Review of Finane (2005) 9: 33 63 Springer 2005 Rationing in IPOs CHRISTINE A. PARLOUR 1 and UDAY RAJAN 2 1 David A. Tepper Shool of Business, Carnegie Mellon University; 2 Stephen M. Ross Shool of Business, University of Mihigan Abstrat. We provide a model of bookbuilding in IPOs, in whih the issuer an hoose to ration shares. Before informed investors submit their bids, they know that, in the aggregate, winning bidders will reeive only a fration of their demand. We demonstrate that this mitigates the winner s urse, that is, the inentive of bidders to shade their bids. It leads to more aggressive bidding, to the extent that rationing an be revenue-enhaning. In a parametri example, we haraterize bid and revenue funtions, and the optimal degree of rationing. We show that, when investors information is diffuse, maximal rationing is optimal. Conversely, when their information is onentrated, the seller should not ration shares. We provide testable preditions on bid dispersion and the degree of rationing. Our model reoniles the doumented anomaly that higher bidders in IPOs do not neessarily reeive higher alloations. 1. Introdution Rationing in IPOs has been extensively doumented. 1 Typially, at the offer prie there is exess demand, and shares are rationed to investors. All investors, both informed and uninformed are rationed. Although explanations for the rationing of uninformed investors have been offered, 2 empirially, we observe that informed investors are also rationed. This is partiularly puzzling, sine informed investors submit prie-ontingent bids. For example, Cornelli and Goldreih (2001) examine the bids and alloations of one bookrunner in the UK. They find that informed bidders (those who submit these prie-ontingent or limit bids) only reeive between 48.2% and 54.2% of the amount they bid for. Further, alloations that are reeived by limit bidders do not appear to depend on the level of their bid (ontingent on it being higher than the offer prie). We are grateful to Peter Boatwright, Bhagwan Chowdhry, Franesa Cornelli, Simon Gervais, Paul Klemperer, Adam Koh, Vlad Mares, Kjell Nyborg, Ann Sherman, Fallaw Sowell, Chester Spatt, and Jeroen Swinkels for helpful omments, and to partiipants at the Summer 2001 Eonometri Soiety Meetings, the 2002 FEEM Aution Design Conferene, and seminars at Arizona State, Carnegie Mellon, Columbia, Illinois, Maryland, Minnesota, Virginia, and Wharton, and espeially to Josef Zehner (the editor) and two anonymous referees. All errors remain our own responsibility. 1 See, for example, Cornelli and Goldreih (2001), and the survey paper by Ritter (1998). 2 One argument is that the investment bank needs to reward informed investors to onvine them to reveal their information (see Benveniste and Spindt, 1989, for example).
34 CHRISTINE A. PARLOUR AND UDAY RAJAN The observed rationing of informed investors is ounterintuitive for two reasons. First, it ontradits the optimal IPO mehanism of Benveniste and Spindt (1989), who postulate that, amongst informed bidders, those with higher signals reeive their full alloation before any shares are given to bidders with lower signals. Seond, it seems to ontradit profit-maximization on the part of the seller. An IPO often proeeds by a bookbuilding proess, during whih a demand urve is generated for the shares to be sold. 3 Yet, faed with this demand urve, the seller frequently hooses a prie below the market-learing prie. Why not hoose a higher prie, redue the degree of rationing, and hene inrease revenue? These onerns have brought IPO alloations to the forefront of poliy debate. 4 Is rationing diret evidene of mispriing? That is, ould a seller simply hoose a higher offer prie and inrease revenue? Further, if the interests of the investment bank and entrepreneur differ, how muh is the entrepreneur hurt by the investment banker s ability to alloate shares? In this paper, we onstrut a stylized version of the bookbuilding proess for a ommon value asset. Our mehanism inludes the standard aution format as a speial ase. Indeed, our view orresponds to that expressed by Benveniste and Busaba (1997) and Sherman (2004), that autions are merely a speial ase of book-building. We onsider different alloation rules the seller may use. We show that it an be revenue-enhaning to ration, rather than hoose a market-learing prie. Rationing shares at the offer prie mitigates informed investors fear of the winner s urse, and may thus inrease seller revenue. Thus, oversubsription and rationing are not prima faie evidene that the issue prie was sub-optimal, and the issuer ould have raised more revenue. Finally, rationing allows for disretionary alloations among investors, whih may have long-term benefits for the seller. This suggests that the urrent debate over disretionary alloations is misplaed to the extent that seller revenue is not hurt by suh alloations. Our model has empirial impliations. First, for a speifi lass of signals, we numerially solve for the optimal degree of rationing. Thus, within our model, we are able to haraterize the observed dispersion of bids and offer pries, and relate these to the alloations. We find that the larger the range of submitted bids and also winning bids, the higher the optimal degree of rationing. Seond, sine rationing an inrease the proeeds of the IPO, money left on the table annot be estimated by the differene between the offer prie and the longterm value of the asset. The offer prie is determined by the bids submitted by investors. We demonstrate that the observed demand urve depends on the mehanism offered by the seller: if investors antiipate a different alloation mehanism (in partiular, a different degree of rationing), they will submit different bids. Thus, 3 A survey of international IPO praties is presented in Sherman (2004). 4 For example, see Ritter and Welh (2002) for a survey of the aademi questions. In the U.S., the SEC has reently expressed in this issue (e.g., Harvey Pitt wants people to look at IPO priing and alloations, Wall Street Journal, Aug 23, 2002).
RATIONING IN IPOS 35 aution must be exerised in performing these thought experiments on the demand urve. The intuition that drives our results is straightforward. Changing the degree of rationing potentially affets two elements of the winner s urse: the expeted onsumption value of the asset onditional on winning, and the prie the winner expets to pay. If informed investors are not rationed, then they only get shares when they are among the highest bidders. Thus, if they win the asset, their ex ante estimate of the value of the asset is higher than that of any bidder who did not get the asset. They take this into aount when they bid; i.e., they optimally redue their bids to avoid this winner s urse. 5 By ontrast, onsider pro rata rationing. Now, an investor an win the asset even when many other investors have higher signals. This auses every bidder to bid higher. However, to inrease the degree of rationing, a seller selets a bidder with a lower estimate of the value of the shares. The optimal degree of rationing is determined by this tradeoff. While we do not expliitly distinguish between issuers (i.e., firms going publi) and underwriters, it is reasonable to suppose that their interests may diverge. 6 Indeed, NASD has reently suggested expliit prohibitions on the preferential alloation of shares to investors, provided in onsideration of future business. This pratie (alled spinning) suggests that investment bankers do provide preferential alloations to some investors. While not all preferential alloations are neessarily bad, it is important to reognize that suh disretion annot exist unless there is oversubsription. Thus, even if rationing is sub-optimal, issuers need not be substantially damaged by these idiosynrati alloations. This view is supported empirially by Ljungqvist and Wilhelm (2002), who onlude that disretionary alloation does not harm issuers. In our model, a bookrunner has k units for sale and ommits to alloate them aross t bidders at the (t +1) st highest bid. In partiular, t may be greater than k,in whih ase the seller rations. For example, with pro rata rationing, all winners may be alloated an amount k. Bidders in our model all reeive signals about the value t of the asset. They then submit sealed bids for the asset. Thus, we are onsidering informed bidders who submit limit bids (in partiular, their bid onsists of a prie they are willing to pay). Our model is, of ourse, a highly stylized desription of an IPO proess. In the U.S., an IPO is typially preeded by a road show, during whih the underwriter makes presentations to groups of buyers in various ities, and often meets with important buyers one-on-one (see, for example, Ritter, 1998). During this road show, the lead investment banker also soliits information from the buyers on quantities they are interested in buying, and the assoiated pries at whih they 5 Nyborg, Rydqvist and Sundaresan (2002) find evidene that bidders ompensate for the winner s urse in Swedish Treasury autions. 6 Biais, Bossaerts and Rohet (2002) present a model in whih institutional investors ollude with bankers against issuers. The reent $100 million settlement between Credit Suisse First Boston and the SEC supports this view.
36 CHRISTINE A. PARLOUR AND UDAY RAJAN are willing to buy. In this book-building proess, a demand urve is onstruted (Ritter, 1998). In our model, we interpret the onstrution of this book or demand urve as analogous to soliiting sealed bids from potential buyers. To the extent that a buyer is unaware of the prie and quantity pair submitted by another buyer, this is equivalent to the simultaneous submission of sealed bids. There are several features of the IPO proess that we omit from our model. First, we fix the demand for eah buyer to be the same. 7 Seond, we onsider a simultaneous game, in whih bidders bid only one. With IPOs, the bookbuilding proess is usually followed by the seller announing a prie at whih shares will be sold. Bidders are then allowed to re-submit quantities they wish to buy at this prie. In other words, they may be given a hane to revise their bids, whih we do not allow in our model. However, Welh (1999) mentions that In reality, an institutional investor who baks out after suh informal requests (espeially if it is lose to the effetive date) may not reeive shares in future offerings; onsequently, suh indiations of interest are pratially firm. A seminal theoretial piee on rationing, the winner s urse, and IPO underpriing is Rok (1986). Though our model ontains both rationing and a winner s urse, it is starkly different. In Rok s model, there are two states of the world. In the good state (i.e., when the value of a share exeeds its offer prie), both informed and uninformed investors demand shares. In the bad state (when the value of a share is less than the offer prie), informed bidders withdraw from the proess, and uninformed bidders obtain an exessive number of shares. Hene, uninformed investors reeive higher alloations in the bad state, and are therefore subjet to the winner s urse. To ompensate them for this asymmetry in alloations aross states, the offer prie must be lowered. Though the probability of reeiving an alloation in the good state is lower, reduing the prie suffiiently inreases their expeted revenue enough to make them willing to partiipate in the mehanism. In our model, all bidders are informed and the degree of rationing is known ex ante (that is, the seller hooses a rationing mehanism before any bids are submitted). Importantly, the seller ommits to a rationing rule that does not depend on the state of the world. In ontrast, in Rok (1986), rationing varies aross states in equilibrium: investors experiene greater rationing in the good state. The variation in ex post rationing hurts Rok s seller (it leads to lower revenue); the ommitment to ex ante rationing in our model benefits the seller. That is, in Rok s model ex post rationing is the embodiment of the winner s urse and generates underpriing, while in our model, ex ante rationing mitigates the winner s urse. The distintion between ex post and ex ante rationing also distinguishes our work from Benveniste and Spindt (1989). 8 In their framework, the underwriter 7 In multi-unit settings, Bak and Zender (1993), building on work by Wilson (1979), show that the uniform prie aution has self-enforing ollusive equilibria, leading to lower revenue than disriminatory autions. 8 Biais and Faugeron-Crouzet (2002) demonstrate that, given a proper hoie of parameters, the Frenh mise en vente mehanism repliates the optimal Benveniste Spindt one.
RATIONING IN IPOS 37 presells some of the issue to informed regular investors. 9 As an inentive to reveal information, investors with good information get the quantity they demand before any shares are alloated to investors with lower signals. Therefore, they predit that, aross all investors who reeive shares, investors with higher bids must be rationed less than investors with lower bids. As mentioned, this ontradits the empirial observations of Cornelli and Goldreih (2001), who argue that alloations are independent of bids above the offer prie. In ontrast, in our model, equilibrium alloations an be ahieved by pro rata rationing, in whih all investors who reeive shares are equally rationed. An alternative is to randomly hoose a subset of potential winning bidders that will reeive shares. In this ase, it is feasible in equilibrium to have some high bidders reeive no shares, though some lower bidders reeive shares. While both Benveniste and Spindt (1989) and the model in this paper onsider finanial assets to be ommon value objets, a ritial differene is that our investors reeive orrelated signals. Benveniste and Spindt provide an optimal IPO mehanism when investors have independent signals. Sine we have orrelated signals, their mehanism is no longer optimal. The optimal mehanism with orrelated signals is exhibited by Cremer and MLean (1988) and MAfee et al. (1989), who show that it is possible for the seller to extrat the full surplus from buyers (i.e., eliminate all underpriing). However, surplus extration mehanisms inlude a strong individual rationality onstraint on the buyers. A seller must have the ability to potentially inflit large punishments on the buyers. Suh mehanisms are not observed in pratie, partly beause the seller s ability to penalize the buyers is limited. Our model draws upon previous work in aution theory, notably Milgrom (1981) and Pesendorfer and Swinkels (1997). Milgrom onsiders an aution for a ommon-value objet, where the seller has t units for sale and eah buyer demands one unit. The t highest bidders are eah given one unit, at the (t + 1) st highest bid. Pesendorfer and Swinkels (1997) show that the t-unit aution has a unique symmetri equilibrium (albeit under onditions that are stronger than those in our paper) and examine properties of the onvergene of the prie to the true value of the objet. We show that the equilibria in this aution are idential to those in our rationing mehanism. To the extent that bidders bid one prie in our model, the framework is different from Wilson (1979). In the latter, bidders submit demand funtions. 10 We assume onstant marginal valuation for eah frational amount of one unit, or flat demand urves. Hene, our bidders submit a single prie. 9 We note that, in our model, we onsider only informed bidders, and hene (unlike Benveniste and Spindt), we provide no preditions on the relative alloations between informed and uninformed bidders. 10 Other papers that onsider models in whih bidders submit demand (or supply) funtions inlude Klemperer and Meyer (1989), Kyle (1989), Bak and Zender (1993), and Kremer and Nyborg (2004).
38 CHRISTINE A. PARLOUR AND UDAY RAJAN In a model with independent signals and (almost) ommon values, Bulow and Klemperer (2002) demonstrate onditions under whih rationing is the optimal mehanism for the seller. A suffiient ondition they identify is that the hazard rate of signals must be dereasing. Under this ondition, the optimal mehanism involves the seller posting a prie, at whih all buyers are willing to buy the good. While this ondition is violated in our example with a uniform distribution in Setion 4, Bulow and Klemperer offer an intuition for our results in terms of a game they introdue alled the Maximum Game. We omment on this in greater detail in Setion 5. The rest of this paper is organized as follows. Setion 2 outlines our general model. Setion 3 examines some properties of bid and revenue funtions in the general ase. Setion 4 disusses a speial ase of the signal distribution that proves more tratable and allows for expliit revenue omparisons, and is followed by a disussion of our results in Setion 5 and some onluding remarks in Setion 6. All proofs are relegated to the Appendix. 2. Model We model the bidding behavior of informed investors in an IPO. The seller wishes to sell k shares to n>kinvestors, eah of whom demands the same number of shares whih we normalize to one. Following Benveniste and Busaba (1997) and Sherman (2004), we model the bookbuilding mehanism as a generalization of a multi-unit ommon value aution. In pratie, of ourse there are many distintive features of bookbuilding, inluding information sharing and gathering. 11 However, one bids have been entered in the book, they are essentially firm. We thus view them as sealed bids. In the bookbuilding mehanism, the seller announes a rationing rule, whih has two omponents. First, the seller hooses a t,wherek t n 1. He promises that the IPO offer prie will be equal to the (t + 1) st highest bid. 12 If t>k, he further announes an alloation rule, to divide the shares aross the t highest bidders. An investor then reeives a private signal about the value of the asset, and submits a bid for one share. Finally, based on the book the seller announes an offer prie and the k shares are alloated to investors. The offer prie thus depends on, and reflets information in, the reeived bids. In addition, the seller must deide on the alloations to be given to investors who bid more than the offer prie. When t = k, eah of the highest k bidders is alloated 11 Spatt and Srivastava (1991) disuss some of these features, inluding the ommuniation between the underwriter and the investors. 12 Consistent with the mehanism design literature (inluding Benveniste and Spindt, 1989), we assume the seller an ommit to a mehanism. Suppose, instead, the seller heated and were to set the offer prie equal to the expeted value of the asset onditional on all bids. This would violate inentive ompatibility for the bidders. No bidder (even a bidder with a high signal) would have an inentive to make a high bid.
RATIONING IN IPOS 39 one share. This effetively represents market-learing. That is, there is no rationing, and the alloation reeived by the winning bidders is 100% of what they requested. In ontrast, for t > k, there is rationing. Eah of the t bidders have indiated a willingness to buy the item at stritly greater than the offer prie. Hene, in the aggregate, potential winners are rationed to a degree 1 k t. DEFINITION 1. Consider some integer t k. In a(k, t)-bookbuilding mehanism: (i) the offer prie is set to the (t + 1) st highest bid. Eah bidder who reeives a positive alloation pays this prie. (ii) the k shares are alloated amongst the t highest bidders. No bidder reeives more than her demand (i.e., one share), and, (iii) onditional on being amongst the t highest bidders, the alloation reeived by an agent is independent of her bid. Many alloation rules are onsistent with this definition, inluding: (i) pro rata rationing: eah investor in the set of potential winners reeives k t shares, (ii) random alloation: the seller randomly hooses k of the t potential winners; eah of these k bidders reeives their full demand (one share), and the others reeive nothing. (iii) disretionary alloation: the seller wishes to reward a set of long-term ustomers, investor 1 through i. These ustomers (preferred investors) reeive their full quota (i.e., one share) if they are in the set of potential winners, and nothing otherwise. Other winners (regular investors) have their alloations redued aordingly, either in pro rata or random fashion. The shares have a ommon value to all investors, V, whih is drawn from an atomless distribution, F( ), on[v l,v h ]. V represents the long-term value of the asset. 13 We emphasize that we do not expet V to be represented by the prie at the end of the first day (or the first week) of trading in the seondary market. Investors have private information about the long-term value of the asset. The information of investor i is represented by a signal, S i. The signals of different investors are onditionally independent (given V ), but all depend on V in the following manner. Eah S i,fori = 1,...,n, is independently drawn from the same atomless distribution G( V = v), with support [v ɛ, v + ɛ] for some ɛ>0. Here, ɛ represents dispersion of opinion about the value of the asset. We assume that v h v l > 2ɛ, so that informed investors beliefs are more preise than the prior over V. 13 Notationally, we use upper ase letters denote random variables, and lower ase ones to denote partiular realizations of random variables. Thus, v denotes a realization of random variable V.
40 CHRISTINE A. PARLOUR AND UDAY RAJAN Hene, given V, the height of the onditional signal distribution at any signal, s, depends only on the position of the signal, relative to the lowest possible value (v ɛ). Therefore, a higher value of v leads to a shift in the support of the signal distribution, but the distribution has the same shape, given the support. Formally, ASSUMPTION 1. (i) G(s v) is an atomless distribution with support [v ɛ, v + ɛ], and density g(s v). (ii) For any pairs (s, v) and ( s,ṽ), ifs v = s ṽ, theng(s v) = G( s ṽ). (iii) (MLRP): g(s v) g(s v ) g(s v) g(s v ) for all s>s, v > v suh that s,s are both in the support of G( v) and G( v ) respetively. Part (ii) of the assumption further implies that g(s v) = g( s ṽ) when s v = s ṽ. Part (iii) is a variant of the Monotone Likelihood Ratio Property. 14 Note that, in our model, ɛ is independent of V. Intuitively, this means that the dispersion of investors opinions over value does not depend on whether V is high or low. That is, ɛ does not depend on the long-term per share prie of the asset. Empirially, ɛ an be inferred from the range of analyst foreasts over the value of the asset. 15 We onsider one-shot equilibria of the bookbuilding game. In pratie, the set of bidders in an IPO varies from issue to issue. For example, in the 39 issues that Cornelli and Goldreih (2001) analyze, only 16.8% bid in at least 10 issues. There are, therefore, several bidders who rarely partiipate in more than one deal. These players will perfore bid as if in the one-shot game. Further, amongst the long-term players, absent expliit ollusion, none knows the speifi set of bidders partiipating in a speifi IPO. Sine long-term bidders also pik and hoose transations, the usual punishments seen in repeated games are diffiult to enfore. Hene, we onsider equilibria of the one-shot game among investors, and ignore repeated game effets in bidding. 2.1. EQUILIBRIUM IN THE BOOKBUILDING MECHANISM Given an alloation rule, a bidder observes her own signal s, and hooses a bid. We onsider a symmetri Bayesian-Nash equilibrium, in whih all bidders hoose the same bid funtion, and a bidder with signal s bids b(s). It is natural to onsider 14 In our model, MLRP annot hold for all s,s,v,v beause a partiular s an lie outside the support of G( v), and hene have a density of zero. 15 Analyst foreasts have been used to proxy for the dispersion of beliefs over asset values in the aounting literature; see, for example, Ajinkya, Atiase, and Gift (1991), and the subsequent literature.
RATIONING IN IPOS 41 equilibria in whih b( ) is stritly inreasing in s; that is, bidders with higher signals submit higher bids. Let Y j,n be a random variable representing the j th highest order statisti of bidders signals, where n signals are drawn. Hene, Y j,n Y j+1,n for all j = 1,...,n 1. We first show that the equilibrium bids (and, by extension, seller revenue) do not depend on the partiular alloation rule used, provided it satisfies the property that the alloations are independent of the atual bids of agents who bid more than the offer prie. The intuition is that, onditional on being amongst the t highest bids, the alloation rule exposes eah agent to a lottery. If the agent does not have one of the t highest bids, she gets zero alloation, with no aess to the lottery. As long as the probability of reeiving an alloation in the lottery is independent of the bids of any agent in the set of potential winners (as required by part (iii) of Definition 1), the behavior of a risk-neutral agent will be unaffeted. PROPOSITION 1. Consider any two alloation rules within a (k, t)-bookbuilding mehanism. These rules result in the same set of equilibria. Proposition 1 implies that all alloation rules allowed for by the mehanism are equivalent (in terms of bids and revenues) to the pro rata rationing rule with no disretionary alloation, though eah of these implies a very different set of final alloations. Thus, if a seller alloates shares in a way that benefits lients with whom he has a long term relationship, this is not to the detriment of the issuer. Within this lass of alloation rules, there are several advantages to pro rata rationing, in partiular. Many exhanges have requirements on the distribution of shares aross investors. 16 Thus, one of the goals of an IPO must be to generate a dispersed shareholder base. It has also been suggested that share dispersion per se inreases the value of a firm. 17 In addition, Brennan and Franks (1996) argue that rationing is used beause urrent owners want to redue the blok size of new shareholders. Markets whih exhibit pro rata rationing inlude Singapore (Koh and Walter, 1989), Israel (Amihud et al., 2003), and the UK (Levis, 1990). Ljungqvist and Wilhelm (2002) desribe the alloation methods used in the UK, Germany, Frane and the US. In Germany, the June 7, 2000, guidelines promulgated by the Federal Ministry of Finane tries to rule out subjetive riteria for alloating shares to retail investors. It reommends that issuers draw lots, alloate pro rata either within ertain order sizes or aross the whole offer, or alloate aording to time priority or some other objetive riteria. One of the mehanisms adopted in Frane, the 16 For example, the NYSE requires at least 500 holders of round lots, while the NASD requires 400. 17 Booth and Chua (1996) onsider a model in whih dispersed ownership inreases seondary market liquidity and hene the value of the firm. Sherman (2000) omments on benefits to the seller and Stoughton and Zehner (1998) on benefits to the issuer.
42 CHRISTINE A. PARLOUR AND UDAY RAJAN offre à prix ferme, has pro rata alloations at a fixed prie. Further, the reent legal troubles of Salomon, Smith Barney over preferential alloation of shares in IPOs 18 enhane the appeal of pro rata rationing, with no disretionary alloation, in the US. We show that the equilibrium bids of the (k, t)- bookbuilding mehanism (in whih the seller has k shares for sale, and rations aross t > k investors), is idential to the orresponding set when the seller atually sells t shares with no rationing. For a fixed t, in either of these mehanisms, a winning bidder has the same information: she knows she has one of the t highest signals. Hene, her bids in the two mehanisms are the same. The symmetri equilibrium of the t-unit ommon-value aution is haraterized by Milgrom (1981). Without loss of generality, onsider the behavior of bidder 1. As Milgrom (1981) shows, bidder 1, with signal s, bids as if her signal is equal to the t th highest (or pivotal) signal amongst the remaining (n 1) bidders. This is the prie at whih she is indifferent between winning and losing the asset. In a symmetri equilibrium, all bidders hoose this strategy. PROPOSITION 2. In a (k, t)-bookbuilding mehanism, (i) the set of equilibria is equivalent to the set of equilibria in the t unit aution. (ii) there is a symmetri equilibrium in whih a bidder with signal s hooses a bidding funtion b(s; t) = E[V S = Y t,n 1 = s]. The offer prie is set by the (t + 1) st highest bid. Sine the bidding funtion is monotone, this is determined by the bidder with the (t + 1) st highest signal, who bids as if she has same signal as the t th highest bidder. In pratie, sine the (t +1) st highest bidder is a losing bidder, the t th highest bidder has an even higher signal. Hene, the expeted value of the asset is always higher than the (t + 1) st highest bid (that is, the offer prie). Therefore, in equilibrium there is no winner s urse. Bidder s rationally antiipate the information ontent in winning, and adjust their bids aordingly. Sine the offer prie is less than the expeted value of the asset, onditional on the (t +1) st highest bid, there is no ex post regret, and eah winner is happy to reeive an alloation. We show that for a fixed k, the bid funtion, b(s; t) is stritly inreasing in t. 19 Sine t is diretly related to the degree of rationing (1 k t ), the bid funtion inreases with rationing. This implies that the seller an benefit from rationing bidders bid more aggressively. There is, of ourse, the obvious 18 See, for example, Ex-Broker Says Salomon Gave IPOs to CEOs to Win Business, Wall Street Journal, July 18, 2002. 19 Exept, of ourse, for the extreme signal values, v l ɛ and v h +ɛ. At these values, whih our with zero probability, the true value of the asset is known with ertainty.
RATIONING IN IPOS 43 ost of hoosing a lower ranked bid to set the offer prie. However, we show later that, under some onditions, the seller an inrease her revenue by rationing. PROPOSITION3. In the (k, t)-bookbuilding mehanism, bid funtions are stritly inreasing in the degree of rationing. That is, b(s; t + 1) > b(s,t) for all s (v l ɛ, v h + ɛ) and all t = k,...,n 2. Intuitively, a higher value of t implies that a bidder may win the objet even if he did not reeive the highest signal. Thus, the winner s urse is mitigated: onditional on winning, a bidder in the (k, t)-bookbuilding mehanism knows only that her signal was among the t highest. Hene, the greater the degree of rationing in the mehanism, the more aggressive is eah bidder. Thus, even when the number of shares being sold is held fixed, the equilibrium bids in the book will vary with the seller s ommitment to rationing. What an we infer about v from realized bids (i.e., the book)? We show that, for low t, bidders shade their bids, in the sense of bidding less than the onditional value of the asset, given their signal. However, for high t, they inrement their bids, by bidding more than this onditional asset value. This effet, therefore, must be taken into aount in inferring bidders beliefs on the onditional value of the asset from an observed book. PROPOSITION 4. Consider the (k, t)-bookbuilding mehanism. For all s (v l ɛ, v h + ɛ), there exists a ˆt(s) suh that, for t ˆt(s), b(s; t) E(V s), andfor t>ˆt(s), b(s; t) > E(V s). In Setion 4, in a parametri model, we haraterize the dispersion of bids submitted to the book and the dispersion of bids that reeive positive alloations, and tie these to the degree of rationing. 3. Bid and Revenue Funtions with Uniform Prior 3.1. BIDS To haraterize the bid funtion further, we assume for the rest of the paper that F( ), the prior over V, is uniform over [v l,v h ]. Beause this prior has finite support, the inferene drawn from any signal depends on its size relative to the endpoints. Consider a signal in the range [v l + ɛ, v h ɛ]. For any suh signal s, the posterior over V has support [s ɛ, s + ɛ]. We term suh signals interior signals. For a signal s in the range [v l ɛ, v l + ɛ), the posterior over V has a trunated support [v l,s + ɛ] beause the lowest possible value of V is v l. Suh signals, and those in the range (v h ɛ, v h + ɛ], are dubbed orner signals. We first show that, for interior signals, bids are linear in signal. In partiular, the bid funtion an be written as the signal plus an adjustment term that ompensates
44 CHRISTINE A. PARLOUR AND UDAY RAJAN for the size of the winner s urse. This adjustment term is inreasing in the number of investors, n, but dereasing in the degree of rationing. PROPOSITION 5. Consider the (k, t)-bookbuilding mehanism. Suppose s [v l + ɛ, v h ɛ]. Then, in equilibrium, the bid funtion is linear in signal. In partiular, b(s; t) = s + ɛ(1 2δ(n,t)), whereδ is inreasing in n, dereasing in t, and is independent of s and ɛ. In other words, regardless of the shape of G( ), the bid funtion over interior signals is linear, and takes the form of the signal plus a onstant (whih may be positive or negative). Further, a hange in t leads to a parallel shift of the bid funtion. The term representing the winner s urse adjustment, δ( ), may be greater or less than 1 2. It is greater than 1 2 for small values of t, and less than 1 for large 2 values of t. Hene, when t is low, bidders bid less than their signal, and, when t is high, they bid more than their signal. For signal ranges in the orners, s [v l ɛ, v l +ɛ) and s (v h ɛ, v h +ɛ],the bid funtion is non-linear. It is still inreasing in n and dereasing in t, but depends on signal as well. In Setion 4, we exhibit the bid funtion over this range for a partiular signal distribution. 3.2. REVENUE We next examine the effet of rationing on revenue. As the degree of rationing inreases, the seller sets the offer prie at the bid of a bidder with a lower signal. Thus, the seller trades off the inrease in the bid funtion against the fat that he is awarding shares to lower signal bidders. In this setion, we haraterize the revenue funtion of the seller. We provide intuition about our results in terms of bidders marginal revenues (along the lines of Bulow and Klemperer, 2002) in Setion 5. Let R(v; t) denote the seller s expeted revenue per share in the (k, t)- bookbuilding mehanism, if the realization of V is v. 20 Theofferprieisthe expeted (t + 1) th highest bid, out of n signal draws. That is, R(v; t) = E [b(y t+1,n ; t) V = v]. As the seller in our model is uninformed, he earns an ex ante revenue of ˆR(t) = v h v l R(v; t)dv. He optimally hooses a t that maximizes ˆR(t). Market-learing orresponds to hoosing t = k, whereas larger values of t imply rationing. A neessary ondition for seller revenue, ˆR(t), to inrease in t (over any range of t) isthatr(v; t) be inreasing in t for some t and some v. Hene, we fous initially on R(v; t). When signals are in the interior, the bid funtion is linear. When V [v l + 2ɛ, v h 2ɛ], all signals are interior. Hene, it follows immediately that the revenue funtion is linear in v over this range. In partiular, the revenue funtion an be written in terms of the winner s urse adjustment, δ(n,t). 20 Sine the number of shares to be sold is fixed at k, maximizing revenue per share is equivalent to maximizing total revenue.
RATIONING IN IPOS 45 As in the proof of Proposition 5, we an define δ(n,t) as follows. First, let x = s (v ɛ). Then, x is a variable defined over the interval [0, 1], whih has the 2ɛ property that H(x) = G(s v) for all s [v ɛ, v + ɛ] (sine the distribution of s given v depends only on s (v ɛ)). Let x represent a single draw of X, andlet X t,n be the t th highest of an independent sample of n draws from X. Then, δ(n,t) is defined as E( x s = X t,n 1 ). Now, the revenue funtion an be written in terms of δ(n,t) and X t+1,n,the (t + 1) st highest order statisti from a sample of n draws of X. PROPOSITION 6. Consider the (k, t)-bookbuilding mehanism, and an interior value of v, sothatv [v l + 2ɛ, v h 2ɛ]. Then, for all t n 1, R(v; t) = v 2ɛ(δ(n,t) E[X t+1,n ]). This form of the revenue funtion illustrates the tradeoff for the seller. For eah value of V in the relevant range, inreasing t leads to a lower winner s urse, and hene a lower adjustment (δ(n,t)) and a higher revenue. However, inreasing t leads to a redued order statisti being used to set the prie of the item, aptured by the E(X t+1,n ) term (whih is delining in t). The linear revenue funtion allows us to determine when rationing inreases revenue. Indeed, the seller ompares the inrease in revenue (in going from a (k, t)- to a (k, t)-bookbuilding mehanism) obtained as a result of the inrease in the bid funtions to the derease in revenue as a result of hoosing the ( t + 1) st bid to set the prie of the item, rather than the (t + 1) st one. The left-hand side of Equation (1) below is the inrease in revenue and the right-hand side is the derease. COROLLARY 6.1. Consider an interior value of v,sothatv [v l + 2ɛ, v h 2ɛ]. Then, for any t, t = k,...,n 2, R(v; t)>r(v; t) if and only if δ(n,t) δ(n, t)>e[x t+1,n ] E[X t+1,n ]. (1) Notie, when V [v l + 2ɛ, v h 2ɛ], the omparison of R(v; t) and R(v; t) is independent of the atual value of V in this range. That is, if (1) holds for some t, t, thenr(v; t)>r(v; t) for all V [v l + 2ɛ, v h 2ɛ]. This ondition is easy to hek for any distribution. While (1) is a neessary ondition for higher degrees of rationing to yield higher revenue, learly there are values of ɛ (given v l and v h ) for whih it is also a suffiient ondition. In partiular, if (1) holds and ɛ is small relative to v h v l,then ex ante revenue must be inreasing in the degree of rationing. To see this, suppose (1) holds for some distribution G and some t > k,sothatr(v; t) > R(v; t) for all V [v l + 2ɛ, v h 2ɛ]. In the orner regions, V [v l,v l + 2ɛ), and V (v h 2ɛ, v h ],wemayhaver(v; t)<r(v; t) over some range. However, the differene, R(v; t) R(v; t) is bounded. Hene, it follows that, if ɛ is small enough relative to (v h v l ),then ˆR( t) > ˆR(t). Thus, if signals are preise relative to the prior over V, (i.e., ɛ small), then (1) is also a suffiient ondition.
46 CHRISTINE A. PARLOUR AND UDAY RAJAN 4. Parametri Signal Distribution From the tradeoff explored in the previous setion, it is lear that the optimal hoie of t depends on the signal distribution. To explore this further, and to provide some eonomi intuition for this tradeoff, we onsider a broad lass of signal distributions that has the uniform as a speial ase. We determine the bid and revenue funtions, and numerially solve for the optimal t, given a signal distribution. This allows us to ompute endogenous bid ranges, and generate empirial preditions. First, assume that the signal distribution is generalized uniform. The extra parameter allows for strit onvexity of the distribution funtion. ASSUMPTION 2. For s [v ɛ, v + ɛ],andsome 1, ( ) s v + ɛ G(s v) =, and g(s v) = 2ɛ 2ɛ ( ) s v + ɛ 1. 2ɛ Figure 1 plots the density and distribution funtions for different values of. When = 1, the distribution is uniform and signals are most diffuse; that is, all signals in the range are equally likely. For > 1 it is stritly onvex. As beomes large, the distribution beomes more onentrated. Indeed, as,the signal distribution onverges to a point mass at v + ɛ. Inreases in thus represent first-order stohasti shifts in the underlying density funtion. For all 1, this distribution satisfies Assumption 1 (MLRP). Given this signal distribution, we determine losed form expressions for the bid funtions. For interior signals, that is s [v l + ɛ, v h ɛ], the bid funtion is linear whereas for orner signals, the bid funtion is more ompliated. PROPOSITION 7. In the (k, t)-bookbuilding mehanism, with generalized uniform signals, (i) for interior signals, s [v l + ɛ, v h ɛ], the equilibrium bid funtion is t 1 n j b(s; t) = s + ɛ 1 1 2. n j (ii) For orner signals, s [v l ɛ, v l + ɛ) or s (v h ɛ, v h + ɛ], the equilibrium bid funtion is b(s; t) = s + ɛ(1 2 δ(s,n,t,ɛ)), where δ(s,n,t,ɛ) = φ( x(s)/2ɛ, 0) φ( x(s)/2ɛ, 1 ) φ(1, 0) φ(x(s)/2ɛ, 0) φ(1, 1 ) φ(x(s)/2ɛ, 1 ) if if s<v l + ɛ s>v h ɛ
RATIONING IN IPOS 47 Figure 1. Density (left) and distribution (right) funtions, for different values of. and φ(x,y) = t 1 i=0 x (n i y) (1 x ) i i! t 1 j=i (n j y). We provide a numerial haraterization of the bid funtions based on the results of Cornelli and Goldreih (2001). They doument that (i) there are an average of 38.9 limit bids and 9.1 step bids 21 per issue (Table III), (ii) the average rationing aross limit and step bids is approximately 50% (Table III), (iii) 21.5% of limit bids and 9.8% of step bids are alloated no shares, sine the pries are below the offer prie (page 2343), and (iv) the mean IPO offer prie is $23.6 (Table I). For our numeri example, therefore, we assume n = 50. As approximately 20% of all bidders get no shares, and those that do are rationed by about 50%, we use t = 40 and k = t = 20. Thus, t = 20 represents the ase of 100% alloation, and t = 40 2 the ase of 50% alloation. We further use v l = 10, v h = 40, and ɛ = 5. Figure 2 plots the bid funtions for the ase of = 1.5whent = 20 and t = 40. The vertial differene between the two funtions represents the differene in the winner s urse adjustment between the two rationing levels. 21 These are the two kinds of bids that are prie-ontingent in their study.
48 CHRISTINE A. PARLOUR AND UDAY RAJAN Figure 2. Bid funtions for = 1.5, t = 20, t = 40. While it is immediate from Proposition 7 that b(s; t) is inreasing in t, the extent of the inrease depends on the winner s urse adjustment and hene. For = 1, inreasing t has a large effet on bids. As gets large, hanging t has a smaller effet on bid funtions. When is large and t is low, there is potentially a large winner s urse. Consider, for example, the ase of = 2andt = k. Ifa bidder wins, sine the distribution funtion is steep at high s, her signal is likely to be further away from other bidder s signals than if were, say, 1. Hene, for low values of t, bidders should shade their bids (relative to their signals) more for higher values of. Whent is high, the reverse logi holds: there is a probability that some bidders have higher signals, and, when the distribution funtion is steep, other bidder s signals an be signifiantly higher than that of the winning bidder. The t that we expet to observe in IPO deals is the t that maximizes seller revenue. Given a value of, i.e., dispersion of investors beliefs, the seller hooses an optimal degree of rationing to maximize ex ante revenue ˆR(t) = v h v l R(v; t)dv. However, as argued before, if rationing leads to higher revenue for all v in the interior, and if ɛ is small relative to (v h v l ), then ex ante revenue ˆR(t) is also higher for the seller with rationing. Thus, we first haraterize the revenue for a given value of v in the interior with a degree of rationing, t. We provide a losed form expression for ondition (1).
RATIONING IN IPOS 49 Figure 3. Revenue differene between t = 40 and t = 20, for = 1. PROPOSITION 8. Suppose v is in the interior; that is, v [v l +2ɛ, v h 2ɛ]. Then, in equilibrium, { t 1 n j (i) R(v; t) = v 2ɛ 1 n j (ii) R(v; t + 1) >R(v; t) if and only if t } n j (n j+ 1 ) ( n t 1 + 1 n t ) t 1 n j 1 n j t n j > n j + 1. (2) Next, we onsider the effets of rationing when V is in the orners; that is, V<v l +ɛ or V>v h ɛ.forv in this range, with positive probability signals will lie outside the interior segment, so that the linear bid funtion no longer applies. In this ase, an analytial examination of the hange in revenue due to rationing is diffiult. Instead, we numerially evaluate the effets of rationing. Figure 3 below provides the differene in revenues, in going from t = 20 to t = 40 for all values of v (inluding the orners) in our numeri example. Note that, as v approahes v l, rationing loses its lustre, and leads to lower revenue. From the seller s point of view, the benefit to rationing is that, by mitigating the winners urse, it indues bidders to bid more aggressively. This effet
50 CHRISTINE A. PARLOUR AND UDAY RAJAN is at its strongest when a bidder s posterior belief over v is diffuse, given his own signal. In our ontext, sine the prior over v is diffuse, a diffuse signal distribution translates to a diffuse posterior belief over v. In general, as v approahes v l or v h, the posterior belief beomes more onentrated (at least, its support shrinks in size), potentially reduing the benefit of rationing. This leads to the downward segment of the urve for values of v lose to v h. This intuition applies both to a posterior that is onentrated beause the signal is lose to one of the endpoints and to a posterior that is onentrated beause the signal distribution was onentrated (large ()). To see this intuition onsider the speial ase of = 1, or G( v) is uniform. Here, the bid and revenue funtions simplify onsiderably. As shown by Harstad and Bordley (1996), the bid funtion for interior signals redues to a funtion of the ratio t, and the revenue funtion for n interior values of v is learly inreasing in t. COROLLARY 8.1. Suppose = 1; that is, the signal distribution is uniform. Then, in equilibrium, (i) for s [v l + ɛ, v h ɛ], b(s; t) = s + ɛ( 2t n 1). (ii) for v [v l + 2ɛ, v h 2ɛ], R(v; t) = v 2ɛ(n t) n(n+1). Clearly, in this ase revenue inreases in t so R(v; t + 1) > R(v; t) for all t = k,...,n 2. That is, the revenue unambiguously inreases as the degree of rationing inreases. Condition (2) redues to 1 > n. This holds for all t, and n+1 thus maximum rationing (t = n 1) is optimal when = 1. This suggests that the oversubsription in IPOs of firms in new industries (where investor beliefs are diffuse) should be high. As inreases, the optimal level of rationing dereases, 22 and when is large, market-learing (t = k) is optimal. We first demonstrate analytially, for interior values of v, that market-learing is optimal when beomes large enough. PROPOSITION 9. Consider interior values v [v l + 2ɛ, v h 2ɛ]. Then, (i) if = 1, maximal rationing is optimal (ii) there exists a >1 suh that, for <, rationing (to some degree) yields higher revenue than market-learing, while for, market-learing is preferable to rationing. Sine further analyti results on the orners are diffiult to obtain, we use our numeri example to ompute the optimal alloation proportion when the seller maximizes ex ante revenue, ˆR(t) = v h v l R(v; t)dv. Figure 4 demonstrates the results. 22 Sine t must be an integer, this derease is in steps, rather than ontinuous.
RATIONING IN IPOS 51 Figure 4. Optimal alloation perentage as hanges Armed with an optimal t for eah, we an determine the endogenous distribution of bids in the book. In partiular, we an ompute the maximum and minimum bids (and hene the range), and the offer prie (and thus the range of bids that reeived a positive alloation). This leads to empirial preditions linking the degree of rationing with observed properties of bids. We emphasize that these omparative statis preditions assume all else held onstant; in partiular, ɛ, n, k, and[v l,v h ]. The optimal value of t is endogenously determined given these parameters and. OBSERVATION 1. The range of submitted bids on the book is smaller if the degree of rationing is smaller. The range of bids reeived (the maximum bid minus the minimum bid) delines with. 23 Hene, this range should be diretly related to the degree of rationing; i.e., inversely related to the alloation perentage aross IPO deals. OBSERVATION 2. The range of bids above the offer prie is smaller if the degree of rationing is smaller. The observed range of bids that reeive a positive alloation (i.e., the maximum bid minus the offer prie) also delines with (sine t 23 This is straightforwardly determined as the differene between the maximum and minimum order statistis.
52 CHRISTINE A. PARLOUR AND UDAY RAJAN delines with ). Hene, this range should also be positively related to the degree of rationing (or alternatively, inversely related to the aggregate alloation perentage). Under what irumstanes would we expet to hange in a preditable manner? Consider a sequene of IPOs of firms in the same industry. One would expet information on the first firm to be diffuse. Sine information about other firms is likely to be orrelated, as more firms go publi, information should beome more preise. 24 Thus, we would expet to inrease with eah subsequent IPO. The same omparative stati holds when omparing IPOs to seasoned equity offerings (SEOs). If a firm has been publily traded for a length of time, information about it is more preise, and hene should be larger. Indeed, Cornelli and Goldreih (2003) find some support for this view. They find that, in their sample, the average elastiity of demand for SEOs is muh larger than the average elastiity of demand for IPOs. This average elastiity measures the proportion of bids within 1% of the offer prie, and hene proxies for the preision in the information of bidders. Given a book with many prie-ontingent bids for a single IPO deal, the parameters and ɛ an be struturally estimated. From this, signals an then be inferred from bids, and hene a more aurate measure of the true value of the asset (whih depends on all signals) an be generated. 5. Disussion It is useful to onsider our results in the framework of Bulow and Klemperer (1996), who onsider an English aution with no reserve prie, and show that, even with affiliated signals, the expeted revenue of the seller is just the expeted marginal revenue of the winning bidder. 25 In our model, different rationing levels alloate the good to different sets of bidders, so variation in seller revenue an be interpreted in terms of variation in marginal revenues aross bidders. It is important to keep in mind, however, that different levels of rationing reveal different information. As Bulow and Klemperer (1996) point out, to determine expeted revenue, eah bidder s marginal revenue must be alulated based on the information the aution will reveal. Reall that in a (k, t)-bookbuilding mehanism, the prie is the (t + 1) st highest bid, so in equilibrium is set by the (t + 1) st highest signal. Let ( Y t+1,n ) denote the onditional distribution of agent i s signal, given that the (t + 1) st highest of all n signals is Y t+1,n,andletψ( Y t+1,n ) be the assoiated density. Then, following Bulow and Klemperer (1996), the marginal revenue of bidder i in our setting is defined as MR i = v 1 (s i Y t+1,n ) v, (3) ψ(s i Y t+1,n ) s i 24 Alti (2002) suggests that suh an information struture should be a feature of observed IPOs. 25 We are grateful both to Paul Klemperer and to a referee for pointing us in this diretion.