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Risk bounds and partial dependence information. (English) Zbl 1383.62251
Ferger, Dietmar (ed.) et al., From statistics to mathematical finance. Festschrift in honour of Winfried Stute. Cham: Springer (ISBN 978-3-319-50985-3/hbk; 978-3-319-50986-0/ebook). 345-366 (2017).
Summary: The evaluation of risks and risk bounds for joint portfolios is an important task in connection with the determination of risk capital as induced by regulatory prescriptions in finance and in insurance. It faces two basic problems. One is induced by the model risk arising from the use of specific but possibly incorrect models. On the other hand risk estimates based only on basic information as for example on the marginal (individual) risk distributions may be too wide to be usable in practise. In this paper we survey some recent endeavor to include partial dependence and structural information in order to obtain reliable and usable improved risk bounds.
For the entire collection see [Zbl 1383.62010].

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
91G70 Statistical methods; risk measures
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[1] C. Bernard and S. Vanduffel. A new approach to assessing model risk in high dimensions. \( Journal of Banking and Finance\) , 58:166-178, 2015.
[2] C. Bernard, X. Jiang, and S. Vanduffel. Note on ’Improved Fréchet bounds and model-free pricing of multi-asset options’ by Tankov (2011). \( Journal of Applied Probability\) , 49(3):866-875, 2012. · Zbl 1259.60022
[3] C. Bernard, Y. Liu, N. MacGillivray, and J. Zhang. Bounds on capital requirements for bivariate risk with given marginals and partial information on the dependence. \( Dependence Modeling\) , 1:37-53, 2013. · Zbl 06297671
[4] C. Bernard, M. Denuit, and S. Vanduffel. Measuring portfolio risk under partial dependence information. \( Social Science Research Network\) , 2014. doi:10.2139/ssrn.2406377.
[5] C. Bernard, L. Rüschendorf, and S. Vanduffel. Value-at-Risk bounds with variance constraints. \( Journal of Risk and Insurance\) , 2015a. Preprint (2013), available at
[6] C. Bernard, L. Rüschendorf, S. Vanduffel, and J. Yao. How robust is the Value-at-Risk of credit risk portfolios? \( Finance \& Stochastics\) , 21:60-82, 2017. doi:
[7] C. Bernard, L. Rüschendorf, S. Vanduffel, and R. Wang. Risk bounds for factor models. \( Social Science Research Network\) , 2016. doi:10.2139/ssrn.2572508.
[8] V. Bignozzi, G. Puccetti, and L. Rüschendorf. Reducing model risk via positive and negative dependence assumptions. \( Insurance: Mathematics and Economics\) , 61(1):17-26, 2015. · Zbl 1314.91242
[9] P. Deheuvels. La fonction de dépendance empirique et ses propriétés. \( Académie royale de Belgique, Bulletin de la classe des sciences\) , 65(5):274-292, 1979. · Zbl 0422.62037
[10] M. Denuit, J. Genest, and É. Marceau. Stochastic bounds on sums of dependent risks. \( Insurance: Mathematics and Economics\) , 25(1):85-104, 1999. · Zbl 1028.91553
[11] M. Denuit, J. Dhaene, and C. Ribas. Does positive dependence between individual risks increase stop loss premiums. \( Insurance: Mathematics and Economics\) , 28:305-308, 2001. · Zbl 1055.91046
[12] P. Embrechts and G. Puccetti. Bounds for functions of dependent risks. \( Finance and Stochastics\) , 10(3):341-352, 2006. · Zbl 1101.60010
[13] P. Embrechts and G. Puccetti. Bounds for the sum of dependent risks having overlapping marginals. \( Journal of Multivariate Analysis\) , 101(1):177-190, 2010. · Zbl 1177.60022
[14] P. Embrechts, A. Höing, and A. Juri. Using copulae to bound the Value-at-Risk for functions of dependent risks. \( Finance and Stochastics\) , 7(2):145-167, 2003. · Zbl 1039.91023
[15] P. Embrechts, G. Puccetti, and L. Rüschendorf. Model uncertainty and VaR aggregation. \( Journal of Banking and Finance\) , 37(8):2750-2764, 2013.
[16] P. Embrechts, G. Puccetti, L. Rüschendorf, R. Wang, and A. Beleraj. An academic response to Basel 3.5. \( Risk\) , 2(1):25-48, 2014.
[17] P. Embrechts, B. Wang, and R. Wang. Aggregation-robustness and model uncertainty of regulatory risk measures. \( Finance and Stochastics\) , 19(4):763 -790, 2015. · Zbl 1327.62326
[18] C. Genest, K. Ghoudi, and L.-P. Rivest. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. \( Biometrika\) , 82(3):543-552, 1995. · Zbl 0831.62030
[19] C. Genest, B. Rémillard, and D. Beaudoin. Goodness-of-fit tests for copulas: A review and a power study. \( Insurance: Mathematics and Economics\) , 44(2):199-213, 2009. · Zbl 1161.91416
[20] D. Hunter. An upper bound for the probability of a union. \( Journal of Applied Probability\) , 13:597-603, 1976. · Zbl 0349.60007
[21] H. G. Kellerer. Measure theoretic versions of linear programming. \( Mathematische Zeitschrift\) , 198(3):367-400, 1988. · Zbl 0627.90067
[22] M. Moscadelli. The modelling of operational risk: experience with the analysis of the data collected by the basel committee. Working Paper 517, Bank of Italy - Banking and Finance Supervision Department, 2004.
[23] G. Puccetti and L. Rüschendorf. Bounds for joint portfolios of dependent risks. \( Statistics \& Risk Modeling\) , 29(2):107-132, 2012a.
[24] G. Puccetti and L. Rüschendorf. Computation of sharp bounds on the distribution of a function of dependent risks. \( Journal of Computational and Applied Mathematics\) , 236(7):1833-1840, 2012b. · Zbl 1241.65019
[25] G. Puccetti and L. Rüschendorf. Sharp bounds for sums of dependent risks. \( Journal of Applied Probability\) , 50(1):42-53, 2013. · Zbl 1282.60017
[26] G. Puccetti and L. Rüschendorf. Asymptotic equivalence of conservative VaR- and ES-based capital charges. \( Journal of Risk\) , 16(3):3-22, 2014.
[27] G. Puccetti, B. Wang, and R. Wang. Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates. \( Insurance: Mathematics and Economics\) , 53(3):821-828, 2013. · Zbl 1290.62019
[28] G. Puccetti, L. Rüschendorf, D. Small, and S. Vanduffel. Reduction of Value-at-Risk bounds via independence and variance information. \( Forthcoming in Scandinavian Actuarial Journal\) , 2015.
[29] L. Rüschendorf. On one sample rank order statistics for dependent random variables. In J. Kozesnik, editor, \( Transactions of the Seventh Prague Conference, Volume B\) , pages 449-456. Springer, 1974.
[30] L. Rüschendorf. Asymptotic distributions of multivariate rank order statistics. \( The Annals of Statistics\) , 4(5):912-923, 1976. · Zbl 0359.62040
[31] L. Rüschendorf. On the minimum discrimination information theorem. \( Statistics \& Decisions, Supplement Issue\) , 1:263-283, 1984.
[32] L. Rüschendorf. Bounds for distributions with multivariate marginals. In K. Mosler and M. Scarsini, editors, \( Stochastic orders and decision under risk\) , volume 19 of \( IMS Lecture Notes\) , pages 285-310. 1991a. · Zbl 0760.60019
[33] L. Rüschendorf. Fréchet bounds and their applications. In G. Dall’Aglio, S. Kotz, and G. Salinetti, editors, \( Advances in Probability Distributions with Given Marginals\) , volume 67 of \( Mathematics and Its Applications\) , pages 151-188. Springer, 1991b.
[34] L. Rüschendorf. Comparison of multivariate risks and positive dependence. \( Journal of Applied Probability\) , 41:391-406, 2004. · Zbl 1049.62060
[35] L. Rüschendorf. Stochastic ordering of risks, influence of dependence, and a.s. constructions. In N. Balakrishnan, I. G. Bairamov, and O. L. Gebizlioglu, editors, \( Advances on Models, Characterization and Applications\) , pages 19-56. Chapman and Hall/CRC, 2005.
[36] L. Rüschendorf. \( Mathematical Risk Analysis\) . Springer Series in Operations Research and Financial Engineering. Springer, 2013. · Zbl 1266.91001
[37] W. Stute. The oscillation behavior of empirical processes: The multivariate case. \( The Annals of Probability\) , 12(2):361-379, 1984. · Zbl 0533.62037
[38] B. Wang and R. Wang. The complete mixability and convex minimization problems with monotone marginal densities. \( Journal of Multivariate Analysis\) , 102(10):1344-1360, 2011. · Zbl 1229.60019
[39] R. Wang. Asymptotic bounds for the distribution of the sum of dependent random variables. \( Journal of Applied Probability\) , 51(3):780-798, 2014. · Zbl 1320.60045
[40] R. Wang, L. Peng, and J. Yang. Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities. \( Finance and Stochastics\) , 17:395-417, 2013. · Zbl 1266.91038
[41] R. C. Williamson and T. Downs. Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds. \( International Journal of Approximate Reasoning\) , 4(2):89-158, 1990. · Zbl 0703.65100
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