Abstract
We study the close relationship between coherent risk measures and convex risk measures. Inspired by the obtained results, we propose a class of coherent risk measures induced by convex risk measures. The robust representation and minimization problem of the induced coherent risk measure are investigated. A new coherent risk measure, the Entropic Conditional Value-at-Risk (ECVaR), is proposed as a special case. We show how to apply the induced coherent risk measure to realistic portfolio selection problems. Finally, by comparing its out-of-sample performance with that of CVaR, entropic risk measure, as well as entropic value-at-risk, we carry out a series of empirical tests to demonstrate the practicality and superiority of the ECVaR measure in optimal portfolio selection.
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References
Acerbi C (2002) Spectral measures of risk: a coherent representation of subjective risk aversion. J Bank Financ 26:1505–1518
Acerbi Tasche D (2002) On the coherence of expected shortfall. J Bank Financ 26:1487–1503
Ahmadi-Javid A (2012) Entropic value-at-risk: a new coherent risk measure. J Optim Theory App 155:1105–1123
Artzner P (1999) Application of coherent risk measures to capital requirements in insurance. N Am Actuar J 3:11–25
Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Financ 9:203–228
Bellini F, Rosazza Gianin E (2008) On Haezendonck risk measures. J Bank Financ 32:986–994
Bellini F, Klar B, Müller A, Rosazza Gianin E (2014) Generalized quantiles as risk measures. Insur Math Econ 54:41–48
Ben-Tal A, Teboulle M (2007) An old-new concept of convex risk measures: an optimized certainty equivalent. Math Financ 17:449–476
Canakgoz NA, Beasley JE (2009) Mixed-integer programming approaches for index tracking and enhanced indexation. Eur J Oper Res 196:384–399
Chen ZP, Wang Y (2008) Two-sided coherent risk measures and their application in realistic portfolio optimization. J Bank Financ 32:2667–2673
Delbaen F (2002) Coherent risk measures on general probability spaces. In: Advances in finance and stochastics. Essays in Honour of Dieter Sondermann. Springer, pp 1–37
Drapeau S, Kupper M (2013) Risk preferences and their robust representation. Math Oper Res 38(1):28–62
Eichhorn A, Römisch W (2005) Polyhedral measures of risk in stochastic programming. SIAM J Optim 16:69–95
Fischer T (2003) Risk capital allocation by coherent risk measures based on one-sided moments. Insur Math Econ 32:135–146
Föllmer H, Schied A (2002) Convex measures of risk and trading constraints. Financ Stoch 6:429–447
Föllmer H, Schied A (2010) Convex and coherent risk measures. In: Cont R (ed) Encyclopedia of quantitative finance. Wiley, pp 1200–1204
Föllmer H, Schied A (2011) Stochastic finance–an introduction in discrete time. Walter de Gruyter and Co., Berlin
Föllmer H, Knispel T (2011) Entropic risk measures: coherence vs. convexity, model ambiguity, and robust large deviations. Stoch Dynam 11:333–351
Fritelli M, Rosazza Gianin E (2002) Putting order in risk measures. J Bank Financ 26:1473–1486
Guastaroba G, Mansini R, Speranza MG (2009) On the effectiveness of scenario generation techniques in single-period portfolio optimization. Eur J Oper Res 192:500–511
Krokhmal P (2007) Higher moment coherent risk measures. Quant Financ 7:373–387
Krzemienowski A (2009) Risk preference modeling with conditional average: an application to portfolio optimization. Ann Oper Res 165:67–95
Laeven RJA, Stadje MA (2013) Entropy coherent and entropy convex measures of risk. Math Oper Res 38:265–293
Rockafellar RT, Uryasev SP (2002) Conditional value-at-risk for general loss distributions. J Bank Financ 26:1443–1471
Wang S, Young V, Panjer H (1997) Axiomatic characterization of insurance prices. Insur Math Econ 21:173–183
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Numbers 71371152 and 11571270). The authors are grateful to the anonymous reviewers and the editor for their constructive comments, which have helped us to improve the paper significantly in both content and style.
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Chen, Z., Hu, Q. On Coherent Risk Measures Induced by Convex Risk Measures. Methodol Comput Appl Probab 20, 673–698 (2018). https://doi.org/10.1007/s11009-017-9584-1
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DOI: https://doi.org/10.1007/s11009-017-9584-1
Keywords
- Coherent risk measure
- Convex risk measure
- Entropic conditional value-at-risk
- Robust representation
- Portfolio selection