Value at Risk Model in Indian Stock Market

8/7/2019 Value at Risk Model in Indian Stock Market

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M P Birla institute of Management - 1 -

Value at Risk model in Indian stock Market

Submitted in partial fulfillment of the requirements of

the M.B.A Degree Course of Bangalore University

By

MEHTA PARTH

(REGD.NO:05XQCM 6046)

Under the Guidance

Of

DR. T.V.NARASIMHA RAO

Faculty

MPBIM

M.P.BIRLA INSTITUTE OF MANAGEMENT

Associate Bharatiya Vidya Bhavan

43, Race Course Road, Bangalore-560001

2005-2007


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DECLARATION

I hereby declare that the dissertation entitled Value at Risk model in Indian

stock Market is the result of work undertaken by me, under the guidance of

Dr.T.V.N.Rao, Associate Professor, M.P.Birla Institute of Management,

Bangalore.

I also declare that this dissertation has not been submitted to any other

University/Institution for the award of any Degree or Diploma.

Place: Bangalore

Date : 13th May 2007 Mehta Parth Bharatbhai


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PRINCIPALS CERTIFICATE

This is to certify that the Research Report entitled Value at Risk model in

Indian stock Market done by Mehta Parth Bharatbhai bearing Registration

No. 05 XQCM 6046 under the guidance ofDr.T.V.N.Rao.

Place: Bangalore (Dr.N.S.Malavalli)

Date: 13th May 2007 Principal

MPBIM, Bangalore


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GUIDES CERTIFICATE

This is to certify that the Research Report entitled Value at Risk model in

Indian stock Market done by Mehta Parth Bharatbhai bearing Registration

No. 05XQCM6046 is a bonafide work done carried under my guidance during

the academic year 2006-07 in a partial fulfillment of the requirement for the

award of MBA degree by Bangalore University. To the best of my knowledge this

report has not formed the basis for the award of any other degree.

Place: Bangalore Dr.T.V.N.Rao

Date : 13th May 2007


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ACKNOWLEDGEMENT

Its my special privilege to extend words of the thanks to all of them who have

helped me and encouraged me in completing the project successfully.

I would thank Dr.T.V.N.Rao for giving me valuable inputs required for completing

this project report successfully. I owe my sincere gratitude to him for spending his

valuable time with me and for his guidance.

It would be improper if I do not acknowledge the help and encouragement by my

friends and well wishers who always helped me directly or indirectly.

My gratitude will not be complete without thanking the almighty god and my

loving parents who have been supportive through out the project.

Mehta Parth Bharatbhai


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TABLE OF CONTENTS

CHAPTERS PARTICULARS PAGE NO.

ABSTRACT 02

1 INTRODUCTION AND THEORETICAL

BACKGROUND

04

2 REVIEW OF LITERATURE 28

3 RESEARCH METHODOLOGY 32

4 PROBLEM STATEMENT 33

5 OBJECTIVE OF THE STUDY 34

6 SAMPLE SIZE AND DATA SOURCES 35

7 TEST OF STATIONARITY 368 AUTO-CORRELATION 38

9 LIMITATIONS OF THE RESEARCH 43

10 DATA ANALYSIS & INTERPRETATIO 44

11 CONCLUSION 53

12 ANNEXTURE 55

13 BIBLIOGRAPHY 64


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immediate past, and autoregressive describes a feedback mechanism that

incorporates past observations into the present. GARCH then is a mechanism

that includes past variances in the explanation of future variances. More

specifically, GARCH is a time-series technique that allows users to model the

serial dependence of volatility.

The data taken here is 10 year S&P CNX Nifty daily Index .Firstly the stationary

of the daily returns is tested with Augmented Dickey-Fuller Test. Then

parameters for the various models are calculated. After forecasting the monthly

variance the results of these competing models are evaluated on the basis of two

categories of evaluation measures symmetric and asymmetric error statistics.

Based on an out of the sample forecasts and a majority of evaluation measures

we find that GARCH (1, 6) method will lead to better volatility forecasts in the

Indian stock market. The same model performed better on the basis of

asymmetric error statistics also. but the other model like GARCH ( 1,1) , GARCH

( 1,2 ) , GARCH ( 3,1 ) are not able to forecast the volatility of the NIFTY index.


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CHAPTER 1

INTRODUCTION AND

THEORETICAL BACKGROUND


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Introduction

Modeling and forecasting volatility of financial time series has been an important

research topic for the last several years. There are two main reasons for the

strong interest in volatility estimates. Since the prices of derivative products do

depend on the volatility of the underlying instrument any pricing of these products

requires volatility forecast. The second reason is related to the concept of

volatility as a measure of market risk. Since the modern banking industry

requires an efficient management of all risks in todays new global financial

architecture, heavy emphasis must be placed on financial market risks. As a

consequence many regulatory requirements (e.g. those initiated by the Bank for

International Settlements) are by now standardized and have introduced many

novel concepts and tools into the management of market, credit and operational

risk. In the case of market risk these developments have led to an uniformly

accepted and applied risk measure called Value-at-Risk (VaR). The VaR of a

portfolio position is defined as the maximum potential loss for this position for a

given holding period and a given confidence level. Alternative specifications of

financial products, increasing availability of financial data and rapid advances incomputer technology have led to the introduction and formulation of various VaR

models that can currently be applied to measure the market risk of a portfolio

analysis.

The VaR concept can be viewed as a generalization of the risk sensitivities


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related to different risk factors. As an example let us quickly look at the market

risk of a simple European call option. If we ignore higher order approximations

the options delta is the sensitivity of the call price with respect to the risk

resulting from a change in the price of the underlying. Hence the delta linearly

measures market risk. This measure, however, is incomplete as long as we do

not know what the volatility of the risk factor is. If we multiply the sensitivity of the

position with the volatility of the risk factor we end up with the VaR, a therefore a

comprehensive measure of market risk. This simple description points out that

the calculation of the VaR is directly related to forecasting volatility of a position.

Only if we have full knowledge about the conditional density it is not necessary to

express percentiles of distributions as multiples of the standard deviation. In that

case we can directly calculate the value at risk. VaR models that are based on

standard distributions (e.g. normal distribution) first estimate the standard

deviation (or covariance matrix) in order to calculate the VaR for a given

confidence level. For that reason good volatility forecasts are an integral part of

sound VaR models.

One of the most widely used volatility models is the GARCH model (Bollerslev,

1986) for which the conditional variance is governed by a linear autoregressive

process of past squared returns and variances. The standard GARCH model

based on a normal distribution captures several stylized facts of asset return

series, like heteroskedasticity (time-dependent conditional variance), volatility

clustering and excess kurtosis. Recent empirical research, however, has found

that there are additional empirical regularities in return data such as negative and

autocorrelated skewness (asymmetry), fat tails and time dependent kurtosis that


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can not be described by the classical GARCH model. For that reason several

alternative specifications have been formulated in the literature.

We take into account the latest developments in conditional volatility research

and propose a generalized model that extends the existing literature in two

directions: the first one is to allow for non-linear dependencies in the conditional

mean and variance, and the second one concerns a non-standard specification

of the conditional density. To estimate nonlinear conditional second moments we

use a neural networkbased approach (i.e., so called recurrent mixture density

networks) for which the conditional mean and variance are modelled by a multi-

layer perceptrons ( see, e.g., Schittenkopf et al. (2000).

With regard to the specification of the conditional distributions, we compare three

different density specifications: 1) a standard GARCH model and its non-linear

generalization with a conditional normaldistribution (heteroskedastic, but neither

skewed nor leptokurtic); 2) a non-linear recurrent GARCH model with a Students

t-distribution (heteroskedastic, not skewed but leptokurtic); and 3) linear and non-

linear recurrent mixture density models, for which the conditional distributions are

approximated by a mixture of gaussians (two components) (heteroskedastic,

skewed and leptokurtic in a time-dependent manner).

These model specifications make clear that our point of interest in this study is

twofold. On the one hand we are interested in forecasting volatilities in order to


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accurately estimate the value at risk of a portfolio. On the other hand we are

concerned with the forecast of conditional distributions that allows the calculation

of VaR directly. Based on these two objectives we empirically evaluate the

forecasting performance of alternative volatility models and apply statistical tests

to discriminate between alternative VaR models. For the latter we apply the

Basle traffic light test, the proportion of failure test and interval tests.

All these tests evaluate the accuracy of a VaR model on the basis of statistical

procedures. Since it is very likely that the statistical criteria do not single out one

model as the best, we alternatively calculate the costs of capital requirements as

induced by a specific VaR model. The rationale behind this approach is the

following. Assume that out of several competing models there are two that

perform equally well with respect to forecasting the value at risk of a portfolio

position, i.e. these two models have two similar statistical characteristics.

The two models, however, can lead to very different costs, as far as the capital

requirements are concerned. Form a banks point of view it is not only necessary

to have a risk management model that correctly predicts the market risk, but one

that additionally uses the least capital possible. Since any capital requirement

incurs opportunity costs for the bank (i.e. capital that is in an unproductive,

regulatory use), it has an interest to cut this requirement down as much as

possible. Hence, VaR models should not only be judged on the basis of their

forecasting power, but also on the basis of their capital costs. This discussion

motivates the structure of our empirical analysis. It is based on return series of

stock indices from three different financial markets. We use return series of the


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Dow Jones Industrial Average (USA), the FTSE 100 (Great Britain) and the

NIKKEI 225 index (Japan) over a period of more than 13 years in order to

evaluate in detail the out-of-sample predictive performance of our models. Our

empirical analysis has the following structure. We predict conditional distributions

and calculate the VaR for each of our models for three different homogeneous

portfolios based on the same stock indices. To evaluate the quality and accuracy

of the VaR models we apply a number of statistical tests specifically designed to

interval forecasts. Among those are regulatory backtesting required as a part of

the capital-adequacy framework (the Basle Committees traffic light); exceptions

testing which examines the frequency with which losses greater than the VaR

estimate are observed together with independence of these events; statistical

test on the accuracy of point estimation of the VaR significance level. The

advantage of these tests is given by the fact that the actual loos of any portfolio

can be measured exactly and hence the VaR forecasts can be evaluated on the

basis of actual observations. As pointed out above, our central focus is also

related to the analysis of the efficiency of VaR measures, as measured by the

costs of capital associated with VaR based regulatory capital requirements

(calculation of the lost interest yield connected with the dynamically computed

model-based capital reserves).


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Theoretical Background

What is VaR?

In recent years value at risk (VaR) has become a very popular measure of

market risk. It is widely used by financial institutions, fund managers, and non

financial corporations to control the market risk in a portfolio of financial

instruments. As discussed by Jorion (1997), it has been adopted by central bank

regulators as the major determinant of the capital banks are required to keep to

cover potential losses arising from the market risks they are bearing.

The VaR of a portfolio is a function of two parameters, a time period and a

confidence level. It equals the dollar loss on the portfolio that will not be

exceeded by the end of the time period with the specified confidence level. If X%

is the confidence level and Ndays is the time period, the calculation of VaR is

based on the probability distribution of changes in the portfolio value over N

days. Specifically VaR is set equal to the loss on the portfolio at the 100-X

percentile point of the distribution. Bank regulators have chosen Nequal to 10days and Xequal to 99%. They set the capital required for market risk equal to

three times the value of VaR calculated using these parameters. In practice the

VaR forNdays is almost invariably assumed to be Ntimes the VaR for one day.

A key task for risk managers has therefore been the development of accurate

and robust procedures for calculating a one-day VaR. One common approach to

calculating VaR involves assuming that daily percentage changes in the


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underlying market variables are conditionally multivariate normal with the mean

percentage change in each market variable being zero. This is often referred to

as the model building approach. If the daily change in the portfolio value is

linearly dependent on daily changes in market variables that are normally

distributed, its probability distribution is also normal. The variance of the

probability distribution, and hence the percentile of the distribution corresponding

to VaR, can be calculated in a straightforward way from the variance-covariance

matrix for the market variables. In circumstances where the linear assumption is

inappropriate, the change in the portfolio value is often approximated as a

quadratic function of percentage changes in the market variables. This allows the

first few moments of the probability distribution of the change in the portfolio 1

We are grateful to the editor Philippe Jorion for many suggestions that improved

this paper.

Value to be calculated analytically so that the required percentile of the

distribution can be estimated.2 An alternative approach to handling non-linearity

is to use Monte Carlo simulation. On each simulation trial daily changes in the

market variables are sampled from their multivariate distribution and the portfolio

is revalued. This enables a complete probability distribution for the daily change

in the portfolio value to be determined. 3 Many market variables have

distributions with fatter tails than the normal distribution. This has led some risk

managers to use historical simulation rather than the model building approach.

Historical simulation involves creating a database consisting of the daily

movements in all market variables over a period of time. The first simulation trial

assumes that the percentage changes in the market variables are the same as


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Introduction to GARCH model

In econometrics, an autoregressive conditional heteroskedasticity (ARCH,

Engle (1982)) model considers the variance of the current error term to be a

function of the variances of the previous time period's error terms. ARCH relates

the error variance to the square of a previous period's error. It is employed

commonly in modeling financial time series that exhibit time-varying volatility.

Specifically, let denote the returns (or return residuals, net of a mean process)

and assume that , where and where the series

are modeled by

and where and

If an autoregressive moving average model (ARMA model) is assumed for the

error variance, the model is a generalized autoregressive conditional

heteroskedasticity (GARCH, Bollerslev(1986)) model.


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In that case, the GARCH(p,q) model (where p is the order of the GARCH terms

and q is the order of the ARCH terms) is given by

IGARCH or Integrated Generalized Autoregressive Conditional

Heteroskedasticity is a restricted version of the GARCH model, where the sum of

the persistent parameters sum up to one.

Generally, when testing for heteroskedasticity in econometric models, the best

test is the White test. However, when dealing with time series data, the best test

is Engle's ARCH test.

Prior to GARCH there was EWMA which has now been superseded by GARCH.

Some people utilise both

ARCH models

The autoregressive conditional heteroskedasticity model was introduced by

Engle (1982) to model the volatility of UK inflation.


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As the name suggests, the model has the following properties:

1) Autoregression - Uses previous estimates of volatility to calculate subsequent

(future) values. Hence volatility values are closely related.

2) Heteroskedasticity - The probability distributions of the volatility varies with the

current value.

In order to introduce ARCH processes, let us assume that we have a time series

of asset price quotes for each time step i. We calculate the fractional change

in the price of the asset between time step i and i+1 using

Furthermore, we are required to determine the long-running historical volatility

(e.g. over several years) denoted by . In the first figure above, is

illustrated by the flat line. We have seen that the volatility rates fluctuate about

this mean long-running mean volatility, therefore, it seems reasonable to

incorporate this quantity in the ARCH model.

Formally, an ARCH(m) process may be expressed mathematically as

where is the volatility at the time step, and are weighting factors that

satisfy


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Here m denotes the number of observations of used to determine . The

most common ARCH(m) process used to model asset price volatility dynamics is

the ARCH(1) model where

or

using the above relation.

GARCH models

Bollerslev (1986) later proposed a more generalised form of the ARCH(m) modelappropriately termed the GARCH(p,q) (General-ARCH) model. The GARCH(p,q)

model may be written as

The p and q denote the number of past observations of and ,


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respectively, used to estimate .

The EWMA model

The Exponentially Weighted Moving Average model (EWMA) is a special case of

the GARCH(1,1) model where . Thus,

Since , we may express the EWMA model as

The EWMA model differs from ARCH and GARCH models since it does not

mean-revert. The preference between these different models is dependent upon

many factors. For example, the asset class, forcasting time frame under

consideration, and the efficiency with which the weighting parameters may be

calibrated to the time series. Whilst the maximum likelihoodestimators method

may be the most obvious method to select for calibration with empirical data,

more efficient algorithms have also been put forward.

Since these volatility forecasting models were introduced, there have been many

alternatives/modifications proposed to these models to better their use in volatility

forecasting.

The great workhorse of applied econometrics is the least squares model. The


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basic version of the model assumes that, the expected value of all error terms, in

absolute value, is the same at any given point. Thus, the expected value of any

given error term, squared, is equal to the variance of all the error terms taken

together. This assumption is called homoskedasticity. Conversely, data in which

the expected value of the error terms is not equal, in which the error terms may

reasonably be expected to be larger for some points or ranges iof the data than

for others, is said to suffer from heteroskedasticity.

It has long been recognized that heteroskedasticity can pose problems in

ordinary least squares analysis. The standard warning is that in the presence of

heteroskedasticity, the regression coefficients for an ordinary least squares

regression are still unbiased, but the standard errors and confidence intervals

estimated by conventional procedures will be too narrow, giving a false sense of

precision. However, the warnings about heteroskedasticity have usually been

applied only to cross sectional models, not to time series models. For example, if

one looked at the cross-section relationship between income and consumption in

household data, one might expect to find that the consumption of low-income

households is more closely tied to income than that of high-income households,

because poor households are more likely to consume all of their income and to

be liquidity-constrained. In a cross-section regression of household consumption

on income, the error terms seem likely to be systematically larger for high-income

than for low-income households, and the assumption of homoskedasticity seems

implausible. In contrast, if one looked at an aggregate time series consumption

function, comparing national income to consumption, it seems more plausible to

assume that the variance of the error terms doesnt changed much over time.


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A recent developments in estimation of standard errors, known as

robust standard errors, has also reduced the concern over heteroskedasticity. If

the sample size is large, then robust standard errors give quite a good estimate

of standard errors even with heteroskedasticity. Even if the sample is small, the

need for a heteroskedasticity correction that doesnt affect the coefficients, but

only narrows the standard errors somewhat, can be debated.

However, sometimes the key issue is the variance of the error terms itself.

This question often arises in financial applications where the dependent variable

is the return on an asset or portfolio and the variance of the return represents the

risk level of those returns. These are time series applications, but it is

nonetheless likely that heteroskedasticity is an issue. Even a cursory look at

financial data suggests that some time periods are riskier than others; that is, the

expected value of error terms at some times is greater than at others. Moreover,

these risky times are not scattered randomly across quarterly or annual data.

Instead, there is a degree of autocorrelation in the riskiness of financial returns.

ARCH and GARCH models, which stand for autoregressive conditional

heteroskedasticity and generalizedautoregressive conditional heterosjedasticity,

have become widespread tools for dealing with time series heteroskedastic

models such as ARCH and GARCH. The goal of such models is to provide a

volatility measure like a standard deviation -- that can be used in financial

decisions concerning risk analysis, portfolio selection and derivative pricing.


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ARCH/GARCH Models

Because this paper will focus on financial applications, we will use financial

notation. Let the dependent variable be labeledt

r, which could be the return on

an asset or portfolio. The mean value m and the variance h will be defined

relative to a past information set. Then, the return r in the present will be equal to

the mean value of r (that is, the expected value of r based on past information)

plus the standard deviation of r (that is, the square root of the variance) times the

error term for the present period.

The econometric challenge is to specify how the information is used to forecast

the mean and variance of the return, conditional on the past information. While

many specifications have been considered for the mean return and have been

used in efforts to forecast future returns, rather simple specifications have proven

surprisingly successful in predicting conditional variances The most widely used

specification is the GARCH(1,1) model introduced by Bollerslev (1986) as a

generalization of Engle(1982). The (1,1) in parentheses is a standard notation in

which the first number refers to how many autoregressive lags appear in the

equation, while the second number refers to how many lags are included in the

moving average component of a variable. Thus, a GARCH (1,1) model for

variance looks like this:


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= + +2

1 1 1t t t t h h h .

This model forecasts the variance of date t return as a weighted average of a

constant, yesterdays forecast, and yesterdays squared error. Of course, if themean is zero, then from the surprise is simply

2

1tr .

Thus the GARCH models are conditionally heteroskedastic but have a constant

unconditional variance.

Possibly the most important aspect of the ARCH/GARCH model is the

recognition that volatility can be estimated based on historical data and that a

bad model can be detected directly using conventional econometric techniques.

A variety of statistical software packages like Eview and others? are available for

implementing GARCH and ARCH approaches.


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Overview of VaR

VaR analysis began in the early 1990s as a way for Wall Street firms to estimate

their daily exposure to trading losses. In 1995 the Basle Capital Accord endorsed

the use of VaR in determining capital requirements for banks, lending credibility

to the practice. The Securities and Exchange Commission also forwarded VaR

as one of three possible methods for the disclosure of derivative exposure by

U.S. corporations. The goal of VaR is to calculate the expected down-side loss

over a specified time period with a specified degree of certainty. A common time

period used for VaR is one day or one month, since it has been used largely by

traders and financial institutions with multi-currency portfolios. Confidence levels

are usually calculated at the 95th and 99th percentiles. A VaR estimate must

include the time period and the degree of confidence. A traders VaR for a

$1,000,000 portfolio of foreign currencies might look like this: the 99% VaR for

one day is $34,950 (the calculation of this number is detailed below).

Part of the basic foundation for VaR comes from modern portfolio theory (MPT).

The calculations implicitly include the volatility-dampening effects ofdiversification when examining a multi-asset or multi-currency portfolio. For an

international bank, this means recognizing that stocks and bonds denominated in

various currencies will not all move in the same direction (and to the same

degree) at once. It also allows a bank to summarize its risk from various assets

into one measure denominated in the banks home currency.


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VaR Methodologies

Various vendors have developed proprietary VaR methodologies. The most

widely available and publicized methodology for setting assumptions is the one

used by J.P. Morgan called RiskMetrics, which was the first major set of

standard, simplifying assumptions. RiskMetrics uses a derivation of the GARCH

(generalized auto-regressive conditional hederoscadasticity) model to estimate

asset volatility and correlation. The method attempts to estimate time-varying

volatility (or correlation) by giving more weight to more recent observations. It is

common for VaR calculations to employ an expected daily return of zero (which

is not much different from the average daily return of 4 basis points over a 250

trading-day year, assuming a nominal return of 10% annually). These methods

use historical data to derive forward assumptions.

The primary disadvantage of historical data is obviously its dependence on

relationships which may change over time. Besides missing structural changes

in markets, such as the collapse of the European Exchange Rate Mechanism in1993, the technique will also not capture the effects of short term shocks, such

as the stock market crash of 1987. In the case of U.S. stocks, the historical

volatility was 1.05% during the 250 trading days prior to the crash on October 19.

That would make the decline one of more than 19 standard deviations.IN

addition, it has become conventional wisdom since 1987 that correlations

increase during times of extreme market declines, and it may be true that


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correlations are lower during positive return periods. In this case, a VaR which

uses historical correlations from a positive period will grossly underestimate the

actual VaR should markets turn lower. However, the primary advantage of

marketbased estimates is that the assumed return distribution can be non-

normal. VaR models sometime use Monte Carlo simulation (a method for

modeling random outcomes) in lieu of historical data, but usually depend on the

assumption of normally distributed returns. Unfortunately, returns are often non-

normally distributed. In the case of U.S. stocks, daily returns are commonly found

to be leptokurtic. Leptokurtic returns have a peaked mean (fortunately, to the

right of zero) and longer, fatter tails than a normal distributions. In practical terms,

observations nearer to and farther from the mean are more common that what a

normal distribution would predict.

Another alternative to the standard GARCH-based volatility prediction is the use

of implied volatility. This involves backing-out the forward volatility expectation

embedded in the market price of options contracts. Unfortunately, options are not

available for all instruments over all desired time horizons (e.g., the implied

volatility from 30 or 90 day options would not be appropriate for use in estimating

a one day VaR). In practice, the use of implied volatility in VaR models is rare.


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Institutional Use

Some large pension plans have established internal VaR programs using

systems developed by outside vendors. Ontario Teachers Plan Board, for

instance, uses a Reuters product to monitor their VaR in-house. Outside VaR

packages can cost around $1 million. Several large global custodians (such as

Bankers Trust and Chase) offer VaR calculation as a value-added service for

around $100,000 a year. Several large plans are said to be studying the use of

VaR, although many have expressed skepticism over the ability of custodians to

carry out the task. Plans considering VaR seem to prefer purchasing a program

to use internally, should they choose to adopt the risk measure.

VaR Concerns

While VaR can be a useful risk measure, we have some concerns with respect to

institutional use. Our primary concern is whether it is a suitable tool for long-term

investors. Researchers have shown that calculated VaRs for even short time

horizons can differ substantially based on which methodology is used to set

assumptions. However, volatility and correlation estimates can be made with a

greater degree of certainty for short-term observations, simply because these

parameters trend. GARCH models (and their cousins) have done a fairly good

job of capturing these trends over short periods of time. However, our confidence


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in ten-year estimates are substantially lower, as no solid methodologies have

been developed to deal with long time periods.

Even if VaRs are calculated correctly, there is the possibility that these numbers

will be relied on too heavily. As mentioned earlier, a common misconception with

VaR is that it quantifies the maximum expected loss over a given time periods.

It actually does quite the opposite. By revealing what is to the right of the value at

risk estimate 99% of the time, VaR gives the break-point for the other 1% of the

observed outcomes. A VaR that correctly predicts that a portfolio will suffer a loss

greater than $1 million during a year only 0.5% of the trading days is a success.

This is true even if the actual loss on that one day was $3 million. A single VaR

estimate should not be used in isolation. Even Philippe Jorion, one of the most

prominent proponents of VaR, suggests that estimates should be stress tested.

Also, a report by the International Securities Market Association noted both the

widespread use of VaR among banks, and the importance of stress-testing

results. The basics of stress-testing involve making VaR estimates based on

higher asset volatilities and/or correlations than those normally assumed. VaR

may also be used in conjunction with more traditional risk measures. These

measures may not be as intuitive as a VaR figure, but can help affirm or

contradict the VaR estimate. Lastly, for pension funds, VaR ignores the important

effects of capital market changes on liabilities.

A VaR methodology for pension plans would be more useful if it targeted

variables such as pension surplus or contributions, rather than simply asset


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values.

An Example

Assumption setting is the most difficult aspect of determining VaR. Once these

assumptions are set, the actual calculation is straight forward. To borrow the

foreign exchange example used earlier, assume that the combined portfolio of

currencies is expected to have a daily volatility of 1.5%. From basic statistics,

one-tailed critical values for 90%, 95%, and 99% confidence levels are 1.28,

1.65, and 2.33 respectively. The 99th percentile VaR of the $1,000,000 portfolio

would be found as follows:

(Portfolio Value) x (Portfolio Standard Deviation) x (Critical Value) = VaR

($1,000,000) x (0.015) x (2.33) = $34,900

The above example assumes a mean of zero, which is standard in VaR

assumption.


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CHAPTER 2

REVIEW OF LITERATURE


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Robert Engle : The Use of ARCH/GARCH Models in Applied

Econometrics , Journal of Economic PerspectivesVolume 15, Number

4Fall 2001Pages 157168

The least squares model assumes that the expected value of all error terms,

when squared, is the same at any given point. This assumption is called

homoskedasticity, and it is this assumption that is the focus of GARCH model.

Data in which the variances of the error terms are not equal, in which the error

terms may reasonably be expected to be larger for some points or ranges of the

data than for others, are said to suffer from heteroskedasticity. The standard

warning is that in the presence of heteroskedasticity, the regression coefficients

for an ordinary least squares regression are still unbiased, but the standard

errors and confidence intervals estimated by conventional procedures will be too

narrow, giving a false sense of precision. Instead of considering this as a

problem to be corrected, GARCH models treat heteroskedasticity as a variance

to be modeled. As a result, not only are the deficiencies of least squares

corrected, but a prediction is computed for the variance of each error term. This

prediction turns out often to be of interest, particularly in applications in finance.

EMPIRICAL ISSUES IN VALUE-AT-RISK

BY DENNIS BAMS 1 AND JACCO L. WIELHOUWER

According to them they have compared four alternative models to calculate VaR


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estimates for the value of a certain portfolio of the bank. Crucial for this

calculation is the underlying return distribution, since it reflects the probability of

extreme returns. A number of issues are important. First, the underlying

probability distribution should be able to reflect the behavior of extreme returns.

Hence, the tail of the distribution should be well modeled. We proposed adopting

a Student-t distribution, since it allows for fatter tails than a normal distribution.

Parametric and Semi-parametric models of

Value-at-Risk: On the way to bias reduction Yan Liu and Richard

Luger Emory University

Due to the existence of the non-linear transformation bias in the VaR estimation

using GARCH model, we propose a new Two-stage VaR model based on our

generalized conditional coverage test. In order to eliminate the non-linear

transformation bias, the first-stage model starts from a parametric conditional

standard deviation model and is then tested by the generalized conditional

coverage test. If the first-stage model passes the test, we will keep using this

model since it is well specialized. If the model fails to pass the test, we will

incorporate some additional variables, selected based on the test, into the

second-stage model, a semi-parametric quantile regression model.

Crucial for the determination of the extreme future market value, and hence for

the VaR, is the distribution function of the return on market value. As allowed by


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the Basle Committee, a normal or lognormal distribution has usually been

assumed for the market return. Recently, alternative distributions have been

proposed that focus more on the tail behavior of the returns. See, for example,

Embrechts, Kluppelberg and Mikosch (1997), McNeil and Frey (1999) and Lucas

and Klaassen (1998) for a discussion. A normal distribution supposedly

underestimates the probability in the tail and hence the VaR result. Popular

alternatives in the financial literature include GARCH-type models which allow for

time-varying volatility, and the Student-t distribution, which allows for more

probability mass in the tail than the normal distribution. For a review of (G)ARCH

models, see Bollerslev, Engle and Nelson (1994). Other papers have focused on

different risk measures and different VaR methods. See, for example, Drudi et al.

(1997), Van Goorbergh and Vlaar (1999) and Jorion (1996).


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CHAPTER 3

RESEARCH METHODOLOGY


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Statement of Problem

Volatility always taken into consideration while taking investment decision.

Generally there are various models are used in forecasting volatility, GARCH is

one of them. We try to use GARCH in forecasting the volatility in the Indian stock

market and try to find out that at what level it is useful.

OBJECTIVES

To find out the forecasting techniques in Indian stock markets.

To ascertain the performance of different GARCH models at different risk

levels


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Study Design

a) Study Type:

The study type is analytical, quantitative and historical. Analyticalbecause

facts and existing information is used for the analysis, Quantitative as

relationship is examined by expressing variables in measurable terms and also

Historicalas the historical information is used for analysis and interpretation.

b) Study population:

Populationis the daily closing prices of NIFTY Index.

c) Sampling frame:

Sampling Frame would be monthly closing prices of NIFTY Index.

d) Sample:

Samplechosen is daily closing values of NIFTY Index from 01-01-1997 to

31-3-2007.

e) Sampling technique:

Deliberate sampling is used because only particular units are selected


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from the sampling frame. Such a selection is undertaken as these units represent

the population in a better way and reflect better relationship with the other

variable.

3.3 SAMPLE SIZE AND DATA SOURCES

In this study S&P CNX Nifty index has been considered as a proxy for the stock

market and accordingly the closing index values were collected from Jan 1,1997

till March 30, 2007.

Here we calculate the monthly variance from the data taken from above

mentiond period and use the first 78 months data to forecasting the remaining 34

months volatility.


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TEST OF STATIONARITY

Dickey-fuller Test for unit root:

Dickey fuller statistic test for the unit root in the time series data rt is regressed

against rt-1 to test for unit root in a time series random walk model.

This is given as:

rt= rt-1 + ut

if is significant equal to 1, then the stochastic variable rt is said to be having unit

root. A series with unit root is said to be un-stationary and does not follow

random walk. There are three most popular dickey-fuller tests for testing unit root

in a series.

The above equation can be rewritten as:

rt= rt-1 + ut

Here = (-1) and here it is tested if is equal to zero. rt is random walk if is

equal to zero. It is possible that time series could behave as a random walk with

a drift. This means that the value of rt may not center to zero and thus a constant

should be added to the random walk equation. A linear trend value could also be

added align with the constant it the equation, which results in a null hypothesis

reflecting stationary deviations from trend.


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The Augmented Dickey-fuller Test:

In conducting the DF test as above, it is assumed that the error term u t was

uncorrelated. But in case the ut are correlated, Dickey and Fuller have developed

a test, known as the augmented Dickey- Fuller ( ADF) test. The ADF test

consists of estimating the following regression:

Yt= 1 + 2t + Yt-1 + i Yt-i+ t

Where, t is a pure whitenoise term and Yt-1 = (Yt-1-Yt-2), Yt-2 = (Yt-2-Yt-3),etc.

The number of lagged difference terms to include is often determined empirically,

the idea being to include enough terms so the error term in above equation is

serially correlated. In ADF we still test whether=0 and the ADF test follow the

same asymptotic distribution as the DF statistic, so the same critical value can be

used.


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AUTO-CORRELATION

The term auto-correlation may be defined as correlation between members of

series of observation ordered in time or space. In the regression context, the

classical linear regression model assumes that such that autocorrelation doesnt

exist in the disturbances ui.

Symbolically

E (ui uj) = 0 (i j)

Put simply, the classical model assumes that the disturbance term relating to any

observation is not influenced by the disturbance term relating to any other

observation. For example: if we are dealing with quarterly time series data

involving the regression of output on labour and capital inputs and if, say there is

a labour strike affecting output in one quarter, there is no reason to believe that

this disruption will be carried out over to the next quarter. That is, if output is

lower this quarter, there is no reason to believe that this disruption will be carried

over to the next quarter.


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GARCH MODELS

The great workhorse of applied econometrics is the least squares model. The

basic version of the model assumes that, the expected value of all error terms, in

absolute value, is the same at any given point. Thus, the expected value of any

given error term, squared, is equal to the variance of all the error terms taken

together. This assumption is called homoskedasticity. Conversely, data in which

the expected value of the error terms is not equal, in which the error terms may

reasonably be expected to be larger for some points or ranges iof the data than

for others, is said to suffer from heteroskedasticity.

It has long been recognized that heteroskedasticity can pose problems in

ordinary least squares analysis. The standard warning is that in the presence of

heteroskedasticity, the regression coefficients for an ordinary least squares

regression are still unbiased, but the standard errors and confidence intervals

estimated by conventional procedures will be too narrow, giving a false sense of

precision. However, the warnings about heteroskedasticity have usually been

applied only to cross sectional models, not to time series models. For example, if

one looked at the cross-section relationship between income and consumption in

household data, one might expect to find that the consumption of low-income

households is more closely tied to income than that of high-income households,

because poor households are more likely to consume all of their income and to

be liquidity-constrained. In a cross-section regression of household consumption

on income, the error terms seem likely to be systematically larger for high-income

than for low-income households, and the assumption of homoskedasticity seems


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implausible. In contrast, if one looked at an aggregate time series consumption

function, comparing national income to consumption, it seems more plausible to

assume that the variance of the error terms doesnt changed much over time.

ARCH stands for autoregressive conditionally heteroskedasticity and these

models are a sophisticated group of time series models initially introduced by

Engle (1982) and ARCH models capture the volatility clustering phenomenon

usually observed in financial time series data. In the linear ARCH (q) model the

time varying conditional variance is postulated to be a linear function of the past

q squared innovations. In other words variance is modeled as a constant plus

a distributed lag on the squared residual terms from earlier periods

rt = + t and t2= +i.t-1

2

Where t~ iidN (0, 1) For stability .

I< 1.0 and theoretically q may assume any

number but generally it is determined based on some information criteria like AIC

or BIC. In financial markets the ARCH (1) model is most oftenly used and this is

a very simple model that exhibits constant unconditional variance but non-

constant conditional variance. Accordingly the conditional variance is modeled as

t2= 0 + 1. t-1

2

i-1

q


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As with simple regression the parameters in ARCH and GARCH models

(discussed next) are estimated at weekly intervals using a rolling window of

weekly 7 year window. The problem with the ARCH models is it involves

estimation of a large number of parameters and if some of the parameters

become negative they lead to difficulties in forecasting. Bollerslev (1986)

proposed a Generalized ARCH or GARCH (p, q) model where volatility at time t

depends on the observed data at t-1, t-2, t-3 .. t-q as well as on volatilities at

t-1, t-2, t-3 ... t-p.

The advantage of GARCH formulation is that though recent innovations enter the

model it involves only estimation of a few parameters hence there will be little

chance that they will ill-behaved. In GARCH there will be two equations

conditional mean equation given below:

rt = + t

and the conditional variance equation shown below,

t2= +i.t-1

2 +i.t-12

i-1 i-1

q p


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the parameters in both the equations are estimated simultaneously using

maximum likelihood methods once a distribution for the innovations t has been

specified generally it is assumed that they are Gaussian.

The simplest and most commonly used member of the GARCH family is the

GARCH (1, 1) model shown below

t2 = + .t-1

2+.t-12

Where,

t2 = variance of the current period

= intercept

t-12 = lag variable of residual

= parameter of error terms lag variable

t-12 = variance of last period

= parameter of lag variance

Following Schwarz Information Criteria and Akiake Information Criteria we found

that the best model in the GARCH (p, q) class for p [1, 5] and q [1, 2] was a


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GARCH (1,1) in the stock market. We also tested for whether the GARCH (1,1)

adequately captured all the persistence in the variance of returns by using Ljung-

Box Q- statistic at the 36th lag of the standardized squared residuals was 37.498

(p = 0.4) indicating that the residuals are not serially correlated.

In our forecasting exercise first we estimated the GARCH parameters using the

estimation period i.e., 1st week of Jan 1997 to last week of March 2004 for Nifty

and then used these parameters to obtain the forecasts for the trading days in 1st

week of April 2004 and these daily forecasts were aggregated to obtain the

forecast for the weeks of April 2004. Then the beginning and end observations

for parameter 4 for conserving space and to maintain the flow the values are not

presented and are available up on request estimation were adjusted by including

the data for 1st week of March 2004 and omitting the data pertaining to 1st week

of Jan 1997. The procedure is repeated for every week using a rolling window of

7 years.

LIMITATIONS OF THE RESEARCH

1. Data considered for ten years only.

2. Sample is restricted to S&P CNX Nifty index.

3. The models are tested on the basis of 3 years forecasted volatility value

only.


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Analysis and Interpretation

Steps followed in the analysis

The data is collected for

NIFTY Index.

Period of data collection January 1st January,1997 to 31st march,2007

The data is converted into log naturals format to way out any spurious

correlations within the data sets.

Then the data is tested for its stationarity using Augmented Dickey fuller

test

The monthly variance of the NIFTY daily closing price found

Out of 120 months, total 77 months used to find out the equation through

which we can forecast the value for next 37 months.

Forecasting the value using which find out the residual value which shows

the minor variation from the forecasted value.


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ADF RESULT

When the daily return of index is been tested for its unit root with four lag variable

the following result is obtained

ADF Test

Statistic -29.994

1% Critical

Value* -3.4437

5% Critical Value -2.8667

10% Critical Value -2.5695

*MacKinnon critical values for rejection of hypothesis of a unit root.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(SER02,2)

Method: Least Squares

Date: 04/30/07 Time: 11:22

Sample(adjusted): 4 601

Included observations: 598 after adjusting endpoints

Variable Coefficient

Std.

Error t-Statistic Prob.

D(SER02(-1)) -2.00722 0.066921 -29.994 0

D(SER02(-1),2) 0.341221 0.038837 8.786002 0C -1.73E-05 0.000764 -0.02267 0.9819

R-squared 0.775773

Mean dependent

var -9.64E-05

Adjusted R-

squared 0.77502

S.D. dependent

var 0.039396

S.E. of

regression 0.018686

Akaike info

criterion -5.11703


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Sum squared

resid 0.207764 Schwarz criterion -5.09499

Log likelihood 1532.993 F-statistic 1029.282

Durbin-Watson

stat 2.171695 Prob(F-statistic) 0

Interpretation

As it can be easily seen from the ADF test, the null hypothesis of unit root can be

rejected as the estimated value is -29.991, which in absolute value is greater

than all the critical value at 1%, 5% and 10% level of significance.

The absence of unit root means the series is stationary, combined with the

phenomenon of volatility clustering implies that volatility can be predicted and the

forecasting ability of the different models can be generalized to other time

periods also.


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REGRESSION ANALYSIS

Dependent variable : Variance

Independent variable : Return^2 , Lag Variance

GARCH ( 1,6 ) Model

Coefficients(a)

ModelUnstandardizedCoefficients

StandardizedCoefficients T Sig.

B Std. Error Beta1 (Constant) 0.000167 4.78E-05 3.491208 0.000814

Return^2 0.007883 0.004951 0.17688 1.592206 0.115601

Lagvariance^2 0.236369 0.111409 0.235694 2.121633 0.037217

A Dependent Variable: Variance

0.000167

Rt-62 1 0.007883

t-12 2 0.236369


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GARCH (1,2)

Dependent Variable: SER43 VarianceMethod: ML ARCH

Date: 05/14/07 Time: 12:50



Convergence achieved after 1 iterations

CoefficientStd.Error

z-Statistic Prob.

Variance Equation

C 9.11E-08 1.45E-07 0.628415 0.5297ARCH(1) 0.133333 0.199803 0.667325 0.5046

GARCH(1) 0.533333 1.354867 0.393642 0.6938

GARCH(2) 0.044444 1.321249 0.033638 0.9732

R-squared -0.85557Mean dependent

var 0.000265

Adjusted R-squared -0.90483

S.D. dependentvar 0.000287

S.E. of regression 0.000397Akaike info

criterion -12.6746

Sum squared resid 1.78E-05 Schwarz criterion -12.5802

Log likelihood 745.4646

Durbin-Watson

stat 0.811168


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GARCH ( 1,1 )

Dependent Variable: variance

Method: ML ARCH

Date: 05/13/07 Time: 13:58




CoefficientStd.Error z-Statistic Prob.

Variance Equation

C 9.10E-08 1.47E-07 0.618426 0.5363

ARCH(1) 0.15 0.196358 0.763913 0.4449

GARCH(1) 0.6 0.095394 6.289721 0


var 0.000264Adjusted R-squared -0.88966


S.E. ofregression 0.000394

Akaike infocriterion -12.6548

Sum squaredresid 1.78E-05 Schwarz criterion -12.5844

Log likelihood 749.6338Durbin-Watson

stat 0.866372


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GARCH ( 2,1 )

Dependent Variable: SER40 Variance

Method: ML ARCH

Date: 05/14/07 Time: 12:42




CoefficientStd.Error

z-Statistic Prob.

Variance

Equation

C 9.72E-08 1.51E-07 0.644293 0.5194

ARCH(1) 0.133333 0.255025 0.522824 0.6011

ARCH(2) 0.044444 0.232875 0.190851 0.8486

GARCH(1) 0.533333 0.110066 4.845583 0


var 0.000264Adjusted R-squared -0.709748


S.E. ofregression 0.000396

Akaike infocriterion -12.6744

Sum squaredresid 1.78E-05 Schwarz criterion -12.5805

Log likelihood 751.7879Durbin-Watson

stat 0.810855


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CONCLUSION

As we try to forecast the NIFTY closing prices for 36 months at different LAG, weget the highest significance at the GARCH ( 1,6 ) level. We took the alpha & Betaand try to forecast the variance for the remaining period. We got the graph for

residual value which is not correlated with the actual variance

We try to forecast NIFTY at different lag like GARCH ( 1,2) , GARCH(1,3) ,GARCH ( 2,1) but we find the value of R^2 like -0.5609, -0.85736 , -0.8557 whichis not significance enough to show forecasting power.

At last we can say that we are not able to forecast the voletility in the NIFTYindex with the help of GARCH model.


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ANNEXTURE

CNX S&P Daily Index

0

500

1000

1500

2000

2500

3000

3500

4000

4500

1/1/1997

7/1/1997

1/1/1998

7/1/1998

1/1/1999

7/1/1999

1/1/2000

7/1/2000

1/1/2001

7/1/2001

1/1/2002

7/1/2002

1/1/2003

7/1/2003

1/1/2004

7/1/2004

1/1/2005

7/1/2005

1/1/2006

7/1/2006

1/1/2007

Correlogram of S&P CNX Nifty monthly return

Date: 05/13/07 Time: 18:41

Sample: 1 2507

Included observations: 119


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Autocorrelation Partial

Correlation

AC PAC Q-

Stat

Prob

.|. | .|. | 1 -

0.00

4

-

0.00

4

0.002

2

0.96

3

.|* | .|* | 2 0.08

3

0.08

3

0.850

9

0.65

3

.|. | .|. | 3 0.03

5

0.03

6

1.002

1

0.80

1

*|. | *|. | 4 -

0.08

1

-

0.08

9

1.833

6

0.76

6

.|. | .|. | 5 -

0.02

9

-

0.03

6

1.938

9

0.85

8

.|* | .|* | 6 0.13

6

0.15

2

4.298

4

0.63

6

*|. | *|. | 7 -

0.070

-

0.059

4.928

7

0.66

9

*|. | *|. | 8 -

0.07

5

-

0.11

3

5.655

8

0.68

6


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.|. | .|. | 9 -

0.01

6

-

0.01

8

5.688

9

0.77

1

.|. | .|* | 1

0

0.06

4

0.12

1

6.233

2

0.79

5

.|. | .|. | 1

1

0.04

7

0.05

7

6.532

8

0.83

6

.|* | .|. | 1

2

0.07

6

0.01

2

7.311

5

0.83

6

.|. | .|. | 1

3

-

0.01

0

-

0.01

9

7.323

9

0.88

5

*|. | .|. | 1

4

-

0.07

2

-

0.04

7

8.027

7

0.88

8

.|. | .|. | 1

5

0.00

2

0.01

3

8.028

5

0.92

3

*|. | *|. | 1

6

-

0.11

8

-

0.13

8

9.980

7

0.86

8

.|* | .|* | 1

7

0.07

8

0.07

8

10.83

0

0.86

5

.|. | .|. | 1

8

-

0.04

2

-

0.01

8

11.07

7

0.89

1


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*|. | *|. | 1

9

-

0.10

0

-

0.09

6

12.50

5

0.86

3

*|. | *|. | 2

0

-

0.06

3

-

0.07

5

13.08

7

0.87

4

.|. | .|. | 2

1

0.03

8

0.06

3

13.29

4

0.89

8

.|. | .|. | 2

2

-

0.03

5

0.00

3

13.47

8

0.91

9

.|* | .|* | 2

3

0.13

8

0.07

0

16.33

6

0.84

1

.|* | .|* | 2

4

0.12

5

0.12

7

18.70

9

0.76

7

.|. | .|* | 2

5

0.04

9

0.08

6

19.07

2

0.79

4

*|. | *|. | 2

6

-

0.06

0

-

0.07

8

19.63

6

0.80

8

*|. | *|. | 2

7

-

0.08

1

-

0.15

8

20.66

7

0.80

2

.|. | .|. | 2

8

-

0.03

5

0.01

1

20.86

5

0.83

1


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.|. | .|. | 2

9

-

0.04

1

-

0.00

1

21.13

0

0.85

4

.|. | .|. | 3

0

0.02

0

0.00

0

21.19

2

0.88

2

.|. | .|. | 3

1

-

0.03

5

-

0.03

0

21.39

2

0.90

1

.|. | .|* | 3

2

0.06

3

0.12

5

22.03

9

0.90

6

.|. | .|* | 3

3

0.05

6

0.10

3

22.56

3

0.91

4

.|. | *|. | 3

4

0.01

4

-

0.10

1

22.59

7

0.93

2

.|* | .|. | 3

5

0.07

3

-

0.02

4

23.50

7

0.93

0

*|. | *|. | 3

6

-

0.06

4

-

0.06

0

24.22

1

0.93

3


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REGRESSION ANALYSIS

Coefficients(a)

ModelUnstandardizedCoefficients

StandardizedCoefficients T Sig.

B Std. Error Beta

1 (Constant) 0.000167 4.78E-05 3.491208 0.000814

Return^2 0.007883 0.004951 0.17688 1.592206 0.115601

Lagvariance^2 0.236369 0.111409 0.235694 2.121633 0.037217

a Dependent Variable: Variance

0.000167

Rt-62 1 0.007883

t-12 2 0.236369

S&P CNX Nifty daily return


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-300

-250

-200

-150

-100

-50

0

50

100

150

200

1 149 297 445 593 741 889 1037 1185 1333 1481 1629 1777 1925 2073 2221 2369 2517 2665 2813Series1


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16151413121110987654321

Lag Number

1.0

0.5

0.0

-0.5

-1.0

ACF

Lower ConfidenceLimit

Upper Confidence Limit

Coefficient

VAR00001


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Bibliography

BOOKS

1. Basic Econometrics: By Damodar N. Gujrati

2. Introductory Econometrics: By Ramu Ramanathan

ECONOMETRICS SOFTWARE PACKAGES

1. Eviews

2. SPSS

References

Bollerslev, T. (1986). A generalized autoregressive conditional

heteroskedasticity. Journal of Econometrics, 31:307 327.

Bollerslev, T. (1987). A conditionally heteroskedastic time series model forspeculative prices and rates of return.

Bollerslev, T., Chou, R., and Kroner, K. (1992). ARCH modelling in finance: A

review of the theory and empirical evidence. Journal of Econometrics, 52:559.


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Christoffersen, P. (1998). Evaluating interval forecast. International Economic

Review, 39(4):841864.

Crouhy, M., Galai, D., and Mark, R. (1998). The new 1998 regulatory framework

for capital adequacy. In Alexander, C., editor, Risk Management and Analysis,

volume 1, chapter 1, pages 137.

John Willey, New York. Dowd, K. (1998). Beyond Value at Risk: the New Science

of Risk Management. John Willey & Sons, England. Duffie, D. and Pan, J.

(1997). An overview of value at risk. Journal of Derivatives, 4:749.

Geman, S., Bienenstock, E., and Doursat, R. (1992). Neural networks and the

bias/variance dilemma. Neural Computation, 4:158.

Hornik, K., Stinchcombe, M., and White, H. (1989). Multilayer feedforward

networks are universal approximators. Neural Networks, 2:359366.

Kupiec, H. (1995). Techniques for verifying the accuracy of risk management

models. Journal of Derivatives, 3:7384.

Lopez, J. (1998). Methods for evaluating value-at-risk estimates. Economic

Policy Review, 4:119124.

Value at Risk Model in Indian Stock Market

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