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Risk Measures and Multivariate Extensions of Breiman's Theorem

Part of: Applications Mathematical economics

Published online by Cambridge University Press:  04 February 2016

Anne-Laure Fougeres  and
Cecile Mercadier
Anne-Laure Fougeres*
Affiliation:
Université Lyon 1
Cecile Mercadier*
Affiliation:
Université Lyon 1
*
∗ Postal address: Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne cedex, France.
∗ Postal address: Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne cedex, France.
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Abstract

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The modeling of insurance risks has received an increasing amount of attention because of solvency capital requirements. The ruin probability has become a standard risk measure to assess regulatory capital. In this paper we focus on discrete-time models for the finite time horizon. Several results are available in the literature to calibrate the ruin probability by means of the sum of the tail probabilities of individual claim amounts. The aim of this work is to obtain asymptotics for such probabilities under multivariate regular variation and, more precisely, to derive them from extensions of Breiman's theorem. We thus present new situations where the ruin probability admits computable equivalents. We also derive asymptotics for the value at risk.

Keywords

Discrete-time modelruin probabilityvalue at riskmultivariate regular variationdependent risk

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 2 , June 2012 , pp. 364 - 384
Copyright
© Applied Probability Trust 

References

Alink, S., Löwe, M. and Wüthrich, M. V. (2004). Diversification of aggregate dependent risks. Insurance Math. Econom. 35, 77–95.Google Scholar
Alink, S., Löwe, M. and Wüthrich, M. V. (2005). Analysis of the expected shortfall of aggregate dependent risks. ASTIN Bull. 35, 25–43.Google Scholar
Arbenz, P., Embrechts, P. and Puccetti, G. (2011). The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables. Bernoulli 17, 562–591.CrossRefGoogle Scholar
Barbe, P., Fougères, A.-L. and Genest, C. (2006). On the tail behaviour of sums of dependent risks. ASTIN Bull. 36, 361–373.Google Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.Google Scholar
Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes. John Wiley, Chichester.CrossRefGoogle Scholar
Cai, J. (2002). Ruin probabilities with dependent rates of interest. J. Appl. Prob. 39, 312–323.Google Scholar
Chen, Y. and Su, C. (2006). Finite time ruin probability with heavy-tailed insurance and financial risks. Statist. Prob. Lett. 76, 1812–1820.Google Scholar
Cossette, H. and Marceau, E. (2000). The discrete-time risk model with correlated classes of business. Insurance Math. Econom. 26, 133–149.Google Scholar
Davis, R. A. and Mikosch, T. (2001). Point process convergence of stochastic volatility processes with application to sample autocorrelation. In Probability, Statistics and Seismology (J. Appl. Prob. Spec. Vol. 38A), ed. Daley, D. J., Applied Probability Trust, Sheffield, pp. 93–104.Google Scholar
De Haan, L. and Resnick, S. I. (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40, 317–337.CrossRefGoogle Scholar
Degen, M., Lambrigger, D. and Segers, J. (2010). Risk concentration and diversification: second-order properties. Insurance Math. Econom. 46, 541–546.Google Scholar
Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.Google Scholar
Embrechts, P., Lambrigger, D. D. and Wüthrich, M. V. (2009a). Multivariate extremes and the aggregation of dependent risks: examples and counter-examples. Extremes 12, 107–127.Google Scholar
Embrechts, P., Nešlehová, J. and Wüthrich, M. V. (2009b). Additivity properties for value-at-risk under Archimedean dependence and heavy-tailedness. Insurance Math. Econom. 44, 164–169.Google Scholar
Foss, S. and Richards, A. (2010). On sums of conditionally independent subexponential random variables. Math. Operat. Res. 35, 102–119.Google Scholar
Goovaerts, M. J. et al. (2005). The tail probability of discounted sums of Pareto-like losses in insurance. Scand. Actuarial J.. 2005, 446–461.Google Scholar
Hult, H. and Samorodnitsky, G. (2008). Tail probabilities for infinite series of regularly varying random vectors. Bernoulli 14, 838–864.CrossRefGoogle Scholar
Kortschak, D. and Albrecher, H. (2009). Asymptotic results for the sum of dependent non-identically distributed random variables. Methodology Comput. Appl. Prob. 11, 279–306.Google Scholar
Mainik, G. and Rüschendorf, L. (2009). On optimal portfolio diversification with respect to extreme risks. Preprint.Google Scholar
Maulik, K., Resnick, S. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699.Google Scholar
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370.Google Scholar
Ng, K. W., Tang, Q. H. and Yang, H. (2002). Maxima of sums of heavy-tailed random variables. ASTIN Bull. 32, 43–55.Google Scholar
Nyrhinen, H. (1999). On the ruin probabilities in a general economic environment. Stoch. Process. Appl. 83, 319–330.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.Google Scholar
Sgibnev, M. S. (1996). On the distribution of the maxima of partial sums. Statist. Prob. Lett. 28, 235–238.Google Scholar
Tang, Q. and Tsitsiashvili, G. (2003a). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108, 299–325.CrossRefGoogle Scholar
Tang, Q. and Tsitsiashvili, G. (2003b). Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6, 171–188.Google Scholar
Wang, D. and Tang, Q. (2006). Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22, 253–272.Google Scholar
Wang, D., Su, C. and Zeng, Y. (2005). Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory. Sci. China Ser. A, 48, 1379–1394.Google Scholar
Wüthrich, M. V. (2003). Asymptotic value-at-risk estimates for sums of dependent random variables. ASTIN Bull. 33, 75–92.Google Scholar
Zhang, Y., Shen, X. and Weng, C. (2009). Approximation of the tail probability of randomly weighted sums and applications. Stoch. Processes Appl. 119, 655–675.Google Scholar
Zhu, C.-H. and Gao, Q.-B. (2008). The uniform approximation of the tail probability of the randomly weighted sums of subexponential random variables. Statist. Prob. Lett. 78, 2552–2558.Google Scholar