The role of actuaries in insurance companies is to estimate the claim reserves as accurately as possible. Known cumulative claims $C_{i,j}$ for $i+j \le I$ are reported in the development triangle, which is used to estimate future cumulative claims $\widehat{C_{i,j}}$ for $i+j >I$, where $i \in \{0, \ldots, I \}$ indicates accident years and $j \in \{ 0, \ldots, J \}$ development years. Actuaries use deterministic and stochastic methods to predict claim amounts. In deterministic methods only a single-point estimate of claim is calculated, while stochastic methods also provide an estimate of the uncertainty of claim reserve by the mean square error of prediction. As the analytical calculation of the mean squared error of prediction is sometimes demanding, actuaries use the bootstrap method. Among the deterministic methods we will consider the Chain-Ladder method, the Bornhuetter-Ferguson method and the Poisson model, and among the stochastic ones, the Mack model, the Overdispersed Poisson model and the bootstrap in Overdispersed Poisson model, and the Exact Bayes model. From the distribution of the total claim reserve we can determine the risk adjustment by the value-at-risk, which is important in the context of the International Financial Reporting Standard 17 - Insurance Contracts.
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