Signaling in Tender O¤er Games
Mike Burkarty Samuel Leez
September, 2009
Abstract We examine whether a bidder can use the terms of the tender o¤er to signal the posttakeover security bene…ts. As atomistic shareholders extract all the gains in security bene…ts, signaling equilibria are subject to a constraint that is absent from bilateral trade models. The buyer (bidder) must enjoy gains from trade that are excluded from bargaining (private bene…ts), but can nonetheless be relinquished and enable shareholders to draw inference about the security bene…ts. Restricted bids and cashequity o¤ers do not satisfy these requirements. Dilution, debt …nancing, probabilistic takeover outcomes and toeholds are all viable signals because they make bidder gains depend on the security bene…ts in a predictable manner. In all the signaling equilibria, lower-valued types must forgo a larger fraction of their private bene…ts and these signaling costs prevent some takeovers. When there is additional private information about the private bene…ts as in the case of two-dimensional bidder types, fully revealing equilibria cease to exist. This does not hold once bidders can o¤er not only cash or equity but also (more) elaborate contingent claims. O¤ers which include options avoid ine ciencies and implement the symmetric information outcome. JEL Classi…cations: G32 Keywords: Signaling, Free-Rider Problem, Means of Payment, Restricted Bids, Twodimensional Types
We thank Patrick Bolton, Franois Desgeorges, Mike Fishman, Alan Morrison and seminar participants at Copenhagen University, the London School of Economics, NHH Bergen, Stockholm School of Economics, University of Amsterdam, the EFA Meeting in Bergen, and the ECGTN Conference in Barcelona for helpful comments. Financial support from the European Corporate Governance Training Network (ECGTN) and the Jan Wallander Foundation is gratefully acknowledged. y Stockholm School of Economics, London School of Economics, CEPR, ECGI and FMG, mike.burkart@hhs.se z Stern School of Business, New York University, slee@stern.nyu.edu
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Introduction
Presumably the best-known friction plaguing the market for corporate control is the freerider problem (Grossman and Hart, 1980; Bradley, 1980): Small shareholders perceive their individual decision as negligible for the tender o¤er outcome, and hence do not tender unless the o¤er price matches at least the post-takeover share value. As a result, they extract all the gains in share value, which in turn may deter potential bidders. Another friction which has arguably received less attention in the takeover literature is asymmetric information. In principle, the bidder as well as the target can possess relevant private information. Contrary to merger negotiations between two management teams, the information advantage in tender o¤ers is likely to be one-sided. Since dispersed target shareholders do not actively monitor the …rm, they seldom possess information not already impounded in the stock price. By contrast, the bidder has typically spent resources to identify the target and to devise post-takeover (restructuring) plans. To succeed, the bidder therefore has to credibly communicate that the o¤er price adequately compensates target shareholders. Otherwise, the o¤er will be rejected even though it may entail a takeover premium. The interaction of information asymmetries and coordination failure makes tender o¤ers distinct from the standard bilateral trade setting. One-sided asymmetric information does not a¤ect the bilateral trade outcome when the informed party makes a take-it-or-leave-it o¤er and is able to match the other party' no-trade payo¤. This is not true in tender o¤er s games. Due to their free-riding behavior, uninformed shareholders have all the bargaining power, even though the informed party (bidder) moves …rst. This paper explores how and when a bidder can signal her private information about the post-takeover security bene…ts to dispersed shareholders. To the best of our knowledge, this question has not yet been systematically analyzed. While it has been shown that separating equilibria can be constructed in tender o¤er games (Hirshleifer and Titman, 1990; Chowdhry and Jegadeesh, 1994), those equilibria require dispersed shareholders to randomize their tendering decisions in a "coordinated" way to produce speci…c takeover probabilities. The present paper derives the general principle lying beneath the existence of separating equilibria, thereby encompassing both previously identi…ed and novel signaling devices. Our analysis begins by considering a tender o¤er game with one-dimensional asymmetric information. Only the bidder knows the post-takeover security bene…ts. In addition, a successful bidder enjoys private bene…ts and cannot commit not to extract these bene…ts. In this setting, an impossibility result obtains: as shareholders extract all the gains in security bene…ts, the bidder cannot reveal her type through the o¤er terms. Thus, neither restricted
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bids nor cash-equity o¤ers are viable signals, which stands out against …ndings from bilateral merger models (e.g., Eckbo et al., 1990). The result exposes a fundamental con between ict incentive-compatibility and free-riding. The incentive-compatibility constraints require that high-valued bidders who have an incentive to mimic low-valued types earn information rents. However, the free-rider behavior precludes that these rents stem from gains in security bene…ts, as they are fully appropriated by the target shareholders. Thus, private bene…ts are a prerequisite for incentive-compatible revealing o¤ers.1 But even if the takeover is associated with private bene…ts, they can only serve a signaling purpose if the bidder is able to commit to relinquish part of the private bene…ts. In addition, the bidder must abandon the private bene…ts in a manner that enables shareholders to infer the post-takeover security bene…ts. Hirshleifer and Titman (1990)' probabilistic separating s equilibrium showcases this principle. In their equilibrium, target shareholders randomize their tendering decision such that bids at lower prices fail with higher probability. The higher failure probability deters high-valued bidders from mimicking low-valued types. Crucially, the deterrence operates exclusively through the risk of forgoing private bene…ts. If bidders lack such bene…ts, or if high-valued bidders have substantially lower private bene…ts, the separating equilibrium breaks down. A principal contribution of this paper is to identify this mechanism as a broad principle for the viability of signaling in tender o¤er games. Accordingly, signaling equilibria can be implemented through dilution, toeholds and debt-…nancing, even when tendering decisions are deterministic. These devices all allow the bidder to choose how much of the proceeds to divert or withhold from target shareholders, and the shareholders to infer the post-takeover security bene…ts. Speci…cally, the bidder can signal low security bene…ts by committing to dilute minority shareholders less, by choosing a smaller toehold, or by raising less debt …nance. While each of these measures reduces her private bene…ts, they allow her to succeed at a lower price. The corresponding signaling equilibria therefore share the implication that the takeover price is positively correlated with the bidder' extraction of private gains (as s exempli…ed by post-takeover dilution, toehold size, or takeover leverage). By contrast, in a tender o¤er game with symmetric information, such a positive correlation is not obtained as the bidder extracts as much private bene…ts as possible irrespective of her type. In these signaling equilibria, lower-valued bidders must forgo more private bene…ts, either through failure or through reducing their level of private bene…t extraction. As a result, the equilibrium outcomes typically exhibit ine ciencies at the "bottom" low-valued bidders :
The result can also be cast in terms of signaling costs. A trustworthy signal must be costly for the bidder. For example, the bidder might voluntarily forgo gains in security bene…ts. Yet, the free-rider problem precludes this possibility by forcing the bidder to forgo all these gains.
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are more prone to fail or do not even submit a bid. In particular, when takeover outcomes are deterministic, only bidders above a cut-o¤ type make a bid in equilibrium, and a lower cut-o¤ value implies more value-improving takeover activity. Furthermore, the signaling equilibria are not robust to the introduction of additional private information. In a setting with two-dimensional bidder types, in which the bidder has additional information about her private bene…ts, signaling is no longer feasible. Essentially, relinquishing a given fraction of private bene…ts is no longer a viable signal because target shareholders are now unable to infer how costly such an o¤er is for the bidder. Another …nding is that bid restrictions, though either insu cient or redundant as a signal, promote takeover activity. This is because smaller transaction sizes mitigate the asymmetric information problem: With fewer traded shares, a bidder gains less from paying a price below the post-takeover share value. This reduces the incentives to mimic low-valued bidders, so that these bidders do not need to sacri…ce as much private bene…ts to credibly reveal their type. Thus, more restricted bids translate into smaller signaling costs. The positive impact of bid restrictions on takeover activity could be taken further if control did not require a majority stake. This suggests another interpretation of the asymmetric information problem in tender o¤er games: control transfers are impaired because control must be transferred along with misvalued cash rights. The appropriate solution is ow therefore to separate votes from cash rights. Indeed, we show that the use of non-voting ow shares or …nancial derivatives can generate signaling equilibria that completely eliminate the impact of asymmetric information. These …nancial instruments allow the bidder to buy the target shares against cash, strip the shares of their votes, repartition the cash rights ow and reissue only those cash rights that she wants to shed. While the …rst two steps give ow the bidder control, the last two steps can be used to self-impose penalties for "lying"about the post-takeover security bene…ts. In particular, call options enable target shareholders to seek "damages"from the bidder ex post if the security bene…ts turn out to be higher than ex ante professed. This makes the bid price de facto contingent on the post-takeover security bene…ts, thereby overcoming the information asymmetry. In practice, derivative claims are indeed used in takeovers; for instance, acquirers sometimes provide target shareholders with upside participation in the form of earn-outs to reduce the payment at the closing of the deal. These earn-outs are triggered only when the target' post-takeover performance exceeds a s certain threshold (Gallant and Ross, 2009). Derivatives allow to implement the symmetric information outcome because the tender o¤er game di¤ers from most other signaling models in corporate …nance. In tender o¤er games, the gains from trade are typically realized upon the transfer of control, as opposed to the transfer of cash rights. Thus, in the market for corporate control, company shares ow 4
represent a bundle of two "goods" cash and voting rights. Separating these goods is , ow bene…cial when frictions in the trade of one impede the trade of the other. The tender o¤er game with privately informed bidders is a corporate …nance application of the informed principal problem (Maskin and Tirole, 1990), with the added feature that the uninformed party su¤ers from a collective action (free-rider) problem. Grossman and Hart (1981) and Shleifer and Vishny (1986) o¤er the …rst analyses of asymmetric information in tender o¤er games. Both papers focus exclusively on pooling equilibria. Hirshleifer and Titman (1990) and Chowdry and Jegadeesh (1994) study tender o¤er games in which takeover outcomes are probabilistic. As we demonstrate, their separating equilibria are applications of a general principle which does not rely on the probabilistic tender o¤er outcome.2 Several papers show that the choice of payment method can overcome asymmetric information problems in mergers (Hansen, 1987; Fishman, 1989; Eckbo et al., 1990; Berkovitch and Naranayan, 1990). Importantly, all of these papers consider bilateral merger negotiations and hence abstract from the free-rider problem. With the exception of Berkovitch and Naranayan, these papers consider two-sided asymmetric information settings in which target shareholders know more either about the share value under the incumbent manager or the takeover synergies. Thus, the settings di¤er from ours in precisely those aspects that are characteristic of tender o¤ers. The same holds true for Brusco et al. (2007) and Ferreira et al. (2007) who study cash-equity o¤ers in a mechanism design framework. The problem they explore becomes rather simple under our informational assumptions, and pure cash o¤ers would always implement the full information outcome. The paper proceeds as follows. The next section presents the basic model with nontransferable private bene…ts. It shows that this model has no signaling equilibria and explains the importance of the free-rider problem for this result. In addition, we demonstrate that neither bid restrictions nor cash-equity o¤ers are viable signals in this setting. Section 3 shows why and how signaling equilibria can be implemented. We …rst develop the principle in an abstract setting and then apply it to familiar variants of the tender o¤er game. We also show that additional private information eradicates fully revealing equilibria. Section 4 demonstrates how the symmetric information outcome can be implemented through the use of derivatives. Section 5 discusses the empirical implications of our analysis. Concluding remarks are in Section 6, and mathematical proofs are in the Appendix.
Signaling can also serve the purpose of deterring potential rivals as in Fishman (1988) and Liu (2008), rather than reducing the information gap between bidder and target shareholders.
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Non-Transferable Private Bene…ts
Our basic setting closely follows existing tender o¤er models with asymmetric information (Shleifer and Vishny, 1986; Hirshleifer and Titman, 1990). There is a widely held …rm that faces a single potential acquirer, henceforth the bidder. If the bidder gains control, she can generate security bene…ts X. The bidder learns her type prior to making the tender o¤er, whereas target shareholders merely know that X is distributed on X = [0; X] according to the continuously di¤erentiable density function g(X). The cumulative distribution function is denoted by G(X). If the takeover does not materialize, the incumbent manager remains in control. The incumbent generates security bene…ts X I which are known to all shareholders and normalized to zero. Thus, we restrict attention to the case of value-improving bids. In addition, control confers exogenous private bene…ts 0 on the bidder. The private bene…ts are only known to the bidder and for simplicity a deterministic function of her type.3 Furthermore, the bidder cannot commit not to extract the private bene…ts once she is in control. As the private bene…ts accrue exclusively to the bidder, they are de facto non-transferable. Our speci…cation of private bene…ts can accommodate various sources of bidder gains, such as dilution (Grossman and Hart, 1980) and toeholds (Shleifer and Vishny, 1986). Though, for the sake of notational simplicity, we subsequently assume that the bidder has no initial stake. As the …rm has a one share - one vote structure, a successful tender o¤er must attract at least 50 percent of the …rm' shares. The tender o¤er is conditional, and therefore becomes s void if less than 50 percent of the shares are tendered. In addition, the bidder can restrict the o¤er to a fraction r 2 [0:5; 1] of the shares. For simplicity, we assume that there are no takeover costs. Hence, the benchmark (full information) outcome is that all takeovers succeed.4 The timing of the model is as follows. In stage 0, the bidder learns her type X. In stage 1, she then decides whether to make a take-it-or-leave-it, conditional, restricted tender o¤er in cash. (Alternative means of payment will be considered later.) If she does not make a bid, the game moves immediately to stage 3. Otherwise, she o¤ers to purchase a fraction r of the outstanding shares at a price rP . In stage 2, the target shareholders non-cooperatively decide whether to tender their shares. Shareholders are homogeneous and atomistic. In stage 3, the incumbent manager
We analyse the more general case with two-dimensional bidder types later in the paper (section 4.2). Like other tender o¤er models exploring the free-rider problem, we assume that the …rm' outstanding s shares of mass 1 are dispersed among an in…nite number of shareholders whose individual holdings are both equal and indivisible. When either of these assumption is relaxed, the Grossman and Hart (1980) result that all the gains in security bene…ts go to the target shareholders becomes diluted (Holmstrm and Nalebu¤, 1992).
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remains in control if the fraction of tendered shares is less than 50 percent. Otherwise, the bidder gains control and pays P unless the o¤er is oversubscribed, in which case she pays rP , and tendering shareholders are randomly rationed. We henceforth refer to the basic model as the tender o¤er game with non-transferable private bene…ts. If target shareholders could coordinate their tendering decisions, they would accept the o¤er whenever the price at least matches the security bene…ts under the incumbent manager. Thus, their reservation price would be independent of the bidder' type, and the bidder would s succeed and appropriate the entire value improvement from the takeover. However, as the shareholders are atomistic and decide non-cooperatively, their reservation price depends on the bidder' type. Each of them tenders at stage 2 only if the o¤ered s price at least matches the expected security bene…ts. Since shareholders condition their expectations on the o¤er terms (r; P ), a successful tender o¤er must satisfy the free-rider condition P E (Xjr; P ). We assume that shareholders do not play weakly dominated strategies. This eliminates failure as an equilibrium outcome when the free-rider condition is strictly satis…ed.5 When the bid price exactly equals the expected post-takeover share value, the target shareholders are strictly indi¤erent between tendering and retaining their shares. That is, they are indi¤erent between these actions irrespective of their beliefs about the takeover outcome, so that the weak dominance criterion does not pin down a tendering strategy. The prevalent way of resolving the indeterminacy when P = E (X jr; P ) is to assume that each shareholder tenders in this case, and hence the bid succeeds with certainty.6 Alternatively, one may assume that strictly indi¤erent shareholders randomize, and that this leads to a probabilistic outcome.7 Subsequently, we focus on deterministic outcomes and examine the conditions under which fully revealing equilibria exist in which (some) bidders signal their type through the chosen o¤er terms. The exception is section 3.1.5 where we consider probabilistic outcomes of the basic model with non-transferable private bene…ts. Finally, to keep focus on the feasibility of signaling, we abstract from pooling equilibrium outcomes
Given a bid is conditional, a shareholder who believes the bid to fail is indi¤erent between tendering and retaining. Imposing this belief on all shareholders and breaking the indi¤erence in favour of retaining supports failure as an equilibrium, irrespective of the o¤ered price (Burkart et al., 2006). To avoid coexistence of success and failure as equilibrium outcomes, it is typically assumed that shareholders tender when they are indi¤erent (e.g., Shleifer and Vishny, 1986). Contrary to our assumption, this precludes failure as the equilibrium outcome for a conditional bid, and hence the existence of an equilibrium when the free-rider condition is violated. 6 A common motivation for this approach is that the bidder could sway the shareholders by raising the price in…nitesimally. Although this argument holds under full information, it does not apply in the asymmetric information setting, as even small price increases a¤ect shareholders'expectations about the post-takeover security bene…ts. 7 Judd (1985) shows that a continuum of i.i.d. variables can generate a stochastic aggregate outcome.
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which exist irrespective of the transferability of private bene…ts.8
2.1
Impossibility of Signaling
Under the assumption that each shareholder tenders in case she is strictly indi¤erent, all shares ( = 1) are tendered in a successful takeover. Accordingly, a successful restricted bid is oversubscribed, and the bidder randomly selects the fraction r among all shareholders whose shares she purchases. The remaining 1 r shareholders cannot sell and become minority shareholders. The bidder' expected pro…t from a bid (r; P ) is s (r; P ) = q(r; P ) [ (X) + r (X P )]
where q(r; P ) denotes the success probability which is equal to 1 for P E (X jr; P ) and 0 otherwise. In a fully revealing equilibrium, the o¤er terms must be distinct across types that make a (successful) bid. This requires that each equilibrium o¤er satis…es the free-rider condition, P (X) X, and the bidder' incentive-compatibility constraint s (X) + r (X) [X for all r 2 [0:5; 1] and P 2 R. Theorem 1 In deterministic tender o¤er games with non-transferable private bene…ts, no fully revealing equilibrium exists. Given that P (X) X, a truthful bidder at best breaks even on the purchased shares, and her expected pro…t cannot exceed (X). However, each type o¤ering her actual security bene…ts cannot be an equilibrium outcome. If a type x would succeed with an o¤er rx, any type X > x would mimic type x to acquire shares at a price below their true value X. This also holds if each type would choose a di¤erent bid restriction r( ). Type X' pro…ts are s higher when buying r(x) shares at a discount compared to buying r(X) shares at their fair price whether r(x) is smaller or larger than r(X). These arguments eliminate P (X) = X combined either with a common r or a type-contingent r( ) as possible equilibria. They also rule out outcomes in which some types o¤er more than their true security bene…ts but less than the highest-valued type' security bene…ts. Successful o¤ers with P (x) 2 (x; X) would s
Pooling outcomes are extensively studied in Shleifer and Vishny (1986), Marquez and Yilmaz (2005) and At et al. (2008).
8
P (X)]
(X) + r (X
P)
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be mimicked by bidders of type X > P (x). Thus, a bidder can credibly signal her type only by o¤ering a su ciently large premium such that P X. Revealing her type with an o¤er P X is, however, not an attractive option for the bidder.9 She can instead make a bid P = X and restrict it to r = 0:5, the minimum fraction required to gain control. The less costly o¤er (0:5; X) succeeds as it satis…es the free-rider condition for all types (and any possible shareholder beliefs). The inexistence result extends to settings where the private bene…ts are not a deterministic function of the bidder' type, but follow some— possibly type-contingent— density s function. Indeed, the constraints in the bidder' maximization problem are not a¤ected by s the non-transferable private bene…ts. They cancel out in the incentive-compatibility constraint and they are not part of the free-rider condition. Also, note that letting bidders choose the fraction of shares that they acquire does not enable them to signal their type. The sole function of the bid restriction is to limit the fraction of shares the bidder purchases for cash. This makes restricted bids in this setting equivalent to bids in which target shareholders are in part compensated through equity. Indeed, it is immaterial whether the bidder makes a partial bid for cash only or acquires all shares in exchange for some cash and 1 r shares in the target …rm under her control. Moreover, control requires that the partial bid is for at least half the shares or that the equity component does not exceed the cash component in the cash-equity o¤er. By virtue of this equivalence, any fully revealing equilibrium in cash-equity o¤ers would also have to exist in restricted cash only o¤ers. Proposition 1 Introducing cash-equity o¤ers into deterministic tender o¤er games with non-transferable private bene…ts does not make fully revealing equilibria feasible. Proposition 1 contrasts with results from bilateral merger models where cash-equity o¤ers can reveal the bidder' type (Hansen, 1987; Berkovitch and Naranayan, 1990; Eckbo et al., s 10 1990). Our basic framework di¤ers in two key respects. First, target shareholders have no private information. Instead, they face a collective action problem, i.e., are unable to
In fact, an incentive-compatible schedule f(r( ); P ( ))g can be constructed which entails that lower-valued bidders o¤er higher (per-share) prices but purchase fewer shares. Bidders abstain from mimicking lowervalued types as they would forego a larger more valuable equity stake. Conversely, bidders do not mimic higher-valued types because they would have to pay a larger total amount for an equity stake that is worth less. 10 Contrary to negotiated mergers, tender o¤ers are usually cash o¤ers. In fact, the mode of acquisition is one of the most important determinants of the payment method (e.g., Martin, 1996). The standard explanation focuses on regulatory delays associated with equity o¤ers, i.e., the greater cost of using equity as a means of payment. Our analysis suggests that means of payment do not help to overcome asymmetric information problems in tender o¤ers.
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coordinate their individual tendering decisions.11 Second, the takeover is not undertaken to combine assets from two …rms but to replace the incumbent managers. How or whether the free-rider problem a¤ects signaling equilibria is studied in the next section, while the role of bidder assets will be explored later in the paper (section 3.1.3).
2.2
Free-Riding and Information Rents
To illustrate the role of the free-rider condition, we digress to a modi…ed setting in which the bidder is able to appropriate part of the security bene…ts. Abstracting from a speci…c extensive form, we assume that the bidder has bargaining power ! 2 (0; 1) such that shareholders, if fully informed, would tender at a price P = (1 !) X. Accordingly, the bidder would under full information appropriate a value improvement !X on the purchased shares. Like the private bene…ts , these gains depend on the bidder type and a successful takeover. But unlike the private bene…ts, the gains are transferable. That is, the bidder can (commit to) leave part of !X to the shareholders. A second, purely simplifying modi…cation is the absence of private bene…ts ( = 0). Given shareholders do not observe the bidder' type, they condition their beliefs on the s o¤er terms and tender only if P (1 !) E (X jr; P ). Consequently, some bidders may not succeed or realize less than the full information pro…t !X. That is, some bidders may have to o¤er more than (1 !) X or set r X. This makes her o¤er less attractive to mimic so that she needs to relinquish less private bene…ts. Paying the shareholders through P ( ) or relinquishing part of ( ) are therefore substitutes, provided that lower-valued types transfer a su ciently large fraction of their private gains. A second source of multiplicity is the freedom of choice with respect to r( ). While the chosen r ( ) a¤ects ( ) and P ( ) through (3), bid restrictions are a redundant signal in the sense that a fully revealing equilibrium can be supported even when the restrictions are uniform, r (X) = r. The reason is that signaling is achieved by transferring private bene…ts, which does not rely on restricting the bid. Though, this is not to say that bid restrictions are irrelevant because they a¤ect the e ciency of the tender o¤er outcome. We will come back to this point in Section 4.1. Finally, the assumption that ( ) is non-decreasing ensures that relinquishing a given fraction of the private bene…ts is more costly for higher-valued types. This condition is suf…cient but not necessary to obtain fully revealing equilibria. When the condition is violated, bidders may …nd it too expensive to separate themselves from higher-valued types so that only pooling equilibria are feasible.14 Though there exist cases with non-monotonic ( ) in which fully revealing equilibria can nonetheless be supported. More importantly, however, the assumption that 0 ( ) 0 seems in general economically plausible. In particular, it is naturally satis…ed in common variants of the tender o¤er game, as we subsequently show. 3.1.1 Dilution
In the …rst extension of the basic tender o¤er game, we let the successful bidder choose what fraction of the …rm' total post-takeover value V 2 V to divert as private bene…ts. s Also, dilution does not dissipate value, so that a successful bid generates security bene…ts X(V ) = (1 )V and private bene…ts (V ) = V . Diversion as a source of private bene…ts was …rst introduced by Grossman and Hart (1980), who assume a uniform dilution rate . Burkart et al. (1998) endogenize the ex post dilution decision by assuming that diversion dissipates value, so that is determined by the
For instance, when ( ) is strictly decreasing, no type X < X is able to separate herself from the highest-valued type X by paying a lower price and relinquishing a larger share of private bene…ts, as this o¤er— if it were preferred by type X over X' o¤er— would be mimicked by X. s
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bidder' post-takeover equity stake. This implies that the bidder can de facto pre-commit s to a dilution rate by choosing what fraction of target shares she acquires. Here, we take a simpler approach and assume that the bidder can commit not to extract more than a fraction 2 [0; ] of V independent of the bid restriction.15 The upper bound 0. In addition, suppose that the bidder can raise debt up to D 2 [0; D] where D 0, the bidder could use claims to such assets to pay target shareholders, and the willingness to do so might reveal her type. Theorem 2 suggests that the viability of such signals requires that the bidder assets appreciate in value as a result of the takeover, and that this appreciation is non-negatively correlated with the value improvement in the target …rm. In addition, these "synergy" gains must be exclusionary in that they do not accrue to non-tendering target shareholders and, as such, do not a¤ect their reservation price (free-rider condition). In this sense, they are equivalent to private bene…ts. Suppose that the bidder owns cash and a …rm that generates security bene…ts Z + X if the bid succeeds and X otherwise. Both the value Z 0 and the parameter > 0 are commonly known and the same for all types. If the bidder is successful, she combines the two …rms in a holding company H. Target shareholders are o¤ered a cash price C( ) and s( ) shares in the holding company, where denotes the fraction of target shares tendered. Furthermore, target shareholders are cash-constrained, and the bidder is unwilling to relinquish majority control of the holding company. These assumptions impose two restrictions on the set of admissible o¤ers, a cash constraint C( ) 0 and a control constraint s( ) 2 [0; 0:5]. Application 3 Sharing exclusionary synergy gains is a viable signal. All types above the cut-o¤ type make a merger bid, and lower types pay more in equity of the merged company. The cash price C and the fraction of shares s that target shareholders receive in equilibrium are inversely related. Bidders who give target shareholders a larger fraction of the post-merger equity pay a smaller amount of cash. Even though X is perfectly correlated with X, the mere fact that the bidder assets are informative about the post-takeover share value is neither necessary nor su cient to obtain a fully revealing equilibrium. The key is that the bidder assets appreciate in value as a result of the takeover, i.e., that the bidder enjoys the exclusionary gains Z. Relinquishing (more) post-merger equity is a means of sharing (more of) these "private bene…ts". Accordingly, the cut-o¤ type is decreasing in Z, and takeover activity— again, absent other signaling devices— vanishes as Z approaches 0. Note also that the value of the gains which the bidder would retain under symmetric information is (X) = Z + X and increasing in her type.
D x, she would pay a cash price x for shares that are worth X. However, ex post she would not capitalize on this gain, as the target shareholders would exercise their options once the actual value improvement becomes known. Conversely, the low-valued type does not mimic the high-valued type because she would pay X for shares that are worth x. Thus, the o¤er schedule is incentive-compatible. Moreover, every bidder succeeds, irrespective of how many shares she acquires or whether she enjoys any private bene…ts. Financial engineering enables the bidder (i) to trade economic ownership void of voting rights and (ii) to issue contingent claims. The …rst step of the transaction consists of acquiring the target shares and stripping them of their votes. In the second step, the bidder re-issues the cash rights, restructured into claims that punish her for "lying"about the security ow bene…ts. The call options which are executed when the post-takeover security bene…ts are higher than professed penalize the pretense of low security bene…ts— ex post when the true value is observed. This makes the o¤er equivalent to the simplest solution to the asymmetric information problem: a bid price which is contingent on the post-takeover share value. 22
The above o¤er transfers, cash aside, only future claims but no actual future cash ows to target shareholders. This is an artefact of the assumption that the post-takeover security bene…ts X are deterministic (i.e., perfectly known by the bidder). Yet, the main intuition carries over to a setting with stochastic cash ows. Let X 2 [0; 1] be a random variable. Suppose that there are n bidder types 2 f1; 2; : : : ; ng, each knowing the probability density function f (X) of her post-takeover cash ows. In addition, we assume that the family of densities ff (X)g 2 satis…es the strong monotone likelihood ratio property (SMLRP). That is, for all 0 > , f 0 (X)=f (X) is strictly increasing. To construct a fully revealing equilibrium, we allow the bidder to pay cash and issue bonds and barrier options.22 The barrier option used to construct the equilibrium is a (cashsettled) "knock-in"(or "up-and-in" call option. This is a latent call option with an exercise ) price of S that becomes activated only once the security bene…ts X exceeds some "trigger" level T > S. When they are exercised, these options dilute the value of the …rm' equity. s Hence, the combination of bonds and knock-in options can alternatively be interpreted as a reduced-form implementation of convertible bonds. To simplify the exposition, we further assume that = 0. Proposition 4 In the tender o¤er game with stochastic post-takeover security bene…ts, a fully revealing equilibrium exists if SMLRP holds. All bidder types make a bid and purchase the target shares for a combination of cash, bonds and knock-in call options. The security design solution in Proposition 4 o¤ers both upside and downside "protection"to the sellers. The security issuer— here, the bidder— primarily wants to signal a high value. To this end, she issues knock-in options that are designed to transfer some high cash realizations to the target shareholders. However, the use of knock-in options makes the ow bidder prone to being mimicked by (even) lower-valued types. Because this casts doubt on the value of the o¤ered options, she must include bonds to separate herself from lower-valued types. Thus, the bidder' need to separate from lower-valued types through o¤ering downs side protection derives endogenously from the bidder' primary intention to separate from s higher-valued types through o¤ering upside protection.23 A recent strand of literature emphasizes the potential adverse e¤ects of using contingent claims to unbundle control and ownership (e.g., Hu and Black, 2006, 2008). This literature
22 A barrier option is a derivative where the option to exercise depends on the (price of the) underlying crossing or reaching a given barrier level. 23 It is instructive to compare this outcome to the security design solutions in Myers and Majluf (1984) and Du e and DeMarzo (1999). There, the issuer is (only) concerned with signaling a high value and therefore issues debt-like securities that are truncated from above. Here, in tender o¤er games, the bidder wants to signal a low value, which is why she (also) issues securities that are truncated from below.
23
contends that re-allocating control void of economic ownership can lead to ine cient decisionmaking ("empty voting" For example, acquiring control but leaving the "upside" to the ). target shareholders could undermine the bidder' incentives to increase the share value. In s a moral hazard setting, where the bidder' post-takeover incentives depend on the nature s of the economic interest that she acquires in the …rm, it would therefore seem both socially and privately suboptimal for the bidder to pay target shareholders with knock-in options. Proposition 4 thus suggests that adverse selection and moral hazard do not yield similar predictions in tender o¤er games. Note that this is not the case in external …nancing models where moral hazard and adverse selection both predict that control and ownership should be bundled and that, if they need to be separated, the controlling party should retain the "upside" 24 In tender o¤er games, unbundling— rather than bundling— control and owner. ship can improve the e ciency of the control allocation. Moreover, conceding the "upside" to outside shareholders can be optimal for the controlling shareholder. Key to the bene…t of unbundling is that the two counterparties, the bidder and the target shareholders, value cash rights and voting rights di¤erently. As a result, asymmetric ow information about one set of rights can impede the (e cient) transfer of the other set of rights. Such situations are by no means con…ned to tender o¤ers. It can arise in any transaction involving bundles of goods, in which the buyer and the seller not only value the goods di¤erently but also have di¤erent levels of information about their quality. Adopting this a view for cash and voting rights can lead to new insights for corporate governance ow and security design.
5
Empirical Implications
Due to the free-rider problem, signaling in tender o¤ers is rather di¤erent from signaling in bilateral merger negotiations. This is most evident in our …nding that cash-equity o¤ers are much less likely to provide the signaling bene…ts that they are associated with in the merger literature (Corollary 1 and Proposition 3). Empirically, this suggests that P1 The use of all-cash or cash-equity o¤ers should be less pronounced in (hostile) tender o¤ers than in negotiated takeovers. P2 The proportion of cash used as a means of payment should be less strongly correlated with the takeover gains in tender o¤ers than in negotiated takeovers.
See e.g., Jensen and Meckling (1976), Leland and Pyle (1977), Myers and Majluf (1984), Innes (1990), Du e and DeMarzo (1999).
24
24
The basic problem in models of bilateral merger negotiations under asymmetric information is a lemon' problem, namely that informed parties are prone to overstate the value s of their assets as e.g., in Myers and Majluf (1984). In contrast, tender o¤ers with privately informed bidders are an example of a "smart buyer"problem, in which the informed bidder has an incentive to understate the value of the asset. One way for the bidder to credibly reveal a lower value is to share more of her private gains from trade, so long as the willingness to do so allows the uninformed seller(s) to infer the true asset value (Theorem 2). Common to such signaling equilibria is the prediction that takeover premia should be positively correlated with the fraction of the takeover surplus that the bidder extracts as private bene…ts (hereafter, relative private bene…ts). By comparison, in the absence of asymmetric information, bidders would appropriate the maximum possible amount of private bene…ts. in which case bid premia and the bidder' relative private bene…ts would either be unrelated s (in the case of dilution or toeholds) or inversely related (in the case of leverage). Since private bene…ts are di cult to measure directly, empirical tests of the relationship between takeover premia and private bene…ts may have to rely on proxies for private bene…t extraction. For such tests, the preceding analysis suggests that P3 Across tender o¤ers, takeover premia should be positively correlated with fewer voluntary governance restrictions for the bidder, greater amounts of bidder debt in bootstrap acquisitions, or larger toehold sizes. A potential caveat is that these predictions may empirically be of second-order importance, as the choice of governance, toeholds or debt may primarily be driven by other factors, such as free cash problems (Jensen, 1986) or aggressive bidding strategies (Chowdhry ow and Nanda, 1993; Bulow et al., 1999). For instance, if the bidder' most important concern is s to deter potential rival bidders, the presence of toeholds might coincide with lower takeover premia, contrary to the above prediction (see e.g., Betton et al., 2008). There exist only few empirical studies regarding the role of debt …nance in takeovers. Schlingemann (2004) …nds no signi…cant relationship between the amount of debt …nancing before a takeover announcement and bidder gains. In contrast, Martynova and Renneboog (2009) …nd that debt …nancing is associated with positive valuation e¤ects, i.e., represents a positive signal to investors.25 For a more in-depth analysis of the e¤ects of takeover leverage, however, one should discriminate between tender o¤ers and negotiated mergers, contested and uncontested tender o¤ers, and bootstrap and non-bootstrap acquisitions. Another way for bidders to signal their true value is to include contingent claims in their tender o¤er (Propositions 3 and 4). The distinctive feature of this security design solution is
25
See also the paper by To¤anin (2005) cited in Betton et al. (2008).
25
that, in order for the bidder to placate target shareholders'fears of being short-changed, the tender o¤er includes claims that become e¤ective if (and only if) the post-takeover target value turns out to be higher than professed at the time of the o¤er. One example for such claims are convertible bonds. Another example are earnouts which trigger payments to the sellers only if— some time after the deal has been closed— certain performance targets are achieved. Our analysis thus suggests that P4 Bidders avoid overpaying for the target by o¤ering option-like claims, such as convertible bonds or earnouts, that pay o¤ in case the target' post-takeover performance exceeds s a certain threshold. The following quote from a practitioner' memo about special purpose acquisition coms panies (SPACs) supports the notion that bidders may provide upside participation in the form of earnouts to reduce the payment at the closing of a deal:26 Using earnouts, the SPAC is able to pay a lesser guaranteed amount to the sellers of the target business on the closing date and will have to pay the additional amounts only if the operating business meets or exceeds expectations. Although the sellers receive a smaller payment on the closing date, they have the chance to receive signi…cantly more than they otherwise would have if they were being paid solely for the current value of the business. Earnout arrangements can therefore bene…t both parties and increase the likelihood of the transaction being approved. There are a few academic studies on the use of derivative securities in control transactions. In his textbook, Gaugham (2002) describes a wide range of derivatives used in mergers and acquisitions, including claims that provide sellers with both downside and upside protection. O cer (2004) studies the use of collars in mergers. Datar et al. (2001) and Ragozzino and Reuer (2009) study earnouts, and Finnerty and Yan (2009) study convertibles. All of these papers report evidence suggesting that acquirers use derivatives to mitigate asymmetric information problems. Though, the working hypotheses almost purely focus on lemon' s problems (sell-side private information) and ignore potential smart buyer problems (buyside private information).
6
Conclusion
This paper analyzes tender o¤ers in which a single bidder is better informed about the posttakeover share value than dispersed target shareholders. Two key features of the tender
26
The quote is taken from Gallant and Ross (2009).
26
o¤er process render this situation very di¤erent from standard bilateral trade models. First, free-riding shareholders have full bargaining power over the value improvement in the target shares, even though the better informed bidder makes a take-it-or-leave-it o¤er. Second, the parties in a tender o¤er bargain both over control (voting rights) and over ownership (cash rights) in the target …rm. That is, unlike other signaling models in …nance, a share ow (trade) represents a (trade of a) bundle of two goods with potentially distinct values. We demonstrate that these di¤erences lead to constraints as well as solutions that are absent in bilateral trade models. Because the bidder is forced to concede all gains in share value to the shareholders, she cannot signal her type by voluntarily giving up such gains. Neither restricted bids nor cash-equity o¤ers are therefore viable signals in tender o¤ers. Instead, the bidder must enjoy private bene…ts that are not only excluded from bargaining but can also be forgone in a manner which allows inference about the post-takeover share value. This is never possible when the bidder has additional private information about the private bene…ts, as in the case of two-dimensional bidder types. In the one-dimensional case, relinquishing private bene…ts is a viable signal because they depend on the security bene…ts in a predictable manner. Dilution, debt …nancing or toeholds can serve this purpose. The underlying principle in all cases is the same: the bidder must forgo (more) private bene…ts to signal a low(er) type. Unfortunately, some low-value bidders may …nd it too costly to signal their type even if the takeover would be e cient. Such ine ciencies can be overcome if the bidder can include derivatives in the tender o¤er terms. Derivatives allow the bidder to separate cash rights from voting rights. ow This separation prevents that the information problems in the trade of cash rights ow spill over into, and thereby impair, the trade of voting rights. As a result, control can be transferred e ciently irrespective of any potential disagreement between the bidder and the target shareholders about the value of the post-takeover cash rights. ow Our analysis has implications for the design of takeover bids. For instance, it suggests that derivatives as a means of payment are an e¤ective signaling device in takeovers, while cashequity o¤ers should play a less prominent role in tender o¤ers than in negotiated mergers. Furthermore, acquiring …rms may signal their quality through self-imposed restrictions on post-takeover decisions or through the amount of takeover leverage. The main theoretical contribution of this paper is to study how the interaction of asymmetric information and collective action problems, in a speci…c market setting, may bear on the optimal design of a trade contract. We believe that there are situations other than tender o¤ers in which such interactions are potentially important as e.g., renegotiations with dispersed bondholders.
27
Proofs
Proof of Proposition 1
Proposition 1 follows from the equivalence of mixed o¤ers and restricted cash-only o¤ers which the subsequent lemma establishes. Consider a bid for r shares that o¤ers a cash price C and t shares in the post-takeover …rm. Lemma 1 Under full information, the restricted mixed o¤er (r; C; t) and the restricted cashonly o¤er (rco ; C co ) with C co = C and rco = r t are payo¤-equivalent. Proof. To succeed, the mixed o¤er must satisfy the free-rider condition C + tX equivalently C=r + (t=r) X X. Given the condition is satis…ed, all shareholders tender, and the bidder' payo¤ is s (X) + r [X Rearranging the free-rider condition (4) to C and the bidder' payo¤ (5) to s (X) + (r t) X C (r t) X (C=r + (t=r) X)] . (5) rX, or (4)
shows that the restricted cash-only o¤er (rco ; C co ) with C co = C and rco = r t is payo¤equivalent for any X. Hence, if a fully revealing equilibrium in mixed o¤ers were to exist, a fully revealing equilibrium in cash-only o¤ers would also exist. As Theorem 1 rules out the latter, a mix of cash and equity is not a viable signal.
Proof of Proposition 2
We …rst characterize properties of an incentive-compatible r-P -schedule. Lemma 2 In a fully revealing equilibrium, r( ), P ( ) and ( ) must be increasing.
Proof. Without loss of generality, consider an arbitrary pair of types, X and x with X > x. A fully revealing schedule f(r( ); P ( ))g must satisfy the non-mimicking constraints r (X) [X P (X)] r (x) [X 28 P (x)] for (x; X) 2 X 2 . (6)
We show by contradiction that (6) requires r (X) > r (x). The non-mimicking constraints for type X and x are C (X) r (X) X C (x) r (x) X and C (x) r (x) x C (X) r (X) x (7)
where C ( ) r ( ) P ( ). For r (X) = r (x), the inequalities hold jointly only if C(X) = C(x), and hence P (X) = P (x), in which case the two o¤ers would be identical. For r (X) C (X) + [r (x) r (X)] x, the constraints cannot hold jointly. Thus, the non-mimicking constraints are violated unless r ( ) is increasing. Given r (X) > r (x), condition (6) implies that the bid price and the bidder' pro…t must s also be increasing in her type. To this end, we rewrite (7) as r (X) [X P (X)] r (x) [X P (x)] and r (X) [x r (x) P (X)] x P (x) .
Given that r (X) =r (x) > 1, the second inequality implies P (X) > P (x). Furthermore, as r (x) [X P (x)] > r (x) [x P (x)], the …rst inequality implies r (X) [X P (X)] r (x) [x P (x)]. Thus, higher types must pay higher prices and make higher pro…ts. Local Optimality. Lemma 2 states necessary conditions for incentive-compatibility. To derive a particular schedule, we assume that r( ) and C( ) are continuously di¤erentiable functions, and rephrase the bidder' optimization problem as a direct truth-telling mechas nism: n o ^ ^ max r(X)X C(X) . (8)
^ X2X
^ In equilibrium, the …rst-order condition must hold at X = X, i.e. r0 (X)X = C 0 (X).
(9)
Quasi-concavity. Condition (9) is su cient to ensure incentive-compatibility if the objective function in (8) is quasi-concave (and out-of-equilibrium beliefs are suitably chosen). Substituting (9) into the derivative of the objective function gives @ h ^ r(X)X ^ @X i ^ C(X) ^ = r0 (X)X
^ ^ ^ C 0 (X)=r 0 (X)X
^ ^ ^ r0 (X)X = r0 (X)(X
^ X).
29
Given that r0 ( ) > 0 (Lemma 2), it follows that the derivative switches sign for all types X 2 (0; X) once (from positive to negative), and the objective function is strictly quasiconcave. Cut-o¤ type. Condition (9) puts a constraint on how equilibrium pro…ts (X) = r(X)X C(X) vary across types. By the envelope theorem, @ (X) = r0 (X)X C 0 (X) + r(X) = r(X). | {z } @X
=0
(10)
That is, the marginal change in pro…ts is given by the bid restriction r(X). Given that bidders have bargaining power !, shareholders always tender at the price P = (1 !)X. As type X buys shares below their true value, she buys all shares and makes X = !X. Since pro…ts decrease at the rate r(X) (condition (10)), the cut-o¤ a pro…t c type X , making zero pro…ts, is de…ned by Z
X
r(u)du = !X.
(11)
Xc
Out-of-equilibrium beliefs. The proposed schedule can be supported as a signaling equilibrium with out-of-equilibrium beliefs that any deviation comes from the highest-valued bidder type X. Under these beliefs, the target shareholders do not tender their shares in ~ response to a deviation bid (~; P ) unless P (1 !) X = P (X). Consider two cases. (i) For r ~ r bidder types X 2 [P (X); X], the deviation bid (~; P (X)) would yield a positive pro…t. Yet, it is dominated by the (r(X); P (X)) = (1; P (X)), the equilibrium bid of the highest type, which we know to be mimicking-proof. Hence, by implication, the deviation is unattractive to these types. (ii) For bidder types X 2 [0; P (X)), the deviation bid would yield a loss and is therefore unattractive to these types.
Proof of Corollary 1
From the de…nition of the cut-o¤ type (equation (11)), it follows that @X c =@! 0, r0 0 and P 0 > 0 that can be supported as a fully revealing equilibrium. Quasi-concavity. Suppose that the proposed schedule satis…es (3) for all X 2 X . This 30
schedule then satis…es quasi-concavity of the objective function. Speci…cally, we show that ^ @ =@ X =
0
^ ^ (X) (X) + r0 (X)[X
^ P (X)] ^ X.
^ ^ r(X)P 0 (X)
(12)
^ is non-negative when X X and non-positive when X Condition (3) implies that ^ ^ r(X)P 0 (X) =
0
^ ^ ^ ^ (X) (X) + r0 (X)[X
^ P (X)].
Substituting the right-hand side into (12) and rearranging yields ^ @ =@ X =
0
^ (X)
h
(X)
i h ^ ^ (X) + r0 (X) X
i ^ X . X and that (X)
(13) ^ (X)
^ ^ The assumption 0 ( ) 0 implies that (X) (X) when X ^ when X X. Given that 0 > 0 and r0 0, it follows that
^ Thus, the proposed schedule makes (X; X) weakly quasi-concave for all X 2 X . This also ^ holds for r0 (X) = 0, in which case all bidder types propose the same bid restriction. Local optimality. Condition (3) is a functional equation for ( ), r ( ) and P ( ) with two degrees of freedom. To derive an example of an incentive-compatible schedule, we set r ( ) = 0:5. Then, condition (3) simpli…es to
0
8 ^ > non-negative for X < X : ^ non-positive for X > X
(X) =
P 0 (X) . 2 (X)
Integrating on both sides over [X; X] yields Z
X 0
(u)du =
X
Z
X
X
P 0 (u) du 2 (u)
,
(X)
(X) =
Z
X
X
P 0 (u) du. 2 (u)
As the highest-valued type does not have to relinquish any private bene…ts [ (X) = 1], (X) = 1 Z
X
X
P 0 (u) du. (u) (X) = 1 RX
X
(14) [ (u)]
1
One possible price schedule is P (X) = X in which case 31
du. As
shareholders receive P (X) + [1 (X)] (X), the free-rider condition is also satis…ed. Cut-o¤ type. The condition (3) puts a constraint on how equilibrium pro…ts vary across types in equilibrium. By the envelope theorem, we have that equilibrium pro…ts must be increasing at the rate @ =@X = (X) 0 (X) + r(X), for any schedule that satis…es (3). Given an equilibrium exists, the cut-o¤ type X c is given by Z
X
Xc
f (u) 0 (u) + r(u)g du = (X).
(15)
Under the proposed equilibrium schedule, bidder types below X c incur a loss under the proposed schedule. Hence, they prefer not making a bid over making the bid prescribed by the proposed schedule. The option of notmaking a bid does not undermine the nonmimicking constraints. Under the proposed schedule, the bidder prefers a loss-making o¤er to o¤ers made by higher-valued types. A fortiori, she also prefers a zero-pro…t o¤er over the latter. Out-of-equilibrium beliefs. The proposed schedule can be supported as a signaling equilibrium under the out-of-equilibrium beliefs that any deviation comes from the highestvalued bidder type, X. Under these beliefs, the target shareholders do not tender their shares ~ ~ in response to a deviation bid (~; ~ ; P ) unless P r X. Any such bid, however, is weakly dominated by 0:5; 1; X , which is the equilibrium bid of the highest type. Since 0:5; 1; X is mimicking-proof, any successful deviation bid is— by implication— unattractive under the proposed out-of-equilibrium beliefs. Bid restriction. The above proves that the bid restriction r is a redundant signal. This follows because, as we have shown, the schedule fr (X); (X); P (X)g = ( n 0:5; 1 RX
X
[ (u)]
1
du; X
f0; 0; 0g
o
for X 2 [X c ; X] for X 2 [0; X c )
can, among others, be supported as a fully revealing equilibrium. By contrast, the private bene…t retention rate and the price P are indispensable as signals. First, if is invariant across types, Theorem 1 applies. Second, a uniform price in a fully revealing equilibrium must satisfy P = X. But then all bidder types X < X prefer the o¤er (0:5; 1; X), which always succeeds irrespective of shareholder beliefs, to any other o¤er with P = X. Hence, they would pool.
32
Application 1 Given bidder of type V 2 V can choose
^ V
h ^ ^ max (V )V + r(V ) (1 with h ^ )V + r (1 (V h ^ (V )V + r (1
2 [0; ], she solves ^ (V )V )
i ^ P (V )
^ Set r(V ) = r, express
as
2 (0; 1), and the objective function simpli…es to ^ (V )V ) i ^ ^ )V P (V ) + (1 (V )) V i h i ^ )) V + r (1 ^) (V )V P (V (X) i ^) P (V
= = h
^ (V ) + r(1
This last h expression is isomorphic to (1) when using the de…nitions X (1 )V , V i ^ ^ and (V ) + r(1 (V )) . Note that V is increasing in the bidder' type. s Application 2 Given bidder of type X 0 2 [X; X] can choose D 2 [0; D], she solves h ^ ^ max D(X) + r(X) X ^ D(X) i ^ P (X)
^ Set r(X) = r, express D as D
and rewrite the objective function as
h i ^ ^ ^ D (X) + r X D (X) P (X) h i ^ + r X D P (X) + (1 ^ ^ D = D (X) (X)) h i h i ^ ^ ^ = (X) + r(1 (X)) D + r X D P (X) This last expression is isomorphic to (1) when using the de…nitions D (X) and h i ^ ^ (X) + r(1 (X)) . Note that D, i.e., ( ), is constant across bidder types. Application 3 Denote the value of the bidder assets by A(I; X) I 0:5 Z + X. The indicator function I 0:5 takes the value 1 if the bid succeeds and 0 otherwise. If the bid succeeds ( 0:5), the holding company is worth H( ; X) = A(1; X) + X. Under full information, shareholders do not tender unless C( ) + s( )H( ; X) X. To ensure a successful merger ( = 1), the
33
bidder must choose s( ) and C( ) such that s( ) X C( ) Z+ X+ X (16)
for all 2 [0:5; 1]. In this case, all shareholders tender their shares whenever they believe that more than half the shares are tendered, and the bidder must ultimately pay C(1) and s(1). To simplify the exposition, we omit and express the bidder' o¤er as a pair (s; C) s which must satisfy the free-rider condition for = 1, i.e., s (X C)=(Z + X + X).27 Note that condition (16) violates neither the cash constraint nor the control constraint if C( ) is chosen su ciently high. For a given cash price C and equity component s, the bidder' pro…t from a successful s merger is therefore (X) = (1 s)H (1; X) C X = (1 s)Z + (1 s) X s X +C 1 s .
X+C Now de…ne 1 s, r = , (X) = Z, and P = s 1 s . The bidder can use s to adjust and r, and she can use C to adjust P . From Theorem 2, it follows that must be increasing which in turn implies that s must be decreasing. The constraint = r results from the fact that the bidder merges the …rms and pays the target shareholders with holding company shares. In this setting, the cut-o¤ type is not necessarily determined by the participation constraint ( 0). As lower types issue more equity, they may also run either into the control constraint s( ) 0:5 or into the cash constraint C( ) 0. The latter may occur because the bidder can in principle become a net issuer, rather than a net purchaser, of …nancial claims. The cash constraint is relevant for bidders for whom A is very large relative to X. Notwithstanding, the participation constraint becomes binding as Z decreases. In particular, Z = 0 is equivalent to (X) = 0, and hence causes signaling breaks down. (It is straightforward to verify that using X (instead of Z) as the synergy gains leads to similar results; in particular, signaling breaks down when = 0.)
Even without a contingent o¤er, there exists a self-ful…lling equilibrium in which the merger succeeds for (C; t) as long as it satis…es the free-rider condition for = 1: If each shareholder believes that all other shareholders tender, she also tenders. Hence, once can alternatively focus on non-contingent o¤ers, and select merger success as the equilibrium outcome whenever it is consistent with the free-rider condition.
27
34
Application 4 ^ The bidder can choose t 2 [0; t] and solves ^ ^ max t(X)X + rt (X)(1 with rt 2
0:5 t ;1 1 t
h ^ t(X)) X
i ^ P (X)
^ . Set rt (X) = rt , express t as t and rewrite the objective function as ^ t (X)X + rt (1 h ^ = (X) + rt (1 h i ^ ^ t (X)) X P (X) i h i ^ ^ (X)) tX + rt X P (X)
This last expression is isomorphic to (1) when using the de…nitions tX (X) and h i ^ ^ (X) + r(1 (X)) . Note that tX, i.e., ( ), is increasing in the bidder' type. s Application 5 The objective function in the probabilistic tender o¤er game is: (r; P ) = q(r; P ) [ (X) + (r; P ) (X = (X) + r (X ^ P) P )]
where q(r; P ) and r q(r; P ) (r; P ). The last expression is isomorphic to (1), except ^ that r can take values below 0:5. Provided that ( ) is a non-decreasing function, Theorem ^ 2 can thus be applied.
Proof of Theorem 3
Consider the type (X; ) and an arbitrary type (X; ) 6= (X; ). In any fully revealing equilibrium, type (X; ) cannot be held to a pro…t lower than because she can always succeed with the bid (r; 1; rX). At the same time, she cannot earn more than because of the free-rider condition. In order for type (X; ) not to mimic type (X; ), the latter type must make an o¤er (r; ; C) which satis…es rX + C, or equivalently C C rX (1 ) . (17)
In addition, a truthful o¤er by (X; ) must also yield a higher pro…t than the "out-ofequilibrium" o¤er 0:5; 1; 0:5X which succeeds irrespective of target shareholder beliefs.
35
That is, her o¤er (r; ; C) must satisfy rX + C C (r
C
0:5(X (1
X) + , or equivalently ) . (18)
0:5) X + 0:5X
The constraints (17) and (18) are simultaneously satis…ed if C C holds. Straightforward manipulations yield (r 0:5) X X (1 )( ). This condition is violated, unless all types with X , there exists a unique X ( 0 ) 2 (0; 1) s.t. f (X) ( > f 0 (X) for all X < X ( 0 ) . X ( 0 )
Proof. By SMLRP, for all
0
Otherwise, if f 0 (X)=f (X) is either always larger or always smaller than 1, it cannot be that F (1) = F 0 (1). This implies the result. Lemma 4 For all
00
8 > < 1 for X < X ( 0 ) : > 1 for X > X ( 0 )
> , there is a unique X ( 0 ) 2 (0; 1) s.t.
>
0
> , X 0 ( 00 )
X ( 0 ).
Proof. Suppose to the contrary that X 0 ( 00 ) < X ( 0 ). By Lemma 3, it then follows that (a) For X 2 (0; X 0 ( 00 )) : (b) For X = X 0 ( ):
00 f f f f f f f f f f
0 (X) 00 (X) 00 (X)
(F)
1 and > 1 and > 1 and
(d) For X = X ( 0 ):
0
(c) For X 2 (X 0 ( 00 ); X ( 0 )) :
0 (X) 00 (X)
0 (X) 00 (X) 0 (X) 00 (X) 0 (X)
(e) For X 2 (X ( ); 1) :
f 0 (X) f (X) f 0 (X) f (X) f 0 (X) f (X) f 0 (X) f (X) f 0 (X) f (X)
<1) <1) 1)
f 00 (X) f (X) f 00 (X) f (X) f 00 (X) f (X) f 00 (X) f (X) f 00 (X) f (X)
<1 1 >1
37
Observe that (i) f 00 (X) = f 0 (X) for X = X 0 ( 00 ) and (ii) f 0 (X) = f (X) for X = X ( 0 ). SMLRP implies that f 00 (X)=f (X) 1 in case (c), and hence that (iii) f 00 (X) = f (X) for X = X ( 0 ). Points (ii) and (iii) together imply that (iv) f 00 (X) = f 0 (X) for X = X ( 0 ). Given that f 00 (X) > f 0 (X) in case (c), points (iv) and (i) can only be reconciled with SMLRP if X 0 ( 00 ) = X ( 0 ). However, this contradicts inequality (F). Main proof. The proof proceeds in two steps. In the …rst step, we compare adjacent types and analyze local incentive-compatibility. In the second step, we show that an o¤er which is locally mimicking-proof is also globally mimicking-proof. Local incentive-compatibility. Consider a type who, for each target share, o¤ers a cash price P , a debt claim with face value D, and a (cash-settled) knock-in call option with exercise price S and trigger level T . Absent private bene…ts, a fully e cient equilibrium requires that the bidder' cash price s is weakly lower than the expected value of the cash rights that she acquires. At the ow same time, the free-rider condition requires that the cash price is weakly higher than the expected value of the transferred cash rights. Both constraints can only be satis…ed ow simultaneously if they are both binding: P = Z
T
(X
D) f (X)dX +
D
Z
1
(S
D)+ f (X)dX.
T
Consequently, every truthful o¤er must yield zero bidder pro…ts. (i) The next higher type + 1 does not mimic i¤ P + Z
T
(X
D) f
+1 (X)dX
+
D
Z
1
(S
D)+ f
+1 (X)dX
0.
T
Substituting for P , the inequality can be written as (S D)
+
Z
1
[f
+1 (X)
f (X)] dX
T
Z
T
[f (X)
f
+1 (X)] (X
D) dX.
(21)
D
Set T = X ( + 1). By Lemma 3, both integrals are then strictly positive for any D 0 such that (21) is satis…ed. (ii) Analogously, the next lower type 1 does not mimic i¤ (S D)
+
Z
1
[f
1 (X)
f (X)] dX
T
Z
T
[f (X)
f
1 (X)] (X
D) dX.
(22)
D
Set D = X 1 ( ). By Lemma 3, the right-hand side is then strictly positive. By Lemma 4, T = X ( + 1) X 1 ( ) so that the left-hand side integral is strictly negative. So, (22) 38
holds. Global incentive-compatibility. We now consider in turn types higher than + 1 and types lower than 1. (i) Given T = X ( + 1) and D = X 1 ( ), consider now the incentive-compatibility constraint of an arbitrary type + > + 1 vis-à-vis type : [S X
1(
)]
+
Z
1
[f + (X)
f (X)] dX Z
X ( +1)
X ( +1)
[f (X)
1(
f + (X)] [X
X
1(
)] dX.
X
)
De…ning (X) [S X
1(
f )]
+
+1 (X) 1
f + (X), write the inequality as [f
+1 (X)
Z
f (X) [f (X)
1(
(X)] dX f
+1 (X)
X ( +1)
Z
X ( +1)
+ (X)] [X
X
1(
)] dX. (23)
X
)
By Lemma 4, X +1 ( + ) X ( + 1) so that (X) > 0 for all X 0,
0
the integral on the left-hand side of (23) is larger than the integral on the left-hand side of (21), and hence strictly positive. We conclude that— for T = X ( + 1) and D = X 1 ( )— there exists a strictly positive price, S > 0, such that no type + > mimics type . (ii) Given T = X ( + 1) and D = X 1 ( ), consider now the incentive-compatibility constraint of an arbitrary type 0 for all X > X 1 ( ). This implies that the right-hand side of (24) is larger than the right-hand side of (22), and hence strictly positive. Turning to the left-hand side, again by Lemma 4, X ( +1) X 1 ( ) X ( 1) so that (X) > 0 for all X > X ( + 1). This implies that the left-hand side integral of (24) is smaller than the left-hand side integral of (22), and hence strictly negative. So, (24) holds. We conclude that— for T = X ( + 1), D = X 1 ( ), and S > 0— no type < mimics type .
40
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