Article Contents
Article Contents

# Robust Markowitz: Comprehensively maximizing Sharpe ratio by parametric-quadratic programming

• * Corresponding author: Su Zhang

The research is supported by the National Natural Science Foundation of China 12071234, National Social Science Fund of China 18BGL063, and Fundamental Research Funds for the Central Universities of China 63202304

• Markowitz formulates portfolio selection and calls the optimal solutions as an efficient frontier. Sharpe initiates Sharpe ratio for frontier portfolios' reward to variability. Finance textbooks assume that there exists a line which passes through a risk-free rate and is tangent to an efficient frontier. The tangent portfolio enjoys the maximum Sharpe ratio.

However, the assumption is over-simplistic because we prove that other situations exist. For example, Sharpe ratio itself may not be even well-defined. We comprehensively maximize Sharpe ratio. In such an area, this paper contributes to the literature. Specifically, we identify the other situations by parametric-quadratic programming which renders complete efficient frontiers by piecewise-hyperbola structure. Researchers traditionally view efficient frontiers by just isolated points. We accomplish handy formulae, so investors can even manually process them.

The COVID-19 pandemic is unleashing crises. Unfortunately, there is quite limited research of portfolio selection for COVID. In such an area, this paper contributes to the practice. Specifically, we originate a counter-COVID measure for stocks and integrate it as a constraint into portfolio-selection models. The maximum-Sharpe-ratio portfolio outperforms stock-market indexes in sample. We launch the models for Dow Jones Industrial Average and discover outperformance out of sample.

Mathematics Subject Classification: Primary: 90C20; Secondary: 91G10.

 Citation:

• Figure 1.  Traditional assumption (upper part) and other unnoticed situations (lower part)

Figure 2.  Portfolio optimization by (ordinary) quadratic programming (upper part) and portfolio optimization by parametric-quadratic programming (lower part)

Figure 3.  Undefined Sharpe ratio and unbounded Sharpe ratio at $\textbf{p}_1$ with $\sigma = 0$

Figure 4.  Touch (instead of tangency) at kink $\textbf{p}_k$

Figure 5.  Touch and out of tangency range

Figure 6.  Computing the tangent portfolio p$_t = (\sigma,E)$ to an efficient-frontier segment

Figure 7.  Categories and variables of the counter-COVID measure for stocks

Table 1.  Categories, variables, and variable descriptions for the counter-COVID measure

 variables variable descriptions management ability emergency response mechanism for COVID management change management adjustment for COVID management warning warning COVID risk in reports return on equity $\frac{\text{net income}}{\text{equity}}$ debt ratio $\frac{\text{total liabilities}}{\text{total assets}}$ cash management quick ratio= $\frac{\text{current assets - inventory}}{\text{current liabilities}}$ total-assets growth $\frac{\text{total assets of this year - total assets of last year}}{\text{total assets of last year}}$ product innovation new product development for COVID business digitization online marketing, business support, and customer service cost management $\frac{\text{operating cost}}{\text{operating income}}$ profit margin $\frac{\text{net income}}{\text{sales}}$ operating-income growth $\frac{\text{operating income of this year - operating income of last year}}{\text{operating income of last year}}$ staff communication remote working and training for employees health and safety practice to reduce health and safety incidents humanistic care good employee relations firm image positive evaluation for corporate image information quantity sufficient information disclosure information quality fast and effective information disclosure social contribution provide social contributions such as donations social promotion publicize knowledge and precautions of COVID prevention

Table 2.  Rating counter-COVID for the 30 component stocks of Dow Jones Industrial Average for the first half of the year 2020

 variables AAPL AMGN AXP BA CAT CRM CSCO CVX DIS DOW GS HD HON management ability 1 1 1 1 1 1 1 1 0 1 1 1 1 management change 1 1 1 1 1 1 1 1 1 1 1 1 1 management warning 1 1 1 1 1 1 1 1 1 1 1 1 1 return on equity 1 1 1 1 1 1 1 1 1 0 1 0 1 debt ratio 0 0 0 0 0 1 0 1 1 0 0 0 0 cash management 0 0 1 0 0 0 0 0 0 0 1 0 0 total-assets growth 0 0 0 1 0 1 0 0 0 0 1 1 1 product innovation 1 1 1 0 1 1 1 0 0 1 1 1 1 business digitization 1 1 1 0 1 1 1 0 0 1 0 1 0 cost management 0 1 0 0 1 1 1 0 0 0 0 1 1 profit margin 1 1 1 0 1 1 1 0 0 0 1 1 1 operating-income growth 1 0 0 0 0 1 0 0 1 0 1 1 0 staff communication 1 1 1 0 1 1 1 1 0 1 1 1 1 health and safety 1 1 1 1 0 1 0 1 0 1 1 1 0 humanistic care 1 1 1 0 0 1 0 1 0 1 1 1 0 firm image 1 1 0 1 0 1 0 1 0 1 0 1 1 information quantity 0 1 1 0 0 1 0 1 0 1 1 1 1 information quality 1 1 1 1 1 1 0 1 0 1 1 1 1 social contribution 1 1 1 1 1 0 0 1 1 1 1 1 1 social promotion 1 1 0 1 0 0 0 1 0 0 1 1 1 sum, counter-COVID, c 15 16 14 10 11 17 9 13 6 12 16 17 14 IBM INTC JNJ JPM KO MCD MMM MRK MSFT NKE PG TRV UNH V VZ WBA WMT 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 15 17 14 13 14 16 18 16 18 15 14 17 14 13 17 15 16
•  [1] A. Almazan, K. C. Brown, M. Carlson and D. A. Chapman, Why constrain your mutual fund manager?, Journal of Financial Economics, 73 (2004), 289-321. [2] D. R. Anderson, D. J. Sweeney, T. A. Williams, J. D. Camm and J. J. Cochran, Statistics for Business and Economics, 13$^{th}$ edition, Cengage Learning, Boston, Massachusetts, USA, 2018. [3] M. J. Best, An algorithm for the solution of the parametric quadratic programming problem, In Applied Mathematics and Parallel Computing, Physica, Heidelberg, (1996), 57–76. [4] Z. Bodie, A. Kane and A. J. Marcus, Investments, 11$^{th}$ edition, McGraw-Hill Education, New York, New York, USA, 2018. [5] R. A. Brealey, S. C. Myers and F. Allen, Principles of Corporate Finance, 12$^{th}$ edition, McGraw-Hill Education, New York, New York, USA, 2017. [6] P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, Springer Series in Statistics. Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4899-0004-3. [7] G. Cornuejols and  R. Tütüncü,  Optimization Methods in Finance, Cambridge University Press, Cambridge, 2007. [8] P. H. Dybvig, Short sales restrictions and kinks on the mean variance frontier, Journal of Finance, 39 (1984), 239-244.  doi: 10.1111/j.1540-6261.1984.tb03871.x. [9] E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9$^{th}$ edition, John Wiley & Sons, New York, New York, USA, 2014. [10] G. W. Evans,  Multiple Criteria Decision Analysis for Industrial Engineering: Methodology and Applications, CRC Press, Boca Raton, Florida, USA, 2016.  doi: 10.1201/9781315381398. [11] C. Goh and X. Yang, Analytic efficient solution set for multi-criteria quadratic programs, European Journal of Operational Research, 92 (1996), 166-181. [12] M. Hanke and S. Penev, Comparing large-sample maximum Sharpe ratios and incremental variable testing, European J. Oper. Res., 265 (2018), 571-579.  doi: 10.1016/j.ejor.2017.08.018. [13] F. S. Hillier and G. J. Lieberman, Introduction to Operations Research, 3$^{rd}$ edition, Holden-Day, Inc., Oakland, Calif., 1980. [14] M. Hirschberger, Y. Qi and R. E. Steuer, Large-scale MV efficient frontier computation via a procedure of parametric quadratic programming, European J. Oper. Res., 204 (2010), 581-588.  doi: 10.1016/j.ejor.2009.11.016. [15] B. I. Jacobs, K. N. Levy and H. M. Markowitz, Portfolio optimization with factors, scenarios, and realistic short positions, Oper. Res, 53 (2005), 586-599.  doi: 10.1287/opre.1050.0212. [16] R. Larson and B. H. Edwards, Calculus, 11$^{th}$ edition, Cengage Learning, Boston, Massachusetts, USA, 2018. [17] O. Ledoit and M. Wolf, Robust performance hypothesis testing with the Sharpe ratio, Journal of Empirical Finance, 15 (2008), 850-859.  doi: 10.1016/j.jempfin.2008.03.002. [18] R. A. Maller, R. Durand and H. Jafarpour, Optimal portfolio choice using the maximum Sharpe ratio, Joumal of Risk, 12 (2010), 49-73.  doi: 10.21314/JOR.2010.212. [19] R. A. Maller, R. B. Durand and P. T. Lee, Bias and consistency of the maximum Sharpe ratio, Journal of Risk, 7 (2005), 103-115.  doi: 10.21314/JOR.2005.117. [20] R. A. Maller, S. Roberts and R. Tourky, The large-sample distribution of the maximum Sharpe ratio with and without short sales, J. Econometrics, 194 (2016), 138-152.  doi: 10.1016/j.jeconom.2016.04.003. [21] H. M. Markowitz, Portfolio Selection, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1959 [22] H. M. Markowitz, The optimization of a quadratic function subject to linear constraints, Naval Res. Logist. Quart., 3 (1956), 111-133.  doi: 10.1002/nav.3800030110. [23] H. M. Markowitz, Portfolio Selection: Efficient Diversification in Investments, John Wiley & Sons, New York, New York, USA, 1959. [24] H. M. Markowitz and G. P. Todd, Mean-Variance Analysis in Portfolio Choice and Capital Markets, Frank J. Fabozzi Associates, New Hope, Pennsylvania, USA, 2000. [25] H. B. Mayo, Investments: An Introduction, 13$^{th}$ edition, Cengage Learning, Mason, Ohio, USA, 2020. [26] R. C. Merton, An analytical derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872. [27] A. Niedermayer and D. Niedermayer, Applying markowitz's critical line algorithm, In Handbook of Portfolio Construction, Springer, New York, New York, (2010), 383–400. doi: 10.1007/978-0-387-77439-8_12. [28] J. D. Opdyke, Comparing Sharpe ratios: So where are the p-values?, Journal of Asset Management, 8 (2007), 308-336.  doi: 10.1057/palgrave.jam.2250084. [29] Y. Qi, Parametrically computing efficient frontiers of portfolio selection and reporting and utilizing the piecewise-segment structure, Journal of the Operational Research Society, 71 (2020), 1675-1690.  doi: 10.1080/01605682.2019.1623477. [30] Y. Qi, data and result for this paper, Mendeley Data, 2021, https://data.mendeley.com/datasets/yjng4g9sy5/1, DOI: 10.17632/yjng4g9sy5.1 [31] Y. Qi, Y. Zhang and S. Ma, Parametrically computing efficient frontiers and reanalyzing efficiency-diversification discrepancies and naive diversification, INFOR Inf. Syst. Oper. Res., 57 (2019), 430-453.  doi: 10.1080/03155986.2018.1533207. [32] F. K. Reilly, K. C. Brown and S. Leeds, Investment Analysis and Portfolio Management, 11$^{th}$ edition, Cengage Learning, Mason, Ohio, USA, 2018. [33] S. A. Ross, R. W. Westerfield, J. Jaffe and B. Jordan, Corporate Finance, 11$^{th}$ edition, McGraw-Hill Education, New York, New York, USA, 2016. [34] M. Rubinstein, Markowitz's "portfolio selection": A fifty-year retrospective, Journal of Finance, 57 (2002), 1041-1045. [35] W. F. Sharpe, Capital asset prices: A theory of market equilibrium, Journal of Finance, 19 (1964), 425-442. [36] W. F. Sharpe, Portfolio Theory and Capital Markets, 1$^{st}$ edition, McGraw-Hill, New York, New York, USA, 1970. [37] W. F. Sharpe, The Sharpe ratio, Journal of Portfolio Management, 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501. [38] W. F. Sharpe, Mutual fund performance, Journal of Business, 39 (1966), 119-138.  doi: 10.1086/294846. [39] N. Shi, M. Lai, S. Zheng and B. Zhang, Optimal algorithms and intuitive explanations for Markowitz's portfolio selection model and Sharpe's ratio with no short-selling, Sci. China Ser. A, 51 (2008), 2033-2042.  doi: 10.1007/s11425-008-0080-5. [40] M. Stein, J. Branke and H. Schmeck, Efficient implementation of an active set algorithm for large-scale portfolio selection, Computers & Operations Research, 35 (2008), 3945-3961.  doi: 10.1016/j.cor.2007.05.004. [41] R. E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, John Wiley & Sons, New York, New York, USA, 1986. [42] S. M. Yiannaki, A systemic risk management model for SMEs under financial crisis, International Journal of Organizational Analysis, 20 (2012), 406-422. [43] C. Zu, X. Yang and C. K. Yu, Sparse minimax portfolio and Sharpe ratio models, Journal of Industrial and Management Optimization, Forthcoming in 2021. doi: 10.3934/jimo.2021111.

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