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.

The traditional reserve estimation by an insurance company is based on the aggregated data. However, new trend is to utilize all the information available and analyse each claim separately. This way the application of claims specific features, such as non-proportional reinsurance or policy limits, is possible. The aim of this thesis is to construct the reserving model based on the individual claims. Following the recent legislative changes, the reserve risk has been redefined from ultimate claim horizon to a one-year risk horizon. Hence, the next task is to setup simulation model to calculate one year horizon reserve risk by updating the estimates based on new observations collected over one year. This is a typical task for Bayesian approach, therefore the model components are estimated using the tools of Bayesian statistics.

[pt] Esta dissertação tem como objetivo principal a apresentação, discussão e implementação de um modelo de espaço de estado, derivado do modelo desenvolvido por De Jong & Zenhwirth, no cenário de estimação de reservas IBNR. O modelo visa obter uma distribuição para as reservas e seu desvio padrão, que permite obter um intervalo de confiança para a estimativa. Também são propostas extensões para o modelo. / [en] The main purpose of this master thesis is the presentation, discussion and implementation of a state space model, derived from the De Jong & Zehnwirth model, on the IBNR Reserve estimation scenario. The model tries to obtain a distribution for the reserves and its standard deviation as well, allowing the cofidence interval estimation. Extensions for the model are also discussed.

Claims reserving and claims process estimation present classical problems in general insurance. The overall reserves are often determined under the assumption of independence among the lines of business. Though, recently modelling of the dependence among multiple lines of business has become crucial issue of reserving process. In this context, copulae provide a useful tool to construct models which go beyond the classical ones in terms of dependence structure. This thesis deals, in particular, with the copula regression model, its properties and possible applications in general insurance. This approach combines GLM modelling of margins and then expressing the dependence structure using copula. The theoretical methods are illustrated on a real dataset.

This presented thesis deals with applications of Tweedie compound Poisson model in non-life insurance pricing and claims reserving. Tweedie models are exponen- tial dispersion models with power mean-variance relationships and compound Poisson distribution is a particular Tweedie model. The interest in Tweedie com- pound Poisson model is motivated by its applications to generalized linear models (GLMs) and generalized estimation equations (GEE). The purpose of this thesis is to construct pricing and claims reserving models in which the response variables follow Tweedie compound Poisson model. Theoretical approaches are applied on the real datasets. 1

In the presented thesis the issue of dependency between response variables within the subjects in the generalized linear models framework is investigated. Reserving in non-life insurance is a key factor for the financial position of a company. The text introduces the basic actuarial notation, terminology and methods. The main part is focused on panel data framework, especially Generalized Linear Mixed Models (GLMM) as well as Generalized Estimating Equations (GEE), and their application on claims reserving. The aim of this thesis is to show the advantages, disadvantages, limitations and the comparison of these approaches on representative datasets, which were chosen according to results obtained from whole database analysis. Significant focus is on model selection and diagnostics used for this purpose. Finally, the obtained results are summarized in tables, figures and the comparison of the methods is provided. Powered by TCPDF (www.tcpdf.org)

This thesis covers stochastic claims reserving in non-life insurance based on individual claims developments. Summarized theoretical methods are applied on data from Czech Insurers' Bureau for educational purposes. The problem of estimation is divided into four parts: oc- curence process generating claims, delay of notification, times between events and payments. Each part is estimated separately based on maximum likelihood theory and final estimates allow us to obtain an estimate of future liabilities distribution. The results are very promis- ing and we believe this method is worth of a further research. Contribution of this work is more rigorous theoretical part and application on data from the Czech market with some new ideas in practical part and simulation. 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

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 study shows that managers of short-term insurers may improve their financial results if they can identify and manage the factors that are critical to their financial results. The development and application of the concept of critical success factors are therefore used as a basis for this study. The study reviews the functions performed by short-term insurers, focusing on the effect these functions have on their cash flows. Selection and pricing of risk are discussed in detail. The underwriting cycle in South Africa, and several possible causes of the cycle are investigated. Reinsurance, claims handling and rilanagement expenses are important components of the cash flows of short-term insurers and are therefore examined in detail. The optimum risk level at various combinations of underwriting and investment income is empirically tested, using the financial results of several insurers. The study investigates different approaches to the measurement of financial success of insurers, and the return on shareholders' funds is found to provide the fairest and most reliable method. Empirical comparisons are made on the financial results of the insurers that participated in the study to distinguish between those that are financially successful and those that are not. To discover what the industry consider to be their critical financial success factors, a postal survey was done of key decision makers in the South African short-term insurance industry. Respondents identified several success factors, but did not include some success factors discovered during the review of the literature. Respondents apparently experienced difficulty in separating strategic issues from operational ones. The survey revealed that the pricing of risk is problematic for short-term insurers. The importance of the investment function is also underestimated by the industry. The study concludes that the combined systematic risk of the investment and underwriting portfolios is a critical success factor, along with the capital base of the insurer, the ability of the insurer to use the leverage provided by using policyholders' funds as free reserves and the size and direction of an insurer's cash flows. / Business Management / D. Com (Business Management)