. Seu cálculo tem sido realizado por métodos, em sua grande maioria, determinísticos, tradicionalmente aplicados a informações de sinistros agrupadas num formato particular intitulado triangulo de run-off. Esta forma de cálculo foi muito usada por décadas por sua simplicidade e pela limitação da capacidade de processamento computacional existente. Hoje, com o grande avanço dessa capacidade, não haveria necessidade de deixar de investigar informações relevantes que podem ser perdidas com agrupamento dos dados. Muitas são as deficiências dos métodos tradicionais apontadas na literatura e o uso de informação detalhada tem sido apontado por alguns artigos como a fonte para superação dessas deficiências. Outra busca constante nas metodologias propostas para cálculo da IBNR é pela obtenção de boas medidas de precisão das estimativas obtidas por eles. Neste ponto, sobre o uso de dados detalhados, há a expectativa de obtenção de medidas de precisão mais justas, já que se tem mais dados. Inspirada em alguns artigos já divulgados com propostas para modelagem desses dados não agrupados esta dissertação propõe um novo modelo, avaliando sua capacidade de predição e ganho de conhecimento a respeito do processo de ocorrência e aviso de sinistros frente ao que se pode obter a partir dos métodos tradicionais aplicados à dados de quantidade para obtenção da quantidade de sinistros IBNR e sua distribuição. / [en] The IBNR reserve is a reserve of paramount importance for insurers. Its calculation has been accomplished by methods, mostly, deterministic, traditionally applied to claims grouped information in a particular format called run-off triangle . This method of calculation was very adequate for decades because of its simplicity and the limited computational processing capacity existing in the past. Today, with the breakthrough of this capacity, no waiver to investigating relevant information that may be lost with grouping data would be need. Many flaws of the traditional methods has been mentioned in the literature and the use of detailed information has been pointed as a form of overcoming these deficiencies. Another frequent aim in methodologies proposed for the calculation of IBNR is get a good measure of the accuracy of the estimates obtained by them and that is another expectation about the use of detailed data, since if you got more data you could get better measures. Inspired by some articles already published with proposals for modeling such not grouped data, this dissertation proposes a new model and evaluate its predictive ability and gain of knowledge about the process of occurrence and notice of the claim against that one can get from the traditional methods applied to data of amount of claims for obtain the amount of IBNR claims and their distribution.