Markowitz Mean-variance Portfolio Research Articles

Enhancing portfolio management using artificial intelligence: literature review.

Building an investment portfolio is a problem that numerous researchers have addressed for many years. The key goal has always been to balance risk and reward by optimally allocating assets such as stocks, bonds, and cash. In general, the portfolio management process is based on three steps: planning, execution, and feedback, each of which has its objectives and methods to be employed. Starting from Markowitz's mean-variance portfolio theory, different frameworks have been widely accepted, which considerably renewed how asset allocation is being solved. Recent advances in artificial intelligence provide methodological and technological capabilities to solve highly complex problems, and investment portfolio is no exception. For this reason, the paper reviews the current state-of-the-art approaches by answering the core question of how artificial intelligence is transforming portfolio management steps. Moreover, as the use of artificial intelligence in finance is challenged by transparency, fairness and explainability requirements, the case study of post-hoc explanations for asset allocation is demonstrated. Finally, we discuss recent regulatory developments in the European investment business and highlight specific aspects of this business where explainable artificial intelligence could advance transparency of the investment process.

Portfolio Construction Based on Optimization Model and Combination of Various Assets

Since the fifties of the last century, the concept of portfolio management is proposed to enrich the portfolio theory by Markowitz et al., as well as the improvement of portfolio theory by later generations, forming a modern portfolio management theory. This paper uses Markowitz's mean variance portfolio theory to perform a portfolio return maximization analysis on selected assets with assumptions that investors are risk-averse according to mean variance theory. Based on mean variance portfolio theory, as well as Python computational simulation modeling, Monte Carlo simulation is used to find the effective set and effective frontier of the portfolio, and the optimal solution of the portfolio is filtered according to the two constraints of maximum Sharpe ratio and minimum volatility. According to the analysis, the portfolio return of the maximum Sharpe ratio is relatively high, and the maximum Sharpe ratio combination is more worthwhile than the minimum volatility combination. Overall, these results provide enlightenment for further exploration and implementation of portfolio theory.

Static Markowitz mean-variance portfolio selection model with long-term bonds

We propose a static Markowitz mean-variance portfolio selection model suitable for long-term zero-coupon bonds. The model uses a multi-factor term structure model of Vasicek (Ornstein-Uhlenbeck) type to compute the portfolio's expected return and its variance in the model. German Government zero-coupon bonds with short to very long time to maturity are considered; the data spans August 2002 to December 2020. The main investment assumption is the re-investment of cash flows of zero-coupon bonds with maturities less than the planning horizon at the current spot interest rate. Solutions for the zero-coupon holding vector and the tangency portfolio are obtained in closed form. Model parameters are estimated under an assumption of modeling ambiguity which takes the form of Knightian uncertainties at the level of the latent factors, allowing the use of a Kalman filter. Different investment strategies are examined on various risk portfolios. Results show that one- and two-factor Vasicek models produce attractive out-of-sample portfolio predictions in terms of the Sharpe ratio especially on long-term investments. It is also noted that a small number of risky bonds can adequately produce very attractive portfolio risk-return profiles.

Construct an Empirical Study on the Concept, Modern Portfolio Theory Using Markowitz Model

In 1952, Harry Markowitz outlined a concept to aid financiers. By examining different combinations of the provided securities, the model aids in the selection of the optimal portfolio. He proposed an assumption of Modern Portfolio Theory, which aids in the creation of an optimal portfolio via the analysis of several portfolios. The Markowitz model is a theoretical framework for investigating the connections between different types of investment risk and return. The Modern Portfolio Theory (MPT) may help risk-averse investors create diversified portfolios that maximize returns without taking on undue risk. A forecasted return on each asset is required for current portfolio theory to be applied. There is no assurance that future results can be predicted based on past performance. Several market and investor presuppositions form the basis of MPT. We will first describe our finding on the "Markowitz optimization riddle" and the "Markowitz mean-variance portfolio theory," and then provide the theoretical proofs of our findings. Returns on investments are treated as random variables in the Markowitz mean-variance portfolio theory. So, picking the right portfolio weighting components is essential.

Markowitz Mean-Variance Portfolio Optimization with Predictive Stock Selection Using Machine Learning

With the advances in time-series prediction, several recent developments in machine learning have shown that integrating prediction methods into portfolio selection is a great opportunity. In this paper, we propose a novel approach to portfolio formation strategy based on a hybrid machine learning model that combines convolutional neural network (CNN) and bidirectional long short-term memory (BiLSTM) with robust input features obtained from Huber’s location for stock prediction and the Markowitz mean-variance (MV) model for optimal portfolio construction. Specifically, this study first applies a prediction method for stock preselection to ensure high-quality stock inputs for portfolio formation. Then, the predicted results are integrated into the MV model. To comprehensively demonstrate the superiority of the proposed model, we used two portfolio models, the MV model and the equal-weight portfolio (1/N) model, with LSTM, BiLSTM, and CNN-BiLSTM, and employed them as benchmarks. Between January 2015 and December 2020, historical data from the Stock Exchange of Thailand 50 Index (SET50) were collected for the study. The experiment shows that integrating preselection of stocks can improve MV performance, and the results of the proposed method show that they outperform comparison models in terms of Sharpe ratio, mean return, and risk.

Markowitz Mean-Variance Portfolio Selection and Optimization under a Behavioral Spectacle: New Empirical Evidence

This paper investigates the robustness of the conventional mean-variance (MV) optimization model by making two adjustments within the MV formulation. First, the portfolio selection based on a behavioral decision-making theory that encapsulates the MV statistics and investors psychology. The second aspect involves capturing the portfolio asset dependence structure through copula. Using the behavioral MV (BMV) and the copula behavioral MV (CBMV), the results show that stocks with lower behavioral scores outperform counterpart portfolios with higher behavioral scores. On the other hand, in the Forex market, the reverse is observed for the BMV approach, while the CBMV remains consistent.

Spectrally-Corrected Estimation for High-Dimensional Markowitz Mean-Variance Optimization

The portfolio problem for high dimensional data when the dimension and size are both large is considered. The traditional Markowitz mean-variance (MV) portfolio by large dimension matrix theory is analyzed, and it is found that the spectral distribution of the sample covariance is the main factor to make the expected return of the traditional MV portfolio overestimate the theoretical MV portfolio. Therefore, a new spectrally corrected method is introduced to correct the spectral elements of the sample covariance to a sample spectrally-corrected covariance, by which the spectrally-corrected portfolio and the corresponding return and risk are provided naturally. Moreover, the limiting behavior of the expected return and risk on the spectrally-corrected MV portfolio is deduced and the superior properties of the spectrally-corrected MV portfolio are illustrated. In simulations, the spectrally-corrected estimates get the best performance in both portfolio return and portfolio risk. The comparisons of their performance by using the S&P 500 data show the superiority of the proposed spectrally-corrected estimates get over the traditional and bootstrap-corrected estimates. Then, the empirical analysis shows that consistent with the theory developed, the proposed spectrally-corrected estimates outperform both the traditional and bootstrap-corrected estimates. Further, the findings show that all risk-averters will get improvement in portfolio returns or risk-adjusted portfolio returns, by adopting our proposed methods.

Portfolio Selection with Regularization

We study the Markowitz mean-variance portfolio selection model under three types of regularizations: single-norm regularizations on individual stocks, mixed-norm regularizations on stock groups, and composite regularizations that combine the single-norm and mixed-norm regularizations. With mixed-norm regularizations incorporated, our model can accomplish group and stock selections simultaneously. Our empirical results using both US and global equity market data show that compared to the classical mean-variance portfolio, almost all regularized portfolios have better out-of-sample risk-adjusted performance measured by Sharpe ratio. In addition, stock selection and group screening accomplished by adding [Formula: see text] and [Formula: see text] regularizations respectively can lead to decreased volatility, turnover rate, and leverage ratio. Yet there are instances in which diversifying across different groups is more favorable, depending on the grouping methods. Moreover, we find a positive correlation between portfolio turnover and leverage. Heavily leveraged portfolios also have high turnover rates and thus high transaction costs.

Impact of ESG Rating of Companies on the Portfolio Performance

Socially responsible investing is becoming more popular among people because people are becoming more concerned about the environment and society. Socially responsible investors screen the company by considering the ESG factors. The question raced is whether socially responsible investing improves the portfolio performance and how the funds perform during uncertain times like the Covid-19 pandemic. Since many critics of ESG funds say that the ESG funds’ performance highly depends on Software and Service company stocks, so the relevance of Software and Service companies in the fund has been analyzed in this research. The portfolios have been formed by using the Markowitz mean-variance portfolio model, and the performance of the minimum variance portfolio has been studied. The fund performance has been analyzed using the Sharpe ratio, and the result concludes that the ESG fund performance with minimum variance has an abnormally high Sharpe Ratio of 10.8. A similar type of performance was identified during the Covid-19 pandemic. The abnormally high Sharpe ratio will encourage investors to move towards socially responsible investing.

Time-Varying Mean-Variance Portfolio Selection under Transaction Costs and Cardinality Constraint Problem via Beetle Antennae Search Algorithm (BAS)

The Markowitz mean-variance portfolio selection is widely acclaimed as a very important investment strategy. A popular option to solve the static mean-variance portfolio selection (MVPS) problem is based on the use of quadratic programming (QP) methods. On the other hand, the static portfolio selection under transaction costs (PSTC) problem is usually approached with nonlinear programming (NLP) methods. In this article, we define and study the time-varying mean-variance portfolio selection under transaction costs and cardinality constraint (TV-MVPSTC-CC) problem as a time-varying nonlinear programming (TVNLP) problem. The TV-MVPSTC-CC also comprises the properties of a moving average. These properties make the TV-MVPSTC-CC an even greater analysis tool suitable to evaluate investments and identify trading opportunities across a continuous-time period. Using the Beetle Antennae Search (BAS) algorithm, we also provide an online solution to the static NLP problem. To the best of our knowledge, this is an innovative approach that incorporates modern meta-heuristic optimization techniques to provide an online, thus more realistic, solution to the TV-MVPSTC-CC problem. In this way, we present an online solution to a time-varying financial problem while eliminating the restrictions of static methods. Our approach is also verified by numerical experiments and computer simulations as an excellent alternative to traditional approaches.

International Portfolio Choice with Frictions: Evidence from Mutual Funds

Using data on international equity portfolio allocations of US mutual funds, we estimate a simple portfolio expression derived from a standard Markowitz mean-variance portfolio model extended with portfolio frictions. The optimal portfolio depends on two benchmark portfolios, the previous month and the buy-and-hold portfolio shares, and a present discounted value of expected future excess returns. We show that equity return differentials are predictable and use the expected return differentials in the mutual fund portfolio regressions. The estimated reduced form parameters are related to the structural model parameters. The estimates imply significant portfolio frictions and a modest rate of risk-aversion. While mutual fund portfolios respond significantly to expected returns, portfolio frictions lead to a weaker and more gradual portfolio response to changes in expected returns. We also document heterogeneity across funds. Global and larger funds face bigger portfolio frictions, while more active funds give relatively less weight to the buy-and- hold portfolio (rebalance more aggressively).

Mean-variance-time: An extension of Markowitz's mean-variance portfolio theory

In finance, investment decisions are commonly based on Markowitz's ex-ante mean-variance (MV) portfolio problem. The static ex-post trading problem, however, is completely absent. In this paper, we propose a theoretical extension of the MV framework by adding a time dimension so that the construction of a portfolio is thought of as an activity that consists of n monetary outcomes, i.e., rates of return on n risk assets, and the portfolio duration time t, which is the investor's optimal trading strategy time. Under a set of axioms, we show the existence and uniqueness of a utility function that represents investors’ preference over different time horizons. The analytical solution over the extended field yields an expression where optimal portfolio duration time depends explicitly on various sources of uncertainty; a key result that distinguishes this paper from the existing literature. We demonstrate empirically that our proposed model can explain many of the observed time-related anomalies of stock returns. Finally, we show that long-term trading strategies are more profitable for rational investors under perfect information.

Large Scale Continuous-Time Mean-Variance Portfolio Allocation via Reinforcement Learning

We propose to solve large scale Markowitz mean-variance (MV) portfolio allocation problem using reinforcement learning (RL). By adopting the recently developed continuous-time exploratory control framework, we formulate the exploratory MV problem in high dimensions. We further show the optimality of a multivariate Gaussian feedback policy, with time-decaying variance, in trading off exploration and exploitation. Based on a provable policy improvement theorem, we devise a scalable and data-efficient RL algorithm and conduct large scale empirical tests using data from the S&P 500 stocks. We found that our method consistently achieves over 10% annualized returns and it outperforms econometric methods and the deep RL method by large margins, for both long and medium terms of investment with monthly and daily trading.

The Kelly Growth Optimal Portfolio with Ensemble Learning

As a competitive alternative to the Markowitz mean-variance portfolio, the Kelly growth optimal portfolio has drawn sufficient attention in investment science. While the growth optimal portfolio is theoretically guaranteed to dominate any other portfolio with probability 1 in the long run, it practically tends to be highly risky in the short term. Moreover, empirical analysis and performance enhancement studies under practical settings are surprisingly short. In particular, how to handle the challenging but realistic condition with insufficient training data has barely been investigated. In order to fill voids, especially grappling with the difficulty from small samples, in this paper, we propose a growth optimal portfolio strategy equipped with ensemble learning. We synergically leverage the bootstrap aggregating algorithm and the random subspace method into portfolio construction to mitigate estimation error. We analyze the behavior and hyperparameter selection of the proposed strategy by simulation, and then corroborate its effectiveness by comparing its out-of-sample performance with those of 10 competing strategies on four datasets. Experimental results lucidly confirm that the new strategy has superiority in extensive evaluation criteria.

Sensitivity Analysis in Applications with Deviation, Risk, Regret, and Error Measures

The envelope formula is obtained for optimization problems with positively homogeneous convex functionals defined on a space of random variables. Those problems include linear regression with general error measures and optimal portfolio selection with the objective function being either a general deviation measure or a coherent risk measure subject to a constraint on the expected rate of return. The obtained results are believed to be novel even for Markowitz's mean-variance portfolio selection but are far more general and include explicit envelope relationships for the rates of return of portfolios that minimize lower semivariance, mean absolute deviation, deviation measures of ${\cal L}^p$-type and semi-${\cal L}^p$ type, and conditional value-at-risk. In each case, the envelope theorem yields explicit estimates for the absolute value of the difference between deviation/risk of optimal portfolios with the unperturbed and perturbed asset probability distributions in terms of a norm of the perturbation.

Improved Constrained Portfolio Selection Model using Particle Swarm Optimization

Objective: The main objective of this study is to improve the extended Markowitz mean-variance portfolio selection model by introducing a new constraint known as expert opinion practicable for portfolio selection in real-life situation. Methods: This new extended model consists of four constraints namely: bounds on holdings, cardinality, minimum transaction lots, and expert opinion. The first three constraints have been presented in other researches in literature. The fourth constraint introduced in this study is an essential parameter in making and guiding a realistic portfolio selection. To solve this new extended model an efficient heuristic method of Particle Swarm Optimization (PSO) was engaged with existing benchmark data in the literature. Results: The outcome of the computational results obtained in this study with the new extended Markowitz mean-variance portfolio selection model proposed in this study and solved with PSO showed an improved performance over existing algorithm in particular GA in different instances of the data set used. Conclusion: The study evolves a new extended portfolio selection model and the findings demonstrate the superiority of PSO performance in solving portfolio selection problem in comparison with GA algorithm.

Weighted Elastic Net Penalized Mean-Variance Portfolio Design and Computation

It is well known that the out-of-sample performance of Markowitz's mean-variance portfolio criterion can be negatively affected by estimation errors in the mean and covariance. In this paper we address the problem by regularizing the mean-variance objective function with a weighted elastic net penalty. We show that the use of this penalty can be motivated by a robust reformulation of the mean-variance criterion that directly accounts for parameter uncertainty. With this interpretation of the weighted elastic net penalty we derive data-driven techniques for calibrating the weighting parameters based on the level of uncertainty in the parameter estimates. We test our proposed technique on U.S. stock return data, and our results show that the calibrated weighted elastic net penalized portfolio outperforms both the unpenalized portfolio and the uniformly weighted elastic net penalized portfolio. This paper also introduces a novel adaptive support split-Bregman approach which leverages the sparse nature of $\e...

Portfolio Theory in Terms of Partial Covariance

It is found that partial correlations between 12 major US equity sector ETFs conditioned on the state of economy (mimicked here by the S&P 500 index) are significantly lower than the Pearson’s correlations. The Markowitz mean-variance portfolio theory is modified in terms of partial covariance. The maximum Sharpe portfolios formed by 12 equity sector ETFs in 2007 – 2015 are examined. With the exclusion of the bear market of 2008, the partial correlation based portfolios (PaCP) are much more diversified than the Pearson’s correlation based portfolios (PeCP). Out-of-sample performance of the maximum Sharpe PeCP and PaCP, and the equal-weight portfolio (EWP) are compared. The results are very sensitive to the model parameters (portfolio calibration window and frequency of portfolio rebalancing). While the PeCP weights change significantly from month to month, the PaCP weights outside the bear market effects are almost constant. PaCP outperforms both EWP and PeCP when the 36-month calibration window and one-month rebalancing frequency are used. We conclude that partial covariance is a promising concept for constructing optimal portfolios.

Index-tracking optimal portfolio selection

This paper considers a portfolio selection problem with multiple risky assets where the portfolio is managed to track a benchmark market barometer, such as the S&P 500 index. A tracking optimization model is formulated and the tracking-efficient (TE) portfolios are shown to inherit interesting properties compared with Markowitz mean-variance (MV) optimal portfolios. In comparison to an MV-portfolio, both the beta and the variance of a TE-portfolio are higher by fixed amounts that are independent of the expected portfolio return. These differences increase with index variance, are convex quadratic in the asset betas, and depend on the asset means and covariance matrix. Furthermore, a TE portfolio is obtained by simply extending an MV portfolio by constant adjustments to portfolio weights, independent of the specified mean return of the portfolio, but dependent on the index variance and asset return parameters. Consequently, at lower thresholds of risk, TE portfolios are better-diversified than MV portfolios.

Portfolio Selection under Condition of Variable Weights

In this paper we generalize the single-period Markowitz Mean-Variance portfolio selection problem. The Markowitz’s model requires that after choosing the number of each security which constructs the portfolio in the beginning of the investment period, these numbers remain constant during and at the end of the investment period. We drop this as-sump- tion and consider an investment model in which the number of each security can vary randomly during the in-vestment period. Indeed we consider a single-period investment with the property that the initial weight of each security is not equal to the final weight of that security. We redefine the notion of the rate of return of each security and show that the return of the investment in a cash account is not certain. We investigate some alternatives among risky securi-ties which acts similar to cash accounts. For this we introduce the notion of free security and relate free securities to a riskless se- curity.