High level quantile approximations of sums of risks

• Autorzy: A. Cuberos , E. Masiello , V. Maume-Deschamps
• Języki publikacji: EN

Abstrakty

EN

The approximation of a high level quantile or of the expectation over a high quantile (Value at Risk (VaR) or Tail Value at Risk (TVaR) in risk management) is crucial for the insurance industry.We propose a new method to estimate high level quantiles of sums of risks. It is based on the estimation of the ratio between the VaR (or TVaR) of the sum and the VaR (or TVaR) of the maximum of the risks. We show that using the distribution of the maximum to approximate the VaR is much better than using the marginal. Our method seems to work well in high dimension (100 and higher) and gives good results when approximating the VaR or TVaR in high levels on strongly dependent risks where at least one of the risks is heavy tailed.

Słowa kluczowe

EN

regularly varying functions; value at risk estimation; risk aggregation

Kategorie tematyczne

• 62H99: None of the above, but in this section
• 62P05: Applications to actuarial sciences and financial mathematics

• Wydawca: De Gruyter Open
• Czasopismo: Dependence Modeling
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2015-06-17

zaakceptowano

2015-10-06

online

2015-10-16

Twórcy

autor

A. Cuberos

• Université de Lyon, Université Lyon 1, Laboratoire SAF EA 2429, SCOR SE

autor

E. Masiello

• Université de Lyon, Université Lyon 1, Institut Camille Jordan ICJ UMR 5208 CNRS

autor

V. Maume-Deschamps

• Université de Lyon, Université Lyon 1, Institut Camille Jordan ICJ UMR 5208 CNRS

Bibliografia

• [1] Albrecher, H., Constantinescu, C., and Loisel, S. (2011). Explicit ruin formulas for models with dependence among risks. Insurance Math. Econom., 48, 65–270.
• [2] Alink, S., Löwe, M., and Wüthrich, M.V. (2004). Diversification of aggregate dependent risks. Insurance Math. Econom., 35(1), 77–95.
• [3] Alink, S., Löwe, M., and Wüthrich M.V. (2005). Analysis of the expected shortfall of aggregate dependent risks. ASTIN Bull., 35(1), 25–43. [Crossref]
• [4] Alink, S., Löwe, M., and Wüthrich M.V. (2007). Diversification for general copula dependence. Stat. Neerl., 61(4), 446–465. [WoS][Crossref]
• [5] 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(2), 562–591. [Crossref][WoS]
• [6] Barbe, P., Fougères, A.-L., and Genest, C. (2006). On the Tail Behavior of Sums of Dependent Risks. ASTIN Bull., 36(2), 361– 373. [Crossref]
• [7] Barbe, P., and McCormick, W.P. (2005). Asymptotic expansions of convolutions of regularly varying distributions. J. Aust. Math. Soc., 78(3), 339–371. [Crossref]
• [8] Bernard, C., Rüschendorf, L., and Vanduffel, S. (2015). Value-at-Risk boundswith variance constraints. J. Risk Insur. Available at http://dx.doi.org/10.2139/ssrn.2342068. [Crossref]
• [9] Bernard, C. and Vanduffel, S. (2015). A new approach to assessing model risk in high dimensions. J. Bank. Finance, 58, 166–178. [WoS]
• [10] Bingham, N.H., Goldie, C.H., and Teugels, J.H. (1989). Regular Variation. Cambridge university press.
• [11] Cénac, P., Loisel, S.,Maume-Deschamps, V., and Prieur, C. (2014). Risk indicatorswith several lines of business: comparison, asymptotic behavior and applications to optimal reserve allocation. Ann. I.S.U.P., 58(3).
• [12] Cossette, H., Côté, M.-P., Mailhot, M., and Marceau, E. (2014). A note on the computation of sharp numerical bounds for the distribution of the sum, product or ratio of dependent risks. J. Multivariate Anal., 130, 1–20. [WoS]
• [13] Dacorogna, M., El Bahtouri, L., and Kratz, M. (2015). Explicit diversification benefit for dependent risks. Preprint.
• [14] Dubey, S.D. (1970). Compound gamma, beta and F distributions. Metrika, 16(1), 27–31. [Crossref]
• [15] Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance, 33. Springer.
• [16] Embrechts, P., Neslehová, J., and Wüthrich, M.V. (2009). Additivity properties for value-at-risk under archimedean dependence and heavy-tailedness. Insurance Math. Econom., 44(2), 164–169.
• [17] Embrechts, P., Puccetti, G., and Rüschendorf, L. (2013). Model uncertainty and VaR aggregation. J. Bank. Finance, 37(8), 2750–2764. [WoS][Crossref]
• [18] Fougères, A.-L., and Mercadier, C. (2012). Riskmeasures andmultivariate extensions of Breiman’s theorem. J. Appl. Probab., 49:364–384. [Crossref]
• [19] Geluk, J. and Tang, Q. (2009). Asymptotic tail probabilities of sums of dependent subexponential random variables. J. Theoret. Probab., 22(4):871–882.
• [20] Kortschak, D., and Albrecher, H. (2009). Asymptotic results for the sum of dependent non-identically distributed random variables. Methodol. Comput. Appl. Probab., 11(3), 279–306. [Crossref][WoS]
• [21] Maume-Deschamps, V., Rullière, D., and Saïd, K. (2015). Impact of dependence on some multivariate risk indicators. Available at https://hal.archives-ouvertes.fr/hal-01171395/document.
• [22] Nguyen, Q.-H., and Robert, C.Y. (2014). New eficient estimators in rare event simulation with heavy tails. J. Comput. Appl. Math., 261, 39–47. [WoS]
• [23] Oakes, D. (1989). Bivariate survival models induced by frailties. J. Amer. Statist. Assoc., 84(406), 487–493. [Crossref]
• [24] Resnick, S.I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer-Verlag New York.
• [25] Weinzierl, W. (2000). Introduction to Monte Carlo Methods. ArXiv preprint hep-ph/0006269.
• [26] Weissman, I. (1978). Estimation of parameters and large quantiles based on the k largest observations. J. Amer. Statist. Assoc., 73(364):812–815.
• [27] Yeh, H-C. (2007). The frailty and the Archimedean structure of the general multivariate Pareto distributions. Bull. Inst.Math. Acad. Sin., 2(3), 713–729.

Identyfikatory

DOI

10.1515/demo-2015-0010