Upper risk bounds in internal factor models with constrained specification sets

Jonathan Ansari; Ludger Rüschendorf

doi:10.1186/s41546-020-00045-y

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January 2020, 5: 3 doi: 10.1186/s41546-020-00045-y

Upper risk bounds in internal factor models with constrained specification sets

Jonathan Ansari ^1,, and Ludger Rüschendorf ^2,

1. Department of Quantitative Finance, Albert-Ludwigs University of Freiburg, Platz der Alten Synagoge 1, KG II, 79098 Freiburg i. Br., Germany

2. Department of Mathematical Stochastics, Albert-Ludwigs University of Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg, Germany

Received April 17, 2019 Published March 2020

Abstract
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For the class of (partially specified) internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models. This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor. The proof of our main comparison result is not standard. It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers. An application to real market data shows considerable improvement over the standard method.

Keywords: Risk bounds, Risk factor model, Supermodular order, Convex order, Convex risk measure, Upper product of bivariate copulas, Comonotonicity.

Citation: Jonathan Ansari, Ludger Rüschendorf. Upper risk bounds in internal factor models with constrained specification sets. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 3-. doi: 10.1186/s41546-020-00045-y

References:

[1]	Aas, K., C. Czado, A. Frigessi, and H. Bakken. (2009). Pair-copula constructions of multiple dependence, Insur. Math. Econ. 44, no. 2, 182–198. Google Scholar
[2]	Ansari, J. (2019). Ordering risk bounds in partially specified factor models, University of Freiburg, Dissertation. Google Scholar
[3]	Ansari, J. and L. Rüschendorf. (2016). Ordering results for risk bounds and cost-efficient payoffs in partially specified risk factor models, Methodol. Comput. Appl. Probab., 1–22. Google Scholar
[4]	Ansari, J. and L. Rüschendorf. (2018). Ordering risk bounds in factor models, Depend. Model. 6.1, 259– 287. Google Scholar
[5]	Bäuerle, N. and A. Müller. (2006). Stochastic orders and risk measures: consistency and bounds, Insur.Math. Econ. 38, no. 1, 132–148. Google Scholar
[6]	Bernard, C. and S. Vanduffel. (2015). A new approach to assessing model risk in high dimensions, J. Bank.Financ. 58, 166–178. Google Scholar
[7]	Bernard, C., L. Rüschendorf, and S. Vanduffel. (2017a). Value-at-Risk bounds with variance constraints, J. Risk. Insur. 84, no. 3, 923–959. Google Scholar
[8]	Bernard, C., L. Rüschendorf, S. Vanduffel, and R. Wang. (2017b). Risk bounds for factor models, Financ.Stoch. 21, no. 3, 631–659. Google Scholar
[9]	Bernard, C., M. Denuit, and S. Vanduffel. (2018). Measuring portfolio risk under partial dependence information, J. Risk Insur. 85, no. 3, 843–863. Google Scholar
[10]	Bignozzi, V., G. Puccetti, and L Rüschendorf. (2015). Reducing model risk via positive and negative dependence assumptions, Insur. Math. Econ. 61, 17–26. Google Scholar
[11]	Cornilly, D., L. Rüschendorf, and S. Vanduffel. (2018). Upper bounds for strictly concave distortion risk measures on moment spaces, Insur Math Econ. 82, 141–151. Google Scholar
[12]	de Schepper, A. and B. Heijnen. (2010). How to estimate the Value at Risk under incomplete information, J. Comput. Appl. Math. 233, no. 9, 2213–2226. Google Scholar
[13]	Demarta, S. and A.J. McNeil. (2005). The t copula and related copulas, Int. Stat. Rev. 73, no. 1, 111–129. Google Scholar
[14]	Denuit, M., C. Genest, and E. Marceau. (1999). Stochastic bounds on sums of dependent risks, Insur. Math.Econ. 25, no. 1, 85–104. Google Scholar
[15]	Embrechts, P. and G. Puccetti. (2006). Bounds for functions of dependent risks, Financ. Stoch. 10, no. 3, 341–352. Google Scholar
[16]	Embrechts, P., G. Puccetti, and L. Rüschendorf. (2013). Model uncertainty and VaR aggregation, J. Banking Financ. 37, no. 8, 2750–2764. Google Scholar
[17]	Embrechts, P., G. Puccetti, L. Rüschendorf, R. Wang, and A. Beleraj. (2014). An academic response to basel 3.5, Risks 2, no. 1, 25–48. Google Scholar
[18]	Embrechts, P., B. Wang, and R. Wang. (2015). Aggregation-robustness and model uncertainty of regulatory risk measures, Financ. Stoch. 19, no. 4, 763–790. Google Scholar
[19]	Föllmer, H. and A. Schied. (2010). Convex and coherent risk measures., Encycl. Quant. Financ., 355–363. Google Scholar
[20]	Goovaerts, M.J., R. Kaas, and R.J.A. Laeven. (2011). Worst case risk measurement: back to the future?Insur. Math. Econ. 49, no. 3, 380–392. Google Scholar
[21]	Hürlimann, W. (2002). Analytical bounds for two Value-at-Risk functionals, ASTIN Bull. 32, no. 2, 235–265. Google Scholar
[22]	Hürlimann, W. (2008). Extremal moment methods and stochastic orders, Bol. Asoc. Mat. Venez. 15, no. 2, 153–301. Google Scholar
[23]	Kaas, R. and M.J. Goovaerts. (1986). Best bounds for positive distributions with fixed moments, Insur.Math. Econ. 5, 87–95. Google Scholar
[24]	McNeil, A.J., R. Frey, and P. Embrechts. (2015). Quantitative Risk Management. Concepts, Techniques and Tools., second edn, Princeton University Press, Princeton. Google Scholar
[25]	Müller, A. (1997). Stop-loss order for portfolios of dependent risks, Insur. Math. Econ. 21, no. 3, 219–223. Google Scholar
[26]	Müller, A. (2013). Duality theory and transfers for stochastic order relations. In: Stochastic orders in reliability and risk, Springer, New York. Google Scholar
[27]	Müller, A. and M. Scarsini. (2001). Stochastic comparison of random vectors with a common copula, Math. Oper. Res. 26, no. 4, 723–740. Google Scholar
[28]	Müller, A. and M. Scarsini. (2006). Stochastic order relations and lattices of probability measures, SIAM J. Optim. 16, no. 4, 1024–1043. Google Scholar
[29]	Müller, A. and D. Stoyan. (2002). Comparison Methods for Stochastic Models and Risks, Wiley, Chichester. Google Scholar
[30]	Nelsen, R.B. (2006). An introduction to copulas, 2nd ed, Springer, New York. Google Scholar
[31]	Nelsen, R.B., J.J. Quesada-Molina, J.A. Rodríguez-Lallena, and M. Úbeda-Flores. (2001). Bounds on bivariate distribution functions with given margins and measures of association, Commun. Stat.Theory Methods 30, no. 6, 1155–1162. Google Scholar
[32]	Puccetti, G. and L. Rüschendorf. (2012a). Bounds for joint portfolios of dependent risks, Stat. Risk. Model.Appl. Financ. Insur. 29, no. 2, 107–132. Google Scholar
[33]	Puccetti, G. and L. Rüschendorf. (2012b). Computation of sharp bounds on the distribution of a function of dependent risks, J. Comput. Appl. Math 236, no. 7, 1833–1840. Google Scholar
[34]	Puccetti, G. and L. Rüschendorf. (2013). Sharp bounds for sums of dependent risks, J. Appl. Probab. 50, no. 1, 42–53. Google Scholar
[35]	Puccetti, G., L. Rüschendorf, D. Small, and S. Vanduffel. (2017). Reduction of Value-at-Risk bounds via independence and variance information, Scand. Actuar. J. 2017, no. 3, 245–266. Google Scholar
[36]	Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process, J. Stat. Plann. Inference 139, no. 11, 3921–3927. Google Scholar
[37]	Rüschendorf, L. (2013). Mathematical Risk Analysis, Springer, New York. Google Scholar
[38]	Rüschendorf, L. (2017a). Improved Hoeffding–Fréchet bounds and applications to VaR estimates. In:Copulas and Dependence Models with Applications. Contributions in Honor of Roger B. Nelsen(M. Úbeda Flores, E. de Amo Artero, F. Durante, and J. Fernández Sánchez, eds.), Springer, Cham. https://doi.org/10.1007/978-3-319-64221-5 12. Google Scholar
[39]	Rüschendorf, L. (2017b). Risk bounds and partial dependence information. In: From Statistics to Mathematical Finance, Springer, Festschrift in honour of Winfried Stute, Cham. Google Scholar
[40]	Rüschendorf, L. and J. Witting. (2017). VaR bounds in models with partial dependence information on subgroups, Depend Model 5, 59–74. Google Scholar
[41]	Shaked, M. and J.G. Shantikumar. (2007). Stochastic Orders, Springer, New York. Google Scholar
[42]	Tian, R. (2008). Moment problems with applications to Value-at-Risk and portfolio management, Georgia State University, Dissertation. Google Scholar

show all references

References:

[1]	Aas, K., C. Czado, A. Frigessi, and H. Bakken. (2009). Pair-copula constructions of multiple dependence, Insur. Math. Econ. 44, no. 2, 182–198. Google Scholar
[2]	Ansari, J. (2019). Ordering risk bounds in partially specified factor models, University of Freiburg, Dissertation. Google Scholar
[3]	Ansari, J. and L. Rüschendorf. (2016). Ordering results for risk bounds and cost-efficient payoffs in partially specified risk factor models, Methodol. Comput. Appl. Probab., 1–22. Google Scholar
[4]	Ansari, J. and L. Rüschendorf. (2018). Ordering risk bounds in factor models, Depend. Model. 6.1, 259– 287. Google Scholar
[5]	Bäuerle, N. and A. Müller. (2006). Stochastic orders and risk measures: consistency and bounds, Insur.Math. Econ. 38, no. 1, 132–148. Google Scholar
[6]	Bernard, C. and S. Vanduffel. (2015). A new approach to assessing model risk in high dimensions, J. Bank.Financ. 58, 166–178. Google Scholar
[7]	Bernard, C., L. Rüschendorf, and S. Vanduffel. (2017a). Value-at-Risk bounds with variance constraints, J. Risk. Insur. 84, no. 3, 923–959. Google Scholar
[8]	Bernard, C., L. Rüschendorf, S. Vanduffel, and R. Wang. (2017b). Risk bounds for factor models, Financ.Stoch. 21, no. 3, 631–659. Google Scholar
[9]	Bernard, C., M. Denuit, and S. Vanduffel. (2018). Measuring portfolio risk under partial dependence information, J. Risk Insur. 85, no. 3, 843–863. Google Scholar
[10]	Bignozzi, V., G. Puccetti, and L Rüschendorf. (2015). Reducing model risk via positive and negative dependence assumptions, Insur. Math. Econ. 61, 17–26. Google Scholar
[11]	Cornilly, D., L. Rüschendorf, and S. Vanduffel. (2018). Upper bounds for strictly concave distortion risk measures on moment spaces, Insur Math Econ. 82, 141–151. Google Scholar
[12]	de Schepper, A. and B. Heijnen. (2010). How to estimate the Value at Risk under incomplete information, J. Comput. Appl. Math. 233, no. 9, 2213–2226. Google Scholar
[13]	Demarta, S. and A.J. McNeil. (2005). The t copula and related copulas, Int. Stat. Rev. 73, no. 1, 111–129. Google Scholar
[14]	Denuit, M., C. Genest, and E. Marceau. (1999). Stochastic bounds on sums of dependent risks, Insur. Math.Econ. 25, no. 1, 85–104. Google Scholar
[15]	Embrechts, P. and G. Puccetti. (2006). Bounds for functions of dependent risks, Financ. Stoch. 10, no. 3, 341–352. Google Scholar
[16]	Embrechts, P., G. Puccetti, and L. Rüschendorf. (2013). Model uncertainty and VaR aggregation, J. Banking Financ. 37, no. 8, 2750–2764. Google Scholar
[17]	Embrechts, P., G. Puccetti, L. Rüschendorf, R. Wang, and A. Beleraj. (2014). An academic response to basel 3.5, Risks 2, no. 1, 25–48. Google Scholar
[18]	Embrechts, P., B. Wang, and R. Wang. (2015). Aggregation-robustness and model uncertainty of regulatory risk measures, Financ. Stoch. 19, no. 4, 763–790. Google Scholar
[19]	Föllmer, H. and A. Schied. (2010). Convex and coherent risk measures., Encycl. Quant. Financ., 355–363. Google Scholar
[20]	Goovaerts, M.J., R. Kaas, and R.J.A. Laeven. (2011). Worst case risk measurement: back to the future?Insur. Math. Econ. 49, no. 3, 380–392. Google Scholar
[21]	Hürlimann, W. (2002). Analytical bounds for two Value-at-Risk functionals, ASTIN Bull. 32, no. 2, 235–265. Google Scholar
[22]	Hürlimann, W. (2008). Extremal moment methods and stochastic orders, Bol. Asoc. Mat. Venez. 15, no. 2, 153–301. Google Scholar
[23]	Kaas, R. and M.J. Goovaerts. (1986). Best bounds for positive distributions with fixed moments, Insur.Math. Econ. 5, 87–95. Google Scholar
[24]	McNeil, A.J., R. Frey, and P. Embrechts. (2015). Quantitative Risk Management. Concepts, Techniques and Tools., second edn, Princeton University Press, Princeton. Google Scholar
[25]	Müller, A. (1997). Stop-loss order for portfolios of dependent risks, Insur. Math. Econ. 21, no. 3, 219–223. Google Scholar
[26]	Müller, A. (2013). Duality theory and transfers for stochastic order relations. In: Stochastic orders in reliability and risk, Springer, New York. Google Scholar
[27]	Müller, A. and M. Scarsini. (2001). Stochastic comparison of random vectors with a common copula, Math. Oper. Res. 26, no. 4, 723–740. Google Scholar
[28]	Müller, A. and M. Scarsini. (2006). Stochastic order relations and lattices of probability measures, SIAM J. Optim. 16, no. 4, 1024–1043. Google Scholar
[29]	Müller, A. and D. Stoyan. (2002). Comparison Methods for Stochastic Models and Risks, Wiley, Chichester. Google Scholar
[30]	Nelsen, R.B. (2006). An introduction to copulas, 2nd ed, Springer, New York. Google Scholar
[31]	Nelsen, R.B., J.J. Quesada-Molina, J.A. Rodríguez-Lallena, and M. Úbeda-Flores. (2001). Bounds on bivariate distribution functions with given margins and measures of association, Commun. Stat.Theory Methods 30, no. 6, 1155–1162. Google Scholar
[32]	Puccetti, G. and L. Rüschendorf. (2012a). Bounds for joint portfolios of dependent risks, Stat. Risk. Model.Appl. Financ. Insur. 29, no. 2, 107–132. Google Scholar
[33]	Puccetti, G. and L. Rüschendorf. (2012b). Computation of sharp bounds on the distribution of a function of dependent risks, J. Comput. Appl. Math 236, no. 7, 1833–1840. Google Scholar
[34]	Puccetti, G. and L. Rüschendorf. (2013). Sharp bounds for sums of dependent risks, J. Appl. Probab. 50, no. 1, 42–53. Google Scholar
[35]	Puccetti, G., L. Rüschendorf, D. Small, and S. Vanduffel. (2017). Reduction of Value-at-Risk bounds via independence and variance information, Scand. Actuar. J. 2017, no. 3, 245–266. Google Scholar
[36]	Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process, J. Stat. Plann. Inference 139, no. 11, 3921–3927. Google Scholar
[37]	Rüschendorf, L. (2013). Mathematical Risk Analysis, Springer, New York. Google Scholar
[38]	Rüschendorf, L. (2017a). Improved Hoeffding–Fréchet bounds and applications to VaR estimates. In:Copulas and Dependence Models with Applications. Contributions in Honor of Roger B. Nelsen(M. Úbeda Flores, E. de Amo Artero, F. Durante, and J. Fernández Sánchez, eds.), Springer, Cham. https://doi.org/10.1007/978-3-319-64221-5 12. Google Scholar
[39]	Rüschendorf, L. (2017b). Risk bounds and partial dependence information. In: From Statistics to Mathematical Finance, Springer, Festschrift in honour of Winfried Stute, Cham. Google Scholar
[40]	Rüschendorf, L. and J. Witting. (2017). VaR bounds in models with partial dependence information on subgroups, Depend Model 5, 59–74. Google Scholar
[41]	Shaked, M. and J.G. Shantikumar. (2007). Stochastic Orders, Springer, New York. Google Scholar
[42]	Tian, R. (2008). Moment problems with applications to Value-at-Risk and portfolio management, Georgia State University, Dissertation. Google Scholar

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