volatility forecasting techniques in the indian capital market

8/7/2019 Volatility Forecasting Techniques in the Indian Capital Market

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Evaluation of the Volatility Forecasting Techniques

M P Birla Institute of Management, Bangalore(Associate Bharatiya Vidya Bhavan)

1


in the Indian Capital Market

Submitted in partial fulfillment of the requirements of

the M.B.A Degree Course of Bangalore University

By

VARUN KUMAR

(REGD.NO:05XQCM 6105)

Under the GuidanceOf

DR. T.V.NARASIMHA RAOFaculty

MPBIM

M.P.BIRLA INSTITUTE OF MANAGEMENTAssociate Bharatiya Vidya Bhavan

43, Race Course Road, Bangalore-560001

2007


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DECLARATION

I hereby declare that the dissertation entitled Evaluation of the

Volatility Forecasting techniques in the Indian Capital 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 : 15th May 2007 Varun Kumar


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PRINCIPALS CERTIFICATE

This is to certify that the Research Report entitled Evaluation of

the Volatility Forecasting techniques in the Indian Capital

Market done by VARUN KUMAR bearing Registration No.

05 XQCM 6105 under the guidance ofDr.T.V.N.Rao.

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

Date : 15th

May 2007 Principal

MPBIM, Bangalore


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GUIDES CERTIFICATE

This is to certify that the Research Report entitled Evaluation of

the Volatility Forecasting techniques in the Indian Capital

Market done by VARUN KUMAR bearing Registration No.

05XQCM6105 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 : 15th May 2007 Professor

MPBIM, Bangalore


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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 am sincerely thankful to Dr. Nagesh Malavalli, Principal M. P.

Birla Institute of Management, Bangalore for granting me the

permission to do this reseach project.

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.

I also wish to express my gratitude to Prof. S. Santhanam for his

valuable guidance and ideas during the project.

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.

Varun Kumar


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TABLE OF CONTENTS

CHAPTERS PARTICULARS PAGE NO.

ABSTRACT 9

1 INTRODUCTION AND

THEORETICAL BACKGROUND

10

2 REVIEW OF LITERATURE 14

3 RESEARCH METHODOLOGY 18

3.1 PROBLEM STATEMENT 19

3.2 OBJECTIVE OF THE STUDY 20

3.3 SAMPLE SIZE AND DATA SOURCES 21

3.4 TEST OF STATIONARITY 22

3.5 AUTO-CORRELATION 24

3.6 COMPPETING MODELS 25

3.7 ERROR MEASUREMENT

TECHNIQUES

30

3.8 LIMITATIONS OF THE RESEARCH 324 DATA ANALYSIS 34

5 FORECASTED RESULTS AND

DISCUSSION

38

6 CONCLUSION 40

7 ANNEXTURE 42

8 BIBLIOGRAPHY 50


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DIAGRAM

No. Page No.

1 Histogram 35

2 S&P CNX Nifty weekly Closing 42

3 Nifty weekly return 42

4 S&P CNX Nifty daily Index 43

5 Nifty Daily return 43

6 Correlogram of Nifty Weekly Variance 45

7 Correlogram of Nifty Weekly Return 46


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ABSTRACT

Volatility forecasting is an important area of research in financial markets and lot of

effort has been expended in improving volatility models since better forecasts translate

in to better pricing of options and better risk management.

In this direction this paper attempts to evaluate the ability of eight different statistical

and econometric volatility forecasting models for predicting the stock price volatility

using S&P CNX Nifty Index return series.

For this purpose ten year S&P CNX Nifty daily Index is taken .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 weekly 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, 1) 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.


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CHAPTER 1

INTRODUCTION AND

THEORETICAL BACKGROUND


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INTRODUCTION AND THEORETICAL BACKGROUND

Volatility forecasting is an important task in financial markets, and it has held the

attention of academics and practitioners over the period of time. Volatility is not the

same as risk. When it is interpreted as uncertainty, it becomes a key input to many

investment decisions and portfolio creations. Investors and portfolio managers have

certain levels of risk which they can bear. A good forecast of the volatility of asset

prices over the investment holding period is a good starting point for assessing

investment risk.

Volatility is the most important variable in the pricing of derivative securities, whose

trading volume has quadrupled in recent years. To price an option, we need to know the

volatility of the underlying asset from now until the option expires. In fact, the market

convention is to list option prices in terms of volatility units. Nowadays, one can buy

derivatives that are written on volatility itself, in which case the definition and

measurement of volatility will be clearly specified in the derivative contracts. In these

new contracts, volatility now becomes the underlying "asset." So volatility forecast and

a second prediction on the volatility of volatility over the defined period is needed to

price such derivative contracts.

Financial risk management has taken a central role since the first Basel Accord was

established in 1996. This effectively makes volatility forecasting a compulsory risk-

management exercise for many financial institutions around the world. Banks and

trading houses have to set aside reserve capital of at least three times that of value-at-

risk (VaR), which is defined as the minimum expected loss with a 1-percent confidence

level for a given time horizon (usually one or ten days). Sometimes, a 5-percent critical

value is used. Such VaR estimates are readily available given volatility forecast, mean

estimate, and a normal distribution assumption for the changes in total asset value.


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When the normal distribution assumption is disputed, which is very often the case,

volatility is still needed in the simulation process used to produce the VaR figures.

Financial market volatility can have a wide repercussion on the economy as a whole.

The incidents caused by the terrorists' attack on September 11, 2001, and the recent

financial reporting scandals in the United States have caused great turmoil in financial

markets on several continents and a negative impact on the world economy. This is

clear evidence of the important link between financial market uncertainty and public

confidence. For this reason, policy makers often rely on market estimates of volatility

as a barometer for the vulnerability of financial markets and the economy. In the United

States, the Federal Reserve explicitly takes into account the volatility of stocks, bonds,

currencies, and commodities in establishing its monetary policy (Sylvia Nasar 1992).

The Bank of England is also known to make frequent references to market sentiment

and option implied densities of key financial variables in its monetary policy meetings.

Volatility is the variability of the asset price changes over a particular period of time

and it is very hard to predict it correctly and consistently. In financial markets volatility

presents a strange paradox to the market participants, academicians and policy makers

without volatility superior returns are can not be earned, since a risk free security offers

meager returns, on the other hand if it is high it will lead to losses for the market

participants and represent costs to the over all economy. Therefore there is no

gainsaying with the statement that volatility estimation is an essential part in most

finance decisions be it asset allocation, derivative pricing or risk management. However

the question as to what model should be used to calculate volatility, there is no unique

answer as different volatility models were proposed in the literature and were being

used by practitioners and these varying models lead to different volatility estimates. In

the past two decades this has been a fertile area for research in financial economics for

both academicians as well as practitioners. Unfortunately most of the work was done in

the context of developed markets in the context of stock and forex markets. This paper

is an attempt to examine the efficacy of the competing volatility forecasting models in

the Indian market.


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The expected volatility of the financial markets is a key variable in many financial

investment decisions. For example, it is a common practice to reduce asset allocation

decisions to a two-dimensional decision problem by focusing solely on the expected

return and risk of an asset or portfolio, with risk being related to the volatility of the

returns.

Considering the significance of the volatility forecasting in asset pricing, option pricing,

in the economic literature, a large number of volatility forecasting techniques have been

evolved over the years and the effort is still on to search for new alternative techniques.

These volatility-forecasting techniques can be categorized into the four classes:

1. HISVOL; for historical volatility models, which includes random walk,

historical averages of squared returns, or absolute returns. Also included in this

category are time series models based on historical volatility using moving

averages, exponential weights, autoregressive models, or even fractionally

integrated autoregressive absolute returns, for example. Note that HISVOL

models can be highly sophisticated. The multivariate VAR realized volatility

model in Andersen et al. (2002) is classified here as a "HISVOL" model. Allmodels in this group model volatility directly omitting the goodness of fit of the

returns distribution or any other variables such as option prices.

2. GARCH; any members of the ARCH, GARCH, EGARCH, and so forth

families are included.

3. ISD; for option implied standard deviation, based on the Black-Scholes model

and various generalizations.

4. SV; for stochastic volatility model forecasts.


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CHAPTER 2

REVIEW OF LITERATURE


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REVIEW OF LITERATURE

Review of literature .means examining and analyzing the various literatures available in

any field either for references purposes or for further research.

Further research can be done by identifying the areas which have not been studied and

in turn undertaking research to add value to the existing literature.

For the purpose of literature review various sources of information have been used.

Sources include books, journals as well as some literature papers.

Dr. S S S Kumar : Forecasting Volatility- Evidence from Indian

Stock And Forex Markets

In his project Dr. Kumar, to forecast volatility of Indian stock and Forex markets,

created models using traditional techniques such as Random walk, Historical mean,

Moving average, Simple regression and EWMA and the modern technique such as

GARCH family model.

After forecasting the volatility of Indian Stock and Forex Market the competing models

are evaluated on the basis of two categories of evaluation measures symmetric and

asymmetric error statistics.

.

Empirical Result:

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

that GARCH (4, 1) and EWMA methods will lead to better volatility forecasts in the

Indian stock market and the GARCH (5, 1) will achieve the same in the forex market.

The same models perform better on the basis of asymmetric error statistics also.


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Vedat Akgiray: - Conditional Heteroscedasticity in Time Series of

Stock Returns: Evidence and Forecasts

This article presents new evidence about the time-series behavior of stock prices. Daily

return series exhibit significant levels of second-order dependence, and they cannot be

modeled as linear white-noise processes. A reasonable return generating process is

empirically shown to be a first-order autoregressive process with conditionally

heteroscedastic innovations. In particular, generalized autoregressive conditional

heteroscedastic GARCH processes fit to data very satisfactorily. Various out-of-sampleforecasts of monthly return variances are generated and compared statistically.

It is discovered that daily series exhibit much higher degrees of statistical dependence

than has been reported in previous studies. This finding is the result of recognizing the

possibility of nonlinear stochastic processes generating security prices. The dependence

structure is then exploited to obtain forecasts of the conditional moments of return

distributions. Forecasts of conditional variances in particular are shown to have

reasonably high accuracy.

Conclusion: Forecasts based on the GARCH model are found to be superior.

Ser-Huang Poon & Clive W. J. Granger: -Forecasting Volatility in

Financial Markets: A Review

This paper has concentrated on two questions: is volatility forecastable? If it is, which

method will provide the best forecasts? To consider these questions, a number of basic

methodological viewpoints need to be discussed, mostly about the evaluation of

forecasts. What exactly is being forecast? Does the time interval (the observation


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CHAPTER 3

RESEARCH METHODOLOGY


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3.1 STATEMENT OF PROBLEM

For many years economists, statisticians, and teachers of finance have been interested

in developing and testing models of forecasting volatility of various financial series

data. There are various traditional forecasting techniques, which are commonly used to

forecast volatility such as Historical Mean, Moving Average, Simple regression etc.

and there are certain modern volatility forecasting techniques such as GARCH family

models etc.

In this project various traditional and modern forecasting techniques are being to

forecast the volatility of Indian stock and Among the selected models, the model which

is able to closely predict the stock price volatility is been found out.


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3.2 OBJECTIVES

Identify the commonly used volatility-forecasting techniques.

Identify the model which could closely forecast the stock price volatility.


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3.3 SAMPLE SIZE AND DATA SOURCES

In this study S&P CNX Nifty index has been considered as a proxy for the stockmarket and accordingly the closing index values were collected from Jan 1,1997 till

March 30, 2007.

Out of the total observations the data pertaining to Jan 1,1997 till March 2004 totaling

126 weekly observations of NIFTY were used for estimation of the model parameters

and the remaining observations will be used for out of sample forecasting also known

as hold out sample. Therefore the first week for which out of sample forecasts are

obtained is 1st

week of April, 2004 and the out of sample forecasts were constructed for

157 weeks till March 2007. The daily observations were converted into continuous

compounded returns in the standard method as the log differences:

Rt = ln (It / It-1)

Where, It stands for the closing index value on dayt;

Following Merton (1980) the weekly volatility is obtained as the sum of the squared

daily returns in that week which is shown below:

2

= Rt2

Where Rt is the daily return on dayt and N is the number of trading days in the week

under question.

N

i =1


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3.4 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.

The Augmented Dickey-fuller Test:

In conducting the DF test as above, it is assumed that the error term ut 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


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Where, tis 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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3.5 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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3.6 COMPETING MODELS

Taking cues from the earlier studies in the international context mentioned in the

literature review this work examines the forecasting capabilities of the following

models:

1. Historical mean

2. Moving average (5 models)

3. Simple regression

4. GARCH models

The models that were considered in this particular study are not exhaustive but cover a

very large variety of models ranging from nave models to the advanced models like

GARCH and the preferred model by practitioners viz.

In the following paragraphs a brief description of all the candidate models are been

given:

1. Historical Mean ModelAssuming the conditional expectation of the volatility constant, this model forecasts

volatility as the historical average of the past observed volatilities

where t = 378..534

2. Moving Average ModelIn the historic mean model the forecast is based on all the available observations

and each observation whether it is very old or immediate is given equal weight this

may lead to stale prices affecting the forecasts. This is adjusted in a moving

averages method which is a traditional time series technique in which the


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volatility is defined as the equally weighted average of realized volatilities in the

past w weeks.

The choice of w is rather arbitrary and in this paper we investigate five models

13, 26, 52, 104 and 260 weeks.

3. Simple RegressionIn this method the familiar regression of actual volatilities on lagged values is run,

in other words it is the first autoregression is performed on the first part of data

which is meant for estimating the parameters and the estimates thus obtained were

used for forecasting the volatility for the next month. Accordingly the first part

involves running the following regression:

2

= + .2

t-1

and are estimated over the 7 year period from Jan 1997 till Mar 2004. Now for

the next forecast the volatility for the first week of April 2004 the parameters and

are reestimated by omitting the most distant past and including the latest actual

volatility observation. This process is repeated and thus the estimation window moves

forward. By following this methodology we actually utilize the time-varying

parameters for each week.


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4. ARCH and GARCHARCH 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-12

Where t ~ iid N (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-12

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:

i-1

q


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rt = + t

and the conditional variance equation shown below,

t2= +i.t-12 +i.t-12

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-12+.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 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.

i-1 i-1

q p


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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.


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3.7 ERROR MEASUREMENT TECHNIQUES

Previous research found that the choice of the best model depends on the error

measurement that depends on the ultimate purpose of the forecasting procedure. A

variety of error measurement techniques are available to evaluate and compare forecast

errors. Consistent with this research, the 47 monthly forecast errors generated from

each model in this study are compared by the Mean Absolute Error (MAE), the Root

Squared Error (RMSE) and the Mean Mixed Error: MME (U) and MME (O).

They are defined as follows:

Symmetric Loss Function:

Asymmetric Loss Function:

Where,

f,t

= Forecasted variance at time t

r,t = Realised variance at time t

n = Forecasted period

U = number of under-predictions and

O = number of over-predictions


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The MAE and RMSE are most commonly used measures to test the predictive

capability of a model. They are not invariant to scale transformation and symmetric, a

property which is not very realistic and inconceivable under some circumstances.

The near zero value indicates a perfect fit. The error statistics reported by the above

measures assume that the underlying loss function is symmetric. From a practical

viewpoint, it is conceivable that many investors will not attribute equal importance to

both over and under-predictions of volatility of similar magnitude. Hence, the

assumption of symmetric is very restrictive. Following the paper of Brailsford and Fall

(1996), and Yu (2002) three statistics are designed to capture potential asymmetric in

the loss functionMean Mixed Error: MME (U) and MME (O). Where O is the

number of over predictions and U is the number of under predictions. MME (U)

penalize more the under predictions and MME (O) penalizes more the over predictions.


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3.8 LIMITATIONS OF THE RESEARCH

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

2. Data considered for ten years only.

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

4. Results arrived at, are generalized for the entire sample.


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CHAPTER 4

DATA ANALYSIS


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4.1 DESCRIPTIVE STATISTICS

The descriptive statistics of the data were presented in Table 1 and figures 1 plot the

return series of Nifty. The mean daily return for nifty was 0.3208% and the annualized

volatility for nifty is around 3.3481%, the series exhibit excess kurtosis indicating that

the unconditional return distributions are not normally. The Jarque-Bera (JB) statistic

confirms that normality is rejected at a p-value of almost 1. The plot of return series in

Figures 1 shows that there is persistence and volatility clustering is a feature of both the

markets which suggests that the volatility is predictable.

Table 1


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HISTOGRAM

Figure 1


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4.2 ADF RESULT

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

ADF Test Statistic -22.0141 1% Critical Value -3.4359

5% Critical Value -2.8632

10% Critical Value -2.5677

Table 2

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 -22.0141, 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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4.3 AUTOCORRELATION

It can be noted from the Figure 6, that there is high correlation of variance at the 1 lag.

This indicates the fact that the current index weekly variance have a close relationship

with the previous week variance value.

Nifty Variance is significantly auto-correlated at lag 1.


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4.4 FORECASTED RESULTS AND DISCUSSION

After testing for stationarity the weekly returns of S&P CNX Nifty data are being usedfor calculating parameters of the competing models. Then the weekly volatility had

been forecasted for three years starting from 1st

week of April 2004 to last week of

March, 2007. Then the forecasted values are tested through the Forecast Error

Statistics, which are explained in earlier section. After calculating the forecast errors

the models are ranked, where the model with low error assigned higher rank and vice

versa.

Table 4 and 5 in annexure present the results of the error statistics explained in the

earlier section. In this table we present the actual statistic errors along with the relative

ranking of that particular method among the competing models from the results we can

make the following observations.

Firstly, based on MAE, the MA1 model has been ranked 1 and GARCH model got

second ranking with a small difference. While on the basis of RMSE there is unanimity

of the superiority of GARCH method. Other models such as MA 5 ranked 3 and Simple

Regression Model ranked 5 and 2 as per MAE and RMSE respectively.

So according to Symmetric Loss function of Forecast Error Statistics, it can be said that

GARCH (1,1) model performed well in comparison of other competing models .And all

the measures in Symmetric loss function indicate historical mean model as the worst

performing model.

On the basis of asymmetric loss error statistics, one can note that only GARCH (1, 1)

model provides unbiased forecasts meaning the probability of over predictions is equal

to the probability of under predictions which is equal to 50% and the null hypothesis of

an equal number of under and over forecasts cannot be accepted for any other model

but for random walk model at conventional levels of significance. In


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stock market the superior ranked models viz., GARCH models have a tendency to

produce over forecasts but with relatively small errors. On the basis of MME (U) the

best fit GARCH (1, 1) emerged better model and Simple regression model ranked 2nd

and on the basis of MME (O) the MA 1 method comes out as the best model while

GARCH (1, 1) ranked 6th

.


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CHAPTER 5

CONCLUSION


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CONCLUSION

Volatility forecasting is an important area of research in financial markets and in this

paper we evaluate the comparative ability of different statistical and econometric

volatility forecasting models in the context of Indian stock.

A total of eight different models were considered in this study and these competing

models are evaluated on the basis of two classes of evaluation measures symmetric

and asymmetric error statistics.

Based on the out of sample forecasts and the number of evaluation measures that rank a

particular method as superior we can infer that GARCH (1, 1) will lead to

improvements in volatility forecasts in the stock market.

This finding is contrary to the findings of Brailsford and Faff (1996) who found no

single method as superior. But the results in stock market are similar to the findings of

Akigray (1989) and McMillan et al (2000). The inferences remain same even on the

basis of asymmetric error statistics i.e., GARCH (1, 1) model when over forecasts are

penalized heavily whereas MA 2 and MA 3 are penalized heavily for under forecasting.


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ANNEXTURE

S&P CNX Nifty Weekly Closing

0

500

1000

1500

2000

2500

3000

3500

4000

4500

1/3/1997

1/3/1998

1/3/1999

1/3/2000

1/3/2001

1/3/2002

1/3/2003

1/3/2004

1/3/2005

1/3/2006

1/3/2007

Figure 2

Nifty Weekly Return

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

1/1

0/1997

1/1

0/1998

1/1

0/1999

1/1

0/2000

1/1

0/2001

1/1

0/2002

1/1

0/2003

1/1

0/2004

1/1

0/2005

1/1

0/2006

1/1

0/2007

Figure 3


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43

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

Figure 4

Nifty Daily Return

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

1/2/1997

7/2/1997

1/2/1998

7/2/1998

1/2/1999

7/2/1999

1/2/2000

7/2/2000

1/2/2001

7/2/2001

1/2/2002

7/2/2002

1/2/2003

7/2/2003

1/2/2004

7/2/2004

1/2/2005

7/2/2005

1/2/2006

7/2/2006

1/2/2007

Figure 5


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44

Table 3

Forecast Error Statistics: Symmetric Loss Function

MAE RANK RMSE RANK

GARCH(1,1) 0.01429112 2 0.018394319 1

Historical Mean Model 0.01630163 8 0.020959297 6

MA 1 0.01333822 1 0.020412488 4

MA 2 0.01559966 7 0.021724413 8

MA 3 0.01509806 6 0.021118691 7

MA 4 0.01463925 4 0.020510997 5MA 5 0.0143535 3 0.019901621 3

Simple Regression Model 0.0147821 5 0.018926 2

Table 4

Forecast Error Statistics: Asymmetric Loss Function

MME(U) RANK MME(O) RANK

GARCH(1,1) 0.03101787 1 0.095092106 6

Historical Mean Model 0.03509557 3 0.100629357 8

MA 1 0.04315484 7 0.070799089 1

MA 2 0.0477235 8 0.081111254 2

MA 3 0.04159458 6 0.084056952 3

MA 4 0.0389779 5 0.087716327 5

MA 5 0.03876587 4 0.086073501 4

Simple Regression Model 0.03186044 2 0.097168602 7


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45

Figure 6: Correlogram of S&P CNX Nifty,

weekly Variance (Jan 1997 to Mar 2007)


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46

Figure 7: Correlogram of S&P CNX Nifty,

weekly return (Jan 1997 to Mar 2007)

Date: 05/03/07 Time: 01:31

Sample: 1 601

Included observations: 534

AutocorrelationPartialCorrelation AC PAC Q-Stat Prob

.|* | .|* | 1 0.089 0.089 4.2454 0.039

.|. | .|. | 2 0.017 0.009 4.3929 0.111

.|. | .|. | 3 0.01 0.008 4.4474 0.217

.|. | .|. | 4 -0.051 -0.053 5.8296 0.212

.|. | .|. | 5 -0.01 -0.001 5.8868 0.317

.|. | .|. | 6 -0.035 -0.033 6.5476 0.365

.|. | .|. | 7 -0.041 -0.034 7.4403 0.385

.|. | .|. | 8 0.049 0.055 8.7672 0.362

.|* | .|* | 9 0.095 0.089 13.719 0.133

.|. | .|. | 10 0.021 0.001 13.955 0.175

.|. | .|. | 11 -0.044 -0.055 15.001 0.182

.|. | .|. | 12 -0.001 0.009 15.001 0.241

.|. | .|. | 13 0.027 0.036 15.409 0.283

.|. | .|. | 14 -0.003 -0.004 15.414 0.35

.|. | .|. | 15 -0.033 -0.03 16.022 0.381

.|. | .|. | 16 -0.003 0.008 16.026 0.451

.|. | .|. | 17 0 -0.007 16.026 0.522

.|. | .|. | 18 0.005 -0.008 16.039 0.59

.|. | .|. | 19 -0.041 -0.04 16.969 0.592

.|. | .|. | 20 -0.029 -0.01 17.427 0.625

.|. | .|. | 21 -0.004 -0.003 17.434 0.684

.|* | .|* | 22 0.089 0.084 21.876 0.467

.|. | .|. | 23 -0.036 -0.054 22.622 0.483

*|. | *|. | 24 -0.087 -0.08 26.897 0.309

.|* | .|* | 25 0.084 0.099 30.837 0.194

.|* | .|* | 26 0.081 0.077 34.556 0.122

.|. | .|. | 27 0 -0.02 34.556 0.15

.|. | .|. | 28 0.013 0.013 34.654 0.18

.|. | .|. | 29 -0.03 -0.015 35.16 0.199

.|. | .|. | 30 -0.011 -0.018 35.224 0.235

.|. | .|. | 31 0.002 -0.011 35.227 0.275

.|. | .|. | 32 -0.04 -0.012 36.154 0.281

.|. | .|. | 33 -0.035 -0.009 36.844 0.295

.|. | .|. | 34 -0.015 -0.04 36.977 0.333

.|. | .|. | 35 0.025 0.001 37.332 0.362


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47

.|. | .|. | 36 -0.017 -0.016 37.498 0.4

*|. | *|. | 37 -0.074 -0.063 40.625 0.314

.|. | .|. | 38 0.022 0.027 40.903 0.344

.|. | .|. | 39 0.003 0 40.91 0.387

.|. | .|. | 40 0.022 0.022 41.182 0.419

.|. | .|. | 41 0.023 0.025 41.487 0.449

*|. | *|. | 42 -0.063 -0.06 43.774 0.396

.|. | .|. | 43 0.035 0.037 44.48 0.409

.|. | .|. | 44 0.014 0.003 44.596 0.447

.|. | .|. | 45 0.007 0.037 44.626 0.488

.|. | .|. | 46 -0.031 -0.014 45.185 0.506

.|. | .|. | 47 0.062 0.047 47.409 0.456

.|. | .|. | 48 0.048 0.016 48.753 0.443

.|. | .|. | 49 -0.042 -0.038 49.812 0.441

.|. | .|. | 50 0.009 0.018 49.856 0.479

.|. | .|. | 51 0.024 0.028 50.193 0.506

.|. | .|. | 52 0.022 0.014 50.483 0.534

.|. | *|. | 53 -0.03 -0.061 51.021 0.552

.|. | .|. | 54 -0.055 -0.045 52.802 0.521

.|. | .|. | 55 0.004 0.029 52.811 0.559

.|. | .|. | 56 0.013 -0.001 52.912 0.593

.|* | .|* | 57 0.081 0.072 56.832 0.481

.|. | .|. | 58 0.019 0.022 57.039 0.511

.|. | .|. | 59 -0.018 -0.008 57.245 0.54

.|. | .|. | 60 0.04 0.018 58.233 0.541

.|. | .|. | 61 -0.007 -0.023 58.262 0.576

*|. | .|. | 62 -0.06 -0.025 60.41 0.533

.|. | .|. | 63 -0.026 -0.002 60.827 0.554

*|. | .|. | 64 -0.059 -0.038 62.967 0.513

.|. | .|. | 65 0.018 0.006 63.155 0.542

.|. | *|. | 66 -0.048 -0.092 64.552 0.527

.|. | .|. | 67 -0.053 -0.035 66.284 0.502

*|. | *|. | 68 -0.097 -0.092 72.025 0.346

.|. | .|* | 69 0.064 0.066 74.513 0.304

.|. | .|. | 70 0.013 -0.001 74.621 0.331

.|. | .|. | 71 0.029 0.049 75.158 0.345

.|. | .|. | 72 -0.032 -0.056 75.798 0.357

.|* | .|. | 73 0.074 0.061 79.22 0.289

.|. | .|. | 74 0.025 0.018 79.602 0.307

*|. | .|. | 75 -0.079 -0.054 83.471 0.235

.|. | .|. | 76 -0.035 -0.001 84.233 0.242

.|. | .|. | 77 -0.034 -0.012 84.961 0.25

.|. | .|. | 78 0.041 0.04 85.996 0.251

.|. | .|. | 79 0.024 -0.008 86.35 0.268

*|. | *|. | 80 -0.076 -0.07 89.985 0.209

*|. | .|. | 81 -0.063 -0.05 92.489 0.18

.|* | .|* | 82 0.07 0.07 95.616 0.144

.|. | .|. | 83 0.001 -0.038 95.616 0.162


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48

.|. | .|. | 84 -0.049 -0.031 97.129 0.155

.|. | .|. | 85 -0.005 -0.012 97.143 0.173

.|. | .|. | 86 -0.027 -0.037 97.62 0.184

.|. | .|. | 87 -0.004 -0.018 97.63 0.205

.|. | .|. | 88 -0.021 -0.009 97.918 0.22

*|. | *|. | 89 -0.11 -0.059 105.73 0.109

.|. | .|. | 90 -0.02 0.011 105.98 0.12

.|. | .|. | 91 0.059 0.024 108.24 0.105

.|. | .|. | 92 0.053 0.063 110.05 0.097

.|. | .|. | 93 -0.038 -0.027 110.97 0.099

.|. | .|. | 94 -0.044 -0.02 112.24 0.097

.|. | .|. | 95 0.02 -0.004 112.5 0.106

.|* | .|* | 96 0.08 0.079 116.73 0.074

.|. | .|. | 97 -0.025 -0.012 117.14 0.08

.|. | .|. | 98 0.011 -0.006 117.21 0.09

.|* | .|. | 99 0.067 0.042 120.14 0.073

.|. | .|. | 100 0.022 -0.013 120.45 0.08

.|. | .|. | 101 0.019 0.01 120.69 0.088

.|. | .|* | 102 0.031 0.071 121.32 0.093

.|. | .|. | 103 -0.026 -0.023 121.75 0.1

.|* | .|* | 104 0.108 0.07 129.55 0.046

.|* | .|. | 105 0.073 0.053 133.13 0.033

.|. | .|. | 106 -0.015 0.024 133.28 0.038

.|. | .|. | 107 0.009 -0.007 133.34 0.043

.|. | .|. | 108 -0.005 -0.021 133.36 0.049

.|. | .|. | 109 -0.025 -0.008 133.77 0.054

.|. | .|. | 110 -0.002 0.001 133.77 0.061

.|. | .|. | 111 0.007 0.064 133.8 0.069

.|. | .|. | 112 0 -0.032 133.8 0.078

.|. | .|. | 113 0.008 -0.025 133.85 0.088

.|. | .|. | 114 0 -0.025 133.85 0.099

*|. | .|. | 115 -0.06 -0.006 136.3 0.085

.|. | .|. | 116 0.028 0.048 136.82 0.091

.|. | .|. | 117 0.017 -0.038 137.02 0.1

.|. | *|. | 118 -0.056 -0.088 139.18 0.089

.|. | .|. | 119 -0.05 -0.019 140.93 0.083

.|. | .|. | 120 -0.042 -0.029 142.12 0.082

.|. | .|. | 121 0.041 0.029 143.28 0.082

.|. | .|. | 122 0.001 -0.007 143.28 0.091

.|. | .|. | 123 -0.01 0.009 143.36 0.101

.|. | .|. | 124 0.008 0.011 143.4 0.112

.|. | *|. | 125 -0.056 -0.058 145.61 0.1

.|. | .|. | 126 0.025 0.012 146.05 0.107

.|. | .|. | 127 -0.005 0.009 146.07 0.118

.|. | .|. | 128 -0.034 -0.002 146.91 0.121

.|. | .|. | 129 0.043 0.038 148.2 0.119

.|. | .|. | 130 0.017 -0.04 148.41 0.129

.|. | .|. | 131 0.033 0.03 149.17 0.132


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49

.|. | .|. | 132 -0.006 -0.027 149.19 0.146

.|. | .|. | 133 -0.041 -0.025 150.42 0.143

.|. | .|. | 134 0.014 0.036 150.55 0.156

.|. | .|. | 135 -0.053 -0.03 152.6 0.143

.|. | .|. | 136 -0.009 -0.001 152.65 0.156

.|. | .|. | 137 -0.035 -0.025 153.53 0.158

.|. | .|. | 138 0.015 0.024 153.7 0.171

.|. | .|. | 139 0.064 0.054 156.67 0.145

.|. | .|. | 140 -0.007 -0.002 156.7 0.159

.|. | .|. | 141 -0.023 0.017 157.08 0.168

.|* | .|. | 142 0.069 0.036 160.6 0.136

.|. | .|. | 143 0.027 -0.007 161.12 0.143

.|. | .|. | 144 0.012 0.031 161.23 0.155

.|. | .|. | 145 -0.039 -0.033 162.34 0.154

.|. | .|. | 146 -0.021 0.025 162.66 0.164

.|. | .|. | 147 0.06 0.03 165.28 0.144

.|. | .|. | 148 0.023 0.007 165.68 0.152

.|. | .|. | 149 -0.01 -0.056 165.76 0.165

.|. | .|. | 150 -0.035 -0.027 166.7 0.166


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BIBLIOGRAPHY

BOOKS

1. Basic Econometrics: By Damodar N. Gujrati

2. Introductory Econometrics: By Ramu Ramanathan

WEBSITES

1. www.nseindia.com

2. www.yahoofinance.com

ECONOMETRICS SOFTWARE PACKAGES

1. Eviews

2. SPSS


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REFERANCES

Dr. S S S Kumar; Associate Professor, Indian Institute of Management, Kozhikode

(2006), Forecasting Volatility Evidence from Indian Stock and Forex Markets

Akgiray, V (1982), Conditional Heteroscedasticity in Time Series of Stock

Returns: Evidence and Forecasts, Journal of Business, Vol. 62, pp. 55-80.

Engle, R F (1982), Autoregressive Conditional Heteroscedasticity with Estimates

of the Variance of United Kingdom Inflation, Econometrica, Vol. 50, pp. 987-1007.

Perry, P (1982), The Time-Variance Relationship of Security Returns: Implication

for the Return-Generating Stochastic Process, Journal of Finance, Vol. 37, pp. 857-

70.

Ser-Huang Poon; Clive W. J. Granger (Jun. 2003) Forecasting Volatility in

Financial Markets: A Review, Journal of Economic Literature, Vol. 41, No. 2, pp.

478-539.

Peter F. Christoffersen; Francis X. Diebold Optimal Prediction under Asymmetric

Loss, Econometric Theory, Vol. 13, No. 6. (Dec., 1997), pp. 808-817.

volatility forecasting techniques in the indian capital market

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