Donner index
2.4 The theoretical background of event-studies .1 Random walk hypothesis
The basic idea is to consider stock prices Pl as following random walks. Note that the time is denoted by l∈{1,2,…,L} to indicate that I regard the estimation period and not the event period denoted t∈{1,2,…,T}. A change in stock prices is only due to public information that can be seen as a white-noise process el. This process of newly available public information possesses the property that there is no autocorrelation.
Putting this feature in other words, it states that it is impossible to draw any conclusions from knowing the public information at l-1 that helps to improve the prediction for the public information occurring in the next period.
l l
l P e
P = −1+ (2.1)
Thereby, the white-noise process el is characterized by a specific covariance function κ(j, l), and the mean function µe(l) is constant over time, but is allowed to differ from zero.
{ }
Note that the variance σe2 remains unchanged over time. Thus, it can be easily seen that taking the first difference from expression (2.1) yields a stationary white-noise process.
2.4.2 Merger announcements – events with great influence
Although the random walk hypothesis was criticized because it is partially refutable,36 I maintain this hypothesis. It is appropriate to describe short-term price fluctuations taking place in a normal environment. These normal situations cover periods in which informed trading is relatively rare because meaningful events, which could have an essential impact on the true value of the underlying stock, do not impend. The public declaration of a merger obviously possesses an enormous impact on the true values of the interacting firms. Jennings (1994) showed that the information asymmetry increases around merger announcements. This is the breeding-ground for informed trading because knowing that a merger is going to occur
36 See, for instance, Fama and French (1988). They uncovered negative autocorrelation of returns in the long-run; thus, the error term el in (2.1) would exhibit autocorrelation.
provides extraordinary profits that exceeds the costs of trading, especially transaction costs.
One can identify two sources of informed trading.
Some insiders, for instance managers of the interacting companies, know with great confidence that a merger is going to be declared. Using their informational advantage, they start to buy stocks, if they expect that the true value exceeds the actual stock price. This insider trading reveals private information through the price process such that the market value narrows to the true value expected by the insiders. This adaptation of the market price to the true value, which is changed by the merger, takes maybe some days. The more market participants belong to the group of insiders the tougher is the competition among these insiders and the faster the adaptation process is finished.37
The other source of informed trading stems from above-average analytical skills that enables to achieve informational advantage from public information. This means that some market participants make more precise predictions about the possibility of a merger than others do.38
I highlight that the event study method can and should be applied in such situations in which the strong market efficiency is relaxed by allowing informed trading. Consequently, the market price Pl does not perfectly reflect all public and private information that exist at time l.
However, the competition among insiders and their trading patterns yield to an adaptation process that guarantees that the private information is reflected in the market price Pl with a time lag. I tackle this problem by constructing a thirty days window surrounding the event day.
Moreover, the presence of informed trading and the described adaptation process are in conflict with my random walk hypothesis. To see this point, consider the second source of informed trading. As it is captured in the variance structure κ(j, l), the public information el-1
does not give a clue about el. However, the variance structure forbids that market participants have the capability to turn el-1 in private information, for instance, they update after observing el-1 their expectation about the probability that a merger occurs. Thus, if informed trading exists that is triggered by an impending considerable event like a merger announcement, the random walk hypothesis failed. This failure is exactly what I try to show in my event-study.
Accordingly, I use the random walk hypothesis as null hypothesis that the event is not meaningful; thus, the merger does not change the fundamental value of the interacting firms.
Based on the random walk, the next section introduces a model that enables to determine the
37 See Kyle (1985) for a theoretical model that describes the strategic behavior of insiders and the competition between them was modeled by Holden and Subrahmanyam (1992).
38 See Kim and Verrecchia (1994) that provide a theoretical model of this sort of informed trading.
normal return. Note that the normal return determines the return that I would have expected if the merger announcement had not occurred. The economic impact of a merger is, thus, the return in the presence of the public declaration minus the normal return. Nevertheless, informed trading has other consequences discussed in chapter three.
2.4.3 The constant-mean-return (CMR) model
Masulis (1980) developed the CMR model that represents the basis of my model. Note that the CMR is nowadays not widely applied because the market model works better under normal circumstances. But finding an appropriate market index on a daily basis for German companies, is difficult in the year 1908, and even available monthly indices39 are not generally accepted.
Taking the first difference of equation (2.1) and dividing by Pl-1 provides the return of stock prices Rl.40 My sample consists of n different stocks i and the estimation window is denoted as l∈{1,2,…,L}; hence, I get the following expression.
il The stock return, like the first difference of equation (2.1), follows a white-noise process;
thereby, eil denotes a white-noise process with mean function that is equal to zero. Thus, µi
represents the mean function of Ril which is supposed to be constant over time. For convenience, I put expression (2.3) in matrix notation; thereby, bold letters indicate matrixes.
l
l µ e
R = + (2.4)
In equation (2.4) all vectors are column vectors with dimension n×1. Note that I maintain the random walk hypothesis and, hence, my null hypothesis that the event has no impact on share prices. Following this logic, I characterize in equation (2.4) the normal return. This is the return I would expect, if the event did not occur. This expression is the core of the constant-mean-return (CMR)41 model. Estimating (2.4) is straightforward.
∑
=39 A already mentioned the `Donnerindex´ - but also the more valid Eube (1998) index is currently not accepted, in general.
40 This model can be expressed in natural logarithms correspondingly. In this case, the first difference is obviously an approximate return. However, the results of my event study are also valid if I use a log linear specification.
41 Although the CMR is the simplest model Brown and Warner (1980,1985) found that it yields very similar result in comparison to more sophisticated models. The variance of the abnormal return is not significantly reduced using a more complex model such as multi-factor models. In chapter four, I provide additional evidence.
Consider L stands for the length of the estimation period, which is the same for all stocks.
And Rl is a n×1 dimensional vector that collects for time point l∈{1,2,…,L} of the estimation window the return of each stock i. Sometimes it is more convenient to use the matrix form (2.6). I will switch between these two notations in the mathematical appendix.
A1 I define matrix A as n×L dimensional matrix that contains all stocks i and for each time l all observed daily returns. I also define the unity vector 1 as being L×1 dimensional. Moreover, I derive the variance of the method of moments estimator directly from expression (2.5).
2
At that point, note that successive returns of stock i are supposed to be uncorrelated over time.
Therefore, it is possible to draw the variance operator under the sum operator. The resulting variance vector σ2 is obviously a n×1 dimensional vector that allows for differences in variances among stocks i and states that the variances remain unchanged over time l∈{1;2;…;L}.42 In comparison to the CMR model proposed by Masulis (1980), I use this variance expression to control for the inaccuracy of estimating normal returns.
2.4.4 Abnormal returns and their statistical properties
Having determined the normal return by the CMR, I can now define the abnormal return εt*
that stems from the merger announcement. The abnormal return is simply the difference between the observed return vector Rt during the event window t∈{1,2,…,T} and the part of this observed return that can be predicted using the CMR model.
( )
+ ∗= t t
t R A ε
R E (2.8)
Note that A is the data matrix containing the daily returns of the estimation period. This data matrix is the ingredient for estimating the mean vector µ like mentioned in (2.6). The conditional mean in equation (2.8) is obviously equal to which follows from taking the conditional expectation of equation (2.4) and replacing the mean vector by its estimate as described in expression (2.6). Therefore, I obtain an estimate for the abnormal return vector.
µˆ
µ R
εˆt∗ = t − ˆ (2.9)
42 These assumptions are very convenient and show that my model share common features with share time series models provided by Dyckman et al. (1984).
Under the null hypothesis that the event has no economic impact, one can now derive the statistical properties of the abnormal returns.
Result 143
Under the null hypothesis, the conditional distribution of the estimated abnormal return vector εt* after having observed the data matrix A and assuming that the abnormal returns are jointly normally distributed44 can be described as follows.
∗
εˆ ~t N
(
0;In×nσ2e +In×nVar( )
µˆ)
(2.10) What I want to show is that the abnormal returns deviate significantly from this conditional distribution; correspondingly, the merger has an essential impact on the market value of the affected firms.2.4.5 Aggregation of abnormal returns over time
Because the adaptation process of market prices caused by the merger announcement takes several days, it seems to be worthwhile using the aggregated value of the abnormal returns as a measure of the impact of mergers. The estimated cumulated abnormal return vector
with dimension n×1 covering the time period from τ
(
τm;τn Cˆ)
m to τn is defined in the following manner.
( ) ∑
=
≡ n ∗ t m
n m;τ
τ τ
τ t
ε
Cˆ ˆ (2.11)
Result 2 offers an appropriate test statistic for the cumulated abnormal return, which is discussed in detail in the mathematical appendix.45
43 This is derived and explained in detail in the mathematical appendix. I deviate from conventional CMR models by taking the variance of the estimated mean vector into account. Thus, I control for an imprecise estimation.
44 It is usually possible to relax this assumption and replace it by a parameter free representation – but the results are almost the same (see Corrado and Zivney, 1992). Moreover, this assumption simplifies the analysis considerably.
45 The mathematical appendix highlights also some interesting insights with respect to an optimal choice of the event window.
Result 2
Assuming that the abnormal returns are jointly normally distributed, like in result one, yields the following test statistic that is t-distributed with T1-2 degrees of freedom. T1 represents the number of the days belonging to the selected event window from τm to τn. The estimated standard deviation appears in the denominator.
( )
element of the covariance matrix Vc, which is derived in the mathematical appendix.
2.4.6 Cumulating abnormal returns over time and over cross sectional units
To assess whether the equally weighted portfolio of the firms, included in the sample, shows systematically higher returns from τm to τn, I use a modified version of cumulated returns.
Accordingly, I now have to aggregate abnormal returns over time and cross sectional data.
The first step is to estimate the sample average of the abnormal returns at time t and to determine the variance of this estimate. For this purpose result 3 provides the results that are discussed in the mathematical appendix in an accessible manner.
Result 3
The estimate for the sample average takes the following form; thereby, 1 is a n×1 dimensional unity vector.
This expression is obviously the arithmetic mean of the abnormal returns at time t. If I assume that the abnormal returns at t are uncorrelated among securities i, I can write the estimated covariance matrix of εt∗ in the following fashion.
( )
n 2 tr(
I σ2e I Var( )
µˆ)
Varεt∗ = − ⋅ n×n + n×n (2.14)
The second step carries out the aggregation over time. I denote the equally weighted cumulated abnormal return covering the time span from τm to τn with the termC
(
τm;τn)
. To test whether this cumulated abnormal return of the whole portfolio is significantly different from zero, I use the following test statistic.Result 446
The standardized C
(
τm;τn)
is approximately standard normally distributed.( )
With these results, I try to uncover the abnormal returns and cumulated abnormal returns that stem from the merger announcement and to assess their significance.
2.5 Empirical results of the event-study