Shrinkage estimation of covariance matrix for portfolio selection on vietnam

MINISTRY OF EDUCATION AND TRAINING THE STATE BANK OF VIETNAMBANKING UNIVERSITY OF HO CHI MINH CITY DOCTORAL DISSERTATION NGUYEN MINH NHAT SHRINKAGE ESTIMATION OF COVARIANCE MATRIX FOR PO

MINISTRY OF EDUCATION AND TRAINING THE STATE BANK OF VIETNAM

BANKING UNIVERSITY OF HO CHI MINH CITY

DOCTORAL DISSERTATION

NGUYEN MINH NHAT

SHRINKAGE ESTIMATION OF COVARIANCE MATRIX

FOR PORTFOLIO SELECTION ON VIETNAM STOCK MARKET

Ho Chi Minh City - 2020

MINISTRY OF EDUCATION AND TRAINING THE STATE BANK OF VIETNAM

BANKING UNIVERSITY OF HO CHI MINH CITY

DOCTORAL DISSERTATION

NGUYEN MINH NHAT

SHRINKAGE ESTIMATION OF COVARIANCE MATRIX FOR PORTFOLIO SELECTION ON VIETNAM STOCK MARKET

Ho Chi Minh City - 2020

Table of Contents

List of Abbreviations

List of Figures

List of Tables

CHAPTER 1: INTRODUCTION

1.1Vietnam stock market overview

1.2Problem statements

1.3Objectives and research questions

1.4Research Methodology

1.5Expected contributions

1.6Disposition of the dissertation

CHAPTER 2: LITERATURE REVIEW

2.1Modern Portfolio Theory Framework

2.1.1Concept of risk and return

2.1.2Assumptions of the modern portfolio theory

2.1.3MPT investment process

2.1.4Critism of the theory

2.2Parameter estimation

2.2.1Expected returns parameter

2.2.2The covariance matrix parameter

2.3Portfolio Selection

2.3.1Mean-Variance Model

2.3.2Global Minimum Variance Model (GMV)

CHAPTER 3: THEORETICAL FRAMEWORK

3.1Basic preliminaries

3.1.1Return

3.1.2Variance

i

3.3The estimators of covariance matrix

3.3.1The sample covariance matrix (SCM)

3.3.2The single index model (SIM)

3.3.3Constant correlation model (CCM)

3.3.4Shrinkage towards single-index model (SSIM)

3.3.5Shrinkage towards Constant correlation Model (SCCM)

3.3.6Shrinkage to identity matrix (STIM)

CHAPTER 4: METHODOLOGY

4.1Input Data

4.2Portfolio performance evaluation methodology

4.3Transaction costs

4.4Performance metrics

4.4.1Sharpe ratio (SR)

4.4.2 Maximum drawdown (MDD)

4.4.3Portfolio turnover (PT)

4.4.4Winning rate (WR)

4.4.5Jensen’s Alpha

4.4.6The statistical significance of the differences between two strategies on the performance measures

4.5VN - Index and 1/N portfolios benchmarks

CHAPTER 5: EMPIRICAL RESULTS

5.1VN – Index and 1/N portfolio performance

5.1.1VN – Index performance

5.1.21/N portfolio performance

5.2Portfolio out – of –sample performance

5.2.1Sample covariance matrix (SCM)

5.2.2Single index model (SIM)

5.2.3Constant correlation model (CCM)

ii 5.2.4 Shrinkage towards single index model (SSIM)

5.2.6 Shrinkage towards identity matrix (STIM)

5.3Summary performances of covariance matrix estimators on out – of – sample CHAPTER 6: CONCLUSIONS AND FUTURE WORKS

6.1Conclusions

6.2Future works

REFERENCES

iii

List of Abbreviations

APT: Arbitrage Pricing Theory

CAPM: Capital Asset Pricing Model

CCM: Constant Correlation Model

DIG: Development Investment Construction Joint Stock CompanyGDP: Gross Domestic Product

GICS: Global Industry Classification Standard

GMV: Global Minimum Variance Model

HOSE: Ho Chi Minh City Stock Exchange

HNX: Ha Noi Stock Exchange

ICF: ICF Cable Joint Stock Company

IPO: Initial Public Offering

MDD: Maximum Drawdown

MLE: Maximum Likelihood Estimator

MV: Mean - Variance

MVO: Mean-Variance Optimization

MPT: Modern Portfolio Theory

OLS: Ordinary Least Squares

PT: Portfolio Turnover

REE: Refrigeration Electrical Engineering Corporation

SAM: Sam Holdings Corporation

SCM: Sample Covariance Matrix

SIM: Single Index Market Model

SSIM: Shrinkage towards Single-index Model

SCCM: Shrinkage towards Constant Correlation Model

STIM: Shrinkage to Identity Matrix

SR: Sharpe Ratio

UPCoM: Unlisted Public Company Market

USD: United States Dollar

VIC: Vingroup Joint Stock Company

VND: Viet Nam Dong

VN - Index: Vietnam stock index

WR: Winning rate

YEG: Yeah1 Group Corporation

List of Figures

Figures Figure 1.1: The performance of investment funds in the period of 2009 – 2019 Figure 1.2: The performance of investment funds in the period of 2017 – 2019 Figure 1.3: Determinants of portfolio performance

Figure 2.1 The MPT investment process

Figure 4.1: The universe of stocks on HOSE from 2013 – 2019

Figure 4.2: The number of listed companies into industry groups on HOSE,

2019

Figure 4.3: The market capitalization of industry groups on HOSE, 2019 Figure 4.4: Back – testing procedure

Figure 5.1: VN-Index’s performance in the period of 2013 – 2019

Figure 5.2: Back-testing results of 1/N portfolio benchmark on out – of –

Figure 5.6: Compare the cumulative return between SIM and VN-Index

Figure 5.7: Back-testing results of CCM on out – of – sample from 1/1/2013 –

31/12/2019

Figure 5.8: Compare the cumulative return between CCM and VN-Index Figure 5.9: Back-testing results of SSIM on out – of – sample from 1/1/2013 –

Figure 5.10: Compare the cumulative return between SSIM and VN-Index Figure 5.11: Back-testing results of SSIM’s shrinkage coefficient (

vii

List of Tables

Tables Table 2.1: Summarized works related to portfolio optimization

Table 4.1: The sample dataset are collected in the period of 2011 – 2019 Table 5.1: The performance of VN – Index in the period of 2013 – 2019 Table 5.2: The performance of the 1/N portfolio benchmark from 1/1/2013 to

31/12/2019

Table 5.3: The performance of SCM from 1/1/2013 to 31/12/2019

Table 5.4: The performance of SIM from 1/1/2013 to 31/12/2019

Table 5.5: The performance of CCM from 1/1/2013 to 31/12/2019

Table 5.6: The performance of SSIM from 1/1/2013 to 31/12/2019

Table 5.7: The performance of SCCM from 1/1/2013 to 31/12/2019

Table 5.8: The performance of STIM from 1/1/2013 to 31/12/2019

Table 5.9: Summary back-testing results of covariance matrix estimators on

out – of – sample

Table 5.10: The movement value of shrinkage coefficient ( )

CHAPTER 1: INTRODUCTION

The target of this chapter is to introduce background information relating to the topic ofthis dissertation as well as to provide its key points such as objective, research questions,research questions and expected contributions

1.1 Vietnam stock market overview

The Vietnamese stock market has been developing for 20 years since the Ho Chi MinhCity Stock Exchange officially came into operation in July 2000 with the first twotickers, REE and SAM, hitting the historic turning point of the Vietnam stock market Sofar, the Vietnam stock market has had flourish development The number of listed andregistered companies trading on two stock exchanges is 1.605 companies, with the stockvolume of 150 billion The market capitalization as of the beginning of 2020 reachednearly 5.7 million billion VND, accounting for 102.74% of GDP, thereby, showing thesignificant role of Vietnam stock market to the economy

Vietnam's stock market operates with three official exchanges, including two listed(HOSE, HNX) and one unlisted (UPCoM) stock exchange In specifically, Ho Chi MinhCity Stock Exchange (HOSE) is considered the largest scale exchange By the end of

2019, HOSE had 382 listed companies; the trading volume reached 8.8 billion shares, theaverage trading value reached more than 4,000 billion VND/session Marketcapitalization on HOSE accounts for 88% of the total market, equivalent to 54.3% ofGDP Enterprises to be listed on the HOSE have to achieve higher standards in terms ofcharter capital, time of operation, performance, information disclosure, and shareholdersstructure Specifically, enterprises registered to list on the HOSE need to have theminimum charter capital contributed at the time of listing registration based on the bookvalue of VND 120 billion, higher than the amount of VND 30 billion on the Ha NoiStock Exchange (HNX) Regarding the time of operation, enterprises must have at leasttwo years of operation as a joint-stock company before the time of registration for listing

on HOSE while on HNX it takes one year Regarding performance criteria, HOSEstipulates that listed enterprises must have profitable business activities in the previoustwo years, one year more than required by HNX Regarding shareholder structure, HOSErequires businesses to have a minimum of 300 non-major shareholders holding at least20% of the voting stock of the company For HNX, this standard is a minimum of 100shareholders holding at least 15% of the shares Especially, HOSE has higher listingstandards for information disclosure Accordingly, the company must disclose all its debts

to internal persons, major shareholders, and related persons

The development of Vietnam's stock market has attracted a growing number of domesticand foreign investors, from 3000 trading accounts in 2000 to 2.5 million accounts in thecurrent period In particularly, there are about 33,000 accounts of foreign organizationsand individuals with a total value of securities held nearly 35 billion USD as of June 30,

2020 During this period, many foreign fund management companies also joinedVietnam'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

However, if you look at the period of 2017-2019, when Vietnam's stock market is facingmany difficulties due to complicated developments from the US-China trade war and therecession of major economies in the world gender has led to disappointing investmentresults of domestic and foreign investment funds, even the portfolio value of theseinvestment funds has fallen significantly more than the overall decline of the market(Figure 1.2).

Figure 1.2: The performance of investment funds in the period of 2017 - 2019

According to research by Brinson, Singer, Beebover (1991), asset allocation activity has a91.5% impact on the investment results of the portfolio, while securities selection,buying, and selling time and other factors affecting only about 9% of the portfolio results(Figure 1.3) However, asset allocation and optimal portfolio selection on Vietnam's stockmarket are relatively new and face many difficulties due to the following main reasons:

2% 2%

5%

Asset Allocation Security Selection Market Timing Other Factors

91%

Figure 1.3: Determinants of portfolio performance

First, the application of quantitative methods in asset allocation and optimal portfolio

selection is quite new in Vietnam's stock market, especially for individual investors Themajority of investors in the Vietnamese stock market are individual investors, who mostlyuse fundamental and technical analysis methods to select stocks The optimal investmentportfolio construction is primarily selected by investors according to their own feelings orsubjective judgments, not based on specific quantitative methods Besides, there are also

a few individual investors actively in using quantitative models to choose optimalportfolios However, these models are traditional models with certain limitations and nothighly applicable

Next, the particularities of the Vietnam stock market make it difficult for investors,

especially for investment funds who want to apply quantitative models in choosingoptimal portfolios The first is data problem Although the Vietnamese stock market hasgone through 20 years of development; however in the early stages, the number ofcompanies participating in the market is not much, and the quality of information stored

in this period is also not guaranteed This affects not only the length of the data but also

its quality Meanwhile, we know that today’s modern methods of portfolio optimizationrequire the size of research data to be large enough and the reliability of the data to beguaranteed Moreover, regulations on the Vietnamese stock market make it difficult forinvestors to build optimal portfolios using quantitative models For example, theregulations of daily trading limit on HOSE (±7%) and HNX (±10%), UPCoM (±15%).This means that the prices of stocks on the market can only fluctuate by a certain marginregardless of how bad or good the market moves The regulation can help stabilizeinvestors' sentiment during periods of deep decline or hot bull market, but it also preventsthe market events from being fully reflected in the prices of stocks, making it difficult forinvestors to forecast the fluctuation of their portfolio in the future.

Furthermore, the delay settlement date of a stock is up to some business days that

influence the testing and adoption of portfolio optimization models Investors whoexecute a buying transaction of stock today ( ), they must be waiting for three businessdays (T+3) to be able to selling this stock on Vietnam stock market Besides, after theysell this stock, they need to wait two more business days (T+2) to start with anotherbuying transaction or accept interest payments to the stock company during these twodays to have money to support their buying transaction immediately These limitations ofVietnam stock market increase risks and incur much transaction costs for investors.Moreover, these restrictions also make it difficult to apply the high – frequency tradingmodels on the Vietnamese stock market Therefore, in the process of building the optimalportfolio selection models, they need to consider and calculate the impact of theserestrictions on their models when applying the models in practice

Moreover, the liquidity risk is also one of the important factors leading to deviations in

the practical application of portfolio optimization models The market size is small andthe number of shares traded during the day is not much, leading to high liquidity risks forinvestors when they want to buy or sell stocks in large quantities Thus, when developingand back-testing the quantitative models, investors need to pay attention to this slippagefactor in the stock trading activity, especially the transactions with large volume ofbuying and selling in their portfolios Otherwise, the theoretical buying and selling prices

could be significantly different from the actual buying and selling prices, which in turnaffects the reliability of the optimal portfolio selection models.

Therefore, the research and selection of an appropriate portfolio optimization method isessential for investors on the Vietnam stock market In the next sections, the author willfocus on presenting suitable methods in selecting the optimal investment portfolios on thestock market in general and the Vietnamese stock market in particular

1.2 Problem statements and research gap

Modern Portfolio Theory (MPT) has been playing an important role in the selection andconstruction of investment portfolios for over 65 years, since it was firstly introduced byHarry Markowitz in his “Portfolio Selection” (1952) article The framework of MPT is toattain as highest return as possible for a certain level of risk through structuring theoptimal weights of various assets (Iyiola, 2012) Although it is broadly employed inpractical investment activities, the main assumptions of MPT model have been facinggreat challenges in recent years One of the main reasons comes from the two main inputs

of MPT that are the mean and the covariance matrix of asset returns

To implement the technique in practice, investors have to estimate the mean and thecovariance matrix of assets’ return At this point, sample mean as well as covariancematrix approaches are usually employed However, these estimators are not really stable

in many cases because of estimation error, this makes the weights of portfolio fluctuatecontinuously over time As a result, the mean – variance portfolios are difficult to beapplied in practice by the portfolio managers Moreover, many well – known empiricalevidences showed that these portfolios underperformed in term of mean and variancemetrics during the out–of–sample period (Michaud, 1989)

In general, there are two ways to overcome the challenges of MPT that are to initiate thenew approaches for estimating the expected return and covariance matrix of assets inportfolio optimization The well-known models have been utilized to estimate expectedreturn parameter such as Capital Asset Pricing Model (CAPM) and Fama – French

models The CAPM, which is considered as one – factor model, describes a linkagebetween systematic risk and expected return of assets Meanwhile, Fama – French statesthat the expected returns of assets should be explained by some other variables besides ofbeta coefficient of CAPM such as size risk, value risk factor, profitability, and investmentfactor In addition, the robust estimators are also applied by the researchers and portfoliomanagers to improve the expected return estimation, for example the truncated/trimmedmean or winsorized mean are employed by Martin, Clark and Green in 2010 Moreover,the other robust estimation methods like M – estimator or S – estimator, Bayes – Steinestimator are developed to solve the non-stationary returns limitation in estimating theexpected return input.

The improvement of estimating assets’ expected returns is one of the ways to remedyMPT’s shortcomings However, the results of Merton’s research (1980) showed that it isnot easy to measure the expected return (µ) In most asset price models, they made anassumption that there is a linkage between the assets’ expected return and the market’sexpected return ( ), in which ( ) is constant overtime Although this assumption makes theassets’ expected return estimation to be easier, it would still take a very long time series

to estimate µ accurately (Merton, 1980) Moreover, all we know that the assumption ofconstant expected return is not reasonable, but if this assumption is relaxed, theestimating µ will be even harder This paves the way for a second research direction,selecting portfolios based on the covariance matrix estimation instead of the expectedreturn estimation The estimation of covariance matrix parameter is an important researchdirection that researchers have paid special attention in the recent period, due to thepotential of this method in improving stability and minimizing risk in selecting theinvestment portfolios

The instability of the mean-variance portfolios comes from estimating the mean assetsreturn Thus, the minimum – variance portfolios have recently been used by manyresearchers and portfolio managers In this method, the investigated portfolios areprimarily based on the estimation of covariance matrix, making them less sensitive to

the estimation errors (Jagannathan and Ma, 2003) Moreover, Jagannathan and Masuggested that “the estimation error in the sample mean is so large that nothing is lost inignoring the mean altogether” This argument also provides detailed empirical evidenceshowing that the minimum-variance portfolio is likely to outperforms on Sharpe’s ratioand other performance metrics in out-of-sample period than any other mean-varianceportfolios (DeMiguel, 2005; Jagannathan and Ma, 2003).

Demiguel (2009) stated that although the minimum-variance portfolio does not depend

on the mean returns estimation, it is still under great impact of estimation error Thesensitivity of the minimum-variance portfolio to estimation error is quite interesting

“These portfolios are based on the sample covariance matrix, which is the maximumlikelihood estimator (MLE) for normally distributed returns Moreover, MLEs aretheoretically the most efficient for the assumed distribution; that is, these estimators havethe smallest asymptotic variance provided the data follows the assumed distribution” Atthis point, a question was raised that why the sample covariance matrix generatesunsuitable portfolios Huber (2004) answered that “the efficiency of MLEs based onassuming normality of returns is highly sensitive to deviations of the asset-returndistribution from the assumed (normal) distribution In particular, MLEs based on thenormality assumption are not necessarily the most efficient for data that depart evenslightly from normality” For portfolio selection, this is very useful as comprehensiveevidence indicates that the empirical return distribution is generally different from normaldistribution

Moreover, although the core of minimum - variance portfolio researches relies solely onhow to estimate reliably the covariance matrix, however, almost traditional approaches ofcovariance matrix estimation such as using the sample covariance matrix (SCM) orordinary least squares (OLS) face many problems in the case of high-dimensionalportfolios Having large dimensionality means that it is easier to get unexpected anduncontrollable errors in some of computational steps, and the sample data may not beenough for the estimation of the true covariance matrix These lead to the estimated

covariance matrix to become ill-conditioned or even singular which is very popular inmatrix computation research Consequently, the portfolios selected from considering thesample covariance matrix often perform poorly and fail in generating profit To solve thisproblem, some new estimators of covariance matrix are conducted by many researchersand portfolio managers There have been many approaches proposed in the literature, andamong them, Ledoit and Wolf (2003) proposed to select the optimal portfolios by usingthe shrinkage estimator of covariance matrix This method is a combination between arough sample covariance matrix and a high-structured target matrix to achieve thebalance between bias and variance The balance can be customized, which is the trade-offbetween bias and estimation errors recognized by shrinkage coefficients The shrinkagetechnique shows theoretically and empirically attractive approach to a high-dimensionalportfolio's covariance estimation problem since it ensures a well-defined covariancematrix is achieved Liu (2014) estimated the covariance matrix by applying the weightedaverage of different shrinkage target matrices, instead of using a single shrinkage targetmatrix as Ledoit and Wolf method Next, by inheriting the potentials and development ofRandom Matrix Theory, Ledoit and Wolf extended their pilot works (Ledoit & Wolf,2017a, 2017b) by using a nonlinear transformation applied for the eigenvaluesconsidering solely the sample data Also, coefficient asymptotically leads to themaximization of the out-of-sample expected utility Then, they performed both numericaland empirical investigation where the out-of-sample behavior of the obtained estimator isanalyzed and it shows remarkable improvements over the simple diversification, and itsrobustness is expressed to the deviations from normality As a matter of fact, DeMiguel et

al (2013) provided an important review paper of shrinkage frameworks and theirpractical application especially for asset optimization, and then they also discussed on anew category of shrinkage-based techniques for the means of return and thecorresponding covariance matrix, as well as, the weights in the asset As a slightenhancement for this research approach, the work of Candelon et al (2012) presentedsuch a kind of double shrinkage adaptation to improve the general stability of the

estimation on even small sample sizes covariance matrices via taking into account a ridgeregression approach to shrink the all the weights towards the equally-weighted asset.

Research gap

Clearly, the selection of covariance matrix estimators influences the performance ofoptimized portfolios The above approaches for covariance matrix estimations pose anopportunity for investors, who usually apply the traditional estimator of samplecovariance matrix, to improve their portfolio performance by altering the new estimators

of covariance matrix in their portfolio optimization models The problem is that there isnot a complete and sure research base regarding the effectiveness of out – of – sampleperformance when making changes to covariance matrix estimations In the other words,there is no solid foundation involved in this field, and portfolio managers would not risktheir money to make investments based on unproven or rigorous research

Moreover, the traditional estimator of covariance matrix is facing many difficulties anddoes not bring the expected results because the development of the financial market hasresulted in the number of investment assets in the market increasing rapidly and muchlarger than the observed sample, from that requiring the new estimators of covariancematrix to be studied and applied Besides, there is still a lot of controversy surroundingthe applicability and effectiveness of covariance matrix estimation methods in differentmarkets

Furthermore, the robust estimators of covariance matrix are mainly applied and tested inthe developed markets; there are not many researches on emerging and developingfinancial market In particular, there is almost no research in Vietnam related to theselection of covariance matrix estimators for optimizing the portfolios, especially in theshrinkage methods Therefore, there is a gap for the author to investigate the level ofinfluence of covariance matrix estimators on the minimum – variances optimizedportfolios, and to test the performance of these estimators on Vietnam stock market

1.3 Objectives and research questions

The objective of this dissertation is to investigate that whether the investors can improvethe performance of minimum – variance optimized portfolios by altering the estimators ofcovariance matrix input Besides, based on the results of out – of – sample portfolioperformance metrics, the dissertation is going to select the suitable estimators ofcovariance matrix for portfolio optimization on Vietnam stock market

In order to achieve the above objectives, this dissertation will attempt to answer theresearch 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 – varianceoptimized 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 inthe portfolio changes?

Question 3: Could the alternation of covariance matrix estimation for portfolio

optimization beat the traditional estimator of covariance matrix and benchmarks of stockmarket on out - of - sample?

1.4 Research Methodology

In order to achieve the research objectives and answer the above questions, the authorneeds to choose an appropriate research method In this dissertation, the author appliedsome research methodology as follows:

First, there are six estimators of covariance matrix used in this study to examine how thechange of covariance matrix estimation affects the optimal portfolio selection Theseestimators of covariance matrix includes the sample covariance matrix (SCM), the singleindex model (SIM), the constant correlation model (CCM), the shrinkage towards single

index model (SSIM), the shrinkage towards constant correlation model (SCCM), and theshrinkage towards identity matrix (STIM) In which, SCM is seen as the traditional orstandard estimator of covariance matrix while SIM and CCM are called as model – basedapproaches; and SSIM, SCCM, STIM are mentioned as shrinkage methods In addition,the minimum – variance optimization is selected for generating the optimal portfoliosthrough the estimated covariance matrices by the above estimators.

Second, to evaluate the feasibility and potential application of the estimators ofcovariance matrix mentioned above, a back-testing process based on the Pythonprogramming language has been developed and applied in this study The back – testingprocess was simulated on the back – testing platform in the previous research of Tran et

al (2020) The statistical properties of estimators of covariance matrix will be examinedand clarified by the back – testing procedure, from that providing insight into whetherwhich estimator will be able to make profit in the reality

Third, through the back – testing process, the portfolio performance metrics which areconsidered as the important criteria for evaluating the portfolios will be estimated.Besides of the basic portfolio evaluation criteria such as portfolio return and volatility ofportfolio, other useful evaluation criteria are also calculated in this research includingportfolio turnover, maximum drawdown, winning rate and Jensen’s Alpha In order tocalculate these portfolio evaluation criteria, the author used a “rolling – horizon”technique that defines as a reactive scheduling method that “solves iteratively thedeterministic problem by moving forward the optimization horizon in every iteration;assuming that the status of the system is updated as soon as the different uncertain or notaccurate enough parameters became to be known, the optimal schedule for the newresulting scenario (and optimization horizon) may be found” Silvente et al.(2015) Therolling – horizon approach allows the investors to update or adjust their input data foroptimal portfolio selection based on currently available information The technique will

be presented more clearly in the next sections Moreover, the input data for the back testing process are weekly stock price series, which will then be converted to weekly

-return during the optimization procedure One more thing, when calculating the portfolioperformance metrics, the transaction costs are also considered at every rebalancing point.Lastly, these estimated performance metrics are applied for comparing the differencesamong the estimators of covariance matrix in selecting the optimal portfolios To makesure that there are significant differences of performance metrics between the two certainestimators, the p – values are computed following the bootstrapping methodology that ismentioned in the research of DeMiguel (2009).

1.5 Expected contributions

After answering the research questions and achieving the research objective, thisdissertation will expect to make some contributions as follows:

First, through empirical research on the Vietnamese stock market, the dissertation has

added concrete evidence that investors can improve their portfolio’s investmentperformance by using the estimation methods to adjust the covariance matrix parameter

in the portfolio optimization These empirical research results show that the model –based estimators of covariance matrix (SIM, CCM) and the shrinkage estimators ofcovariance matrix (SSIM, SCCM, STIM) give much superior results compared to thetraditional sample covariance matrix (SCM) on almost tested portfolios (N = 50, 100,

200, 350) and most portfolio performance metrics such as portfolio return, level of risk,portfolio turnover, maximum drawdown, winning rate and Jensen’s Alpha In particular,this superiority is more evident when the number of stocks considered in the portfoliotends to increase

Second, among the estimators of covariance matrix mentioned in this dissertation, the

shrinkage estimators of covariance matrix show outstanding results compared to otherestimators and benchmarks of market on almost the portfolio evaluation criteria,especially in the case of high – dimensional portfolios Besides, the shrinkage towardsconstant correlation model (SCCM) reflects the most efficient level of optimal portfolio

selection compared to the shrinkage towards single index model (SSIM) and shrinkagetowards identity matrix (STIM).

Third, another new point of the dissertation is to consider the performance of the

estimators of covariance matrix under the influence of dimension of covariance matrixand the effect of transaction costs in computing the portfolio performance on out – of –sample Specifically, the dissertation has evaluated the effectiveness of the estimationmethods when the number of stocks in the portfolio changes from N = 50 to N = 350while the transaction costs considered at every rebalancing point is 0.3%

Fourth, one more point is that the dissertation uses a variety of criteria to measure the

effectiveness of a portfolio In order to have multi-dimensional perspectives and areasonable assessment of the effectiveness of estimation methods in the selection of theoptimal portfolio, the dissertation employs more portfolio performance metrics such asSharpe ratio, portfolio turnover, maximum drawdown, winning rate or Jensen’s Alpha toevaluate the performance of the selected portfolios, instead of using the commonperformance metrics including return and variance criteria like the previous researches.This can be seen as the author's effort in analyzing the effectiveness of the covariancematrix estimation methods compared to previous studies

The final contribution of this dissertation is that all estimators of covariance matrix havebeen experimented on Vietnam stock market where is considered as an emerging market.Although, there are many researchers and financial practitioners using the estimationmethods to optimize their investment portfolios on the developed financial market such

as US or European, however, there are very few researches employing this method on theemerging market, especially on Vietnam stock market The empirical results of this studywill support the researchers and investors to see clearly the different performance of theestimators of covariance matrix between the emerging financial market and the developedfinancial market

1.6 Disposition of the dissertation

This dissertation includes 6 chapters, in which Chapter 1 focuses on the problemstatements, objectives of the study, research questions, research methodology as well asexpected contributions of the research The next chapters are presented as follows:

Chapter 2, Literature Review, provides an overall review about relevant researches

regarding portfolio optimization before developing a specific theoretical framework andmethodology in the next chapter

Chapter 3, Theoretical Framework, presents the foundation theory which underlies this

dissertation First, some basic preliminaries and portfolio optimization problem will beintroduced in this chapter, and then theory regarding the covariance matrix estimationswill be developed

Chapter 4, Methodology, deploys the basic methodology used for answering research

questions in the dissertation

Chapter 5, Empirical Results, withdraws the empirical results in the dissertation based on

the back – testing performance of covariance matrix estimators on out – of – sample

Chapter 6, Conclusions and future works, make some conclusion on findings of the

research as well as discuss about future works in the next researches

CHAPTER 2: LITERATURE REVIEW

The purpose of the chapter is to provide a deep-dive literature review related withportfolio optimization as well as motivating the theoretical framework and methodology

of this dissertation

2.1 Modern Portfolio Theory Framework

Harry Markowitz is recognized as one of the pioneers in contributing his theory tofinancial economics In 1990, Markowitz deserved to receive the Nobel Prize for hiscontributions related to the portfolio selection method The “Portfolio Selection” articlewas first published in 1952 in the journal “The Journal of Finance” and then written in abook titled “Portfolio Selection: Efficient Diversification” in 1959 Markowitz'sgroundbreaking work shaped what we today call modern portfolio theory (MPT)

The MPT is a method of investment for selecting and building investment portfolios thatseeks to maximize the expected return on the portfolio for a given amount of portfoliorisk or to reduce the risk of the expected return level by carefully choosing the weights ofdifferent assets Although the MPT is commonly applied in the investment sector, thebasic assumptions of the MPT remain controversial in recent years

The MPT which is an improvement on the classical quantitative models plays a pivotalrole in the mathematical modeling of finance This framework motivates thediversification to protect the investment portfolios from market risk as well as specificrisk of company Sometimes, the theory is also called Portfolio Management Theorybecause it supports the investors to classify, evaluate and measure both the expected riskand return The main idea of this theory is its quantification of relationship between riskand return, and assuming that investors have to be compensated for taking risks

The diversification concept of MPT is to select the investment portfolios that have lowerrisk than any security in the portfolios The diversification can lower the investment risk

no matter what the correlation between security returns is positive or negative More

technically, the MPT considers a security return as “a normally distributed function” andrisk as “a standard deviation of return” The portfolio’s return will be determined as theweighted combination of the securities’ returns The total variance of the portfolio returnwill be reduced if the correlation among the securities’ returns is not perfectly positively.Moreover, the assumptions of MPT state that “investors are rational and market isefficient”.

Investing can be seen as a trade-off between the expected return and the risk In general,

an asset with higher expected return is typically riskier (Taleb, 2007) The MPT explainsthe way for choosing a portfolio with maximum expected return for a given level of risk.And, this model may also clarify how to get a portfolio with the lowest possible risk for acertain amount of expected return

2.1.1 Concept of risk and return

Return

Return could be seen as a foundation motivation and an important reward for anyinvestment project Returns can be defined in terms of realized return (the receivedreturn) and expected return (the return anticipated by investors over future investmentperiods) When we refer to the expected return, it shows us that this is a forecasted orestimated return and is likely to happen or not Meanwhile, the realized returns obtained

in the past help the investors to calculate cash inflows such as: “dividends, interest,incentives, capital gains and so on” Moreover, an investor can estimate the total profit orloss over a given investment period and considered as a percentage of return on theamount of initial investment This is seen as a cumulative return of an investment Withrespect to investment in securities, the return includes the dividends and the capital gain

or loss at the point of selling of such securities

In investment activities, risk is defined as the unpredictability of the return of aninvestment in the future In other words, risk is understood as the possibility of profit infact different from expectations Risk refers to the probability that the real outcome(return) from an investment will not be equivalent to the anticipated outcome Risk withregard to a business can be defined as the probability that the actual outcome of afinancial decision can vary from the estimated one Investments with greater returnvolatility are perceived to be riskier than investments with lower return volatility

Moreover, risk and uncertainty need to be clearly distinguished “Risk is stated as asituation where the possibility of occurring or non-occurring of an event can bequantified and measured, while uncertainty is a situation where this possibility cannot bemeasured Therefore, risk is a situation where probabilities can be assigned to an event onthe basis of facts and figures available regarding the decision On the other hand,uncertainty can be seen as a situation where either the facts and figures are not available,

or the probabilities cannot be assigned” (Iyiola, 2012)

As we know, no investor is able to predict with certainty the outcome of an investment.However, statistical estimation methods can be used to determine the risk of an estimatedreturn, thereby measuring the difference between the expected return and the actual return

on an investment Therefore, the statistical methods such as standard deviation andvariance are usually used to estimate the risk of an investment

2.1.2 Assumptions of the modern portfolio theory

The modern portfolio theory’s framework contains a lot of assumptions relating to investorsand markets Markowitz developed his optimal portfolio selection theory based on the

following main assumptions: First, investors are identical, risk averse, and rational Second, investors seek to minimize risk of portfolios while maximizing expected returns of the portfolios Third, investors select portfolios solely on the basis of their expected returns and risk, where the latter is measured as the variance of portfolio returns Fourth, asset returns are stationary over time Fifth, an investor knows all assets’ prices considered

for investment, and update his/her portfolio according to changes in asset prices

immediately and costlessly Sixth, asset prices are exogenous (no investor’s choices affect

asset prices) Seventh, all assets considered for investments are infinitely liquid, thus

trades of any size can be made on those assets Eighth, investors can take negative or

“short” positions on assets Ninth, investors can borrow and lend without risk and at the

same interest rate Tenth, investors incur no transaction costs (e.g taxes, brokerage fees,

bid-ask spreads, foreign exchange commissions) Finally, investors will allocate their

entire budget to portfolio (no savings)

In MPT equation, some of these premises are explicit such as the use of normal

distributions for model returns Some are tacit including tax indifference and transaction

fees None of these premises are completely valid, and in some degree each compromises

the MPT The key assumption of the MPT is that the market theory is efficient

2.1.3 MPT investment process

Fabozzi et al (2002) contended that the most common function of MPT is asset allocation

Initially, investors must evaluate the assets in which they may invest, and any restrictions that

they will face The next step is to get estimates of the investable securities' returns,

correlations and volatility Then the predictions are used in an optimization process and in

practice a result that suits individual expectations is eventually enforced (Figure 2.1)

Expected

Return

Covariance matrix

2.1.4 Critism of the theory

The MPT has been widely criticized given its theoretical relevance; its simplisticassumptions are a prevailing bias Some critics doubt its effectiveness as an investmentstrategy, since in many cases its financial market model does not suit reality In recentyears, the MPT's basic underlying premises have been strongly questioned by such fields

as behavioral economics Efforts to apply the theoretical framework of this model to theconstruction of the optimal portfolio in practice face many difficulties, due to theinstability of the input parameters in the optimization problem Recent studies show thatthese types of instabilities will disappear when a “regularizing constraint” or “penaltyterm” is incorporated into the optimization process

The market is not really modeled by the MPT

The metrics as risk, return and correlation applied by modern portfolio theory aredependent on “ex-ante” values that include mathematical assumptions about the future(where expected value is explicit in the calculations, while implicit in definitions ofvariance and covariance) In reality, investors need to replace forecasts relied onhistorical asset return and volatility measurements for those values in equations Usually,when a new event which was not previously reflected in historical data happens in thefuture, such expected values will fail to predict correctly the forthcoming movement ofportfolio’s relating metrics More practically, an investor faces many difficulties toestimate key parameters from ex-post data of market as the MPT tries to model riskincluding the possibility of losses, but not mention the reason why losses happen Thereare no structural, probabilistic measures in the risk measurement used Contrary to otherengineering approaches to risk management, this leads to a significant difference

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 otherconsequences More specifically, its full reliance on asset prices makes the MPT more

vulnerable in cases where the market does not reflect the normal standards because ofproblems stemming from “information asymmetry, externalities, and public goods” Inaddition, a business could have strategic or social objectives which affect its investmentdecisions, and may have personal goals for an individual investor In both casesknowledge that is not historical returns is important as indicated by MPT.

The MPT does not take cognizance of its own effect on asset prices

Diversification decreases unyielding risk, but at the expense of rising the systemic risk.Diversification can make an investor select securities without analyzing theirfundamentals, since he/she is only focused on removing the unsystematic risk of theportfolio (Chandra, 2003) This artificially enhanced demand drives up the securities’price which would be of little fundamental interest if evaluated individually Theconsequence is that the entire portfolio is more costly and a positive portfolio returnprobability is small This implies portfolio risk continues to grow

2.2 Parameter estimation

The phenomenon of mean – covariance optimization is claimed of having estimationerror In the study of Markowitz (1952), he stated that the theoretical soundness of theproposed portfolio selection approach was more emphasized, which results in the factthat MTP practical application was less focused To implement MTP realistically, themeans and covariance of assets returns need estimating because they are not previouslyknown Thereafter, the estimated results are employed to find solutions for optimizationproblems of investor (Elton et al 2012) A great number of studies conducted previouslyconclude that this brings about the most crucial disadvantage of the mean-varianceapproach: there might be estimation error when using inappropriate moments for plugin(Michaud (1989); Chopra and Ziernba (1993)) At this point, the optimizer is notconscious that the data inputs are only statistical estimates and there are not anycertainties, which causes the flaw

The classical statistical procedure of estimating assets returns and covariance is to use ex-post returns to calculate the assets’ respective sample estimates The assumption of thisapproach is that historical data can partially describe the future tendency of assets’ price.Nevertheless, some previous researches have reported this ordinary procedure’sproblems DeMiguel (2009) confirmed that using sample estimates as input parametersdecides not necessarily result in a mean- variance optimized portfolios that outperform anequally – weighted one In this line, Jobson and Korkie (1980) also showed a relativeresults Besides, Best and Grauer (1991) indicated that estimation error is consequentlytransferred into the weights of optimized portfolios, which causes a deviation betweenestimated optimal weights and the true ones.

Chopra and Ziernba (1993) suggested that the error of expected returns have more impact

on the out-of-sample performance of optimal portfolios when compared to that ofcovariance matrix They also potentially explained the reason why there was disparity inpeople’s concentration between expected return vector and the covariance matrix in thepast Not to be outdone, since the financial markets was introduced, forecasting reins hasstarted to be all the rage beyond the concept of capital allocation as it was Nonetheless,Michaud (2012) rejected the Chopra and Ziernba (1993)’s findings when noted that there

is a recurring widespread errors in the research observing the importance of estimationerror which is relative to the covariance matrix in return In more details, they indicatedthat paper by Chopra and Ziemba (1993) was relied on a limited in-the-sample studywhich is thereby not bearing when concerning the effects of error in estimating the strictout-of-sample MV optimization On contrary, they found that in fact, the estimation error

in covariance matrix may dominate the portfolio optimization process when the number

of assets increases

Therefore, it is obvious that the estimation error should be considered as one of the mostimportant aspect of mean-variance optimization To obtain a more potential optimizedportfolio in out-of-sample period, lessening such error is highly necessary, whicheventually significantly benefits assets managers who employ MVO Before some

scholars emphasized the important role of predicting covariance matrix in MVO, therehad been a great number of extensive literatures focusing on forecasting assets return.Thus, the demand for comprehensive researches on this approach and making comparisonbetween results of different methods has been increasing recently.

The following parts will (1) include an overview of different solutions for forecastingexpected returns and (2) deliver literatures review that refer to the method of reducingestimation error in covariance matrix

2.2.1 Expected returns parameter

Foreign researches on expected return estimation

The concept of Capital Asset Pricing Model (CAPM) was first proposed by WilliamSharpe in 1964 It is based on the foundation of MPT and also known as one-factormodel for forecasting the expected return of assets Specifically, it is the relationshipbetween systematic risk and assets’ expected return that is explained by this model.However, in 1992, Eugene Fama and Kenneth French discovered that the beta coefficientused in CAPM model did not precisely describe the expected return of Americansecurities between 1963 and 1990 Thus, they began to observe the movement of twoclasses of stocks which tended to outperform the whole market The two groups includedthe small caps and the stocks that have high Book to Market Equity ratio Thereafter, theyadded two more factors which can measure the sensitivity of the portfolio to two classes

of stocks to CAPM and formed the three-factor model in 1993 This model then wasproven effective when has been through various empirical tests in developed stockmarkets and emerging ones The three-factor model of Fama – French implies that return

of a specific portfolio is relied market returns, size risk and value risk factors A decadelater, they added two new factors in the model including profitability and investmentfactors to form a 5-factor model in 2014 which can predict the movement of expectedreturn

Moreover, the researchers also employed robust estimators to improve the expected

return estimation For examples, instead of using the sample mean m ∑ to estimatepopulation expected returns, the sample truncated/trimmed mean or winsorized can beused (Martin et al 2010) The trimmed mean is calculated after removing the k% mostextreme values, while the winsorized mean is estimated after replacing those values withthe next k% most extreme values However, both estimators are more efficient todeviations from distributional assumptions on assets return, so if the returns are notstationary, not only both estimators but also the other robust estimators like M – estimator

or S – estimator could not provide the true estimates of expected returns

To improve performance in the face of non-stationary returns and to automate theselection of tuning parameters, researchers have turned to Shrinkage Estimators Theshrinkage approach begins with the observation that, much like uncertainty in actualexpected returns (asset risk), uncertainty in the estimate of expected returns (estimationrisk) implies a loss of investor utility (Jorion, 1985) Thus, the optimization problemshould minimize utility loss from selecting a portfolio based on sample estimates, instead

of true values In this line of thought, the solution is clearly not to estimate each asset’sexpected return individually, but to select an estimator that minimizes utility loss fromaggregate parameter uncertainty Jorion (1986) suggests using a Bayes–Stein

estimator that shrinks each asset’s sample

Through the simulation, this estimator

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accurately (Merton, 1980) Moreover, all we know that the assumption of constantexpected return is not reasonable, but if this assumption is relaxed, the estimating µ will

be even harder This paves the way for a second research direction, selecting portfoliosbased on the covariance matrix estimation instead of the expected return estimation

Local researches on expected return estimation

The local studies mainly developed in the direction that selects the optimized portfoliosbased on estimating the expected return parameter Phuong Nguyen (2012) used thesingle-factor model (SIM) to measure risks and to determine expected returns of stocks inthe construction industry Linh Ho (2013) also applied this model to measure the risksand expected returns of real estate stocks listed on the Ho Chi Minh Stock Exchange(HOSE) Moreover, Truong and Duong (2014) relies on CAPM and Fama - French three-factor models to optimize the investment portfolios on HOSE Tram Le (2014) alsoemployed Fama - French three-factor model to measure risks and estimate expectedreturns of stocks on Vietnam stock market Besides, Nguyen Tho (2010) applied thearbitrage pricing theory (APT) to examine the stock price behavior of an emerging stockmarket including Vietnam and Thailand stock markets

2.2.2 The covariance matrix parameter

Foreign researches on covariance matrix estimation

This is an important research direction that researchers have paid special attention to inthe recent period, due to the potential of this method in improving stability andminimizing risks in selecting the investment portfolios The traditional covariance matrixestimation for portfolio optimization is to use the sample covariance matrix (SCM).However, Michaud (1989) has proved that this estimation method brings manyshortcomings and limitations that contain a lot of statistical errors and become ill –conditioned when the number of samples is comparable with the number of assets.Michaud described this phenomenon as "Markowitz enigma" Frankfurter, Phillips andSeagle (1971) also confirmed that the sample covariance matrix estimation for MPT

model does not bring superior results compared to an equally weighted portfolio selectionthat DeMiguel (2009) called as the nạve 1/N portfolio.

Based on the single-index model given by Sharpe (1964), the researchers estimated thecovariance matrix to select the optimal portfolio The covariance matrix estimated fromthe one-factor model will be calculated as follows ∑ ∑ The SIM has

three advantages over standard MPT approach that the covariance matrix is estimated by

sample First, SIM requires estimating only 2N+1 parameters to construct the covariance

matrix versus the N(N+1)/2 parameters required by the standard approach (exceeds 2N

+1 for N ≥ 4, N is the number of assets) Second, whenever the investors add a new asset

to their sample, they only need to estimate its β and to update their precision matrix,versus estimating the asset’s variance and covariance with every other asset in theirsample (N + 1 parameters) and inverting their new covariance matrix to update their

precision matrix Third, the SIM approach only requires T > 2 observations to estimate β

and for every asset (as well as ) and thus to estimate the precision matrix On thecontrary, the standard MPT approach requires T > N (T is the sample size) In theresearch of Senneret et al., (2016), there is ample evidence that the SIM approachproduces portfolios with much less sensitive to estimation error than the standardapproach does, therefore performing better across a range of risk and return metrics.However, both the standard approach and SIM method are vulnerable to estimation errorfrom using the sample mean returns vector (SIM is also exposed to estimation errorfrom ̂ )

To overcome the limitation of the SIM, Elton and Gruber (1973) proposed the otherapproach that is called Constant Correlation Model (CCM) The model assumes that allstocks have the same correlation, equal to the sample (historical) mean correlation TheCCM implicitly assumes that the historical correlation matrix only contains informationabout the average correlation for future periods, but no information about pairwisecorrelations’ deviation from that average Clearly, this is a strong assumption in thismodel Elton et al (2009) summarizes the literature’s findings that the CCM is moreaccurate forecasts of future covariance matrices than the sample covariance matrix and

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SIM, from that producing portfolio with superior performance than competing models,across a range of metrics However, this CCM model still faces many problems inestimating large dimensional covariance matrix for portfolio choice.

Ledoit và Wolf (2003) introduced a new approach to estimate the covariance matrix forportfolio selection that is called shrinkage method The approach is the weightedcombination between “the sample covariance matrix and a target matrix of the samedimensions The objective is to reach a weighted average that is closest to the truecovariance matrix according to an intuitively appealing criterion”, (Kwan, 2011) Thisapproach becomes more suitable (i) the less the bias introduced by the shrinkage targetmatrix, (ii) the larger the noise in the data, and (iii) the larger N/T In general, the idea is

and equal to the mean correlation, while Ledoit and Wolf (2003b) suggest ∑

index model) that the known feature that stock returns have a factor – model structure.However, Ledoit and Wolf (2004) proposed a target matrix that does not based anybenefit from the application – specific knowledge like the two previous shrinkage

methods The target matrix is an Identity matrix ∑

the elements of the principal diagonal are ones and all other elements are zeros” With theintroduction of this kind of this target matrix, Ledoit and Wolf tried to answer thequestion of whether an investor can choose an optimal portfolio if they are not using anythe domain knowledge in finance field They find that “all three improved estimators ofthe covariance matrix dominated the sample covariance matrix, but there was no clearwinner among the three If anything, shrinkage to the constant-correlation matrix wasbest for portfolio sizes N ≤ 100, whereas shrinkage to the single-factor matrix was bestfor portfolio sizes N ≥ 225”

In an important survey, Bai and Shi (2011) summarized some of notable contributionswidely applied in high-dimensional estimation of covariance matrix such as shrinking,observable and implicit variables, even from Bayesian approach, and random matrixtheory Whereas, Yang et al (2014) suggested a kind of hybrid covariance matrixestimation approach based on robust M-estimation and a shrinkage calculation of Ledoitand Wolf (2004) Ikeda and Kubokawa (2016), on the other hand, found a class ofgenerally weighted estimators involving a linear combination of sample covariancematrices with rule-based estimators and linear shrinkage estimators with no additionaland special factors under the component scheme Konno (2009) proposed an estimationapproach for large-dimensional covariance matrices having complex types of multivariatenormal distributions when the dimensions of variables were greater than the number ofobserved samples Unbiased risk estimates for certain groups of global covariancematrices were derived by considering these techniques under real and complex invariantquadratic forms of loss functions In another scenario, Chen et al (2010) adopted theshrinkage approach, and suggested an estimator of sample covariance relied on reducingthe mean square error in Gaussian samples.

In the recent time, the newer work has focused on non – linear shrinkage estimators thatcontinue to answer the question whether it is “possible to generalize and improve linearshrinkage to the identity matrix in the absence of financial knowledge or in other wordswhether the investors can be totally ignorant about the true covariance matrix and still dobetter than generic linear shrinkage” (Ledoit and Wolf, 2018) The non – linear shrinkageestimators do not try to predict the true covariance matrix, this shrinkage methodconsiders the eigenvalues distribution of covariance matrix instead So the non-linearshrinkage method shows the flexibility in estimating eigenvalues of covariance matrixcompared to the linear shrinkage method, which can only estimate specific eigenvaluevalues via the linear function Ledoit and Wolf (2012, 2015) proposed the QuESTfunction to estimate the covariance matrix in the non-linear shrinkage method with theassumption that the number of investment assets and the observed data sample are

approaching to infinity, this method is called an indirect nonlinear shrinkage Lam (2016)continued to improve the non-linear shrinkage method by introducing a new approachNERCOME (Nonparametric Eigenvalue-Regularized Covariance Matrix Estimator) byseparating the observation sample into two separate parts, partly for calculatingeigenvectors and the rest for calculating eigenvalues related to the above eigenvectors.Ledoit and Wolf (2018) once again developed the applicability of non-linear shrinkagemethod by combining the power of 3 components which are the fast processing speed oflinear shrinkage method, the accuracy of the QuEST function and the transparency level

of the NERCOME method LW called this method as a direct nonlinear shrinkage.However, these are still new methods and have not been applied much in practice, due tothe controversy surrounding the applicability of these methods

Local researches on covariance matrix estimation

Vo Quy and Nguyen Hoang (2011) applied the Markowitz portfolio theory for estimatingthe risk tolerance of investors in HOSE and based on the known level of risk to optimizeportfolios Huyen Nguyen (2015) with the topic of applying modern financial theory tomeasure risks in stock investment on Vietnam stock market However, these portfoliooptimization models mainly base on the traditional method that applies the samplecovariance matrix (SCM) to estimate the covariance matrix for portfolio selection.According to the author's research, up to now, there have not been many studies inVietnam to optimize the securities portfolio through changing the covariance matrixestimation in modern portfolio theory