Loss reserving in P&C insurance, is the common practice of estimating the insurer’sliability from future claims it will have to pay out on. In the recent years, it has beenpopulartoexploretheoptionsofforecastingthislosswiththehelpofmachinelearningmethods. This is mainly attributed to the increase in computational power, openingup opportunities for handling more complex computations with large datasets. ThemainfocusofthispaperistoimplementandevaluatearecurrentneuralnetworkcalledthedeeptrianglebyKuoformodellingpaymentsofindividualreportedbutnotsettledclaims. The results are compared with the traditional Chain Ladder method and abaseline model on a simulated dataset provided by Wüthrich’s simulation machine.The models were implemented in Python using Tensorflow’s functional API. Theresults show that the recurrent neural network does not outperform the Chain Laddermethod on the given data. The recurrent neural network is weak towards the sparseand chaotic nature of individual claim payments and is unable to detect a stablesequential pattern. Results also show that the neural network is prone to overfitting,whichcantheoreticallybecompensatedwithlargerdatasetbutcomesatacostintermsof feasibility. / Reservsättninginomskadeförsäkringhandlaromattberäknaframtidakostnaderavenförsäkringsgivare. Under de senaste åren har det blivit allt populärare att undersökatillämpningen av olika statistiska inlärningsmetoder inom reservsättning. Den häruppsatsensyftartillattimplementeraochutvärderaettåterkommandeneuraltnätverksom kallas för ”deeptriangle by Kuo” för att modellera utbetalningar av individuellarapporterade men icke­färdigbetalda försäkringsfordringar. Resultaten kommer attjämföras med den traditionella Chain Ladder metoden samt en grundmodell på ettsimulerat dataset som tillhandahålls av ”Wüthrichs simulation machine”. Modellernaimplementeras i Python med hjälp av Tensorflows Functional API. Resultatet är attdetåterkommandeneuralanätverketinteöverträffarChainLaddermetodenmeddengivna datan. Det återkommande neurala nätverket har svårigheter för att känna igenmönster i datamängder som individuella skadebetalningar eftersom datamängden tillsin natur är spridd och kaotisk. Resultaten visar också att det neurala nätverket ärbenäget att överanpassa, vilket teoretiskt kan kompenseras med en större datamängdmen som i sin tur bidrar till en risk för ogenomförbarhet.

This thesis deals with the application of stochastic claims reserving methods to given data with some trends in claims development. It describes the chain ladder method and the generalized linear models as its stochastic framework. Some simple functions are suggested for smoothing the origin and development period coefficients from the estimated model. The extrapolation is also considered for estimation of the unobserved tail values. The residual bootstrap is used for the reparameterized model in order to get the predictive distribution of the estimated reserve together with its standard deviation as a measure of volatility. Solvency capital requirement in one year time horizon is also calculated. 1

In practice there is a long tradition of actuaries calculating reserve estimates according to deterministic methods without explicit reference to a stochastic model. For instance, the chain-ladder was originally a deterministic reserving method. Moreover, the actuaries often make ad hoc adjustments of the methods, for example, smoothing of the chain-ladder development factors, in order to fit the data set under analysis. However, stochastic models are needed in order to assess the variability of the claims reserve. The standard statistical approach would be to first specify a model, then find an estimate of the outstanding claims under that model, typically by maximum likelihood, and finally the model could be used to find the precision of the estimate. As a compromise between this approach and the actuary's way of working without reference to a model the object of the research area has often been to first construct a model and a method that produces the actuary's estimate and then use this model in order to assess the uncertainty of the estimate. A drawback of this approach is that the suggested models have been constructed to give a measure of the precision of the reserve estimate without the possibility of changing the estimate itself. The starting point of this thesis is the inconsistency between the deterministic approaches used in practice and the stochastic ones suggested in the literature. On one hand, the purpose of Paper I is to develop a bootstrap technique which easily enables the actuary to use other development factor methods than the pure chain-ladder relying on as few model assumptions as possible. This bootstrap technique is then extended and applied to the separation method in Paper II. On the other hand, the purpose of Paper III is to create a stochastic framework which imitates the ad hoc deterministic smoothing of chain-ladder development factors which is frequently used in practice.

The subject matter of this master thesis is the introduction to the claims reserving methodology applied in the general insurance with the focus on the agragated data represented in the form of triangle schemes. First the basic deterministic methods are to be presented including the Chain ladder method as the most known and widely used tool in claims reserving. Next we will concentrate on the stochastic approaches. The method of bootstrapping is to be described more in detail as it is the main topic of this thesis. Finally the accuracy of the prediction of several specific models and algorithms is to be examined with the goal of their overall comparison (using randomly generated input data).

The aim of this thesis is to describe calculation methods for variability esti- mation of claims reserve in non-life insurance. The thesis focuses on three main categories of models: Mack's stochastic Chain-Ladder, generalized linear models and bootstrap. Both the theoretical and also the empirical parts are included. Empirical part is devoted to application of all the models described above on both real and simulated data. 1

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.

[pt] O principal objetivo desta dissertação é realizar a previsão da reserva IBNR. Para isto foi desenvolvido um modelo estatístico de distribuições combinadas que busca uma adequada representação dos dados. A reserva IBNR, sigla em inglês para Incurred But Not Reported, representa o montante que as seguradoras precisam ter para pagamentos de sinistros atrasados, que já ocorreram no passado, mas ainda não foram avisados à seguradora até a data presente. Dada a importância desta reserva, diversos métodos para estimação da reserva IBNR já foram propostos. Um dos métodos mais utilizado pelas seguradoras é o Método Chain Ladder, que se baseia em triângulos run-off, que é o agrupamento dos dados conforme data de ocorrência e aviso de sinistro. No entanto o agrupamento dos dados faz com que informações importantes sejam perdidas. Esta dissertação baseada em outros artigos e trabalhos que consideram o não agrupamento dos dados, propõe uma nova modelagem para os dados não agrupados. O modelo proposto combina a distribuição do atraso no aviso da ocorrência, representada aqui pela distribuição log-normal truncada (pois só há informação até a última data observada); a distribuição da quantidade total de sinistros ocorridos num dado período, modelada pela distribuição Poisson; e a distribuição do número de sinistros ocorridos em um dado período e avisados até a última data observada, que será caracterizada por uma distribuição Binomial. Por fim, a quantidade de sinistros IBNR foi estimada por método e pelo Chain Ladder e avaliou-se a capacidade de previsão de ambos. Apesar da distribuição de atrasos do modelo proposto se adequar bem aos dados, o modelo proposto obteve resultados inferiores ao Chain Ladder em termos de previsão. / [en] The main objective of this dissertation is to predict the IBNR reserve. For this, it was developed a statistical model of combined distributions looking for a new distribution that fits the data well. The IBNR reserve, short for Incurred But Not Reported, represents the amount that insurers need to have to pay for the claims that occurred in the past but have not been reported until the present date. Given the importance of this reserve, several methods for estimating this reserve have been proposed. One of the most used methods for the insurers is the Chain Ladder, which is based on run-off triangles; this is a format of grouping the data according to the occurrence and the reported date. However this format causes the lost of important information. This dissertation, based on other articles and works that consider the data not grouped, proposes a new model for the non-aggregated data. The proposed model combines the delay in the claim report distribution represented by a log normal truncated (because there is only information until the last observed date); the total amount of claims incurred in a given period modeled by a Poisson distribution and the number of claims occurred in a certain period and reported until the last observed date characterized by a binomial distribution. Finally, the IBNR reserve was estimated by this method and by the chain ladder and the prediction capacity of both methods will be evaluated. Although the delay distribution seems to fit the data well, the proposed model obtained inferior results to the Chain Ladder in terms of forecast.

This thesis deals with a description of three claims reserving methods - with stochastic models for Chain ladder, Bornhuetter/Ferguson and multiplicative method. There are mentioned their assumptions, parameter estimates, their properties and formulas for loss reserves in the first part. The second part of the text is devoted to formulas for the mean squared error of prediction and its estimate. Finally, a numerical example shows comparison of these methods.