Download IMPACT OF TIME VARYING DISTRIBUTIONAL PARAMETERS ON

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. Moreover, multi-period
forecasted DCC-GARCH(p,q) covariance matrixes might be implemented to the portfolio
optimization process instead of the rolling window method to see how the weights in the
efficient portfolios are changing over time.
83
TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE
Reference List
Akgiray, V. (1989). Conditional Heteroscedasticity in Time Series of Stock Returns:
Evidence and Forecasts. The Journal of Business, 62(1), 55-80.
Alexander, C. (2008). Moving Average Models for Volatility and Correlation, and
Covariance Matrices. In Handbook of finance. John Wiley & Sons.
Benett, R., & Fillebeen, T., (2015). GARPFRM: Global Association of Risk Professionals:
Financial Risk Manager. R package version 0.1.0/r221. Retrieved March 23, 2015
from http://r-forge.r-project.org/R/?group_id=1791
Best, M. J., & R. R. Grauer. (1991). On the Sensitivity of Mean-Variance-Efficient Portfolios
to Changes in Asset Means: Some Analytical and Computational Results. The Review
of Financial Studies, 4:315–42.
Black, F., & Litterman, R. (1992). Global Portfolio Optimization. Financial Analysts
Journal, 48(5), 28-43.
Bloomberg terminal. Information about indexes and data sets. Retrieved from Bloomberg
terminal in February 2015.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of
Econometrics, 31, 307-327.
Bower, B., & Wentz, P. (2005). Portfolio Optimization: MAD vs. Markowitz. The RoseHulman Undergraduate Mathematics Journal, 6.
Britten-Jones, M. (1999). The Sampling Error in Estimates of Mean-Variance Efficient
Portfolio Weights. The Journal of Finance, 54(2), 655-671.
Brodie, J., Daubechies, I., Mol, C., Giannone, D., & Loris, I. (2009). Sparse and stable
Markowitz portfolios. Proceedings of the National Academy of Sciences, 1226712272.
84
TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE
Brooks, C. (2008). Introductory econometrics for finance (2nd ed.). Cambridge: Cambridge
University Press.
Campbell, J., & Lo, A. (1997). The econometrics of financial markets. Princeton, N.J.:
Princeton University Press.
Cha, H., & Jithendranathan, T. (2009). Time-varying correlations and optimal allocation in
emerging market equities for the US investors. International Journal of Finance &
Economics, 14, 172-187.
Corvalan, A. (2005). Well diversified efficient portfolios. Banco Central de Chile.
Crawley, M. (2012). The R Book (2nd ed., p. 1080). John Wiley & Sons.
Damghani, B. (2013). The Non-Misleading Value of Inferred Correlation: An Introduction to
the Cointelation Model. Wilmott, 50-61.
Dancey, C., & Reidy, J. (2004). Statistics without maths for psychology: Using SPSS for
Windows (3rd ed.). New York: Prentice Hall.
Ederington, L., & Guan, W. (2007). Time Series Volatility Forecasts for Option Valuation
and Risk Management. AFA 2008 New Orleans Meetings Paper. Retrieved March 20,
2015, from http://ssrn.com/abstract=956345
Engle, R. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the
Variance of United Kingdom Inflation. Econometrica, 50(4), 987-1007.
Engle, R. (2002). Dynamic Conditional Correlation - A simple class of multivariate GARCH
models. Forthcoming Journal of Business and Economic Statistics 2002.
Engle, R., & Sheppard, K. (2001). Theoretical and empirical properties of Dynamic
Conditional Correlation Multivariate GARCH. NBER Working Paper No. 8554.
Retrieved April 14, 2015, from http://www.nber.org/papers/w8554
85
86
TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE
EY, (2015). Estimating Risk-Free Rates for Valuations [PDF document]. Retrieved April 10,
2015
from
http://www.ey.com/Publication/vwLUAssets/EY-estimating-risk-free-
rates-for-valuations/$FILE/EY-estimating-risk-free-rates-for-valuations.pdf
Ghalanos, A., (2014). rmgarch: Multivariate GARCH models. R package version 1.2-8.
Retrieved March 9, 2015 from http://CRAN.R-project.org/package=rmgarch
Ghalanos, A., (2014). rugarch: Univariate GARCH models. R package version 1.3-4.
Retrieved March 9, 2015 from http://CRAN.R-project.org/package=rugarch
Graves, S., (2014). FinTS: Companion to Tsay (2005) Analysis of Financial Time Series. R
package version 0.4-5. Retrieved March
20, 2015 from
http://CRAN.R-
project.org/package=FinTS
Green, R., & Hollifield, B. (1992). When Will Mean-Variance Efficient Portfolios Be Well
Diversified? The Journal of Finance, 47(5), 1785-1809.
Hansen, P. R., & Lunde, A. (2005). A Forecast Comparison Of Volatility Models: Does
Anything Beat A GARCH(1,1)? Journal of Applied Econometrics, 873-889.
Hodges, S. D., & R. A. Brealey. (1978). Portfolio Selection in a Dynamic and
UncertinWorld, in James H. Lorie and R. A. Brealey (eds.), Modern Developments in
Investment Management (2nd ed.). Hinsdale, IL: Dryden Press.
Horasanl, M., & Fidan, N. (2007). Portfolio Selection by Using Time Varying Covariance
Matrices. Journal of Economic and Social Research, 9(2), 1-22.
Yin, G., & Zhou, X. (2004). Markowitz's mean-variance portfolio selection with regime
switching: From discrete-time models to their continuous-time limits. IEEE
Transactions On Automatic Control, 49(3), 349-360.
J. P. Morgan (1996). RiskMetrics: Technical document (4th ed.). Morgan Guaranty Trust
Company, New York.
Kaplan, P. D. (1998). Asset Allocation Models Using the Markowitz Approach.
TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE
Konno, H., & Yamazaki, H. (1991). Mean-Absolute Deviation Portfolio Optimization Model
and Its Applications to Tokyo Stock Market. Management Science, 37(5), 519-531.
Litterman, R. (2003). Modern Investment Management: An Equilibrium Approach. New
York: Wiley.
Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91.
Merton, R. (1973). An Intertemporal Capital Asset Pricing Model. Econometrica, 41, 867888.
Michaud, R. (1989). The Markowitz Optimization Enigma: Is ‘Optimized’ Optimal?
Financial Analysts Journal, 31-42.
Mora, M., Franco, J., & Preciado, L. (2010). Optimal Portfolio Allocation for Latin American
Stock Indices. Cuadernos De Administración, 23(40), 191-214.
Mossin, J. (1968). Optimal Multiperiod Portfolio Policies. The Journal of Business, 41, 215229.
Mwamba, M., & Mwambi, J. (2010). An alternative to portfolio selection problem beyond
Markowitz’s: Log Optimal Growth Portfolio. University of Johannesburg.
Peters, T. (2008). Forecasting the covariance matrix with the DCC GARCH model.
Mathematical Statistics Stockholm University, ISSN 0282-9169.
Peterson, B., G., & Carl, P., (2014). PerformanceAnalytics: Econometric tools for
performance and risk analysis. R package version 1.4.3541. Retrieved February 28,
2015 from http://CRAN.R-project.org/package=PerformanceAnalytics
Ruppert, D. (2011). Statistics and data analysis for financial engineering. New York:
Springer.
Said, S., & Dickey, D. (1984). Testing for Unit Roots in Autoregressive-Moving Average
Models of Unknown Order. Biometrika, 71(3), 599-607.
87
TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE
Samuelson, P. (1969). Lifetime Portfolio Selection By Dynamic Stochastic Programming.
The Review of Economics and Statistics, 51, 239-246.
Steinbach, M. (2001). Markowitz Revisited: Mean-Variance Models in Financial Portfolio
Analysis. SIAM Review, 43(1), 31-85.
Taylor, S. (1986). Modelling financial time series. John Wiley & Sons.
Trapletti, A., & Hornik, K., (2013). tseries: Time Series Analysis and Computational Finance.
R package version 0.10-32. Retrieved February 29, 2015 from http://CRAN.Rproject.org/package=tseries
Tse, Y. (1991). Stock returns volatility in the Tokyo stock exchange. Japan and the World
Economy, 3(3), 285-298.
Wei, T., (2013). corrplot: Visualization of a correlation matrix. R package version 0.73.
Retrieved March 2, 2015 from http://CRAN.R-project.org/package=corrplot
Wuertz, D., Setz, T., Chalabi, Y., & Rmetrics Core Team, (2014). fPortfolio: Rmetrics Portfolio Selection and Optimization. R package version 3011.81. Retrieved February
28, 2015 from http://CRAN.R-project.org/package=fPortfolio
Zeileis, A., & Grothendieck, G., (2005). zoo: S3 Infrastructure for Regular and Irregular
Time Series. Journal of Statistical Software, 14(6), 1-27. Retrieved February 28, 2015
from http://www.jstatsoft.org/v14/i06/
Zhou, X., & Li, D. (2000). Continuous-Time Mean-Variance Portfolio Selection: A
Stochastic LQ Framework. Applied Mathematics and Optimization, 19-33.
Zivot, E. (2006, May 30). Diagnostics for Constant Parameters [PDF document]. Retrieved
February 23, 2015 from
http://faculty.washington.edu/ezivot/econ424/rollingslides.pdf
Zivot, E. (2011, August 11). Functions for portfolio analysis [R script]. Retrieved March 15,
2015 from http://faculty.washington.edu/ezivot/econ424/portfolio.r
88
TIME VARYING PARAMETERS IMPACT ON PORTFOLIO PERFORMANCE
Zivot, E. (2012, May 8). R examples for lectures on multivariate GARCH models [R script].
Retrieved March 20, 2015 from
http://faculty.washington.edu/ezivot/econ589/econ589multivariateGarch.r
Zivot, E. (2013, August 1). Examples used in Portfolio Theory with Matrix Algebra Chapter
[R script]. Retrieved February 28, 2015 from
http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.r
Zivot, E. (2013, August 7). Portfolio Theory with Matrix Algebra [PDF document]. Retrieved
February 7, 2015
http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf
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
Zivot, E. (2013, August 21). Rolling Estimation of Efficient Portfolios [PDF document].
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