Global Minimum Variance Portfolio Research Articles

Shrinkage and thresholding approaches for expected utility portfolios: An analysis in terms of predictive ability

In this paper, we estimate Expected Utility Portfolios (EUPs) in high-dimensional, low-sample settings using various covariance matrix estimation methods, including shrinkage and thresholding-based methods. We perform synthetic experiments comparing these methods, using Average Out-of-Sample Variance (AOV) for Global Minimum Variance (GMV) portfolios and Average Out-of-Sample Utility (AOU) for EUPs. Additionally, we propose a practical method for fund managers to select optimal models based on empirical data, relying on AOV and AOU performance measures. The results indicate that shrinkage-based methods outperform thresholding-based ones in high-dimensional settings, with non-linear shrinkage being particularly effective.

Portfolio Optimization Based on U.S. Stock

In the financial domain, portfolio optimization is of immense significance due to its potential for risk diversification and return enhancement. Particularly in today's society where inflation is on the rise, a well-constructed portfolio can effectively assist investors in maintaining or increasing real assets. This study selected 10 U.S. stocks and analyzed their data for the past decade. Utilizing Monte Carlo simulation, the study identified the efficient frontier and constructed three types of portfolios: equal-weighted (1/N) portfolio, global minimum variance portfolio (hereinafter referred to as GMVP), and maximum Sharpe ratio portfolio. The findings show that Microsoft and Nvidia have the greatest investment weights in the portfolio with the maximum Sharpe ratio. In the GMVP, JNJ and KO have the highest investment weights. By contrasting the three portfolios' combined performance with the S&P 500 index, both the maximum Sharpe ratio and the 1/N portfolios outperformed the benchmark index, while the GMVP performed below the benchmark. The findings of this study may offer insights into portfolio management for investors looking to increase their real assets.

Does green improve portfolio optimisation?

Our study uses the GARCH-EVT-copula model to develop out-of-sample forecasts for diverse asset classes, including a green asset. To construct optimal portfolios, we apply four different portfolio allocation techniques: equal weighting, minimum variance, global minimum variance (GMV), and certainty equivalence tangency (CET) criteria. The results demonstrate that the GMV portfolio outperforms other portfolios in risk measures. Further, backtesting evidence shows that the portfolio containing a green asset performs better than the benchmark for short horizons. The results have implications for fund managers and policymakers since green asset provides valuable diversification benefits and further the cause of sustainable development.

An Empirical Study on the Markowitz portfolio

Markowitz's portfolio model laid the foundation of the modern portfolio theory (MPT). As an intuitive and effective quantitative method, the Markowitz model has long been considered one of the most successful approaches in financial modeling. This paper tries to illustrate the process of Markowitz modeling and demonstrate the usefulness of the Markowitz theory empirically. Three high-tech companies, Apple Inc. (AAPL), Alphabet Inc. (GOOG), and Microsoft Corporation (MSFT) investigated to build the Markowitz model. We first retrieve the dataset from the website of yahoo finance and provide the descriptive statistics for the three companies, respectively. Then, the global minimum variance (GMV) portfolio is constructed to identify the boundary point of the efficient frontier. The optimized portfolio model at the given risk level is constructed according to the Markowitz theory. Also, the random portfolio weights under the budget constraint are generated for comparison. To evaluate the performance, we provide the Sharpe ratio for the constructed Markowitz portfolio.

Brazilian REITs: Are They an Opportunity for Diversification and Performance?

We study the dynamic diversification benefits of Brazilian real estate investment trusts (BR-REITs) by analyzing their complex dynamic relationship with stock and government bond indices. We estimate the assets’ conditional variance–covariance matrix using a vector autoregressive multivariate dynamic conditional correlation GARCH model with Student’s t distribution (VAR-DCCt MGARCH) and a global minimum variance portfolio. We find that BR-REITs provide strong risk-adjusted performance and persistent diversification benefits for Brazilian investors. Furthermore, these diversification benefits are not impacted by the market performance of stocks and bonds. Nevertheless, our results show that the BR-REITs’ conditional correlations with stocks and bonds are dynamic; thus, BR-REIT investors should employ dynamic investment strategies that account for time variation in REITs’ diversification benefits. Our results suggest that Brazilian investors, particularly long-term investors, would secure significant performance and diversification benefits, lowering their portfolios’ short-term volatility, by investing in BR-REITs.

Is the empirical out-of-sample variance an informative risk measure for the high-dimensional portfolios?

The main contribution of this paper is the derivation of the asymptotic behavior of the out-of-sample variance, the out-of-sample relative loss, and of their empirical counterparts in the high-dimensional setting, i.e., when both ratios p/n and p/m tend to some positive constants as m→∞ and n→∞, where p is the portfolio dimension, while n and m are the sample sizes from the in-sample and out-of-sample periods, respectively. The results are obtained for the traditional estimator of the global minimum variance (GMV) portfolio and for the two shrinkage estimators introduced by Frahm and Memmel (2010) and Bodnar et al. (2018). We show that the behavior of the empirical out-of-sample variance may be misleading in many practical situations, leading, for example, to a comparison of zeros. On the other hand, this will never happen with the empirical out-of-sample relative loss, which seems to provide a natural normalization of the out-of-sample variance in the high-dimensional setup. As a result, an important question arises if the out-of-sample variance can safely be used in practice for portfolios constructed from a large asset universe.

Portfolio Selection under Systemic Risk

AbstractThis paper proposes a modified Sharpe ratio to construct optimal portfolios under systemic events. The portfolio allocation problem is solved analytically under the absence of short‐selling restrictions and numerically when short‐selling restrictions are imposed. This approach is made operational by embedding it in a multivariate dynamic setting using dynamic conditional correlation and copula models. We evaluate the out‐of‐sample performance of our portfolio empirically over the period 2007 to 2020 using ex post final wealth paths and systemic risk metrics against mean–variance, equally weighted, and global minimum variance portfolios. Our portfolio outperforms all competitors under market distress and remains competitive in noncrisis periods.

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.

A Regularized K-means Autoregressive Shrinkage Method to Portfolio Optimization

Estimation is always an issue of portfolio optimization. To address this issue this paper started from addressing global minimum variance (GMV) portfolio, which gives the weights as the function of estimated covariance matrix of the asset net returns. The inverse operation of the matrix always leads to ill-conditioned matrix due to the noise in the return series. Therefore, regularized sample covariance matrix is often used to address this problem. In this work, a regularized autoregressive shrinkage method for covariance matrix estimation with the historical data is developed, which regards the future covariance matrix as the function of the historical covariance matrix. K-means clustering algorithm is also added to enhance the performance of the regularized autoregressive model. The results demonstrate that the performance of GMV portfolio can be enhanced with regularization under autoregressive shrinkage model.

‘Safe Assets’ during COVID-19: A Portfolio Management Perspective

The pandemic crisis of COVID-19 hit the financial markets like a shockwave on 16 March 2020. This paper attempts to capture which ‘safe assets’ asset managers could have fled during the first wave of the pandemic. From an investment manager’s perspective, candidate assets are stocks, bonds, exchange rates, commodities, gold, and (gold-backed) cryptocurrencies. Empirical tests of the ‘Safe-Haven’ hypothesis are conducted, upon which the selection of assets is performed. The methodological framework hinges on the Global Minimum Variance Portfolio with Monte Carlo simulations, and the routine is performed under Python. Other optimization techniques, such as risk parity and equal weighting, are added for robustness checks. The benchmark portfolio hits a yearly profitability of 7.2% during such a stressful event (with 3.6% downside risk). The profitability can be enhanced to 8.4% (even 14.4% during sub-periods) with a careful selection of ‘Safe assets’. Besides short- to long-term U.S. bonds, we document that investors’ exposure to Chinese, Argentinian, and Mexican stocks during COVID-19 could have been complemented with Swiss and Japanese currencies, grains, physical gold mine ETFs, or gold-backed tokens for defensive purposes.

Real-Time Portfolio Management System Utilizing Machine Learning Techniques

There are 1641 companies listed on the National Stock Exchange of India. It is undoubtedly infeasible for a retail investor to invest in all the stocks. It is a well-known fact that the portfolio’s return is an average return of all its constituent stocks, and risk will be less than or equal to the maximum risk of all the portfolio components. This paper is unique as it elaborates on the entire portfolio selection, optimization, and management process. Portfolio selection is accomplished through the K-Means algorithm. Optimization is achieved utilizing the genetic algorithm, and a sliding window is applied for portfolio management. Four different ways of portfolio calculation, namely, equally-weighted portfolio, global minimum variance portfolio, market cap-weighted portfolio, and maximum Sharpe ratio portfolio, are applied. The results depict that all three optimized portfolios outperform the Nifty index. The dataset for the study is obtained from globaldatafeeds.in.

Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio

In this paper, new results in random matrix theory are derived which allow us to construct a shrinkage estimator of the global minimum variance (GMV) portfolio when the shrinkage target is a random object. More specifically, the shrinkage target is determined as the holding portfolio estimated from previous data. The theoretical findings are applied to develop theory for dynamic estimation of the GMV portfolio, where the new estimator of its weights is shrunk to the holding portfolio at each time of reconstruction. Both cases with and without overlapping samples are considered in the paper. The non-overlapping samples corresponds to the case when different data of the asset returns are used to construct the traditional estimator of the GMV portfolio weights and to determine the target portfolio, while the overlapping case allows intersections between the samples. The theoretical results are derived under weak assumptions imposed on the data-generating process. No specific distribution is assumed for the asset returns except from the assumption of finite $4+\varepsilon$, $\varepsilon>0$, moments. Also, the population covariance matrix with unbounded spectrum can be considered. The performance of new trading strategies is investigated via an extensive simulation. Finally, the theoretical findings are implemented in an empirical illustration based on the returns on stocks included in the S\&P 500 index.

Interdependence, contagion and speculative bubbles in cryptocurrency markets

After detecting several bubbles during 2015–2022, this study investigates the impact of the two biggest bubbles – those of 2017 and 2021 – on interdependence and contagion among cryptocurrencies. Interdependence declines during these bubbles relative to the post-bubble periods, and there is strong evidence of contagion over the whole sample and in the post-2021 bubble period. To illustrate their impact, optimal weights, volatility, and expected shortfall of a global minimum variance portfolio are examined. While volatility is higher during bubbles, the expected shortfall is stronger in the post-bubble periods. My results provide useful information for risk management and derivative pricing.

K-Covariance: An Approach of Ensemble Covariance Estimation and Undersampling to Stabilize the Covariance Matrix in the Global Minimum Variance Portfolio

A covariance matrix is an important parameter in many computational applications, such as quantitative trading. Recently, a global minimum variance portfolio received great attention due to its performance after the 2007–2008 financial crisis, and this portfolio uses only a covariance matrix to calculate weights for assets. However, the calculation process of that portfolio is sensitive with outliers in the covariance matrix, for example, a sample covariance matrix estimation or linear shrinkage covariance matrix estimations. In this paper, we propose the use of an undersampling technique and ensemble learning to stabilize the covariance matrix by reducing the impacts of outliers on the output of a covariance estimation. Experimenting on an emerging stock market using three performance metrics shows that our approach significantly improves the sample covariance matrix and also a linear shrinkage to the single-index model to a level of two shrinkage estimations, a shrinkage to identity matrix and shrinkage to constant correlation model.

A Bayesian Graphical Approach for Large-Scale Portfolio Management with Fewer Historical Data

Managing a large-scale portfolio with many assets is one of the most challenging tasks in the field of finance. It is partly because estimation of either covariance or precision matrix of asset returns tends to be unstable or even infeasible when the number of assets $p$ exceeds the number of observations $n$. For this reason, most of the previous studies on portfolio management have focused on the case of$ p < n$. To deal with the case of $p > n$, we propose to use a new Bayesian framework based on adaptive graphical LASSO for estimating the precision matrix of asset returns in a large-scale portfolio. Unlike the previous studies on graphical LASSO in the literature, our approach utilizes a Bayesian estimation method for the precision matrix proposed by Oya and Nakatsuma (2022) so that the positive definiteness of the precision matrix should be always guaranteed. As an empirical application, we construct the global minimum variance portfolio of $p = 100$ for various values of n with the proposed approach as well as the non-Bayesian graphical LASSO approach, and compare their out-of-sample performance with the equal weight portfolio as the benchmark. In this comparison, the proposed approach produces more stable results than the non-Bayesian approach in terms of Sharpe ratio, portfolio composition and turnover. Furthermore, the proposed approach succeeds in estimating the precision matrix even if $n$ is much smaller than $p$ and the non-Bayesian approach fails to do so.

Predicting the Global Minimum Variance Portfolio

We propose a novel dynamic approach to forecast the weights of the global minimum variance portfolio (GMVP) for the conditional covariance matrix of asset returns. The GMVP weights are the population coefficients of a linear regression of a benchmark return on a vector of return differences. This representation enables us to derive a consistent loss function from which we can infer the GMVP weights without imposing any distributional assumptions on the returns. In order to capture time variation in the returns’ conditional covariance structure, we model the portfolio weights through a recursive least squares (RLS) scheme as well as by generalized autoregressive score (GAS) type dynamics. Sparse parameterizations and targeting toward the weights of the equally weighted portfolio ensure scalability with respect to the number of assets. We apply these models to daily stock returns, and find that they perform well compared to existing static and dynamic approaches in terms of both the expected loss and unconditional portfolio variance.

Modeling and forecasting realized portfolio weights

We propose direct multiple time series models for predicting high dimensional vectors of observable realized global minimum variance portfolio (GMVP) weights computed based on high-frequency intraday returns. We apply Lasso regression techniques, develop a class of multiple AR(FI)MA models for realized GMVP weights, suggest suitable model restrictions, propose M-type estimators and derive the statistical properties of these estimators. In the empirical analysis for portfolios of 225 stocks from the S&P 500 we find that our direct models effectively minimize either statistical or economic forecasting losses both in- and out-of-sample as compared to relevant alternative approaches.

Portfolio Optimization Using a Consistent Vector-Based MSE Estimation Approach

This paper is concerned with optimizing the weights of the global minimum-variance portfolio (GMVP) in high-dimensional settings where both observation and population dimensions grow at a bounded ratio. Optimizing the GMVP weights is highly influenced by the data covariance matrix estimation. In a high-dimensional setting, it is well known that the sample covariance matrix is not a proper estimator of the true covariance matrix since it is not invertible when we have fewer observations than the data dimension. Even with more observations, the sample covariance matrix may not be well-conditioned. This paper determines the GMVP weights based on a regularized covariance matrix estimator to overcome the abovementioned difficulties. Unlike other methods, the proper selection of the regularization parameter is achieved by minimizing the mean squared error of an estimate of the noise vector that accounts for the uncertainty in the data mean estimation. Using random-matrix-theory tools, we derive a consistent estimator of the achievable mean squared error that allows us to find the optimal regularization parameter using a simple line search. Simulation results demonstrate the effectiveness of the proposed method when the data dimension is larger than, or of the same order of, the number of data samples.

Another look at portfolio optimization with mental accounts

Das et al. (2010, 2018)[11,12] numerically solve the portfolio optimization with mental accounts (POMA) problem, which maps the mean-variance theory and mean-variance utility into a behavioral portfolio theory. We derive a POMA closed-form solution based on the maximum Sharpe ratio and minimum value-at-risk (VaR) rule. The extension offers an alternate equivalence between the POMA problem, the mean-VaR model, and the generalized Sharpe measure. From the manageable VaR-measure perspective, our evidence indicates that many efficient portfolios are statistically equivalent to the global minimum variance portfolio under the estimation risk.

Reactive global minimum variance portfolios with k-BAHC covariance cleaning

We introduce a covariance cleaning method which works well in the very high-dimensional regime, i.e. when there are many more assets than data points per asset. This opens the way to unconditional reactive portfolio optimization when there are not enough points to calibrate dynamical conditional covariance models, which happens, for example, when new assets appear in a market. The method is a k-fold boosted version of the Bootstrapped Average Hierarchical Clustering cleaning procedure for correlation and covariance matrices. We apply this method to global minimum variance portfolios and find that k should increase with the calibration window length. We compare the performance of k-BAHC with other state-of-the-art covariance cleaning methods, including dynamical conditional covariance (DCC) with non-linear shrinkage. Generally, we find that our method yields better Sharpe ratios after transaction costs than competing unconditional covariance filtering methods, despite requiring a larger turnover. Finally, k-BAHC yields better Global Minimum Variance portfolios with long–short positions than DCC in a non-stationary investment universe.