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A Theory of the Risk for Empirical CVaR with Application to Portfolio Selection

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Abstract

When decisions are based on empirical observations, a trade-off arises between flexibility of the decision and ability to generalize to new situations. In this paper, we focus on decisions that are obtained by the empirical minimization of the Conditional Value-at-Risk (CVaR) and argue that in CVaR the trade-off between flexibility and generalization can be understood on the ground of theoretical results under very general assumptions on the system that generates the observations. The results have implications on topics related to order and structure selection in various applications where the CVaR risk-measure is used. A study on a portfolio optimization problem with real data demonstrates our results.

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References

  1. Vapnik V, The Nature of Statistical Learning Theory, Springer Science & Business Media, New York, 2013.

    MATH  Google Scholar 

  2. Calafiore G and Campi M C, Uncertain convex programs: Randomized solutions and confidence levels, Mathematical Programming, 2005, 102(1): 25–46.

    Article  MathSciNet  Google Scholar 

  3. Calafiore G C and Campi M C, The scenario approach to robust control design, IEEE Transactions on Automatic Control, 2006, 51(5): 742–753.

    Article  MathSciNet  Google Scholar 

  4. Campi M and Garatti S, The exact feasibility of randomized solutions of uncertain convex programs, SIAM Journal on Optimization, 2008, 19(3): 1211–1230.

    Article  MathSciNet  Google Scholar 

  5. Campi M C and Garatti S, A sampling-and-discarding approach to chance-constrained optimization: Feasibility and optimality, Journal of Optimization Theory and Applications, 2011, 148(2): 257–280.

    Article  MathSciNet  Google Scholar 

  6. Ramponi F A, Consistency of the scenario approach, SIAM Journal on Optimization, 2018, 28(1): 135–162.

    Article  MathSciNet  Google Scholar 

  7. Garatti S, Campi M, and Carè A, On a class of interval predictor models with universal reliability, Automatica, 2019, 110: 108542.

    Article  MathSciNet  Google Scholar 

  8. Ramponi F A and Campi M C, Expected shortfall: Heuristics and certificates, European Journal of Operational Research, 2018, 267(3): 1003–1013.

    Article  MathSciNet  Google Scholar 

  9. Campi M C and Garatti S, Wait-and-judge scenario optimization, Mathematical Programming, 2018, 167(1): 155–189.

    Article  MathSciNet  Google Scholar 

  10. Carè A, Garatti S, and Campi M C, The wait-and-judge scenario approach applied to antenna array design, Computational Management Science, 2019, 16: 481–499.

    Article  MathSciNet  Google Scholar 

  11. Carè A, Garatti S, and Campi M C, Scenario min-max optimization and the risk of empirical costs, SIAM Journal on Optimization, 2015, 25(4): 2061–2080.

    Article  MathSciNet  Google Scholar 

  12. Rockafellar R T and Uryasev S, Optimization of conditional value-at-risk, Journal of Risk, 2000, 2: 21–42.

    Article  Google Scholar 

  13. Ben-Tal A and Teboulle M, An old-new concept of convex risk measures: The optimized certainty equivalent, Mathematical Finance, 2007, 17(3): 449–476.

    Article  MathSciNet  Google Scholar 

  14. Mansini R, Ogryczak W, and Speranza M G, Conditional value at risk and related linear programming models for portfolio optimization, Annals of Operations Research, 2007, 152(1): 227–256.

    Article  MathSciNet  Google Scholar 

  15. Artzner P, Delbaen F, Eber J M, and Heath D, Coherent measures of risk, Mathematical Finance, 1999, 9(3): 203–228.

    Article  MathSciNet  Google Scholar 

  16. Pflug G C, Some remarks on the Value-at-Risk and the Conditional Value-at-Risk, Probabilistic Constrained Optimization: Methodology and Applications (Ed. by Uryasev S P), Boston, MA: Springer US, 2000, 272–281.

    Chapter  Google Scholar 

  17. Gotoh J Y and Takeda A, CVaR Minimizations in Support Vector Machines, John Wiley & Sons, Ltd, 2016, 233–265.

  18. Chan T C, Mahmoudzadeh H, and Purdie T G, A robust-CVaR optimization approach with application to breast cancer therapy, European Journal of Operational Research, 2014, 238(3): 876–885.

    Article  MathSciNet  Google Scholar 

  19. Nasir H A, Care A, and Weyer E, A randomised approach to flood control using Value-at-Risk, 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 3939–3944.

  20. Filippi C, Guastaroba G, and Speranza M, Conditional value-at-risk beyond finance: A survey, International Transactions in Operational Research, 2020, 27(3): 1277–1319.

    Article  MathSciNet  Google Scholar 

  21. Hull J C, Options, Futures and Other Derivatives, Pearson, 2019.

  22. David H A and Nagaraja H N, Order Statistics, Wiley, New York, 2003.

    Book  Google Scholar 

  23. Carè A, Garatti S, and Campi M C, A coverage theory for least squares, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2017, 79(5): 1367–1389.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Information Engineering, University of Brescia, via Branze 38, 25123, Brescia, Italy

    Giorgio Arici, Marco C. Campi, Algo Carè, Marco Dalai & Federico A. Ramponi

Authors
  1. Giorgio Arici
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  2. Marco C. Campi
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  3. Algo Carè
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  4. Marco Dalai
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  5. Federico A. Ramponi
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Corresponding authors

Correspondence to Giorgio Arici, Marco C. Campi, Algo Carè, Marco Dalai or Federico A. Ramponi.

Additional information

This research was partially supported by Regione Lombardia under Grant MoSoRe E81B19000840007.

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Arici, G., Campi, M.C., Carè, A. et al. A Theory of the Risk for Empirical CVaR with Application to Portfolio Selection. J Syst Sci Complex 34, 1879–1894 (2021). https://doi.org/10.1007/s11424-021-1229-3

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  • DOI: https://doi.org/10.1007/s11424-021-1229-3

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