Session: 47

AFIR/ERM

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Summary

The absolute deviation of the expected return on a portfolio from its required economic risk capital according to the expected shortfall method identifies with an expected shortfall deviation from the mean return, called portfolio shortfall risk. The natural risk contribution of each portfolio asset to the portfolio shortfall risk is called shortfall risk of the asset. Replacing the variance as a measure of risk in the classical portfolio selection model by the shortfall risk defines mean-shortfall portfolio selection. For some legitimated cases of mean-variance portfolio selection, namely the multivariate elliptical return distributions, both approaches lead to the same conclusions. An important situation, for which the alternative approach appears tractable under more general return distributions, is discussed.

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Session: 13

General Insurance

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Summary

We propose a copula based statistical method of fitting joint cumulative returns between a market index and a stock from the index family to daily data. Modifying the method of inference functions for margins (IFM method), we perform two separate maximum likelihood estimations of the univariate marginal distributions, assumed to be normal inverse gamma mixtures with kurtosis parameter equal to 6, followed by a minimization of the bivariate chi-square statistic associated to an adequate bivariate version of the usual Pearson goodness-of-fit test. Our copula fitting results for daily cumulative returns between the Swiss Market Index and a stock in the index family for an approximate one-year period are quite satisfactory. The best overall fits are obtained for the new linear Spearman copula, as well as for the Frank and Gumbel-Hougaard copulas. Finally, a significant application to covariance estimation for the linear Spearman copula is discussed.

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Session: 96

ASTIN

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Summary

Following the axiomatic approach to measures of statistical quantities initiated by van Zwet(1964) and developed by several other authors, we present a general axiomatic system for the measure of the quantities risk and price. We argue that risk and insurance price are closely related through the notion of risk loading, viewed as function of the measure of risk, and that risk should be closely related to the measures of scale, skewness and kurtosis. We consider "universal" measures of scale and risk, which can be adjusted for skewness and kurtosis. Concerning the measure of price, the distortion pricing principle introduced by Denneberg(1990), studied further by Wang(1996a/b), and justified axiomatically as insurance price in a competitive market setting by Wang et al.(1997), is a measure of price for our more general axiomatic system. Our presentation includes numerous examples, some of which have so far not been encountered in actuarial science.

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Werner Hürlimann, born 1953, has studied mathematics and physics at ETHZ, where he obtained his PhD in 1980 with a thesis in algebra.

After postdoctoral fellowhips at Yale University and at the Max Planck Institute in Bonn he became 1984 an actuary at Winterthur Life and Pensions. 

He has been visiting associate professor in actuarial science at the University of Toronto during the academic year 1988-89. 

He has written more than hundred papers, published in refereed journals or presented at International Colloquia.

His current interests in Actuarial Science and Finance encompass Multivariate Models of Risk Management, Portfolio Theory, Pricing Theoy, Ordering of Risks, etc.

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