Markowitz Mean-variance Research Articles

ARX-GARCH Probabilistic Price Forecasts for Diversification of Trade in Electricity Markets—Variance Stabilizing Transformation and Financial Risk-Minimizing Portfolio Allocation

In this paper, we propose dynamic, short-term, financial risk management strategies for small electricity producers and buyers that trade in the wholesale electricity markets. Since electricity is mostly nonstorable, financial risk coming from extremely volatile electricity prices cannot be reduced by using standard finance-based approaches. Instead, a short-term operational planing and a proper trade diversification might be used. In this paper, we analyze the price risk in terms of the Markowitz mean–variance portfolio theory. Hence, it is crucial to forecast properly the variance of electricity prices. To this end, we jointly model day-ahead and intraday or balancing prices from Germany and Poland using ARX-GARCH type models. We show that using heteroscedastic volatility significantly improves probabilistic price forecasts according to the pinball score, especially if variance stabilizing transformation is applied prior to a model estimation. The price forecasts are then used for construction of dynamic diversification strategies that are based on volatility-type risk measures. We consider different objectives as well as a buyer’s and a seller’s perspective. The proposed strategies are applied for the diversification of trade among different markets in Germany and Poland. We show that the objective of the strategy can be achieved using the proposed approach, but the risk minimization is usually related to lower profits. We find that risk minimization is especially important for a seller in both markets, while for a buyer a profit maximization objective leads to a more optimal risk–return trade-off.

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.

Inflation and portfolio selection

This study proposes and tests a portfolio selection model with inflation allocation lines (IAL) for corresponding capital allocation line (CAL) and utilities in several scenarios of crisis. The model is based on Markowitz's mean-variance (MV) theory, with modification of Tobin's portfolio utility function, and Sharpe's (1964) portfolio theory. The model introduces inflation as a significant factor. According to study results, empirically tested with least squares (OLS) and quantile regression models, the study verifies that under conditions of low and moderate inflation the investor chooses an optimal portfolio which generates the highest real returns (including borrowed funds). For the case of severe recession, the investor chooses a minimum variance portfolio.

A Stochastic Approximation Method for Simulation-Based Quantile Optimization

We present a gradient-based algorithm for solving a class of simulation optimization problems in which the objective function is the quantile of a simulation output random variable. In contrast with existing quantile (quantile derivative) estimation techniques, which aim to eliminate the estimator bias by gradually increasing the simulation sample size, our algorithm incorporates a novel recursive procedure that only requires a single simulation sample at each step to simultaneously obtain quantile and quantile derivative estimators that are asymptotically unbiased. We show that these estimators, when coupled with the standard gradient descent method, lead to a multitime-scale stochastic approximation type of algorithm that converges to an optimal quantile value with probability one. In our numerical experiments, the proposed algorithm is applied to optimal investment portfolio problems, resulting in new solutions that complement those obtained under the classical Markowitz mean-variance framework. History: Accepted by Alice E. Smith, Editor-in-Chief; Bruno Tuffin, Area Editor for Simulation. Funding: The work of Y. Peng was supported in part by the National Natural Science Foundation of China (NSFC) [Grants 72022001, 92146003, and 71901003], and by the Key Research and Development Programof Beijing Municipal Science and Technology Commission. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.1214 .

Research On Finding Optimal Portfolios Based on Markowitz Mean-Variance Model -- Constructing Efficient Frontier and Portfolio with Highest Sharpe Ratio Using 4 Assets in Chinese Market

Markowitz mean-variance model is the foundation of modern investment science, and how to apply this model to construct optimal portfolios is always of interest. This paper uses the Markowitz mean-variance model and four typical assets in Chinese market to construct efficient frontier with highest expected return rate at each risk level and lowest risk at each expected return rate level, and the portfolios with the highest Sharpe ratio. There are three main results in this paper. Firstly, the efficient frontier has the U shape and V shape rotated clockwise in the case of excluding and including risk-free asset respectively. Secondly, the theoretical efficient frontier obtained from mathematical formulas approximately coincide with those from the simulation experiments. Thirdly, the portfolio with the highest Sharpe ratio always occurs in the intersection point of the efficient frontier and its tangent line that passes through the risk-free asset. The results from this paper provide a convenient method of using mathematical formula for investor to construct optimal portfolios in real practice.

Research on the investment value of China’s medical sector in the context of COVID-19

COVID-19 has slowed the growth of, the global economy, which has certain practical significance. Consequently, this study seeks to analyze the investment opportunities in the medical sector before and after the COVID-19 outbreak. In this study, the Markowitz mean–variance (MV) model, capital asset pricing model (CAPM), and correlation models are constructed based on the principle of Markowitz MV and correlation analysis. Simultaneously, statistical analysis is used to verify the analysis, and the MATLAB statistical tool is used to build the model. The results show that the actual expected yield of China’s medical sector is significantly higher than that calculated by the CAPM before and after the pandemic, and that the investment value of the medical sector is undervalued by the market. From the perspective of risk, China’s medical sector has a stable systemic risk premium. Based on the above analysis, when building investment portfolios in the post-pandemic era, investors should appropriately allocate stocks in the medical and pharmaceutical sectors to improve the portfolio income and diversify the investment risk.

A Swarm Intelligence Optimization Algorithm for Cryptocurrency Portfolio Optimization

In recent years, cryptocurrency has been widely adopted and seen as an alternative investment tool for investors. However, which cryptocurrency to invest in and how much to invest becomes a problem. Since there is a conflict of multiple criteria, portfolio optimization (PO) is needed to solve the problem. In this study, an Artificial Bee Colony (ABC) algorithm has been developed based on Markowitz's mean-variance model (M-MVM). With this method, the portfolio of cryptocurrencies has been tried to be optimized. Hourly data of 12 cryptocurrencies between 01.09.2020 and 01.04.2021 were used as data. It has been observed that the ABC algorithm achieves good results in the solution of the problem in a reasonable time. In addition, the method was tested with different parameter values and different risk-averse coefficient values (λ).

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.

Behavioral finance and the architecture of the asset management industry

The standard portfolio approach assumes that investors maximize expected utility, and that the Markowitz mean–variance standard portfolio optimization approach can be applied. Behavioral research, however, indicates that investors’ behavior with respect to risk or uncertainty is not consistent with expected utility. Notably, decision makers transform probabilities. Most of the academic literature integrating those aspects, however, focuses on so-called financial market anomalies but does not focus on the structural impact those behavioral aspects might have, notably in terms of industrial organization of the asset management industry. This paper provides a survey of the asset management industry and argues that the architecture of the asset management industry is inconsistent with the standard expected utility framework. The origin of the industrial organization of the asset management industry stems from the convexity of the flow–performance relationship, which is due to prospect theory effects. The convexity of the flow–performance relationship explains the inflation of mutual funds and fund families, a phenomenon inconsistent with the Mutual Fund Separation Theorem.

The Interaction of Major Crypto-assets, Clean Energy, and Technology Indices in Diversified Portfolios

Cryptocurrencies have gained high interest from media, regulatory authorities, and both retailer and institutional investors especially in the COVID-19 pandemic followed by a tremendous academic interest. Classification of cryptos as an asset is one important issue while another crucial topic is the significantly high return and volatility fluctuations which does not make life easier for portfolio investors. High energy consumption to produce cryptos and carbon emission issues are also not developing the favor of cryptocurrencies. Moreover, in financial literature recent studies show that prices of renewable energy stocks have long-term cointegrating relationship with technology companies. In this context first we employ an asymmetric VAR-GARCH model to study spillover effects between major crypto-currencies, clean energy, and technology indices. Using daily data of the two major cryptocurrencies for the period of 1/1/2016 and 30/09/2021, we relate risk and return of different mean-variance portfolio strategies to Bitcoin (BTC), Etherium (ETH), S&P Global Clean Energy Index (SPGCE) and MSCI World Information Technology Index (MSCIWIT). Secondly, we apply the Markowitz mean-variance framework to assess risk-return benefits of cryptocurrency-portfolios. Our main goal is to offer optimal portfolio allocation approaches including cyptocurrencies with traditional financial assets. We will combine cryptocurrencies, clean energy, and technology indices to maximize return and Sharpe ratio. Furthermore, we will use our asymmetric VAR-GARCH models results to understand and cross check the Markowitz portfolio allocation results in details.

L1-Regularization in Portfolio Selection with Machine Learning

In this work, we investigate the application of Deep Learning in Portfolio selection in a Markowitz mean-variance framework. We refer to a l1 regularized multi-period model; the choice of the l1 norm aims at producing sparse solutions. A crucial issue is the choice of the regularization parameter, which must realize a trade-off between fidelity to data and regularization. We propose an algorithm based on neural networks for the automatic selection of the regularization parameter. Once the neural network training is completed, an estimate of the regularization parameter can be computed via forward propagation. Numerical experiments and comparisons performed on real data validate the approach.

A combinatorial optimization approach to scenario filtering in portfolio selection

Recent studies stressed the fact that covariance matrices computed from empirical financial time series appear to contain a high amount of noise. This makes the classical Markowitz Mean–Variance Optimization model unable to correctly evaluate the performance associated to selected portfolios. Since the Markowitz model is still one of the most used practitioner-oriented tool, several filtering methods have been proposed in the literature to overcome the problem. Among them, the two most promising ones refer to the Random Matrix Theory and to the Power Mapping strategy. The basic idea of these methods is to transform the estimated correlation matrix before applying the Mean–Variance Optimization model. However, experimental analysis shows that these two strategies are not always effective when applied to real financial datasets.In this paper we propose a new filtering method based on Quadratic Programming. We develop a Mixed Integer Quadratic Programming model, which is able to filter those observations that may affect the performance of the selected portfolio. We discuss the properties of this new model and test it on some real financial datasets. We compare the out-of-sample performance of our portfolios with the one of the portfolios provided by the two above mentioned alternative filtering methods giving evidence that our method outperforms them. Although our model can be solved efficiently with standard optimization solvers, the computational burden increases for large datasets. To solve also these problems, we propose a heuristic procedure, which, on the basis of our empirical results, shows to be both efficient and effective.

Optimal Allocation of Retirement Portfolios

A retiree with a savings account balance, but without a pension, is confronted with an important investment decision that has to satisfy two conflicting objectives. Without a pension, the function of the savings is to provide post-employment income to the retiree. At the same time, most retirees want to leave an estate to their heirs. Guaranteed income can be acquired by investing in an annuity. However, that decision takes funds away from investment alternatives that might grow the estate. The decision is made even more complicated because one does not know how long one will live. A long life expectancy may require more annuities, and a short life expectancy could promote more risky investments. However there are very mixed opinions about both strategies. A framework has been developed to assess consequences and the trade-offs of alternative investment strategies. We propose a stochastic programming model to frame this complicated problem. The objective is to maximize expected estate value, subject to cash outflow constraints. The model is motivated by the Markowitz mean-variance approach, but with risk measured by CVaR and additional sophisticated constraints. The cash outflow shortages are penalized in the objective function of the problem. We use the kernel method to build position adjustment functions that control how much is invested in each asset. These adjustments nonlinearly depend upon asset returns in previous years. A case study was conducted using two variations of the model. The parameters used in this case study correspond to a typical retirement situation. The case study shows that if the market forecasts are pessimistic, it is optimal to invest in an annuity. The case study results, codes, and data are posted on our website.

A tail-revisited Markowitz mean-variance approach and a portfolio network centrality

A measure for portfolio risk management is proposed by extending the Markowitz mean-variance approach to include the left-hand tail effects of asset returns. Two risk dimensions are captured: asset covariance risk along risk in left-hand tail similarity and volatility. The key ingredient is an informative set on the left-hand tail distributions of asset returns obtained by an adaptive clustering procedure. This set allows a left tail similarity and left tail volatility to be defined, thereby providing a definition for the left-tail-covariance-like matrix. The convex combination of the two covariance matrices generates a “two-dimensional” risk that, when applied to portfolio selection, provides a measure of its systemic vulnerability due to the asset centrality. This is done by simply associating a suitable node-weighted network with the portfolio. Higher values of this risk indicate an asset allocation suffering from too much exposure to volatile assets whose return dynamics behave too similarly in left-hand tail distributions and/or co-movements, as well as being too connected to each other. Minimizing these combined risks reduces losses and increases profits, with a low variability in the profit and loss distribution. The portfolio selection compares favorably with some competing approaches. An empirical analysis is made using exchange traded fund prices over the period January 2006–February 2018.

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.

Virtual Power Plant (VPP) scheduling with uncertain multiple Locational Marginal Prices

Aggregation of heterogeneous distributed energy resources (DERs) and prosumers for profit maximisation by virtue of the market interface has evolved into virtual power plants (VPPs). The VPP operator optimises its aggregated resources to maximise profit by participating in the wholesale electricity market. VPP resources may be geographically dispersed and connected to a transmission grid at various nodes having different locational marginal prices (LMPs). These LMPs vary throughout the day and exhibit mutual correlation. Considering this, the current study presents a profit maximisation framework for VPP operators working under multiple LMPs, with their correlated uncertainties. This work also provides VPP resource scheduling for profit maximisation, considering optimal trading at various LMP nodes for a constrained internal network. This decision is devised using Markowitz's mean-variance (M-V) criterion under expedient correlation between volatile LMPs. A numerical case study on the proposed model with the PJM market demonstrates that LMP correlation alters the role of the VPP operator. This affects its trading and scheduling decisions as a buyer/seller at various time intervals to improve its profit–risk trade-off and consequently the market interfacing of the VPP operator. The proposed model is relevant in practical market conditions for day-ahead/medium-term VPP planning while managing market uncertainty.

Hierarchical Risk Parity as an Alternative to Conventional Methods of Portfolio Optimization: (A Study of Tehran Stock Exchange)

One of the most critical investment issues faced by different investors is choosing an optimal investment portfolio and balancing risk and return in a way that, maximizes investment returns and minimize the investment risk. So far, many methods have been introduced to form a portfolio, the most famous of the Markowitz approach. The Markowitz mean-variance approach is widely known in the world of finance and, it marks the foundation of every portfolio theory. The mean-variance theory has many practical drawbacks due to the difficulty in estimating the expected return and covariance for different asset classes. In this study, we use the Hierarchical Risk Parity (HRP) machine learning technique and compare the results with the three methods of Minimum Variance (MVP), Uniform Distribution (UNIF), and Risk Parity (RP). To conduct this research, the adjusted price of 50 listed companies of the Tehran Stock Exchange for 2018-07-01 to 2020-09-29 has been used. 70% of the data are considered as in-sample and the remaining 30% as out-of-sample. We evaluate the results using four criteria: Sharp, Maximum Drawdown, Calmer, Sortino. The results show that the MVP and, UNIF approach within the in-sample and, the UNIF and HRP approach out-of-sample have the best performance in sharp measure.

Time-varying mean–variance portfolio selection problem solving via LVI-PDNN

It is widely acclaimed that the Markowitz mean–variance portfolio selection is a very important investment strategy. One approach to solving the static mean–variance portfolio selection (MVPS) problem is based on the usage of quadratic programming (QP) methods. In this article, we define and study the time-varying mean–variance portfolio selection (TV-MVPS) problem both in the cases of a fixed target portfolio’s expected return and for all possible portfolio’s expected returns as a time-varying quadratic programming (TVQP) problem. The TV-MVPS also comprises the properties of a moving average. These properties make the TV-MVPS an even greater analysis tool suitable to evaluate investments and identify trading opportunities across a continuous-time period. Using an originally developed linear-variational-inequality primal–dual neural network (LVI-PDNN), we also provide an online solution to the static QP problem. To the best of our knowledge, this is an innovative approach that incorporates robust neural network techniques to provide an online, thus more realistic, solution to the TV-MVPS problem. In this way, we present an online solution to a time-varying financial problem while eliminating static method limitations. It has been shown that when applied simultaneously to TVQP problems subject to equality, inequality and boundary constraints, the LVI-PDNN approaches the theoretical solution. Our approach is also verified by numerical experiments and computer simulations as an excellent alternative to conventional MATLAB methods.