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Properties, formulations, and algorithms for portfolio optimization using Mean-Gini criteria

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Abstract

We study an extended set of Mean-Gini portfolio optimization models that encompasses a general version of the mean-risk formulation, the Minimal Gini model (MinG) that minimizes Gini’s Mean Differences, and the new risk-adjusted Mean-Gini Ratio (MGR) model. We analyze the properties of the various models, prove that a performance measure based on a Risk Adjusted version of the Mean Gini Ratio (RAMGR) is coherent, and establish the equivalence between maximizing this performance measure and solving for the maximal Mean-Gini ratio. We propose a linearization approach for the fractional programming formulation of the MGR model. We also conduct a thorough evaluation of the various Mean-Gini models based on four data sets that represent combinations of bullish and bearish scenarios in the in-sample and out-of-sample phases. The performance is (i) analyzed with respect to eight return, risk, and risk-adjusted criteria, (ii) benchmarked with the S&P500 index, and (iii) compared with their Mean-Variance counterparts for varying risk aversion levels and with the Minimal CVaR and Minimal Semi-Deviation models. For the data sets used in our study, our results suggest that the various Mean-Gini models almost always result in solutions that outperform the S&P500 benchmark index with respect to the out-of-sample cumulative return. Further, particular instances of Mean-Gini models result in solutions that are as good or better (for example, MinG in bullish in-sample scenarios, and MGR in bearish out-of-sample scenarios) than the solutions obtained with their counterparts in Mean-Variance, Minimal CVaR and Minimal Semi-Deviation models.

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Notes

  1. FORS is an abbreviation for the initials of the surnames of the authors of the paper—Fabozzi, Ortobelli, Rachev, and Shalit [see note #3 in Ortobelli et al. (2009)].

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Authors and Affiliations

  1. Department of Decision Sciences, George Washington University, 2201 G St NW, Funger Hall 415, Washington, DC, USA

    Ran Ji, Miguel A. Lejeune & Srinivas Y. Prasad

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  1. Ran Ji
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  2. Miguel A. Lejeune
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  3. Srinivas Y. Prasad
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Correspondence to Ran Ji.

Appendix

Appendix

1.1 Model comparisons

See Tables 3, 4, 5, 6, 7, 8 and 9.

Table 3 Performance of \(\mathbf{MV}_{\lambda }\) models as a function of risk aversion \(\lambda \)—part I
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Table 4 Performance of \(\mathbf{MV}_{\lambda }\) models as a function of risk aversion \(\lambda \)—part II
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Table 5 Performance of \(\mathbf{MG}_{\lambda }\) models as a function of risk aversion \(\lambda \)—part I
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Table 6 Performance of \(\mathbf{MG}_{\lambda }\) models as a function of risk aversion \(\lambda \)—part II
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Table 7 Statistical results of cumulative returns differences
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Table 8 MinCVaR and MinSD models
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Table 9 Results for normality tests
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Ji, R., Lejeune, M.A. & Prasad, S.Y. Properties, formulations, and algorithms for portfolio optimization using Mean-Gini criteria. Ann Oper Res 248, 305–343 (2017). https://doi.org/10.1007/s10479-016-2230-4

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