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An optimization–diversification approach to portfolio selection

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Abstract

The classical approaches to optimal portfolio selection call for finding a feasible portfolio that optimizes a risk measure, or a gain measure, or a combination thereof by means of a utility function or of a performance measure. However, the optimization approach tends to amplify the estimation errors on the parameters required by the model, such as expected returns and covariances. For this reason, the Risk Parity model, a novel risk diversification approach to portfolio selection, has been recently theoretically developed and used in practice, mainly for the case of the volatility risk measure. Here we first provide new theoretical results for the Risk Parity approach for general risk measures. Then we propose a novel framework for portfolio selection that combines the diversification and the optimization approaches through the global solution of a hard nonlinear mixed integer or pseudo Boolean problem. For the latter problem we propose an efficient and accurate Multi-Greedy heuristic that extends the classical single-threaded greedy approach to a multiple-threaded setting. Finally, we provide empirical results on real-world data showing that the diversified optimal portfolios are only slightly suboptimal in-sample with respect to optimal portfolios, and generally show improved out-of-sample performance with respect to their purely diversified or purely optimized counterparts.

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Notes

  1. See also http://www.thierry-roncalli.com/riskparity.html and references therein.

  2. Second Sevilla Workshop on Mixed Integer NonLinear Programming, 2015 ; EURO 2015—27th European Conference on Operational Research in Glasgow; Conference Innovations in Insurance, Risk- and Asset Management in Munich, 2017; 41st Annual Meeting of AMASES in Cagliari, 2017; XVIII Workshop on Quantitative Finance in Milan, 2017; ICMFII 2018—7th International Conference in Multidimensional Finance, Insurance and Investment in Chania, 2018.

References

  1. Anagnostopoulos, K., Mamanis, G.: The mean–variance cardinality constrained portfolio optimization problem: an experimental evaluation of five multiobjective evolutionary algorithms. Expert Syst. Appl. 38, 14208–14217 (2011)

    Google Scholar 

  2. Anderson, R.M., Bianchi, S.W., Goldberg, L.R.: Will my risk parity strategy outperform? Financ. Anal. J. 68(6), 75–93 (2012)

    Google Scholar 

  3. Asness, C.S., Frazzini, A., Pedersen, L.H.: Leverage aversion and risk parity. Financ. Anal. J. 68(1), 47–59 (2012)

    Google Scholar 

  4. Bacon, C.R.: Practical Portfolio Performance Measurement and Attribution, vol. 568. Wiley, Hoboken (2011)

    Google Scholar 

  5. Bai, X., Scheinberg, K., Tutuncu, R.: Least-squares approach to risk parity in portfolio selection. Quant Finance 16(3), 357–376 (2016)

    MathSciNet  Google Scholar 

  6. Bertrand, P., Lapointe, V.: How performance of risk-based strategies is modified by socially responsible investment universe? Int. Rev. Financ. Anal. 38, 175–190 (2015)

    Google Scholar 

  7. Best, M., Grauer, R.: Sensitivity analysis for mean–variance portfolio problems. Manage. Sci. 37(8), 980–989 (1991)

    MATH  Google Scholar 

  8. Best, M.J., Grauer, R.R.: On the sensitivity of mean–variance-efficient portfolios to changes in asset means: some analytical and computational results. Rev. Financ. Stud. 4(2), 315–342 (1991)

    Google Scholar 

  9. Boudt, K., Carl, P., Peterson, B.: Asset allocation with conditional value-at-risk budgets. J. Risk 15, 39–68 (2013)

    Google Scholar 

  10. Bruni, R., Cesarone, F., Scozzari, A., Tardella, F.: Real-world datasets for portfolio selection and solutions of some stochastic dominance portfolio models. Data Brief 8, 858–862 (2016)

    Google Scholar 

  11. Bruni, R., Cesarone, F., Scozzari, A., Tardella, F.: On exact and approximate stochastic dominance strategies for portfolio selection. Eur. J. Oper. Res. 259(1), 322–329 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Cesarone, F., Colucci, S.: Minimum risk versus capital and risk diversification strategies for portfolio construction. J. Oper. Res. Soc. 69(2), 183–200 (2018). https://doi.org/10.1057/s41274-017-0216-5

    Article  Google Scholar 

  13. Cesarone, F., Lampariello, L., Sagratella, S.: A risk-gain dominance maximization approach to enhanced index tracking. Finance Res. Lett. 29, 231–238 (2018). https://doi.org/10.1016/j.frl.2018.08.001

    Article  Google Scholar 

  14. Cesarone, F., Moretti, J., Tardella, F., et al.: Optimally chosen small portfolios are better than large ones. Econ. Bull. 36(4), 1876–1891 (2016)

    Google Scholar 

  15. Cesarone, F., Mottura, C., Ricci, J.M., Tardella, F.: On the stability of portfolio selection models. SSRN. http://ssrn.com/abstract=3420081 (2019)

  16. Cesarone, F., Scozzari, A., Tardella, F.: A new method for mean-variance portfolio optimization with cardinality constraints. Ann. Oper. Res. 205, 213–234 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Cesarone, F., Tardella, F.: Equal risk bounding is better than risk parity for portfolio selection. J. Glob. Optim. 68(2), 439–461 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Chan-Lau, J.A.: Frontier equity markets: risk parity lessons for asset allocation. J. Altern. Invest. 16(4), 28–36 (2014)

    Google Scholar 

  19. Chaves, D., Hsu, J., Li, F., Shakernia, O.: Risk parity portfolio vs. other asset allocation heuristic portfolios. J. Invest. 20(1), 108–118 (2011)

    Google Scholar 

  20. Chopra, V., Ziemba, W.T.: The effect of errors in means, variances, and covariances on optimal portfolio choice. J. Portf. Manage 19, 6–11 (1993)

    Google Scholar 

  21. Choueifaty, Y., Froidure, T., Reynier, J.: Properties of the most diversified portfolio. J. Invest. Strateg. 2(2), 49–70 (2013)

    Google Scholar 

  22. Clarke, R., De Silva, H., Thorley, S.: Risk parity, maximum diversification, and minimum variance: an analytic perspective. J. Portf. Manag. 39(3), 39–53 (2013)

    Google Scholar 

  23. Corkery, M., Cui, C., Grind, K.: Fashionable “risk parity” funds hit hard. Wall St. J. (2013)

  24. Dagher, V.: New allocation funds redefine idea of ’balance’. Wall St. J. (2012)

  25. DeMiguel, V., Garlappi, L., Uppal, R.: Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy? Rev. Financ. Stud. 22, 1915–1953 (2009)

    Google Scholar 

  26. Guastaroba, G., Mansini, R., Ogryczak, W., Speranza, M.G.: Linear programming models based on omega ratio for the enhanced index tracking problem. Eur. J. Oper. Res. 251(3), 938–956 (2016)

    MathSciNet  MATH  Google Scholar 

  27. Holmström, K., Göran, A., Edvall, M.: User’s guide for TOMLAB/CPLEX. https://tomopt.com/docs/TOMLAB_CPLEX.pdf (2009)

  28. Jensen, M.C.: The performance of mutual funds in the period 1945–1964. J Finance 23(2), 389–416 (1968)

    Google Scholar 

  29. Keating, C., Shadwick, W.F.: A universal performance measure. J Perform. Meas. 6(3), 59–84 (2002)

    Google Scholar 

  30. Kondor, I., Pafka, S., Nagy, G.: Noise sensitivity of portfolio selection under various risk measures. J. Bank. Finance 31(5), 1545–1573 (2007)

    Google Scholar 

  31. Konno, H., Tanaka, K., Yamamoto, R.: Construction of a portfolio with shorter downside tail and longer upside tail. Comput. Optim. Appl. 48(2), 199–212 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Lee, W.: Risk-based asset allocation: a new answer to an old question? J. Portf. Manag. 37, 11–2 (2011)

    Google Scholar 

  33. Lohre, H., Opfer, H., Orszag, G.: Diversifying risk parity. J. Risk 16(5), 53–79 (2014)

    Google Scholar 

  34. Maillard, S., Roncalli, T., Teiletche, J.: The properties of equally weighted risk contribution portfolios. J. Portf. Manag. 36, 60–70 (2010)

    Google Scholar 

  35. Markowitz, H.: Portfolio selection. J. Financ. 7, 77–91 (1952)

    Google Scholar 

  36. Markowitz, H.M.: Portfolio selection: efficient diversification of investments. Cowles Foundation for Research in Economics at Yale University, Monograph 16. Wiley, New York (1959)

  37. Meucci, A., Santangelo, A., Deguest, R.: Risk budgeting and diversification based on optimized uncorrelated factors. Risk 11(29), 70–75 (2015)

    Google Scholar 

  38. Ortobelli, S., Shalit, H., Fabozzi, F.J.: Portfolio selection problems consistent with given preference orderings. Int. J. Theor. Appl. Finance 16(05), 1350029 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Pflug, G.C., Pichler, A., Wozabal, D.: The 1/N investment strategy is optimal under high model ambiguity. J. Bank. Financ. 36, 410–417 (2012)

    Google Scholar 

  40. Qian, E.: Risk parity portfolios: efficient portfolios through true diversification. Panagora Asset Manag. Technical report: https://www.panagora.com/assets/PanAgora-Risk-Parity-Portfolios-Efficient-Portfolios-Through-True-Diversification.pdf (2005)

  41. Qian, E.: Risk parity and diversification. J. Invest. 20(1), 119–127 (2011)

    Google Scholar 

  42. Rachev, S., Biglova, A., Ortobelli, S., Stoyanov, S.: Different approaches to risk estimation in portfolio theory. J. Portf. Manag. 31(1), 103–112 (2004)

    Google Scholar 

  43. Rachev, S., Jašić, T., Stoyanov, S., Fabozzi, F.J.: Momentum strategies based on reward-risk stock selection criteria. J. Bank. Finance 31(8), 2325–2346 (2007)

    Google Scholar 

  44. Rachev, S., Ortobelli, S., Stoyanov, S., Fabozzi, F.J., Biglova, A.: Desirable properties of an ideal risk measure in portfolio theory. Int. J. Theor. Appl. Finance 11(01), 19–54 (2008)

    MathSciNet  MATH  Google Scholar 

  45. Rachev, S., Stoyanov, S., Fabozzi, F.: Advanced stochastic models, risk assessment, and portfolio optimization: the ideal risk, uncertainty, and performance measures, vol. 149. Wiley, Hoboken (2008)

    Google Scholar 

  46. Roncalli, T.: Introduction to Risk Parity and Budgeting. Chapman & Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton (2014)

    MATH  Google Scholar 

  47. Sharpe, W.F.: Mutual fund performance. J. Bus. 39(1), 119–138 (1966)

    Google Scholar 

  48. Sharpe, W.F.: The sharpe ratio. J. Portf. Manag. 21(1), 49–58 (1994)

    Google Scholar 

  49. Sorensen, E.H., Alonso, N.F.: The resale value of risk-parity equity portfolios. J. Portf. Manag. 41, 23–32 (2015)

    Google Scholar 

  50. Spinu, F.: An algorithm for computing risk parity weights. SSRN. http://ssrn.com/abstract=2297383 (2013)

  51. Stoyanov, S.V., Rachev, S.T., Fabozzi, F.J.: Optimal financial portfolios. Appl. Math. Finance 14(5), 401–436 (2007)

    MathSciNet  MATH  Google Scholar 

  52. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)

    MathSciNet  MATH  Google Scholar 

  53. Tu, J., Zhou, G.: Markowitz meets talmud: a combination of sophisticated and naive diversification strategies. J. Financ. Econ. 99(1), 204–215 (2011)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Economia Aziendale, Università degli Studi Roma Tre, Rome, Italy

    Francesco Cesarone

  2. Facoltà di Economia, Università degli Studi Niccolò Cusano, Rome, Italy

    Andrea Scozzari

  3. Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza, Sapienza Università di Roma, Rome, Italy

    Fabio Tardella

Authors
  1. Francesco Cesarone
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  2. Andrea Scozzari
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  3. Fabio Tardella
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Correspondence to Francesco Cesarone.

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Cesarone, F., Scozzari, A. & Tardella, F. An optimization–diversification approach to portfolio selection. J Glob Optim 76, 245–265 (2020). https://doi.org/10.1007/s10898-019-00809-7

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