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Published online by Cambridge University Press:  21 December 2023

Ludger Rüschendorf
Affiliation:
Albert-Ludwigs-Universität Freiburg, Germany
Steven Vanduffel
Affiliation:
Vrije Universiteit Brussel
Carole Bernard
Affiliation:
Grenoble Ecole de Management
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Chapter
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Model Risk Management
Risk Bounds under Uncertainty
, pp. 309 - 319
Publisher: Cambridge University Press
Print publication year: 2024

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References

Aas, K., Czado, C., Frigessi, A., and Bakken, H. 2009. Pair-copula constructions of multiple dependence. Insurance: Math. Econom., 44(2), 182198.Google Scholar
Abdous, B., Genest, C., and Rémillard, B. 2005. Dependence properties of meta-elliptical distributions. Pages 1–15 of: Duchesne, P., and Rémillard, B. (eds), Statistical Modelling and Analysis for Complex Data Problems. Springer.Google Scholar
Acciaio, B., Beiglböck, M., Penkner, F., and Schachermayer, W. 2016. A model-free version of the fundamental theorem of asset pricing and the super replication theorem. Math. Finance, 26, 233251.CrossRefGoogle Scholar
Actuarial Association of Europe (AAE). 2017. Comments template on discussion paper on the review of specific items in the Solvency II delegated regulation. Technical report. Actuarial Association of Europe.Google Scholar
Albrecher, H., Constantinescu, C., and Loisel, S. 2011. Explicit ruin formulas for models with dependence among risks. Insurance: Math. Econom., 48(2), 265270.Google Scholar
Ansari, J., and Rüschendorf, L. 2021. Ordering results for elliptical distributions with applications to risk bounds. J. Multivariate Anal., 182, Art.–Id. 104709.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. 1999. Coherent measures of risk. Math. Finance, 9(3), 203228.CrossRefGoogle Scholar
Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D. 1972. Statistical Inference under Order Restrictions: The Theory and Application of Isotonic Regression. Wiley.Google Scholar
Barrieu, P., and Scandolo, G. 2015. Assessing financial model risk. Eur. J. Oper. Res., 242(2), 546556.CrossRefGoogle Scholar
Basel Committee on Banking Supervision. 2010 (Oct.). Developments in Modelling Risk Aggregation. Bank for International Settlements.Google Scholar
Basu, S., and DasGupta, A. 1997. The mean, median, and mode of unimodal distribution: A characterization. Theory Probab. Appl., 41(2), 210223.CrossRefGoogle Scholar
Bäuerle, N., and Müller, A. 2006. Stochastic orders and risk measures: Consistency and bounds. Insurance: Math. Econom., 38, 132148.Google Scholar
Beiglböck, M., Henry-Labordère, P., and Penkner, F. 2013. Model-independent bounds for option prices – a mass transport approach. Finance Stoch., 17(3), 477501.CrossRefGoogle Scholar
Benson, F. 1949. A note on the estimation of mean and standard deviation from quantiles. J. R. Stat. Soc., Ser. B, Stat. Methodol., 11, 91100.Google Scholar
Bernard, C., and McLeish, D. 2016. Algorithms for finding copulas minimizing convex functions of sums. Asia-Pacific J. Oper. Res., 33(5), Art.–Id. 1650040.CrossRefGoogle Scholar
Bernard, C., and Vanduffel, S. 2014. Mean-variance optimal portfolios in the presence of a benchmark with applications to fraud detection. Eur. J. Oper. Res., 234(2), 469480.CrossRefGoogle Scholar
Bernard, C., and Vanduffel, S. 2015. A new approach to assessing model risk in high dimensions. J. Banking Finance, 58, 167178.CrossRefGoogle Scholar
Bernard, C., Bondarenko, O., and Vanduffel, S. 2018b. Rearrangement algorithm and maximum entropy. Ann. Oper. Res., 261(1–2), 107134.CrossRefGoogle Scholar
Bernard, C., Bondarenko, O., and Vanduffel, S. 2021. A model-free approach to multivariate option pricing. Rev. Deriv. Res., 24(2), 121.CrossRefGoogle Scholar
Bernard, C., Boyle, P., and Vanduffel, S. 2014a. Explicit representation of cost-efficient strategies. Finance, 35(2), 555.CrossRefGoogle Scholar
Bernard, C., Pesenti, S. M., and Vanduffel, S. 2023. Robust distortion risk measures. Mathematical Finance, https://doi.org/10.1111/mafi.12414.CrossRefGoogle Scholar
Bernard, C., Denuit, M., and Vanduffel, S. 2018a. Measuring portfolio risk under partial dependence information. J. Risk Insurance, 85(3), 843863.CrossRefGoogle Scholar
Bernard, C., Jiang, X., and Vanduffel, S. 2012. A note on ‘Improved Fréchet bounds and model-free pricing of multi-asset options’ by Tankov (2011). J. Appl. Probab., 49(3), 866875.CrossRefGoogle Scholar
Bernard, C., Jiang, X., and Wang, R. 2014b. Risk aggregation with dependence uncertainty. Insurance: Math. Econom., 54, 93108.Google Scholar
Bernard, C., Kazzi, R., and Vanduffel, S. 2020. Range value-at-risk bounds for uni-modal distributions under partial information. Insurance: Math. Econom., 94(1), 924.Google Scholar
Bernard, C., Liu, Y., MacGillivray, N., and Zhang, J. 2013. Bounds on capital requirements for bivariate risk with given marginals and partial information on the dependence. Depend. Model., 1, 3753.Google Scholar
Bernard, C., Rüschendorf, L., and Vanduffel, S. 2017c. Value-at-risk bounds with variance constraints. J. Risk Insurance, 84(3), 923959.CrossRefGoogle Scholar
Bernard, C., Rüschendorf, L., Vanduffel, S., and Wang, R. 2017b. Risk bounds for factor models. Finance Stoch., 3, 631659.CrossRefGoogle Scholar
Bernard, C., Rüschendorf, L., Vanduffel, S., and Yao, J. 2017a. How robust is the value-at-risk of credit risk portfolios? Eur. J. Finance, 23(6), 507534.CrossRefGoogle Scholar
Bignozzi, V., Puccetti, G., and Rüschendorf, L. 2015. Reducing model risk via positive and negative dependence assumptions. Insurance: Math. Econom., 61, 1726.Google Scholar
Boudt, K., Jakobsons, E., and Vanduffel, S. 2018. Block rearranging elements within matrix columns to minimize the variability of the row sums. 4OR, 16(1), 3150.CrossRefGoogle Scholar
Brighi, B., and Chipot, M. 1994. Approximated convex envelope of a function. SIAM J. Numer. Anal., 31(1), 128148.CrossRefGoogle Scholar
Burgert, C., and Rüschendorf, L. 2006. Consistent risk measures for portfolio vectors. Insurance: Math. Econom., 38, 289297.Google Scholar
Carhart, M. M. 1997. On persistence in mutual fund performance. J. Finance, 52(1), 5782.CrossRefGoogle Scholar
Chamberlain, G., and Rothschild, M. 1983. Arbitrage, factor structure, and mean-variance analysis on large asset markets. Econometrica, 51, 12811304. http://doi.org/10.3386/w0996CrossRefGoogle Scholar
Chernoff, H., and Reiter, S. 1954. Selection of a distribution function to minimize an expectation subject to side conditions. Technical Report. Stanford University: Applied Mathematics and Statistics Labs.CrossRefGoogle Scholar
Chong, K.-M. 1974. Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications. Canad. J. Math., 26, 13211340.CrossRefGoogle Scholar
Chong, K.-M., and Rice, N. M. 1971. Equimeasurable Rearrangements of Functions. Queen’s Papers in Pure and Applied Mathematics, vol. 28. Queen’s University.Google Scholar
Choquet, G. 1954. Theory of capacities. Ann. Inst. Fourier, 5, 131295.CrossRefGoogle Scholar
Coffman, E. G., and Yannakakis, M. 1984. Permuting elements within columns of a matrix in order to minimize maximum row sum. Math. Oper. Res., 9(3), 384390.CrossRefGoogle Scholar
Connor, G., and Korajczyk, R. A. 1993. A test for the number of factors in an approximate factor model. J. Finance, 48(4), 12631291.CrossRefGoogle Scholar
Cont, R., Deguest, R., and Scandolo, G. 2010. Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance, 10(6), 593606.CrossRefGoogle Scholar
Cope, E., Mignola, G., Antonini, G., and Ugoccioni, R. 2009. Challenges and pitfalls in measuring operational risk from loss data. J. Oper. Risk, 4(4), 855863.Google Scholar
Cornilly, D., Puccetti, G., Rüschendorf, L., and Vanduffel, S. 2022. Fair allocation of indivisible goods with minimum inequality or minimum envy. Eur. J. Oper. Res., 297(2), 741752.CrossRefGoogle Scholar
Cornilly, D., Rüschendorf, L., and Vanduffel, S. 2018. Upper bounds for strictly concave distortion risk measures on moment spaces. Insurance: Math. Econom., 82, 141151.Google Scholar
Credit Suisse First Boston. 1997. CreditRisk+: A credit risk management framework. Technical report. Credit Suisse First Boston Bank.Google Scholar
Cuberos, A., Masiello, E., and Maume-Deschamps, V. 2015. High level quantile approximations of sums of risks. Depend. Model., 3(1), 141158.Google Scholar
Cuesta-Albertos, J. A., Rüschendorf, L., and Tuero-Diaz, A. 1993. Optimal coupling of multivariate distributions and stochastic processes. J. Multivariate Anal., 46, 335361.CrossRefGoogle Scholar
Czado, C. 2010. Pair copula constructions of multivariate copulas. Pages 93–109 of: Jaworski, P., Durante, F., Härdle, W. K., and Rychlik, T. (eds), Copula Theory and Its Applications. Lect. Notes Stat., vol. 198. Springer.CrossRefGoogle Scholar
Dacorogna, M. M., Elbahtouri, L., and Kratz, M. 2016. Explicit diversification benefit for dependent risks. SCOR Papers, 38(1), 125.Google Scholar
das Gupta, S., Olkin, I., Savage, L. J., Eaton, M. L., Perlman, M., and Sobel, M. 1972. Inequalities on the probability content of convex regions for elliptically contoured distributions. Pages 241–265 of: Le Cam, L. M., Neyman, J., and Scott, E. L. (eds), Proc. 6th Berkeley Sympos. Math. Statist. Probab., vol. 2. Berkeley, University of California Press.Google Scholar
Day, P. W. 1972. Rearrangement inequalities. Canad. J. Math., 24, 930943.CrossRefGoogle Scholar
de Schepper, A., and Heijnen, B. 2010. How to estimate the value at risk under incomplete information. J. Comput. Appl. Math., 233(9), 22132226.CrossRefGoogle Scholar
de Vylder, F. E. 1982. Best upper bounds for integerals with respect to measures allowed to vary under conical and integral constraints. Insurance: Math. Econom., 1(2), 109130.Google Scholar
de Vylder, F. E. 1996. Advanced Risk Theory: A Self-contained Introduction. Éd. de L’Univ. de Bruxelles.Google Scholar
Deelstra, G., Diallo, I., and Vanmaele, M. 2008. Bounds for Asian basket options. J. Comput. Appl. Math., 218(2), 215228.CrossRefGoogle Scholar
Denuit, M. 1999. The exponential premium calculation principle revisited. Astin Bull., 29(2), 215226.CrossRefGoogle Scholar
Denuit, M., Genest, C., and Marceau, É. 1999. Stochastic bounds on sums of dependent risks. Insurance: Math. Econom., 25(1), 85104.Google Scholar
Denuit, M., Lefèvre, C., and Shaked, M. 1998. The s-convex orders among real random variables, with applications. Math. Inequal. Appl., 1(4), 585613.Google Scholar
Dhaene, J., Laeven, R. J. A., Vanduffel, S., Darkiewicz, G., and Goovaerts, M. J. 2008. Can a coherent risk measure be too subadditive? J. Risk Insurance, 75(2), 365386.CrossRefGoogle Scholar
Dhaene, J., Tsanakas, A., Valdez, E. A., and Vanduffel, S. 2012. Optimal capital allocation principles. J. Risk Insurance, 79(1), 128.CrossRefGoogle Scholar
Dhaene, J., Vanduffel, S., Goovaerts, M., Olieslagers, R., and Koch, R. 2003. On the computation of the capital multiplier in the Fortis credit economic capital model. Belg. Actuar. Bull., 3(1), 5057.Google Scholar
Dhaene, J., Vanduffel, S., Goovaerts, M. J., Kaas, R., Tang, Q., and Vyncke, D. 2006. Risk measures and comonotonicity: A review. Stoch. Models, 22(4), 573606.CrossRefGoogle Scholar
Dubey, S. D. 1970. Compound gamma, beta and F distributions. Metrika, 16(1), 2731.CrossRefGoogle Scholar
Ekern, S. 1980. Increasing n-th degree risk. Economics, 6(4), 329333.Google Scholar
El Ghaoui, L., Oks, M., and Oustry, F. 2003. Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Oper. Res., 51(4), 543556.CrossRefGoogle Scholar
Embrechts, P., and Puccetti, G. 2006a. Aggregating risk capital, with an application to operational risk. Geneva Risk Insur. Rev., 31(2), 7190.CrossRefGoogle Scholar
Embrechts, P., and Puccetti, G. 2006b. Bounds for functions of dependent risks. Finance Stoch., 10(3), 341352.CrossRefGoogle Scholar
Embrechts, P., and Puccetti, G. 2010. Risk aggregation. Pages 111–126 of: Jaworski, P., Durante, F., Härdle, W. K., and Rychlik, T. (eds), Copula Theory and Its Applications. Springer.Google Scholar
Embrechts, P., Höing, A., and Juri, A. 2003. Using copulae to bound the value-at-risk for functions of dependent risks. Finance Stoch., 7(2), 145167.CrossRefGoogle Scholar
Embrechts, P., McNeil, A., and Straumann, D. 1999. Correlation: Pitfalls and alternatives. Risk Mag., 12, 6971.Google Scholar
Embrechts, P., McNeil, A., and Straumann, D. 2002. Correlation and dependence in risk management: Properties and pitfalls. Pages 176–223 of: Dempster, M., and Moffatt, H. (eds), Risk Management: Value-at-Risk and Beyond. Cambridge, Cambridge University Press.Google Scholar
Embrechts, P., Puccetti, G., and Rüschendorf, L. 2013. Model uncertainty and VaR aggregation. J. Banking Finance, 37(8), 27502764.CrossRefGoogle Scholar
Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R., and Beleraj, A. 2014. An academic response to Basel 3.5. Risks, 2(1), 2548.CrossRefGoogle Scholar
Embrechts, P., Wang, B., and Wang, R. 2015. Aggregation-robustness and model uncertainty of regulatory risk measures. Finance Stoch., 19(4), 763790.CrossRefGoogle Scholar
Engle, R. F., Ng, V. K., and Rothschild, M. 1990. Asset pricing with a factor-ARCH covariance structure: Empirical estimates for treasury bills. J. Econometrics, 45(1), 213237.CrossRefGoogle Scholar
European Central Bank (ECB). 2017. Guide for the targeted review of internal models (TRIM). Technical report. European Central Bank.Google Scholar
Everett, H. III 1963. Generalized Lagrange multiplier method for solving problems of optimum allocation of resources. Oper. Res., 11(3), 399417.CrossRefGoogle Scholar
Fama, E. F., and French, K. R. 1993. Common risk factors in the returns on stocks and bonds. J. Financial Econom., 33(1), 356.CrossRefGoogle Scholar
Fan, K., and Lorentz, G. G. 1954. An integral inequality. Amer. Math. Monthly, 61, 626631.CrossRefGoogle Scholar
Föllmer, H., and Schied, A. 2004. Stochastic Finance. An Introduction in Discrete Time. 2nd revised and extended edn. Berlin, de Gruyter.CrossRefGoogle Scholar
Frank, M. J., and Schweizer, B. 1979. On the duality of generalized infimal and supremal convolutions. Rend. Mat., VI. Ser., 12, 123.Google Scholar
Fréchet, M. 1951. Sur les tableaux de corrélation dont les marges sont données. Ann. l’Université de Lyon, Section A, 14, 5377.Google Scholar
Gaffke, N., and Rüschendorf, L. 1981. On a class of extremal problems in statistics. Math. Operationsforsch. Stat., Ser. Optimization, 12(1), 123135.Google Scholar
Genest, C., Marceau, É., and Mesfioui, M. 2002. Upper stop-loss bounds for sums of possibly dependent risks with given means and variances. Statist. Probab. Lett., 57, 3334.CrossRefGoogle Scholar
Gordy, M. B. 2000. A comparative anatomy of credit risk models. J. Banking Finance, 24(1), 119149.CrossRefGoogle Scholar
Gordy, M. B. 2003. A risk-factor model foundation for ratings-based bank capital rules. J. Financial Intermed., 12(3), 199232.CrossRefGoogle Scholar
Gumbel, E. J. 1954. The maxima of the mean largest value and of the range. Ann. Math. Stat., 25, 7684.CrossRefGoogle Scholar
Haaf, H., Reiss, O., and Schoenmakers, J. 2004. Numerically stable computation of CreditRisk+. Pages 69–77 of: Gundlach, M., and Lehrbass, F. (eds), CreditRisk+ in the Banking Industry. Springer.Google Scholar
Hardy, G. H., Littlewood, J. E., and Pólya, G. 1952. Inequalities. 2nd edn. Cambridge.Google Scholar
Haus, U.-U. 2015. Bounding stochastic dependence, complete mixability of matrices, and multidimensional bottleneck assignment problems. Oper. Res. Lett., 43, 7479.CrossRefGoogle Scholar
Hoeffding, W. 1940. Maßstabinvariante Korrelationstheorie. Ph.D.Thesis, Schr. Math. Inst. u. Inst. Angew. Math. Univ. Berlin 5, 181233 (1940) and Berlin: Dissertation.Google Scholar
Hofert, M. 2020. Implementing the rearrangement algorithm: An example from computational risk management. Risks, 8(2), 47.CrossRefGoogle Scholar
Hofert, M., Memartoluie, A., Saunders, D., and Wirjanto, T. 2017. Improved algorithms for computing worst value-at-risk. Stat. Risk Model., 34(1–2), 1331.CrossRefGoogle Scholar
Hürlimann, W. 2002. Analytical bounds for two value-at-risk functionals. Astin Bull., 32(2), 235265.CrossRefGoogle Scholar
Ingersoll, J. E. 1984. Some results in the theory of arbitrage pricing. J. Finance, 39(4), 10211039.CrossRefGoogle Scholar
Joe, H. 1997. Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability, vol. 73. London, Chapman & Hall.Google Scholar
Jorion, P. 2006. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd edn. New York, McGraw-Hill.Google Scholar
Jouini, E., Schachermayer, W., and Touzi, N. 2006. Law invariant risk measures have the Fatou property. Pages 49–71 of: Kusuoka, S., and Yamazaki, A. (eds), Advances in Mathematical Economics, vol. 9 Springer.CrossRefGoogle Scholar
Kaas, R., and Goovaerts, M. J. 1986. Best bounds for positive distributions with fixed moments. Insurance: Math. Econom., 5(1), 8792.Google Scholar
Kaas, R., Dhaene, J., and Goovaerts, M. J. 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econom., 27(2), 151168.Google Scholar
Kantorovich, L. V. 1942. On the translocation of masses. C. R. (Dokl.) Acad. Sci. URSS, Ser., 37, 199201.Google Scholar
Karlin, S. 1968. Total Positivity, vol. I. Stanford, Stanford University Press.Google Scholar
Karlin, S., and Novikoff, A. 1963. Generalized convex inequalities. Pac. J. Math., 13, 12511279.CrossRefGoogle Scholar
Karlin, S., and Shapley, L. S. 1953. Geometry of Moment Spaces, vol. 12. American Mathematical Society.Google Scholar
Karlin, S., and Studden, W. J. 1966. Tchebycheff Systems: With Applications in Analysis and Statistics, vol. 15. Interscience Publishers.Google Scholar
Kellerer, H. G. 1988. Measure theoretic versions of linear programming. Math. Z., 198, 367400.CrossRefGoogle Scholar
Khintchine, A. Y. 1938. On unimodal distributions. Izv. Nauchno-Issle., Inst. Mat. Mekhhaniki, 2(2), 17.Google Scholar
Kusuoka, S. 2001. On law invariant coherent risk measures. Pages 83–95 of: Kusuoka, S., et al. (eds), Advances in Mathematical Economics. Springer.CrossRefGoogle Scholar
Lai, T. L., and Robbins, H. 1978. A class of dependent random variables and their maxima. Z. Wahrscheinlichkeitstheor. Verw. Geb., 42, 89111.CrossRefGoogle Scholar
Lai, T. L., and Robbins, M. 1976. Maximally dependent random variables. Proc. Nat. Acad. Sci. USA, 73, 286288.CrossRefGoogle ScholarPubMed
Levy, H., and Kroll, Y. 1978. Ordering uncertain options with borrowing and lending. J. Finance, 33(2), 553574.CrossRefGoogle Scholar
Lewbel, A. 1991. The rank of demand systems: Theory and nonparametric estimation. Econometrica, 59, 711730.CrossRefGoogle Scholar
Li, J. Y.-M. 2018. Clossed-form solutions for worst-case law invariant risk measures with application to robust portfolio optimization. Oper. Res., 66(6), 15331541.CrossRefGoogle Scholar
Li, L., Shao, H., Wang, R., and Yang, J. 2018. Worst-case range value-at-risk with partial information. SIAM J. Financial Math., 9(1), 190218.CrossRefGoogle Scholar
Lo, A. W. 1987. Semi-parametric upper bounds for option prices and expected payoffs. J. Financial Econom., 19, 373387.CrossRefGoogle Scholar
Lorentz, G. G. 1953. An inequality for rearrangements. Amer. Math. Monthly, 60, 176179.CrossRefGoogle Scholar
Lowenstein, R. 2008. Long-term capital management: It’s a short term memory. New York Times, 6(1), 13.Google Scholar
Lux, T., and Papapantoleon, A. 2017. Improved Fréchet–Hoeffding bounds on d-copulas and applications in model-free finance. Ann. Appl. Probab., 27(6), 36333671.CrossRefGoogle Scholar
Lux, T., and Papapantoleon, A. 2019. Model-free bounds on value-at-risk using partial dependence information. Insurance: Math. Econom., 86, 7383.Google Scholar
Lux, T., and Rüschendorf, L. 2018. VaR bounds with two-sided dependence information on the copula. Math. Finance, 29, 9671000.CrossRefGoogle Scholar
Luxemburg, W. A. J. 1967. Rearrangement invariant Banach function spaces. Queen’s Papers Pure Appl. Math., 10, 83144.Google Scholar
Mainik, G., and Rüschendorf, L. 2010. On optimal portfolio diversification with respect to extreme risks. Finance Stoch., 14, 593623.CrossRefGoogle Scholar
Makarov, G. D. 1981. Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed. Theory Probab. Appl., 26, 803806.CrossRefGoogle Scholar
Marshall, A. W., Olkin, I., and Arnold, B. C. 2011. Inequalities: Theory of Majorization and Its Applications. 2nd edn. Springer Series in Statistics. Springer.CrossRefGoogle Scholar
McNeil, A. J., Frey, R., and Embrechts, P. 2005. Quantitative Risk Management. Princeton Series in Finance. Princeton, Princeton University Press.Google Scholar
McNeil, A. J., Frey, R., and Embrechts, P. 2015. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press.Google Scholar
Meilijson, I., and Nadas, A. 1979. Convex majorization with an application to the length of critical paths. J. Appl. Probab., 16, 671677.CrossRefGoogle Scholar
Mesfioui, M., and Quessy, J.-F. 2005. Bounds on the value-at-risk for the sum of possibly dependent risks. Insurance: Math. Econom., 37(1), 135151.Google Scholar
Moscadelli, M. 2004 (July). The Modelling of Operational Risk: Experience with the Analysis of the Data Collected by the Basel Committee. Available at http://ideas.repec.org/p/bdi/wptemi/td_517_04.html.CrossRefGoogle Scholar
Moynihan, R., Schweizer, B., and Sklar, A. 1978. Inequalities among operations on probability distribution functions. Pages 133–149 of: Beckenbach, E. F. (ed), Allgemeine Ungleichungen 1. Intern. Ser. Num. Math., vol. 41.Google Scholar
Müller, A., and Scarsini, M. 2001. Stochastic comparison of random vectors with a common copula. Math. Oper. Res., 26, 723740.CrossRefGoogle Scholar
Müller, A., and Scarsini, M. 2005. Archimedean copulae and positive dependence. J. Multivariate Anal., 93(2), 434445.CrossRefGoogle Scholar
Müller, A., and Stoyan, D. 2002. Comparison Methods for Stochastic Models and Risks. Chichester, John Wiley & Sons Ltd.Google Scholar
Natarajan, K., Sim, M., and Uichanco, J. 2010. Tractable robust expected utility and risk models for portfolio optimization. Math. Finance, 20(4), 695731.CrossRefGoogle Scholar
Nelsen, R. B. 2006. An Introduction to Copulas. Properties and Applications. 2nd edn. Lect. Notes Stat., vol. 139. Springer.Google Scholar
Nelsen, R. B., Quesada Molina, J. J., Rodríguez Lallena, J. A., and Úbeda Flores, M. 2004. Best-possible bounds on sets of bivariate distribution functions. J. Multivariate Anal., 90(2), 348358.CrossRefGoogle Scholar
Németh, A. B. 2003. Characterization of a Hilbert vector lattice by the metric projection onto its positive cone. J. Approx. Theo., 123(2), 295299.CrossRefGoogle Scholar
Nešlehová, J., Embrechts, P., and Chavez-Demoulin, V. 2006. Infinite-mean models and the LDA for operational risk. J. Oper. Risk, 1, 325.CrossRefGoogle Scholar
Oakes, D. 1989. Bivariate survival models induced by frailties. J. Amer. Statist. Assoc., 84(406), 487493.CrossRefGoogle Scholar
Office of the Superintendent of Financial Institutions (OSFI). 2014. Quantitative Impact Study No. 4: General – Aggregation and Diversification – Supplementary Information. Online document. www.osfi-bsif.gc.ca/Eng/Docs/qis4_ir_sup.pdf.Google Scholar
Pan, X., Qiu, G., and Hu, T. 2016. Stochastic orderings for elliptical random vectors. J. Multivariate Anal., 148, 8388.CrossRefGoogle Scholar
Panjer, H. H. 1981. Recursive evaluation of a family of compound distributions. Astin Bull., 12(1), 2226.CrossRefGoogle Scholar
Panjer, H. H. 2001. Measurement of Risk, Solvency Requirements and Allocation of Capital Within Financial Conglomerates. University of Waterloo, Institute of Insurance and Pension Research.Google Scholar
Prudential Regulation Authority (PRA). 2018. Model risk management principles for stress testing. Supervisory statement. Bank of England, Prudential Regulation Authority.Google Scholar
Puccetti, G. 2005. Bounds for Functions of Univariate and Multivariate Risks. Ph.D. Thesis, University of Pisa, Italy.Google Scholar
Puccetti, G., and Rüschendorf, L. 2012a. Bounds for joint portfolios of dependent risks. Stat. Risk Model., 29, 107132.CrossRefGoogle Scholar
Puccetti, G., and Rüschendorf, L. 2012b. Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math., 236, 18331840.CrossRefGoogle Scholar
Puccetti, G., and Rüschendorf, L. 2013. Sharp bounds for sums of dependent risks. J. Appl. Probab., 50(1), 4253.CrossRefGoogle Scholar
Puccetti, G., and Rüschendorf, L. 2014. Asymptotic equivalence of conservative VaR-and ES-based capital charges. J. Risk, 16(3), 322.CrossRefGoogle Scholar
Puccetti, G., Rüschendorf, L., and Manko, D. 2016. VaR bounds for joint portfolios with dependence constraints. Depend. Model., 4(1), 368381.Google Scholar
Puccetti, G., Rüschendorf, L., Small, D., and Vanduffel, S. 2017. Reduction of value-at-risk bounds via independence and variance information. Scand. Actuar. J., 3, 245266.Google Scholar
Puccetti, G., Wang, B., and Wang, R. 2012. Advances in complete mixability. J. Appl. Probab., 49(2), 430440.CrossRefGoogle Scholar
Puccetti, G., Wang, B., and Wang, R. 2013. Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates. Insurance: Math. Econom., 53(2), 821828.Google Scholar
Rachev, S. T., and Rüschendorf, L. 1994. Solution of some transportation problems with relaxed and additional constraints. SIAM Contr. Optim., 32, 673689.CrossRefGoogle Scholar
Rachev, S. T., and Rüschendorf, L. 1998a. Mass Transportation Problems. Vol. I: Theory. Springer.Google Scholar
Rachev, S. T., and Rüschendorf, L. 1998b. Mass Transportation Problems. Vol. II: Applications. Springer.Google Scholar
Ross, S. A. 1976. The arbitrage theory of capital asset pricing. J. Econom. Theory, 13(3), 341360.CrossRefGoogle Scholar
Rüschendorf, L. 1980. Inequalities for the expectation of Δ-monotone functions. Z. Wahrscheinlichkeitstheor. Verw. Geb., 54, 341354.CrossRefGoogle Scholar
Rüschendorf, L. 1981a. Characterization of dependence concepts for the normal distribution. Ann. Inst. Stat. Math., 33, 347359.CrossRefGoogle Scholar
Rüschendorf, L. 1981b. Sharpness of Fréchet bounds. Z. Wahrscheinlichkeitstheor. Verw. Geb., 57, 293302.CrossRefGoogle Scholar
Rüschendorf, L. 1982. Random variables with maximum sums. Adv. Appl. Probab., 14, 623632.CrossRefGoogle Scholar
Rüschendorf, L. 1983a. On the multidimensional assignment problem. Methods Oper. Res., 47, 107113.Google Scholar
Rüschendorf, L. 1983b. Solution of a statistical optimization problem by rearrangement methods. Metrika, 30, 5562.CrossRefGoogle Scholar
Rüschendorf, L. 1984. On the minimum discrimination information theorem. Stat. Dec., Suppl. issue 1, 1, 263283.Google Scholar
Rüschendorf, L. 1985. The Wasserstein distance and approximation theorems. Z. Wahrscheinlichkeitstheor. Verw. Geb., 70, 117129.CrossRefGoogle Scholar
Rüschendorf, L. 1991. Bounds for distributions with multivariate marginals. Pages 285–310 of: Mosler, K., and Scarsini, M. (eds), Stochastic Order and Decision under Risk, vol. 19. IMS Lecture Notes.CrossRefGoogle Scholar
Rüschendorf, L. 2004. Comparison of multivariate risks and positive dependence. J. Appl. Probab., 41, 391406.CrossRefGoogle Scholar
Rüschendorf, L. 2005. Stochastic ordering of risks, influence of dependence and a.s. constructions. Pages 19–56 of: Balakrishnan, N., Bairamov, I. G., and Gebizlioglu, O. L. (eds), Advances on Models, Characterizations and Applications. Chapman & Hall/CRC Press.Google Scholar
Rüschendorf, L. 2013. Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer.CrossRefGoogle Scholar
Rüschendorf, L. 2017a. Improved Hoeffding–Fréchet bounds and applications to var estimates. Pages 181–202 of: Úbeda Flores, M., et al. (eds), Copulas and Dependence Models with Applications. Springer.Google Scholar
Rüschendorf, L. 2017b. Risk bounds and partial dependence information. Pages 345–366 of: Ferger, D., González Manteiga, W., Schmidt, T., and Wang, J.-L. (eds), From Statistics to Mathematical Finance. Springer.Google Scholar
Rüschendorf, L., and Uckelmann, L. 2002. Variance minimization and random variables with constant sum. Pages 221–222 of: Cuadras, C. M., et al. (eds), Distributions With Given Marginals and Statistical Modelling. Springer.Google Scholar
Rüschendorf, L., and Witting, J. 2017. VaR bounds in models with partial dependence information on subgroups. Depend. Model., 5(1), 5974.CrossRefGoogle Scholar
Rustagi, J. S. 1976. Variational Methods in Statistics. Mathematics in Science and Engineering, vol. 121. Elsevier, Amsterdam.Google Scholar
Rustagi, J. S. 1957. On minimizing and maximizing a certain integral with statistical applications. Ann. Math. Stat., 28, 309328.CrossRefGoogle Scholar
Salmon, F. 2009. Recipe for disaster: The formula that killed Wall Street. Wired Magazine, 17(3), Available online at www.wired.com/2009/02/wp–quant/.Google Scholar
Santos, A. A. P., Nogales, F. J., and Ruiz, E. 2013. Comparing univariate and multivariate models to forecast portfolio value-at-risk. J. Financial Econom., 11(2), 400441.CrossRefGoogle Scholar
Scaillet, O. 2005. A Kolmogorov–Smirnov type test for positive quadrant dependence. Can. J. Stat., 33(3), 415427.CrossRefGoogle Scholar
Shaked, M., and Shantikumar, J. G. 2007. Stochastic Orders. New York, Springer.CrossRefGoogle Scholar
Sharpe, W. F. 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. J. Finance, 19(3), 425442.Google Scholar
Sklar, A. 1973. Random variables, joint distribution functions, and copulas. Kybernetika, 9, 449460.Google Scholar
Snijders, T. A. B. 1984. Antithetic variates for Monte Carlo estimation of probabilities. Stat. Neerl., 38, 5573.CrossRefGoogle Scholar
Strassen, V. 1965. The existence of probability measures with given marginals. Ann. Math. Stat., 36, 423439.CrossRefGoogle Scholar
Tankov, P. 2011. Improved Fréchet bounds and model-free pricing of multi-asset options. J. Appl. Probab., 48, 389403.CrossRefGoogle Scholar
Tsanakas, A. 2009. To split or not to split: Capital allocation with convex risk measures. Insurance: Math. Econom., 44(2), 268277.Google Scholar
Tsanakas, A., and Desli, E. 2005. Measurement and pricing of risk in insurance markets. Risk Anal., 25(6), 16531668.CrossRefGoogle ScholarPubMed
Vandendorpe, A., Ho, N.-D., Vanduffel, S., and Van Dooren, P. 2008. On the parame-terization of the CreditRisk+ model for estimating credit portfolio risk. Insurance: Math. Econom., 42(2), 736745.Google Scholar
Vanduffel, S., Shang, Z., Henrard, L., Dhaene, J., and Valdez, E. A. 2008. Analytic bounds and approximations for annuities and Asian options. Insurance: Math. Econom., 42(3), 11091117.Google Scholar
Vanmaele, M., Deelstra, G., Liinev, J., Dhaene, J., and Goovaerts, M. J. 2006. Bounds for the price of discrete arithmetic Asian options. J. Comput. Appl. Math., 185(1), 5190.CrossRefGoogle Scholar
Wang, B., and Wang, R. 2011. The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal., 102, 13441360.CrossRefGoogle Scholar
Wang, B., and Wang, R. 2015. Extreme negative dependence and risk aggregation. J. Multivariate Anal., 136, 1225.CrossRefGoogle Scholar
Wang, B., and Wang, R. 2016. Joint mixability. Oper. Res., 41(3), 808826.Google Scholar
Wang, B., Peng, L., and Yang, J. 2013. Bounds for the sum of dependent risks and worst value-at-risk with monotone marginal densities. Finance Stoch., 17(2), 395417.CrossRefGoogle Scholar
Wang, R. 2014. Asymptotic bounds for the distribution of the sum of dependent random variables. J. Appl. Probab., 51(3), 780798.CrossRefGoogle Scholar
Wang, R. 2015. Current open questions in complete mixability. Probab. Surv., 12, 1332.CrossRefGoogle Scholar
Wang, S. 1996. Premium calculation by transforming the layer premium density. Astin Bull., 26(1), 7192.CrossRefGoogle Scholar
Wei, G., and Hu, T. 2002. Supermodular dependence ordering on a class of multivariate copulas. Stat. Probab. Lett., 57, 375385.CrossRefGoogle Scholar
Whitt, M. 1976. Bivariate distributions with given marginals. Ann. Stat., 4, 12801289.CrossRefGoogle Scholar
Williamson, R. C., and Downs, T. 1990. Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds. Internat. J. Approx. Reason, 4(2), 89158.CrossRefGoogle Scholar
Wirch, J. L., and Hardy, M. R. 1999. A synthesis of risk measures for capital adequacy. Insurance: Math. Econom., 25(3), 337347.Google Scholar
Yeh, H.-C. 2007. The frailty and the Archimedean structure of the general multivariate Pareto distributions. Bull. Inst. Math. Acad. Sin. (N.S.), 2(3), 713729.Google Scholar
Yin, C. 2021. Stochastic ordering of multivariate elliptical distributions. J. Appl. Probab., 58(2), 551568.CrossRefGoogle Scholar

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