Submitted By:

Rakesh Das…………. 139

Sheweta Gaggar……. 147

Shikha Jain…………. 148

Tanmay Gupta……… 155

ECONOMETRICS PROJECT Estimation of β & Factors

Determining Stock Return

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EXECUTIVE SUMMARY

The main aim of the report is to estimate the Security Characteristic Line for two different

stocks listed on the Bombay Stock Exchange. One of the stocks is Reliance Industries and the

other Tata Consultancy Services. The report shows how the beta, which is an estimate of

returns on a stock when the market changes by a unit percentage, of each stock can be

calculated. Historical beta is calculated by regression of the stock return and market return for

a particular time period unit. The analysis takes weekly data for a 52 week period, starting 2nd

March, 2009 to 29th

Feb, 20010.

The findings of the report can be used by equity analysts to predict the returns on RIL stock

based on the movement of the SENSEX. This value of β can also be used to predict the cost

of equity using the CAPM model. The stock price of Reliance Industries moves in the same

direction by 1.0878%, when the market index moves by 1%. The value of β for Reliance

Industries as published by the Bombay Stock Exchange is 1.18 while the value obtained by

our model in 1.0878% which is in congruence with the literature value to quite an extend.

.71/.47

On the other hand we took into consideration another stock namely TCS. We took into

consideration numerous independent variables which affect the TCS stock returns, starting

from the sensex index then incorporating the industry segment by including the returns on

TCS followed by incorporating the exchange rate, the FII inflows, the inflation and with each

of these the model went on improving as elaborated underneath.

Thus the project basically tracks the movement of these stocks and the factors influencing

them i.e. the sensitivity of the stocks, in a nutshell.

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Table of Contents

EXECUTIVE SUMMARY ....................................................................................................... 2

INTRODUCTION ..................................................................................................................... 5

Beta ........................................................................................................................................ 5

Advantages of Beta ................................................................................................................ 6

Disadvantages of Beta............................................................................................................ 6

REVIEW OF LITERATURE .................................................................................................... 8

ECONOMIC MODEL ............................................................................................................... 9

METHODOLOGY .................................................................................................................. 10

Collection of Data ................................................................................................................ 10

Estimating Beta .................................................................................................................... 10

Tests performed ................................................................................................................... 10

OLS ...................................................................................................................................... 11

AUTOCORRELATION ...................................................................................................... 12

STATIONARITY ................................................................................................................ 13

Unit root test .................................................................................................................... 14

HETEROSKEDASTICITY ................................................................................................. 14

Goldfeld–Quandt Test for Heteroscedasticity ................................................................. 15

THE ELASTICITY OF THE PRICE OF A STOCK AND ITS BETA .............................. 15

Problems with Beta .......................................................................................................... 15

Elasticity of price of a stock ............................................................................................. 16

Theorem ........................................................................................................................... 16

Illustrations ...................................................................................................................... 17

ANALYSIS & RESULTS ....................................................................................................... 19

TEST FOR STATIONARITY ............................................................................................. 19

Analysis of the independent variable Sensex................................................................... 23

OLS & AUTOCORRELATION.......................................................................................... 24

OLS using EViews ........................................................................................................... 24

Checking for AR(1) Autocorrelation using excel : .......................................................... 25

Estimating ρ using excel .................................................................................................. 25

Eliminating AR(1) Auto Correlation ............................................................................... 26

TEST FOR HETEROSKEDASTICTY ............................................................................... 27

ELASTICITY MODEL FOR STOCK BETA ..................................................................... 30

ANALYSIS FOR TCS:........................................................................................................ 31

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Results after Inclusion of the INFOSYS stock returns .................................................... 32

Results after inclusion of the FII inflows component: ..................................................... 32

Results after inclusion of the Foreign Exchange Rate factor ........................................... 33

Results after the inclusion of the Inflation parameter ...................................................... 33

Testing for AR (1) Autocorrelation: ................................................................................ 34

OLS using EViews ........................................................................................................... 34

Limitations ....................................................................................................................... 35

INFERENCES ......................................................................................................................... 37

Interpretation of β of Reliance Industries ............................................................................ 37

Factors that determine the change in returns of the TCS stock ........................................... 37

REFERENCES ........................................................................................................................ 39

APPENIDX .............................................................................................................................. 40

Estimation of ρ using Excel ................................................................................................. 40

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INTRODUCTION

For a financial analyst, one of the main concerns in analyzing a security for investment is the

riskiness of the security relative to the overall market. One way of measuring the volatility of

a specific stock is to compare the return on the stock versus that of the entire market. The

comparison of a stock's return to the market return is referred to as the beta (β) of the stock.

Beta

Beta is a measure of a stock's volatility in relation to

the market. By definition, the market has a beta of 1.0,

and individual stocks are ranked according to how

much they deviate from the market. One calculates

beta using regression analysis.

• A beta of 1 indicates that the stock's price will move

with the market.

• A beta greater than 1 indicates that the security's

price will be more volatile than the market.

• A beta less than 1 means that it will be less volatile than the market.

Thus, beta is the tendency of a stock's returns to respond to swings in the market. For

example, if a stock's beta is 1.2 it's theoretically 20% more volatile than the market.

High-beta stocks are supposed to be riskier but provide a potential for higher returns; low-

beta stocks pose less risk but also lower returns. For an investor, a stock‟s β is an indication

of the returns that the investor must expect from the stock. Higher the β, higher is the

expected return.

Beta is a key component for the capital asset pricing model (CAPM), which is used to

calculate cost of equity. The cost of capital represents the discount rate used to arrive at the

present value of a company's future cash flows. All things being equal, the higher a

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company's beta is, the higher its cost of capital discount rate. The higher the discount rate, the

lower the present value placed on the company's future cash flows. In short, beta can impact a

company's share valuation.

Advantages of Beta

To followers of CAPM, beta is a useful measure. A stock's price variability is important to

consider when assessing risk. Indeed, if you think about risk as the possibility of a stock

losing its value, beta has appeal as a proxy for risk.

Besides, beta offers a clear, quantifiable measure, which makes it easy to work with. Sure,

there are variations on beta depending on things such as the market index used and the time

period measured, but broadly speaking, the notion of beta is fairly straightforward. It's a

convenient measure that can be used to calculate the costs of equity used in a valuation

method that discounts cash flows.

Disadvantages of Beta

However, if you are investing in a stock's fundamentals, beta has plenty of shortcomings.

For starters, beta doesn't incorporate new information. Also at the same time, many

technology stocks, such as Google, are so new to the market they have insufficient price

history to establish a reliable beta.

Another troubling factor is that past price movements are very poor predictors of the future.

Betas are merely rear-view mirrors, reflecting very little of what lies ahead.

Furthermore, the beta measure on a single stock tends to flip around over time, which makes

it unreliable. Granted, for traders looking to buy and sell stocks within short time periods,

beta is a fairly good risk metric. But for investors with long-term horizons, it's less useful.

Also, the trouble is that beta, as a proxy for risk, doesn't distinguish between upside and

downside price movements. For most investors, downside movements are risk while upside

ones mean opportunity. Beta doesn't help investors tell the difference. For most investors,

that doesn't make much sense.

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Since beta is evaluated as the covariance between the stock returns and index returns, scaled

down by the variance of the index returns and the index volatility is time-varying , thus beta

is not constant over a period of time and thus the inefficiency of the CAPM for estimating the

expected returns using beta. The constancy nature of beta raises doubts about the suitability

of using it as a measure of the sensitivity of the security‟s return corresponding to market

returns.

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REVIEW OF LITERATURE

Estimating the β of a listed security is a routine practice in equity research. Research firms

like Bloomberg International always quote the historical β of a security in their forecasts.

OLS is the generally used technique to estimate the β of a stock. The following are a set of

references that were used to arrive at our topic:

The Elasticity of the price of a stock and its Beta – Cyriac Anthony & E.S. Jeevanand

Stock Beta & Volatility – http://www.money-zine.com/Investing/Stocks/Stock-Beta-and-

Volatility/

Performing online search, we narrowed on two companies. One, with a large market

capitalization and the other, with a small market capitalization. Reliance Industries Limited

has a market capitalization of 14.84%, while Tata Consultancy Services is 1.78%.1

We also found the already calculated beta values on RIL and TCS for the one year period

(Mar 09 to Feb 10). Below are the values found:

TCS= 0.71

RIL= 1.182

1 Source: http://www.bseindia.com/about/abindices/betavalues.asp#

2 Source: same as above

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ECONOMIC MODEL

In estimating the β value of a stock i that is listed on the SENSEX using the Ordinary Least

Square Method (OLS), the following economic model will be used:

y = f(x)

where, y is the return of the stock i and x is the market return.

The straight line showing the relationship between the rate of return of a stock and the rate of

market return is known as the Security’s Characteristic Line (SCL). The SCL is given by

the following equation:

y = βx + α

The slope of the characteristic line β is called the security‟s beta. This measures the

systematic risk of the stock.

Alpha, α, the intercept of the y-axis, measures the excess return (above the expected return)

of the stock.

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METHODOLOGY

Collection of Data

We chose TCS and RIL as the two companies that are actively traded on the BSE SENSEX.

To estimate β, a 1-year historical time series data of the week closing price of RIL and TCS

and the week closing value of the SENSEX was collected3. The time-series starts from March

2, 2009 to Feb 28, 2010 Using this data, the return on the stock (the dependent or Y variable)

was regressed against the return on the market (the independent or X variable).

After collecting the data, since the values given were absolute, returns on the stock and the

sensex were calculated. So,

Return on RIL stock on March 9, 2009 would be

Price of RIL stock, this week – Price of RIL stock, previous week

Similarly, returns on RIL stock were calculated for each period and the same was followed to

find returns on TCS stock and the sensex. The tests were performed for the data thus got.

Estimating Beta

To estimate Beta, we used the list of returns for the two stock and returns for the sensex. For

our analysis, we took weekly returns. Next, with the help of Eviews as well as excel, with the

sensex returns as the x variable(independent) and the stock returns as the y

variable(dependent), the equation of the line was found. The slope of the fitted line from the

linear least-squares calculation is the estimated Beta. The y-intercept is the alpha.

Tests performed

1. OLS for each stock to get regression lines

3 Source: http://www.bseindia.com/histdata/hindices.asp

=

Price of RIL stock, previous week i.e March 2, 2009

* 100

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2. Check for auto-correlation

3. Unit Root test : Test for stationarity

4. Test for checking heteroskedasticity

5. Other tests

a. Elasticity of price of stock

For TCS, after arriving at the OLS regression coefficients by using independent variable as

the returns on sensex and dependent variable as returns on TCS stock, the model was

improved on by adding more variables that the return on TCS stock could be dependent on.

These include the industry sentiment, exchange rate, inflation and the FII inflows.

OLS

The least square method is used to compute estimations of parameters and to fit data. The

ordinary least square method is widely used to find or estimate the numerical values of the

parameters to fit a function to a set of data and to characterize the statistical properties of

estimates. In the standard formulation, a set of N pairs of observations {Yi,Xi} is used to find

a function giving the value of the dependent variable (Y ) from the values of an independent

variable (X). With one variable and a linear function, the prediction is given by the following

equation:

Y = a + bX (1)

This equation involves two free parameters which specify the intercept (a) and the slope (b)

of the regression line. The least square method defines the estimate of these parameters as the

values which minimize the sum of the squares (hence the name least squares) between the

measurements and the model (i.e., the predicted values). This amounts to minimizing the

expression:

(where E stands for “error” which is the quantity to be minimized).

Taking the derivative of with respect to a and b and setting them to zero gives the

following set of equations (called the normal equations):

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Solving these 2 equations gives the least square estimates of a and b as:

OLS can be extended to more than one independent variable (using matrix algebra) and to

non-linear functions.

AUTOCORRELATION

In statistics, the autocorrelation function (ACF) of a random process describes the correlation

between the process at different points in time. Let Xt be the value of the process at time t

(where t may be an integer for a discrete-time process or a real number for a continuous-time

process). If Xt has mean μ and variance σ2 then the definition of the ACF is

where, E is the expected value operator.

If the function R is well-defined its value must lie in the range [−1, 1], with 1 indicating

perfect correlation and −1 indicating perfect anti-correlation.

In regression analysis using time series data, autocorrelation of the residuals ("error terms", in

econometrics) is a problem. Autocorrelation violates the OLS assumption that the error terms

are uncorrelated. While it does not bias the OLS coefficient estimates, the standard errors

tend to be underestimated (and the t-scores overestimated).

The traditional test for the presence of first-order autocorrelation is the Durbin–Watson

statistic, computed from the residuals.

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Most regression applications calculate it automatically and present it as one of the standard

regression diagnostics.

STATIONARITY

A common assumption in many time series techniques is that the data are stationary.

A stationary process has the property that the mean, variance and autocorrelation structure do

not change over time. Stationarity can be simply defined as a flat looking series, without

trend, constant variance over time, a constant autocorrelation structure over time and no

periodic fluctuations (seasonality).

If the time series is not stationary, we can often transform it to stationarity with one of the

following techniques.

1. We can difference the data. That is, given the series Zt, we create the new series

The differenced data will contain one less point than the original data. Although you

can difference the data more than once, one difference is usually sufficient.

2. If the data contain a trend, we can fit some type of curve to the data and then model

the residuals from that fit. Since the purpose of the fit is to simply remove long term

trend, a simple fit, such as a straight line, is typically used.

3. For non-constant variance, taking the logarithm or square root of the series may

stabilize the variance. For negative data, you can add a suitable constant to make all

the data positive before applying the transformation. This constant can then be

subtracted from the model to obtain predicted (i.e., the fitted) values and forecasts for

future points.

The above techniques are intended to generate series with constant location and scale.

Although seasonality also violates stationarity, this is usually explicitly incorporated into the

time series model.

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Unit root test

A unit root test tests whether a time series variable is non-stationary using an autoregressive

model. The most famous test is the Augmented Dickey-Fuller test.

A simple AR(1) model is yt = ρyt − 1 + ut, where yt is the variable of interest, t is the time

index, ρ is a coefficient, and ut is the error term. A unit root is present if | ρ | = 1. The model

would be non-stationary in this case. Naturally it would be even more non-stationary if |ρ|>1

The regression model can be written as Δyt = (ρ − 1)yt − 1 + ut = δyt − 1 + ut,

This model can be estimated and testing for a unit root is equivalent to testing δ = 0. Since the

test is done over the residual term rather than raw data, it is not possible to use standard t-

distribution to as critical values. Therefore this statistic τ has a specific distribution simply

known as the Dickey Fuller table.

ADF estimates a complex model and more general which adds a time trend.

HETEROSKEDASTICITY

In statistics, a sequence or a vector of random variables is heteroskedastic if the random

variables have different variances. The complementary concept is called homoskedasticity.

When using some statistical techniques, such as ordinary least squares (OLS), a number of

assumptions are typically made. One of these is that the error term has a constant variance.

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This will be true if the observations of the error term are assumed to be drawn from identical

distributions. Heteroskedasticity is a violation of this assumption.

For example, the error term could vary or increase with each observation, something that is

often the case with cross-sectional or time series measurements. Heteroskedasticity is often

studied as part of econometrics, which frequently deals with data exhibiting it.

Goldfeld–Quandt Test for Heteroscedasticity

1. Order the observations according to the magnitude of the X thought to be related.

2. Divide observations into two groups, one with low values of X and one with high,

omitting some central observations.

3. Run two separate regressions

4. Calculate F test

5. Should be unity for homoskedasticity.

THE ELASTICITY OF THE PRICE OF A STOCK AND ITS BETA

Estimating the expected return on investments to be made in the stock market is a challenging

job before an ordinary investor. Different market models and techniques are being used for

taking suitable investment decisions. The past behavior of the price of a security and the

share price index plays a very important role in security analysis. The straight line showing

the relationship between the rate of return of a security and the rate of market return is known

as the security‟s characteristic line. The slope of the characteristic line is called the security‟s

beta. The concept of beta introduced by Markowitz(1959) is being widely used to measure

the systematic risk involved in an investment. Ordinary least square (OLS) method is used by

researchers and practitioners for estimating the characteristic line. However Beta is unable is

to explain the stock sensitivity entirely and the concept of elasticity needs to be introduced.

Problems with Beta

As beta is the slope of a straight line, it is always a constant. However another school of

thought believes that beta is time varying. Black (1976), for instance linked beta to leverage

which changes owing to changes in the stock price. Mandelker & Rhee (1984) related beta to

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decisions by the firm and thus a varying measure. The relationship between macro-economic

variables and the firm‟s beta points to the varying character of beta. Since beta is evaluated as

the covariance between the stock returns and index returns, scaled down by the variance of

the index returns and the index volatility is time-varying , thus beta is not constant over a

period of time and thus the inefficiency of the CAPM for estimating the expected returns

using beta. The constancy nature of beta raises doubts about the suitability of using it as a

measure of the sensitivity of the security‟s return corresponding to market returns. Hence the

need for a concept that reflects instantaneous changes of the market is necessary. To measure

the market sensitivity, the concept of elasticity is thus more useful.

Elasticity of price of a stock

The term „elasticity‟ is a technical term used mainly by economists to describe the degree of

responsiveness of the endogenous variable in an economic model with respect to the changes

in the exogenous variable of the model. It measures the percentage change in the endogenous

variable when the exogenous variable is increased or decreased by 1 %. So the concept of

elasticity will be useful to measure the sensitivity of the price of a stock corresponding to

market movements. If Y = f (X) is the functional relationship between X and Y, then the

elasticity of Y with respect to X is given by

η =(X/Y)*(Dy/DX)

Theorem

The elasticity of price of a security with respect to price index is a constant „k‟ if and only if

the relationship between the price Y of the security and the price index X is of the form

Y = C (X^k) where C > 0, k > 0……………….(i)

On integration of both sides,

log Y = k log X + log C

Also we know that

Y = a + b X……………………….……………(ii)

Therefore using (i) and (II)

We get that

η = b X/(a + b X)

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Note that the value of η is not a constant. It depends on the value of X. So, η varies when X

varies. This means that the sensitivity of the price of a stock is not the same at all levels of the

index. Further, the value of η depends on both the parameters a and b.

Illustrations

Case (i). η = 1.

This is the case when the price

return of a stock is the same as

that of the market return. This

means that the price of a

security increases (decreases)

by 1 % when the share price

index increases (decreases) by

1 %. In this case, a = 0. Since

the intercept is zero, the

regression line passes through

the origin.

Case (2). η >1.

This is the case when the price return of a stock is more than proportional to market return.

Since the price of a stock and the market index are generally positively correlated, the slope

of the regression line „b‟ is positive. Therefore, η >1 only if „a‟ is negative.

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Case (3). η <1.

This is the case when the price return of a stock is less than proportional to market return.

That is, the price of the security increases (decreases) by less than1 % when the share price

index increases (decreases) by 1%. Also, η <1 only if „a‟ is positive

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ANALYSIS & RESULTS

TEST FOR STATIONARITY

From the definition of a random process, we know that all random processes are composed of

random variables, each at its own unique point in time. Because of this, random processes

have all the properties of random variables, such as mean, correlation, variances, etc.. When

dealing with groups of signals or sequences it will be important for us to be able to show

whether of not these statistical properties hold true for the entire random process. To do this,

the concept of stationary processes has been developed. The general definition of a stationary

process is a random process where all of its statistical properties do not vary with time

.Processes whose statistical properties do change are referred to as nonstationary.

Understanding the basic idea of stationarity will help you to be able to follow the more

concrete and mathematical definition to follow. Also, we will look at various levels of

stationarity used to describe the various types of stationarity characteristics a random process

can have.

A stationary time series is one whose mean,variance and auto correlation do not change

over time.

A non stationary time series is one whose mean,variance and autocorrelation may vary with

time.

Unit Root Test: This is the test used to determine whether a given variable is stationary or

not. This model can be estimated and testing for a unit root is equivalent to testing δ = 0.

Since the test is done over the residual term rather than raw data, it is not possible to use

standard t-distribution to as critical values. Therefore this statistic τ has a specific distribution

simply known as the Dickey Fuller table.

The null hypothesis is: Process is nonstationary

Alternative hypothesis is defined as: Process is stationary

Analysis for the dependent variable i.e. RIL returns

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The critical value for 1percent is coming out to be -4.15.

After the unit root test was performed the t statistic for the augmented dickey fuller test

statistic came out to be -854 which was much lower than -4.15.

Null hypothesis is rejected.

Hence it is concluded that the percent change for stock price for RIL is stationary data.

Analysis for the dependent variably i.e. TCS stock returns:

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Critical value for one percent is -4.18 and the t statistic for the augmented dickey fuller test

statistic was found out to be -6.54.

Conclusion: Percent change in stock price for TCS is stationary.

This is as expected as the mean ,variance of the percent change in the stock price is assumed

to follow a trend.

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Analysis of the independent variable Sensex

Critical value -4.15 and the t statistic for the augmented dickey fuller test statistic -9.2

The value of -9.2 is much lower than the minimum range of -4.15.Thus it can be inferred that

the movement of sensex over time assumes a pattern in which the variance and mean of the

index of 30 blue chip companies does not change over time.

Hence The percentage change in price movement of sensex is highly stationary.

Note: since the dependent variable i.e. the RIL stock returns and the TCS stock returns are

stationary in nature so immaterial of whether the other independent variables are stationary or

non-stationary there shall exist no co-integration and hence the no need for the same

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OLS & AUTOCORRELATION

The Percentage Return on RIL stock was regressed against the percentage return on the

SENSEX index using Ordinary Least Squares method.

OLS using EViews

The EViews output for the regression is shown below.

The regressed relation from EViews is given by:

Y = 1.0567*X + 0.006177

Where Y - %return on RIL stock

X - %return on the SENSEX index

The Beta of the RIL stock obtained from this equation is 1.0567. Before we interpret this β,

we need to test the time series data for autocorrelation.

The value of R-squared is 0.7338 which means the 73.8% of the variation in the return on the

RIL stock is explained by the estimated equation.

The value of the Durbin-Watson statistic for the regression is :

DW statistic = 2.4196

In order to check for positive or negative autocorrelation, we need to find out the critical

values for the DW statistic. The critical value depends on the number of elements in the

sample and the number of independent variables. From The Durbin-Watson table of critical

values, the critical values are given by:

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dL = 1.50

dU = 1.59

The critical values for negative autocorrelation are obtained by doing 4-dL and 4-dU.

Based on the above values, we can identify autocorrelation using the following rules

The calculated DW value of 2.4196 is close to the indeterminate range towards Negative

autocorrelation.

Checking for AR(1) Autocorrelation using excel :

In order to graphically check for autocorrelation, a plot of the residual with respect to time

was obtained using excel.

The residual plot shows that the residuals are largely random. However, for some residual

values, there is a negative correlation with the previous value.

Estimating ρ using excel

Positive

Autocorrelation

Negative

Autocorrelation No Autocorrelation Indeter-

minate

Indeter-

minate

0 2 4 1.5 1.59 2.41 2.5

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The autocorrelation can be modeled using the following relation

et = ρ*et-1 + vt

On regressing et on et-1 we get the following relation

et = -0.21*et-1 - 8E-05

Hence, ρ = -0.2

A ρ of -0.2 indicates weak negative autocorrelation

The figure below shows a plot of et on et-1. This shows evidence of weak negative

autocorrelation. If there was no autocorrelation, the value of ρ would have been 0 and the

scatter of the residuals would be totally random.

Eliminating AR(1) Auto Correlation

AR(1) auto correlation can be modeled as

Yt = β1 + β2*Xt + ut

ut = ρ*ut-1 + vt

In order to eliminate autocorrelation, we substitute the residual equation in the equation for y

to obtain

the following equation

ttttt XXYY 12211 )1(

β1, β2 and ρ are estimated using EViews by adding AR(1) as a variable. The following is the

OLS output given by EViews:

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From the above result, the coefficients can be estimated as :

β1 = C = 0.006078

β2 = Coefficient of X = 1.0878

ρ = AR(1) = -0.2215

The estimate of ρ obtained from EViews is very close to the estimate of ρ obtained earlier

using excel (-0.2)

Using the above estimates we can construct the new equation as :

Yt = 0.0074 – 0.22Yt-1 + 1.0878Xt + 0.2393Xt-1

Where,

Yt - % Return on RIL stock

Xt - % Return on SENSEX

The value of DW stat after adding AR(1) is 2.009661. This means that the new equation does

not have any autocorrelation.

Using this equation the estimated β of Reliance Industries is 1.0878.

βRelianceIndustries = 1.0878

TEST FOR HETEROSKEDASTICTY

Heteroskedasticity does not cause the OLS coefficient estimates to be biased, but it results in

an increase in the variance of the Beta distribution. Thereby, the OLS underestimates true

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variance and overestimates t-statistics in case of heteroskedasticity. Sometimes it also results

in making insignificant variables appear to be statistically significant.

There are several methods to detect heteroskedasticity. To detect heteroskedasticity in the

estimated OLS coefficient, we used the Goldfeld-Quandt test. The returns on the stock of RIL

were taken.

The data for RIL returns was divided into two groups, one with 20 low values of returns on

RIL stock and the other with 20 high values of return on RIL stock. The 13 central

observations were omitted.

The null hypothesis taken is

Two separate regressions were run on Eviews.

RSS1 = 73.60489

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Now,

where, n1 ,n2 are the number of observations in the lower and upper regressions.

k is the number of parameters (independent variables) in the model.

Here, k = 1

n1 = n2 = 20

Therefore,

= 1.676563

The given F values at significance levels 1%, 5%, 10% are as below

F(19,19) critical 1% 3.03

F(19,19) critical 5% 2.17

RSS2 = 123.4032

123.4032 / 20-1

73.60489 / 20-1

F =

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F(19,19) critical 10% 1.82

Thus, it can be seen that the calculated F statistic is less than the given F values. Thus we

accept the null hypothesis that the variances are equal.

The data for returns on RIL stock is homoskedastic.

Since the data is homoskedastic, no remedial measures are necessary.

ELASTICITY MODEL FOR STOCK BETA

We had log RIL as the dependent variable and log sensex (i.e. the ril stock prices and the

sensex prices) the independent variable. The equation obtained is:

Log ril= log sensex*1.732649 – 3.978453

As seen above the R-squared value is quite high of the order 97.8% meaning that 97.8% of

the percentage variation in y that is being explained by the model.

So the value of elasticity is 1.732649 henceforth meaning thereby that if the value of x

changes by 1 % then y changes by 1.73%. It should be noted that elasticity depends on x and

varying x we ll get different values of the elasticity.

Page 31 of 42

ANALYSIS FOR TCS:

In the first case we took into consideration the return on the sensex as the independent

variable and the return on TCS stocks as the dependent variable. Now we ran an OLS and

found the regression equation for the same and the results are as stated underneath:

The value of R^2 is .215494 which is quite low and that means we require more number of

variables to explain the movement of the TCS returns and we incorporated the returns on

INFOSYS stocks to take into account the general industry sentiment which will obviously

affect the stock prices and hence the likelihood of an increased R^2 value. Also it should be

noted that the value of the Durbin Watson stat is 2.131054 which indicates that there exists

no auto-correlation.

Page 32 of 42

Results after Inclusion of the INFOSYS stock returns The E-views result table is as indicated below:

As seen from above the R-squared value has increased significantly to 41.6076% which is

quite n improvement and suggests that our model has improved further. To further improve

upon this model we include the variable FII inflows because it is likely to effect the stock

movement

Results after inclusion of the FII inflows component:

Page 33 of 42

As seen the value of the R-squared has further improved to 43.2977% which is an

improvement from the previous value indicating that the model has further improved. To

further improve upon the model we incorporate another variable namely Exchange rate

movements because it is likely to affect an export oriented industry like IT.

Results after inclusion of the Foreign Exchange Rate factor

As seen the value of the R-squared improves marginally to 43.3016% which is a slight

improvement from the previous value. To further improve the model we incorporate another

variable namely the inflation rate that ll again affect the stock prices.

Results after the inclusion of the Inflation parameter

Page 34 of 42

As seen the value of R- squared has again improved marginally to 43.3021% which is a

marginal improvement in the model.

Testing for AR (1) Autocorrelation: The Log of RIL stock was regressed against the log of the SENSEX index using Ordinary

Least Squares method.

OLS using EViews The EViews output for the regression is shown below.

The regressed relation from EViews is given by :

LogY = 1.0567*LogX + 0.006177

Where Y - RIL stock price

X - SENSEX index

The elasticity of the RIL stock obtained from this equation is 1.732649. Before we interpret

this , we need to test the time series data for autocorrelation.

The value of R-squared is .978035 which means the 97.8% of the variation in the RIL stock

is explained by the estimated equation.

The value of the Durbin-Watson statistic for the regression is:

DW statistic = .821234

As seen from this value a strong positive autocorrelation exists and the need for removing the

same. After performing AR1 the output is as illustrated below where the value of the Durbin-

Watson stat has improved to 2.405771 and hence the auto correlation removed.

Page 35 of 42

The equation finally obtained is as stated underneath:

ρ = 0.9510

β1 = -1.035478

β2 = 1.0452

Log(RIL) = -0.05073 + 0.951Log(Ril)t-1 + 1.0451Log(Sensex)t - 0.9939Log(Sensex)Xt-1

Limitations

One thing worth noting is the fact that the impact on the TCS stock returns of the various

parameters namely the sensex returns, the Infosys returns, the exchange rate, the inflation rate

and the FII inflows could be much more marked but because we have taken the data for 53

weeks from 2nd

March, 2007 to 29TH

Feb, 2008 the influence is not so significant. Also some

factors that cannot be quantified do affect the stock prices like the existing market sentiment,

internal decisions of the company, foreign deals made( especially important in export

intensive companies like TCS) etc. Thus the model fairly explains the TCs stock movements

taking into account all the above stated parameters.

Validating our model:

In order to validate the goodness of our model we considered the returns for the RIL stock

and the returns on the sensex for approximately the past 20 days. We compared the value

Page 36 of 42

obtained from our model to the value that was provided from the literature and found the

error term which was quite insignificant hence validating the goodness of our model. Then

we plotted the actual values and the fitted values on the same se of axis and obtained the

following scatter diagram.

The scatter diagram above shows that the fitted values are close to the actual values and

hence the estimated equation is a good fit

Page 37 of 42

INFERENCES

The aim of the project was to estimate the Security Characteristic Line for 2 different stocks

listed on the Bombay Stock Exchange. One of the stocks was Reliance Industries and the

other was Tata Consultancy Services.

The time series analysis for RIL included tests for stationarity, autocorrelation and

Hetereoskedasticity. Both the series were found to be stationary in nature using the unit root

test. One of the reasons could be that the series were derived by finding the first difference

between the stock prices and the SENSEX figures. The final equation obtained for the RIL

stock does not have any autocorrelation and is Homoskedastic in nature.

Interpretation of β of Reliance Industries

βRelianceIndustries is a measure of the systematic risk of the Reliance Industries stock. It means

that when the market index moves by 1%, the stock price of Reliance Industries also moves

in the same direction by 1.0878%. The value of β for Reliance Industries as published by the

Bombay Stock Exchange is 1.18. The value of β can be used by equity analysts to predict the

returns on RIL stock based on the movement of the SENSEX. This value of β can also be

used to predict the cost of equity using the CAPM model.

βRelianceIndustries is however not a stable estimate of the sensitivity of the RIL stock. Since beta

is evaluated as the covariance between the stock returns and index returns, scaled down by

the variance of the index returns and the index volatility is time-varying, beta is not constant

over a period of time. A different measure, the elasticity of the stock η would be a better

measure of sensitivity. To find η, we regressed the log of RIL stock prices on the log of

SENSEX index figures. The slope of the regression line so obtained is the elasticity

coefficient η. Elasticity of the stock indicates the % change in stock price for a 1% change in

the index. Elasticity depends on the value of the index. Hence, it means that the sensitivity of

the price of the stock is not same at different levels of the index.

Factors that determine the change in returns of the TCS stock

The exercise to estimate βTCS was similar to the one used for estimating βRelianceIndustries.

However, the R2 value obtained in this case was only 0.2153. In order to increase the value of

R2, we explored adding several other independent variables. The first one that we added was

Page 38 of 42

the return on INFOSYS stock during the same period. The idea behind adding this variable

was to find out if the performance of the IT sector in general had an influence on the return of

the TCS stock. After adding the Return on INFOSYS stock, the R2 value improved by 0.21 to

0.42. This shows that the return on the TCS stock is influenced by the performance of other

companies in the IT sector. To improve the R2 value further, other variables like FII inflows,

Dollar-rupee exchange rate and the domestic inflation rate were added to the model one by

one. However, there was no significant improvement in value of R2. This implies that the

return on stock is determined by many factors other than the market and industry

performance. These factors could include company specific factors like the number of

deals/projects in the pipeline or certain qualitative factors like market sentiments. The time

frame used for our analysis was one year. Certain factors like exchange rate fluctuation or FII

inflows may not have a significant impact during the period of one year. A larger time frame

might be needed to evaluate the impact of these factors.

Page 39 of 42

REFERENCES

1. http://www.investopedia.com/articles/stocks/04/113004.asp

2. http://www.bseindia.com/histdata/hindices.asp

3. http://www.wikipedia.com

4. Basic econometrics by Damodar N. Gujarati & Sangeetha, Fourth Edition

5. The Elasticity of the price of a stock and its Beta – Cyriac Anthony & E.S. Jeevanand

6. Stock Beta & Volatility – http://www.money-zine.com/Investing/Stocks/Stock-Beta-

and-Volatility/

Page 40 of 42

APPENIDX

Estimation of ρ using Excel

x Estimated y e(t) e(t-1)

e(t) - e(t-1)

(e(t)-e(t-1)) ^ 2 e(t)^2

-0.0001 0.0061 -

0.0052 0.0000

0.0032 -0.0311 0.0169 -

0.0052 0.0221 0.0005 0.0003

-0.0573 0.0789 -

0.0178 0.0169 -0.0347 0.0012 0.0003

0.0174 -0.0108 0.0029 -

0.0178 0.0207 0.0004 0.0000

0.0119 -0.0113 0.0034 0.0029 0.0005 0.0000 0.0000

-0.0075 0.0496 -

0.0105 0.0034 -0.0139 0.0002 0.0001

0.0109 0.0467 0.0459 -

0.0105 0.0563 0.0032 0.0021

0.0598 0.0070 -

0.0090 0.0459 -0.0548 0.0030 0.0001

0.0502 0.0081 0.0207 -

0.0090 0.0296 0.0009 0.0004

0.0101 -0.0043 0.0092 0.0207 -0.0115 0.0001 0.0001

-0.0091 0.0450 0.0236 0.0092 0.0145 0.0002 0.0006

-0.0008 0.0088 0.0073 0.0236 -0.0163 0.0003 0.0001

-0.0449 0.0233 -

0.0096 0.0073 -0.0169 0.0003 0.0001

0.0070 -0.0306 -

0.0209 -

0.0096 -0.0113 0.0001 0.0004

-0.0454 0.0136 -

0.0016 -

0.0209 0.0193 0.0004 0.0000

0.0055 0.0289 -

0.0146 -

0.0016 -0.0130 0.0002 0.0002

0.0568 0.0196 -

0.0217 -

0.0146 -0.0071 0.0001 0.0005

0.0714 0.0288 -

0.0223 -

0.0217 -0.0005 0.0000 0.0005

0.0538 0.0280 0.0054 -

0.0223 0.0277 0.0008 0.0000

-0.0179 0.0264 0.0405 0.0054 0.0351 0.0012 0.0016

0.0543 -0.0163 0.0054 0.0405 -0.0351 0.0012 0.0000

-0.0798 -0.0005 -

0.0344 0.0054 -0.0399 0.0016 0.0012

Page 41 of 42

0.0320 -0.0127 0.0180 -

0.0344 0.0524 0.0027 0.0003

-0.0190 -0.0455 0.0135 0.0180 -0.0045 0.0000 0.0002

0.0008 0.0273 -

0.0150 0.0135 -0.0285 0.0008 0.0002

0.0687 0.0717 0.0326 -

0.0150 0.0476 0.0023 0.0011

-0.0144 0.0249 -

0.0240 0.0326 -0.0566 0.0032 0.0006

0.0495 0.0071 0.0302 -

0.0240 0.0541 0.0029 0.0009

0.0314 0.0712 0.0467 0.0302 0.0165 0.0003 0.0022

0.0190 0.0525 -

0.0430 0.0467 -0.0897 0.0080 0.0018

-0.0231 0.0356 0.0461 -

0.0430 0.0891 0.0079 0.0021

0.0247 0.0446 -

0.0112 0.0461 -0.0573 0.0033 0.0001

0.0153 -0.0431 0.0051 -

0.0112 0.0163 0.0003 0.0000

-0.0668 0.1075 -

0.0172 0.0051 -0.0222 0.0005 0.0003

0.0322 0.0464 -

0.0388 -

0.0172 -0.0217 0.0005 0.0015

0.0042 -0.0504 0.0588 -

0.0388 0.0977 0.0095 0.0035

0.0024 0.0504 0.0008 0.0588 -0.0580 0.0034 0.0000

0.0145 -0.0392 0.0168 0.0008 0.0160 0.0003 0.0003

0.0054 0.0348 -

0.0208 0.0168 -0.0376 0.0014 0.0004

0.0139 0.0391 -

0.0423 -

0.0208 -0.0215 0.0005 0.0018

0.0157 0.0096 0.0071 -

0.0423 0.0493 0.0024 0.0001

-0.0190 -0.0396 -

0.0207 0.0071 -0.0278 0.0008 0.0004

-0.0345 0.0638 0.0039 -

0.0207 0.0246 0.0006 0.0000

0.0340 0.0313 -

0.0011 0.0039 -0.0049 0.0000 0.0000

-0.0369 0.0134 0.0343 -

0.0011 0.0354 0.0013 0.0012

-0.0331 -0.0858 -

0.0192 0.0343 -0.0535 0.0029 0.0004

-0.0504 -0.0301 -

0.0378 -

0.0192 -0.0186 0.0003 0.0014

Page 42 of 42

-0.1125 -0.0007 -

0.0253 -

0.0378 0.0125 0.0002 0.0006

0.0132 -0.0389 -

0.0083 -

0.0253 0.0170 0.0003 0.0001

-0.0194 0.0455 0.0242 -

0.0083 0.0325 0.0011 0.0006

-0.0043 -0.0385 -

0.0235 0.0242 -0.0477 0.0023 0.0006

0.0129 0.0202 -

0.0085 -

0.0235 0.0150 0.0002 0.0001

0.0307 -0.0902 0.0050 -

0.0085 0.0135 0.0002 0.0000

SUM 0.076049339 0.031429526

LINEST Result 1.056732 0.006176575 0.089114 0.003436643 0.733845 0.024824689 140.6176 51

0.086658 0.031429526

Durbin-Watson test

d 2.419678244