Skewness preference and measurement of abnormal returns, a comparative evaluation of current vs proposed event study paradigm

Typically the market model linear characteristic line model or a variant o f it is used as the underlying return generating process in the computation o f event specific abnormal returns

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Evaluation of Current Vs Proposed Event Study Paradigm

Major: Interdepartmental Area o f Business (Finance)

Under the Supervision o f Professor John M Geppert

Lincoln, Nebraska

December, 2002

Copyright 2003 by Mishra, Suchismita

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Skewnes Preference and Measurement of Abnormal Returns: A Comparative

Fvalnainn of Curraftt v-s-gg°Posed Event Study-Earadigm

U N IV E R SIT Y ! O F

Suchismita Mishra, Ph.D.

University o f Nebraska, 2002

Advisor: John M Geppert

If asset returns have systematic skewness, expected returns should include rewards for accepting this risk Many recent empirical studies such as by Kraus and Litzenberger (1976), Sears and Wei (1988), Harvey and Sddique (2000) etc have shown that the pricing o f assets can be better explained by the three-moment capital asset pricing model that accounts for systematic skewness, rather than the traditional two-moment CAPM Event studies in finance are concerned with abnormal returns after removing an estimate o f the portion o f total return that represents the premium for bearing risk Till date event studies have used the return generating models that are consistent with some form o f the traditional capital asset pricing model Typically the market model (linear characteristic line model) or a variant o f it is used as the underlying return generating process in the computation o f event specific abnormal returns We investigate the possible implications o f recognizing skewness preference for event study by using the quadratic characteristic lines model (QCL), which is the return generating model consistent with the three moment CAPM We replicate the pioneer event study on stock split by Fama, Fisher, Jensen and Roll (FFJR) (1969) on a new data set using their methodology as well as other methodologies used by other event studies First we test the

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to the given data set With the market model we obtain the same intertemporal trend in the abnormal return as reported by FFJR Using QCL the same trend is maintained but the level o f the values o f the abnormal returns are statistically significantly different than that obtained using the market model.

I would like to begin by thanking my elder brother Dr Subhendu K Mishra and

my husband Mr Rohit Singh My elder brother is the reason I am here today with my Ph.D I honor my husband’s support and sacrifice for my success in the graduate school

He has done everything he could to take care o f me during the grueling years o f my doctoral study Rohit, I love you

I would like to thank my brother Dr Sandip K Mishra, who was there for me through out my education in India I must admit both my brothers are one o f the most wonderful gifts I have

To my Mom (Mrs, Kumudini Mishra) and Dad (Mr Nanda Kishore Mishra), a thanks is never enough to express my feeling o f gratitude towards you Your sacrifices made me successful and my Ph.D belongs to you

I would like to thank my dissertation committee Dr John Geppert, Dr Gordon Karels, Dr Manferd Peterson, Dr James Schmidt and Dr Richard Defusco To Dr Geppert my committee chair, thank you for your patience, guidance and support Dr Karels, you showed me how the world o f academics looks like You taught me how to become better scholar, by thinking critically and accepting challenges Dr Peterson you are the one responsible for my joining the finance Ph D program I can never thank you enough for that Dr Defusco, thank you for your guidance and I look forward to working with you Dr Schmidt, your in-depth teaching o f econometrics has enabled Ph.D

students like me to be able to conduct research In addition to the members o f my committee, I would like to thank all the faculty members in the finance department at UNL for your guidance and willingness to provide support in various issues concerning research I have immensely enjoyed my time at UNL and I must admit whenever I go back there, I feel like I am going back to a family

Looking back on my graduate school experience I realize any achievement in a life is not solely due to the personal effort o f the individual but also because o f the effort and support o f the people who stand by the person To all o f you, thanks for caring and thanks for helping me to achieve my goal

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Chapter 1 Introduction and Purpose o f Study

I Introduction 1

II Theoretical Justification and Background 3

Chapter 2 Literature Review 9

I Introduction 9

II A Short Description o f the Event Study 10

HI Model Specification 11

IV Empirical Application o f the Market Model and Specification Error Biases 14

V The Single Index Market Model Versus Multifactor Models: The Issue o f The Associated Pricing Mechanism 18

VI Beyond the Two-Moment CAPM 20

VII Recent Support for the Three-Moment CAPM 24

VIII Purpose o f This Study 28

Chapter 3 Methodological Framework 30

I A Brief Survey of Event Study Methodologies 30

II The FFJR Methodology 35

III Event Study with the QCL Model 38

IV Tests o f significance 38

V Comparison o f the QCL and the FFJR Market Model Results 40

VI Some Additional Graphical Comparisons 40

VII Suggested Preliminary Tests o f the Appropriateness o f the QCL Model 41

V m Data 43

Chapter 4 An Analytical Evaluation o f the Expected Results with the QCL Model 44

II The Information Content o f Stock Splits and the Signaling

Hypothesis 62III Analysis o f the Residuals for the Market Model

Chapter 1 Introduction and Purpose of Study

(i) Introduction

No theoretical development in finance had such a profound effect on the academics and professionals as the Capital Asset Pricing Model (CAPM) developed by Sharpe (1964), Lintner (1965) and its attendant return generating process (Markowitz, 1959) The CAPM evolved over time with improvements advanced by Black (1972) and Rubeinstein (1983) Furthermore, its (the CAPM’s) underlying return generating model (commonly referred to as the market model) also became a very useful tool in studying the abnormal performance o f securities and portfolios In this context it is appropriate to say that the market model and the development and use o f CAPM are intertwined

The original development of the CAPM is based on the assumption that investors make their decisions based solely on the expected rate o f return and the variance That is, implicitly it is assumed that the investor either has a quadratic utility function and the rate

o f return follows a normal probability distribution (see Tobin, 1958) or the investor’s utility function belongs to HARA (hyperbolic absolute risk version) class (Rubinstein, 1973) with cut-off point taken at the second power in the Taylor’s expansion o f the utility function However, the empirical findings o f Fisher and Lorie (1970) and Ibbotson and Sinquefield (1976) have shown that the rates o f return distributions are skewed to the right Also, Friend and Blume (1975), using the data provided by the Internal Revenue Service, empirically showed that the investor’s utility function does belong to the HARA class These empirical findings seem to negate not only the assumption o f normal

probability distribution but also the choice o f second power as the cut-off point in the expansion o f the utility function These apparent discrepancies in the theory and empirical findings prompted Arditti (1971), Jean (1973) and others to argue that some improvement or modifications are needed in the original two moment asset pricing paradigm

Kraus and Litzenberger (1976) were able to successfully reconcile these apparent discrepancies in theory and empirical findings By incorporating the investor’s preference for positive skewness they developed the three moment CAPM Furthermore, they

empirically showed the three moment CAPM explained asset prices more adequately than the two moment CAPM did

In empirical applications such as the study o f the abnormal performances o f securities, the market model is used quite frequently as the return generating model (Peterson, 1989) The implicit assumption is that the assets are priced by the two moment CAPM However, if the empirical findings show that the assets are priced by the three moment CAPM then the underlying return generating process should be consistent with this asset pricing model Since, the underlying return generating process for the three moment CAPM is the quadratic characteristic lines model1, then the abnormal

performance should also be measured using this model If this is the case, it will be interesting to empirically examine whether the abnormal returns obtained using the quadratic characteristic lines model will result in an outcome different than the one obtained using the market model in event studies in finance Note that throughout in this dissertation we use the term "event study" to delineate only those studies where assets are assumed to be priced by theoretically developed equilibrium pricing model and the

abnormal returns pertaining to an event is measured by a return generating process commensurate with the assumed equilibrium asset pricing model2 To achieve this objective we replicate the original study by Fama, Fisher, Jensen and Roll (FFJR, 1969)

on stock splits with the market model (that they used) as well as with the quadratic

1 The QCL is given by

t ~Rft = CQi + Cu(Rmt - Rj } ) + C2i(.Rmt - R m)~ + e it where Rit, R^ and Rml are the rates o f return on the security i, the risk-free rate, and the market rate o f return Coi,C u and C2Il-sate the regression parameters.

1 Even the simplest o f all the market adjusted returns model where the ex post abnormal return on any security i is given by the difference between its return and that on the market portfolio is based on the

assumption that securities have a systematic risk o f unity (Brown and Warner, 1980)

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characteristic lines model The results we obtained for the market model using our data set are strikingly similar to the original FFJR3 study But the results using the quadratic characteristic lines model are statistically significantly different than the one obtained using the market model.

In what follows we provide the theoretical justification and background o f our approach and the organization o f this dissertation

(ii) Theoretical Justification and Background

The Sharpe (1964) and Lintner (1965) capital asset pricing model (CAPM) based

on two parameters (mean and variance) has numerous applications in the finance literature: performance measurement, tests o f security market efficiency etc The two- moment CAPM assumes that the underlying return generating process is the single index model This single index represents the entire economy (the market), hence the more

popular name for the single index model is the market model.

Tests o f the CAPM are based on the assumption that the market model is the appropriate underlying return-generating process (Gibbons, 1982; Stambaugh, 1982;

Shanken, 1985) Ex ante both the CAPM and the market model are single period models

To test the validity o f the CAPM in ex post form, however, the CAPM is treated as a

cross sectional model (assuming in equilibrium all the firms in the economy are priced by

the CAPM for a given period) The market model in its ex post form is treated as a time

series model for every individual firm in the economy Thus, the CAPM is a cross- sectional model that seeks to explain the price o f an asset in terms o f the risk-free return available, the return on the market portfolio, and the beta (P ) factor or the relative

3 We intentionally use a different data set to check whether the findings o f FFJR (1969) still hold In other words we wanted to make sure that the findings are not just unique to the data set FFJR used.

volatility of the asset compared to the market The market model is thus supposed to provide a proxy for the estimate of this parameter.4

Several studies such as Blume and Friend (1970, 1973), Black, Jensen and Scholes (1972), Blume and Husick (1973), Fama and Macbeth (1973 ), Basu (1977), Reinganum (1981), Litzenberger and Ramaswamy (1979 ) and Banz (1981 ) etc have appeared in the literature regarding the validity o f the CAPM, but no single empirical work has provided evidence that the two-moment CAPM correctly represents the pricing

o f assets in the economy In general, all the above cited studies found for aggressive

stocks ( p > 1) the CAPM tends to underestimate the required rate o f return For

defensive stocks (/? < 1), the CAPM overestimates the required rate o f return (Copeland and Weston, 1992, Pg:215) To correct for this overestimation/underestimation problem, Kraus and Litzenberger (1976) argue that the investor’s risk preference should be

measured not only by the variance of the underlying stock, but by variance and a preference for positive skewness.5 Their study extends the Sharpe-Lintner two-moment CAPM by incorporating the effect o f skewness on valuation Kraus and Litzenberger empirically show that there is no underestimation or overestimation problem in asset valuation, as is the case with the two-moment CAPM The extension o f the two-moment CAPM to a three-moment CAPM incorporates the behavioral assumptions o f preference for positive skewness

Empirical studies by Kane (1982), Sears, and Wei (1988) and Harvey and Siddique (2000) also support a preference for positive skewness.6,7 Thus, if the three-

4 Note that estimated from the market model is an input in the security market line (SML) form o f the CAPM

5 Note that one o f the sufficient conditions for the two-moment CAPM to be valid is that investors base their choices on expected return and the variance o f the underlying rate o f return probability distribution In the case o f the three-moment CAPM, however, it is assumed that investors base their preferences on expected rate o f return, variance, and preference for positive skewness o f the underlying rate o f return probability distribution.

6 The two-moment CAPM can be obtained by assuming that an investor has a quadratic utility function and that the underlying rates o f return distribution is Gaussian (normal) The assumption o f normality can be

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moment CAPM is the appropriate pricing mechanism, the return-generating process must

be the quadratic characteristic lines model (QCL) and not the linear characteristic lines model (LCL) What would be the effect, if any, if one uses the QCL rather than the LCL

to compute abnormal returns in event studies? This is the research question we address in this dissertation In general, event studies assume that the two-moment CAPM is the appropriate pricing mechanism and that the correct model to ascertain the effect o f an event on the rates o f return is the LCL

In this study we examine whether the use o f the QCL will result in a different outcome than that obtained using the LCL Our study is performed in a comparative framework We select stock split as our event because Fama, Fisher, Jensen, and Roll’s (FFJR hereafter) (1969) seminal event study was done on stock splits To facilitate comparison, we use the logarithmic form o f the market model (as used by FFJR) on our data set Also, we repeat the FFJR event study methodology for selecting an event and

discarded if one assumes that the investor's utility function belongs to the HARA class o f utility functions Note that in the development o f the three-moment CAPM the assumption o f quadratic utility will not be valid by construction Therefore, here also it is assumed that the investor’s utility function belongs to the HARA class The only difference between the two moment and the three moment CAPM is that in the case

o f the earlier the expansion o f the utility function is done to the quadratic term, whereas in the case o f the former the expansion is carried out to the cubic term (Rubinstein, 1973).

7 As mentioned earlier, the expansion o f any of the utility functions from the HARA class by Taylor’s theorem can be carried out to any number of terms, as all are infinite series In the CAPM literature to date, however, the expansion is done either up to the quadratic (for the two-moment CAPM) or to the cubic levels (for the three-moment CAPM) As demonstrated by Rubinstein (1973) and Stephens and Proffitt (1991), it is easy to derive an n-moment (n>3) CAPM As Kraus and Litzenberger observe, there is no behavioral justification beyond three moments One cannot explain how an investor will view the fourth moment (kurtosis) or any o f the higher central moments.

estimation window.8 We will also empirically obtain and compare the results using widely practiced standard event study procedures.9

FFJR hypothesize that investors might interpret a stock split as a message about future changes in the firm’s expected cash flows (and therefore as a message about a change in dividends) A dividend increase shows the manager’s confidence about maintaining the firm’s cash flow at a higher level in the long run To test this hypothesis, their sample is divided into firms that increased their dividend beyond the average for the market after the split and those who paid lower dividends The FFJR results show that the stocks in the dividend increase class have positive abnormal returns, whereas the

cumulative average returns (CARS) for the split stocks with poor dividend performance

decline until about a year after the split This FFJR hypothesis is known as the signaling

hypothesis (Lakonishok and Lev, 1987) In this dissertation we examine whether this

hypothesis still holds when positive skewness preference is recognized and whether there

is a significant difference in the market reactions for the dividend increase and dividend decrease groups Will this hypothesis hold if QCL is used as the appropriate process to measure the abnormal returns? Broadly speaking we address this question in this dissertation

Kraus and Litzenberger (1976) also provide conditions under which the QCL10 will be the appropriate return-generating process compared to the LCL:11

8 For a detailed description o f the event studies procedures, see Chapter 3 FFJR eliminate the 30-month period surrounding the effective split date for firms subsequently announcing dividend decreases Only IS months preceding the split month for splits followed by dividend increases are eliminated Their event window is 60 months surrounding the split date both for the dividend increase and decrease class, thereby leading to an overlapping estimation and event window.

9 The more standard practice is to separate the event and the estimation window to eliminate any announcement effect on the parameter estimation o f the normal performance measure (Campbell, Lo, and MacKinley, 1997).

10 A detailed analytical examination o f these conditions is provided in Chapter 4 o f this dissertation.

11 The results are in terms o f CARs.

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(a) The parameter associated with the squared market term (in the deviation from mean form) is significantly different from zero, and

(b) The return on the market portfolio is asymmetrically distributed

In a preliminary check, we find that these conditions hold for our data set The parameter described above (in (a)), for each security in our sample is significantly different from zero and the return on the market is asymmetric as well

Having ascertained that the conditions for QCL hold, we find that the C ARs obtained using QCL dominate the CARs obtained using LCL in event time and CAR space for the dividend increase sub-sample Furthermore, the standardized abnormal returns for the QCL model are significantly different than those obtained using the LCL model For the dividend decrease group, the CARs for the FFJR model dominate the CARs for the QCL model The standardized abnormal returns for the QCL model also are significantly different than those o f the LCL model Neither the FFJR model nor the QCL paradigm reveals any statistically significant abnormal return for the dividend decrease group Using QCL we do find support for the dividend hypothesis12 o f the FFJR study Also the extent o f investor reaction obtained using QCL is statistically significantly different than that obtained using the FFJR methodology

Our study examines the validity o f the Kraus and Litzenebrger conditions for QCL to be the appropriate return generating process in an event study framework

According to their argument if these conditions hold then skewness preference exists and thus QCL and LCL results should be significantly different If these conditions do not hold, then the results o f LCL and QCL should be identical Thus our hypothesis of interest is if the Kraus and Litzenberger conditions are valid for our data set then the

12 FFJR argue that a large price increase at the time o f stock split is due to altered expectations concerning future dividends rather than to any intrinsic effects o f the splits themselves.

abnormal returns adjusted for skewness preference via QCL should be significantly different than the market model abnormal returns.

The rest o f this study is organized as follows Chapter 2 reviews the literature on the market model and its applications in event studies and the literature on the three- moment CAPM and the QCL model Chapter 3 describes the FFJR and standard event studies methodologies Chapter 4 develops a conceptual framework on the relationship between QCL and LCL The chapter explores the possibilities o f surmising a priori results that the QCL will obtain, given that we know the result that has been obtained using LCL Chapter 5 presents the empirical results Chapter 6 summarizes our findings and explores the possibilities for further research

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Chapter 2 Literature Review

In this dissertation we use the QCL model to repeat FFJR’s (1969) pioneering event study on stock splits The purpose o f this empirical study is twofold First, we test

if QCL is a better fitting return-generating model than the market model Then we compare the pattern o f abnormal returns using QCL with that o f the FFJR methodology Our intent is to see if the results using QCL are significantly different than the market model If we find the abnormal returns with QCL to be statistically significantly different, then further refinements in event study methodology may be needed For example, it may

be prudent to check data for the appropriate return-generating model before conducting

an event study

This chapter reviews the literature on the market model and its applications in event study techniques The chapter also examines the literature on the three-moment CAPM and the QCL model We focus our review on the empirical findings and

theoretical developments o f the three-moment CAPM done by Kraus and Litzenberger and further refinements and additions by others The theoretical underpinnings for the current research are found in the literature in the following areas:

(a) The Event Study,(b) Model Specification,(c) Empirical Application of the Market Model and Specification Error Biases,

(d) The Single Index Market Model Versus Multifactor Models: The Issue

o f The Associated Pricing Mechanism,(e) Beyond the Two-Moment CAPM,(f) Recent Support for the Three-Moment CAPM

(ii) A Short Description o f the Event Study

Event study methodology is a frequently used analytical tool in financial research The goal o f an event study is to determine whether security holders earn abnormal returns because o f specific corporate events such as merger announcements, earnings

announcements, stock splits, etc The excess return, or abnormal return (Peterson, 1989),

is the difference between the observed return and the predicted return assuming that returns stem from a given return-generating process Assuming that the participants exhibit rational behavior in the market place, the effect o f an event will be reflected immediately in asset prices The event’s economic impact can be measured using asset prices observed over a relatively short time period

Most event studies rely exclusively on Sharpe’s (1963) return-generating model The original market model and its various different specifications have been used to obtain abnormal returns in event studies We describe the original form and frequently used specifications o f the market model below

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(iii) Model Specification

The original return-generating process proposed by Markowitz (1959) and Sharpe

(1963) is a statistical hypothesis that states that the expected rate o f return on a stock i in

time period / is a linear function o f the expected rate o f return on a global market

portfolio The ex ante form o f the market model hypothesizes a stochastic linear process

that generates security returns and is given by:

where ctj and pi are the intercept and the slope, respectively, o f the straight line defined on

the [ E(Rmt), £(/?„) ] plane, and Rit and Rmt are the rates o f return on the iIh security and

the market portfolio during the /th time period

In ex ante!ex post form, the market model can be written as:

where the left side is the expected return on security i in period t, given the market rate o f return The testable ex post form o f the model may be written as:

where S.t is the random error term or the residual portion that is unexplained by the

regression of the z',h stock on the market rate o f return during the /th time period

The market model assumes that investors are single period, risk-averse, and that

they maximize the expected utility of terminal period wealth In its original ex ante form the market model is a single period model But in the ex post form it is assumed that the slope and intercept terms are constant over the time period so that estimates o f a and P

can be obtained The market model is a statistical model, but it can be theoretically related to the CAPM (Fama, 1976; Gibbons, 1982)

Equation (3) is a Type IV regression o f Rit on Rml that is used to empirically obtain estimates o f a and Pi (Press, 1972) In this simple linear regression model, the

independent variable is known and assumed to be nonstochastic But in the market

model, the independent variable is random, not nonstochastic Therefore, equation (3) is a regression equation o f Type IV;13 that is, equation (3) is an error-in-variables regression, not a Type I (the functional regression usually seen in statistical textbooks) regression Estimation o f alpha and beta, however, is treated as a simple linear regression equation o f Type I The stochastic random term is presumed to follow wide-sense stationarity

assumptions:

(a) E (eit) = 0(zero mean) (b) \a r(e it) = c r c (homoscedastic) (c) cov(£it,ei ti.k) = 0 for all Ar^Oand (d) cow(eu,R mt) = 0

The sits are not assumed to be independent; rather, they are assumed to be

uncorrelated—there is no linear relationship between any two error terms No assumption regarding the distribution o f error terms is made (Eventually the error terms are assumed

to be normally distributed to facilitate hypothesis testing.) The covariance between

Rm and s it is zero, market returns are observed without error The above assumptions

imply that the conditional expectation o f Rit given Rmt is E(Rit | Rmr) = a { + PtRmt, the ex

ante/ex post specification o f the model.

There are several re-specifications o f the market model that are frequently used in empirical research:

(a) The logarithmic form o f the price relatives is specified as:

loge rjt = a, + /?, loge lt +eit, (4)

where rJt and /, are the price relatives for security j and the equally weighted market portfolio For small values o f rJt and I, , the natural

logarithm o f the security price relative is approximately the rate o f return

13 Unlike the simple linear regression where the independent variable is assumed to be a known variable, in the case o f a Type IV regression the independent variable is a stochastic random variable Hence,

o f

If it were a Type I regression, <JJ would equal (Tei.

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in period t This specification o f the market model is used in FFJR’s

seminal event study

(b) The standard specification of the market model is:

n, = R» ~ Rf< and rmt =Rm - R fn

where Rfi is the risk free rate during the period t.

(d) Black’s specification of the market model argues that Rfi should be

replaced by R q , , where R0t is the return on a zero beta portfolio.

These re-specifications o f the market model assume that in equilibrium the correct underlying pricing mechanism is the SML o f the two-moment CAPM, i.e.:

The underlying return-generating process o f the CAPM is specified by one o f the above four return-generating processes The market model and CAPM will be equal if the following conditions hold:

(c) The investment horizons in both the models are the same, and all sufficient conditions o f the CAPM hold.

Fama (1973) and Subrahmanyan and Stapleton (1983) report these conditions Fama (1973) and Subrahmanyam and Stapleton (1983) also show that a linear market model is sufficient for deriving the CAPM This makes the market model the true underlying return-generating process for the CAPM

Several other ad hoc return-generating processes have appeared in the literature (Chang, 1991; Ndubizu, Arize and Chandy, 1989) These are based on the assertion that the rate o f return on assets (i.e., the left side in the market model) can be better described not only by market factors, but also by industry and other factors For example Benjamin (1966) argues for the inclusion o f size, etc No theoretically viable equilibrium pricing mechanism can be ascribed to these ad hoc models

(iv) Empirical Application o f the Market Model and Specification Error Biases

The market model is a common specification o f the return-generating process for assets (Fama, Fisher, Jensen and Roll, 1969; Smith, 1977; Dodd, 1980; etc.) The model

also is referred to as the one factor market model, the single index model, and the LCL.

Usually the ordinary least squares (OLS) technique is used to estimate the parameters o f the single index market model to conduct event studies in finance (Fama, Fisher, Jensen and Roll, 1969; Smith, 1977; Dodd, 1980 etc.) Under general conditions, the ordinary least squares method is a BLUE (best linear unbiased) estimate o f the market model parameters

FFJR (1969) use the market model to study stock splits and their implications for market efficiency The FFJR model assumes that alpha and beta are constant in the pre-

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and post-event dates The study uses the logarithmic specification o f the simple single index market model [equation (4)] to test the effect o f stock splits on the rate o f return o f

a security The price relative for t h e / h security for month t is rJt, /, is the link relative to Fisher’s (1966) combination investment performance index, and uJt is the random

disturbance term satisfying the assumptions of the linear regression model The natural logarithm o f the security price relative is the rate o f return (with continuous

compounding) for the month in question The log of the market index relative is approximately the monthly rate o f return on a portfolio that includes equal dollar amounts

o f all securities in the market Using the available time series on Rjt and I, , the parameters a ] and are calculated

Based on these estimates o f the parameters, FFJR calculate abnormal returns and cumulative abnormal returns to determine if stock splits affect stock price returns

disproportionately.13 Their evidence indicates that stocks in the dividend increase class (the sample o f firms that increase dividends beyond the average for the market following

a split) have positive abnormal returns following the split According to the FFJR study, the market’s reaction to the split is the reaction to its dividend implications Thus, abnormal return around stock splits, when followed by an increase in dividends beyond the market average, is due to improved performance prospects

Brown and Warner (1980) employ stock returns data to examine various methodologies employed in event studies to measure security price performance They compare various abnormal performance measures such as mean-adjusted returns, market- adjusted returns, and market- and risk-adjusted returns The probabilities o f Type I and Type II errors are assessed for each abnormal performance measure using both

parametric and nonparametric tests In addition, Brown and Warner examine the distributional properties o f test statistics generated by each methodology Brown and

15 The exact computation methods are given in Chapter 3.

Warner conclude that tests that use risk-adjusted returns are no more powerful than tests that use returns not adjusted for systematic risk.

The Brown and Warner conclusion, however, is based on the assumption that the market model residual method is the appropriate method o f risk adjustment In order to investigate the robustness o f their findings, Brown and Warner also simulate other risk adjustment methods These methodologies include the Fama-Macbeth (1973) residuals method (which consists o f using the SML instead o f the market model itself) and a control portfolio technique in which the return on a portfolio o f sample securities is compared to that o f another portfolio with the same systematic risk Brown and Warner (pg.249) conclude: “beyond a simple one factor market model, there is no evidence that more complicated methodologies convey any benefit.”

Note that Brown and Warner study is a simulation model where hypothetical event month has been generated and abnormal performance has been artificially introduced to test whether various methodologies used in event studies will add anything new to already existing and known empirical findings As Brown and Warner conclude (see page: 249, Brown and Warner, 1980) the goal o f their research is not to formulate the 'best' event studies methodology, but to compare various methodologies that have been widely used and provide a useful basis for discriminating between alternative procedures Also Brown and Warner (1980) in all their empirical work on simulating the various return generating processes implicitly assumed that the underlying asset pricing model is the two moment CAPM16

We digress from the Brown and Warner study and assume that the three moment CAPM is the appropriate pricing mechanism Our goal in this research is to apply the

16 For the mean adjusted returns, market adjusted returns, and the market and risk adjusted returns the consistency o f Sharpe’s two moment CAPM and for the Fama-Macbeth residuals the consistency o f Black’s zero beta version o f the CAPM have been discussed in the Brown and Warner (1980).

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QCL in a well known event study that uses LCL and examine whether it substantially changes the results (i.e the result is statistically significantly different) In other words,

we want to test for the conditions for the QCL to hold given by Kraus and Litzenberger (1976), and then see whether the results o f an existing event study using LCL is going to change significantly17

Brown and Warner (1985) extended their 1980 study using daily stock returns (The 1980 study uses monthly returns.) Daily data in event studies pose potential problems The daily stock return for an individual security departs more from normality than do monthly returns There may also be potential for bias o f the OLS beta due to nonsynchronous trading For securities with trading delays different than those o f the market, OLS beta estimates may be biased Similarly, for securities with trading frequencies different than those o f the market index, OLS beta estimates will differ The study also looks at another potential problem involving the estimation o f variance in hypothesis tests Autocorrelation in daily excess returns and variance increase in the days around an event can be o f concerns

Brown and Warner’s (1985) study with daily data reinforces their previous conclusions with monthly data: methodologies based on the OLS market model and on standard parametric tests are well specified under a variety o f conditions

Cheng and Lee (1986) apply the specification error test developed by Ramsey (1969) and Ramsey and Schmidt (1976) to determine whether alternative market models (i.e., the Sharpe-Lintner specification, the Black specification, or a standard one factor market model are empirically appropriate for estimating a company’s beta coefficient and rate o f return They find that the simplest model (the one factor market model) has the smallest percentage o f specification error They also find that the risk-retum relationship

is biased within misspecified models and the systematic risk estimates are random with

17 The simulation approach which Brown and Warner (1980) is beyond the scope o f this dissertation

misspecified models These findings are consistent with those o f Brown and Warner (1980).

Both Brown and Warner and Cheng and investigate event study methodologies associated with the traditional two-moment CAPM They do not look at methodologies based on other pricing processes

(v) The Single Index Market Model Versus Multifactor Models:

The Issue of The Associated Pricing Mechanism

Tests o f market efficiency are joint tests o f market efficiency, an underlying equilibrium model, and a related market model Brenner (1977) believes that different market models should be used to test the efficient market hypothesis (EMH) and that the difference should be analyzed to see whether results for EMH are conditional upon the validity of the market model employed For example, Brenner (1977) compares the one- factor model with a two-factor model to assess which has less specification error The two-factor model is:

R U ~ a j + b j R mt + Cj R kt + £ i, (8)

where R kl represents the second market factor If an incorrect model is used (i.e., if a

two-factor model is used when the one-factor model is appropriate), Brenner states that specification error is committed Brenner concludes that one may be better off using the one-factor model; the inclusion o f erroneous variables is a more serious problem than imposing erroneous constraints on parameters

The multifactor models are ad hoc, as they do not have a pricing mechanism associated with them Ross’s (1977) arbitrage pricing theory (APT) is an exception, as he defines a multifactor return-generating process along with the pricing mechanism Ross’s return-generating process may not be suitable for event studies as a multifactor model because APT neither specifies the number of factors nor identifies the factors (Campbell,

Lo, and MacKinley, 1997)

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A growing body o f literature identifies many non-market risk factors such as book-to-market, size, and price to earning ratios that appear to be priced In particular Fama and French (1993, 1995) show that the importance o f small minus big (SMB, i.e., the difference between the return on a portfolio of small size stocks and the return on the portfolio o f large size stocks) and high minus low (HML, i.e., the difference between the return on a portfolio o f high book-to-market value stocks and the return on a portfolio o f low book-to-market value stocks) in explaining the cross-section o f stock returns.

According to Elton and Gruber (1995), Fama and French have converted the size component from a direct measure to a return concept by constructing a portfolio to capture this influence In the Fama and French portfolio technique, size enters the return- generating process as well as the pricing equation This technique allows researchers to investigate both the time-series as well as the cross-sectional properties o f non-market factors such as size and book-to-market ratios According to Elton and Gruber: "Which of these approaches (measuring the beta directly from size or estimating it from regressing a portfolio) is better awaits further empirical investigation." (Elton and Gruber, pg 386, 1995)

There is substantial debate regarding the economic meaning o f SMB and HML Daniel and Titman (1997) show that SMB and HML capture the co-movements o f stocks with similar characteristics; the characteristics and not the co-movements explain cross- sectional return variation The issue o f survivorship bias in the data used to test the multifactor models cannot be overlooked (Kothari, Shanken, and Sloan, 1995) In this context a new set o f literature (Harvey and Siddique, 2000; Chung, Johnson and Schill, 2001) can be referred to which argue that the non-market risk factors may be merely proxies for measures o f market risk not captured by the CAPM

According to Campbell, Lo, and MacKinley (1997), the gains from employing multifactor models for event studies may be limited Because the marginal explanatory power o f additional factors beyond the market factor is small, there is little reduction in

the variance o f the abnormal return There also is no good reason to use an economic model instead o f a statistical model in an event study (Campbell, Lo, and MacKinley, 1997).

The single factor market model and its variations (with the exception o f the return-generating process underlying the APT) have been used extensively in the literature on event studies The main advantages o f the single factor model (market model) are its amenability to event studies and its associated equilibrium model, i.e., the traditional CAPM

(vi) Beyond the Two-Moment CAPM

Rubinstein (1973) developed the first multi-moment CAPM Jean (1971) observes, however, that there is no behavioral justification for a multi-moment CAPM beyond three-moments In a two-moment world, investors make decisions based on the expected rate o f return (reward) and variance (risk) The underlying assumption typically

is that the probability distribution o f the rates o f return is normal and that the investor has

a quadratic utility function But Arrow (1971) argues that the desirable properties for an investor’s utility function are

(a) A positive marginal utility for wealth, i.e., nonsatiety with respect to wealth,(b) Decreasing marginal utility for wealth, i.e., risk aversion, and

(c) Non-increasing absolute risk aversion, i.e., risk assets are not inferior goods.The third assumption o f non-increasing absolute risk aversion implies a preferencefor positive skewness (See Kraus and Litzenberger, 1973, for the proof.)18 According to

18 u is the utility o f wealth and w is the end o f period wealth for the investor Denoting u ‘ as the i"1 derivative with respect to w , the non-increasing absolute risk aversion,

/ d w = [-u'u" + (u*)2] /(«') 5 0 ,

is a sufficient condition for u" > 0 , implying preference for positive skewness This follows because the

necessary condition for

Kraus and Litzenberger, the utility functions satisfying the desired attributes (a), (b), and(c) are logarithmic, power, and negative exponential utility functions (utility functions belonging to the HARA class) It is mathematically possible to obtain a multi-moment CAPM by expanding any o f these utility functions belonging to the HARA class by the Taylor series expansion.19 Kraus and Litzenberger show that investors not only dislike

risk, but also they prefer positive skewness Ceteris paribus, investors should prefer

portfolios with larger probabilities o f large pay-offs This assertion may result from Fisher and Lorie’s (1970) empirical finding that over a long period o f time (40 years) the rates o f return in the U.S market exhibit a pattern o f positive skewness.20

Studies by Friend and Blum (1970), Black, Jensen, and Scholes (1972), Miller and Scholes (1972), Fama and Macbeth (1973), and Blum and Friend (1973) suggest that the slope o f the CAPM is lower and the intercept is higher than predicted by traditional theory According to Kraus and Litzenberger these findings may result from the

misspecification o f the CAPM by the omission o f systematic skewness

Kraus and Litzenberger empirically test whether skewness is incorporated in the valuation o f assets In the three moment specification they find that the constant term is equal to the riskless rate o f interest, a finding not shared by any o f the two-moment CAPM-based empirical studies They also confirm the presence o f a significant price for systematic skewness in their pricing model

The assumptions underlying the three-moment CAPM have the following properties:

(a) A positive preference for increase in wealth ( u f > 0), (b) An aversion to risk ( u" < 0 ), and

A skewness preference must be acknowledged (Arditti, 1967) unless we concede that risk assets can logically be considered to be inferior goods.

19 Fama (1965) finds that the rates o f return probability distribution for individual asset are not normal, but belong to a Paretian distribution with the characteristic exponent oc = 1.72 For a normal distribution °c should be equal to 2.0

20 Ibbotson and Sinquefield (1976) confirm Fisher and Lorie’s findings.

(c) Preference for positive skewness (« " > 0).

The third assumption o f the three-moment CAPM o f non-increasing absolute risk aversion (risk assets are not inferior goods) requires that the skewness preference must be acknowledged in pricing assets

Using the Taylor series expansion, the expected utility o f end o f period wealth

u(w) may be expanded as follows:

fi[u(»0)] = « (£ (» )) + i [ U'( £ ( S ) ) k ’ + i [ U'( £ ( K ) ) K 1 (9)

where a j = E[w - £(vv)], m j = E[w - £(vv))]3 are the second and third central moments

The parameters b{ and b2 are the market price o f beta reduction and the market

price o f gamma reduction, respectively, and are given by:

21 Statistical skewness is

whereas co-skewness is y , ■

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=(dE(w)ldcrw)cTm, and ={dE(w)/d/w^K, (13)Equation (11) is the three-moment CAPM According to Kraus and Litzenberger, risk can be separated into systematic risk, unsystematic risk, and positive skewness risk Due to a positive theory o f valuation, there is no need to include terms o f fourth or

higher-moment as no a priori behavioral arguments for general investor attitude toward

fourth and higher moments have been made

The return-generating process corresponding to their three-moment CAPM is given by:

** — R-ft = Q + ~ R-fl) "*■ C^XR-mt ~ R m) + Sit • 0 ^ )

Rit, Rj, and Rml are security /’ s return, the riskless return, and the market return during

period t, respectively The error term e it is assumed to be homoscedastic, has an expected

value o f zero, is independent o f the excess rate o f return on the market portfolio

(Rmt - R fi) , and is independent o f the squared deviation of the excess return on the

market portfolio from its expected value (Rml - Rm)2,22 Kraus and Litzenberger also

provide expressions for the security’s beta (systematic covariance) and gamma

(systematic skewness) in terms o f Cu and C2i.

The expressions are:

For those risk assets whose returns are described by the market model or the LCL,

C2i = 0 and thus /?, = y t - If all risk assets had LCL, then the three-moment CAPM would

reduce to the traditional form o f the Sharpe-Lintner CAPM This holds even when the

22

The statistical significance o f C2i plays a significant role in our research.

rate o f return on the market portfolio is asymmetrically distributed According to Kraus

and Litzenberger, if C2i * 0 for some i, the traditional theory (Sharpe-Lintner CAPM)

would not hold when the rate o f return on the market portfolio is asymmetrically

distributed Given that C2i * 0 and that the return on the market portfolio is

asymmetrically distributed, the equilibrium model is the three-moment CAPM Thus, the appropriate return-generating process is the QCL and not the LCL model

Kraus and Litzenberger derive and test a three-moment CAPM and find that skewness explains empirical anomalies o f the two-moment CAPM such as an intercept term significantly different from zero a slope less than the difference between the return

on the market portfolio minus the risk free rate in the two-moment CAPM The derivation by Kraus and Litzenberger will be a tool we use to examine the link between LCL and QCL

(vii) Recent Support fo r the Three-Moment CAPM

Sears and Wei (1988) provide a nonlinear formulation o f the Kraus and Litzenberger three-moment CAPM that shows a direct link between the two-moment and three-moment CAPMs Sears and Wei provide theoretical conditions under which

skewness preference is consistent with the two-moment CAPM empirical results TheKraus and Litzenberger model, according to Sears and Wei, can be written as:

K 2 = [{dw / dmw) /( d w /d a w) \m w / a w) , The market’s marginal rate o f substitution

between skewness and the risk times the risk- adjusted skewness o f the market portfolio

a 1 m , m i m = Second and third central moments about the

market portfolio’s return

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\ v , a 2w ,m 3w = First moment about origin, second, and third

central moments about end o f period wealth.The linear empirical version o f the three-moment CAPM is given by:

where b0 is the constant and is hypothesized to be zero.

The empirical two-moment CAPM is given by the following:

(20)

where b *o and b 'l are hypothesized, respectively, to be equal to zero and (Rm —Rf )

Under the hypothesis that the three-moment CAPM is correct, Sears and Wei

derive the expression for b \ as follows:

In equation (23), K 3 is an elasticity coefficient that measures the relative

importance o f skewness compared to the standard deviation Investors theoretically

display some preference for positive skewness vis-a-vis risk if AT3 < 0 and when mm> 0

or if AT3 > 0 when mm < 0 According to Sears and Wei, equation (23) provides analytical

support for Kraus and Litzenberger’s heuristic rationale o f skewness preference and their

empirical results because if a > 1 when K 3 < 0 (m m > 0), there will be a specification bias in the two-moment model because b ’ < (Rm - R f ) and bQ’ > 0 In this case the

empirical results o f the two-moment CAPM are consistent with a market preference for

positive skewness when m m > 0 Skewness preference also implies a specification bias in the two-moment model when b x < ( R m — R f ) and bQ > 0 if K 3 > 0 (rnm < 0) and a

< 1 A preference for a positive skewness when m n < 0 requires higher f t s to be

associated with proportionately smaller y s The nonlinear formulation o f Sears and Wei

b'x =Cov[Ri - R f ),/3i]/V a r(fi)

= Cov[(b0 +blPi +b2yi)Pi]/Var(fii)

= (Rm- R f m + aK 3)/(\ + K 3)

where a = Cov(^(.,y(.) / Var{fi.) is the slope o f the regression o f against /?

They also show that

K 3 = ~{dam / <Tm) I(dmm / mm).

(22)

(23)

not only provides a direct link between the two-moment CAPM and the three-moment CAPM, but also provides the theoretical conditions under which skewness preference is consistent with two-moment CAPM empirical results The Sears and Wei formula provides additional information on skewness preference by deriving the elasticity coefficient relating risk to skewness.

A study by Kian-Guam Lim (1989) tests the three-moment CAPM o f Kraus and Litzenberger using Hansen’s generalized method o f moments (GMM) With the inclusion

o f skewness in asset valuation, the distribution o f returns no longer can be normal Also, there is no obvious multivariate distribution for returns that exhibit co-skewness Because the GMM approach does not impose any distributional assumptions on asset returns, it provides an appropriate tool to test the three-moment CAPM The study uses monthly stock returns and finds some evidence of systematic skewness being priced

Sengupta and Zheng (1997) analyze the conditional return series for mutual funds and the S & P 500 to test whether there is persistence in skewness Three groups o f statistical models o f market volatility are estimated: ARCH, generalized ARCH, and a nonlinear logistic model for persistence in conditional variance Empirical estimates o f the statistical models are used to test for skewness persistence They also test the impact

o f skewness on the conditional variance

Sengupta and Zheng uncover three important results:

(a) Conditional skewness appears to be more persistent than conditional variance;

(b) Mutual fund skewness is more affected by the overall market index and also by its own variance, and

(c) The impact o f variance on skewness is negative in most cases, implying the presence o f asymmetry

Harvey and Siddique (2000) develop a model that incorporates conditional skewness (unlike Kraus and Litzenberger who consider unconditional skewness) to test if

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systematic skewness is priced in the market They find that including conditional skewness helps explain the cross-sectional variation of the expected return across assets and that conditional skewness is still significant when factors based on size and book-to- market are included They also show that systematic skewness commands a risk premium

o f almost 3.60 percent per year

Harvey (2001) investigates the effect o f skewness in international markets He examines a comprehensive list of 18 different risk factors that potentially impact international equity returns and finds that world beta and co-skewness reasonably capture

a cross-section o f the average returns world market

The studies cited above generally concern the equilibrium structure o f returns when skewness preference is taken into account Several studies show that portfolio selection also can be affected by the skewness o f the distribution o f returns Kane (1982) looks at the impact o f skewness on risk taking The derivation of skewness preference functions and an appropriate skewness measure indicate the extent to which the third moment affects the optimal allocation o f investments to risky assets Kane (1982) finds that if co-moments o f second order are negative or small relative to covariance, the loss

o f skewness from diversification may be more rapid than the reduction in variance, which reduces the attractiveness o f diversification

Simkowitz and Beedles (1978) and Conine and Tamarkin (1981) explain the low diversification of many investors’ portfolios by a preference for positive skewness Lai (1991) and Chunhachinda, Dandapani, Hamid, and Prakash (1997) find that incorporating skewness into the investor’s portfolio decision causes a major change in the construction

o f the optimal portfolio They show that investors trade expected return for skewness, which is consistent with Sear and Wei’s (1988) theoretical findings

( viii) Purpose o f This Study

To summarize, the traditional CAPM assumes that only the mean and the variance

o f the return matter in asset pricing This assumption means that both upside and downside risks are considered equally likely by investors But the evidence shows that investors prefer positive skewness Kraus and Litzenberger, Brennan (1979), and He and Leland (1993) show that if the market portfolio’s return has a constant mean and

volatility, the average investor has a power utility function Because this function has a positive third derivative, it implies that a preference for skewness is positively valued by investors The traditional equilibrium model based on mean and variance may not be sufficient as an asset pricing model A mean-variance-skewness equilibrium model (i.e., the three-moment CAPM) may be a more appropriate pricing model

This study proposes to replicate FFJR’s 1969 pioneering event study23 on stock splits using the QCL model associated with Kraus and Litzenberger’s three-moment CAPM If we obtain statistically significantly different results using QCL than with LCL (given that the conditions for QCL to hold are satisfied), it will at least provide some preliminary checks on the data for the choice o f the appropriate retum-generating model when conducting an event study

Stock splits as signaling devices have received substantial attention in the literature Taken at its face value, a stock split is just a finer slicing o f a given cake—the total value o f the firm—and it should not have an effect on the firm’s investors

(Lakonishok and Lev, 1987)

FFJR (1969) and other studies (Lakonishok and Lev, 1987; Pilotte and Manuel, 1996) find, however, that stock splits are more than cosmetic changes They determine that stock splits result in statistically significant stock price revaluations around the ex- dividend dates Stock splits present a conundrum to finance theorists These seeming non-

23 The FFFJR study led to the use o f CAR method in event studies.

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events (they are purely cosmetic changes) lead to significant price changes Grinblatt, Masulis, and Titman (1984) show that even a clean stock split—where no other firm- specific event coincides with the split announcement—generates a positive abnormal return o f close to 3.0 percent upon announcement and an additional 1.0 percent abnormal return on the ex-day Numerous hypotheses have attempted to explain the persistence o f splits and the associated market reaction, but none is strongly supported by data These hypotheses fall into two classes: signaling and optimal price According to the signaling hypothesis, given asymmetric information between managers and investors, the former might use financial decisions such as stock splits to convey favorable information to the latter (Ross, 1977; Leland and Pyle, 1977) The FFJR (1969) findings support the signaling hypothesis The Grinblatt (1984), Bar-Yosef and Brown (1977), and Lakonishok and Lev (1987) findings also support the FFJR conclusions In this research

we want to explore whether these well-established results will be significantly different if skewness preference is recognized Our goal is not to establish further evidence for the signaling hypothesis, but to compare the market model and the QCL We use one established result around a stock-split event to empirically explore which return- generating process appropriately represents the actual observed return

Chapter 3

Methodological Framework

(i) A B rief Survey o f Event Study Methodologies

The event study is a frequently used tool in financial research An event study investigates the market’s response to some well-defined event by observing security prices An event generally is related to the release o f information to market participants through the financial press, through a corporate release, via specific corporate actions, or through governmental actions An important issue in event studies is specifying the

proper focal date (called the event date) to be used in the study The following aspects

must be addressed in any event study

(i.a) Identifying the Time Parameter

Normal returns for a security are those returns observed when there is no event Normal returns generally are estimated outside the period surrounding an event date There are three broad ways to choose the estimation period

(a) An estimation period prior to the event period is chosen for applications in which the determinants o f the normal return are not expected to change due to the event

(b) The estimation period may follow the event period in applications where the determinants o f the normal return are expected to change

(c) In applications where the determinants are supposed to change due to the event, the estimation period represents some average o f pre- and post-event information Hence, an estimation period lying on both sides o f the event date can be chosen Fama, Fisher, Jensen, and Roll (1969), Dodd and Leftwich (1980), and Wayne and Partch (1986) offer some applications where the estimation period lies on both sides o f the event period

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