FOUR FACTOR PRICING MODEL IN INDIAN EQUITY MARKET

PROJECT REPORT

Rohit Dhanda

2748

Under guidance of

Dr. V.K Vasal PH.D. (Accounting)(Internal Supervisor)

&

CA Aarsh DuaManager, Corporate Finance

HCL Technologies Ltd.(External Supervisor)

Executive Summary

The aim of this project is to test the efficiency of Carhart four factor model in Indian equity market over the period of 3 years from April 2011-March 2014. The comparative performances of capital asset pricing model (CAPM), Fama-French three factor model, and Fama-French four factor model is also examined. Each of these three models is regressed on 5 different sets of portfolios, i.e., monthly excess returns of five portfolios each of size, B/M, momentum and six portfolios each of size-B/M and size-momentum are used as dependent variables in time-series regressions. Confronted with the excess returns of the portfolios, the Carhart 4 factor model outperforms both CAPM and Fama French three factor model based on the results of adjusted R-squared values and minimum pricing error in the models.

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Contents

Certificate

Acknowledgement

Executive Summary..................................................................................................................1

List of Tables.............................................................................................................................3

1. Introduction......................................................................................................................4

1.1 Objective.........................................................................................................................4

1.2 Project structure.............................................................................................................4

2. Literature review...............................................................................................................5

2.1 Theoretical Background..................................................................................................5

2.2 Research paper...............................................................................................................7

3. Data and Methodology.....................................................................................................8

3.1 Portfolios formation for calculating SMB, HML and WML..............................................8

3.2 Calculation of four factors (Rm – Rf, SMB, HML and WML)..............................................9

3.3 Calculation of four factors free from correlation..........................................................10

3.4 Portfolios formation for dependent variables of regression.........................................11

3.5 Regression analysis.......................................................................................................11

4. Findings and discussion...................................................................................................12

5. Summary and Conclusion................................................................................................18

Bibliography........................................................................................................................... 19

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List of Tables

Correlation matrix of original 4 factors 9Correlation matrix after auxiliary regression 9Regression summary for CAPM model 11Regression summary for Fama French 3 factor model 13Regression summary for Carhart 4 factor model 14Comparison of goodness of fit for all three models 15

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1. Introduction Asset pricing has always been one of the main areas of modern financial economics. It can be claimed that the introduction of capital asset pricing model (CAPM) by Sharpe (1964), Lintner (1965), and Black (1972) made a breakthrough in the area of financial economics. Even today, it is apparent that the CAPM is one of the most widely used models among academicians and practitioners. The fact that CAPM can be used in performance evaluation, estimating the cost of capital, selecting portfolios, and measuring abnormal returns etc., is one of the main reasons why this model is so much appreciated. Despite its popularity and success, since its introduction there have always been criticism, with claims that CAPM is not sufficient to explain the variations in excess returns. In line with this argument Fama and French (1992, 1993 and 1996) showed that there is a relationship between size and average return on one side, and B/M and average return on the other side. Moving from this claim, they laid the foundations of their three factor model by adding two more risk factors to CAPM. Fama-French model gained big importance in modern finance as CAPM. Carhart constructed his own 4 factor model (1997) using Fama and French 3 factor model and Jagadeesh and Titman’s (1993) one year momentum anomaly and found that adding momentum factor noticeably reduced pricing errors.

1.1 Objective The main aim of this project is to examine how well CAPM, Fama and French 3 factor model and Carhart 4 factor model capture average returns for portfolios formed on size, value, momentum, size-value and size-momentum. For examining the efficiency of these models, hypotheses to be tested are significance of intercept term, coefficients of all four factors and goodness of fit (adjusted R2) for all models.

1.2 Project structureIn line with this objective, the rest of the project is organized as follows: The next chapter gives some theoretical background in detail regarding the development of four factor model starting from the invention of CAPM. The second section gives main points of the international studies for Fama-French models and Carhart 4 factor model

The third chapter explains the data and methodology used in this project. Data collection, portfolio formation, and factor construction methods are described in detail.

Chapter four continues with regression results of CAPM, three factor and Carhart four factor model for all portfolios formed on basis of size, value, momentum, size-value and size momentum. In chapter five key findings are summarized.

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2. Literature review 2.1 Theoretical Background One of the main attempts of the financial economics has been to describe, predict or assess the relation between risk and return since 1950‟s. After Markowitz introduced his renowned and famous mean-variance model in 1952, many models were developed based on his theorem. One of the most important models based on his theorem was CAPM (Capital Asset Pricing Model) which was introduced by Sharpe (1964), Lintner (1965), and Black (1972). Since its introduction, it still continues to constitute one of the cornerstones of modern finance theory. It is widely used in performance evaluation, estimating the cost of capital, selecting portfolios, and measuring abnormal returns. To be able to comprehend CAPM better, we should examine some details about the development and assumptions of the model. In his paper „‟Capital Asset Prices: Theory of Market Equilibrium under Conditions of Risk‟‟ (1964), William Sharpe put forward an argument to construct a relation between average return and standard deviation. He claimed that in equilibrium there will be a simple linear relationship between the expected return and standard deviation of return for efficient combinations of risky assets. (Sharpe, 1964). This relationship was described by beta, which implied the systematic risk. Each individual asset or portfolio has a beta value, which shows the riskiness of that asset or portfolio relative to the riskiness of the market. In other words, this beta shows the level of responsiveness to the movements in market. The assumptions underlying CAPM are as follows:

1. All investors are single-period expected utility of terminal wealth maximizers who choose among alternative portfolios on the basis of mean and variance (or standard deviation) of return.

2. All investors can borrow or lend an unlimited amount at an exogenously given risk free rate of interest and there are no restrictions on short sales of any asset.

3. All investors have identical subjective estimates of the means, variances and covariance of return among all assets.

4. All assets are perfectly divisible and perfectly liquid, i.e., all assets are marketable and there are no transaction costs.

5. There are no taxes. 6. All investors are price takers. 7. The quantities of all assets are given (Jensen, 1972).

CAPM can be described by the following equation:

Expected Rate of Return:

r=R f+β (Rm−R f )

Where,

Rf is the risk-free interest rate is what an investor would expect to receive from a risk-free investment.

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β is a stock beta is used to mathematically describe the relationship between the movements of an individual stock versus the entire market. It is sensitivity of the expected excess asset returns to the expected excess market returns

Rm is the expected market return is the return the investor would expect to receive from a broad stock market indicator such as the S&P 500 Index, BSE 500 Index, NIFTY Index etc.

To explain the pricing anomalies not captured by CAPM, Fama French (1993), developed a three factor asset pricing model which states that the expected return on a portfolio in excess of the risk free rate is explained by the sensitivity of its return to three factors:

(i) the excess return on a broad market portfolio, (ii) the difference between the return on a portfolio of small stocks and the return on a

portfolio of big stocks (SMB) and (iii) the difference between the return on a portfolio of high-book-to-market stocks and

the return on a portfolio of low-book-to- market stocks (HML), where the last two are mimicking size and value factors respectively. They then added two factors to CAPM to reflect a portfolio's exposure to these two classes:

r=R f+β (Km−R f )+ βs . SMB+βv .HML+α

r is the portfolio's expected rate of return, Rf is the risk-free return rate, and Km is the return of the market portfolio. The "three factor" β is analogous to the classical β but not equal to it, since there are now two additional factors to do some of the work.

SMB stands for "Small [market capitalization] Minus Big" and

HML for "High [book-to-market ratio] Minus Low";

They measure the historic excess returns of small caps over big caps and of value stocks over growth stocks. These factors are calculated with combinations of portfolios composed by ranked stocks (B/M ranking, Cap ranking) and available historical market data.

The Carhart four-factor model is an extension of the Fama–French three-factor model including a momentum factor, also known in the industry as the MOM factor (momentum). Momentum in a stock is described as the tendency for the stock price to continue rising if it is going up and to continue declining if it is going down. The MOM can be calculated by subtracting the equal weighted average of the highest performing firms from the equal weighed average of the lowest performing firms, lagged one month (Carhart, 1997). A stock is showing momentum if its prior 12-month average of returns is positive.

EXRt=αc+βmkt EXM K T t+βHMLHM Lt+ βSMB SMBt+βUMDUMDt+e t

Where, EXRt is the monthly return to the asset of concern in excess of the monthly t-bill rate. We typically use these three models to adjust for risk. In each case, we regress the excess returns of the asset on an intercept (𝛼c) and some factors on the right hand side of the equation that attempt to control for market-wide risk factors. The right hand side risk factors are: the monthly return of the CRSP value-weighted index less the risk free rate (EXM KTt), monthly premium of the book-to-market factor (HML) the monthly premium of

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the size factor (SMB), and the monthly premium on winners minus losers (UMD) from Fama-French (1993) and Carhart (1997). SMB is a zero-investment portfolio that is long on small capitalization (cap) stocks and short on big cap stocks. Similarly, HML is a zero-investment portfolio that is long on high book-to-market (B/M) stocks and short on low B/M stocks, and UMD is a zero-cost portfolio that is long previous 12-month return winners and short previous 12-month loser stocks.

2.2 Research paperEugene F. Fama, Kenneth R. French (2011), “Size, value, and momentum in international stock returns”, Journal of Financial Economics

This paper examines international stock returns, with two goals. The first is to detail the size, value, and momentum patterns in average returns for developed markets. Their main contribution is evidence for size groups. Most prior work on international returns focuses on large stocks. Sample covers all size groups, and tiny stocks (microcaps) produce challenging results. Their second goal is to examine how well Fama and French 3 factor model and Carhart 4 factor model capture average returns for portfolios formed on size and value or size and momentum. They examined local versions of the models in which the explanatory returns (factors) and the returns to be explained are from the same region. For perspective on whether asset pricing is integrated across regions, they also examine models that use global factors to explain global and regional returns.

In our project, this paper is replicated on Indian equity market as here efficiency of traditional CAPM, Fama French 3 factor and Carhart 4 factor model is examined on portfolios formed on size, value, momentum, size-value, size-momentum.

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3. Data and Methodology

Following data has been collected from the ACE Equity database for the period of 2010-2014 of the BSE 500 companies

Market capitalization (year-end) Price/book value (year-end) Share price (month end closing price)

The above data is collected for 359 companies whose market capitalization and price to book values data is available on ACE Equity.

The data for risk-free rate (91 day T-bill) is collected from RBI website. The weekly 91 day T-bill data is converted to monthly data by taking average of all weeks for a month and then annual rate is converted to monthly rate of return by dividing by 12.

Market rate of return is calculated on monthly returns of BSE 500 closing values. The data is collected from BSE website.

The portfolios are made from April to March every year.

The methodology consists of 5 parts:

1. Portfolios formation for calculating SMB, HML and WML2. Calculation of Rm – Rf, SMB, HML and WML 3. Calculation of four factors free from correlation4. Portfolios formation for dependent variables of regression, i.e., size sorted, value

sorted, momentum sorted, size-value sorted and size-momentum sorted portfolios.5. Regression analysis.

3.1 Portfolios formation for calculating SMB, HML and WML

These factors are calculated by differently sorting all the companies on basis of size, value and 1 year returns.

Portfolio for calculating SMBAccording to the research paper by Fama-French, all the companies are divided into 2 groups, big and small, on the basis of median of year end market capitalization. The companies with market capitalization greater than median are termed as big companies and rest are termed as small companies.This process should be done for all the 3 years. Thus we get two portfolios of big and small on basis of each year market capitalization.

Portfolio for calculating HMLAccording to the research paper by Fama-French, all the companies are divided into 3 groups, high book to market, neutral and low book to market ratios, on the breakpoints

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of 30-40-30 percentiles of year end book to market ratios. Top 30%ile are high B/M companies, next 40%ile companies are neutral and last 30% are low B/M companies1.This process should be done for all the 3 years. Thus we get three portfolios of high, neutral and low B/M on basis of each year book to market ratios.

Portfolio for calculating WMLAll the companies are divided into 2 groups, winner and losers, on the basis of median of momentum returns for previous 1 year. These returns are calculated from previous year march to current year march. The companies with market capitalization greater than median are termed as winner companies and rest are termed as loser companies.This process should be done for all the 3 years. Thus we get different portfolios of winners and losers on basis of each year market capitalization.

3.2 Calculation of four factors2 (Rm – Rf, SMB, HML and WML)

Find the monthly returns for all the companies by using formula:R1 = (P1 – P0)/P0

Where R1 is return for 1st monthP1 is stock price for 1st monthP0 is stock price for month 0.

Thus using this formula monthly returns for all the companies are calculated.

For SMB calculation, average returns for all the small companies and large companies are calculated. The difference between average return of small companies and avg. return for large companies is SMB factor for each month.

SMB = Avg. return of small companies – Avg. return of large companies

For HML calculation, difference between average returns of high B/M ratios companies and low B/M ratio companies are calculated.

HML = Avg. return of high B/M companies – avg. return of low B/M companies Similarly, WML is difference between average return of winner companies and loser

companies.WML = Avg. return of winners – avg. return of losers

Market rate of return (RM) is calculated by using;

R1 = (P1 – P0)/P0

Where R1 is return for 1st month

1 Companies with negative book to market ratio are also included in low book to market companies for calculating HML.2 All of 4 factors are independent of each other, i.e., SMB is not impacted by value and momentum and similarly HML and WML depend only on value and momentum returns respectively.

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P1 is closing value of BSE 500 index for 1st monthP0 is closing value of BSE 500 index for month 0.Risk free rate of return (RF) is weekly value issued by RBI, so in order to calculate monthly value the average of all weekly values is taken for each month. Then these monthly values are divided by 12 for getting the monthly returns.

Rm – Rf is calculated by taking difference of Rm and Rf calculated from above steps.

Thus all these factors are calculated for the period of 3 years from April 2011 to March 2014. These are used in right hand side of regression as independent variable for portfolios formed on basis of size, value, momentum, size-value and size-momentum.

3.3 Calculation of four factors free from correlation The 4 factors calculated above are checked for correlation, as we are doing multivariate regression, thus in order to have data free from multi-collinearity, the correlation among these 4 factors should be minimum.

Correlation matrix of original 4 factors

rm-rf smb hml wmlrm-rf 1smb -0.0113279 1hml 0.53519232 -0.44375 1wml -0.7321066 -0.40423 -0.18781 1

Thus, initially there is high correlation among the factors.

In order to remove this, Auxiliary Regression3 is used. It is method of removing the correlation by regressing the dependent variable with independent variables and the residuals of regression are the parts of dependent variable not explained by independent variables. These residuals are then used in place of that dependent variable.

In this project, HML and WML are having high correlation with Rm - Rf and SMB. Thus both HML and WML are regressed with Rm - Rf and SMB and the residuals are taken. The correlation matrix after auxiliary regression is given below:

Correlation matrix after auxiliary regression

rm-rf smb resid hml resid wmlrm-rf 1smb -0.01133 1resid hml -1.5E-16 -2.2E-16 1resid wml 1.04E-16 -7.7E-17 0.05983677 1

3 Auxiliary regression concept is taken from Gujarati Damodar N. Basic Econometrics [Book]: Tata McGraw-Hill, 2007.

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Hence now all values are very small except -0.01133 and 0.0598. These values are tested for 5% level of significance and t test is performed.

H0 : The correlation coefficient is insignificant for SMB and Rm - Rf.

H1 : The correlation coefficient is insignificant for SMB and Rm - Rf.

Since t critical for two tailed test is 2.03 for 34 degrees of freedom and t value for coefficient is t= -0.066. Thus, |t statistic| < t criticalHence, null hypothesis is not rejected and correlation coefficient is insignificant.

H0 : The correlation coefficient is insignificant for resid HML and resid WML.

H1 : The correlation coefficient is insignificant for resid HML and resid WML.

Since t critical for two tailed test is 2.03 for 34 degrees of freedom and t value for coefficient is t = 0.349. Thus, |t statistic| < t criticalHence, null hypothesis is not rejected and correlation coefficient is insignificant.Hence these 4 factors are free from correlation and thus regression results from these will be free from multi-collinearity.

3.4 Portfolios formation for dependent variables of regression

The 5 portfolios are made for each of size, value and momentum based on equal percentile of all portfolios.2*3 portfolios on basis of size- value and size momentum are made where size is divided by median, value and momentum by 30%ile – 70%ile.Six portfolio made from size – value are big-high b/m, big-neutral, big-low b/m, small-high b/m, small-neutral, small-low b/m.Similarly 6 portfolios made from size-momentum are; winner-big, neutral-big, loser-big, winner-small, neutral-small and loser-small. The average returns for each portfolio calculated separately. The average returns for each of them are used as dependent variable for regression with the 4 factors.

3.5 Regression analysisThe excess returns of portfolios sorted on size, value, momentum, size-value and size momentum are regressed with Rm-Rf (excess market return) to get the results for capital asset pricing model.

For Fama and French 3 factor model testing, excess returns of portfolios are regressed with Rm-Rf (excess market return), SMB (small minus big) and HML (high book to market minus low book to market).

For testing efficiency of Carhart 4 factor model, the excess returns of portfolios are regressed with all 4 factors namely, Rm-Rf (excess market return), SMB (small minus big), HML (high book to market minus low book to market) and WML (winners minus losers).

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The results of regression of regression are discussed in next section.

4. Findings and discussion

From the regression of average returns for all portfolios, the following hypotheses are tested:

H0: The intercept term is insignificant.H1: The intercept term is significant and thus there is pricing error.

With 5% level of significance, the portfolios with p value of intercept less than 0.05 are having insignificant intercept and thus have pricing errors. Here pricing error is found in small (size) portfolio, winner portfolios in momentum sorted, winner-small and winner-big portfolio in size-momentum sorted portfolios.

H0: The coefficient RM-Rf is insignificant.H1: The coefficient RM-Rf is significant.

Coefficient of RM-Rf is significant in all the case, as the p value is zero in all cases. This means that market factor explains the average portfolio returns. Although market factor is significant but it is unable to explain size effect as there has been no substantial difference between beta coefficient of small and large stock portfolios which indicates that market risk of small firms is not substantially larger than that of large firms.The market model results show that the intercept value is low for the high B/M portfolio as compared to the low B/M portfolio, suggesting that low B/M stocks generate higher CAPM based risk adjusted extra normal returns during the study period.

Regression summary for CAPM model

p values

Portfolio intercept (a)

coefficient Rm-Rf (b)

(a) (b) Adj. R square

size sorted

Big 8.64E-05 1.181 0.9642 0 0.954p2 0.00019 1.263 0.9508 0 0.908p3 0.002017 1.245 0.715 0 0.846p4 0.000401 1.362 0.9492 0 0.865Small 0.02754 1.138 0.0001 0 0.693

value (B/M) sorted

High b/m 0.005922 1.656 0.3284 0 0.854P2 0.001995 1.365 0.7024 0 0.831P3 0.006016 1.270 0.2679 0 0.866

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P4 0.006256 1.003 0.0604 0 0.884low b/m 0.018999 0.836 0.0607 0 0.292

momentum sorted

Winner 0.013667 0.941 0.0024 0 0.762p2 0.008861 0.838 0.0144 0 0.859p3 0.001263 1.134 0.7729 0 0.874p4 0.011387 1.291 0.2311 0 0.484Loser 0.004033 1.927 0.5793 0 0.856

size b/m sorted

big - high b/m -0.004944 1.737 0.381 0 0.838big - neutral 0.000633 1.357 0.850 0 0.919big - low b/m 0.003479 0.862 0.262 0 0.867small - high b/m 0.008296 1.493 0.206 0 0.831small - neutral 0.010911 1.073 0.113 0 0.739small - low b/m 0.034784 0.961 0.051 0 0.136

size momentum

sorted

winner - big 0.007 0.893 0.0376 0 0.850neutral - big -0.002 1.111 0.5336 0 0.924loser - big -0.002 1.864 0.6488 0 0.853winner - small 0.019 0.937 0.0049 0 0.702neutral - small 0.008 1.101 0.2451 0 0.770loser- small 0.020 1.584 0.0504 0 0.518

For Fama and French 3 factor model, the results obtained are as follows;

Apart from previous 2 hypotheses, 2 new hypotheses are tested;

H0: The coefficient SMB is insignificant.H1: The coefficient SMB is significant.

SMB is significant factor in asset pricing under 5% level of significance. Although it is insignificant for big stocks and loser stocks. Coefficient of SMB increases as we move from big to small stocks. SMB increases as we move from high b/m stocks to low b/m stocks. These patterns can also be seen in size-sorted portfolios.

H0: The coefficient HML is insignificant.H1: The coefficient HML is significant.

HML coefficient is significant for all portfolios except winner-big portfolio of size-momentum sorted and big-low B/M of size-value sorted. HML coefficient decreases as we move from high b/m to low b/m in value sorted portfolios and similar pattern seen in size-value sorted portfolios.

Thus these factors are significant and adjusted R2 is also increased which implies Fama and French 3 factor model is superior to capital asset pricing model.

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Regression summary for Fama French 3 factor model

Portfoliointercept

(a)coeffi cient Rm-Rf (b)

smb (s) hml (h) (a) (b) (s) (h)

Big 0.003278 1.212 -0.218 0.198 0.1023 0 0.0845 0.0005p2 -0.000593 1.256 0.053 0.128 0.89 0 0.8037 0.298p3 -0.000479 1.221 0.170 0.361 0.92 0 0.3553 0p4 -0.004864 1.358 0.450 0.366 0.1246 0 0.0001 0Small -0.011833 0.625 3.279 -1.078 0.1969 0 0.0001 0.0241

High b/m 0.002968 1.658 0.205 0.620 0.2985 0 0.0526 0P2 -0.001344 1.367 0.231 0.473 0.6792 0 0.0685 0P3 0.00163 1.272 0.304 0.343 0.6286 0 0 0P4 0.005392 1.004 0.060 0.205 0.1384 0 0.3868 0.0001low b/m -0.004122 0.847 1.603 -0.705 0.3886 0 0 0

Winner 0.010084 0.943 0.248 0.177 0.0241 0 0.0127 0.0012p2 0.006942 0.838 0.133 0.182 0.0138 0 0.0048 0.0002p3 -0.001755 1.135 0.209 0.283 0.5045 0 0.0004 0p4 -0.012435 1.302 1.651 -0.419 0.0446 0 0.0001 0.0283Loser 0.001396 1.928 0.183 0.704 0.7484 0 0.1907 0

big - high b/m -0.002391 1.736 -0.177 0.484 0.6447 0 0.4591 0.0012big - neutral 6.24E-05 1.357 0.039 0.263 0.9824 0 0.7329 0.0002big - low b/m 0.00332 0.862 0.011 -0.038 0.3454 0 0.8242 0.2343small - high b/m 0.003643 1.496 0.322 0.603 0.1756 0 0.0003 0small - neutral 0.003548 1.077 0.510 0.468 0.242 0 0 0small - low b/m -0.00874 0.981 3.017 -1.090 0.1525 0 0 0

winner - big 0.0063 0.893 0.049 0.016 0.086 0 0.3993 0.6775neutral - big -0.0018 1.111 0.012 0.185 0.405 0 0.8711 0.0001loser - big -0.0009 1.863 -0.101 0.443 0.855 0 0.6873 0.0046winner - small 0.0128 0.940 0.432 0.343 0.027 0 0.0008 0neutral - small 0.0029 1.104 0.363 0.449 0.372 0 0.0001 0loser- small -0.0110 1.599 2.121 -0.272 0.170 0 0 0.1552

size sorted

value (B/M) sorted

momentum sorted

size b/m sorted

size momentum

sorted

p values

Next we regressed excess portfolio returns to four factors of Carhart 4 factor model, adding WML (winner minus loser) factor and the results obtained are as follows;

The hypotheses tested are similar to previous regressions and the new hypothesis to be tested is;

H0: The coefficient WML is insignificant.H1: The coefficient WML is significant.

It is insignificant for winner-big, neutral-big, big-low B/M and small-high B/M portfolios and for rest it is significant. The WML factor changes from positive to negative values as we move from winner to loser portfolios.

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Regression summary for Carhart 4 factor model

Portfoliointercept

(a)Rm-Rf

(b)smb (s) hml (v) wml(m) (a) (b) (s) (v) (m)

Big 0.0026 1.21 -0.17 0.19 -0.24 0.13 0 0.061 0 0.003p2 -0.0014 1.25 0.11 0.12 -0.30 0.74 0 0.455 0.07 0.002p3 -0.0007 1.22 0.19 0.36 -0.09 0.88 0 0.271 0 0.306p4 -0.0084 1.28 0.60 0.30 -0.01 0.03 0 0.000 0 0.837Small -0.0142 0.60 3.44 -1.09 -0.90 0.05 0 0 0 0.018

High b/m 0.0030 1.66 0.20 0.63 -0.31 0.26 0 0 0 0.005P2 -0.0013 1.37 0.23 0.49 -0.39 0.66 0 0 0 0.007P3 0.0016 1.27 0.30 0.34 0.05 0.64 0 0 0 0.676P4 0.0054 1.00 0.06 0.20 0.07 0.13 0 0.405 0 0.465low b/m -0.0041 0.85 1.60 -0.69 -0.60 0.28 0 0 0 0

Winner 0.0136 0.84 0.19 0.12 0.37 0.00 0 0.035 0.02 0.004p2 0.0069 0.84 0.13 0.17 0.25 0.01 0 0.004 0 0.003p3 -0.0018 1.14 0.21 0.28 0.04 0.51 0 0.001 0 0.697p4 -0.0124 1.30 1.65 -0.38 -1.33 0.00 0 0 0 0Loser 0.0014 1.93 0.18 0.72 -0.47 0.70 0 0.006 0 0.004

big - high b/m -0.0024 1.74 -0.18 0.51 -0.85 0.62 0 0.0095 0 0big - neutral 0.0001 1.36 0.04 0.27 -0.31 0.98 0 0.512 0 0.0007big - low b/m 0.0033 0.86 0.01 -0.04 0.14 0.33 0 0.832 0.17 0.262small - high b/m 0.0036 1.50 0.32 0.61 -0.14 0.18 0 0 0 0.130small - neutral 0.0035 1.08 0.51 0.46 0.23 0.20 0 0 0 0.087small - low b/m -0.0087 0.98 3.02 -1.05 -1.19 0.04 0 0 0 0

winner - big 0.0063 0.893 0.049 0.011 0.152 0.091 0 0.427 0.80 0.300neutral - big -0.0018 1.111 0.012 0.189 -0.138 0.410 0 0.814 0 0.100loser - big -0.0009 1.863 -0.100 0.471 -0.932 0.837 0 0.123 0 0.000winner - small 0.0128 0.940 0.431 0.329 0.453 0.005 0 0 0 0.001neutral - small 0.0029 1.104 0.363 0.441 0.265 0.346 0 0 0 0.036loser- small -0.0110 1.599 2.121 -0.230 -1.354 0.003 0 0 0 0.000

p values

size - momentum

sorted

momentum sorted

value (B/M) sorted

size sorted

size b/m sorted

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Comparison of goodness of fit for all three models

Portfolio CAPMFama French 3 factor model

Carhart 4 factor model

Big 0.954 0.967 0.976p2 0.908 0.913 0.923p3 0.846 0.922 0.920p4 0.865 0.962 0.964Small 0.693 0.762 0.820

High b/m 0.854 0.961 0.968P2 0.831 0.923 0.937P3 0.866 0.936 0.934P4 0.884 0.912 0.910low b/m 0.292 0.906 0.943

Winner 0.762 0.794 0.818p2 0.859 0.896 0.911p3 0.874 0.928 0.926p4 0.484 0.825 0.951Loser 0.856 0.956 0.968

big - high b/m 0.838 0.892 0.938big - neutral 0.919 0.946 0.956big - low b/m 0.867 0.860 0.861small - high b/m 0.831 0.962 0.962small - neutral 0.739 0.920 0.926small - low b/m 0.136 0.904 0.961

winner - big 0.850 0.842 0.843neutral - big 0.924 0.943 0.944loser - big 0.853 0.889 0.939winner - small 0.702 0.831 0.867neutral - small 0.770 0.909 0.917loser- small 0.518 0.881 0.973

size sorted

value (B/M) sorted

momentum sorted

size b/m sorted

size - momentum

sorted

adjusted R square

It is observed that as compared to big stocks, small stocks have larger unexplained portion as they have small adjusted R2.

Thus from significance of SMB (small minus big), HML (high minus low), WML (winner minus loser) and higher adjusted R2 it is evident that Carhart 4 factor model is better model than Fama and French 3 factor model and Capital asset pricing model (CAPM).

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Size effect

CAPM results show that the extra normal returns (after adjusting for market risk) is 2.7% per month for small stock and 0.008% per month for large stock portfolios. Small stock portfolios earn statistically significantly positive extra risk adjusted returns confirming the size effect.

There has been no substantial difference between beta coefficient of small and large stock portfolios which indicates that market risk of small firms is not substantially larger than that of large firms. This is the reason why CAPM fails to explain size effect.

Adjusted R2 is low for small stock portfolios vis-a-vis large stocks showing that the portfolios of small stocks have a very large unexplained variation in their returns. Fama-French 4 factor regressions show that both SMB coefficients are significant.

However these factors only partially explain the size effect, as the small size portfolio still provides an abnormal return of -1.4% per month which is statistically significant. Thus FF 4 factor model fails for small companies.

Value effect

The market model results show that the intercept value is low for the low B/M portfolio as compared to the high B/M portfolio, suggesting that low P/B stocks generate higher CAPM based risk adjusted extra normal returns during the study period. However, CAPM is unable to absorb cross sectional differences on value sorted portfolios.

The h coefficient is negative (-0.686) for low BE/ME and positive (0.629) for high BE/ME confirming the presence of value effect. The value effect is very high and is very significant in explaining the portfolios.

Momentum effect

CAPM results show that intercepts for winner portfolios are statistically significant. My findings confirm that market factor does not explain momentum. This could be attributed to the fact that there is very small difference in betas of the corner portfolios

The intercept of the winner portfolio is significant and provides an abnormal return of 1.367% per month. The FF model fails to capture momentum owing to the fact that loser portfolio tends to load more heavily on value factor compared to winners portfolio which is in contrast to risk theory.

Winner portfolio should have comprised of more distressed high B/M stocks for providing a risk explanation. So winner stocks are growth stocks. Further there is no significant difference between the sensitivity of winner and loser portfolios to the size factor. WML factor is significant under 5% level of significance in Indian market returns.

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5. Summary and Conclusion

In this project we examine the efficiency of three pricing models, Capital asset pricing model, Fama and French 3 factor model and Carhart 4 factor model in explaining the average returns of the different portfolios formed on the basis of size, value, momentum, size-value and size-momentum for Indian companies listed on BSE 500 index.

From the findings, since WML (winner minus loser) factor in Carhart 4 factor model is coming to be statistically significant and coefficients of other three factors for this model are also statistically comparable to those from other 2 models, thus it is concluded that Carhart 4 factor model is better than CAPM and Fama and French 3 factor model.

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Bibliography

Carhart Mark M. (1997), “On Persistence in Mutual Fund Performance”, the Journal of Finance

Sehgal Sanjay, Subramaniam Srividya, and Laurence Porteu De La Morandiere (2012), “A Search for Rational Sources of Stock Return Anomalies: Evidence from India”, International Journal of Economics and Finance Vol. 4

Eugene F. Fama, Kenneth R. French (2011), “Size, value, and momentum in international stock returns”, Journal of Financial Economics, SciVerse Science Direct website

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