ỦY BAN NHÂN DÂN THÀNH ĐOÀN TP HỒ CHÍ MINH THÀNH PHỐ HỒ CHÍ MINH TRUNG TÂM PHÁT TRIỂN SỞ KHOA HỌC VÀ CÔNG NGHỆ KHOA HỌC VÀ CÔNG NGHỆ TRẺ CHƯƠNG TRÌNH KHOA HỌC VÀ CÔNG NGHỆ CẤP THÀNH PHỐ BÁO CÁO TỔNG HỢ[.]
INTRODUCTION
Background
Modern Portfolio Theory (MPT), introduced by Harry Markowitz in 1952, has significantly influenced investment portfolio construction for over 65 years This framework aims to optimize expected returns while managing risk through the strategic allocation of asset weights Despite its widespread application, MPT's foundational assumptions are increasingly being questioned, particularly regarding the reliance on mean returns and the covariance matrix of asset returns.
To effectively implement investment portfolios, investors must estimate the mean and covariance matrix of asset returns Traditionally, the sample mean and covariance matrix have been utilized, but these estimators often lack stability due to estimation errors, leading to continuous fluctuations in portfolio weights Consequently, mean-variance portfolios pose significant challenges for portfolio managers in practical application Furthermore, substantial empirical evidence indicates that these portfolios tend to perform poorly in out-of-sample mean and variance scenarios (Michaud, 1989).
The instability of mean-variance portfolios arises from the challenges in accurately estimating mean asset returns, prompting researchers to increasingly favor minimum-variance portfolios These portfolios rely solely on covariance matrix estimation, making them less sensitive to estimation errors (Chan et al., 1999; Jagannathan and Ma, 2003) Jagannathan and Ma argue that the significant estimation error in sample means justifies ignoring the mean altogether, a claim supported by extensive empirical evidence demonstrating that minimum-variance portfolios often outperform traditional portfolios in out-of-sample scenarios.
2 other mean-variance portfolio on Sharpe ratio and other performance metrics (DeMiguel, 2005; Jagannathan and Ma, 2003)
According to Demiguel (2009), the minimum-variance portfolio is not reliant on mean return estimations; however, it is significantly affected by estimation errors, highlighting its sensitivity to such inaccuracies.
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 due to its minimal asymptotic variance However, Huber (2004) highlights a critical issue: the efficiency of MLEs is highly sensitive to deviations from the normal distribution As a result, when asset-return distributions diverge even slightly from normality, MLEs may not provide the most efficient estimates This insight is particularly valuable for portfolio selection, as extensive evidence suggests that the empirical distribution of returns often deviates from the normal distribution, leading to potentially unsuitable portfolios.
The effectiveness of minimum-variance portfolios hinges on accurately estimating the covariance matrix; however, traditional methods like the sample covariance matrix (SCM) and ordinary least squares (OLS) struggle with high-dimensional portfolios The increased dimensionality can lead to unexpected errors during computation, and insufficient sample data may hinder the accurate estimation of the true covariance matrix As a result, the estimated covariance matrix can become ill-conditioned or singular, a common issue in matrix computation research This often leads to poor portfolio performance and a failure to generate profits A promising solution to this challenge is the use of shrinkage estimators in covariance matrix estimation, first introduced by Ledoit and Wolf in 2003.
This approach emphasizes controlling the tradeoff between bias and variance in covariance matrix estimation by utilizing a predefined target matrix F and a mathematically derived shrinkage intensity σ from the sample covariance matrix The target matrix F for Ledoit-Wolf (LW) shrinkage can be derived from various models, such as a single index model, a constant correlation model, or an identity matrix The findings by Ledoit and Wolf (2003) are particularly valuable for investment managers, raising the question of whether they should reconsider their traditional covariance matrix estimation methods The choice of estimation technique significantly impacts mean-variance optimized portfolios However, there is a lack of research on the out-of-sample performance associated with different covariance matrix estimation techniques, leading to a precarious situation where investment managers may hesitate to risk capital based on unproven frontier research.
There is a lack of research in Vietnam focusing on estimation techniques for the covariance matrix, particularly regarding shrinkage methods for optimized portfolio selection This presents an opportunity for the author to implement shrinkage methods to estimate the covariance matrix for minimum-variance portfolio optimization in the Vietnamese stock market.
Purpose, research questions and expected contribution
This study aims to utilize shrinkage estimators, which are linear combinations of sample covariance matrices and a target matrix, specifically the identity matrix, to estimate the covariance matrix This estimation method will be applied to the minimum variance portfolio optimization process to create stable and reliable stock portfolios within the Vietnam stock market Furthermore, the author will evaluate the effectiveness of shrinkage methods in comparison to traditional approaches.
4 covariance matrix estimation, but also the performance of portfolios benchmark based on out – of – sample of the performance metrics
This author will attempt to achieve the purpose of this study by answering the following research questions:
Question 1: How do the shrinkage estimators of covariance matrix perform on out – of – sample in selecting the minimum – variance optimized portfolios?
Question 2: How does the shrinkage intensity work among three linear shrinkage estimations for portfolio selection?
This study aims to contribute to the understanding of covariance estimation techniques, particularly their robustness, thereby aiding financial practitioners in employing minimum-variance optimization in quantitative models It will evaluate the effectiveness of shrinkage estimators in selecting minimum-variance optimized portfolios, using performance metrics such as Sharpe ratio, variance, turnover, maximum drawdown (MDD), and Jensen’s Alpha Successful performance of these estimators suggests that investors can construct optimal portfolios even without extensive financial knowledge Additionally, this research will explore the application of shrinkage methods in the Vietnam stock market, an emerging market, addressing the gap in existing literature which predominantly focuses on developed markets like the US and Europe.
Disposition of the study
This study comprises five chapters, with Chapter 1 addressing the problem statements, the study's purpose, research questions, and anticipated contributions Subsequent chapters delve into the detailed findings and analyses related to the research objectives.
Chapter 2, Literature Review, examines key research in portfolio optimization, laying the groundwork for the theoretical framework and methodology to be presented in the following chapter.
Chapter 3, Theoretical Framework, outlines the foundational theories guiding this study, beginning with an introduction to basic preliminaries and the portfolio optimization problem, followed by a discussion on covariance matrix estimations.
Chapter 4, Methodology, presents the general methodology employed for investigating the research questions in this study
Chapter 5, Empirical Results and Conclusions, provides the empirical findings in this study and concludes the study and presents some suggestions for further research
LITERATURE REVIEW
Modern Portfolio Theory Framework
Harry Markowitz is recognized as a pioneer in financial economics and corporate finance, notably receiving the Nobel Prize in 1990 for his influential contributions His seminal article, “Portfolio Selection,” published in 1952 in The Journal of Finance and later expanded in his book “Portfolio Selection: Efficient Diversification” (1959), laid the groundwork for Modern Portfolio Theory (MPT).
The Modern Portfolio Theory (MPT) is an investment framework designed to optimize portfolio returns for a specific level of risk, or to minimize risk for a desired return by strategically selecting asset weights Despite its widespread use in the financial sector, the fundamental assumptions of MPT have faced increasing scrutiny and debate in recent years.
The Modern Portfolio Theory (MPT) enhances traditional investment models by emphasizing the importance of diversification to mitigate both market and specific company risks Often referred to as Portfolio Management Theory, it aids investors in classifying, evaluating, and measuring expected risks and returns A key aspect of this theory is its quantification of the risk-return relationship, along with the premise that investors should receive compensation for taking on risk.
The concept of diversification in Modern Portfolio Theory (MPT) focuses on creating investment portfolios that exhibit lower risk than individual securities within those portfolios Diversification effectively reduces investment risk regardless of whether the correlation between security returns is positive or negative MPT treats security returns as normally distributed and defines risk through the standard deviation of these returns Consequently, the overall return of a portfolio is calculated as a weighted combination of the returns from its constituent securities, influencing the total variance of the portfolio's return.
7 will be reduced if the correlation between the securities’ returns is not perfectly positively Moreover, MPT assumes that an investor is rational and market is efficient
Investing involves balancing risk and expected returns, where assets with higher expected returns typically carry greater risks (Taleb, 2007) The Modern Portfolio Theory (MPT) guides investors in selecting portfolios that maximize expected returns for a specific risk level, or conversely, minimize risk for a desired level of expected return.
Parameter estimation
Capital Asset Pricing Model (CAPM) which was first introduced by William Sharpe in
In 1964, the Capital Asset Pricing Model (CAPM) was established as a one-factor model to forecast asset returns, highlighting the relationship between systematic risk and expected returns However, in 1992, finance professors Eugene Fama and Kenneth French found that the beta coefficient of CAPM failed to adequately explain the expected returns of U.S securities from 1963 to 1990 They identified two stock categories that consistently outperformed the market: small-cap stocks and those with a high Book-to-Market Equity ratio To enhance the CAPM model, Fama and French incorporated these two factors, resulting in the introduction of their three-factor model in 1993, which has since been validated in both developed and emerging markets The Fama-French three-factor model posits that the return of a specific portfolio or stock is influenced by three key factors: market return, size risk, and value risk.
(2014) continued to introduce a 5-factor model to predict the expected return of portfolio with two new factors that are profitability and investment factors
Moreover, the researchers also employed robust estimators to improve the expected return estimation For examples, instead of using the sample mean m = 𝟏
To estimate expected population returns, the sample truncated/trimmed mean or Winsorized mean can be utilized (Martin, Clark, and Green, 2010) The trimmed mean is derived by removing the k% most extreme values, while the Winsorized mean replaces those extremes with the next k% most extreme values Both estimators are particularly efficient in addressing deviations from distributional assumptions regarding asset returns However, if the returns are non-stationary, neither the trimmed nor Winsorized means, nor other robust estimators such as M-estimator or S-estimator, will 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 diminish investor utility Therefore, the optimization problem should focus on minimizing utility loss associated with selecting a portfolio based on sample estimates rather than true values Instead of estimating each asset's expected return individually, it is more effective to choose an estimator that reduces utility loss from overall parameter uncertainty Jorion (1986) recommends employing a Bayes–Stein estimator, which adjusts each asset's sample mean towards the grand mean.
N ∑ m N 1 i Through the simulation, this estimator reduce risk of portfolio and outperforms portfolios constructed by m T
Enhancing the estimation of expected returns on assets addresses some limitations of Modern Portfolio Theory (MPT) However, Merton's 1980 research indicates that accurately predicting these expected returns is challenging Most asset pricing models assume a consistent relationship between the expected return of assets and the market's expected return, which is treated as a constant over time.
Estimating the expected return of assets can be simplified by making nine key assumptions, yet accurately doing so still requires extensive time series data (Merton, 1980) While the assumption of a constant expected return is widely recognized as unrealistic, relaxing this assumption complicates the estimation process further This leads to an alternative research direction focused on selecting portfolios based on covariance matrix estimation rather than relying solely on expected return estimation.
Recent research has increasingly focused on improving investment portfolio stability and minimizing risks through advanced methods Traditionally, portfolio optimization relies on the sample covariance matrix (SCM), but Michaud (1989) highlighted significant shortcomings of this approach, including statistical errors and poor conditioning 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 SCM estimation does not yield better results than an equally weighted portfolio, which DeMiguel (2009) referred to as the naïve 1/N portfolio.
According to Sharpe's single-index model (1964), researchers can estimate the covariance matrix to optimize portfolio selection The covariance matrix derived from this one-factor model is calculated as ∑ SIM = ββ^T σ_m^2 + ∑ ε The single-index model offers three key advantages over the traditional modern portfolio theory (MPT) approach Firstly, it requires estimating only 2N+1 parameters for the covariance matrix, compared to the N(N+1)/2 parameters needed by the standard method, which becomes more complex as N increases (N ≥ 4) Secondly, when investors introduce a new asset, they only need to estimate its β and σ_ε to update their precision matrix, rather than recalculating the asset's variance and covariance with every other asset in the sample, which involves estimating N + 1 parameters and inverting the entire covariance matrix.
The SIM approach requires only T > 2 observations to estimate β, 𝜎 𝜀, and 𝜎 𝑚 2 for each asset, allowing for the estimation of the precision matrix, while the traditional MPT approach demands T > N Research by Senneret et al (2016) demonstrates that the SIM method yields portfolios that exhibit significantly less sensitivity to estimation errors compared to the standard method, resulting in superior performance across various risk and return metrics Nevertheless, both the standard and SIM approaches to portfolio selection are susceptible to estimation errors arising from the sample mean returns vector 𝑚 𝑇, with SIM also facing potential errors from 𝜎̂ 𝑚 2.
Elton and Gruber (1973) introduced the Constant Correlation Model (CCM) to address the limitations of the SIM, positing that all stocks share the same correlation, equal to the historical mean This model suggests that the historical correlation matrix reflects only the average correlation for future periods, neglecting the variations in pairwise correlations, which is a significant assumption Elton et al (2009) highlighted that the CCM provides more accurate forecasts of future covariance matrices compared to the sample covariance matrix and SIM, leading to better portfolio performance across various metrics Nevertheless, the CCM still encounters challenges in estimating large-dimensional covariance matrices for effective portfolio selection.
In 2003, Ledoit and Wolf introduced the shrinkage method for estimating the covariance matrix in portfolio selection, which combines the sample covariance matrix with a target matrix of the same dimensions This method aims to achieve a weighted average that closely approximates the true covariance matrix based on a clear and intuitive criterion Its effectiveness increases when the bias from the shrinkage target matrix is minimal, when data noise is significant, and when the ratio of observations to variables (N/T) is high The core concept involves estimating the covariance matrix using a formula that incorporates both the sample covariance and the shrinkage target.
𝑡𝑎𝑟𝑔𝑒𝑡 , where ẟ ∗ is determined by a data-driven algorithm and ∑̂
Ledoit and Wolf (2003a) propose the Constant Correlation Model (∑̂ 𝐶𝐶𝑀), which is based on the assumption that all stocks share the same correlation, equal to the average correlation Additionally, Ledoit and Wolf (2003b) offer alternative suggestions for the target matrix.
The Single Index Model (SIM) highlights that stock returns exhibit a factor-model structure However, Ledoit and Wolf (2004) introduced a target matrix that does not rely on specific knowledge benefits, unlike previous shrinkage methods This target matrix is represented as an identity matrix, denoted as ∑̂.
The identity matrix (IM) is a square matrix characterized by ones on its principal diagonal and zeros elsewhere In exploring the potential for optimal portfolio selection without relying on financial domain knowledge, Ledoit and Wolf's research reveals that while all three enhanced estimators of the covariance matrix outperform the sample covariance matrix, there is no definitive best option Specifically, their findings suggest that for portfolio sizes of N ≤ 100, shrinkage to the constant-correlation matrix is most effective, while for larger portfolios of N ≥ 225, shrinkage to the single-factor matrix yields better results.
Recent research has explored non-linear shrinkage estimators to determine if linear shrinkage can be generalized and enhanced for the identity matrix without requiring financial expertise, allowing investors to outperform basic linear shrinkage methods despite a lack of knowledge about the true covariance matrix (Ledoit and Wolf, 2018) Unlike linear shrinkage, which estimates specific eigenvalues through a linear function, non-linear shrinkage focuses on the distribution of eigenvalues, offering greater flexibility in covariance matrix estimation Ledoit and Wolf (2012, 2015) introduced the QuEST function for this non-linear shrinkage approach, assuming that the number of investment assets and observed data samples approaches infinity Lam (2016) further advanced this methodology by proposing a new technique.
NERCOME (Nonparametric Eigenvalue-Regularized Covariance Matrix Estimator) enhances covariance matrix estimation by dividing the observation sample into two parts: one for calculating eigenvectors and the other for eigenvalues Ledoit and Wolf (2018) advanced the non-linear shrinkage method by integrating the rapid processing of linear shrinkage, the precision of the QuEST function, and the clarity of the NERCOME approach, coining this innovation as direct nonlinear shrinkage Despite these advancements, the practical application of these methods remains limited, largely due to ongoing debates regarding their effectiveness.
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 can lead to significant estimation errors, particularly with 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, literature on portfolio selection has diverged into two main paths: one adhering to the traditional mean-variance model and the other embracing the more contemporary global minimum-variance model.
Research indicates that optimizing an investment portfolio benefits more from estimating the covariance matrix, as it is less susceptible to estimation errors compared to expected returns The shrinkage model offers a more accurate covariance matrix by effectively balancing the strengths and weaknesses of traditional estimators Various target matrices, including linear and non-linear options, have been introduced and tested across different market conditions to identify the most effective approach The subsequent step involves applying the estimated covariance matrix to enhance portfolio performance.
In the pursuit of an optimal portfolio, researchers and portfolio managers commonly utilize mean-variance and global minimum-variance models Historical literature indicates that the global minimum-variance model often demonstrates superior performance compared to other methods.
THEORETICAL FRAMEWORK
Basic preliminaries
The return of a financial security, represented as R, reflects the gain or loss over a specific time period For a particular security i, the return is denoted as \( R_i \) The price of the security at time t is indicated as \( S_t \) To calculate the return for a non-dividend security over the time interval [t,T], where T is greater than t, a specific formula is applied.
As for a security that pays dividends over the time period [t,T], the return is calculated as:
In minimum-variance optimization, variance is a crucial concept as it serves as the primary statistic for assessing the risk and volatility of asset returns Defined as the expected value of the squared differences from the mean, variance plays a key role in evaluating a portfolio consisting of multiple assets For a portfolio with n assets, the calculation of variance can be derived systematically.
To summarize, the variance of the portfolio is given by:
Portfolio Optimization
One of the key methods for selecting an optimal investment portfolio is through modern portfolio theory, which emphasizes the identification of a global minimum variance portfolio (GMVP) within a universe of N assets This approach involves determining stock weights represented as w = (𝑤1; 𝑤2;…; 𝑤𝑁), aimed at minimizing risk while maximizing returns.
The total of the stock weight will be equivalent to one (𝚺 𝑖=1 𝑁 𝑤 𝑖 = 1), with the terms of
𝑤 𝑖 > 0 implies that there is no short selling The problem of portfolio selection is describes as:
In which: 1 denotes a vector of ones, and Σ is the covariance matrix of N stocks The theoretical approach to the problem (1) is feasible:
The solution (2) involves the inverse of the covariance matrix, typically derived from the sample covariance matrix However, this approach can be problematic as the sample covariance is often ill-conditioned and may not be invertible, especially in high-dimensional portfolios To address this issue, shrinkage estimators should be employed to adjust the covariance matrix parameter in equation (1), leading to an improved solution in equation (2).
Estimating the covariance matrix
Linear shrinkage estimation combines the sample covariance matrix (SCM) with a high-structure target matrix through a weight known as shrinkage intensity (σ), represented by the formula: ∑ Shrinkage = (1 − σ)S SCM + σ ∑ target (0 ≤ σ ≤ 1) This method leverages the strengths of both the SCM and the target matrix while mitigating their weaknesses By optimizing the weights between these two covariance matrices, the estimated covariance matrix can closely resemble the true covariance matrix The linear shrinkage approach is defined by three key elements: the sample covariance matrix, the target matrix, and the shrinkage intensity.
Assuming that 𝑟 𝑖,𝑡 , 𝑟 𝑗,𝑡 are the historical returns of assets i and j at the time period t The historical average returns of asset i (𝑟̅ 𝑖 ) and asset j (𝑟̅ 𝑗 ) in the period [1, T] will be calculated as follows:
The equation that is used to calculate the sample covariance between any two assets i, j is:
From the equation (2), the sample covariance matrix (Σ̂ 𝑆𝐶𝑀 ) that shows the relationship among N assets in the portfolio is identified as follows: Σ̂ = [ 𝑆𝐶𝑀
This paper examines three shrinkage target matrices: the single-index model (SIM), constant correlation model (CCM), and identity matrix (IM) The identification of these shrinkage target matrices will be detailed in the following sections.
This model assumes that the assets’ returns are significantly influenced by the market return Thus, the estimated return of an asset i (𝑟̂ 𝑖,𝑡 ) will be identified through a following regression model:
The estimated market return (𝑟̂ 𝑚) is influenced by random error (𝜀 𝑖,𝑡) and coefficients (𝛼 𝑖 and 𝛽 𝑖) derived from regression analysis This model operates under specific assumptions, including the independence of the error term (𝜀 𝑖,𝑡), with Cov[𝜀 𝑖 ,𝜀 𝑗 ] equal to 0 and Cov[𝑟̂ 𝑚 ,𝜀 𝑖 ] also equal to 0 Additionally, the error term (𝜀 𝑖,𝑡) is assumed to follow a normal distribution, characterized by a variance of Var[𝜀 𝑖 ] = 𝜎 𝜀 2 𝑖 and an expected value of E[𝜀 𝑖 ] = 0.
The variance and covariance of estimated asset returns i, j are measured by these following formulas:
The covariance matrix calculated by SIM is as follows:
Where 𝜎̂ 𝑚 = Var[𝑟̂ 𝑚 ] is the estimated market variance; Σ̂ 𝜀 is the matrix of regression’s error with the shape of N x N diagonal matrix
The covariance matrix derived from the Constant Correlation Model (CCM) assumes that all stock pairs within the portfolio share identical correlations, which are equal to the average correlation Consequently, the constant correlation matrix is computed accordingly.
First, the sample correlations between stocks i, j is calculated by:
In which: S is the sample covariance matrix and 𝑠 𝑖𝑗 is the element of the matrix S
Second, the average of sample correlations is calculated by the equation:
Finally, constant correlation matrix (C) is defined as:
An identity matrix (IM) is a square matrix characterized by having all its main diagonal elements equal to one, while all other elements are zero Shrinkage towards an identity matrix differs significantly from shrinkage towards other target matrices, as the latter necessitates careful selection based on specific known features of the true covariance matrix.
Ledoit and Wolf utilized the factor-model structure of stock returns to develop a covariance matrix through shrinkage towards the single-index model (SSIM) They also leveraged the positive average correlation of stock returns to estimate the covariance matrix in the shrinkage towards the constant correlation model (SCCM) In contrast, the shrinkage towards the identity matrix (STIM) serves as a generic approach that lacks the advantages of application-specific knowledge The STIM method addresses whether investors can optimize their portfolios without financial expertise.
The optimal shrinkage coefficient (ẟ ∗), identified by Ledoit and Wolf, represents a balance between the Shrinkage Covariance Matrix (SCM) and the target shrinkage matrix A higher shrinkage coefficient signifies a greater influence of shrinkage methods on the covariance matrix estimation, which is crucial for portfolio selection performance The projected covariance matrix is directly affected by the shrinkage coefficient, highlighting the importance of determining an optimal value between 0 and 1 A low shrinkage coefficient suggests minimal flaws in the SCM calculation, while a high coefficient indicates increased errors.
The shrinkage coefficient is determined using the Quadratic Loss function, defined as L(α) = || ẟF + (1-ẟ)S - ∑||² In this equation, F represents the shrinkage target matrix, S is the sample covariance matrix, and ∑ denotes the true covariance matrix The optimal shrinkage coefficient is achieved when L(α) reaches its minimum value.
METHODOLOGY
Input Data
The weekly stock prices are gathered for the optimization process, with weekly returns calculated for all stocks, accounting for adjusted dividends and capital changes due to stock splits The dataset D(t) is split into two segments: W(t), which serves as the in-sample period for estimating the covariance matrix and initializing the first portfolio, and V(t), the out-of-sample period used to evaluate the effectiveness of the estimation methods.
Regarding more information, the cumulative data points observed in this analysis are D(t)
= 416, referring to 468 weeks from January 2011 to January 2019 The initialization period W(t) = 104 weeks refers to the two-year duration from January 2011 to January
The study analyzes a data set spanning 312 weeks from January 2013 to January 2019, focusing on companies listed on the Ho Chi Minh City Stock Exchange (HOSE) that have been publicly traded for more than two years A total of 350 companies are included in the analysis, with data sourced from HOSE and denominated in VND The VNINDEX, representing the Vietnam Stock Index, serves as the reference index for the single-index model employed in the research.
Back-Testing Process
This research evaluates the efficiency of covariance matrix shrinkage methods through a back-testing process, utilizing a platform established by Tran et al (2020) The back-testing process aids in assessing the feasibility and potential application of future estimations based on a series of portfolio price values.
Step 1: Dividing observations D(t) into two parts W and V Therein, W is considered as initial stage to estimate covariance matrix, usually call in-the-sample process and V is considered as testing stage of methodologies in portfolio selection, usually called out-of-sample process In our study, based on the policies and settings of Vietnam stock market (for example, three days are required for selling or buying stocks), we choose weekly trading other than daily trading Hence, the total observation is D(t) 416, each data point equal to unit of time is week Therein, initial stage W = 104 weeks within 2 years and testing stage V = 312 weeks
Step 2: Using the data in initial part W to estimate covariance matrix and use this matrix as input in the portfolio optimization for selecting the optimal portfolios And then, the optimal portfolios will be tested on data point 𝑡 𝑤+1 based on the portfolio performance criteria
Step 3: Carrying out replacing data 𝑡 1 with data point 𝑡 𝑤+1 in the initial part W to create 𝑊 1 , and then continue the optimal portfolio selection process and evaluate results of the selection as in step 2 on data point 𝑡 𝑤+2 This process is repeated during testing process V and end at data point 𝑡 𝑣+𝑤
Step 4: Calculating and extracting the results during testing stage V The portfolio performance criteria are applied to evaluate portfolio selection process V including: average return of portfolio, volatility of portfolio, portfolio turnover, maximum
This study evaluates drawdown, winning rate, and Jensen’s Alpha while incorporating transaction costs into the testing procedure Each portfolio adjustment based on optimal results incurs transaction costs, estimated at 0.3% of the total buying or selling value, reflecting the standard rates applied by most stock firms on the Vietnam equity exchange.
The testing process is presented in the diagram below (see Figure 1):
Performance Metrics
Performance metrics serve as essential criteria for assessing the efficiency of optimal portfolios, primarily focusing on portfolio return and risk Portfolio return is defined as the overall gain or loss generated by the portfolio.
The risk of a portfolio is assessed through the volatility of its returns, which can be quantified by the variance of those returns (Nguyen, 2019) In addition to standard performance metrics, this paper also evaluates other important criteria, including the Sharpe ratio, maximum drawdown, portfolio turnover, winning rate, and Jensen’s Alpha.
The Sharpe ratio is a key metric for investors, measuring profit relative to each unit of risk in a portfolio; a higher ratio indicates more effective investment performance (Sharpe, 1964).
Where: 𝑅 𝑝 is the average return of portfolio; 𝑅 𝑓 is the risk-free rate for the evaluated period and 𝜎 𝑝 is the standard deviation of portfolio’s returns
The maximum drawdown is a crucial metric for assessing portfolio efficiency, as it indicates the level of risk associated with a portfolio in challenging market conditions It can be calculated using specific formulas to quantify potential losses during downturns.
The portfolio value at time t, denoted as 𝑉 𝑖,𝑡, is optimized based on strategy i A lower Maximum Drawdown (MDD) is appealing to investors, as it indicates a less risky investment strategy.
This indicator reflects the stability of a portfolio as it transitions according to an optimal strategy Investors typically favor lower turnover rates, as this indicates reduced liquidity risks and lower transaction costs The portfolio turnover for a given strategy is defined accordingly.
In which: T is the number of times of portfolio change, 𝑤 𝑖,𝑗,𝑡+1 is the weight of asset j optimized in line with strategy i time t+1
The winning rate reflects the proportion of successful trades made by investors, but a high winning rate does not guarantee profitability Instead, it enhances the likelihood of successful investments, making a higher win rate advantageous for portfolios.
Jensen's alpha is a key metric that measures a portfolio's actual return against its theoretical expected return, as defined by the Capital Asset Pricing Model (CAPM) This calculation utilizes the Beta coefficient and the average market return to assess performance The formula for Jensen's alpha is α = Rp – [Rf + β(Rm - Rf)], where Rp represents the portfolio return, Rf is the risk-free rate, and Rm is the market return.
Where: 𝑅 𝑝 is the average return of portfolio, 𝑅 𝑓 is the risk free rate, β is beta coefficient of portfolio and often estimated by Ordinary Least Square (OLS) regression (Le et al
2018) and 𝑅 𝑚 is the average market return.
VNINDEX and 1/N portfolios benchmark
To effectively assess portfolio performance, various benchmarks are utilized, including indexes that gauge diverse investments and market segments These benchmarks, which encompass both broad market and specific sector measurements, are essential for the ongoing evaluation and adjustment of portfolios In Vietnam, the Vietnam Stock Index (VNINDEX) serves as a crucial benchmark, representing a capitalization-weighted index of all companies listed on the Ho Chi Minh City Stock Exchange, with an initial base index value of 100 established in July.
28, 2000 Therefore, to evaluate the performance of the optimized portfolios estimated from different covariance matrix, this study used VNINDEX as benchmark to compare
The 1/N portfolio serves as a crucial benchmark for evaluating individual investments and investment portfolios DeMiguel (2009) tested 14 models on seven empirical datasets to identify optimized portfolios, but none consistently outperformed the 1/N rule in terms of Sharpe ratio, certainty-equivalent return, or turnover This indicates that the potential benefits of optimal portfolio selection are not yet fully realized in practice Consequently, the 1/N portfolio is a robust standard against which all portfolio selection strategies should be measured In this research, the author utilized the 1/N portfolio as a secondary benchmark to evaluate the performance of the optimized portfolios.
EMPIRICAL RESULTS
VNINDEX and 1/N portfolio performance
From 2013 to 2019, the VNINDEX experienced significant growth, rising from 464 to 880 points, with an average annual growth rate of 13.58% This period marked a notable success for the Vietnam stock market, particularly when compared to previous downturns and regional counterparts The growth was primarily driven by Vietnam's stable macroeconomic environment and the overall development of global stock markets Notably, 2017 stood out as an exceptional year, with the VNINDEX achieving nearly 50% annual profit However, this period was also characterized by considerable fluctuations, especially in the last six months.
On December 27, 2018, the stock market experienced a significant decline, marking the first negative growth of the VNINDEX in five years, with a drop of nearly 10% This downturn was primarily attributed to the instability caused by the US-China trade war Throughout this five-year period, the VNINDEX exhibited an average volatility of 16.73%, indicating relatively high fluctuations in the market.
Figure 5.2: The annual return and turnover of 1/N portfolio benchmark
Over a five-year period, the 1/N portfolio benchmark demonstrates impressive performance, achieving an average annual return of 17.9% This strategy maintains equal weight across stocks, resulting in lower portfolio volatility at 11.44% and minimal daily turnover of just 1.3% Additional performance metrics for the 1/N portfolio benchmark will be provided in the following table.
Table 5.1: The performance of the 1/N portfolio benchmark from
Portfolio out – of –sample performance
Table 5.2: Backtesing results of sample covariance matrix on out – of – sample from
The backtesting process conducted on out-of-sample data revealed that the average annual return of the sample covariance matrix was approximately 7.8% Notably, the years 2013 and 2014 recorded exceptional annual returns exceeding 20%.
2018 was the worse that lost over 10% The sharpe ratio of the method was not high that was at 0.49 times The maximum drawdown was relative high about 17.29% One of the
Figure 5.3: The sample covariance matrix’s annual return Figure 5.4: The sample covariance matrix’s daily turnover
Figure 5.5: The sample covariance matrix’s maximum drawdown Figure 5.6: The sample covariance matrix’s sharpe ratio
5.1.2 Shrinkage towards Single index model
Table 5.3: Backtesting results of Shrinkage towards Single index model on out – of – sample from 1/1/2013 – 1/1/2019
Figure 5.8: The Shrinkage towards Single index model’s annual return Figure 5.9: The Shrinkage towards Single index model’s daily turnover
Figure 5.10: The Shrinkage towards Single index model’s maximum drawdown Figure 5.11: The Shrinkage towards Single index model’s sharpe ratio
The backtesting results on the out-of-sample data revealed a remarkable average annual return of 21.22% for the shrinkage towards the single index model Notably, 2014 experienced the highest annual return exceeding 30%, while 2018 recorded a loss of 2% The method achieved a high Sharpe ratio of approximately 2, indicating strong risk-adjusted performance Additionally, the portfolio weight turnover was around 2.74%, and Jensen’s Alpha reached an impressive 13.9%, highlighting the portfolios' superior returns Furthermore, the maximum drawdown showed significant improvement, recorded at just 7.5%.
Figure 5.12: Shrinkage intensity of Shrinkage towards Single index model
The shrinkage intensity coefficient of SSIM ranges from 5% to 50%, indicating that higher shrinkage intensity significantly impacts covariance matrix estimation Notably, the influence of shrinkage methods on estimating the covariance matrix was more pronounced during the period of 2017-2018 compared to 2013-2016.
5.1.3 Shrinkage towards Constant correlation model
Table 5.4: Backtesting results of Shrinkage towards constant correlation model on out – of – sample from 1/1/2013 – 1/1/2019
The backtesting procedure on out-of-sample data revealed that the shrinkage towards constant correlation model achieved an impressive average annual return of approximately 22.37% Notably, the highest annual return occurred in 2014, exceeding 30%, while 2018 saw a decline to around 2% Despite this downturn, the model outperformed the market, as the VNINDEX experienced a nearly 10% loss, and the 1/N portfolio benchmark mirrored this poor performance Additionally, portfolios based on the sample covariance matrix incurred significant losses of over 12% This model marked a significant achievement by generating profits in a challenging fiscal environment.
Figure 5.14: The Shrinkage towards constant correlation model’s annual return
Figure 5.15: The Shrinkage towards constant correlation model’s daily turnover
Figure 5.16: The Shrinkage towards constant correlation model’s maximum drawdown Figure 5.17: The Shrinkage towards constant correlation model’ s sharpe ratio
In 2018, the investment method achieved a high Sharpe ratio of 1.94, indicating strong risk-adjusted returns The portfolio weight turnover was approximately 2.36%, while Jensen’s Alpha reached an impressive 14.05%, highlighting the superior returns generated by the portfolios Additionally, the maximum drawdown showed significant improvement, recorded at just 6.88%.
Figure 5.18: Shrinkage intensity of SCCM
The shrinkage intensity coefficient of SCCM ranges from 25% to 65%, indicating that the impact of the shrinkage method on estimating the covariance matrix is more pronounced in the period from 2017 to 2018 compared to the period from 2013 to 2016.
Table 5.5: Backtesting results of Shrinkage towards Identity Matrix on out – of – sample from 1/1/2013 – 1/1/2019
The backtesting procedure on out-of-sample data revealed an impressive average annual return of 20.42% with a high Sharpe ratio of 2.05, attributed to a low average annual volatility of 7.4% The portfolio weight turnover was approximately 2.18%, while Jensen’s Alpha demonstrated a notable positive value of 13.07%, indicating superior portfolio returns Additionally, the model maintained a low maximum drawdown of 7.99%.
Figure 5.20: The Shrinkage towards identity matrix’s annual return
Figure 5.21: The Shrinkage towards identity matrix’s sharpe ratio
Figure 5.22: The Shrinkage towards identity matrix’s daily turnover Figure 5.23: The Shrinkage towards identity matrix’s maximum drawdown
Figure 5.24: Shrinkage intensity of STIM
The shrinkage intensity coefficient of STIM varies significantly, ranging from over 10% to 60% Additionally, the influence of the shrinkage method on covariance matrix estimation was notably greater during the period of 2017-2018 compared to 2013-2016.
5.3 Summary of out – of –sample performance
Approach Method Return Risk SR Daily
Table 5.6 presents a summary of backtesting results for various covariance matrix estimation techniques applied to out-of-sample data from January 1, 2013, to January 1, 2019 The table includes p-values that assess the statistical significance of the performance metric differences between each covariance matrix estimator and the sample covariance matrix estimator.
This section presents the results derived from the backtesting procedure detailed in Section 4.2, utilizing weekly return data outlined in Section 4.1 The estimations are based on a moving window of the past 104 weeks, with all portfolios rebalanced on a weekly basis.
Shrinkage-based portfolios consistently outperform benchmark portfolios, achieving average annual returns exceeding 20% Notably, the Shrink-CCM portfolio stands out with an impressive return of 22.37% Additionally, these portfolios exhibit lower average annual volatility, approximately 7.9%, compared to 13% for benchmark portfolios, highlighting their superior risk-adjusted performance.
Shrinkage-based portfolios significantly outperform traditional portfolios, exhibiting Sharpe ratios that are approximately twice as high as the average Sharpe ratios of benchmark portfolios, which stand around 0.7 Additionally, the maximum drawdown (MDD) of shrinkage-based portfolios is notably lower at about 7.5%, compared to 19% for benchmark portfolios, with the Shrink-CCM portfolio achieving an impressive MDD of just 6.88% This indicates that investors in shrinkage-based portfolios face the least potential maximum accumulated loss.
Shrinkage-based portfolios exhibit the highest positive Jensen’s Alpha values, exceeding 13% This suggests that these portfolios generate returns surpassing risk-adjusted benchmarks Despite numerous studies indicating that most portfolios struggle to outperform market returns, shrinkage-based strategies stand out for their superior performance.
Empirical results indicate minimal differences among linear shrinkage methods, with SCCM demonstrating efficiency in identifying highly profitable portfolios, while STIM excels in finding highly stable ones This supports Ledoit and Wolf's conclusion that no single shrinkage method is superior Additionally, the shrinkage intensity of SCCM significantly impacts the estimation of the covariance matrix, varying from 25% to 65%, more so than STIM and SSIM methods.
Empirical evidence indicates that the shrinkage method for covariance matrix in portfolio optimization yields promising results for investors in the Vietnam stock market This approach enables the creation of optimal portfolios that achieve higher profits with reduced risk compared to traditional SCM methods Notably, during challenging market conditions, the shrinkage method continues to provide safe scenarios, effectively safeguarding asset portfolios against maximum losses Additionally, the portfolio turnover associated with the shrinkage method enhances overall investment performance.