forecasting value-at-risk by nasir ali khan

7/29/2019 Forecasting Value-At-risk by Nasir Ali Khan

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FORECASTING VALUE-AT-RISK BYUSING GARCHMODELS

BY

NASIRALI KHAN

B.S.(ACTUARIAL SCIENCE &RISKMANAGEMENT)

UNIVERSITY OF KARACHI

February 2007


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ACKNOWLEDGEMENT

Acknowledge is due to Department of Statistics, University of Karachi for support of this

project.

We wish to express our appreciation to Mr. Usaman Shahid who served as advisor, and

has major influence in guiding us in the correct direction of implementing the project.

Specially, I thanks to my taecher Mr. Uzair Mirza for supporting and helping me in

problems I faced during this period.

I would also like to thanks my friends and colleagues who have been of great help by

providing advices to improve our work.


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ABSTRACT: The variance of a portfolio can be forecasted using asingle index model or the covariance matrix of the portfolio. Using

univariate and multivariate conditional volatility models, this paper

evaluates the performance of the single index and portfolio models in

forecasting Value-at-Risk (VaR) of a portfolio by using GARCH-type

models, suggests that which model have lesser number of violations,

and better explains the realized variation.


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IV

TABLE OF CONTENTS

1.0 Introduction 11.1 Stochastic Process

1.1.1 Stationary time series1.1.2 Non-stationary time series1.2 Value-at-Risk1.3 Conditional volatility model1.4 Methodology

2.0 Conditional Hetroscedastic Model .102.1 Introduction2.2 Assumption for OLS Regression2.3 Hetroscedasticiy2.4 Hetroscedastic Model and its specification

2.4.1 ARCH Model2.4.2 GARCH Model2.4.3 EGARCH Model2.4.4 GJR Model

2.5 Parameter Estimation2.6 Dynamic Conditional Correlations

3.0 Data 203.1 Introduction3.2 KSE-100 Index3.3 Sample Portfolio

4.0 Pre-Estimation Analysis ...264.1 Introduction4.2 Ljung-Box-Pierce Q-Test4.3 Engle ARCH Test4.4 Analysis

5.0 Applying Models 325.1 Introduction5.2 Estimation of Parameter and Conditional Volatility5.3 Post Estimation Analysis

6.0 Testing of Models ...406.1 Introduction6.2 Linear Regression Approach6.3 Back Testing

7.0 Conclusion44


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8.0 Appendices ..45A. PlotsB. ACF & PACF Plots of RetornsC. ACF & PACF Plots of Squared RetornsD. ACF & PACF Plots of Absolute RetornsE.

Ljung-Box Peirce Q-TestF. Engles ARCH Test

G. Parameter EstimationH. Linear Regression ApproachI. Back TestingJ. MATLAB Code

9.0 Biblography .79


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LIST OF FIGURE

3.1 Daily prices, returns and squared returns of KSE-100 index ........213.2 Price trend of six selected stocks ......233.3 Plot of returns and squared returns of sample portfolio ........244.1 ACF and PACF plot of log returns of KSE-100 index .....284.2 ACF.and PACF plot of squared returns of KSE-100 index ......294.3 ACF and PACF plot of absolute returns of KSE-100 index .....294.4 LBQ-test and ARCH-test of KSE-100 index returns ....304.5 LBQ-test and ARCH-test of KSE-100 index squared returns .....315.1 Estimated coefficient with iterations for GARCH(1,1) specification ...345.2 Comparison of return, innovations and conditional volatilities ........365.3 Standardized innovations of GARCH(1,1) ...375.4 ACF plot of squared standardized innovations .375.5 LBQ-test and ARCHtest for innovations 386.1 forecasted VaR and violations of upper and lower band...42


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LIST OF TABLES

3.1

portfolio components with their respective sectors, index weights and portfolio weights22

3.2 Descriptive statistics of index and sample stocks returns 256.1 Number and percentage of violation of VaR bands .....42


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1.0 INTRODUCTION

Conditional variance of the portfolio or stock returns is one of the key ingredients

required to calculate the Value-at-Risk (VaR) of a portfolio or any individual stock.

Conditional volatility models used to estimate the conditional variance of the portfolio

returns either by using a multivariate volatility model to forecast the conditional variance

of each asset in the portfolio, as well as the conditional co variances between the assets

pair, in order to calculate the forecasted portfolio conditional variance and called theportfolio model; or fitting univaiate volatility model to the portfolio returns, called single

index model. Secondly, compare each estimated conditional variance and VaR from

different hetroscedastic models and suggests which one model is better explains.

In this document, I compare the performance of the KSE-100 index as single index and

portfolio, which covers 45% market capitalization of KSE-100 index and contains its six

highly traded components, taking as sample. I also compare single index model with

same portfolio model. There are different criteria are used to compare the forecasting

performance of the various conditional volatility models and methods considered,

namely: (1) the linear regression approach of Pagan and Schwert (1990) and (2)

backtesting method of Crnkovic and Drachman (1996), applied in J. P. Morgans

RiskMetrics technical document

Engle (2000) proposed a Dynamic Conditional Correlation (DCC) multivariate GARCH

model which models the conditional variances and correlations using a single step

procedure and which parameterizes the conditional correlations directly in a bivariate

GARCH model. In this approach, a univariate GARCH model is fitted to a product of

two return series. Parameters or model coefficients of GARCH model can be estimated

by log likelihood estimation.

The need to model the variance of a financial portfolio accurately has become especially

important following the 1995 amendment to the Basel Accord, whereby banks were

permitted to use internal models to calculate their Value-at-Risk (VaR) thresholds (see


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Jorion (2000) for a detailed discussion of VaR). This amendment was in response to

widespread criticism that the Standardized approach, which banks used to calculate

their VaR thresholds, led to excessively conservative forecasts. Excessive conservatism

has a negative impact on the profitability of banks as higher capital charges are

subsequently required.

Although the amendment to the Basel Accord was designed to reward institutions with

superior risk management systems, a back testing procedure, whereby the realized returns

are compared with the VaR forecasts, was introduced to asses the quality of the internal

models. In cases where the internal models lead to a greater number of violations than

could reasonably be expected, given the confidence level, the bank is required to hold a

higher level of capital. If a banks VaR forecasts were violated more than 9 times in any

financial year, the bank may be required to adopt the Standardized approach. The

imposition of such a penalty is severe as it affects the profitability of the bank directly

through higher capital charges has a damaging effect on the banks reputation, and may

lead to the imposition of a more stringent external model to forecast the banks VaR.

1.1 STOCHASTIC PROCESSStochastic or random process is a collection of random variable ordered in time. If we let

P denote a random variable, and if it is continuous, we denote as P(t), but if it is discrete,

we denoted it as Pt. Economic data and Assets prices, such as daily stock prices, bond

prices, foreign exchange rate, GDP, inflation rates etc. are examples of stochastic process

and in the discrete form. As we work on equity prices, IfPtdenote a stocks price at time

t, where t=1,2,3,.n. stochastic process are two types, stationary and non-stationary

stochastic process.

1.1.1

STATIONARY TIME SERIES

The foundation of time series is stationary. A time series {rt} is said to be strictly

stationary if the joint distribution of (rt1,.,rtk) is identical to that of (rt1+t,.,rtk+t) for all

t, where kis arbitrary positive integer and (t1,.tk) is a collection of k positive integers. In

other words, a stochastic process is said to be stationary if its mean and variance are

constant over time and the value of covariance between the two time periods depends


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only on the distance or gap or the lag between two time periods and not the actual time at

which covariance is computed. A time series {rt} is weakly stationary if both the mean of

rt and covariance rt and rt-l, where l is an arbitrary integer. More over rt is weakly

stationary if:

( )

( ) ( )

( ) ( )( )[ ] kkttktttt

t

PPEPPCov

PEPVar

PE

==

==

=

++,

2

1.1.2 NON-STATIONARY TIME SERIESIf a time series is not stationary in the sense just defined, it is called non-stationary time

series. In other words, a non-stationary time series will have a time-varying mean or

time-varying variance or both. The classic example of non-stationary time series is

Random Walk Model (RWM). It is often said that asset prices, such as stock prices or

exchange rates, follows a random walk; that is they are non-stationary. There are two

types of random walks:

1. Random walk without drift2. Random walk with drift

RANDOM WALK WITHOUT DRIFTLet rt be a white noise with mean 0 and variance 2. Then the series Pt is said to be

random walk if:

ttt rPP += 1

In random walk model shows that the value ofPat time is equal to its value at time t-1

plus a random shock; thus it is an AR(1). Believer in the efficient capital market

hypothesis argue that stock prices are essentially random and no scope for profitable in

the stock markets.


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In general, if the process started at some time 0 with value ofY0, we have

+=tt rPP 0

( ) ( ) 00 PrPEPE tt =+=

( ) 2tPV t =

RANDOM WALK WITH DRIFTLet

ttt rPP ++= 1 Where is known as drift parameter.

We ca also write as

tttt rPP += It shows thatPtdrift upward or downward, depending on m being positive or negative. It

is also anAR(1) model.

Where

( ) tPPE t += 0 ( ) 2tPV t =

RWM with drift the mean as well as the variance increases over time, again violating the

conditions of weak stationary. RWM, with or without drift, is a non-stationary stochastic

process.

1.2 VALUE-AT-RISK (VaR)When using VaR measure, we are interested in making a statement of the following form.For example:

We are P% certain that we will not lose more than $V in the next N days.


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Vis the VaR of the portfolio. It is a function of two parameters: Nis the time horizon,

and, the confidence level. It is the loss level overNdays that the manager isP% certain

will not be exceeded. In general, when an N day is the time horizon, and P% is the

confidence level, VaR is the loss corresponding to the (100-P)th percentile of the

percentile of the distribution of the change in the value of the portfolio over the next Ndays.

VaR is an attractive measure because it is easy to understand. In essence, it asks the

simple question how bad can things get? All senior mangers want answered this

question. A measure that deals with the problem we have just mentioned is conditional

VaR (C-VaR). C-VaR if things do get bad, how much can we expect to lose? C-VaR is

expected loss during an N-days period conditional that we are in the (100-X)% left tail of

the distribution.

In theory, VaR has two parameters. These are N, the time horizon measured in days, and

X, and the confidence interval. In practice, analyst almost invariability set N = 1 in the

first instance. The usual assumption is

N-day VaR = 1-day VaR (N)

This formula is exactly true when the changes in the value of the portfolio on the

successive days have independent identical normal distribution with mean zero. In other

cases, it is an approximation.

There are different methodologies to calculate the VaR, most popular are Historical

simulation, Variance Covariance, Monte Carlo simulation, J. P. Morgans

RiskMetrics Methodology etc. Different methodologies have different approaches and

different inputs to calculate the VaR of the portfolio, but volatility is the main ingredient

to estimate the VaR of an asset. In this technical report, estimate volatility by using

RiskMetrics. Actually, this technical report focused on C-VaR or forecasting VaR, so

RiskMetrics is specially design for C-VaR. however, volatility is the main ingredient tocalculate VaR and conditional VaR depends upon conditional volatility.

RiskMetrics uses historical time series analysis to derive estimates of volatilities and

correlations on a large set of financial instruments. It assumes that the distribution of past

returns can be modeled to provide us with a reasonable forecast of future returns over

different horizons. While RiskMetrics assumes conditional normality of returns, we have


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refined the estimation process to incorporate the fact that most markets show kurtosis and

leptokurtosis. We will be publishing factors to adjust for this effect once the RiskMetrics

customizable data engine becomes available on the Reuters Web. These volatility and

correlation estimates can be used as inputs to:

Analytical VaR models Full valuation models.

Users should be in a position to estimate market risks in portfolios of foreign exchange,

fixed income, equity and commodity products.

1.3 CONDITIONAL VOLATILITY MODELSAs in technical document, J. P. Morgans RiskMetrics (1995) introduced EWMA

approach. RiskMetrics uses the exponentially weighted moving average model (EWMA)

to forecast variances and covariances (volatilities and correlations) of the multivariate

normal distribution. This approach is just as simple, yet an improvement over the

traditional volatility forecasting method that relies on moving averages with fixed, equal

weights. This latter method is referred to as the simple moving average (SMA) model.

To compute exponentially weighted (standard deviation) volatility, formula given as:

=

+ )1=

T

t

t

t

t rr1

12

1 )(( (1.1)

Consequently, under the EWMA model in Eq. (1.1) the conditional variance or rt is

proportional to the time horizon k. The conditional standard deviations of a k-period

horizon return kat+1

EWMA is a special form of heteroscedastic model IGARCH(1,1) with intercept equal to

zero, todays GARCH-type models have gained the most attention, because that time

series realization of returns often show time-dependent volatility. This idea was first give

in Engles (1982) ARCH (Auto Regressive Conditional Heteroscedasticity) model, which

is base on the specification of conditional densities at consecutive periods with a time


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dependent volatility process. Volatility compute by these models is easy; it is two-step

regression model on assets returns. Then we will also work on GARCH models, which

are advance shape of ARCH model, which firstly introduced by Bolerslev, T. in his

journal of Econometrics, Generalized Auto Regressive Conditional Hetroscedasticity in

1986, other are GJR, EGARCH, IGARCH, and etc. the detail discussions about eachmodel will be in next chapter. EWMA is special form of IGARCH(1,1) without drift or

with intercept 0.

1.4 METHODOLOGYRisk is often measured in terms of price changes. These changes can take a variety of

forms such as absolute price change, relative price change, and log price change. When aprice change is defined relative to some initial price, it is known as a return. RiskMetrics

measures change in value of a portfolio (often referred to as the adverse price move) in

terms of log price changes also known as continuously compounded returns.

For any asset, we know the price, for example at time tis Ptof an asset, and its current

price isPt+1, so the absolute return,Ra, for an asset at time tfrom t+1 is given as:

tta PPR = + 1 (1.2)

Where a in the subscript ofR is referring to absolute, relative price change or return, Rr,

relative to pricePt:

t

ttr

P

PPR

= + 1

(1.3)

Then the log price change or continuously compounded return, rt+1, of a security is

defined to be the natural logarithm of its gross return.


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=

+=

+

t

t

r

P

Pr

Rr

1ln

)1ln(

( ) ( )ttt PPr lnln 11 = ++ (1.4)

We firstly compose the portfolio, give weights each with respect its market capitalization

and KSE-100 index acts as single or index, which discuss in detail in chapter 3. In

chapter 4, now we check each time series of stocks return that model which we have

discussed in chapter 2 are applicable or not. For this, we use three methods, ACF and

PACF, Ljung-Box-Perice Q-Test and Engles ARCH Test; apply on returns and squared

returns and absolute. In chapter 5, by using MATLAB 7.0 commands, estimate the

parameter or coefficients of each of the given models, MATLAB commands and code are

given in the appendix J that used for this paper and then test the outputs. In the next

chapter, assessing the each model by to approaches, linear regression approach and back

testing. In regression approach, apply simple linear regression on each model variance by

taking independent variable, squared returns as dependent, check coefficients, and

coefficient of determination under the assumptions of OLS model. On the other hand,back testing on each model and confirm the number of VaR violations. Suppose that the

financial position is a long position so that loss occurs when there is a big price drop. If

the probability is set to 5%, then RiskMeterics uses 1.65t+1 to measure the risk of the

portfolio-that is, it uses the one sided 5% quantile of a normal distribution with mean and

variance 0 and t+1. The actual 5% quantile is -1.65t+1, but the negative sign is ignored

with the understanding that it signifies a loss. Consequently, if the standard deviation is

measured in percentage, the daily VaR of the portfolio under RiskMeterics is:

VaR = (Amount of Position) (1.65t+1)

In addition, that of a k-day horizon is

VaR = (Amount of Position) (1.65t+1)


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Where the argument (k) of VaR is used to denote the time horizon. We have

VaR = k x VaR

This is referred to as the square root of time rule in VaR calculation under RiskMetreics.


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2.0 CONDITIONAL HETEROSCEDASTIC

MODELS

2.1 INTRODUCTIONEconometricians are typically to determine how much one variable will change in

response to a change in some other variable. Increasingly however, econometricians are

being asked to forecast and analyze the size of the errors of the model. In this case, the

questions are about volatility, and the standard tools have become the ARCH/GARCH

models.

2.2 ASSUMPTIONS FOR OLSREGRESSIONOLS (ordinary least square) regression analysis is the great workhorse of economists and

statisticians and forms a central tool in financial modeling. The well-proved technique

relies on a set of four basic underlying assumptions to produce linear models that are the

Best Unbiased Linear Estimate (BLUE). Furthermore, it is necessary to add a fifth

assumption of homoscedasticity to obtain results that are also statistically consistent. It is

assume that there are linear parameters, meaning that there is a linear relationship

between the dependent and the explanatory variables. Mathematically this relationship is

expresses as a function of the form:

uXY ++= 10 (2.1)

Where 0 is the intercept, 1 the slope of the function and where u represents an error

term containing all the factors affecting Yother than the specified independent variable.

Second, it is intuitively a necessity that the sample to be analyzed must consist of a

random sample of the relevant population to yield an unbiased result. Mathematically (xi,


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yi): i = 1, 2...n.Third, a zero conditional mean is assumed. This means that the linear

model will be the single line that minimizes the value of the sum of all error terms

forming an average of the positive and negative errors. In other words, this means that the

average error term of the function should always be zero as the negative and positive

errors cancel each other out. This can be refined into the assumption that the averagevalue ofu does not depend on the value ofXas for any value ofXthe average value ofu

will be equal to the average value of u in the entire population, which is zero.

Mathematically it can Econometricians are typically to determine how much one variable

will change in response to a change in some other variable. Increasingly however,

econometricians are being asked to forecast and analyze the size of the errors of the

model. In this case, the questions are about volatility, and the standard tools have become

the ARCH/GARCH models be expressed as:

( ) ( ) 0| == uExuE

Fourth, it is assumed a sample variation in the independent variables X. This means that

two independent variables cannot be equal to the same constant. This is however not an

assumption that is likely to fail in an interesting statistical analysis as a completely

homogenous population is not the typical target for statistical analysis. The assumption is

defined asxi, i = 1, 2...n. These assumptions assure an unbiased result where the sample

n is equal to the populationn.

Finally, we assume homoscedasticity to obtain a consistent result. This assumption states

that the value of the variance of error term u conditional on the explanatory variable Xis

constant. In other words, the pattern of distribution of error terms at any given value ofX

will show the same distribution with a mean around the samplenX. This is expressed as:

2)|( =xuVar

2.3 HETROSCEDASTICITYThe effect of a violation of this assumption is that we still have a BLUE model, however it is

no longer consistent and, as a result the regression output in terms of test statistics can no

longer be reliable. This is because the variance that is in the heart of these statistics is no

longer constant and will hence be false.


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In cross sectional data the distribution of the error term u should constant distribution in

relation to the function uXY ++= 10 . Heteroscedasticity in time series will take a

graphically different appearance is that the variance, or the volatility, will vary according

to time. Engle (1982) and others with him have looked at the properties of the volatility

of financial markets and there is wide recognition of the presence of heteroscedasticity in

the distribution of returns. Mandelbrot (2002) describes this phenomenon as a clustering

of volatility where a period of high volatility is likely to be followed by another period of

high volatility and opposite. The sometimes calm and sometimes turbulent volatilities

observed in financial markets.

Research has found out that a relationship between volatility from one period to the next

one exists. The presence of this heteroscedastic relationship may be used when modeling

and forecasting future volatility of financial markets. The range and complexity ofmethods applied to this problem is vast.

2.4 HETERSCEDASTIC MODELS AND ITS SPECIFICATIONSimply using all past information on past price movements does in fact utilize the

heteroscedastic properties of financial markets to some extent. By using the formula

( )=

=N

t

t rrN 1

2

1

1

(2.2)

The assumption that all past prices have an equal relevance in the shaping of the

Volatility of the future is applied. Intuitively this assumption is too crude as more recent

volatility is likely to have more relevance than that of several years ago and should hence

be given a relatively higher weight in the calculation. A simple way to counter this

problem is done by only using the last 30 days to calculate the historical volatility and

this model is actually widely used by actors in the financial markets. The model weighs

volatility older than 30 days as zero and puts equal weight on the volatility of the last 30

days. This model is however still crude and more sophisticated models are frequently

used moving into the area covered by models such as the Exponential Weighted

Volatility models (EWMA) as shown in Eq (1.1).


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Models that can be used to estimate the conditional variance of a portfolio directly by

modeling the historical stock or portfolio returns or indirectly by modeling the

conditional variance of each asset and the conditional correlation of each pair of assets

(namely, the portfolio model) are shown in this section. Financial returns are typically

modeled as a stationaryAR(1) process.

2.4.1 ARCH MODELPrior to the ARCH model introduced by Engle (1982), the most common way to forecast

volatility was to determine the standard deviation using a fixed number of the most recent

observations. As we know that the variance is not constant, i.e. homoscedastic, but rather

a heterocedastic process, it is unattractive to apply equal weights considering we know

recent events are more relevant. Moreover, it is not beneficial to assume zero weights for

observations prior to the fixed timeframe. The ARCH model overcomes these

assumptions by letting the weights be parameters to be estimated thereby determining the

most appropriate weights to forecast the variance. An ARCH(1) model, where the

conditional variance depends only on one lagged square error, is given by

2

110

2

+= tt (2.3)

We can capture more of the dependence in the conditional variance by increasing the

number of lags,p , giving us an ARCH(p ) model

22

22

2

110

2

ptpttt ++++= L (2.4)

2.4.2 GARCH MODELBollerslev in 1986 proposed a useful extension known as the Generalized Autoregressive

Conditional Heteroscedasticity (GARCH) model. For a log return rt. Let at be the mean

corrected log return. Iftfollows a GARCH(u, v) model is:


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ttt = = =

++=u

i

v

j

jtjitit

1 1

22

0

2

(2.5)

Since, we know that r t is log return of daily stock prices with mean, =0 and we also

know that tis mean corrected return and can be written as:

= tt r

Where t is a random variable with mean 0and variance 1, and assumed to be standard

normal or standardized student-t distribution. If v=0, then equation (2.5) reduces to

ARCH(u). Simplest form of GARCH model is GARCH(1,1), and written as

2

1

2

10

2

++= ttt (2.6)

Where, ( ) 1,1,0


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( ) {tsStudent

GaussianEE

jt

jt

jt

'

/2

2

2

1

2

=

=

As the range oflog (t) is the real number line, the EGARCH model does not require anyparametric restrictions to ensure that the conditional variances are positive. Furthermore,

the EGARCH specification is able to capture several stylized facts, such as small positive

shocks having a greater impact on conditional volatility than small negative shocks, and

large negative shocks having a greater impact on conditional volatility than large positiveshocks. Such features in financial returns and risk are often cited in the literature to

support the use of EGARCH to model the conditional variances.

2.4.4 GJRMODELGlosten, Jagannathan and Runkle (1992) extended the GARCH model to capture possible

asymmetries between the effects of positive and negative shocks of the same magnitude

on the conditional variance through changes in the debt-equity ratio. The GJR(u,v) model

is given by:

( ) =

=

+++=v

i

iti

u

j

ttjtjt D11

2

11

2

0 (2.8)

Where ttt = and indicator or dummy variable,D (1), is defined as

( ) 01 00{>= t

if

iftD

For the case u = 1, a0>0, 1>0, 1+1 >0, 10 are sufficient conditions to ensure a

strictly positive conditional variance, t>0. The indicator variable distinguishes between

positive and negative shocks, where the asymmetric effect ( 1 > 0 ) measures the

contribution of shocks to both short run persistence (1 +1/ 2 ) and long run persistence

(1 +1 +1/ 2). Several important theoretical results are relevant for the GARCH model.


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Ling and McAleer (2002) established the necessary and sufficient conditions for strict

stationarity and ergodicity, as well as for the existence of all moments, for the univariate

GARCH( u,v ) model, and Ling and McAleer (2003) demonstrated that the QMLE for

GARCH( u, v ) is consistent if the second moment is finite, E(2)


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=

2

2

2

122

1i

i

em

it

(2.9)

Where m is the number of observations.

Maximizing an expression is equivalent to maximizing the logarithm of the expression.

Taking the logarithm of the expression in equation (2.7) and ignoring constant

multiplicative factors, it can be seen that we wish to maximize.

=

m

i t

it

1

2

22ln

An iterative search is used to find the parameters in the model that maximize the

expression in equation (2.9).

2.6 DYNAMIC CONDITIONAL CORRELATIONA new class of multivariate GARCH estimators which can best be viewed as a

generalization of Bollerslev(1990)s constant conditional correlation estimator. In

ttt RDDH =

Where tit hdiagD ,=

Where Ht is conditional covariance matrix and R is a correlation matrix containing the

conditional correlations as can directly be seen from rewriting this equation as:

[ ] RDHDE tttttt ==

11

1 '

Since ttt D 1=


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The expressions for h are typically thought of as univariate GARCH models; however,

these models could certainly include functions of the other variables in the system as

predetermined variables or exogenous variables. A simple estimate of R is the

unconditional correlation matrix of the standardized residuals.

Engles (2000) proposes an estimator called Dynamic Conditional Correlation orDCC.

The dynamic correlation model differs only in allowingR to be time varying:

tttt DRDH = Parameterizations ofR have the same requirements thatHdid except that the conditional

variances must be unity. The matrixRtremains the correlation matrix.

Probably the simplest specification for the correlation matrix is the exponential smoother,

which can be expressed as:

=

=

= =1

1

,

1

1

,

,

1

1 ,

,t

t

itj

t

t

iti

ij

t

t iti

tij

A geometrically weighted average of standardized residuals. Clearly, these equations will

produce a correlation matrix at each point in time. A simple way to construct this

correlation is through exponential smoothing. In this case the process followed by the

( ) 1,1,1,, 1 += tijtjtitij qq

tjjtii

tij

tijqq

q

,,

,

, =


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A natural alternative is suggested by the GARCH(u, v) model.

= =

++=u

m

v

n

ntnmtmtmtij qq

1 1

0,

Matrix versions of these estimators can be written as:

( )( ) 111 '1 += tttt QQ

and 111 )'()1( ++= tttt QSQ where Sis the unconditional correlation matrix and Qt is covariance matrix.

So, GARCH models can also be used for updating covariance estimates and forecasting

the future level of covariance. For example, the GARCH(1,1) model for updating a

covariance and simply write as is

Cov t= 0 +1ri,t-1r j,t-1 +Covt-1

In addition, the long-term average covariance is 0/ (11+1).


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3.0 DATA

3.1 INTRODUCTIONFor this study, I select KSE-100 Index stocks of Karachi Stock Exchange. My main

objective is not only to forecast conditional volatilities or to suggest which model better

explains the given data, but I also want to study the behavior of our local markets. Wecan also implement on the other financial and economic time series data, such are foreign

exchange rate, bonds or fixed income securities, etc. but for ARCH/GARCH model

require large sample size data and it cannot easily available.

3.2 KSE-100INDEXKarachi Stock Exchange is the biggest and liquid exchange and has been declared as the

Best Performing Stock Market of the World for the year 2002. As on June 30, 2006,658 companies were listed with the market capitalization of Rs. 2,801.182 billion (US $

46.69) having listed capital of Rs. 495.968 billion (US $ 8.27 billion). KSE began with a

50 shares index. On November 1, 1991 the KSE-100 was introduced and remains to this

date the most generally accepted measure of the Exchange. The KSE-100 is a capital-

weighted index and consists of 100 companies representing about 90 percent of market

capitalization of the Exchange. KSE-100 index is weighted average of 100 stock prices

on Karachi Stock Exchange and is therefore good benchmark for KSE. The KSE-100

Index closed at 9989.41 on June 30, 2006. Karachi Stock Exchange recently introduced

KSE-30 index, last year.

As we see in figure 3.1, KSE-100 index have increasing trend from Jan 2004 to Oct 2006.

There some fluctuation in KSE-100 between Jan and Jul in both year 2004 and 2005,

KSE-100 rapidly increases and drop, known as March Crisis. KSE-100 index average

returns are zero throughout the graph; there are huge fluctuations and deviations between


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Jan and Jul in 2004 and 2005. The main purpose of square returns plot to analyze the

variation in the returns of index.

Figure 3.1: Daily prices, returns and squared returns of KSE-100 index.

The data used for this study consists of stock prices from six different companies and the

price of the index itself, selection based on market capitalization. The prices are corrected

for stock splits and dividend. The daily closing prices were retrieved dated from 22 Dec

2003 to Oct 2006. This means that the dataset consists of 692 observations for each

company and the index itself. The mixture of six different stocks called Portfolio.

3.3 SAMPLE PORTFOLIOOGDC is on number one of the list of KSE-100 index, because of its huge market

capitalization, its weight is 20%, KSE-100 compose on the basis of Market capitalization

and its market capitalization is about Rs. 50, 707, 946, 000. On the second number,

Pakistan Telecommunication, the biggest telephone service provider, only fixed landline

telephone service provider in the country and have large customers. Their subsidiaries are


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Ufone (Cell phone service Provider), Paknet (Internet service provider) and V-PTCL

(based on WLL technology). National Bank is the biggest bank in Pakistan, which

management is completely own by the public sector. The MCB Bank Limited is one of

the largest banks in Pakistan. MCB, advised by Merrill Lynch, became the fourth

Pakistani company (the other three being Hubco, PTCL and Chakwal Cement - they allhave been de listed) to list on the London Stock Exchange when it raised $150 million

floating global depositary receipts. MCB Bank Ltd. ranks amongst the Leaders in the

commercial banking industry. MCB has been one of the most profitable banks of 2005,

registering an increase of over 250% in net profits. POL is a petroleum exploration and

production company. The company also own and operates a network of pipelines for

transportation of crude oil to Attock Refinery Limited in Rawalpindi. POL has two

subsidiaries. One subsidiary CAPGAS markets LPG and another subsidiary Attock

Chemical produces sulphuric acid. Pakistan State Oil (PSO) is the oil market leader in

Pakistan. The well-established infrastructure, built at par with international standards,

representing 82% of countrys storage, provides PSO an edge over its competitors. PSO

is currently enjoying over 73% share of Black Oil market and 59% share of White Oil

market. It is engaged in import, storage, distribution and marketing of various petroleum

products including mogas, high speed diesel (HSD), fuel oil, jet fuel, kerosene, liquefied

petroleum gas (LPG), compressed natural gas (CNG) and petrochemicals. PSO also

enjoys around 35% market participation in lubricants.

Table 3.1: Sample portfolio components with their respective sectors, index weights and portfolio weights.


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Figure 3.2: price trend of six selected stocks.

As we see in figure 3.2, stocks price prices moves in horizontal direction form jan 2004 toJan 2005 as well as KSE-100 also move in the same in the same direction. Nevertheless, after

Jan 2004 all stock prices rapidly increased, PTC and MCB have very small effect of it.

Stocks that have Massive change in prices all are from oil and gas sector, except NBP.

Therefore, this gain in prices is not due to demand and supply or change in prices of

petroleum product in domestic and international market, this is rumor which made by some

big market player. If compare both figures 3.1 and 3.3, portfoliossquared returns are greater

as compare to square return of KSE-100 index. Therefore, portfolio, which composed, is

more volatile than KSE-100 index, because KSE-100 is hundred shares portfolio, more

diversified and of course, it has less return. However, trend in both are identical, sampleportfolio contain contains those stocks which are actively trade in a market and these stocks

have huge average daily trading volume.


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Figure 3.3: Plot of returns and squared returns of sample portfolio.

We assume log returns are random variable or white noise, which are normally distributed

with mean 0 and variance, say 2. As we see in table 3.2 that all mean of log returns are

significantly close to zero and some exactly equal to zero, like PTC and PSO. Variances of

stocks log returns are approximately same, .001. We know that kurtosis of normal

distribution is 3. POL log returns are not normally distributed, because its kurtosis is much

greater than normal kurtosis and other kurtosis are significantly close to 3. All medians are

close to its corresponding means. All returns are negatively skewed, but close to symmetric

except POL is more negatively skewed, Skewness is -4.42. Finally, I suggest that all log

returns are normally distributed with mean and median close to zero, all variances are similar

and symmetric except POL returns. All of six stocks are good correlated to KSE-100 index.

As KSE-100 20% depends on OGDC and its correlation is close to zero, 0.877. In addition,

correlations among stocks are normally between 0.5 and 0.7. as we compare figure 3.1 and

3.3, price trend of KSE-100 and portfolio are same and same movement of returns and

squared returns. So we say that portfolio follow the same trend and behavior as KSE-100 and

portfolio is true sample of KSE-100 index.


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Table 3.2: Descriptive statistics of index and sample stocks returns.

Table 3.3: Correlations among stocks and index.


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4.0 PRE-ESTIMATION ANALYSIS

4.1 INTRODUCTIONTo justify modeling the returns by a GARCH model, the presence needs to be detected

first. To detect the presence of a GARCH process, some qualitative and quantitative

checks can be performed on the dataset. To check it qualitatively, plots will be made ofthe sample autocorrelation function (ACF) and the partial-autocorrelation function

(PACF) on the returns, looking for signs of correlation. For quantitative checks, two tests

will be employed, Ljung-Box-Pierce Q-Test and Engle's ARCH Test.

4.2 LJUNG-BOX-PIERCE Q-TESTThe Ljung-Box Pierce Q-Test (LBQ Test) can verify, at least approximately, if asignificant correlation is present or not. It performs a lack-of-fit hypothesis test for model

misspecification, which is based on the Q-statistic.

( )=

+=L

k

k

kN

rNNQ

1

2

)(2

(4.1)

WhereN= sample size,L = number of autocorrelation lags included in the statistic, andr

2k is the squared sample autocorrelation at lag k. Once you fit a univariate model to an

observedtime series, you can use the Q-statistic as a lack-of-fit test for a departure from

randomness.The Q-test is most often used as a post estimation lack-of-fit test applied to

the fitted innovations (i.e., residuals). In this case, however, you can also use it as part of

the prefit analysis because the default model assumes that returns are just a simple


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constant plus a pure innovations process. Under the null hypothesis of no serial

correlation, the Q-test statistic is asymptotically Chi-Square distributed.

4.3 ENGLE'S ARCHTESTAs for Engle's ARCH Test, the ARCH test also tests the presence of significant evidence

in support of GARCH effects (i.e. heteroskedasticity). It tests the null hypothesis that a

time series of sample residuals consists of independent identically distributed (i.i.d.)

Gaussian disturbances, i.e., that no ARCH effects exist. Given sample residuals obtained

from a curve fit (e.g., a regression model), this test tests for the presence of uth order

ARCH effects by regressing the squared residuals on a constant and the lagged values of

the previousM. squared residuals. Under the null hypothesis, the asymptotic test statistic,

T(R2) , where Tis the number of squared residuals included in the regression andR2is the

sample multiple correlation coefficients, is asymptotically chi square distributed with M

degrees of freedom. When testing for ARCH effects, a GARCH(u,v) process is locally

equivalent to an ARCH(u+v) process.

All the analysis about pre estimation analysis were did in MATLAB 7.0, Q-statistics or

Engles ARCH-statistics, p-value and critical values at 95% confidence level for 10, 15

and 20 lags are generated in MATLAB. Both functions return identical outputs. The first

output, H, is a Boolean decision flag. H = 0 implies that no significant correlation exists(i.e., do not reject the null hypothesis). H = 1 means that significant correlation exists

(i.e., reject the null hypothesis). The remaining outputs are the P-value (p-Value), the test

statistic (Stat), and the critical value of the Chi-Square distribution (Critical Value).

4.4 ANALYSISAs we can see ACF and PACF plots of KSE-100 index in figure 4.1, 4.2 and 4.3 , which

shows that there are no autocorrelation or no significant serial correlation and

independent. PACF of log returns are give different result, at lag 1 returns are good

correlated and till lag 4 PACF shows significant positive serial correlation. Both these

plots are useful preliminary identification tools as they provide some indication of the

broad correlation characteristics of the returns. Form ACF of log returns, we suggest that

no auto correlation is present in returns data. ACF of squared returns are shows that there


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are serial correlation and dependent. We deduce that squared returns are variance process

of the returns and PACF of squared returns give the same result. ACF of absolute returns

also shows in the figure 4.3that is significantly serially correlated and dependent. ACF

and PACF of returns, squared an absolute returns of each of the six stocks are give in theappendix. ACFs of each of the stock shows the similar phenomena as KSE-100. ACF of

each of the stock shows that returns are not correlated and independent. While ACFs of

stock returns are correlated except POL and NBP. However, ACF of absolute return of all

six stocks shows that there is serial correlation and dependent. Lastly, I suggest that all

are serially dependent and volatility models, i.e. ARCH and GARCH model are

applicable and attempt to capture such dependence in the return series.

Figure 4.1: ACF and PACF plot of log returns of KSE-100 index.


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Figure 4.2: ACF and PACF plot of squared returns of KSE-100 index.

Figure 4.3: ACF and PACF plot of absolute returns of KSE-100 index.


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Using LBQ-test, I can verify, at least approximately, that no significant correlation is

present in the raw returns when tested for up to 10, 15, and 20 lags of the ACF at the 0.05

level of significance. Under the null hypothesis H0: No correlation and Ha: correlation is

present.

Figure 4.4: MATLAB output for LBQ-test and ARCH-test of KSE-100 index returns.

From the above MATLAB out, we observed that H, is a Boolean decision flag, is equal to

1, which indicates that significant exist (i.e. rejects the null hypothesis that correlation is

present. However, P-value at any lags of 10, 15 and 20 are less than .05 and all Q-

statistics are exceeds its corresponding critical values. Finally, I suggest that correlation is

present in log returns of KSE-100 at 5% level of significance. From the above figure 4.4of the MATLAB output for KSE-100 index. In the above output H=1, p-value less 0.05

and equal to zero, ARCH test statistics exceeds its critical value. Therefore, ARCH test

strongly rejects the null hypothesis that there is no ARCH/GARCH effect in given return

of KSE-100. Finally, log returns have an ARCH effect at significance level of 5% and

given time series has no random sequence of Gaussian disturbance.


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Figure 4.4: MATLAB output for LBQ-test and ARCH-test of KSE-100 index squared returns.

Same analysis for the squared returns of KSE-100. All H is equal to 1, simply we can

serial correlation is present in the squared returns. We further analyzed that P-value for

any lag are zero, null hypothesis (H0) cannot be accept at any significance level. Q-

statistics at lags 10, 15 and 20 exceeds its corresponding critical values. Therefore, serial

correlation is present in the squared returns of KSE-100.


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5.0 APPLYING MODELS

5.1 INTRODUCTIONIn the previous chapter, we analyzed the nature of time series. First, we do qualitative test

by plotting ACF and PACF of each return or squared return series and check the quality

of time series, that they are serially correlated or independent. For further analysis, wequantify the preceding qualitative checks for correlation (ACF and PACF) using formal

hypothesis tests, such as the Ljung-Box-Pierce Q-test and Engle's ARCH test. Ljung-

Box-Pierce Q-test implemented to check the randomness of the time series data and

suggested that correlation is significant or not. In Engle's ARCH test, check the presence

of ARCH effects. In this chapter, we will estimate the parameters of the ARCH/GARCH

models, which discussed in the second chapter. The parameters of GARCH models can

be estimated by maximum likelihood estimation technique (MLE), which is discussed in

the as chapter 2. Practically, I use packages, like MATLAB 7.0 and Eveiws for parameter

estimation and its analysis. The presence of heteroscedasticity, shown in the previous

chapter, indicates that GARCH modeling is appropriate. Use the estimation MATLAB

function to estimate the model parameters. I cannot discuss every model for each of the

time series. For illustration, I talk about KSE-100 index return series.

5.2 ESTIMATION OF PARAMETER AND CONDITIONAL VOLATILITYBy using MATLAB commands and enter the return series as input and estimates theparameters of a conditional variance specification of GARCH, EGARCH, or GJR form.

The estimation process infers the innovations (i.e., residuals) from the return series, and

fits the model specification to the return series by maximum likelihood. MATLAB

commands related to GARCH are give in Appendix J. Estimated coefficients and


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MATLAB output for GARCH(1,1) specification with iteration are given in the below

figure:


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Figure 5.1: MATLAB output shows estimated coefficient with iterations for GARCH(1,1) specification.

Generalized form of GARCH(u,v) can be write as:

2

11

2

110

2

++= ttt

Where LV =0


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Therefore, is the weight assigned to long run average variance rate, VL, 1 is a weight

assigned to 2 1t and 1 is to2

1t . Because weight must be sum to 1.

111 =++ Then 111 =

And11

0

1

=LV

The estimation process implies that the GARCH(1,1) conditional variance model that

best fits the observed data is

2

1

2

1

2 71255.02582.000001008.0 ++= ttt

It follow that VL= 0.000344. In other words, the long-run average variance pre day

implied by the model is 0.000344. This corresponds to volatility of (0.000344) =

0.01856 or 1.856% per day.

Same analysis for the portfolio, which have different types of components or stocks, as

we discuss in chapter 3. Portfolio, which I compose, contains six stocks and different

stocks have different characteristic and volatility fashion. Therefore, I fit the above model

on each time series or stocks return at a time. Then combine all the volatilities, and

measure for whole portfolio. In figure, notice that both the innovations (top plot) and the

returns (bottom plot) exhibit volatility clustering.

5.3 POST ESTIMATION ANALYSISAlthough the figure 5.2,shows that the fitted innovations exhibit volatility clustering, if

you plot the standardized innovations (the innovations divided by their conditional

standard deviation), they appear generally stable with little clustering as compare to

figure 5.3, within -0.1 to 0.4. If you plot the ACF of the squared standardized

innovations, they also show no correlation. Now compare the ACF of the squared


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standardized innovations in figure 5.4 to the ACF of the squared returns prior to fitting

the GARCH(1,1) figure 4.2 (chapter 4). The comparison shows that the model

sufficiently explains the heteroscedasticity in the raw returns. Compare the results below

of the Q-test and the ARCH test with the results of these same tests in the pre estimation

analysis (chapter 4). In the pre estimation analysis, both the Q-test and the ARCH testindicate rejection (H = 1 with P-value = 0) of their respective null hypotheses, showing

significant evidence in support of GARCH effects. In the post estimation analysis, using

standardized innovations based on the estimated model, these same tests indicate

acceptance (H = 0 with highly significant P-values) of their respective null hypotheses as

shown in figure 5.4 and 5.5 and confirm the explanatory power of the GARCH(1,1)

model.

Figure 5.2: comparison of return, innovations and conditional volatilities from GARCH(1,1) model.


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Figure 5.3: standardized innovations of GARCH(1,1).

Figure 5.4: ACF plot of squared standardized innovations.


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Figure 5.5: LBQ-test and ARCHtest for innovations.


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6.0 Testing of Models

6.1 INTRODUCTIONAfter applying different models, next step is to check which is appropriate. In last

chapter, we show step of estimation of the model and show only GARCH(1,1) for the

purpose of illustration. There are different ARCH/GARCH-type model can be applied onthe data which are discussed in the chapter 2, i.e. ARCH, GARCH, EGARCH, GJR

models, some of them shown in the appendix with their estimated coefficients. There are

different approaches to check which model better explains our stock exchange data. Each

model can be applied with different parameters, i.e. u and v, but in limited way, each

parameter cannot be greater than three for ARCH and simple GARCH models and two

for EGARCH and GJR models. However, I apply different models with different

combinations of parameters; they have different characteristics and volatility forecasts,

analyze them and identify the causes of difference in them. Two different criteria are used

to compare the forecasting performance of the various conditional volatility models and

methods considered, namely: (1) the linear regression approach of Pagan and Schwert

(1990); (2) RiskMetrics approach ofCrnkovic and Drachman (1996).

6.2 LINEAR REGRESSION APPROACHPagan and Schwert (1990) proposed a procedure whereby the volatility forecasts are

regressed on the realized volatility. In this paper, the squared portfolio returns are used as

a proxy for the realized volatility. The auxiliary regression equation is given by:

ttt eFVRV ++=


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WhereRVtis the realized volatility and FVtis the forecasted volatility. In this auxiliary

equation, the intercept, , should be equal to zero and the slope, , should be equal to 1.

The coefficient of determination,R2, is a measure of forecasting performance, and the t-

ratio of the coefficients is a measure of the bias.

TablesH-1 and H-2 in appendix give the estimates and test statistics for the single index

and portfolio models. Based on the R2 criterion, the portfolio EGARCH(2,1) model

performs the best, with R2 = 0.29079 and single index EGARCH(1,2), which R2 =

0.34885 with little difference. The worst performing models are the KSE-100 (single)

index ARCH(1) and portfolio EGARCH(2,2) models, which have R2

= 0.02845 and

0.18765 respectively. On the basis of coefficients, intercept terms of all models are same

to each other or close to zero. However, on th basis of slope, KSE-100s EGARCH(2,1)

and portfolios EGARCH(2,2) with slopes 0.92954 and 1.69581 respectively. In all

cases, the single index models outperform the portfolio models based on R2 , which

suggests that the index model approach leads to superior forecasts of the conditional

variance of the single index compared with their portfolio counterparts.

6.3 BACK TESTINGThe purpose of this section is not to offer a review of the quantitative measures for VaR

model comparison. There is a growing literature on such measures and we refer the

reader to Crnkovic and Drachman (1996) for the latest developments in that area. Instead,

we present simple calculations that may prove useful for determining the appropriateness

of the models.

6.3.1 PORTFOLIOFor back testing, I use same index (KSE-100) and portfolio, which contains six stocks

with different weights. Using daily prices for the period December 22, 2003 throughOctober 4, 2006 (a total of 962 observations), we construct 1-day VaR forecasts over the

most recent 962 days of the sample period. We then compare these forecasts to their

respective realized profit/loss (P/L) which are represented by 1-day returns.


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=

=6

1

,,

i

tiitp rwr

Where ri,trepresents the log return of the ith stock. The Value-at-Risk bands are based onthe portfolios standard deviation. The formula for the portfolios variance at t is:

Where is the 2i,t variance of the ith return series made for time tand ij,t is the correlationbetween the ith and jth returns for time t andwiis the weight for the ithstock.

6.3.2 ASSESSING THE MODELSThe first measure of model performance is a simple count the number of times that the

VaR estimates under predict future losses (gains). Each day it is assumed that there is a

5% chance that the observed loss exceeds the VaR forecast. For the sake of generality,

lets define a random variable X(t) on any day tsuch that X(t) = 1 if a particular days

observed lossis greater than its corresponding VaR forecast and X(t) = 0 otherwise. We

can write the distributionofX(t) as follows

( ) 1,0)()05.01(05.00)(1)(

05.0|)( =

= tXtXtX

tXfe.w

Now, suppose we observeX(t) for a total ofTdays, t= 1,2,..., T, and we assume that the

X(t)s are independent over time. In other words, whether a VaR forecast is violated on a

particular day is independent of what happened on other days. The random variable X(t)

is said to follow a Bernoulli distribution whose expected value is 0.05.The total number

of VaR violations over the time period Tis given by

= = >

+=6

1

6

1

,

2

,

22

, 2i i ij

tijjitiitp www


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=

=T

t

T tXX1

)(

The expected value of, i.e., the expected number of VaR violations over T days, is T

times 0.05. For example, if we observe T= 961 days of VaR forecasts, then the expected

number of VaR violations is (961) (0.05) = 48; hence one would expect to observe one

VaR violation every 961 days. What is convenient about modeling VaR violations

according to Eq. 6.1 is that the probability of observing a VaR violation overTdays is

same as the probability of observing a VaR violation at any point in time, t. Therefore,

we are able to use VaR forecasts constructed over time to assess the appropriateness of

the models for this portfolio of 6 stocks.

For example, if we forecast VaR for KSE-100 by the GARCH(1,1) conditional variancemodel . If true probability of violation is 5%, then realized VaR violations are:

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

KSE

SD-1.65

SD+1.65

Figure 6.1: forecast VaR for KSE-100 by the GARCH(1,1) and violations of upper and lower band.

Table 6.1: No. and percentage of violations of upper and lower band.


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At confidence level of 95%, we expect that there is 5% chances that VaR band of each

side and the expected number of violation is 48 days out of 961 days, violation of lower

band closer to expected, of upper band lesser than expected. VaR violation for other

models is given in Appendix I.


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7.0 CONCLUSIONThe Endeavour to examine if GARCH-type models were the better model in describing

return series for VaR was done statistically and empirically. The dataset was first tested

by various statistical testing methods to see if GARCH modelling was suitable. Only for

NBP and POL, the tests showed showed that there is no significant correlation.

The other companies including the KSE-100 index contained correlation in its returns orsquared returns, which meant that a GARCH process was found and modeling with

GARCH was appropriate. After testing the dataset, the models were set up and run; the

parameters were estimated for each of the model with their conditional volatility. As the

conditional volatility is the main ingredient for forecasting VaR and its depends on

Conditional variance. Then we check the quality of our estimated parameter and

volatility. First test the innovations of each, that there are any kind of correlation is

present or not. I found that there is no significant correlation and ARCH effect is not

present. In the next step, check the conditional volatility by applying linear regression

approach that each model can explains market variation. Therefore, models for single-

index that are good fitted and better explain the market variation as compare to portfolio,

its coefficients of determination are high, beta coefficient close to 1 and a smaller amount

of standard errors. In back testing, test the number of violations of VaR band for each

model at 95% confidence level. I observed that there is huge number of violations of

portfolio VaR and the number of violation of KSE index of its VaR band, close and

within its expected violation.

In whole document, we assume that KSE-100 index as single index. However, I alsocompose six and twenty components of single index as well as portfolio of twenty stocks

and applying univariate and multivariate model on single index and portfolio

respectively. It is observed that single index of six and twenty stocks has less number of

violation and close to its expected violations as compare to its portfolios. In addition, it is

also observed that the number of stocks increases in the portfolio, so the number of

violation also increases as we can see in appendix figure


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APPENDIX A:


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APPENDIX B:ACF AND PACF OF RETURNS


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APPENDIX C:ACF AND PACF OF SQUARED

RETURNS


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APPENDIX D:ACF AND PACF OF ABSOLUTE

RETURNS


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APPENDIX E: LJUNG-BOX-PIERCE Q-TEST OF

RETURNS AND SQUARED RETURNS


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APPENDIX F:ENGLES ARCHTEST


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APPENDIX G:PARAMETER ESTIMATION

(1)

(2)

(3)


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(4)

(5)

(6)


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(7)

(8)

(9)


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APPENDIX H:LINEAR REGRESSION APPROACH

Table H-1: linear regression approach for single index model (KSE-100)


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Table H-2: linear regression approach for portfolio by using multivariate model


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APPENDIX I:BACK TESTING

Table I-1: No. and percentage of violations of upper and lower VaR band of single index model(KSE-100)

-0.1000000

-0.0800000

-0.0600000

-0.0400000

-0.0200000

0.0000000

0.0200000

0.0400000

0.0600000

0.0800000

0.1000000

KSE

SD-1.65

SD+1.65

Figure I-1: violations of upper and lower VaR band of single index model (KSE-100)


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Table I-2: No. and percentage of violations of upper and lower VaR band of Portfolio

-0.10000

-0.08000

-0.06000

-0.04000

-0.02000

0.00000

0.02000

0.04000

0.06000

0.08000

port_ret

SD-1.65

SD+1.65

Figure I-2: violations of upper and lower VaR band of Portfolio


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-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

SD-1.65

SD+1.65

Portfolio (20)

Figure I-3: violations of upper and lower VaR band of Portfolio of 20 components by using multivariate

model (example)

Table I-2: No. and percentage of violations of upper and lower VaR band of Portfolio of 20 components by

using multivariate model (example)

-0.10000

-0.08000

-0.06000

-0.04000

-0.02000

0.00000

0.02000

0.04000

0.06000

0.08000

0.10000

PORTFOLIO (6)

SD-1.65

SD+1.65

Figure I-4: violations of upper and lower VaR band of Portfolio of 6 components by using univariate

model (example)


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Table I-4: No and percentage of violations of upper and lower VaR band of Portfolio of 6 components by

using univariate model (example)

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Portfoio (20)

SD-1.65

SD+1.65

Figure I-4: violations of upper and lower VaR band of Portfolio of 20 components by using univariate

model (example)

Table I-4: No and percentage of violations of upper and lower VaR band of Portfolio of 20 components byusing univariate model(example)


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8.0 BIBLIOGRAPHY

Tsay, Ruey S.Analysis of Financial Time Series, First Edition. John Wiley & Sons, INC

(2002)

Gujrati, Demodar N. Basic Econometrics, Fourth Edition. McGraw Hill International

(2003)

Longerstaey, Jacques, and Peter Zangari. RiskMetricsTechnical Document. 4th

Edition. (New York: Morgan Guaranty Trust Co., 1996)

Engle, Robert. Journal of Economic PerspectivesVolume 15, Number 4Fall 2001

Pages 157168

Veiga, Bernardo da & McAleer, Michael Single Index and Portfolio Models for

Forecasting Value-at-Risk Thresholds. School of Economics and Commerce, University of

Western Australia (January 2005)

Hull, John.C. Options, Futures, and Other Derivatives, fifth edition, Prentice Hall

International. (2003).

forecasting value-at-risk by nasir ali khan

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