Abstract
The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can represent bailout costs. More recently, a framework has been introduced, where institutions are supplied with capital allocations before aggregation. This yields an interpretation that is particularly useful for regulatory purposes. In each framework, the set of all feasible capital allocations leads to a multivariate risk measure. In this paper, we present dual representations for scalar systemic risk measures as well as for the corresponding multivariate risk measures concerning capital allocations. Our results cover both frameworks: aggregating after allocating and allocating after aggregation. As examples, we consider the aggregation mechanisms of the Eisenberg–Noe model as well as those of the resource allocation and network flow models.
Similar content being viewed by others
References
Amini, H., Filipovic, D., Minca, A.: Systemic risk and central clearing counterparty design. Swiss Finance Institute Research Paper No. 13–34, SSRN e-prints, 2275376 (2015)
Ararat, Ç., Hamel, A.H., Rudloff, B.: Set-valued shortfall and divergence risk measures. Int. J. Theor. Appl. Finance 20(5), 1750026 (48 pages) (2017)
Armenti, Y., Crépey, S., Drapeau, S., Papapantoleon, A.: Multivariate shortfall risk allocation. SIAM J. Financ. Math. 9(1), 90–126 (2018)
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)
Biagini, F., Fouque, J.-P., Fritelli, M., Meyer-Brandis, T.: A unified approach to systemic risk measures via acceptance sets. Math. Finance 29(1), 329–367 (2019)
Biagini, F., Fouque, J.-P., Fritelli, M., Meyer-Brandis, T.: On fairness of systemic risk measures. arxiv:1803.09898 (2019)
Boţ, R.I., Grad, S.-M., Wanka, G.: Generalized Moreau–Rockafellar results for composed convex functions. Optimization 58(7), 917–933 (2009)
Brunnermeier, M.K., Cheridito, P.: Measuring and allocating systemic risk. Risks 7(2), 46: 1-19 (2019)
Chen, C., Iyengar, G., Moallemi, C.C.: An axiomatic approach to systemic risk. Manag. Sci. 59(6), 1373–1388 (2013)
Cifuentes, R., Shin, H.S., Ferrucci, G.: Liquidity risk and contagion. J. Eur. Econ. Assoc. 3(2–3), 556–566 (2005)
Eisenberg, L., Noe, T.H.: Systemic risk in financial systems. Manag. Sci. 47(2), 236–249 (2001)
Farkas, W., Koch-Medina, P., Munari, C.: Measuring risk with multiple eligible assets. Math. Financ. Econ. 9(1), 3–27 (2015)
Feinstein, Z., Rudloff, B., Weber, S.: Measures of systemic risk. SIAM J. Financ. Math. 8(1), 672–708 (2017)
Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6(4), 429–447 (2002)
Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. De Gruyter Textbook Series, 3rd edn. Walter de Gruyter GmbH & Co. KG, Berlin/New York (2011)
Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1(1), 66–95 (2010)
Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C.: Set optimization—a rather short introduction. In: Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C. (eds.) Set Optimization and Applications—The State of the Art. From Set Relations to Set-Valued Risk Measures, pp. 65–141. Springer, Berlin (2015)
Harris, T.E., Ross, F.S.: Fundamentals of a Method for Evaluating Rail Net Capacities, Research Memorandum RM-1573. The RAND Corporation, Santa Monica (1955)
Hoffmann, H., Meyer-Brandis, T., Svindland, G.: Risk-consistent conditional systemic risk measures. Stoch. Process. Appl. 126(7), 2014–2037 (2016)
Kabanov, Y., Mokbel, R., El Bitar, K.: Clearing in financial networks. Theory Probab. Appl. 62(2), 311–344 (2017)
Kromer, E., Overbeck, L., Zilch, K.: Systemic risk measures over general measurable spaces. Math. Methods Oper. Res. 84(2), 323–357 (2016)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rockafellar, R.T., Wets, R.J.-B.: Variational analysis, Grundlehren der mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998). (Corrected third printing 2010)
Rogers, L.C.G., Veraart, L.A.M.: Failure and rescue in an interbank network. Manag. Sci. 59(4), 882–898 (2013)
Schrijver, A.: On the history of the transportation and maximum flow problems. Math. Program. Ser. B 91(3), 437–445 (2002)
Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant No. 1321794 and the OeNB anniversary fund, Project Number 17793. Part of the manuscript was written when the first author visited Vienna University of Economics and Business. The authors would like to thank an anonymous referee for useful comments and suggestions that helped improving the manuscript. The first author would like to thank Fabio Bellini, Zachary Feinstein and Daniel Ocone for fruitful discussions. The authors would like to thank Cosimo-Andrea Munari and Maria Arduca for pointing out an issue in an earlier version of the paper, as well as Alexander Smirnow and Jana Hlavinova for pointing out a simplification in the proof of the second part of Theorem 3.2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ararat, Ç., Rudloff, B. Dual representations for systemic risk measures. Math Finan Econ 14, 139–174 (2020). https://doi.org/10.1007/s11579-019-00249-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11579-019-00249-7
Keywords
- Systemic risk
- Risk measure
- Financial network
- Dual representation
- Convex duality
- Penalty function
- Relative entropy
- Multivariate risk
- Shortfall risk