Source: J. Appl. Probab.
Volume 49, Number 2
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.
Alink, S., Löwe, M. and Wüthrich, M. V. (2004). Diversification of aggregate dependent risks. Insurance Math. Econom. 35, 77–95.
Alink, S., Löwe, M. and Wüthrich, M. V. (2005). Analysis of the expected shortfall of aggregate dependent risks. ASTIN Bull. 35, 25–43.
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.
Barbe, P., Fougères, A.-L. and Genest, C. (2006). On the tail behaviour of sums of dependent risks. ASTIN Bull. 36, 361–373.
Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.
Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes. John Wiley, Chichester.
Cai, J. (2002). Ruin probabilities with dependent rates of interest. J. Appl. Prob. 39, 312–323.
Chen, Y. and Su, C. (2006). Finite time ruin probability with heavy-tailed insurance and financial risks. Statist. Prob. Lett. 76, 1812–1820.
Cossette, H. and Marceau, E. (2000). The discrete-time risk model with correlated classes of business. Insurance Math. Econom. 26, 133–149.
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. D. J. Daley, Applied Probability Trust, Sheffield, pp. 93–104.
De Haan, L. and Resnick, S. I. (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40, 317–337.
Mathematical Reviews (MathSciNet): MR478290
Degen, M., Lambrigger, D. and Segers, J. (2010). Risk concentration and diversification: second-order properties. Insurance Math. Econom. 46, 541–546.
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.
Mathematical Reviews (MathSciNet): MR652832
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.
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.
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.
Foss, S. and Richards, A. (2010). On sums of conditionally independent subexponential random variables. Math. Operat. Res. 35, 102–119.
Goovaerts, M. J. \et (2005). The tail probability of discounted sums of Pareto-like losses in insurance. Scand. Actuarial J. 2005, 446–461.
Hult, H. and Samorodnitsky, G. (2008). Tail probabilities for infinite series of regularly varying random vectors. Bernoulli 14, 838–864.
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.
Mainik, G. and Rüschendorf, L. (2009). On optimal portfolio diversification with respect to extreme risks. Preprint.
Maulik, K., Resnick, S. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370.
Ng, K. W., Tang, Q. H. and Yang, H. (2002). Maxima of sums of heavy-tailed random variables. ASTIN Bull. 32, 43–55.
Nyrhinen, H. (1999). On the ruin probabilities in a general economic environment. Stoch. Process. Appl. 83, 319–330.
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
Mathematical Reviews (MathSciNet): MR900810
Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.
Sgibnev, M. S. (1996). On the distribution of the maxima of partial sums. Statist. Prob. Lett. 28, 235–238.
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.
Tang, Q. and Tsitsiashvili, G. (2003b). Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6, 171–188.
Wang, D. and Tang, Q. (2006). Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22, 253–272.
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.
Wüthrich, M. V. (2003). Asymptotic value-at-risk estimates for sums of dependent random variables. ASTIN Bull. 33, 75–92.
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.
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. \endharvbibliography