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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

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.

[2]

Ansari, J. (2019). Ordering risk bounds in partially specified factor models, University of Freiburg, Dissertation.

[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.

[4]

Ansari, J. and L. Rüschendorf. (2018). Ordering risk bounds in factor models, Depend. Model. 6.1, 259– 287.

[5]

Bäuerle, N. and A. Müller. (2006). Stochastic orders and risk measures: consistency and bounds, Insur.Math. Econ. 38, no. 1, 132–148.

[6]

Bernard, C. and S. Vanduffel. (2015). A new approach to assessing model risk in high dimensions, J. Bank.Financ. 58, 166–178.

[7]

Bernard, C., L. Rüschendorf, and S. Vanduffel. (2017a). Value-at-Risk bounds with variance constraints, J. Risk. Insur. 84, no. 3, 923–959.

[8]

Bernard, C., L. Rüschendorf, S. Vanduffel, and R. Wang. (2017b). Risk bounds for factor models, Financ.Stoch. 21, no. 3, 631–659.

[9]

Bernard, C., M. Denuit, and S. Vanduffel. (2018). Measuring portfolio risk under partial dependence information, J. Risk Insur. 85, no. 3, 843–863.

[10]

Bignozzi, V., G. Puccetti, and L Rüschendorf. (2015). Reducing model risk via positive and negative dependence assumptions, Insur. Math. Econ. 61, 17–26.

[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.

[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.

[13]

Demarta, S. and A.J. McNeil. (2005). The t copula and related copulas, Int. Stat. Rev. 73, no. 1, 111–129.

[14]

Denuit, M., C. Genest, and E. Marceau. (1999). Stochastic bounds on sums of dependent risks, Insur. Math.Econ. 25, no. 1, 85–104.

[15]

Embrechts, P. and G. Puccetti. (2006). Bounds for functions of dependent risks, Financ. Stoch. 10, no. 3, 341–352.

[16]

Embrechts, P., G. Puccetti, and L. Rüschendorf. (2013). Model uncertainty and VaR aggregation, J. Banking Financ. 37, no. 8, 2750–2764.

[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.

[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.

[19]

Föllmer, H. and A. Schied. (2010). Convex and coherent risk measures., Encycl. Quant. Financ., 355–363.

[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.

[21]

Hürlimann, W. (2002). Analytical bounds for two Value-at-Risk functionals, ASTIN Bull. 32, no. 2, 235–265.

[22]

Hürlimann, W. (2008). Extremal moment methods and stochastic orders, Bol. Asoc. Mat. Venez. 15, no. 2, 153–301.

[23]

Kaas, R. and M.J. Goovaerts. (1986). Best bounds for positive distributions with fixed moments, Insur.Math. Econ. 5, 87–95.

[24]

McNeil, A.J., R. Frey, and P. Embrechts. (2015). Quantitative Risk Management. Concepts, Techniques and Tools., second edn, Princeton University Press, Princeton.

[25]

Müller, A. (1997). Stop-loss order for portfolios of dependent risks, Insur. Math. Econ. 21, no. 3, 219–223.

[26]

Müller, A. (2013). Duality theory and transfers for stochastic order relations. In: Stochastic orders in reliability and risk, Springer, New York.

[27]

Müller, A. and M. Scarsini. (2001). Stochastic comparison of random vectors with a common copula, Math. Oper. Res. 26, no. 4, 723–740.

[28]

Müller, A. and M. Scarsini. (2006). Stochastic order relations and lattices of probability measures, SIAM J. Optim. 16, no. 4, 1024–1043.

[29]

Müller, A. and D. Stoyan. (2002). Comparison Methods for Stochastic Models and Risks, Wiley, Chichester.

[30]

Nelsen, R.B. (2006). An introduction to copulas, 2nd ed, Springer, New York.

[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.

[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.

[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.

[34]

Puccetti, G. and L. Rüschendorf. (2013). Sharp bounds for sums of dependent risks, J. Appl. Probab. 50, no. 1, 42–53.

[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.

[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.

[37]

Rüschendorf, L. (2013). Mathematical Risk Analysis, Springer, New York.

[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.

[39]

Rüschendorf, L. (2017b). Risk bounds and partial dependence information. In: From Statistics to Mathematical Finance, Springer, Festschrift in honour of Winfried Stute, Cham.

[40]

Rüschendorf, L. and J. Witting. (2017). VaR bounds in models with partial dependence information on subgroups, Depend Model 5, 59–74.

[41]

Shaked, M. and J.G. Shantikumar. (2007). Stochastic Orders, Springer, New York.

[42]

Tian, R. (2008). Moment problems with applications to Value-at-Risk and portfolio management, Georgia State University, Dissertation.

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.

[2]

Ansari, J. (2019). Ordering risk bounds in partially specified factor models, University of Freiburg, Dissertation.

[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.

[4]

Ansari, J. and L. Rüschendorf. (2018). Ordering risk bounds in factor models, Depend. Model. 6.1, 259– 287.

[5]

Bäuerle, N. and A. Müller. (2006). Stochastic orders and risk measures: consistency and bounds, Insur.Math. Econ. 38, no. 1, 132–148.

[6]

Bernard, C. and S. Vanduffel. (2015). A new approach to assessing model risk in high dimensions, J. Bank.Financ. 58, 166–178.

[7]

Bernard, C., L. Rüschendorf, and S. Vanduffel. (2017a). Value-at-Risk bounds with variance constraints, J. Risk. Insur. 84, no. 3, 923–959.

[8]

Bernard, C., L. Rüschendorf, S. Vanduffel, and R. Wang. (2017b). Risk bounds for factor models, Financ.Stoch. 21, no. 3, 631–659.

[9]

Bernard, C., M. Denuit, and S. Vanduffel. (2018). Measuring portfolio risk under partial dependence information, J. Risk Insur. 85, no. 3, 843–863.

[10]

Bignozzi, V., G. Puccetti, and L Rüschendorf. (2015). Reducing model risk via positive and negative dependence assumptions, Insur. Math. Econ. 61, 17–26.

[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.

[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.

[13]

Demarta, S. and A.J. McNeil. (2005). The t copula and related copulas, Int. Stat. Rev. 73, no. 1, 111–129.

[14]

Denuit, M., C. Genest, and E. Marceau. (1999). Stochastic bounds on sums of dependent risks, Insur. Math.Econ. 25, no. 1, 85–104.

[15]

Embrechts, P. and G. Puccetti. (2006). Bounds for functions of dependent risks, Financ. Stoch. 10, no. 3, 341–352.

[16]

Embrechts, P., G. Puccetti, and L. Rüschendorf. (2013). Model uncertainty and VaR aggregation, J. Banking Financ. 37, no. 8, 2750–2764.

[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.

[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.

[19]

Föllmer, H. and A. Schied. (2010). Convex and coherent risk measures., Encycl. Quant. Financ., 355–363.

[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.

[21]

Hürlimann, W. (2002). Analytical bounds for two Value-at-Risk functionals, ASTIN Bull. 32, no. 2, 235–265.

[22]

Hürlimann, W. (2008). Extremal moment methods and stochastic orders, Bol. Asoc. Mat. Venez. 15, no. 2, 153–301.

[23]

Kaas, R. and M.J. Goovaerts. (1986). Best bounds for positive distributions with fixed moments, Insur.Math. Econ. 5, 87–95.

[24]

McNeil, A.J., R. Frey, and P. Embrechts. (2015). Quantitative Risk Management. Concepts, Techniques and Tools., second edn, Princeton University Press, Princeton.

[25]

Müller, A. (1997). Stop-loss order for portfolios of dependent risks, Insur. Math. Econ. 21, no. 3, 219–223.

[26]

Müller, A. (2013). Duality theory and transfers for stochastic order relations. In: Stochastic orders in reliability and risk, Springer, New York.

[27]

Müller, A. and M. Scarsini. (2001). Stochastic comparison of random vectors with a common copula, Math. Oper. Res. 26, no. 4, 723–740.

[28]

Müller, A. and M. Scarsini. (2006). Stochastic order relations and lattices of probability measures, SIAM J. Optim. 16, no. 4, 1024–1043.

[29]

Müller, A. and D. Stoyan. (2002). Comparison Methods for Stochastic Models and Risks, Wiley, Chichester.

[30]

Nelsen, R.B. (2006). An introduction to copulas, 2nd ed, Springer, New York.

[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.

[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.

[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.

[34]

Puccetti, G. and L. Rüschendorf. (2013). Sharp bounds for sums of dependent risks, J. Appl. Probab. 50, no. 1, 42–53.

[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.

[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.

[37]

Rüschendorf, L. (2013). Mathematical Risk Analysis, Springer, New York.

[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.

[39]

Rüschendorf, L. (2017b). Risk bounds and partial dependence information. In: From Statistics to Mathematical Finance, Springer, Festschrift in honour of Winfried Stute, Cham.

[40]

Rüschendorf, L. and J. Witting. (2017). VaR bounds in models with partial dependence information on subgroups, Depend Model 5, 59–74.

[41]

Shaked, M. and J.G. Shantikumar. (2007). Stochastic Orders, Springer, New York.

[42]

Tian, R. (2008). Moment problems with applications to Value-at-Risk and portfolio management, Georgia State University, Dissertation.

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