A Theory of Arbitrage Capital

Transcription

1 A Theory of Arbitrage Caital Viral V. Acharya NYU-Stern, CEPR, and NBER Hyun Song Shin Princeton University Tanju Yorulmazer Federal Reserve Ban of New Yor We resent a model of equilibrium allocation of caital for arbitrage. If asset rices may fall low enough, it is rofitable to carry liquid caital to acquire assets in such states. Set against this, eeing caital in liquid form entails costs in terms of foregone rofitable investments. This trade-off generates occasional fire sales and limited arbitrage caital as robust henomena. With learning-by-doing effects, arbitrage caital moves in to acquire assets only if fire sales are stee. However, once arbitrage caital finds it rofitable to acquire assets, it requires similar returns elsewhere, inducing contagious fire-sale rices even for unrelated assets. (JEL G21, G28, G38, E58, D62) Our understanding of financial crises has been enhanced by a large and raidly growing emirical literature that has documented the incidence and severity of fire sales by distressed arties in a wide range of asset classes. 1 Indeed, it would not be too much of an exaggeration to say that fire sales have been a defining feature of most financial crises, including the most recent crisis of The term fire sale carries the connotation that assets are being sold at rices that are below some benchmar, fair fundamental rice that would revail in the absence of a crisis. However, the notion that assets are We would lie to than Paolo Fulghieri (editor), Douglas Gale, Yrjo Kosinen, an anonymous referee, and the articiants of the FIRS Conference in Florence for helful comments. All errors are our own. The views exressed here are those of the authors and do not necessarily reresent the views of the Federal Reserve System or the Federal Reserve Ban of New Yor. Send corresondence to Viral Acharya, Deartment of Finance, Stern School of Business, New Yor University, 44 West 4th Street, Room 9-84, New Yor, NY 10012, USA; telehone: (212) vacharya@stern.nyu.edu. 1 Fire sales have been shown to exist in distressed sales of aircrafts by Pulvino (1998), in cash auctions in banrutcies by Stromberg (2000), in creditor recoveries during industry-wide distress esecially for industries with high asset-secificity by Acharya, Bharath, and Srinivasan (2007), in equity marets when mutual funds engage in sales of similar stocs by Coval and Stafford (2007), and in an international setting in which foreign direct investment increases during emerging maret crises to acquire assets at stee discounts in the evidence by Krugman (1998), Aguiar and Goinath (2005), and Acharya, Shin, and Yorulmazer (2012). ß The Author Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, lease journals.ermissions@ou.com. doi: /rcfs/cfs006 Advance Access ublication January 18, 2013

2 A Theory of Arbitrage Caital being sold at rices below their fundamental value begs an imortant question. How can fire sales tae lace in a world in which arbitrage caital waits on the sidelines to tae advantage of artificially low rices? If there were such arbitrageurs who wait on the sidelines, would they not comete with each other as soon as the crisis eruts, roviding a cushion for rices? As well as these ositive questions on the nature of the equilibrium outcome, there are also imortant normative questions on the social value of arbitrage caital. Is the equilibrium rovision of arbitrage caital at the efficient level? If not, is the efficient level higher or lower than the equilibrium level? Our aer tries to answer these questions in a setting that is simle and transarent enough to hel uncover the underlying economic mechanisms at wor. We show that the answers to both the ositive and normative sets of questions rely on the interlay between two underlying allocative mechanisms in the economy. One oerates at the ex ost stage, and has to do with the efficient allocation of assets to those economic agents that can generate most value from them. The second oerates at the ex ante stage, and has to do with how much of the economy s resources are set aside in the form of idle arbitrage caital that waits on the sidelines. In our baseline setu, engaging in roduction entails greater exertise to insiders through learning-by-doing effects, and these insiders are the natural holders of the assets in the sense of being able to generate greater value from them comared with outsiders who tae over distressed assets of illiquid insiders. The greater is the discount in the value realized by outsiders, the more severe must be the fire sale before outsiders enter to cushion the distress. Thus, for a finite ool of arbitrage caital at the ex ost stage, there are states of the world with ossibly stee rice discounts, which then create the incentive to hold unroductive idle arbitrage caital at the ex ante stage. In equilibrium, the two choices whether to invest in rofitable activities or to set aside funds for arbitrage in the future earn the same rate of return when viewed ex ante. As a consequence, limited rovision of arbitrage caital and fire sales emerge as robust features of the equilibrium. However, there are two imortant normative consequences of fire sales. First, there is ex ost inefficiency due to assets being held by outsiders in some states of the world. Second, there is also ex ante inefficiency due to the ex ante allocation of resources to arbitrage caital. There are also imlications for the deth of fire sales following long eriods of low liquidity ris (good times). During eriods when liquidity ris is low, illiquid rojects are more attractive relative to holding cash balances. Hence, a higher fraction of agents choose to become insiders and there is less liquid caital ut aside. As a result, when liquidity shocs do materialize fire-sale effects in asset rices are more severe as there was less liquid caital ut aside for arbitrage. This can otentially exlain 63

3 Review of Cororate Finance Studies / v 2 n why crises that erut after good times are associated with sharer, more severe fire sales. 2 The icture that emerges from our analysis is that rivate incentives lead to the over-rovision of arbitrage caital relative to its socially otimal level. Intuitively, in states in which there are few liquid insiders, asset rices must fall so that the maret clears. Nevertheless, there is allocative efficiency as assets remain in the hands of insiders. The resence of arbitrage caital interferes with the efficient allocation. Ex ost, there is inefficiency if arbitrageurs are less efficient than insiders in deloying assets. Imortantly, even if arbitrageurs are as efficient as insiders, setting aside arbitrage caital ex ante imlies assing over rofitable investment oortunities. Somewhat counterintuitively erhas, relative to the cometitive outcome, the social otimum features lower asset rices ex ost and greater rofitable investments ex ante. Because our welfare results arise from the interlay between the ex ost asset allocation and the ex ante rovision of arbitrage caital, one imortant message from our aer is that the oening of caital marets in the ex ost eriod where liquid insiders can raise new funding from outsiders does not eliminate the inefficiency, although there is some mitigation of the inefficiency. We demonstrate this feature in an extension of our framewor, where we allow insiders to raise additional funding from outsiders by selling financial claims. Because outsiders have the choice between acquiring hysical assets or buying financial claims sold by insiders, arbitrage caital is allocated in such a way that the returns are equalized. The result is that fire-sale discounts in rices for acquisition of assets must equal that for rovision of external finance, giving rise to a sillover or contagion from illiquidity in the maret for real assets to that for financing of these assets. From the welfare standoint, ex ost efficiency of allocation is restored as arbitrageurs simly fund asset urchases of insiders, but they mae same rofits ex ante because of discounts they charge in funding insiders. As a result, it remains rofitable for there to be some arbitrage caital in equilibrium and there continues to be some ex ante inefficiency in investment decisions. In another extension of our benchmar model, we show that contagion can result across different asset classes whenever the rovision of arbitrage caital in these asset marets is from a common ool; returns on different investments in the ortfolio of an arbitrageur must be the same. 2 Conversely, as economic times worsen, more caital is set aside for arbitrage. For instance, according to the article titled Cashing in on the crash in the Economist on August 23, 2007, vulture funds raised $15.1 billion in the first seven months of 2007, more than the $13.9 billion in all of 2006, to tae advantage of fire sales due to exected distress in financial marets. The same article oints out that although some hedge funds suffered, the others, such as Citadel, Ellington, and Marathon Asset Management, had the ready cash. The article highlights Citadel s strategy of eeing more than a third of its assets in cash or liquid securities, allowing the comany to tae advantage of fire sales when oortunities arise. 64

4 A Theory of Arbitrage Caital The fact that the quantity of arbitrage caital is limited imlies that these returns are ositive in all marets where arbitrageurs allocate caital. 3 This channel of contagion oerates even though the fundamentals of two assets are indeendent and the existence of fire sales in one asset can give rise to fire sales in the other. Fire sales are not, of course, a new idea. The idea that asset rices may contain liquidity discounts when otential buyers are financially constrained and assets are not easily redeloyable were discussed by Williamson (1988) and Shleifer and Vishny (1992). This early literature suggests that firms, whose assets tend to be secific (i.e., whose assets cannot be readily redeloyed by firms outside of the industry) are liely to exerience lower liquidation values because they may suffer from fire-sale discounts in cash auctions for asset sales, esecially when firms within an industry become simultaneously financially or economically distressed. Since then, fire sales have often figured in models of crises (Allen and Gale 1994, 1998; among others). Intimately tied to the notion of fire sales is the idea that arbitrageurs wanting to buy assets at stee discounts may also face financing frictions due to rincial-agent roblems. The resulting limits of arbitrage (Shleifer and Vishny 1997) can entrench fire-sale rices for a eriod of time once they materialize. 4 Our contribution relative to this earlier literature is to focus on the ex ante decisions of investors, which much of the literature taes as given, and thereby to exlain the origins of the limited nature of arbitrage caital as an equilibrium henomenon. In this sense, our wor is closest to the analysis by Allen and Gale (2004) of the ortfolio choice of bans between holding safe versus risy assets. Gorton and Huang (2004), another closely related aer, also consider the equilibrium ortfolio choice of firms, deriving that it is socially inefficient to hold large quantities of safe assets required to avoid fire sales, and study in this context the role of government bailouts during crises. Our framewor is tractable and facilitates cris conclusions on ey comarative statics and welfare questions. In articular, our result that arbitrage caital is endogenously lower in good times and therefore that crises arising directly after good times feature deeer fire-sale discounts, is noteworthy and is one which (to our nowledge) has not been discussed so far in the literature. Another advantage of our tractable framewor is to oen u for scrutiny the arbitrageurs access to different real and financial marets, and 3 For (aarent) dislocations between different caital marets and the effect of liquidations in one maret on rices in another, see an excellent discussion of the large body of extant emirical evidence in Duffie and Strulovici (2012). For similar evidence in an international setting, see Rigobon (2002) and Kaminsy and Schmuler (2002), and the discussion in Pavlova and Rigobon (2007). The literature by and large attributes such dislocations to investment-style restrictions or limited arbitrage caital. 4 Mitchell, Pedersen, and Pulvino (2007) rovide comelling eisodic evidence for the fact that caital aears to be slow moving when it enters marets affected by fire-sale discounts in rices. 65

5 Review of Cororate Finance Studies / v 2 n thereby identify a channel of contagion that relies urely on the limited nature of arbitrage caital. Our results on this front are closest to Gromb and Vayanos (2007) and Duffie and Strulovici (2012). Gromb and Vayanos (2007) consider arbitrageurs exloiting fire-sale oortunities across marets, and this equilibrates returns they can earn in different marets. Duffie and Strulovici (2012) study a similar setting, taing the financing friction of arbitrageurs as given. In both of these aers, the quantity of equilibrium arbitrage caital is exogenous, whereas our central theoretical concern is to endogenize the quantity of arbitrage caital and illustrate that its limited quantity as well as its limited exertise mae fire sales a robust equilibrium henomenon. 5 Ramini and Viswanathan (2010) consider a dynamic contracting setu for exloring which firms mae investments and which reserve debt caacities for the future. This question is related to our analysis. Although we model fire sales as an outcome from maret clearing when buyers are financially constrained, Ramini and Viswanathan model asset rices as being temorarily low because of low cash-flow realizations. Acharya, Shin, and Yorulmazer (2011) also consider gains from acquiring assets at fire-sale rices that mae it attractive for firms (in their model, bans) to hold liquid assets ex ante but that when such fire-sale states are not too liely, bans hold too little liquidity because of the asset-substitution roblem. Their focus is on analyzing how government or central ban interventions to resolve baning crises (that may be desirable ex ost) affect ex ante liquidity in otentially adverse ways. 6 Bolton, Santos, and Scheinman (2011) and Diamond and Rajan (2011) resent models wherein once an adverse state of the world arises, decisions of individual bans affect when assets are sold immediately or with delay. Immediate sales can increase returns to holding cash and lead to otentially excessive cash hoarding ex ante. In contrast to these models, our aer is more in the sirit of Allen and Gale (1994, 1998) and Gorton and Huang (2004) in that once the adverse state arises, there is an immediate (fire) sale of assets. Our study is also related to the seminal wor of Kiyotai and Moore (1997) on credit cycles. In Kiyotai and Moore (1997) and Krishnamurthy (2003), the underlying asset cannot be ledged because of inalienable human caital. Krishnamurthy (2003) differs from Kiyotai and Moore (1997) in that all contingent claims on aggregate variables are subject to collateral constraints. However, land can be 5 Note that contagion has been derived in many other settings through ortfolio flows (Kodres and Pritser 2002) or utility-based assumtions (Kyle and Xiong 2001). 6 Huang and Wang (2010) also show that cometitive maret forces fail to lead to an efficient suly of liquidity. The maret rovision of liquidity is generally too low when the robability of a liquidity event is small and is too high when the robability of a liquidity event is large. 66

6 A Theory of Arbitrage Caital ledged and has value both as a roductive asset and as collateral. Caballero and Krishnamurthy (2001) emloy a Holmstrom and Tirole (1998) aroach with exogenous liquidity shocs and allow firms to ost collateral in a manner similar to Kiyotai and Moore (1997). Finally, there are models in which there are ecuniary externalities from fire sales of assets. Lorenzoni (2008), for examle, considers a cometitive model of intermediaries in which ignoring these externalities leads to excessive borrowing ex ante and excessive volatility ex ost. In our model, there are no externalities from fire sales and the result is that the cometitive equilibrium features too much setting aside of idle caital ex ante for the urose of undertaing arbitrage ex ost. 1. Model The time line of the model is rovided in Figure 1. There are three dates indexed by t 2 f0, 1, 2g. There is a unit measure of ris-neutral agents. Each agent is endowed with one unit of the consumtion good at t ¼ 0. There are two tyes of assets in the economy. There is a storage technology, referred to as cash, that allows any agent to transfer one unit of the consumtion good from date t to date t þ 1. In this sense, the ris-free interest rate is zero. There is also an (artially) illiquid investment oortunity, referred to as the asset, that agents can undertae at date 0. The investment oortunity is indivisible and needs the full unit of the consumtion good as the inut. The investment ays a return of R t at t ¼ 1, 2. For simlicity we assume that R t ¼ R for t ¼ 1, 2 and R 4 1=2. 7 Agents decide whether or not to undertae the illiquid investment at date 0. Those who decide to invest are nown as insiders. Those who choose to ee their wealth in the storage technology are nown as arbitrageurs. Insiders can be hit by a liquidity shoc at t ¼ 1, in which case they need to consume at t ¼ 1 and want to convert their return of R at date 2 into cash at date 1. We call insiders that are hit by a liquidity shoc at t ¼ 1asilliquid insiders. Insiders that are not hit by the liquidity shoc, denoted as liquid insiders, and the arbitrageurs consume at t ¼ 2. 8 The robability that an insider is hit by the liquidity shoc is given by. However, we allow aggregate uncertainty in the economy. When 7 Alternatively, we could assume that the illiquid roject ays off only at t ¼ 2, that is, R 1 ¼ 0. This would not change our results qualitatively. However, in our setu, insiders that are not hit by the liquidity shoc can use the return at t ¼ 1 for acquiring the assets of insiders that get hit by a liquidity shoc, and this adds richness to our model. 8 We could assume that the arbitrageurs can get hit by a liquidity shoc at t ¼ 1 as well. This would not change our results qualitatively as long as arbitrageurs are less liely to be hit by the liquidity shoc comared with the insiders. However, it would mae it less attractive to become an arbitrageur because the only source of rofit for arbitrageurs is the fire sale of the assets of illiquid insiders. 67

7 Review of Cororate Finance Studies / v 2 n t = 0 t = 1 States A fraction, w, of agents choose to become arbitrageurs, whereas a fraction (1 - w) choose to become insiders. Returns from the risy investments are realized. A roortion,, of insiders fails. < Price is the full rice,. All assets are urchased by surviving insiders. Price is decreasing in but is still above the threshold value of arbitrageurs,. All assets are urchased by surviving insiders. Insiders invest in risy rojects using their own caital. Failed insiders are auctioned to surviving insiders and arbitrageurs. < > Price is the threshold value of arbitrageurs,. Arbitrageurs acquire some failed insiders assets. Price is below the threshold value of arbitrageurs,, and is decreasing in. Arbitrageurs acquire some failed insiders assets. Figure 1 Time line of the model viewed from date 0, the robability itself is uncertain. Nature first draws from a nown density fðþ that is continuous over ½0, 1Š and then determines the realizations for each agent as indeendent and identically distributed (i.i.d.) draws from coin tosses where the robability of a liquidity shoc is fixed at. By the law of large numbers, the roortion of illiquid insiders is exactly, but this roortion is uncertain at the time of the investment. This aggregate uncertainty lays a ey role in our model. 1.1 Insiders and arbitrageurs We will denote the roortion of agents who choose to become arbitrageurs by w. We assume that, by the nature of the investment, insiders benefit from learning-by-doing, so that insiders become roficient in managing the asset. This is an imortant feature of our model, as the role of arbitrageurs is double sided. Although they stand on the sidelines ready to urchase the assets of illiquid insiders, they are not natural holders of the asset, and there are social costs as a result of their ownershi of the assets, as we will describe below. Proortion of insiders turn out to be illiquid, and by date 1, the identity of the liquid and illiquid insiders is nown. Illiquid insiders want to consume at t ¼ 1 and try to convert their return of R at t ¼ 2 into cash at t ¼ 1. They do this by utting u their future return into the asset maret in return for cash at t ¼ 1, where the otential buyers of these assets are liquid insiders and arbitrageurs. 68

8 A Theory of Arbitrage Caital However, learning-by-doing matters for the terminal return from the asset. Between date 1 and date 2 (i.e., after the realization of the liquidity shocs but before the realization of the return at t ¼ 2), the return from the asset deends on who holds the asset. If the asset is held by an insider, the asset can be managed well because of the exertise gained by the insider in the initial eriod of roduction. In the hands of the insider, the terminal value of the asset at date 2 is given by ¼ R. However, if the asset is run by an arbitrageur between date 1 and 2, the uee of the asset is not as good so that the terminal value of the asset is given by, where ¼ R, ð1þ and 2½0, RŠ is assumed to be the loss in the asset value in the hands of the arbitrageurs. Our assumtion that 5 is motivated by the emirical evidence that during major financial crises, the entry of outside investors taes lace only when incumbents face severe financial distress and lac the resources to tae over failing rivals. The resolution of distressed bans is erhas the clearest illustration of the comarative advantage of insiders in managing the assets. When faced with the imminent failure of a ban, regulators turn to other (healthier) bans to tae over the ailing rival. It is only when otential acquiring bans are themselves distressed that regulators turn to outside investors. For these reasons, we believe that the case in which 5 is the natural one to examine in our model. Nevertheless, it should be emhasized that our framewor is rich enough to accommodate the alternative scenario, where 4 so that the outsiders have greater exertise and are able to generate more value than the insiders. We examine this case in subsection and show that the welfare results are made more subtle. The imortant oint is that even with greater exertise ex ost, the overall welfare calculation must also tae account of the ex ante inefficiency associated with idle arbitrage caital, and the greater exertise of outsiders may not eliminate the inefficiency of arbitrage caital. Returning to our benchmar model, we continue under the assumtion that 5. A natural regularity condition in this context is that the ex ante roductivity of investment justifies the cost of investment, namely, R 4 1=2 as assumed earlier. This condition states that the investment is not unjustified even under the most otimistic scenario in which all the assets of illiquid insiders end u in the hands of liquid insiders. 1.2 Welfare The welfare function is defined as the unweighted sum of the ex ante ayoffs of all agents. In our framewor, the welfare function is the 69

9 Review of Cororate Finance Studies / v 2 n exectation at date 0 of the total consumtion across all agents both insiders and arbitrageurs. Note that welfare taes account only of the total consumtion rather than the distribution of consumtion across agents. Denote by y a the mass of assets held by arbitrageurs from t ¼ 1to t ¼ 2. Then the total assets held by the insiders are ð1 wþ y a. Thus, interim welfare can be written as ðw, Þ ¼ð1 wþ½ð1 Þ2R þ ðr þ Þ Š þ w y a ¼ ð1 wþ2r þ w y a : We can write ex ante welfare as the exectation of (2) with resect to the realization of the aggregate shoc, that is, ¼ E ½ð1 wþ2r þ w y a Š: ð3þ From (3) we see that ex ante welfare is decreasing in y a for any fixed value of w. The ey question is how behaves as a function of w. Because R 4 1=2, we now that the risy asset has a higher return than the safe asset. Hence, ex ante welfare is decreasing in w so that w ¼ 0is the unique socially otimal level of w. This gives us the following roosition. Proosition 1 For R 4 1=2, w¼ 0 is the unique socially otimal level of arbitrage caital. As noted already, any outcome in which the asset ends at date 2 in the hands of an arbitrageur entails a welfare cost of and is hence not socially otimal. Two conditions are necessary and sufficient for the social otimum, namely, that the interim outut at date 1 is maximized and that no assets end in the hands of arbitrageurs. When w ¼ 0, both conditions are satisfied. There are no arbitrageurs at date 1 and so all the assets end in the hands of insiders only. Meanwhile, for R 4 1=2, the interim outut is maximized when w ¼ 0. Hence, the outcome with no arbitrageurs is the unique socially otimal level of arbitrage caital. In the next section, we characterize the equilibrium and show that the social otimum is never an equilibrium outcome. Thus, there is a strict searation between the socially otimal outcome and the equilibrium outcome. ð2þ 2. Benchmar Equilibrium Having characterized the socially otimal outcome, we now consider the cometitive equilibrium. In the benchmar model, we assume that liquid insiders cannot raise additional funding, and must rely solely on the 70

10 A Theory of Arbitrage Caital resources from the successful roject. Thus, until further notice, we oerate under the following assumtion. 9 Assumtion 1 The insiders cannot raise outside caital, and each liquid insider only has R units of cash that can be used to urchase distressed assets. The ayoffs of the model are determined through a cometitive auction of illiquid insiders assets at the interim date. We will then solve the model bacward, by first considering the sale of illiquid insiders assets and the resulting asset rices, and next, analyzing the ex ante choice to become insiders or arbitrageurs. 2.1 Asset sales and liquidation rices We ee trac of two ey features in the urchase of illiquid insiders assets. First, arbitrageurs, using arbitrage caital, and the liquid insiders, using their first-eriod return, comete to urchase these assets. Second, liquid insiders may not have enough resources to acquire all illiquid insiders assets. To focus on the interlay between these two features, we model asset sales as follows. 1. All illiquid insiders assets are ooled and cometitively auctioned to the liquid insiders and arbitrageurs as described below. 2. The liquid insiders and arbitrageurs submit a demand schedule y i ðþ that secifies the quantity demanded for illiquid insiders assets for each rice. The index i belongs in ½0, ð1 wþð1 ÞŠ if i is a liquid insider, whereas i 2½1 w,1š if i is an arbitrageur. 3. We assume that insiders cannot raise additional financing. Hence, the resources available to each liquid insider for urchasing illiquid insiders assets is the ayoff R at t ¼ 1 from the asset. 4. The rice clears the maret, where assets allocated to liquid insiders and arbitrageurs add u at most to the roortion of illiquid insiders: 10 Z ð1 wþð1 Þ 0 Z 1 y i ðþdi þ y i ðþdi ð1 wþ: 1 w ð4þ 5. We in down the rice by focusing on the symmetric case in which all liquid insiders submit the same schedule, that is, y i ðþ ¼yðÞ for all i 2½0, ð1 wþð1 ÞŠ, and all arbitrageurs 9 We relax this assumtion in Section Because no insider asset is scraed, the equation holds with equality in equilibrium. 71

11 Review of Cororate Finance Studies / v 2 n submit identical schedules, that is, y i ðþ ¼y a ðþ for all i 2½1 w,1š: To solve for the cometitive allocation, we first derive the demand schedule for liquid insiders. The exected rofit of a liquid insider from the asset urchase is yðþ½ Š: The liquid insider wishes to maximize this rofit subject to the budget constraint: yðþ R: ð5þ Hence, for 5, liquid insiders are willing to urchase the maximum amount of assets using their resources. Thus, the otimal demand schedule for liquid insiders is yðþ ¼ R : ð6þ For 4, the demand is yðþ ¼0, and for ¼, yðþ is infinitely elastic. In other words, as long as urchasing assets is rofitable, a liquid insider wishes to use all its resources to urchase assets. We can derive the demand schedule for arbitrageurs in a similar way. Note that, arbitrageurs value these assets at. For 5, arbitrageurs are willing to suly all their funds for the asset urchase. Thus, their demand schedule is y a ðþ ¼ 1 : ð7þ For 4, the demand is y a ðþ ¼0, and for ¼, y a ðþ is infinitely elastic. Next, we characterize how illiquid insiders assets are allocated and the rice function that results. The equilibrium rice function is also illustrated in Figure 2. Lemma 1 The equilibrium rice function for liquidated assets ðþ is given as follows: 11 ðþ ¼ 8 for ð1 ÞR >< for 2ð, Š for 2ð,, ð8þ Š >: ð1 ÞR þ w ð1 wþ for 4 11 Note that deends on w as well as, that is, we have ð, wþ. To simlify notation, we use ðþ. 72

12 A Theory of Arbitrage Caital * ( ) w (1 w) 1 Figure 2 Price function where,, and are given, resectively, by Equations (27), (28), and (29) in the Aendix. Arbitrageurs acquire assets whenever 4. Finally, as arbitrage caital w increases, the rice wealy increases, that is, d dw 0: We now that in the absence of financial constraints, the efficient outcome is to sell all assets to liquid insiders. However, liquid insiders may not be able to ay the threshold rice of for all assets. If rice falls further, buying these assets becomes rofitable for arbitrageurs and they articiate in the auction, resulting in misallocation of assets whenever 4 0. Secifically, in the first region, that is, for, the number of failures is small and liquid insiders have enough liquidity to acquire assets at the full rice. For moderate roortion of failures, that is, for 2ð, Š, however, liquid insiders can no longer ay the full rice for all assets but can still ay at least the threshold value of, below which arbitrageurs have a ositive demand. In this region, liquid insiders use all available funds and the rice falls as the roortion of failures increases. This effect comes from cash-in-the-maret ricing, as in Allen and Gale (1994, 1998), and is ain to the industry equilibrium hyothesis of Shleifer and Vishny (1992), who argue that when industry eers of a firm in distress are financially constrained, the eers may not be able to ay a rice for assets of the distressed firm that equals the value of these assets to them. However, as the roortion of illiquid insiders increases even further, liquid insiders cannot ay the threshold rice of for all assets and 73

13 Review of Cororate Finance Studies / v 2 n rofitable otions emerge for arbitrageurs. Hence, arbitrageurs are willing to suly their funds for the asset urchase. With the injection of arbitrageurs funds, rices can be sustained at until a critical roortion of failures. In the extreme, the number of failures may be so large that even the injection of arbitrageur caital is not enough to sustain the rice at : This can be considered as an aggregate shortage of liquidity in that there is cash-in-the-maret ricing even when all liquidity in the economy is channeled for asset urchases. Note that the resulting rice function is downward-sloing in the roortion of illiquid insiders in two searate regions. In the first downward-sloing region, arbitrageurs have not yet entered the maret ð 2ð, ŠÞ and there is cash-in-the-maret ricing given the limited funds of liquid insiders. In the second downward-sloing region ( 4 ), even the funds of arbitrageurs are not enough to sustain the rice at, their highest valuation of assets. 2.2 Inefficiency of equilibrium Insiders exected rofit, denoted by EðÞ, consists of rofit from their own investments, rofit from asset urchases, and the amount they recover for their assets when they are hit by the liquidity shoc at t ¼ 1, which can be derived using the rice in Equation (8). In articular, we have that the rofit of each insider is EðÞ ¼ E ð1 ÞR þ þ R 1, ð9þ where E denotes exectation over : In contrast, the only source of rofit for arbitrageurs is the asset urchase at fire-sale rices. Therefore, we have that the rofit of each arbitrageur is Eð a Þ¼ E max 0, 1 : ð10þ In the cometitive equilibrium, the caital allocation (characterized by arbitrage caital w) must be such that the two ayoffs are equalized at the ex ante stage so that EðÞ ¼ Eð a Þ, ð11þ as otherwise, there is an incentive for some insiders to become arbitrageurs instead, or vice-versa. Then, the following roosition formally characterizes agents choices. Under a technical condition that the distribution fðþ not converge to zero too raidly as goes to 1 (or in other words, that there is a sufficiently thic tail that there will be a large 74

14 A Theory of Arbitrage Caital number of failures and arbitrageurs will mae rofits), we obtain the following roosition. Proosition 2 Let fðþ be a continuous robability distribution over ½0, 1Š with an order of less than one. In the cometitive equilibrium, a roortion w 2 0, 1þ of agents choose to become arbitrageurs, where w satisfies the indifference equation in (11). Further, 5 1 so that there are states of the world in which 5. 1 Formally, for w ¼ 0, we have = ¼ =R 1 1, which converges to þ1 as converges to one. Even though fðþ can converge to zero as converges to one, as long as fðþ has an order of less than one, we have that in the limit the exected arbitrage rofits are too large when there are no other arbitrageurs in the maret: lim = fðþ ¼þ1. And, this!1 cannot be an equilibrium. For distribution f that has an order of greater than one and lim fðþ ¼0, we have lim = fðþ ¼0 so that for w ¼ 0!1!1 we have Eð a Þ5 þ1: Under such robability distributions, it is ossible to have no arbitrage caital ðw ¼ 0Þ in equilibrium. For examle fðþ ¼cð1 Þ a, for 2½0, 1Š and a 4 1, would give such a result. Nevertheless, the case of interest for our analysis is the one in which there is some arbitrage caital in equilibrium, and the technical condition in the Proosition shows that this is robustly the case. Note also that there cannot be an equilibrium with just arbitrageurs (w ¼ 1) as there would be no rofits for arbitrageurs because of a lac of any fire sales. Hence, we focus for the rest of our analysis on the equilibrium when the fractions of agents who choose to become insiders and arbitrageurs are bounded away from 0. An imortant imlication is that cash-inthe-maret rices are robust to the endogenous choice of arbitrage caital. That is, there will always be states of nature for which the rice falls not only below the fundamental value of but also below, the value arbitrageurs attach to these assets. In these states, there is an aggregate shortage of liquidity as all caital with insiders and arbitrageurs is not sufficient to ee the asset rices above or equal to, which is necessary for efficient ex ost allocation of assets. This is a robust feature of our model. For there to be arbitrage caital in equilibrium, there must be states of the world in which arbitrageurs mae rofits. In these states rices are below the arbitrageurs valuation of assets. And, this is indeed the case in equilibrium. 75

15 Review of Cororate Finance Studies / v 2 n Comarative statics Next, we formalize some comarative statics features of our model in the benchmar case under Assumtion 1. We have the following relationshis between the level of arbitrage caital and (1) liquidity ris roxied by the aggregate distribution of liquidity shocs and (2) asset secificity. Proosition 3 Equilibrium level of arbitrage caital w satisfies two features: 1. Suose f and g are two robability densities for, where f dominates g in the sense of first-order stochastic dominance. Let w f and w g be the equilibrium level of arbitrage caital under densities f and g, resectively. Then, w f 4 w g. R 2. Let ^w ¼ : For w 5 ^w, as the difference of exertise RþR þ 2 between insiders and arbitrageurs widens the equilibrium roortion of arbitrageurs decreases, that is, dw d 5 0. Consider (2) first. As the difference between the exertise levels of insiders and arbitrageurs widens (i.e., as insiders assets become more secific), the return arbitrageurs mae from these assets decreases. In turn, the region over which arbitrageurs enter the maret shrins. Thus, asset secificity reinforces fire-sale discounts in rices further. Next, consider (1). During good times, it is more liely that illiquid rojects erform well. The decrease in liquidity ris has two effects on agents choice that go in the same direction. First, the exected return from being an insider increases. Also, the roortion of illiquid insiders decreases, which limits the fire-sale oortunities for arbitrageurs. Hence, during good times, we would exect a higher fraction of agents to become insiders and tae illiquid rojects and a smaller fraction to set aside caital for arbitrage. Furthermore, from the rice function in Equation (8), we now that as the fraction of arbitrageurs w decreases, we observe bigger deviations in the rice of illiquid insiders assets from the fundamental value of. Hence, a corollary of Proosition 3 is that when adverse liquidity shocs arise during good times, fire-sale effects in asset rices are more severe, resulting in lower asset rices and higher rice volatility. This result is a novel contribution of our analysis and rovides one exlanation for why crises that follow good times are associated with greater asset rice deterioration Acharya and Viswanathan (2011) build an alternative exlanation in a model in which there is greater entry of oorly caitalized institutions when fundamentals are stronger, but in their model insiders serve as arbitrageurs and there is no arbitrage caital set aside in equilibrium. 76

16 A Theory of Arbitrage Caital Corollary 1 Adverse liquidity shocs during good times measured by high values of result in bigger deviations in the rice of illiquid insiders assets from the fundamental value of, that is, ð ðþþ increases. 2.4 Discussion: Efficiency of arbitrage caital We have the star contrast between Proosition 2, which states that the equilibrium level of arbitrage caital is strictly ositive, and Proosition 1, which states that the socially otimal level of arbitrage caital is zero. In fact, the result that zero arbitrage caital is socially otimal holds even when arbitrageurs are as efficient as insiders ( ¼ 0) in running the assets. This is because even though there is no ex ost allocation inefficiency in this case, rofitable oortunities are assed ex ante as caital remains idle waiting for arbitrage oortunities that do not create any social welfare. Thus, for arbitrage caital to have social value, there has to be aeal to other rationales. One candidate is ris aversion, which would introduce a motive to reduce the rice fluctuations across states of the world (Allen and Gale 2004, 2005). Nevertheless, even with ris aversion, the ex ante gains from ris-sharing have to be sufficiently large that it swams the roductive inefficiency. Any resumtion that arbitrage caital has social value must thus be justified. An alternative channel through which arbitrage caital may have value is to moderate amlifying effects of financial distress whenever some fragility exists in the economic system that triggers snowball effects (e.g., due to maring-to-maret constraints as in Cifuentes, Ferucci, and Shin 2005). In such a context, mitigating the initial shocs through the cushioning effect of arbitrage caital could have substantial welfare benefits. However, as with the case for ris-aversion, the final assessment should be based on a comarison of the magnitudes, and any resumtion one way or the other would be unjustified. Finally, another rationale for arbitrage caital is that the arbitrageurs are exerts in managing distressed assets. In such a setu, arbitrageurs are willing to ay a higher rice for illiquid insiders assets, and they will be the first to acquire these assets. In this case, although investing in the liquid asset yields lower returns comared with the risy investment, it allows arbitrageurs as tae-over exerts to acquire illiquid insiders assets and generate higher returns from these distressed assets comared with the insiders. We setch this version below Arbitrageurs as tae-over exerts One otential interretation of arbitrageurs can be that they may be exerts in taing over and managing distressed assets, in which case, 77

17 Review of Cororate Finance Studies / v 2 n they value illiquid insiders assets higher than liquid insiders. Hence, arbitrageurs are willing to ay a higher rice for illiquid insiders assets, and they will be the first to acquire these assets. However, when arbitrageur funds are limited, for sufficiently large roortion of failures, rices fall, and even though insiders are inefficient in managing distressed assets, they will acquire some of these assets. Formally, let arbitrageurs generate a return of ^ ¼ þ, with 4 0, from illiquid insiders assets. Hence, for 5 ^, arbitrageurs are willing to suly all their funds for the asset urchase and their demand schedule is y a ðþ ¼ 1 : For 4 ^, their demand is y aðþ ¼0, and for ¼ ^, y a ðþ is infinitely elastic. In the absence of financial constraints, the efficient outcome is to sell the assets to arbitrageurs. However, when arbitrage caital is limited, arbitrageurs may not be able to ay the rice of ^ for all assets and some of the illiquid insiders assets get acquired by liquid insiders. If the roortion of illiquid insiders is sufficiently small, arbitrageurs have enough funds to ay the full rice ^ for all assets. More secifically, for a, where w a ¼ ð1 wþ ^, ð12þ the auction rice is a ¼ ^ and each arbitrageur is allocated a share y a ð ^Þ ¼ ð1 wþ w. For moderate values of, arbitrageurs cannot ay the rice ^ for all assets but can still ay at least, below which insiders have a ositive demand. Formally, for 2ð a, a Š, where w a ¼ ð1 wþ, ð13þ the rice is set at a ¼ w ð1 wþ, and all assets are acquired by arbitrageurs. For 4 a, arbitrageurs cannot ay for all assets, and liquid insiders suly their funds for the asset urchase. With the injection of insiders funds, rices can be sustained at until some critical roortion of failures a a : We obtain the resulting rice function (also see Figure 3): 8 ^ for a w >< ð1 wþ for 2ð a, a Š a ðþ ¼ : ð14þ for 2ð a, a Š >: ð1 ÞR þ w ð1 wþ for 4 a Note that as the roortion w of agents that choose to become arbitrageurs increases, the boundaries a, a and a increase, as well as the 78

18 A Theory of Arbitrage Caital a* ( ) ˆ w (1 w) a a 1 Figure 3 Price with arbitrageurs as taeover exerts rice a in the second and the fourth regions. Hence, as w increases, the rice wealy increases, that is, we have d a dw 0:13 The social lanner maximizes the exected total outut generated by the economy, ¼ Ewþð1 ½ wþð1 ÞR þ y I þ y A ^ Š, ð15þ where y I and y A reresent the units of illiquid insiders assets acquired by insiders and arbitrageurs, resectively, and y I þ y A ¼ð1 wþ. 14 Note that the condition R 4 1=2 is sufficient for the risy investment to have a higher exected return than the investment in the safe asset. Although investing in the liquid asset yields lower returns comared with the risy investment, it allows arbitrageurs tae-over exerts acquire illiquid insiders assets and generate higher returns from these distressed 13 Note that the equilibrium level of arbitrage caital is determined by the same condition h as in the i benchmar model, with the differencethat the exected rofit for arbitrageurs is Eð a Þ¼E 1 ð ^ Þ,andin equilibrium, a roortion w ^ 2 0, 1þ ^ of agents choose to become arbitrageurs, where w satisfies the indifference equation in (11). 14 Furthermore, we obtain: 8 8 ð1 wþ for a 0 for a >< ð1 wþ for 2ð a, a Š >< 0 for 2ð a, a Š y A ¼ w for 2ð a, a Š and y I ¼ ð1 wþ w for 2ð a, a Š : >: wð1 wþ >: ð1 wþ ð1 wþð1 ÞRþw for 4 2 ð1 ÞR a ð1 wþð1 ÞRþw for 4 a 79

19 Review of Cororate Finance Studies / v 2 n assets comared with the insiders. Hence, for sufficiently high values of, that is, when arbitrageurs are sufficiently more efficient in running distressed assets comared with insiders, the foregone exected outut from investing in the liquid assets is comensated by the efficiency gain from the taeover exertise of arbitrageurs. In that case, it would be socially otimal to set aside some arbitrage caital to acquire distressed assets. Although it is lausible in some contexts that arbitrageurs are able to generate more value than the insiders as in the extension we just considered, we note that such an assumtion runs counter to the intuition that insiders gain by learning-by-doing so that they are the natural holders of the assets. As mentioned already, Acharya, Shin, and Yorulmazer (2012) exhibit evidence that during emerging maret crises, foreign arbitrageurs resell, or fli, the assets they acquired during the fire sale to local insiders, suggesting they were temorary owners of assets due to aggregate shortage of liquidity rather than ermanent owners due to exertise. Imortantly, even with greater exertise for arbitrageurs, the overall welfare calculation must also tae into account the ex ante inefficiency associated with idle arbitrage caital, and the greater exertise of arbitrageurs may not eliminate the inefficiency of arbitrage caital. We now return to our benchmar model in which arbitrageurs are inefficient relative to insiders and generalize it to a setting with caital marets. 3. Introducing a Caital Maret The inefficiency identified in the benchmar equilibrium reflects Assumtion 1, which stated that the liquid insiders cannot raise outside caital and each only has R units of the consumtion good that can be used to urchase distressed assets. It might aear that the inefficiency is somehow fragile to the introduction of a caital maret, where we relax Assumtion 1 and allow liquid insiders to sell claims to the arbitrageurs. However, this is not the case. It turns out that even when insiders can ledge all of their cash flows fully to arbitrageurs and raise external financing at date 1, the inefficiency ersists. On the other hand, the introduction of a caital maret mitigates the inefficiency, in a way to be made more recise below. To see this, we relax Assumtion 1 and allow liquid insiders to generate funds from arbitrageurs against the assets they acquire: Assumtion 2 The liquid insiders can raise outside caital by issuing shares to arbitrageurs and deloy it along with R units of cash that each liquid insider has to urchase assets. 80

20 A Theory of Arbitrage Caital In articular, liquid insiders issue shares, which is a claim on the return at t ¼ 2 from a unit of illiquid insiders assets they acquire or their own return at t ¼ 2, to generate funds er unit of share issued. In general, we can assume that because of various imerfections, such as asymmetric information, moral hazard, etc., insiders may not be able to fully ledge their future cash flows (à la Holmstrom and Tirole 1998). However, we consider here the case with full ledgeability as this gives liquid insiders the best chance to urchase liquidated assets and stacs the odds against arbitrage caital being attractive ex ante. 15 We denote as qðþ the rice of equity share in liquid insiders, urchased by arbitrageurs. Hence, when the roortion of illiquid insiders is, the amount of funding available with the liquid insiders for the urchase of assets, including funds that can be generated against returns from urchased assets, is given as: LðÞ ¼ ð1 wþð1 Þ½R þ sqðþš, ð16þ where s is the units of shares issued by each liquid insider, which must be less than or equal to 1 þ m, the sum of the units of assets m acquired by each liquid insider and one for their own return at date 2. Clearly, this total liquidity available with the liquid insiders for asset urchases is higher comared with the benchmar case. As a result, the region over which we observe cash-in-the-maret ricing is smaller,that is, it starts at a larger roortion of failures, comared with the benchmar case (Lemma 1). Now, we have two marets: one for assets of illiquid insiders and one for shares of liquid insiders. To find the equilibrium rices and allocations in these two marets, we formally state the otimization roblem that liquid insiders and arbitrageurs face. If a liquid insider issues s units of shares at the rice qðþ and urchases m units of assets at the rice ðþ, it maes an exected rofit of mð ðþþ sð qðþþ: Note that in any equilibrium, qðþ cannot exceed. Thus, we have qðþ, and liquid insiders issue equity just enough for the asset urchase, not more. Using this, we can state a liquid insider s maximization roblem as: max m, s s:t: mð ðþ Þ sð qðþþ s qðþþr m ðþ s 1 þ m ð17þ 15 Details of the case with artial ledgeability are available uon request. 81

21 Review of Cororate Finance Studies / v 2 n For qðþ ðþ, liquid insiders cannot mae ositive rofits by issuing equity to urchase assets. Thus, when qðþðþ, s ¼ 0 and m ¼ R ðþ : And, when qðþ 4 ðþ, liquid insiders mae ositive rofits from asset urchase using the funds they generate by issuing equity. Hence, they would lie to issue as much equity as ossible, that is, s ¼ 1 þ m: We can state each arbitrageur s maximization roblem in a similar way: x ðþ þ z qðþ max x, z s:t: x ðþþz qðþ 1, ð18þ where x and z reresent the units of assets and the roortion of shares in liquid insiders urchased by arbitrageurs, resectively. When the share rice of liquid insiders, qðþ, is low comared with the rice of illiquid insiders assets, ðþ, arbitrageurs refer to urchase shares of liquid insiders. However, if ðþ instead becomes low comared with qðþ, then arbitrageurs may refer to acquire the assets directly. When ðþ 4, arbitrageurs do not want to urchase assets and xðq, Þ ¼0: When ðþ 5, arbitrageurs choose x to maximize: 1 xðþ x ðþ þ ð qðþþ qðþ ¼ x ðþ qðþ þ qðþ 1 : ð19þ Thus, if ðþ 5 and qðþ4 ðþ, then arbitrageurs use all their funds for the asset urchase, that is x ¼ 1 ðþ : When ðþ 5 and qðþ5 ðþ, arbitrageurs use all their funds for the equity urchase, that is, z ¼ 1 qðþ, and when qðþ ¼ ðþ, arbitrageurs are indifferent between the equity and the asset urchase. In equilibrium, demand for shares of liquid insiders and assets of illiquid insiders should equal their suly. Hence, we have the maret clearing conditions: ð1 wþð1 Þs ¼ wz ðequity maretþ, ð20þ ð1 wþð1 Þm þ wx ¼ð1 wþ ðasset maretþ: ð21þ We concentrate on the equilibrium where the articiation of arbitrageurs in the equity maret is maximum, which results in the maximum rice for assets. However, even in this case, we show that for a large roortion of failures, the share rice of liquid insiders falls below. 82