Significant changes in the insurance and financial markets are giving increasing attention to the need for developing a standard framework for risk measurement. Recently, there has been growing interest among insurance and investment experts to focus on the use of a tail conditional expectation because it shares properties that are considered desireable and applicable in a variety of situations. In particular, it satisfies requirements of a "coherent" risk measure in the spirit developed by Artzner, et al. (1999). In this paper, we derive explicit formulas for computing tail conditional expectations for elliptical distributions, a family of symmetric distributions which includes the more familiar normal and student-t distributions. We extend this investigation to multivariate elliptical distributions allowing us to model combinations of correlated risks. We are able to exploit properties of these distributions naturally permitting us to decompose the conditional expectation so that we are able to allocate contribution of individual risks to the aggregated risks. This is meaningful in practice particularly in the case of computing capital requirements for an institution who may have several lines of correlated business and is concerned of fairly allocating the total capital to these constituents. *
In a previous investigation we studied some asymptotic properties of the sample mean location on submanifolds of Euclidean space. The sample mean location generalizes least squares statistics to smooth compact submanifolds of Euclidean space.In this paper these properties are put into use. Tests for hypotheses about mean location are constructed and confidence regions for mean location are indicated. We study the asymptotic distribution of the test statistic. The problem of comparing mean locations for two samples is analyzed.Special attention is paid to observations on Stiefel manifolds including the orthogonal group O( p) and spheres S k&1 , and special orthogonal groups SO( p). The results also are illustrated with our experience with simulations. 1998Academic Press
In this paper we consider the important circumstances involved when risk managers are concerned with risks that exceed a certain threshold. Such conditions are well-known to insurance professionals, for instance in the context of policies involving deductibles and reinsurance contracts. We propose a new premium called tail variance premium (TVP) which answers the demands of these circumstances. In addition, we suggest a number of risk measures associated with TVP. While the well-known tail conditional expectation risk measure provides a risk manager with information about the average of the tail of the loss distribution, tail variance risk measure (TV) estimates the variability along such a tail. Furthermore, given a multivariate setup, we offer a number of allocation techniques which preserve different desirable properties (sub-additivity and fulladditivity, for instance). We are able to derive explicit expressions for TV and TVP, and risk capital decomposition rules based on them, in the general framework of multivariate elliptical distributions. This class is very popular among actuaries and risk managers because it contains distributions with marginals whose tails are heavier than those of normal distributions. This distinctive feature is desirable when modeling financial datasets. Moreover, according to our results, in some cases there exists an optimal threshold, such that by choosing it, an insurance company minimizes its risk.
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