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 ACADEMIC SUPERVISOR PGS.TS NGUYEN DUC TRUNG Ho Chi Minh City - 2021 i MINISTRY OF EDUCATION AND TRAINING THE STATE BANK OF VIETNAM BANKING UNIVERSITY OF HO CHI MINH CITY NGUYEN MINH NHAT SHRINKAGE ESTIMATION OF COVARIANCE MATRIX FOR PORTFOLIO SELECTION ON VIETNAM STOCK MARKET DOCTORAL DISSERTATION Major: Banking & Finance Code: 9.34.02.01 ACADEMIC SUPERVISOR PGS.TS NGUYEN DUC TRUNG Ho Chi Minh City - 2021 ii DECLARATION I, Nhat Minh Nguyen, declare that the work in this dissertation titled “Shrinkage estimation of covariance matrix for portfolio selection on Vietnam Stock Market” has been composed by myself and that it has not been submitted, in whole or in part, in any previous application for a degree Except where states otherwise by reference or acknowledgment, the work presented is entirely my own Ho Chi Minh City, January 2021 Signature i ACKNOWLEDGEMENTS First and foremost, I would like to thank God for continual guidance during this research I would like to thank my wonderful supervisor, Prof Trung D Nguyen for allowing me to undergo this research under his guidance I would like to thank him for all his invaluable advice, pushing me and especially pointing me in the right direction to have contact with other researchers who have been beneficial to me I would also like to thank Tuan Tran and An Mai, who acted as my great colleagues and supported me towards the required background on Markowitz portfolio optimization and its application in industry I am also grateful to my colleagues at the Banking University of Ho Chi Minh City for providing a supporting research atmosphere I would also like to thank my family for their encouragement throughout my research experience To my wife, Trang Nguyen, thank you for being a great example to me, for your constant love, support and motivation throughout the years Lastly, I would like to thank my parents for supporting and providing me with the necessary funding to undertake this research work I am forever grateful and to them I dedicate this dissertation ii TĨM TẮT Tối ưu hóa danh mục đầu tư chứng khốn ln tốn thú vị nhà đầu tư thị trường Các nhà đầu tư cố gắng xây dựng danh mục đầu tư đáp ứng lợi nhuận kỳ vọng họ đồng thời hạn chế rủi ro xảy danh mục đầu tư Họ chấp nhận mức độ rủi ro cao bù đắp mức lợi nhuận kỳ vọng hợp lý, hai danh mục đầu tư có mức lợi nhuận kỳ vọng danh mục đầu tư mang lại rủi ro danh mục đầu tư lựa chọn Lý thuyết danh mục đầu tư đại (MPT) thường nhà đầu tư lựa chọn để giải toán Lý thuyết lần đề xuất Harry Markowitz đạt giải thưởng Nobel năm 1952, kể từ thu hút ý nhiều nhà nghiên cứu lĩnh vực thống kê, kinh tế đặc biệt lĩnh vực đầu tư tài Mặc dù tiếp cận rộng rãi, lý thuyết tồn hạn chế định khiến khơng đạt kết mong đợi thực tế Hạn chế đến từ không ổn định ước tính lợi nhuận kỳ vọng ma trận hiệp phương sai, hai biến số quan trọng mô hình MPT để lựa chọn danh mục đầu tư tối ưu Sự không ổn định hai yếu tố đầu vào dẫn đến tính chất khơng ổn định danh mục đầu tư khuyến nghị, làm cho danh mục đầu tư ln chịu chi phí giao dịch cao không tạo lợi nhuận kỳ vọng cho nhà đầu tư Điều giải thích nhà nghiên cứu nhà quản lý danh mục đầu tư giành nhiều thời gian nghiên cứu để cải thiện khả dự báo lợi nhuận kỳ vọng ước lượng ma trận hiệp phương sai danh mục đầu tư Xu hướng trước đây, họ chủ yếu tập trung vào hướng nghiên cứu thứ xây dựng mơ hình để dự báo lợi nhuận kỳ vọng danh mục đầu tư, nhiên theo nhiều nghiên cứu đặc biệt nghiên cứu tiếng từ Merton Michaud cho thấy khó để dự báo lợi nhuận kỳ vọng tài sản danh mục đầu tư, việc dự báo thường mang lại mức độ sai số lớn ảnh hưởng đến kết lựa chọn danh mục đầu tư tối ưu Trong thời gian gần đây, hướng nghiên cứu thứ hai ước lượng ma trận hiệp phương sai danh mục đầu tư nhà nghiên cứu nhà quản lý danh mục quan tâm, tiềm iii phương pháp việc giảm mức độ sai số mơ hình cải thiện kết danh mục đầu tư lựa chọn Bên cạnh đó, phương pháp ước lượng ma trận hiệp phương sai truyền thống gặp phải nhiều khó khăn không mang lại kết kỳ vọng phát triển thị trường tài dẫn đến số lượng tài sản đầu tư thị trường tăng cách nhanh chóng lớn nhiều lần so với mẫu quan sát, từ địi hỏi phương pháp ước lượng cần phải nghiên cứu ứng dụng Tuy nhiên, có nhiều tranh cãi xung quanh tính ứng dụng hiệu phương pháp ước lượng ma trận hiệp phương sai thị trường khác Thêm nữa, phương pháp ước lượng ma trận hiệp phương sai chủ yếu áp dụng kiểm định thị trường phát triển, chưa có nhiều nghiên cứu thị trường tài chưa phát triển Tại Việt Nam, theo khảo lược tác giả khơng có nghiên cứu sâu phương pháp ước lượng ma trận hiệp phương sai để tối ưu danh mục đầu tư, nhà nghiên cứu Việt Nam chủ yếu thực lựa chọn danh mục đầu tư thông qua theo hướng ước lượng lợi nhuận kỳ vọng tài sản danh mục đầu tư Các nghiên cứu ước lượng ma trận hiệp phương sai để tối ưu danh mục đầu tư Việt Nam xoay xung quanh việc sử dụng phương pháp ước lượng truyền thống không mang lại kết mong đợi số lượng tài sản đầu tư bắt đầu tăng nhanh thị trường tài Đó lý dẫn đến việc tác giả lựa chọn đề tài: “Shrinkage estimation of covariance matrix for portfolio selection on Vietnam stock market” làm chủ đề nghiên cứu cho luận án Mục tiêu nghiên cứu luận án muốn xem xét thay đổi yếu tố ma trận hiệp phương sai tác động đến kết lựa chọn danh mục đầu tư thơng qua tìm hiểu xem liệu nhà đầu tư cải thiện hiệu hoạt động danh mục đầu tư việc điều chỉnh ma trận hiệp phương sai mơ hình tối ưu hóa với phương sai nhỏ hay khơng Đồng thời, dựa kết nghiên cứu thực nghiệm, luận án lựa chọn phương pháp ước lượng ma trận hiệp phương sai phù hợp thị trường chứng khoán Việt Nam iv Table of Contents List of Abbreviations viii List of Figures x List of Tables xii CHAPTER 1: INTRODUCTION 1.1 Vietnam stock market overview 1.2 Problem statements 1.3 Objectives and research questions 12 1.4 Research Methodology 12 1.5 Contributions of the research 14 1.6 Disposition of the dissertation 16 CHAPTER 2: LITERATURE REVIEW 17 2.1 Modern Portfolio Theory Framework 17 2.1.1 Assumptions of the modern portfolio theory 18 2.1.2 MPT investment process 19 2.1.3 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 29 2.3.1 Mean-Variance Model 29 2.3.2 Global Minimum Variance Model (GMV) 31 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 v 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 57 4.3 Transaction costs 60 4.4 Performance metrics 61 4.4.1 Sharpe ratio (SR) 61 4.4.2 Maximum drawdown (MDD) 62 4.4.3 Portfolio turnover (PT) 62 4.4.4 Winning rate (WR) 63 4.4.5 Jensen’s Alpha 63 4.4.6 The statistical significance of the differences between two strategies on the performance measures 64 4.5 VN - Index and 1/N portfolios benchmarks 65 CHAPTER 5: EMPIRICAL RESULTS & DISCUSSION 67 5.1 VN – Index and 1/N portfolio performance 67 5.1.1 VN – Index performance 67 5.1.2 1/N portfolio performance 70 5.2 Portfolio out – of –sample performance 73 5.2.1 Sample covariance matrix (SCM) 73 5.2.2 Single index model (SIM) 77 5.2.3 Constant correlation model (CCM) 80 5.2.4 Shrinkage towards single index model (SSIM) 83 5.2.5 Shrinkage towards constant correlation model (SCCM) 91 5.2.6 Shrinkage towards identity matrix (STIM) 96 vi 5.3 Summary performances of covariance matrix estimators on out – of – sample 100 5.4 Conclusion and future works 106 5.4.1 Conclusion 106 5.4.2 Future works 112 REFERENCES vii 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 viii 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 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 107 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 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: 108 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 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 109 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 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 110 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 Seventh, the shrinkage approaches, especially the SCCM, have proven that they are well suited for optimal portfolio selection under bad stock market conditions Under these conditions, the market is often in a strong downtrend, leading to a loss in the portfolio's investment results Because the shrinkage method tends to select low volatility portfolios, so it can help investors get through tough market times The empirical results have shown that the shrinkage method has the lowest loss during the downtrend of 2018 – 2019; the SCCM even has a positive profit value during this period Eighth, 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, 111 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 Moreover, the selection of optimal portfolio based on the estimators of covariance matrix can be a reference channel for regulators in building a portfolio representing the market next to the VN-Index 5.4.2 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 112 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 controversy around the method Moreover, the author only consider the dimension of covariance matrix with the maximum number of shares N = 350 in this research, so in the future when Vietnam's financial market develops, researchers may increase the considered number of stocks in portfolio to have better assessing the difference between the estimators of covariance matrix such as model – based and shrinkage approaches and the traditional sample covariance matrix Furthermore, the trading frequency used in the back-testing is weekly In the future, when the Vietnamese financial market develops, the company's shares can be bought and sold on 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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?...MINISTRY OF EDUCATION AND TRAINING THE STATE BANK OF VIETNAM BANKING UNIVERSITY OF HO CHI MINH CITY NGUYEN MINH NHAT SHRINKAGE ESTIMATION OF COVARIANCE MATRIX FOR PORTFOLIO SELECTION ON VIETNAM STOCK. .. 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