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\title{\bf Merger negotiations with stock market feedback\thanks{
%An early draft of this paper (without the structural analysis below) was entitled "Markup pricing revisited". We are grateful for the comments of seminar
%%\citep[without the structural analysis below]{BettEckbThor08b}
%participants at Boston University, Dartmouth College, HEC Montreal, Lille University, Norwegian School of Economics and Business administration, Southern Methodist University, Texas Tech University, University of Arizona, University of British Columbia, University of Colorado, University of Connecticut, University of Notre Dame, University of Georgia, University of Oregon, University of Stavanger, York University and the annual meetings of the Financial Management Association (FMA), FMA European Meetings, the European Financial Management Association, and the Northern Finance Association.
sbett@jmsb.concordia.ca; b.espen.eckbo@dartmouth.edu; rex@mail.cox.smu.edu; karin.thorburn@nhh.no.}}
\author{\bf Sandra Betton\\
John Molson School of Business, Concordia University\\[0.25cm]
\bf B. Espen Eckbo\\
Tuck School of Business at Dartmouth\\[0.25cm]
%Norwegian School of Economics\\[0.25cm]
\bf Rex Thompson\\
Cox School of Business, Southern Methodist University\\[0.25cm]
\bf Karin S. Thorburn\\
Norwegian School of Economics \\}
\date{May 2011}% \\[0.50cm]
%JEL classifications: G3, G34 \\[0.25cm]
%Keywords: Takeover bidding, offer premium, market anticipation, runup, markup pricing}
\maketitle
\begin{abstract}
\noindent
Merger negotiations routinely occur amidst economically significant a target stock price runups. Since the source of the runup is unobservable (is it a target stand-alone value change and/or deal anticipation?), feeding the runup back into the offer price risks ``paying twice" for the target shares. We present a novel structural empirical analysis of this runup feedback hypothesis. We show that rational deal anticipation implies a nonlinear relationship between the runup and the offer price markup (offer price minus runup). Our large-sample tests confirm the existence of this nonlinearity and reject the feedback hypothesis for the portion of the runup not driven by the market return over the runup period. Also, rational bidding implies that bidder takeover gains are increasing in {\it target} runups, which our evidence supports. Bidder toehold acquisitions in the runup period are shown to fuel target runups, but lower rather than raise offer premiums. We conclude that the parties to merger negotiations interpret market-adjusted target runups as reflecting deal anticipation.
%Negotiating a takeover while observing a target stock price runup presents the bidder with a dilemma: does the information in the runup justify correcting the planned offer? While the true cause of the runup is unobservable, runups driven by deal anticipation require no corrective actions while runups driven by target stand-alone value increases do. We present a structural empirical analysis which illuminates how bidders tend to resolve this dilemma in practice. Recognizing that rumors about synergistic takeovers jointly affect the takeover likelihood and the expected takeover premium, we show that deal anticipation implies a highly nonlinear relationship between runups and premiums. Large-sample tests confirm the existence of this nonlinearity in the data. Deal anticipation also implies that {\it bidder} announcement returns should be increasing in target runups, which the data also confirms. Finally, we show that bidder toehold acquisitions in the runup period, which fuel target runups, lower rather than raise offer premiums. We conclude that the parties to merger negotiations interpret market-adjusted target runups as reflecting deal anticipation while, at the same time, correcting the offer for market-induced target runups.
\end{abstract}
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\section{Introduction\label{sec:intro}}
There is growing interest in the existence of informational feedback loops in financial markets. A feedback loop exists if economic agents take corrective actions based on information inferred from security prices. We analyze this phenomenon in the rich context of merger negotiations and takeover bidding. Our setting is one where a bidder is in the process of finalizing merger negotiations, and where the target management demands an increase in the planned offer price to reflect a recent runup in the target's stock price. The target argues that the runup reflects an increase in its market value as a {\it stand-alone} entity (i.e. without a control-change) and therefore justifies correcting the already planned offer.
The problem for the bidder is that the runup may alternatively reflect rumor-induced market anticipation of the pending deal, and so a markup of the bid risks "paying twice". After all, takeover bids are frequently preceded by media speculations, occasionally fueled by disclosures of large open market purchases of target shares (toeholds). %\citep{MikkRuba85,JarrPoul89}.
Also, there is large-sample evidence that target runups tend to be reversed absent a subsequent control change---when all bids fail and the target remains independent \citep{BettEckbThor09}. This reversal is inconsistent with runups representing increase in target stand-alone values. Last, but not the least, bidders should be weary of target incentives to overstate the case for offer price markups regardless of its true source.
Absent a clearly identifiable source for the runup, a rational (Bayesian) response may be to assign some positive probability to both the deal anticipation and stand-alone scenarios. For example, since general market movements during the negotiations may cause changes in the target's stand-alone value, it may be reasonable to expect some impact of runups on bid premiums. However, the issue is complex, as there are a number of alternative bid strategies. For example, bidders may choose to ignore the runup initially, leaving it to competing bidders to ``prove" undervaluation, and walk away if the final premium becomes too high. More complex bidding strategies emerge if one assumes that the market is better informed than the bidder \citep{BondGoldPres10}, or if the seller reserve price is stochastic \citep{KhorDodo07}. The latter is reminiscent of our context where merger negotiations are linked to the information content of the target stock price runup.
%\footnote{We do not derive optimal bid strategies involving target runups in this paper \citep{KyleVila91,BagnLipm96,RaviSpie99Bris02}.}
We pose the following type of questions: Do takeover rumors which cause target stock price runups feed back into higher offer prices? What are testable implications of a scenario where the bidder ``pays twice" by transferring the runup to the target in the absence of changes in the target stand-alone value during the runup period? Do toehold purchases in the runup period fuel runups and make the takeover more costly for the bidder?
To address these issues, we use a rational economic bidding model which permits takeover rumors to simultaneously affect the probability of a bid and the expected value of the offer conditional on a bid being made. We begin by proving that the relationship between runups and offer price markups is highly nonlinear under deal anticipation. Moreover, the model implies that bidder takeover gains are increasing in the {\it target} runups. These pricing implications have been overlooked by previous empirical research on takeovers, and they reverse an influential conclusion by \cite{Schw96} that target runups tend to lead to costly offer markups.
%\footnote{We show that the predicted nonlinear fit is superior to a linear projection in terms of reducing residual serial correlation in the data. This conclusion holds for several alternative definitions of markup and runup, and it is robust to a number of controls for bidder-, target- and deal characteristics.}
We also prove that with rational bidding, feeding the runup back into the offer price implies a {\it positive} relationship between runups and offer price markups. The intuition is that, as synergy signals improve (causing larger runups), bidder deal benefits increase faster than the deal probability declines. Our evidence rejects this implication of the feedback hypothesis as the projection of markups on runups yields a significantly negative slope. Bids do, however, appear to be corrected for the portion of the target runup which is driven by the (exogenous) market return over the runup period. Since a correction for the market return does not alter the sharing of synergy gains, it does not make the takeover more expensive for the bidder.
Finally, we present evidence on the effect of bidder open-market purchases of target shares (toeholds) during the runup period (short-term toeholds).
%Previous large-sample studies of toehold bidding do not separate short-term-- from long-term toeholds. %\citep{BettEckb00,BettEckbThor09}.
Short-term toehold purchases are interesting in our context because they may create takeover rumors and fuel target runups.
%The net effect of a toehold-driven increase in runup is unclear, however, because toeholds may also enhance the bidder's bargaining power by deterring rival bidders and thus lower offer premiums \citep{BuloHuanKlem99}.
%\citep{MikkRuba85,BagnLipm96},\citep{RaviSpie99,Bris01}, attract target resistance \citep{GoldQian05,BettEckbThor09}
We find that runups are in fact greater for takeovers with toehold acquisitions in the runup period. Nevertheless, as reported previously \citep{BettEckb00,BettEckbThor09}, toeholds reduce rather than increase offer premiums, possible because toeholds improve the bidder's bargaining position \citep{BuloHuanKlem99}. We find no evidence that toeholds acquired during the runup period increase the cost of the takeover.\footnote{While not pursued here, our lack of evidence of a feedback loop from (conditional) target runups to offer prices suggests also that {\it expected} (unconditional) runups have little if any impact on the expected value of bidding so long as producing synergy gains also requires unique bidder resources. For an empirical analysis of ex ante effects of market feedback loops, see \cite{EdmaGoldJian09}.}
The rest of the paper is organized as follows. Section 2 lays out the dynamics of runups and markups as a function of the information arrival process surrounding takeover events, and it discusses predictions of the deal anticipation hypothesis. Section 3 performs our empirical analysis of the projections of markups on runups based on the theoretical structure from Section 2. Section 4 shifts the focus to the relationship between target runups and bidder takeover gains, developing both theory and tests. Section 5 examines toehold purchases in the runup period, while Section 6 concludes the paper.
\section{Projections of markups on runups: Theory}
This section analyzes the information arrival process around takeovers, and how the information in principle affects offer prices and, possibly, feeds back into offer price corrections. It is instructive to begin by providing a sense of the economic importance of the target runup in our data. Figure \ref{fig:runup} illustrates the takeover process which begins with the market receiving a rumor of a pending takeover bid, resulting in a runup $V_R$ of the target stock price. In our vernacular, $V_R$ is the market feedback to the negotiating parties prior to finalizing the offer price.
%\footnote{For other types of models of information arrival processes around takeovers, see \cite{MalaThom85}, \cite{LaneThom88}, and \cite{EckbMaksWill90}.}
Since the exact date of the rumor is largely unobservable, $V_R$ is measured over a {\it runup period} which we in our empirical analysis take to be the two calendar months prior to the first public bid announcement. Figure \ref{fig:runup} shows an average target abnormal (market risk adjusted) stock return of about 10\% when cumulated from day -42 through day -2.\footnote{Our sample selection procedure is explained in section \ref{sec:data} below.}
In the theory below, we define the expected offer price markup as $V_P-V_R$, where $V_P$ denotes the expected final offer premium. In Figure \ref{fig:runup}, this is shown as the target revaluation over the three-day announcement period (day -1 through day +1). The initial offer announcement does not fully resolve all uncertainty about the outcome of the takeover (it may be followed by a competing offer or otherwise rejected by target shareholders), and so $V_P$ is the expected final offer premium conditional on a bid having been made. The average two-day target announcement return is about 25\% in the full sample of takeovers.
The challenge for the negotiating parties is to interpret the information in this runup: does it justify correcting (marking up) the already planned bid? In some cases, the runup may reflect a known change in stand-alone value which naturally flows through to the target in the form of a higher offer premium. In other cases, the target management may have succeeded in arguing that the runup is driven by stand-alone value changes when it is not (and so feeding the runup back into the offer price amounts to ``paying twice"). The point of our analysis is not to rationalize a specific bargaining outcome but to derive testable implications for the relationship between runups, markups, and bidder returns.
We begin by analyzing the case where the negotiating parties agree that the target runup is driven by deal anticipation only. This is followed by the presence of a known target stand-alone value change in the runup period. Finally, the analysis covers the feedback hypothesis where the offer price is marked up with the target runup even in the absence of a target stand-alone value change.
\subsection{Pricing implications of deal anticipation\label{sec:anticipation}}
Suppose the market receives a signal $s$ which partially reveals the potential for synergy gains $S$ from a takeover. $S$ is known to the bidder and the target, while the market only knows the distribution over $S$ given the signal. The bid process involves a known sharing rule for how the synergy gains will be split between target and bidder, and a known bidding cost $C$. Depending on the sharing rule for the synergy gains and bidding costs, there exists a threshold, $K$, in $S$ above which the benefit to the bidder of making a bid is positive.
%\footnote{The cost $C$ include things such as fees, litigation risk, and, quite possibly, the opportunity cost of expected synergy gains from a better business combination than the target under consideration. The question of whether or not bids to targets are set so that targets share in the cost of extending bids is an interesting empirical question. We use a benefit function for bidders and targets which allow bidders and targets to share the bidding costs, as the case may be.}
$B(S,C)$ denotes the benefit to the target of takeover, i.e., its portion of the total synergy gains $S$ net of the target's portion of the bidding cost $C$. We assume that $B(S,C)=0$ if no bid takes place (which occurs when $S20\%$ new equity}), and a hostile (vs. friendly or neutral) target reaction ({\it Hostile}), respectively, are all negative and significant. Finally, contests starting with a tender offer are more likely to succeed, as are contests announced in the 1990s and the 2000s. The dummy variable indicating an all-cash bid generates a significantly negative coefficient only when controlling for the time period (Column 2).
There are a total 6,103 targets with available data on the characteristics used in the probit estimation. For each of these, we multiply the markup with the estimated success probability computed using the second model in Table \ref{tab:prob} (which includes the two decade dummies). This "expected markup" is then used in the nonlinear projection (3) reported in Table \ref{tab:nonlinear}. Interestingly, the nonlinear form are now again able to remove the significant linear residual serial correlation from 0.027 (t=2.11) to an insignificant 0.016 (t=1.25) with the nonlinear estimation.\footnote{Table \ref{tab:prob}, in columns 3-6, also show the coefficients from probit estimations of the probability that the initial control bidder wins the takeover contest. The pseudo-$R^2$ is somewhat higher than for this success probability, ranging from 22\% to 28\%. Columns 3 and 4 use the same models as the earlier estimations of contest success, while columns 5 and 6 add a variable capturing the percent of target shares owned by the initial control bidder at the time of the bid ({\it Toehold size}). Almost all explanatory variables generate coefficients that are similar in size, direction, and significance level to the ones in the probit regressions of contest success. The reason is that in the vast majority of successful contests, it is the initial bidder who wins control of the target. The only difference between the probability estimations is that the existence of a target poison pill does not substantially affect the likelihood that the initial bidder wins. The larger the initial bidder toehold, however, the greater is the probability that the initial bidder wins.}
\subsubsection{Information known before the runup period\label{sec:before}}
Up to this point, we have assumed that the market imparts a vanishingly small likelihood of a takeover into the target price before the beginning of the runup period (day -42 in Figure \ref{fig:runup}). However, the market possibly receives information prior to the runup period that informs both the expected bid if a bid is made and the likelihood of a bid. To illustrate, consider the case where the market has a signal $z$ at time zero. During the runup, the market receives a second signal $s$ and, finally, a bid is made if $s+z$ exceeds a threshold level of synergy gains.
Working through the valuations, we have one important change. Define $\pi(z)E(B|z)$ as the expected value of takeover prospects given $z$ and a diffuse prior on $s$. We then have that, at time zero in our model (event day -42 in our empirical analysis), $V_0 = \pi(z)E(B|z),$ and the runup and the bid premium would now be measured relative to $V_0$. Instead of $V_R$, the runup measured over the runup period is now
\begin{equation}
\label{eq:info1}
V_R-V_0= \pi(s+z)E_{s+z}[B(S,C)]+T|s+z,bid]+[1-\pi(s+z)]T-\pi(z)E(B|z),
\end{equation}
and the premium is
\begin{equation}
\label{eq:info2}
V_P-V_0=E_{s+z}[B(S,C)+T|s+z,bid]-\pi(z)E(B|z)]-\pi(z)E(B|z).
\end{equation}
Setting aside the influence of $T$, for an investigation into the nonlinear influence of prior anticipation, one would want to add back $V_0$ to both the runup and the bid premium. Since the influence of $V_0$ is a negative one-for-one on both quantities, markups are not affected.
In order to unwind the influence of a possibly known takeover signal $z$ prior to the runup period, we use the following three deal characteristics defined in Table \ref{tab:def}: $Positive\ toehold$, $Toehold\ size$, and the negative value of $52-week\ high$. The positive toehold means that the bidder at some point in the past acquired a toehold in the target. There is empirical evidence that toehold acquisitions announced through 13(d) filing with the SEC cause some market anticipation of a future takeover \citep{MikkRuba85}. Moreover, it is reasonable to assume that the signal is increasing in the size of the toehold. Moreover, \cite{BakePanWurg10} report an impact of the target's 52-week high on takeover premiums.
Using these variables, model (4) reported in Table \ref{tab:nonlinear} implements two multivariate adjustments to model (1). The first adjustment, as dictated by eq. (\ref{eq:info1}), augments the runup by adding $R_0$, where $R_0$ is the projection of the total runup ($\frac{P_{-2}}{P_{-42}}-1$) on $Positive\ toehold$, $Toehold\ size$, and the negative value of $52-week\ high$. The second adjustment is to use as dependent variable the "residual markup $U_P$, which is the residual from the projection of the total markup, $\frac{OP}{P_{-2}}-1$, on the deal characteristics used to estimate the success probability $\pi$ in Table \ref{tab:prob}, \underline{excluding} $Positive\ toehold$, $Toehold\ size$, and $52-week\ high$ which are used to construct the augmented runup.
Model (4) in Table \ref{tab:nonlinear} shows the linear and nonlinear projections of the residual markup on the augmented runup. The linear slope remains negative and highly significant (slope = -0.36, t = -12.1). The serial correlation of the ordered residuals from the linear projection is 0.052 with t = 4.03. After the nonlinear fit, the serial correlation drops to 0.031 with a t of 2.45. In this experiment the shape looks similar to the other nonlinear fits except that the right tail tips upward slightly. Thus, this evidence also supports the presence of a deal anticipation effect in the runup measured over the runup period.
\subsubsection{Projections using abnormal stock returns}
The last two projections in Table \ref{tab:nonlinear} use cumulative abnormal stock returns $(CAR)$ to measure both the markup, $CAR(-1,1)$, and the runup, $CAR(-42,-2)$. In projection (5), $CAR$ is estimated using the market model amd in projection (6) we use the CAPM. The parameters of the return generating model are estimated on stock returns from day -297 through day -43. The $CAR$ uses the model prediction errors over the event period (day -42 through day +1).
In projection (5), the linear residual serial correlation is a significant 0.039 (t=3.10), which is almost unchanged in the nonlinear form. Thus, we can reject the linearity of the projection. However, our specific nonlinear fit fails to remove the serial correlation. Interestingly, notice from Panel B of Figure \ref{fig:fit} that the shape of our nonlinear form looks very much like the form in Panel A, in which the nonlinear form does succeed in eliminating the residual serial correlation.
\section{Target runups and rational bidding}
We have so far examined the relationship between offer markups and target runups. We now turn to implications of rational bidding for the relationship between bidder takeover gains and target runups given that bids are made. These implications turn out to be important for the overall conclusion of our empirical analysis.
\subsection{Bidder valuations: Theory}
Let $\nu$ denote bidder valuations, again measured in excess of stand-alone valuation at the beginning of the runup period. Valuation equations for the bidder are:
%\footnote{During the runup period, any effect on bidders might be diluted by a lack of resolution about which potential bidder sees the highest synergy gains and thus which potential bidder will reap the bidder's gains, if any, from the takeover. In the special case where there is only a single, known bidder, the observation of $s$ results in a runup in the bidder's value and, as with targets, a bid resolves additional uncertainty and causes a valuation response for bidders.}
\begin{equation}
\label{eq:bvp}
\nu_R=\int^{\infty}_K (S-C-B(S,C))g(S)dS,
\end{equation}
where $\nu_R$ has the same interpretation as $V_R$ for targets. At the moment of a bid announcement, but without knowing precisely what the final bid is, we again have that
\begin{equation}
\label{eq:bvr}
\nu_P=\frac{1}{\pi(s)}\int^\infty_K (S-C-B(S,C))g(S)dS.
\end{equation}
The observed valuation of the bidder after the bid is announced includes an uncorrelated random error around the expectation in equation (\ref{eq:bvr}) driven by the resolution of $S$ around its conditional expectation.
\hopp
\noindent{\bf Proposition 4 (rational bidding):} {\it Let $G$ denote the bidder net gains from the takeover ($G=S-C-B$). For a fixed benefit function $G$, rational bidding behavior implies the following:
\hopp (i) Bidder and target synergy gains are positively correlated: $Cov(G,B)>0$.
\hopp(ii) Bidder synergy gains and target runup are positively correlated: $Cov(G,V_R)>0$.
\hopp (iii) The sign of the correlation between $G$ and target markup $V_P-V_R$ is ambiguous.}
\hopp
\noindent{\bf Proof:} See Appendix A.
\hopp
Rational bidding inour context means that the bidder conditions the bid on the correct value of $K$. Figure \ref{fig:bidtheory} shows the theoretical relation between the bidder expected benefit $\nu_P$ and the target runup $V_R$ for the uniform case with $\theta=0.5$ and $\gamma=1$. In panels A and B, the bidder rationally adjusts the bid threshold $K$ to the scenario being considered: In Panel A, there is no transfer of the runup to the target, and so $K=\frac{\gamma C}{\theta}$ as in equations (\ref{eq:tvr}) and (\ref{eq:tvp}). In Panel B, the bidder transfers the runup, but also rationally adjusts the synergy threshold to $K^*=\frac{\gamma C+V_R}{\theta}$ (as in Section \ref{sec:paytwice} above). In either case, the bidder expected benefit $\nu_P$ is increasing and concave in the target runup.
In Panel C of Figure \ref{fig:bidtheory}, the bidder transfers the target runup but fails to rationally adjust the bid threshold from $K$ to $K^*$. In this case, the bidder expected benefit is declining in $V_R$ except at the very low end of the synergy signals which create very small runups. Panel $C$ of Figure \ref{fig:bidtheory} joins Figure \ref{fig:paytwice} and suggest that a powerful test of the feedback hypothesis is to test whether (1) the correlation between $\nu_P$ and $V_R$ is negative and (2) that the projection of $V_P-V_R$ on $V_R$ is positive. We have already shown that the data rejects the second of these two implications, and now turn to the first.
\subsection{Are bidder gains increasing in target runups?}
Proposition 4 and Panels A and B of Figure \ref{fig:bidtheory} show that, with rational market pricing and bidder behavior, bidder takeover gains $\nu_P$ are increasing in the target runup $V_R$. $\nu_P$ is decreasing in the target runup only if bidders fails to rationally compute the correct bid threshold level $K$. In this section we test this proposition empirically using the publicly traded bidders in our sample.
We estimate $\nu_P$ using the prediction errors from a market model regression estimated over the period from day -297 through day -43 relative to the initial offer announcement date. The prediction errors are then cumulated over the period -42 through +1 relative to the initial bid announcement date. The best linear projection of $\nu_P$ on $V_R$ produces a significantly positive slope coefficient of 0.045 with a t-value of 3.45 (the intercept is -0.019, t=-6.16). The linear residual serial correlation is an insignificant 0.021 (t=1.27). After fitting the nonlinear model, the residual serial correlation drops to 0.016 (t=0.99).
A second model wherein $\nu_P$ is projected on the augmented target runup described in Section \ref{sec:before} (to account for information about merger activity prior to the runup period), produces an almost identical linear intercept and slope along with a similar nonlinear shape (see Panel B of Figure \ref{fig:bidderfit}).
Both Panels A and B of Figure \ref{fig:bidderfit} show that the nonlinear fits are upward sloping and concave in $V_R$. This is as predicted by our theory, and the empirical shapes in Figure \ref{fig:bidderfit}) have a striking visual similarity to the theoretical projections in panels A and B of Figure \ref{fig:bidtheory}. The positive and monotone relationship between $\nu_P$ and $V_R$ rejects irrational bidding and is consistent with deal anticipation in the runup. It rejects {\it a fortiori} the negative relation shown in Panel C of Figure \ref{fig:bidtheory} implied by the case where the bidder agrees to a full transfer of the runup to the target without rationally adjusting the bid threshold $K$.
Changing focus to bidder announcement returns $\nu_P-\nu_R$, it is common in the merger literature to show the results of linear cross-sectional regressions on deal-specific characteristics. Table \ref{tab:bidder} does the this for our total sample of 3,624 publicly traded initial bidders. Our regressions contain the variables $Net\ Runup$ and $Runup$ which are new to the literature. The dependent variable is the bidder markup $\nu_P-\nu_R$, i.e., bidder abnormal return over the three-day window centered on the announcement day. The first three columns focus on the effect of the target net runup (the runup net of the average market return), while the last three columns focuses on the effect of the target raw runup on bidder returns. The regressions in columns 2 and 5 include year fixed effects, while columns 3 and 6 add a dummy indicating that the initial bidder ultimately wins the contest. The p-values use \cite{Whit80}'s heteroscedasticity-consistent standard errors.
$Net \ Runup$ receives a positive coefficient in all specifications in Table \ref{tab:bidder}, ranging from 0.11 to 0.13 and with a p-value between 0.036 and 0.063. The coefficient on $Runup$ is 0.13 and significant at the 5\%-level in all specifications, suggesting that the positive effect on bidder returns is robust to the choice of runup-estimate and the inclusion of year dummies. In sum, we conclude that bidder announcement returns are non-decreasing (and likely increasing) in the target runup.\footnote{Several of the other explanatory variables in Table \ref{tab:bidder} receive significant coefficients. As expected, bidder returns increase in the relative size of the target, and all-cash bids are associated with higher bidder returns, while all-stock bids tend to generate lower bidder returns. Moreover, acquirer announcement returns are on average higher the lower the liquidity of the target stock ($Turnover$) and the larger the bidder toehold ($Toehold \ size$).}
\section{Effects of toehold purchases in the runup period}
\subsection{Hypothetical toehold costs/benefits}
In theory, bidders have an incentive to acquire target shares in the target (a toehold) prior to making the bid. Bidder toehold benefits include a reduction in the number of target shares that must be acquired at the full takeover premium, and the expected profits from selling the toehold should a rival bidder win the target.
However, as documented by \cite{BettEckbThor09}, toeholds are rare, indicating that such bidder toehold benefits are largely offset by toehold costs. The takeover literature identifies several potential sources of toehold costs, ranging from market illiquidity \citep{RaviSpie99} and information disclosure \citep{JarrBrad80,Bris02} to target resistance costs \citep{GoldQian05,BettEckbThor09}. We add to this literature by considering effects of toeholds purchased in the runup period---which we call ``short-term toeholds". Specifically, we are interested in the possibility that short-term toehold acquisitions fuel costly target stock price runups.
An open-market purchase of a short-term toehold may provide rivals with time and information to prepare competing bids. There are several ways in which information in the toehold purchase reaches the market. First, if the target stock is illiquid, the purchase may cause abnormal movements in the stock price and attract investor scrutiny. Second, block trades and abnormal trading volume may also trigger media speculations that a firm is in play. For example, \cite{JarrPoul89} show that media speculations result in significant share price runups. Third, the information in Schedule 13(d) filings discloses the toehold purchaser's intentions with the target, causing price effects \citep{MikkRuba85}. Fourth, the toehold purchase may trigger a pre-merger notification under the 1976 Hart-Scott-Rodino Antitrust Improvements Act.%\footnote{As of October 2005, pre-merger notification is required for transactions of \$212 million or more, and for transactions exceeding \$53 million if one party has assets or revenues of at least \$106 million and the other party of at least \$11 million. Merger-notifications are typically made public with a two-month or longer delay.}
Thus, a decision to acquire a short-term toehold must be weighted against the odds that the toehold creates costly competition. Competition among bidders raises the target's bargaining power and increases the target's share ($1-\theta$) of the total synergy. This increase in turn manifests itself through a higher markup and total offer premium, and in reduced bidder gains. However, this is not the only possibility: because the expected toehold benefits allow the acquirer to bid more aggressively, toeholds may also {\it deter} competition \citep{BuloHuanKlem99} and allow the bidder to win more often. In this case, the target's share of the synergy gains are decreasing in the toehold.
%\footnote{\cite{BettEckboThor09} develops a model in which some target resist toehold bidders because toeholds are costly to targets. Their model gives rise to a toehold threshold defined by the condition that the expected bidder toehold benefit equals the expected bidder cost of target rejection. In equilibrium, bidders approach the target either with zero toehold (to avoid resistance) or with toeholds exceeding the threshold size. This is consistent with the fact that (i) toeholds are large when they occur, and (ii) the overall toehold frequency is low (about 15\%) except in hostile bids where it is high (around 50\%).}
The effect of a toehold purchase on expected bidder gains is therefore an open empirical question. If toeholds increase the bidder's bargaining power, the offer premium should be decreasing in the toehold (whether the toehold is held long- or short term prior to the offer). To address this issue, and to examine short-term toehold effects in the runup period, we use SDC to collect block trades during the runup period for our sample targets. We also record whether the block is purchased by the bidder or some other investor. Presumably, the potential for a block trade to fuel a costly target runup is greatest when the buyer is a potential acquirer.
\subsection{Toeholds, runups, and offer premiums (H4)}
Our toehold data is summarized in Table \ref{tab:toe}. We sample short-term toeholds in SDC over the period 1980-2008 by selecting "acquisitions of partial interest", where the bidder seeks to own less than 50\% of the target shares. As shown in Panel A, over the six months preceding bid announcement [-126,0], the initial control bidders acquire a total of 136 toeholds in 122 unique target firms. Of these stakes, 104 toeholds in 94 different targets are purchased over the 42 trading days leading up to and including the day of the announcement of the initial control bid. Thus, less than 2\% of our initial control bidders acquire a toehold in the runup period. For 98\% of the target firms, the initial control bidder does not buy any short-term toehold. The typical short-term toehold acquired by the initial bidder in the runup period is relatively large, with a mean of 12\% (median 9\%).
The timing of the toehold purchase during the runup period is also interesting. Two-thirds of the initial control bidders' toehold acquisitions are announced on the day of or the day before the initial control bid [-1,0]. Since the SEC allows investors ten days to file a 13(d), we infer that these toeholds must have been purchased sometime within the 10-day period preceding and including the offer announcement day. For these cases, the target stock-price runup does not contain information from a public Schedule 13(d) disclosure (but will of course still reflect any market microstructure impact of the trades). The remaining short-term toeholds are all traded and disclosed in the runup period.
Panels B and C of Table \ref{tab:toe} show toehold purchases by rival control bidders (appearing later in the contest) and other investors. Rival bidders acquire a toehold in the runup period for only 3 target firms. The average size of these rival short-term toeholds are 7\% (median 6\%). Other investors, not bidding for control in the contest, acquire toeholds in 73 target firms (1\% of target firms) during the 42 days preceding the control bid. The announcement of 21\% (18 of 85) of these toeholds coincide with the announcement of the initial control bid, suggesting that rumors may trigger toehold purchases by other investors. Overall, there are few purchases of toeholds in the two-month period leading up to the initial control bid.
Table \ref{tab:runup} shows determinants of target net runups (columns 1-2) and initial offer premiums (columns 3-6) as a function of toeholds.\footnote{Because the toehold decision is endogenous, we developed and tested a \cite{Heck79} correction for endogeneity by including the estimated Mill's ratio \citep{Madd83} in Table \ref{tab:runup}. See \cite{BettEckbThor08b} for details of this analysis. As shown there, the coefficient on the Mill's ratio is not statistically significant, and it is therefore not included here.} The dummy variables $Stake \ bidder$ and $Stake \ other$ indicate toehold purchases by the initial control bidder and any other bidder (including rivals), respectively, in the runup window through day 0.
Notice first that short-term toehold purchases by investors other than the initial bidder have a significantly positive impact on the net runup in both regressions. Furthermore, short-term toehold purchases by the initial bidder also increase the net runup, but with less impact on the runup: the coefficient for $Stake \ bidder$ is 0.05 compared to a coefficient for $Stake \ other$ of 0.12. While short-term toeholds tend to increase the runup, the total bidder toehold has the opposite effect. The variable $Toehold \ size$, which measures the fraction of the target's shares held by the initial bidder at the time of bid announcement, enters with a negative and significant sign. In sum, the evidence shows that only the short-term toehold purchases have a positive impact on target runups.\footnote{Several of the other explanatory variables for the target net runup receive significant coefficients. The smaller the target firm ($Target\ size$) and the greater the relative drop in the target stock price from its 52-week high (52-$week\ high$), the higher the runup. Moreover, the runup is higher when the acquirer is publicly traded and in tender offers, and lower for horizontal takeovers. The inclusion of year-fixed effects in the second column does not change any of the results.}
Table \ref{tab:runup} also shows coefficient estimates from OLS regressions for the initial offer premium. In addition to the explanatory variables from the regression model for the net runup, columns 3 and 4 further include the market return during the runup period. The offer premium is decreasing in $Toehold\ size$, a result also reported earlier by \cite{BettEckb00} and \cite{BettEckbThor09}.\footnote{Note that the dummy for short-term toehold purchases have no separate impact on offer premiums, irrespectively of whether the purchase is by the initial control bidder or another investor.}
%In other words, while short-term toehold acquisitions tend to increase the runup, the bidder adjusts for this effect in determining the offer premium.
Offer premiums are also decreasing in $Target\ size$ and in 52-$week\ high$, both of which are highly significant. Moreover, as noted earlier in Section \ref{sec:nonlinear}, offer premiums increase with the market return over the runup period, with a highly significant coefficient of 0.926 on $Market\ runup$.\footnote{Offer premiums are also higher in tender offers and when the acquirer is publicly traded. The greater offer premiums paid by public over private bidders is also reported by \cite{BargSchlStulZutt07}.}
Finally, the two last regression models in columns 5 and 6 of Table \ref{tab:runup} add the net runup. $Net\ runup$ is highly significant but its inclusion does not affect the size or significance of the other variables. Not surprisingly, inclusion of the net runup increases the regression $R^2$ substantially, from 8\% to 34\%.
In sum, despite fueling target runups, toehold bidding conveys a bidder benefit by lowering the total offer premium in the deal.
\section{Conclusions}
There is growing interest in financial market feedback loops, in which economic agents may take corrective action based on information in stock price changes. We study this phenomenon in the rich context of offer price bargaining where the negotiating parties observe an economically significant target stock price runup. Since the true source of the runup (target stand-alone value change and/or deal anticipation) is unobservable, it is subject to interpretation by the bargaining parties which may result in a correction of the planned offer price. The problem for the bidder is that correcting the offer price when the runup reflects deal anticipation means ``paying twice" for the target shares. Thus the outcome of this feedback loop is important for the incentive to make bids and, therefore, for the efficiency of the takeover process.
Earlier papers have examined this issue empirically by estimating the slope coefficient in linear cross-sectional regressions of the offer price markup (the offer price minus the runup) on the runup. Causal intuition suggests that this slope coefficient should be negative one under the deal anticipation hypothesis, as the runup substitutes dollar for dollar for the offer price markup. By extension, finding a coefficient greater than negative one has been taken to mean that runups tend to result in costly offer price markups.
%\footnote{``The evidence...suggests that, all else equal, the [pre-bid target stock price] runup is an added cost to the bidder."} \citep[p.190]{Schw96}.}
Our structural empirical analysis leads to different conclusions. First, we show that rational market pricing implies a theoretical slope coefficient involving the conditional takeover probability ($\pi(s)$) when markups are projected on runups. This means that the projection is generally nonlinear and non-monotonic in the synergy signal ($s$), and so linear projections will produce slope coefficients that are strictly greater than negative one---and need not even be different from zero. We also show that when bidders mark up offers with target runups that are caused by deal anticipation (the feedback hypothesis), the implied relation between markups and runups is positive.
Our large-sample projection of offer price markups on runups yields a highly significant linear slope coefficients of $-0.2$. Using a residual serial correlation test, we also reject linearity and show that the nonlinear form of the projection corresponds closely to the theoretical expectation. Thus, the data is consistent with the deal anticipation hypothesis for runups. Moreover, our finding of a significantly negative slope coefficient estimate directly rejects the hypothesis that bidders feed back the runup into the offer price. Another interesting discovery is that offer premiums tend to be marked up by the {\it market return} over the runup period. It appears that the market component of the runup is interpreted by the negotiations as an exogenous change in the target's stand-alone value (which may be transferred to the target without the risk of ``paying twice").
We also prove that market efficiency coupled with rational bidding implies a positive relationship between bidder takeover gains and {\it target} runups. This is true even if the outcome of merger negotiations is to feed the runup back into the offer price. With endogenous target runups there still exists an equilibrium with observed bids---but these necessarily have relatively large expected synergy gains to finance the runup transfer. Conversely, if bidders fail to rationally adjust their bid thresholds with the cost of the runup, the relationship between observed bidder gains and target runups will be negative. The estimated relationship is significantly positive in our data, which rejects bidder irrationality.
Finally, we investigate the effect of bidder toeholds in the target. Bidding theories suggest that toeholds increase bidder bargaining power and thus affect offer premiums. Consistent with previous toehold studies, we find that bids which involve toeholds on average command lower premiums than bids without toeholds. Our analysis separate out toehold purchases in the runup period and find that these tend to fuel fuel target runups. However, there is no evidence that the greater target runup causes offer premium markups. A consistent explanation is that bidders are able to convince the target that the extra runup reflects deal anticipation triggered by the toehold purchase rather than a change in the target stand-alone value.
%The parties negotiating a merger must determine whether a contemporaneous target stock price runup justifies a markup of the offer price. Since the true cause of the runup cannot be determined with certainty, it is subject to interpretation. It is in the target's interest to argue that the runup reflects a surprise increase in the firm's stand-alone value, thus justifying an offer price markup. The bidder, on the other hand, will argue that the runup is driven by takeover rumors and thus represent simply the capitalized value of the already planned offer premium. Under this view, marking up the offer is tantamount to ``paying twice" for the target. With runups averaging as much as 12\% in our data, the consequence of misinterpreting the runup as increased stand-alone value when it is truly a result of market anticipation is of fundamental important for the efficiency of the takeover process. The resolution of this conflict is the subject of this paper.
%The tradition in the takeover literature has been to view offer premiums as exogenous to the runup---with no feedback from the runup to the premium. This view is generally supported by the widespread speculations and rumors often preceding takeover bids, and from the fact that most of the runup occur very close to (within ten days of) the offer date itself. Moreover, empirical research which shows that target runups tend to be reversed when takeover bids fail (and the target remains independent) gives further credence to the deal anticipation hypothesis for the initial runup.
%We provide a novel structural analysis of the true relation between premiums and runups under various assumptions about the source of the target runup. ......
%Our empirical analysis strongly supports that projections of offer price markups on runups is both nonlinear and of a form consistent with the anticipation theory. We also test directly whether runups tend to reduce bidder takeover gains. We find that bidder gains are {\it increasing} in the target runup. This finding contradicts the theory that runups lead to corrective markups. It is, however, as predicted by the deal anticipation hypothesis under the reasonable assumption that that total synergy gains varies in the cross-section of takeovers.
%Finally, we investigate large block purchases of target stock during the runup period (short-term target toehold acquisitions). We find that
%Finally, if short-term toehold purchases reveal valuable information to potential rival bidders, one would expect a positive effect on the rival entry probability. We find instead that total toeholds (the sum of short-term and long-term toeholds) {\it reduce} the likelihood of attracting rival bidders. Moreover, both short-term and total toeholds increase the initial bidder's probability of winning. Toehold bidding also reduces offer prices, possibly because it deters competition.
\newpage
\appendix
\section{Appendix: Proofs of lemma 1 and Proposition 4}
%This appendix presents proofs of equation (\ref{eq:UVP}) and propositions 2 and 3.
%Equation (\ref{eq:UVP}) assumes a uniform distribution for the synergy $S$ given the signal $s$: $S\sim U[s-\Delta,\ s+\Delta]$. Given the uniform, the density $g(S)$ is a constant, $g(s)=\frac{1}{s+\Delta-(s-\Delta)}=\frac{1}{2\Delta}$. Moreover, the takeover probability $\pi(s)=Prob[s\ge K]=\frac{s+\Delta-K}{2\Delta}$. Since the actual bid is $B(S,C)=(1-\theta)S-(1-\gamma)C$, the expected bid is
%\begin{eqnarray}
%V_P&=&\frac{1}{\pi(s)}\int_K^{s+\Delta} B(S,C)g(S)dS \nonumber \\
%&=&\frac{2\Delta}{s+\Delta -K}\int_K^{s+\Delta}[(1-\theta)S-(1-\gamma)C]\frac{1}{2\Delta}dS \nonumber \\
%&=&\frac{1}{s+\Delta-K}\left\{\frac{(1-\theta)S^2}{2}-(1-\gamma)CS\right\}^{s+\Delta}_K \nonumber \\
%&=&\frac{1}{s+\Delta-K}\left\{\frac{1-\theta}{2}[s+\Delta-K][s+\Delta+K]-(1-\gamma)C[s+\Delta-K]\right\} %\nonumber \\
%&=& \frac{1-\theta}{2}(s+\Delta+K)-(1-\gamma)C.
%\end{eqnarray}
%Noting that $K=\frac{\gamma C}{\theta}$ yields the expression for $V_P$ in equation (\ref{eq:UVP}). Moreover, %the expression for $V_R$ in equation (\ref{eq:UVP}) is $V_R=\pi(s)V_P=\frac{s+\Delta-K}{2\Delta}V_P$.
%\endproof
%\hopp
%\begin{quote}
\noindent {\bf lemma 1 (linear projection):} {\it With deal anticipation, and as long as the takeover probability $\pi$ is a function of the synergy gains $S$, a linear projection of $\ V_P-V_R$ on $V_R$ yields a slope coefficient that is strictly greater than -1, and the coefficient need not be different from zero.}
%\end{quote}
\hopp
\noindent{\bf Proof:}
For the first part of the lemma, it suffices to show that the maximum negative slope in the projection of
the expected markup on the runup is greater than -1. Differentiation of equation (\ref{eq:RATIO}) shows that the slope becomes more negative as $s$ increases. The most negative slope is at the maximum $s$ that still causes
uncertainty in the bid. This point is reached when $s=(\gamma/\theta)C+\Delta$. Substitution into the ratio
$(V_P-V_R)/V_R$ yields $-A/[A+(1-\theta)\Delta]$, where $A=(1-\theta)\Delta+ (\gamma/\theta)C$. Since $A\ge0$,
this ratio must be greater than -1. For the second part of the lemma, note that equation (\ref{eq:RATIO})
equals zero at the point where $s=(1-\gamma)/(1-\theta)C$. Such a point is viable whenever
$(1-\gamma)/(1-\theta)C+\Delta\ge \gamma C/\theta$, and $0<\theta<1$. There always exists a $\Delta$ for which
this is true.
\endproof
\hopp
\noindent{\bf Proposition 4 (rational bidding):} {\it Let $G$ denote the bidder synergy gains from the takeover. For a fixed benefit function $G$, rational bidding behavior implies the following:
\hopp (1) Bidder and target synergy gains are positively correlated: $Cov(G,B)>0$:
\hopp(ii) Bidder synergy gains and target runup are positively correlated: $Cov(G,V_R)>0$
\hopp (iii) The sign of the correlation between $G$ and target markup $V_P-V_R$ is ambiguous.}
\hopp
\noindent{\bf Proof:}
For the first part (i), recall that we have assumed that, if a bid is made, the bidder and target share in
the synergy gains ($1<\theta<1$), implying $0< \frac{\partial B(S,C)}{\partial S}<1$. It follows immediately
that both the bidder and target gains increase in $S$ throughout the entire range of $S$ wherein bids are
possible. This includes ranges over which bids are certain given the signal, $s$. In the case of our closed form example above, write out the target gain, $B=(1-\theta)S-(1-\gamma)C$, and the bidder gain, $G=\theta S-\gamma
C$. Clearly, both $B$ and $G$ are increasing (and linear) in $S$.\footnote{In the example, and measuring
$Cov(G,B)$ as the product of the derivatives of $G$ and $B$ w.r.t. $S$, $Cov(G,B)=\theta(1-\theta).$ This
means that the expected ``slope coefficient" of a projection of $G$ on $B$ equals $\theta/(1-\theta)$.}
To prove the rest of Proposition 4 it is necessary to work with the conditional distribution of $s$ given $S$,
which we denote $f(s|S)$. Knowledge of $f(s|S)$ is required to determine the expected value of the runup for a
given observed $S$, revealed when the bid is made. When $S$ is revealed through the bid, $s$ is random in the
sense that many signals could have been received prior to the revelation of $S$. For (ii), the covariance
between the target runup and the bidder gains is the covariance between the expected runup, at a given $S$, and
the bidder gain, at the same $S$. This covariance is measured by the product of derivatives so it suffices to
show that the derivative of the expected runup is always positive to prove the second part of the proposition.
To prove the last part (iii) of the proposition, it must be shown that the derivative of the expected markup is
not always less than $1-\theta$ for all $S$.
While proof of (ii) and (iii) can be generalized, we focus on the case where the prior distribution of $S$ is diffuse and the posterior distribution of $S$, given $s$ is uniform (our closed-form example). With diffuse prior, the law of inverse probability implies that $f(s|S)$ is
proportional to the posterior distribution of $S$ given $s$, or $f(s|S)\sim U(S-\Delta,\ S+\Delta)$.
We now have
\begin{equation}
\label{eq:prop}
E_s(V_R)\propto \int_{S-\Delta}^{S+\Delta}V_R(s)f(s|S)ds
\end{equation}
Since $f(s|S)$ is a constant in our case, differentiation by $S$ through Leibnitz rule gives the simple form:
\begin{equation}
\frac{\partial E_s(V_R)}{\partial S} = V_R(s=S+\Delta)-V_R(s=S-\Delta)
\end{equation}
By inspection, $V_R(s)$ in equation (\ref{eq:UVP}) is increasing in $s$. This establishes that the
$Cov(G, E(V_R))>0$ for all viable bids including bids which are certain.
Part (iii) of proposition 4 relates to $Cov(G,\ E_s(V_R))$. Since the target markup equals $B-E_s(V_R)$,
we need to evaluate the sign of the derivative of this difference with respect to $S$.
Define $E[M(S)] = B-E_s(V_R)$. Applying similar logic,
\begin{equation}
\frac{\partial E(M)}{\partial S}=(1-\theta)-[V_R(S+\Delta)-V_R(S-\Delta)]
\end{equation}
Inspection of Figure \ref{fig:uniform} clearly shows that the ``slope" of the difference in runups at
$S+\Delta$ an $S-\Delta$ depends on $S$ and need not be less than $1-\theta$,
the slope of $V_P$ in the figure. Thus the covariance between bidder gain and expected target markup need
not be positive in a sample of data drawn over any range of $S$. If the range of $S$ happens to cover
(uniformly) the entire range of viable bids that are uncertain, there is no clear covariance between bidder gain
and expected target markup.\endproof
%The derivatives of target benefit and expected target runup are both equal to $1-\theta$ in this situation.\endproof
%
%********************************
%figure 1: runup
%figure 2: uniform
%figure 3: normal
%figure 4: standalone
%figure 5: fit
\newpage
\begin{figure}[htbp]
\begin{center}
\leavevmode
\caption{The takeover information arrival process and average target runup in event time}
\label{fig:runup}
\begin{minipage}{\textwidth}\footnotesize
The runup period is measured from day -42 through day -1 relative to the first public offer for the target (day 0). The offer announcement period is day -1 through 0. The figure plots the percent average target abnormal (market risk adjusted) stock return in the runup and announcement periods for the total sample of 6,150 U.S. public targets (1980-2008). The average abnormal return in the runup period is 12\% while the announcement-period return (markup) is 25\%.
%\vspace{4mm}
\end{minipage}
\begin{minipage}{\linewidth}\footnotesize
\begin{center}
\includegraphics[width=18cm,angle=0]{Figure1-targetrunup}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\newpage
\begin{figure}[htbp]
\begin{center}
\leavevmode
\caption{\bf Theoretical markup projections with only deal anticipation in the runup.}
\label{fig:uniform}
\begin{minipage}{\textwidth}\footnotesize
$V_R$ is the expected target runup, $V_P$ is the expected offer price, $V_P-V_R$ is the offer price markup, and $\pi(s)$ is the probability of a takeover given the synergy signal $s$ in the runup period. The synergy $S$ given $s$ is distributed uniform. Panel A shows the target valuations as a function of $s$, while Panel B shows the theoretical projection of the markup on the target runup. The takeover benefit function has target and bidder equally sharing synergy gains ($\theta=0.5)$, while bidder bears the entire bid cost ($\gamma=1)$. The expected markup approaches zero as the anticipated deal probability $\pi(s)$ approaches one.
%\vspace{4mm}
\end{minipage}
\begin{minipage}{\linewidth}\footnotesize
\begin{center}
\includegraphics[width=18cm,angle=0]{Figure2ab-anticipation}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\newpage
\begin{figure}[htbp]
\begin{center}
\leavevmode
\caption{\bf Theoretical markup projections with stand-alone value change $T$ in the runup.}
\label{fig:standalone}
\begin{minipage}{\textwidth}\footnotesize
The target stand-alone value changes by a known value $T$ in the runup period. The figure shows that sample variation in $T$ flattens the projection of markup on runup. The solid line is the average expected markup computed as the vertical summation of expected markups occurring across sub-samples with different changes in target stand-alone value $T$. Dashed lines are relations within a sub-sample having the same change in target stand-alone value. Uncertainty in the signal $s$ is uniform, and benefit sharing is as in Figure \ref{fig:uniform}. $\pi(s)$ is the probability of a takeover given the synergy signal $s$.
%\vspace{4mm}
\end{minipage}
\begin{minipage}{\linewidth}\footnotesize
\begin{center}
\includegraphics[width=20cm,angle=0]{Figure3-standalone}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\newpage
\begin{figure}[htbp]
\begin{center}
\leavevmode
\caption{\bf Theoretical markup projections with runup transferred to target (``paying twice").}
\label{fig:paytwice}
\begin{minipage}{\textwidth}\footnotesize
All parameters are as in Figure \ref{fig:uniform}, except that the bid now also includes the target runup $V_R$. This changes the conditional probability of a takeover from $\pi(s)$ to $\pi^*(s,V_R)$ (right-side vertical axis of Panel B). While the signal scale in Panel A and B are the same, Panel A shows the projection of markup on runup \underline{without} transferring $V_R$ to the target. This serves to illustrate that the effect of ``paying twice" is to eliminate relatively low-synergy takeovers from the sample, leaving only those bidders with sufficient synergy gains to pay for the transfer (and still find it beneficial to make a bid). As a result, given the sample of high-synergy bidders, the probability $\pi(s)$ quickly reaches one in Panel A where the bidder does not need to transfer $V_R$. In Panel B, the probability $\pi^*(s,V_R)$ never exceeds 0.5, and it is only 0.37 when $\pi(s)=1$.
%\vspace{4mm}
\end{minipage}
\begin{minipage}{\linewidth}\footnotesize
\begin{center}
\includegraphics[width=17cm,angle=0]{Fig4ab-runuptransfer}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\newpage
\begin{figure}[htbp]
\begin{center}
\leavevmode
\caption{\bf Empirical markup projections for the total sample of 6,150 bids, 1980-2008.}
\label{fig:fit}
\begin{minipage}{\textwidth}\footnotesize
In Panel A, the markup is measured as $\frac{OP}{P_{-2}}-1$, where $OP$ is the offer price and $P_{-2}$ is the target stock price on day -2 relative to the first offer announcement date, and the runup is $\frac{P_{-2}}{P_{-42}}-1$. In Panel B, the markup is the Marked Model $CAR(-1,1)$ and the runup is $CAR(-42,-2)$. A flexible form (equation \ref{eq:beta} in the text) is used to contrast linear fit with best fit.
%\vspace{4mm}
\end{minipage}
\begin{minipage}{\linewidth}\footnotesize
\begin{center}
\includegraphics[width=16cm,angle=0]{Figure5-markuprunup}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\newpage
\begin{figure}[htbp]
\begin{center}
\leavevmode
\caption{\bf Theoretical projections of bidder merger gains $\nu_P$ on target runup $V_R$ with and without feeding $V_R$ back into the offer price.}
\label{fig:bidtheory}
\begin{minipage}{\textwidth}\footnotesize
The market receives a synergy signal $s$ in the runup period resulting in the conditional expected synergy to be embedded into the stock prices of the bidder and the target. Uniform case with target and bidder shares equally synergy gains ($\theta=0.5)$ and bidder bears all bid cost ($\gamma=1)$. In Panel A, the bidder does {\underline{not}} transfer the runup $V_R$ to the target. In Panel B, bidder transfers $V_R$ {\underline{and}} rationally adjusts the minimum bid threshold to $K^*=\frac{\gamma C+V_R}{\theta}$. In Panel C, bidder also transfers $V_R$ to the target {\underline{but does not}} adjust the minimum bid threshold to $K^*=$ (it remains at $K=\frac{\gamma C}{\theta}$). Thus, in both Panel B and C the bidder ``pays twice", but only in Panel C does the bidder fail to take this extra takeover cost into account ex ante.
%\vspace{4mm}
\end{minipage}
\begin{minipage}{\linewidth}\footnotesize
\begin{center}
\includegraphics[width=17cm,angle=0]{Figure6-biddertheory}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\newpage
\begin{figure}[htbp]
\begin{center}
\leavevmode
\caption{\bf Empirical projections of bidder gains on target runups, 1980-2008.}
\label{fig:bidderfit}
\begin{minipage}{\textwidth}\footnotesize
Bidder takeover gains $(\nu_P$) is measured as the Market Model bidder $CAR(-42,1)$ relative to the first announcement date of the offer. In Panel A, the target runup is $\frac{P_{-2}}{P_{-42}}-1$, where $P_{-2}$ is the target stock price on day -2 relative to the first offer announcement date. Panel B uses the augmented target runup (defined in the text and in Table \ref{tab:nonlinear}). A flexible form (equation \ref{eq:beta} in the text) is used to contrast linear fit with best fit. Sample of 3,689 public bidders.
%\vspace{4mm}
\end{minipage}
\begin{minipage}{\linewidth}\footnotesize
\begin{center}
\includegraphics[width=15cm,angle=0]{Figure7-bidderfit}
\end{center}
\end{minipage}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%% figures end here &&&&&&&&&&&&&&&&&&&&&&
%%%%%%%%%%%%%%%% TABLES &&&&&&&&&&&&&&&&&&&&&&
% tab:selection
% tab:sample
% tab:desc
% tab:def
% tab:linear
% tab:markup
% tab:bidder
% tab:runup
% tab:rival
\newpage
\begin{table}[htbp]
\caption{\bf Sample selection} \label{tab:selection} \centering
\vspace{4mm}
\begin{minipage}{\textwidth}\footnotesize
Description of the sample selection process.
An initial bid is the first control bid for the target in 126 trading days (six months).
Bids are grouped into takeover contests, which end when there are no new control bids for the target in 126 trading days.
All stock prices $p_i$ are adjusted for splits and dividends,
where $i$ is the trading day relative to the date of announcement (day 0). \vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize
\rule{\textwidth}{1pt} \tabcolsep 4pt
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}llrr} \hline \\
&& Number of & Sample \\
Selection criteria & Source & exclusions & size \vspace{2mm}\\
\hline \\
\\
All initial controlbids in SDC (FORMC = M, AM) for US public targets \\
during the period 1/1980-12/2008 & SDC & & 13,893 \vspace{2mm} \\
Bidder owns $<$50\% of the target shares at the time of the bid & SDC & 46 & 13,847 \vspace{2mm} \\
Target firm identified in CRSP and listed on NYSE, AMEX or NASDAQ & CRSP & 4,138 & 9,109 \vspace{2mm} \\
Deal value $>$ \$10 million & SDC & 1,816 & 7,293 \vspace{2mm} \\
Target stock price on day -42 $>$ \$1 & CRSP & 191 & 7,102 \vspace{2mm} \\
Offer price available &SDC & 239 & 6,863 \vspace{2mm} \\
Target stock price on day -2 available & CRSP & 6 & 6,857 \vspace{2mm} \\
Target announcement returns [-1,1] available & CRSP & 119 & 6,738 \vspace{2mm} \\
Information on outcome and ending date of contest available & SDC & 324 & 6,414 \vspace{2mm} \\
Contest shorter than 252 trading days & SDC & 264 & 6,150 \vspace{2mm} \\
\hline
\\
Final sample &&& 6,150 \\
\\
\hline
\end{tabular*}
\rule{\textwidth}{1pt}
\end{minipage}
\end{table}
\newpage
\begin{table}[htbp]
\caption{\bf Sample size, offer premium, markup, and runup, by year} \label{tab:sample} \centering
\begin{minipage}{\textwidth}\footnotesize
The table shows the mean and median offer premium, markup, target stock-price runup and net runup for the sample of 6,150 initial control bids for U.S. publicly traded target firms in 1980-2008. The premium is $(OP/P_{-42})-1$, where $OP$ is the price per share offered by the initial control bidder and $P_i$ is the target stock price on trading day $i$ relative to the takeover announcement date ($i=0$), adjusted for splits and dividends. The markup is $(OP/P_{-2})-1$, the runup is $(P_{-2}/P_{-42})-1$ and and the net runup is $(P_{-2}/P_{-42})- (M_{-2}/M_{-42})$, where $M_{i}$ is the value of the equal-weighted market portfolio on day $i$. \vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize
\rule{\textwidth}{1pt} \tabcolsep 1pt
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}lccccccccc}
\hline \\
& Sample& \mc{2}{c}{Offer premium} & \mc{2}{c}{Markup} & \mc{2}{c}{Runup} & \mc{2}{c}{Net runup} \\
& size & \mc{2}{c}{$\frac{OP}{P_{-42}}-1$} & \mc{2}{c}{$\frac{OP}{P_{-2}}-1$} & \mc{2}{c}{$\frac{P_{-2}}{P_{-42}}-1$} & \mc{2}{c}{$\frac{P_{-2}}{P_{-42}}- \frac{M_{-2}}{M_{-42}}$} \\
\cline{2-2} \cline{3-4} \cline{5-6} \cline{7-8} \cline{9-10} \\
Year & $N$ & mean & median & mean & median & mean & median & mean & median \\
\hline \\
1980 & 10 & 0.70 & 0.69 & 0.53 & 0.34 & 0.15 & 0.19 & 0.10 & 0.12 \vspace{1mm} \\
1981 & 35 & 0.60 & 0.48 & 0.40 & 0.36 & 0.15 & 0.13 &0.16 & 0.14 \vspace{1mm} \\
1982 & 48 & 0.53 & 0.48 & 0.34 & 0.32 & 0.15 & 0.10 & 0.13 & 0.09 \vspace{1mm} \\
1983 & 58 & 0.49 & 0.50 & 0.33 & 0.35 & 0.12 & 0.12 & 0.10& 0.08 \vspace{1mm} \\
1984 & 115 & 0.51 & 0.43 & 0.45 & 0.32 & 0.07 & 0.05 &0.06& 0.06 \vspace{1mm} \\
1985 & 161 & 0.40 & 0.34 & 0.26 & 0.22 & 0.11 & 0.10 &0.08&0.06 \vspace{1mm} \\
1986 & 209 & 0.40 & 0.36 & 0.26 & 0.23& 0.12 & 0.09 &0.08& 0.06 \vspace{1mm} \\
1987 & 202 & 0.39 & 0.36 & 0.32 & 0.25& 0.07 & 0.07 &0.06& 0.03 \vspace{1mm} \\
1988 & 270 & 0.56 & 0.47 & 0.35 & 0.29 & 0.15 & 0.10 &0.12& 0.08 \vspace{1mm} \\
1989 & 194 & 0.54 & 0.43& 0.39 & 0.30 & 0.11 & 0.07 &0.07 & 0.03 \vspace{1mm} \\
1990 & 103 & 0.53 & 0.49 & 0.49 & 0.41 & 0.05 & -0.00& 0.05& -0.01 \vspace{1mm} \\
1991 & 91 & 0.55 & 0.46 & 0.40 & 0.33& 0.12 & 0.09 &0.08&0.05 \vspace{1mm} \\
1992 & 106 & 0.57 & 0.51 & 0.40 & 0.35 & 0.13 & 0.08 &0.11&0.08 \vspace{1mm} \\
1993 & 146 & 0.48 & 0.43 & 0.36 & 0.33& 0.10 & 0.07 &0.08&0.05 \vspace{1mm} \\
1994 & 228 & 0.44 & 0.42 & 0.34 & 0.31 & 0.08 & 0.07 &0.07&0.07 \vspace{1mm} \\
1995 & 290 & 0.47 & 0.39 & 0.33 & 0.27 & 0.11 & 0.09 &0.06&0.04 \vspace{1mm} \\
1996 & 319 & 0.40 & 0.37 & 0.27 & 0.24 & 0.11 & 0.07 &0.07&0.04 \vspace{1mm} \\
1997 & 434 & 0.41 & 0.38 & 0.26 & 0.23& 0.13 & 0.12 &0.09&0.08 \vspace{1mm} \\
1998 & 465 & 0.46 & 0.37 & 0.37 & 0.26& 0.08 & 0.07 &0.05&0.03 \vspace{1mm} \\
1999 & 496 & 0.55 & 0.45 & 0.37 & 0.30 & 0.15 & 0.11 &0.12&0.08 \vspace{1mm} \\
2000 & 415 & 0.53 & 0.45 & 0.38 & 0.34 & 0.13 & 0.06 &0.12&0.08 \vspace{1mm} \\
2001 & 270 & 0.55 & 0.46 & 0.40 & 0.34& 0.11 & 0.08 &0.12& 0.09 \vspace{1mm} \\
2002 & 154 & 0.52 & 0.36 & 0.42 & 0.32 & 0.09 & 0.03 &0.12&0.06 \vspace{1mm} \\
2003 & 189 & 0.47 & 0.34 & 0.30 & 0.23& 0.13 & 0.08 &0.09& 0.05 \vspace{1mm} \\
2004 & 195 & 0.30 & 0.26 & 0.24 & 0.21 & 0.06 & 0.04 &0.03&0.02 \vspace{1mm} \\
2005 & 230 & 0.30 & 0.27 & 0.25 & 0.21& 0.05 & 0.04 &0.04& 0.03 \vspace{1mm} \\
2006 & 258 & 0.31 & 0.27 & 0.25 & 0.21 & 0.05 & 0.03&0.03&0.02 \vspace{1mm} \\
2007 & 284 & 0.31 & 0.28 & 0.29 & 0.23& 0.02 & 0.02 &0.00&0.00 \vspace{1mm} \\
2008 & 175 & 0.34 & 0.30 & 0.40 & 0.34 & -0.04 & -0.04 &0.01&0.00 \\ \\
\hline \\
Total& 6,150 & 0.45 & 0.38 & 0.33 & 0.27 & 0.10 & 0.07 & 0.08 & 0.05 \\ \\
\hline
\end{tabular*}
\rule{\textwidth}{1pt}
\end{minipage}
\end{table}
\newpage
\begin{table}[htbp]
\caption{\bf Linear and nonlinear projections of markups on runups} \label{tab:nonlinear} \centering
\begin{minipage}{\textwidth}\footnotesize
The linear projections is a simple OLS regression of the markup on the runup. The nonlinear projection is
$$Markup= \alpha+\beta[(r-min)^{(v-1)} (max-r)^{w-1}/\Lambda(v,w)(max-min)^{v+w-1}]+\epsilon,$$
where $\Lambda(v,w)$ is the beta distribution with shape parameters $v$ and $w$, $r$ is the runup, $max$ and $min$ are respectively the maximum and minimum runups in the data, $\alpha$ is an overall intercept, $\beta$ is a scale parameter, and $\epsilon$ is a residual error term. The projection uses a starting values $v=1, w=2$ (for which the beta density is linear downward sloping) and the OLS estimates of $a$ and $b$ (for $\alpha$ and $\beta$), followed by a least squares fit over all four parameters to identify a best non-linear shape. For all of the projections in this table, the resulting form of the non-linearity corresponds closely to that shown in Figure \ref{fig:fit} for projection (1), and are thus consistent with the general concave then convex shape shown in the theoretical Figure \ref{fig:uniform}. First-order residual serial correlation is calculated after ordering the data by runup. Using the t-statistic (in parentheses), a significant positive residual serial correlation rejects the hypothesis that the true projection is linear and is consistent with deal anticipation driving a portion of the runup.
\vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize
\rule{\textwidth}{1pt} \tabcolsep 1pt
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}cccclcc}
\hline \\
&&Markup measure & Runup measure &Linear projection & Linear residual &Nonlinear residual \\
&N& $V_P-V_R$& $V_R$ & $V_P-V_R=a+bV_R$& serial correlation& serial correlation\\ \hline \\
(1)&6,150&Total markup & Total runup & $a=0.36$ &0.030 &0.015\\
&&$\frac{OP}{P_{-2}}-1$ & $\frac{P_{-2}}{P_{-42}}-1$& $b=-0.24 \ (-11.9)$ & (2.36) & (1.15) \\ \\
(2)&5,035$^a$ &Total markup & Total runup & $a=0.36$ &0.045 &0.027\\
&&$\frac{OP}{P_{-2}}-1$ & $\frac{P_{-2}}{P_{-42}}-1$& $b=-0.22 \ (-10.1)$ & (3.21) & (2.19) \\ \\
(3)&6,103& Expected markup$^b$ & Total runup & $a=0.31$ &0.027 &0.016\\
&&$\pi[\frac{OP}{P_{-2}}-1]$ & $\frac{P_{-2}}{P_{-42}}-1$& $b=-0.17 \ (-9.5)$ & (2.11) & (1.25) \\ \\
%(3)&6103$^b$&Total markup & Total runup & $a=$ & &\\
%&&$\frac{OP}{P_{-2}}-1$ & $\frac{P_{-2}}{P_{-42}}-1$& $b=- \ (-)$ & () & () \\ \\
%(2) &Total offer markup& Net runup &$a=$ & & \\
%&$\frac{OP}{P_{-2}}-1$& $\frac{P_{-2}}{P_{-42}}-\frac{M_{-2}}{M_{-42}}$& $b=- \ (-)$ & () & ()\\ \\
%(4)& Adjusted offer markup & Net runup & $a=$ &&\\
%&$\pi[\frac{OP}{P_{-2}}-1]$ & $\frac{P_{-2}}{P_{-42}}-\frac{M_{-2}}{M_{-42}}$& $b=-\ (-)$ & () & () \\ \\
%(3) &6150& Adjusted offer markup & $CAPM $& $a=0.39$ & 0.047 & 0.036 \\
%&&$[\frac{OP}{P_{-2}}-1]-\alpha$ & $CAR(-42, -2)$& $b=-0.28 \ (-6.39)$ & (3.66) & (2.84)\\ \\
%(6) &Adjusted offer markup& $CAPM$&$a=0.33$ & 0.040 & 0.032 \\
%&$\pi[\frac{OP}{P_{-2}}-1]-\alpha$& $CAR(-42, -2)$& $b=-0.20 \ (-5.60)$ & (3.12) & (2.46)\\ \\
(4) &6,150& Residual markup$^c$ & Augmented runup$^d$& $a=0.36$ & 0.052 & 0.031 \\
&&$U_P$& $(\frac{P_{-2}}{P_{-42}}-1)+R_0$& $b=-0.21 \ (-12.1)$ & (4.03) & (2.45)\\ \\
(5) &6,150& Market Model$^e$& Market Model$^e$& $a=0.22$ & 0.039 & 0.038 \\
&& $CAR(-1,1)$& $CAR(-42, -2)$& $b=-0.09 \ (-6.7)$ & (3.10) & (2.95)\\ \\
%(7) & Market Model& Market Model& $a=$ & & \\
%& $CAR(-1,126)$& $CAR(-42, -2)$& $b=- \ ()$ & () & ()\\ \\
%(6) &6,150& CAPM$^f$& CAPM$^f$& $a=0.23$ & 0.021& 0.016 \\
%&& $CAR(-1,1)$& $CAR(-42, -2)$& $b=-0.16 \ (-11.5)$ & (1.63) & (1.23)\\ \\
\hline
\end{tabular*}
\rule{\textwidth}{1pt}
\noindent $^a$This projection is for the subsample where the initial bid in the contest ultimately leads to a control change in the target (successful targets).
\noindent $^b$This projection is for the subsample with available data on the target--, bidder-- and deal characteristics used to estimate the probability $\pi$ of bid success in Table \ref{tab:prob}. The projection includes the effect of these variables by multiplying the total markup with the estimated value of $\pi$.
%\noindent $^c$This projection is for the subsample with available data on the target--, bidder-- and deal characteristics used to estimate the probability $\pi$ of bid success in Table \ref{tab:prob}. The projection includes the effect of these variables by adding these on the right-hand-side of the projection.
\noindent$^c$ Residual markup, $U_P$, is the residual from the projection of the total markup, $\frac{OP}{P_{-2}}-1$, on the deal characteristics used to estimate the success probability $\pi$ in Table \ref{tab:prob}, excluding $Positive\ toehold$, $Toehold\ size$, and $52-week\ high$ which are used to construct the augmented runup. Variable definitions are in table \ref{tab:def}.
\noindent $^d$ The enhancement $R_0$ in the augmented runup adds back into the runup the effect of information that the market might use to anticipate possible takeover activity {\it prior} to the runup period. $R_0$ is the projection of the total runup ($\frac{P_{-2}}{P_{-42}}-1$) on the deal characteristics $Positive\ toehold$, $Toehold\ size$, and the negative value of $52-week\ high$, all of which may affect the prior probability of a takeover (prior to the runup period). The augmented runup is the total runup {\underline{plus}} $R_0$. Variable definitions are in table \ref{tab:def}.
\noindent $^e$ Target cumulative abnormal stock returns (CAR) are computed using the estimated Market Model parameters: $r_{it}=\alpha + \beta r_{mt}+ u_{it},$ where $r_{it}$ and $r_{mt}$ are the daily returns on stock $i$ and the value-weighted market portfolio, and $u_{it}$ is a residual error term. The estimation period is 252 trading days prior to day -42 relative to the day of the announcement of the initial bid.
%\noindent $^f$ The CAPM $CAR$ is computed by cumulated the abnormal stock returns which have been estimated using excess returns and constraining the intercept in the Market Model to be zero.
\end{minipage}
\end{table}
%%% Table for the probability of successful contest %%%
\newpage
\begin{table}[htbp]
\caption{\bf Probability of contest success} \label{tab:prob} \centering
\begin{minipage}{\textwidth}\footnotesize
The table shows coefficient estimates from logit regressions for the probability that the contest is successful (columns 1-2) and that
the initial control bidder wins (columns 3-6).
P-values are in parenthesis. The sample is 6,103 initial control bids for public US targets,
1980-2008, with a complete set of control variables (defined in Table \ref{tab:def}).
\vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize
\rule{\textwidth}{1pt} \tabcolsep 1pt
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}lcccccc}
\hline \\
Dependent variable: & \mc{2}{c}{Contest successful} & \mc{4}{c}{Initial control bidder wins} \vspace{1mm} \\
\cline{2-3} \cline{4-7}
\\
Intercept & 1.047 & 0.909 & 0.657 & 0.455 & 0.626 & 0.437 \\
& (0.000) & (0.000)& (0.000) & (0.005) & (0.000) & (0.007)
\vspace{1mm} \\
{\bf Target characteristics} \vspace{1mm} \\
$Target \ size$ & 0.137 & 0.085 & 0.148 & 0.094 & 0.150 & 0.096 \\
& (0.000)& (0.005) & (0.000) & (0.001) & (0.000) & (0.001)
\vspace{1mm} \\
$NYSE/Amex$ &-0.365 & -0.269 & -0.435 & -0.330 & -0.433 & -0.329 \\
& (0.000)& (0.005) & (0.000) & (0.000)& (0.000) & (0.000)
\vspace{1mm} \\
$Turnover$ &-0.017 & -0.019 & -0.017 & -0.019 & -0.017 & -0.019 \\
& (0.002) & (0.001) & (0.002) & (0.001) & (0.003)& (0.001)
\vspace{1mm} \\
$Poison \ pill$ & -0.578 & -0.513 & -0.506 & -0.436 & -0.406 & -0.341 \\
&(0.028)& (0.053) & (0.063) & (0.114) & (0.138)& (0.219)
\vspace{1mm} \\
$52-week \ high$ &1.022 & 1.255 & 0.864 & 1.117 & 0.868 & 1.120 \\
& (0.000) & (0.000) & (0.000) & (0.000)& (0.000)& (0.000)
\vspace{1mm} \\
{\bf Bidder characteristics} \vspace{1mm} \\
$Toehold$ & -0.819 & -0.688 & -0.978 & -0.833 & -1.589 & -1.419 \\
& (0.000) & (0.000) & (0.000) & (0.000) & (0.000)& (0.000)
\vspace{1mm} \\
$Toehold \ size$ & & &&& 0.039 & 0.038 \\
& & &&& (0.000) & (0.000)
\vspace{1mm} \\
$Acquirer \ public$ & 0.833 & 0.804 & 0.938 & 0.900 & 0.952 & 0.915 \\
& (0.000) & (0.000) & (0.000)& (0.000)& (0.000)& (0.000)
\vspace{1mm} \\
$Horizontal$ & 0.248 & 0.211 & 0.276 & 0.226 &0.281 & 0.232 \\
& (0.020) & (0.050) & (0.006)& (0.025)& (0.005)& (0.022)
\vspace{1mm} \\
$> 20\% \ new \ equity $ & -0.585 & -0.577 & -0.531 & -0.522 & -0.536 & -0.526 \\
& (0.000) & (0.000) & (0.000)& (0.000)& (0.000)& (0.000) \vspace{1mm} \\
$Premium$ & 0.343 & 0.371 & 0.334 & 0.365 & 0.350 & 0.380 \\
& (0.001) & (0.000) & (0.001) & (0.000)& (0.000)& (0.000)
\vspace{1mm} \\
{\bf Deal characteristics} \vspace{1mm} \\
$Tender \ offer$ &2.173 & 2.307 & 1.912 & 2.053 & 1.945 & 2.085 \\
&(0.000)& (0.000) & (0.000) & (0.000)& (0.000) & (0.000)
\vspace{1mm} \\
$Cash$ & -0.148& -0.276 & -0.105 &-0.224 & -0.114 & -0.236 \\
&(0.119)& (0.005)& (0.236)& (0.014) & (0.199)& (0.010)
\vspace{1mm} \\
$Hostile$ & -2.264 & -2.149 & -3.086 & -2.980 & -2.994 & -2.893 \\
& (0.000) & (0.000) & (0.000)& (0.000)& (0.000)& (0.000)
\vspace{1mm} \\
$1990s$ & & 0.435& & 0.566 & & 0.548 \\
& & (0.000) & & (0.000)& & (0.000)
\vspace{1mm} \\
$2000s$ & & 0.775 & & 0.824 & & 0.816 \\
& & (0.000) & & (0.000)& & (0.000)
\vspace{1mm} \\
\\
Pseudo-$R^2$ (Nagelkerke) & 0.208 & 0.219 & 0.263 & 0.276 & 0.269 & 0.281 \vspace{1mm}\\
$\chi^2$ & 755.1 & 795.8 & 1074.0 & 1129.3 & 1098.5 & 1151.8 \\
%& (0.000) & (0.000)& (0.000) & (0.000)& (0.000) & (0.000) \vspace{2mm} \\
%Number of cases ($N$) & 6103 & 6103 & 6103 & 6103 & 6103 & 6103 \\
\\
\hline
\end{tabular*}
\rule{\textwidth}{1pt}
\end{minipage}
\end{table}
%%%%%%%%%%%%%%%% TABLE Variable definitions %%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{table}[htbp]
\caption{\bf Variable definitions} \label{tab:def} \centering
\vspace{4mm}
\begin{minipage}{\textwidth}\footnotesize
Variable definitions. All stock prices $P_i$ are adjusted for splits and dividends,
where $i$ is the trading day relative to the date of announcement ($i=0$), and, if missing,
replaced by the midpoint of the bid/ask spread.
\vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize\tabcolsep 3pt \rule{\textwidth}{1pt}
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}p{1.0in}p{4.5in}p{0.9in}} \hline \\
Variable & Definition & Source \vspace{2mm}\\
\hline \\
\multicolumn{2}{l}{\bf A. Target characteristics} \vspace{2mm} \\
{\it Target size} & Natural logarithm of the target market
capitalization in \$ billion on day -42 & CRSP \vspace{1mm} \\
{\it Relative size} & Ratio of target market capitalization to
bidder market capitalization on day -42 & CRSP \vspace{1mm} \\
{\it NYSE/Amex} & The target is listed on NYSE or Amex vs. NASDAQ (dummy) & CRSP \vspace{1mm} \\
{\it Turnover} & Average daily ratio of trading volume to total shares oustanding over the 52 weeks ending on day -43& CRSP \vspace{1mm} \\
{\it Poison pill} & The target has a poison pill (dummy) & SDC \vspace{1mm} \\
{\it 52-week high} & Change in the target stock price from the highest price $P_{high}$ over the 52-weeks ending on day -43, $P_{-42}/P_{high}-1$ & CRSP \\ \\
\multicolumn{2}{l}{\bf B. Bidder characteristics} \vspace{2mm} \\
{\it Toehold} & The acquirer owns shares in the target when announcing the bid (dummy) & SDC \vspace{1mm} \\
{\it Toehold size} & Percent target shares owned by the acquirer when announcing
the bid & SDC \vspace{1mm} \\
{\it Stake bidder} & The initial bidder buys a small equity stake in the target during the runup period (dummy) & SDC \vspace{1mm} \\
{\it Stake other} & Another investor buys a small equity stake in the target during the runup period (dummy) & SDC \vspace{1mm} \\
\\
{\it Acquirer public } & The acquirer is publicly traded (dummy) & SDC \vspace{1mm} \\
{\it Horizontal} & The bidder and the target has the same primary 4-digit SIC code (dummy) & SDC \vspace{1mm} \\
{\it >20\% new equity} & The consideration includes a stock portion which exceeds 20\% of the acquirer's shares oustanding (dummy) & SDC \\ \\
\multicolumn{2}{l}{\bf C. Contest characteristics} \vspace{2mm}
\\
{\it Premium} & Bid premium defined as $(OP/P_{-42})-1$, where $OP$ is the offer price. \vspace{1mm} & SDC,CRSP \\
{\it Markup} & Bid markup defined as $(OP/P_{-2})-1$, where $OP$ is the offer price. \vspace{1mm} & SDC,CRSP \\
{\it Runup} & Target raw runup defined as $(P_{-2}/P_{-42})-1$ \vspace{1mm} & CRSP \\
{\it Net runup} & Target net runup defined as $(P_{-2}/P_{-42})-(M_{-2}/M_{-42})$, where $M_{i}$ is the value of the equal-weighted market portfolio on day $i$. \vspace{1mm} & CRSP \\
{\it Market runup} & Stock-market return during the runup period defined as $M_{-2}/M_{-42}-1$, where $M_{i}$ is the value of the equal-weighted market portfolio on day $i$. \vspace{1mm} & CRSP \\
{\it Tender offer} & The initial bid is a tender offer (dummy) & SDC \vspace{1mm} \\
{\it All cash} & Consideration is cash only (dummy) & SDC \vspace{1mm} \\
{\it All stock} & Consideration is stock only (dummy) & SDC \vspace{1mm} \\
{\it Hostile} & Target management's response is hostile vs. friendly or neutral (dummy) & SDC \vspace{1mm} \\
{\it Initial bidder wins} & The initial bidder wins the contest (dummy) & SDC \vspace{1mm} \\
{\it 1990s} & The contest is announced in the period 1990-1999 (dummy) & SDC \vspace{1mm} \\
{\it 2000s} & The contest is announced in the period 2000-2008 (dummy) & SDC \vspace{2mm} \\
\end{tabular*}
\rule{\textwidth}{1pt}
\end{minipage}
\end{table}
\newpage
\begin{table}[htbp]
\caption{\bf Projections of bidder announcement returns on target runup} \label{tab:bidder} \centering
\begin{minipage}{\textwidth}\footnotesize
The table shows OLS estimates of bidder announcement cumulative
abnormal returns $BCAR[-1,+1]$, from a market model estimated over day -293 through -43.
$Net \ Runup = (P_{-2}/P_{-42})-(M_{-2}/M_{-42})$ and $Runup=(P_{-2}/P_{-42})-1$, where $P_i$ is the
target stock price and $M_i$ is the value of the equal-weighted market portfolio on day $i$.
The p-values (in parenthesis) use White's (1980) heteroscedasticity-consistent standard errors.
The sample is 3,624 initial control bids for public US targets, 1980-2008, by U.S. public bidders. All variables are defined in Table \ref{tab:def}.\vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize
\rule{\textwidth}{1pt} \tabcolsep 1pt
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}lcccccc}
\hline \\
Intercept & -0.016 & -0.011 & -0.020 & -0.016 & -0.011 & -0.020 \\
& (0.000) & (0.931) & (0.000) & (0.000) & (0.982) & (0.000) \vspace{2mm} \\
$Net \ Runup$ & 0.011& 0.013 & 0.011 \\
& (0.055) & (0.036) & (0.063)
\vspace{2mm} \\
$Runup$ &&&& 0.013 & 0.013 & 0.013 \\
&&&& (0.028) & (0.029) & (0.032)
\\ \\
\mc{4}{l}{\bf Other target characteristics} \\ \\
$Relative \ size$ & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 &0.000 \\
& (0.015)&(0.012)& (0.008) & (0.013) & (0.010) &(0.007)
\vspace{2mm}\\
$NYSE/Amex$ & -0.002 & -0.002 & -0.002 & -0.002 & -0.002 & -0.002 \\
& (0.401) & (0.534) & (0.393) & (0.387) & (0.514) & (0.380)
\vspace{2mm} \\
$Turnover$ &-0.001 & -0.001 & -0.001 & -0.001 & -0.001 & -0.001 \\
& (0.000)& (0.000) & (0.000)& (0.000)& (0.000) & (0.000)
\vspace{2mm} \\
\\
\bf Bidder characteristics \\ \\
$Toehold \ size$ & 0.001 & 0.000 & 0.001 & 0.001 & 0.000 & 0.001 \\
& (0.049) & (0.071) & (0.043) & (0.046)& (0.067)& (0.042)
\vspace{2mm} \\
$Horizontal$ & 0.003& 0.004 & 0.003 &0.004& 0.004 & 0.003 \\
& (0.157) & (0.138)& (0.172)& (0.144)& (0.130) & (0.156)
\vspace{2mm} \\
\mc{4}{l}{\bf Deal characteristics} \\ \\
$Cash$ & 0.024 & 0.025 & 0.024 &0.024& 0.025 & 0.024 \\
& (0.000) & (0.000)& (0.000) & (0.000)& (0.000)& (0.000)
\vspace{2mm} \\
$Stock$ & -0.007& -0.008 & -0.007 &-0.007& -0.008 & -0.007 \\
&(0.025)& (0.013) & (0.025) &(0.021) & (0.012) & (0.021)
\vspace{2mm} \\
$Hostile$ & -0.000 & -0.002 & 0.003 &-0.000& -0.002 & 0.003 \\
&(0.988) & (0.745) & (0.643) &(0.977) & (0.741)& (0.674)
\vspace{2mm} \\
$Initial \ bidder \ wins$ &&& 0.006 &&& 0.005 \\
&&& (0.211) &&& (0.240) \vspace{2mm} \\
\\
year dummies & no & yes & no & no & yes & no \\
\\
Adjusted $R^2$ & 0.062 & 0.076 & 0.062 &0.062 & 0.076 & 0.063 \vspace{2mm} \\
F-value & 19.10 &5.74 & 17.31 & 19.12 & 5.74 & 17.32 \\
& (0.000) & (0.000) & (0.000) & (0.000) & (0.000)& (0.000) \\
\\
Sample size, $n$ & 3,623 & 3,623 & 3,623 & 3,624 & 3,624 & 3,624 \\
\\
\hline
\end{tabular*}
\rule{\textwidth}{1pt}
\end{minipage}
\end{table}
%%%% Table describing the toeholds %%%%%
\newpage
\begin{table}[htbp]
\caption{\bf Description of toeholds purchased in the target firm} \label{tab:toe} \centering
\begin{minipage}{\textwidth}\footnotesize
The table shows toehold acquisitions made by the initial control bidder, a rival control bidder,
and other investors. Stake purchases are identified from records of completed partial acquisitions in SDC.
The initial control bid is announced on day 0.
The sample is 6,150 initial control bids for U.S. publicly traded targets, 1980-2008.
\vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize
\rule{\textwidth}{1pt} \tabcolsep 1pt
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}llccccc}
\hline \\
&& \mc{3}{c}{Target stake } && Total toehold \\
&& \mc{3}{c}{announced in window} && on day 0 \\
\cline{3-5} \\
&& [-126,0] & [-42,0] & [-1,0] && \vspace{2mm} \\
\hline \\
{\bf A: Toehold acquired by initial control bidder} \\ \\
Number of toehold purchases && 136 & 104 & 70 \vspace{2mm} \\
Number of firms in which at least one stake is purchased && 122 & 94 & 63 && 648 \vspace{1mm} \\
In percent of target firms && 2.0\% & 1.5\% & 1.0\% && 10.5\% \vspace{2mm} \\
Size of toehold (\% of target shares) when toehold positive: & mean & 12.2\% & 11.7\% & 12.7\% && 15.5\% \\
&median & 9.9\% & 9.3\% & 9.4\% && 9.9\% \\
\\
{\bf B: Toehold acquired by rival control bidder} \\ \\
Number of toehold purchases && 7 & 3 & 1 \vspace{2mm} \\
Number of firms in which at least one stake is purchased && 6 & 3 & 1 && \vspace{1mm} \\
In percent of target firms && 0.1\% & 0.05\% & 0.02\% && n/a \vspace{2mm} \\
Size of toehold (\% of target shares) when toehold positive: & mean & 9.4\% & 7.0\% & 4.9\% \\
&median & 9.1\% & 6.2\% & 4.9\% \\
\\
{\bf C: Toehold acquired by other investor} \\ \\
Number of toehold purchases && 235 & 85 & 18 \vspace{2mm} \\
Number of firms in which at least one stake is purchased && 196 & 73 & 15 && \vspace{1mm} \\
In percent of target firms && 3.2\% & 1.2\% & 0.2\% && n/a \vspace{2mm} \\
Size of toehold (\% of target shares) when toehold positive: & mean & 6.8\% & 8.7\% & 10.1\% \\
&median & 5.4\% & 6.3\% & 7.6\% \\
\\
\hline
\end{tabular*}
\rule{\textwidth}{1pt}
\end{minipage}
\end{table}
%%%%%%%%%%% Table 8: determinants of the runup %%%%%%%%%%%
\newpage
\begin{table}[htbp]
\caption{\bf Target runup and initial offer premium} \label{tab:runup} \centering
\begin{minipage}{\textwidth}\footnotesize
The table shows coefficient estimates from OLS regressions of the target net runup and the initial offer premium.
The net runup is $(P_{-2}/P_{-42})-(M_{-2}/M_{-42})$ and the offer premiums is $(OP/P_{-42})-1$, where $P_i$ is the target stock
price and $M_i$ is the value of the equal-weighted market portfolio on trading day $i$ relative to the announcement of the initial control bid, and $OP$ is the price offered by
the initial control bidder.
P-values are in parenthesis. The sample is 6,100 initial control bids for public US targets,
1980-2008, with a complete set of control variables (defined in Table \ref{tab:def}).
\vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize
\rule{\textwidth}{1pt} \tabcolsep 1pt
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}lcccccc}
\hline \\
Dependent variable: & \mc{2}{c}{Target net runup} & \mc{4}{c}{Initial offer premium} \\
&\mc{2}{c}{$\frac{P_{-2}}{P_{-42}}-\frac{M_{-2}}{M_{-42}}$ } & \mc{4}{c}{$\frac{OP}{P_{-42}}-1$} \vspace{1mm} \\
\cline{2-3} \cline{4-7}
\\
Intercept & 0.116 & 0.282 & 0.616 & 1.073 & 0.494 & 0.778 \\
& (0.000) & (0.012) & (0.000)& (0.000)& (0.000)& (0.000)
\vspace{1mm} \\
\mc{7}{l}{\bf Target characteristics} \vspace{1mm} \\
$Target \ size$ & -0.015 & -0.012 & -0.054 & -0.048 & -0.039 & -0.035 \\
& (0.000)& (0.000) & (0.000)& (0.000)& (0.000)& (0.000)
\vspace{1mm} \\
$NYSE/Amex$ & 0.007 & 0.003 & 0.017 & 0.011 & 0.010 & 0.008 \\
& (0.330) & (0.650) & (0.239) & (0.442)& (0.422)& (0.529)
\vspace{1mm} \\
$Turnover$ & 0.000 & 0.000 & -0.001 & 0.000 & 0.001 & 0.000 \\
& (0.754) & (0.986) & (0.561) & (0.775)& (0.589)& (0.698)
\vspace{1mm} \\
$52-week \ high$ & -0.042 & -0.029 & -0.214 & -0.175 & -0.169 & -0.146 \\
& (0.000) & (0.018) & (0.000) & (0.000) & (0.000)& (0.000)
\vspace{1mm} \\
\mc{7}{l}{\bf Bidder characteristics} \vspace{1mm} \\
$Acquirer \ public$ & 0.032 & 0.032 & 0.046 & 0.052 & 0.012 & 0.018 \\
& (0.000) & (0.000) & (0.001) & (0.000) & (0.305)& (0.136)
\vspace{1mm} \\
$Horizontal$ & -0.015 & -0.013 & -0.009 & -0.002 & 0.007 & 0.012 \\
& (0.036) & (0.065) & (0.536) & (0.891)& (0.555)& (0.324)
\vspace{1mm} \\
$Toehold \ size$ & -0.001 & -0.002&-0.003 &-0.004 & -0.002 & -0.002 \\
& (0.002) & (0.000)& (0.000)& (0.000)& (0.014) & (0.004)
\vspace{1mm} \\
$Stake \ bidder$ &0.050& 0.056 & -0.029&-0.012 & -0.082 & -0.072 \\
& (0.043)&(0.024) &(0.560) &(0.804) & (0.051)& (0.088)
\vspace{1mm} \\
$Stake \ other$ &0.125&0.126& 0.089 & 0.093 & -0.044 & -0.040 \\
&(0.000) &(0.000)&(0.100) & (0.084) & (0.340)& (0.382)
\vspace{1mm} \\
\mc{7}{l}{\bf Deal characteristics} \vspace{1mm} \\
$Market \ runup$ &&& 0.924 & 1.054 & 0.815 & 0.926 \\
&&& (0.000) & (0.000) & (0.000)& (0.000)
\vspace{1mm} \\
$Net \ runup$ &&& && 1.077 & 1.068 \\
&&&& & (0.000)& (0.000)
\vspace{1mm} \\
$Tender \ offer$ & 0.037 & 0.028 & 0.094 & 0.076 & 0.055 & 0.046 \\
&(0.000) & (0.000) & (0.000) & (0.000)& (0.000)& (0.001)
\vspace{1mm} \\
$All \ cash$ & -0.009 & 0.000 & -0.024 & -0.002 & -0.014 & -0.001 \\
&(0.209)& (0.948)& (0.112) & (0.914)& (0.278)& (0.949)
\vspace{1mm} \\
$All \ stock $ & 0.003 & 0.000 & -0.005 & -0.008 & -0.007 & -0.008 \\
& (0.725) & (0.976) & (0.755) & (0.631) & (0.600) & (0.578)
\vspace{1mm} \\
$Hostile$ & -0.009 & -0.011 & -0.005 & -0.008 & 0.005 & -0.004 \\
& (0.521) & (0.425) & (0.865)& (0.773)& (0.825)& (0.874)
\vspace{1mm} \\
Year fixed effects & no & yes & no & yes& no & yes \vspace{1mm} \\
Adjusted $R^2$ & 0.025 & 0.038 & 0.077 & 0.092 & 0.339 & 0.346 \\
$F-value$ & 13.1 & 6.86& 37.4 & 15.7 & 209.2 & 76.0 \vspace{2mm} \\
%Number of cases & 6,100 & 6,100 & 6,100 & 6,100 & 6,100 & 6,100
\hline
\end{tabular*}
\rule{\textwidth}{1pt}
\end{minipage}
\end{table}
\newpage \singlespacing\normalsize
\bibliography{jdefs,norli,eckbo,mabig,restructuring}
%\bibliographystyle{myapsr}
\bibliographystyle{jf}
\processdelayedfloats
\end{document}
\newpage
\begin{table}[htbp]
\caption{\bf Linear projections of markups on runups} \label{tab:linear} \centering
\begin{minipage}{\textwidth}\footnotesize
The table shows coefficient estimates from OLS regressions of the markup:
$$Markup_i = a + bRunup_i + cX_i + e_i,$$ where $X$ is a vector of explanatory variables.
In the first three columns (raw returns), $Markup = (OP/P_{-2})-1$ and $Runup=(P_{-2}/P_{-42})-1$, where $OP$ is the offer price and $P_i$ is the target stock price on day $i$. $Rival \ runup$ is the average runup across the target firm's industry rivals, defined as all firms in CRSP with the same 4-digit SIC code. In columns 4 and 5 (\$ returns), $Markup = (OP-P_{-2})*shrout$ and $Runup = (P_{-2}-P_{-42})*shrout$, where $shrout$ is the number of shares outstanding. The p-values (in parenthesis) use White's (1980) heteroscedasticity-consistent standard errors.
The sample is 6,857 initial control bids for public US targets,
1980-2008. All variables are defined in Table \ref{tab:def}.
\vspace{4mm}
\end{minipage}
\begin{minipage}{\textwidth}\footnotesize
\rule{\textwidth}{1pt} \tabcolsep 1pt
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}lcccccc}
\hline \\
Markup estimate: & \mc{4}{c}{Percent raw returns} & \mc{2}{c}{Dollar raw returns} \vspace{1mm} \\
\cline{2-5} \cline{6-7}
\\
Intercept & 0.360 & 0.361 & 0.556 & 0.567 & 229.2 & -1,129 \\
& (0.000) & (0.000)& (0.000) & (0.000) & (0.000)& (0.000) \vspace{2mm} \\
$Runup$ & -0.251 & -0.249 & -0.278 & -0.280 & -0.208 & -0.246 \\
& (0.000) & (0.000)& (0.000)&(0.000) & (0.414)& (0.343)
\vspace{2mm} \\
$Rival \ runup$ & & -0.038 & &-0.022 & \\
&& (0.062) && (0.220) & &
\vspace{2mm} \\
\mc{5}{l}{\bf Target characteristics} \vspace{2mm} \\
$Target \ size$ &&& -0.045 & -0.048 & & 312.1 \\
&&& (0.000)& (0.000) && (0.002) \vspace{1mm} \\
$NYSE/Amex$ &&&0.023& 0.022 && -7.740 \\
&&& (0.024)& (0.041)&& (0.756) \vspace{1mm} \\
$Turnover$ &&&0.002 & 0.002 && -10.646 \\
& && (0.011) & (0.003) && (0.274) \vspace{2mm} \\
\mc{5}{l}{\bf Bidder characteristics} \vspace{2mm} \\
$Toehold$ && &-0.029 & -0.028 && -94.85 \\
&&& (0.035) & (0.047) && (0.089) \vspace{1mm} \\
$Acquirer \ public$ &&& 0.021 & 0.027 && -210.2 \\
&&& (0.080) & (0.008) && (0.333) \vspace{1mm} \\
{\bf Deal characteristics} \vspace{2mm} \\
$Cash$ &&& 0.002 & 0.004 && -18.57 \\
&& & (0.874)& (0.716) && (0.320) \vspace{1mm} \\
$Stock$&& & -0.019 & -0.026 && 187.0 \\
&& & (0.157) & (0.029) && (0.354) \vspace{1mm} \\
$Hostile$ &&& 0.010 & 0.018 && -60.15 \\
&&& (0.431) & (0.189) && (0.357) \vspace{2mm} \\
Adjusted $R^2$ & 0.024 & 0.028 &0.056 & 0.067 & 0.002 & 0.024 \vspace{1mm}\\
F-value & 93.4 & 46.5 & 20.0 & 26.3 & 0.67 & 30.1 \\
& (0.000) & (0.000)& (0.000) & (0.000) & (0.414) & (0.000)
\vspace{2mm} \\
Sample size ($N$) & 6,857 & 6,498 & 6,781 & 6,436 & 6,857 & 6,781 \\
\\
\hline
\end{tabular*}
\rule{\textwidth}{1pt}
\end{minipage}
\end{table}
\subsection{Incorporating information known before the runup period\label{sec:before}}
Up to this point, we have assumed that the market imparts a vanishingly small likelihood of a takeover into the target price before the beginning of the runup period. An interesting extension is to imagine that the market possesses information at time zero that informs both the expected bid if a bid is made and the likelihood of a bid. Consider the case where the market has a signal $z$ at time zero. During the runup, the market receives a second signal $s$ and, finally, a bid is made if $s+z$ exceeds a threshold level of synergy gains.
Working through the valuations, we have one important change. Define $\pi(z)E(B|z)$ as the expected value of takeover prospects given $z$ and a diffuse prior on $s$. We then have that, at time zero in our model,
$V_0 = \pi(z)E(B|z),$ and the runup and the bid premium would now be measured relative to $V_0$. The runup is
\begin{equation}
\label{eq:info1}
V_R-V_0= \pi(s+z)E_{s+z}[B(S,C)]+T|s+z,bid]+[1-\pi(s+z)]T-\pi(z)E(B|z),
\end{equation}
and the premium is
\begin{equation}
\label{eq:info2}
V_P-V_0=E_{s+z}[B(S,C)+T|s+z,bid]-\pi(z)E(B|z)]-\pi(z)E(B|z).
\end{equation}
Setting aside the influence of $T$, what are now the natural surrogates for the runup and the markup for empirical work? For an investigation into the nonlinear influence of prior anticipation, one would want to add back $V_0$ to both the runup and the bid premium. The influence of $V_0$ is a negative one-for-one on both quantities. Thus markups are not affected. In order to unwind the influence of a known signal prior to time 0, one would need a measure of the signal and data to determine how the signal relates to deal characteristics and frequencies.
\newpage
\begin{figure}[htbp]
\begin{center}
\leavevmode
\caption{\bf The expected change in target valuations through the takeover process
under deal anticipation, assuming no change in target stand-alone value and that
the takeover rumor produces a synergy signal $s$ that is distributed normal.}
\label{fig:normal}
\begin{minipage}{\textwidth}\footnotesize
$V_R$ is the expected target runup, $V_P$ is the expected total target gain, and $V_P-V_R$ is the markup.
Figure A shows the target valuations as a function of the signal $s$.
Figure B shows the the theoretical projection of the markup on the runup. The benefit function has target and bidder
equally sharing synergy gains, while bidder bears a larger share of bid costs. Expected markup approaches
zero as deal anticipation approaches one.
\vspace{4mm}
\end{minipage}
\begin{minipage}{\linewidth}\footnotesize
\begin{center}
%\includegraphics[width=15cm,angle=0]{Figure3ab-normal}
\end{center}
\end{minipage}
\end{center}
\end{figure}
Another case of interest is when the runup contains $T$ but the bargaining outcome is always such that the bidder does {\it not} have to raise the premium accordingly. This is equivalent to the bidder receiving a price discount of $T$ from the target, which lowers the bidder's threshold for making a bid from $K$ to $K^*=\frac{\gamma C}{\theta}-T$. Using the adjusted integral limit, we have that the runup is now
\begin{equation}
\label{eq:tvrt4}
V_{RT}^*=\pi(s)^*E_s[B(S,C)-T|s,bid]+[1-\pi(s)]T.
%V_P-V_R&=&[(1-\pi(s)]\{E_s[B(S,C)|s,bid\}]-[1-\pi(s)]T.
\end{equation}
Combining equations (\ref{eq:tvrt4}) and (\ref{eq:tvp}) yields
\begin{equation}
\label{eq:markup3}
V_{P}-V^*_{RT}=\frac{1-\pi^*(s)}{\pi^*(s)}[V_{RT}-T].
\end{equation}
Eq. (\ref{eq:markup3}) is identical in form to the projection in eq. (\ref{eq:markup2}) for the case with {\it full} markup of $T$. However, the markup values differ because the conditional takeover probability $\pi^*(s)<\pi(s)$ since $K^*