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

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 i 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 ii Table of Contents List of Abbreviations iv List of Figures vi List of Tables viii CHAPTER 1: INTRODUCTION 1.1 Vietnam stock market overview 1.2 Problem statements 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 i 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 ii 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 iii 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 iv VIC: Vingroup Joint Stock Company VND: Viet Nam Dong VN - Index: Vietnam stock index WR: Winning rate YEG: Yeah1 Group Corporation v List of Figures Figures Pages 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 18 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, 2019 53 Figure 4.3: The market capitalization of industry groups on HOSE, 2019 54 Figure 4.4: Back – testing procedure 57 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 – sample from 1/1/2013 – 31/12/2019 Figure 5.3: Back-testing results of SCM on out – of – sample from 1/1/2013 – 31/12/2019 Figure 5.4: Compare the cumulative return between SCM and VN-Index Figure 5.5: Back-testing results of SIM on out – of – sample from 1/1/2013 – 31/12/2019 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 69 72 73 75 76 78 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 vi 31/12/2019 Figure 5.10: Compare the cumulative return between SSIM and VN-Index Figure 5.11: Back-testing results of SSIM’s shrinkage coefficient ( ) on out – of – sample from 1/1/2013 - 31/12/2019 Figure 5.12: Back-testing results of SCCM on out – of – sample from 1/1/2013 – 31/12/2019 Figure 5.13: Compare the cumulative return between SCCM and VN-Index Figure 5.14: Back-testing results of SCCM’s shrinkage coefficient ( ) on out – of – sample from 1/1/2013 - 31/12/2019 Figure 5.15: Back-testing results of STIM on out – of – sample from 1/1/2013 – 31/12/2019 Figure 5.16: Compare the cumulative return between STIM and VN-Index Figure 5.17: Back-testing results of STIM’s shrinkage coefficient ( ) on out – of – sample from 1/1/2013 - 31/12/2019 vii 85 86 89 91 92 94 95 96 List of Tables Tables Pages 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 31/12/2019 67 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 out – of – sample Table 5.10: The movement value of shrinkage coefficient ( ) viii 97 – 98 102 CHAPTER 6: CONCLUSIONS AND FUTURE WORKS 6.1 Conclusions Optimizing the securities portfolio is always an interesting problem for investors in the market These investors attempt to build a portfolio that meets their expected return and has limited risk They only accept a higher level of risk when compensated by a reasonable expected return If two portfolios have the same expected return, the portfolio with lower risk would be selected Modern Portfolio Theory (MPT) that was firstly introduced by Harry Markowitz in 1952 is usually applied to address above issue Although this theory has been applied widely, this theory still has some limitations, leading to unexpected results in real life These limitations mainly come from the instability of expected return and estimated covariance matrix, which are the significant parameters in MPT for portfolio selection This leads the portfolio from MPT model to fluctuate continuously over time and to suffer high transaction costs when applying in practice Moreover, with the rapid development of the current financial market, the application of this theory has become more and more difficult because the number of stocks in the market quickly increased and even exceeded the number of observed samples The superiority between the number of stocks compared to the amount of observed samples makes it difficult for traditional methods to choose the optimal portfolio, since there is not enough information required to make a decision From there, we can see that the research and development of portfolio selection models is an urgent issue for investors, portfolio managers and researchers The difficulty in choosing an optimal portfolio is even more complicated when placed in the context of the Vietnamese stock market This difficulty stems from specific characteristics of emerging markets such as the Vietnamese market For example, investors' lack of understanding in the application of models, data problem, regulations of the state financial agency such as daily trading limit, delay settlement date…all make it 105 difficult not only to build and develop optimal portfolio selection models but also to test these strategies in practice In that context, the dissertation has shown that the alternations of covariance matrix for minimum – variance portfolio optimization can be an effective solution for the optimal portfolio selection in the financial market in general and the Vietnamese stock market in particular In this research, the author selected five estimators of covariance matrix to investigate the effectiveness of minimum – variance optimized portfolios through alternation of covariance matrix estimations The five estimators which are single index model (SIM), constant correlation model (CCM), shrinkage towards single index model (SSIM), shrinkage towards constant correlation model (SCCM) and shrinkage towards identity matrix (STIM) are divided by two types of approaches The first approach is model – based as SIM and CCM and the second one is shrinkage approach as SSIM, SCCM, and STIM In order to prove that investors can improve their investment efficiency through adjusting the covariance matrix for portfolio optimization, the dissertation needs to take the following important steps: First, the input data that are the weekly price series of stocks are collected and checked carefully The whole dataset was taken directly from Ho Chi Minh City Stock Exchange (HOSE) and tested with the other data sources in fully There are a total of 382 companies listed on HOSE as of the end of 2019, but there is only 350 companies satisfy the liquidity and listed time requirements The collection period is from 2011 to 2019 corresponding to 468 weekly points, in which the period of 2011 – 2013 is considered as in the sample and the remaining period of 2013 – 2019 is selected as out – of – sample Next, the portfolio performance evaluation methodology needs to be clearly identified after the data has been fully collected and processed To evaluate the efficiency of covariance matrix estimation methods, a back-testing process is built and applied in this research from using a back-testing platform in Tran et al.(2020) Back-testing process supports the author in appraising the possibility and potential application of near future 106 estimation, with the series of price value in portfolio Based on the back-testing system, the author compares the different policies or covariance matrix estimations using a “rolling-horizon” procedure Besides, transaction costs are also considered in the back – testing process Moreover, the performance metrics are also used to compare the performances among the estimators of covariance matrix These performance metrics are portfolio return, volatility, sharpe ratio, portfolio turnover, maximum drawdown, winning rate and Jensen’s Alpha Besides, in order to determine that the difference of performance metrics between two estimators is significant, the p – values are computed following the bootstrapping methodology applied by DeMiguel (2009) Furthermore, based on the back – testing results of performance metrics on the out – of – sample, the author will compare the effectiveness of each covariance matrix estimator The content of discussion and analysis will turn around three questions that are raised above including how the robust estimators of covariance matrix perform on out – of – sample compared to the estimator of traditional covariance matrix; how the estimators of covariance matrix work on the minimum – variance optimized portfolios when the number of stocks in portfolio changes; and which the estimator of covariance matrix in this research will show the best results on performance metrics of minimum – variance portfolios such as portfolio return, level of risk, portfolio turnover, maximum drawdown, winning rate and Jensen’s Alpha on Vietnam stock market, especially in the case of high – dimensional portfolios The answers for these questions are summarized as follows: First, the robust estimators of covariance matrix perform on out – of – sample better than the estimator of traditional covariance matrix in selecting the minimum – variance optimized portfolios In particular, the estimators of SIM and CCM, which are model – based approaches, and the estimators of SSIM, SCCM and STIM, which are shrinkage approaches, give better results than the traditional estimator of SCM across all tested 107 portfolios and most performance metrics This is considered as one of the most important conclusions for investors as well as portfolio managers, because it again affirms the rationale in choosing the optimal portfolio based on the adjustment of the covariance matrix parameter This rationality not only brings efficiency in developed markets as previous studies by Ledoit and Wolf, but it also shows efficiency in emerging markets as the Vietnamese financial market From there, it opens a clear research direction for investors in building the methods for selecting the optimal portfolios on the stock market Second, the superiority of model – based methods over the traditional SCM takes place when N = 100 and the degree of dominance is increasing when N approaches to 350 stocks The shrinkage methods also show superiority in portfolio optimization over the traditional SCM method and benchmarks when N tends to increase The shrinkage method begins to clearly outperform the performance of SCM when N is larger than 100 stocks This conclusion helps investors and portfolio managers to see that if the market size exceeds the number of shares N = 100, they must consider applying new estimators of covariance matrix because at this point the traditional sample covariance matrix is no longer effective in choosing the optimal portfolio Third, the SIM and CCM which are model – based approaches have shown their very good ability in the selection of optimal portfolios and somewhat outperform the shrinkage method when the number of stocks is N = 50,100, 200 However, the shrinkage method demonstrates its superiority in the case of high - dimensional portfolios When N soared to reach 350 stocks, the model-based approaches was completely defeated by the shrinkage methods such SSIM, SCCM, and STIM This conclusion enables portfolio managers to discover that the application of model-based covariance matrix estimation is only effective when the market size is less than 200 stocks; if the size of the market increases the number of shares to 350, the portfolio managers should consider using the shrinkage of covariance matrix method because at this time the shrinkage method will bring more efficiency in choosing the optimal portfolio 108 Fourth, in the model-based approaches, the estimator of constant correlation model (CCM) is much more effective than the one of single index model (SIM) for portfolio optimization This result is consistent with research results of Elton et al (2009), it shows that the assumption of all stocks have the same correlation, which is equal to the sample mean correlation, will be more reasonable than that of stocks’ price is mainly influenced by the market return on the Vietnamese stock market This characteristic is one of the important points that investors should pay attention to when investing in the Vietnamese stock market, besides the market authorities can also refer to better implementation of their management and policy recommendations Fifth, the SCCM shows the best performance on out – of – sample among the shrinkage methods when clearly outperforming the other two methods across all tested portfolios (N = 50, 100, 200, 350) and on most portfolio performance metrics; meanwhile, the SSIM and STIM methods not have much difference in the selection of optimal portfolios This result is slightly different from the studies of Ledoit & Wolf (2003, 2004) and Ledoit & Wolf (2017, 2018); these studies conclude that the SCCM and SSIM estimation methods gives much better optimal results than the STIM estimation method, in which the SCCM method will be the best choice if the number of shares in the portfolio N ≤ 100, otherwise the SSIM method will be the most reasonable choice when N ≥ 225 Sixth, in the case of high – dimensional portfolio, the SCCM shows that it is the best estimator of covariance matrix for portfolio optimization on Vietnam stock market The performance of SCCM completely outperforms the traditional estimator of sample covariance matrix, benchmarks such as VN – Index and 1/N portfolio strategy, model – based approaches and other shrinkage estimators on most back – testing performance metrics However, the difference between SCCM and two other shrinkage estimators such as SSIM and STIM tends to decrease when the number of stocks in portfolio soars; especially it will be the strong competition between SCCM and STIM The SCCM method has many advantages in creating highly profitable portfolios, but the STIM method is capable of creating safer portfolios 109 Seventh, in the two selected benchmarks, the 1/N portfolio strategy showed much superiority to the VN - Index on most of the criteria for measuring the effectiveness of a portfolio This shows that the VN-Index's representativeness for the changing trend in the market is not really effective The main reason comes from the fact that this index is strongly influenced by industry groups and companies with large capitalization in the market, leading to deviations in the forecast of market volatility Therefore, investors should pay attention when choosing the VN - Index as an index that predicts the changing trend of the market, while also posing a problem for market managers in building a more reasonable index that represents the change of the whole market In this dissertation, the effectiveness of conventional sample covariance matrix in estimating parameters for portfolio optimization is challenged by other newer approaches Particularly, the problem that whether the performance of minimum-variance optimized portfolios can be enhanced by the use of another covariance matrix estimator is examined by evaluating the performance of SCM and potential alternative estimators (which are SIM, CCM, SSIM, SCCM, and STIM) in a practical back-testing procedure, in which other factors in the minimum-variance optimization were remained equal Generally, most of the empirical results support that the alternation of covariance matrix estimations for portfolio optimization brings a lot of benefits for portfolio managers and investor in practical way They can achieve more monetary benefits by employing the estimators of covariance matrix on the Vietnamese stock market Thus, apart from contributing to the available knowledge about optimizing investment portfolio, this research also provides evidence of the covariance matrix estimation on Vietnam stock market For an emerging market that is significantly attracting capital inflow like Vietnam, this evidence can partially give investors who are investing and going to invest in this market more confidence when using the estimators to optimize their portfolio 110 6.2 Future works Under the scope of this dissertation, the author has only investigated about the performance of model – based and shrinkage estimators against the traditional one In the future, the researchers can select the new approaches in estimating covariance matrix for portfolio optimization In particular, in the shrinkage approach, the researchers can consider to select the new shrinkage target matrices to combine with the sample covariance matrix in generating the estimated covariance matrix, besides of using these target matrices mentioned in this dissertation such as single – index model (SIM) or constant correlation model (CCM) In addition, researchers can also change the combination way between the shrinkage target matrix and sample covariance matrix for estimating the covariance matrix There are two development trends for this research direction First, we can combine the sample covariance matrix and some shrinkage target matrices at the same time for estimating the covariance matrix, instead of using single shrinkage target matrix as this research This approach was initiated by Liu (2014) and is called generalized multivariate shrinkage However, this approach faces many difficulties in determining the optimal shrinkage coefficient for each estimator Second, a non – linear shrinkage estimator is one of the approaches that researchers are interested and developing in the recent time This approach try to answer the question that whether it is possible to generalize and improve linear shrinkage in the absence of financial knowledge The non – linear shrinkage estimator not predict the true covariance matrix for portfolio optimization, this shrinkage method considers the eigenvalues distribution of covariance matrix instead However, the problem of this approach is how to balance the processing speed, the level of transparency and the accuracy of method Besides, it is still a new method and has not been applied much in practice because of 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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?... 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...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