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Forecasting volatility
Richard Minkah
U.U.D.M. Project Report 2007:7
Examensarbete i matematik, 20 poäng
Handledare och examinator: Maciej Klimek
Februari 2007
Department of Mathematics
Uppsala University
i
Dedications
To the almighty God, Jesus Christ and the Holy Spirit for the guidance and care.
To my cherished Mum, Agnes Owusu and Uncle F.K. Owusu.
You provided me with parental care and financial support to ensure my pursuit for
higher education become real.
To Miss Theodora Donkor.
Though you were far away, your persistent telephone calls and the thought of you gave
me the enthusiasm to carry on with my academic work.
ii
Acknowledgement
I am especially grateful to my supervisor Professor Maciej Klimek for his helpful
comments, suggestions and corrections towards the realization of this work.
I am also indebted to all the lecturers who taught me in the entire program especially
Dag Jonsson, Professor Johan Tysk, Professor Ingemar Kaj and Silvelyn Zwanzig.
I am also thankful to Mr Ernest Amartey-Vondee and Rev. Ernest Koranteng for their
material support.
Finally, I wish to express heartfelt thanks to my course mates, Bernard Mawah and
Francis Atsu for your help and encouragement throughout my stay in Sweden.
iii
Abstract
FORECASTING VOLATILITY
Volatility plays a very important role in any financial market around the world.
Accurate forecasting of volatility is essential for asset and derivative pricing models
and other financial applications. The goal of any volatility model is to be able to
forecast volatility. In this paper, we examine the forecasting ability of three widely
used time series volatility models namely, the Historical Variance, The Generalized
AutoRegressive Conditional Heteroscedastic (GARCH) Model and the RiskMetrics
Exponential Weighted Moving Average. The characteristics of these volatility models
are explored using data on the Standard &Poor’s (S&P) 500 Index, Dow Jones
Industrial Average (DJIA), OMX Swedish Stock Exchange (OMXS30) index, Dow
Jones-AIG Commodity Index (DJ-AIGCI), The 3 Months US Treasury Bill Yield and
the Ghanaian Cedi and the US Dollar (CEDI/USD) exchange rate.
Keywords: Exponentially weighted Moving Average; GARCH; Historical Variance;
Volatility
Examiner: Professor Maciej Klimek
iv
Table of Contents
Dedications............................................................................................................. i
Acknowledgement.................................................................................................. ii
Abstract................................................................................................................... iii
List of Tables.......................................................................................................... vi
List of figures......................................................................................................... vii
1 Introduction………………………………………………………………. 1
2 Time Series Concept……………………………………………………… 3
2.1 Stationarity………………........…………………………………… 3
2.1.1 Nonstationarity…........……………………………………. 5
2.2 Autocorrelation………................…………………………………. 5
2.2.1 The Ljung-Box Q-Statistics……………………………….. 6
3 Statistical and Probability Foundations………………………………… 8
3.1 Financial Price Changes and returns………………………………. 8
3.1.1 Return Aggregation………………………………………... 9
3.2 Modelling financial prices and returns...…………………………... 10
4 Volatility Modelling and Forecasting........................................................ 12
4.1 Historical Variance........................................................................... 13
4.2 The RiskMetrics’ Exponential Weighted Moving
Average………………………………………….………………... 14
4.2.1 Estimating the Parameters of the RiskMetrics
Model……………………………………………………… 16
4.2.2 Determining the Decay Factor ……………………………. 17
4.3 The Generalized Autoregressive Conditional
Heteroscedastic (GARCH) model………………………………… 17
4.3.1 Estimation of the GARCH(1,1) Model. .............................. 18
4.3.2 The GARCH(1,1) k-Period Volatility Forecast.................... 19
4.4 Measuring Forecasting Performance................................................ 20
5 Data Analysis............................................................................................... 22
5.1 Description of Data………………………………………………. 22
5.2 Fitting the GARCH(1,1) Model...................................................... 28
5.3 Fitting the RiskMetrics Exponential Weighted Moving Average
(EWMA).......................................................................................... 43
5.4 Comparing the Volatility Models.................................................... 50
5.5 Conclusions……………………………………………………….. 56
References 57
v
Appendix
A.1 Aggregation Property of the Normal Distribution............................ 58
A.2 Test for Conditional Normality.….…………............……....…...... 58
A.3 The RiskMetrics k-period (day) Volatility Forecast......................... 59
vi
List of Tables
5.1 summary statistics of the returns…………………………………………. 28
5.2 Ljung-Box Q test for returns……………………………………………... 35
5.3 Ljung-Box Q test for squared returns…………………………………….. 35
5.4 Estimated parameters of the GARCH(1,1) model for various return
series............................................................................................................ 36
5.5 Unconditional mean and volatility estimates of the GARCH(1,1)............. 36
5.6 Ljung-Box Q-Statistics for squared standardised residuals using
GARCH(1,1)............................................................................................... 36
5.7 RiskMetrics volatility Estimates................................................................. 43
5.8 Ljung-Box Q-Statistics for squared standardized residuals using
RiskMetrics EWMA …............................................................................... 50
5.9 In-Sample Root Mean Squared Forecast Errors of the volatility models.... 50
5.10 Out-Sample Root Mean Squared Forecast Errors for a 20-day Forecasting
Horizon......................................................................................................... 55
5.11 Out-Sample Root Mean Squared Forecast Errors for a 60-day Forecasting
Horizon......................................................................................................... 55
5.12 Out-Sample Root Mean Squared Forecast Errors for a 120-day Forecasting
Horizon......................................................................................................... 56
vii
List of Figures
2.1 Simulated stationary time series…………………………………………… 4
2.2 A non-stationary time series of SEK/CEDI foreign exchange rate………. 5
2.3 Plot of a sample autocorrelation coefficients for SEK/CEDI foreign
exchange rate……………………………………………………………… 6
5.1 Time series plots of S&P500, DJIA and the OMXS30 index..................... 23
5.2 Time series plots of DJ-AIGCI, 3 Months US T-Bill and CEDI/USD
Exchange rate............................................................................................. 24
5.3 Plots of returns for the S&P500, DJIA and the OMXS30.......................... 26
5.4 Plots of returns for the DJ-AIGCI, 3 Months US Treasury Bill and the
CEDI/USD.................................................................................................. 27
5.5 Normal probability plot of the returns of the S&P500, DJIA, and the
OMXS30.................................................................................................... 29
5.6 Normal probability plot of the returns of the DJ-AIGCI, 3 Months US
Treasury Bill, CEDI/USD......................................................................... 30
5.7 Plots of autocorrelation coefficients for the returns of the S&P500,
DJIA and OMXS30................................................................................... 31
5.8 Plots of autocorrelation coefficients for the returns of the DJ-AIGCI,
3 Months US T-Bill, CEDI/USD............................................................ 32
5.9 Plots of autocorrelation coefficients for the squared returns of the
S&P500, DJIA and the OMXS30............................................................. 33
5.10 Plots of autocorrelation coefficients for the squared returns of the
DJ-AIGCI, 3 Months US Treasury Bill and CEDI/USD......................... 34
5.11 Estimated conditional volatility using GARCH(1,1) for the S&P500,
DJIA and OMXS30................................................................................. 37
5.12 Estimated conditional volatility using GARCH(1,1) for the DJ-AIGCI,
3 Months US T-Bill and CEDI/USD........................................................ 38
5.13 Plot of the standardised residuals for S&P500, DJIA and OMXS30
using GARCH(1,1)................................................................................... 39
5.14 Plot of the standardised residuals for DJ-AIGCI, 3 Months US T-Bill
and CEDI/USD using GARCH(1,1).......................................................... 40
5.15 Plots of autocorrelation coefficients for the squared standardised
residuals for S&P500, DJIA and OMXS30 using GARCH(1,1).............. 41
5.16 Plots of autocorrelation coefficients for the squared standardised
residuals for the DJ-AIGCI, 3 Months US T-Bill and CEDI/USD
using GARCH(1,1)...................................................................................... 42
5.17 Plot of estimated conditional volatility using RiskMetrics EWMA for the
S&P500, DJIA and OMXS30.................................................................... 44
5.18 Plot of estimated volatility using RiskMetrics EWMA model for the DJ-
AIGCI, 3 Months US T-Bill and CEDI/USD............................................ 45
5.19 Normal probability plots for the standardised residuals of the S&P500,
DJIA and OMXS30 using RiskMetrics EWMA model............................. 46
5.20 Normal probability plots for the standardised residuals of the DJ-AIGCI,
3 Months US T-Bill and CEDI/USD using RiskMetrics EWMA model.... 47
5.21 Plots of autocorrelation coefficients for the squared standardised residuals
for the S&P500, DJIA and OMXS30 using RiskMetrics EWMA model… 48
5.22 Plots of autocorrelation coefficients for the squared standardised
residuals for the DJ-AIGCI, 3 Months US T-Bill and CEDI/USD using
RiskMetrics EWMA model………………………....................................... 49
viii
5.23 GARCH(1,1) and EWMA estimators on the S&P500, DJIA and
OMXS30 return series.......................................................................... 51
5.24 GARCH(1,1) and EWMA estimators on the DJ-AIGCI, 3 Months
US T-Bill and CEDI/USD return series.................................................. 52
5.25 Volatility Forecast on S&P500, DJIA and OMXS30 return series......... 53
5.26 Volatility Forecast on DJ-AIGCI, 3 Months US T-Bill and CEDI/USD
return series............................................................................................ 54
1
Chapter 1
Introduction
Volatility has become an indispensable topic in financial markets for risk managers,
portfolio managers, investors, academicians and almost all that have something to do
with the financial markets. Forecasting accurately future volatility and correlations of
financial asset returns is essential to derivatives pricing, optimal asset allocation,
portfolio risk management, dynamic hedging and as an input for Value-at-Risk
models.
The importance of volatility forecasting was highlighted when in 2003 Professor R.F
Engle was awarded a Noble price for his outstanding contribution in modelling
volatility dynamics.
“The advantage of knowing about risks is that we can change our behaviour to avoid
them. Of course, it is easily observed that to avoid all risks would be impossible; it
might entail no flying, no driving, no walking, eating and drinking only healthy foods
and never being touched by sunshine. Even a bath could be dangerous. I could not
receive this prize if I sought to avoid all risks. There are some risks we choose to take
because the benefits from taking them exceed the possible costs. Optimal behaviour
takes risks that are worthwhile. This is the central paradigm of finance; we must take
risks to achieve rewards but not all risks are equally rewarded. Both the risks and the
rewards are in the future, so it is the expectation of loss that is balanced against the
expectation of reward. Thus we optimize our behaviour, and in particular our
portfolio, to maximize rewards and minimize risks”1.
Volatility has a central role in the derivatives pricing theory. The Black-Scholes model
has volatility as the only parameter among strike price, time to expiration, interest rate,
and strike price that has to be forecasted. The underlying assets’ volatility is needed in
the pricing of an option and there are options with volatility as the underlying assets.
The 1996 and 1999 Basel Accord makes it compulsory for financial institutions to
incorporate financial risk exposure in calculating the basic capital requirements. This
makes volatility forecasting an obligatory task for all financial institutions.
Volatility and correlations are estimated from historical data on asset returns or from
observed option prices since it cannot be observed directly. Any volatility model
should be able to forecast volatility. Time series models are used on historical data on
asset returns to forecast volatility and correlations whiles implied volatility uses
observed option prices.
The common assumed model for logarithmic returns of an asset is the (Multi) normal
distribution. Real data from the financial markets have being found to violate this
assumption, with the leptokurtotic (fat-tailed) a more appealing distribution. The non-
stationarity in the financial market data introduces some complication in the
variance/covariance matrix forecasting. In this work, we focus on three time series
models for estimating volatility namely Historical Variance, Exponential Weighted
1 Engle 2003, Noble Lecture, page 326
2
Moving Average (EWMA) and the Generalised AutoRegressive Moving Average
(GARCH). We investigate the forecasting ability of these three models.
The paper is organised as follows. In chapter 2, we explore the concept of time series,
including definitions, stationarity and autocorrelations. In chapter 3, we will describe
the statistical and probability foundations underpinning the various volatility models.
This involves description of financial price changes, return aggregation and the
modelling of the price changes. We then introduce the volatility models namely, the
Historical Variance, The RiskMetrics Exponential Weighted Moving Average and the
Generalised Autoregressive Conditional Heteroscedastic. Finally, in chapter 5, we fit
the Standard & Poor’s (S&P) 500 Stock Price Index, Dow Jones Industrial Average
(DJIA), OMXS30 Stock Price Index, Dow Jones-AIG Commodity Index (DJ-AIGCI),
3 Months US Treasury Bill Yield and the Ghanaian Cedi and the US Dollar Exchange
Rate (CEDI/USD) by the various models. We will perform some diagnostics on the
fits. The In-Sample and Out-of-Sample forecasting ability of the various models are
compared and we make our conclusions.
3
Chapter 2
Time Series Concept
A time series is a set of observations { , 0, 1, 2,...}tx t = ± ± , each one being recorded at a
specific time t. There are two kinds of time series data namely discrete-time and
continuous-time series. In a discrete-time time series, the set of times at which
observations are made is discrete, as is the case for example, when observations are
made at fixed time intervals. However, in a continuous-time series, observations are
recorded continuously over some time interval, e.g. [0,1].
An important part of time series analysis is the selection of a suitable probability
model for the data. A time series model for observed data { tx } is a specification of a
stochastic process { tX } of which { tx } is postulated to be a realization. The term time
series is also used with respect to the stochastic process { tX }.
2.1 Stationarity
Stationarity has two forms namely (weak) stationarity and strict stationarity.
In simple terms, a time series { tX } is said to be stationary if it has statistical property
similar to those of the “time shifted” series { t hX + }, for each integer h and t∈ℤ .
Definition 2.1 (The joint distribution function)
The joint distribution function of random variables 1 2, ..., TX X X is given by
1 2, ,..., 1, 2 1 1 2 2( ,... ) ( , ..., )TX X X T T TF x x x P X x X x X x= ≤ ≤ ≤ (1)
where 1, 2 ,... Tx x x ∈ℝ .
Definition 2.2 (The mean function)
Let { }tX X= be a time series. The mean function of { tX } is
( ) ( )X tt E Xµ = , (2)
where ( )E is the mathematical expectation.
Definition 2.3 (The covariance function)
The covariance function is computed to give a summary of the dependence between
any two random variables used in modelling a time series. The covariance function of
a time series { }tX with variance ( )tVar X < ∞ is given by
4
( , ) [( ( ))( ( ))]X s X t Xcov s t E X s X tµ µ= − − (3)
for all integers s and t.
Definition 2.4 (Stationarity )
The time series { }tX is said to be (weakly) stationary if it has a finite variance,
constant first moment (mean), and is such that the cov( , )s t depends only on ( )t s− i.e.
(a) 2( ) ,tE X t< ∞ ∀
(b) ( ) ,tE X tµ= ∀
(c) cov ( , ) cov ( , ), , ,X Xs t s h t h s t h= + + ∀
where , ,s t h belongs to the set of integers.
Definition 2.5 (Strict stationarity)
The time series { }tX is said to be strictly stationary if the
joint distribution of 1 2
( , ,..., )pt t tX X X is the same as that of
1 2( , ,..., )
pt h t h t hX X X+ + + for
any choice of time instances 1 2, ,..., pt t t and any increment h.
21-Mar-1994 10-May-1994 29-Jun-1994 18-Aug-1994-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Simulated Stationary TimeSeries
dates
Returns
Figure 2.1 Simulated stationary time series.
The figure 2.1 above shows how the stationary time series fluctuates around its mean,
due to property (b) in the definition. In this case, the mean is 0.0094.
5
2.1.1 Nonstationarity
Nonstationary time series can occur in many ways, e.g. if the mean or variance
depends on t. Figure 2.2 is a plot of the foreign exchange rate of the Ghanaian Cedi
and the Swedish Krona which illustrates a nonstationary time series model.
21-Mar-1994 03-Aug-1995 15-Dec-1996 29-Apr-1998 11-Sep-1999 23-Jan-2001 07-Jun-2002 20-Oct-2003 03-Mar-2005 16-Jul-20060
200
400
600
800
1000
1200
1400
Exchange rate
SEK/CEDI Foreign exchange rate
Figure 2.2 A nonstationary time series of SEK/CEDI foreign exchange rate
2.2 Autocorrelation
The autocorrelation of a given time series { }tX , measures the correlation of the series
across time. The standard correlation coefficient between two given random variables
X and Y is given by
2
xy
xy
x y
σρ
σ σ= (4)
Where iσ is the standard deviation of the random variable i and 2
xyσ the covariance
between X and Y. The variance of a random variable X, is a measure of the spread of
the data around its mean. The mathematical expression for the variance is given by
2 2[( ) ]XE Xσ µ= − . (5)
The kth order autocorrelation coefficient, ( )kρ for a time series { tX , 1,2,...,t T= } is
given by the expression
2 2
, ,
2( )
t t k t t k
t t k t
kσ σ
ρσ σ σ
− −
−
= = , (6)
6
provided that we assume (weak) stationarity of { tX }. ( )kρ is normally estimated by
using a given sample of { tX }. Let { tx } be a given sample, then the estimate of the
autocorrelation coefficient at lag k is given by
{ }1
2
1
( )( ) /( 1)
ˆ( )
{( ) }/( 1)
T
t t k
t k
T
t
t
x x x x T k
k
x x T
ρ−
= +
=
− − − −=
− −
∑
∑ (7)
where 1
1 T
t
t
x xT =
= ∑ , the sample mean.
The autocorrelation function ( ) [ 1,1]kρ ∈ − , an autocorrelated time series has the
sample autocorrelation significantly different from zero. For significantly large
amount of historical returns, up to 5% of the sample estimate of the autocorrelation
function should fall outside the interval 1.96 1.96
ˆ( ) [ , ]kT T
ρ ∈ − 2. Figure 2.3 illustrates
the plot of a sample autocorrelation coefficients showing bounds.
0 2 4 6 8 10 12 14 16 18 20-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF)
Figure 2.3 Plot of a sample autocorrelation coefficients for SEK/CEDI foreign
exchange rate.
2.2.1 The Ljung-Box Q-Statistics
Apart from the visual inspection of the plotted autocorrelation function, the Ljung-Box
Q-Statistic is used for quantification of the autocorrelation. It test for high order serial
correlation in the residuals. The Ljung-Box Q-Statistic3 is defined by
2 Brockwell & Davis, 2002
3 Ljung and Box, 1978
7
2
1
( 2)h
j
j
Q n nn j
ρ
=
= +−∑ (8)
Where n is the number of observations, h is the largest lag and jρ is the sample
autocorrelation function at lag j, of an appropriate time series.
Under the null hypothesis that a times series is not autocorrelated, the Ljung-Box Q-
Statistic is distributed as chi-squared with h degrees of freedom.
8
Chapter 3
Statistical and Probability Foundations
In this section, we present the statistical and probability foundations upon which the
Historical Variance, RiskMetrics EWMA and the GARCH models are based on.
3.1 Financial Price Changes and returns
Price changes are often used as a measure of risk and a variety of these changes exist.
Among them are absolute, relative and log price changes. A return on a portfolio is a
price change defined relative to some initial price.
Let tP be the price of a security at date t, where t is usually taken as one business day
but can be a week or a month etc. The relative price change or percent return is
defined as
1
1
t tt
t
P PR
P
−
−
−=
(9)
If the gross return on a security is just1 tR+ , then the logarithmic return (or
continuously compounded return), tr , of a security is defined as
ln(1 )t tr R= +
1
1
ln t
t
t t
P
P
p p
−
−
=
= − (10)
where ln( )t tp P= is the natural logarithm of tP .
Returns are preferred over prices in this work because returns have more attractive
statistical properties than prices.
Similarly for multiple-day (k-days) horizon, the relative price change is defined as
,t t k
t t k
t k
P PR
P
−+
−
−= . (11)
The k-days gross return ,1 t t kR ++ can be expressed in terms of the 1-day returns as
, 1 2 11 (1 )(1 )(1 )...(1 )t t k t t t t kR R R R R+ − − − −+ = + + + +
1 2 1
1 2 3
. . ...t t t t k
t t t t k
P P P P
P P P P
− − − −
− − − −
= t
t k
P
P−
= (12)
9
3.1.1 Return Aggregation
The logarithmic return for the multiple-day is equal to the sum of the one day returns.
We consider two types of aggregation namely temporal and cross section.
In temporal aggregation, multiple-day returns are constructed from one day returns by
summing across time. i.e.
, ln tt t k
t k
Pr
P+
−
=
[ ],
1 2 1
1 2 1
ln(1 )
ln (1 ).(1 ).(1 ).(1 )
...
t t k
t t t t k
t t t t k
R
R R R R
r r r r
+
− − − −
− − − +
= +
= + + + +
= + + +
(13)
However, in the cross section approach aggregation is done across the individual
returns. Consider a portfolio that consists of N instruments. Let ir and iR (i=1,2,…N)
be respectively the continuously compounded and percent returns. We assign weights
iw to the ith instrument in the portfolio and with a condition of no short sales
1
1N
i
i
w=
=∑ . If the initial value of this portfolio is 0P , and the price after one period is
1P , then by using discrete compounding we derive the usual expression for a portfolio
return as follows.
1 1 0 1 2 0 2 0. .(1 ) . .(1 ) ... . .(1 )N NP w P R w P R w P R= + + + + + + . (14)
Noting that
1 0
0
p
P PR
P
−= (15)
then
1 1 2 2. . ... .p N NR w R w R w R= + + + . (16)
RiskMetrics uses logarithmic returns as the basis in all computations and the
assumption that a portfolio return is a weighted average of logarithmic returns. i.e.
1
.N
pt i it
i
r w r=
≅∑ (17)
The justification is the fact that log(1 )x x+ ≈ for x close to 0. E.g. if | | 2%x ≤ then
| log(1 ) | 0.0002x x+ − ≤ (i.e. 0.02%). For daily data 2% change would be considered
as very large.
10
3.2 Modelling financial prices and returns
In an attempt to measure the future price changes in a portfolio’s value, a risk
measurement model is used to describe the behaviour of such movements in prices. In
order to achieve this, the future prices of the underlying assets of the portfolio are
forecasted using past price changes. This task demands that we model the following
1. The temporal dynamics of return, i.e. model the evolution of returns over time
2. The distribution of returns at any point in time.
The random walk is one of the widely used class of models to characterize the
development of price returns. The fundamental random walk model for single price
assets such as commodities, foreign exchange and equities is modelled as
1t t tP Pµ σε−= + + , (18)
where tε ’s are independent and identically distributed (iid) N(0,1). We note that
there is a positive probability of getting a negative price from this price movement. In
order to guarantee non-negativity of prices, we model the log price tp as a random
walk with independent and identically distributed (iid) normally distributed changes
1t t tp pµ σε−= + + , tε ~iid N(0,1). (19)
The use of log prices, implies that the model has continuously compounded returns,
i.e. t tr µ σε= + and hence an expression for prices can be derived as
( )
1t
t tP P eµ σε+
−= (20)
The assumption of tε ~iid N(0,1) and with 1tP− , ( )teµ σε+
both positive, tP then follows
the lognormal distribution.
The above models assume a constant variance in price changes, which in practice is
flawed in most financial time series data. We can relax this assumption to let the
variance vary with time in our adopted model. i.e.
1t t t tp pµ σ ε−= + + , tε ~N(0,1). (21)
For fixed income instruments, prices and yields are observed and we model the yield
rather than prices. This is because of the pull to par phenomenon of the price return
i.e. the unique feature of bonds such that the price approaches its face value as the time
of maturity approaches. At maturity the price volatility converges to zero.
Let tY be the yield on a bond at period t and the log yield is modelled as
1t t t ty yµ σ ε−= + + , tε ~ N(0,1). (22)
11
All the random walk models presented imply certain movement of financial prices
over time. Stationary and nonstationary time series properties apply to these
movements of price changes.
RiskMetrics review of historical observations of financial return distributions by
researchers have been summarized into four basic observations.
� Financial return distributions have “fat tails”. This means that extreme price
movements occur more frequently than implied by the normal distribution.
� The peak of the return distribution is higher and narrower than that predicted by
the normal distribution. These characteristics (usually referred to as “thin waist”)
along with fat tails is a characteristic of the leptokurtotic distribution.
� Returns have small autocorrelations.
Squared returns usually have significant autocorrelation.4
The summary above illustrates the failure of the normal distribution to accurately
model returns. The other category of distributions that do not depend on the iid
assumptions and treat volatility as time-dependent process accounts for these
shortcomings. An example is the GARCH model, which we take a detail look at in
chapter 4.
The RiskMetrics model of financial returns which we make use in this work has it that
� The variances of returns change over time-a phenomenon called
heteroscedasticity.
� There is an autocorrelation between return covariances and posses
dynamic features.
� The importance of the assumption about the normal distribution for the
returns are
1. A complete description of the shape of the distribution is achieved by
specifying only the mean and variance (or covariance in case of two or
more return series).
2. Portfolio returns are the weighted sum of the underlying returns and this
stem from the fact that, the sum of the multivariate normal returns is also
normally distributed.
The assumptions underlying RiskMetrics variance/covariance methodology for N set
of securities i=1,2,…,N are
, , ,i t i t i tr σ ε= , ,i tε ~ N(0,1)
tε ~ (0, )tMVN R , 1, 2, ,[ , ,..., ]t t t N tε ε ε ε= (23)
where tR is an N N× time dependent correlation matrix. For any fixed i, the random
variables { ,i tε } are assumed to be independent. The variances 2
,i tσ and correlation
between returns ,ij tρ are time-dependent.
4 RiskMetrics, 1996, Page 64-65
12
Chapter 4
Volatility Modelling and Forecasting
Volatility of financial markets changes over time. Consequently, forecasting of
volatility can be of practical importance. In recent years a number of related formal
models for time-varying variance have been developed. In this section, we will discuss
the use of these models to predict volatilities of asset returns.
There are many time-series models for forecasting market volatilities. Among the most
popularly used ones by market professionals and receive many textbook attention are
the
1. The Historical Variance
2. The Exponential Weighted Moving Average (EWMA).
3. The Generalized Autoregressive Conditional Heteroscedastic (GARCH) model.5
( Ederington and Guan, 2005)
These three estimation models belong to the so called Linear Squared Deviation
(LSD) class of estimators because the forecast variance is a linear combination of the
squared deviation of recent returns from their expected value.
The historical variance assigns weights of zero to squared deviations prior to a chosen
cut-off date and an equal weight to observations after the cut-off date. On the other
hand, the weighting scheme for GARCH(1,1) and the Exponential Weighted Moving
Average are such that weights decline exponentially in both models i.e. the weight
attached to observation at time t-(i+1) is a fixed proportion, β , of the weight attached
to observation at time t-i.
4.1 Historical Variance
In finance, the sample standard deviation, σ , or the sample variance, 2σ , of return is
used as a simple forecast of volatility of returns, tr , over the future period [t+1,t+h]6.
The k-days period historical variance is calculated as
1
2 2
0
1ˆ ( ) ( )
1
k
t t i
i
k r rk
σ−
−=
= −− ∑ (24)
where the sample mean return 1
0
1 k
t i
i
r rk
−
−=
= ∑ is the estimate of the mean µ . The
estimation of the expected return by the sample mean reduces the degree of freedom
by one resulting in the division of the squared deviations in (24) by (k-1).
5 Ederington and Guan, 2005 6 Poon and Granger, 2002
13
If the mean, µ , is known then the k-period historical variance is given by
1
2 2
0
1ˆ ( ) ( )
k
t t i
i
k rk
σ µ−
−=
= −∑ . (25)
An important issue that arises in the estimation of the historical variance is the noisy
estimate of the mean return. This is from the fact that the mean logarithmic return
depends on the range (length) of the return series in the sense that:
1 1(ln ln ) ln lnt t t t k k
r P P P Pr
k k k
− − +− −
= = =∑ ∑. (26)
Thus the mean return does not take into account the price movements or the number of
prices within the period. Most of the time, mean is set to zero to get a better forecast7.
Multiplying the variance by N, the number of trading days in a year and taking the
square root gives the annualised volatility
2ˆNσ σ= ⋅
⌢. (27)
Usually we take N=250. The value σ⌢ is the best estimator for the volatility from the
available price data and the volatility of any period of length, k, can be estimated from
this value.
The weighting scheme of the historical variance estimator is an assignment of zero
weight to the squared deviations before and at the time t-k while observations after
time t-n are assigned a weight of 1 k . A common convention is to set the length of the
period used in calculating the historical volatility, k, and that of the period of the
forecast, s, to be equal8. According to Figlewski (2004), a much longer period reduces
the forecasting errors.
4.2 The RiskMetrics’ Exponential Weighted Moving Average
The RiskMetrics’ Exponential Weighted Moving Average model (EWMA) will be
used to forecast the variances and covariances (volatilities and correlations) of the
multivariate normal distribution.
The RiskMetrics EWMA model uses historical observations to capture the dynamic
features of the volatility. It assigns the highest weight to the latest observations and the
least to the oldest observations in the volatility estimate. The assignment of these
weights enables volatility to react to large return (jump) in the market and following a
jump, the volatility declines exponentially as the weight of the jump falls.
The EWMA estimates the volatility for a given sequence of k returns as
7 Figlewski, 2004 8 Ederington and Guan, 2005
14
12
0
(1 ) ( )k
i
t i
i
r rσ λ λ−
−=
= − −∑ , (28)
where (0 1)λ λ< < is the decay factor. This parameter determines the relative weights
that are assigned to returns (observations) and the effective amount of data used in
estimating volatility. The latest return has weight (1 )λ− and the second latest
(1 )λ λ− and so on. The oldest return appears with weight 1(1 ) kλ λ −− .
We assume the sample mean r is zero and that infinite amounts of data are available.
Then by using the recursive feature of the exponential weighted moving average
(EWMA) estimator, the one -day variance forecast is
2 2 2
1, 1| 1, | 1 1,(1 )t t t t trσ λσ λ+ −= + − , (29)
where 2
1, 1|t tσ + denotes 1-day time 1t + forecast given information up to time t. Taking
the square root of both sides of (29) we get the one day volatility forecast as
2 2
1, 1| 1, | 1 1,(1 )t t t t trσ λσ λ+ −= + −. (30)
A simple proof of equations (29) and (30) are illustrated below.
2 2
1, 1| 1,
0
(1 ) i
t t t i
i
rσ λ λ∞
+ −=
= − ∑
( )2 2 2 2
1, 1, 1 1, 2(1 ) ...t t tr r rλ λ λ− −= − + + +
( )2 2 2 2
1, 1, 1 1, 2 1, 3(1 ) (1 )t t t tr r r rλ λ λ λ− − −= − + − + +
2 2
1, | 1 1,(1 )t t trλσ λ−= + −
and the volatility is obviously given by (30). For two return series, the EWMA
estimate of covariance for a given sequence of k returns is given by
1
1,2 1, 1 2, 2
0
(1 ) ( )( )k
i
k i k i
i
r r r rσ λ λ−
− −=
= − − −∑ . (31)
If we assume 1 2 0r r= = , then just as before it can be shown that
1
1,2 1, 2,
0
(1 )k
i
k i k i
i
r rσ λ λ−
− −=
= − ∑ (32)
Similar to the expression for the variance forecast equation (29), the covariance
forecast can be written in recursive form. The one-day covariance forecast between
any two return series, 1,tr and 2,tr made at time t is
15
2 2
12, 1| 12, | 1 1, 2,(1 ) .t t t t t tr rσ λσ λ+ −= + − . (33)
The corresponding one-day correlation forecast for the two returns is given by
2
12, 1|
12, 1|
1, 1| 2, 1|
t t
t t
t t t t
σρ
σ σ+
++ +
= . (34)
In managing risk, one may be interested in a longer horizon other than just one day
and hence we should construct an EWMA model over multiple horizon. The forecasts
of the variance and the covariance for k-period (i.e. over k-days) are respectively
2 2
1, | 1, 1|t k t t tkσ σ+ += or 1, | 1, 1|t k t t tkσ σ+ += (35)
and
12, |
2 2
12, 1|t k t t tkσ σ+ += 9
. (36)
The correlation forecast does not depend on the forecasting horizon. i.e.
2
12, 1|
12, | 12, 1|
1, 1| 2, 1|
t t
t k t t t
t t t t
σρ ρ
σ σ+
+ ++ +
= = (37)
It is observed that multiple day forecasts are simply multiples of one-day forecasts.
4.2.1 Estimating the Parameters of the RiskMetrics Model
The two estimation issues that arise in computation of estimates for the RiskMetrics
volatilities and covariances are the sample mean and the exponential decay factor λ .
In practice, the RiskMetrics model assumes zero mean for the sample. The largest
sample size available should be used to reduce the standard error. Choosing a suitable
decay factor is a necessity in forecasting volatility and correlations. One essential
issue is this estimation is the determination of an effective number of days (k) used in
forecasting. This is postulated in the RiskMetrics model to be determined by the
assumed tolerance level
(1 ) t
t k
α λ λ∞
=
= − ∑ . (38)
Expanding the summation we get
9 We illustrate simple Proof of Equations (35) and (36) in appendix A.3.
16
2(1 )[1 ...]kλ λ λ λ α− + + + = . (39)
Taken the natural logarithms of both sides, we find k as
ln
lnk
αλ
= . (40)
4.2.2 Determining the Decay Factor
The forecast of the variance of returns at time t+1, made one period earlier is defined
as 2 2
1 1|( )t t t tE r σ+ += where tE denotes the conditional expectation based on information
up to time t. Similarly for two return series, 1, 1tr + and 2, 1tr + , the forecast at time t+1 of
the covariance between the two return series made one period earlier is 2
1, 1 2, 1 12, 1|( )t t t t tE r r σ+ + += . In general, these results hold for any forecast made at
time , 1t j j+ ≥ .
The forecast error of the variance is defined as
2 2
1| 1 1|( ) ( )t t t t trε λ σ λ+ + += − (41)
with an expected value of zero. i.e.
2 2
1| 1 1|( ) ( ) 0t t t t t t tE E rε σ+ + += − = (42)
A natural consideration in choosing is to minimize average squared errors. We apply
this to daily forecasts of variance and according to RiskMetrics, an
appropriateλ should be chosen such that the root mean square (RMSE) of the errors
12 2 2
1 1|
0
1ˆ[ ( )]
k
v t i t i t i
i
RMSE rk
σ λ−
− + − + −=
= −∑ (43)
is minimized.
Similarly, the RMSE expression for the covariance forecast can be derived. The
covariance forecast error is
2
12, 1| 1, 1 2, 1 12, 1|t t t t t tr rε σ+ + + += − (44)
and its conditional expected value is
2
12, 1| 1, 1 2, 1 12, 1|( ) ( ) 0t t t t t t t tE E r rε σ+ + + += − = . (45)
The RMSE of the covariance forecast is given by
17
12 2
1, 1 2, 1 12, 1|
0
1ˆ[ ( )]
k
c t i t i t i t
i
RMSE r rk
σ λ−
− + − + − +=
= −∑ . (46)
In principle, we can find a set of optimal decay factors, one for each covariance can be
determined such that the estimated covariance matrix is symmetric and positive
definite. RiskMetrics present a method for choosing one optimal decay factor to be
used in estimation of the entire covariance matrix. They found 0.94λ = to be the
optimal for one-day forecast and 0.97λ = for one month (25 trading days) forecast.10
4.3 The Generalized Autoregressive Conditional
Heteroscedastic (GARCH) model
Generalized ARCH, or GARCH, framework developed by Bollerslev (1986) explains
variance by two distributed lags, one on past squared residuals to capture high
frequency effects, and the second on lagged values of the variance itself, to capture
longer term influences. These enable volatility clustering to be captured and the
leptokurtosis nature of the unconditional distribution of returns although it is a simple
model.
Let tψ be the information set (σ -field) of all information through time t. A
stochastic process { }tX is a GARCH(p, q) if
1( | )=0t tE X ψ − (47)
and
1( | )=t t tVar X hψ − (48)
with
2
0
1
q p
t i t i j t j
i j
h X hα α β− −=
= + +∑ ∑ (49)
where 00, 0, 0q p α> ≥ > and 0iα > for i=1,…,q, 0, 1,...,j j pβ > = . These
conditions are needed to guarantee that the conditional variance 0th > . It is usually
also assumed that t/tX h is i.i.d with mean 0 and variance 1 (“strong GARCH”).
The simplest and the most commonly used GARCH process is the GARCH(1,1)
process for which
10 RiskMetrics, 1996
18
2
0 1 1 1 1t t th X hα α β− −= + + (50)
where 0 10, 0α α> ≥ and 1 0β > .
The intuitive forecasting strategy of the GARCH (1,1) model is that the estimated
volatility at a given date is a combination of the long run variance and the variance
expected for last period, adjusted to incorporate the size of the last period's observed
shock11.
Every GARCH (p, q) is defined recursively and as earlier stated conditions are needed
to guarantee the existence of stationary solutions. We take a look at such condition for
the GARCH(1,1) process. We divide (47) by the square root of the conditional
variance of tX and obtain:
1| , 1, 2,...,tt
t
Xt T
hψ − = (51)
Then tZ defined as , 1, 2,...,tt
t
XZ t T
h= = , should be independent and identically
distributed. Suppose that the process is stationary. Then
2
1 1( ) ( ) ( )t t tE X E h E h h− −= = = (52)
In order to find the unconditional variance, we take the unconditional expectation of
both sides of equation (50). Solving for h we find
0
1 11h
αα β
=− −
. (53)
It can be shown that that if 1 1 1α β+ < and (0,1)tZ N∼ then tX is (weakly) stationary.
The sum, 1 1α β+ , is called the persistence parameter. A persistence value that is close
to one is describe as high and it implies a shock in the return series will decay slowly.
A low persistence on the other hand leads to a fast decay of the shock to its long run
variance.12
4.3.1 Estimation of the GARCH(1,1) Model
In this section we consider maximum likelihood estimation of the parameters of the
GARCH(1,1) as indicated in Bollerslev, 1986. The joint density of the observations
, 1, 2,...,t
X t T= can be written as the product of the conditional densities conditioning
on the previous observations:
1 2 1 2 1 1, ,..., 1 2 | , ,..., 1 2 1 1
2
( , ,..., ) ( | , ,..., ) ( )T j j
T
X X X T X X X X j j x
j
F x x x f x x x x f x− −
=
= ∏ . (54)
11 Figlewski, 2004 12 Jorion, 1997
19
The marginal density of 1X is dropped for simplicity. With the conditional normal
assumption, the conditional density of , 2,...,kX k T= , conditioning on 1 1,..., kX X − is
given by
1 2 1
2
| , ,..., 1, 2 1
1( | ,..., ) exp
22k k
kX X X X k k
kk
xf x x x x
hhπ− −
= −
(55)
and the conditional likelihood function, given 1X and 1h is:
2 1 10 1 1 ,..., | , 2 1 1( , , ) ( ,..., | , )kX X X h kL f x x x hα α β =
2
**2
1exp
22
Tk
j jj
x
hhπ=
= −
∏ (56)
Where * 2 *
0 1 1 1 1j t jh X hα α β− −= + + are obtained recursively. Substituting 1h by its
expected value 1 0 1 1( ) 1E h α α β= − − we find the log likelihood function as
* 2 *
0 1 1
2
1( , , | , ) log
2
T
j j j
j
l X h h x hα α β=
= − +
∑ (57)
where 1( ,..., )TX X X ′= and 1( ,..., )Th h h ′= 13 .
4.3.2 The GARCH(1,1) k-period Volatility Forecast
The k-day volatility forecast can also be found by using the GARCH model. Suppose
the model is estimated using daily returns on a stock. Similar to equation (13),
multiple-day returns are constructed from one day returns by summing across time
, 1 ...t t k t t Tr r r r+ += + + + . (58)
Under the condition that returns are uncorrelated across days, the multiple-day
variance as of 1t − is given by
2 2 2 2
, 1 1 1 1 1| ( | ) ( | ) ... ( | )t t k t t t t t T tE r E r E r E rψ ψ ψ ψ+ − − + − − = + + + . (59)
The forecast of the variance k-period ahead is derived as follows: Let 1 10,s q α β> = +
and 1tψ − -information at time t. Then
13 Bollerslev (1986)
20
2 2
0 1 1 -1 1( 1)t s t s t s t sh qX h Zα α+ + − + + −= + + −
2
0 -1 1 -1 1( 1)t s t s t sqh h Zα α+ + + −= + + − . (60)
1( | ) 1t s tE Z ψ+ − = (61)
so
0 -1( | ) ( | )t s t t s tE h qE hψ α ψ+ += + . (62)
Therefore
0 -1( | ) ( | )t k t t k tE h qE hψ α ψ+ += +
( )0 0 -2
2
0 0 -2
( | )
( | )
t k t
t k t
q qE h
q q E h
α α ψ
α α ψ+
+
= + +
= + +
.
.
.
2 1
0 0 0 +1... (h | )k k
t tq q q Eα α α ψ− −= + + + +
0
1
0 0 0
1
1
... ( | )
kt
k k
t t
hq
q
q q q E h
α
α α α ψ−
−−
= + + + +��������� �����
1 10 1 1
1 1
1 ( )( ) .
1 ( )
kk
thα β
α α βα β
− += + +
− + (63)
Note that if 1 1 1α β+ < , then
0
1 1
lim ( | )1
t k tk
E hα
ψα β+→∞
=− −
(64)
The volatility forecast over the future period from 1t + to t k+ denoted by ,G tσ , is an
average of the expected volatility on each day from t to t k+ i.e.
,
1
1( | )
k
G t t k t
i
E hk
σ ψ+=
= ∑ (65)
where the expected values are given by (63).
4.4 Measuring Forecasting Performance
The In-Sample and Out-of-Sample forecasting ability of the various volatility models
21
will be measured by the Root Mean Squared Forecasting Errors (RMFSE). In the In-
Sample comparison of the various forecasting models, the entire data sets for each
return series will be used. To compare the out-of-sample forecasting performance of
the variuous volatility model, we will split the sample into two parts. The first, which
is used to estimate the parameters of the GARCH(1,1) model contains at least two
thirds of the entire sample. The remaining part of the sample will be used to test the
forecasting ability of the volatility models. The parameters of the GARCH(1,1)
model will be estimated each time with a rolling constant sample size. Thus for a
forecasting horizon of x, at each forecasting date we add x amount of new data and
subtract x amount of the oldest data.
The realised volatility at each forecast date is calculated from the expression
2
,
1
1 N
R t t i
i
rN
σ +=
= ∑ . (66)
Let 2
,F tσ be the forecast of the variance given by one of the volatility models, then the
RMSFE for a model is given by equation
2
, ,
1( )F t R t
t s
RMSFEn
σ σ∈
= −∑ . (67)
n and s denotes the number of forecasts and the set of times at which ex ante forecasts
are produced respectively in the above expression14.
14 Xu and Taylor (1995)
22
Chapter 5
Data Analysis
In this section we evaluate the performance of the three models on different financial
time series taking from various stocks. The dataset that we discuss here are DJ-AIG
Commodity Index, S&P500 index, DJ-Industrial Average, OMXS30 Index, Ghanaian
Cedi and the US dollar exchange rate i.e. CEDI/USD and the Yield on the US 3
Months Treasury Bill. We first describe the dataset above and the historical variance
in section 5.1. Then, we fit the GARCH(1,1) model and the RiskMetrics EWMA
model to the datasets in section 5.2 and section 5.3 respectively. Lastly, in section 5.4,
we compare the various forecasting models and make our conclusions in section 5.5.
5.1 Description of Data
The S&P500 consist of weighted market value of 500 of the most widely held stocks
in the US stock market. This is considered as a more representative index with
coverage of over 70% of the US equity market. The S&P500 dataset has a total of
3828 observations from September 19, 1991 to November 22, 2006.
The DJ-AIG Commodity Index (DJ-AIGCI) is a rolling index composed of 19
physical commodities’ future contracts traded on the US stock exchanges. According
to Investopedia, the primary goal of the DJ-AIGCI is to provide a diversified
commodities index with weightings based on the economic significance of individual
components, while maintaining low volatility and sufficient liquidity. The DJ-AIGCI
data is from January 2, 1991 to January 31, 2006 with 3771 observations.
The Dow Jones Industrial Index Average (DJIA) comprises of stocks of 30 leading
industrial companies in the US. The index is the most widely quoted US stock index
and the oldest despite its small number of companies. The DJIA dataset being
considered is the historical price data between September 4, 1990 to November 22,
2006 with a total of 4092 observations.
The OMXS30 was introduced by the Swedish exchange for options and forwards. It
is a market value weighted index of 30 leading stocks on the OMX Stock Exchange
in Stockholm and accounts for about 70% of trading conducted on the exchange. Data
is from January 3, 2000 to November 20, 2006 and comprises of 1721 observations.
CEDI/USD foreign exchange rate from April 23, 1994 to November 5, 2006 with
total observations of 4613 and finally, the yield on 3 months US Treasury bills from
August 23, 1989 to August 22, 2000 with 2612 observations are considered.
The time series plot of the S&P500, DJIA and OMXS30 are shown in figure 5.1
whilst that of DJ-AIGCI, 3 months US Treasury bill yield and CEDI/USD exchange
rate are shown in figure 5.2. All of the plots exhibit clustering of small or large
movements in price, a feature of volatility process of assets prices. Engle and Patton
(2000) states that, this feature was reported in the earlier works by Mandelbort
23
09/91 08/93 07/95 06/97 05/99 04/01 03/03 02/05 01/07200
400
600
800
1000
1200
1400
1600S&P500 Index
Date
Price
09/90 08/92 07/94 06/96 05/98 04/00 03/02 02/04 01/06 12/072000
4000
6000
8000
10000
12000
14000
Date
Price
DJ Industrial Average
01/00 11/00 09/01 07/02 05/03 03/04 01/05 11/05 09/06 07/07400
600
800
1000
1200
1400
1600OMXS30 Index
Date
Price
Figure 5.1: Time series plots of S&P500 index, DJIA and OMXS30 index.
24
01/91 12/92 11/94 10/96 09/98 08/00 07/02 06/04 05/0660
80
100
120
140
160
180
200
Date
Price
DJ-AIGCI
08/90 07/92 06/94 05/96 04/98 03/00 02/022
3
4
5
6
7
83 Months US Treasury Bill
Date
Yield
03/23/94 08/05/95 12/17/96 05/01/98 09/13/99 01/25/01 06/09/02 10/22/03 03/05/05 07/18/06 11/30/070
0.2
0.4
0.6
0.8
1
1.2x 10
-3 CEDI/USD Exchange Rate
Date
Rate
Figure 5.2: Time series plots of DJ-AIGCI, 3 Months US T-Bill and CEDI/USD
Exchange rate.
25
(1963), Fama (1965) and numerous other studies such as Chou (1988), Schwert
(1989) and Baillie et al (1996). We base the volatility dynamics on each series’
exclusive history but Engle and Patton (2000) reports that, financial assets prices are
affected by the market around it as well as deterministic events such as company
announcements and macroeconomics announcements.
This is evident in the plot of the CEDI/USD exchange rate. In 1992, Ghana made a
transition from military rule to constitutional rule. The military junta at the time
transformed itself into a political party and subsequently won the election. Many
businesses did not have much confidence in the economy and thus the local currency,
the cedi, continued to depreciate against all the other major trading currencies. In
2000, the opposition won the elections and brought about good fiscal policies. One of
the significant one was the removal of government subsidies from petroleum products
which was the major determining factor in the strength of the cedi. Hence the
Ghanaian Cedi’s fall against the US dollar in the 2000’s has been stable as compared
to the 1990’s.
The Matlab function price2ret converts the price index, yield and exchange rate to
continuously compounded returns. Figure 2.6 shows the plots of the returns on the
S&P500, DJIA and OMXS30 with similar plots of DJ-AIGCI, 3 months US T-Bill
and CEDI/USD exchange rate over the various time intervals shown in figure 2.7.
Clearly, it is evident from the plots that volatility of returns changes over time.
The Matlab functions mean, var, skewness and kurtosis applied to returns gave the
sample statistics mean, variance, skewness and kurtosis respectively of the various
return series15. These summary statistics are shown in table 5.1. It shows that the
S&P500, DJIA and DJ-AIGCI have small positive average returns of close to one
twenty fifth of a percent for S&P500, DJIA and about 0.01% for DJ-AIGCI per day.
On the other hand, OMXS30, 3 months US T-bill and CEDI/USD have negative daily
average returns of 0.008% for 3 months US T-bill and one twentieth of a percent for
OMXS30 and CEDI/USD per day. These are evident from the Figure 5.1 and 5.2,
with the positive mean return series having a slightly increasing trend and a
decreasing trend for the negative mean return series.
The daily variance for S&P500, DJIA, DJ-AIGCI, CEDI/USD, 3 Months US T-Bill
and OMXS30 are 1.0008, 0.9689, 0.6123, 1.9044, 1.4858 and 2.6588 respectively.
These implies an average annualized volatility of 15.88%, 15.63%, 12.42%, 26.36%,
19.11% and 25.88% respectively for S&P500, DJIA, DJ-AIGCI, CEDI/USD, 3
Months US T-Bill and OMXS30.
The skewness coefficient of the OMXS30 is 0.0775 which indicates that it is
slightly skewed to the right. However the rest of the series have negative skewness
coefficients which suggest a left skewed distribution, which is in conformity with
common feature of asset returns. Lastly, the kurtosis of all the return series are greater
than that of the normal distribution of 3. This implies the various return series have
heavier tails than that suggested by the normal distribution. These are also shown in
the normal probability plots of the return series in figure 5.5 and figure 5.6.
15 Returns were multiplied by 100 to shorten the decimal places
26
09/91 08/93 07/95 06/97 05/99 04/01 03/03 02/05 01/07-6
-4
-2
0
2
4
6
8
Date
Returns
S&P500 Index
09/90 08/92 07/94 06/96 05/98 04/00 03/02 02/04 01/06 12/07-8
-6
-4
-2
0
2
4
6
8DJ Industrial Average
Date
Returns
01/00 11/00 09/01 07/02 05/03 03/04 01/05 11/05 09/06 07/07-10
-8
-6
-4
-2
0
2
4
6
8
10OMXS30 Index
Date
Returns
Figure 5.3: Plots of returns for S&P500, DJIA and OMXS30.
27
01/91 12/92 11/94 10/96 09/98 08/00 07/02 06/04 05/06-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06DJ-AIG Composite Index
Date
Returns
08/90 07/92 06/94 05/96 04/98 03/00 02/02-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.153 Months US T-Bill Yield
Date
Returns
03/94 08/95 12/96 05/98 09/99 01/01 06/02 10/03 03/05 07/06 11/07-20
-15
-10
-5
0
5
10
15
20
25CEDI/USD Exchange Rate
Date
Returns
Figure 5.4: Plots of returns for DJ-AIGCI, 3 Months US Treasury Bill, CEDI/USD.
28
Estimates S&P500 DJIA DJ-
AIGCI
CEDI/USD US 3
Months
T-Bill
OMXS30
Mean 0.0336 0.0379 0.0146 -0.0486 -0.0084 -0.0053
Variance 1.0008 0.9689 0.6123 1.9044
1.4858
2.6588
Skewness -0.1161 -0.2134 -0.3019
-0.3021
-0.2382 0.0775
Kurtosis 7.1872 7.8196
9.6007
39.6932 18.0065 5.6187
Table 5.1: summary statistics of the returns.
All the figures show deviations from normality by its deviation from the straight
line16.
The Matlab function autocorr gives the autocorrelation plots of the various return
series and are presented in figures 5.7 and 5.8. With the exception of the CEDI/USD,
all the other returns show very weak serial dependence since almost all the
autocorrelation coefficients lay within the 95% approximate confidence limits.
However, an inspection of the autocorrelation plots of squared returns in figure 5.9
and figure 5.10 indicates that most of the autocorrelation coefficients are above the
95% confidence limits. This means that there are considerable serial dependence in
the return series17.
The Ljung-Box Q-statistics for the returns and squared returns of the various series
are presented in table 5.2 and table 5.3 respectively. The tests show that, raw returns
for S&P500 and DJ-AIGCI have no significant autocorrelation up to the 20th lag.
Raw returns for DJIA, 3M-US-T-Bill and OMXS30 however have some weak
autocorrelation in it. However CEDI/USD raw returns show very significant
autocorrelation. A look at the Ljung-Box Q test for squared returns shows strong
evidence of autocorrelations among squared returns for each series. A property of
financial time series reported in the earlier works of Miller (1979).
5.2 Fitting the GARCH(1,1) Model.
In this section, we estimate the parameters 0α , 1α and 1β of the GARCH(1,1) model
and the in-sample estimates of the volatility. The Matlab command garchfit, models
the return series as GARCH(1,1) and estimates the parameters 0α , 1α and 1β via
maximum likelihood. The estimates of the parameters of the various series are shown
in table 5.4. The return series, CEDI/USD, has the highest persistence value which is
16 See Appendix A.1.2 for description of the normal probability plot 17 These buttress earlier works of several people including Miller (1979) and also in RiskMetrics (1996).
29
-4 -3 -2 -1 0 1 2 3 4-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Standard Normal Quantiles
Quantiles of Input Sample
QQ Plot of Sample Data versus Standard Norma
(Returns of S&P500 Index)l
-4 -3 -2 -1 0 1 2 3 4-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Standard Normal Quantiles
Quantiles of Input Sample
QQ Plot of Sample Data versus Standard Normal
(DJIA)
-4 -3 -2 -1 0 1 2 3 4-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Standard Normal Quantiles
Quantiles of Input Sample
QQ Plot of Sample Data versus Standard Normal
(OMXS30)
Figure 5.5: Normal probability plot of the returns of the S&P500, DJIA, and the
OMXS30.
30
-4 -3 -2 -1 0 1 2 3 4-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Standard Normal Quantiles
Quantiles of Input Sample
QQ Plot of Sample Data versus Standard Normal
(DJ-AIGCI Returns)
-4 -3 -2 -1 0 1 2 3 4-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Standard Normal Quantiles
Quantiles of Input Sample
QQ Plot of Sample Data versus Standard Normal
(3 Months US Treasury Bill)
-4 -3 -2 -1 0 1 2 3 4-20
-15
-10
-5
0
5
10
15
20
25
Standard Normal Quantiles
Quantiles of Input Sample
QQ Plot of Sample Data v ersus Standard Normal
(CEDI/USD)
Figure 5.6: Normal probability plot of the returns of the DJ-AIGCI, 3 Months US T-
Bill and CEDI/USD.
31
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF)
S&P500 Index
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF)
(DJIA Returns)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF)
(OMXS30)
Figure 5.7: Plots of autocorrelation coefficients for the returns of the S&P500, DJIA
and the OMXS30.
32
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF)
(DJ-AIGCI)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF)
(Returns of 3 Month US Treasury Bill)
0 2 4 6 8 10 12 14 16 18 20-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF)
(CEDI2USD Foreign Exchange Rate)
Figure 5.8: Plots of autocorrelation coefficients for the returns of the DJ-AIGCI, 3
Months US T-Bill and CEDI/USD.
33
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF) of Squared Returns
(S&P500 Index)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF) of Squared Returns
(DJIA)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF) of Squared Returns
(OMXS30 Index)
Figure 5.9: Plots of autocorrelation coefficients for the squared returns of the
S&P500, DJIA and OMXS30.
34
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF) of Squared Returns
(DJ-AIGCI)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF) of Squared Returns
(3 Months US Treasury Bill)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function (ACF) of Squared Returns
(CEDI/USD Exchange Rate))
Figure 5.10: Plots of autocorrelation coefficients for the squared returns of the DJ-
AIGCI, 3 Months US T-Bill and CEDI/USD.
35
close to the close to the condition under which the unconditional expectation of th is
infinite.
Returns S&P500 DJIA DJ-
AIGCI
CEDI/USD US 3
months
T-Bill
OMXS30
Hypothesis 0.000 1.000 0.000 1.000 1.000 1.000
p-Value 0.2166 0.025 0.217 0.000 0.000 0.000
Q-statistics 24.615 34.159 24.615 732.100 115.335 50.417
Critical
value
31.4104 31.410 31.410 31.410 31.410 31.410
Table 5.2: Ljung-Box Q test for returns.
Returns S&P500 DJIA DJ-AIGCI CEDI/USD US 3
months T-
Bill
OMXS30
Hypothesis 1.000 1.000 1.000 1.000 1.000 1.000
p-Value 0.000 0.000 0.000 0.000 0.000 0.000
Q-statistics 176.838 1356.900 193.461 879.750 262.389 725.242
Critical
value
31.4104 31.410 31.410 31.410 34.410 31.410
Table 5.3: Ljung-Box Q test for squared returns.
On the other hand all other return series are quite persistence with a value of about
0.99. This indicates that the volatility process returns to its mean after some time.
The ratio of 0α to the difference between 1 and the sum of 1α and 1β which
represents the unconditional mean of the GARCH(1,1) process are presented in table
5.5. The same table shows the corresponding annualized volatilities of the various
return series. The time series plot of the estimated series of conditional variance th ,
for the various returns are shown in Figure 5.11 and 5.12. It is observed from the
various diagrams that there exist periods of high and low estimated volatilities. It is
noticed from table 5.4, that the 1β values are close to 1 whiles the 0α and 1α have
very small values. From equation (54), it can be deduced that th and 1th − are inclined
to be close to each other. This closeness of 1th − to th implies that large values of th
will be clustered together and so are the small values.
An examination of the plots of the standardised residuals after fitting the
GARCH(1,1) model for the return series indicates that the residuals are generally
stable with some clustering. However, an examination of the autocorrelation plots of
the squared standardised residuals shows no serial correlations. The plots of the
standardised residuals for the various series are shown in fig 2.11 and fig 2.12
whereas the corresponding plots of the autocorrelation of squared standardised
residuals are shown in fig 2.13 and 2.14.
36
Paramete
-r
S&P500 DJIA OMXS3
0
DJ-
AIGCI
3 Months
US T-
Bill
CEDI/USD
Constant
0α
1α
1β
Persistence
( 1α + 1β )
-0.055868
0.0056142
0.065924
0.92953
0.9954
-0.056236
0.008634
0.072898
0.91975
0.9927
0.079051
0.018501
0.10142
0.8946
0.9960
0.007293
0.004463
0.057274
0.93681
0.9941
-0.01274
0.027685
0.071047
0.91806
0.9891
-0.037098
0.0009160
0.056881
0.94312
.9999
Table 5.4: Estimated parameters of the GARCH(1,1) model for various series.
Series Unconditional Mean of
variance
Annualized Volatility (%)
S&P500 1.217 17.51
DJIA 1.1781 17.23
OMXS30 4.625 34.14
DJ-AIGCI 0.7543 13.79
3 Months US T-Bill 2.5413 25.31
CEDI/USD 9.1600 57.82
Table 5.5: Unconditional mean and volatility estimates of the GARCH(1,1).
In addition to the graphical outputs, the Ljung-Box Q-statistics for the squared
standardised residuals are presented in table 5.6. We compare the results in table 5.6
with the results of the same test in table 5.3. We notice that there was a rejection of
the null hypotheses (Hypothesis=1 with p-value=0) indicating a significant evidence
in support of GARCH effects. However, the same test on the standardised residuals
based on the GARCH(1,1) indicates acceptance of the null hypotheses18. Thus the
autocorrelations have been significantly removed by fitting the GARCH(1,1) model
to the return series.
Returns S&P500 DJIA DJ-AIGCI CEDI/USD US 3
months T-
Bill
OMXS30
Hypothesis 0.000 0.000 0.000 0.000 0.000 0.000
p-Value 1.000 0.927 1.000 1.000 1.000 1.000
Q-statistics 0.510 11.681 1.917 1.724 1.234 0.388
Critical
value
31.4104 31.410 31.410 31.410 34.410 31.410
Table 5.6: Ljung-Box Q-Statistics for squared standardised residuals using
GARCH(1,1).
18 Hypothesis=0 with very significant p-values
37
09/91 08/93 07/95 06/97 05/99 04/01 03/03 02/05 01/070
0.5
1
1.5
2
2.5
3
Volatilities
(S&P500)
Volatility
Date
09/90 08/92 07/94 06/96 05/98 04/00 03/02 02/04 01/06 12/070
0.5
1
1.5
2
2.5
3
Volatilities
(DJIA)
Volatility
Date
01/00 11/00 09/01 07/02 05/03 03/04 01/05 11/05 09/06 07/070.5
1
1.5
2
2.5
3
3.5
4
4.5
Volatilities
(OMXS30)
Date
Volatility
Figure 5.11: Estimated conditional volatility using GARCH(1,1) for the S&P500,
DJIA and OMXS30.
38
01/91 12/92 11/94 10/96 09/98 08/00 07/02 06/04 05/060
0.5
1
1.5
2
2.5
3
Volatilities
(DJ-AIGCI)
Volatility
Time (t)
08/90 07/92 06/94 05/96 04/98 03/00 02/020.5
1
1.5
2
2.5
3
3.5
4
4.5
Volatilities
(3 Months US T-Bill)
Volatility
Time (t)
03/23/94 08/05/95 12/17/96 05/01/98 09/13/99 01/25/01 06/09/02 10/22/03 03/05/05 07/18/06 11/30/070
1
2
3
4
5
6
7
CEDI/USD
Volatility
Time ( t)
Figure 5.12: Estimated conditional volatility using GARCH(1,1) for the DJ-AIGCI, 3
Months US T-Bill and CEDI/USD.
39
09/91 08/93 07/95 06/97 05/99 04/01 03/03 02/05 01/07-5
-4
-3
-2
-1
0
1
2
3
4
5
Standardized Residuals
(S&P500)
Date
Residuals
09/90 08/92 07/94 06/96 05/98 04/00 03/02 02/04 01/06 12/07-5
-4
-3
-2
-1
0
1
2
3
4
5
Standardized Residuals
(DJIA)
Date
Residual
01/00 11/00 09/01 07/02 05/03 03/04 01/05 11/05 09/06 07/07-6
-5
-4
-3
-2
-1
0
1
2
3
4
Standardized Residuals
(OMXS30)
Date
Residuals
Figure 5.13: Plot of the standardised residuals for S&P500, DJIA and OMXS30 using
GARCH(1,1).
40
01/91 12/92 11/94 10/96 09/98 08/00 07/02 06/04 05/06-8
-6
-4
-2
0
2
4
6
Standardized Residuals
(DJ-AIGCI)
Date
residual
08/90 07/92 06/94 05/96 04/98 03/00 02/02-10
-5
0
5
Standardized Residuals
(3 Months US T-Bill)
Time (t)
Residual
03/23/94 08/05/95 12/17/96 05/01/98 09/13/99 01/25/01 06/09/02 10/22/03 03/05/05 07/18/06 11/30/07-15
-10
-5
0
5
10
15
Standardised Residuals
(CEDI/USD)
Time ( t)
Residuals
Figure 5.14: Plot of the standardised residuals for DJ-AIGCI, 3 Months US T-Bill
and CEDI/USD using GARCH(1,1).
41
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
sample ACF
ACF of the Squared Standardized Residuals
(S&P500)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample ACF
Sample ACF of the Squared Standardized Residuals
(DJIA)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample ACF
ACF of the Squared Standardized Residuals
(OMXS30)
Figure 5.15: Plots of autocorrelation coefficients for the squared standardised
residuals for S&P500, DJIA and OMXS30 using GARCH(1,1).
42
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample ACF
ACF of the Squared Standardized Residuals
(DJ-AIGCI)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample ACF
Sample ACF of the Squared Standardized Residuals
(3 Months US T-Bill)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
ACF
Sample ACF of the Squared Standardized Residuals
(CEDI/USD)
Figure 5.16: Plots of autocorrelation coefficients for the squared standardised
residuals for the DJ-AIGCI, 3 Months US T-Bill and CEDI/USD using GARCH(1,1).
43
5.3 Fitting the RiskMetrics Exponential Weighted Moving Average (EWMA)
In this section, we discuss some empirical results of the RiskMetrics EWMA model.
The only parameter we make use is the decay factor, λ , which RiskMetrics sets to
0.94λ = . The estimates of the variance and the corresponding annualised volatilities
for the various return series are presented in table 5.7.
The conditional variance, th , for the various return series are presented in figure 5.17
and figure 5.18. Similar to the GARCH(1,1) plots, it is observed from the various
diagrams that there exist periods of high and low estimated volatilities.
The normal probability plots of the standardised residuals for the various return series
after fitting the RiskMetrics EWMA model are presented in figure 5.19 and figure
5.20. All the figures show significant deviation from the standard normal. A visual
inspection of the autocorrelation plots of the squared standardised residuals indicates
that at the 20th lag, only DJ-AIGCI has its autocorrelations significantly removed as a
result of fitting the RiskMetrics EWMA model. The Ljung-Box Q-Statistic of the
various squared standardised residuals are presented in table 5.8. It also buttress the
observation from the graphical plots. All the return series except DJ-AIGCI have
hypothesis=1 with p-value=0 indicating a rejection of the null hypotheses. Thus there
is still some autocorrelation among the standardised squared returns for those series
after fitting the RiskMetrics EWMA model.
Series RiskMetrics Estimates of
Variance
Annualized
Volatility (%)
S&P500 1.0001 15.88
DJIA 0.9708 15.64
OMXS30 2.7077 26.12
DJ-AIGCI 0.6123 12.42
3 Months US T-Bill 1.4241 22.80
CEDI/USD 1.8924 25.80
Table 5.7: RiskMetrics volatility Estimates.
44
09/91 08/93 07/95 06/97 05/99 04/01 03/03 02/05 01/070
0.5
1
1.5
2
2.5
3
Date
Volatility
RiskMetrics Volatility Estimates
(S&P500)
09/90 08/92 07/94 06/96 05/98 04/00 03/02 02/04 01/06 12/070
0.5
1
1.5
2
2.5
3
RiskMetrics Volatility Estimates
(DJIA)
Date
Volatility
01/00 11/00 09/01 07/02 05/03 03/04 01/05 11/05 09/06 07/070
0.5
1
1.5
2
2.5
3
3.5
4
RiskMetrics Volatility Estimates
(OMXS30 Index)
Date
Volatility
Figure 5.17: Plot of estimated conditional volatility using RiskMetrics EWMA for the
S&P500, DJIA and OMXS30.
45
01/91 12/92 11/94 10/96 09/98 08/00 07/02 06/04 05/060
0.5
1
1.5
2
2.5
3
RiskMetrics Estimate of Volatilities
(DJ-AIGCI)
Date
Volatility
08/90 07/92 06/94 05/96 04/98 03/00 02/020
0.5
1
1.5
2
2.5
3
3.5
4
4.5
RiskMetrics Volatility Estimates
(3 Months US T-Bill)
Date
Volatility
03/94 08/95 12/96 05/98 09/99 01/01 06/02 10/03 03/05 07/06 11/070
1
2
3
4
5
6
7
Date
Volatility
RiskMetrics Volatility Estimates
(CEDI/USD)
Figure 5.18: Plot of estimated volatility using RiskMetrics EWMA model for the DJ-
AIGCI, 3 Months US T-Bill and CEDI/USD.
46
-4 -3 -2 -1 0 1 2 3 4-50
-40
-30
-20
-10
0
10
Standard Normal Quantiles
Sample Quantiles
QQ Plot of Sample Data versus Standard Normal
(S&P500)
-4 -3 -2 -1 0 1 2 3 4-60
-50
-40
-30
-20
-10
0
10
Standard Normal Quantiles
Sample Quantiles
QQ Plot of Sample Data versus Standard Normal
(DJIA)
-4 -3 -2 -1 0 1 2 3 4-30
-25
-20
-15
-10
-5
0
5
10
Standard Normal Quantiles
Sample Quantiles
QQ Plot of Sample Data versus Standard Normal
(OMXS30)
Figure 5.19: Normal probability plots for the standardised residuals of the S&P500,
DJIA and OMXS30 using RiskMetrics EWMA model.
47
-4 -3 -2 -1 0 1 2 3 4-160
-140
-120
-100
-80
-60
-40
-20
0
20
Standard Normal Quantiles
Sample Quantiles
QQ Plot of Sample Data versus Standard Normal
(DJ-AIGCI)
-4 -3 -2 -1 0 1 2 3 4-80
-70
-60
-50
-40
-30
-20
-10
0
10
Standard Normal Quantiles
Sample Quantiles
QQ Plot of Sample Data versus Standard Norma
(3 Months US T-Bill)l
-4 -3 -2 -1 0 1 2 3 4-120
-100
-80
-60
-40
-20
0
20
Standard Normal Quantiles
Sample Quantiles
QQ Plot of Sample Data versus Standard Normal
(CEDI/USD)
Figure 5.20: Normal probability plots for the standardised residuals of the DJ-AIGCI,
3 Months US T-Bill and CEDI/USD using RiskMetrics EWMA model.
48
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample ACF
ACF of Squared Standardised Residuals
(S&P500)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample AC
ACF of Squared Standardised Residuals
(DJIA)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample AC
Sample ACF of Squared Standardised Residuals
(OMXS30)
Figure 5.21: Sample ACF plot of the squared standardised residuals for the S&P500,
DJIA and OMXS30 using RiskMetrics EWMA model.
49
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample AC
ACF of Squared Standardised Residuals
(DJ-AIGCI)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample AC
Sample ACF of Squared Standardised Residuals
(3 Months US T-Bill)
0 2 4 6 8 10 12 14 16 18 20-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample ACF
ACF of Squared Standardised Residuals
(CEDI/USD)
Figure 5.22: Sample ACF plots of the squared standardised residuals for the DJ-
AIGCI, 3 Months US T-Bill and CEDI/USD using RiskMetrics EWMA model.
50
Returns S&P500 DJIA DJ-AIGCI CEDI/USD US 3
months T-
Bill
OMXS30
Hypothesis 1.000 1.000 0.000 1.000 1.000 1.000
p-Value 0.000 0.000 1.000 0.000 0.000 0.028
Q-statistics 85.534 60.809 2.314 675.949 294.798 33.721
Critical
value
31.4104 31.410 31.410 31.410 34.410 31.410
Table 5.8: Ljung-Box Q-Statistics for squared standardized residuals using
RiskMetrics EWMA.
5.4. Comparing the Volatility Models
The RiskMetrics EWMA model can be viewed as a special case of the GARCH(1,1)
with 0 0α = and 1 1 1α β+ = . With 0 10, 0α α> ≥ and 1 0β > as a condition for the
existence GARCH(1,1), it implies that the estimated variance of the RiskMetrics
EWMA closely mimics that of GARCH(1,1). Figure 5.23 and figure 5.24 shows the
absolute difference between the in-sample forecast of the variances by GARCH(1,1)
and the RiskMetrics EWMA. Most of the differences are close to zero indicating that
the volatility estimates by the two models at any time are close to each other.
The in-sample Root Mean Squared Forecast Errors (RMSFEs) which provides a
statistical measure of the forecasting performance of the volatility models are
presented in table 1.9. The lowest RMSFE which indicate better forecast of volatility
for each return series is shown in bold. In comparing the RMSFEs for the models in
the six markets, RiskMetrics EWMA comes first with the lowest RMSFEs in all the
markets, followed by GARCH and lastly the Historical Variance.
Market Forecasting
model S&P500 DJIA DJ-AIGCI CEDI/USD
US 3
months
T-Bill
OMXS30
Historical
Variance
1.0004 0.9844 1.7934 1.3800 1.2189 1.6306
GARCH 0.6841 0.6813 1.7782 1.3731 1.0690 1.0903
RiskMetrics
EWMA
0.4468 0.6387 1.6739 1.2830 0.9682 1.0234
Table 5.9: In-Sample Root Mean Squared Forecast Errors of the volatility models.
51
09/91 08/93 07/95 06/97 05/99 04/01 03/03 02/05 01/070
0.2
0.4
0.6
0.8
1
1.2
1.4
Absolute Value of the Difference Between Volatility Estimates
(S&P500)
Date
Differences
0 200 400 600 800 1000 1200 1400 1600 18000
0.5
1
1.5
2
2.5
3
3.5
4
Absolute Value of the Difference Between Volatility Estimates
(DJIA)
Date
Differences
01/00 11/00 09/01 07/02 05/03 03/04 01/05 11/05 09/06 07/070
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Absolute Value of the Difference Between Volatility Estimates
(OMXS30)
Date
Differences
Figure 5.23: GARCH(1,1) and EWMA estimators on the S&P500, DJIA and
OMXS30 return series.
52
01/91 12/92 11/94 10/96 09/98 08/00 07/02 02/04 05/060
0.2
0.4
0.6
0.8
1
1.2
1.4
Absolute Value of the Difference Between Volatility Estimates
(DJ-AIGCI)
Date
Differences
08/90 07/92 06/94 04/96 03/98 02/00 01/020
0.5
1
1.5
Absolute Value of the Difference Between Volatility Estimates
(3 Months US T-Bill)
Date
Differences
03/94 08/95 12/96 05/98 09/99 01/01 06/02 10/03 03/05 07/06 11/070
0.5
1
1.5
2
2.5
3
3.5
4
Absolute Value of the Difference Between Volatility Estimates
(CEDI/USD)
Date
Differences
Figure 5.24: GARCH(1,1) and EWMA estimators on the DJ-AIGCI, 3 Months US T-
Bill and CEDI/USD return series.
53
0 100 200 300 400 500 6000
5
10
15
20
25
Period
Volatilities
600-Period Volatility Forecast
(S&P500)
GARCH
EWMA
Historical Variance
0 100 200 300 400 500 6000
5
10
15
20
25
30
35
40
Period
Volatilities
600-Period Volatility Forecast
(DJIA)
GARCH
EWMA
Historical Variance
0 100 200 300 400 500 6000
5
10
15
20
25
30
35
40
Period
Volatilities
600-Period Volatility Forecast
(OMXS30)
GARCH
EWMA
Historical Variance
Figure 5.25: Volatility Forecast on S&P500, DJIA and OMXS30 return series.
54
0 100 200 300 400 500 6000
2
4
6
8
10
12
14
16
18
20
Period
Volatilities
600-Period Volatility Forecast
(DJ-AIGCI)
GARCH
EWMA
Historical Variance
0 100 200 300 400 500 6000
5
10
15
20
25
30
Period
Volatilities
600-Period Volatility Forecast
(3 Months US T-Bill)
GARCH
EWMA
Historical Variance
0 100 200 300 400 500 6000
5
10
15
20
25
30
35
Period
Volatilities
600-Period Volatility Forecast
(CEDI/USD)
GARCH
EWMA
Historical Variance
Figure 5.26: Volatility Forecast on DJ-AIGCI, 3 Months US T-Bill and CEDI/USD
return series.
55
Next, we consider the out-of-sample forecasting ability of the various models. The
600-period volatility forecasts on the various return series are presented in figure 5.25
and figure 5.26. The graphs indicate an increasing trend for the volatility forecasts
and buttress the scaling of volatility forecasts with time.
The Forecasts of average volatility over 20, 60 and 120 forecasting horizons are
constructed from each f the forecasting models. We use 600 trading days’ data for
each return series. A total of 30, 10 and 5 forecasts are generated for each forecasting
horizon respectively. The RMSFEs for 20, 60 and 120 forecasts are reported in tables
5.10, 5.11 and 5.12 respectively. In the 20 horizon, the GARCH(1,1) forecast best in
four markets with the Historical Variance and the RiskMetrics EWMA coming first
in each of the remaining two markets. However the story is different in the longer
horizons of 60 and 120. In each of this case, the Historical Volatility dominates in
four of the six markets. The GARCH(1,1) forecasts best in two markets in the 120
horizon whereas in the 60 horizon GARCH(1,1) and RiskMetrics EWMA performs
best in S&P500 and CEDI/USD respectively.
Market Forecasting
model S&P500 DJIA DJ-
AIGCI
CEDI/USD US 3
months
T-Bill
OMXS30
Historical
Variance 0.1743 0.1958 0.2852 1.5554 08318 0.4284
GARCH 0.1792 0.1912 0.2652 1.2221 0.7303 0.4241
RiskMetrics
EWMA
0.2885 0.2966 0.3889 1.1343 1.0337 0.4681
Table 5.10: Out-Sample Root Mean Squared Forecast Errors for a 20-day
Forecasting Horizon.
Market Forecasting
model S&P500 DJIA DJ-
AIGCI
CEDI/USD US 3
months
T-Bill
OMXS30
Historical
Variance
0.1328 0.1329 0.2036 1.1162 0.6870 0.3520
GARCH 0.1202 0.1439 0.2436
1.0995 0.7003 0.3871
RiskMetrics
EWMA
0.2454 0.2516 0.3577 0.9733 0.9824 0.3685
56
Table 5.11: Out-Sample Root Mean Squared Forecast Errors for a 60-day
Forecasting Horizon.
Market Forecasting
model S&P500 DJIA DJ-AIGCI CEDI/USD
US 3
months
T-Bill
OMXS30
Historical
Variance 0.0304 0.0274 0.1515 1.1069 0.7587 0.2420
GARCH 0.2099 0.2131 0.1745
0.6503 0.6863 0.7677
RiskMetrics
EWMA
0.2131 0.2168
0.3308 0.8119 0.8613 0.3225
Table 5.12: Out-Sample Root Mean Squared Forecast Errors for a 120-day
Forecasting Horizon.
5.5 Conclusions
We have compared the forecasting ability of the Historical Variance, The
GARCH(1,1) and the RiskMetrics Exponentially Weighted Moving Average volatility
models.
The In-Sample volatility forecasts by the complex models i.e. GARCH(1,1) and
RiskMetrics EWMA outperforms the simple Historical Variance. This finding is in
conformity with numerous other studies and financial time series literature19.
The Out-of-Sample forecasting accuracy comparisons are also in conformance with
earlier studies and volatility forecasting literatures. We found that for shorter
forecasting horizons, the GARCH(1,1) performs better whereas at longer horizons the
simple Historical Variance outperforms all in most markets. The complex models have
more parameters and thus add to the estimation errors and its forecasts are consistently
poor Out-of-Sample.
19 E.g Ederington & Guan(2005), Figlewski (1997), Poon & Granger (2002)
57
References
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Heteroscedasticity. Journal of Econometrics 31, 307-327.
• Brockwell P.J. and Davis R.A. (2002). Introduction to Time Series and
Forecasting, second edition, Springer-Verlag, New York.
• Campbell, John Y., Burton Malkiel, Martin Lettau, and Yexiao Xu. (2001).
Have Individual Stocks Become More Volatile? An Empirical Exploration of
Idiosyncratic Risk. Journal of Finance, 56, 1-43
• Ederington, L.H and Guan, W. (2005). Forecasting Volatility. Journal of
Futures Markets. 25, 465-490.
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Practice. Nobel Lecture. nobelprize.org/nobel_prizes/economics/laureates/2003/engle-
lecture.pdf
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58
Appendix
A.1 Aggregation Property of the Normal Distribution
The aggregation property of the normal distribution implies that the sum of jointly
normal random variables is also normal. This property underlines the RiskMetrics
assumption that portfolio’s return is the weighted sum of the underlying assets,(see
equation (17)).
Let , , 1,2,...,i tr i N= be the returns of the underlying assets of a portfolio then the
return on the portfolio is given by
, ,
1
N
p t i i t
i
r w r=
=∑ . (A1.1)
The portfolio returns has mean
,
1
N
p t i i
i
wµ µ=
=∑ ,
where ,( )i i tE rµ = . RiskMetrics models each return , , 1,2,...,i tr i N= as a random walk
as in equation (19) i.e.
, , ,i t i i t i tr µ σ ε= + . (A1.2)
Assume that the errors ,i tε are distributed as multivariate normal
( , )t tMVN Rε µ∼ , 1, 2, ,, ,...,t t N tε ε ε ε = (A1.3)
where tR is the correlation matrix of the errors , , 1, 2,...,i t i Nε = . In view of the
property that the sum of jointly normal random variables is normal, the portfolio return
,p tr is normally distributed with mean ,p tµ and variance 2
,p tσ given by
2
,
, 1
N
p t i j ij
i j
wwσ σ=
=∑ (A1.4)
A.2 Test for Conditional Normality
The quantile-quantile plot or for short Q-Q plot is used as a graphical check on the
deviation of the data from normality. The chart shows a plot of the quantiles of the
standardized distribution of the observed returns against the quantiles of the standard
normal distribution. A distribution is normal if the plot is a straight line whereas a
deviation from normality is characterize by deviation from the straight line.
59
In addition to the graphical inspection, a quantitative method can be used to calculate
the correlation coefficient of the Q-Q plot,
1
2 2
1 1
( )( )
( ) ( )
T
j j
j
QT T
j j
j j
q q r r
q q r r
ρ =
= =
− −
=
− −
∑
∑ ∑ (A2.1)
At 5% significant level, one needs 0.999Qρ ≥ for normality test to hold under a large
sample size, T.
A.3 The RiskMetrics k-period (day) Volatility Forecast
We derive here the k-period volatility forecast for a single asset whose return is
conditional normal as a proof of equations (35) and (36). This derivation can be
generalized for an n-dimensional case where the assets follow a conditional
multivariate normal distribution.
For the RiskMetrics model, the one-day log-price return ( , 1)t t + is assumed to follow
the dynamics
1 1 1t t tr σ ε+ + += (A3.1)
where
2 2 2
1 (1 )t t trσ λσ λ+ = + − (A3.2)
and [0,1]λ∈ . By definition, 1tσ + is measurable w.r.t tψ , and the error terms tε are
assumed to be N(0,1) and independent of the information at time t. The k-day return is
given by
,
1
k
t t k t i
i
r r+ +=
=∑ , (A3.3)
hence from equation (A3.1) it follows that
,
1
k
t t k t i t i
i
r σ ε+ + +=
=∑ . (A3.4)
The expected return for the k-days at time t is
60
,
1
( | ) ( | )k
t t k t t i t i t
i
E r Eψ σ ε ψ+ + +=
=∑
1
1
[ ( | ) | ]k
t i t i t i t
i
E E σ ε ψ ψ+ + + −=
=∑
( )1
1
[| ( | ) | ]k
t i t i t i t
i
E Eσ ε ψ ψ+ + + −=
=∑
1
[ ( | )]k
t i t i t
i
E Eσ ε ψ+ +=
=∑
0= (A3.5)
and the corresponding variance is
( )2
, , ,( | ) [ ( | )] |t t k t t t k t t k t tVar r E r E rψ ψ ψ+ + += −
2
,( | )t t k tE r ψ+=
2
1
[ ] |k
t i t
i
E r ψ+=
=
∑
, 1
( | )k
t i t i t j t j t
i j
E σ ε σ ε ψ+ + + +=
=∑ . (A3.6)
It is obvious that for i j≠ , the expectation vanishes. Assume for example that
1 i j≤ < , then
1( | ) [ ( | ) | ]t i t i t j t j t t i t i t j t j t j tE E Eσ ε σ ε ψ σ ε σ ε ψ ψ+ + + + + + + + + −=
( )1[ ( | ) | ]t i t i t j t j t j tE Eσ ε σ ε ψ ψ+ + + + + −=
[ ( ) | ]
0
t i t i t j t j tE Eσ ε σ ε ψ+ + + +=
= (A3.7)
This holds for i j> also because of symmetry.
For 1i j= ≥ , we have
2 2 2 2
1( | ) [ ( | ) | ]t i t i t t i t i t i tE E Eσ ε ψ σ ε ψ ψ+ + + + + −=
( )2 2
1
2 2
2
[ ( | ) | ]
[ ( | )]
[ | ].
t i t i t i t
t i t i t
t i t
E E
E E
E
σ ε ψ ψ
σ ε ψ
σ ψ
+ + + −
+ +
+
=
=
=
(A3.8)
Substituting this result into equation (A3.6), we get
2
,
1
( | ) [ | ].k
t t k t t i t
i
Var r Eψ σ ψ+ +=
=∑ (A3.9)
61
From the definition of 2
1tσ + it follows that
2 2 2
1 1[ | ] [ (1 ) | ]t i t t i t i tE E rσ ψ λσ λ ψ+ + − + −= + −
2 2
1 1
2 2
1 1
2
1
[ | ] (1 ) ( | )]
[ | ] (1 ) [ | ]
[ | ],
t i t t i t
t i t t i t
t i t
E E r
E E
E
λ σ ψ λ ψ
λ σ ψ λ σ ψ
σ ψ
+ − + −
+ − + −
+ −
= + −
= + −
=
(A3.10)
and by recursion it follows that
2 2 2
1( | ) [ | ] ,t i t t i t tE r Eψ σ ψ σ+ + += = (A3.11)
for all 0i ≥ . By substituting the above expression into equation (A3.9), finally gives
2
, 1
1
( | ) [ | ]k
t t k t t t
i
Var r Eψ σ ψ+ +=
=∑
2
1
1
2
1.
k
t
i
tk
σ
σ
+=
+
=
=
∑ (A3.12)
The k-period volatility estimate is then given by taking square root of the variance
| 1|t k t t tkσ σ+ += . (A3.13)