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Validation of aggregated risks models

  • Michel Dacorogna (a1), Laila Elbahtouri (a2) and Marie Kratz (a3)
Abstract

Validation of risk models is required by regulators and demanded by management and shareholders. Those models rely in practice heavily on Monte Carlo (MC) simulations. Given their complexity, the convergence of the MC algorithm is difficult to prove mathematically. To circumvent this problem and nevertheless explore the conditions of convergence, we suggest an analytical approach. Considering standard models, we compute, via mixing techniques, closed form formulas for risk measures as Value-at-Risk (VaR) VaR or Tail Value-at-Risk (TVaR) TVaR on a portfolio of risks, and consequently for the associated diversification benefit. The numerical convergence of MC simulations of those various quantities is then tested against their analytical evaluations. The speed of convergence appears to depend on the fatness of the tail of the marginal distributions; the higher the tail index, the faster the convergence. We also explore the behaviour of the diversification benefit with various dependence structures and marginals (heavy and light tails). As expected, it varies heavily with the type of dependence between aggregated risks. The diversification benefit is also studied as a function of the risk measure, VaR or TVaR.

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Copyright
Corresponding author
*Correspondence to: ESSEC Business School, CREAR Risk Research Center, Avenue Bernard Hirsch, BP50105, 95021 Cergy-Pontoise, France. E-mail: kratz@essec.edu
References
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Albano, M., Amdeberhan, T., Beyerstedt, E. & Moll, V.H. (2011). The integrals in Gradshteyn and Ryzhik. Part 19: the error function. SCIENTIA Series A: Mathematical Sciences, 21, 2542.
Albrecher, H., Constantinescu, C. & Loisel, S. (2011). Explicit ruin formulas for models with dependence among risks. Insurance: Mathematics and Economics, 48(2), 265270.
Bürgi, R., Dacorogna, M.M. & Iles, R. (2008). Risk aggregation, dependence structure and diversification benefit. In D. Rösch & H. Scheule (Eds.), Stress Testing for Financial Institutions (pp. 265–306), Riskbooks, Incisive Media, London.
Busse, M., Dacorogna, M.M. & Kratz, M. (2014). The impact of systemic risk on the diversification benefits of a risk portfolio. Risks, 2, 260276.
Constantinescu, C., Hashorva, E. & Ji, L. (2011). Archimedean copulas in finite and infinite dimensions – with application to ruin problems. Insurance: Mathematics and Economics, 49, 487495.
Dacorogna, M. (2017). Approaches and techniques to validate internal model results. Available online at the address https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2983837 [accessed May 2017].
Dacorogna, M., Elbahtouri, L. & Kratz, M. (2015–2016). Explicit diversification benefit for dependent risks, ESSEC WP 1522 (2015) & SCOR Paper 38.
Emmer, S., Kratz, M. & Tasche, D. (2015). What is the best risk measure in practice? A comparison of standard measures. Journal of Risk, 18(2), 3160.
Feller, W. (1966). An Introduction to Probability Theory and its Applications (Vol. II). Wiley, New-York.
Gradshteyn, I.S. (1988). Tables of Integrals, Series, and Products, Academic Press, San Diego.
Hashorva, E. & Ji, L. (2014). Random shifting and scaling of insurance risks. Risks, 2, 277288.
Kohl, K.T. & Moll, V.H. (2011). The integrals in Gradshteyn and Ryzhik. Part 20: hypergeometric functions. SCIENTIA Series A: Mathematical Sciences, 21, 4354.
Kratz, M. (2014). Normex, a new method for evaluating the distribution of aggregated heavy tailed risks. Application to risk measures. Extremes 17(4), Special issue on Extremes and Finance (Guest Ed. P. Embrechts), 661–691.
Marshall, A.W. & Olkin, I. (1988). Families of Multivariate Distributions. Journal of the American Statistical Association, 83, 834841.
Oakes, D. (1989). Bivariate Survival Models Induced by Frailties. Journal of the American Statistical Association, 84, 487493.
Sarabia, J., Gómez-Déniz, E., Prietoa, F. & Jordá, V. (2017). Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions. Available online at the address https://arxiv.org/abs/1705.00289 [accessed May 2017].
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Annals of Actuarial Science
  • ISSN: 1748-4995
  • EISSN: 1748-5002
  • URL: /core/journals/annals-of-actuarial-science
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