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.

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.

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

The thesis is about the arrangement of the last known claim values into the run-off triangle. This diagram is used in non-life insurance, mainly in methods for calculating technical claims reserves. Individual methods will be described in detail and consecutively applied on real data. The real data are a set of data with long tail. We are differentiating between easier deterministic and stochastic methods, which are more demanding for calculation. The results will be compared by basic statistical parameter of the analyzed data and at the end the best method will be chosen for the data.