Mean-variance Portfolio Optimization Research Articles

The Smirnov Property for Weighted Lebesgue Spaces

We establish lower norm bounds for multivariate functions within weighted Lebesgue spaces, characterised by a summation of functions whose components solve a system of nonlinear integral equations. This problem originates in portfolio selection theory, where these equations allow one to identify mean-variance optimal portfolios, composed of standard European options on several underlying assets. We elaborate on the Smirnov property—an integrability condition for the weights that guarantees the uniqueness of solutions to the system. Sufficient conditions on weights to satisfy this property are provided, and counterexamples are constructed, where either the Smirnov property does not hold or the uniqueness of solutions fails.

An adapted Black Widow Optimization Algorithm for Financial Portfolio Optimization Problem with cardinalty and budget constraints

Financial Portfolio Optimization Problem (FPOP) is a cornerstone in quantitative investing and financial engineering, focusing on optimizing assets allocation to balance risk and expected return, a concept evolving since Harry Markowitz’s 1952 Mean-Variance model. This paper introduces a novel meta-heuristic approach based on the Black Widow Algorithm for Portfolio Optimization (BWAPO) to solve the FPOP. The new method addresses three versions of the portfolio optimization problems: the unconstrained version, the equality cardinality-constrained version, and the inequality cardinality-constrained version. New features are introduced for the BWAPO to adapt better to the problem, including (1) mating attraction and (2) differential evolution mutation strategy. The proposed BWAPO is evaluated against other metaheuristic approaches used in portfolio optimization from literature, and its performance demonstrates its effectiveness through comparative studies on benchmark datasets using multiple performance metrics, particularly in the unconstrained Mean-Variance portfolio optimization version. Additionally, when encountering cardinality constraint, the proposed approach yields competitive results, especially noticeable with smaller datasets. This leads to a focused examination of the outcomes arising from equality versus inequality cardinality constraints, intending to determine which constraint type is more effective in producing portfolios with higher returns. The paper also presents a comprehensive mathematical model that integrates real-world constraints such as transaction costs, transaction lots, and a dollar-denominated budget, in addition to cardinality and bounding constraints. The model assesses both equality/inequality cardinality constraint versions of the problem, revealing that the inequality constraint tends to offer a wider range of feasible solutions with increased return potential.

Enhancing mean–variance portfolio optimization through GANs-based anomaly detection

Enhancing mean–variance portfolio optimization through GANs-based anomaly detection

Portfolio optimization for sustainable investments

In mean-variance portfolio optimization, multi-index models often accelerate computation, reduce input requirements, facilitate understanding, and allow easy adjustment to changing conditions more effectively than full covariance matrix estimation in many situations. In this paper, we develop a multi-index model-based portfolio optimization approach that takes into account aspects of the environment, social responsibility and corporate governance (ESG). Investments in assets related to ESG have recently grown, attracting interest from both academic research and investment fund practice. Various literature strands in this area address the theoretical and empirical relation among return, risk and ESG. Our portfolio optimization approach is flexible enough to take these literature strands into account and does not require large-scale covariance matrix estimation. An extension of our approach even allows investors to empirically discriminate among the literature strands. A case study demonstrates the application of our portfolio optimization approach.

Dynamic Mean–Variance Portfolio Optimization with Value-at-Risk Constraint in Continuous Time

Recognizing the importance of incorporating different risk measures in the portfolio management model, this paper examines the dynamic mean-risk portfolio optimization problem using both variance and value at risk (VaR) as risk measures. By employing the martingale approach and integrating the quantile optimization technique, we provide a solution framework for this problem. We demonstrate that, under a general market setting, the optimal terminal wealth may exhibit different patterns. When the market parameters are deterministic, we derive the closed-form solution for this problem. Examples are provided to illustrate the solution procedure of our method and demonstrate the benefits of our dynamic portfolio model compared to its static counterpart.

Machine Learning for liquidity classification and its applications to portfolio selection

Liquidity refers to the ease of asset conversion into cash, playing a crucial role in investment decisions for achieving optimal returns. This study proposes a novel stock liquidity classification method using machine learning algorithms, trained, and tested on ten years of Brazilian stock market (B3) data. Achieving an accuracy of 99.2%, the classifier, when integrated with the mean-variance portfolio optimization model, reduces portfolio uncertainty by preventing an average of 11.5% of illiquid asset sales.

Mean Variance Complex-Based Portfolio Optimization

Mean-Variance (MV) is a method that collects several assets using appropriate weight intending to maximize profits and to reduce risk. Stock market conditions are very volatile, mean variance method does not reach stock market fluctuation well because MV method is only limited to one time period. This study proposes a mean variance complex-based approach that transforms real returns into complex returns by using Hilbert transform to construct an optimal mean-variance portfolio based on complex returns and then find its dynamic asset allocation. The results show that with the same risk tolerance, the mean variance complex-based approach outperforms MV method in profits, losses, and portfolio performance tests.

3D Investing: Jointly Optimizing Return, Risk, and Sustainability

Traditional mean-variance portfolio optimization is based on the premise that investors only care about risk and return. However, some investors also have non-financial objectives such as sustainability goals. We show how the traditional approach can readily be extended to mean-variance-sustainability optimization and explain why this 3D investing approach is ex-ante Pareto-optimal. We illustrate its efficacy empirically in several studies, including carbon footprint and sustainable development goal objectives. Importantly, we highlight conditions under which a 3D optimization approach is superior to a naïve 2D approach augmented with sustainability constraints.

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.