Risk aggregation, dependence structure and diversification benefit

Bürgi, Roland and Dacorogna, Michel M and Iles, Roger (2008): Risk aggregation, dependence structure and diversification benefit. Forthcoming in:

Abstract

Insurance and reinsurance live and die from the diversification benefits or lack of it in their risk portfolio. The new solvency regulations allow companies to include them in their computation of risk-based capital (RBC). The question is how to really evaluate those benefits.

To compute the total risk of a portfolio, it is important to establish the rules for aggregating the various risks that compose it. This can only be done through modelling of their dependence. It is a well known fact among traders in financial markets that "diversification works the worst when one needs it the most''. In other words, in times of crisis the dependence between risks increases. Experience has shown that very large loss events almost always affect multiple lines of business simultaneously. September 11, 2001, is an example of such an event: when the claims originated simultaneously from lines of business which are usually uncorrelated, such as property and life, at the same time that the assets of the company were depreciated due to the crisis on the stock markets.

In this paper, we explore various methods of modelling dependence and their influence on diversification benefits. We show that the latter strongly depend on the chosen method and that rank correlation grossly overestimates diversification. This has consequences on the RBC for the whole portfolio, which is smaller than it should be when correctly accounting for tail correlation. However, the problem remains to calibrate the dependence for extreme events, which are rare by definition. We analyze and propose possible ways to get out of this dilemma and come up with reasonable estimates.

• Item Type: MPRA Paper
• Language: English
• Keywords: Risk-Based Capital, Hierarchical Copula, Dependence, Calibration
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C43 - Index Numbers and Aggregation; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General

References

Acerbi, C. and Tasche, D. (2002). On the coherence of expected shorfall. Journal of Banking \& Finance, 26(7):1487--1503.

Armstrong, M. (2003). Copula catalogue, part 1: Bivariate {A}rchimedean copulas. \url{http://www.cerna.ensmp.fr/Documents/MA-CopulaCatalogue.pdf}.

Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D.(1999). Coherent risk measures. Mathematical Finance, 9:203--228.

Berg, D. (2007). Copula goodness-of-fit testing: an overview and power comparison. Statistical Research Report, University of Oslo.

Bürgi, R. and Müller, U. (2008). Aggregating risks in hierarchical copula trees. To be published.

Demarta, S. and McNeil, A.~J. (2005). The {T} copula and related copulas. International Statistical Review, 73(1):111--129.

Embrechts, P., Lindskog, F., and McNeil, A. (2003). Handbook of Heavy Tailed Distributions in Finance, chapter Modelling Dependence with Copulas and Applications to Risk Management. Elsevier.

Embrechts, P., McNeil, A., and Straumann, D. (2002). Risk Management: Value at Risk and Beyond, chapter Correlation and Dependence in Risk Management: Properties and Pitfalls. Cambridge University Press.

Fang, K.~T. and Zhang, Y.~T. (1990). Generalized Multivariate Analysis. Springer.

Genest, C. and Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, pages 347--368.

Nelsen, R.~B. (1999). An Introduction to Copulas. Springer.

Savu, C. and Trede, M. (2006). Hierarchical {A}rchimedean copulas. \url{http://www.uni-konstanz.de/micfinma/conference/Files/papers/Savu_Trede.% pdf}.

Savu, C. and Trede, M. (2008). Goodness-of-fit tests for parametric families of {A}Archimedean copulas. Quantitative Finance, 8(2):109--116.

Sklar, A. (1958). Fonctions de répartition à n dimensions et leurs marges. Publications de l'Institut de Statistique de L'Universit\'e de Paris, 8:229--231.

Tasche, D. (2002). Expected shortfall and beyond. Journal of Banking \& Finance, 26(7):1519--1533.

Wang, S.~S. (1998). Aggregation of correlated risk portfolios: Models and algorithms. Proceedings of the Casualty Actuarial Society}, pages 848--939.

Whelan, N. (2004). Sampling from {A}rchimedean copulas. Quantitative Finance, 4:339--352.