Abstract
The analysis of loss data is of utmost interest in many branches of the financial and insurance industries, in structural engineering and in operation research, among others. In the financial industry, the determination of the distribution of losses is the first step to take to compute regulatory risk capitals; in insurance we need the distribution of losses to determine the risk premia. In reliability analysis one needs to determine the distribution of accumulated damage or the first time of occurrence of a composite event, and so on. Not only that, but in some cases we have data on the aggregate risk, but we happen to be interested in determining the statistical nature of the different types of events that contribute to the aggregate loss. Even though in many of these branches of activity one may have good theoretical descriptions of the underlying processes, the nature of the problems is such that we must resort to numerical methods to actually compute the loss distributions. Besides being able to determine numerically the distribution of losses, we also need to assess the dependence of the distribution of losses and that of the quantities computed with it, on the empirical data. It is the purpose of this note to illustrate the how the maximum entropy method and its extensions can be used to deal with the various issues that come up in the computation of the distribution of losses. These methods prove to be robust and allow for extensions to the case when the data has measurement errors and/or is given up to an interval.
Highlights
Introduction and PreliminariesThe problem of determining the distribution of aggregate losses begins by developing a model for the random variable in whose distribution we are interested
For each loss type h, we shall suppose that Nh is an integer-valued random variable modeling the frequency of losses in a given time lapse, and that for each h, the random variable Xh,k is a positive, continuous random variable, modeling the occurrence of the individual n-th loss of that type
Since the probability density of a positive random variable can be recovered from its Laplace transform, here we examine how maximum entropy methods invert numerically the Laplace transform
Summary
The problem of determining the distribution of aggregate losses begins by developing a model for the random variable in whose distribution we are interested. When all that we have is empirical data about aggregate losses, and no models at hand, the direct approach may not be applied In this case, we must recur to numerical procedures to determine the density of a continuous random variable S from an empirical sample {s1 , ..., s M }. This is a rather important issue, because many quantities of interest depend heavily of the tail behavior of the density and cannot be estimated with a small amount of data in the tail of the distribution, if it exists at all. We shall illustrate the sample variability of some risk premia that depend strongly on the tail of the distribution
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
