×

Bivariate lower and upper orthant value-at-risk. (English) Zbl 1304.91097

Summary: Value-at-risk (VaR) is an important risk measure widely used in actuarial science and quantitative risk management. P. Embrechts and G. Puccetti [J. Multivariate Anal. 97, No. 2, 526–547 (2006; Zbl 1089.60016)] have introduced the multivariate lower and upper orthant VaR. The practical applications of these risk measures is very promising, especially in actuarial science and quantitative risk management. Our objective is to study in details the multivariate lower and upper orthant VaR in the bivariate setting, their properties and their applications. In particular, new characterizations of the bivariate lower and upper orthant VaR and desirable properties are given, such as translation invariance, positive homogeneity and comonotonic additivity. Lower and upper confidence regions for random vectors are developed and used to provide new results on the convexity conditions and to suggest capital allocation techniques. We provide bounds on functions of random pairs and derive interesting relations with existing results. We motivate the use of the bivariate lower and upper ortant VaR for risk allocation, to represent bivariate ruin probabilities and for risk comparison. Practical illustrations and examples of the results are presented throughout the article.

MSC:

91B30 Risk theory, insurance (MSC2010)

Citations:

Zbl 1089.60016
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balakrishnan N, Lai CD (2009) Continuous bivariate distributions. Springer · Zbl 1267.62028
[2] Cai J, Li H (2005) Multivariate risk model of phase type. Insur Math Econ 36:137–152 · Zbl 1122.60049 · doi:10.1016/j.insmatheco.2004.11.004
[3] Cai J, Li H (2007) Dependence properties and bounds for ruin probabilities in multivariate compound risk models. J Multivar Anal 98:757–773 · Zbl 1280.91090 · doi:10.1016/j.jmva.2006.06.004
[4] Chan WS, Yang H, Zhang L (2003) Some results on ruin probabilities in a two-dimensional risk model. Insur Math Econ 32:345–358 · Zbl 1055.91041 · doi:10.1016/S0167-6687(03)00115-X
[5] Cherubini U, Luciano E (2001) Value at risk trade-off and capital allocation with copulas. Econ Notes 30(2):235–256 · doi:10.1111/j.0391-5026.2001.00055.x
[6] Cherubini U, Luciano E, Vecchiato W (2004) Copula Methods in Finance. Wiley, London · Zbl 1163.62081
[7] Denuit M, Genest C, Marceau E (1999) Stochastic bounds on sums of dependent risks. Sci Agric 25:85–104 · Zbl 1028.91553
[8] Di Bernardino E, Lalo T, Maume-Deschamps V, Prieur C (2011) Plug-in estimation of level sets in a non-compact setting with application in multivariate risk theory. ESAIM Prob Stat · Zbl 06282472
[9] Embrechts P, Hoeing A, Juri A (2003) Using Copulae to bound the value-at-risk for functions of dependent risks. Finance Stoch 7(2):145–167 · Zbl 1039.91023 · doi:10.1007/s007800200085
[10] Embrechts P, Puccetti G (2006a) Bounds for functions of multivariate risks. J Multivar Anal 97(2):526–547 · Zbl 1089.60016 · doi:10.1016/j.jmva.2005.04.001
[11] Embrechts P, Puccetti G (2006b) Bounds for functions of dependent risks. Finance Stoch 10:341–352 · Zbl 1101.60010 · doi:10.1007/s00780-006-0005-5
[12] Frank MJ, Schweizer B (1979) On the duality of generalized infimal and supremal convolutions. Rendiconti di Matematica 12:1–23 · Zbl 0467.39004
[13] Frees EW, Valdez EA (1998) Understanding relationships using copulas. N Am Actuar J 2(1):1–25 · Zbl 1081.62564 · doi:10.1080/10920277.1998.10595667
[14] Guégan D, Hassani B (2012) Multivariate VaRs for operational risk capital computation : a vine structure approach. Working paper
[15] Johnson N, Kotz S, Balakrishnan N (1997) Discrete multivariate distributions. Wiley, London · Zbl 0868.62048
[16] Jouini E, Meddeb M, Touzi N (2004) Vector-valued coherent risk measures. Finance Stoch 8(4):531–552 · Zbl 1063.91048
[17] Lee SCK, Lin XS (2012) Modeling dependent risks with multivariate Erlang mixtures. ASTIN Bull 42(1):153–180 · Zbl 1277.62255
[18] Lehmann EL (1966) Some concepts of dependence. Ann Math Stat 37:1137–1153 · Zbl 0146.40601 · doi:10.1214/aoms/1177699260
[19] Lindskog F, McNeil A (2003) Common poisson shock models: applications to insurance and credit risk modelling. ASTIN Bull 33(2):209–238 · Zbl 1087.91030 · doi:10.2143/AST.33.2.503691
[20] Makarov GD (1982) Estimates for the distribution function of the sum of two random variables when the marginal distributions are fixed. Theory Probab Appl 26:803–806 · Zbl 0488.60022 · doi:10.1137/1126086
[21] McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Princeton University Press, Princeton · Zbl 1089.91037
[22] Mesfioui M, Quessy J-F (2005) Bounds on the value-at-risk for the sum of possibly dependent risks. Insur Math Econ 37:135–151 · Zbl 1115.91032 · doi:10.1016/j.insmatheco.2005.03.002
[23] Nelsen RB (2006) An introduction to Copulas. Springer, Berlin · Zbl 1152.62030
[24] Rüschendorf L (1982) Random variables with maximum sums. Adv Appl Prob 14:623–632 · Zbl 0487.60026 · doi:10.2307/1426677
[25] Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de statistique de l’Université de Paris 8:229–231 · Zbl 0100.14202
[26] Williamson RC, Downs T (1990) Probabilistic arithmetic I: numerical methods for calculating convolutions and dependency bounds. Int J Approx Reason 4:89–158 · Zbl 0703.65100 · doi:10.1016/0888-613X(90)90022-T
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.