10. Event Studies

8/7/2019 10. Event Studies

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10 Event Studies

Reference: Bodie, Kane, and Marcus (2005) 12.3 or Copeland, Weston, and Shastri

(2005) 11

Reference (advanced): Campbell, Lo, and MacKinlay (1997) 4

More advanced material is denoted by a star (). It is not required reading.

10.1 Basic Structure of Event Studies

The idea of an event study is to study the effect (on stock prices or returns) of a special

event by using a cross-section of such events. For instance, what is the effect of a stock

split announcement on the share price? Other events could be debt issues, mergers and

acquisitions, earnings announcements, or monetary policy moves.

The event is typically assumed to be a discrete variable. For instance, it could be a

merger or not or if themonetary policy surprise was positive (lowerinterestthan expected)

or not. The basic approach is then to study what happens to the returns of those assets

that have such an event.Only news should move the asset price, so it is often necessary to explicitly model

the previous expectations to define the event. For earnings, the event is typically taken to

be the earnings announcement minus (some average of) analysts forecast. Similarly, for

monetary policy moves, the event could be specified as the interest rate decision minus

previous forward rates (as a measure of previous expectations).

The abnormal return of asset i in period t is

ui;t D Ri;t Rnormali;t ; (10.1)

where Rit is the actual return and the last term is the normal return (which may differ

across assetsand time). The definition of the normal returnis discussed in detail in Section

10.2. These returns could be nominal returns, but more likely (at least for slightly longer

horizons) real returns or excess returns.

Suppose we have a sample of n such events (assets). To keep the notation (reason-

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timefirm 1 firm 2

1011 0 1

Figure 10.1: Event days and windows

ably) simple, we normalize the time so period 0 is the time of the event. Clearly the

actual calendar time of the events for assets i and j are likely to differ, but we shift the

time line for each asset individually so the time of the event is normalized to zero for

every asset.

The (cross-sectional) average abnormal return at the event time (time 0) is

Nu0 D .u1;0 C u2;0 C ::: C un;0/ =n DPn

iD1ui;0=n: (10.2)

To control for information leakage and slow price adjustment, the abnormal return is often

calculated for some time before and after the event: the event window (often 20 days

or so). For lead s (that is, s periods after the event time 0), the cross sectional average

abnormal return is

Nus D .u1;s C u2;s C ::: C un;s/ =n DPn

iD1ui;s=n: (10.3)

For instance, Nu2 is the average abnormal return two days after the event, and Nu1 is for

one day before the event.

The cumulative abnormal return (CAR) of asset i is simply the sum of the abnormal

return in (10.1) over some period around the event. It is often calculated from the be-

ginning of the event window. For instance, if the event window starts at w, then the

q-period (day?) car for asset i is

cari;q D ui;w C ui;wC1 C : : : C ui;wCq1: (10.4)

The cross sectional average of the q-period car is

carq D

car1;q C car2;q C ::: C carn;q

=n DPn

iD1cari;q=n: (10.5)

Example 10.1 (Abnormal returns for day around event, two firms) Suppose there are

two firms and the event window contains 1 day around the event day, and that the

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abnormal returns (in percent) are

Time Firm 1 Firm 2 Cross-sectional Average

1 0:2 0:1 0:05

0 1:0 2:0 1:5

1 0:1 0:3 0:2

We have the following cumulative returns


1 0:2 0:1 0:05

0 1:2 1:9 1:55

1 1:3 2:2 1:75

10.2 Models of Normal Returns

This section summarizes the most common ways of calculating the normal return in

(10.1). The parameters in these models are typically estimated on a recent sample, the

estimation window, that ends before the event window. In this way, the estimated be-

haviour of the normal return should be unaffected by the event. It is almost always as-

sumed that the event is exogenous in the sense that it is not due to the movements in the

asset price during either the estimation window or the event window. This allows us to

get a clean estimate of the normal return.

The constant mean return model assumes that the return of asset i fluctuates randomly

around some mean i

Ri;t D i C i;t with E.i;t / D Cov.i;t ; i;ts/ D 0: (10.6)

This mean is estimated by the sample average (during the estimation window). The nor-

mal return in (10.1) is then the estimated mean. Oi so the abnormal return becomes Oi;t .

The market model is a linear regression of the return of asset i on the market return

Ri;t D iCiRm;tC"it with E."i;t / D Cov."i;t ; "i;ts/ D Cov."i;t ; Rm;t/ D 0: (10.7)

Notice that we typically do not impose the CAPM restrictions on the intercept in (10.7).

The normal return in (10.1) is then calculated by combining the regression coefficients

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time

event windowestimation window(for normal return)

firm i

Figure 10.2: Event and estimation windows

with the actual market return as O i C OiRm;t , so the the abnormal return becomes O"it .

Recently, the market model has increasingly been replaced by a multi-factor model

which uses several regressors instead of only the market return. For instance, Fama and

French (1993) argue that (10.7) needs to be augmented by a portfolio that captures the

different returns of small and large firms and also by a portfolio that captures the different

returns of firms with high and low book-to-market ratios.

Finally, yet another approach is to construct a normal return as the actual return on

assets which are very similar to the asset with an event. For instance, if asset i is a

small manufacturing firm (with an event), then the normal return could be calculated as

the actual return for other small manufacturing firms (without events). In this case, the

abnormal return becomes the difference between the actual return and the return on the

matching portfolio. This type of matching portfolio is becoming increasingly popular.

All the methods discussed here try to take into account the risk premium on the asset.

It is captured by the mean in the constant mean mode, the beta in the market model, and

by the way the matching portfolio is constructed. However, sometimes there is no data

in the estimation window, for instance at IPOs since there is then no return data before

the event date. The typical approach is then to use the actual market return as the normal

returnthat is, to use (10.7) but assuming that i D 0 and i D 1. Clearly, this does not

account for the risk premium on asset i , and is therefore a fairly rough guide.

Apart from accounting for the risk premium, does the choice of the model of the

normal return matter a lot? Yes, but only if the model produces a higher coefficient ofdetermination (R2) than competing models. In that case, the variance of the abnormal

return is smaller for the market model which the test more precise (see Section 10.3 for

a discussion of how the variance of the abnormal return affects the variance of the test

statistic). To illustrate this, consider the market model (10.7). Under the null hypothesis

that the event has no effect on the return, the abnormal return would be just the residual

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in the regression (10.7). It has the variance (assuming we know the model parameters)

Var.ui;t / D Var."it / D .1R2/ Var.Ri;t /; (10.8)

where R2 is the coefficient of determination of the regression (10.7).

Proof. (of (10.8)) From (10.7) we have

Var.Ri;t / D 2i Var.Rm;t /CVar."it /:

We therefore get

Var."it / D Var.Ri;t / 2i Var.Rm;t/

D Var.Ri;t / Cov.Ri;t ; Rm;t/2

= Var.Rm;t /

D Var.Ri;t / Corr.Ri;t ; Rm;t/2 Var.Ri;t /

D .1R2/ Var.Ri;t /:

The second equality follows from the fact that i D Cov.Ri;t ; Rm;t/= Var.Rm;t /, the

third equality from multiplying and dividing the last term by Var .Ri;t / and using the

definition of the correlation, and the fourth equality from the fact that the coefficient

of determination in a simple regression equals the squared correlation of the dependent

variable and the regressor.This variance is crucial for testing the hypothesis of no abnormal returns: the smaller

is the variance, the easier it is to reject a false null hypothesis (see Section 10.3). The

constant mean model has R2 D 0, so the market model could potentially give a much

smaller variance. If the market model has R2 D 0:75, then the standard deviation of

the abnormal return is only half that of the constant mean model. More realistically,

R2 might be 0.43 (or less), so the market model gives a 25% decrease in the standard

deviation, which is not a whole lot. Experience with multi-factor models also suggest that

they give relatively small improvements of the R2 compared to the market model. For

these reasons, and for reasons of convenience, the market model is still the dominating

model of normal returns.

High frequency data can be very helpful, provided the time of the event is known.

High frequency data effectively allows us to decrease the volatility of the abnormal return

since it filters out irrelevant (for the event study) shocks to the return while still capturing

the effect of the event.

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10.3 Testing the Abnormal Return

In testing if the abnormal return is different from zero, there are two sources of sampling

uncertainty. First, the parameters of the normal return are uncertain. Second, even if

we knew the normal return for sure, the actual returns are random variablesand they

will always deviate from their population mean in any finite sample. The first source

of uncertainty is likely to be much smaller than the secondprovided the estimation

window is much longer than the event window. This is the typical situation, so the rest of

the discussion will focus on the second source of uncertainty.

It is typically assumed that the abnormal returns are uncorrelated across time and

across assets. The first assumption is motivated by the very low autocorrelation of returns.

The second assumption makes a lot of sense if the events are not overlapping in time, so

that the event of assets i and j happen at different (calendar) times. In contrast, if the

events happen at the same time, the cross-correlation must be handled somehow. This is,

for instance, the case if the events are macroeconomic announcements or monetary policy

moves. An easy way to handle such synchronized events is to form portfolios of those

assets that share the event timeand then only use portfolios with non-overlapping events

in the cross-sectional study. For the rest of this section we assume no autocorrelation or

cross correlation.

Let 2i D Var.ui;t / be the variance of the abnormal return of asset i . The variance of

the cross-sectional (across the n assets) average, Nus in (10.3), is then

Var. Nus/ D

21 C 22 C ::: C

2n

=n2 D

PniD1

2i =n

2; (10.9)

since all covariances are assumed to be zero. In a large sample (where the asymptotic

normality of a sample average starts to kick in), we can therefore use a t -test since

Nus= Std. Nus/ !d N.0;1/: (10.10)

The cumulative abnormal return over q period, cari;q , can also be tested with a t -test.

Since the returns are assumed to have no autocorrelation the variance of the car i;q

Var.cari;q/ D q2i : (10.11)

This variance is increasing in q since we are considering cumulative returns (not the time

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average of returns).

The cross-sectional average cari;q is then (similarly to (10.9))

Var.carq/ D

q21 C q22 C ::: C q

2n

=n2 D q

PniD1

2i =n

2; (10.12)

if the abnormal returns are uncorrelated across time and assets.

Figures 4.2ab in Campbell, Lo, and MacKinlay (1997) provide a nice example of an

event study (based on the effect of earnings announcements).

Example 10.2 (Variances of abnormal returns) If the standard deviations of the daily

abnormal returns of the two firms in Example 10.1 are 1 D 0:1 and and2 D 0:2 , then

we have the following variances for the abnormal returns at different days


1 0:12 0:22

0:12 C 0:22

=4

0 0:12 0:22

0:12 C 0:22

=4

1 0:12 0:22

0:12 C 0:22

=4

Similarly, the variances for the cumulative abnormal returns are


1 0:12

0:22

0:12

C 0:22

=40 2 0:12 2 0:22 2

0:12 C 0:22

=4

1 3 0:12 3 0:22 3

0:12 C 0:22

=4

Example 10.3 (Tests of abnormal returns) By dividing the numbers in Example 10.1 by

the square root of the numbers in Example 10.2 (that is, the standard deviations) we get

the test statistics for the abnormal returns


1 2

0:5 0:4

0 10 10 13

1 1 1:5 1:8

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Similarly, the variances for the cumulative abnormal returns we have


1 2 0:5 0:4

0 8:5 6:7 9:8

1 7:5 6:4 9:0

(I have not double checked the calculations...)

10.4 Quantitative Events

Some events are not easily classified as discrete variables. For instance, the effect of

positive earnings surprise is likely to depend on how large the surprise isnot just if there

was a positive surprise. This can be studied by regressing the abnormal return (typically

the cumulate abnormal return) on the value of the event (xi )

cari;q D a C bxi C i : (10.13)

The slope coefficient is then a measure of how much the cumulative abnormal return react

to a change of one unit of xi .

Bibliography

Bodie, Z., A. Kane, and A. J. Marcus, 2005, Investments, McGraw-Hill, Boston, 6th edn.

Campbell, J. Y., A. W. Lo, and A. C. MacKinlay, 1997, The Econometrics of Financial

Markets, Princeton University Press, Princeton, New Jersey.

Copeland, T. E., J. F. Weston, and K. Shastri, 2005, Financial Theory and Corporate

Policy, Pearson Education, 4 edn.

Fama, E. F., and K. R. French, 1993, Common Risk Factors in the Returns on Stocks

and Bonds, Journal of Financial Economics, 33, 356.

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10. Event Studies

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