Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-15T13:25:25.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Measurement and Modelling of Dependencies in Economic Capital

Published online by Cambridge University Press:  21 February 2012

R. A. Shaw ,
A. D. Smith  and
G. S. Spivak
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper covers a number of different topics related to the measurement and modelling of dependency within economic capital models. The scope of the paper is relatively wide. We address in some detail the different approaches to modelling dependencies ranging from the more common variance-covariance matrix approach, to the consideration of the use of copulas and the more sophisticated causal models that feature feedback loops and other systems design ideas.

There are many data and model uncertainties in modelling dependency and so we have also endeavoured to cover topics such as spurious relationships and wrong-way risk to highlight some of the uncertainties involved.

With the advent of the internal model approval process under Solvency II, senior management needs to have a greater understanding of dependency methodology. We have devoted a section of this paper to a discussion of possible different ways to communicate the results of modelling to the board, senior management and other interested parties within an insurance company.

We have endeavoured throughout this paper to include as many numerical examples as possible to help in the understanding of the key points, including our discussion of model parameterisation and the communication to an insurance executive of the impact of dependency on economic capital modelling results.

The economic capital model can be seen as a combination of two key components: the marginal risk distribution of each risk and the aggregation methodology which combines these into a single aggregate distribution or capital number. This paper is concerned with the aggregation part, the methods and assumptions employed and the issues arising, and not the determination of the marginal risk distributions which is equally of importance and in many cases equally as complex.

Keywords

Archimedean CopulaBivariate CopulaCausal ModelCoefficient Of Tail DependenceCopulaCopula VinesCorrelationDependencyEconomic CapitalElliptical DistributionsGaussian CopulaHalf SpaceKendall TauLinear correlationPearson CorrelationPositive Semi-DefiniteSpearman CorrelationSpurious RelationshipT CopulaTail Concentration FunctionTail CorrelationTail DependencyTime SeriesVariance-Covariance Matrix
Type
Sessional meetings: papers and abstracts of discussions
Information
British Actuarial Journal , Volume 16 , Issue 3 , September 2011 , pp. 601 - 699
Copyright
Copyright © Institute and Faculty of Actuaries 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

The following list includes, not only the works referred to in the paper, but also other publications which would be of use to readers.Google Scholar
Aas, K., Bakken, H., Czado, C., Frigessi, A. (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 44(2), 182–198, ISSN 0167-6687.Google Scholar
Anscombe, F.J. (1973). Graphs in Statistical Analysis. The American Statistician, 27(1), 17–21.Google Scholar
Barnett, J., Kreps, R., Major, J., Venter, G. (2007). Multivariate Copulas for Financial Modelling. Casualty Actuarial Society, 1(1), 103–199.Google Scholar
Basel Committee Of Banking Supervision. (2009). Range of practices and issues in economic capital frameworks. Bank for International Settlements. Available at http://www.bis.org/publ/bcbs152.htmGoogle Scholar
Bedford, T., Cooke, R. (2001). Probabilistic Risk Analysis: Foundations and Methods. Cambridge University Press. ISBN: 0521773202.Google Scholar
Brooks, D., Care, R.J., Chaplin, M.B., Kaufman, A.M., Morgan, K.A., Roberts, D.N., Skinner, J.M., Huntington-Thresher, D.J.K., Tuley, P.J., Wong, D.L. (2009). Actuarial Aspects of Internal Models for Solvency II. British Actuarial Journal, 15(2), 367–482.Google Scholar
Budden, M., Hadavas, P., Hoffman, L. (2008). On the Generation of Correlation Matrices. Applied Mathematics E-Notes, 8, pp. 279–282.Google Scholar
CEIOPS. (2008). QIS4 Technical Specifications (MARKT/2505/08). Brussels.Google Scholar
CEIOPS. (2010a). Calibration of Market Risk Module (formerly CP70). CEIOPS’ Advice for Level 2 Implementing Measures on Solvency II.Google Scholar
CEIOPS. (2010b). Equity risk sub-module (formerly CP69). CEIOPS’ Advice for Level 2 Implementing Measures on Solvency II.Google Scholar
CEIOPS. (2010c). SCR Standard Formula Article 111(d) Correlations (formerly CP74). CEIOPS’ Advice for Level 2 Implementing Measures on Solvency II.Google Scholar
CEIOPS. (2010d). Tests and Standards for Internal Model Approval (formerly CP56). CEIOPS’ Advice for Level 2 Implementing Measures on Solvency II.Google Scholar
CRO FORUM. (2009). Calibration Proposals. Available at: http://www.thecroforum.org/publication/calibration_principles/Google Scholar
Dorey, M., Joubert, P., Vencatasawmy, C. (2005). Modelling Dependencies – an Overview. Staple Inn Actuarial Society. Available at: http://www.sias.org.uk/siaspapers/listofpapers/view_paper?id=DependenciesGoogle Scholar
Embrechts, P., Frey, R., McNeil, A. (2005). Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton Series in Finance, Princeton University Press. ISBN 0-691-12255-5.Google Scholar
Embrechts, P., Lindskog, F., McNeil, A.J. (2003). Modelling Dependence with Copulas and Applications to Risk Management. In Handbook of heavy tailed distributions in finance, edited by Rachev ST, published by Elsevier/North-Holland, Amsterdam. Available at: http://www.ma.hw.ac.uk/~mcneil/ftp/DependenceWithCopulas.pdfGoogle Scholar
Embrechts, P., McNeil, A., Straumann, A. (1999). Correlation and Dependence in Risk Management: Properties and Pitfalls. Preprint, ETH Zurich. Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.73.2469&rep=rep1&type=pdfGoogle Scholar
Embrechts, P., McNeil, A., Straumann, D. (2002). Correlation and Dependence in Risk Management. Chapter in Risk Management: Value at Risk and Beyond, Cambridge University Press. Available at: www.ma.hw.ac.uk/~mcneil/ftp/pitfalls.pdfGoogle Scholar
Fang, K.T., Kots, S., Ng, K.W. (1990). Symmetric Multivariate and Related Distributions. Chapman & Hall, London.Google Scholar
Frankland, R., Smith, A.D., Wilkins, T., Varnell, E., Holtham, A., Biffis, E., Eshun, S., Dullaway, D. (2008). Modelling Extreme Market Events: A Report of the Benchmarking Stochastic Models Working Party. British Actuarial Journal, 15, 99–201.Google Scholar
Groupe Consultatif. (2005). Diversification. Technical paper by the Solvency II Groupe Consultatif Working Group.Google Scholar
Guy Carpenter & Co. (2007). Enterprise Risk Analysis for Property & Liability Insurance Companies.Google Scholar
Insurance ERM. (2009). Trying to find the sting in the tail. Available at: http://www.insuranceerm.com/analysis/trying-to-find-the-sting-in-the-tail.htmlGoogle Scholar
International Actuarial Association (IAA). (2004). A Global Framework for Insurer Solvency Assessment. Report by the Insurer Solvency Assessment Working Party of the International Actuarial Association.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar
Kat, H.M. (2002). The Dangers of Using Correlation to Measure Dependence. ISMA Centre, The University of Reading. Available at: http://www.icmacentre.ac.uk/pdf/discussion/DP2002-23.pdfGoogle Scholar
Mcneil, A., Smith, A.D. (2009). Reverse Stress Testing. Paper presented to the 2009 Life Convention; Institute & Faculty of Actuaries.Google Scholar
Mikosch, T. (2005). Copulas: Tales and Facts. University of Copenhagen. Available at: http://www.math.ku.dk/~mikosch/Preprint/Copula/s.pdfGoogle Scholar
Nelsen, R. (1999). An Introduction to Copulas. Springer, NY.Google Scholar
Noether, G.E. (1986). Why Kendall Tau? In The Best of Teaching Statistics. Teaching Statistics. Available at: http://www.rsscse.org.uk/ts/bts/noether/text.htmlGoogle Scholar
Rousseeuw, P.J., Ruts, I. (1999). The depth function of a population distribution. Available at: ftp://ftp.win.ua.ac.be/pub/preprints/99/Depfun99.pdfGoogle Scholar
Rousseeuw, P.J., Ruts, I., Tukey, J.W. (1999). The Bagplot: A Bivariate Boxplot. The American Statistician, 53, 382–387. Available at: ftp://ftp.win.ua.ac.be/pub/preprints/99/Bagbiv99.pdfGoogle Scholar
Schmidt, R. (2005). Tail Dependence. In Statistical Tools for Finance and Insurance. Springer.Google Scholar
Shaw, J. (2007). Beyond VAR and Stress Testing in VAR: Understanding and Applying Value at Risk, pp. 221–224. Risk Publications, London.Google Scholar
Shaw, R., Spivak, G. (2009). Correlations and dependencies in economic capital models. Paper presented to the 2009 Risk and Investment Conference. Institute and Faculty of Actuaries.Google Scholar
Sheldon, T.J., Smith, A.D. (2005). Do I really need all those sims? An Introduction to Malorie-Leslie Techniques. Presentation to the 2005 Life Convention, Cardiff; Institute and Faculty of Actuaries.Google Scholar
Smith, A.D. (2002). Dependent Tails. Paper to the 2002 General Insurance Convention; Institute and Faculty of Actuaries. Available at: http://www.actuaries.org.uk/research-and-resources/documents/dependent-tailsGoogle Scholar
Smith, A.D. (2009). Correlations and Dependency. In Solvency II Handbook, Risk Publications.Google Scholar
The Math Works. (2009). Copulafit. Matlab.Google Scholar
Tsanakas, A., Smith, A.D. (2007). High Dimensional Modelling and Simulation with Asymmetric Normal Mixtures. Available at: www.ssrn.com/abstract=1005894Google Scholar
Tufte, E.R. (2006). The Cognitive Style of PowerPoint: Pitching Out Corrupts Within. Cheshire. Connecticut: Graphics Press.Google Scholar
Tukey, J. (1975). Mathematics and the picturing of data. Proceedings of the International Congress of Mathematics; Vancouver, pp. 523–531.Google Scholar
Venter, G. (2002). Tails of Copulas. Proceedings of CAS LXXXIX, pp. 68–113.Google Scholar
West, G. (2009). Better approximations to cumulative normal functions. Wilmott Magazine. July 2009. Available at: www.wilmott.com/pdfs/090721_west.pdfGoogle Scholar
Wikipedia. (2009). Spurious relationship. Available at: www.wikipedia.org/wiki/Spurious_relationshipGoogle Scholar
Wikipedia. (2010). Feedback. Available at: www.wikipedia.org/wiki/FeedbackGoogle Scholar
Wikipedia. (2010). Positive Feedback. Available at: www.wikipedia.org/wiki/Reinforcing_loopGoogle Scholar