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IMPACT OF TIME VARYING DISTRIBUTIONAL PARAMETERS ON PORTFOLIO PERFORMANCE A Thesis Presented to the Faculty of ISM University of Management and Economics in Partial Fulfillment of the Requirements for the Degree of Master of Financial Economics by Kristina Barauskaitė May 2015 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE ABSTRACT The aim of this paper is to examine the impact of time-varying distributional parameters on portfolio performance. Paper is related to other literature such as Tse (1991) and Horasanl & Fidan (2007). In this work here is used the portfolio formed from 1 ETF and 6 Indexes. First, the evidences are provided that portfolio distributional parameters (volatilities, correlations, and means) are varying over time. Secondly, Markowitz efficient frontier with chosen efficient portfolio is used as a benchmark. Using EWMA and DCCGARCH(1,1) methods two efficient frontiers with time-varying covariance matrixes are calculated. Near to that, 24 months rolling window method is applied for Markowitz, EWMA and DCC-GARCH(1,1) estimated efficient portfolios to see how the portfolios weights are changing over time. The findings of this paper are as following: EWMA model provides the least risky efficient portfolios while Markowitz and DCC-GARCH(1,1) models give very similar results. Rolling EWMA efficient portfolio suggests how the portfolio weights should be amended over time. This is the most important at the times of the turbulences in the financial markets as it may help to protect the portfolio from the big losses. However, it is not straightforward how often the portfolio should be rebalanced. The further research, related to this paper, could examine if more advanced GARCH(p,q) forms together with DCC model can outperform the DCC-GARCH(1,1) model and maybe can also beat the EWMA model which could lead to even better impact of time-varying distributional parameters on portfolio performance. Keywords: portfolio optimization, portfolio management, time-varying distributional parameters, EWMA, DCC-GARCH(1,1), rolling portfolios, risk management 2 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 3 TABLE OF CONTENTS ABSTRACT .................................................................................................................................... 2 1. INTRODUCTION ...................................................................................................................... 7 2. LITERATURE REVIEW ......................................................................................................... 10 2.1. Modern Portfolio Theory ................................................................................................... 10 2.2. Critique about Modern Portfolio Theory ........................................................................... 12 2.3. Alternative Optimal Portfolio Composition Methods ........................................................ 14 2.3.1. Portfolio Optimization Using Equally Weighted Data ................................................ 15 2.3.2. Portfolio Optimization Using Time-Varying Parameters ............................................ 17 3. METHODOLOGY ................................................................................................................... 21 3.1. Sample Definition .............................................................................................................. 21 3.2. Data Analysis Tests and Normalization ............................................................................. 24 3.2.1. Normality Tests ........................................................................................................... 24 3.2.2. Stationarity Test ........................................................................................................... 25 3.2.3. Data Normalization Process ........................................................................................ 25 3.3. Markowitz Optimal Portfolio Using Matrix Algebra ......................................................... 26 3.4. Improvement of Markowitz Approach: Rolling Portfolios ................................................ 30 3.5. Time-varying Parameters: Exponentially Weighted Moving Average. ............................. 33 3.6. Time varying Parameters: GARCH Models ...................................................................... 35 3.6.1. Testing for ARCH Effect ............................................................................................. 35 3.6.2. GARCH(1.1) Model .................................................................................................... 36 3.6.3. The Dynamic Conditional Correlation (DCC) Model ................................................. 37 3.6.3.1 Forecast of Dt (GARCH(1,1)) ............................................................................... 38 3.6.3.2 Forecast of Rt ......................................................................................................... 39 4. EMPIRICAL RESEARCH RESULTS ..................................................................................... 41 4.1. Data Analysis and Testing.................................................................................................. 41 4.1.1. Descriptive Statistics ................................................................................................... 41 4.1.2. Normality Testing and Data Normalization ................................................................ 43 4.1.3. Testing for Stationarity ................................................................................................ 46 4.1.4. Assets Correlations and Returns .................................................................................. 47 4.2. Markowitz Efficient Frontier and Optimal Portfolio ......................................................... 48 4.3. Rolling Parameters and Efficient Portfolios....................................................................... 51 4.3.1. Rolling Global Minimum Variance Portfolio .............................................................. 53 4.3.2. Rolling Efficient Portfolio with 0.8% Monthly Returns ............................................. 54 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 4 4.4. EWMA Efficient Frontier and Rolling Portfolio ............................................................... 55 4.4.1. EWMA Efficient Frontier and Optimal Portfolio........................................................ 55 4.4.2. EWMA Lambda Coefficient ....................................................................................... 58 4.4.3. EWMA Rolling Efficient Portfolio with 0.8% Monthly Returns ................................ 59 4.5. GARCH Efficient Frontier and Rolling Portfolio .............................................................. 60 4.5.1. Lagrange Multiplier (LM) Test for Assets Returns ..................................................... 61 4.5.2. GARCH Efficient Frontier .......................................................................................... 61 4.5.3. GARCH Rolling Efficient Portfolio with 0.8% Monthly Returns .............................. 65 5. DISCUSSION ........................................................................................................................... 68 5.1. Final Research Results ....................................................................................................... 68 5.1.1. Efficient Frontiers ........................................................................................................ 68 5.1.2 Adjusting Weights for Efficient Portfolio with 0.8% Monthly Returns....................... 70 5.2. Research Linkage to the Existing Literature ...................................................................... 73 5.3. Limitations of Research ..................................................................................................... 75 5.4. Suggestions for Further Research ...................................................................................... 77 6. CONCLUSIONS....................................................................................................................... 80 Reference List ............................................................................................................................... 84 APPENDICES .............................................................................................................................. 90 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 5 List of Figures Figure 1. Correlation Matrix ......................................................................................................... 47 Figure 2. Returns of Individual Indexes and ETF......................................................................... 48 Figure 3. Efficient Frontier (Markowitz Bullet) ........................................................................... 51 Figure 4. Rooling mean and sd for TLT ....................................................................................... 52 Figure 5. Time-varying correlations between ASX and SPGSAG Indexes ................................. 52 Figure 6. Rolling man and sd of GMV ......................................................................................... 53 Figure 7. Rolling GMV portfolio weights .................................................................................... 53 Figure 8. Rolling sd of Markowitz target return portfolio ............................................................ 54 Figure 9. Rolling weights of Markowitz target return portfolio ................................................... 54 Figure 10. Efficient Frontier (EWMA model) .............................................................................. 58 Figure 11. Rolling sd of EWMA target return portfolio ............................................................... 59 Figure 12. Rolling weights of EWMA target return portfolio ...................................................... 59 Figure 13. Covariance Estimation and Forecast of DCC-GARCH(1,1) ....................................... 62 Figure 14. Efficient Frontier (DCC-GARCH(1,1) model) ........................................................... 65 Figure 15. Rolling sd of GARCH target return portfolio .............................................................. 66 Figure 16. Rolling weights of GARCH target return portfolio..................................................... 66 Figure 17. Markowitz, EWMA and GARCH efficient frontiers .................................................. 69 Figure 18. Rolling weights of EWMA efficient portfolio with target monthly returns of 0.8% .. 71 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 6 List of Tables Table 1. Assets in the portfolio: Indexes and ETF ....................................................................... 22 Table 2. Summary statistics for the period 2005.01.31 – 2014.12.31 .......................................... 42 Table 3. Normality tests for original data ..................................................................................... 44 Table 4. Normality tests for data without outliers ........................................................................ 45 Table 5. Summary statistics of data without some outliers .......................................................... 46 Table 6. Adf.test for stationarity ................................................................................................... 47 Table 7. Time-invariant covariance matrix ................................................................................... 49 Table 8. Markowitz Efficient portfolios characteristics ............................................................... 49 Table 9. EWMA covariance matrix .............................................................................................. 56 Table 10. EWMA Efficient portfolios characteristics .................................................................. 56 Table 11. Lagrange Multiplier (LM) test for assets returns .......................................................... 61 Table 12. GARCH covariance matrix ........................................................................................... 63 Table 13. GARCH Efficient portfolios characteristics ................................................................. 63 Table 14. Risk level of efficient portfolios with target monthly returns of 0.8% ......................... 70 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 1. INTRODUCTION One of the main aims for all investors is to maximize their portfolio returns with specific level of risk. Modern portfolio theory (MPT), introduced by Harry Markowitz in 1952, is trying to maximize portfolio returns for a given level of risk or to minimize risk for the given level of returns. Since H. Markowitz introduced Modern Portfolio Theory there have been a lot of criticisms about it from different authors such as J. Campbell and A. Lo (1997), B. Damghani (2013), J. Brodie and others (2009), R. Michaud (1989) and others, because MPT is not fitting the reality in many ways. One of the main criticisms about MPT is that portfolio distributional parameters are constant and does not change in time, while in real world it is seen that these parameters are changing over time. Because of that, it is important to measure if time varying distributional parameters can improve assets allocation and lead to better returns after constructing the optimal portfolio. Usually MPT is being used to construct optimal portfolio, but in order to add timevarying distributional parameters to portfolio optimization process different authors are using other techniques: Cha & Jithendranathan (2009) and many others use Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) method, Tse (1991) and Horasanl & Fidan (2007) compares few techniques that use time-varying volatility: GARCH and exponentially-weighted moving average (EWMA) methods. Many authors are talking about the impact of time-varying volatility and correlations to the portfolio allocation and how these parameters can be added into the optimization models. However, it is important to mention, that the mean as well as the volatilities and correlations is time varying. 7 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE The goal of these theses is to understand the dynamics of time varying mean, volatility and correlations and the ways in which the portfolio allocation can be adapted to account for all these time-varying parameters. Main question of thesis is: what kind of impact the time varying distributional parameters do on portfolio performance? Main objectives of thesis are as following: To describe how time varying distributional parameters can be added while constructing the optimal portfolio; To analyze how these parameters change the portfolio optimization process; To identify what kind of impact it does on the portfolio performance; To analyze which technique is giving the best results together with time varying distributional parameters. General problem of this thesis is changes in distributional parameters and their impact on the investment portfolio performance. Specific problem is to choose good allocation of assets in the portfolio while facing the impact of time varying distributional parameters. Also, there are different methods how to build portfolio and the result of each of the method could be different so there is a need to compare different methods while constructing the optimal portfolio together with changes in distributional parameters. Concept of the research is planned as following: in order to choose the best combination of assets in the portfolio there is a need to evaluate time varying distributional parameters and their impact on portfolio performance. In this research, optimal portfolios are constructed using three techniques: Markowitz frontier, EWMA and GARCH. Markowitz frontier is constructed by keeping distributional parameters constant. EWMA and GARCH models are constructed by implementing time-varying volatility. Moreover, time-varying 8 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE means are calculated and three optimal portfolios are optimized by them. In the end the impact of time varying distributional parameters on portfolio performance is discussed and explained and conclusions are drawn. Results of this thesis should be the most important to the portfolio managers and advisors as same as for all investors who seek to choose the best asset allocation in their investment portfolio while facing market ups and downs together with other changes in the investment world and between assets. Moreover, results of this research should lead to better understanding what kind of impact the time-varying distributional parameters have on portfolio performance and its returns. The paper is organized as follows. Next chapter provides literature review about portfolio optimization process: starting from Markowitz mean-variance approach and going through more advanced methods. Near to that, in this chapter there is provided thesis contribution to the existing knowledge. Chapter 3 analyzes methodology which is being used in this paper and 4th chapter presents the results of empirical research. In the last chapters there are presented discussion and conclusions about the findings of this research. 9 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 2. LITERATURE REVIEW All investors know that in order to reach better returns with lower risks there is used the diversification of assets, asset classes and other financial instruments. However, it is important to know how to combine different assets to one portfolio that it would reach highest returns with excepted level of risk. Nowadays there are a lot of different techniques which help to combine assets and suggest weights. In this section we are looking through existing literature about different portfolio optimization methods and their critique. Even if this thesis focuses on the time-varying parameters and their impact to the portfolio performance, it is important to take a look at the early stages of the portfolio optimization theory and how it was implemented. Starting from H. Markowitz (1952) the forefather of modern portfolio theory the analysis of the works of other authors who showed critique to Markowitz work and offered other methods which can be used to combine optimal portfolios in order to reach better portfolio results is made. 2.1. Modern Portfolio Theory Harry Markowitz (1952) in his paper considers the rule that all investors are seeking for expected returns and try to avoid variance as much as possible. He demonstrates geometrically connections between “beliefs and choice of portfolio according to the "expected returns-variance of returns" rule.” (pg.77). Author suggests that investors should diversify their investments and the same way maximize expected return. According to Markowitz (1952) returns between securities are inter-correlated and diversification is not eliminating variance fully. “The portfolio with maximum expected return is not necessarily the one with minimum variance. There is a rate at which the investor can gain expected return by taking on variance, or reduce variance by giving up expected 10 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE return” (H.Markowitz, pg. 79). He developed mean-variance analysis in order to select stocks for investment portfolio. While constructing the portfolio, all assets are considered as asset classes rather than single securities. The best way to reduce portfolio risk is to avoid correlation between assets inside the portfolio. Mean-variance analysis is a very strong tool which can help to find a lot of possibilities how to reduce portfolio risk using diversification process (P. D. Kaplan, 1998). R. O. Michaud (1989) marked some benefits of Markowitz mean-variance optimization process. First, what author says is that mean-variance optimization process is a good framework that helps to satisfy client needs when talking about investment portfolios. Secondly, this mean-variance optimizer can be used as a tool to control different parts of risk. Moreover, author suggests that behavior of the investor can be reflected within meanvariance optimizer by choosing certain level of risks and benchmark. Last but not least, according to R. O. Michaud (1989) a lot of information can be processed within portfolio optimizer. This optimizer is very important for huge companies which needs to know how new information impacts their portfolios results as fast as possible. Many authors wrote articles based on Markowitz theory. From the other hand, there are many authors who disagree with H. Markowitz or add something to his theory. Even H. Markowitz (1952), himself knew that there are a lot of gaps in his theory and in his final pages wrote: I believe that better methods, which take into account more information, can be found. I believe that what is needed is essentially a "probabilistic" reformulation of security analysis. I will not pursue this subject here, for this is "another story." It is a story of which I have read only the first page of the first chapter (H. Markowitz, 1952, pg. 91). 11 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 2.2. Critique about Modern Portfolio Theory Many authors tried to extend H. Markowitz theory or to put some critique on it. As M. A. Mora, J. B. Franco, and L. B. Preciado (2010) stretched that the most common critique from different authors about H. Markowitz method is that it is difficult to estimate expected returns and covariances and to understand how these parameters impact the weights of resulting portfolio. Near to that, same authors add that “the optimization eventually suggests portfolio weights that tend to be concentrated in a few assets and are prone to change abruptly when the optimization is effected in a different, albeit close, period” (Mora et al., 2010, pg. 193). F. Black and R. Litterman (1992) raised two main problems related to Markowitz approach. First problem is related to the difficulty to estimate expected returns. Markowitz method requires providing returns of all assets. However, investors do not have clear view about assets returns in some of the markets and because they need to use some extra assumptions, unclear historical returns provide misleading information for the future returns. Second problem which is raised by F. Black and R. Litterman (1992) is that weights of assets in the optimal portfolio are too much reliable on the assumptions which are being used while calculating expected returns. At the same time R. C. Green and B. Hollifield (1992) write that the biggest gap of Markowitz optimization process is lack of diversification while choosing weights for assets in optimal portfolio. They mark that Markowitz method can reach for extreme positions if there are a large number of assets in the portfolio. These authors analyze if there is any difference between mean-variance portfolio and portfolio which is correctly diversified. According to Britten Jones (1999), from financial researches point of view Markowitz mean-variance analysis is a key to many asset pricing theories and their tests. From the other 12 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE hand, practitioners have stated that they are facing difficulties while trying to implement mean-variance analysis. For A. Corvalan (2005) one of the main problems of Markowitz efficient portfolio is also the diversification process: “When investors impose the short sales constraint, the MV (mean-variance) solution puts all wealth in a few assets” (A. Corvalan, 2005, pg. 1). One of the reasons to improve Markowitz approach is disadvantages of optimization process while trying to use it with portfolios which have possibility to short. R. O. Michaud (1998) determined that fluctuations and uncertainty are the main two weaknesses of H. Markowitz theory. Moreover, the author notes that changes in volatilities, expected returns, and covariances can perform changes in the optimal portfolio composition. M. J. Best and R. R. Grauer (1991) and S. D. Hodges and R. A. Brealey (1978), these authors also wrote and discussed the problems which appear while working with H. Markowitz approach. They discussed negative side of Markowitz model as same as other authors such as R. Litterman (2003). However, mean-variance optimization process inspired many extensions and applications. According to G. Yin and X. Y. Zhou (2004), for many years there have been a lot of trials to extend portfolio selection that the model would not be static and could work as the model for several periods or as the continuous time model. They said: “However, the research works on dynamic portfolio selections have been dominated by those of maximizing expected utility functions of the terminal wealth, which is in spirit different from the original Markowitz’s model” (G. Yin and X. Y. Zhou, 2004, pg. 349). In the next section there is described several different methods how to choose assets and weights for optimal portfolio. Models such as Mean-Absolute Deviation (MAD) model, portfolio optimization using utility function, portfolio optimization using log optimal growth portfolio and others are mentioned and analyzed below. 13 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 2.3. Alternative Optimal Portfolio Composition Methods One of the alternatives proposed instead of the Markowitz method is portfolio optimization using Mean-Absolute Deviation model. Mean-Absolute Deviation (MAD) model is just one of the alternatives to the Minimum-Variance (MV) model. MAD was introduced by Konno Hiroshi and Yamazaki Hiroaki (1991). The main difference between MV and MAD is that in MAD model there is not used the assumption that stock returns are normally distributed. According to B. Bower and P. Wentz (2005), MAD is easier to use than Markowitz model because it does not require building a covariance matrix while Markowitz approach does. Moreover, MAD model also minimize the risk only by minimizing mean absolute deviation. Another alternative to mean-variance approach is portfolio optimization using utility function. There are many authors who analyze multi-period utility models such as J. Mossin (1968), P. A. Samuelson (1969) and others. Continuous-time utility model is analyzed by Merton (1973) who analyzes how to maximize a utility using market factors modelled as a diffusion process instead of Markowitz chain process. One more alternative is portfolio optimization using Log Optimal Growth (LOG) portfolio introduced by M. M. John and S. Mwambi (2010). According to these authors, this model is constructed on the framework of mean-variance model. While in the mean-variance model there is only one assumption about normal distribution of the returns, in this LOG model there are two assumptions which say that asset prices follow a Geometric Brownian Motion and those prices are log-normally distributed. X. Y. Zhoud and D. Li (2000) introduced stochastic linear-quadratic (LQ) control framework which allows them to look to the continuous-time mean-variance optimization problem. 14 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE There are many other techniques which can be used as alternative ways to get optimal portfolios. One more example of the alternative portfolio optimization can be shown while analyzing the portfolio optimization by regularization of Markowitz portfolio construction. J. Brodie, I. Daubechies, C. D. Mol, D. Giannone, and I. Loris (2009) suggest expanding “Markowitz objective function by adding penalty term proportional to the sum of the absolute values of the portfolio weights” (pg. 12267). Penalty term is used in order to stabilize the optimization process and encourage spare portfolios. Authors analyze the case with no-shortpositions portfolios that includes several active assets. They implement their method using two benchmarks. Those benchmark data sets were taken from Fama and French. Empirical evidence is found that optimal spare portfolios perform better than the equally weighted portfolios: their variance is smaller and they act better only by having few active positions. It is proven that adding penalty term to Markowitz objective functions is a positive instrument for portfolio optimization tasks. As J. Brodie et al. (2009) write: This penalty forces our optimization scheme to select, on the basis of the training data, few assets forming a stable and robust portfolio, rather than being “distracted” by the instabilities because of collinearities and responsible for meaningless artifacts in the presence of estimation errors (pg. 12272). 2.3.1. Portfolio Optimization Using Equally Weighted Data As mentioned before, in H. Markowitz (1952) introduced mean-variance optimization process there is a need to estimate expected returns, variances and covariances. It is possible to do so by taking ex-post parameters and weighting it equally for all periods to estimate exante portfolio parameters. 15 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE According to C. Alexander (2008) equally weighted historical data is the first method which was accepted all over the world and used while forecasting volatilities and correlations of financial assets returns. “For many years, it was the market standard to forecast average volatility over the next h days by taking an equally weighted average of squared returns over the previous h days. This method was called the historical volatility forecast.” (C. Alexander, 2008, pg. 3). Author adds that nowadays there are many other forecasting techniques which can be used together with historical data sets and it would be a huge mistake to call equally weighed method the only one historical method. However, there are some problems related to the results when data from different periods is weighted equally. According to M. Horasanh and N. Fidan (2007), equally weighted data composition does not show the dynamic structure of the market. C. Alexander (2008) sums up the limits of Markowitz approach in four shortcomings: first, volatility and correlation forecast for all horizon is equal to the current estimate of volatility. The main reason for that is the underlying assumption which says that “returns are independent and identically distributed”; secondly, in this method there is only one choice that has to be made by investor or manager who is using this approach: which data points to use in the data window; third, after extreme market move this model creates a pattern of that extreme move and because of this reason the forecast of volatility and correlation can misrepresent the reality; finally, forecast bias strongly depends on the size of the model data window (pg. 4). J. P. Morgan bank in 1996 started the Risk MetricsTM and software which popularized exponentially weighted moving average (EWMA) method instead of equally weighted data method between financial analysts (C. Alexander, 2008). 16 17 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 2.3.2. Portfolio Optimization Using Time-Varying Parameters According to C. Alexander (2008), an exponentially weighted moving average (EWMA) helps to avoid risks which are being faced while using equally weighted data by simply putting higher weights on the recent observations: “Thus as extreme returns move further into the past as the data window slides along, they become less important in the average” (C. Alexander, 2008, pg. 11). Same author adds that EWMA method is extremely useful when looking for shortterm forecasts (for a day or a week). However, there are two major problems when looking for long-term forecasts. First of all, there is the underlying assumption in the EWMA model which says that: “returns are independent and identically distributed (Alexander, 2008, pg. 14). Second problem related to EWMA is that there is a choice for a user to decide what value he/she will add to the smoothing constant λ. C. Alexander (2008) says the following: The forecasts produced depend crucially on this decision, yet there is no statistical procedure to choose λ. Often an ad hoc choice is made; for example, the same λ is taken for all series and a higher lambda is chosen for a longer-term forecast (pg. 14). In moving average models it is assumed that returns are independent and identically distributed. Moreover, standard errors and confidence intervals can be used because it is assumed that returns are normally distributed. As C. Alexander (2008) writes: “empirical observations suggest that returns to financial assets are hardly ever independent and identically, let alone normally distributed” (pg. 14). Because of all the reasons more practitioners started to forecast using generalized autoregressive conditional heteroskedasticity (GARCH) models. According to M. C. Steinbach (2001), after analyzing assets and their parameters it got clear that volatilities of assets returns as same as other factors are changing over time and it has led to the development of the GARCH models. TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE “There is no doubt that such models produce superior volatility forecasts. It is only in GARCH models that the term structure volatility forecasts converge to the long run average volatility—the other models produce constant volatility term structures” (P. Alexander, 2008, pg. 14). V. Akgiray (1989) presents some new proof how stock prices behaves over the time. Author compares several techniques by modeling and forecasting stock returns. In this research data is taken from the Center of Research in Security Prices (CRSP) tapes. It contains 6030 daily returns from the period January 1963 – December 1986. After analyzing the data author looks how benchmark forecast, EWMA, autoregressive conditional heteroskedasticity (ARCH) and GARCH forecasts fit the real data. Both GARCH and ARCH models show that they are able to imitate very closely the real volatility of the stock market. If comparing both models, GARCH model outperform ARCH model. As author expected, historical averages do not follow short-term changes in volatility and EWMA model is not capable to model transitory changes in volatility. V. Akgiray compares forecasts using a list of different statistics: mean error (ME), root mean squared error (RMSE), mean absolute error (MAE) and mean absolute percent error (MAPE). Based on this statistics, results show that GARCH(1,1) process shows the best fit and best forecast for the daily data. However, author agree that GARCH(1,1) model is the best for the daily data and different results are found for weekly and monthly data. Y. K. Tse (1991) analyzes volatility of stock returns in the Tokyo Stock Exchange. Author estimates and forecasts the structures of returns volatility. He looks to the period from 1986 till 1989. Author uses ARCH/GARCH model to forecast volatility of returns and compares it with benchmark value, naïve forecast and EWMA forecast. From the research results it is seen that EWMA gives the best forecast results while ARCH/GARCH forecast reacts slower to the changes in the market. 18 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE M. Horasanl and N. Fidan (2007) compare the performance of Markowitz, EWMA and GARCH methods which are applied while constructing optimal portfolios from the daily data of fifteen securities from Istanbul Stock Exchange XU030 index. Data is taken from the period August 9th 2005 – December 30th 2005. The results show that EWMA method will always let investor to have less risky portfolio while GARCH (together with BEKK) forecasts react slower to the changes in volatility while comparing it with EWMA and Markowitz models. H. J. Cha and T. Jithendranathan (2009) in their research use generalized autoregressive conditional heteroskedasticity (GARCH) model in order to overcome the problem of assets misallocation in the portfolio optimization process which occurs while using unconditional estimates of correlation. To construct variances and covariances which are not constant over time authors use Dynamic Conditional Correlation (DCC) model established by Engle (2002). DCC model has a procedure of two steps which is used to estimate variances and correlations. Authors use weekly data of 20 indices from January 1996 to December 2004. The main aim of their work is to estimate the diversification benefits for investors while diversifying portfolios to emerging markets. From the other hand, authors show that Dynamic Conditional Correlation (DCC) model has also shown to improve the results of the portfolio optimization process. As discussed in this chapter, impact of time-varying variances and correlations is widely analyzed and importance of this impact to the portfolio optimization process is being shown in different markets and segments. Some authors suggest that the best method for portfolio optimization is GARCH. Some of them disagree and suggest that EWMA reacts faster to the changes in the market and gives better optimal portfolio than GARCH (especially while talking about weekly or monthly results). 19 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE However, it is important to remind that EWMA and GARCH models intend to replicate time-varying volatility and correlations. Near to that, the mean as well as the volatilities and correlations are time varying. This thesis contributes to the existing knowledge by comparing how different methods (EWMA, GARCH) together with timevarying parameters help to improve portfolio optimization process. Researches which are analyzed in this chapter showed that for a daily data the best method which adds time varying volatilities to optimization process is GARCH, but for a weekly or monthly data results are different. This thesis focuses how EWMA and GARCH models combine optimal portfolios from a monthly data and what results each of them brings comparing it with stable Markowitz approach. Moreover, in this thesis time-varying mean as same as time-varying variances and correlations is being added into the optimization process and analyzes is being made if portfolio optimization for all these time-varying parameters can improve asset allocation in the portfolio and lead to better portfolio results. 20 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 3. METHODOLOGY In this part of Master Thesis sample data and research process together with applied methods are explained. Firstly, the data which is used for this research is described and data analysis tests to check this data are presented. Secondly, research process in steps is described and applied methods are explained. The research process is divided into four parts. In the beginning Markowitz approach is used to construct time-invariant efficient frontier with constant distributional parameters: mean, variance and correlations. In the second part of the research rolling portfolio (first extension of the Markowitz approach) is presented. In this part it is shown that distributional parameters are time-varying and because of this reason portfolio weights are also changing over time which leads to different portfolio compositions. In the third part more advanced method is presented. Exponentially weighted moving average method is applied to construct EWMA efficient frontier. Finally, Generalized Auto Regressive Conditional Heteroskedasticity method is applied as one more alternative to forecast volatilities, correlations and covariances which are used while constructing GARCH rolling efficient frontier. In the end all portfolios are compared and conclusions about time-varying distributional parameters impact on portfolio performance are made. 3.1. Sample Definition In this research portfolio is constructed from 6 indexes and 1 ETF: Bloomberg European 500 Index (BE500 Index), Swiss Market Index (SMI Index), MSCI AC Asia Ex. Japan Index (MXASJ Index), Dow Jones Precious Metals Index (DJGSP Index), Australian Stock Exchange All Ordinaries Index (AS30-ASX Index), IShares 20+ Year Treasury Bond ETF (TLT US Equity), and S&P GSCI Agricultural Index Spot CME (SPGSAG Index). All 7 21 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE variables are described below (see table 1). Information about variables is taken from Bloomberg terminal. Table 1. Assets in the portfolio: Indexes and ETF Index name Description Bloomberg This Index is a free float capitalization-weighted index of the 500 most highly European 500 capitalized European companies. The index was developed with a base value of Index 120.33 as of December 31, 1996. Index rebalances semi-annually in January and July. Swiss Market This Index is an index of the largest and most liquid stocks traded on the Geneva, Index Zurich, and Basle Stock Exchanges. The index has a base level of 1500 as of June 1988. MSCI AC This Index is a free-float weighted equity index. It captures large and mid-cap Asia Ex. representation across 2 out of 3 Developed Markets countries in Asia (Hong Kong Japan Index and Singapore, but excluding Japan) and 8 Emerging Markets countries in Asia (India, China, South Korea, Indonesia, Malaysia, Taiwan, the Philippines and Thailand). This index covers approximately 85% of the free float-adjusted market capitalization in each country. It was developed with a base value of 100 as of December 31, 1987. Dow Jones This Index was created to represent the performance of US-trading stocks of Precious companies engaged in the exploration and production of gold, silver and platinum Metals Index group metals. Australian This Index is a capitalization weighted index. The index is made up of the largest Stock 500 companies as measured by market cap that are listed on the ASX. The index was Exchange All developed with a base value of 500 as of 1979 and is calculated by ASX/S&P. The Ordinaries groups of this index were discontinued on July5th, 2002. Index IShares 20+ This ETF is an exchange-traded fund incorporated in the USA. The ETF seeks to Year Treasury track the investment results of an index composed of U.S. Treasury bonds with Bond ETF remaining maturities greater than twenty years. S&P GSCI The S&P GSCI Agriculture Index is designed as a benchmark for investment in the Agricultural commodity markets and as a measure of commodity market performance over time. In. Spot CME Note. Information about Indexes and ETF is taken from Bloomberg Terminal in February 2015. 22 23 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE For this research monthly data from the period 2005-01-31 – 2014-12-31 is chosen (120 observations). Data is collected using Bloomberg terminal. Logarithmic monthly returns are calculated using collected monthly last prices of Indexes and ETF. Logarithmic returns (𝑟𝑡 ) are calculated by using formula: 𝑃𝑡+1 𝑟𝑡 = ln ( 𝑃𝑡 ) (3.1) In this formula 𝑃𝑡+1 stands for monthly closing price of the Index or ETF at time 𝑡 + 1 and 𝑃𝑡 stands for monthly closing price of the Index or ETF at time 𝑡. According to C. Brooks (2008), logarithmic returns might be called as continuously compounded returns. This means that frequency of compounding of the returns is not important and it is easier to compare returns on different assets. After calculating logarithmic monthly returns of variables, descriptive statistics of each of variables is calculated (see Table 2 in the 4th part of this paper). Summary statistics table presents time series means, medians, standard deviations, minimums, maximums, kurtosis and skewness. Results of summary statistics are analyzed in the next chapter. It is important to mention that there is no reason for specific period of time or securities (indexes) which are chosen for this research. The most important thing related to the preparation for the research is that sample data would satisfy the normality and stationarity assumptions because it is the most important part in order to get meaningful results of the whole research. For this purpose data analysis tests must be applied. TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 3.2. Data Analysis Tests and Normalization 3.2.1. Normality Tests Chosen variables are analyzed using different data analysis tests. Histogram, Quantilequantile (QQ) plot, Shapiro-Wilk, Doornik-Hansen, Lilliefors and Jarque-Bera tests are performed on the collected data. QQ plot is a graphical method to see if data has normal distribution. According to M. Crawley (2012), this method ranks sample from given data-set versus identical amount of ranked quantiles from normally distributed sample. When given data-set is normally distributed the graph of QQ plot is showing a straight line. If data is not normally distributed the line in the graph can be in different forms. Histogram is also a graphical method which shows how data is spread out. Normally distributed data has the shape of the Bell curve or close to that. M. Crawley (2012) in his book write that Shapiro-Wilk test is hypothesis testing test which can be used in order to see if the data which is chosen to be tested is coming from normal distribution. In this test null hypothesis says that the chosen data is normally distributed. Null hypothesis is rejected when p-value is less than 0.05. If p-value is greater than 0.05 we fail to reject null hypothesis and can assume that data is normally distributed. Doornik-Hansen, Lilliefors and Jarque-Bera tests as same as Shapiro-Wilk test are hypothesis testing tests which provide the conclusions if data is coming from normal distribution. All these tests have null hypothesis which claim that data is normally distributed (C. Brooks, 2008). 24 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 3.2.2. Stationarity Test For models such as GARCH, stationarity is one of the main assumptions as normally distributed data for Markowitz approach. However, literature suggests that if data is normally distributed it will be stationary, but not always vice versa. Augmented Dickey-Fuller (ADF) test of stationarity (unit root) is performed. Unit root test looks if time series data is consistent with a unit root process. If non-stationarity of the data is not under the account it can lead to wrong tests results and interpretations. Null hypothesis of this test claims that data has a unit root. Null hypothesis is rejected if p-value is lower than chosen alpha (0.05, 0.01) and it can be assumed that data is stationary without a unit root (S. Said and D. Dickey, 1984). 3.2.3. Data Normalization Process In this research data normalization is made. The different literature provides many solutions how to normalize the data. The logarithm transformation and the square root transformations are two the most used transformations that can help to normalize the data (D. Ruppert, 2011). One more way to normalize the data is to check whether the data has any outliers as they could be changed to the mean of the variable. According to M. Crawley (2012), outliers can be measured by the following rule of thumb which says that outlier is a value which is 1.5 times interquartile range above the third quartile or below the first quartile. Interquartile is difference between the first and the third quartiles. 25 26 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 3.3. Markowitz Optimal Portfolio Using Matrix Algebra As mentioned before, all investors are reaching for expected returns and try to avoid variance as much as possible. Markowitz (1952) mean-variance approach can be used to construct optimal portfolio from chosen assets by diversifying risk and choosing specific weights for each of the assets in the portfolio. This method can help to create efficient portfolio with a specific level of returns by minimizing portfolio variance or maximize portfolio returns with a given level of risk. In order to apply Markowitz mean-variance model there is a need to calculate means, variances and covariances as inputs of the model. For these calculations equally weighted scheme is used. According to Eric Zivot (2013), when the research is done while analyzing the big portfolios, the algebra of portfolio expected returns, variances and covariances become difficult. Matrix (linear) algebra is a great simplification of calculations. The following column vectors show returns and weights of the assets in the portfolio: 𝑅1 𝑅2 𝑥1 𝑅3 𝑅1 𝑅 = 𝑅4 = ⋮ , 𝑥 = ⋮ 𝑥7 𝑅7 𝑅5 𝑅6 (𝑅7 ) (3.2.) 𝑅1 Where ⋮ represents returns of each of 7 variables used in the portfolio and R is a 𝑅7 𝑥1 single vector for multiple returns. ⋮ represents weights of each of the variables in the optimal 𝑥7 portfolio. 27 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE In the constant expected return model (CER) “all returns are jointly normally distributed and this joint distribution is completely characterized by means, variances and covariances of the returns” (E. Zivot, 2013, pg. 4). These values are described using matrix notations below. 7x1 vector of portfolio expected returns is: µ1 𝑅1 𝐸[𝑅1 ] 𝐸[𝑅] = 𝐸 [( ⋮ )] = ( ⋮ ) = ( ⋮ ) = µ µ7 𝑅7 𝐸[𝑅7 ] (3.3.) 7x7 covariances matrix of returns is given below. This covariance matrix is symmetric. In this matrix there are used asset variances 𝑣𝑎𝑟(𝑅𝑖 ) = 𝜎𝑖2 where “i” represents each variable in the portfolio (from 1 to 7) and covariances between assets 𝑐𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) = 𝜎𝑖𝑗 where “i” and “j” also represents each variable in the portfolio (from 1 to 7). Important to mention that when “i=j”: 𝜎𝑖𝑗 = 𝜎𝑖2 . 𝜎12 𝜎21 𝜎31 𝑣𝑎𝑟(𝑅) = 𝜎41 𝜎51 𝜎61 (𝜎71 𝜎12 𝜎22 𝜎32 𝜎42 𝜎52 𝜎62 𝜎72 𝜎13 𝜎23 𝜎32 𝜎43 𝜎53 𝜎63 𝜎73 𝜎14 𝜎24 𝜎34 𝜎42 𝜎54 𝜎64 𝜎74 𝜎15 𝜎25 𝜎35 𝜎45 𝜎52 𝜎65 𝜎75 𝜎16 𝜎26 𝜎36 𝜎46 𝜎56 𝜎62 𝜎76 𝜎17 𝜎27 𝜎37 𝜎47 = ∑ 𝜎57 𝜎67 𝜎72 ) (3.4) Based on the matrix algebra above we can get return on the portfolio: 𝑅𝑝,𝑥 𝑅1 = (𝑥1 , … , 𝑥7 ) × ( ⋮ ) = 𝑥1 𝑅1 + ⋯ + 𝑥7 𝑅7 𝑅7 (3.5) Expected return on the portfolio: µ𝑝,𝑥 µ1 = 𝑥 𝐸[𝑅] = 𝑥 µ = (𝑥1 , … , 𝑥7 ) × ( ⋮ ) = 𝑥1 µ1 + ⋯ + 𝑥7 µ7 µ7 ′ ′ (3.6) 28 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Variance of the portfolio is: 2 𝜎𝑝,𝑥 𝜎12 𝜎21 𝜎31 ′ ( ) = 𝑣𝑎𝑟(𝑥′𝑅) = 𝑥 ∑𝑥 = 𝑥1 , … , 𝑥7 × 𝜎41 𝜎51 𝜎61 (𝜎71 𝜎12 𝜎22 𝜎32 𝜎42 𝜎52 𝜎62 𝜎72 𝜎13 𝜎23 𝜎32 𝜎43 𝜎53 𝜎63 𝜎73 𝜎14 𝜎24 𝜎34 𝜎42 𝜎54 𝜎64 𝜎74 𝜎15 𝜎25 𝜎35 𝜎45 𝜎52 𝜎65 𝜎75 𝜎16 𝜎26 𝜎36 𝜎46 𝜎56 𝜎62 𝜎76 𝜎17 𝜎27 𝜎37 𝑥1 𝜎47 ×( ⋮ ) (3.7) 𝑥7 𝜎57 𝜎67 𝜎72 ) In this model the condition that portfolio weights must be equal to one can be expressed as following: 1 𝑥 ′ 1 = (𝑥1 , … , 𝑥7 ) × ( ⋮ ) = 𝑥1 + ⋯ + 𝑥7 = 1 (3.8) 1 In this research it is assumed that investor seeks to minimize risk for a specific level of return. According to E. Zivot (2013), this assumption makes asset allocation a bit simpler and we can look only to the efficient portfolios. Following the Markowitz when we have target level of expected return and want to minimize the portfolio risk our equations look as following: 2 min 𝜎𝑝,𝑥 = 𝑥 ′ ∑𝑥 𝑠. 𝑡 𝑥 (3.9) µ 𝑝 = 𝐱’µ = µ 𝑝,𝑜 , 𝑎𝑛𝑑 𝑥′ 1 = 1 Based on E. Zivot (2013), in this research efficient frontier of the portfolios is created using two specific efficient portfolios: global minimum variance portfolio and efficient portfolio with target expected return which is equal to the highest expected return between all 7 variables in the portfolio. α below is any constant and define portfolio z (linear combination of two specific efficient portfolios (x and y) mentioned above): 29 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 𝛼𝑥𝐴 + (1 − 𝛼)𝑦𝐴 𝑧 = 𝛼𝑥 + (1 − 𝛼)𝑦 = (𝛼𝑥𝐵 + (1 − 𝛼)𝑦𝐵 ) 𝛼𝑥𝐶 + (1 − 𝛼)𝑦𝐶 (3.10) From (3.10) equation it is seen that portfolio ‘z’ is a minimum variance portfolio with expected return and variance which can be written as following: µ 𝑝,𝑧 = 𝐳’µ = 𝛂µ 𝑝,𝑥 + (1 − 𝛼)µ 𝑝,𝑦 σ2 𝑝,𝑧 = 𝐳’ ∑ 𝒛 = α2 𝜎2𝑝,𝑥 + (1 − 𝛼)2 𝜎2𝑝,𝑦 + 2𝛼(1 − 𝛼)𝜎𝑥𝑦 (3.11) (3.12) Portfolio ‘z’ belongs to efficient frontier when µ 𝑝,𝑧 ≥ µ 𝑝,𝑚 , where µ 𝑝,𝑚 stands for expected return on the global minimum variance portfolio (E. Zivot, 2013). In this research efficient frontier is created by taking the 5 steps provided by E. Zivot (2013). As E. Zivot (2013) writes, efficient frontier can be created as following: 1. Compute the global minimum variance portfolio (m) by solving (3.13) and 2 compute µ 𝑝,𝑚 = 𝒎’µ and 𝜎𝑝,𝑚 = 𝒎′∑𝒎. 𝑚𝑖𝑛(𝑚) 𝜎2𝑝,𝑥 = 𝑚′ ∑𝑚 𝑠. 𝑡. 𝑚′ 1 = 1 2. (3.13) Compute the efficient portfolio x with target expected return equal to the maximum expected return of the assets under consideration (3.9) with µo = max{ µ1, 2 …, µ7} and compute µ 𝑝,𝑥 = 𝒙’µ and 𝜎𝑝,𝑥 = 𝒎′∑𝒎. 3. Compute cov(Rp,m, , Rp,x) = σmx = m’Σx 4. Create an initial grid of α values {1, 0.9, …, -0.9, -1} compute the frontier portfolios “z” using (3.10) and compute their expected returns and variances using (3.10), (3.11) and (3.12), respectively. 5. Plot µ 𝑝,𝑧 against 𝜎 𝑝,𝑧 and adjust the grid of values to create a nice plot (pg.21). Steps mentioned above help to combine the efficient frontier from selected risky assets. This frontier is often called “Markowitz bullet”. (E. Ziwot, 2013). Moreover, short- 30 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE selling is allowed for Markowitz efficient portfolios as same as for other methods which are used in this research. In this research Markowitz efficient frontier is used as a benchmark for other models. Tangency portfolio is one of the efficient portfolios with the highest possible Sharp ratio. Sharp ratio presents how much additional return investor gets for one additional unit of risk. In order to get tangency portfolio there is a need to have risk-free rate – 𝑟𝑓 (E. Zivot, 2013). As E. Zivot (2013) writes, “the tangency portfolio solves the constrained maximization problem” (p. 23) which can be written as following: max 𝑡 𝑡 ′ µ−𝑟𝑓 (𝑡 ′ ∑ 𝑡)1/2 = µ 𝑝,𝑡 −𝑟𝑓 𝜎𝑝,𝑡 𝑠. 𝑡. 𝑡′1 = 1 (3.14) Where t is tangency portfolio, rf is a chosen risk free rate, 𝑡 ′ µ is equal to µ 𝑝,𝑡 and (𝑡 ′ ∑ 𝑡)1/2 is equal to 𝜎𝑝,𝑡 . Without risk free rate there would be impossible to find tangency portfolio. According to EY (2015), “Government bond yields are frequently used as a proxy for risk-free rates” (pg. 1). In this paper, here is decided to use average risk-free rate of the European countries that the tangency portfolios would be calculated (results are presented in 4th section figures 3, 10 and 14). European countries average risk-free rate is taken from EY (2015) paper and is equal to 1.4% per year (0.12% monthly risk-free rate). Risk-free rate is taken from Europe region as it is decided that this model is being constructed for the investor who is located in Europe. 3.4. Improvement of Markowitz Approach: Rolling Portfolios In the above section it was described how, from selected variables, the Markowitz efficient frontier is created. Markowitz frontier is combined while keeping that means and 31 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE variances are constant over time. This section allows extending time-invariant Markowitz approach by showing that means, variances and covariances are time-varying parameters. Before computing rolling parameters, there is a need to choose the appropriate time window. Many authors raised the issue relate to the size of time windows. This issue is discussed in technical analysis literature as same as in other works such as E. Zivot and J. Wang (2006) or C. Alexander (2008). However, there is no special formula or rules how to select the time window. These windows are chosen due to the length of the time-series. For the data sets which have the fast time-scales (are collected in short intervals) the smaller windows can be applied. Larger windows can be applied for the longer interval data. Moreover, too small rolling window leads to the noisy results when parameters are changing too dramatically from one period to another. At the same time, the larger rolling windows estimate the smother changes. E. Zivot (2013) performs the analyses of the monthly data of 100 parameters and uses 24 months rolling windows to see how the parameters are changing over time. Because there is no special formula or rules how to select time window for the calculations, based on E. Zivot (2013) work, in this research there is applied the 24 month rolling time window while taking into account that too small time window can have too much noise and vice versa with too big time window (in the next chapter, the results with shorter time window are provided too as a comparison to the chosen 24 months rolling window). It is important to stretch that time window size can have significant influence to the final research results. According to E. Zivot (2006), rolling parameters can be computed as following: Rolling mean formula: 1 µ̂𝑡 (𝑛) = 𝑛 ∑𝑛−1 𝑖=0 𝑅𝑡−𝑖 = 1 𝑛 (𝑅𝑡 + 𝑅𝑡−𝑖 + ⋯ + 𝑅𝑡−𝑛+1 ) 𝑡 = 𝑛, 𝑛 + 1, … , 𝑇 (3.15) 32 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Where n is the size of data window, Rt is return at given time period. Within this method means are calculated from different time frames. Rolling variance and standard deviations are computed as following: 1 𝜎̂𝑡2 (𝑛) = 𝑛−1 ∑𝑛−1 ̂ 𝑡 (𝑛))2 𝑖=0 (𝑅𝑡−𝑖 − µ 𝜎̂𝑡 (𝑛) = √𝜎̂𝑡2 (𝑛) (3.16) (3.17) 𝑡 = 𝑛, 𝑛 + 1, … , 𝑇 Where n is the size of data window, Rt-1 is return at given time period and µ̂𝑡 (𝑛) is rolling mean. Rolling covariances and correlations are computed as following: 1 𝜎̂𝑗𝑘,𝑡 (𝑛) = 𝑛−1 ∑𝑛−1 ̂ 𝑗 (𝑛))(𝑅𝑘𝑡−𝑖 − µ̂𝑘 (𝑛)) 𝑖=0 (𝑅𝑗𝑡−𝑖 − µ 𝜌̂𝑗𝑘,𝑡 (𝑛) = 𝜎̂ ̂𝑗𝑘,𝑡 (𝑛) 𝜎 ̂𝑘𝑡 (𝑛) 𝑗𝑡 (𝑛)𝜎 (3.18) (3.19) 𝑡 = 𝑛, 𝑛 + 1, … , 𝑇 After calculating these rolling parameters they are plotted over time (figures are presented in 4.3 section of this paper). If means, variances, covariance and correlations results at time t ≈ t+1 ≈ at T it is concluded that data is constant over time. Moreover, after combining separate rolling distributional parameters and using them for expected returns (µ) and covariance matrix (∑) the rolling efficient portfolios are constructed and plotted. The efficient frontier is constructed in the same way as it is written in the section above. The only difference is that this time the model inputs are rolling over time. Results show if there are any variations in resulting portfolio weights (graphs are provided in 4th section of this paper). In this part of the research the first steps are taken to show that distributional parameters are time-varying and they can have significant impact on the portfolio weights and final portfolio optimization results. 33 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 3.5. Time-varying Parameters: Exponentially Weighted Moving Average. According to C. Brooks (2008), exponentially weighted moving average is one of the extensions how to measure historical volatility. This method put more weights on recent observations and it let these current observations to make a bigger influence on the forecasted volatility comparing it with older observations. In EWMA model the latest data has the highest weights and weights for previous data decline exponentially over time. The same author, C. Brooks (2008) mentions two advantages of EWMA model comparing it with simple historical models and simple moving average (MA) model which puts the same weights to the all data points and uses estimations of the rolling windows. The first advantage is that in the real world volatility is affected more by recent events comparing it with some event in the past and EWMA at the same time gives more attention to those recent events. At the same time simply moving average model weights recent event as same as event in the past and this can lead to misleading too low volatility forecast results if, for example, specific shock suddenly drops out of the sample or vice versa if specific shock is in the sample for a long period of time. The second advantage is that “the effect on volatility of a single given observation declines at an exponential rate as weights attached to recent events fall” (C. Brooks, 2008, pg. 384). Exponentially weighted moving average model can be computed in several ways. One of them is the following: ∞ 𝜎𝑡2 = (1 − 𝜆) ∑ 𝜆𝑗 (𝑟𝑡−𝑗 − 𝑟̅ )2 𝑗=0 Where 𝜎𝑡2 is the estimate of the variance for the period t. 𝜎𝑡2 also becomes forecast for the future volatility for all periods. Other input 𝑟̅ is average return which is estimated over the observations. (3.20) TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Input λ is the decay factor which shows how much weight is given to recent observations comparing it with older once. In many studies λ is used between 0.9-0.96 as it is recommended by RiskMetrics (1996). However, in this research different Lambdas are used to see how it impacts the final results. RiskMetrics (1996) as same as other authors in their calculations assume that 𝑟̅ is equal to zero. According to C. Brooks (2008), there are couple important limitations of EWMA models: The first limitation is about weights. Even if there are several ways how to compute EWMA, it is very important to remember that weights will sum up to less than one. This is because weights sum up to one when there is infinite sum of observations (3.19). As in the real data there is finite sum of observations the weights sum up to less than one. It is the most important to the small samples as it can have a huge impact on the final results. Secondly, many time-series models (for example GARCH) “will have forecasts that tend towards the unconditional variance of the series as the prediction horizon increases. This is a good property for a volatility forecasting model to have, since it is well known that volatility series are ‘mean-reverting’ ” (C. Brooks, 2008, pg. 385). According to the same authors C. Brooks (2008), this means that if volatility is higher than the historical average of the same volatility, it should get back to the historical average level. From the other hand, if at the moment the volatility series are lower comparing them with historical average, they should have tendency to get back up to the volatility average level. However this structure is common for GARCH volatility forecasting models but is not applicable to the method such as exponentially weighted moving average. In this research EWMA model is used to construct EWMA covariance matrix which is applied as an input to the model of the portfolio optimization processes (graphs and tables 34 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE are provided in 4.4 section of this paper). EWMA is chosen as one of the additional models while building the efficient frontier because different authors such as Tse (1991) or Horasanl & Fidan (2007) in their works confirms that this method provides the best portfolio optimization results. However, other authors such as Cha & Jithendranathan (2009) and others provides results that Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) method is more advanced and it should be used in the portfolio optimization process. However, it is important to mention that selected data or other issues such as time frame could influence different results of the different researchers. According to the C. Alexander (2008), while talking about EWMA and GARCH models it can be concluded that in EWMA method the same smoothing constant λ is being used for all the returns. Moreover, this constant λ is chosen subjectively. When using GARCH method, covariance matrixes reflect the time-varying volatilities and correlations without any inputs which can be added subjectively. 3.6. Time varying Parameters: GARCH Models 3.6.1. Testing for ARCH Effect Lagrange Multiplier (LM) test for autoregressive conditional heteroskedasticity is performed. This test checks if there is ARCH effect in the data. GARCH models are needed and fit the data if ARCH effect is confirmed (4.5.1 part of this paper presents the results of Lagrange Multiplier (LM) test for assets returns). ARCH LM test is hypothesis testing test. In this test null hypothesis says that the data has no ARCH effect. Null hypothesis is rejected when p-value is less than chosen confidence interval (0.05, 0.01) and it is assumed that data has ARCH effect. 35 36 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 3.6.2. GARCH(1.1) Model Bollerslev (1986) and Taylor (1986) individually developed Generalized ARCH (GARCH) model. It is one more alternative for volatility modeling and forecasting. This model is chosen for this research as an alternative and more advanced method comparing it with EWMA. According to RiskMetrics (1996), GARCH type models are the most popular between academics and practitioners. The main reason for this is that time varying volatility has an impact on a returns of time series data. Even though this idea was first developed by Engle (1982) in his work about Auto Regressive Conditional Heteroskedasticity (ARCH) model, due to ARCH model limitations GARCH models are used more widely. According to C. Brooks (2008), the main limitations which stopped practitioners of using ARCH model are: 1. It is not clear how many lags should be used for the squared residuals (q); 2. Conditional variance estimations must be positive; 3. If model has a lot of parameters in conditional variance equation it is most likely that some of the values will be negative which limits the usage of ARCH model (pg. 392). In GARH model the conditional variances are dependent on their own previous lags. Looking at the simplest case of the conditional variances equation (also known as GARCH (1,1) model) we get the following: 2 2 𝜎𝑡2 = 𝛼0 + 𝛼1 𝑢𝑡−1 + 𝛽𝜎𝑡−1 (3.21) Where 𝜎𝑡2 is a conditional variance. This is estimation for one period ahead variance which is calculated from any relevant information from the past. Long term average value depends on 𝛼0 . 37 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 2 𝛼1 𝑢𝑡−1 : has information about historical volatility. 2 𝛽𝜎𝑡−1 : this part of the formula presents fitted variance from the model during the previous period (C. Brooks, 2008, pg. 392). According to C. Brooks (2008), “Thus the GARCH(1,1) model, containing only three parameters in the conditional variance equation, is a very parsimonious model, that allows an infinite number of past squared errors to influence the current conditional variance” (pg. 393394). GARCH (1,1) model can be extended to GARCH(p,q) model where current conditional variance depends on “q” lags of squared error and “p” lags of conditional variance (C. Brooks, 2008). GARCH (p,q) looks as following: 2 2 2 2 2 2 𝜎𝑡2 = 𝛼0 + 𝛼1 𝑢𝑡−1 + 𝛼2 𝑢𝑡−2 + ⋯ + 𝛼𝑞 𝑢𝑡−𝑞 + 𝛽1 𝜎𝑡−1 + 𝛽2 𝜎𝑡−2 + ⋯ + 𝛽𝑝 𝜎𝑡−𝑝 (3.22) 2 2 𝜎𝑡2 = 𝛼0 + ∑𝑞𝑖=1 𝛼𝑖 𝑢𝑡−𝑖 + ∑𝑝𝑗=1 𝛽𝑗 𝜎𝑡−𝑗 (3.23) As different authors confirm, the general GARCH (1,1) method is enough to capture volatility clustering in the data. P. R. Hansen and A. Lunde (2005) estimated 330 different models to see if any of them can outperform GARCH(1,1) and their final results are that none of those models outperformed GARCH(1,1). 3.6.3. The Dynamic Conditional Correlation (DCC) Model GARCH(1,1,) model estimates univariate parameters. In this research there is a need to forecast multivariate covariance matrix. For this reason multivariate Dynamic Conditional Correlation (DCC) model is used. DCC model compose the covariance matrix as following: 𝐻𝑡 = 𝐷𝑡 𝑅𝑡 𝐷𝑡 (3.24) 38 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Where, as T. Peters (2008) writes: Ht is estimated covariance matrix; Dt is a diagonal matrix of time varying standard variation from univariate GARCH; Rt is the conditional correlation matrix. DCC model is different from the other because in this model R is time varying parameter. According to T. Peters (2008) forecasted covariance matrix within DCC model can be computed from forecasted inputs of the covariance matrix: “The forecast of the diagonal matrix of time varying standard variation from univariate GARCH – processes Dt and the forecast of the conditional correlation matrix of the standardized disturbances Rt can be calculated separately” (T. Peters, 2008, pg. 19). 3.6.3.1 Forecast of Dt (GARCH(1,1)) When volatility for time t is known, the forecast for period t+k is derived from the (3.21) formula. If k=1 the volatility equal to: 2 ℎ𝑡+1 = 𝜎𝑡+1 = 𝛼0 + 𝛼1 𝑢𝑡2 + 𝛽𝜎𝑡2 (3.25) According to L. H. Ederington and W. Guan (2007), expected volatility at time t+k based on the forecast for t+1 is equal to: 𝑗 𝑘−1 𝑗 𝑘−1 ℎ𝑡+𝑘 = 𝛼0 ∑𝑘−2 ℎ𝑡+1 = 𝛼0 ∑𝑘−1 [𝛼1 𝑢𝑡2 + 𝑗=0 (𝛼1 + 𝛽) + (𝛼1 + 𝛽) 𝑗=0 (𝛼1 + 𝛽) + (𝛼1 + 𝛽) 𝛽𝜎𝑡2 ] (3.26) Where k is the forecast horizon and j is the data in the past. However this method has some limitations too. According to different authors, GARCH(1,1) model is criticized due to the too short memory especially related to high frequency data (T. Peters, 2008). 39 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 3.6.3.2 Forecast of Rt According to R. F. Engle and K. Sheppard (2001), DCC evaluation process is nonlinear process and is equal to: ′ ] + 𝛽𝑄𝑡+𝑟−1 𝑄𝑡+𝑘 = (1 − 𝛼 − 𝛽)𝑄̅ + 𝛼[𝜀𝑡+𝑘−1 𝜀𝑡+𝑘−1 (3.27) Authors use assumptions that 𝑄̅ ≈ 𝑅̅ and 𝐸𝑡 [𝑄𝑡+1 ] ≈ 𝐸𝑡 [𝑅𝑡+1 ]. With these assumptions, the forecast of Rt+k can be expressed as following: 𝑗 𝑘−1 ̅ 𝐸𝑡 [𝑅𝑡+𝑘 ] = ∑𝑘−2 𝑅𝑡+1 𝑗=0 (1 − 𝛼 − 𝛽)𝑅 (𝛼 + 𝛽) + (𝛼 + 𝛽) (3.28) In this forecast of conditional correlation matrix the influence of 𝑅𝑡+1 becomes lower for each future step with the ratio of (𝛼 + 𝛽). Forecast of Dt and forecast of Rt combined together into the covariance matrix formula (3.32) provide the results of forecasted DCC covariance matrix which is used in this research while modeling GARCH efficient frontier (graphs and tables with DCCGARCH(1,1) forecasted covariance matrix and GARCH efficient frontier are provided in 4.5.2 section of this paper) All models which are described in this section: Rolling portfolios, EWMA and GARCH models, they all are used as the extensions of the time invariant optimization model. The main idea is that the portfolio optimization process takes into account that the correlation matrix is not constant leading so different performance as compared to what is expected under a time-invariant correlation matrix (4.3., 4.4.3 and 4.5.3 parts of this paper provide graphical results of time-varying efficient portfolios parameters and weights). In the optimization process estimation and a forecast of variances and correlations are implemented in order to improve the allocation. TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE At the end of the research EWMA and GARCH optimal portfolios and efficient frontiers are compared with the benchmark (Markowitz efficient frontier). The conclusions about time varying distributional parameters impact on portfolio performance are made and discussion part, together with limitations of this research, is provided. In the next chapter the most significant results of the research are presented. Research is done using R Programing software. The main parts of the final script are presented in Appendix 17. 40 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 4. EMPIRICAL RESEARCH RESULTS In this part of the paper, here are provided the main results of the performed empirical research. This research is done based on the methodology which is explained in the 3rd chapter of this work. Empirical research results are presented by splitting it to 5 main parts: In the first part, there is described the analysis of the data together with summary statistics. Normality and stationarity tests are performed and presented too. In the second part, Markowitz efficient frontier is built using time-invariant parameters. Moreover, the optimal portfolio with chosen target monthly returns of 0.8% is constructed. In the third part, rolling portfolios are presented together with time varying correlations, volatilities, and optimal portfolio weights. The fourth part presents EWMA efficient frontier which is created using EWMA covariance matrix. Rolling portfolio technique is implemented and time varying volatilities together with time varying weights for EWMA optimal portfolio with target returns are presented. The fifth part presents GARCH efficient frontier which is constructed using DCCGARCH(1,1) forecasted correlation matrix. Rolling portfolio technique is adopted and time varying volatilities together with time varying weights for GARCH optimal portfolio with target returns are presented. 4.1. Data Analysis and Testing 4.1.1. Descriptive Statistics Before starting the research, data analysis is performed. Log returns are calculated for each of 7 variables using (3.1) formula and Summary statistics is calculated (see table 2). 41 42 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Table 2. Summary statistics for the period 2005.01.31 – 2014.12.31 Summary Statistics Stat./Variables TLT SMI BE500 MXASJ Mean Median Standard Deviation Kurtosis Skewness Minimum Maximum Count (observations) 0,004628 0,001153 0,006452 0,006558 0,00239 0,009649 0,006855 0,0104 0,0003479 0,010130 0,003730 0.01408 0,005677 0,007432 0,052436 4.160839 0,7973 -0,1073 0,1900 0,03594 0,042712 3.667949 4.641301 -0,5574 -0,8769 -0,11530 -0,1369 0,097680 0,1217 0,055542 4.365923 -0,7536 -0,1766 0,154200 0,09198 4.662336 -0,56798 -0,3874 0,2136 0.05772 4.291862 -0.8356 -0.2239 0.1189 0,065231 3.198309 -0,18344 -0,18120 0,18560 120 120 120 120 120 120 120 DJGSP ASX SPGSAG If looking at all 7 variables from table 2, variables have 120 observations each and the highest mean value belongs to MSCI AC Asia Ex. Japan Index (MXASJ) which is equal to 0.006855 and the lowest mean value of 0.00239 belongs to Bloomberg European 500 Index (BE500 Index). At the same time, the highest standard deviation belongs to Dow Jones Precious Metals Index (DJGSP Index) and is equal to 0.09198. The highest standard deviation means that values of DJGSP Index are more spread out from the mean on the average when comparing it with other Indexes and ETF. Australian Stock Exchange All Ordinaries Index (ASX Index) has the lowest minimum value of -0.2239. The highest maximum value is equal to 0.2136 and belongs to Dow Jones Precious Metals Index (DJGSP Index). In this research, the normality of data set is very important issue. If mean and median for the same variable are close to each other it could be a sign that data is normally distributed. In table 2 it is seen that most of the data has bigger gaps between mean and median. Australian Stock Exchange All Ordinaries Index (ASX Index) has the biggest difference between these two parameters: mean equals to 0.0037 while the median is equal to 0.014. Almost the same mean and median are for Swiss Market Index (SMI Index) and is equal to 0.0065 and 00066. TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Two very important parameters for normally distributed data are skewness and kurtosis. According to M. Crawley (2012), skewness measures if distributions have long, drawn-out tails on one side or another. Normal distributions should have skewness equal to 0. From table 2, it is seen that all 6 indexes have negative skewness (skew to the left) and IShares 20+ Year Treasury Bond ETF has positive skewness (skew to the right). Kurtosis measures non-normality which is related to heaviness in the tails. Normal distribution should have kurtosis equal to 3. All 7 variables have kurtosis larger than 3 (highest 4.66 is for DJGSP Index). There is a need to test whether ETF and Indexes data sets satisfy the normality assumption, because for each of the variables the conclusions from the summary statistics can be drawn as following: the mean of the parameter is significantly different than the median of the same parameter as same as Skewness is not equal to 0 and Kurtosis is not equal to 3. These results question the normality of the data sets. 4.1.2. Normality Testing and Data Normalization For normality testing, Doornik-Hansen test, Shapiro-Wilk test, Lilliefors test and Jarque-Bera test are performed together with QQ-plot and histogram. Normality testing graphical results for QQ-plots and histograms are presented in Appendix 1. Table 3 presents 4 normality tests which are performed for each of the variables. All 4 tests rejects NULL hypothesis ( p-value < 0.05) that data is normally distributed for 5 variables: TLT ETF, SMI Index, BE500 Index, MXASJ index, and ASX Index. For DJGSP Index, only Lilliefors test does not reject NULL hypothesis that data is normally distributed (p-value = 0.18), but data normalization process is still needed because other 3 tests reject NULL hypothesis. Only for SPGSAG Index all 4 test cannot reject NULL hypothesis (pvalue > 0.05 and 0.1) that data is normally distributed. 43 44 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Moreover, graphical tests (see Appendix 1) show that data sets are not really bellshaped and have the outliers too. Table 3. Normality tests for original data Normality tests Doornik-Hansen test Shapiro-Wilk W Lilliefors test Jarque-Bera test Total Tests results (p-value > 0.05) TLT (p-values) SMI (p-values) BE500 (p-values) MXASJ (p-values) DJGSP (p-values) ASX (p-values) SPGSAG (pvalues) 0.00266 0.0406 0.0011 0.0045 0.0026 0.0016 0.4572 0.0021 0.02 3.342e-005 0.0452 0.05 0.0109 3.78e-005 0.03 2.31e-007 0.0007 0 1.662e-005 0.0127 0.18 1.872e-005 0.0011 0.01 7.408e-006 0.7159 ~0.2 0.604 0/4 0/4 0/4 0/4 1/4 0/4 4/4 As it is concluded that 6 out of 7 variables are not normally distributed the normalization of the data is performed. It is important to mention that log transformation and square-root transformation, both do not perform well and do not help to normalize the data. It was chosen to take the biggest outliers from the data sets and check whether it is helping to normalize the data. By using the rule of thumb, it is calculated which points in the data are outliers (Appendix 2). One of the methods to take out outliers is to change them to the data mean. In Appendix 2 there is presented which outliers are taken out from the data and changed to the variables means. It is important to mention that there is no need to take out all the outliers from the data that it would become normally distributed. Moreover, elimination of all outliers from the data would change the final results dramatically. In this research 1 outlier is changed to the mean for SMI index as same as for DJGSP Index, 2 outliers are changed for TLT ETF, 3 outliers are changed for ASX Index and 5 outliers are changed for BE500 Index. After taking out the biggest outliers the same 4 normality tests are performed. In table 4 the results are shown. 45 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Table 4. Normality tests for data without outliers TLT (p-values) 0.1158 SMI (p-values) 0.235 BE500 (p-values) 0.0262 MXASJ (p-values) 0.0819 DJGSP (p-values) 0.9721 ASX (p-values) 0.2305 0.2223 0.3023 0.3293 0.1528 0.6726 0.0775 Lilliefors test 0.11 0.13 0.45 0.04 0.22 0.09 Jarque-Bera test 0.084 0.2257 0.0587 0.0847 0.9359 0.2907 4/4 4/4 3/4 3/4 4/4 4/4 Normality tests Doornik-Hansen test Shapiro-Wilk W Total Tests results (p-value > 0.05) After taking out the outliers the normality tests show different results. This time all 4 tests cannot reject NULL hypothesis that data is normally distributed for 4 out of 6 variables: TLT ETF and 3 indexes: SMI, DJGSP and ASX (all p-values > 0.05). For two indexes: BE500 and MXASJ 3 out of 4 tests cannot reject NULL hypothesis that data is normally distributed. Taking out the outliers helps to normalize the data. Because critical values were changed to the mean values, it is important to take a look at the summary statistics after the normalization. In table 5 below, it is seen that mean and median for each of the variables now are much closer to each other than comparing it to the statistics before taking out the outliers. Mean and median are almost equal for TLT ETF. The same results are reached for SMI Index and MXASJ Index. The biggest gap between these parameters left for DJGSP Index (mean equal to 0.003579 and median equal to 0.010130). As mean is not totally equal for median in each of 7 variables data sets, it influences the results of skewness and kurtosis. Even if results got closer to the 3 for kurtosis and 0 for skewness than comparing it with results before normalization of the data it still have some positive or negative effects. 46 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Table 5. Summary statistics of data without some outliers Summary Statistics for Normal Data Stat./Variables TLT SMI BE500 Mean 0.001763 0.007466 0.0075 Median 0.001153 0.007184 Kurtosis 3.3421 Skewness 0.4504 MXASJ DJGSP ASX SPGSAG 0.0111 0.003579 0.00792 0.005677 0.009649 0.0104 0.010130 0.01408 0.007432 3.1932 3.9508 3.7219 2.8133 2.8116 3.1983 -0.3609 -0.1437 -0.3009 -0.04175 -0.3401 -0,1834 To summarize the normality testing and normalization of the data it can be concluded that for SPGSAG index data NULL hypothesis for normally distributed data is not rejected and the normalization process is not required. Outliers are taken out from other 6 variables and it helps to normalize the data. For the further research normally distributed variables are used. However, all 6 Indexes have fatter tails on the right and ETF has fatter tail on the left side of normal distribution. 4.1.3. Testing for Stationarity For methods such as GARCH, which is applied in this research, the assumption that data is normally distributed is not enough. Data should be stationary and do not have unit roots. For this reason ADF stationarity test is performed. In table 6 the results of ADF test is shown. TLT ETF and 5 indexes except SMI index strongly rejects NULL hypothesis that unit-root exist under 0.05 and 0.01 confidence intervals and assume the stationarity of the data. SMI index rejects NULL hypothesis under the 0.05 confidence interval and also assumes stationarity of the data. 47 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Table 6. Adf.test for stationarity Test for stationarity (p-value < 0.05) Adf test (p-values) TLT SMI BE500 MXASJ DJGSP ASX SPGSAG <0.01 0.0364 <0.01 <0.01 <0.01 <0.01 <0.01 ADF test is performed on normally distributed data that leads to the good test results because normally distributed data is also stationary. 4.1.4. Assets Correlations and Returns After data is checked to be normally distributed and stationary the correlation matrix is performed to see how chosen assets affect one another. Figure 1 shows correlation matrix. None of the pairs of 7 variables have strong negative correlation. Based on C. Dancey and J. Reidy’s (2004) categorization of the correlation results, moderate negative correlation is between TLT ETF and BE500 Index (-0.41). Weak negative correlation is between TLT ETF and MXASJ, ASX and SMI Indexes (-0.19, -0.25 and 0.16). It Figure 1. Correlation Matrix can be concluded that almost no correlation is between these pairs: TLT and DJGSP; TLT and SPGSAG and DJGSP and SMI Indexes (correlations are between -0.1 and 0.1). 5 pairs have moderate positive correlation: MXASJ and SMI; MXASJ and BE500; ASX and SMI; ASX and BE500, and SMI and BE500. While one indexes pair (MXASJ and ASX) has strong positive correlation equal to 0.75. Even if chosen indexes are from different regions or fields they are still correlating between each other more positively. ETF is acting more independent and does not have strong negative or positive correlations with any of the indexes. TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 3 presents pictures of assets returns and plotted correlations. It is seen that each of assets has its own peaks and different time brings different results. Optimal portfolio allocation and diversification among assets can help to reach the best results. Figure 2 below presents individual assets returns together with their riskiness. MXASJ index brings the highest returns while DJGSP Index brings one of the lowest returns with the highest risk and TLT ETF brings lowest returns. Indexes SMI and BE500 have close enough returns to each other with almost the same amount of risk and a little bit overlap each other in the graph. Moreover, these two Indexes have the lowest risk level between chosen assets. ASX Index brings similar returns as BE500 Index and SMI Index, only the risk level is much higher comparing to the same two Indexes. Figure 2. Returns of Individual Indexes and ETF 4.2. Markowitz Efficient Frontier and Optimal Portfolio When data is already tested and concluded to fit the models which are used in this research, Markowitz efficient frontier with time invariant distributional parameters is constructed. In the beginning variance-covariance matrix is built (see Table 7). The covariance matrix is unstandardized correlation matrix. It is used together with assets returns to find efficient portfolios and efficient frontier. 48 49 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Table 7. Time-invariant covariance matrix TLT SMI BE500 MXASJ DJGSP ASX SPGSAG TLT 0.0022413 -0.000257 -0.000651 -0.000426 0.0001379 -0.000593 0.0001214 SMI -0.000257 0.0011661 0.0007628 0.0008769 0.0001658 0.0009716 0.000328 BE500 -0.000651 0.0007628 0.0011517 0.0010027 0.0004096 0.0011784 0.0004835 MXASJ -0.000426 0.0008769 0.0010027 0.0023354 0.0013322 0.0018349 0.0005624 DJGSP 0.0001379 0.0001658 0.0004096 0.0013322 0.0071866 0.0011508 0.0013466 ASX -0.000593 0.0009716 0.0011784 0.0018349 0.0011508 0.002534 0.0007436 SPGSAG 0.0001214 0.000328 0.0004835 0.0005624 0.0013466 0.0007436 0.0042551 In this research short-selling is allowed which means that portfolio owner can borrow shares and sell it when needed. However, the assumption is made that short-selling is free and does not have any costs which would have negative impact on portfolio returns. Assets returns together with covariance matrix are used in the model and different efficient portfolios are calculated. In table 8 below there are presented different portfolios and assets weights which are calculated with the created model (see Appendix 17 for R script). Table 8. Markowitz Efficient portfolios characteristics Assets Weights Portfolio/Assets Return Risk TLT SMI BE500 MXASJ DJGSP ASX SPGSA G Equally weighted 0.00643 0.03 1/7 1/7 1/7 1/7 1/7 1/7 1/7 Global min.variance Tangency portfolio Effic.portf with same return as "ASX" Effic.portf with same return as "MXASJ" Chosen eff.port. with target return = 0.8% 0.00535 0.0079 0.021 0.0267 0.3417 0.1885 0.1565 0.2107 0.518 0.431 0.0004 0.4447 0.0272 -0.0451 -0.069 -0.2728 0.0252 0.043 0.0079 0.026 0.1926 0.2093 0.4329 0.4329 -0.0432 -0.267 0.0425 0.011 0.042 0.0084 0.2743 0.3288 0.9671 -0.1302 -0.5124 0.064 0.008 0.027 0.1882 0.2108 0.4308 0.4457 -0.0453 -0.2732 0.043 From table 8 it is seen that BE500 Index has the biggest weight (51.8%) in Global minimum variance portfolio. BE500 Index has the lowest volatility comparing it with other 6 variables and it could be the reason why this Index weight reaches almost 52% in Global TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE minimum variance portfolio. In this portfolio one short-sell is required for ASX Index with weight of -6.9%. Already from table 8 above it is seen that equally weighted portfolio drops out of efficient portfolios frontier because it has monthly returns of 0.006 and risk level of 0.03, while efficient portfolio with same returns as ASX Index gives better returns (0.0079) with lower amount of risk (0.026). In the efficient portfolio with same returns as MXASJ Index a lot of weight is given to one MXASJ Index (96.7%). As MXASJ Index has higher returns comparing it to other indexes and ETF (see Figure 2) it could influence the weights of this portfolio and it may explain why MXASJ Index creates the biggest part of this portfolio. Tangency portfolio is also calculated. Monthly risk-free rate of 0.12% is used (based on EY (2005) paper). As it is mentioned in the 3rd part of this paper, tangency portfolio is the portfolio with maximum Sharp ratio. This ratio measures the additional returns which investor gets for additional unit of risk. Tangency portfolio is the optimal portfolio for the investor. In this model this portfolio brings 0.79% monthly returns with 2.67% of risk. Based on the calculated tangency portfolio, the portfolio of risky assets for the investor with target monthly returns is chosen. There is created optimal portfolio which brings monthly returns equal to 0.008 (0.8%). This portfolio has risk of 2.7% and weights are as following: BE500 Index and MXASJ Index together create the biggest part of portfolio (43% and 44%), SMI Index has weight of 0.21, TLT ETF has weight of 0.188 and SPGSAG has a small 0.043 weight. Moreover, there are two short-selling positions for DJGSP Index (0.045) and for ASX Index (-0.273) in this portfolio. After efficient portfolios are calculated the efficient frontier called Markowitz bullet frontier is formed. Figure 3 presents the Markowitz bullet efficient frontier with Global minimum variance portfolio, Tangency portfolio, and portfolio with target monthly returns of 50 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 0.8% which is overlapping the tangency portfolio (find the additional graph of Markowitz efficient frontier in Appendix 4). Figure 3. Efficient Frontier (Markowitz Bullet) In Figure 3 together with Markowitz efficient frontier there are presented risks and returns of each individual Indexes and ETF. Any portfolio on the Markowitz efficient frontier is the efficient portfolio for the specific level of returns (with the lowest possible risk level). Markowitz theory suggests that it is not possible to create less risky portfolios from these assets than portfolios that are located on Markowitz efficient frontier. In the next section rolling portfolio technique is implemented. Rolling means, standard deviations and correlations together with rolling portfolio weights are presented and impact on Markowitz time-invariant efficient portfolio is discussed. 4.3. Rolling Parameters and Efficient Portfolios Rolling parameters are created using 24 months rolling windows. In this research there are 120 monthly observations of 10 years. Model with different size windows is tested to see which size rolling windows fits data the best. In the beginning it was tried to use 12 month rolling windows, but the results were too noisy and weights of the efficient portfolio 51 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE and parameters were changing too dramatically (see example graphs in Appendix 5). Because of this reason and based on E. Zivot (2013) work, 24 month rolling windows is applied. First of all time varying means and standard deviations for each Index and ETF monthly log returns are performed. In Figure 4 there are plotted TLT ETF log returns and rolling mean together with rolling standard deviation. From this figure it is clearly seen that mean as same as standard deviation is not constant and Figure 4. Rooling mean and sd for TLT changes over time. The same results are reached for 6 Indexes (see figures in Appendix 6). Secondly, the rolling parameters method is applied to calculate correlations between all 7 variables. Figure 5 present results of rolling correlations between ASX and SPGSAG Indexes. Correlations between ASX and SPGSAG Indexes Figure 5. Time-varying correlations between ASX and SPGSAG Indexes As it is seen from figure 5, correlation between ASX and SPGSAG Indexes is not constant over time. Correlation is time-varying and fluctuates between positive moderate correlation and correlation close to zero which means that at that time the Indexes are not 52 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE correlating significantly with each other. Time-varying correlations are presented for pairs of all 7 variables. The results of all correlations are the same: all correlations are time-varying and those changes might have impact to the portfolio weights (find all time-varying correlations in Appendix 7). Because all parameters (means, standard deviations and correlations) are not constant over time, their impact on global minimum variance portfolio and efficient portfolio with monthly returns of 0.008 are measured. 4.3.1. Rolling Global Minimum Variance Portfolio Rolling windows method is applied for Global minimum variance (GMV) portfolio. Figure 6 represents rolling returns and rolling standard deviation of Markowitz global minimum variance portfolio. Global minimum variance portfolio returns are changing over time. With 24 months rolling window, returns decreases to negative values in December 2007 and becomes positive again only in December 2009. Financial crises should be the reason for this drop. The lowest returns of -0.00739 are reached in January 2009. Risk of Global minimum variance portfolio is also time-varying. The highest risk of the portfolio of 0.0225 is reached in December 2010 and the lowest risk is reached in May 2013 and is equal to 0.012. Figure 6. Rolling man and sd of GMV Figure 7. Rolling GMV portfolio weights 53 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 54 Markowitz Global minimum variance portfolio risk and returns characteristics are changing over time. Together with these characteristics portfolio weights are also changing. In Figure 7 weights of Global minimum variance portfolio are presented. In Appendix 8 there are presented graph and data how weights of Global minimum variance portfolio are changing with 24 months rolling window. From Figure 7 above it is seen that portfolio weights are changing over time. Weights for BE500 Index, DJGSP Index, MXASJ Index, ASX Index, SMI Index, and SPGSAG Index are changing between long and short-selling positions. Only TLT ETF keeps long positions through all the period. 4.3.2. Rolling Efficient Portfolio with 0.8% Monthly Returns Rolling windows method is also applied for efficient portfolio with 0.8% monthly returns. The main idea of this portfolio is that it should keep stable monthly returns for all investment period. Figure 8 presents rolling standard deviation of this efficient portfolio. Returns are stable through all the period at level of 0.8% per month, but standard deviation is changing. Using 24 months rolling windows standard deviation reaches the peak of 5% in Figure 8. Rolling sd of Markowitz target return portfolio Figure 9. Rolling weights of Markowitz target return portfolio TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE October 2009. Weights in the efficient portfolio with stable 0.8% monthly returns, as same as in Global minimum variance portfolio, are time varying. These weights are presented in Figure 9. BE500 Index and TLT ETF often have the biggest weights in rolling portfolios. In the middle of 2009 MXASJ Index has weights for almost 50% of the portfolios. ASX Index is being short-sell the most with the lowest weight of -72% in August 2011. SPGSAG Index weights change over time from around -20% till around +20% of portfolio. SMI Index weights change quite dramatically: from 40% weight in 2007 it drops till almost -40% weights in 2008. In 2010 the weights of this index reach +30%. After the short-selling period in 2011 and 2012, SMI index performs better and keep weights from 20% to 50% until the end of 2014. In Appendix 9 there is presented more information about weights of this efficient portfolio and how they are changing while using 24 months rolling windows. Rolling portfolios method provided evidence that time-varying means, standard deviations and correlations make changes in efficient portfolios weights a same as means and standard deviations. As volatilities are time varying the special methods such as EWMA and GARCH can be used to take these time varying volatilities into portfolio construction process. 4.4. EWMA Efficient Frontier and Rolling Portfolio 4.4.1. EWMA Efficient Frontier and Optimal Portfolio The first method which estimates time varying volatilities and is used in this research is EWMA. EWMA is used to create covariance matrix in a slightly different way when it is done in Markowitz model. 55 56 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Multivariate EWMA covariance estimation method together with lambda of 0.8 is used to build EWMA covariance matrix (see table 9). Table 9. EWMA covariance matrix TLT SMI BE500 MXASJ DJGSP ASX SPGSAG TLT 0.0018088 -2.07E-04 -0.000524 -0.000342 1.08E-04 -0.000476 0.0001003 SMI -0.000207 9.16E-04 0.0005964 0.0006818 4.66E-05 0.0007745 0.0002795 BE500 -0.000524 5.96E-04 0.0009149 0.0007899 2.58E-04 0.0009448 0.0004039 MXASJ -0.000342 6.82E-04 0.0007899 0.0018589 9.74E-04 0.001473 0.0004738 DJGSP 0.0001083 4.66E-05 0.0002579 0.0009736 5.56E-03 0.0008795 0.0011067 ASX -0.000476 7.75E-04 0.0009448 0.001473 8.80E-04 0.0020445 0.0006127 SPGSAG 0.0001003 2.79E-04 0.0004039 0.0004738 1.11E-03 0.0006127 0.0034383 EWMA covariance matrix, together with assets returns, is used in efficient portfolios calculations. These efficient portfolios are called EWMA efficient portfolios that at the end of the research it would be easier to compare different methods and their results. Table 10 below presents different EWM efficient portfolios. Table 10. EWMA Efficient portfolios characteristics Assets Weights Portfolio/Assets EWMA Global min.variance portf. EWMA Tangency portfolio Effic.portf with same return as "ASX" and EWMA Effic.portf with same return as "MXASJ" and EWMA EWMA Eff.port with targer return of 0.8% Return Risk TLT SMI BE500 MXASJ DJGSP ASX SPGSA G 0.00537 0.0185 0.3341 0.1726 0.5176 0.0043 0.0388 -0.0842 0.0168 0.0079 0.0234 0.1840 0.2294 0.4336 0.4344 -0.0278 -0.2854 -0.0317 0.0079 0.024 0.1844 0.2293 0.4338 0.4335 -0.0276 -0.2850 0.0316 0.011 0.037 -0.0021 0.2999 0.3294 0.9678 -0.1102 -0.5349 0.0501 0.008 0.0237 0.1799 0.231 0.4313 0.4463 -0.0296 -0.291 0.0321 EWMA Global minimum variance portfolio has one short-sell position of –8.4% for ASX Index. BE500 Index has weight of 51.7% and TLT ETF has 33.4% weight in this portfolio. MXASJ Index has the smallest weight of only 0.4% in EWMA Global minimum variance portfolio. TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE EWMA tangency portfolio presents the optimal portfolio for the investor which has the best risk and return ratio. As same as in Markowitz model, tangency portfolio brings monthly returns of 0.79%. This time tangency portfolio has 2.34% of risk. EWMA Efficient portfolio with same returns as ASX Index has two main assets in it. BE500 Index and MXASJ Index have 43% weights. Two short-sell positions in this efficient portfolio belongs to DJGSP Index (-2.8%) and ASX Index (-28.5%). In this portfolio SPGSAG Index has the lowest weight of 3.2%. EWMA efficient portfolio with same returns as MXASJ Index performs very similar to the same portfolio constructed under Markowitz approach with simple constant covariance matrix. In this portfolio weight of 96.8% is given to MXASJ Index. The last EWMA efficient portfolio in this table has constant monthly returns of 0.8% (chosen investor portfolio). This portfolio risk is equal to 2.37% and weights are allocated between assets as following: BE500 and MXASJ Indexes have weights of 43.1% and 44.6%, SMI Index has 23.1% weight while ASX Index has short-sell position of -29.1%. TLT ETF makes almost 18% of this EWMA efficient portfolio. The efficient frontier is calculated and plotted together with EWMA Global minimum variance portfolio and EWMA efficient portfolio with target monthly returns of 0.8% as same as EWMA tangency portfolio. In Figure 10 it is seen that chosen investor portfolio with target monthly returns of 0.8% and EWMA tangency portfolio are overlapping each other (see Figure 10). 57 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Figure 10. Efficient Frontier (EWMA model) In Figure 10 there is presented EWMA efficient frontier which is calculated using EWMA estimated covariance matrix. This frontier suggests all possible portfolios that have the lowest risk for the specific level of returns. EWMA global minimum variance portfolio and EWMA portfolio with target monthly returns of 0.8% as same as EWMA tangency portfolio are just a few of efficient portfolios in this frontier. A list of EWMA efficient portfolios and their weights is provided in Appendix 10. 4.4.2. EWMA Lambda Coefficient For EWMA model Lambda is chosen by comparing different Lambdas. Lambda which provides the best efficient frontier within this model is chosen for the model. Literature suggests that Lambda of 0.91 should fit this kind of data, but after different Lambdas are used and compared, Lambda of 0.8 is chosen. There is no literature evidence that 0.8 Lambda is the best for this type of data. This Lambda is chosen because with Lambda = 0.8 assets returns from the crises period have slightly lower effect on the final results than comparing it with 0.91 Lambda. Moreover, 0.8 Lambda is high enough that results would not be too optimistic (see figure of EWMA efficient frontiers with different Lambdas in appendix 11). 58 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 59 4.4.3. EWMA Rolling Efficient Portfolio with 0.8% Monthly Returns After EWMA efficient frontier is created and weights for efficient portfolio with monthly target returns of 0.8% are fixed it can be checked how this portfolio is changing over time. The rolling portfolios method as same as for Markowitz efficient portfolios is applied. The same 24 months rolling windows are used. Figure 11 presents rolling standard deviation of EWMA efficient portfolio with monthly target returns of 0.8%. While portfolio returns are constant, portfolio standard deviation is changing over time (Figure 11). Standard deviation fluctuates from 1.85% up to 3.36%. The peak of standard deviation is reached in October 2009. From the beginning of 2010 until the middle of 2011 there is a calm period without any rises in the standard deviation of this portfolio. The similar stable period starts at the beginning of 2013 and continues till 2015. Figure 11. Rolling sd of EWMA target return portfolio Figure 12. Rolling weights of EWMA target return portfolio Figure 12 presents time varying weights of Indexes and ETF in EWMA efficient portfolio with target monthly returns of 0.8%. BE500 Index plays the important part in this portfolio. Most of the time weights for this Index reach from 40% to 70% of the portfolio. The highest weight of BE500 Index of 91% is reached in December 2012. The lowest weight TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE of the same BE500 Index of almost 17% is from August 2011. Average BE500 Index weight in this portfolio is equal to 54.9%. TLT ETF average weight in this portfolio is equal to 32.9%. TLT ETF, as same as BE500 Index, does not have any short-sell positions in this rolling portfolio. 85% of the time ASX Index has only short-sell positions. These short-sell positions changes from -1% to -27%. ASX Index has only a few positive weights and the average weight of ASX Index in this portfolio is equal to -8%. Almost 1/3 of the time SMI Index has also short-sell positions. The highest short-sell position of 89% belongs to SMI Index in October 2009. Due to the big short-selling positions this Index average weight reaches only 6.5%. MXASJ Index has similar 6.7% average weight in this portfolio. In October 2012 MXASJ Index weight is equal to -54.5%. MXASJ Index weights are varying up to 44%. SPGSAG Index does not have any big long or short positions. Weights of this Index fluctuates from -18.8% up to 32.8%. DJGSP Index performs within the lowest weights between all the assets in the portfolio. Weights are changing from -13% to 17% of the portfolio value. In this EWMA efficient portfolio with 0.8% monthly target returns this DJGSP Index weights are equal to 1.9% on the average. In the Appendix12 there are presented more statistical information about EWMA portfolio weights and how they are changing over time while using 24 months rolling windows. 4.5. GARCH Efficient Frontier and Rolling Portfolio GARCH method is chosen as alternative and more advanced method for modeling time varying volatilities, correlations, and covariances. According to Therese Peters (2008) the main difference between GARCH(1,1) and exponential weighted moving average model 60 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE is that “in the GARCH case the parameters need to be estimated and mean reversion has been incorporated in the model” (p. 9). 4.5.1. Lagrange Multiplier (LM) Test for Assets Returns Before starting modeling the data using GARCH, Lagrange Multiplier (LM) test for autoregressive conditional heteroskedasticity is performed. Table 11 presents the results of this test. Models data strongly rejects NULL hypothesis which says that there is no ARCH effect under 0.05 and 0.01 confidence intervals (p-value = 6.661e-16) and assume that GARCH models are needed for this data. Table 11. Lagrange Multiplier (LM) test for assets returns ARCH test Lagrange Multiplier (LM) test Returns for all Indexes and ETF (p-value < 0.05) 6.661e-16 4.5.2. GARCH Efficient Frontier DCC-GARCH(1,1) model is used to estimate and forecast conditional covariance matrix which is used when constructed GARH efficient frontier. The main difference between all 3 efficient frontiers is the way how the covariances matrixes are calculated. Markowitz efficient frontier uses time invariant covariance matrix (Table 7), EWMA efficient frontier is constructed while covariance matrix is estimated with EWMA method (Table 9) and GARCH efficient frontier is constructed using forecasted covariance matrix. When using DCC-GARCH(1,1) method, as it is explained in the 3rd part of the thesis, the forecasted covariance matrix is combined. As an example, Figure 13 presents forecasted 61 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE covariances between some Indexes such as DJGSP and MXASJ Indexes or SPGSAG and ASX Indexes (more examples of forecasted covariances can be found in Appendix 13). Figure 13. Covariance Estimation and Forecast of DCC-GARCH(1,1) Figure 13 shows how DCC-GARCH(1,1) model estimates the parameters and forecasts the results for 3 months ahead. Gray line presents the estimated parameters and red line presents the forecasted part of the parameters. It is seen that for each of the pair the covariances are changing within every period ahead and those changes may have a significant impact to the final results of this model. In this research, GARCH efficient frontier is combined while using variancecovariances matrix which is forecasted only one period ahead. As further the parameter is forecasted the less impact the historical data does on the final results. There is a possibility that final results of GARCH efficient frontier would be significantly different if this efficient frontier would be modeled with variance-covariance matrix which is forecasted for more than one period ahead. But this observation is not under the examination of this work. 62 63 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Table 12 provides GARCH covariance matrix which is forecasted for 1 period ahead. The biggest difference between Markowitz covariance matrix, EWMA covariance matrix, and GARCH is that two first matrixes are calculated using historical data while the last one – GARCH covariance matrix is forecasted from estimated data. Table 12. GARCH covariance matrix TLT SMI BE500 MXASJ DJGSP ASX SPGSAG TLT 0.0011975 -0.000135 -0.000422 -0.000223 -7.51E-05 -0.000374 0.0001952 SMI -0.000135 0.0011823 0.000844 0.0009045 0.0004478 0.0008574 0.0002661 BE500 -0.000422 0.000844 0.0012102 0.0010267 0.0007094 0.0011254 0.0004427 MXASJ -0.000223 0.0009045 0.0010267 0.0024085 0.0015777 0.0018456 0.0005187 DJGSP -7.51E-05 0.0004478 0.0007094 0.0015777 0.0073109 0.0012108 0.0010574 ASX -0.000374 0.0008574 0.0011254 0.0018456 0.0012108 0.0026666 0.0007893 SPGSAG 0.0001952 0.0002661 0.0004427 0.0005187 0.0010574 0.0007893 0.0044227 GARCH covariance matrix is used in the calculations of the efficient portfolios. These efficient portfolios are called GARCH efficient portfolios that it would be easier to compare them with already existing Markowitz and EWMA efficient portfolios. Table 13 presents GARCH efficient portfolios. Table 13. GARCH Efficient portfolios characteristics Assets Weights Portfolio/Assets Return Risk TLT SMI BE500 MXAS J DJGSP ASX SPGSA G GARCH Global min.variance portf. 0.0045 5 0.0196 0.4792 0.0787 0.4288 -0.0343 0.0133 0.0184 0.0159 GARCH Tangency portoflio 0.0080 4 0.0280 4 0.2129 0.182 0.3591 0.4527 -0.0822 -0.1839 0.0594 0.0079 0.028 0.2215 0.1787 0.3613 0.4369 -0.0791 -0.1774 0.0581 0.011 0.042 -0.0209 0.2727 0.2978 0.8804 -0.1661 -0.3616 0.0977 0.008 0.028 0.2157 0.1809 0.3598 0.4475 -0.0812 -0.1817 0.059 GARCH Effic.portf with same return as "ASX" GARCH Effic.portf with same return as "MXASJ” GARCH Eff.port with targer return of 0.8% TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE In GARCH Global minimum variance portfolio TLT ETF and BE500 Index have the biggest weights of 48% and 43%. Other Indexes has significantly lower weights than TLT ETF and BE500 Index. SMI Index has 7.8% weight in this portfolio while 3 other Indexes: DJGSP, ASX and SPGSAG have weights around only 1.5%. One low weight short-sell position of -3.4% in GARCH Global minimum variance portfolio belongs to MXASJ Index. GARCH tangency portfolio presents the optimal portfolio for the investor which has the best risk and return ratio. As same as in Markowitz and EWMA models, tangency portfolio brings monthly returns of 0.79%. GARCH tangency portfolio has 2.8% of risk. The main asset in GARCH efficient portfolio with same returns as ASX Index is MXASJ Index with weight equal to 43.4%. BE500 Index also has a similar weight of 36%. TLT ETF and SMI Index have weights of 22% and 17.8%. The biggest short-sell position of -17.7% belongs to ASX Index. Assets in GARCH efficient portfolio with same returns as MXASJ Index have the following weights: weight of 88% belongs to MXASJ Index, 29.7% and 27% weights belong to BE500 and SMI Indexes while SPGSAG Index has weight equal to 9.7%. ASX Index has the biggest short-sell position of -36% of this efficient portfolio. The last GARCH efficient portfolio in this table, as same as in Markowitz and EWMA models, has constant monthly returns of 0.8%. This portfolio has risk equal to 2.8%. MXASJ Index has 44.7% weight, BE500 Index has almost 36% weight and TLT ETF and SMI Index have 21.5% and 18% of total portfolio weights. SPGSAG Index forms a small part (only 6%) of the portfolio while DJGSP and ASX Indexes have positions of short-sell of -8% and -18%. GARCH Efficient frontier is calculated and plotted together with GARCH Global minimum variance portfolio and GARCH efficient portfolio with monthly returns of 0.8% which is overlapping with GARCH tangency portfolio (see Figure 14). 64 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Figure 14. Efficient Frontier (DCC-GARCH(1,1) model) GARCH efficient frontier is calculated using GARCH forecasted covariance matrix. This frontier is made from all possible efficient portfolios that are calculated under GARCH. GARCH global minimum variance portfolio is portfolio with lowest possible risk. However, returns of this efficient portfolio are also the lowest comparing it with others efficient portfolios. A list of GARCH efficient portfolios and their weights is provided in Appendix 14. 4.5.3. GARCH Rolling Efficient Portfolio with 0.8% Monthly Returns After GARCH efficient frontier is created and weights for efficient portfolio with monthly target returns of 0.8% are fixed it can be checked how this portfolio is changing over time. A rolling portfolios method as same as for Markowitz and EWMA efficient portfolios is applied. The same 24 months rolling window is used. Figure 15 presents rolling standard deviation of GARCH efficient portfolio with monthly target returns of 0.8%. 65 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Figure 15. Rolling sd of GARCH target return portfolio Figure 16. Rolling weights of GARCH target return portfolio GARCH efficient portfolio returns are constant and do not change in time. This causes bigger fluctuations in portfolio standard deviation. The standard deviation has several peaks over the time. The highest standard deviation of 3.4% is reached in October 2009. Other peaks are a little bit lower, up to 3.1% in March 2008, August 2012 and October 2012. The average standard deviation of rolling GARCH efficient portfolio is equal to 2.35%. Figure 16 presents time varying weights of Indexes and ETF in GARCH efficient portfolio with target monthly returns of 0.8%. TLT ETF and BE500 Index both do not have any short-sell positions in this portfolio. TLT ETF weight fluctuates over time from almost 20% up to 76%. The lowest TLT ETF weight is applied in August 2012 while the highest weight is applied in October 2009. BE500 Index weight in this portfolio changes from 5% in August 2011 up to almost 85% in October 2009. On the average BE500 Index weight is equal to 47% and TLT ETF average weight is equal to 43%. SMI Index has the highest short-sell positions in this portfolio. Weights of this Index fall dramatically in October 2009 to -91%. Without this short-sell position the lowest shortsell weight of SMI Index is -56%. The highest weight of the same Index is equal to almost 66 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 71% in October 2012. The lowest MXASJ Index weight is equal to -55% in October 2012. The highest MXASJ Index weight of 45.6% is from March 2008. The average weight of SPGSAG Index in this portfolio is equal to only 5%. Almost 1/3 of a time this model suggests to short-sell SPGSAG Index and that is the reason for low average weight. The maximum SPGSAG Index weight is equal to almost 33% in March 2012. DJGSP and ASX Indexes do not have any big long or short positions. DJGSP weight fluctuates from -14% up to 17% and ASX Index weight changes from -20% till 24.6%. However, average weights of these two Indexes do not reach 1% of this total rolling GARCH efficient portfolio. In Appendix 15 there are presented more graphical and statistical information about GARCH portfolio weights and how they are changing over time using 24 months rolling windows. In this part of the thesis the most important empirical research results are explained. In the next chapter the comparison of the final results and discussions about Markowitz, EWMA and GARCH efficient frontiers and their efficient portfolios are presented. Moreover, the discussion and comparison of this research findings and results from the other works, presented in the literature review chapter, together with limitations of this research, and suggestions for the further research are provided. 67 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 5. DISCUSSION In the previous section there are presented 3 different efficient frontiers with their efficient portfolios. All of them are constructed using different methods: Markowitz efficient frontier is constructed by keeping distributional parameters invariant while EWMA and GARCH efficient frontiers are constructed with time-varying covariance matrixes which are combined from time varying correlations. Efficient portfolios with 0.8% monthly returns are found and the rolling window method is used to see how weights in those portfolios are changing over time. In this part of Master Thesis the final results of the research are summarized and conclusions about the best model are drawn. Moreover, the linkage between the final results of this research and existing literature is provided. Limitations of this research together with the suggestions for the further research are summarized and provided too. 5.1. Final Research Results 5.1.1. Efficient Frontiers In figure 17, there are plotted all 3 final efficient frontiers of this research: the black line presents Markowitz efficient frontier, blue line presents the efficient frontier for which covariance matrix is constructed using EWMA method; green line presents the efficient frontier for which covariance matrix is forecasted using DCC-GARCH(1,1) method. 68 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Figure 17. Markowitz, EWMA and GARCH efficient frontiers If comparing EWMA and GARCH efficient frontiers with Markowitz efficient frontier it is seen that EWMA efficient frontier gives the best results while GARCH and Markowitz frontiers intersect with each other. EWMA efficient frontier suggests the efficient portfolios with the lowest risk at any level of expected returns. It is important to mention that this EWMA efficient frontier is constructed with Lambda = 0.8. If Lambda is equal to 0.91 (as it is recommended by RiskMetrics (1996)), EWMA efficient frontier is still suggesting the portfolios with the lowest risk levels (Appendix 16 presents EWMA efficient frontier (when Lambda=0.91) together with Markowitz and GARCH efficient frontiers). GARCH and Markowitz efficient frontiers look more similar. If talking about Global minimum variance portfolios it is seen that GARCH suggests less risky portfolio with lower returns comparing it to Markowitz Global minimum variance portfolio. GARH efficient frontier is slightly steeper than Markowitz frontier. Moreover, GARCH efficient frontier suggests a bit better portfolios (with lower risk levels) when expected returns are higher than ~1.14% per month. 69 70 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE If talking about the portfolios which have the same level of expected returns, the conclusion can be drawn that less risky portfolios are constructed while using EWMA covariance matrix. Portfolios marked as brown dots on all of the frontiers (see Figure 17) present the efficient portfolios with target monthly returns of 0.8%. Table 14 presents risk levels of these efficient portfolios. Table 14. Risk level of efficient portfolios with target monthly returns of 0.8% Efficient portfolios Markowitz efficient portfolio with target monthly returns of 0.8% GARCH efficient portfolio with target monthly returns of 0.8% EWMA efficient portfolio with target monthly returns of 0.8% Risk 2.7% 2.8% 2.3% As it is seen in table 14, when target monthly returns of 0.8% are set for the portfolios, the following risk levels of those efficient portfolios are calculated: 2.8% for GARCH efficient portfolio, 2.7% for Markowitz efficient portfolio and 2.3% for EWMA efficient portfolio. EWMA method diversifies the risk in the best way by offering the lowest risky portfolios for any level of expected returns. From the other hand if comparing Markowitz and GARCH efficient portfolios, it is hard to say if Markowitz or GARCH model is working better. It belongs on the personal decision about the riskiness of the efficient portfolio or what level of expected return this portfolio should bring. 5.1.2 Adjusting Weights for Efficient Portfolio with 0.8% Monthly Returns The final results of this research confirm that efficient portfolios which are constructed with exponentially weighted moving average method are giving better results TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE than efficient portfolios which are constructed while using DCC-GARCH(1,1) and Markowitz methods. In the 4th part of this work the rolling portfolio method is applied and analyzed with all of the three models. As EWMA efficient frontier brings the best results, the rolling portfolio method together with EWMA method should be used as the best combination to get less risky portfolios over time. In this research, the 24 months rolling windows are used to see how the weights in the portfolio should be adjusted over time that portfolio would have the same expected returns. Rolling weights of the least risky EWMA efficient portfolio with monthly target returns of 0.8% are plotted and explained briefly in the 4th part of this work (pg. 58-59). The deeper analysis of these time-varying weights is done and the main conclusions are provided below (see Figure 18). Figure 18. Rolling weights of EWMA efficient portfolio with target monthly returns of 0.8% As it is explained in section 4.3.3 and also presented in figure 18, BE500 Index holds the biggest part of this portfolio even when the weights are changing over time. Figure 18 presents how the weights of the portfolio inputs are changing over time and how this research 71 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE suggests the investors to allocate their portfolio weights that it would keep stable target monthly returns of 0.8% in every period of time with the lowest possible level of risk. If looking more carefully to figure 18, it is seen that there are two periods when portfolio weights are changing more dramatically. First period is around 2009 and the second period starts in 2011 and continues until the end of 2012. Both of these periods are related to some unstable periods in the financial markets which are related to the Global Financial crisis in 2008, the European Sovereign Debt crisis which took place in 2011, and the credit rating downgrades of USA and Japan in 2011. For those shocking periods in the financial markets this model suggests changing weights of this portfolio inputs more dramatically. This example of these changing weights of the portfolio with target returns proves that this model reacts to the turbulence of the financial markets and allocates the weights accordingly that the target returns or target risk level would be stabilized without any big losses. The importance of the time-varying parameters is the most important especially when shocks appear in the financial markets. If looking at the more stable periods such as over the year 2013, it is seen that model is not suggesting chancing weights a lot and composition of the portfolio inputs remains similar over the time. However, it is not straight forward to say how often the portfolio should be rebalanced. One of the strategies that the investor could use is to rebalance the portfolio weights more often in the times of the shocks in the financial markets and to keep the portfolio more stable in the calm periods. Moreover, it is important to mention that the size of the rolling window is being selected manually and there are no rules or methods how to select the best size of this rolling window. As it is shown in this research, too small rolling window brings too much noise and weights of the assets are changing too dramatically. 72 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Separate research is needed to analyze the importance of the size of the rolling window and how the size of this window influences the final results of the portfolio weights. All 3 models are compared and conclusions are drawn that EWMA efficient frontier brings the best results and the weights of the portfolio with 0.8% monthly returns should be changed over time as it is presented in figure 18 that the target level of returns would be stabilized during the shocks in the financial markets. Moreover, this research provides the evidence that portfolio optimization with timeinvariant parameters is leading to wrong portfolio allocation and forgone returns of the investment portfolio which investor would face especially when the shocks in the financial markets occur. Time-varying parameters show to improve the portfolio optimization process and help the investor to react to the changing environment in the financial markets. Near to that, time-varying nature of the portfolio parameters is helping investor to avoid a wrong portfolio allocation which would take place if time-invariant parameters are used in portfolio optimization process. Time-varying nature helps to avoid big losses when the shocks appear and help the investor to control portfolio performance over time. In the next section, the final results of this work are compared with the existing literature which is discussed in the literature review part. 5.2. Research Linkage to the Existing Literature The main aim of this work is to understand the dynamics of time varying parameters and the ways in which the portfolio allocation can be adapted to account for all these timevarying parameters. This work is the most related to Y. K. Tse (1991) and M. Horasanl and N. Fidan (2007) works. All these authors compare different techniques which can be used together with time varying parameters. The findings of this research agree with both of these 73 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE works that exponentially weighted moving average technique is letting to reach better allocation of the portfolio. Y. K. Tse (1991) suggests that GARCH model reacts slower to the changes in the market. The final results of this research are showing the same results that GARCH model performs worse than EWMA model. However, this research cannot agree with the statement of M. Horasanl and N. Fidan (2007) which suggests that GARCH model brings worse results than Markowitz efficient frontier. In this research GARCH model acts pretty much the same as Markowitz model and it is hard to say which one of them is performing better. Both works uses GARCH(1,1) technique, but the final forecasted covariance matrixes are calculated using different methods. Different results related to GARCH efficient frontier of this research and M. Horasanl and N. Fidan (2007) work might be because those authors uses BEKK model to forecast covariance matrix while in this work the covariance matrix is constructed using DCC-GARCH(1,1) model. V. Akgiray (1989) in his work compares several techniques by modeling and forecasting stock returns and concludes that GARCH(1,1) model is the best for the daily data. This author also provides evidence that different results are reached if model is constructed from monthly data. These V. Akgiray (1989) findings also go along with the other work of H. J. Cha and T. Jithendranathan (2009) which shows that when DCC model is used in estimating correlations and covariances, the results of the portfolio optimization process are improving while using a weekly data. The findings of this research are a little bit different than H. J. Cha and T. Jithendranathan (2009) because in this research it is hard to say if DCC model improved the portfolio optimization as GARCH and Markowitz frontiers are too similar. In this research monthly data is used while Cha and Jithendranathan use weekly data. As V. Akgiray (1989) 74 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE suggests that GARCH works better for data of short-periods, the conclusion can be drawn that monthly data can be the reason why GARCH efficient frontier does not outperform the Markowitz model. However, interesting point is related to the fact that M. Horasanl and N. Fidan (2007) in their work use daily data but their GARCH model performs worse that in this research if comparing it with Markowitz models. According to V. Akgiray (1989), GARCH should work better with shorter-periods data. However, the results are different while comparing this research with the work of M. Horasanl and N. Fidan. Of course it could be the reason of different securities or different time frames, but the biggest difference, as already mentioned, should be regarding the methods which were used to construct the forecasted covariance matrixes. Differences between BEKK and DCC models might be the main reason related to the different results which are reached in this research and in the M. Horasanl and N. Fidan (2007) work. If looking at the rolling portfolios and the size of the rolling window, E. Zivot (2013) uses 24 months rolling windows to see how portfolio weights are changing over time. In this research the same size rolling window is used because smaller rolling window brings too much noise and weights of the Indexes are changing too dramatically. However, there are no specific rules or methods in the literature how to choose the size of the rolling window even if this decision has a huge impact on the final results of the rolling portfolios. That is the main reason why many authors criticize this method of the rolling window. 5.3. Limitations of Research This research, as same as many other works, has its own limitations. The list of the most important limitations from this work with the explanations is provided below: 75 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Monthly returns. In this research, here are used monthly returns of Indexes and ETF. Different results are reached when combining models with daily, weekly or monthly returns. This research fits only for the monthly data and for weekly or daily data the research should be repeated that the differences between the results could be compared. Inflation. The impact of the inflation over the period for the returns of the portfolio and its’ inputs is ignored. Short-selling. In this research the short-selling is allowed. However, the assumption is made that short-selling is free and does not have any costs which would have negative impact on the returns of the portfolio. Separate discussion can be held according the short-selling positions of the portfolio. There are different short-selling strategies and investor himself can choose one of them. Those strategies are not analyzed in this work which leaves the place for the further research and discussion. Size of the rolling window. As it is explained in the research, the size of the rolling window is chosen based on the E. Zivot (2013) work which is done while using 100 observations of the monthly data. The smaller 12 months rolling window is tried and the conclusion that it brings too much noise is made. There are no specific rules how to choose the size of the rolling window. However, this decision has a huge impact on the changes in the portfolio weights over time. Lambda coefficient in exponentially weighted moving average model. Lambda coefficient which is offered by the literature of 0.91 is ignored. Lambda is chosen to be 0.8 that the previous crises would have less significant impact while estimating the parameters for EWMA covariance matrix which is used to build EWMA efficient frontier. However, model with the Lambda equal to 0.91 is calculated and the results still confirm that EWMA model brings the best results. Model constructed with 76 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Lambda equal to 0.8 let the efficient portfolios have lower risk comparing it with portfolios which are constructed with Lambda equal to 0.91. DCC-GACH(1,1) model. Different authors conclude that univariate GARCH(1,1) model is the best and the other univariate GARCH models do not outperform GARCH(1,1). In this research, there are ignored other GARCH(p,q) models. Only GARCH(1,1) is considered for this data. The results show that efficient frontier constructed with DCC-GARCH(1,1) model is very similar to the Markowitz efficient frontier. Other GARCH(p,q) compositions are ignored and it is unknown if the final results would be significantly different comparing it with Markowitz model and DCCGARCH(1,1) model. 5.4. Suggestions for Further Research This research is done under the investigation of the time-varying distributional parameters and their impact on the investment portfolio performance. There are many parts of this research that could be updated or changed, or improved, but for that the new research should be done. The most important issues are summarized and written below. These issues might be investigated in the further research: Different portfolio combinations. In this work, here the portfolio is constructed from the monthly returns of 6 Indexes and 1 ETF. However, the same research can be performed with more ETFs, separate assets, bonds or other financial instrument which might help to improve the risk diversification within the portfolio especially at the time of the shocks in the financial markets. Portfolios without short-selling positions and portfolios with short-selling positions for which the investor is being charged. The price of the short-selling can be taken into the consideration to see how it influences the final results of the portfolio returns. 77 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Moreover, not all of the investors are willing to take up the short-selling positions. The model can be adjusted for those investors who do not want to short-sell the assets. Portfolio returns witch fail to be normally distributed. In the real life, there are a lot of examples when the returns of the assets are not normally distributed and the data after the transformation processes (when the data from non-normality is being transformed to normally distributed data) does not present the situation in the real world anymore. There are methods which are used with data which is not normally distributed. Results with such a technique could be compared with the results of this thesis. This might help to improve the final results of the efficient portfolios. Estimated Lambda parameter. Lambda is the most important parameter in EWMA model. In the literature, there can be found some suggestions how to calculate the best Lambda for the specific data. In the further research, there can be considered the estimation of the Lambda parameter. If Lambda parameter is estimated for a specific data, in such a case there is no risk that chosen Lambda is not fitting for the data. Calculated Lambda could be compared with Lambda which is recommended by the RiskMetrics (1996) and it might help to improve the final results of the exponentially weighted moving average model. DCC-GARCH(p,q) model. GARCH efficient frontier can be constructed using the different univariate GARCH model than GARCH(1,1). Instead of using DCCGARCH(1,1) model, the other more advanced GARCH(p.q) models can be applied while estimating and forecasting correlations and volatilities with DCC model. It might help to improve the allocation of the final GARCH efficient frontier which might outperform the Markowitz efficient frontier and might challenge the EWMA model. 78 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Moreover, multi-period forecasted DCC-GARCH(p,q) covariance matrixes might be implemented to the GARCH portfolio optimization process. These covariance matrixes can be used as alternative to the rolling window method to see how the weights in the efficient portfolios are changing over time. Longer/shorter rolling windows. As literature suggests, the size of the rolling window should be adjusted depending on the length of the analyzed period. The same model can be analyzed with the smaller or bigger rolling windows and the changes in the portfolio weights can be compared that conclusions about the proportion of the rolling window among the whole research period could be drawn. Moreover, the deeper research is needed to answer the question how often the portfolio should be rebalanced. 79 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 6. CONCLUSIONS In this thesis, here is provided the analysis of time-varying parameters such as mean, volatilities, and correlations and the ways in which the portfolio allocation can be adapted to account for all these time-varying parameters. Many authors have analyzed the time-varying parameters already. However, the common conclusions are not reached yet. There is no straight forward answer which method is the best to model the time-varying parameters in the portfolio optimization process. In this research the following main question has been raised: what kind of impact the time varying distributional parameters do on portfolio performance. Therefore 2 different methods: EWMA and DDC-GARCH(1,1) methods were employed in the portfolio optimization process and results were compared with traditional Markowitz approach. Rolling window technique was applied that the impact of time-varying means would be added into the model too. The objectives listed below were raised in this thesis and the following conclusions are made from the performed research: 1. By analyzing different works and their methodologies it was described how time varying distributional parameters can be added while constructing the optimal portfolios and what results are reached by other authors. The following conclusions are done: Some authors use GARCH models to work with time-varying parameters and confirm that GARCH models are providing the best results as it is the only method which forecast parameters instead of using only historical data. However, others claim that GARCH models such as GARCH(1,1) together with BEKK are reacting slower to the changes in the market and EWMA model provides better portfolio results when working with time-varying volatilities. Therefore, in this research the contribution to the existing literature was made by comparing EWMA and DCC-GARCH(1,1) models which were employed in the portfolio optimization process and results were 80 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE compared with traditional Markowitz approach. DCC-GARCH(1,1) model was chosen instead of BEKK-GARCH(1,1) model because literature suggests that DCC model performs better comparing it with BEKK and other models. 2. The portfolio optimization processes with and without time-varying parameters were analyzed and described. The biggest differences between these processes were stretched. From this part of the research the following conclusions are done: The biggest differences between models with and without time-varying distributional parameters are in the way the covariance matrixes are constructed. Markowitz model uses time-invariant covariance matrix while EWMA model estimate covariance matrix with the chosen smoothing parameter which gives lower weights for the observations further in the past, and DCC-GARCH(1,1) model forecasts covariance matrix from the estimated historical parameters. Moreover, for the implementation of time-varying means, the 24 months rolling window method was employed to see how the portfolio weights are changing due to the changes in the means. 3. The impact of time-varying distributional parameters on the portfolio performance was identified. This process leads to the following findings: Distributional parameters: volatilities, means, and correlations as same as covariances are not stable over time and, depending on the time period, bring different results. When those varying parameters are inserted into the model, the results are significantly different from the traditional Markowitz model. Varying volatilities and correlations might change the riskiness and/or the returns of the efficient portfolios. Time-varying means change the portfolio weights over time. The changes in portfolio weights are the most important at the times of the turbulences in the financial markets as it might help to protect the investment portfolio from the big losses. 81 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE 4. Markowitz, EWMA and DCC-GARCH(1,1) techniques, which were used in this research, together with their final results were compared that the method which brings the best results for the portfolio optimization process would be found. The comparison of these methods leads to such findings: This thesis supports the findings of those works which concludes that EWMA model is the model which brings the best results for the efficient portfolios. However, this research denies the idea, which is found in the literature, that GARCH model reacts slower to the changes in the financial markets than EWMA and Markowitz models. In this research, the EWMA model reacts in the better way, but efficient frontiers which are constructed with DCC-GARCH(1,1) and Markowitz methods are very similar and it is hard to say which is giving better results as it belongs on the particular level of expected returns or risk. GARCH model results are different from the results found in the literature because in this work the final GARCH covariance matrix was constructed with DCC model. In the literature it is found that the same GARCH covariance matrix is constructed using BEKK model. Moreover, the evidence which is found in the literate, that DCC method performs better than the BEKK model, is confirmed. Finally, the thesis contributed to theoretical and empirical research by estimating Markowitz, EWMA and GARCH efficient portfolios for the investment portfolio which was constructed from 6 Indexes and 1 ETF. Investor can always have the least risky portfolios if EWMA model is used in the portfolio optimization process. Moreover, rolling portfolio method helps to see how the efficient portfolio weights are changing when different shocks appear in the financial markets. This work provides evidence that efficient portfolios must be rebalanced regarding the inputs changes over time. However, how often to rebalance, is not straight forward to determine and is left for the investor to decide by himself. 82 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE To finalize the overall work, the conclusion is drawn that this research helps to understand how time-varying parameters are changing the portfolio optimization process. Financial advisors or investors should use the EWMA model in their portfolio optimization process that the best results would be reached. Further research could examine if more advanced GARCH(p,q) forms together with DCC model can outperform the DCCGARCH(1,1) model and maybe can also beat the EWMA model. 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Retrieved February 21, 2015 from http://faculty.washington.edu/ezivot/econ424/RollingPortfoliosPowerPoint.pdf Zivot, E., & Wang, J. (2006). 9. Rolling Analysis of Time Series. In Modeling Financial Time Series with S-PLUS (2nd ed.). New York, NY: Springer. 89 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE APPENDICES Appendix 1(1st page) Normality testing graphical results for QQ-plots and histograms for data with outliers 1. TLT ETF Q-Q plot and histogram 2. SMI Index Q-Q plot and histogram 3. BE500 Index Q-Q plot and histogram 90 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 1(2nd page) 4. MXASJ Index Q-Q plot and histogram 5. DJGSP Index Q-Q plot and histogram 6. ASX Index Q-Q plot and histogram 7. SPGSAG Index Q-Q plot and histogram 91 92 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 2 (1st page) Table A1. Calculations of Data outliers: Rule of thumb 1st Qu TLT SMI BE500 MXASJ DJGSP ASX SPGSA G 3rd Qu Interquartile (1.5 *Interquartile) -0.03016 0.03184 0.06200 -0.01062 -0.01707 -0.01379 -0.05038 -0.02612 0.03278 0.04340 0.02836 0.04543 0.03975 0.05354 0.05749 0.10787 0.04147 0.06759 0.093 0.0651 0.068145 0.08031 0.161805 0.101385 0.04639 0.07630 -0.02991 0.11445 Outlier is Outlier is less than more than -0.12316 0.12484 -0.07572 0.09788 -0.08522 0.096505 -0.09410 0.12006 -0.21219 0.219295 -0.12751 0.142855 -0.14436 0.16084 Table A2. Log returns of 6 Indexes and 1 ETF are lsited below. Yellow marked outliers which are changed to the means of the returns, grey marked outliers who are left in the data without excluding them: Date TLT SMI BE500 MXASJ DJGSP ASX SPGSAG 2006-05-31 -0.0972 2007-11-30 -0.10661 2008-01-31 -0.12294 -0.16838 -0.11582 2008-03-31 -0.10516 -0.18125 2008-06-30 -0.10539 -0.1412 2008-09-30 -0.12517 -0.14375 -0.15977 2008-10-31 -0.13691 -0.17655 -0.38736 -0.22394 2008-11-28 0.131597 -0.09932 2009-02-27 -0.11525 -0.10339 2009-04-30 0.121734 0.154203 2011-08-31 -0.11562 -0.10909 2011-09-30 0.190045 -0.14802 2012-05-31 0.151617 2012-07-31 0.18562 2013-04-30 -0.21996 2015-01-30 0.163099 93 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 3 Returns and correlations of Indexes and ETF: 94 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 4 Figure below presents Markowitz efficient frontier with efficient portfolios with same returns as ASX and MXASJ Indexes as same as Global minimum variance portoflio and equally weighted portoflio Table A3. A list of Markowitz efficient portfolios and their weights: Portfolio 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Returns 0.01110 0.01080 0.01049 0.01019 0.00989 0.00959 0.00928 0.00898 0.00868 0.00838 0.00807 0.00777 0.00747 0.00717 0.00687 0.00656 0.00626 0.00596 0.00566 0.00535 St.dev. 0.0418 0.0402 0.0385 0.0370 0.0354 0.0339 0.0324 0.0310 0.0296 0.0283 0.0270 0.0259 0.0248 0.0238 0.0230 0.0222 0.0216 0.0212 0.0210 0.0209 TLT 0.0084 0.0260 0.0435 0.0611 0.0786 0.0961 0.1137 0.1312 0.1488 0.1663 0.1838 0.2014 0.2189 0.2365 0.2540 0.2715 0.2891 0.3066 0.3242 0.3417 SMI 0.2743 0.2681 0.2619 0.2557 0.2495 0.2433 0.2371 0.2309 0.2247 0.2185 0.2123 0.2061 0.2000 0.1938 0.1876 0.1814 0.1752 0.1690 0.1628 0.1566 BE500 0.3288 0.3388 0.3487 0.3587 0.3686 0.3786 0.3885 0.3985 0.4084 0.4184 0.4283 0.4383 0.4482 0.4582 0.4681 0.4781 0.4881 0.4980 0.5080 0.5179 Weights MXASJ 0.9671 0.9162 0.8653 0.8144 0.7635 0.7127 0.6618 0.6109 0.5600 0.5091 0.4583 0.4074 0.3565 0.3056 0.2547 0.2039 0.1530 0.1021 0.0512 0.0004 DJGSP -0.1302 -0.1219 -0.1136 -0.1053 -0.0970 -0.0887 -0.0805 -0.0722 -0.0639 -0.0556 -0.0473 -0.0390 -0.0308 -0.0225 -0.0142 -0.0059 0.0024 0.0107 0.0189 0.0272 ASX -0.5124 -0.4890 -0.4657 -0.4424 -0.4190 -0.3957 -0.3723 -0.3490 -0.3257 -0.3023 -0.2790 -0.2556 -0.2323 -0.2090 -0.1856 -0.1623 -0.1389 -0.1156 -0.0923 -0.0689 SPGSAG 0.0639 0.0619 0.0599 0.0578 0.0558 0.0537 0.0517 0.0497 0.0476 0.0456 0.0435 0.0415 0.0394 0.0374 0.0354 0.0333 0.0313 0.0292 0.0272 0.0252 95 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 5 To noisy rolling parameters with 12 months rolling windows: 96 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 6 Time varying means and standard deviations of 6 Indexes with 24 months rolling windows: TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 7 (1st page) Figures present time-varying correlations between portfolio inputs. Names of each correlation in the figures a made from the first letters of the Indexes and ETF: t – TLT ETF; sm – SMI Index; b – BE500 Index; d – DJGSP Index; m – MXASJ Index a – ASX Index sp – SPGSAG Fr example: “t.a” presents correlation between TLT ETF and ASX Index; “sm.d” presents correlation between SMI and DJGSP Indexes, and etc. 97 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 7 (2nd page) 98 99 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 8 (1st page) Figure of rolling Global minimum variance portfolio weights using 24 months rolling windows: Table A4. Rolling Global minimum variance portfolio returns, standard deviations and weights: Weights Date Returns St.dev. 2006-12-31 0.0120478 0.01515 TLT 0.1711 SMI 0.461668 BE500 0.69349 MXASJ -0.475767 DJGSP -0.0150151 ASX 0.0117904 SPGSAG 0.152744 2007-01-31 0.0108659 0.01514 0.1308 0.468485 0.66025 -0.541009 -0.0166503 0.0427144 0.255441 2007-03-03 0.0105182 0.01547 0.1317 0.448386 0.67222 -0.532258 -0.0346824 0.1155160 0.199071 2007-03-31 0.0121109 0.01513 0.1041 0.365429 0.82438 -0.554162 -0.0083491 0.0638039 0.204776 2007-05-01 0.0093391 0.01455 0.1728 0.308415 0.76704 -0.451417 0.0055175 0.0194744 0.178176 2007-05-31 0.0066717 0.01594 0.2307 0.190400 0.70001 -0.360385 0.0001592 0.0750258 0.164048 2007-07-01 0.0023791 0.01752 0.4114 0.116516 0.51955 -0.167330 0.0505816 -0.0010130 0.070301 2007-07-31 0.0007655 0.01736 0.4649 0.038695 0.59777 -0.206900 0.0757134 0.0001813 0.029633 2007-08-31 0.0007471 0.01688 0.4119 0.135973 0.42903 -0.203960 0.0873882 0.0803204 0.059306 2007-10-01 0.0025417 0.01626 0.4259 -0.077366 0.59867 -0.185567 0.0842929 0.0998705 0.054224 2007-10-31 0.0010075 0.01573 0.4163 0.138324 0.36000 -0.134519 0.0834201 0.0811014 0.055369 2007-12-01 -0.0005486 0.01444 0.4183 -0.014171 0.58239 -0.137698 0.1256026 -0.0472705 0.072832 2007-12-31 -0.0012391 0.01393 0.3544 0.115948 0.53191 -0.132841 0.1524677 -0.1025781 0.080684 2008-01-31 -0.0011051 0.01379 0.3457 0.095993 0.54945 -0.114749 0.1264962 -0.0971709 0.094278 2008-03-02 -0.0044712 0.01553 0.3834 0.104991 0.65072 -0.142757 0.1540043 -0.1941298 0.043755 2008-03-31 -0.0026825 0.01469 0.4834 0.022357 0.60427 -0.114709 0.1079799 -0.1494979 0.046150 2008-05-01 -0.0019136 0.01297 0.4158 0.086850 0.59897 -0.085576 0.1482068 -0.1949505 0.030672 2008-05-31 -0.0017483 0.01343 0.4875 0.169249 0.32177 -0.008895 0.0991796 -0.0740812 0.005262 2008-07-01 -0.0040468 0.01294 0.5118 0.225892 0.25188 -0.016322 0.1014372 -0.0644854 -0.010172 2008-07-31 -0.0039852 0.01400 0.3256 0.326029 0.42810 0.040056 0.1840457 -0.2755160 -0.028337 2008-08-31 -0.0039934 0.01390 0.3494 0.277464 0.45081 0.057858 0.1536286 -0.2634381 -0.025680 2008-10-01 -0.0029762 0.01468 0.2755 0.290957 0.50709 0.053191 0.1267380 -0.2621026 0.008635 100 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 8 (2nd page) Weights Date Returns St.dev. 2008-10-31 -0.0024466 0.01487 TLT 0.2430 SMI 0.287956 BE500 0.54769 MXASJ 0.059544 DJGSP 0.0948984 ASX -0.2554698 SPGSAG 0.022383 2008-12-01 -0.0026390 0.01473 0.2336 0.272909 0.58034 0.066850 0.0860241 -0.2638818 0.024168 2008-12-31 -0.0049572 0.01446 0.1887 0.323885 0.60871 0.096119 0.1076381 -0.3453555 0.020267 2009-01-31 -0.0073890 0.01578 0.2037 0.521344 0.34931 0.054252 0.1303796 -0.2687182 0.009730 2009-03-03 -0.0067632 0.01607 0.2042 0.519476 0.37787 0.059729 0.1302438 -0.3023817 0.010897 2009-03-31 -0.0067276 0.01637 0.2239 0.556024 0.32114 0.006702 0.1533851 -0.2628373 0.001715 2009-05-01 -0.0053620 0.01678 0.2848 0.502125 0.26423 0.026340 0.1087369 -0.2121517 0.025930 2009-05-31 -0.0061466 0.01673 0.2887 0.521824 0.25694 -0.018953 0.1091589 -0.1629728 0.005259 2009-07-01 -0.0024020 0.02050 0.3510 0.423624 0.20812 -0.005292 0.0829923 -0.1113734 0.050959 2009-07-31 -0.0015896 0.02087 0.3690 0.414662 0.16853 0.089144 0.0694335 -0.1796993 0.068978 2009-08-31 -0.0010867 0.02081 0.3732 0.422798 0.18969 0.094970 0.0656217 -0.2110775 0.064810 2009-10-01 -0.0020463 0.02113 0.3710 0.436076 0.16553 0.099523 0.0595635 -0.2063233 0.074672 2009-10-31 -0.0022218 0.02119 0.3799 0.448162 0.16076 0.101422 0.0461056 -0.2178261 0.081519 2009-12-01 -0.0006028 0.02158 0.3850 0.444161 0.16992 0.076052 0.0622896 -0.1999527 0.062577 2009-12-31 0.0007828 0.02104 0.3651 0.508987 0.15329 0.058333 0.0748691 -0.2063463 0.045788 2010-01-31 0.0025218 0.02176 0.3721 0.467418 0.22941 0.058790 0.0723516 -0.2260186 0.025955 2010-03-03 0.0049588 0.01968 0.3917 0.335897 0.02924 0.119343 0.0421206 -0.0491026 0.130835 2010-03-31 0.0053403 0.01987 0.3820 0.376485 0.05761 0.099930 0.0474725 -0.0768125 0.113304 2010-05-01 0.0075932 0.01999 0.3643 0.359248 0.06456 0.100655 0.0406218 -0.0483018 0.118864 2010-05-31 0.0140552 0.02018 0.3831 0.331202 -0.09863 0.234880 0.0390653 0.0431212 0.067265 2010-07-01 0.0114652 0.02017 0.3641 0.170994 -0.02398 0.095496 0.0274505 0.1982830 0.167679 2010-07-31 0.0112962 0.02163 0.3695 0.113496 0.06987 0.073686 0.0088555 0.1962620 0.168311 2010-08-31 0.0108185 0.02208 0.3310 0.129698 0.06443 0.042728 0.0195615 0.2165134 0.196055 2010-10-01 0.0106841 0.02210 0.3238 0.135271 0.06336 0.035140 0.0198146 0.2161779 0.206452 2010-10-31 0.0115727 0.02235 0.3030 0.176492 0.08472 0.026169 0.0084751 0.1904236 0.210751 2010-12-01 0.0130578 0.02255 0.3244 0.137921 0.08192 0.064620 -0.0019122 0.1940653 0.199033 2010-12-31 0.0107582 0.02141 0.3557 0.123959 0.22580 -0.072529 -0.0242748 0.2162145 0.175122 2011-01-31 0.0114906 0.01705 0.3782 -0.127056 0.51916 -0.048716 0.0073320 0.1240994 0.147012 2011-03-03 0.0098955 0.01919 0.4016 -0.215071 0.46358 0.033220 -0.0007315 0.1747365 0.142644 2011-03-31 0.0075744 0.01707 0.3578 -0.192602 0.53207 0.162854 -0.0516725 0.0643203 0.127241 2011-05-01 0.0084346 0.01619 0.4097 -0.212707 0.49603 0.125582 -0.0957120 0.1400901 0.137042 2011-05-31 0.0055763 0.01871 0.4207 -0.287002 0.67075 0.112291 -0.0535899 0.0506123 0.086264 2011-07-01 0.0043022 0.01716 0.3615 -0.203694 0.68522 0.182890 -0.0397448 -0.0400792 0.053953 2011-07-31 0.0049689 0.01743 0.3460 -0.126035 0.58037 0.197529 -0.0672570 0.0264728 0.042952 2011-08-31 0.0016572 0.01855 0.3663 -0.138849 0.66744 0.188992 -0.0555782 -0.0115942 -0.016757 2011-10-01 0.0040286 0.01753 0.3626 -0.148389 0.75093 0.208903 -0.0269782 -0.1224572 -0.024650 2011-10-31 0.0041306 0.01747 0.3775 -0.148008 0.73495 0.198101 -0.0432777 -0.0915110 -0.027721 101 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 8 (3rd page) Weights Date Returns St.dev. 2011-12-01 0.0040159 0.01724 TLT 0.2988 SMI -0.151754 BE500 0.84810 MXASJ 0.282695 DJGSP 0.0185295 ASX -0.2618920 SPGSAG -0.034440 2011-12-31 0.0056279 0.01822 0.3009 -0.108054 0.84529 0.234044 0.0125370 -0.2629357 -0.021800 2012-01-31 0.0066965 0.01819 0.3347 -0.147721 0.83729 0.212459 -0.0055354 -0.2051919 -0.026034 2012-03-02 0.0026168 0.01611 0.3306 -0.093887 0.88725 0.187651 -0.0111906 -0.1853200 -0.115056 2012-03-31 0.0022576 0.01616 0.3493 -0.107127 0.86387 0.176491 -0.0253183 -0.1400585 -0.117108 2012-05-01 0.0020439 0.01615 0.3523 -0.114825 0.86692 0.187878 -0.0232398 -0.1541345 -0.114893 2012-05-31 0.0018752 0.01611 0.3513 -0.106832 0.85959 0.208660 -0.0181915 -0.1805255 -0.113984 2012-07-01 0.0020637 0.01628 0.3327 -0.084766 0.89919 0.245026 0.0139231 -0.2723599 -0.133711 2012-07-31 0.0018249 0.01614 0.3320 -0.059707 0.85845 0.249736 0.0212619 -0.2769584 -0.124827 2012-08-31 0.0026476 0.01597 0.3421 -0.029998 0.83894 0.226326 0.0181517 -0.2712612 -0.124260 2012-10-01 0.0036207 0.01482 0.3813 -0.004762 0.83094 0.170474 -0.0053309 -0.1906956 -0.181892 2012-10-31 0.0046619 0.01540 0.3790 -0.026335 0.83952 0.150753 0.0080544 -0.1746382 -0.176309 2012-12-01 0.0063039 0.01454 0.4261 -0.096428 0.90573 0.170631 -0.0135040 -0.1823394 -0.210181 2012-12-31 0.0063676 0.01457 0.4170 0.023911 0.78643 0.097792 0.0044636 -0.0968142 -0.232809 2013-01-31 0.0065019 0.01508 0.3954 0.027223 0.81164 0.121154 0.0232010 -0.1659972 -0.212609 2013-03-03 0.0094990 0.01288 0.3986 0.139368 0.79941 -0.030301 0.0251113 -0.1057346 -0.226472 2013-03-31 0.0089825 0.01210 0.3844 0.229390 0.76112 -0.094953 0.0516605 -0.0846598 -0.246968 2013-05-01 0.0079462 0.01191 0.3745 0.258992 0.75175 -0.043688 0.0529200 -0.1638272 -0.230635 2013-05-31 0.0085961 0.01241 0.3668 0.390422 0.69413 -0.068030 0.0431779 -0.2056539 -0.220809 2013-07-01 0.0108356 0.01276 0.3758 0.361232 0.71972 -0.045262 0.0215871 -0.2192471 -0.213829 2013-07-31 0.0093942 0.01378 0.3504 0.459517 0.59100 -0.038180 0.0489991 -0.1938990 -0.217882 2013-08-31 0.0098108 0.01426 0.3346 0.475264 0.53284 -0.019411 0.0605331 -0.1984320 -0.185382 2013-10-01 0.0109861 0.01470 0.3353 0.522359 0.50720 -0.023341 0.0572174 -0.2272323 -0.171486 2013-10-31 0.0101642 0.01470 0.3486 0.495439 0.54687 -0.039183 0.0579455 -0.2420104 -0.167615 2013-12-01 0.0092243 0.01342 0.3365 0.582537 0.43308 0.023101 0.0431640 -0.2752421 -0.143139 2013-12-31 0.0102234 0.01383 0.3105 0.575507 0.41505 0.081640 0.0332835 -0.2709773 -0.145046 2014-01-31 0.0105411 0.01389 0.3135 0.574658 0.41359 0.095880 0.0299881 -0.2761882 -0.151384 2014-03-03 0.0075964 0.01516 0.3050 0.423829 0.40162 0.176400 0.0255391 -0.2259303 -0.106494 2014-03-31 0.0087037 0.01477 0.2908 0.484529 0.40140 0.141184 0.0318556 -0.2491989 -0.100549 2014-05-01 0.0102548 0.01577 0.2799 0.528002 0.42476 0.043387 0.0400710 -0.2488170 -0.067351 2014-05-31 0.0097347 0.01575 0.2726 0.524422 0.42810 0.041010 0.0453240 -0.2443002 -0.067123 2014-07-01 0.0052391 0.01537 0.2599 0.425117 0.32217 0.144277 0.0367914 -0.1861933 -0.002102 2014-07-31 0.0077786 0.01656 0.2261 0.379325 0.37197 0.182651 0.0199662 -0.1703172 -0.009648 2014-08-31 0.0077465 0.01670 0.2159 0.343120 0.37980 0.173192 0.0237790 -0.1361214 0.000375 2014-10-01 0.0070898 0.01674 0.2145 0.309628 0.35454 0.218456 0.0096797 -0.1372190 0.030446 2014-10-31 0.0084652 0.01761 0.1891 0.306895 0.34469 0.224851 -0.0061966 -0.0710291 0.011738 2014-12-01 0.0082447 0.01751 0.2144 0.294754 0.34342 0.201426 -0.0034616 -0.0742552 0.023726 2014-12-31 0.0100183 0.01924 0.3663 -0.032685 0.68717 0.030814 -0.0439891 -0.0372837 0.029646 102 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 9 (1st page) Figure presents weights of rolling efficient portfolio with target monthly returns of 0.8% (using 24 months rolling windows): Table A5. Rolling efficient portfolio with target monthly returns (0.8%) st.dev and weights: Weights Date Return St.Dev TLT SMI BE500 MXASJ DJGSP ASX SPGSAG 2006-12-31 0.008 0.01696 0.371487 0.22557 0.765021 -0.468185 0.0225137 0.005972 0.077625 2007-01-31 0.008 0.01645 0.319554 0.27038 0.764739 -0.503175 0.0284658 -0.034730 0.154768 2007-03-03 0.008 0.01625 0.270859 0.31093 0.758505 -0.524511 0.0079849 0.016844 0.159383 2007-03-31 0.008 0.01664 0.312232 0.25946 0.788517 -0.511637 0.0303810 -0.033195 0.154239 2007-05-01 0.008 0.01467 0.234442 0.28737 0.741322 -0.429447 0.0173258 -0.009537 0.158529 2007-05-31 0.008 0.01602 0.172555 0.21521 0.725414 -0.377973 -0.0128104 0.091753 0.185847 2007-07-01 0.008 0.01980 0.133790 0.27096 0.491052 -0.057389 -0.0032557 0.025806 0.139032 2007-07-31 0.008 0.01907 0.177936 0.31105 0.392343 -0.046156 -0.0158306 0.017720 0.162943 2007-08-31 0.008 0.01866 0.135863 0.39007 0.271299 -0.041299 -0.0056816 0.066824 0.182919 2007-10-01 0.008 0.01736 0.260484 -0.07812 0.613440 -0.029331 0.0220128 0.107685 0.103827 2007-10-31 0.008 0.01865 0.132463 0.22723 0.365674 0.051088 -0.0244002 0.058699 0.189250 2007-12-01 0.008 0.01894 0.090294 0.02931 0.629274 0.071488 -0.0096795 -0.051707 0.241020 2007-12-31 0.008 0.01903 0.103152 -0.09289 0.759293 0.050964 -0.0042633 -0.059844 0.243588 2008-01-31 0.008 0.01878 0.104403 -0.12882 0.792025 0.069450 -0.0382172 -0.051412 0.252570 2008-03-02 0.008 0.02703 0.122390 -0.37289 1.091511 0.229541 -0.2192496 -0.098957 0.247657 2008-03-31 0.008 0.02174 0.494596 -0.40855 0.793165 0.160275 -0.1975584 -0.017122 0.175191 2008-05-01 0.008 0.02175 0.349354 -0.36147 0.838305 0.091466 -0.1195111 0.047515 0.154337 2008-05-31 0.008 0.02106 0.334904 -0.33336 0.878163 0.067486 -0.0836374 -0.007053 0.143499 2008-07-01 0.008 0.02354 0.303655 -0.36118 0.999263 0.215732 -0.1318387 -0.175266 0.149640 2008-07-31 0.008 0.02333 0.295024 -0.30432 0.955411 0.201092 -0.1100423 -0.180406 0.143240 103 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 9 (2nd page) Weights Date Return St.Dev TLT SMI BE500 MXASJ DJGSP ASX SPGSAG 2008-08-31 0.008 0.02328 0.311170 -0.34013 0.973889 0.216235 -0.1346587 -0.172976 0.146469 2008-10-01 0.008 0.02319 0.353263 -0.35636 0.965700 0.242558 -0.1192183 -0.210549 0.124603 2008-10-31 0.008 0.02289 0.446757 -0.35547 0.861580 0.187289 -0.0586951 -0.167295 0.085837 2008-12-01 0.008 0.02396 0.487611 -0.35031 0.784092 0.198001 -0.0206636 -0.173534 0.074805 2008-12-31 0.008 0.02522 0.403294 -0.34321 0.893599 0.383124 0.0242698 -0.452470 0.091393 2009-01-31 0.008 0.03426 0.669987 -0.10903 0.015605 0.425690 0.0291227 -0.153400 0.122025 2009-03-03 0.008 0.02887 0.589873 0.05784 -0.037272 0.391251 0.0302079 -0.187714 0.155818 2009-03-31 0.008 0.02768 0.542545 0.04632 0.056812 0.435362 0.0074346 -0.248585 0.160109 2009-05-01 0.008 0.02951 0.453039 0.04999 0.146631 0.514646 0.0966215 -0.399771 0.138840 2009-05-31 0.008 0.03419 0.566129 -0.09491 0.006998 0.533551 0.0590789 -0.183991 0.113141 2009-07-01 0.008 0.03532 0.555660 -0.14497 0.038833 0.591416 0.0756348 -0.203569 0.086993 2009-07-31 0.008 0.03767 0.506301 -0.27254 0.240865 0.200989 0.1700322 0.186538 -0.032186 2009-08-31 0.008 0.03609 0.514327 -0.20294 0.369494 0.183895 0.1691297 0.066274 -0.100175 2009-10-01 0.008 0.05021 0.655527 -0.63467 0.131686 0.268887 0.1266367 0.470118 -0.018187 2009-10-31 0.008 0.03554 0.408705 -0.14863 0.216084 0.190552 0.2100529 0.190321 -0.067082 2009-12-01 0.008 0.03337 0.351508 -0.01680 0.250647 0.151911 0.2073845 0.165023 -0.109677 2009-12-31 0.008 0.03498 0.393195 0.13084 0.084842 0.195321 0.2071020 0.184603 -0.195908 2010-01-31 0.008 0.02741 0.363379 0.40817 0.263905 0.037523 0.2027134 -0.064385 -0.211309 2010-03-03 0.008 0.02105 0.391254 0.25837 -0.029318 0.132308 0.0836500 0.087504 0.076232 2010-03-31 0.008 0.02094 0.388831 0.29975 -0.014916 0.120359 0.0844362 0.047961 0.073577 2010-05-01 0.008 0.02002 0.368833 0.34835 0.049815 0.114091 0.0452829 -0.039566 0.113194 2010-05-31 0.008 0.02154 0.349816 0.30566 0.155177 0.027700 0.0277987 -0.002911 0.136755 2010-07-01 0.008 0.02102 0.343348 0.23803 0.166080 -0.030232 0.0306807 0.077007 0.175086 2010-07-31 0.008 0.02220 0.359020 0.15789 0.233248 -0.032247 0.0026822 0.100128 0.179276 2010-08-31 0.008 0.02250 0.331767 0.16161 0.210043 -0.041943 0.0112928 0.128712 0.198523 2010-10-01 0.008 0.02247 0.339706 0.15145 0.208682 -0.032114 0.0113503 0.133022 0.187898 2010-10-31 0.008 0.02279 0.338691 0.14317 0.261622 -0.051934 0.0149895 0.111017 0.182450 2010-12-01 0.008 0.02327 0.374000 0.08496 0.313952 -0.032028 0.0151903 0.088731 0.155198 2010-12-31 0.008 0.02174 0.390644 0.08608 0.360940 -0.132837 -0.0051773 0.159455 0.140892 2011-01-31 0.008 0.01798 0.436224 -0.19135 0.709101 -0.092892 0.0245649 0.015219 0.099136 2011-03-03 0.008 0.01947 0.433534 -0.24364 0.581563 -0.001933 0.0103361 0.105352 0.114786 2011-03-31 0.008 0.01709 0.347838 -0.17905 0.502635 0.168417 -0.0529283 0.080416 0.132668 2011-05-01 0.008 0.01622 0.418416 -0.23187 0.533868 0.119158 -0.0908408 0.121202 0.130068 2011-05-31 0.008 0.01978 0.335813 -0.15473 0.459637 0.166752 -0.0590971 0.129948 0.121680 2011-07-01 0.008 0.01971 0.282234 -0.04704 0.295974 0.232927 -0.0725724 0.173648 0.134825 2011-07-31 0.008 0.01991 0.281361 -0.12684 0.371483 0.240883 -0.0361381 0.100948 0.168303 2011-08-31 0.008 0.03154 0.088177 0.12991 0.665901 0.429474 0.1170892 -0.722284 0.291728 2011-10-01 0.008 0.02510 0.085528 -0.10663 0.897923 0.408745 0.2010669 -0.619365 0.132733 104 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 9 (3rd page) Weights Date Return St.Dev TLT SMI BE500 MXASJ DJGSP ASX SPGSAG 2011-10-31 0.008 0.03046 -0.007076 -0.09365 1.080363 0.332426 0.3873425 -0.873222 0.173820 2011-12-01 0.008 0.03184 0.390069 -0.05248 0.462539 -0.099756 0.0617647 -0.005848 0.243713 2011-12-31 0.008 0.02113 0.135936 -0.12354 0.911257 0.352552 0.1917672 -0.541553 0.073580 2012-01-31 0.008 0.01904 0.233454 -0.18417 0.974978 0.340356 0.0736557 -0.459947 0.021673 2012-03-02 0.008 0.02365 0.081752 -0.07503 1.100684 0.470683 0.1164138 -0.842206 0.147702 2012-03-31 0.008 0.03086 0.292365 -0.07521 0.746711 0.280954 -0.1944671 -0.338391 0.288036 2012-05-01 0.008 0.03110 0.495005 -0.02102 0.451074 -0.117214 -0.3508996 0.360472 0.182580 2012-05-31 0.008 0.02794 0.469190 -0.12052 0.623866 -0.199303 -0.3021350 0.441289 0.087617 2012-07-01 0.008 0.02686 0.433488 -0.03966 0.626306 -0.105529 -0.2198232 0.262523 0.042696 2012-07-31 0.008 0.03011 0.392916 -0.19455 1.016739 -0.218943 -0.2547022 0.234024 0.024517 2012-08-31 0.008 0.03801 0.158513 0.46999 0.847954 -0.062645 -0.0854649 -0.561907 0.233563 2012-10-01 0.008 0.02919 0.486288 0.53830 0.752909 -0.489896 -0.2366927 0.274662 -0.325571 2012-10-31 0.008 0.01869 0.437101 0.04200 1.019184 -0.055667 -0.1357597 -0.053334 -0.253523 2012-12-01 0.008 0.01511 0.473695 -0.12571 1.010843 0.118810 -0.0748845 -0.140610 -0.262146 2012-12-31 0.008 0.01509 0.459103 0.03330 0.845860 0.027655 -0.0462868 -0.031002 -0.288633 2013-01-31 0.008 0.01542 0.430646 0.01010 0.874992 0.099628 -0.0265427 -0.137215 -0.251613 2013-03-03 0.008 0.01322 0.372552 0.10169 0.772102 0.037639 0.0562536 -0.151512 -0.188725 2013-03-31 0.008 0.01226 0.369274 0.20524 0.748922 -0.055246 0.0690890 -0.111877 -0.225404 2013-05-01 0.008 0.01191 0.375151 0.25890 0.754748 -0.044299 0.0515753 -0.164563 -0.231511 2013-05-31 0.008 0.01248 0.361533 0.38556 0.669261 -0.061734 0.0563860 -0.196348 -0.214654 2013-07-01 0.008 0.01359 0.356765 0.37902 0.566478 0.000770 0.0698512 -0.197810 -0.175074 2013-07-31 0.008 0.01397 0.356191 0.40462 0.585493 -0.035480 0.0645923 -0.186467 -0.188954 2013-08-31 0.008 0.01457 0.345285 0.40277 0.531638 -0.018148 0.0805825 -0.191061 -0.151067 2013-10-01 0.008 0.01548 0.339141 0.39565 0.502079 -0.015222 0.0889440 -0.200019 -0.110578 2013-10-31 0.008 0.01499 0.365178 0.41833 0.568395 -0.073604 0.0819972 -0.220050 -0.140249 2013-12-01 0.008 0.01352 0.344755 0.54158 0.434745 0.010580 0.0560048 -0.263457 -0.124208 2013-12-31 0.008 0.01412 0.304583 0.48732 0.393854 0.122175 0.0527640 -0.242934 -0.117765 2014-01-31 0.008 0.01428 0.306401 0.45844 0.390390 0.169930 0.0479996 -0.249761 -0.123396 2014-03-03 0.008 0.01518 0.305413 0.44632 0.403379 0.163526 0.0208166 -0.229777 -0.109673 2014-03-31 0.008 0.01481 0.291888 0.44476 0.395241 0.167863 0.0377390 -0.240040 -0.097454 2014-05-01 0.008 0.01601 0.314217 0.44794 0.399809 0.080086 0.0594075 -0.243982 -0.057478 2014-05-31 0.008 0.01593 0.301795 0.44283 0.423484 0.063117 0.0559840 -0.240183 -0.047021 2014-07-01 0.008 0.01570 0.238280 0.50243 0.350741 0.134246 0.0282223 -0.193422 -0.060496 2014-07-31 0.008 0.01656 0.224534 0.38787 0.371444 0.182661 0.0191587 -0.171217 -0.014447 2014-08-31 0.008 0.01670 0.214727 0.35139 0.380590 0.172047 0.0227226 -0.136575 -0.004903 2014-10-01 2014-10-31 0.008 0.008 0.01679 0.01762 0.208417 0.191956 0.34013 0.28836 0.349144 0.348508 0.224496 0.220818 0.0007248 -0.0015269 -0.137735 -0.068120 0.014820 0.020007 2014-12-01 2014-12-31 0.008 0.008 0.01751 0.01956 0.213247 0.367576 0.28735 -0.11865 0.342696 0.713190 0.200680 -0.007399 -0.0001419 -0.0203395 -0.072434 -0.010529 0.028606 0.076154 105 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 10 Table A6. A list of EWMA efficient portfolios and their weights: Weights Portfolio Returns St.dev. TLT SMI BE500 MXASJ DJGSP ASX SPGSAG 1 0.01110 0.03732 -0.00206 0.29992 0.32940 0.96784 -0.11029 -0.53488 0.05007 2 0.01080 0.03585 0.01564 0.29322 0.33931 0.91712 -0.10244 -0.51116 0.04833 3 0.01050 0.03440 0.03333 0.28652 0.34921 0.86641 -0.09460 -0.48744 0.04658 4 0.01019 0.03297 0.05103 0.27982 0.35912 0.81569 -0.08675 -0.46372 0.04483 5 0.00989 0.03158 0.06872 0.27312 0.36902 0.76498 -0.07891 -0.44000 0.04308 6 0.00959 0.03021 0.08641 0.26642 0.37893 0.71426 -0.07106 -0.41628 0.04133 7 0.00929 0.02888 0.10411 0.25972 0.38884 0.66355 -0.06322 -0.39256 0.03958 8 0.00899 0.02759 0.12180 0.25302 0.39874 0.61283 -0.05537 -0.36884 0.03783 9 0.00869 0.02635 0.13950 0.24632 0.40865 0.56211 -0.04753 -0.34512 0.03608 10 0.00839 0.02516 0.15719 0.23962 0.41855 0.51140 -0.03968 -0.32140 0.03433 11 0.00809 0.02404 0.17488 0.23292 0.42846 0.46068 -0.03184 -0.29768 0.03258 12 0.00778 0.02299 0.19258 0.22622 0.43836 0.40997 -0.02400 -0.27396 0.03083 13 0.00748 0.02202 0.21027 0.21952 0.44827 0.35925 -0.01615 -0.25024 0.02908 14 0.00718 0.02114 0.22797 0.21282 0.45817 0.30854 -0.00831 -0.22652 0.02734 15 0.00688 0.02037 0.24566 0.20612 0.46808 0.25782 -0.00046 -0.20280 0.02559 16 0.00658 0.01971 0.26335 0.19942 0.47799 0.20711 0.00738 -0.17909 0.02384 17 0.00628 0.01919 0.28105 0.19272 0.48789 0.15639 0.01523 -0.15537 0.02209 18 0.00598 0.01881 0.29874 0.18602 0.49780 0.10568 0.02307 -0.13165 0.02034 19 0.00568 0.01857 0.31643 0.17932 0.50770 0.05496 0.03092 -0.10793 0.01859 20 0.00537 0.01849 0.33413 0.17262 0.51761 0.00425 0.03876 -0.08421 0.01684 106 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 11 Lambda testing for EWMA model. Figure presents results with different Lambdas: Light blue line presents EWMA efficient frontier if Lambda = 0.91; Black line presents EWMA efficient frontier if Lambda = 0.8; Orange line presents EWMA efficient frontier if Lambda = 0.7 107 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 12 (1st page) Table A7. St.dev. and weights of rolling EWMA efficient portfolio with target returns of 0.8%: Date Returns St.Dev SMI 0.0660641 BE500 0.5703357 Weights MXASJ -0.042611 DJGSP 0.027662 ASX -0.026352 SPGSAG -0.006415 2006-12-31 0.008 0.0189869 TLT 0.4113169 2007-01-31 0.008 0.0186642 0.3797527 0.1133551 0.5557297 -0.021213 0.035133 -0.066481 0.003723 2007-03-03 0.008 0.0185973 0.3687005 0.1295498 0.5418081 -0.011407 0.0356595 -0.081057 0.0167459 2007-03-31 0.008 0.0187939 0.3936793 0.1103482 0.5350402 -0.006321 0.0273084 -0.083572 0.0235166 2007-05-01 0.008 0.0185375 0.3112025 0.1902131 0.5128955 0.0110835 0.0422318 -0.08427 0.016644 2007-05-31 2007-07-01 0.008 0.008 0.0192373 0.0202483 0.2388186 0.2001091 0.2356358 0.2209231 0.4985613 0.4601952 0.0625905 0.1496694 0.0420469 0.0574039 -0.105295 -0.124746 0.0276419 0.0364456 2007-07-31 0.008 0.0198142 0.2285272 0.2127645 0.4451707 0.1660336 0.0344306 -0.154816 0.0678892 2007-08-31 2007-10-01 0.008 0.008 0.0209067 0.0196063 0.2044605 0.270092 0.1755056 0.0955013 0.4113476 0.4850331 0.2207713 0.1691922 0.0223303 0.031151 -0.137324 -0.101253 0.1029092 0.0502833 2007-10-31 0.008 0.0218605 0.2166537 0.022036 0.4899433 0.2652761 0.0007229 -0.113636 0.1190038 2007-12-01 0.008 0.0238364 0.2032478 -0.060377 0.4876481 0.327753 -0.022241 -0.097788 0.1617582 2007-12-31 0.008 0.0234653 0.2799563 -0.200347 0.5893346 0.304188 -0.042709 -0.086386 0.1559629 2008-01-31 2008-03-02 0.008 0.008 0.0243807 0.0278225 0.2388418 0.2570633 -0.150443 -0.293491 0.4818949 0.4933964 0.3203158 0.4453628 -0.012202 -0.071954 -0.069191 -0.073987 0.190783 0.24361 2008-03-31 0.008 0.0237575 0.3185838 -0.241787 0.5839511 0.3234459 -0.062369 -0.075613 0.1537885 2008-05-01 0.008 0.0229173 0.3030805 -0.207792 0.5558071 0.2698805 -0.030935 -0.012151 0.1221108 2008-05-31 2008-07-01 0.008 0.008 0.0228625 0.0242572 0.3098227 0.3114202 -0.271616 -0.33633 0.6936178 0.7300881 0.1608719 0.2785824 -0.011854 -0.038242 -0.035859 -0.114472 0.1550161 0.1689529 2008-07-31 0.008 0.0234862 0.3354954 -0.28384 0.6870004 0.2224943 -0.050705 -0.089542 0.1790968 2008-08-31 0.008 0.0235957 0.3848494 -0.406887 0.781648 0.1740411 -0.035945 -0.021047 0.1233396 2008-10-01 0.008 0.0232992 0.4285947 -0.421048 0.8213759 0.2111765 -0.037035 -0.073577 0.0705121 2008-10-31 0.008 0.0222061 0.4633969 -0.359491 0.7976773 0.1297133 0.0083972 -0.072777 0.0330836 2008-12-01 0.008 0.0224465 0.4798328 -0.353462 0.7680489 0.1252009 0.0344884 -0.084906 0.0307966 2008-12-31 0.008 0.0234364 0.4665581 -0.381228 0.7478301 0.2234787 0.0467477 -0.16283 0.0594437 2009-01-31 0.008 0.0273556 0.5304878 -0.48299 0.6564173 0.3653673 0.0755206 -0.215996 0.0711927 2009-03-03 0.008 0.0255305 0.4821905 -0.341405 0.5192941 0.3456747 0.0724327 -0.1701 0.0919131 2009-03-31 0.008 0.0236679 0.4297414 -0.300976 0.5853547 0.3536128 0.0354905 -0.188516 0.0852923 2009-05-01 0.008 0.0246654 0.3976907 -0.311952 0.5923868 0.3939649 0.0743312 -0.23262 0.0861987 2009-05-31 0.008 0.0266347 0.4553946 -0.431148 0.583534 0.441232 0.075106 -0.171195 0.0470767 2009-07-01 0.008 0.0263488 0.4258648 -0.449382 0.6463813 0.4301327 0.0754353 -0.139976 0.0115438 2009-07-31 0.008 0.0261433 0.4349889 -0.483839 0.7135649 0.2208222 0.1005475 0.0711619 -0.057246 2009-08-31 0.008 0.0242094 0.4519573 -0.378054 0.7600577 0.163829 0.0792005 0.031321 -0.108311 2009-10-01 0.008 0.0336127 0.5762963 -0.88991 0.8665757 0.2447298 0.0826631 0.2356554 -0.11601 2009-10-31 0.008 0.0261751 0.4021166 -0.440514 0.6903054 0.1883501 0.1244044 0.0971651 -0.061828 2009-12-01 0.008 0.0244562 0.3516912 -0.285703 0.6332308 0.1923213 0.0932729 0.1009509 -0.085764 2009-12-31 0.008 0.0264666 0.3879723 -0.14392 0.4432716 0.2329332 0.077812 0.1900704 -0.18814 108 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 12 (2nd page) Date Returns St.Dev SMI 0.1135962 BE500 0.3999186 Weights MXASJ 0.06245 DJGSP 0.0970699 2010-01-31 0.008 0.022801 TLT 0.3735307 ASX 0.12415 SPGSAG -0.170715 2010-03-03 0.008 0.0190699 0.3348317 0.1427016 0.4727309 0.032 2010-03-31 0.008 0.0194557 0.3466709 0.128722 0.4343807 0.0433905 0.0592747 -0.003497 -0.038042 0.077345 0.0139269 -0.044436 2010-05-01 0.008 0.018839 0.355765 0.1386921 0.4641257 0.0732557 0.0576687 -0.076878 -0.012629 2010-05-31 0.008 0.0185224 0.3380448 0.176028 0.492693 0.0275207 0.0418252 -0.084378 0.008266 2010-07-01 0.008 0.0185164 0.3368128 0.166245 0.4977908 0.0234039 0.0414055 -0.077146 0.0114885 2010-07-31 0.008 0.0185023 0.3349448 0.1725612 0.5029894 0.0152422 0.0414833 -0.081255 0.014034 2010-08-31 0.008 0.0186308 0.325042 0.1856919 0.4504951 0.0465262 0.0519915 -0.071893 0.0121464 2010-10-01 2010-10-31 0.008 0.0190067 0.2951892 0.2125701 0.3866014 0.0796937 0.0637585 -0.067238 0.0294247 0.008 0.0188051 0.2975545 0.2268695 0.4037254 0.0600732 0.0496504 -0.065443 0.0275702 2010-12-01 0.008 0.0186524 0.3028072 0.2167579 0.4399205 0.0333678 0.0435411 -0.065384 0.0289894 2010-12-31 0.008 0.018541 0.3160182 0.1948909 0.4804734 0.0151281 0.038591 -0.068894 0.0237919 2011-01-31 0.008 0.0187777 0.3835106 0.1415062 0.5743592 -0.009272 0.0373029 -0.126868 -0.000539 2011-03-03 0.008 0.0184944 0.3356161 0.1719755 0.5189756 0.0034775 0.0387792 -0.085123 0.0162989 2011-03-31 0.008 0.0185987 0.3029625 0.2083112 0.4752376 0.0014529 0.0457549 -0.058789 0.0250699 2011-05-01 0.008 0.0184991 0.3394717 0.1610555 0.5315391 0.0054176 0.0389246 -0.09119 0.0147813 2011-05-31 0.008 0.0186137 0.3016199 0.2288041 0.4581313 0.002643 0.0433846 -0.063142 0.0285593 2011-07-01 0.008 0.0198291 0.2534012 0.3506703 0.2584749 -0.00166 0.0510198 0.0030511 0.0850426 2011-07-31 0.008 0.0203886 0.2695456 0.2488505 0.2913086 0.0242749 0.0634092 -0.041842 0.1444532 2011-08-31 0.008 0.0248286 0.2454207 0.4007076 0.1696591 0.0444269 0.0782194 -0.216262 0.277828 2011-10-01 0.008 0.0232382 0.1920783 0.2106301 0.3669871 -0.005006 0.1098119 -0.096693 0.2221916 2011-10-31 0.008 0.0278017 0.2359658 0.1967769 0.402016 -0.211251 0.1766127 -0.097813 0.2976925 2011-12-01 0.008 0.0228018 0.3814247 0.2703374 0.3023048 -0.139161 0.0694104 -0.07751 0.1931934 2011-12-31 0.008 0.0210259 0.2849148 0.1351457 0.3854847 -0.006119 0.1170665 -0.049466 0.132974 2012-01-31 0.008 0.0218613 0.2665638 -0.128223 0.7602306 0.1243505 0.0449733 -0.217438 0.1495428 2012-03-02 0.008 0.0235759 0.2559902 0.0809894 0.5969585 0.0787075 -0.003715 -0.27325 0.2643198 2012-03-31 0.008 0.0262185 0.2520222 0.2250472 0.5078576 0.0039091 -0.093661 -0.216835 0.3216594 2012-05-01 0.008 0.0275646 0.3125442 0.2591382 0.3828047 -0.148999 -0.134599 0.0006272 0.3284831 2012-05-31 0.008 0.024863 0.2596568 0.0786873 0.6764522 -0.23153 -0.081773 0.0653113 0.2331949 2012-07-01 0.008 0.0208254 0.3143406 0.2413975 0.3785632 -0.101828 -0.028821 0.0364323 0.1599154 2012-07-31 0.008 0.0241618 0.2450218 0.1008844 0.6838698 -0.245033 -0.060937 0.0676898 0.2085042 2012-08-31 0.008 0.0271723 0.1677824 0.4200632 0.5808547 -0.289885 -0.044501 -0.112049 0.2777352 2012-10-01 0.008 0.0292634 0.2824874 0.698299 0.438362 -0.545113 -0.086793 0.1105569 0.1022 2012-10-31 0.008 0.022712 0.3084562 0.3187167 0.7140164 -0.1951 -0.076657 -0.108015 0.0385825 2012-12-01 0.008 0.0240817 0.4095905 0.1229078 0.9110637 -0.083254 -0.103075 -0.179574 -0.077659 2012-12-31 0.008 0.0226564 0.3503684 0.4103561 0.5814549 -0.162477 -0.051361 -0.037717 -0.090624 2013-01-31 0.008 0.0214839 0.3722008 0.3276203 0.4695145 0.0117626 -0.073913 -0.046565 -0.06062 2013-03-03 0.008 0.0188786 0.339881 0.2612556 0.4986021 -0.033213 0.0092848 -0.068749 -0.007061 2013-03-31 0.008 0.018917 0.3535362 0.2213194 0.5052309 -0.017697 -0.006705 -0.058154 0.0024705 109 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 12 (3rd page) Date Returns St.Dev SMI 0.2074449 BE500 0.635782 Weights MXASJ -0.000507 DJGSP -0.040673 ASX -0.139647 2013-05-01 0.008 0.0198735 TLT 0.3477862 SPGSAG -0.010186 2013-05-31 0.008 0.0196759 0.3382461 0.2577817 0.5762092 0.005827 -0.033716 -0.150571 0.006223 2013-07-01 2013-07-31 0.008 0.008 0.0192483 0.02009 0.3262578 0.2954056 0.2116767 0.3109595 0.6343502 0.58577 -0.022502 0.0064637 -0.005941 -0.022199 -0.137043 -0.146139 -0.006799 -0.03026 2013-08-31 0.008 0.0189678 0.304299 0.2472491 0.5416889 0.0029457 0.0036555 -0.099164 -0.000674 2013-10-01 0.008 0.0188126 0.3172004 0.2377462 0.5371138 0.0058239 0.0097153 -0.10724 -0.00036 2013-10-31 2013-12-01 0.008 0.008 0.0188701 0.0196485 0.3110069 0.2852784 0.2359263 0.2732108 0.5389971 0.603661 0.0281697 0.0254277 0.0043511 -0.007689 -0.125196 -0.163028 0.0067446 -0.016861 2013-12-31 0.008 0.0195519 0.3000922 0.2914291 0.5931887 -0.003871 -0.005567 -0.164046 -0.011227 2014-01-31 0.008 0.0193053 0.3019716 0.2929593 0.5751941 -0.029586 0.0044936 -0.139696 -0.005336 2014-03-03 0.008 0.0194474 0.3038188 0.2908203 0.5728388 -0.034981 -0.007914 -0.121297 -0.003285 2014-03-31 0.008 0.0190759 0.3010389 0.2665011 0.5788646 -0.031543 0.0108688 -0.129839 0.0041092 2014-05-01 0.008 0.0187569 0.3063085 0.2258822 0.5634926 -0.012184 0.0202605 -0.109572 0.0058116 2014-05-31 0.008 0.0191172 0.2973851 0.257596 0.5873897 -0.006828 0.0189226 -0.132262 -0.022204 2014-07-01 0.008 0.0200437 0.2912015 0.2727231 0.6389521 0.0156871 0.019761 -0.162137 -0.076188 2014-07-31 0.008 0.0193669 0.3101544 0.2583839 0.5752115 0.0227956 0.0159649 -0.130184 -0.052326 2014-08-31 0.008 0.0189134 0.3260301 0.2290279 0.5645986 0.0146876 0.0152901 -0.125548 -0.024087 2014-10-01 0.008 0.0190449 0.3310056 0.2275581 0.550398 0.0330631 -0.003365 -0.121424 -0.017235 2014-10-31 0.008 0.0187427 0.3304322 0.2119026 0.5508553 0.0262483 0.0146358 -0.128024 -0.006051 2014-12-01 0.008 0.0185993 0.3359738 0.1968502 0.5349113 0.0198217 0.0208939 -0.112831 0.0043804 2014-12-31 0.008 0.0186175 0.331875 0.1514287 0.4942123 -0.026125 0.0518496 -0.039037 0.0357962 110 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 13 Examples of DCC-GARCH(1,1) forecasted covariances: Red lines present forecasted covariances for 3 months ahead; grey lines present estimated covariances from historical data. 111 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 14 Table A8. A list of GARCH efficient portfolios and their weights: Weights Portfolio Returns St.dev. TLT SMI BE500 MXASJ DJGSP ASX SPGSAG 1 0.01110 0.04241 -0.02097 0.27271 0.29783 0.88036 -0.16607 -0.36160 0.09774 2 0.01075 0.04067 0.00536 0.26250 0.30472 0.83222 -0.15663 -0.34160 0.09343 3 0.01041 0.03895 0.03168 0.25229 0.31162 0.78408 -0.14720 -0.32160 0.08912 4 0.01006 0.03725 0.05801 0.24209 0.31851 0.73594 -0.13776 -0.30160 0.08481 5 0.00972 0.03558 0.08433 0.23188 0.32541 0.68780 -0.12832 -0.28160 0.08050 6 0.00938 0.03395 0.11066 0.22167 0.33230 0.63966 -0.11888 -0.26160 0.07619 7 0.00903 0.03236 0.13698 0.21146 0.33919 0.59152 -0.10945 -0.24160 0.07188 8 0.00869 0.03081 0.16331 0.20126 0.34609 0.54338 -0.10001 -0.22160 0.06757 9 0.00834 0.02931 0.18963 0.19105 0.35298 0.49524 -0.09057 -0.20160 0.06327 10 0.00800 0.02787 0.21596 0.18084 0.35987 0.44710 -0.08113 -0.18160 0.05896 11 0.00765 0.02650 0.24228 0.17063 0.36677 0.39896 -0.07169 -0.16160 0.05465 12 0.00731 0.02522 0.26861 0.16043 0.37366 0.35082 -0.06226 -0.14160 0.05034 13 0.00696 0.02402 0.29493 0.15022 0.38056 0.30268 -0.05282 -0.12160 0.04603 14 0.00662 0.02294 0.32126 0.14001 0.38745 0.25454 -0.04338 -0.10160 0.04172 15 0.00627 0.02198 0.34758 0.12981 0.39434 0.20640 -0.03394 -0.08160 0.03741 16 0.00593 0.02116 0.37391 0.11960 0.40124 0.15826 -0.02451 -0.06160 0.03310 17 0.00559 0.02051 0.40023 0.10939 0.40813 0.11012 -0.01507 -0.04160 0.02880 18 0.00524 0.02002 0.42656 0.09918 0.41503 0.06198 -0.00563 -0.02161 0.02449 19 0.00490 0.01973 0.45288 0.08898 0.42192 0.01384 0.00381 -0.00161 0.02018 20 0.00455 0.01963 0.47921 0.07877 0.42881 -0.03430 0.01325 0.01839 0.01587 112 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 15 (1st page) Figure presents weights of rolling GARCH efficient portfolio with target monthly returns of 0.8% (using 24 months rolling windows): Table A9. St.dev. and weights of rolling GARCH eff. portfolio with target monthly returns 0.8%: Weights Date Returns St.Dev 2006-12-31 0.008 0.01978 2007-01-31 0.008 2007-03-03 0.008 2007-03-31 TLT SMI BE500 MXASJ DJGSP ASX SPGSAG 0.43120 0.13434 0.40149 -0.01466 0.01542 0.00120 0.03096 0.02001 0.40170 0.16553 0.37953 -0.00485 0.01395 0.00587 0.03829 0.02015 0.38970 0.17565 0.38112 -0.00642 0.01362 0.02090 0.02542 0.008 0.01996 0.40710 0.14552 0.41025 -0.02457 0.01810 0.02488 0.01871 2007-05-01 0.008 0.02115 0.32070 0.19349 0.40364 0.00148 0.02056 0.02809 0.03200 2007-05-31 0.008 0.02272 0.25050 0.22482 0.39963 0.06506 0.00651 -0.00227 0.05579 2007-07-01 0.008 0.02415 0.21710 0.18792 0.36240 0.16139 0.02470 -0.02253 0.06905 2007-07-31 0.008 0.02318 0.26320 0.17209 0.34276 0.18863 -0.00042 -0.06949 0.10321 2007-08-31 0.008 0.02425 0.24530 0.13094 0.33114 0.22519 -0.00958 -0.05393 0.13094 2007-10-01 0.008 0.0223 0.34140 -0.00375 0.41808 0.19524 0.00052 -0.02247 0.07096 2007-10-31 0.008 0.02504 0.26760 -0.04772 0.45298 0.26823 -0.02802 -0.05026 0.13721 2007-12-01 0.008 0.0271 0.24710 -0.11789 0.47088 0.32562 -0.04570 -0.05270 0.17272 2007-12-31 0.008 0.02603 0.36020 -0.29413 0.59318 0.30487 -0.06099 -0.05758 0.15448 2008-01-31 0.008 0.02738 0.30030 -0.22749 0.47636 0.32307 -0.02822 -0.04088 0.19682 2008-03-02 0.008 0.03104 0.31270 -0.36991 0.52208 0.45632 -0.08245 -0.07764 0.23893 2008-03-31 0.008 0.0259 0.41600 -0.33806 0.59129 0.31921 -0.07642 -0.05514 0.14315 2008-05-01 0.008 0.02518 0.39390 -0.30362 0.55714 0.27144 -0.04766 0.01032 0.11852 113 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 15 (2nd page) Weights Date Returns St.Dev BE500 MXASJ DJGSP 2008-05-31 0.008 0.02494 0.41280 -0.37697 0.69712 0.15608 -0.02418 -0.01160 0.14672 2008-07-01 0.008 0.02649 0.41350 -0.44348 0.74226 0.26953 -0.05110 -0.08792 0.15724 2008-07-31 0.008 0.02529 0.44730 -0.38481 0.68979 0.21229 -0.05967 -0.06594 0.16108 2008-08-31 0.008 0.02469 0.51600 -0.48246 0.76691 0.15143 -0.03957 -0.00827 0.09599 2008-10-01 0.008 0.02364 0.56860 -0.44890 0.75584 0.15383 -0.03724 -0.03517 0.04305 2008-10-31 0.008 0.02166 0.58920 -0.29731 0.65028 0.05257 0.00240 -0.01033 0.01320 2008-12-01 0.008 0.02154 0.59830 -0.26724 0.61039 0.03905 0.01870 -0.01081 0.01160 2008-12-31 0.008 0.02275 0.60880 -0.34757 0.62945 0.11224 0.02714 -0.06107 0.03104 2009-01-31 0.008 0.0258 0.68530 -0.45423 0.57988 0.20446 0.05106 -0.10052 0.03408 2009-03-03 0.008 0.0245 0.63870 -0.34353 0.47826 0.19720 0.05000 -0.07227 0.05158 2009-03-31 0.008 0.02363 0.58710 -0.33960 0.53632 0.23255 0.02041 -0.09253 0.05574 2009-05-01 0.008 0.02536 0.56770 -0.40191 0.55370 0.29321 0.05099 -0.13119 0.06753 2009-05-31 0.008 0.02663 0.62860 -0.48480 0.55166 0.31303 0.05268 -0.08934 0.02820 2009-07-01 0.008 0.0271 0.59790 -0.53582 0.61950 0.33391 0.05152 -0.07164 0.00463 2009-07-31 0.008 0.02692 0.60440 -0.56017 0.68035 0.16529 0.07544 0.08966 -0.05495 2009-08-31 0.008 0.02454 0.61730 -0.43358 0.68513 0.10201 0.04873 0.07109 -0.09063 2009-10-01 0.008 0.03353 0.76050 -0.91448 0.84482 0.18718 0.06843 0.17141 -0.11789 2009-10-31 0.008 0.02759 0.57010 -0.54942 0.66820 0.14768 0.09975 0.12205 -0.05834 2009-12-01 0.008 0.02684 0.49480 -0.42713 0.62141 0.17443 0.07039 0.15001 -0.08395 2009-12-31 0.008 0.02889 0.54560 -0.25844 0.40037 0.20626 0.05111 0.24619 -0.19108 2010-01-31 0.008 0.02444 0.53830 0.02071 0.30934 0.01652 0.06665 0.21411 -0.16565 2010-03-03 0.008 0.02042 0.47850 0.05127 0.38166 -0.00374 0.03474 0.09960 -0.04208 2010-03-31 0.008 0.02051 0.49810 0.04009 0.35508 -0.00223 0.04844 0.09787 -0.03737 2010-05-01 0.008 0.01969 0.49280 0.06224 0.40959 -0.01084 0.02027 0.02105 0.00491 2010-05-31 0.008 0.01964 0.47590 0.07751 0.44151 -0.04472 0.01189 0.01797 0.01996 2010-07-01 0.008 0.01964 0.47660 0.08290 0.44112 -0.04617 0.01140 0.01483 0.01934 2010-07-31 0.008 0.01964 0.47750 0.07863 0.44426 -0.04491 0.01035 0.01558 0.01863 2010-08-31 0.008 0.01992 0.46340 0.10423 0.33447 0.01970 0.03126 0.03471 0.01224 2010-10-01 0.008 0.02094 0.39980 0.16291 0.23396 0.06997 0.04938 0.04179 0.04221 2010-10-31 0.008 0.02072 0.39000 0.20220 0.23497 0.05406 0.03039 0.04733 0.04103 2010-12-01 0.008 0.02055 0.38230 0.20558 0.26553 0.02434 0.02250 0.05230 0.04744 2010-12-31 0.008 0.0203 0.39030 0.18294 0.31172 0.00182 0.01255 0.05781 0.04281 2011-01-31 0.008 0.01969 0.45070 0.09971 0.40892 -0.02784 0.01362 0.03086 0.02405 2011-03-03 0.008 0.02021 0.38890 0.13062 0.38099 0.00014 0.01168 0.04495 0.04269 2011-03-31 0.008 0.02072 0.35610 0.21389 0.31226 -0.03815 0.02993 0.07918 0.04684 2011-05-01 0.008 0.01981 0.43740 0.15638 0.35547 -0.03911 0.01280 0.04902 0.02806 2011-05-31 0.008 0.02063 0.36510 0.25265 0.27901 -0.03688 0.02279 0.06621 0.05109 2011-07-01 2011-07-31 2011-08-31 0.008 0.008 0.008 0.02183 0.02213 0.0271 0.34370 0.38020 0.36730 0.34240 0.20265 0.34809 0.12588 0.18814 0.05132 -0.04087 -0.01287 -0.00555 0.03163 0.04991 0.06826 0.10187 0.04371 -0.10266 0.09536 0.14823 0.27327 2011-10-01 0.008 0.02635 0.26640 0.16464 0.27180 -0.03828 0.09955 -0.00763 0.24353 TLT SMI ASX SPGSAG 114 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 15 (3rd page) Weights Date Returns St.Dev SMI BE500 MXASJ 2011-10-31 0.008 0.03039 0.36810 0.11742 0.30353 -0.24615 0.16972 -0.00260 0.29000 2011-12-01 0.008 0.02244 0.53940 0.15222 0.25955 -0.14229 0.05102 0.01471 0.12541 2011-12-31 0.008 0.02265 0.42060 0.05278 0.30159 -0.04586 0.10148 0.03782 0.13157 2012-01-31 0.008 0.02465 0.37010 -0.27680 0.75101 0.11385 0.02252 -0.14417 0.16350 2012-03-02 0.008 0.02651 0.34800 -0.01714 0.55005 0.06270 -0.02201 -0.20127 0.27963 2012-03-31 0.008 0.02943 0.31830 0.19103 0.44333 -0.00930 -0.11195 -0.15845 0.32701 2012-05-01 0.008 0.03023 0.39360 0.26587 0.30494 -0.16567 -0.14075 0.02731 0.31468 2012-05-31 0.008 0.0279 0.30820 0.05291 0.64813 -0.24238 -0.09805 0.09675 0.23447 2012-07-01 0.008 0.02202 0.42550 0.20664 0.29496 -0.12505 -0.03771 0.09229 0.14335 2012-07-31 0.008 0.02721 0.29050 0.07677 0.64885 -0.25827 -0.07872 0.10850 0.21240 2012-08-31 0.008 0.03104 0.19780 0.43147 0.48263 -0.31230 -0.06571 -0.02251 0.28866 2012-10-01 0.008 0.03184 0.37180 0.70897 0.28610 -0.55242 -0.10237 0.20053 0.08739 2012-10-31 0.008 0.0252 0.41260 0.26676 0.62636 -0.22605 -0.11370 -0.00156 0.03555 2012-12-01 0.008 0.02567 0.56080 0.00210 0.81221 -0.10307 -0.13124 -0.06032 -0.08047 2012-12-31 0.008 0.02439 0.48410 0.33873 0.45913 -0.19147 -0.08378 0.08090 -0.08761 2013-01-31 0.008 0.02242 0.50570 0.23760 0.37618 -0.02571 -0.08754 0.04951 -0.05576 2013-03-03 0.008 0.01997 0.48140 0.16443 0.40313 -0.06573 -0.01317 0.03502 -0.00503 2013-03-31 0.008 0.01977 0.48850 0.11040 0.42058 -0.04431 -0.01011 0.02901 0.00597 2013-05-01 0.008 0.02136 0.48480 0.11492 0.55073 -0.03272 -0.07313 -0.03082 -0.01382 2013-05-31 0.008 0.02119 0.47360 0.17548 0.48416 -0.02936 -0.06777 -0.03893 0.00286 2013-07-01 0.008 0.02122 0.45420 0.13026 0.58060 -0.06349 -0.05447 -0.03486 -0.01226 2013-07-31 0.008 0.02277 0.39460 0.27416 0.49833 -0.02737 -0.07729 -0.03003 -0.03244 2013-08-31 0.008 0.02117 0.39900 0.21945 0.45848 -0.03062 -0.04910 0.00669 -0.00387 2013-10-01 0.008 0.02072 0.42990 0.20080 0.45142 -0.02915 -0.04124 -0.00480 -0.00694 2013-10-31 0.008 0.02097 0.41330 0.19971 0.45797 0.00796 -0.05152 -0.03269 0.00531 2013-12-01 0.008 0.02266 0.36580 0.24025 0.54132 0.00097 -0.06956 -0.06077 -0.01796 2013-12-31 0.008 0.02221 0.40060 0.26099 0.51840 -0.04333 -0.06373 -0.05918 -0.01376 2014-01-31 0.008 0.0218 0.40300 0.27466 0.49328 -0.08186 -0.04953 -0.03181 -0.00777 2014-03-03 0.008 0.02174 0.40890 0.25818 0.48638 -0.08095 -0.05804 -0.01003 -0.00447 2014-03-31 0.008 0.02158 0.39310 0.25056 0.51240 -0.08836 -0.04355 -0.02846 0.00435 2014-05-01 0.008 0.02118 0.38590 0.20865 0.51530 -0.06274 -0.03598 -0.01414 0.00296 2014-05-31 0.008 0.02201 0.37970 0.24180 0.53461 -0.05138 -0.03871 -0.02909 -0.03692 2014-07-01 0.008 0.02359 0.38290 0.23035 0.59187 -0.01815 -0.04104 -0.04473 -0.10123 2014-07-31 0.008 0.02201 0.42120 0.21620 0.50064 -0.00761 -0.03903 -0.01612 -0.07528 2014-08-31 0.008 0.02087 0.45630 0.16901 0.49253 -0.01865 -0.03510 -0.02224 -0.04185 2014-10-01 0.008 0.0207 0.46430 0.15369 0.46526 0.00286 -0.04676 -0.01537 -0.02397 2014-10-31 0.008 0.02041 0.46620 0.14155 0.47673 0.00067 -0.03371 -0.03398 -0.01744 2014-12-01 0.008 0.0199 0.48040 0.11327 0.45109 -0.01194 -0.01692 -0.01416 -0.00171 2014-12-31 0.008 0.01964 0.47790 0.07329 0.42154 -0.04428 0.01915 0.03043 0.02194 TLT DJGSP ASX SPGSAG 115 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 16 Markowitz and GARCH efficient frontiers together with EWMA efficient frontier (when Lambda = 0.91). 116 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE Appendix 17(1st page) Below the main R script of this thesis is provided. In this script there are used information and formulas from the following literature: 1. Zivot, E. (2011, August 11). Functions for portfolio analysis [R script]. Retrieved March 15, 2015 from http://faculty.washington.edu/ezivot/econ424/portfolio.r 2. Zivot, E. (2012, May 8). R examples for lectures on multivariate GARCH models [R script]. Retrieved from http://faculty.washington.edu/ezivot/econ589/econ589multivariateGarch.r 3. Zivot, E. (2013, August 1). Examples used in Portfolio Theory with Matrix Algebra Chapter [R script]. Retrieved from http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.r 4. Zivot, E. (2013, August 21). Examples of rolling analysis of portfolios [R script]. Retrieved February 9, 2015 from http://faculty.washington.edu/ezivot/econ424/rollingPortfolios.r Moreover, some parts of the script rely on the scripts available at these 4 internet pages listed above. setwd("C:/Users/User/Desktop/ISM/Master Thesis/Duomenys") data <- read.table("MasterDaNorm.txt", header=T,) names(data) # Load the packages that are used in this part library(zoo) library(fPortfolio) library(tseries) library("corrplot") library(PerformanceAnalytics) TLT = data$TLT SMI = data$SMI BE500 = data$BE500 MXASJ = data$MXASJ DJGSP = data$DJGSP ASX = data$ASX SPGSAG = data$SPGSAG ##################### Data Summary and Data Checking######################### #######################(Without Chosen Outliers)############################## summary(TLT) summary(SMI) summary(BE500) TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE summary(MXASJ) summary(DJGSP) summary(ASX) summary(SPGSAG) ########Testing for variables normality and stationarity after taking out needed outliers – the same model used for all 7 inputs (example with TLT ETF)##### windows(7,5) plot(TLT, type = "l") qqnorm(TLT) qqline(TLT) hist(TLT, breaks=20) shapiro.test(TLT) skew(TLT) kur(TLT) #Stationarity adf.test(TLT, alternative='stationary') # no unit-root #############Means, correlations, covariance matrixes for Markowitz########## asset.names <- c("TLT", "SMI", "BE500", "MXASJ", "DJGSP", "ASX", "SPGSAG") returns=data.frame(TLT, SMI, BE500, MXASJ, DJGSP, ASX, SPGSAG) #set time frame td = seq(as.Date("2005-01-31"), as.Date("2014-12-31"), by="months") # estimate parameters of constant expected return model si.z = zoo(returns,td) ret.z = si.z ret.mat = si.z n.obs = nrow(ret.mat) # plot returns over full sample windows() plot(ret.z, main="", plot.type="single" ,ylab="Returns", cex.axis = 1.5, cex.lab = 1.5, lwd=2, col=1:7) abline(h=0) legend(x="bottomright", legend=colnames(returns), col=1:7, lwd=2, cex=1) pairs(ret.mat, col="slateblue1", pch=16, cex=0.5) nobs = nrow(ret.mat) muhat.vals = colMeans(ret.mat) mu.vec = muhat.vals sigmahat.vals = apply(ret.mat,2,sd) cov.mat = var(ret.mat) cor.mat = cor(ret.mat) returns_correlation=cor.mat returns_covariance=cov.mat sigma.mat <- returns_covariance 117 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE #plot correlation matrix windows() corrplot(returns_correlation,order = "AOE", col=col1(20), cl.length = 21, addCoef.col="grey", method="color", type="lower") sd.vec = c(sd(TLT),sd(SMI),sd(BE500),sd(MXASJ), sd(DJGSP), sd(ASX), sd(SPGSAG)) #plot all 7 inputs windows() plot(sd.vec, mu.vec,pch=16, col="green", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) text(sd.vec, mu.vec, labels=asset.names, pos=2, cex = .8) ############################Part 1. Markowitz model########################## # Equally weighted portfolio x.vec.Eq = rep(1,7)/7 names(x.vec.Eq) = asset.names sum(x.vec.Eq) # Compute mean, variance and std deviation mu.p.x.Eq = crossprod(x.vec.Eq,mu.vec) sig2.p.x.Eq = t(x.vec.Eq)%*%returns_covariance%*%x.vec.Eq sig.p.x.Eq = sqrt(sig2.p.x.Eq) mu.p.x.Eq # compute portfolio returns sig.p.x.Eq # compute standard deviation of portfolio # GlobalMin.VarPortfolio one.vec = rep(1, 7) sigma.inv.mat = solve(returns_covariance) top.mat = sigma.inv.mat%*%one.vec bot.val = as.numeric((t(one.vec)%*%sigma.inv.mat%*%one.vec)) m.mat = top.mat/bot.val m.mat[,1] mu.px.Gl = as.numeric(crossprod(m.mat, mu.vec)) mu.px.Gl sig2.px.Gl = as.numeric(t(m.mat)%*%sigma.mat%*%m.mat) sig.px.Gl = sqrt(sig2.px.Gl) sig.px.Gl # Efficient portfolio with same mean as "ASX" top.mat = cbind(2*sigma.mat, mu.vec, rep(1, 7)) mid.vec = c(mu.vec, 0, 0) bot.vec = c(rep(1, 7), 0, 0) Ax.mat = rbind(top.mat, mid.vec, bot.vec) bsmi.vec = c(rep(0, 7), mu.vec["ASX"], 1) z.mat = solve(Ax.mat)%*%bsmi.vec y.vec = z.mat[1:7,] y.vec 118 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE # compute mean, variance and std deviation of portfolio above mu.py = as.numeric(crossprod(y.vec, mu.vec)) mu.py sig2.py = as.numeric(t(y.vec)%*%sigma.mat%*%y.vec) sig.py = sqrt(sig2.py) sig.py # Efficient portfolio with same mean as "MXASJ" top.mat = cbind(2*sigma.mat, mu.vec, rep(1, 7)) mid.vec = c(mu.vec, 0, 0) bot.vec = c(rep(1, 7), 0, 0) Ax.mat = rbind(top.mat, mid.vec, bot.vec) bmxasj.vec = c(rep(0, 7), mu.vec["MXASJ"], 1) z.mat = solve(Ax.mat)%*%bmxasj.vec x.vec = z.mat[1:7,] x.vec # compute mean, variance and std deviation of portfolio above mu.px = as.numeric(crossprod(x.vec, mu.vec)) mu.px sig2.px = as.numeric(t(x.vec)%*%sigma.mat%*%x.vec) sig.px = sqrt(sig2.px) sig.px # find efficient portfolio from two efficient portfolios a = 0.5 z.vec = a*x.vec + (1-a)*y.vec z.vec # compute mean, variance and std deviation sigma.xy = as.numeric(t(x.vec)%*%sigma.mat%*%y.vec) mu.pz = as.numeric(crossprod(z.vec, mu.vec)) sig2.pz = as.numeric(t(z.vec)%*%sigma.mat%*%z.vec) sig.pz = sqrt(sig2.pz) mu.pz sig.pz #find efficient portfolio with er = 0.008 a.01 = (0.008 - mu.py)/(mu.px - mu.py) a.01 z.01 = a.01*x.vec + (1 - a.01)*y.vec z.01 #compute mean, var and sd mu.pz.01 = a.01*mu.px + (1-a.01)*mu.py sig2.pz.01 = a.01^2 * sig2.px + (1-a.01)^2 * sig2.py + 2*a.01*(1-a.01)*sigma.xy sig.pz.01 = sqrt(sig2.pz.01) mu.pz.01 sig.pz.01 119 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE #compute efficient portfolios as convex combinations of global min portfolio and efficient portfolio with same mean as MXASJ Index a = seq(from=1, to=-1, by=-0.1) n.a = length(a) z.mat = matrix(0, n.a, 7) mu.z = rep(0, n.a) sig2.z = rep(0, n.a) sig.mx = t(m.mat)%*%sigma.mat%*%x.vec for (i in 1:n.a) { z.mat[i, ] = a[i]*m.mat + (1-a[i])*x.vec mu.z[i] = a[i]*mu.px.Gl + (1-a[i])*mu.px sig2.z[i] = a[i]^2 * sig2.px.Gl + (1-a[i])^2 * sig2.px + 2*a[i]*(1-a[i])*sig.mx } #plot Markowitz Efficient frontier with different points (select from below) cex.val = 1.5 windows() plot(sqrt(sig2.z), mu.z, type="b", ylim=c(0.001, 0.02), xlim=c(0.015, 0.078), pch=16, col="black", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) legend(x="topleft",legend=c("Markowitz efficient frontier"), lwd=2, col=c("black")) points(sd.vec, mu.vec, pch=8, cex=2, lwd=2, col=2:8) text(sd.vec, mu.vec, labels=asset.names, pos=1:4, cex = 0.5) points(sig.px.Gl, mu.px.Gl, pch=16, col="orange", cex=1.5) text(sig.px.Gl, mu.px.Gl, labels="Global.min", pos=2, cex = 0.8) points(sig.pz.01, mu.pz.01,pch=16, col="brown", cex=1.5) text(sig.pz.01, mu.pz.01, labels="Returns0.08", pos=2, cex = 0.8) points(sig.px, mu.px, pch=16, col="red", cex=1.5) text(sig.px, mu.px, labels="Port.with MXASJ returns", pos=2, cex = 0.8) points(sig.p.x.Eq, mu.p.x.Eq, pch=16, col="yellow", cex=1.5) text(sig.p.x.Eq, mu.p.x.Eq, labels="Equal weighted", pos=4, cex = 0.8) points(sig.py, mu.py, pch=16, col="blue", cex=1.5) text(sig.py, mu.py, labels="Port.with ASX returns .", pos=2, cex = 0.8) ##########################Part 2. Rolling Portfolios######################### options(digits=4) cex.val = 1.5 # compute rolling means and standard deviations over 24 month windows – the same model used for all 7 inputs (example with TLT ETF) roll.muhat.tlt = rollapply(returns[,"TLT"], width=24, FUN=mean, align="right") roll.sigmahat.tlt = rollapply(returns[,"TLT"],width=24, FUN=sd, align="right") 120 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE # data frames td = seq(as.Date("2005-01-31"), as.Date("2014-12-31"), by="months") td1 = seq(as.Date("2006-12-31"), as.Date("2014-12-31"), by="months") #plot returns together with rolling means and standard deviations – the same model used for all 7 inputs (example with TLT ETF) par(mfrow=c(2,2)) # TLT windows() plot(td, ret.z[,"TLT"], type = "line", pch=16, col="black", cex = cex.val, ylab=expression(mu[p]), xlab=expression(time)) lines(td1, roll.sigmahat.tlt, pch=16, col="orange", cex = cex.val, ylab=expression(mu[p]), xlab=expression(time)) lines(td1 , roll.muhat.tlt, pch=16, col="blue", cex = cex.val, ylab=expression(mu[p]), xlab=expression(time)) abline(h=0) legend(x="bottomright",legend=c(" TLT Monthly returns", "Rolling sd","Rolling mean" ), lwd=2, col=c("black","orange","blue")) par(mfrow=c(1,1)) # compute rolling correlations over 24 months windows my.panel <- function(...) { lines(...) abline(h=0) } roll.cor = function(x) { cor.hat = cor(x) cor.vals = cor.hat[lower.tri(cor.hat)] names(cor.vals) = c("t.sm","t.b","t.m","t.d","t.a","t.sp","sm.b", "sm.m","sm.d", "sm.a", "sm.sp", "b.m", "b.d", "b.a", "b.sp", "m.d", "m.a", "m.sp", "d.a", "d.sp", "a.sp") return(cor.vals) } roll.cor.vals = rollapply(ret.z, width=24, by.column=FALSE, FUN=roll.cor, align="right") # plot rolling correlations (change column numbers (1:21) in the brackets near to the roll.cor.vals to get plots for different correlations) windows() plot(roll.cor.vals[], panel=my.panel, main="", lwd=1.5, col="blue", ylim=c(-1,1)) # global minimum variance portfolio gmin.4 = globalMin.portfolio(er=muhat.vals,cov.mat=cov.mat) gmin.4 # The same as in the beginning (checking) # efficient portfolio with target return = 0.008 eport.01 = efficient.portfolio(er=muhat.vals,cov.mat=cov.mat, target.return=0.008) eport.01 # compute efficient frontier ef.4 = efficient.frontier(er=muhat.vals,cov.mat=cov.mat) 121 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE ef.4 #Rolling parameters rollGmin = function(x) { mu.hat = colMeans(x) cov.hat = var(x) gmin = globalMin.portfolio(er=mu.hat,cov.mat=cov.hat) ans = c(gmin$er,gmin$sd,gmin$weights) names(ans)[1:2] = c("er","sd") return(ans) } rollefficient = function(x,target=0.008) { mu.hat = colMeans(x) cov.hat = var(x) eport = efficient.portfolio(er=mu.hat, cov.mat=cov.hat, target.return=target) ans = c(eport$er,eport$sd,eport$weights) names(ans)[1:2] = c("er","sd") return(ans) } # rolling 24-month global minimum variance portfolios roll.gmin = rollapply(ret.z, width=24, by.column=FALSE,align="right", FUN=rollGmin) colnames(roll.gmin) # plot rolling weights in global min var portfolio windows() plot(roll.gmin[,3:9],main="", ylim=c(-0.8, 0.9), plot.type="single", col=1:7, lwd=2, ylab="weight", xlab="time") abline(h=0) legend(x="bottomright",legend=colnames(ret.z), lty=rep(1,7),col=1:7,lwd=1.5) windows() chart.StackedBar(roll.gmin[,3:9], ylab="weights") # plot rolling means and sds of global min variance portfolio windows() plot(roll.gmin[,1:2],plot.type="single",ylab="percent", main="", col=c("black","blue"),lwd=2) abline(h=0) legend(x="bottomright",legend=c("Rolling mean","Rolling sd"), lty=rep(1,2),col=c("black","blue"),lwd=2) # rolling efficient portfolios with target = 0.008 roll.eport = rollapply(ret.z, width=24, by.column=F,align="right", FUN=rollefficient) colnames(roll.eport) # plot rolling weights of efficient portfolio with target = 0.008 windows() plot(roll.eport[,3:9],main="", plot.type="single", ylab="weight", col=1:7,lwd=2) abline(h=0) legend(x="bottomright",legend=colnames(ret.z), 122 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE lty=rep(1,7),col=1:7,lwd=2) # stacked bar chart windows() chart.StackedBar(roll.eport[,3:9]) #plot rolling sd of efficient portfolios with target = 0.008 windows() plot(roll.eport[,1:2], plot.type="single", ylab="percent", main="", ylim=c(0, 0.05), col=c("black","blue"),lwd=2) abline(h=0) legend(x="topleft",legend=c("Target er=0.008","Rolling sd"), lty=rep(1,2),col=c("black","blue"),lwd=2) #####################Part 3. EWMA efficient frontier######################### #package for EWMA model part library(GARPFRM) #install.packages("GARPFRM", repos="http://R-Forge.R-project.org") lambda = 0.8 returns.xts <- as.xts(ret.z) #calculate EWMA covariance matrix with chosen Lambda cov_mv <- EWMA(returns.xts, lambda, initialWindow = 119, type="covariance") cov.mat.ewma <- as.matrix(cov_mv$estimate) # EWMA GlobalMin.Var Portfolio one.vec.ewma = rep(1, 7) sigma.inv.mat.ewma = solve(cov.mat.ewma) top.mat.ewma = sigma.inv.mat.ewma%*%one.vec.ewma bot.val.ewma = as.numeric((t(one.vec.ewma)%*%sigma.inv.mat.ewma%*%one.vec.ewma)) m.mat.ewma = top.mat.ewma/bot.val.ewma m.mat.ewma[,1] # compute expected return, variance and sd with EWMA mu.px.Gl.ewma = as.numeric(crossprod(m.mat.ewma, mu.vec)) mu.px.Gl.ewma sig2.px.Gl.ewma = as.numeric(t(m.mat.ewma)%*%cov.mat.ewma%*%m.mat.ewma) sig.px.Gl.ewma = sqrt(sig2.px.Gl.ewma) sig.px.Gl.ewma # Efficient portfolio with same mean as "ASX" and EWMA top.mat.ewma = cbind(2*cov.mat.ewma, mu.vec, rep(1, 7)) mid.vec.ewma = c(mu.vec, 0, 0) bot.vec.ewma = c(rep(1, 7), 0, 0) Ax.mat.ewma = rbind(top.mat.ewma, mid.vec.ewma, bot.vec.ewma) bsmi.vec.ewma = c(rep(0, 7), mu.vec["ASX"], 1) z.mat.ewma = solve(Ax.mat.ewma)%*%bsmi.vec.ewma y.vec.ewma = z.mat.ewma[1:7,] y.vec.ewma # compute mean, variance and std deviation of portfolio above 123 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE mu.py.ewma = as.numeric(crossprod(y.vec.ewma, mu.vec)) mu.py.ewma sig2.py.ewma = as.numeric(t(y.vec.ewma)%*%cov.mat.ewma%*%y.vec.ewma) sig.py.ewma = sqrt(sig2.py.ewma) sig.py.ewma # Efficient portfolio with same mean as "MXASJ" and EWMA top.mat.e = cbind(2*cov.mat.ewma, mu.vec, rep(1, 7)) mid.vec.e = c(mu.vec, 0, 0) bot.vec.e = c(rep(1, 7), 0, 0) Ax.mat.e = rbind(top.mat.e, mid.vec.e, bot.vec.e) bmxasj.vec.e = c(rep(0, 7), mu.vec["MXASJ"], 1) z.mat.e = solve(Ax.mat.e)%*%bmxasj.vec.e x.vec.e = z.mat.e[1:7,] x.vec.e # compute mean, variance and std deviation of portfolio above mu.px.e = as.numeric(crossprod(x.vec.e, mu.vec)) mu.px.e sig2.px.e = as.numeric(t(x.vec.e)%*%cov.mat.ewma%*%x.vec.e) sig.px.e = sqrt(sig2.px.e) sig.px.e # find efficient portfolio from two efficient portfolios a.ewma = 0.5 z.vec.ewma = a.ewma*x.vec.e + (1-a.ewma)*y.vec.ewma z.vec.ewma # compute mean, variance and std deviation sigma.xy.ewma = as.numeric(t(x.vec.e)%*%cov.mat.ewma%*%y.vec.ewma) mu.pz.ewma = as.numeric(crossprod(z.vec.e, mu.vec)) sig2.pz.ewma = as.numeric(t(z.vec.e)%*%cov.mat.ewma%*%z.vec.e) sig.pz.ewma = sqrt(sig2.pz.ewma) mu.pz.ewma sig.pz.ewma #find efficient portfolio with er = 0.008 a.01ewma = (0.008 - mu.py.ewma)/(mu.px.e - mu.py.ewma) #??????? a.01ewma z.01ewma = a.01ewma*x.vec.e + (1 - a.01ewma)*y.vec.ewma z.01ewma # compute mean, var and sd mu.pz.01e = a.01ewma*mu.px.e + (1-a.01ewma)*mu.py.ewma sig2.pz.01e = a.01ewma^2 * sig2.px.e + (1-a.01ewma)^2 * sig2.py.ewma + 2*a.01ewma*(1a.01ewma)*sigma.xy.ewma sig.pz.01e = sqrt(sig2.pz.01e) mu.pz.01e sig.pz.01e effi.EWMA = efficient.frontier(er=muhat.vals,cov.mat=cov.mat.ewma) #Compute efficient portfolios 124 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE a = seq(from=1, to=-1, by=-0.1) n.a = length(a) z.mat = matrix(0, n.a, 7) mu.ze = rep(0, n.a) sig2.ze = rep(0, n.a) sig.mx = t(m.mat.ewma)%*%cov.mat.ewma%*%x.vec.e #?????? for (i in 1:n.a) { z.mat[i, ] = a[i]*m.mat.ewma + (1-a[i])*x.vec.e mu.ze[i] = a[i]*mu.px.Gl.ewma + (1-a[i])*mu.px.e sig2.ze[i] = a[i]^2 * sig2.px.Gl.ewma + (1-a[i])^2 * sig2.px.e + 2*a[i]*(1-a[i])*sig.mx } #compute EWMA frontier tangent portfolio risk.free <- 0.0012 #Average EU risk-free rate tangency.ewma <- tangency.portfolio(mu.vec, cov.mat.ewma,risk.free) mu.tang.ewma <- tangency.ewma$er sd.tang.ewma <- tangency.ewma$sd weights.tang.ewma <- tangency.ewma$weights #plot EWMA efficient frontier cex.val = 1.5 windows() plot(sqrt(sig2.ze), mu.ze, type="b", ylim=c(0.001, 0.02), xlim=c(0.0, 0.078), pch=16, col="black", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) legend(x="topleft",legend=c("EWMA efficient frontier. Lambda=0.7", "EWMA efficient frontier. Lambda=0.8", "EWMA efficient frontier. Lambda=0.91"), lwd=2, col=c("orange", "black", "lightblue")) points(sd.tang.ewma, mu.tang.ewma, pch=16, cex=2, col="purple") text(sd.tang.ewma, mu.tang.ewma, labels="Tangency port. ", pos=2, cex = 0.8) sr.tangE = (mu.tang.ewma - risk.free)/sd.tang.ewma abline(a=risk.free, b=sr.tangE, col="grey") points(sig.px.Gl.ewma, mu.px.Gl.ewma, pch=16, col="yellow", cex=1.5) text(sig.px.Gl.ewma, mu.px.Gl.ewma, labels="EWMA Global.min", pos=4, cex = 0.8) points(sig.pz.01e, mu.pz.01e,pch=16, col="red", cex=1.5) text(sig.pz.01e, mu.pz.01e, labels="EWMA portf with returns0.008", pos=4, cex = 0.8) #2 functions(lines) below used to create Apendix 10 (EWMA eff.frontiers with different Lambdas) #lines(sqrt(sig2.ze), mu.ze, type="b", ylim=c(0.001, 0.02), xlim=c(0.0, 0.078), # pch=16, col="lightblue", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) #lines(sqrt(sig2.ze), mu.ze, type="b", ylim=c(0.001, 0.02), xlim=c(0.0, 0.078), # pch=16, col="orange", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) # Rolling weigths with EWMA rollefficient = function(x,target=0.008) { mu.hat = colMeans(x) 125 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE cov.hat = cov.mat.ewma eport = efficient.portfolio(er=mu.hat, cov.mat=cov.hat,target.return=target) ans = c(eport$er,eport$sd,eport$weights) names(ans)[1:2] = c("er","sd") return(ans) } # rolling EWMA efficient portfolios with target monthly returns 0.008 roll.eport = rollapply(ret.z, width=24, by.column=F,align="right", FUN=rollefficient) colnames(roll.eport) # plot rolling weights of EWMA efficient portfolios with target = 0.008 windows() plot(roll.eport[,3:9],main="", plot.type="single", ylab="weight", col=1:7,lwd=2) abline(h=0) legend(x="bottomright",legend=colnames(ret.z), lty=rep(1,7),col=1:7,lwd=2) windows() chart.StackedBar(roll.eport[,3:9]) #plot rolling EWMA sd of efficient portfolios with target monthly returns = 0.008 windows() plot(roll.eport[,1:2], plot.type="single", ylab="percent", main="", ylim=c(0, 0.04), col=c("black","blue"),lwd=2) abline(h=0) legend(x="topleft",legend=c("Target er=0.008","Rolling sd"), lty=rep(1,2),col=c("black","blue"),lwd=2) ############################4. GARCH efficient frontier ####################### #pakage for ARCH test library(FinTS) # packages for DCC estimation and forecast library(rugarch) library(rmgarch) #test to see if GARCH is needed ArchTest(returns) # univariate normal GARCH(1,1) for each series garch11.spec = ugarchspec(mean.model = list(armaOrder = c(0,0)), variance.model = list(garchOrder = c(1,1), model = "sGARCH"), distribution.model = "norm") # dcc specification - GARCH(1,1) for conditional correlations dcc.garch11.spec = dccspec(uspec = multispec( replicate(7, garch11.spec) ), dccOrder = c(1,1), distribution = "mvnorm") dcc.garch11.spec dcc.fit = dccfit(dcc.garch11.spec, data = data.frame(returns,row.names=td)) dcc.fit # forecasting conditional volatility and correlations. Forecasting covariance matrix for 3 periods ahead dcc.fcst = dccforecast(dcc.fit, n.ahead=3) 126 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE #plot estimated and forecasted covariances (change "series" numbers to get different covariances) windows(5,7) plot(dcc.fcst, which=3, series=3:7) # show forecasts of GARCH cov.mat. dcc.fcst@mforecast$H #use 1 period ahead forecasted covariance matrix from dcc.fcst@mforecast$H cov.GARCH <- matrix(c( 0.0011975, -0.0001350, -0.0004217, -0.0002227, -0.0000751, 0.0003739, 0.0001952, -0.0001350, 0.0011823, 0.0008440, 0.0009045, 0.0004478, 0.0008574, 0.0002661, -0.0004217, 0.0008440, 0.0012102, 0.0010267, 0.0007094, 0.0011254, 0.0004427, -0.0002227, 0.0009045, 0.0010267, 0.0024085, 0.0015777, 0.0018456, 0.0005187, -0.0000751, 0.0004478, 0.0007094, 0.0015777, 0.0073109, 0.0012108, 0.0010574, -0.0003739, 0.0008574, 0.0011254, 0.0018456, 0.0012108, 0.0026666, 0.0007893, 0.0001952, 0.0002661, 0.0004427, 0.0005187, 0.0010574, 0.0007893, 0.0044227), nrow = 7) rownames(cov.GARCH) <- asset.names colnames(cov.GARCH) <- asset.names cov.GARCH #Compute Global Min-Var portfolio with GARCH covariance matrix gmin.portGARCH <- globalMin.portfolio(mu.vec, cov.GARCH) gmin.portGARCH one.vec = rep(1, 7) sigma.inv.matG = solve(cov.GARCH) top.matG = sigma.inv.matG%*%one.vec bot.valG = as.numeric((t(one.vec)%*%sigma.inv.matG%*%one.vec)) m.matG = top.matG/bot.valG m.matG[,1] mu.GARCH = as.numeric(crossprod(m.matG, mu.vec)) mu.GARCH sig2.px.GlG = as.numeric(t(m.matG)%*%cov.GARCH%*%m.matG) sig.px.GlG = sqrt(sig2.px.GlG) sig.px.GlG eport.01GARCH = efficient.portfolio(er=muhat.vals,cov.mat=cov.GARCH, target.return=0.008) eport.01GARCH #Compute weights of GARCH eff.frontier ef.4GARCH = efficient.frontier(er=muhat.vals,cov.mat=cov.GARCH) 127 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE ef.4GARCH #Compute portfolio with with same mean as "ASX" and GARCH top.matG = cbind(2*cov.GARCH, mu.vec, rep(1, 7)) mid.vecG = c(mu.vec, 0, 0) bot.vecG = c(rep(1, 7), 0, 0) Ax.matG = rbind(top.matG, mid.vecG, bot.vecG) bsmi.vecG = c(rep(0, 7), mu.vec["ASX"], 1) z.matG = solve(Ax.matG)%*%bsmi.vecG y.vecG = z.matG[1:7,] y.vecG # compute mean, variance and std deviation of portfolio above mu.pyG = as.numeric(crossprod(y.vecG, mu.vec)) mu.pyG sig2.pyG = as.numeric(t(y.vecG)%*%cov.GARCH%*%y.vecG) sig.pyG = sqrt(sig2.pyG) sig.pyG #Compute portfolio with with same mean as "MXASJ" and GARCH top.matG = cbind(2*cov.GARCH, mu.vec, rep(1, 7)) mid.vecG = c(mu.vec, 0, 0) bot.vecG = c(rep(1, 7), 0, 0) Ax.matG = rbind(top.matG, mid.vecG, bot.vecG) bmxasj.vecG = c(rep(0, 7), mu.vec["MXASJ"], 1) z.matG = solve(Ax.matG)%*%bmxasj.vecG x.vecG = z.matG[1:7,] x.vecG # compute mean, variance and std deviation of portfolio above mu.pxG = as.numeric(crossprod(x.vecG, mu.vec)) mu.pxG sig2.pxG = as.numeric(t(x.vecG)%*%cov.GARCH%*%x.vecG) sig.pxG = sqrt(sig2.pxG) sig.pxG # find efficient portfolio from two efficient portfolios above and GARCH model a = 0.5 z.vecG = a*x.vecG + (1-a)*y.vecG z.vecG # compute mean, variance and std deviation sigma.xyG = as.numeric(t(x.vecG)%*%cov.GARCH%*%y.vecG) mu.pzG = as.numeric(crossprod(z.vecG, mu.vec)) sig2.pzG = as.numeric(t(z.vecG)%*%cov.GARCH%*%z.vecG) sig.pzG = sqrt(sig2.pzG) mu.pzG sig.pzG #find efficient portfolio with er = 0.008 and GARCH model a.01G = (0.008 - mu.pyG)/(mu.pxG - mu.pyG) a.01G z.01G = a.01G*x.vecG + (1 - a.01G)*y.vecG 128 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE z.01G # compute mean, var and sd mu.pz.01G = a.01G*mu.pxG + (1-a.01G)*mu.pyG sig2.pz.01G = a.01G^2 * sig2.pxG + (1-a.01G)^2 * sig2.pyG + 2*a.01G*(1a.01G)*sigma.xyG sig.pz.01G = sqrt(sig2.pz.01G) mu.pz.01G sig.pz.01G #Compute GARCH efficient portfolios a = seq(from=1, to=-1, by=-0.1) n.a = length(a) z.matG = matrix(0, n.a, 7) mu.zG = rep(0, n.a) sig2.zG = rep(0, n.a) sig.mxG = t(m.matG)%*%cov.GARCH%*%x.vecG for (i in 1:n.a) { z.matG[i, ] = a[i]*m.matG + (1-a[i])*x.vecG mu.zG[i] = a[i]*mu.GARCH + (1-a[i])*mu.pxG sig2.zG[i] = a[i]^2 * sig2.px.GlG + (1-a[i])^2 * sig2.pxG + 2*a[i]*(1-a[i])*sig.mxG } #computer GARCH frontier tangent portfolio risk.free <- 0.0012 #Average EU risk-free rate tangency.garch <- tangency.portfolio(mu.vec, cov.GARCH,risk.free) mu.tang.garch <- tangency.garch$er sd.tang.garch <- tangency.garch$sd weights.tang.garch <- tangency.garch$weights #plot GARCH efficient frontier cex.val = 1.5 windows() plot(sqrt(sig2.zG), mu.zG, type="b", ylim=c(0.001, 0.02), xlim=c(0.0, 0.078), pch=16, col="black", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) legend(x="topleft",legend=c("GARCH efficient frontier"), lwd=2, col=c("black")) points(sd.tang.garch, mu.tang.garch, pch=16, cex=2, col="purple") text(sd.tang.garch, mu.tang.garch, labels="Tangency port.", pos=2, cex = 1) sr.tangG = (mu.tang.garch - risk.free)/sd.tang.garch abline(a=risk.free, b=sr.tangG, col="grey") points(sig.px.GlG, mu.GARCH, pch=16, col="yellow", cex=1.5) text(sig.px.GlG, mu.GARCH, labels="GARCH Global.min", pos=4, cex = 0.8) points(sig.pz.01G, mu.pz.01G,pch=16, col="red", cex=1.5) text(sig.pz.01G, mu.pz.01G, labels="GARCH portf with returns0.008", pos=4, cex = 0.8) # Rolling weights with GARCH rollefficient = function(x,target=0.008) { mu.hat = colMeans(x) 129 TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE cov.hat = cov.GARCH eport = efficient.portfolio(er=mu.hat, cov.mat=cov.hat, target.return=target) ans = c(eport$er,eport$sd,eport$weights) names(ans)[1:2] = c("er","sd") return(ans) } # rolling GARCH efficient portfolios with target monthly returns = 0.008 roll.eport = rollapply(ret.z, width=24, by.column=F,align="right", FUN=rollefficient) colnames(roll.eport) # plot rolling weights of GARCH efficient portfolio with target monthly returns = 0.008 windows() plot(roll.eport[,3:9],main="", plot.type="single", ylab="weight", col=1:7,lwd=2) abline(h=0) legend(x="bottomright",legend=colnames(ret.z), lty=rep(1,7),col=1:7,lwd=2) windows() chart.StackedBar(roll.eport[,3:9]) #plot rolling sd of GARCH efficient portfolio with target monthly returns = 0.008 windows() plot(roll.eport[,1:2], plot.type="single", ylab="percent", main="", ylim=c(0, 0.04), col=c("black","blue"),lwd=2) abline(h=0) legend(x="topleft",legend=c("Target er=0.008","Rolling sd"), lty=rep(1,2),col=c("black","blue"),lwd=2) #######################All 3 models on the same graph:######################## ###################Markowitz, EWMA and GARCH Eff.Frontiers################ cex.val = 1.5 windows() plot(sqrt(sig2.z), mu.z, type="b", ylim=c(0.001, 0.02), xlim=c(0.015, 0.078), pch=16, col="black", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) points(sig.px.Gl, mu.px.Gl, pch=16, col="orange", cex=1.5) text(sig.px.Gl, mu.px.Gl, labels=" Global.min.portf", pos=4, cex = 1) points(sig.pz.01, mu.pz.01,pch=16, col="brown", cex=1.5) text(sig.pz.01, mu.pz.01, labels=" Eff.portf with ret=0.8%", pos=4, cex = 1) #EWMA lines(sqrt(sig2.ze), mu.ze, type="b", ylim=c(0.001, 0.0105), xlim=c(0.02, 0.04), pch=16, col="blue", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) points(sig.pz.01e, mu.pz.01e, pch=16, col="brown", cex=1.5) #text(sig.pz.01e, mu.pz.01e, labels="Eff.portf with ret=0.8%", pos=2, cex = 1) points(sig.px.Gl.ewma, mu.px.Gl.ewma, pch=16, col="orange", cex=1.5) #text(sig.px.Gl.ewma, mu.px.Gl.ewma, labels="Global.min.portf", pos=2, cex = 1) #GARCH lines(sqrt(sig2.zG), mu.zG, type="b", ylim=c(0.001, 0.0105), xlim=c(0.02, 0.04), pch=16, col="green", cex = cex.val, ylab=expression(mu[p]), xlab=expression(sigma[p])) points(sig.pz.01G, mu.pz.01G,pch=16, col="brown", cex=1.5) points(sig.px.GlG, mu.GARCH, pch=16, col="orange", cex=1.5) legend(x="topleft",legend=c("EWMA frontier","Markowitz frontier", "GARCH frontier"), lwd=2, col=c("blue","black","green")) 130