FOREX FORECASTING BY USING NGARCH MODEL
GAN LONG FATT
This report submitted in partial fulfillment of the
requirements for the award of the degree of
Master of Science (Mathematics).
Faculty of Science
Universiti Teknologi Malaysia
November 2009
iii
To my beloved family, for your love and support.
To my friends, for your wits, intelligence and guidance in life.
iv
ACKNOWLEDGEMENT
First and foremost, I would like to extend my heartfelt gratitude to my
supervisor, Prof. Dr. Zuhaimy Hjh Ismail for his guidance and support that he has given
me throughtout the duration of this research.
My fellow postgraduate students should also be recognized for their support. My
sincere appreciation also extends to all my colleagues and other who have provided
assistance at various occasions. Their views and opinions are helpful indeed.
Unfortunately it is impossible to list all of them in this limited space.
Last but not least, I would like to thank my family members who have given me
their undying support.
v
ABSTRACT
Foreign Exchange (Forex) is the market where a nation's currency trade with
another. Flexible exchange rate started in 1973 since Bretton Woods Agreement was
breakdown and it because the fluctuating exchange rate moves more drastically then
before. Thus, forecasting exchange rates have become very important and challenge
research issue for both academic and industrial. In this study used exchange rate selling
prices of RM/USD. The daily data cover the period from 03/05/2007 to 29/05/2009
exchange rate from Bank Negara Malaysia and it was volatility and moves through the
times. Thus, NGARCH model will be introduced in this study to forecast the selling
prices for RM/USD exchange rate in future and GARCH(1,1) as a benchmark. All the
data was analyzed by using Microsoft Office Excel 2007 software. The forecast
performance value of RMSE of NGARCH (1, 1) is 0.022809308 smaller than RMSE of
GARCH(1, 1) which value is 0.23439891. Therefore, the NGARCH model is more
accuracy than GARCH model.
vi
ABSTRAK
Pertukaran Wang Asing (FOREX) adalah pasaran di mana sebuah negara
perdagangan mata wang dengan satu negara yang lain. Pada 1973, keruntuhan Bretton
Woods Agreement menyebabkan keseluruhan dunia mula menerima kadar pertukaran
yang fleksibel. Oleh itu, kadar naik turun pertukaran sering bergerak lebih drastik
daripada dahulu. Oleh itu, kadar-kadar pertukaran peramalan telah menjadi amat penting
dan mencabar dalam kedua-dua bidang akademik dan perindustrian. Dalam kajian ini,
harga jualan kadar pertukaran RM/USD telah digunakan. Tempoh data yang digunakan
daripada Bank Negara Malaysia dari 03/05/2007 hingga 29/05/2009 dimana kemeruapan
dan bergerak melalui masa-masa itu. Oleh itu, model NGARCH akan diperkenalkan
dalam kajian ini untuk meramalkan harga jualan untuk RM/ USD kemudian hari dan
GARCH(1,1) seperti satu penanda aras. Semua data itu telah dianalisis dengan
menggunakan perisian Microsoft Office Excel 2007. Prestasi ramalan NGARCH (1, 1)
di mana nilai RMSE adalah 0.022809308 lebih kecil daripada GARCH (1, 1) di mana
value sama dengan 0.23439891. Oleh itu, model NGARCH adalah lebih tepat daripada
model GARCH.
ix
TABLE OF CONTENTS
CHAPTER
I
TITLE
PAGE
TITLE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xii
LIST OF SYMBOLS
xiv
LIST OF APPENDICES
xv
INTRODUCTION
1.0
Introduction
1
1.1
History of Foreign Exchange
2
1.2
Problem Statement
4
1.3
Objective of the Study
4
1.4
Scope of the Study
5
1.5
Organization of the Research
5
ix
II
III
IV
LITERATURE REVIEW
2.0
Introduction
7
2.1
Literature of ARCH and GARCH Model
7
2.2
Literature of NGARCH Model
9
METHODOLOGY
3.0
Introduction
11
3.1
Data
11
3.2
Stylized Facts of Asset Return
12
3.3
Generalised ARCH Models
14
3.3.1
NGARCH (1, 1) model
14
3.3.2
GARCH (1, 1) model
16
3.4
Maximum likelihood method
20
3.5
3.6
Microsoft Office Excel 2007
Forecast Performance
21
25
RESULT AND DISCUSSION
4.0
Introduction
26
4.1
The Exchange Rate Series
26
4.2
Stationary Series
30
4.3
Estimation
32
4.3.1. Parameters Estimation of NGARCH (1, 1)
32
4.3.2. Parameters Estimation of GARCH (1, 1)
38
Forecasting
43
4.4
ix
4.4.1 Forecasting future value by using
44
NGARCH (1, 1)
4.4.2Forecasting future value by using GARCH
45
(1,1)
4.5
Comparison of NGARCH (1,1) and GARCH
(1,1) Model
47
47
4.5.1 Comparison of Forecasting Model
V
CONCLUSION
5.0
Introduction
49
5.1
Result and Comparison
50
5.2
Suggestion for Further Research
51
REFERENCES
52
APPENDICES
55
x
LIST OF TABLES
TABLE NO.
4.1
TITLE
Data analysis of first order difference exchange
rate selling prices series
PAGE
31
4.2
Starting value of coefficient parameters of
NGARCH
32
4.3
Value of parameters coefficient for
NGARCH(1,1)
33
4.4
Standardized return for NGARCH(1,1) Model
37
4.5
Starting value of parameters coefficient for
GARCH(1,1) model
38
4.6
The coefficient parameters of GARCH(1,1)
39
4.7
Standardized Return for GARCH(1,1) Model
43
4.8
Comparison the real value against forecast value
by using NGARCH(1,1)
44
Comparison the real value against forecast value
4.9
by using GARCH(1,1)
45
xi
LIST OF TABLES
TABLE NO.
TITLE
PAGE
4.10
RMSE value of NGARCH(1,1)
47
4.11
RMSE value of GARCH(1,1)
48
5.1
The plot of actual prices against forecast prices
by NGACH(1,1) and GARCH(1,1)
53
xii
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
3.1
Estimation of unknown parameters- Step 1
Estimation of unknown parameters- Step 2(ii)
22
3.2
3.3
3.4
4.1
4.2
4.3
Estimation of unknown parameters- Step
2(iii)
23
24
Estimation of unknown parameters- Step 3(i)
The dynamics of the RM/USD exchange
rate selling prices in the period 2007-2009
Daily log returns for the RM/USD exchange
rate
The squared innovation return in the period
24
27
28
29
2007-2009
4.4
First Order Difference Selling Prices of
RM/USD
30
4.5
Answer Report from Solver for
NGARCH(1,1)
34
4.6
Conditional Variance
Model
35
4.7
Conditional
Standard
NGARCH(1,1) Model
4.8
Answer Report from Solver for GARCH(1,1)
40
4.9
Conditional Variance for GARCH(1,1) Model
41
4.10
Conditional Standard Deviation for
GARCH(1,1) Model
42
for NGARCH(1,1)
Deviation
for
36
xiii
LIST OF FIGURES
FIGURE NO.
4.11
5.1
TITLE
The plot of actual prices against forecast
prices by NGARCH(1,1) and GARCH(1,1)
model
PAGE
46
The plot of actual prices against forecast
prices by NGARCH(1,1) and GARCH(1,1)
model
50
xiv
LIST OF SYMBOLS
-
Logarithm return for daily closing prices
C
-
The observed assets prices
, , -
Conditional variance
-
Coefficient parameter of GARCH model
-
the correlation between returns and variance
-
iid standard normal random variable 0,1
-
the constant one period riskless domestic interest rate
-
the constant one period riskless foreign interest rate
-
the constant risk premium (the reward for investing in the foreign
currency)
xv
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
Foreign Exchange Daily Data for the selling
prices of RM/USD.
52
NO.
A
CHAPTER I
INTRODUCTION
1.0
Introduction
Foreign Exchange (Forex) is the market where a nation's currency trade with
another. In 1973, Bretton Woods Agreement was breakdown and it makes most of
countries in the world started to accept flexible exchange rate. Therefore, fluctuating of
exchange rate often move more drastically than before. Thus, forecasting exchange rates
have become very important and challenge research issue for both academic and
industrial.
Due to the fact that Forex forecasting is of practical as well as theoretical
importance, a large number of methods and techniques (including linear and nonlinear)
were introduced to beat random walk model in Forex market. With increasing
development of time series forecasting, researchers and investors are hoping that the
mysteries of the Forex market can be solve. In last two decades, extensive research has
been carried out on the returns of time series data Generalized Autoregressive
Conditional Heteroscedastic (GARCH) models. Many previous studies have been
discussed for FOREX forecasting.
In this study used exchange rate United State Dollar (USD) versus Ringgit
Malaysia (RM). The daily data cover the period from 03/05/2007 to 29/05/2009
2
exchange rate from Bank Negara Malaysia and it was volatility and moves through the
times. Thus, GARCH models will be introduced in this study to forecast the exchange
rate in future.
A GARCH process is a form of stochastic process that is commonly used in
finance and economics. These models are known as univariate time series models and is
very useful in analyze and forecast the volatility. These models were developing in order
to model and forecast the variance of financial and economic time series over time.
GARCH models are simple and easy to handle, it take care of clustered errors,
nonlinearities and changes in the econometrician’s ability to forecast. GARCH is a time
series modeling technique that used past variances and past variance forecasts to forecast
future variances.
1.1
History of Foreign Exchange
At 1870s, Western Europe was acceptance with their countries currency set par
value in term of gold which is the gold standard. Since governments agreed to buy or
sell gold on demand with anyone at its own fixed parity rate, therefore the value of each
currency in terms of gold are fixed.
Most central banks supported their currency with convertibility to gold before
World War I (WWI). Although paper money could always be exchanged for gold, in
reality this did not occur often, fostering the sometimes disastrous notion that there was
not necessarily a need for full cover in the central reserves of the government.
3
Occasionally, inflation and political instability are causes by the ballooning
supply of paper money without gold cover. Foreign exchange controls were introduced
to prevent market forces from punishing monetary irresponsibility to protect local
national interests.
In July 1994, the Bretton Woods agreement was reached on the initiative of the
USA in the latter stages of World War II (WWII). The agreement established a US
dollar based monetary system. Other international institutions such as the IMF, the
World Bank and GATT (General Agreement on Tariffs and Trade) were created in the
same period as the emerging victors of WWII searched for a way to avoid the
destabilizing monetary crises which led to the war. The Bretton Woods agreement
resulted in a system of fixed exchange rates that partly reinstated the gold standard,
fixing the US dollar at USD35/oz and fixing the other main currencies to the dollar - and
was intended to be permanent.
In 1973, Bretton Woods Agreement was breakdown and the dollar was no longer
suitable as the sole international currency at a time when it was under severe pressure
from increasing US budget and trade deficits. Foreign exchange trading was start
develop in the following decades into all around the world. Restrictions on capital flows
have been removed in most countries and most of countries in the world started to accept
flexible exchange rate.
4
1.2
Problem Statement
GARCH models take care of the kurtosis of the frequency and the volatility
clustering of the data. These two characteristics are very important in financial time
series. It can provide the accuracy of the forecasts of variances and covariance in assets
returns. It was ability to model time-varying conditional variances. The exchange rate
will be forecast by GARCH in this research. In fact, GARCH model can also be apply in
such field like risk management, portfolio management and asset allocation, option
pricing, foreign exchange and the structure of interest rates.
In this research, we can easy found the highly significant of GARCH in the
equity market, at present and for the future. GARCH models can use to examine the
relationship between the long-term and short-term of the interest rates. Moreover,
GARCH models can be using in analysis the time-varying risk premiums and the Forex
market which consist the highly persistent periods of volatility and clustering.
1.3
Objectives of the Study
The objectives of this study are:
i. To explore NGARCH and GARCH model of forecasting exchange rate by using
Microsoft Excel 2007.
ii. To explore the NGARCH model in predict the Forex data.
iii. To predict the daily exchange rate for RM/ USD selling prices.
5
1.4
Scope of the Study
This research is focuses on examination of the Forex market by using NGARCH
model. The GARCH models will be introduced as the benchmark to this study. There
cover 2 years of exchange rate data will be forecast by using NGARCH model for more
understanding for the reader. Hopefully this study will increase the understanding and
further exploration about the GARCH model.
1.5
Organization of the Research
This research is organized into five chapters.
Chapter 1 gives the introduction of this study. It begins with the overview of
FOREX market, GARCH models appoarch, the objectives and the scopes.
Chapter 2 and 3 describes the introduction of GARCH models with its
specifications. The details and principles will also be focusing. It will present the
GARCH family with their consistency and the appropriate model for the data will be
selected.
Chapter 4 describes introduction of case study and analyze the data of Forex
market which RM/USD. The data consists of daily observations of buying and selling
price of exchange rate against RM (Ringgit Malaysia) over the periods 2007 – 2009.
Between, this chapter will identify the variables that can influence the moving of Forex
and its future. These variables will be included in the models so that the results in predict
the exchange rate will more consistency and accurately.
6
Chapter 5 gives a summary of the whole study. Some conclusions are drawn and
finally, some thoughts on possible directions in which future research in this area might
be pursued are offered.
Chapter II
LITERATURE REVIEW
2.0
Introduction
This chapter contains a literature review of the study of justify the stated
objectives. Section 2.1 contains the literature of ARCH and GARCH model. Section 2.2
studies the literature of NGARCH model.
2.1
Literature of ARCH and GARCH Model
Financial time series such as stock price or exchange rate usually are available on
very high frequencies such as minute by minute. However, the econometrician uses
highly aggregated data such as daily or weekly returns. Recently, many empirical studies
have found different estimating approaches around the volatility problem (Bates, 1996;
Kim & Kim, 2004; Watanabe, 1999). As alternative to the historical and implied
approach, numerous models are devised that correspond to the stochastic volatility
process characteristic. One widespread approach is ARCH or generalized ARCH
(GARCH), devised by Engle (1982) and Bollerslev (1986).
8
Engle (1982) developed the Autoregressive Conditional Heteroskedasticity
(ARCH) model to characterize the observed serial correlation in asset price volatility.
Suppose we assume that a price follows a random walk ,
where ~ 0, . The variance of the error term depends upon , and the objective
of the model is to characterize the way in which this variance changes over time. The
ARCH model assumes that this dependence can be captured by and autoregressive
process of the form
where the restrictions 0, !
0 for " 0, 1, , # ensure that the predicted
variance is always nonnegative. This specification illustrates clearly how current levels
of volatility will be influenced by the past, and how periods of high or low price
fluctuation will tend to persist.This volatility approach is calculated using observations
of historical daily asset prices and considering both the conditional and unconditional
variance in the estimation process. Afterward introduced by Engle, it follow by various
extensions of the ARCH model were made by many researchers.
Bollerslev (1986) and Taylor (1986), independently of each other, suggested the
generalized ARCH (GARCH) model. GARCH models have been found to be suitable
for describing stock returns data, such as volatility clustering and thick-tailed behavior.
The GARCH model has been used to characterize patterns of volatility in US dollar
foreign exchange markets (Baillie and Bollerslev 1989 and 1991). However, initial
investigations into the explanatory power of out of sample forecasts produced
disappointing results (West and Cho, 1995). Jorion (1995) found that volatility forecasts
for several major currencies from the GARCH model were outperformed by implied
volatilities generated from the Black Scholes option pricing model.
9
In 1991, Nelson was developed the exponential GARCH (EGARCH) model to
analyze the effects of positive and negative stock market news, with negative shocks
affecting volatility differently from positive shocks of equal magnitude. Moreover, the
possibility of leverage, whereby negative shocks increase volatility while positive
shocks decrease volatility, is present in the EGARCH formulation.
In 1998, Andersen and Bollerslev were demonstrated that one can significantly
improve the forecasting power of GARCH model by measuring volatility as the sum of
intraday squared returns. This measure is referred to as integrated or realized volatility.
In theory, if the true underlying price path is a diffusion process it is possible to obtain
progressively more accurate estimates of the true volatility by increasing the frequency
of intraday observation. Of course, there are practical limits to this; micro structural
effects begin to degrade accuracy beyond a certain point.
Ding, Granger and Engle (1993), Glosten, Jagannathan and Runkle (1993) and
Hentschel (1995) were extensions of the leverage effect and other GARCH. Bollerslev,
Chou and Kroner (1992) and the work of Duan (1997) enclose the existing GARCH
models into a common system known as the increased GARCH (p,q) process.
2.2
Literature of NGARCH Model
GARCH option pricing model can be describe by using the standard discretetime GARCH specification. Particularly, the GARCH process known as the non-linear
asymmetric GARCH(1,1) (NGARCH(1,1)) process is restrict from the first appeared in
Engle and Ng (1993). Our pricing results can, nevertheless, be extended to all
GARCH(1,1) specifications with little effort. Since NGARCH (1,1) process fulfill the
leverage effect which is an important feature of asset returns, hence it chosen over the
standard linear GARCH(1,1) model. In, other hand, GARCH models such as the
EGARCH process by Nelson (1991) and the GJR-GARCH process by Glosten et al.
10
(1993) can also capture the leverage effect. According to Duan (1997) concluded that
the NGARCH model is better although Engle and Ng (1993) conclude the GJR-GARCH
model performs best. Since there is no conclusive evidence in favor of a particular
GARCH specification, our choice of the NGARCH model is better viewed as for the
demonstration purpose for the time being.
In the main paper (Petra POSEDEL, 2006), the author was state that the
NGARCH model could be used for option pricing which as an alternative to the Black
Scholes model. The author was shows that introducing heteroscedasticity results in
better fitting of the empirical distribution of foreign currency than the Brownian model.
For simulation, the NGARCH model show it is possible to described the empirically
observed facts from foreign currency option markets.
CHAPTER III
METHODOLOGY
3.0
Introduction
The time series is the observations of variable Y that become available over time,
which are often recorded at fixed time intervals. Monthly inventory levels, weekly
exchange rate, gold rate and sequence of annual sales figures are examples of time
series. Generally, time series required special methods for the analysis as the
observations are related to one another which called autocorrelation. This produces
patterns of variability that can be used to forecast future values.
This chapter contains a methodology of the study of justify the stated objectives.
We will first discuss Generalised Autoregressive Conditional Heteroskedasticity
(GARCH) model which is more flexible than the approach ARCH in Section 3.1. In
Section 3.2, we will discuss the Maximum Likelihood method (MLE) to estimate the
unknown parameters. The software used in this study will be discuss in Section 3.3.
3.1
Data
In this research, the measurements of the data from 3rd May 2007 to 29th May
2009 are used in the model development and simulation and the remainder is kept for the
12
purpose of accuracy evaluation. These data consists of daily exchange rate in Bank
Negara Malaysia (BNM). The data set obtained from the website of BNM.
3.2
Stylized Facts of Asset Return
Movements in asset prices or equivalently asset returns is the causes of market
risk, the first step is by defining returns and then give an overview of the characteristics
of typical asset returns.
By defining the daily geometric or “log” return on an asset as the change in the
logarithm of the daily closing price of the asset,
$%& ' $%&
(3.1)
By using log return it can easily calculate the compounded return at the K−day
horizon simply as the sum of the daily returns.
:) ln&) ' ln& )
, ln&- ' ln&- -.
)
, ln&- – ln&- -.
(3.2)
-
13
Below are the following lists of so-called stylized facts, which apply to most
stochastic returns.
(i.)
It
have
a
very
little
&0 , 1 2 0
autocorrelation
in
daily
return.
for 3 1, 2, 3, . . . , 100
In other words, returns are almost impossible to predict from their own
past.
(ii.)
The standard deviation of returns completely dominates the mean of
returns at short horizons such as daily. It is not possible to statistically
reject a zero mean return.
(iii.)
Variance, measured, for example, by squared returns, displays positive
correlation with its own past. This is most evident at short horizons such
as daily or weekly.
(iv.)
The negative correlation value between variance and returns is the
leverage effect.
(v.)
Correlation between assets appears to be time varying. Importantly, the
correlation between assets appears to increase in highly volatile down
markets and extremely so during market crashes.
(vi.)
As the return-horizon increases, the unconditional return distribution
changes and looks increasingly like the normal distribution.
14
3.3
Generalised ARCH Models
GARCH is a short form of Generalized Autoregressive Conditional
Heteroscedasticity. Heteroscedasticity know as time-varying variance which means
volatility. Conditional implies a dependence on the observations of the immediate past,
and autoregressive describes a feedback mechanism that incorporates past observations
into the present. Therefore, GARCH model is a mechanism that includes past variances
in the explanation of future variances. Besides that, GARCH is a time series modeling
technique that uses past variances and past variance forecasts to forecast future
variances.
3.3.1
NGARCH (1, 1) model
The GARCH(1,1) is the simplest and most robust of the family of
volatility models. However, the model can be extended and modified in many
ways. NGARCH is the modification among of GARCH model.
With modify the GARCH models so that the weight given to the return
depends on whether the return is positive or negative in the following simple
manner:
– 7 2 ' 7 which is referred to as the NGARCH (nonlinear GARCH) model.
(3.3)
15
The dynamics of the time series of return is described with a
nonlinear-in mean, asymmetric GARCH (1,1) model (Engle and Ng, 1993):
8 ln 9
&
:
&
' ' (3. 4)
' (3. 5)
where are i.i.d. standard normal random variables N (0,1) and
; 0, 0, 0, and 1 < 1
(3. 6)
where
the constant one period riskless domestic interest rate
the constant one period riskless foreign interest rate
the constant risk premium (the reward for investing in the foreign
currency)
the correlation between returns and variance.
For the tomorrow’s variance, , is known at the end of today’s time .
Let us denote with = > ? the expected return in time . Therefore, the
expected return and variance of the return based on the information
available until time can be write as
= > ? ' ' (3.7)
16
@A > ? (3.8)
The unconditional variance for NGARCH can be define as
?
8 = >
B
CDE F
(3. 9)
From relation Equation (3.7) and Equation (3.8) it follows that the
forecast of the variance is directly given by the model with . If observe the
forecast of daily returns variance for k periods ahead, using the recursive
specification of the asymmetric GARCH model (3.5) it follows that
?
= >' >
1 ?- (3. 10)
where
?
= = >represents the expected value of the future
variance for horizon G.
The expression 1 is the persistence of the model. If the
value of 1 is near 1 then shocks in the market persist through a long
timeG H ∞. In that case the time series will say has a long memory. In other
word, when 1 is small, it show that shocks in returns die out more
quickly in time.
3.3.2
GARCH (1, 1) model
17
The simplest generalized autoregressive conditional heteroskedasticity
(GARCH) model is GARCH (1, 1) model. The (1,1) in parentheses is a standard
notation in which the first number refers to how many autoregressive lags, or
ARCH terms, appear in the equation, while the second number refers to how
many moving average lags are specified, which here is often called the number
of GARCH terms.
The dynamic of the time series of return is described with a
generalized autoregressive conditional heteroskedasticity (GARCH) model can
be written as
8 ln 9
&
:
&
' ' (3.11)
(3.12)
where are independently and identically (i. i. d) standard normal variables
0,1 and
; 0, ; 0, ; 0, and < 1
(3.13)
so that next period forecast of variance is a blend of last period forecast and last
period’s squared return.
For the tomorrow’s variance, , is known at the end of today’s time .
Let us denote with = > ? the expected return in time . Therefore, the
18
expected return and variance of the return based on the information
available until time can be write as
= > ? ' ' @A > ? (3.14)
(3.15)
The unconditional, or long-run average, variance, , can define as
?
8 = >
= > ? = > ?
so that
/1 ' ' (3.16)
The GARCH model, in turn, implicitly relies on . This can be seen by
solving for in the long-run variance equation and substituting it into the
dynamic variance equation. We get
1 ' ' ' ' (3.17)
19
Thus, tomorrow’s variance is a weighted average of the long-run
variance, today’s squared return, and today’s variance. Put differently,
tomorrow’s variance is the long-run average variance with something added
(subtracted) if today’s squared return is above (below) its long-run average, and
something added (subtracted) if today’s variance is above (below) its long-run
average.
It is often useful not only to forecast next period’s variance of returns, but
also to make an k-step ahead forecast, especially if the goal is to price an option
with k- steps to expiration using the volatility model. Again starting from the
, it can derive the forecast for next period’s
GARCH(1,1) equation for variance,
?
= >' = >-
' 2? = >-
' ?
= >-
- ' ? = > - ' ?
= >-
? ' ,
so that
= >? ' 2 - = >
? ' - J
– K
(3.18)
where
?
conditional expectation, refers to taking the expectation using all the
= >-
information available at the end of day t , which includes the squared return on
day t itself.
20
From the above equation = >? H as G H ∞ so as the forecast
horizon goes to infinity, the variance forecast approaches the unconditional
variance. From the G-step ahead variance forecast, as the persistence of
the model determines how quickly the variance forecast converges to the
unconditional variance.
In Section 3.3.1 and 3.3.2, we have mentioned the domestic and foreign interest
rate in Equation (3.4), (3.11) and (3.14) which are and . In this study, for illustration
purposes, and are taken as constants of known values. It is possible to consider and as time series and in that case we estimate them from annual report from BNM.
On the annual level the respective rates are approximately 3.3% and 2.9%. The
analysis of course can be done for any other choice of these values. Since the frequency
of data is at a daily level, we transform the annual interest rate into a daily interest rate
by dividing by 252 the average number of business days in one year.
For , this is the constant risk premium i.e. the reward for investing in the foreign
currency. We notice that a negative value of decreases the mean value RM/USD
exchange return which indicates the appreciation of the domestic currency. Furthermore,
if we are observing the USD/RM exchange rate a positive value of the parameter will
explain the appreciation of the domestic currency.
3.4
Maximum likelihood method
21
Maximum Likelihood Estimation (MLE) is a popular statistical method used for
fitting a statistical model to data, and providing estimates for the model’s parameters.
Let denote with 7 , , , , for GARCH (1,1) model and NGARCH (1,1)
model which are sets of unknown parameters.
For GARCH (1, 1) and NGARCH (1, 1) model with Normal conditional returns,
the likelihood function is
$ √MNO
exp '
Y
E
E SO TU TV WNOXY NOXY
NOE
(3.19)
and thus the joint likelihood of the model is
Z[ ∏[. $
∏[.
√MNO
exp '
(3.20)
Y
E
E SO TU TV WNOXY NOXY
NOE
(3.21)
Since the ln Z function is monotonically increasing function of Z, with maximize
the log of the likelihood function
Z[
∑[ >' ln2^ [ .
'
Y E
9SO _TU TV WNOXY ENOXY `:
ln '
?
NOE
(3.22)
under condition given by constraint (3.5) which for NGARCH (1,1) model and
constraint (3.16) which for GARCH (1,1) model into the Equation (3.22).
22
The maximization of the function Z [ by the parameters of the model is done by
using the Solver in Microsoft Office Excel. By using the Solver we can get the value of
the unknown parameters.
3.5
Microsoft Office Excel 2007
Microsoft Office Excel 2007 is an electronic spreadsheet program that can be
used for storing, organizing and manipulating data. Normally spreadsheets are often
used to store financial data instance performing basic mathematical operations such as
summing columns and rows of figures, finding values such as profit or loss calculating
repayment plans for loans or mortgages and finding the average, maximum, or minimum
values in a specified range of data.
In this study, we will use the add-in function in Microsoft office Excel 2007 to
add in the Solver to estimate the parameters which are , , , , and . To add in the
Solver in Microsoft Excel 2007, go to Excel Option>> Add-Ins >> Manage: Excel Addins >> Solver add-in >> Ok.
Here is the application of Microsoft Office Excel 2007 software in discussing the
parameters estimated. Firstly, insert all the data in sequences of Date. After that key in
all related function to get period return, square return, conditional variance and
conditional standard deviation which as shown in Figure 3.1.
23
Step 1: Insert all the data in sequences.
Figure 3.1: Estimation of unknown parameters- Step 1
To estimate the unknown parameters, we need to assume a number for
, , , , and under the condition given in constraint Equation (3.5) and constraint
Equation (3.16).
Step 2:
i.)
Put the cursor at Target Cell that is cell G1.
ii.)
Click Data>> Solver.
iii.)
Key in all the data as Figure 3.3.
24
Figure 3.2: Estimation of unknown parameters- Step 2(ii)
25
Figure 3.3: Estimation of unknown parameters- Step 2(iii)
Step 3:
(i.)
Click Solve and a dialog will pop up as Figure 3.4.
Figure 3.4: Estimation of unknown parameters- Step 3(i)
(ii.)
Click all the Report and click Ok.
(iii.)
All the report will show in other 3 new sheets.
3.6
Forecast Performance
When reach the forecasting step, we would compare RMSE, values from
NGARCH (1, 1) and GARCH(1,1) models. RMSE is the short form of Root Mean
Square Error which is generating from Mean Square Error (MSE).
RMSE which has the same units as the quantity being estimated; for an unbiased
estimator, the RMSE is the square root of the variance, known as the standard error. In
26
statistical modelling, the RMSE is a measure of the differences between values predicted
by a model or an estimator and the values actually observed from the thing being
modeled or estimated. RMSE is a alternative form for the standard deviation, calculate
using the formula below (Equation 3.21).
∑[.e ' ef abc= d
%
where
e the actual value of the data
ef = the predicted value of the data
% number of observation
Obviously, in a forecast time series, the smaller forecast performances are
preferred. A model which has a small RMSE is said that is a better model then others.
Chapter IV
RESULT AND DISCUSSION
4.0
Introduction
This chapter contains a result and discussion of the study. We will firstly discuss
the results by application of Microsoft Office Excel 2007 for GARCH and NGARCH
model in Section 4.1. In Section 4.2, we will discuss the results of comparison between
GARCH and NGARCH model and time series methods are presented.
4.1
The Exchange Rate Series
The in-sample period of exchange rate data from 3rd May 2007 to 30th April 2009
will be plotted with Microsoft Office Excel. The exchange rate series is shown in the
figure below
28
Selling prices of RM/USD from
03/05/07 - 29/05/09
3.8
3.7
3.6
3.5
3.4
3.3
3.2
Selling prices
of RM/USD
3.1
3
2.9
3/5/2007
1/6/2007
2/7/2007
31/7/2007
29/8/2007
28/9/2007
30/10/2007
29/11/2007
2/1/2008
5/2/2008
7/3/2008
8/4/2008
8/5/2008
9/6/2008
8/7/2008
6/8/2008
5/9/2008
8/10/2008
7/11/2008
9/12/2008
12/1/2009
16/2/2009
18/3/2009
16/4/2009
2.8
Figure 4.1: The dynamics of the RM/USD exchange
rate selling prices in the period 2007-2009
In figure 4.1, the time series graph has been plotted from 3rd January 2007 to 30th
April 2009. The exchange rate for RM/USD selling prices trend corresponding to the
analyzed period and indicates that the selling prices of RM/USD have mainly fluctuated
in the range of about RM3.4 to RM 3.7. During the period in end of the year 2007 until
29
July 2008, there occurs a breakdown. The selling prices of RM/USD are around RM3.1
to RM3.3 when the globalization economic crisis. After July 2008, the selling prices of
RM/USD fluctuations are increase with quite equilibration that is around RM3.4 to RM
3.7.
Next, we plot the daily log returns prices of the RM/USD exchange rate.
Daily log returns for the RM/USD exchange
rate
0.02000000
0.01500000
0.01000000
0.00500000
log returns
for RM vs
USD
0.00000000
-0.00500000
-0.01000000
-0.01500000
3/5/2007
6/6/2007
10/7/2007
13/8/2007
17/9/2007
22/10/2007
26/11/2007
2/1/2008
12/2/2008
17/3/2008
21/4/2008
27/5/2008
30/6/2008
1/8/2008
5/9/2008
13/10/2008
17/11/2008
22/12/2008
30/1/2009
10/3/2009
13/4/2009
-0.02000000
Figure 4.2: Daily log returns for the RM/USD
exchange rate
30
Figure 4.2 show the returns on the RM/USD exchange rate with the average
volatility. In Figure 4.2, it shows the difference with respect to first order for exchange
rate selling prices series. The difference series is stationary because most of the price
values are located around mean of zero. But, there are some spikes found in the figure
that representing those high volatility periods.
Squared innovation return in the period
2007-2009
0.00035
0.00030
0.00025
Squared
innovation
return
0.00020
0.00015
0.00010
0.00005
3/5/2007
6/6/2007
10/7/2007
13/8/2007
17/9/2007
22/10/2007
26/11/2007
2/1/2008
12/2/2008
17/3/2008
21/4/2008
27/5/2008
30/6/2008
1/8/2008
5/9/2008
13/10/2008
17/11/2008
22/12/2008
30/1/2009
10/3/2009
13/4/2009
0.00000
Figure 4.3: The squared innovation return in the
period 2007-2009
31
Figure 4.3 describes the changes in the RM/USD exchange rate prices in the
period from 2007-2009. From the figure, it can be immediately noticed that some
periods have very different volatilities, which can also seen from the graph of the
squared innovation returns.
4.2
Stationary Series
Hereby, we provide the graph plot to show the stationary of this data series. First
lagged difference from the original data time series which has giving the stationary
series is shown in the figure below.
First Order Difference Selling Prises
0.08
0.06
0.04
0.02
0
First Order
Difference
Selling
Prises
-0.02
-0.04
3/5/2007
1/6/2007
2/7/2007
31/7/2007
29/8/2007
28/9/2007
30/10/2007
29/11/2007
2/1/2008
5/2/2008
7/3/2008
8/4/2008
8/5/2008
9/6/2008
8/7/2008
6/8/2008
5/9/2008
8/10/2008
7/11/2008
9/12/2008
12/1/2009
16/2/2009
18/3/2009
16/4/2009
-0.06
Figure 4.4: First Order Difference Selling prices of RM/USD.
32
In Figure 4.4, it shows the difference with respect to first order for selling prices
series. The difference series is stationary because most of the price values are located
around mean of zero. But, there are some spikes found in the figure that representing
those high volatility periods.
Let us check with its normality distribution test statistics including mean,
median, maximum and minimum values, standard deviation, skewness, kurtosis, range,
confidence interval of first order difference exchange rate selling prices series as Table
4.1.
Table 4.1: Data analysis of first order difference exchange rate selling prices series
Data Analysis
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Range
Minimum
Maximum
Sum
Count
Largest(1)
Smallest(1)
Confidence Level(95.0%)
0.0003
0.0723
0.0005
0.0000
1.0142
0.0002
0.1040
-0.0485
0.0555
0.1400
493.0000
0.0555
-0.0485
0.0013
From Table 4.1, the mean value is 0.0003 and standard deviation 1.0172 which
gives a standard normal distribution of 0,1. The median is equal to 0 because most
of the values after first lagged difference fall within the intervals of zero. The skewness
and kurtosis values are -0.0455 and 1.1584respectively which shown the distribution is
slightly asymmetric and highly leptokurtosis.
33
4.3
Estimation
4.3.1
Parameters Estimation of NGARCH (1, 1)
In this section, estimation of the parameters in NGARCH (1,1) and GARCH
(1,1) model will be discussed. In this study, we have five unknown parameter need to
estimate which are , , , and . The parameter estimation of the model will be done
by Microsoft Office Excel 2007 software. Firstly, set the unknown parameters for
NGARCH (1,1) model as follow:
Table 4.2: Starting value of coefficient Parameters of NGARCH(1,1)
Starting value
ω
6.42866 x10^-07
α
0.05000000
β
0.90000000
ρ
0.50000000
λ
0.50000000
α(1+ρ2)+β
0.96250000
MLE
1928.63450700
After define the unknown parameters, by using Solver in Microsoft Office excel
the five unknown parameters was found. The result was shown as Table 4.3.
34
Table 4.3: Value of parameters coefficient for NGARCH(1, 1)
NGARCH (1,1)
ω
6.42866 x10-07
α
0.05000000
β
0.90000000
ρ
0.50000000
λ
0.50000000
α(1+ρ2)+β
0.96250000
MLE
1928.63450700
In Table 4.3, it shown that the values of unknown parameters are unchanged after
applies by Solver. From the Figure 4.4, it was shown that the MLE =1928.63450700,
6.42866 m 10n , 0.0500, 0.9000, 0.5000, 0.5000 and
1 0.9625. All the values we get are under condition as Equation (3.14).
The expression 1 is called persistence of the NGARCH (1,1) model. From
expression it follows that the shocks in the market persist through a long time which
define as G H ∞ since the value of 1 is near 1. In this case, we will say
that the time series has a long memory.
35
Figure 4.5: Answer Report from Solver for NGARCH(1,1)
From the obtained result, the optimal parameters imply the following variance
dynamics:
6.42866 x10n 0.5 ' 0.5 0.9
36
The NGARCH (1, 1) model can be written into conditional mean and conditional
variance as
1 = > ? 0.0131% ' 0.0115% 0.5 ' 2
@A > ? The following figures present the graphical plot for conditional variance and
conditional standard deviation. The values for conditional standard deviation are taking
square roots from conditional variance.
Conditional Variance of selling prices
RM/USD
0.000070
0.000060
0.000050
0.000040
0.000030
conditional
variance prices
0.000020
0.000010
3/5/2007
7/6/2007
12/7/2007
16/8/2007
21/9/2007
29/10/2007
4/12/2007
14/1/2008
22/2/2008
31/3/2008
6/5/2008
11/6/2008
16/7/2008
20/8/2008
25/9/2008
4/11/2008
10/12/2008
19/1/2009
27/2/2009
6/4/2009
0.000000
Figure 4.6:
Conditional Variance for NGARCH(1,1) model
37
In Figure 4.5, conditional variance graph is plotted. There are lesser spikes
compared to conditional standard deviation graph but it does point out some of the high
volatile clusters in the series. However, the extraordinary long spikes are the high
volatile periods in the data series.
Conditional Standard Deviation for selling
pricess RM/USD
0.009
0.008
0.007
0.006
0.005
0.004
Conditional
Standard
Deviation
0.003
0.002
0.001
3/5/2007
1/6/2007
2/7/2007
31/7/2007
29/8/2007
28/9/2007
30/10/2007
29/11/2007
2/1/2008
5/2/2008
7/3/2008
8/4/2008
8/5/2008
9/6/2008
8/7/2008
6/8/2008
5/9/2008
8/10/2008
7/11/2008
9/12/2008
12/1/2009
16/2/2009
18/3/2009
16/4/2009
0
Figure 4.7:
Conditional standard deviation for NGARCH(1,1) model
A conditional standard deviation graph is drawn by Microsoft Office Excel
shown in Figure 4.6. The extraordinary long spikes are the high volatile periods of the
series. With NGARCH(1,1) model, the volatility clustering will be detected.
38
After we estimate the parameters, next step will be diagnostic checking on the
adequacy for NGARCH (1, 1) model. It can be done through when checking on the
autocorrelation and partial autocorrelation of standardized return approximately zero.
Table 4.4: Standardized Return for NGARCH (1, 1) Model
ACF
Standardized
return
Lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.140728
0.037413
0.056746
0.14629
0.043647
0.044736
-0.004524
0.082708
0.015038
0.048333
-0.019805
0.019213
-0.042582
0.039118
-0.005254
-0.022726
-0.032651
-0.002665
0.132111
-0.032387
PACF
Standardized
return
0.140728
0.017965
0.050054
0.134068
0.004114
0.03158
-0.028206
0.067415
-0.013301
0.038724
-0.034765
0.005673
-0.052113
0.039027
-0.007224
-0.029667
-0.013452
-0.012182
0.152998
-0.076134
t-Statistics
3.12468
0.81473
1.23407
3.17162
0.92754
0.94904
-0.09579
1.75135
0.31646
1.01695
-0.41583
0.40326
-0.89345
0.81945
-0.10992
-0.47542
-0.68271
-0.05567
2.75975
-0.66634
Q-Statistics
9.8232
10.5189
12.1226
22.8026
23.7552
24.7581
24.7684
28.2103
28.3243
29.5047
29.7033
29.8905
30.8124
31.592
31.6061
31.8703
32.4169
32.4205
41.4062
41.9474
In the Table4.4, ACF and PACF of square return are approximately zero. The
Ljung-Box q-statistic also provide the similar evidence with r-value that
NGARCH(1,1) model is adequate. Once again, we can conclude that the model is
adequate.
39
4.3.2
Parameters Estimation of GARCH (1, 1)
For the GARCH (1,1) model, as the same step as NGARCH (1,1) model, the
starting value was set in Microsoft Excel 2007 as Table 4.4.
Table 4.5 Starting values of parameters coefficient for GARCH model
Starting value
ω
0.001144629
α
0.05000000
β
0.90000000
ρ
0.50000000
λ
0.50000000
α+β
0.95000000
MLE
1928.63450700
Refer to Table 4.5, we have generated the coefficient parameter of GARCH (1,
1) model by Microsoft Office Excel software. The estimated parameter coefficients by
GARCH (1, 1) model gives 9.59927 m 10n , 0.049997, 0.8999, 0.5,
0.4999, and 0.95. The coefficients which estimate by using Solver are all
under condition as shown at Equation (3.5). From expression it follows that the shocks
in the market persist through a long time which defines as G H ∞ since the value of
is near 1. In this case, we will say that the time series has a long memory.
40
Table 4.6: The coefficient parameters of GARCH (1,1)
GARCH(1,1)
ω
9.59927 m 10n
α
0.049997095
β
0.899985239
ρ
0.5
λ
0.499998103
α+β
MLE
0.95000006
1824.590093
From the obtained result, the optimal parameters imply the following variance
dynamics:
9.59927 m 10'7 0.049997095 0.899985239
The GARCH (1, 1) model can be written into conditional mean and conditional
variance as
1 = > ? 0.0131% ' 0.0115% 0.5 ' 2
@A > ? 41
Figure 4.8: Answer Report from Solver for GARCH(1,1)
The following figures present the graphical plot for conditional variance and
conditional standard deviation. The values for conditional standard deviation are taking
square roots from conditional variance.
42
Conditional Variance of Selling Prices
RM/USD
0.0200000000
0.0150000000
0.0100000000
0.0050000000
0.0000000000
-0.0050000000
3/5/2007
30/5/2007
26/6/2007
23/7/2007
17/8/2007
14/9/2007
11/10/2007
9/11/2007
6/12/2007
7/1/2008
6/2/2008
6/3/2008
3/4/2008
30/4/2008
29/5/2008
25/6/2008
22/7/2008
18/8/2008
15/9/2008
14/10/2008
11/11/2008
9/12/2008
8/1/2009
10/2/2009
10/3/2009
6/4/2009
-0.0100000000
coditional variance
Figure 4.9:
Conditional Variance for GARCH(1,1) model
In Figure 4.8, conditional variance graph is plotted. There is having a consistent
pattern of the graph. All the point is lay around the zero. However, the extraordinary
long spikes are the high volatile periods in the data series.
Next, we will plot a conditional standard deviation graph as in Figure 4.9. The
extraordinary long spikes are the high volatile periods of the series. With GARCH(1,1)
model, the volatility clustering will be detected.
43
Conditional Standard Deviation of Selling
Prices RM/USD
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
3/5/2007
25/5/2007
18/6/2007
10/7/2007
1/8/2007
23/8/2007
17/9/2007
9/10/2007
1/11/2007
26/11/2007
18/12/2007
15/1/2008
12/2/2008
5/3/2008
28/3/2008
21/4/2008
14/5/2008
6/6/2008
30/6/2008
22/7/2008
13/8/2008
5/9/2008
29/9/2008
23/10/2008
17/11/2008
10/12/2008
6/1/2009
30/1/2009
25/2/2009
20/3/2009
13/4/2009
0
Conditional Standard Deviation
Figure 4.10: Conditional Standard Deviation for GARCH(1,1) model
After we estimate the parameters, next step will be diagnostic checking on the
adequacy for GARCH (1, 1) model. It can be done through when checking on the
autocorrelation and partial autocorrelation of standardized return approximately zero.
44
Table 4.7: Standardized Return for GARCH (1, 1) Model
Lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
ACF
PACF
t-Statistics
Q-Statistics
Standardized
Standardized
Return
Return
0.126154
0.126154
2.80108
7.8939
0.067216
0.052131
1.46924
10.1394
0.061714
0.047992
1.34311
12.0363
0.164783
0.151545
3.57318
25.5872
0.03369
-0.008346
0.71233
26.1547
0.057214
0.037489
1.20846
27.795
-0.00347
-0.0304
-0.07307
27.8011
0.097608
0.075134
2.05556
32.5949
-0.00794
-0.034447
-0.16579
32.6267
0.020398
0.006077
0.42591
32.837
-0.016821
-0.020946
-0.35109
32.9802
0.082243
0.063771
1.71616
36.4119
-0.045604
-0.058119
-0.94599
37.4692
0.035716
0.035559
0.73953
38.1191
-0.005767
-0.005564
-0.11928
38.1361
0.000185
-0.026913
0.00383
38.1361
-0.040096
-0.021409
-0.82928
38.9603
-0.021521
-0.032311
-0.44448
39.1983
0.122863
0.1547
2.53655
46.97
-0.028929
-0.079511
-0.5896
47.4018
In the Table4.6, ACF and PACF of square return are approximately zero. The
Ljung-Box q-statistic also provide the similar evidence with r-value that GARCH(1,1)
model is adequate. Once again, we can conclude that the model is adequate.
4.4
Forecasting
45
4.4.1
Forecasting future value by using NGARCH (1,1)
Future values of selling prices of RM/USD forecast by NGARCH (1, 1) model
using Microsoft Office Excel. The duration of forecast is started from 4th May 2009 to
29th May 2009. The value of forecast is according to static forecasting approach which
performs the series of one-step ahead forecast of exchange rate selling prices of
RM/USD.
Table 4.6 was shown that the real value against forecast value by using
NGARCH (1, 1) model from 04th May 2009 to 29th May 2009. Refer to Table 4.6, we
can saw that the forecasted value by using Microsoft Office Excel 2007 are quite near
with the real value. It was shown that the accuracy of NGARCH (1,1) model.
Table 4.8: Comparison the real value against forecast value by using NGARCH
(1,1)
Real Value
4/5/2009
5/5/2009
6/5/2009
7/5/2009
8/5/2009
11/5/2009
12/5/2009
13/5/2009
14/5/2009
15/5/2009
18/5/2009
19/5/2009
20/5/2009
21/5/2009
22/5/2009
25/5/2009
26/5/2009
27/5/2009
28/5/2009
29/5/2009
Forecast value
3.5315
3.52
3.543
3.5225
3.535
3.503
3.525
3.513
3.555
3.545
3.57
3.543
3.5495
3.53
3.499
3.49
3.494
3.491
3.525
3.51
3.577201432
3.544552717
3.532512534
3.555118745
3.534100038
3.546214326
3.513711947
3.535397216
3.523001541
3.564776342
3.554423973
3.579181541
3.551822906
3.558066013
3.538263211
3.506952152
3.497708115
3.50150677
3.49830334
3.532188084
46
4.4.2
Forecasting future value by using GARCH (1,1)
The duration of forecast is started from 4th May 2009 to 29th May 2009. Future
values of selling prices of RM/USD forecast by GARCH (1, 1) model by using
Microsoft Office Excel 2007. The value of forecast is according to static forecasting
approach which performs the series of one-step ahead forecast of exchange rate selling
prices of RM/USD.
Table 4.7 was shown that the real value against forecast value by using GARCH
(1, 1) model from 04th May 2009 to 29th May 2009. Refer to Table 4.7, we can saw that
the forecasted value by using Microsoft Office Excel 2007 are different with the real
value. It was shown the inaccuracy by using the GARCH (1,1) model.
Next, the actual and forecast daily exchange rates of RM/USD by selling prices
by GARCH (1, 1) model are being plotted
Table 4.9: Comparison the real value against forecast value by using GARCH (1,1)
Real Value
4/5/2009
5/5/2009
6/5/2009
7/5/2009
8/5/2009
11/5/2009
12/5/2009
13/5/2009
14/5/2009
15/5/2009
18/5/2009
19/5/2009
20/5/2009
21/5/2009
22/5/2009
Forecast value
3.5315
3.52
3.543
3.5225
3.535
3.503
3.525
3.513
3.555
3.545
3.57
3.543
3.5495
3.53
3.499
3.84689862
3.68975006
3.67806037
3.82447825
3.70173947
3.71452066
3.78165449
3.71898288
3.70666294
3.83559469
3.75083626
3.77658264
3.82234327
3.7645709
3.74443873
47
25/5/2009
26/5/2009
27/5/2009
28/5/2009
29/5/2009
3.77554286
3.71052522
3.71466336
3.76662235
3.75299947
3.49
3.494
3.491
3.525
3.51
3.9
3.8
3.7
3.6
real value
NGARCH modle
3.5
GARCH value
3.4
4/5/2009
5/5/2009
6/5/2009
7/5/2009
8/5/2009
11/5/2009
12/5/2009
13/5/2009
14/5/2009
15/5/2009
18/5/2009
19/5/2009
20/5/2009
21/5/2009
22/5/2009
25/5/2009
26/5/2009
27/5/2009
28/5/2009
29/5/2009
3.3
Figure 4.11: The plot of actual prices against forecast prices by NGARCH(1,1) and
GARCH(1,1) model
In Figure 4.11, the actual and forecast daily exchange rates of RM/USD by
selling prices by NGARCH (1, 1) and GARCH(1,1) model are being plotted. The trend
of forecast prices has followed tight to the actual selling prices of RM/USD for 1 month
out-sample period. The trend of forecast prices by using GARCH(1,1) model has far
away to the actual selling prices of RM/USD for 1 month out-sample period.
48
4.5
Comparison of NGARCH (1,1) and GARCH(1,1) Model
One of the study objectives is to compare the forecast performances by two
univariate time series models. We will compare the NGARCH (1, 1) and GARCH (1,1)
models in terms of their RMSE values in forecast performances in forecasting stage.
4.5.1
Comparison of Forecasting Model
When reach the forecasting step, we would compare RMSE, values from
NGARCH(1, 1) and GARCH(1,1) models. Obviously, the smaller forecast performances
are preferred. When the relation of actual values and forecast values are closer to each
others, a smaller forecast performance will be obtained.
Table 4.10: RMSE value of NGARCH(1,1)
Real Value
Forecast NGARCH
3.5315
3.507201432
3.52
3.504552717
3.543
3.502512534
3.5225
3.505118745
3.535
3.544100038
3.503
3.516214326
3.525
3.503711947
3.513
3.555397216
3.555
3.503001541
3.545
3.534776342
3.57
3.554423973
3.543
3.519181541
3.5495
3.521822906
3.53
3.518066013
3.499
3.498263211
3.49
3.496952152
3.494
3.497708115
3.491
3.49150677
Error
0.024298568
0.015447283
0.040487466
0.017381255
0.009100038
0.013214326
0.021288053
0.042397216
0.051998459
0.010223658
0.015576027
0.023818459
0.027677094
0.011933987
0.000736789
0.006952152
0.003708115
0.00050677
Error Square
0.00059042
0.000238619
0.001639235
0.000302108
8.28E-05
0.000174618
0.000453181
0.001797524
0.00270384
0.000104523
0.000242613
0.000567319
0.000766022
0.00014242
5.43E-07
4.83E-05
1.38E-05
2.57E-07
49
3.51830334
3.532188084
3.525
3.51
Average
RMSE
0.00669666
0.022188084
0.018281523
4.48E-05
0.000492311
0.000520265
0.022809308
Table 4.11: RMSE value of GARCH(1,1)
Real Value
Forecast value
Error
3.5315
3.84689862
3.52
3.68975006
3.543
3.67806037
3.5225
3.82447825
3.535
3.70173947
3.503
3.71452066
3.525
3.78165449
3.513
3.71898288
3.555
3.70666294
3.545
3.83559469
3.57
3.75083626
3.543
3.77658264
3.5495
3.82234327
3.53
3.7645709
3.499
3.74443873
3.49
3.77554286
3.494
3.71052522
3.491
3.71466336
3.525
3.76662235
3.51
3.75299947
Average
RMSE
0.31539862
0.16975006
0.13506037
0.30197825
0.16673947
0.21152066
0.25665449
0.20598288
0.15166294
0.29059469
0.18083626
0.23358264
0.27284327
0.2345709
0.24543873
0.28554286
0.21652522
0.22366336
0.24162235
0.24299947
0.22914837
Error Square
0.09947629
0.02881508
0.0182413
0.09119086
0.02780205
0.04474099
0.06587153
0.04242895
0.02300165
0.08444527
0.03270175
0.05456085
0.07444345
0.05502351
0.06024017
0.08153472
0.04688317
0.0500253
0.05838136
0.05904874
0.05494285
0.23439891
Table 4.10 and 4.11 shown the value of RMSE by using NGARCH(1,1)
and GARCH(1,1). From the table, value of RMSE of NGARCH(1,1) is 0.022809308
while value of GARCH(1,1) is 0.23439891. From this two value, we can noticed that the
value of NGARCH(1,1) is smaller than GARCH(1,1). Therefore, we conclude that
NGARCH(1,1) model is a better forecast model for daily exchange rate selling prices for
RM/USD than GARCH (1, 1) model.
Chapter V
CONCLUSION
5.0
Introduction
This chapter will summarize the study and conclusions are made based on
the analysis and the results of the forecasting in section 5.1. In Section 5.2, we will give
suggestions for further study based on this study.
51
5.1
Result and Comparisons
Table 5.1:The plot of actual prices against forecast prices by NGARCH(1,1) and
GARCH(1,1) model
Real Value
Forecast value
Forecast value
NGARCH (1,1)
GARCH (1,1)
4/5/2009
3.5315
3.577201432
0.00383756
5/5/2009
3.52
3.544552717
0.00368927
6/5/2009
3.543
3.532512534
0.00354839
7/5/2009
3.5225
3.555118745
0.00341464
8/5/2009
3.535
3.534100038
0.00328772
11/5/2009
3.503
3.546214326
0.00316735
12/5/2009
3.525
3.513711947
0.00305327
13/5/2009
3.513
3.535397216
0.00294522
14/5/2009
3.555
3.523001541
0.00284296
15/5/2009
3.545
3.564776342
0.00274625
18/5/2009
3.57
3.554423973
0.00265486
19/5/2009
3.543
3.579181541
0.00256856
20/5/2009
3.5495
3.551822906
0.00248714
21/5/2009
3.53
3.558066013
0.00241039
22/5/2009
3.499
3.538263211
0.00233812
25/5/2009
3.49
3.506952152
0.00227011
26/5/2009
3.494
3.497708115
0.00220619
27/5/2009
3.491
3.50150677
0.00214617
28/5/2009
3.525
3.49830334
0.00208986
29/5/2009
3.51
3.532188084
0.00203710
52
Refer to the Table 5.1, the prediction by using time series method which is
NGARCH (1, 1) followed tight to the actual selling prices of RM/USD for 1 month outsample period. Whereas, GARCH (1,1) is far away to the actual selling prices.
GARCH(1,1) is a model able to capture the volatility y the conditional variance
of being non-constant. Whereas, Nonlinear GARCH (NGARCH) is a negative return
increases variance by more than a positive return of the same magnitude is the better
estimate and forecast model.
5.2
Suggestions for Further Research
This study have explored two approaches that were the NGARCH (1, 1) and
GARCH (1, 1) in forecasting the Foreign Exchange for the selling prices of RM/USD
for daily data from 03rd May 2007 to 29th May 2009.
NGARCH is a model which modify from GARCH model. It know as a GARCH
model with leverage effect which known as negative return increases variance by more
than a positive return of the same magnitude. NGARCH also known as a alternative to
the Black Scholes Model. (Petra POSEDEL, 2006).
Further research can be done by using this research as a benchmark to forecast
the future values of the Foreign Exchange for the currency selling prices of RM/USD for
daily data. Complex method that being done to forecast the future values can be
compared to the result of this stud y so that the better method in
forecasting can be identify.
53
REFERENCE
List of reference
Andersen, Torben and Tim Bollerslev. 1998. “Answering the skeptics: Yes, standard
volatility models do provide accurate forecasts.” International Economic Review
39: 885-905.
Baillie, R. and T. Bollerslev. 1989. “The Message in Daily Exchange Rates: A
Conditional- Variance Tale.” Journal of Business and Economic Statistics 7:
297-305.
Bera, A. K., and Higgins, M. L. 1993, “ARCH Model: Properties, Estimation and
Testing,” Journal of Economic Surveys, Vol. 7,
No. 4, 307-366.
Baillie, R. and T. Bollerslev. 1991. “Intra-Day and Inter-Market Volatility in Foreign
Exchange Rates.” Review of Economic Studies 58: 565-585.
Bates, D., 1996. Jumps and stochastic volatility: Exchange rate process implicit in
Deutsche mark options. Review of Financial Studies, 9, 69-107
Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal
of Econometrics 31, 307-327.Ding, Z., Granger, C. W. J., & Engle, R. F. 1993.
A long memory property of stock market returns and a new model. Journal of
Empirical Finance, 1, 83-106.
Duan, J.-C., 1995. The GARCH option pricing model. Mathematical Finance 5, 13-32.
54
Duan, J.-C., 1997. Augmented GARCH(p, q) process and its diffusion limit. Journal of
Econometrics79, 97-127.
Engle, R. Ng, V., 1993. Measuring and testing of the impact of reason volatility. Journal
of Finance 48, 1749-1778.
Engle, R., 1982. Autoregressive conditional heteroscedasticity with estimates of the
variance of UK inflation. Econometrica 50, 987-1108.
Glosten, L. R., Jagannathan, R., & Runkle, D.,1993. On the relation between the
expected value and the volatility of the nominal excess return on stocks. Journal
of Finance, 48, 1779-1801.
Hentschel, L., 1995. All in the family: Nesting symmetric and asymmetric GARCH
models. Journal of Financial Economics, 39, 71-104.
Kim, I. J., & Kim, S., 2004. Empirical comparison of alternative stochastic volatility
option pricing models: Evidence from Korean KOSPI 200 index options market.
Pacific-Basin Finance Journal, 12, 117–142.
Nelson, D. (1991). Conditional heteroskedasticity in asset returns: A new approach.
Econometria, 59, 347–370.
Nelson, D. B., 1991. Conditional heteroscedasticity in assets returns: A new approach.
Econometrica, 55, 703-708
Petra POSEDEL, 2006. Analysis of the exchange rate and pricing foreign currency
options on the Croatian Market: The NGARCH model as alternative to the
Blacks Scholes Model, Financial Theory and Practice 30 (4), 347-368
Taylor, S., 1986. Modelling financial time series. New York: John Wiley & Sons.
Watanabe, T., 1999. A non-linear filtering approach to stochastic volatility models with
an application to daily stock returns. Journal of Applied Econometrics, 14, 101–
121.
55
West, Kenneth D. and Dongchul Cho. 1995. “The Predictive Ability of Several Models
of Exchange Rate Volatility.” Journal of Econometrics 69(2): 367-91.
56
APPENDIX A
Foreign Exchange Daily Data for the selling prices of RM/USD.
Ringgit/USD
Date
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
3/5/2007
4/5/2007
7/5/2007
8/5/2007
9/5/2007
10/5/2007
11/5/2007
14/5/2007
15/5/2007
16/5/2007
17/5/2007
18/5/2007
21/5/2007
22/5/2007
23/5/2007
24/5/2007
25/5/2007
28/5/2007
29/5/2007
30/5/2007
31/5/2007
1/6/2007
4/6/2007
5/6/2007
6/6/2007
7/6/2007
8/6/2007
11/6/2007
12/6/2007
13/6/2007
14/6/2007
15/6/2007
US Dollar
Selling Prices
3.4235
3.421
3.42
3.4155
3.4105
3.4025
3.409
3.403
3.4033
3.4025
3.4015
3.403
3.393
3.3935
3.39
3.387
3.392
3.3895
3.3875
3.4005
3.406
3.3995
3.3965
3.401
3.4185
3.436
3.455
3.4595
3.4615
3.472
3.465
3.461
57
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
18/6/2007
19/6/2007
20/6/2007
21/6/2007
22/6/2007
25/6/2007
26/6/2007
27/6/2007
28/6/2007
29/6/2007
2/7/2007
3/7/2007
4/7/2007
5/7/2007
6/7/2007
9/7/2007
10/7/2007
11/7/2007
12/7/2007
13/7/2007
16/7/2007
17/7/2007
18/7/2007
19/7/2007
20/7/2007
23/7/2007
24/7/2007
25/7/2007
26/7/2007
27/7/2007
30/7/2007
31/7/2007
1/8/2007
2/8/2007
3/8/2007
6/8/2007
7/8/2007
8/8/2007
9/8/2007
10/8/2007
13/8/2007
3.4385
3.435
3.434
3.449
3.4535
3.4625
3.4665
3.49
3.472
3.4565
3.447
3.4425
3.453
3.4575
3.453
3.441
3.435
3.4515
3.456
3.4475
3.446
3.447
3.4525
3.4445
3.429
3.419
3.404
3.418
3.4305
3.4645
3.47
3.4555
3.473
3.476
3.4655
3.475
3.4665
3.4645
3.4555
3.478
3.4765
58
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
14/8/2007
15/8/2007
16/8/2007
17/8/2007
20/8/2007
21/8/2007
22/8/2007
23/8/2007
24/8/2007
27/8/2007
28/8/2007
29/8/2007
30/8/2007
3/9/2007
4/9/2007
5/9/2007
6/9/2007
7/9/2007
10/9/2007
11/9/2007
12/9/2007
13/9/2007
14/9/2007
17/9/2007
18/9/2007
19/9/2007
20/9/2007
21/9/2007
24/9/2007
25/9/2007
26/9/2007
27/9/2007
28/9/2007
1/10/2007
2/10/2007
3/10/2007
4/10/2007
5/10/2007
8/10/2007
9/10/2007
10/10/2007
3.4775
3.493
3.509
3.5205
3.5025
3.4975
3.5005
3.47
3.492
3.48
3.4985
3.506
3.5055
3.499
3.506
3.5085
3.5105
3.5035
3.517
3.5185
3.5065
3.492
3.4815
3.4865
3.493
3.457
3.4475
3.4465
3.434
3.437
3.4275
3.421
3.4185
3.397
3.4
3.4105
3.424
3.407
3.39
3.403
3.388
59
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
11/10/2007
12/10/2007
16/10/2007
17/10/2007
18/10/2007
19/10/2007
22/10/2007
23/10/2007
24/10/2007
25/10/2007
26/10/2007
29/10/2007
30/10/2007
31/10/2007
1/11/2007
2/11/2007
5/11/2007
6/11/2007
7/11/2007
9/11/2007
12/11/2007
13/11/2007
14/11/2007
15/11/2007
16/11/2007
19/11/2007
20/11/2007
21/11/2007
22/11/2007
23/11/2007
26/11/2007
27/11/2007
28/11/2007
29/11/2007
30/11/2007
3/12/2007
4/12/2007
5/12/2007
6/12/2007
7/12/2007
10/12/2007
3.379
3.3735
3.387
3.388
3.374
3.364
3.3795
3.38
3.3715
3.3635
3.351
3.3425
3.349
3.3435
3.332
3.3455
3.3495
3.354
3.3335
3.323
3.341
3.354
3.343
3.363
3.3845
3.368
3.3815
3.369
3.384
3.376
3.357
3.37
3.377
3.373
3.36
3.363
3.3545
3.3445
3.3445
3.333
3.326
60
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
11/12/2007
12/12/2007
13/12/2007
14/12/2007
17/12/2007
18/12/2007
19/12/2007
21/12/2007
24/12/2007
26/12/2007
27/12/2007
28/12/2007
31/12/2007
2/1/2008
3/1/2008
4/1/2008
7/1/2008
8/1/2008
9/1/2008
11/1/2008
14/1/2008
15/1/2008
16/1/2008
17/1/2008
18/1/2008
21/1/2008
22/1/2008
24/1/2008
25/1/2008
28/1/2008
29/1/2008
30/1/2008
31/1/2008
4/2/2008
5/2/2008
6/2/2008
11/2/2008
12/2/2008
13/2/2008
14/2/2008
15/2/2008
3.3165
3.326
3.313
3.3175
3.3325
3.351
3.35
3.352
3.34
3.342
3.341
3.326
3.308
3.3105
3.308
3.285
3.285
3.274
3.274
3.258
3.2595
3.258
3.2595
3.283
3.2835
3.274
3.302
3.2705
3.244
3.245
3.238
3.241
3.237
3.231
3.2335
3.233
3.239
3.2455
3.236
3.2395
3.232
61
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
18/2/2008
19/2/2008
20/2/2008
21/2/2008
22/2/2008
25/2/2008
26/2/2008
27/2/2008
28/2/2008
29/2/2008
3/3/2008
4/3/2008
5/3/2008
6/3/2008
7/3/2008
10/3/2008
11/3/2008
12/3/2008
13/3/2008
14/3/2008
17/3/2008
18/3/2008
19/3/2008
21/3/2008
24/3/2008
25/3/2008
26/3/2008
27/3/2008
28/3/2008
31/3/2008
1/4/2008
2/4/2008
3/4/2008
4/4/2008
7/4/2008
8/4/2008
9/4/2008
10/4/2008
11/4/2008
14/4/2008
15/4/2008
3.222
3.222
3.222
3.222
3.2161
3.2175
3.215
3.2045
3.202
3.19
3.2015
3.192
3.183
3.17
3.175
3.204
3.203
3.185
3.164
3.171
3.192
3.182
3.1685
3.1955
3.201
3.2015
3.182
3.1975
3.21
3.189
3.182
3.189
3.191
3.1945
3.189
3.188
3.185
3.154
3.154
3.163
3.1635
62
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
16/4/2008
17/4/2008
18/4/2008
21/4/2008
22/4/2008
23/4/2008
24/4/2008
25/4/2008
28/4/2008
29/4/2008
30/4/2008
2/5/2008
5/5/2008
6/5/2008
7/5/2008
8/5/2008
9/5/2008
12/5/2008
13/5/2008
14/5/2008
15/5/2008
16/5/2008
20/5/2008
21/5/2008
22/5/2008
23/5/2008
26/5/2008
27/5/2008
28/5/2008
29/5/2008
30/5/2008
2/6/2008
3/6/2008
4/6/2008
5/6/2008
6/6/2008
9/6/2008
10/6/2008
11/6/2008
12/6/2008
13/6/2008
3.1645
3.152
3.1485
3.1455
3.143
3.1325
3.1365
3.151
3.1565
3.155
3.1595
3.163
3.165
3.149
3.1535
3.209
3.21
3.2175
3.22
3.253
3.275
3.255
3.255
3.231
3.213
3.218
3.224
3.25
3.245
3.2405
3.245
3.228
3.227
3.2345
3.262
3.26
3.265
3.271
3.273
3.2755
3.2755
63
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
16/6/2008
17/6/2008
18/6/2008
19/6/2008
20/6/2008
23/6/2008
24/6/2008
25/6/2008
26/6/2008
27/6/2008
30/6/2008
1/7/2008
2/7/2008
3/7/2008
4/7/2008
7/7/2008
8/7/2008
9/7/2008
10/7/2008
11/7/2008
14/7/2008
15/7/2008
16/7/2008
17/7/2008
18/7/2008
21/7/2008
22/7/2008
23/7/2008
24/7/2008
25/7/2008
28/7/2008
29/7/2008
30/7/2008
31/7/2008
1/8/2008
4/8/2008
5/8/2008
6/8/2008
7/8/2008
8/8/2008
11/8/2008
3.2755
3.257
3.244
3.262
3.26
3.2655
3.267
3.2655
3.258
3.2665
3.268
3.267
3.275
3.267
3.269
3.267
3.261
3.254
3.2445
3.248
3.237
3.216
3.225
3.23
3.244
3.238
3.243
3.243
3.245
3.253
3.2656
3.26
3.264
3.264
3.263
3.267
3.27
3.278
3.285
3.298
3.3185
64
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
12/8/2008
13/8/2008
14/8/2008
15/8/2008
18/8/2008
19/8/2008
20/8/2008
21/8/2008
22/8/2008
25/8/2008
26/8/2008
27/8/2008
28/8/2008
29/8/2008
2/9/2008
3/9/2008
4/9/2008
5/9/2008
8/9/2008
9/9/2008
10/9/2008
11/9/2008
12/9/2008
15/9/2008
16/9/2008
17/9/2008
18/9/2008
19/9/2008
22/9/2008
23/9/2008
24/9/2008
25/9/2008
26/9/2008
29/9/2008
30/9/2008
3/10/2008
6/10/2008
7/10/2008
8/10/2008
9/10/2008
10/10/2008
3.335
3.318
3.332
3.343
3.337
3.3385
3.333
3.344
3.342
3.369
3.392
3.374
3.3775
3.391
3.424
3.441
3.431
3.4595
3.421
3.454
3.46
3.471
3.46
3.45
3.455
3.449
3.462
3.4645
3.42
3.404
3.423
3.43
3.4345
3.444
3.46
3.4775
3.4845
3.485
3.5
3.505
3.514
65
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
13/10/2008
14/10/2008
15/10/2008
16/10/2008
17/10/2008
20/10/2008
21/10/2008
22/10/2008
23/10/2008
24/10/2008
28/10/2008
29/10/2008
30/10/2008
31/10/2008
3/11/2008
4/11/2008
5/11/2008
6/11/2008
7/11/2008
10/11/2008
11/11/2008
12/11/2008
13/11/2008
14/11/2008
17/11/2008
18/11/2008
19/11/2008
20/11/2008
21/11/2008
24/11/2008
25/11/2008
26/11/2008
27/11/2008
28/11/2008
1/12/2008
2/12/2008
3/12/2008
4/12/2008
5/12/2008
9/12/2008
10/12/2008
3.5125
3.493
3.511
3.526
3.527
3.525
3.527
3.541
3.573
3.5775
3.595
3.575
3.555
3.565
3.523
3.544
3.523
3.5455
3.555
3.541
3.58
3.591
3.599
3.594
3.599
3.601
3.611
3.626
3.63
3.632
3.6235
3.6235
3.6225
3.62
3.631
3.6415
3.6415
3.6385
3.639
3.631
3.62
66
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
11/12/2008
12/12/2008
15/12/2008
16/12/2008
17/12/2008
18/12/2008
19/12/2008
22/12/2008
23/12/2008
24/12/2008
26/12/2008
30/12/2008
31/12/2008
2/1/2009
5/1/2009
6/1/2009
7/1/2009
8/1/2009
9/1/2009
12/1/2009
13/1/2009
14/1/2009
15/1/2009
16/1/2009
19/1/2009
20/1/2009
21/1/2009
22/1/2009
23/1/2009
28/1/2009
29/1/2009
30/1/2009
3/2/2009
4/2/2009
5/2/2009
6/2/2009
10/2/2009
11/2/2009
12/2/2009
13/2/2009
16/2/2009
3.59
3.552
3.57
3.557
3.5335
3.485
3.47
3.4825
3.487
3.477
3.482
3.485
3.4665
3.474
3.482
3.501
3.5
3.5315
3.545
3.5775
3.58
3.576
3.595
3.582
3.572
3.608
3.613
3.614
3.626
3.612
3.6025
3.61
3.62
3.614
3.623
3.61
3.594
3.615
3.613
3.618
3.6185
67
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
17/2/2009
18/2/2009
19/2/2009
20/2/2009
23/2/2009
24/2/2009
25/2/2009
26/2/2009
27/2/2009
2/3/2009
3/3/2009
4/3/2009
5/3/2009
6/3/2009
10/3/2009
11/3/2009
12/3/2009
13/3/2009
16/3/2009
17/3/2009
18/3/2009
19/3/2009
20/3/2009
23/3/2009
24/3/2009
25/3/2009
26/3/2009
27/3/2009
30/3/2009
31/3/2009
1/4/2009
2/4/2009
3/4/2009
6/4/2009
7/4/2009
8/4/2009
9/4/2009
10/4/2009
13/4/2009
14/4/2009
15/4/2009
3.635
3.659
3.6605
3.672
3.667
3.673
3.671
3.676
3.695
3.7275
3.7115
3.721
3.722
3.722
3.703
3.695
3.692
3.698
3.698
3.675
3.6685
3.658
3.647
3.649
3.625
3.633
3.631
3.617
3.637
3.6495
3.651
3.63
3.59
3.555
3.585
3.634
3.619
3.619
3.618
3.598
3.6195
68
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
16/4/2009
17/4/2009
20/4/2009
21/4/2009
22/4/2009
23/4/2009
24/4/2009
27/4/2009
28/4/2009
29/4/2009
30/4/2009
4/5/2009
5/5/2009
6/5/2009
7/5/2009
8/5/2009
11/5/2009
12/5/2009
13/5/2009
14/5/2009
15/5/2009
18/5/2009
19/5/2009
20/5/2009
21/5/2009
22/5/2009
25/5/2009
26/5/2009
27/5/2009
28/5/2009
29/5/2009
3.59
3.605
3.6325
3.65
3.644
3.643
3.597
3.605
3.62
3.601
3.5635
3.5315
3.52
3.543
3.5225
3.535
3.503
3.525
3.513
3.555
3.545
3.57
3.543
3.5495
3.53
3.499
3.49
3.494
3.491
3.525
3.51