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Risk aggregation in the presence of discrete causally connected random variables

Published online by Cambridge University Press:  26 August 2014

Peng Lin ,
Martin Neil  and
Norman Fenton
Peng Lin*
Affiliation:
Phd candidate, School of Electronic Engineering and Computer Science, Queen Mary, University of London, UK
Martin Neil
Affiliation:
Agena Ltd and Professor of Computer Science and Statistics, School of Electronic Engineering and Computer Science, Queen Mary, University of London, UK
Norman Fenton
Affiliation:
CEO, Agena Ltd and Professor of Risk Information Management, School of Electronic Engineering and Computer Science, Queen Mary, University of London, UK
*
*Correspondence to: Peng Lin, Department of EECS, Queen Mary, University of London, UK. Tel: +44 (0) 20 78828027; Fax: +44 (0) 870 131 8460; E-mail: p.lin@qmul.ac.uk
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Abstract

Risk aggregation is a popular method used to estimate the sum of a collection of financial assets or events, where each asset or event is modelled as a random variable. Applications include insurance, operational risk, stress testing and sensitivity analysis. In practice, the sum of a set of random variables involves the use of two well-known mathematical operations: n-fold convolution (for a fixed number n) and N-fold convolution, defined as the compound sum of a frequency distribution N and a severity distribution, where the number of constant n-fold convolutions is determined by N, where the severity and frequency variables are independent, and continuous, currently numerical solutions such as, Panjer’s recursion, fast Fourier transforms and Monte Carlo simulation produce acceptable results. However, they have not been designed to cope with new modelling challenges that require hybrid models containing discrete explanatory (regime switching) variables or where discrete and continuous variables are inter-dependent and may influence the severity and frequency in complex, non-linear, ways. This paper describes a Bayesian Factorisation and Elimination (BFE) algorithm that performs convolution on the hybrid models required to aggregate risk in the presence of causal dependencies. This algorithm exploits a number of advances from the field of Bayesian Networks, covering methods to approximate statistical and conditionally deterministic functions to factorise multivariate distributions for efficient computation. Experiments show that BFE is as accurate on conventional problems as competing methods. For more difficult hybrid problems BFE can provide a more general solution that the others cannot offer. In addition, the BFE approach can be easily extended to perform deconvolution for the purposes of stress testing and sensitivity analysis in a way that competing methods do not.

Keywords

Risk aggregationBayesian factorisation and eliminationConvolutionDeconvolutionCausal dependency modellingBayesian NetworksDynamic discretisation
Type
Papers
Information
Annals of Actuarial Science , Volume 8 , Issue 2 , September 2014 , pp. 298 - 319
Copyright
© Institute and Faculty of Actuaries 2014 

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References

AgenaRisk. (2014). AgenaRisk. Available online at the address http://www.agenarisk.com/ (accessed 23 December 2014).Google Scholar
Arbenz, P. & Canestraro, D. (2012). Estimating copulas for insurance from scarce observations, expert opinion and prior information: a Bayesian approach. ASTIN Bulletin, 42(1), 271–290.Google Scholar
Black, F. & Litterman, R.B. (1991). Asset allocation: combining investor views with market equilibrium. The Journal of Fixed Income, 1(2), 7–18.CrossRefGoogle Scholar
Brechmann, E.C. (2014). Hierarchical Kendall copulas: properties and inference. Canadian Journal of Statistics, 42(1), 78–108.Google Scholar
Bruneton, J.-P. (2011). Copula-based hierarchical aggregation of correlated risks. The behaviour of the diversification benefit in Gaussian and lognormal trees. arXiv:1111.1113 [q-Fin]. Retrieved from http://arxiv.org/abs/1111.1113 (accessed 20 February 2014).Google Scholar
Cooper, G.F. & Herskovits, E. (1992). A Bayesian method for the induction of probabilistic networks from data. Machine Learning, 9(4), 309–347.CrossRefGoogle Scholar
Cowell, R.G., Verrall, R.J. & Yoon, Y.K. (2007). Modeling operational risk with Bayesian networks. Journal of Risk and Insurance, 74(4), 795–827.Google Scholar
Embrechts, P. (2009). Copulas: a personal view. Journal of Risk and Insurance, 76(3), 639–650.CrossRefGoogle Scholar
Fishman, G. (1996). Monte Carlo: Concepts, Algorithms, and Applications. New York: Springer.Google Scholar
Fenton, N. & Neil, M. (2012). Risk Assessment and Decision Analysis with Bayesian Networks. Boca Raton, Florida: CRC Press.Google Scholar
Heckman, P.E. & Meyers, G.G. (1983). The calculation of aggregate loss distributions (pp. 22–61). Presented at the Proceedings of the Casualty Actuarial Society. Retrieved from http://casact.net/pubs/proceed/proceed83/83022.pdf (accessed 23 December 2013).Google Scholar
IMF. (2009). IMF global financial stability report – navigating the financial challenges ahead. Washington, DC.Google Scholar
Jensen, F.V. & Nielsen, T.D. (2009). Bayesian Networks and Decision Graphs. New York: Springer.Google Scholar
Laeven, L. & Valencia, F. (2008). Systemic Banking Crises: A New Database (SSRN Scholarly Paper No. 1278435). Social Science Research Network, Rochester, NY.Google Scholar
Lauritzen, S.L. (1996). Graphical Models. Oxford University Press.Google Scholar
Marquez, D., Neil, M. & Fenton, N. (2010). Improved reliability modeling using Bayesian networks and dynamic discretization. Reliability Engineering & System Safety, 95(4), 412–425.CrossRefGoogle Scholar
McNeil, A.J., Frey, R. & Embrechts, P. (2010). Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton, New Jersey: Princeton University Press.Google Scholar
Meucci, A. (2008). Fully Flexible Views: Theory and Practice (SSRN Scholarly Paper No. 1213325). Social Science Research Network, Rochester, NY.Google Scholar
Neil, M., Chen, X. & Fenton, N. (2012). Optimizing the calculation of conditional probability tables in hybrid Bayesian networks using binary factorization. IEEE Transactions on Knowledge and Data Engineering, 24(7), 1306–1312.Google Scholar
Neil, M. & Fenton, N. (2008). Using Bayesian networks to model the operational risk to information technology infrastructure in financial institutions. Journal of Financial Transformation, 22, 131–138.Google Scholar
Neil, M., Tailor, M. & Marquez, D. (2007). Inference in hybrid Bayesian networks using dynamic discretization. Statistics and Computing, 17, 219–233.CrossRefGoogle Scholar
Nelsen, R.B. (2007). An Introduction to Copulas. New York: Springer.Google Scholar
Panjer, H.H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin, 1(12), 22–26.CrossRefGoogle Scholar
Pearl, J. (1993). [Bayesian analysis in expert systems]: comment: graphical models, causality and intervention. Statistical Science, 8(3), 266–269.Google Scholar
Politou, D. & Giudici, P. (2009). Modelling operational risk losses with graphical models and copula functions. Methodology and Computing in Applied Probability, 11(1), 65–93.CrossRefGoogle Scholar
R. (2013). The R Project for Statistical Computing. Available online at the address http://www.r-project.org/ (accessed 23 December 2013).Google Scholar
Rebonato, R. (2010). Coherent Stress Testing: A Bayesian Approach to the Analysis of Financial Stress. West Sussex, UK: John Wiley & Sons.Google Scholar
Spiegelhalter, D.J. & Lauritzen, S.L. (1990). Sequential updating of conditional probabilities on directed graphical structures. Networks, 20(5), 579–605.Google Scholar