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Tail Variance Premium with Applications for Elliptical Portfolio of Risks

  • Edward Furman (a1) and Zinoviy Landsman (a2)
Abstract

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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References
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Anderson, T.W. (1984) An Introduction to Multivariate Statistical Analysis, New York: Wiley.
Artzner, P., Delbaen, F., Eber, J.M., and Heath, D. (1999) Coherent Measures of Risk, Mathematical Finance, 9, 203228.
Bawa, V.S. (1975) “Optimal Rules for Ordering Uncertain Prospects”, Journal of Financial Economics, 2, 95121.
Bian, G. and Tiku, M.L. (1997) Bayesian Inference based on Robust Priors and MMLEstimators: Part 1, Symmetric Location-Scale Distributions, Statistics, 29, 317345.
Bingham, N.H. and Kiesel, R. (2002) Semi-Parametric Modelling in Finance: Theoretical Foundations, Quantitative Finance, 2, 241250.
Cambanis, S., Huang, S., and Simons, G. (1981) On the Theory of Elliptically Contoured Distributions, Journal of Multivariate Analysis, 11, 368385.
Embrechts, P. and Landsman, Z. (2004) “Multivariate Generalization of Stein’s Lemma for Elliptical Class of Distributions”, Technical Report N04-07-1, Actuarial Research Center, University of Haifa, Israel, http://uhda.haifa.ac.il
Embrechts, P., McNeil, A., and Straumann, D. (2001) Correlation and Dependence in Risk Management: Properties and Pitfalls, Risk Management: Value at Risk and Beyond, edited by Dempster, M. and Moffatt, H.K., Cambridge University Press.
Fang, K.T., Kotz, S. and Ng, K.W. (1987) Symmetric Multivariate and Related Distributions, London: Chapman & Hall.
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Volume 2, New York: Wiley.
Frees, E.W. (1998) Relative Importance of Risk Sources in Insurance Systems, North American Actuarial Journal, 2, 3452.
Furman, E. and Landsman, Z. (2005) Risk Capital Decomposition for a Multivariate Dependent Gamma Portfolio, Insurance: Mathematics and Economics, 37, 635649.
Gupta, A.K. and Varga, T. (1993) Elliptically Contoured Models in Statistics, Netherlands: Kluwer Academic Publishers.
Hürlimann, W. (2001a) Analytical Evaluation of Economic Risk Capital for Portfolio of Gamma Risks, ASTIN Bulletin, 31(1), 107122.
Hürlimann, W. (2001b) Financial Data Analysis with Two Symmetric Distributions, ASTIN Bulletin, 31(1), 187211.
Hürlimann, W. (2003) Conditional Value-at-Risk Bounds for Compound Poison risks and a Normal Approximation, Journal of Applied Mathematics, 3, 141153.
Hürlimann, W. (2004) Multivariate Frechet Copulas and Conditional Value-at-Risk, International Journal of Mathematics and Mathematical Sciencies, 7, 345364.
Hult, H. and Lindskog, F. (2002) Multivariate Extremes, Aggregation and Dependence in Elliptical Distributions, Advances in Applied Probability, 34, 587608.
Kaas, R., Goovaerts, M.J., Dhaene, J. and Denault, M. (2001) Modern Actuarial Risk Theory, Kluwer Academic Publishers.
Kelker, D. (1970) Distribution Theory of Spherical Distributions and Location-Scale Parameter Generalization, Sankhya, 32, 419430.
Kotz, S. (1975) Multivariate Distributions at a Cross-Road, Statistical Distributions in Scientific Work, 1, edited by Patil, G.K. and Kotz, S., D. Reidel Publishing Company.
Kotz, S., Balakrishnan, N. and Johnson, N.L. (2000) Continuous Multivariate Distributions, New York: Wiley.
Landsman, Z. (2006) On the generalization of Stein’s Lemma for elliptical class of distributions, Statistics and Probability Letters, 76, 10121016.
Landsman, Z. and Valdez, E. (2003) Tail Conditional Expectation for Elliptical Distributions, North American Actuarial Journal, 7(4).
Landsman, Z. and Valdez, E. (2005) Tail Conditional Expectation for Exponential Dispersion Models, ASTIN Bulletin, 35(1), 189209.
Landsman, Z. and Sherris, M. (2005). An Insurance and Asset Pricing Model for Non-Normal Distributions and Incomplete Markets, Technical report, Actuarial research Center, University of Haifa, http://arc.haifa.ac.il
Loève, M. (1977) Probability Theory 1, Springer-Verlag, New York Heidelberg Berlin.
MacDonald, J.B. (1996) Probability Distributions for Financial Models, Handbook of Statistics, 14, 427461.
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management, Princeton University Press, Princeton and Oxford.
Owen, J. and Rabinovitch, R. (1983) On the Class of Elliptical Distributions and their Applications to the Theory of Portfolio Choice, The Journal of Finance, 38(3), 745752.
Panjer, H. (Editor), Boyle, P., Cox, S., Dufresne, D., Mueller, H., Pedersen, H., Pliska, S., Sherris, M., Shiu, E. and Tan, K. (1998) Financial Economics: With Applications to Investments, Insurance and Pensions, Actuar. Foundation, Schaumburg.
Panjer, H.H. (2002) Measurement of Risk, Solvency Requirements, and Allocation of Capital within Financial Conglomerates, Institute of Insurance and Pension Research, University of Waterloo Research Report 01-15.
Ross, S.A. (1978) Mutual Fund Separation in Financial Theory - the Separating Distributions, Journal of Economic Theory, 17, 256286.
Schmidt, R. (2002) Tail Dependence for Elliptically Contoured Distributions, Mathematical Methods of Operations Research, 55, 301327.
Tobin, J. (1958) Liquidity Preference as Behavior Toward Risk, Review of Economic Studies, 25, 6586.
Valdez, E. and Chernich, A. (2003) Wang’s Capital Allocation Formula for Elliptically-Contoured Distributions, Insurance: Mathematics & Economics, 33, 517532.
Valdez, E. (2004) Tail Conditional Variance for Elliptically Contoured Distributions, working paper, University of New South Wales, Sydney, Australia.
Wang, S. (1998) An Actuarial Index of the Right-Tail Risk, North American Actuarial Journal, 2, 88101.
Wang, S. (2002) A Universal Framework for Pricing Financial and Insurance Risks, ASTIN Bulletin, 32, 213234.
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ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
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