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Estimating Copulas for Insurance from Scarce Observations, Expert Opinion and Prior Information: A Bayesian Approach

Published online by Cambridge University Press:  09 August 2013

Philipp Arbenz  and
Davide Canestraro
Davide Canestraro
Affiliation:
SCOR SE, Zurich Branch, General Guisan – Quai 26, CH-8022 Zürich, Switzerland, E-mail: [email protected]
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Abstract

A prudent assessment of dependence is crucial in many stochastic models for insurance risks. Copulas have become popular to model such dependencies. However, estimation procedures for copulas often lead to large parameter uncertainty when observations are scarce. In this paper, we propose a Bayesian method which combines prior information (e.g. from regulators), observations and expert opinion in order to estimate copula parameters and determine the estimation uncertainty. The combination of different sources of information can significantly reduce the parameter uncertainty compared to the use of only one source. The model can also account for uncertainty in the marginal distributions. Furthermore, we describe the methodology for obtaining expert opinion and explain involved psychological effects and popular fallacies. We exemplify the approach in a case study.

Keywords

CopulasExpert judgmentInsuranceDependence measureBayesian inferenceCorrelationRisk management
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 1 , May 2012 , pp. 271 - 290
Copyright
Copyright © International Actuarial Association 2012

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References

Bernardo, J. and Smith, A. (1994) Bayesian Theory. Wiley, Chichester.Google Scholar
Blum, P., Dias, A. and Embrechts, P. (2002) The art of dependence modelling: the latest advances in correlation analysis. In Lane, M., editor, Alternative Risk Strategies, pages 339–356. Risk Books, London.Google Scholar
Böcker, K., Crimmi, A. and Fink, H. (2010) Bayesian risk aggregation: Correlation uncertainty and expert judgement. In Böcker, K., editor, Rethinking Risk Measurement and Reporting Uncertainty. RiskBooks, London.Google Scholar
Bühlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and its Applications. Springer, Berlin.Google Scholar
CEIOPS (2010) QIS5 Technical Specifications. Technical report, Committee of European Insurance and Occupational Pensions Supervisors.Google Scholar
Clemen, R., Fischer, G. and Winkler, R. (2000) Assessing dependence: Some experimental results. Management Science, 46(8), 1100–1115.Google Scholar
Clemen, R. and Winkler, R. (1999) Combining probability distributions from experts in risk analysis. Risk Analysis, 19(2), 187–203.Google Scholar
Cooke, R. (1991) Experts in Uncertainty: Opinion and Subjective Probability in Science. Oxford University Press, New York.CrossRefGoogle Scholar
Cooke, R. and Goossens, L. (2000) Procedures guide for structural expert judgement in accident consequence modelling. Radiation Protection Dosimetry, 90(3), 303–309.CrossRefGoogle Scholar
Cooke, R., Mendel, M. and Thijs, W. (1988) Calibration and information in expert resolution. Auto-matica, 24(1), 87–94.Google Scholar
Daneshkhah, A. (2004) Uncertainty in probabilistic risk assessment: A review. Working paper, The University Of Sheffield. www.sheffield.ac.uk/content/1/c6/03/09/33/risk.pdf.Google Scholar
Donnelly, C. and Embrechts, P. (2010) The devil is in the tails: actuarial mathematics and the sub-prime mortgage crisis. ASTIN Bulletin, 40(1), 1–33.Google Scholar
Eddy, D. (1982) Probabilistic reasoning in clinical medicine: Problems and opportunities. In Kahneman, D., Slovic, P., and Tversky, A., editors, Judgment under uncertainty: Heuristics and biases, pages 249–267. Cambridge University Press, Cambridge.Google Scholar
Embrechts, P., McNeil, A. and Straumann, D. (2002) Correlation and dependence in risk management: Properties and pitfalls. In Dempster, M., editor, Risk Management: Value at Risk and Beyond, pages 176–223. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
FOPI (2006) Technical document on the Swiss Solvency Test. Technical report, Swiss Federal Office of Private Insurance.Google Scholar
Genest, C., Gendron, M. and Bourdeau-Brien, M. (2009) The advent of copulas in finance. The European Journal of Finance, 15(7–8), 609–618.CrossRefGoogle Scholar
Gokhale, D. and Press, S. (1982) Assessment of a prior distribution for the correlation coefficient in a bivariate normal distribution. Journal of the Royal Statistical Society, Series A, 145, 237–249.CrossRefGoogle Scholar
Hall, J. (2010) The Total Cost of Fire in the United States. National Fire Protection Association, Quincy, Massachusetts.Google Scholar
Haug, S., Klüppelberg, C. and Peng, L. (2011) Statistical models and methods for dependence in insurance data. Journal of the Korean Statistical Society, 40, 125–139.CrossRefGoogle Scholar
Hofert, M. (2010) Sampling Nested Archimedean Copulas with Applications to CDO Pricing. PhD thesis, University of Ulm.Google Scholar
Jouini, M. and Clemen, R. (1996) Copula models for aggregating expert opinions. Operations Research, 44(3), 444–57.Google Scholar
Kahneman, D. and Tversky, A. (1973) On the psychology of prediction. Psychological review, 80(4), 237–251.CrossRefGoogle Scholar
Kahneman, D. and Tversky, A. (1982) On the study of statistical intuitions. Cognition, 11, 123–141.CrossRefGoogle Scholar
Kallen, M. and Cooke, R. (2002) Expert aggregation with dependence. In Probabilistic Safety Assessment and Management, pages 1287–1294. International Association for Probabilistic Safety Assessment and Management.Google Scholar
Kynn, M. (2008) The “heuristics and biases” bias in expert elicitation. Journal of the Royal Statistical Society: Series A (Statistics in Society), 171(1), 239–264.CrossRefGoogle Scholar
Lambrigger, D., Shevchenko, P. and Wüthrich, M. (2007) The quantification of operational risk using internal data, relevant external data and expert opinions. Journal of Operational Risk, 2(3), 3–27.Google Scholar
McNeil, A., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press, Princeton.Google Scholar
Meyer, M. and Booker, J. (2001) Eliciting and Analyzing Expert Judgment: A Practical Guide. Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar
Morgan, M. and Henrion, M. (1992) Uncertainty: A Guide to Dealing With Uncertainty in Quantitative Risk and Policy Analysis. Cambridge University Press, Cambridge.Google Scholar
Mosleh, A., Bier, V. and Apostolakis, G. (1988) A critique of current practice for the use of expert opinions in probabilistic risk assessment. Reliability Engineering and system safety, 20(1), 63–85.Google Scholar
Nelsen, R. (2006) An Introduction to Copulas. Springer, New York, 2nd edition.Google Scholar
O'Hagan, A., Buck, C., Daneshkhah, A., Eiser, J., Garthwaite, P., Jenkinson, D., Oakley, J. and Rakow, T. (2006) Uncertain Judgements: Eliciting Experts' Probabilities. Wiley, Chichester.CrossRefGoogle Scholar
Ouchi, F. (2004) A literature review on the use of expert opinion in probabilistic risk analysis. Working Paper 3201, World Bank Policy Research.Google Scholar
Price, H. and Manson, A. (2002) Uninformative priors for Bayes' theorem. Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 617(1), 379–391.CrossRefGoogle Scholar
Robert, C. and Casella, G. (2005) Monte Carlo Statistical Methods. Springer, New York.Google Scholar
Van der Vaart, A. (1998) Asymptotic Statistics. Cambridge University Press, Cambridge.Google Scholar
Winkler, R. (1968) The consensus of subjective probability distributions. Management Science, 15, B61–B75.CrossRefGoogle Scholar