Shrinkage estimation of covariance matrix for portfolio selection on Vietnam stock market

urpass CCM’s. 93 Source: Results calculated by back-testing system Figure 5.13: Compare the cumulative return between SCCM and VN-Index The results of back-testing procedure on out-of-sample reveal that the average return of SSCM-based portfolio is pretty high at 18.81%, which is far higher than the figure of VN-Index (12.73%) and 1/N portfolio (15.27%) with N=350. SCCM is one of rare approaches with an average positive return in 2018; this leads the cumulative return of SCCM in the period of 2013 – 2019 to be very high at nearly 275% that is completely superior than VN-Index’s and 1/N strategy’s. Moreover, 1/N strategy and VN-Index has significant return fluctuation at 10.15% and 16.02%, while the figure for SCCM is only at 7.48%. At N=200 and above, the Sharpe ratio of SCCM obviously dominates that of the two benchmarks (ex. with N=350, Sharpe ratio of SCCM is 1.72 times, compared to 1.02 times of 1/N and 0.55 times of VN-Index). This ensures a high compensation rate for each unit of risk taken for the SCCM – based portfolio investor who hold more than 200 stocks. Besides, regarding the loss level in the worst scenario of the market, the SCCM method still provides lower MDD in almost every portfolio size: while MDD of 1/N strategy is 19.71% and of VN-Index is 27.13%, the result for 100-stock SCCM portfolio is only at 15.29%, not to mention that this figure will be improved when N rises to 350. When comparing the Jensen’s alpha metric, the SCCM performance starts to exceed 1/N portfolio at N=200 (9.49% compared to 6.01%). When it comes to 350-stock portfolio, whose size equals to 1/N one, the superior return reaches up to 11.36%. However, the 94 portfolio turnover and winning rate metrics of 1/N are still better than SCCM’s. This means 1/N portfolio holder can save more transaction cost and has better possibility of earning positive return, but this figure is trivial. Generally, with 5/7 metrics better, it is undeniable that SCCM approach outperform the benchmarks. Back-testing results N=350 N=200 N=100 N=50 Shrinkage coefficient Source: Results calculated by back-testing system Figure 5.14: Back-testing results of SCCM’s shrinkage coefficient ( ) on out – of – sample from 1/1/2013 - 31/12/2019 The change of SCCM’s shrinkage coefficient has many similarities with that of SSIM that is when the number of shares in the portfolio increase, the fluctuating area of shrinkage coefficient tends to increase. For example, the fluctuating area of SCCM’s shrinkage coefficient are 0.3 – 0.7; 0.25 – 0.65; 0.25 – 0.55; and 0.2 – 0.5 corresponding to N = 350, 200, 100, 50. The higher value of shrinkage coefficient, the more it shows the importance of the shrinkage target matrix in determining the estimated covariance matrix for portfolio optimization because it is more involved in the estimation process of covariance matrix. The shrinkage target matrix of CCM has done its job well in minimizing the estimation errors arising in the traditional SCM method, especially when N is high. In particular of N = 350, there are times when SCCM's shrinkage coefficient has reached very high values up to 0.7, which means that the shrinkage target matrix has affected 70% of the estimation of the covariance matrix. Moreover, the value of SCCM’s shrinkage coefficient tends to be higher than that of SSIM. This will lead the performance of SCCM to be better than that of SSIM in generating the return of portfolio and minimizing the risk of portfolio. 95 5.2.6 Shrinkage towards identity matrix (STIM) Table 5.8: The performance of STIM from 1/1/2013 to 31/12/2019 Source: Results calculated by back-testing system STIM demonstrates the efficiency when the sample size is rising, in line with the fluctuation of stock numbers in the sense of an increasingly developing market. Therefore, considering the broad scale of the portfolio, the covariance matrix of the portfolio that is determined through the STIM back-testing method tends to be stronger in terms of volatility than other methods. Specifically, when analyzing the portfolio's highest loss level in the 2013–2019 period, we can see that, for a limited sample size, STIM has not really outperformed the rest of the models in Maximum Drawdown (MDD) but is not so special. Nevertheless, when sample size grew over 200 and stopped at 8.43% when N=350, MDD showed great progress. Performance metrics N=350 N=200 N=100 N=50 Average annual return Number of stocks (N) Performance metrics Average annual return Average annual volatility Sharpe ratio (Times) Portfolio Turnover (Daily) MDD Winning rate Alpha N = 50 10.59% 11.16% 0.56 2.02% (20.62%) 53.78% 1.92% N = 100 10.58% 9% 0.65 2.28% (18.35%) 53.2% 2.92% N = 200 11.82 7.4% 0.95 2.36% (14.23%) 55.21% 4.88% N = 350 16.55% 6.52% 1.71 2.26% (8.43%) 58.08% 9.59% 96 Average annual volatility Sharpe ratio MDD Portfolio turnover (daily) Source: Results calculated by back-testing system Figure 5.15: Back-testing results of STIM on out – of – sample from 1/1/2013 – 31/12/2019 In more specific, average annual return of all assets given by STIM was superior to SCM's when N was over 200 and much higher with N=350 (16.55% compare to 10.28%). In terms of volatility metrics, almost every portfolio size that has STIM model-estimated covariance matrix has better results than those that use SCM. In fact, even with a limited number of stocks in the portfolio, the SCM model outweighs STIM for MDD even with a portfolio of 350 stocks, using the STIM model results in absolute excellent performance relative to the MDD of SCM portfolios (8.43% compared to 23.68%). STIM's portfolio turnover far outperforms the SCM benchmark. The SCM approach has a relatively high daily turnover of 7.04%/ day, equivalent to 35.2 %/ week, indicating the low volatility in the weight of the properties. This means that after each reevaluation, composition in the SCM-based portfolio will shift dramatically. At this point, the STIM strategy offers a much better degree of portfolio change stability than the SCM process, with an average 97 turnover of only 2.26% per day, equal to around 15.82% of the trading value of the portfolio per week. The STIM method's superiority benefit is also demonstrated in the Jensen's Alpha during the 2013–2019 period. For the portfolio size larger than 100, STIM model gives higher superior return than the SCM methods. Source: Results calculated by back-testing system Figure 5.16: Compare the cumulative return between STIM and VN-Index To evaluate STIM back-testing procedure more intensively, comparing with benchmarks are also made. At N=50, N=100 and N=200, both VN-Index and 1/N portfolio show better performance than STIM in terms of higher return. But, the average annual return of the STIM was very high about 16.55%, which is much higher than the benchmark of VN- Index’s (12,73%) and 1/N portfolio (15,27%), at N=350. Especially, in case of N=350 – equivalent to the sample of VN-Index and 1/N portfolio – STIM-based portfolio outperforms VN-Index and 1/N strategy in most of metrics like average annual return, annual volatility, Sharp’s ratio, MDD, and Jensen’s Alpha. On the other hand, STIM- based portfolio turnover is still a bit lower than 1/N, which means holding 1/N portfolio can help investors save more transaction cost and avoid liquidity risk. Compared to 1/N strategy and VN-index benchmarks, winning rate of STIM is a little bit lower. Thus, when holding a STIM-based portfolio, the investors have less potential opportunity to earn positive rate of return than holding 1/N portfolio. 98 Back-testing results N=350 N=200 N=100 N=50 Shrinkage coefficient Source: Results calculated by back-testing system Figure 5.17: Back-testing results of STIM’s shrinkage coefficient ( ) on out – of – sample from 1/1/2013 - 31/12/2019 When observing the shrinkage coefficient of the STIM method in the period from 2013 to 2019, we can see that the coefficient changes at an average level when it distributes in the area with values from 0.05 to 0.55. Although the value of the coefficient of shrinkage is only at the average level, however, depending on the different stages, the shrinkage coefficient has relatively different fluctuations. During the period from the beginning of 2013 to the end of 2016, the shrinkage coefficient is relatively low, as it only fluctuates around the area with values from 0.05 to 0.25. This shows that there is not much error in the sample covariance matrix estimation and the covariance matrix estimated by the STIM during this period is strongly influenced by the SCM. However, in the period from the end of 2016 to the beginning of 2018, there was a drastic change in the shrinkage coefficient, which quickly increased from 0.25 to 0.55. This movement begins to show that there have been errors in the SCM estimation and the estimated covariance matrix by the STIM requires more involvement from the shrinkage target matrix. In the period of 2018 - 2019, when the errors in the SCM method showed signs of decrease, the shrinkage coefficient changed in the opposite direction, this coefficient decreased from 0.55 to 0.3 at the end of the year 2019. In addition, we also see that the moving area of STIM’s shrinkage coefficient tends to increase as the number of stocks in the portfolio increases. In particular, when the amount of portfolio stocks is 350, the value of STIM’s shrinkage coefficient moves in range from 99 0.1 – 0.6 while the ranges of coefficient are 0.1 – 0.5, 0.1 – 0.4, and 0.05 – 0.35 corresponding to N = 200, N = 100, and N = 50. 5.3 Summary performances of covariance matrix estimators on out – of – sample Approaches Covariance matrix estimators Performance metrics Average annual return Average annual volatility SR (Times) Daily Portfolio Turnover MDD WR Alpha Benchmarks VN-Index 12.73% 16.02% 0.55 - (27.13%) 54.42% - 1/N portfolio 15.27% 10.15% 1.02 0.7% (19.71%) 58.72% 6.01% N = 50 Standard SCM 11.5% (1.00) 11.35% (1.00) 0.63 (1.00) 2.41% (1.00) (19.06%) 53.21% 2.91% (1.00) Model - based SIM 10.71% (0.00) 13.27% (0.00) 0.5 (0.00) 0.97% (0.00) (19.63%) 55.16% (0.11%) (0.00) CCM 13.35% (0.00) 12.6% (0.00) 0.7 (0.00) 1.4% (0.00) (19.29%) 54.24% 3.26% (0.00) Shrinkage SSIM 11.17% (0.03) 11.23% (0.02) 0.6 (0.02) 2.25% (0.03) (20.08%) 53.61% 2.52% (0.03) SCCM 11.53% (0.04) 11.13% (0.015) 0.64 (0.02) 2% (0.02) (20.39%) 53.96% 2.66% (0.02) STIM 10.59% (0.00) 11.16% (0.01) 0.56 (0.00) 2.02% (0.00) (20.62%) 53.78% 1.92% (0.00) N = 100 Standard SCM 10.97% (1.00) 9.67% (1.00) 0.58 (1.00) 3.36% (1.00) (17.9%) 52.4% 2.83% (1.00) Model - based SIM 12.28% (0.00) 11.57% (0.00) 0.67 (0.00) 0.96% (0.00) (18.26%) 56% 2.26% (0.00) CCM 16.28% (0.00) 10.56% (0.00) 1.03 1.68% (0.00) (15.62%) (0.00) 54.64% (0.00) 6.93% (0.00) Shrinkage SSIM 10.87% (0.03) 9.93% (0.025) 0.65 (0.02) 2.87% (0.0) (15.795) 53.04% 3.33% (0.01) Table 5.9: Summary back-testing results of covariance matrix estimators on out – of – sample from 1/1/2013 – 31/12/2019 100 Source: Results calculated by back-testing system Notes: “p – values that measure the statistical significance of the differences among performance metrics of a particular covariance matrix estimator and those of sample covariance matrix estimator in the case of different number of stocks”. SCCM 13.56% (0.00) 9.27% (0.00) 0.89 (0.00) 2.22% (0.00) (15.29%) 54.36% 5.59% (0.00) STIM 10.58% (0.02) 9% (0.01) 0.65 (0.00) 2.28% (0.00) (18.35%) 53.2% 2.92% (0.01) N = 200 Standard SCM 11.16% (1.00) 8.66% (1.00) 0.76 (1.00) 4.93% (1.00) (14%) 54.41% 4.6% (1.00) Model - based SIM 14.14% (0.00) 9.91% (0.00) 0.94 (0.00) 0.96% (0.00) (17.53%) 57.91% 4.87% (0.00) CCM 19.52% (0.00) 9% (0.00) 1.5 (0.00) 1.97% (0.00) (11.08%) 58.14% 10.96% (0.00) Shrinkage SSIM 12.7% (0.00) 7.8% (0.00) 1 (0.00) 3.34% (0.00) (11.59%) 55.9% 5.92% (0.00) SCCM 16.98% (0.00) 8.05% (0.00) 1.41 (0.00) 2.62% (0.00) (8.38%) 56.8% 9.49% (0.00) STIM 11.82 (0.02) 7.4% (0.00) 0.95 (0.00) 2.36% (0.00) (14.23%) 55.21% 4.88% (0.00) N = 350 Standard SCM 10.28% (1.00) 8.97% (1.00) 0.68 (1.00) 7.04% (1.00) (23.68%) 55.39% 4.19% (1.00) Model - based SIM 17.33% (0.00) 8.94% (0.015) 1.34 (0.00) 1% (0.00) (17.05%) 59.63% 8.26% (0.00) CCM 19.69% (0.00) 8.61% (0.01) 1.59 (0.00) 1.69% (0.00) (12.13%) 57.79% 11.13% (0.00) Shrinkage SSIM 16.18% (0.00) 6.8% (0.00) 1.59 (0.00) 3.45% (0.00) (7.87%) 57.34% 9.4% (0.00) SCCM 18.81% (0.00) 7.48% (0.00) 1.72 (0.00) 2.63% (0.00) (8.2%) 57.7% 11.36% (0.00) STIM 16.55% (0.00) 6.52% (0.00) 1.71 (0.00) 2.26% (0.00) (8.43%) 58.08% 9.59% (0.00) 101 Table 5.9 summarizes the out – of – sample performance metrics for the different estimators of covariance matrix, the VN-Index and 1/N portfolio benchmarks and p- values are presented in parenthesis. The performance metrics of portfolios calculated from the sample covariance matrix are set up as the standard/traditional approach, so that the other estimators are compared to it. The difference between two estimators is recognized as significant when the value of p is lower than 5%. In order to save space, the values of p are less than 1% denoted by 0.00. Based on the back-testing results of covariance matrix estimators on out – of – sample, we can draw some important points as follows: First, the 1/N portfolio strategy is still considered to be a tough benchmark that other strategies need to overcome. The 1/N strategy has shown its superiority on most performance metrics compared to VN-Index’s, especially its sharpe ratio is twice as high as VN-Index’s. The limitation of VN-Index benchmark comes from the fact that this index tends to be influenced by a group of industries and large market capitalization companies, resulting in low diversification and high volatility. Meanwhile, the 1/N portfolio strategy shows higher portfolio diversification and better profitability as well as lower risk compared to VN – Index’s. Besides, the 1 / N portfolio strategy also shows a great advantage with the lowest portfolio turnover that make the strategy save the transaction costs and reduce the liquidity risk in trading. In addition, the high winning rate is also a highlight of this strategy. Second, the traditional estimator of sample covariance matrix is not highly effective for portfolio optimization. The back–testing results of SCM on portfolios with different number of stocks (N=50,100,200, 350) all did not perform as well as performance of benchmarks. In which, the best back-testing performance of SCM was achieved when N = 200, however, most its performance metrics cannot pass the performance of 1/N portfolio benchmark, except for the portfolio volatility criteria. 102 Moreover, the performance of SCM is completely ineffective compared to other approaches such as model-based or shrinkage approaches, and the difference is larger when the number of stocks in portfolio increases. For example, when N = 50, there is not much difference of average annual return between SCM and model- based, shrinkage approaches, however, when N = 350, the difference is 8.23% and 6.9% respectively. The same goes for all other criteria in the portfolio such as average annual volatility, sharpe ratio, portfolio turnover, maximum drawdown, winning rate, and alpha coefficient. Third, the model-based approaches give better results than benchmarks and are more efficient than the traditional SCM method. The superiority of model - based methods over benchmarks begins to manifest at N = 200 and is reflected on most performance metrics, except for portfolio turnover and winning rate metrics. Meanwhile, 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. 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 happened across all tested portfolios and most performance metrics. In particular, criteria reflecting the profitability of the portfolio such as average annual return, sharpe ratio or Jensen's Alpha all show that the CCM method gives much better results than SIM method. Besides, the CCM method also shows a more stable level of profit and is safer than the SIM method, even in cases where the market has bad developments. That's why the average annual volatility and maximum drawdown of CCM is much lower than these of SIM on all tested portfolios. Although the portfolio turnover and winning rate criteria of the SIM approach have better results than the CCM approach, the advantages from these two criteria do not help the SIM method have better performance than the CCM method. Fourth, the shrinkage methods also show superiority in portfolio optimization over the traditional SCM method and benchmarks when N tends to increase. When the number of 103 shares in the portfolio was N = 50, the shrinkage method did not differ much from the SCM, and even worse than benchmarks. However when N was larger than 100 stocks, the shrinkage method began to clearly outperform the performance of SCM and when N approached to 200 stocks, the shrinkage methods outperformed the benchmarks on most criteria. Moreover, the shrinkage method also demonstrates its superiority in the case of high - dimensional portfolios. This is clearly reflected when comparing shrinkage method with model - based method. 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, when N soared to reach 350 stocks, the model-based approaches was completely defeated by the shrinkage methods such SSIM, SCCM, and STIM. We can see that the model-based method is still very impressive in its ability to generate high profits for the portfolio at N = 350, but the shrinkage method does even better by maintaining profit growth while minimizing the volatility of portfolio. The average annual volatility of shrinkage method is only about 6.9%, which is very low if compared to the value of 8.8% in the model – based method. This makes the performance metrics such as sharpe ratio and alpha of shrinkage method completely superior to these of model – based approaches. In addition, the average maximum drawdown of shrinkage approach also performs very well, showing that maximum loss of portfolio is half that of model – based approaches. Although the criteria of portfolio turnover and winning rate of shrinkage method are somewhat weaker than the model - based method, in general the shrinkage method shows much better performance than the model - based method in the case of high – dimensional portfolios. Fifth, among the shrinkage methods, the SCCM clearly outperforms the other two methods across all tested portfolios and on most portfolio performance metrics. In particular, the average annual return of SCCM on four tested portfolios is 15.22% which is much higher than 12.73% of SSIM and 12.38% of STIM. In which, the highest annual 104 return of SCCM is up to 18.81% when the number of stocks in portfolio is 350. The extremely low volatility is considered one of the best advantages of shrinkage portfolios and among three shrinkage methods, the STIM’s portfolio volatility is the best on out – of – sample back-testing results. However, when considering portfolio return over a risk unit, the SCCM still delivers the best results. The average sharpe ratio of SCCM on four tested portfolios is 1.16 times which is the higher than that of SSIM and STIM with values of 0.96 times and 0.97 times respectively. The highest sharpe ratio of SCCM is 1.72 times when N = 350. Moreover, the alpha coefficient of SCCM also achieved the biggest positive value of three shrinkage methods. Meanwhile, the performance criteria of portfolio turnover, maximum drawdown and winning rate are not much different among these shrinkage methods. In the case of high – dimensional portfolio (N = 350), the SCCM still shows the best results overall, however the difference among these shrinkage methods tends to decrease, especially between the SCCM and STIM. Table 5.10: The movement value of shrinkage coefficient ( ) Number of stocks Movement area of shrinkage coefficient SSIM SCCM STIM N = 50 0.025 – 0.3 0.25 – 0.5 0.05 – 0.35 N = 100 0.05 – 0.35 0.25 – 0.55 0.1 – 0.4 N = 200 0.05 – 0.45 0.27 – 0.65 0.1 – 0.52 N = 350 0.05 – 0.55 0.3 – 0.7 0.1 – 0.62 Source: Results calculated by back-testing system Lastly, the superiority of SCCM over SSIM and STIM methods can be explained by the value of shrinkage coefficient. The table 5.10 shows that the movement areas of SCCM’s shrinkage coefficient are always higher than that of SSIM and STIM on all tested portfolios. The higher shrinkage coefficient helps the SCCM method to be able to adjusting the estimated covariance matrix more and generating the profit more compared to the SSIM and STIM. 105 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 dateall make it 106 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 107 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 108 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. 109 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 do 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. 110 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. 111 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 do 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 112 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. REFERENCES Bai, J., and Shi, S. (2011). Estimating high dimensional covariance matrices and its applications. Annals of Economics and Finance, 12: 199–215. Broadie, M. (1993). Computing efficient frontiers using estimated parameters. Annals of operations research, 45:21-58. Best, J., and Grauer, R. (1991). Sensitivity analysis for mean-variance portfolio problems. Management Science, 37(8):980–989. Bengtsson, C., and Holst, J. (2002). On portfolio selection: Improved covariance matrix estimation for Swedish asset returns. Journal of Portfolio Management, 33(4): 55-63. Chan, L., Jason, K., and Josef, L.(1999). On portfolio optimization: Forecasting covariances and choosing the risk model. The Review of Financial Studies, 12(5):937–974. Chandra, S.(2003). Regional Economy Size and the Growth-Instability Frontier: Evidence from Europe. J. Reg. Sci., 43(1): 95-122. doi:10.1111/1467-9787.00291. Chekhlov, A., and Uryasev, S. (2005). Drawdown measure in portfolio optimization. International Journal of Theoretical and Applied Finance, 8(1):13-58. Chen, Y., Ami W., Yonina E., and Alfred, H. (2010). Shrinkage algorithms for MMSE covariance estimation. IEEE Transactions on Signal Processing, 58: 5016–29. Chopra, V., and Ziemba, W.(1993). The effect of errors in means, variances, and covariances on optimal portfolio choice. Journal of Portfolio Management, 19(2):6- 11. Clarke, Roger G., Harindra De Silva, and Steven Thorley. (2006). Minimum-variance portfolios in the U.S. equity market. Journal of Portfolio Management, 33: 10–24. Cohen, J., and Pogue, J. (1967). An empirical evaluation of alternative portfolio selection models. The Journal of Business, 40(2):166–193. De Nard, G., Ledoit, O., and Wolf, M. (2019). Factor models for portfolio selection in large dimensions: The good, the better and the ugly. Journal of Financial Econometrics. Forthcoming. DeMiguel, V., and Francisco N., (2009). Portfolio selection with robust estimation. Operations Research, 57(3):560–577. Efron, B., R. Tibshirani. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York. Edwin, J., Martin, G., and Manfred, P. (1997). Simple rules for optimal portfolio selection: the multi group case. Journal of Financial and Quantitative Analysis, 12(3):329–345. Elton, Edwin J., and Martin J Gruber. (1973). Estimating the dependence structure of share prices: implications for portfolio selection. The Journal of Finance, 28(5):1203– 1232. Elton, Edwin J., Martin J Gruber., Stephen B., and William N Goetzmann.(2009). Modern portfolio theory and investment analysis. John Wiley & Sons. Ernest, Chan P. (2009). Quantitative trading: How to Build Your Own Algorithmic Trading Business. John Wiley & Sons. Ernest, Chan P. (2013). Algorithmic Trading: Winning Strategies and Their Rationale. John Wiley & Sons. Fabozzi, F., Gupta, F., and Markowitz, H.(2002). The legacy of modern portfolio theory. The Journal of Investing, 11(3): 7–22. Fama, E., and French, K. (1993). Common risk factors in the returns on stocks and bonds. The Journal of Financial Economics, 33: 3–56. Fama, E., and French, K. (2015). A five-factor asset pricing model. Journal of Financial Economics, 16: 1–22. Fama, E., and French, K. (2016). Dissecting Anomalies with a Five-Factor Model. Review of Financial Studies, 29: 69–103. Fernández, A., and Gómez, S. (2007). Portfolio selection using neural networks. Computers and Operations Research, 34: 1177–91. Frankfurter, G., Phillips, H., and Seagle, J. (1971). Portfolio Selection: The Effects of Uncertain Means, Variances, and Covariances. Journal of Financial and Quantitative Analysis, 6(5):1251 – 1262. George Derpanopoulos (2018). Optimal financial portfolio selection. Working paper. Haugen, Robert A., and Nardin L. Baker. (1991). The efficient market inefficiency of capitalization-weighted stock portfolios. Journal of Portfolio Management, 17: 35– 40. Huber, P. J. (2004). Robust Statistics. John Wiley & Sons, New York. Ikeda, Y., and Kubokawa, T. (2016). Linear shrinkage estimation of large covariance matrices using factor models. Journal of Multivariate Analysis, 152: 61–81. Iyiola, O., Munirat, Y., and Christopher, N.(2012). The modern portfolio theory as an investment decision tool. Journal of Accounting and Taxation, 4(2):19-28. Jagannathan, R., and Ma, T., (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. The Journal of Finance, 58(4):1651–1683. James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1, pages 361– 380. Jensen, M.(1968). The performance of mutual funds in the period 1945 – 1964. Journal of Finance, 23(2):389-416. Jobson, J., Korkie, B. (1980). Estimation for Markowitz efficient portfolios. Journal of the American Statistical Association, 75(371): 544 – 554. Jorion, P. (1985). International Portfolio Diversification with Estimation Risk. Journal of Business, 58: 259–78. Jorion, P. (1986). Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21(3):279–292. Kwan, Clarence C.Y. (2011). An introduction to shrinkage estimation of the covariance matrix: A Pedagogic Illustration. Spreadsheets in Education (eJSiE), 4(3): article 6. Kempf, A., and Memmel, C. (2006). Estimating the global minimum variance portfolio. Schmalenbach Business Review, 58:332-48. Konno, Y. (2009). Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss. Journal of Multivariate Analysis, 100: 2237–53. Lam, C. (2016). Nonparametric eigenvalue-regularized precision or covariance matrix estimator. Annals of Statistics, 44(3):928–953. Liagkouras, K., and Kostas M. (2018). Multi-period mean-variance fuzzy portfolio optimization model with transaction costs. Engineering Applications of Artificial Intelligence, 67: 260–69. Ledoit, O. and Peche, S. (2011). Eigenvectors of some large sample covariance matrix ensembles. Probability Theory and Related Fields, 150(1–2):233–264. Ledoit, O. and Wolf, M. (2003a). Honey, I shrunk the sample covariance matrix. Journal of Portfolio Management, 30(4):110–119. Ledoit, O. and Wolf, M. (2003b). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10(5):603–621. Ledoit, O. and Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2):365–411. Ledoit, O. and Wolf, M. (2008). Robust performance hypothesis testing with the Sharpe ratio. Journal of Empirical Finance, 15(5):850–859. Ledoit, O. and Wolf, M. (2012). Nonlinear shrinkage estimation of large-dimensional covariance matrices. Annals of Statistics, 40(2):1024–1060. Ledoit, O. and Wolf, M. (2015). Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions. Journal of Multivariate Analysis, 139(2):360–384. Ledoit, O. and Wolf, M. (2017a). Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks. Review of Financial Studies, 30(12):4349–4388. Ledoit, O. and Wolf, M. (2017b). Numerical implementation of the QuEST function. Computational Statistics & Data Analysis, 115:199–223. Ledoit, O. and Wolf, M. (2018b). Optimal estimation of a large-dimensional covariance matrix under Stein’s loss. Bernoulli, 24(4B). 3791–3832. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7:77–91. Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments. John Wiley & Sons. Martin, D., Clark, A., and Christopher G. (2010). Robust portfolio construction. Handbook of Portfolio Construction. Springer, 337–380. Merton, R. (1969). Life time portfolio selection under uncertainty: The continuous-time case. The review of Economics and Statistics, 51: 247–57. Merton, R. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3: 373–413. Merton, R. (1972). An analytic derivation of the efficient portfolio frontier. Journal of Financial and Quantitative Analysis, 7: 1851–72. Merton, C. (1980). On estimating the expected return on the market: An exploratory investigation. Journal of financial economics, 8(4):323–361. Michaud, Richard O. (1989). The Markowitz optimization enigma: Is ”optimized” optimal. Financial Analysts Journal, 45(1):31–42. Michaud, Richard O., Esch, D., and Michaud, R. (2012). Deconstructing Black- Litterman: How to get the Portfolio you already knew you wanted. Available at SSRN 2641893. Nick Radge (2006). Adaptive analysis for Australian stocks. Wrightbooks. Nguyen, Tho. (2010). Arbitrage Pricing Theory: Evidence from an Emerging Stock Market, Working Papers 03, Development and Policies Research Center (DEPOCEN), Vietnam. Okhrin, Yarema, and Wolfgang Schmid. (2007). Comparison of different estimation techniques for portfolio selection. Asta Advances in Statistical Analysis, 91: 109–27. Pogue, A. (1970). An extension of the Markowitz portfolio selection model to include variable transactions’ costs, short sales, leverage policies and taxes. The Journal of Finance, 25: 1005–27. Ross, S. A. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3):341–360. Samuelson, A. (1969). Life time portfolio selection by dynamic stochastic programming. The Review of Economics and Statistics, 51: 239–46. Sch¨afer, J. and Strimmer, K. (2005). A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. Statistical Applications in Genetics and Molecular Biology, 4(1). Article 32. Soleimani, H., Golmakani, H., and Mohammad S. (2009). Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm. Expert Systems with Applications, 36: 5058– 63. Sharpe, William F. (1963). A simplified model for portfolio analysis. Management science, 9(2):277–293. Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The journal of finance, 19(3):425–442. Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, pages 197–206. University of California Press. Stein, C. (1986). Lectures on the theory of estimation of many parameters. Journal of Mathematical Sciences, 34(1):1373–1403. Taleb, N. (2007). The Black Swan: The Impact of the Highly Improbable, Random House, 80-120. Truong, L., and Duong, T. (2014). Fama and French three-factor model: Empirical evidences from the Ho Chi Minh Stock Exchange. Journal of Science Can Tho University, 32:61 – 68. Vo, Q., and Nguyen, H. (2011). Measuring risk tolerance and establishing an optimal portfolio for individual investors in HOSE. Economic Development Review, 33-38. Xue, Hong-Gang, Cheng-Xian Xu, and Zong-Xian Feng. (2006). Mean-variance portfolio optimal problem under concave transaction cost. Applied Mathematics and Computation, 174: 1–12. Yang, L., Romain, C., and Matthew, M. (2014). Minimum variance portfolio optimization with robust shrinkage covariance estimation. Paper presented at 2014 48th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, November 2–5; pp. 1326–30.