Multidimensional frequency models can be used for modeling number of claims from different branches which are somehow dependent on each other. As in the one-dimensional case Poisson distribution and negative binomial distribution are primarily used for modeling multidimensional claim counts data, only they are extended to higher dimensions. The generalization of multi- dimensional distributions is often done using so-called shock variables, where one random variable is included in all dimensions of a random vector which models claim counts. The more comprehensive approach to modeling dependence uses copulas. Comparison of these models is done on a simulated data of number of claims from two different car insurance guarantees.

In this thesis, we consider several statistical techniques applicable to claim frequency models of an MTPL portfolio with a focus on overdispersion. The practical part of the work is focused on the application and comparison of the models on real data represented by an MTPL portfolio. The comparison is presented by the results of goodness-of-fit measures. Furthermore, the predictive power of selected models is tested for the given dataset, using the simulation method. Hence, this thesis provides a combination of the analysis of goodness-of-fit results and the predictive power of the models.

The costs related to events of losses within non-life insurance are stochastic and a prerequisite of running a successful insurance business is to predict risks and future costs. From both a business- and regulatory perspective, it is of high interest to have a genuine understanding of the precision and the sensitivity of the estimated costs and future risks. This thesis aims to provide an alternative procedure of how to estimate the costs related to the future and, above all, the variance, in the case of dealing with inhomogeneous and decreasing risk. The procedure is based on a separate modeling of the claim frequency and the claim severity, that later can be combined to yield a total cost distribution for a determined time period. The claim severities are modeled based on a parametric and a non-parametric approach and the claim frequencies are modeled with the resampling method bootstrap and by the use of scenarios. The thesis is made in collaboration with the insurance company, Anticimex Insurance, who has contributed with the data as well as expert knowledge related to the actuarial field. The results of the thesis show that the procedure is successful for evaluating estimated total costs distributions and their first and second moments, even in the case of inhomogeneous and decreasing risk. / Kostnader som uppkommer på grund av skador inom skadeförsäkring är stokasiska och en förusättning för att kunna bedriva ett framgångsrikt försäkringsbolag är att kunna prediktera risk och framtida kostnader. Utifrån ett såväl försäkrings- som reglatoriskt perspektiv är det av stor vikt att ha en gedigen förståelse av både precisionen och känsligheten i de skattade estimaten. Denna uppsats syftar till att ta fram ett alternativt tillvägagångssätt till hur kostnader relaterade till framtiden ska predikteras, med fokus på att utvärdera variationen i estimaten, vid fallet av en inhomogen och avtagande risk. Tillvägagångssättet bygger på en uppdelning mellan antalet skador och kostnaden för skador, vilka modelleras separat för att sedan kombineras och ge en totalkostnadsfördelning för den avsedda tidsperioden. De historiska kostnaderna modelleras utifrån ett parametriskt- och ett ickeparametriskt tillvägagångssätt. Skadefrekvensen modelleras med hjälp av bland annat samplingsmetoden bootstrap samt genom användandet av scenarier. Uppsatsen görs i samarbete med skadeförsäkringsbolaget, Anticimex Försäkringar, vilka har bidragit med data och expertkunskap inom det aktuariella området. Arbetets resultat visar att det föreslagna tillvägagångssättet är en framgångsrik strategi för att utvärdera de första två momenten av de predikterade totalkostnadsfördelningarna, även vid fallet av en inhomogen och avtagande risk.

This diploma thesis deals with the mathematical models in general insurance. The aim of this thesis is to analyse selected mathematical models that are widely used in general insurance for the estimation of insurance portfolio statistics, pricing and the regulatory capital requirement calculation. Claim frequency models, claim severity models, aggregate loss models and generalized linear models are analysed. This thesis consists of a theoretical and a practical part. The theoretical part contains description of selected models. Described models are then applied to a real dataset in the practical part. The real dataset modelling was performed using the statistical software R. It has been proved that maximum likelihood parameter estimations are of better quality than the method of moments or quantile method estimations. The results of aggregate loss distribution computational methods are comparable. This comparability is mostly caused by a large number of observations. In the context of tariff analysis it was found that the most significant factors are driver's age and the driver's area of residence.