Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T23:06:16.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Modeling and Comparing Dependencies in Multivariate Risk Portfolios

Published online by Cambridge University Press:  29 August 2014

Nicole Bäuerle  and
Alfred Müller
Nicole Bäuerle*
Affiliation:
Abteilung Mathematik VII, Operations Research, Universität Ulm
Alfred Müller*
Affiliation:
Institut für Wirtschaftstheorie, und Operations Research, Universität Karlsruhe
*
Abteilung Mathematik VII, (Operations Research), Helmholtzstr. 18, Universität Ulm, D-89069 Ulm, Germany
Institut für Wirtschaftstheorie und, Operations Research, Kaiserstr. 12, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we investigate multivariate risk portfolios, where the risks are dependent. By providing some natural models for risk portfolios with the same marginal distributions we are able to compare two portfolios with different dependence structure with respect to their stop-loss premiums. In particular, some comparison results for portfolios with two-point distributions are obtained. The analysis is based on the concept of the so-called supermodular ordering. We also give some numerical results which indicate that dependencies in risk portfolios can have a severe impact on the stop-loss premium. In fact, we show that the effect of dependencies can grow beyond any given bound.

Keywords

Dependent risksStop-loss premiumSupermodular orderStop-loss orderMajorizationComonotonicityExchangeable Bernoulli random variables
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 28 , Issue 1 , May 1998 , pp. 59 - 76
Copyright
Copyright © International Actuarial Association 1998

References

Arnold, B.C., Villasenor, J.A. (1986). Lorenz ordering of means and medians. Statistics and Probability Letters 4, 47–49.CrossRefGoogle Scholar
Bäuerle, N. (1997a). Inequalities for stochastic models via supermodular orderings. Communications in Statistics – Stochastic Models 13, 181–201.CrossRefGoogle Scholar
Bäuerle, N. (1997b). Monotonicity results for MR/GI/1 queues. Journal of Applied Probability 34, 514–524.CrossRefGoogle Scholar
Bäuerle, N., Rieder, U. (1997). Comparison results for Markov-modulated recursive models. Probability in the Engineering and Informational Sciences 11, 203–217.CrossRefGoogle Scholar
Chow, Y.S., Teicher, H. (1978). Probability Theory. Springer, New York.CrossRefGoogle Scholar
Dhaene, J., Goovaerts, M.J. (1996). Dependency of risks and stop-loss order. ASTIN Bulletin 26, 201–212.CrossRefGoogle Scholar
Dhaene, J., Goovaerts, M.J. (1997). On the dependency of risks in the individual life model. Insurance: Mathematics and Economics 19, 243–253.Google Scholar
Feller, W. (1966). An introduction to Probability Theory and Its Applications. Volume II. Wiley, New York.Google Scholar
Goovaerts, M.J., Kaas, R., van Heerwaarden, A.E., Bauwelinckx, T. (1990). Effective Actuarial Methods. Insurance Series vol. 3, North Holland.Google Scholar
Heilmann, W.-R. (1986). On the impact of independence of risks on stop loss transforms. Insurance: Mathematics and Economics 5, 197–199.Google Scholar
Hürlimann, W. (1993). Bivariate distributions with diatomic conditionals and stop-loss transforms of random sums. Statistics and Probability Letters 17, 329–335.CrossRefGoogle Scholar
Lefèvre, C., Utev, S. (1996). Comparing sums of exchangeable Bernoulli random variables. Journal of Applied Probability 33, 285–310.CrossRefGoogle Scholar
Madsen, R.W. (1993). Generalized binomial distributions. Communications in Statistics – Theory and Methods 22, 3065–3086.CrossRefGoogle Scholar
Marshall, A.W., Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Müller, A. (1996). Orderings of risks: A comparative study via stop-loss transforms. Insurance: Mathematics and Economics 17, 215–222.Google Scholar
Müller, A. (1997). Stop-Loss Order for Portfolios of Dependent Risks. Insurance: Mathematics and Economics 21, 219–223.Google Scholar
Schmeidler, D. (1986). Integral Representation without additivity. Proceedings of the American Mathematical Society 97, 255–261.CrossRefGoogle Scholar
Shaked, M., Shanthikumar, J.G. (1994). Stochastic Orders and their Applications. Academic Press, London.Google Scholar
Shaked, M., Shanthikumar, J.G. (1997). Supermodular Stochastic Orders and Positive Dependence of Random Vectors. Journal of Multivariate Analysis 61, 86–101.CrossRefGoogle Scholar
Shaked, M., Tong, Y.L. (1985). Some Partial Orderings of Exchangeable Random Variables by Positive Dependence. Journal of Multivariate Analysis 17, 333–349.CrossRefGoogle Scholar
Tchen, A.H. (1980). Inequalities for Distributions with given marginals. Annals of Probability 8, 814–827.CrossRefGoogle Scholar
Tong, Y.L. (1980). Probability Inequalities in multivariate Distributions. Academic Press, New York.Google Scholar
Tong, Y.L. (1989). Inequalities for a class of positively dependent random variables with a common marginal. Annals of Statistics 17, 429–435.CrossRefGoogle Scholar
Wang, S. (1997). Aggregation of Correlated Risk Portfolios: Models and Algorithms. Preprint.Google Scholar
Yaari, M.E. (1987). The dual theory of choice under risk. Econometrica 55, 95–115.CrossRefGoogle Scholar