Mean-variance Portfolio Optimization Research Articles

The Number of Stocks Before (n) and After Portfolio Optimization (k): The Heuristic k ≈ sqrt(n)

This study focuses on the relationship between the number of securities (n) pre-selected for mean-variance portfolio optimization and the number of optimal securities (k). We propose a heuristic k ≈ square root (n) based on empirical research optimizing different sized (n) portfolios. That is, a sample selection of 20-30 securities should yield a portfolio of about five optimal securities, and an initial sample of 500 securities, should result in an optimal portfolio of about 22. We focus on the tangent portfolio that maximizes the return-to-risk ratio. The heuristic finds its support, rationale, and logic in the numerical properties and statistical nature of the optimization. More specifically, the heuristic seems to originate in the dynamic convergence patterns observable in many statistical processes, especially in standard deviations. It is also supported by available results in the literature. Our “square root” heuristic functions as part of the wider family of approximation around the power law, where some variables (authors, securities, people) receive a disproportionate share of a given collection of items – see, for example, Pareto’s principle, Zipf’s law, Lotka’s law, Price’s square root law, Simon’s law, etc. The heuristic provides assistance not only in anticipating the number of optimal securities chosen by the mean-variance optimizer but also in suggesting selectivity in the effort of pre-selecting securities prior to the optimization and in sharpening portfolio-based approaches to investing in general. In sum, the heuristic k ≈ sqrt(n) seems helpful at all levels of portfolio management.

Mean-variance Portfolio Optimization by LSTM-based Predictions

With the progress of machine learning, its application fields are gradually increasing, especially in the field of quantitative finance, which is particularly outstanding, the Portfolio optimization combine with time series prediction and machine learning has brought great development prospects for investors. This study mainly employed the LSTM model and Mean-Variance Model to predict stock return and build an optimal combination respectively. This study selects relatively high weight stocks on the NASDAQ index, 'AAPL', 'AMZN',' ASML', 'AVGO', 'GOOG', from December 31, 2019, to July 1, 2023. First, the study obtained the predicted stock prices of five stocks through LSTM Model and based calculated the predicted returns on a rolling basis. Second, based on the modern investment theory, this study uses the predicted rate of return to construct the optimal daily investment ratio through Mean-Variance Model. Finally, this study compared cumulative return of optimal portfolios with the NASDAQ within the same time frame. This study draws a conclusion that hybrid model which combine the stock price forecasting with asset allocation can indeed bring excess returns.

A monotone numerical integration method for mean–variance portfolio optimization under jump-diffusion models

We develop a highly efficient, easy-to-implement, and strictly monotone numerical integration method for Mean-Variance (MV) portfolio optimization. This method proves very efficient in realistic contexts, which involve jump-diffusion dynamics of the underlying controlled processes, discrete rebalancing, and the application of investment constraints, namely no-bankruptcy and leverage.A crucial element of the MV portfolio optimization formulation over each rebalancing interval is a convolution integral, which involves a conditional density of the logarithm of the amount invested in the risky asset. Using a known closed-form expression for the Fourier transform of this conditional density, we derive an infinite series representation for the conditional density where each term is strictly positive and explicitly computable. As a result, the convolution integral can be readily approximated through a monotone integration scheme, such as a composite quadrature rule typically available in most programming languages. The computational complexity of our proposed method is an order of magnitude lower than that of existing monotone finite difference methods. To further enhance efficiency, we propose an implementation of this monotone integration scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix structure.The proposed monotone numerical integration scheme is proven to be both ℓ∞-stable and pointwise consistent, and we rigorously establish its pointwise convergence to the unique solution of the MV portfolio optimization problem. We also intuitively demonstrate that, as the rebalancing time interval approaches zero, the proposed scheme converges to a continuously observed impulse control formulation for MV optimization expressed as a Hamilton–Jacobi–Bellman Quasi-Variational Inequality. Numerical results show remarkable agreement with benchmark solutions obtained through finite differences and Monte Carlo simulation, underscoring the effectiveness of our approach.

Quantifying the Impact of Impact Investing

We propose a quantitative framework for assessing the financial impact of any form of impact investing, including socially responsible investing; environmental, social, and governance (ESG) objectives; and other nonfinancial investment criteria. We derive conditions under which impact investing detracts from, improves on, or is neutral to the performance of traditional mean-variance optimal portfolios, which depends on whether the correlations between the impact factor and unobserved excess returns are negative, positive, or zero, respectively. Using Treynor–Black portfolios to maximize the risk-adjusted returns of impact portfolios, we derive an explicit and easily computable measure of the financial reward or cost of impact investing as compared with passive index benchmarks. We illustrate our approach with applications to biotech venture philanthropy, a semiconductor research and development consortium, divesting from “sin” stocks, ESG investments, and “meme” stock rallies such as GameStop in 2021. This paper was accepted by Agostino Capponi, finance. Funding: Research funding from the National Key Research and Development Program of China [Grant 2022YFA1007900], the National Natural Science Foundation of China [Grant 12271013], Peking University’s Fundamental Research Funds for the Central Universities, and the Massachusetts Institute of Technology Laboratory for Financial Engineering is acknowledged. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.01168 .

Partially egalitarian portfolio selection

We propose a new portfolio optimization framework, partially egalitarian portfolio selection (PEPS). Inspired by the celebrated LASSO regression and its recent variant partially egalitarian LASSO (PELASSO) developed in [1] in the context of the forecast combinations problem in econometrics in [1], we regularize the mean-variance portfolio optimization of Markowitz by adding two regularizing terms that essentially zero out portfolio weights of some of the assets in the portfolio and select and shrink portfolio weights of the remaining assets towards equal weights to hedge against parameter estimation risk. We solve our PEPS formulations by applying Gurobi 9.0 mixed integer optimization (MIO) solver that allow us to tackle large-scale portfolio problems. We test our PEPS portfolios against an array of classical portfolio optimization strategies on a number of datasets in the US equity markets. The PEPS portfolios exhibit the highest out-of-sample Sharpe ratios in all instances considered.

Application of ARIMA in Mean-Variance Portfolio Optimization

The stock market often carries investment risks, and portfolio investment can to some extent reduce investment risk, helping investors to achieve certain goal in the financial market. In this paper, stock data from March 15th, 2022, to March 20th, 2023, is collected, and the ARIMA prediction model is ap-plied in the mean-variance framework for constructing optimized portfolios. The results are summarized as follows. First, the data passed the white noise test and was used to conduct ARIMA prediction. The residual sequence of the prediction results is stable, and the model performance is good. Then, the minimum-variance model and the maximum Sharpe ratio model are imple-mented according to the predicted data. Ping An Insurance has the largest weight in the minimum variance model, and Baosteel has the largest weight in the maximum Sharpe ratio model. Overall, the result in this paper provides insightful point for financial investors.

A Low-Cost Alternating Projection Approach for a Continuous Formulation of Convex and Cardinality Constrained Optimization

We consider convex constrained optimization problems that also include a cardinality constraint. In general, optimization problems with cardinality constraints are difficult mathematical programs which are usually solved by global techniques from discrete optimization. We assume that the region defined by the convex constraints can be written as the intersection of a finite collection of convex sets, such that it is easy and inexpensive to project onto each one of them (e.g., boxes, hyper-planes, or half-spaces). Taking advantage of a recently developed continuous reformulation that relaxes the cardinality constraint, we propose a specialized penalty gradient projection scheme combined with alternating projection ideas to compute a solution candidate for these problems, i.e., a local (possibly non-global) solution. To illustrate the proposed algorithm, we focus on the standard mean-variance portfolio optimization problem for which we can only invest in a preestablished limited number of assets. For these portfolio problems with cardinality constraints, we present a numerical study on a variety of data sets involving real-world capital market indices from major stock markets. In many cases, we observe that the proposed scheme converges to the global solution. On those data sets, we illustrate the practical performance of the proposed scheme to produce the effective frontiers for different values of the limited number of allowed assets.

Revisit the Use of Index Returns as the Proxy of Market Return in CAPM

This study compares the performance of the capital asset pricing model (CAPM), which uses a mean-variance optimal portfolio, and the Jakarta Composite Index (JCI) as the market return proxy. The data samples are monthly stock returns of Kompas 100, LQ45, and IDX30 index from September 2012-August 2022. The expected returns of assets are calculated using the CAPM with rolling regression methodology based on each index's 2-5 years period. The mean-squared errors for each sample group are calculated to determine the CAPM performance. Although the market indexes have a sub-optimal risk and return profile according to their position on the efficient frontier diagram, this study finds that the JCI as the market proxy results in better CAPM performance than the optimal portfolio. However, the beta results from using JCI as the market proxy are also consistently higher than those using an optimal portfolio, leading to the overestimation bias of the asset risk. Since most studies in CAPM testing use the market index as the market return proxy, particularly JCI in Indonesia, this study provides new insight into the challenges of using the optimal portfolio as the market return proxy.

Futuristic portfolio optimization problem: wavelet based long short-term memory

PurposeThis paper aims to propose an improved version of portfolio optimization model through the prediction of the future behavior of stock returns using a combined wavelet-based long short-term memory (LSTM).Design/methodology/approachFirst, data are gathered and divided into two parts, namely, “past data” and “real data.” In the second stage, the wavelet transform is proposed to decompose the stock closing price time series into a set of coefficients. The derived coefficients are taken as an input to the LSTM model to predict the stock closing price time series and the “future data” is created. In the third stage, the mean-variance portfolio optimization problem (MVPOP) has iteratively been run using the “past,” “future” and “real” data sets. The epsilon-constraint method is adapted to generate the Pareto front for all three runes of MVPOP.FindingsThe real daily stock closing price time series of six stocks from the FTSE 100 between January 1, 2000, and December 30, 2020, is used to check the applicability and efficacy of the proposed approach. The comparisons of “future,” “past” and “real” Pareto fronts showed that the “future” Pareto front is closer to the “real” Pareto front. This demonstrates the efficacy and applicability of proposed approach.Originality/valueMost of the classic Markowitz-based portfolio optimization models used past information to estimate the associated parameters of the stocks. This study revealed that the prediction of the future behavior of stock returns using a combined wavelet-based LSTM improved the performance of the portfolio.

A LINEAR-PROGRAMMING PORTFOLIO OPTIMIZER TO MEAN–VARIANCE OPTIMIZATION

In the Markowitz mean–variance portfolio optimization problem, the estimation of the inverse covariance matrix is not trivial and can even be intractable, especially when the dimension is very high. In this paper, we propose a linear-programming portfolio optimizer (LPO) to solve the Markowitz optimization problem in both low-dimensional and high-dimensional settings. Instead of directly estimating the inverse covariance matrix [Formula: see text], the LPO method estimates the portfolio weights [Formula: see text] through solving an [Formula: see text]-constrained optimization problem. Moreover, we further prove that the LPO estimator asymptotically yields the maximum expected return while preserving the risk constraint. To offer a practical insight into the LPO approach, we provide a comprehensive implementation procedure of estimating portfolio weights via the Dantzig selector with sequential optimization (DASSO) algorithm and selecting the sparsity parameter through cross-validation. Simulations on both synthetic data and empirical data from Fama–French and the Center for Research in Security Prices (CRSP) databases validate the performance of the proposed method in comparison with other existing proposals.

Nonconvex multi-period mean-variance portfolio optimization

AbstractIn this paper, we address the problem of long-term investment by exploring optimal strategies for allocating wealth among a finite number of assets over multiple periods. Based on the classical Markowitz mean-variance philosophy, we develop a new portfolio optimization framework which can produce sparse portfolios. The sparsity of the portfolio at each and across periods is characterized by the possibly nonconvex penalties. For the constructed nonconvex and nonsmooth constrained model, we propose a generalized alternating direction method of multipliers and its global convergence to a stationary point can be guaranteed theoretically. Moreover, some numerical experiments are conducted on several datasets generated from practical applications to illustrate the effectiveness and advantage of the proposed model and solving method.

Mean–variance portfolio selection under no-shorting rules: A BSDE approach

This paper revisits the mean–variance portfolio selection problem in continuous-time within the framework of short-selling of stocks is prohibited via backward stochastic differential equation approach. To relax the strong condition in Li et al. (Li et al. 2002), the above issue is formulated as a stochastic recursive optimal linear–quadratic control problem. Due to no-shorting rules (namely, the portfolio taking non-negative values), the well-known “completion of squares” no longer applies directly. To overcome this difficulty, we study the corresponding Hamilton–Jacobi–Bellman (HJB, for short) equation inherently and derive the two groups of Riccati equations. On one hand, the value function constructed via Riccati equations is shown to be a viscosity solution of the HJB equation mentioned before; On the other hand, by means of these Riccati equations and backward semigroup, we are able to get explicitly the efficient frontier and efficient investment strategies for the recursive utility mean–variance portfolio optimization problem.

Stock market optimization amidst the COVID-19 pandemic: Technical analysis, K-means algorithm, and mean-variance model (TAKMV) approach

The Philippine stock market, just like most of its neighbors in the region, was seriously impacted by the global pandemic COVID-19. Investors remain hopeful while continuing to seek great ones in the damaged market. This paper developed a methodology for portfolio selection and optimization with the use of technical analysis, machine learning techniques, and portfolio optimization model. The combined methods of technical analysis, K-means clustering algorithm, and mean-variance portfolio optimization model will result in the development of the proposed TAKMV method. The study aims to integrate these three important analyses to identify portfolio investments. This paper uses the average annual risk and annual rate of return data for the years 2018 and 2020 to form the clusters and assessed the stocks that correspond to the investor's technical strategy such as Moving Average Convergence/Divergence (MACD) and Hybrid MACD with Arnaud Legoux Moving Average (ALMA). This paper solved the risk minimization problem on selected shares of the companies, based on the mean-variance portfolio optimization model. There are 230 and 239 companies for 2018 and 2020, respectively, listed in Philippine Stock Market, and all simulations were performed in MATLAB environment platform. Results showed that MACD strategy dominates the MACD-ALMA strategy in terms of the number of assets with a positive annual rate of return. The MACD works efficiently in the pre-COVID-19 condition while MACD-ALMA works efficiently during-COVID-19 condition, regardless of the number of assets with a positive annual rate of return. The results also show that the maximum expected portfolio return (RP) can be achieved using the MACD and MACD-ALMA in the pre-and during-COVID-19 conditions, respectively. The MACD-ALMA shows an advantage during high-risk market conditions and can also provide maximum RP. The performance of the TAKMV method was validated by applying its results and comparing it to the next year's historical price. The 2018 results were compared to 2019 data and the 2020 results were compared to 2021 data. For consistency, the comparison was applied to the same company per portfolio. Simulation results show that the MACD strategy is more effective compared to MACD-ALMA.

A Mayfly algorithm for cardinality constrained portfolio optimization

Portfolio optimization is an essential issue in quantitative investing, which aims to find the best set of portfolios by allocating the proportion of assets. One of the most widely studied portfolio optimization models is the cardinality constrained mean–variance model, which incorporates real-world constraints on the number of selected assets and lower and upper bounds on the proportion of each asset. This paper presents a novel metaheuristic algorithm based on the Mayfly algorithm to solve the cardinality constrained mean–variance portfolio optimization problem. To better adapt to this problem, we design and introduce some new features to the proposed algorithm, including (1) a new cardinality constraint handling strategy; (2) a new local search strategy; and (3) changes to the crossover operator. We have designed comparison experiments for the proposed metaheuristic and evaluated its performance using five commonly used performance metrics. The experimental results show that the proposed approach achieves competitive performance on datasets of different sizes. The results also demonstrate the feasibility of this approach in solving the cardinality constrained mean–variance portfolio optimization problem.

A Privacy-preserving mean–variance optimal portfolio

Following strong regulations such as the European General Data Protection Regulation (GDPR), privacy protection in the financial sector has recently emerged as an urgent issue. To manage the privacy risk in robo-advisor, a representative fintech service, we propose a novel framework that allows robo-advisors to offer the optimal portfolio while complying with the privacy of their customers by encrypting individual risk aversion with homomorphic encryption (HE). By introducing an HE-friendly method for constrained optimization, our model can find a mean–variance quadratic programming solution even with inequality constraints. This study makes two main findings through empirical evaluation (i) our model can approximate optimal solution at an acceptable level of accuracy loss and the cost of preserving privacy, and (ii) the number of assets and the degree of correlation between assets affect the accuracy loss.

BOUNDED STRATEGIES FOR MAXIMIZING THE SHARPE RATIO

Bernard et al. [(2019) Optimal strategies under omega ratio, European Journal of Operational Research 275 (2), 755–767] use convex ordering arguments to determine the bounded payoff for maximizing the omega ratio. However, it appears difficult to apply such reasoning to estimate the bounded payoff for maximizing the Sharpe ratio. As a proposed solution, this paper uses a Lagrange multiplier method to derive the bounded payoff for maximizing the Sharpe ratio. In contrast to the optimal strategy in Bernard & Vanduffel [(2014) Mean–variance optimal portfolios in the presence of a benchmark with applications to fraud detection, European Journal of Operational Research 234 (2), 469–480], the optimal strategy in this paper is bounded from below. It can protect investors from substantial losses when they invest in payoffs with a maximized Sharpe ratio.

Optimising the Investor's Portfolio through Modern Portfolio Theory: Empirical Evidence from Pakistan

Diversification is considered a way to lower investment risk by using a variety of investment avenues. The objective of this study is to compare the portfolio performance based on mean-variance optimisation with a naively diversified portfolio for textile spinning stocks in the Pakistan country. The data period of study spans from January 2015 to April 2022. The result shows that portfolios based on naïve diversification strategies outperformed the mean-variance portfolio optimisation in terms of risk and return for the textile sector in Pakistan. Of all the 9 variance-covariance estimators, the portfolio of sample and single index covariance technique is optimal for estimating variance-covariance matrices based on low RMSE (root mean square error) in Pakistan. This study extends the debate on mean-variance portfolio theory and naïve diversification strategy from the developed economy to the emerging equity market of Pakistan. This research contributes by providing the framework to the potential investors regarding their investments in the textile sector in Pakistan.

Application of Maximum Sharpe Ratio and Minimum Variance Portfolio Optimization for Industries

Since the start of 2022, expectations regarding the continuous raising of interest rates by the Fed to curb the 40-year-high inflation has been negatively impacting the US stock market, leading to volatility spikes. Thus, portfolio optimization plays an important role to risk-averse investors. This paper investigated an optimal portfolio strategy that aims to maximize the Sharpe ratio to outperform a common benchmark, the S&P500. Eleven US stocks were selected, one from each of the eleven market sectors, based on market capitalization. Then, mean-variance portfolio optimization was conducted to obtain an optimal distribution of stocks weighing in Maximum Sharpe Ratio Portfolio (MSRP) and Global Minimum Variance Portfolio (GMVP), and backtesting was adopted to construct portfolio performance for MSRP, GMVP and Equal-Weight Portfolio (1/N) strategies by using historical data from the first eight months of 2022. The Sharpe ratio was calculated to compare the performance between the three strategies---MSRP, GMVP and 1/N---and the S&P500. Empirical results, supported by a robustness test, suggested that MSRP strategy outperformed S&P500 with the highest Sharpe ratio, which indicated that MSRP achieved a higher return per unit of risk regardless of market volatility presented in different time intervals. Therefore, this research provides insights into portfolio investment decision-making for risk-averse investors.

Online risk-based portfolio allocation on subsets of crypto assets applying a prototype-based clustering algorithm

Mean-variance portfolio optimization models are sensitive to uncertainty in risk-return estimates, which may result in poor out-of-sample performance. In particular, the estimates may suffer when the number of assets considered is high and the length of the return time series is not sufficiently long. This is precisely the case in the cryptocurrency market, where there are hundreds of crypto assets that have been traded for a few years. We propose enhancing the mean-variance (MV) model with a pre-selection stage that uses a prototype-based clustering algorithm to reduce the number of crypto assets considered at each investment period. In the pre-selection stage, we run a prototype-based clustering algorithm where the assets are described by variables representing the profit-risk duality. The prototypes of the clustering partition are automatically examined and the one that best suits our risk-aversion preference is selected. We then run the MV portfolio optimization with the crypto assets of the selected cluster. The proposed approach is tested for a period of 17 months in the whole cryptocurrency market and two selections of the cryptocurrencies with the higher market capitalization (175 and 250 cryptos). We compare the results against three methods applied to the whole market: classic MV, risk parity, and hierarchical risk parity methods. We also compare our results with those from investing in the market index CCI30. The simulation results generally favor our proposal in terms of profit and risk-profit financial indicators. This result reaffirms the convenience of using machine learning methods to guide financial investments in complex and highly-volatile environments such as the cryptocurrency market.

Integrating prediction in mean-variance portfolio optimization

In quantitative finance, prediction models are traditionally optimized independently from their use in the asset allocation decision-making process. We address this limitation and present a stochastic optimization framework for integrating regression prediction models in a mean-variance optimization (MVO) setting. Closed-form analytical solutions are provided for the unconstrained and equality constrained MVO case. For the general inequality constrained case, we make use of recent advances in neural-network architecture for efficient optimization of batch quadratic programs. To our knowledge, this is the first rigorous study of integrating prediction in a mean-variance portfolio optimization setting. We present several simulations, using both synthetic and global futures data, and demonstrate the benefits of the integrated approach in comparison to the decoupled alternative.