An analysis of the Danish Fire Insurance data

Specialeforsvar ved Christopher de la Motte Olsen 

Titel: An Analysis of the Danish Fire Insurance data

Abstract: This master thesis is an investigation of the Danish Fire Insurance data from the period $1980-2002$, where data represents $90\%$ of the Danish insurance market. The data consists of three components: building loss, content loss and loss of profit, the sum of these components is denoted as the total loss. Initially, we prepare the data first by indexation of the losses and second by simulating new losses in order to obtain a more homogeneous portfolio. We then deduce some results from the univariate extreme value theory where we focus on the estimation of the tail-distribution. This collapses with the estimation of the shape parameter $\xi$. For each of the three components, their respective $\xi$-estimate is positive, which implies that all of the components belong to the maximum domain of attraction of the Fréchet distribution. With the extension to the multivariate case, it is not uncommon to transform the variates such that they are tail-equivalent to a standard Pareto distribution. To do so, we use the estimated tail distribution from the univariate case; hence, the tail distribution is also important for the multivariate analysis. The notion of a regularly varying vector is the key point for the multivariate case. Results for this lead, among other things, to estimation of the angular measure which determines whether variates seem asymptotically dependent or independent. Here, the fact that the components building and content losses seem asymptotically independent, comes as a surprise to us. Lastly, we estimate the exponent measure with help from the methodology introduced by Cătălin Stărică. This measure satisfies the homogeneity property which we make use of when estimating the probability of failure sets. The Stărică plot can here give an indication of potential bias. 

Vejleder: Thomas Mikosch
Censor: Yuri Goegebeur, SDU