INTRODUCTION
Vietnam stock market overview
The Vietnamese stock market has experienced significant growth over the past 20 years since the Ho Chi Minh City Stock Exchange commenced operations in July 2000, marked by the introduction of its first two tickers, REE and SAM Currently, there are 1,605 companies listed and registered across the two stock exchanges, with a trading volume of 150 billion shares As of early 2020, the market capitalization reached nearly 5.7 million billion VND, representing 102.74% of the country's GDP, highlighting the vital role of the Vietnamese stock market in the national economy.
Vietnam's stock market consists of three official exchanges: two listed exchanges, the Ho Chi Minh City Stock Exchange (HOSE) and the Hanoi Stock Exchange (HNX), along with one unlisted exchange, UPCoM Among these, HOSE is recognized as the largest exchange in terms of scale.
As of 2019, the Ho Chi Minh Stock Exchange (HOSE) featured 382 listed companies, with a trading volume of 8.8 billion shares and an average trading value exceeding VND 4,000 billion per session HOSE's market capitalization represented 88% of the total market, accounting for 54.3% of Vietnam's GDP Companies seeking to list on HOSE must meet stringent criteria, including a minimum charter capital of VND 120 billion, significantly higher than the VND 30 billion required on the Hanoi Stock Exchange (HNX) Additionally, these enterprises must have operated as joint-stock companies for at least two years prior to their listing registration.
The listing requirements on the HOSE (Ho Chi Minh Stock Exchange) are more stringent compared to the HNX (Hanoi Stock Exchange) HOSE mandates that listed companies must demonstrate profitable business activities for the past two years, which is one year longer than HNX's requirement Additionally, HOSE requires a minimum of 300 non-major shareholders who collectively hold at least 20% of the company's voting stock, while HNX requires only 100 shareholders holding at least 15% Furthermore, HOSE enforces higher standards for information disclosure, necessitating that companies reveal all debts to internal parties, major shareholders, and related individuals.
Vietnam's stock market has seen significant growth, increasing from just 3,000 trading accounts in 2000 to 2.5 million accounts today Notably, there are approximately 33,000 accounts held by foreign organizations and individuals, representing a total securities value of nearly USD 35 billion as of June 30.
2020 During this period, many foreign fund management companies also joined Vietnam's stock market Investment results show that investment funds in the period of
2009 - 2019 have relatively good investment results compared to the average growth rate of Vietnam's stock market (Figure 1.1)
Figure 1.1: The performance of investment funds in the period of 2009 – 2019
Between 2017 and 2019, Vietnam's stock market encountered significant challenges due to the complexities of the US-China trade war and the recession of major global economies These factors resulted in disappointing investment outcomes for both domestic and foreign investment funds, with their portfolio values declining even more sharply than the overall market downturn.
Figure 1.2: The performance of investment funds in the period of 2017 – 2019
Research by Brinson, Singer, and Beebower (1991) highlights that asset allocation significantly influences investment outcomes, accounting for 91.5% of portfolio results, whereas factors like security selection and timing contribute only 9% In Vietnam's stock market, the concepts of asset allocation and optimal portfolio selection are still emerging and encounter several challenges.
V E IL VOF V CBF -T CF F T S E S S IA M V N I nde x V F M V F 1 PYN V CB F -BC F VFM V N 30 V F M V F 4 V N 30 Inde x S S I-S CA V a n E ck
Figure 1.3: Determinants of portfolio performance From Brinson et al (1991)
The use of quantitative methods for asset allocation and optimal portfolio selection is relatively new among individual investors in Vietnam's stock market Most investors rely on fundamental and technical analysis to make stock selections, often basing their investment portfolios on personal feelings or subjective judgments rather than quantitative approaches While a small number of individual investors do utilize quantitative models for portfolio optimization, these models tend to be traditional and have significant limitations in their applicability.
The unique characteristics of the Vietnam stock market pose challenges for investors, particularly for investment funds seeking to utilize quantitative models for optimal portfolio selection One significant issue is the data quality; despite 20 years of market development, the initial phase featured a limited number of companies and unreliable information This lack of robust data impacts both the quantity and reliability of historical market insights.
Asset AllocationSecurity SelectionMarket TimingOther Factors
Modern portfolio optimization methods demand extensive and reliable research data, yet investors in the Vietnamese stock market face challenges due to regulatory constraints Daily trading limits on HOSE (±7%), HNX (±10%), and UPCoM (±15%) restrict stock price fluctuations, which can stabilize investor sentiment during market volatility However, these regulations hinder the accurate reflection of market events in stock prices, complicating future portfolio fluctuation predictions for investors.
The Vietnam stock market imposes significant delays in stock transactions, requiring investors to wait three business days (T+3) after a purchase before selling, and an additional two business days (T+2) before initiating new buying transactions or receiving interest payments These delays increase risks and transaction costs for investors, complicating the implementation of high-frequency trading models Consequently, when developing optimal portfolio selection models, it is essential for investors to account for these restrictions to ensure practical applicability and effectiveness.
Liquidity risk significantly impacts the practical application of portfolio optimization models A small market size and limited daily trading volumes create high liquidity risks for investors, particularly when executing large buy or sell orders Therefore, when developing and back-testing quantitative models, it is crucial for investors to consider slippage in stock trading activities, especially for high-volume transactions within their portfolios Failing to account for this factor can lead to discrepancies between theoretical and actual buying and selling prices.
7 could be significantly different from the actual buying and selling prices, which in turn affects the reliability of the optimal portfolio selection models
Choosing the right portfolio optimization method is crucial for investors in the Vietnam stock market This article will explore effective strategies for selecting optimal investment portfolios, with a specific focus on the unique dynamics of the Vietnamese market.
Problem statements
Modern Portfolio Theory (MPT), introduced by Harry Markowitz in 1952, has significantly influenced investment portfolio selection and construction for over 65 years Its primary objective is to maximize returns for a given level of risk by optimizing the asset weights within a portfolio Despite its widespread application in investment practices, MPT's foundational assumptions have encountered substantial challenges in recent years, particularly concerning the reliance on the mean and covariance matrix of asset returns.
To effectively apply the mean-variance technique, investors must estimate the mean and covariance matrix of asset returns, typically using sample mean and covariance approaches However, these estimators often suffer from instability due to estimation errors, leading to continuous fluctuations in portfolio weights Consequently, portfolio managers find it challenging to implement mean-variance portfolios in practice Furthermore, numerous empirical studies, such as Michaud (1989), have demonstrated that these portfolios tend to underperform in terms of mean and variance metrics during out-of-sample periods.
To address the challenges of Modern Portfolio Theory (MPT), two key strategies can be employed: developing new methodologies for estimating the expected return and covariance matrix of assets in portfolio optimization Prominent models, including the Capital Asset Pricing Model (CAPM) and the Fama-French model, are commonly used to estimate the expected return parameters.
The Capital Asset Pricing Model (CAPM) is a one-factor model that establishes a relationship between systematic risk and the expected return of assets However, the Fama-French model suggests that expected returns should also consider additional variables beyond the CAPM's beta coefficient, including size risk, value risk, profitability, and investment factors To enhance the accuracy of expected return estimations, researchers and portfolio managers utilize robust estimators such as truncated/trimmed means or winsorized means, as demonstrated by Martin, Clark, and Green in 2010 Furthermore, advanced robust estimation methods like M-estimators, S-estimators, and Bayes-Stein estimators have been developed to address the challenges posed by non-stationary returns in expected return calculations.
Improving the estimation of expected returns on assets addresses some limitations of Modern Portfolio Theory (MPT) However, Merton's research (1980) indicates that accurately measuring expected returns is challenging Many asset pricing models assume a constant relationship between an asset's expected return and the market's expected return, simplifying the estimation process but requiring extensive time series data for accuracy This assumption of constant expected returns is often unrealistic, and relaxing it complicates the estimation further Consequently, a promising alternative research direction involves selecting portfolios based on covariance matrix estimation rather than expected returns Recent research has focused on estimating covariance matrix parameters, highlighting its potential to enhance stability and reduce risk in investment portfolio selection.
The instability of mean-variance portfolios arises from the challenges in accurately estimating mean asset returns Consequently, minimum-variance portfolios have gained popularity among researchers and portfolio managers This approach focuses primarily on estimating the covariance matrix of the assets, which reduces sensitivity to return fluctuations.
Jagannathan and Ma (2003) highlighted significant estimation errors in financial models, suggesting that the sample mean's estimation error is so substantial that it may be disregarded entirely Their findings, supported by empirical evidence, indicate that minimum-variance portfolios tend to outperform traditional mean-variance portfolios in terms of Sharpe's ratio and other performance metrics during out-of-sample periods (DeMiguel, 2005; Jagannathan and Ma, 2003).
According to Demiguel (2009), the minimum-variance portfolio is significantly influenced by estimation errors, despite not relying on mean return estimations This sensitivity to estimation error highlights a crucial aspect of portfolio management.
Portfolios constructed using the sample covariance matrix rely on the maximum likelihood estimator (MLE) for normally distributed returns, which is theoretically the most efficient estimator with the smallest asymptotic variance when the data adheres to the normality assumption However, Huber (2004) highlights a critical issue: the efficiency of MLEs is highly sensitive to any deviations from this normal distribution Consequently, MLEs may not be the most efficient choice when asset-return distributions slightly diverge from normality This insight is particularly relevant for portfolio selection, as extensive evidence suggests that empirical return distributions often differ significantly from normal distributions.
The effectiveness of minimum-variance portfolio research hinges on accurately estimating the covariance matrix However, traditional methods like the sample covariance matrix (SCM) and ordinary least squares (OLS) encounter significant challenges in high-dimensional portfolios Increased dimensionality heightens the risk of unexpected errors during computation, and insufficient sample data can hinder the estimation of the true covariance matrix, resulting in unreliable estimates.
The covariance matrix often becomes ill-conditioned or singular, leading to poor portfolio performance and profit generation To address this issue, researchers and portfolio managers have developed new covariance matrix estimators Notably, Ledoit and Wolf (2003) introduced a shrinkage estimator that combines a rough sample covariance matrix with a structured target matrix, allowing for a customizable balance between bias and variance through shrinkage coefficients This shrinkage technique offers a robust solution for high-dimensional portfolio covariance estimation, ensuring a well-defined covariance matrix Liu (2014) further enhanced this by employing a weighted average of multiple shrinkage target matrices Building on Random Matrix Theory, Ledoit and Wolf (2017a, 2017b) applied a nonlinear transformation to eigenvalues derived from sample data, maximizing out-of-sample expected utility Their numerical and empirical investigations demonstrated significant improvements over simple diversification, showing robustness against deviations from normality Additionally, DeMiguel et al (2013) reviewed shrinkage frameworks for asset optimization and introduced new shrinkage-based techniques for return means and covariance matrices Candelon et al (2012) contributed to this research by proposing a double shrinkage adaptation to enhance stability.
11 estimation on even small sample sizes covariance matrices via taking into account a ridge regression approach to shrink the all the weights towards the equally-weighted asset
The choice of covariance matrix estimators significantly affects the performance of optimized portfolios Investors who typically rely on the traditional sample covariance matrix can enhance their portfolio outcomes by incorporating alternative covariance matrix estimators into their optimization models However, the lack of comprehensive research on the out-of-sample performance of these new estimators creates uncertainty Consequently, portfolio managers may hesitate to invest based on unverified or insufficiently rigorous research, limiting their willingness to adopt these innovative approaches.
The traditional covariance matrix estimator struggles to deliver accurate results due to the rapid increase in the number of investment assets in financial markets, often exceeding the available sample size This situation necessitates the exploration and application of new covariance matrix estimators Additionally, there remains significant debate regarding the applicability and effectiveness of various covariance matrix estimation methods across different markets.
Robust estimators of the covariance matrix have primarily been studied in developed markets, with limited research conducted in emerging and developing financial markets, particularly in Vietnam There is a notable lack of studies focusing on the selection of covariance matrix estimators for portfolio optimization in this region, especially concerning shrinkage methods This presents an opportunity for further investigation into how different covariance matrix estimators impact minimum-variance optimized portfolios and to assess their performance within the Vietnamese stock market.
Objectives and research questions
This dissertation aims to explore whether investors can enhance the performance of minimum-variance optimized portfolios by modifying the estimators of the covariance matrix input Additionally, it will identify the most suitable covariance matrix estimators for portfolio optimization in the Vietnam stock market based on out-of-sample portfolio performance metrics.
In order to achieve the above objectives, this dissertation will attempt to answer the research questions as follows:
Question 1: How do the robust estimators of covariance matrix perform on out – of – sample performance metrics such as portfolio return, level of risk, portfolio turnover, maximum drawdown, winning rate and Jensen’s Alpha in selecting minimum – variance optimized portfolios?
Question 2: How do the estimators of covariance matrix affect the out – of – sample performance of minimum – variance optimized portfolios when the number of assets in the portfolio changes?
Question 3: Could the alternation of covariance matrix estimation for portfolio optimization beat the traditional estimator of covariance matrix and benchmarks of stock market on out - of - sample?
Research Methodology
To effectively meet the research objectives and address the outlined questions, the author must select a suitable research method In this dissertation, various research methodologies have been employed to achieve these aims.
This study investigates the impact of different covariance matrix estimators on optimal portfolio selection, utilizing six key methods: the sample covariance matrix (SCM), the single index model (SIM), the constant correlation model (CCM), and the shrinkage towards single index model.
The article discusses three shrinkage methods for estimating covariance matrices: the Shrinkage Covariance Matrix (SCM), the Shrinkage towards Constant Correlation Model (SCCM), and the Shrinkage towards Identity Matrix (STIM) SCM is regarded as the traditional estimator, while SCCM and STIM represent model-based approaches The minimum-variance optimization technique is employed to create optimal portfolios based on the covariance matrices estimated by these methods.
To assess the feasibility and potential applications of the covariance matrix estimators discussed, this study implemented a back-testing process using Python This process was modeled after the back-testing platform established in the previous research by Tran et al (2020) Through this back-testing procedure, the statistical properties of the covariance matrix estimators will be analyzed, offering insights into their real-world profitability.
In this study, the back-testing process is utilized to estimate key portfolio performance metrics essential for portfolio evaluation Alongside fundamental criteria such as portfolio return and volatility, additional metrics including portfolio turnover, maximum drawdown, winning rate, and Jensen’s Alpha are calculated The author employs a "rolling-horizon" technique, which is a reactive scheduling method that iteratively optimizes the portfolio by adjusting the optimization horizon as new, uncertain parameters become known This approach enables investors to refine their input data for optimal portfolio selection based on the most current information The input data for the back-testing process consists of weekly stock price series, which are subsequently transformed into weekly returns.
14 return during the optimization procedure One more thing, when calculating the portfolio performance metrics, the transaction costs are also considered at every rebalancing point
In conclusion, the estimated performance metrics are utilized to compare the variances among different covariance matrix estimators for optimal portfolio selection To ensure that significant differences exist between the two specific estimators, p-values are calculated using the bootstrapping methodology outlined in DeMiguel's 2009 research.
Contributions of the research
After answering the research questions and achieving the research objective, this dissertation will expect to make some contributions as follows:
Empirical research on the Vietnamese stock market demonstrates that investors can enhance their portfolio performance by utilizing estimation methods to adjust the covariance matrix parameter in portfolio optimization The findings indicate that model-based estimators (SIM, CCM) and shrinkage estimators (SSIM, SCCM, STIM) significantly outperform the traditional sample covariance matrix (SCM) across nearly all tested portfolios (N = 50, 100).
Research indicates that portfolio performance metrics, including portfolio return, risk level, portfolio turnover, maximum drawdown, winning rate, and Jensen’s Alpha, demonstrate significant superiority, particularly as the number of stocks in the portfolio increases.
In this dissertation, shrinkage estimators of the covariance matrix demonstrate superior performance compared to other estimators and market benchmarks across various portfolio evaluation criteria, particularly for high-dimensional portfolios Notably, the shrinkage towards the constant correlation model (SCCM) indicates the most efficient level for achieving optimal portfolio performance.
15 selection compared to the shrinkage towards single index model (SSIM) and shrinkage towards identity matrix (STIM)
The dissertation introduces a novel perspective by examining how the performance of covariance matrix estimators is affected by the dimensionality of the covariance matrix and the impact of transaction costs on out-of-sample portfolio performance It specifically evaluates the effectiveness of various estimation methods as the number of stocks in the portfolio varies from N = 50 to N = 350, while incorporating a transaction cost of 0.3% at each rebalancing point.
The dissertation enhances the evaluation of portfolio effectiveness by utilizing a diverse set of performance metrics, including the Sharpe ratio, portfolio turnover, maximum drawdown, winning rate, and Jensen’s Alpha This multi-dimensional approach allows for a more comprehensive assessment of estimation methods in selecting the optimal portfolio, moving beyond traditional metrics such as return and variance used in previous research This effort underscores the author's commitment to analyzing the efficacy of covariance matrix estimation methods in comparison to earlier studies.
This dissertation's key contribution is the evaluation of covariance matrix estimators within the Vietnam stock market, an emerging market While numerous researchers and financial professionals have applied these estimation methods to optimize investment portfolios in developed markets like the US and Europe, there is a notable scarcity of studies focusing on emerging markets, particularly Vietnam The empirical findings of this research will provide valuable insights for researchers and investors, highlighting the performance differences of covariance matrix estimators between emerging and developed financial markets.
Disposition of the dissertation
This dissertation comprises five chapters, with Chapter 1 outlining the problem statements, study objectives, research questions, research methodology, and the contributions of the research The subsequent chapters are structured accordingly.
Chapter 2, Literature Review, provides an overall review about relevant researches regarding portfolio optimization before developing a specific theoretical framework and methodology in the next chapter
Chapter 3, Theoretical Framework, establishes the foundational theory for this dissertation by first introducing essential preliminaries and the portfolio optimization problem, followed by an exploration of covariance matrix estimation theories.
Chapter 4, Methodology, outlines the fundamental approach employed to address the research questions in this dissertation It provides a clear presentation of the input data, the portfolio performance evaluation methodology, and the relevant performance metrics.
Chapter 5 presents the empirical results derived from back-testing the performance of covariance matrix estimators on out-of-sample data The findings of this research are analyzed, leading to key conclusions about the effectiveness of these estimators Additionally, the chapter discusses potential avenues for future research, highlighting areas for further exploration and development in this field.
LITERATURE REVIEW
Modern Portfolio Theory Framework
Harry Markowitz is a key figure in financial economics, renowned for his pioneering contributions to portfolio selection theory He was awarded the Nobel Prize in 1990 for his significant work in this area His influential article "Portfolio Selection," published in 1952 in "The Journal of Finance," laid the foundation for modern portfolio theory (MPT), which he further expanded upon in his 1959 book, "Portfolio Selection: Efficient Diversification."
The Modern Portfolio Theory (MPT) is an investment strategy aimed at maximizing expected returns for a specific level of risk or minimizing risk for a desired return by strategically allocating asset weights Despite its widespread use in the investment industry, the foundational assumptions of MPT have faced growing criticism in recent years.
The Modern Portfolio Theory (MPT) enhances classical quantitative models and is essential in financial mathematical modeling It emphasizes diversification to safeguard investment portfolios against market and specific company risks Often referred to as Portfolio Management Theory, it aids investors in classifying, evaluating, and measuring expected risk and return Central to this theory is the quantification of the relationship between risk and return, asserting that investors deserve compensation for the risks they undertake.
The concept of diversification in Modern Portfolio Theory (MPT) involves selecting investment portfolios that exhibit lower risk than any individual security within them This strategy effectively reduces investment risk, regardless of whether the correlation between the returns of the securities is positive or negative.
According to Modern Portfolio Theory (MPT), a security's return is viewed as a normally distributed function, while risk is defined as the standard deviation of that return The overall return of a portfolio is calculated as a weighted combination of the returns of its individual securities Additionally, the total variance of the portfolio return decreases when the correlations among the securities' returns are not perfectly positive MPT also assumes that investors act rationally and that the market operates efficiently.
Investing involves balancing expected returns against associated risks, with higher expected returns generally indicating greater risk (Taleb, 2007) The Modern Portfolio Theory (MPT) offers a framework for selecting a portfolio that maximizes expected returns for a specific risk level, while also identifying the lowest risk portfolio for a given expected return Expected return refers to the forecasted or estimated return on an investment, while risk is defined as the unpredictability of future returns, highlighting the potential for actual profits to differ from expectations Investments characterized by higher return volatility are considered riskier compared to those with lower volatility.
2.1.1 Assumptions of the modern portfolio theory
Modern portfolio theory, developed by Markowitz, is built on several key assumptions about investors and markets It posits that all investors are identical, exhibiting rational behavior and averse to risk.
Investors aim to optimize their portfolios by minimizing risk while maximizing expected returns, selecting them based on anticipated returns and the variance of those returns They assume that asset returns remain stationary over time and have complete knowledge of asset prices, allowing for immediate and costless portfolio adjustments in response to price changes Additionally, asset prices are considered exogenous, meaning investor choices do not influence them, and all investment assets are infinitely liquid, enabling trades of any size Furthermore, investors have the option to engage in short selling, allowing for a more flexible investment strategy.
Investors can engage in borrowing and lending without facing risk, maintaining a consistent interest rate Additionally, they do not incur any transaction costs such as taxes, brokerage fees, bid-ask spreads, or foreign exchange commissions Ultimately, investors allocate their entire budget to their portfolio, with no portion set aside for savings.
The Modern Portfolio Theory (MPT) relies on several premises, both explicit and tacit, which influence its validity Explicit assumptions include the use of normal distributions for model returns, while tacit ones encompass tax indifference and transaction fees However, none of these premises are entirely accurate, leading to compromises in the effectiveness of MPT A fundamental assumption of the MPT is that market theory operates efficiently.
According to Fabozzi et al (2002), the primary function of Modern Portfolio Theory (MPT) is asset allocation Investors must first assess potential investment assets and any associated restrictions Subsequently, they need to estimate the returns, correlations, and volatility of the investable securities These predictions are then utilized in an optimization process to achieve a final result that aligns with individual expectations.
Investor Objectives Figure 2 1: MPT investment process From Fabozzi, F., Gupta, F., & Markowitz (2002)
The Modern Portfolio Theory (MPT) has faced significant criticism for its simplistic assumptions and questionable effectiveness as an investment strategy, as many argue that its financial market model does not accurately reflect reality Recent challenges from behavioral economics have further scrutinized the MPT's foundational principles Additionally, practical applications of the MPT in constructing optimal portfolios encounter difficulties due to unstable input parameters in the optimization process However, recent studies indicate that these instabilities can be mitigated by incorporating a "regularizing constraint" or "penalty term" into the optimization framework.
The market is not really modeled by the MPT
Modern portfolio theory relies on metrics such as risk, return, and correlation, which are based on "ex-ante" values that include mathematical assumptions about future performance Investors often need to substitute these forecasts with historical data on asset returns and volatility, as unexpected future events can render these predictions inaccurate This reliance on past data complicates the estimation of key parameters, as MPT models risk without addressing the underlying causes of potential losses Unlike other engineering approaches to risk management, MPT lacks structural and probabilistic measures in its risk assessments, highlighting a significant gap in its methodology.
The personal, environmental, strategic, or social dimensions of investment decisions are not considered in this theory
The principle aims only to optimize risk-adjusted returns regardless of other consequences More specifically, its full reliance on asset prices makes the MPT more
Markets can be vulnerable when they do not align with standard norms due to issues like information asymmetry, externalities, and public goods Additionally, businesses may pursue strategic or social objectives that influence their investment choices, while individual investors may have personal goals In these scenarios, understanding factors beyond historical returns becomes crucial, as highlighted by Modern Portfolio Theory (MPT).
The MPT does not take cognizance of its own effect on asset prices
Diversification reduces unsystematic risk but increases systemic risk, leading investors to select securities without thorough fundamental analysis This focus on minimizing unsystematic risk can inflate demand and prices of securities that may not hold individual value As a result, the overall cost of the portfolio rises, diminishing the likelihood of positive returns and ultimately contributing to increased portfolio risk.
Parameter estimation
Mean-covariance optimization is often criticized for its estimation error, as highlighted in Markowitz's (1952) study, which prioritized the theoretical foundations of portfolio selection over practical applications To realistically implement mean-variance optimization (MTP), accurate estimates of asset return means and covariances are essential, as these values are not known a priori Subsequent studies, including those by Elton et al (2012), indicate that these estimated values are used to solve investor optimization problems However, significant research, such as that by Michaud (1989) and Chopra and Ziernba (1993), reveals a critical drawback: the potential for estimation errors when inappropriate statistical moments are used This oversight occurs because the optimizer may not recognize that the data inputs are merely statistical estimates, leading to flawed decision-making.
The traditional method for estimating asset returns and covariance relies on ex-post returns to derive sample estimates, based on the assumption that historical data can provide insights into future asset price trends However, research has highlighted significant issues with this approach DeMiguel (2009) found that using sample estimates as input does not guarantee that mean-variance optimized portfolios will outperform equally-weighted portfolios Similarly, Jobson and Korkie (1980) presented related findings, while Best and Grauer (1991) emphasized that estimation errors can distort the weights of optimized portfolios, leading to discrepancies between estimated optimal weights and actual values.
Chopra and Ziemba (1993) posited that errors in expected returns significantly influence the out-of-sample performance of optimal portfolios more than errors in the covariance matrix They suggested this disparity in focus on expected return vectors versus covariance matrices could explain historical trends in investment strategies However, Michaud (2012) contested their findings, highlighting that widespread estimation errors in research emphasize the importance of the covariance matrix in return analysis Michaud argued that Chopra and Ziemba's study was limited to in-sample data, which inadequately addressed the effects of estimation errors on out-of-sample mean-variance optimization Furthermore, they concluded that as the number of assets increases, errors in the covariance matrix may play a more dominant role in the portfolio optimization process.
Estimating errors play a crucial role in mean-variance optimization (MVO), making it essential to address these inaccuracies for achieving a more effective optimized portfolio during out-of-sample periods Reducing estimation errors is vital, as it greatly benefits asset managers utilizing MVO strategies.
Numerous scholars have highlighted the crucial role of predicting the covariance matrix in Mean-Variance Optimization (MVO), leading to a significant body of literature dedicated to forecasting asset returns Consequently, there is a growing demand for comprehensive research that compares the outcomes of various forecasting methods.
This article provides an overview of various solutions for forecasting expected returns while also presenting a literature review focused on methods for reducing estimation errors in the covariance matrix.
The Capital Asset Pricing Model (CAPM), introduced by William Sharpe in 1964, is a one-factor model that forecasts the expected return of assets based on the relationship between systematic risk and expected return However, in 1992, Eugene Fama and Kenneth French found that the beta coefficient in CAPM did not accurately reflect the expected returns of American securities from 1963 to 1990 They identified two stock classes, small caps and those with a high Book to Market Equity ratio, that consistently outperformed the market This led to the development of the three-factor model in 1993, which incorporated these stock classes along with market returns, size risk, and value risk, proving effective in both developed and emerging stock markets In 2014, Fama and French expanded the model to include two additional factors—profitability and investment—resulting in a five-factor model that enhances predictions of expected return movements.
Moreover, the researchers also employed robust estimators to improve the expected return estimation For examples, instead of using the sample mean m ∑ to
To estimate expected returns in a population of 24, researchers can utilize the trimmed or winsorized mean, as suggested by Martin et al (2010) The trimmed mean is derived by excluding the k% most extreme values, while the winsorized mean replaces these extreme values with the next k% most extreme values Both methods offer enhanced efficiency when dealing with deviations from distributional assumptions regarding asset returns However, if the returns are non-stationary, these estimators, along with other robust methods like M-estimators or S-estimators, may fail to yield accurate estimates of expected returns.
To enhance performance amid non-stationary returns and automate the selection of tuning parameters, researchers have adopted Shrinkage Estimators This approach recognizes that both uncertainty in actual expected returns and estimation risk can lead to a decrease in investor utility Therefore, the optimization process should focus on minimizing utility loss from portfolio selection based on sample estimates rather than true values Instead of estimating each asset’s expected return individually, the goal is to choose an estimator that reduces utility loss from aggregate parameter uncertainty Jorion (1986) recommends employing a Bayes–Stein estimator, which adjusts each asset’s sample mean towards the overall grand mean.
The simulation conducted by this estimator effectively reduces portfolio risk and outperforms traditional portfolio constructions While enhancing the estimation of assets' expected returns is recognized as a key strategy to address the shortcomings of Modern Portfolio Theory (MPT), Merton's research further supports this approach.
Estimating the expected return on assets is challenging, as many pricing models assume a consistent relationship between expected returns and the market over time This limitation leads to an alternative research approach that focuses on selecting portfolios based on covariance matrix estimation rather than relying solely on expected return estimation.
Local studies in Vietnam have primarily focused on optimizing investment portfolios by estimating expected return parameters Phuong Nguyen (2012) utilized the single-factor model (SIM) to assess risks and determine expected returns for stocks in the construction sector Similarly, Linh Ho (2013) applied this model to evaluate risks and expected returns of real estate stocks listed on the Ho Chi Minh Stock Exchange (HOSE) Additionally, Truong and Duong (2014) employed the Capital Asset Pricing Model (CAPM) and the Fama-French three-factor model to enhance investment portfolio optimization on HOSE.
In 2014, researchers utilized the Fama-French three-factor model to assess risks and forecast expected returns of stocks within the Vietnamese stock market Additionally, Nguyen Tho (2010) implemented the arbitrage pricing theory (APT) to analyze stock price movements in emerging markets, specifically focusing on Vietnam and Thailand.
Recent research has increasingly focused on improving stability and minimizing risks in investment portfolio selection, particularly through enhanced methods of covariance matrix estimation Traditional approaches, such as the sample covariance matrix (SCM), have been criticized for their shortcomings, as highlighted by Michaud (1989), who noted that these methods can produce significant statistical errors and become ill-conditioned when the number of samples is similar to the number of assets, a phenomenon he termed the "Markowitz enigma." Additionally, Frankfurter, Phillips, and Seagle (1971) found that the SCM does not outperform an equally weighted portfolio, which DeMiguel (2009) referred to as the naïve 1/N portfolio.
Utilizing the single-index model proposed by Sharpe (1964), researchers estimated the covariance matrix to identify the optimal portfolio The covariance matrix derived from this one-factor model is calculated using a specific formula The single-index model (SIM) offers three key advantages over the traditional Modern Portfolio Theory (MPT) approach, primarily because the covariance matrix is estimated based on sample data Notably, SIM simplifies the process by requiring the estimation of only 2N+1 parameters to construct the covariance matrix.
Portfolio Selection
The mean-variance model is a widely used approach for portfolio selection, allowing investors to optimize asset weights based on the mean and covariance matrix of asset returns However, the reliance on estimated sample means and covariance matrices often leads to significant estimation errors, particularly in sample means As a result, many investors prefer the global minimum-variance model, which focuses solely on the covariance matrix for determining asset weights Consequently, the literature on portfolio selection has diverged into two main streams: those adhering to the traditional mean-variance model and those embracing the more recent global minimum-variance model.
The return on securities is considered as a “random variable with Gaussian distribution” under the standard mean-variance model introduced by Markowitz (1952) Accordingly,
The normal (Gaussian) distribution posits that asset returns rely solely on mean and variance, a concept pioneered by Markowitz in 1959 with his mean-variance model for portfolio selection Two decades later, Merton expanded on this model by incorporating short-sale options in portfolio selection Over the years, these foundational models have been utilized in numerous studies, as summarized in Table 2.1, which highlights significant research in this area.
Table 2.1 : Summarized works related to portfolio optimization Author Year Paper/book/Thesis Title Summarized works
Markowitz 1952 “Portfolio Selection” Standard mean – variance model
Samuelson 1969 “Lifetime portfolio selection by dynamic stochastic programming”
Applying a discrete time multi–period model
“Lifetime portfolio selection under uncertainty: The continuous – time case”
“An extension of the Markowitz portfolio selection model to include variable transaction costs, short sales, leverage policies and taxes”
Mean – variance portfolios with transaction costs
Merton 1972 “An analytic derivation of the efficient portfolio frontier”
Allowing short – sale in mean – variance portfolio selection
Gomez 2007 “Portfolio selection using neural networks”
Applying neural networks for portfolio optimization
(Source: Risk and Financial Management, 2019)
The traditional mean-variance model by Markowitz is static, as investors make decisions only at the beginning of the investment period and wait for it to conclude This limitation prompted the evolution of the mean-variance model, with Samuelson (1969) introducing a discrete time multi-period consumption investment model aimed at enhancing investors' final expected utility Furthermore, Merton (1969, 1971) contributed groundbreaking research on continuous time models, focusing on maximizing expected returns within a defined planning period.
Pogue (1970) introduced the mean-variance portfolio concept while accounting for transaction costs To align with real market conditions, Xue et al (2006) developed a mean-variance model that incorporates "concave transaction costs." More recently, Liagkouras and Metaxiotis proposed a multi-stage fuzzy portfolio optimization algorithm that also addresses transaction costs.
In 2018, advancements were made to enhance the traditional mean-variance model, which originally faced limitations Fernández and Gómez (2007) incorporated cardinality constraints to ensure specific asset investments while restricting allocations to individual assets Similarly, Soleimani et al (2009) further refined the model by adding minimum trading lot sizes and considering market capitalization, thereby improving its applicability in investment strategies.
2.3.2 Global Minimum Variance Model (GMV)
The Global Minimum Variance (GMV) portfolio represents an optimized investment strategy on the efficient frontier, specifically designed to minimize variance Research by Haugen and Backer (1991) assessed the efficiency of capitalization-weighted portfolios, revealing that these portfolios often fall short of efficiency unless under highly restrictive conditions, even in an efficient market They concluded that investors rationally optimize the trade-off between risk and expected return Additionally, Chopra and Ziemba (1993) demonstrated that employing the GMV model can significantly reduce errors in variance and covariance calculations, yielding results up to ten times more accurate compared to traditional expected return methods.
Research has shown that the Global Minimum Variance (GMV) portfolio outperforms the traditional mean-variance model established by Markowitz (Chan et al., 1999) Jagannathan and Ma (2003) highlighted that GMV portfolios exhibit more stable asset weights due to reduced estimation errors in covariance compared to the mean-variance approach Additionally, Kempf and Memmel (2006) supported the notion that GMV portfolios yield superior out-of-sample results compared to tangent portfolio theory Furthermore, DeMiguel and Nogales (2009) noted that the GMV model's reliance solely on covariance matrices makes it less susceptible to estimation errors than conventional models.
Recent research has highlighted the growing popularity of the Global Minimum Variance (GMV) portfolio as an effective tool for optimizing investment portfolios The GMV density function, developed by Okhrin and Schmid (2006), enhances the understanding of portfolio weight distributions Additionally, Clarke et al (2006) emphasized that stock weights on the left side of the effective boundary in the minimum variance model are unaffected by expected safe returns Consequently, to achieve an optimized portfolio, it is essential to eliminate the equilibrium expectation and focus solely on the covariance matrix.
Research indicates that for optimizing investment portfolios, estimating the covariance matrix is more effective than predicting expected returns, as it is less susceptible to estimation errors The shrinkage model offers a more accurate covariance matrix compared to traditional estimators by leveraging their strengths and mitigating their weaknesses Various target matrices, both linear and non-linear, have been tested for effectiveness across different market conditions to identify the most superior approach The next phase involves applying the estimated covariance matrix to variance models to determine the optimal portfolio, with mean-variance and global minimum-variance models being the most commonly utilized by researchers and portfolio managers Historical studies suggest that the global minimum-variance model tends to outperform other methods.
THEORETICAL FRAMEWORK
Basic preliminaries
This region includes a short overview to fundamental principles and techniques for optimizing minimum variance
The return of a stock, denoted as R, represents its gain or loss over a specific time period For a stock priced at time t, the return during a non-dividend period from t to T (where T is greater than t) can be calculated to assess the stock's performance.
With respect to a stock which pay dividends in the term [t, T], the return is computed as:
At time t, the effects of certain factors on stock returns are unknown, making the return of a stock a random variable The estimated outcome of this random variable can be expressed as E[Return].
A portfolio can consist of several stocks when determining the portfolio allocation, where the weight of stock i, , in the portfolio is computed respectively:
Thus, we have this for an asset universe of size n:
The expected return of this kind of portfolio, , - shall then be calculated by:
E[ ] = E[∑ ] = ∑ , ] = ∑ = à Here, = [ ,…, ] and à = , ,…, ] T , these are vector notations where w, à є R nx1
Variance plays a crucial role in minimum variance optimization, serving as the primary statistic for measuring the risk and volatility of asset returns It is calculated by evaluating the expected squared differences from the mean For a portfolio containing n assets, the overall variance can be determined using a specific formula that accounts for the asset universe.
In short, the portfolio volatility is indicated:
Portfolio Optimization
Mean-variance optimization, rooted in Markowitz's traditional portfolio theory, faces significant challenges due to its sensitivity to estimation errors in asset return means and covariance matrices (Jorion, 1985) Research indicates that the accuracy of covariance matrix estimation surpasses that of expected returns, with studies demonstrating that minimum-variance portfolios often outperform other mean-variance strategies (Jorion, 1986; Jagannathan and Ma, 2003) This dissertation focuses on enhancing the efficiency of the global minimum variance portfolio (GMVP), which relies solely on covariance matrix assessments.
In a portfolio consisting of N risky assets with weights represented as w = (w1, w2, , wN), it is essential that the weights sum to one, denoted by the constraint w'1 = 1, where 1 is the vector of ones Furthermore, the condition w > 0 indicates that short selling is not permitted, aligning with the regulations of the Vietnam stock market The expected return of the portfolio is calculated using the formula ̂ = ∑ ̂ ̂, while the portfolio variance is determined by the equation ∑̂.
Portfolio optimization problem is solved with Markowitz’s MPT theory through linear programming following statistics function:
In which: “1 denotes a vector of ones, and Σ is the covariance matrix of N stocks” The theoretical approach to the problem (3.2.1) is feasible:
To determine the optimal weights for a minimum variance portfolio, the estimated covariance matrix is inputted into a quadratic optimization program The solution, derived from the inverse of the covariance matrix, is typically based on the sample covariance matrix However, this approach can be problematic, as sample covariance matrices are often ill-conditioned and may not be invertible, especially in high-dimensional portfolios To address this issue, employing shrinkage estimators to adjust the covariance matrix parameters can lead to improved solutions.
The estimators of covariance matrix
To effectively assess portfolio risk, investors must understand the risk levels of individual assets and their return correlations Covariance serves as a key metric for this analysis According to Ledoit and Wolf (2003b), the standard sample covariance matrix (SCM) is a preferred maximum likelihood estimator under normality assumptions, making it a "best-unbiased" estimation tool However, its reliance on data poses limitations, particularly in small sample sizes where its effectiveness diminishes.
Noise data can lead to overpowering results in financial analysis, as highlighted by various studies indicating that the Sample Covariance Matrix (SCM) performs inconsistently across different samples To address this issue, investors may consider expanding their sample data by increasing the viewing window; however, this approach can introduce redundant data that offers limited insights for future forecasting Additionally, when the number of assets under consideration significantly exceeds the number of historical observations, the sample covariance matrix risks becoming ill-conditioned, rendering it non-invertible This presents a critical concern for portfolio optimization, as noted by Bengtsson and Holst (2002).
3.3.1 The sample covariance matrix (SCM)
First, assuming denotes the historical return for asset i at time period t Then, the historical average returns in the period [1, T] of asset i ( ̅) will be determined as follows: ̅ = ∑ (3.3.1)
Next, the equation that is used to calculate the sample covariance between any two assets i, j is formulated:
From the equation (3.3.2), the sample covariance matrix ( ̂ ) that shows the relationship among N assets in the portfolio is identified as follows: ̂ [ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂
Moreover, the covariance matrix Σ will be determined according to:
Where: “σ denotes as a column vector of standard deviations; diag(σ) is a matrix with the elements of σ on the main diagonal; C is a correlation matrix”
Investors can derive various covariance matrices by altering the method used to calculate the correlation of asset returns in the correlation matrix C In this context, the estimation of σ, denoted as ̂, represents the sample standard deviation of historical asset returns It's important for investors to understand that ̂ remains unaffected by the chosen method of correlation estimation.
Therefore, the estimated covariance matrices obtained by historical correlation are:
An expression of is described by:
Where “ is the correlation between asset returns and The estimation of , denoted as ̂ , is computed by using the pairwise sample correlations of the historical asset returns”
3.3.2 The single index model (SIM)
The Single-Index Model (SIM), developed by Sharp in 1963, estimates asset returns and the covariance matrix by recognizing that the returns of individual assets are influenced by overall market performance This one-factor model aims to mitigate the volatility associated with asset returns by employing regression analysis to align asset returns with market returns The fundamental equation of SIM expresses asset returns in relation to market movements.
The estimated return of asset i at time i is represented by ̂, while ̂ denotes the estimated market return The model incorporates random error and employs OLS regression to derive the parameters α and β It assumes that the random error is independent, with Cov[ , ] = 0, and uncorrelated with the market return, represented by Cov[ ̂ , ].
= 0, the error must follow normal distribution Var[ ] = and E[ ] = 0
Thus, with the construction of asset returns based on SIM, the variance - covariance of asset returns and those of model is expressed as:
• The variance of estimated asset return i:
• The covariance of estimated asset return i, j:
• The estimated covariance matrix of SIM:
The estimated market variance, denoted as Var[ ̂ ], is derived from the vector of coefficients β obtained through SIM's regression analysis involving N assets Additionally, ̂ represents the regression error matrix, structured as an N x N diagonal matrix It is crucial to note that risks are indicated by variance, leading to the assumption that the market's variance must be greater than zero, expressed as ̂ > 0.
This study utilizes a single index model (SIM) with market return as the index, contrasting with Cohen and Pogue's (1967) approach, which focused on industry returns While the SIM can be expanded into a model with multiple asset factors, George Derpanopoulos (2018) argues that increasing complexity in the SIM tends to introduce more noise than valuable information, primarily because it becomes challenging to attribute stock covariances to factors beyond the market.
In 1895, Karl Pearson developed the Pearson Product-moment Correlation Coefficient (PPMCC), a method for measuring the correlation between variables This statistical tool is extensively utilized across various disciplines, including economics, mathematics, and the sciences.
Formula to compute Pearson correlations is:
The Pearson correlation is commonly used to estimate the covariance matrix, often referred to as the Constant Correlation Model (CCM) According to Elton and Gruber (1973), this model assumes that all stocks share identical correlations, which are equal to the average correlation, allowing for the calculation of covariances.
The estimation of the covariance matrix with constant correlation can be expressed using a specific formula In this context, the sample covariance matrix of asset returns is represented as S, which includes various elements Additionally, the sample correlations among different stocks are provided, highlighting the relationships between their returns.
√ The average of sample correlations is calculated as: ̅ ( ) ∑ ∑
Finally, constant correlation matrix C is defined as: and ̅ √
3.3.4 Shrinkage towards single-index model (SSIM)
Ledoit and Wolf (2003) introduced a shrinkage method for estimating the covariance matrix that enhances stability This approach merges the sample covariance matrix \( S \) with a structured covariance estimator \( F \), derived from the Single-Index Model, represented as \( F = \hat{F} \) The method involves a convex linear combination of the two matrices, expressed as \( \delta F + (1−\delta) S \), where \( \delta \) serves as the shrinkage constant This technique effectively redistributes the weights between the more stable estimator \( F \) and the sample covariance matrix \( S \) using the shrinkage constant \( \delta \).
By using the definition of Frobenuis norm, constant shrinkage estimator is calculated through following equation:
This equation means that using Frobenuis norm to scale the different between the True population covariance matrix Σ and the shrunk covariance matrix ∑̂ = δF + (1−δ)S Here, the shrinkage constant δ as variable
From equation (3.3.9), it gives the risk function calculated as:
Thus, R(δ) is the risk of equation (3.3.9) The mission of investors is to minimize risk For doing this mission, the first and second derivative of R(δ) must be calculated:
As a results of second derivative, (ẟ) are always larger than 0 Therefore, minimizing the risk R(δ) is solved by (ẟ) = 0 Then the shrinkage constant will be found as follows:
∑ ∑ ( ) ( ) (3.3.10) Where Ledoit and Wolf (2003a) showed that:
= ∑ ∑ ,√ - is the sum of asymptotic variances ρ = ∑ ∑ ,√ √ - is the sum of asymptotic covariance and γ = ∑ ∑ ( ) 2
The equation (3.3.10) will turn into:
The optimal shrinkage intensity, or shrinkage coefficient, identified by Ledoit and Wolf, represents a balance between a sample covariance matrix and a shrinkage target matrix A higher shrinkage coefficient indicates a greater influence of shrinkage methods on covariance matrix estimation Since the performance of portfolio selection relies heavily on this estimation, and the estimated covariance matrix is governed by the shrinkage intensity, it is clear that the essence of shrinkage methods lies in estimating the shrinkage coefficient or determining the optimal shrinkage intensity.
3.3.5 Shrinkage towards Constant correlation Model (SCCM)
Ledoit and Wolf (2003b) proposed a method utilizing the Constant Correlation Model (CCM) as the target matrix for the shrinkage technique Their findings indicated that applying shrinkage towards CCM outperformed shrinkage towards the Sample Inverse Matrix (SIM) and offered greater ease of implementation.
The shrunk covariance matrix, represented as ∑̂ = δ + (1−δ)S, is derived by replacing the single index F's covariance matrix with a new constant correlation covariance matrix Here, δ signifies the shrinkage constant, which will be detailed later in this section This approach redistributes the weight of the covariance matrix by blending a more stable sample covariance matrix with the sample covariance matrix S, utilizing the shrinkage constant δ for optimal results.
An objective must be chosen according to which the shrinkage coefficient is optimal
Existing shrinkage estimators in finite-sample statistical decision theory, including the method by Frost and Savarino (1986), fail when the sample size (N) meets or exceeds the number of parameters (T) due to their reliance on the inverse of the covariance matrix To address this issue, a new loss function is proposed that avoids dependence on this inverse, offering a more intuitive solution: it measures the quadratic distance between the true and estimated covariance matrices using the Frobenius norm Building on this framework, Ledoit and Wolf (2003b) developed a method to effectively estimate the shrinkage intensity.
The Frobenius norm of the N × N symmetric matrix Z with entries ( ), in which i, j
Based on the Frobenius norm, the difference between a shrinkage estimator of covariance matrix and a true covariance matrix will be calculated; and the quadratic loss function is identified as follows:
The shrinkage coefficient δ will be found through minimizing the expected value of the loss function:
Assuming that N, T are fixed and approach to infinity respectively, Ledoit and Wolf
(2003) prove that “the optimal value δ asymptotically behaves like a constant over T (up to higher-order terms)” and κ denoted as the constant is shown the following formula:
Where: π is the sum of “asymptotic variances of the entries of sample covariance matrix” scaled by √ :
∑ ∑ [√ ] ρ is the sum of “asymptotic covariance of the entries of the shrinkage target with the entries of sample covariance matrix” also scaled by √ :
∑ ∑ [√ √ ] γ which denotes the misspecification of the (population) shrinkage target is determined as follows: