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

Yue Qi; Tongyang Liu; Su Zhang; Yu Zhang

doi:10.3934/jimo.2021235

Article Contents

2023, Volume 19, Issue 2: 1426-1446. Doi: 10.3934/jimo.2021235

This issue Previous Article A novel separate chance-constrained programming model to design a sustainable medical ventilator supply chain network during the Covid-19 pandemic Next Article A multi-objective lion swarm optimization based on multi-agent

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

1.
China Academy of Corporate Governance & Department of Financial Management, Business School, Nankai University, 94 Weijin Road, Tianjin, 300071, China
2.
Department of Financial Management, Business School, Nankai University, 94 Weijin Road, Tianjin, 300071, China

^* Corresponding author: Su Zhang
^* Corresponding author: Su Zhang

Received: July 2021

Revised: October 2021

Early access: January 2022

Published: February 2023

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

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Abstract

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.

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

Citation:

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

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

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

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

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Figure 5. Touch and out of tangency range

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

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

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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

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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

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