Shrinkage estimation of covariance matrix for portfolio selection on vietnam

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 F

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 iv

List of Figures vi

List of Tables viii

CHAPTER 1: INTRODUCTION 1

1.1 Vietnam stock market overview 1

1.2 Problem statements 6

1.3 Objectives and research questions 11

1.4 Research Methodology 11

1.5 Expected contributions 13

1.6 Disposition of the dissertation 13

CHAPTER 2: LITERATURE REVIEW 16

2.1 Modern Portfolio Theory Framework 16

2.1.1 Concept of risk and return 17

2.1.2 Assumptions of the modern portfolio theory 18

2.1.3 MPT investment process 19

2.1.4 Critism of the theory 20

2.2 Parameter estimation 21

2.2.1 Expected returns parameter 23

2.2.2 The covariance matrix parameter 25

2.3 Portfolio Selection 30

2.3.1 Mean-Variance Model 30

2.3.2 Global Minimum Variance Model (GMV) 32

CHAPTER 3: THEORETICAL FRAMEWORK 34

3.1 Basic preliminaries 34

3.1.1 Return 34

3.1.2 Variance 35

3.2 Portfolio Optimization 36

3.3 The estimators of covariance matrix 37

3.3.1 The sample covariance matrix (SCM) 38

3.3.2 The single index model (SIM) 39

3.3.3 Constant correlation model (CCM) 41

3.3.4 Shrinkage towards single-index model (SSIM) 42

3.3.5 Shrinkage towards Constant correlation Model (SCCM) 44

3.3.6 Shrinkage to identity matrix (STIM) 47

CHAPTER 4: METHODOLOGY 51

4.1 Input Data 51

4.2 Portfolio performance evaluation methodology 55

4.3 Transaction costs 59

4.4 Performance metrics 60

4.4.1 Sharpe ratio (SR) 60

4.4.2 Maximum drawdown (MDD) 61

4.4.3 Portfolio turnover (PT) 61

4.4.4 Winning rate (WR) 62

4.4.5 Jensen’s Alpha 62

4.4.6 The statistical significance of the differences between two strategies on the performance measures 63

4.5 VN - Index and 1/N portfolios benchmarks 64

CHAPTER 5: EMPIRICAL RESULTS 66

5.1 VN – Index and 1/N portfolio performance 66

5.1.1 VN – Index performance 66

5.1.2 1/N portfolio performance 69

5.2 Portfolio out – of –sample performance 72

5.2.1 Sample covariance matrix (SCM) 72

5.2.2 Single index model (SIM) 76

5.2.3 Constant correlation model (CCM) 79

5.2.4 Shrinkage towards single index model (SSIM) 82

5.2.5 Shrinkage towards constant correlation model (SCCM) 90

5.2.6 Shrinkage towards identity matrix (STIM) 95

5.3 Summary performances of covariance matrix estimators on out – of – sample 99

CHAPTER 6: CONCLUSIONS AND FUTURE WORKS 105

6.1 Conclusions 105

6.2 Future works 111

REFERENCES

List of Abbreviations

APT: Arbitrage Pricing Theory

CAPM: Capital Asset Pricing Model

CCM: Constant Correlation Model

DIG: Development Investment Construction Joint Stock Company GDP: 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

Figure 1.1: The performance of investment funds in the period of 2009 – 2019 2

Figure 1.2: The performance of investment funds in the period of 2017 – 2019 3

Figure 4.1: The universe of stocks on HOSE from 2013 – 2019 52

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

Figure 4.3: The market capitalization of industry groups on HOSE, 2019 54

Figure 5.1: VN-Index’s performance in the period of 2013 – 2019 64

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

Figure 5.3: Back-testing results of SCM on out – of – sample from 1/1/2013 –

Figure 5.4: Compare the cumulative return between SCM and VN-Index 73

Figure 5.5: Back-testing results of SIM on out – of – sample from 1/1/2013 –

Figure 5.6: Compare the cumulative return between SIM and VN-Index 76

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

Figure 5.8: Compare the cumulative return between CCM and VN-Index 79

Figure 5.9: Back-testing results of SSIM on out – of – sample from 1/1/2013 – 82

31/12/2019

Figure 5.10: Compare the cumulative return between SSIM and VN-Index 85

Figure 5.11: Back-testing results of SSIM’s shrinkage coefficient ( ) on out

Figure 5.12: Back-testing results of SCCM on out – of – sample from

Figure 5.13: Compare the cumulative return between SCCM and VN-Index 91

Figure 5.14: Back-testing results of SCCM’s shrinkage coefficient ( ) on out

Figure 5.15: Back-testing results of STIM on out – of – sample from 1/1/2013

Figure 5.16: Compare the cumulative return between STIM and VN-Index 95

Figure 5.17: Back-testing results of STIM’s shrinkage coefficient ( ) on out

List of Tables

Table 2.1: Summarized works related to portfolio optimization 29 Table 4.1: The sample dataset are collected in the period of 2011 – 2019 51

Table 5.1: The performance of VN – Index in the period of 2013 – 2019 66

Table 5.2: The performance of the 1/N portfolio benchmark from 1/1/2013 to

Table 5.3: The performance of SCM from 1/1/2013 to 31/12/2019 70

Table 5.4: The performance of SIM from 1/1/2013 to 31/12/2019 74

Table 5.5: The performance of CCM from 1/1/2013 to 31/12/2019 77

Table 5.6: The performance of SSIM from 1/1/2013 to 31/12/2019 80

Table 5.7: The performance of SCCM from 1/1/2013 to 31/12/2019 88

Table 5.8: The performance of STIM from 1/1/2013 to 31/12/2019 93

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

Table 5.10: The movement value of shrinkage coefficient ( ) 102

CHAPTER 1: INTRODUCTION

The target of this chapter is to introduce background information relating to the topic of this 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 Minh City Stock Exchange officially came into operation in July 2000 with the first two tickers, REE and SAM, hitting the historic turning point of the Vietnam stock market So far, the Vietnam stock market has had flourish development The number of listed and registered companies trading on two stock exchanges is 1.605 companies, with the stock volume of 150 billion The market capitalization as of the beginning of 2020 reached nearly 5.7 million billion VND, accounting for 102.74% of GDP, thereby, showing the significant 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 Minh City 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, the average trading value reached more than 4,000 billion VND/session Market capitalization on HOSE accounts for 88% of the total market, equivalent to 54.3% of GDP Enterprises to be listed on the HOSE have to achieve higher standards in terms of charter capital, time of operation, performance, information disclosure, and shareholders structure Specifically, enterprises registered to list on the HOSE need to have the minimum charter capital contributed at the time of listing registration based on the book value of VND 120 billion, higher than the amount of VND 30 billion on the Ha Noi Stock Exchange (HNX) Regarding the time of operation, enterprises must have at least two 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, HOSE stipulates that listed enterprises must have profitable business activities in the previous two years, one year more than required by HNX Regarding shareholder structure, HOSE requires businesses to have a minimum of 300 non-major shareholders holding at least 20% of the voting stock of the company For HNX, this standard is a minimum of 100 shareholders holding at least 15% of the shares Especially, HOSE has higher listing standards 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 domestic and foreign investors, from 3000 trading accounts in 2000 to 2.5 million accounts in the current period In particularly, there are about 33,000 accounts of foreign organizations and 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 joined Vietnam's stock market Investment results show that investment funds in the period of

2009 - 2019 have relatively good investment results compared to the average growth rate

of Vietnam's stock market (Figure 1.1)

Figure 1.1: The performance of investment funds in the period of 2009 - 2019

However, if you look at the period of 2017-2019, when Vietnam's stock market is facing many difficulties due to complicated developments from the US-China trade war and the recession of major economies in the world gender has led to disappointing investment results of domestic and foreign investment funds, even the portfolio value of these investment 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

a 91.5% 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 stock market are relatively new and face many difficulties due to the following main reasons:

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 The majority of investors in the Vietnamese stock market are individual investors, who mostly use fundamental and technical analysis methods to select stocks The optimal investment portfolio construction is primarily selected by investors according to their own feelings or subjective judgments, not based on specific quantitative methods Besides, there are also a few individual investors actively in using quantitative models to choose optimal portfolios However, these models are traditional models with certain limitations and not highly 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 choosing optimal portfolios The first is data problem Although the Vietnamese stock market has gone through 20 years of development; however in the early stages, the number of companies 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

91%

5%

2% 2%

Asset Allocation Security Selection Market Timing Other Factors

its quality Meanwhile, we know that today’s modern methods of portfolio optimization require the size of research data to be large enough and the reliability of the data to be guaranteed Moreover, regulations on the Vietnamese stock market make it difficult for investors to build optimal portfolios using quantitative models For example, the regulations 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 margin regardless of how bad or good the market moves The regulation can help stabilize investors' sentiment during periods of deep decline or hot bull market, but it also prevents the market events from being fully reflected in the prices of stocks, making it difficult for investors 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 who execute a buying transaction of stock today ( ), they must be waiting for three business days (T+3) to be able to selling this stock on Vietnam stock market Besides, after they sell this stock, they need to wait two more business days (T+2) to start with another buying transaction or accept interest payments to the stock company during these two days to have money to support their buying transaction immediately These limitations of Vietnam stock market increase risks and incur much transaction costs for investors Moreover, these restrictions also make it difficult to apply the high – frequency trading models on the Vietnamese stock market Therefore, in the process of building the optimal portfolio selection models, they need to consider and calculate the impact of these restrictions 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 and the number of shares traded during the day is not much, leading to high liquidity risks for investors when they want to buy or sell stocks in large quantities Thus, when developing and back-testing the quantitative models, investors need to pay attention to this slippage factor in the stock trading activity, especially the transactions with large volume of buying 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 turn affects the reliability of the optimal portfolio selection models

Therefore, the research and selection of an appropriate portfolio optimization method is essential for investors on the Vietnam stock market In the next sections, the author will focus on presenting suitable methods in selecting the optimal investment portfolios on the stock 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 and construction of investment portfolios for over 65 years, since it was firstly introduced by Harry Markowitz in his “Portfolio Selection” (1952) article The framework of MPT is to attain as highest return as possible for a certain level of risk through structuring the optimal weights of various assets (Iyiola, 2012) Although it is broadly employed in practical investment activities, the main assumptions of MPT model have been facing great 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 the covariance matrix of assets’ return At this point, sample mean as well as covariance matrix approaches are usually employed However, these estimators are not really stable

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

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

models The CAPM, which is considered as one – factor model, describes a linkage between systematic risk and expected return of assets Meanwhile, Fama – French states that the expected returns of assets should be explained by some other variables besides of beta coefficient of CAPM such as size risk, value risk factor, profitability, and investment factor In addition, the robust estimators are also applied by the researchers and portfolio managers to improve the expected return estimation, for example the truncated/trimmed mean 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 – Stein estimator are developed to solve the non-stationary returns limitation in estimating the expected return input

The improvement of estimating assets’ expected returns is one of the ways to remedy MPT’s shortcomings However, the results of Merton’s research (1980) showed that it is not easy to measure the expected return (µ) In most asset price models, they made an assumption that there is a linkage between the assets’ expected return and the market’s expected return ( ), in which ( ) is constant overtime Although this assumption makes the assets’ 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 of constant expected 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 portfolios based on the covariance matrix estimation instead of the expected return estimation The estimation of covariance matrix parameter is an important research direction that researchers have paid special attention in the recent period, due to the potential of this method in improving stability and minimizing risk in selecting the investment portfolios

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

the estimation errors (Jagannathan and Ma, 2003) Moreover, Jagannathan and Ma suggested that “the estimation error in the sample mean is so large that nothing is lost in ignoring the mean altogether” This argument also provides detailed empirical evidence showing that the minimum-variance portfolio is likely to outperforms on Sharpe’s ratio and other performance metrics in out-of-sample period than any other mean-variance portfolios (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 The sensitivity of the minimum-variance portfolio to estimation error is quite interesting

“These portfolios are based on the sample covariance matrix, which is the maximum likelihood estimator (MLE) for normally distributed returns Moreover, MLEs are theoretically the most efficient for the assumed distribution; that is, these estimators have the smallest asymptotic variance provided the data follows the assumed distribution” At this point, a question was raised that why the sample covariance matrix generates unsuitable portfolios Huber (2004) answered that “the efficiency of MLEs based on assuming normality of returns is highly sensitive to deviations of the asset-return distribution from the assumed (normal) distribution In particular, MLEs based on the normality assumption are not necessarily the most efficient for data that depart even slightly from normality” For portfolio selection, this is very useful as comprehensive evidence indicates that the empirical return distribution is generally different from normal distribution

Moreover, although the core of minimum - variance portfolio researches relies solely on how to estimate reliably the covariance matrix, however, almost traditional approaches of covariance matrix estimation such as using the sample covariance matrix (SCM) or ordinary least squares (OLS) face many problems in the case of high-dimensional portfolios Having large dimensionality means that it is easier to get unexpected and uncontrollable errors in some of computational steps, and the sample data may not be enough 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 in matrix computation research Consequently, the portfolios selected from considering the sample covariance matrix often perform poorly and fail in generating profit To solve this problem, some new estimators of covariance matrix are conducted by many researchers and portfolio managers There have been many approaches proposed in the literature, and among them, Ledoit and Wolf (2003) proposed to select the optimal portfolios by using the shrinkage estimator of covariance matrix This method is a combination between a rough sample covariance matrix and a high-structured target matrix to achieve the balance between bias and variance The balance can be customized, which is the trade-off between bias and estimation errors recognized by shrinkage coefficients The shrinkage technique shows theoretically and empirically attractive approach to a high-dimensional portfolio's covariance estimation problem since it ensures a well-defined covariance matrix is achieved Liu (2014) estimated the covariance matrix by applying the weighted average of different shrinkage target matrices, instead of using a single shrinkage target matrix as Ledoit and Wolf method Next, by inheriting the potentials and development of Random Matrix Theory, Ledoit and Wolf extended their pilot works (Ledoit & Wolf, 2017a, 2017b) by using a nonlinear transformation applied for the eigenvalues considering solely the sample data Also, coefficient asymptotically leads to the maximization of the out-of-sample expected utility Then, they performed both numerical and empirical investigation where the out-of-sample behavior of the obtained estimator is analyzed and it shows remarkable improvements over the simple diversification, and its robustness 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 their practical application especially for asset optimization, and then they also discussed on a new category of shrinkage-based techniques for the means of return and the corresponding covariance matrix, as well as, the weights in the asset As a slight enhancement for this research approach, the work of Candelon et al (2012) presented such 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 ridge regression approach to shrink the all the weights towards the equally-weighted asset

Research gap

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

of covariance matrix in their portfolio optimization models The problem is that there is not a complete and sure research base regarding the effectiveness of out – of – sample performance 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 risk their money to make investments based on unproven or rigorous research

Moreover, the traditional estimator of covariance matrix is facing many difficulties and does not bring the expected results because the development of the financial market has resulted in the number of investment assets in the market increasing rapidly and much larger than the observed sample, from that requiring the new estimators of covariance matrix to be studied and applied Besides, there is still a lot of controversy surrounding the applicability and effectiveness of covariance matrix estimation methods in different markets

Furthermore, the robust estimators of covariance matrix are mainly applied and tested in the developed markets; there are not many researches on emerging and developing financial market In particular, there is almost no research in Vietnam related to the selection of covariance matrix estimators for optimizing the portfolios, especially in the shrinkage methods Therefore, there is a gap for the author to investigate the level of influence of covariance matrix estimators on the minimum – variances optimized portfolios, 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 improve the performance of minimum – variance optimized portfolios by altering the estimators of covariance matrix input Besides, based on the results of out – of – sample portfolio performance metrics, the dissertation is going to select the suitable estimators of covariance matrix for portfolio optimization on Vietnam stock market

In order to achieve the above objectives, this dissertation will attempt to answer the research questions as follows:

Question 1: How do the robust estimators of covariance matrix perform on out – of –

sample performance metrics such as portfolio return, level of risk, portfolio turnover, maximum drawdown, winning rate and Jensen’s Alpha in selecting minimum – variance optimized portfolios?

Question 2: How do the estimators of covariance matrix affect the out – of – sample

performance of minimum – variance optimized portfolios when the number of assets in the portfolio changes?

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

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

1.4 Research Methodology

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

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

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

Second, to evaluate the feasibility and potential application of the estimators of covariance matrix mentioned above, a back-testing process based on the Python programming language has been developed and applied in this study The back – testing process 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 examined and clarified by the back – testing procedure, from that providing insight into whether which estimator will be able to make profit in the reality

Third, through the back – testing process, the portfolio performance metrics which are considered as the important criteria for evaluating the portfolios will be estimated Besides of the basic portfolio evaluation criteria such as portfolio return and volatility of portfolio, other useful evaluation criteria are also calculated in this research including portfolio turnover, maximum drawdown, winning rate and Jensen’s Alpha In order to calculate these portfolio evaluation criteria, the author used a “rolling – horizon” technique that defines as a reactive scheduling method that “solves iteratively the deterministic 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 not accurate enough parameters became to be known, the optimal schedule for the new resulting scenario (and optimization horizon) may be found” Silvente et al.(2015) The rolling – horizon approach allows the investors to update or adjust their input data for optimal 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 portfolio performance metrics, the transaction costs are also considered at every rebalancing point Lastly, these estimated performance metrics are applied for comparing the differences among the estimators of covariance matrix in selecting the optimal portfolios To make sure that there are significant differences of performance metrics between the two certain estimators, the p – values are computed following the bootstrapping methodology that is mentioned in the research of DeMiguel (2009)

1.5 Expected contributions

After answering the research questions and achieving the research objective, this dissertation 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 investment performance 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 of covariance matrix (SSIM, SCCM, STIM) give much superior results compared to the traditional 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 portfolio tends to increase

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

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

selection compared to the shrinkage towards single index model (SSIM) and shrinkage towards 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 matrix and the effect of transaction costs in computing the portfolio performance on out – of – sample Specifically, the dissertation has evaluated the effectiveness of the estimation methods when the number of stocks in the portfolio changes from N = 50 to N = 350 while 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 a reasonable assessment of the effectiveness of estimation methods in the selection of the optimal portfolio, the dissertation employs more portfolio performance metrics such as Sharpe ratio, portfolio turnover, maximum drawdown, winning rate or Jensen’s Alpha to evaluate the performance of the selected portfolios, instead of using the common performance 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 covariance matrix estimation methods compared to previous studies

The final contribution of this dissertation is that all estimators of covariance matrix have been experimented on Vietnam stock market where is considered as an emerging market Although, there are many researchers and financial practitioners using the estimation methods 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 the emerging market, especially on Vietnam stock market The empirical results of this study will support the researchers and investors to see clearly the different performance of the estimators of covariance matrix between the emerging financial market and the developed financial market

1.6 Disposition of the dissertation

This dissertation includes 6 chapters, in which Chapter 1 focuses on the problem statements, objectives of the study, research questions, research methodology as well as expected 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 and methodology 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 be introduced in this chapter, and then theory regarding the covariance matrix estimations will 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 with portfolio 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 to financial economics In 1990, Markowitz deserved to receive the Nobel Prize for his contributions related to the portfolio selection method The “Portfolio Selection” article was first published in 1952 in the journal “The Journal of Finance” and then written in a book titled “Portfolio Selection: Efficient Diversification” in 1959 Markowitz's groundbreaking work shaped what we today call modern portfolio theory (MPT)

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

The MPT which is an improvement on the classical quantitative models plays a pivotal role in the mathematical modeling of finance This framework motivates the diversification to protect the investment portfolios from market risk as well as specific risk of company Sometimes, the theory is also called Portfolio Management Theory because it supports the investors to classify, evaluate and measure both the expected risk and return The main idea of this theory is its quantification of relationship between risk and 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 lower risk 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” and risk as “a standard deviation of return” The portfolio’s return will be determined as the weighted combination of the securities’ returns The total variance of the portfolio return will 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 is efficient”

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 explains the 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 a certain 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 any investment project Returns can be defined in terms of realized return (the received return) and expected return (the return anticipated by investors over future investment periods) When we refer to the expected return, it shows us that this is a forecasted or estimated 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 or loss over a given investment period and considered as a percentage of return on the amount of initial investment This is seen as a cumulative return of an investment With respect to investment in securities, the return includes the dividends and the capital gain

or loss at the point of selling of such securities

Risk

In investment activities, risk is defined as the unpredictability of the return of an investment in the future In other words, risk is understood as the possibility of profit in fact 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 with regard to a business can be defined as the probability that the actual outcome of a financial decision can vary from the estimated one Investments with greater return volatility are perceived to be riskier than investments with lower return volatility

Moreover, risk and uncertainty need to be clearly distinguished “Risk is stated as a situation where the possibility of occurring or non-occurring of an event can be quantified and measured, while uncertainty is a situation where this possibility cannot be measured Therefore, risk is a situation where probabilities can be assigned to an event on the 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 estimated return, thereby measuring the difference between the expected return and the actual return

on an investment Therefore, the statistical methods such as standard deviation and variance 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 investors and 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)

Risk – Return Efficient Frontier

Optimal Investor

2.1.4 Critism of the theory

The MPT has been widely criticized given its theoretical relevance; its simplistic assumptions are a prevailing bias Some critics doubt its effectiveness as an investment strategy, since in many cases its financial market model does not suit reality In recent years, 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 the construction of the optimal portfolio in practice face many difficulties, due to the instability of the input parameters in the optimization problem Recent studies show that these types of instabilities will disappear when a “regularizing constraint” or “penalty term” 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 are dependent on “ex-ante” values that include mathematical assumptions about the future (where expected value is explicit in the calculations, while implicit in definitions of variance and covariance) In reality, investors need to replace forecasts relied on historical 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 the future, such expected values will fail to predict correctly the forthcoming movement of portfolio’s relating metrics More practically, an investor faces many difficulties to estimate key parameters from ex-post data of market as the MPT tries to model risk including the possibility of losses, but not mention the reason why losses happen There are no structural, probabilistic measures in the risk measurement used Contrary to other engineering 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 other consequences 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 of problems stemming from “information asymmetry, externalities, and public goods” In addition, a business could have strategic or social objectives which affect its investment decisions, and may have personal goals for an individual investor In both cases knowledge 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 their fundamentals, since he/she is only focused on removing the unsystematic risk of the portfolio (Chandra, 2003) This artificially enhanced demand drives up the securities’ price which would be of little fundamental interest if evaluated individually The consequence is that the entire portfolio is more costly and a positive portfolio return probability is small This implies portfolio risk continues to grow

2.2 Parameter estimation

The phenomenon of mean – covariance optimization is claimed of having estimation error In the study of Markowitz (1952), he stated that the theoretical soundness of the proposed portfolio selection approach was more emphasized, which results in the fact that MTP practical application was less focused To implement MTP realistically, the means and covariance of assets returns need estimating because they are not previously known Thereafter, the estimated results are employed to find solutions for optimization problems of investor (Elton et al 2012) A great number of studies conducted previously conclude that this brings about the most crucial disadvantage of the mean-variance approach: there might be estimation error when using inappropriate moments for plugin (Michaud (1989); Chopra and Ziernba (1993)) At this point, the optimizer is not conscious that the data inputs are only statistical estimates and there are not any certainties, 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 this approach is that historical data can partially describe the future tendency of assets’ price Nevertheless, some previous researches have reported this ordinary procedure’s problems DeMiguel (2009) confirmed that using sample estimates as input parameters decides not necessarily result in a mean- variance optimized portfolios that outperform an equally – weighted one In this line, Jobson and Korkie (1980) also showed a relative results Besides, Best and Grauer (1991) indicated that estimation error is consequently transferred into the weights of optimized portfolios, which causes a deviation between estimated 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 of covariance matrix They also potentially explained the reason why there was disparity in people’s concentration between expected return vector and the covariance matrix in the past Not to be outdone, since the financial markets was introduced, forecasting reins has started 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 estimation error which is relative to the covariance matrix in return In more details, they indicated that paper by Chopra and Ziemba (1993) was relied on a limited in-the-sample study which is thereby not bearing when concerning the effects of error in estimating the strict out-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 most important aspect of mean-variance optimization To obtain a more potential optimized portfolio in out-of-sample period, lessening such error is highly necessary, which eventually significantly benefits assets managers who employ MVO Before some

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

The following parts will (1) include an overview of different solutions for forecasting expected returns and (2) deliver literatures review that refer to the method of reducing estimation 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 William Sharpe in 1964 It is based on the foundation of MPT and also known as one-factor model for forecasting the expected return of assets Specifically, it is the relationship between systematic risk and assets’ expected return that is explained by this model However, in 1992, Eugene Fama and Kenneth French discovered that the beta coefficient used in CAPM model did not precisely describe the expected return of American securities between 1963 and 1990 Thus, they began to observe the movement

of two classes of stocks which tended to outperform the whole market The two groups included the small caps and the stocks that have high Book to Market Equity ratio Thereafter, they added 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 was proven effective when has been through various empirical tests in developed stock markets 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 decade later, they added two new factors in the model including profitability and investment factors to form a 5-factor model in 2014 which can predict the movement of expected return

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

return estimation For examples, instead of using the sample mean m ∑ to estimate population expected returns, the sample truncated/trimmed mean or winsorized can be used (Martin et al 2010) The trimmed mean is calculated after removing the k% most extreme values, while the winsorized mean is estimated after replacing those values with the next k% most extreme values However, both estimators are more efficient to deviations from distributional assumptions on assets return, so if the returns are not stationary, 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 the selection of tuning parameters, researchers have turned to Shrinkage Estimators The shrinkage approach begins with the observation that, much like uncertainty in actual expected returns (asset risk), uncertainty in the estimate of expected returns (estimation risk) implies a loss of investor utility (Jorion, 1985) Thus, the optimization problem should 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’s expected return individually, but to select an estimator that minimizes utility loss from aggregate parameter uncertainty Jorion (1986) suggests using a Bayes–Stein estimator that shrinks each asset’s sample mean to the grand mean ̅ ∑ Through the simulation, this estimator reduce risk of portfolio and outperforms portfolios constructed by .

Although improving the estimating assets’ expected return is considered as one of the effective ways to solve the MPT’s flaw, Merton’s research (1980) stated that it is difficult

to estimate the expected return (µ) Most assets price models assume that there is an existence of relationship between the expected return of assets and market ( ) while remained constant overtime Even though this assumption makes the assets’ 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 of constant expected 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 portfolios based 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 portfolios based on estimating the expected return parameter Phuong Nguyen (2012) used the single-factor model (SIM) to measure risks and to determine expected returns of stocks in the construction industry Linh Ho (2013) also applied this model to measure the risks and 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) also employed Fama - French three-factor model to measure risks and estimate expected returns of stocks on Vietnam stock market Besides, Nguyen Tho (2010) applied the arbitrage pricing theory (APT) to examine the stock price behavior of an emerging stock market 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 in the recent period, due to the potential of this method in improving stability and minimizing risks in selecting the investment portfolios The traditional covariance matrix estimation for portfolio optimization is to use the sample covariance matrix (SCM) However, Michaud (1989) has proved that this estimation method brings many shortcomings 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 and Seagle (1971) also confirmed that the sample covariance matrix estimation for MPT

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

Based on the single-index model given by Sharpe (1964), the researchers estimated the covariance matrix to select the optimal portfolio The covariance matrix estimated from the 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 their sample (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 the contrary, the standard MPT approach requires T > N (T is the sample size) In the research of Senneret et al., (2016), there is ample evidence that the SIM approach produces portfolios with much less sensitive to estimation error than the standard approach does, therefore performing better across a range of risk and return metrics However, both the standard approach and SIM method are vulnerable to estimation error from using the sample mean returns vector (SIM is also exposed to estimation error from ̂ )

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

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

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

to estimate ∑̂ = (1 - ) + ∑̂ , where is determined by a data-driven algorithm and ∑̂ is chosen through some prior belief about asset return covariance

In terms of target matrices, Ledoit and Wolf (2003a) recommend ∑̂ (Constant Correlation Model) that based on the prior belief that all stocks have the same correlation and equal to the mean correlation, while Ledoit and Wolf (2003b) suggest ∑̂ (Single 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 any benefit from the application – specific knowledge like the two previous shrinkage methods The target matrix is an Identity matrix ∑̂ that is “a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros” With the introduction of this kind of this target matrix, Ledoit and Wolf tried to answer the question of whether an investor can choose an optimal portfolio if they are not using any the domain knowledge in finance field They find that “all three improved estimators of the covariance matrix dominated the sample covariance matrix, but there was no clear winner among the three If anything, shrinkage to the constant-correlation matrix was best for portfolio sizes N ≤ 100, whereas shrinkage to the single-factor matrix was best for portfolio sizes N ≥ 225”

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

In the recent time, the newer work has focused on non – linear shrinkage estimators that continue to answer the question whether it is “possible to generalize and improve linear shrinkage to the identity matrix in the absence of financial knowledge or in other words whether the investors can be totally ignorant about the true covariance matrix and still do better than generic linear shrinkage” (Ledoit and Wolf, 2018) The non – linear shrinkage estimators do not try to predict the true covariance matrix, this shrinkage method considers the eigenvalues distribution of covariance matrix instead So the non-linear shrinkage method shows the flexibility in estimating eigenvalues of covariance matrix compared to the linear shrinkage method, which can only estimate specific eigenvalue values via the linear function Ledoit and Wolf (2012, 2015) proposed the QuEST function to estimate the covariance matrix in the non-linear shrinkage method with the assumption 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 approach NERCOME (Nonparametric Eigenvalue-Regularized Covariance Matrix Estimator) by separating the observation sample into two separate parts, partly for calculating eigenvectors and the rest for calculating eigenvalues related to the above eigenvectors Ledoit and Wolf (2018) once again developed the applicability of non-linear shrinkage method by combining the power of 3 components which are the fast processing speed of linear 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 to the 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 estimating the risk tolerance of investors in HOSE and based on the known level of risk to optimize portfolios Huyen Nguyen (2015) with the topic of applying modern financial theory to measure risks in stock investment on Vietnam stock market However, these portfolio optimization models mainly base on the traditional method that applies the sample covariance 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 in Vietnam to optimize the securities portfolio through changing the covariance matrix estimation in modern portfolio theory

2.3 Portfolio Selection

The mean-variance model is usually used for portfolio selection, in which, based on the mean and covariance matrix of asset return, investor can structure the assets weights their portfolio to the most optimal percentages However, the estimation of sample means and covariance matrix used to replace the true mean and covariance matrices often involve estimations errors Moreover, the errors in estimating sample means are much significant than that of estimating sample covariance Hence, people tend to use the global minimum-variance model which only bases the asset's weight on estimating the covariance matrix Thus, previous literatures on portfolio selection have been divided into two directions: one follow traditional mean-variance model and others follow the newer, global minimum-variance model

2.3.1 Mean-Variance Model

The return on securities is considered as a “random variable with Gaussian distribution” under the standard mean-variance model introduced by Markowitz (1952) Accordingly, the normal (Gaussian) distribution assumes that assets returns are only dependent on the mean and variance Markowitz (1959) was the pioneer in developing the mean-variance model published in his book relating to portfolio selection 20 years after, Merton (1972) continued the study about this model but it can allow short-sale in portfolio selection Throughout the years the models described above have been applied in several studies The table 2.1 will summarize some ground-breaking papers: