Cash-in-the-Market Pricing and Optimal Bank Bailout Policy 1

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1 Cash-in-the-Maret Pricing and Otimal Ban Bailout Policy 1 Viral V. Acharya 2 London Business School and CEPR Tanju Yorulmazer 3 Ban of England J.E.L. Classification: G21, G28, G38, E58, D62. Keywords: Ban regulation, Systemic ris, Baning crises, Time inconsistency, Too many to fail, Herding This Draft: July, We are grateful to Mar Flannery, Douglas Gale, Denis Gromb, Jose Liberti, Enrico Perotti, Hyun Shin, Raghu Sundaram and Lucy White (discussant) for useful discussions. We would lie to than seminar articiants at London Business School, University of Vienna and articiants of the Financial Fragility and Ban Regulation Conference at the Ban of Portugal for comments on this version, and seminar articiants at London Business School, Ban of England, Cass Business School at the City University of London, Judge Institute of Management at Cambridge, Norwegian School of Management, Tilburg University and University of Amsterdam for comments on the twoban version of this aer titled Too Many to Fail: An Analysis of Time-Inconsistency in Ban Closure Policies. All errors remain our own. 2 Contact: Deartment of Finance, London Business School, Regent s Par, London NW1 4SA, England. Tel: +44 (0) Fax: +44 (0) e mail: vacharya@london.edu. Acharya is also a Research Affiliate of the Centre for Economic Policy Research (CEPR). 3 Contact: FIRD HO-3, Ban of England, Threadneedle Street, London, EC2R 8AH, Tel: +44 (0) Fax: +44 (0) e mail: Tanju.Yorulmazer@banofengland.co.u.

2 Cash-in-the-Maret Pricing and Otimal Ban Bailout Policy Abstract As the number of ban failures increases, the set of assets available for acquisition by the surviving bans enlarges but the total amount of available liquidity within the surviving bans falls. This results in cash-in-the-maret ricing for liquidation of baning assets. At a sufficiently large number of ban failures, and in turn, at a sufficiently low level of asset rices, there are too many bans to liquidate and inefficient users of assets who are liquidity-endowed may end u owning the liquidated assets. In order to avoid this allocation inefficiency, it may be ex-ost otimal for the regulator to bail out some failed bans. Ex ante, this gives bans an incentive to herd by investing in correlated assets, thereby maing aggregate baning crises more liely. These effects are robust to allowing the surviving bans to issue equity and allowing the regulator to rice-discriminate against outsiders in the maret for ban sales. J.E.L. Classification: G21, G28, G38, E58, D62. Keywords: Ban regulation, Systemic ris, Baning crises, Time inconsistency, Too many to fail, Herding 1

3 1 Introduction In this aer, we develo a theory of otimal bailout olicy during baning crises in a framewor wherein asset liquidations can result in allocation inefficiencies. We also study the effect of bailout olicies on ex-ante incentives of bans. The wide-sread belief that rescuing troubled bans can create moral hazard and can give bans incentives to tae excessive ris dates bac to Bagehot (1873): Any aid to a resent bad ban is the surest mode of reventing the establishment of a good ban. However, emirical evidence suggests that regulatory actions taen in resonse to baning roblems vary significantly. In many eisodes, actions taen by regulators aear to deend on whether the roblems arise from idiosyncratic reasons secific to articular institutions or from aggregate reasons with otential threats to the whole system, as documented in Santomero and Hoffman (1999) and Kasa and Siegel (2003). Hoggarth, Reidhill and Sinclair (2004) also study resolution olicies adoted in 33 baning crises over the world during They document that when faced with individual ban failures authorities have usually sought a rivate-sector resolution where the losses have been assed onto existing shareholders, managers and sometimes uninsured creditors, but not to taxayers. However, government involvement has been an imortant feature of the resolution rocess during systemic crises: At early stages, liquidity suort from central bans and blanet government guarantees have been granted, usually at a cost to the budget; ban liquidations have been very rare and creditors have rarely made losses. We argue in this aer that this difference in regulatory actions arises from the fact that resolution otions oen for an isolated failure of a single institution are different from those available when facing a systemic failure. When only a few bans fail, these bans can be acquired by the surviving bans. However, regulators cannot commit not to intervene when the crisis is systemic. In articular, for a large number of failures, the liquidity of surviving bans enables them to acquire all failed bans only at fire-sale rices. 1 The resulting cashin-the-maret ricing, as in Allen and Gale (1994, 1998), maes it more liely that investors outside the baning sector, who are liquidity-endowed but otentially not the most efficient users of these assets, will end u urchasing some failed bans assets. Thus, when the baning crisis is systemic in nature, there are too many (bans) to liquidate and bailing out some of the failed bans may be otimal in order to avoid allocation inefficiencies. However, this ex-ost otimal bailout olicy induces bans to herd ex ante in order to increase the lielihood of being bailed out. For examle, they may lend to similar industries or bet on common riss such as interest and mortgage rates. This in turn increases 1 This effect 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. 2

4 the ex-ante lielihood of exeriencing systemic baning crises. The regulator s roblem is thus one of time-inconsistency. Its ex-ost otimal bailout olicy is not ex-ante otimal. Or said differently, the ex-ante otimal olicy would involve not rescuing bans in systemic crises, but this is time-inconsistent. We formalize these ideas in a framewor wherein the ex-ante and the ex-ost otimal bailout olicies and the cash-in-the-maret ricing are endogenously derived. We consider a two-eriod model with n bans, a regulator, and outside investors who could urchase baning assets were they to be liquidated. Examining a setting with an arbitrary number of bans enables us to exlore the richness of the cash-in-the-maret rice function. Each ban invests either in a common industry or in a ban-secific industry. This decision affects the correlation of ban returns and in turn the lielihood that bans fail together. The regulator designs closure and bailout olicies in order to maximize the total outut generated by the baning sector net of any costs associated with closures and bailouts. If failed bans are closed, then their assets are sold to surviving bans and outsiders at maret-clearing rices. Instead, if failed bans are bailed out, then their owners are allowed to continue oerating the bans. For simlicity, we assume that deosits are fully insured (at least in the first eriod). The immediacy of funds required for deosit insurance, net of the roceeds received from liquidating the failed bans, entails fiscal costs for the regulator. The regulatory olicies are rationally anticiated by bans and deositors. Three central assumtions drive our results: (i) bans have access to limited liquidity in articular, we assume that surviving bans only have their first-eriod rofits to acquire the failed bans assets (we relax this assumtion later and allow ledgeability of future cash flows), (ii) bans are more efficient users of baning assets than outsiders as long as ban owners tae good rojects, 2 and (iii) there is a ossibility of moral hazard in that ban owners derive rivate benefits from bad rojects; hence, bans tae good rojects only if ban owners retain a large enough share in ban rofits. If the return from the first-eriod investment of the ban is low, then the ban is in default. The surviving bans, if any, use their first-eriod rofits to urchase failed bans assets. U to a critical number of ban failures, liquidity with the surviving bans is enough to urchase all the baning assets at their fundamental rice: surviving bans comete with each other and their surlus is eroded to zero. Beyond this critical number of failures, additional assets cannot be absorbed by the available liquidity of surviving bans at the fundamental rice. Thus, the maret-clearing rice declines with each additional failure. Under assumtion (ii), outsiders are not as efficient users of baning assets as the surviving bans. Hence, they do 2 James (1991) studies the losses from ban failures in the United States during the eriod 1985 through mid-year 1988, and documents that there is significant going concern value that is reserved if the failed ban is sold to another ban (a live ban transaction) but is lost if the failed ban is liquidated by the Federal Deosit Insurance Cororation (FDIC). 3

5 not enter the maret unless rices fall sufficiently. In other words, there is limited maret articiation. If the rice declines sufficiently, then the liquidity-endowed outsiders enter the maret and urchase some of the assets. This gives rise to an allocation inefficiency. The regulator decides whether to allow a rivate-sector resolution, that is, to let the surviving bans and/or outsiders urchase failed bans assets, or to intervene in the form of bailing out some or all of the failed bans. In a bailout, a failed ban is not liquidated and its existing owners are allowed to continue oerating the ban. Bailouts are thus associated with an oortunity cost for the regulator: the regulator incurs a higher fiscal cost to ay off the romised deosits since no roceeds are collected through asset sales. Since bailouts are costly, the regulator does not intervene as long as failed bans are sold to the surviving bans. However, when asset rices decline sufficiently, it is otimal for the regulator to revent sales to outsiders by bailing out bans until the fiscal cost of a bailout exceeds the misallocation cost from liquidating the marginal ban to outsiders. With limited outsider funds, when the number of failures is sufficiently large, even the articiation of outsiders in the maret for asset sales is not enough to sustain the rice at the threshold value of outsiders and there is a further decline in the rice as the number of failures increase. In this range, the entire liquidity in the maret, that is, all funds with the surviving bans and the outsiders, are collected through the sale of failed bans. If the regulator decides to bail out a ban in these cases, the roceeds from the asset sale are not affected and bailouts do not entail any additional fiscal costs. Thus, the regulator bails out more failed bans and rices are sustained at the threshold level for outsiders. Crucially, the states where the number of failures is high are always associated with welfare losses either fiscal costs through bailouts or misallocation costs through liquidations to outsiders. Ex ante, the regulator wishes to imlement a low correlation between bans investments in order to minimize the lielihood that many bans fail, and simultaneously imlement closure olicies that are ex-ost otimal. The regulator can imlement such a welfare-maximizing outcome only if it can commit to sufficiently diluting the share of ban owners when they are bailed out. By so doing, the regulator can mae bailout subsidies small enough that bans have incentives to secialize in order to cature the surlus from buying assets at cash-inthe-maret rices. However, assumtion (iii) imlies that such a dilution may not always be feasible. If the moral hazard due to rivate benefits is sufficiently high, then excessive dilution of a ban s equity leads ban owners to choose bad rojects and this generates continuation values that are worse than liquidation values. In this case, the only credible mechanism through which the regulator can imlement low correlation is committing to liquidate a sufficiently large number of failed bans. In general, this is ex-ost inefficient and thus lacs commitment. In a first extension, we allow bans to issue equity against their future rofits. Equity 4

6 issuance has a ositive effect on rices of failed bans assets. This imroves welfare since assets are urchased by surviving bans over a larger range of ban failures. Also, higher rices increase roceeds from asset sales and this alleviates the fiscal cost for the regulator from roviding deosit insurance funds. Imortantly, the rice for shares of surviving bans follows an interesting attern. When the number of failures is large, cash-in-the-maret ricing results in the rice of failed bans assets falling below the threshold value of outsiders. Since urchasing failed bans assets at such rices becomes rofitable for outsiders, in equilibrium they must be comensated for urchasing shares in surviving bans. As a result, share rice of surviving bans also falls below their fundamental value. Thus, limited funds within the whole system and the resulting cash-in-the-maret ricing affects not only the rice of failed bans assets but also the rice of shares of surviving bans. In a second extension, we chec the robustness of our results by allowing for unlimited outsider funds. In this case, when the number of failures is sufficiently large, the articiation of outsiders in the maret for asset sales is sufficient to sustain the rice at the threshold level for outsiders. Thus, with unlimited outsider funds, we observe fewer bailouts. This, in turn, gives stronger incentives for bans to choose the low correlation. Also, with unlimited outsider funds, surviving bans do not have to comensate outsiders for the share urchase and the share rice of surviving bans stays at its fundamental value. Finally, in a third extension, we investigate the welfare imlications of allowing the regulator to rice-discriminate against outsiders in the asset sales (or equivalently, to give liquidity to the surviving bans). To mitigate the misallocation costs during systemic crises, the regulator may allow only the surviving bans to buy the failed bans assets, ossibly at rices that are lower than what the outsiders would be willing to ay. However, the regulator uses the roceeds from asset sales to cover some of the (immediate) funds needed for deosit insurance. Thus, while rice discrimination revents welfare losses from misallocation of assets, it exacerbates the fiscal burden during aggregate crises. It may thus be otimal for the regulator to let some assets be liquidated to outsiders. 3 Our aer is related to the baning literature that has focused on otimal ban closure olicies. Mailath and Mester (1994) and Freixas (1999) discuss the time-inconsistency of closure olicy in a single-ban model. Penati and Protoaadais (1988) assume that the regulator rovides insurance to uninsured deositors when the number of baning failures is large, and illustrate that this leads bans to invest inefficiently in common marets so as to attract deosits at a lower cost. Mitchell (1997) considers an argument along the lines of the signal-jamming model of Rajan (1994) to show that if the regulator bails out bans when 3 Santomero and Hoffman (1998) document that during the resolution of Savings and Loans crisis, the Federal Savings and Loan Insurance Cororation (FSLIC) initially tried to merge failed thrifts with a stronger thrift, but these attemts failed due to bidder scarcity. This eventually led FSLIC to oen the bidding for the failed thrifts to non-thrifts. 5

7 they fail together, then bans coordinate on disclosing their losses and delay classifying bad loans by rolling them over. Perotti and Suarez (2002) consider a dynamic model where selling failed bans to surviving bans (reducing cometition) increases the charter-value of surviving bans and gives bans ex-ante incentives to stay solvent. However, in contrast to our model, their aer does not examine the effect of closure olicies on inter-ban correlation. Diamond and Rajan (2003) show that a ban failure can cause aggregate liquidity shortages and regulatory intervention may be otimal. The focus of their aer is on demonstrating the liquidity channel of contagion and the difficulty of resolving it ex ost, but not on its imlications for the ex-ante investment choices of bans. Finally, some of the ideas resented in this aer are motivated by the analysis in Acharya (2001) and Acharya and Yorulmazer (2004). In our oinion, the strongest differentiating oint of the current aer is its modelling generality in allowing for n bans (rather than a single ban or two bans as in the cited aers) and endogenously deriving the cash-in-the-maret ricing as well as the ex-ante and the ex-ost otimal bailout olicies. This lends the model an element of richness that can be exloited to address other issues of interest such as the rivate versus the social otimum levels of liquidity buffers of bans, and an in-deth welfare analysis of various otions available to regulators to resolve and restructure failed bans. Acharya and Yorulmazer (2004), which considers a two-ban version of this model, also comare too-big-to-fail to too-many-to-fail and show that herding incentives are stronger for small bans than for large bans. The remainder of the aer is structured as follows. Section 2 and Section 3 resent the model and the analysis. Section 4 considers extensions of the benchmar model. Section 5 concludes. Proofs not contained in the text are contained in the Aendix. 2 Model The benchmar model is outlined in Figure 1. We consider an economy with three dates t = 0, 1, 2, n bans, ban owners, deositors, outside investors, and a regulator. Each ban can borrow from a continuum of deositors of measure 1. Ban owners, as well as deositors, are ris-neutral, and obtain a time-additive utility u t where u t is the exected wealth at time t. Deositors receive a unit of endowment at t = 0 and t = 1. Deositors also have access to a reservation investment oortunity that gives them a utility of 1 er unit of investment. In each eriod, that is at date t = 0 and t = 1, deositors choose to invest their good in this reservation oortunity or in their ban. Deosits tae the form of a simle debt contract with maturity of one eriod. In articular, the romised deosit rate is not contingent on investment decisions of the ban or on realized returns. Finally, the disersed nature of deositors is assumed to lead to a collective-action roblem, resulting in a run on a ban that fails to ay the romised return to its deositors. 6

8 In other words, the deosit contract is hard and cannot be renegotiated. 4 In order to ee the model simle and yet cature the fact that there are limits to equity financing due to associated costs (for examle, due to asymmetric information as in Myers and Majluf, 1984), we do not consider any ban financing other than deosits. We relax this assumtion artly in an extension in Section 4.1. Bans require one unit of wealth to invest in a risy technology. The risy technology is to be thought of as a ortfolio of loans to firms in the cororate sector. The erformance of the cororate sector determines its random outut at date t + 1. We assume that all firms in the sector can either reay fully the borrowed ban loans or they default on these loans. In case of a default, we assume for simlicity that there is no reayment. Suose R t is the romised return on a ban loan at time t. We denote the random reayment on this loan as R t, R t {0, R t }. The robability that the return from these loans is high in eriod t is α t : { Rt with robability α t, R t = (1) 0 with robability 1 α t. We assume that the returns in the two eriods are indeendent but allow the robability, as well as the level of the high return, to be different in the two eriods. This hels isolate their effect on our results. There is a otential for moral hazard at the level of an individual ban. If the ban chooses a bad roject, then when the return is high, it cannot generate R t but only (R t ) and its owners enjoy a non-ecuniary benefit of B <. Therefore, for the ban owners to choose the good roject, aroriate incentives have to be rovided by giving them a minimum share of the ban s rofits. We denote the share of ban owners as θ. If r t is the cost of borrowing deosits, then the incentive-comatibility constraint is: α t θ(r t r t ) α t θ((rt ) r t ) + B. (IC) (2) We have assumed that the ban is able to ay the romised return of r t when the investment had the high return irresective of whether the roject is good or bad. The left hand side of the (IC ) constraint is the exected rofit for the ban from the good roject when it has a share of θ of the rofit. On the right hand side, we have the exected rofit from the bad roject when ban owners have a share of θ, lus the non-ecuniary benefit of choosing the bad roject. Using this constraint, we can show that ban owners need a minimum share of θ = B to choose the good roject.5 We assume that at t = 0, the entire share of the ban 4 For a micro foundation for why ban deosits would be designed as a hard contract, see Calomiris and Kahn (1991) and Diamond and Rajan (2001). 5 See Hart and Moore (1994) and Holmstrom and Tirole (1998) for models with similar incentivecomatibility constraints. 7

9 rofits belongs to the ban owners, and therefore, there is no moral hazard to start with. In addition to bans and deositors, there are ris-neutral outside investors who have limited funds amounting to w (an assumtion we relax in Section 4) to urchase baning assets were these assets to be sold. Outsiders do not have the sills to generate the full value from baning assets. In articular, outsiders are inefficient users of baning assets relative to the ban owners, rovided ban owners oerate good rojects. This can be considered a metahor for some form of exertise or learning-by-doing effect for maing and administering loans. It is also a simle way of introducing barriers to entry in the baning sector. To cature this formally, we assume that outsiders cannot generate R t in the high state but only (R t ). Thus, when the baning assets are sold to outsiders, there may be a social welfare loss due to a misallocation of the assets. We also assume that > so that outsiders can generate more than what the bans can generate from bad rojects. The notion that outsiders may not be able to use the baning assets as efficiently as the existing ban owners is ain to the notion of asset-secificity, first introduced in the cororatefinance literature by Williamson (1988) and Shleifer and Vishny (1992). In summary, this literature suggests that firms, whose assets tend to be secific, that is, 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 get simultaneously into financial or economic distress. 6 In the evidence of such secificity for bans and financial institutions, James (1991) shows that the liquidation value of a ban is tyically lower than its maret value as an ongoing concern. In articular, his emirical analysis of the determinants of the losses from ban failures reveals a significant difference in the value of assets that are liquidated and similar assets that are assumed by acquiring bans. Finally, there is a regulator who emloys olicy instruments such as bailouts and sale of failed bans assets through auctions with the objective of maximizing the total outut generated by the baning sector net of any costs associated with these olicy otions. These olicies are assumed to be rationally anticiated by bans and deositors. Below we describe these olicies informally. The formal descrition follows in the model analysis. We assume that deosits are fully insured in the first eriod. The rovision of immediate funds to ay off failed deosits, net of any roceeds from the sale of failed bans assets, entails fiscal costs for the regulator (assumed to be exogenous to the model). The fiscal costs of roviding funds to the baning sector with immediacy can be lined to a variety of sources, 6 There is strong emirical suort for this idea in the cororate-finance literature, as shown, for examle, by Pulvino (1998) for the airline industry, and by Acharya, Bharath, and Srinivasan (2003) for the entire universe of defaulted firms in the US over the eriod 1981 to 1999 (see also Berger, Ofe, and Swary (1996) and Stromberg (2000)). 8

10 most notably, (i) distortionary effects of tax increases required to fund deosit insurance and bail outs; and, (ii) the liely effect of huge government deficits on the country s exchange rate, manifested in the fact that baning crises and currency crises have often occurred as twins in many countries (esecially, in emerging maret countries). Ultimately, the fiscal cost we have in mind is one of immediacy: Government exenditures and inflows during the regular course of events are smooth, relative to the otentially raid growth of off-balance-sheet contingent liabilities such as deosit-insurance funds, costs of ban bailouts, etc. 7 Note that the second eriod is the last eriod in our model and there is no further investment oortunity. As a result, our analysis is not affected by whether deosits are insured for the second investment or not. If the ban return from the first-eriod investment is high, then the ban oerates one more eriod and maes the second-eriod investment. If the return is low, then there is a run on the ban. We assume that the surviving bans (if any) use their first-eriod rofits to urchase failed bans assets. The regulator decides whether to bail out some or all of the failed bans or to let the surviving bans (if any) and/or outsiders urchase failed bans assets. When a ban is bailed out, the regulator may dilute the equity share of ban owners. Proceeds from the sale of failed bans reduce the costs of roviding deosit insurance. Hence, bailouts are associated with an oortunity cost for the regulator. These costs are also a art of the regulator s objective function. Deending on the first eriod returns, some of the bans (say out of n) fail. Since bans are identical at t = 0, we denote the ossible states at t = 1 with, the number of ban failures. 2.1 Correlation of ban returns A crucial asect of our model is the correlation of ban returns. At t = 0, bans borrow deosits and then choose the comosition of loans in their resective ortfolios. This choice determines the level of correlation between the returns from their resective investments. We refer to this correlation as inter-ban correlation. We suose that there is a common industry that all bans can access and there are n other industries, one for each ban, such that only ban i can access industry i (region, set of 7 See, for examle, the discussion on fiscal costs associated with baning collases and bailouts in Calomiris (1998). Hoggarth, Reis and Saorta (2001) find that the cumulative outut losses have amounted to a whoing 15-20% annual GDP in the baning crises of the ast 25 years. Cario and Klingebiel (1996) argue that the bailout of the thrift industry cost $180 billion (3.2% of GDP) in the US in the late 1980s. They also document that the estimated cost of bailouts were 16.8% for Sain, 6.4% for Sweden and 8% for Finland. Honohan and Klingebiel (2000) find that countries sent 12.8% of their GDP to clean u their baning systems whereas Claessens, Djanov and Klingebiel (1999) set the cost at 15-50% of GDP. 9

11 customers, etc.). To focus on the effect of inter-ban correlation, we assume that the returns from the common and the n secific industries have the same return structure and they are indeendent. That is, the return from industry i, denoted by R it, is given as: { Rit with robability α t R it = (3) 0 with robability 1 α t where i {1, 2,..., n} denotes ban-secific industries and i = c denotes the common industry. Each ban chooses whether to invest a unit of wealth in the ban-secific industry or in the common industry, that is, x i {0, 1}. 8 The vector of choices (x 1,..., x n ) determines the joint robability distribution of ban returns. If bans in equilibrium choose to lend to firms in the common industry, then they are assumed to be erfectly correlated, that is, the correlation of bans returns is ρ = 1. However, if they choose different industries, then their returns are indeendent, that is, ρ = 0. Note that the individual robability of each ban succeeding or failing (α t and 1 α t at time t, resectively) is indeendent of the inter-ban correlation. We focus on symmetric ure strategy Nash equilibrium and hence denote x i s simly as x. This gives us the following robabilities for the number of ban failures at t = 1. When x = 0, bans invest in indeendent industries and we obtain a Binomial distribution for the number of ban failures: Pr() = C(n, ) α n 0 (1 α 0 ) for {0, 1,...n}, (4) where C(n, ) is the number of combinations of objects from a total of n. When x = 1, bans invest in the common industry and we obtain: α 0 for = 0 Pr() = 0 for {1,..., n 1}. (5) 1 α 0 for = n 3 Analysis We analyze the model roceeding bacwards from the second eriod to the first eriod. The romised deosit rate at t = 0, 1 is denoted by r t. We assume throughout that R t > r t for t = 0, 1. 8 The case where bans invest both in the common asset as well as in the ban-secific asset gives rise to qualitatively similar results, but is technically quite involved. The two-ban version of this more realistic modelling of ban investments is contained in the Addendum to Acharya and Yorulmazer (2004). 10

12 The surviving bans oerate for another eriod at t = 1. Since, the returns from each eriod s investments are assumed to be indeendent, the robability of having the high return for each ban is equal to α 1. As this is the last eriod there is no further investment oortunity. The exected ayoff to the ban from its second-eriod investment, E(π 2 ), is thus E(π 2 ) = α 1 R 1 r 1. (6) Note that this ayoff is indeendent of inter-ban correlation. For a ban to continue oerating for another eriod, it needs to ay its old deositors r 0 and it needs an additional one unit of wealth for the second investment. A failed ban cannot generate the needed funds, (1 + r 0 ), from its deositors at t = 1: Its deositors are endowed with only one unit of wealth at t = 1. Anticiating this, deositors run on the ban and the ban fails. An imortant ossibility is that the surviving bans and/or outsiders may urchase the assets of failed bans. Next, we investigate sales of failed bans assets and the resulting asset rices. 3.1 Asset sales and liquidation values In examining the urchase of failed bans assets, several interesting issues arise. First, surviving bans and outsiders may comete with each other if there are enough resources with them to acquire all failed bans assets. Second, unless the game for asset acquisition is secified with reasonable restrictions, an abundance of equilibria arises. Third, surviving bans in fact may not have enough resources to acquire all failed bans. Hence, the regulator may find it otimal to intervene to sell as many failed bans assets as ossible to surviving bans, and bail out some others. First we examine asset sales and rices without bailouts. In the next section, we investigate the regulator s bailout olicy and its effect on rices. To ee the analysis tractable and, at the same time, reasonable, we mae the following assumtions: (i) The regulator ools all failed bans assets and auctions these assets to the surviving bans and the outsiders. Assets of a failed ban can be acquired artially, and when a ortion of these assets are acquired, the urchasing ban can also access the same ortion ( branches ) of the failed ban s deositors. In essence, the ban can be sold in arts. This assumtion of artial ban sales is motivated by the literature on share auctions (Wilson, 1979) and simlifies the analysis substantially. When only a art of the total failed bans assets are sold and the remaining are bailed out, the assets to be sold are chosen randomly. (ii) Denoting the surviving bans as i {1, 2,..., (n )} and the outsiders as i = 0, each surviving ban and outsiders submit a schedule y i () for the amount of assets they are 11

13 willing to urchase as a function of the rice at which a unit of the baning asset (inclusive of associated deosits) is being auctioned, where y i () 0,. (iii) We assume that surviving bans cannot raise additional financing from the marets, an assumtion we later relax in Section 4.1. Hence, the resources available with each surviving ban for urchasing failed bans assets, denoted by l, equal the first-eriod rofits, that is, l = (R 0 r 0 ). (iv) The regulator cannot rice-discriminate in the auction. In Section 4.3, we relax this assumtion and show that even when the regulator has the otion to rice-discriminate and revent outsiders from articiating in the auction, he may rationally choose not to do so. (v) The regulator determines the auction rice so as to maximize the exected outut of the baning sector, subject to the natural constraint that ortions allocated to surviving bans and outsiders add u at most to the number of failed bans, that is, n i=0 y i(). Given the allocation inefficiency of selling assets to outsiders, it turns out that if the surviving bans and the outsiders ay the same rice for the failed bans assets, the regulator allocates the maximum amount he can to the surviving bans. (vi) We focus on the symmetric outcome where all surviving bans submit the same schedule, that is, y i () = y() for all i {1, 2,..., (n )}. First, we derive the demand schedule for surviving bans. Let = α 1 (R 1 r 1 ) = E(π 2 ), which is the exected rofit for a surviving ban from the risy asset in the second eriod. Note that the exected rofits of a surviving ban from the asset urchase can be calculated as: y(). (7) The surviving ban wishes to maximize these rofits subject to the resource constraint y() l. (8) Hence, for <, surviving bans are willing to urchase the maximum amount of failed bans assets using their resources. Thus, otimal demand schedule for surviving bans is y() = l. (9) For >, the demand is y() = 0, and for =, y() is indeterminate. In words, as long as urchasing ban assets is rofitable, a surviving ban wishes to use u all its resources to urchase failed bans assets. For a formal roof of this result that taes into account the correlation structure of assets, see the Aendix. 12

14 We can derive the demand schedule for outsiders in a similar way. Note that, outsiders can generate only (R 1 ) in the high state. Let = α 1 ((R 1 ) r 1 ) = α 1, the exected rofit for the outsiders from the risy asset in the second eriod. For <, outsiders are willing to suly all their funds for the asset urchase. Thus, otimal demand schedule is y 0 () = w. (10) For >, the demand is y 0 () = 0, and for =, y 0 () is indeterminate. Thus, for >, there is limited articiation in the maret for baning assets. Next, we analyze how the regulator otimally allocates the failed bans assets and the rice function that results. We now that in the absence of financial constraints, the efficient outcome is to sell the failed bans assets to surviving bans. However, the surviving bans may not be able to ay the threshold rice of for all failed bans assets. If rices fall further, these assets become rofitable for the outsiders and they articiate in the auction. The regulator cannot set > since in this case we have y() = y 0 () = 0. If, and the number of failed bans is sufficiently small, the surviving bans have enough funds to ay the full rice for all the failed bans assets. More secifically, for, where ( ) nl = floor, (11) l + and floor(z) is the largest integer smaller than or equal to z, the regulator sets the auction rice at =. At this rice, surviving bans are indifferent between any quantity of assets urchased. Hence, the regulator can allocate a share y( ) = to each surviving ban. (n ) For moderate values of, surviving bans cannot ay the full rice for all failed bans assets but can still ay at least the threshold value of, below which outsiders have a ositive demand. Formally, for { + 1,..., }, where = floor ( nl l + ), the regulator sets the rice at = (12) ( ) (n )l, and again, all baning assets are acquired by the surviving bans. Note that, in this region, surviving bans use all available funds and the rice falls as the number of failures increases. This effect is basically the cash-in-themaret ricing as in Allen and Gale (1994, 1998) and is also 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. 13

15 For >, the surviving bans cannot ay the threshold rice of for all failed bans assets and rofitable otions emerge for outsiders. At this oint, outsiders have a ositive demand and are willing to suly their funds for the asset urchase. With the injection of outsider funds, rices can be sustained at until some critical number of failures. However, for >, even the injection of outsiders funds is not enough to sustain the rice at. Formally, for { + 1,..., }, where ( ) nl + w = floor, l + the regulator sets the rice at. At this rice, outsiders are indifferent between any quantity ( ) l of assets urchased. Hence, the regulator can allocate a share of y() = to each surviving ( ) ban and the rest, y 0 () = (n )l, to outsiders. And beyond this oint, that is, >, the rice is again strictly decreasing in and is given by (n ) l + w () =, ( and y( l ) = and y 0 ( ) = ) ( w ). The resulting rice function is bacward-bending or downward-sloing in the number of failed bans in two searate regions. In the first downward-sloing region, outsiders have not yet entered the maret ( { + 1,..., }) and there is cash-in-the-maret ricing given the limited liquidity of surviving bans. In the second downward-sloing region, even the liquidity of outsiders is not enough to sustain the rice at, their highest valuation of baning assets. Thus, there is cash-in-the-maret ricing in this region given the limited liquidity of the entire set of maret layers bidding for failed assets, that is, of surviving bans as well as outsiders. This rice function is formally stated in the following roosition and is illustrated in Figure 2. Proosition 1 In the absence of bailouts, the rice of failed bans assets as a function of the number of failed bans is as follows: for () = (n )l for { + 1,..., } for { + 1,..., } (n )l+w for > (13) (14). (15) 14

16 3.2 Bailouts To summarize the result from the revious analysis, when the number of ban failures is sufficiently small,, all failed bans assets are resolved through a urchase by surviving bans. Since this allocation entails no welfare losses, the regulator does not have any incentive to intervene ex ost. In contrast, if >, then some of these assets are urchased by outsiders who are not the most efficient users. Hence, the regulator comares the misallocation cost resulting from asset sales to outsiders with the cost of bailing out failed bans. Since the misallocation cost is constant at (α 1 ) er unit of failed bans assets, the regulator bails out failed bans as long as the marginal cost of a bailout is less than this misallocation cost. Note that for a failed ban to continue oerating, it needs a total of (r 0 + 1) units. Since available deosits for a ban amount to only one unit (the t = 1 endowment of its deositors), the ban cannot oerate unless the regulator injects r 0 at t = 1. Under our assumtion of full deosit insurance, the regulator does inject r 0 at t = 1, so that a bailout is equivalent to the regulator granting ermission to a failed ban s owners to oerate one more eriod. In order to analyze the regulator s decision to bail out or liquidate failed bans, we mae the following assumtions: (i) The regulator incurs a fiscal cost of f(c) when it injects c units of funds into the baning sector. We assume this cost function is strictly increasing and convex (ossibly linear): f > 0 and f 0. We also assume that f (c) > 1, for all c, which has the natural interretation that for every additional unit of funds needed, the cost of these funds increases by more than one. We do not model this cost for which we have in mind fiscal and oortunity costs to the regulator from roviding funds with immediacy to the baning sector (see footnote 7). (ii) If the regulator decides not to bail out a failed ban, the existing deositors are aid bac r 0 through deosit insurance and the failed ban s assets are sold at the maret-clearing rice. Thus, when the regulator bails out b of the failed bans, the fiscal cost incurred is f(r 0 ( b) ( b)) as roceeds from sale of the remaining ( b) bans are ( b) ( b). The crucial difference between bailouts and asset sales from an ex-ost standoint is that roceeds from asset sales lower the fiscal cost from immediate rovision of deosit insurance, whereas bailouts roduce no such roceeds. In other words, bailouts entail an oortunity cost to the regulator in fiscal terms. (iii) The regulator can tae an equity share in the bailed out ban(s). Let β be the share the regulator taes in a bailed out ban. If the bailed out ban has a high return from the second investment (which has a robability of α 1 ), then the regulator gets bac β(r 1 r 1 ) at t = 2. Ex ost, such dilution of a bailed-out ban s equity is merely a transfer from the ban owners to the regulator. However, as argued before, if the regulator taes a share greater 15

17 than (1 θ), then the ban owners are left with a share of less than θ, the critical share below which the ban chooses the bad roject. Since sales to outsiders generate a higher ayoff comared to that from a bailed-out ban that chooses a bad roject ( > ), the regulator never taes a share greater than (1 θ). As the value of this stae will be realized in the future, it is assumed to not affect the cost of roviding deosit insurance with immediacy. We characterize the otimal bailout olicy under these assumtions. The regulator s objective is to maximize the total exected outut of the baning sector net of any bailout or liquidation costs. As argued before, the regulator never intervenes when. When { + 1,..., }, the rice for failed bans assets is and the marginal cost of bailing out the b th ban is g(, b) = f(r 0 ( b)) f(r 0 ( b + 1)). (16) This marginal cost is (at least wealy) increasing in b. 9 Hence, there is a maximum number of bans, denoted by b(), u to which the bailout costs are smaller than misallocation costs. Formally, b() satisfies the following conditions: g(, b) α 1 < g(, b + 1). (17) The maximum number of bans that can be acquired by the surviving bans is (n )y(), where y() = l. Thus, the regulator bails out b () = min { b(), (n )y() } bans. Finally, when >, all the funds of the surviving bans and the outsiders, a total of (n )l + w, are collected through the sale of the failed bans assets. Hence, if the regulator decides to bail out a ban, the roceeds from the asset sale are not affected and bailouts do not incur any (oortunity) fiscal cost. Thus, the regulator first bails out b() failed bans where b() is the maximum number of bailouts such that the regulator can collect all the available liquidity in the system, (n )l + w, by selling the remaining b() bans. Note that when the regulator bails out b() failed bans, the rice for the remaining b() failed bans assets reaches. Formally, we have (n )l + w = ( b()), (18) which gives us b() = 9 In articular, (n )l + w. (19) g b = f (r 0 ( b)) f (r 0 ( b + 1)). Note that, ( r 0 ( b) ) > ( r 0 ( b + 1) ), and since f 0, we have g b 0. 16

18 From this oint on, the outsider funds are sufficient to sustain the rice for asset sales of remaining b() bans at the level and an additional bailout decreases the roceeds from asset sales by. Hence, the regulator s decision to bail out additional bans is similar to that for the case where { + 1,..., }. In articular, the regulator bails out a total of b() + b() bans, until the marginal bailout cost starts exceeding the misallocation cost. Thus, b() is given by the condition: h(, b) α 1 < h(, b + 1), (20) where h(, b) is defined as: h(, b) = f(r 0 (( b()) b)) f(r 0 (( b()) b + 1)). (21) Note that the marginal bailout cost h(, b) reflects the fact that with b() + b bailouts, the regulator receives roceeds from asset sales amounting to ( b()) b). To summarize, the regulator s otimal bailout olicy b () is such that min { b(), (n )y() } for { + 1,..., } b () = } min { b() + b(), (n )y() for >. (22) Bans are chosen randomly between the three otions of being sold to surviving bans, bailed out, or liquidated to outsiders, and the regulator taes a share of β in all bailed-out bans. We state this closure/bailout olicy formally in a roosition: Proosition 2 Under the ex-ost otimal bailout olicy, (i) When, surviving bans urchase all failed bans assets and the regulator does not intervene. (ii) When >, the regulator bails out b () of the failed bans, where b () is defined by conditions (17), (20) and (22). The bans to be bailed out are chosen randomly with equal robability. The otimal bailout olicy has the intuitive roerty that in states with a large number of ban failures, there are too many (bans) to liquidate and the regulator is forced to bail out some of the failed bans. In articular, irresective of the fiscal cost function, it is always otimal to bail out u to b() bans in the region of high ban failures, >, or, in other words, in the region of cash-in-the-maret ricing due to limited liquidity of the 17

19 entire maret for baning assets. Bailouts in this region entail no oortunity costs for the regulator but hel avoid misallocation costs. 10 With bailouts, the rice never falls below the reservation rice of outsiders,. The resulting rice function is illustrated in Figure 3. It differs from Figure 2, the no bail-out case, along an imortant dimension. First, there is only one downward-sloing region in Figure 3 comared to Figure 2. In effect, the rice function is the same as one would obtain were the outsiders to have unlimited funds (see Section 4.2 and Figure 9). Note that the bailout olicy b () is not always monotone increasing in over the entire range. This is because the marginal cost of bailout, g(, b), is strictly increasing in if the fiscal cost function is strictly convex: with more ban failures, the fiscal cost of deosit insurance is higher, and given the convexity of this cost function, incurring the oortunity cost of bailouts becomes more severe. In other words, b() is decreasing in when the cost function is convex. A similar argument alies to the marginal cost h(, b) and the b(). The general behavior of the bailout olicy b () is illustrated in Figure 4. Nevertheless, note that b () has the roerty that bailouts occur only when ban failures are sufficiently large in number ( > ), and, if large enough to reach the second cash-in-the-maret rice region ( > ), then the oortunity cost of bailouts becomes zero (u to bailouts of b() bans). In order to derive and exloit a closed-form exression for the otimal bailout olicy, it is useful to consider a linear cost function: f(c) = F c, F > 0. One analytical advantage of the linear cost function is that the bailout olicy b () is now always monotone increasing. Secifically, we obtain that b () = (n )l, for >, if the cost arameter F is sufficiently small, and b () = b(), for only >, otherwise. In words, when the fiscal cost arameter is small, outsiders are et entirely out of the maret for bans sales: as many failed bans as can be acquired by the surviving bans are sold and the remaining (if any) are bailed out. When the fiscal cost arameter is large, outsiders articiate in asset sales until the number of ban failures is high enough that the second cash-in-the-maret region is reached. At this oint, all incremental bans that fail are bailed out. The resulting bailout olicy is given by the following corollary and is illustrated in Figure 5. Corollary 1 With a linear fiscal cost function, the regulator bails out b () of the failed bans when >, where b () is defined as follows: 10 In our model, the level of asset rices have no mar-to-maret effect on collateral values and future liquidity of surviving bans. In a richer setting with such a collateral channel for the amlification of fire-sale rices, bailouts would be otimal ex ost not just to avoid the allocation inefficiency but also to revent reciitous declines in the maret rices of baning assets. 18

20 (i) When F α 1 b () =, (ii) When F > α 1 (n )l,. (23) 0 for { + 1,..., } b () =. (24) b() for > where b() is given by equation (19). 3.3 Choice of inter-ban correlation In this section, we analyze bans ex-ante choice of correlation. First, we derive bans exected rofits when they invest in idiosyncratic industries and the common industry, that is, when the inter-ban correlation ρ equals 0 and 1, resectively. We show that the level of correlation chosen by bans deends on the exected rofit bans mae from the urchase of failed bans assets when they survive and the exected subsidy they receive through bailouts when they fail. In Proosition 3, we formally state the conditions under which bans choose to invest in the common industry. In the first eriod, all bans are identical. Hence, we consider a reresentative ban. Formally, the objective of each ban is to choose the level of inter-ban correlation ρ at date 0 that maximizes E(π 1 (ρ)) + E(π 2 (ρ)), (25) where discounting has been ignored since it does not affect the results. The exected ayoff to the ban at date 0 from its first-eriod investment, E(π 1 ), is E(π 1 ) = α 0 (R 0 r 0 ), (26) which does not deend on the level of inter-ban correlation. Hence, bans only tae into account the second-eriod rofits when choosing ρ. Note that when bans invest in the same industry, if the return is low, then all bans fail together and the regulator bails out (randomly iced) b (n) of them taing an equity stae of β in the bailed-out bans. Thus, the exected rofit from the second-eriod investment is given as: E(π 2 (1)) = α 0 E(π 2 ) + (1 α 0 ) (b (n)/n) (1 β)e(π 2 ). (27) 19