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Aggregation of log-linear risks

Part of: Stochastic processes

Published online by Cambridge University Press:  30 March 2016

Paul Embrechts ,
Enkelejd Hashorva  and
Thomas Mikosch
Paul Embrechts
Affiliation:
ETH Zürich and Swiss Finance Institute, Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland. Email address: [email protected].
Enkelejd Hashorva
Affiliation:
Faculty of Business and Economics (HEC Lausanne), University of Lausanne, 1015 Lausanne, Switzerland
Thomas Mikosch
Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark.
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Abstract

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In this paper we work in the framework of a k-dimensional vector of log-linear risks. Under weak conditions on the marginal tails and the dependence structure of a vector of positive risks, we derive the asymptotic tail behaviour of the aggregated risk and present an application concerning log-normal risks with stochastic volatility.

Keywords

Risk aggregationlog-linear modelsubexponential distributionGumbel max-domain of attraction

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
Part 5. Finance and econometrics
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 203 - 212
Copyright
Copyright © Applied Probability Trust 2014 

References

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
Asmussen, S., and Rojas-Nandayapa, L. (2008). Asymptotics of sums of lognormal random variables with Gaussian copula. Statist. Prob. Lett./ 78, 2709–2714.CrossRefGoogle Scholar
Asmussen, S., Blanchet, J., Juneja, S., and Rojas-Nandayapa, L. (2011). Efficient simulation of tail probabilities of sums of correlated lognormals. Ann. Operat. Res./ 189, 5–23.CrossRefGoogle Scholar
Barndorff-Nielsen, O., Kent, J. and Sörensen, M. (1982). Normal variance-mean mixtures and z# distributions. Internat. Statist. Rev./ 50, 145–159.CrossRefGoogle Scholar
Bluhm, C., Overbeck, L., and Wagner, C. (2002). em An Introduction to Credit Risk Modeling. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). em Modelling Extremal Events. Springer, Berlin.CrossRefGoogle Scholar
Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). Model uncertainty and VaR aggregation. J. Bank. Finance 37, 2750–2764.CrossRefGoogle Scholar
Farkas, J., and Hashorva, E. (2014). Tail approximation for reinsurance portfolios of Gaussian-like risks. To appear in Scand. Actuar. J./ Google Scholar
Foss, S., and Richards, A. (2010). On sums of conditionally independent subexponential random variables. Math. Operat. Res./ 35, 102–119.CrossRefGoogle Scholar
Foss, S., Korshunov, D., and Zachary, S. (2009). Convolutions of long-tailed and subexponential distributions. J. Appl. Prob./ 46, 756–767.CrossRefGoogle Scholar
Foss, S., Korshunov, D., and Zachary, S. (2013). em An Introduction to Heavy-tailed and Subexponential Distributions, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Gulisashvili, A., and Tankov, P. (2014). Tail behavior of sums and differences of log-normal random variables. To appear in Bernoulli.Google Scholar
Hashorva, E. (2012). Exact tail asymptotics in bivariate scale mixture models. Extremes 15, 109–128.CrossRefGoogle Scholar
Hashorva, E. (2013). Exact tail asymptotics of aggregated parametrised risk. J. Math. Anal. Appl./ 400, 187–199.CrossRefGoogle Scholar
Kortschak, D., and Hashorva, E. (2013). Efficient simulation of tail probabilities for sums of log-elliptical risks. J. Comput. Appl. Math./ 247, 53–67.CrossRefGoogle Scholar
Mainik, G., and Embrechts, P. (2013). Diversification in heavy-tailed portfolios: properties and pitfalls. Ann. Actuarial Sci./ 7, 26–45.CrossRefGoogle Scholar
Mitra, A., and Resnick, S. I. (2009). Aggregation of rapidly varying risks and asymptotic independence. Adv. Appl. Prob./ 41, 797–828.CrossRefGoogle Scholar
Piterbarg, V. I. (1996). em Asymptotic Methods in the Theory of Gaussian Processes and Fields (Trans. Math. Monogr. 148). American Mathematical Society, Providence, RI.Google Scholar
Puccetti, G. and Rüschendorf, L. (2013). Sharp bounds for sums of dependent risks. J. Appl. Prob./ 50, 42–53.CrossRefGoogle Scholar
Resnick, S. I. (1987). em Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Rootzén, H. (1987). A ratio limit theorem for the tails of weighted sums. Ann. Prob./ 15, 728–747.Google Scholar
Tang, Q., and Tsitsiashvili, G. (2004). Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Prob./ 36, 1278–1299.CrossRefGoogle Scholar