a theory of failed bank resolution: technological change and political economics
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Journal of Financial Stability 9 (2013) 612– 627
Contents lists available at ScienceDirect
Journal of Financial Stability
journal homepage: www.elsevier.com/locate/jfstabil
theory of failed bank resolution: Technological change and political economics
obert DeYounga,∗, Michal Kowalikb, Jack Reidhill c
University of Kansas, 1300 Sunnyside Avenue, Lawrence, KS 66045, United StatesFederal Reserve Bank of Kansas City, 1 Memorial Drive, Kansas City, MO 64198, United StatesFederal Deposit Insurance Corporation, 550 17th Street, NW, Washington, DC 20219, United States
r t i c l e i n f o
rticle history:eceived 8 August 2011eceived in revised form 21 June 2012ccepted 14 September 2012vailable online 3 October 2012
EL classification:21
a b s t r a c t
We model the failed bank resolution process as a repeated game between a utility-maximizing govern-ment resolution authority (RA) and a profit-maximizing banking industry. Limits to resolution technologyand political/economic pressure create incentives for the RA to bail out failed complex banks; the inabil-ity of the RA to credibly commit to closing these banks creates an incentive for bank complexity. Wesolve the game in mixed strategies and find equilibrium conditions remarkably descriptive of govern-ment responses to actual and potential large bank insolvencies during the recent financial crisis. Thecentral role of the technology constraint in this model highlights a crucial determinant of failed bank
28
eywords:ank failuresailed bank resolution
resolution policy that has been overlooked in the theory literature to date; without improved resolutiontechnologies, future bank bailouts are inevitable. The effects of political pressure in this model remind usthat regulatory reform (e.g., Dodd-Frank) is only as good as the regulators that implement the reform.
© 2012 Elsevier B.V. All rights reserved.
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. Introduction
Government bailouts of large insolvent financial institutionsas one of the most critical and controversial events of the recent
nternational financial crisis. While the details of these bailoutsiffered, the underlying policy motivations were the same: to pre-ent the financial troubles at single institutions from spreading tother parts of the financial system, thus avoiding a collapse of creditarkets and disastrous macro-economic consequences. By guaran-
eeing that creditors of these institutions suffered few if any losses,olicymakers struck an implicit bargain with the financial system:reserve financial market liquidity today at the cost of increasinghe moral hazard incentives of financial market participants in theuture. In other words, policymakers traded market discipline inxchange for market liquidity.
We explore the implications of this policy tradeoff for the riskomposition of the banking industry. In our theory model, we stress
crucial determinant of policy that has received scant attention
n the previous literature: the limited set of failed bank resolu-ion technologies that can leave regulators with little choice but toail out systemically important banks. Our study is timely, as the∗ Corresponding author. Tel.: +1 785 864 1806; fax: +1 785 864 5328.E-mail addresses: [email protected] (R. DeYoung), [email protected]
M. Kowalik), [email protected] (J. Reidhill).
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572-3089/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.jfs.2012.09.003
echnology sets of bank resolution authorities—most notably theederal Deposit Insurance Corporation (FDIC)—are in the processf expanding. For example, the Wall Street Reform and Consumerrotection Act of 2010 (a.k.a. the Dodd-Frank Act) expands theDIC’s resolution authority beyond insolvent banks, and gives thegency “orderly liquidation authority” to place systemically impor-ant financial companies of all types into receivership and liquidatehem. Dodd-Frank also mandates that financial institutions per-orm more of their derivatives trading through centralized clearingouses and requires systemically important financial firms to fileliving wills” with the FDIC—changes that improve the FDIC’s abilityo accurately value the assets, and better understand the produc-ion processes, of troubled complex banking firms.
But having authority to resolve insolvent banks is not equiva-ent to actually wielding that authority. Our model also emphasizeshe likelihood that mounting political and/or economic pressureuring a financial crisis can lead even a well-armed regulator toailout systemically important banks. The public relations messagehat accompanied Dodd-Frank was clear and seemingly unequiv-cal. At bill’s signing, President Obama said “The American peopleill never again be asked to foot the bill for Wall Street’s mistakes.
here will be no more taxpayer-funded bailouts. Period.” Like most
eclarative statements, this one contains some wiggle room: rulingut “taxpayer-funded bailouts” does not rule out bailouts funded byome other third party and hence does not by itself reduce moralazard incentives. Dodd-Frank provides a resolution mechanism
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wtomade for preserving market liquidity while still imposing at leasta modicum of discipline on depositors. Kaufman and Seelig (2002)proposed a combination of quick access to insured deposit funds
R. DeYoung et al. / Journal of F
n which losses are borne by stockholders and unsecured credi-ors at the insolvent firm, with losses larger than this shared acrosshe entire banking industry. But this new structure is untested, andegulatory credibility will not be established until a firm previouslyonsidered “too big to fail” is closed and liquidated without creating
crisis in financial markets.Our model is a straightforward, repeated game between a
tility-maximizing resolution authority that chooses between clos-ng and bailing out failed banks, and an expected profit-maximizinganking industry that chooses between simple (transparent, easyo unwind) and complex (opaque, difficult to unwind) loan pro-uction processes. The regulator values resolutions that generateoth market discipline and market liquidity, but it is forced torade the former for the latter (i.e., choose a bail out) when itsesolution technology is insufficient to close a failed complexank without imposing spillover costs on the macro-economy.he key innovation in our model is the inclusion of a technologyonstraint—a realistic condition not considered in previous mod-ls of bank resolution—and tightening or loosening this constraintrovides key results. In equilibrium, insufficient resolution tech-ology, combined with short-run political or economic pressure,upport a too-complex-to-fail (TCTF) resolution policy; this inabil-ty of regulators to credibly commit to closing failed complex banksncourages continued or increased bank complexity. These condi-ions are remarkably descriptive of government responses to actualnd potential large bank insolvencies before and during the recentnancial crisis. Improvements in resolution technology have overime allowed the FDIC to close increasingly large and complexanks, but economic and political pressures during the financialrises resulted in resolution choices (e.g., allowing already TCTFanks to acquire insolvent TCTF banks) that exacerbated the gapetween bank complexity and the ability of regulators to closeailed complex banks. In the end, a deeper technological toolboxan be useless if regulators favor preserving short-run liquidity overmposing long-run discipline.
It is important to state what our model is not about. The banks inur model do not choose to be risky or safe; rather, they choose toe either complex or simple, where complexity is unrelated to therobability that a bank fails, but makes a bank difficult for regula-ors to unwind should it fail. Thus, the regulator in our model is nothoosing its strategy in order to minimize moral hazard incentiveshat make banks more prone to take pre-failure risks, but rather toeduce the post-failure complexity that makes it necessary to bailut failed banks. These distinctions set our paper apart from mostf the theoretical literature on bank failure regulation.
Because the U.S. has the longest history of deposit insurance andailed bank resolution, we couch our discussion of bank resolutionuthority in terms of the FDIC; nevertheless, our findings have clearmplications for bank failure resolution outside the U.S. The remain-er of the paper unfolds as follows. Section 2 reviews the historicalradeoff between preserving liquidity and imposing discipline inailed bank resolution policy in the U.S. Section 3 describes theechniques used by the FDIC to resolve failed banks and how thisechnology set has evolved over time, including during and afterhe financial crisis. (A substantially more detailed discussion of the
aterial in Sections 2 and 3, along with an historical appendix, arevailable in the longer working paper version of this study.) Section
presents our theory model and analyzes its main results. Section 5iscusses the implications of our analysis for bank resolution policy.
. Market liquidity versus market discipline
Commercial banks play a central role in our economy, butheir inherent fragility requires special regulatory attention. Absent e
al Stability 9 (2013) 612– 627 613
ppropriate regulation, depositors and short-term creditors mayithdraw their funds from banks experiencing declines in asset
uality, prompting reductions in economic liquidity beyond theroubled banks themselves. Bank failures can also reduce liquid-ty by disrupting borrower access to credit (e.g., Ashcraft, 2005).epending on the size and/or number of the affected banks, theseisruptions to market liquidity can be debilitating for the econ-my at large.1 Repeated banking panics in the United States duringhe nineteenth and early twentieth centuries led to the creationf the FDIC in 1934, a new federal agency with the mandate tonsure bank deposits and the power to seize and quickly resolveailed banks. Deposit insurance reduced incentives for small depos-tors to run and precipitate bank failure, and bypassing lengthyankruptcy proceedings for failed banks reduced disruptions toepositor liquidity, borrower liquidity and payments.
The potential cost of preserving liquidity in this fashion ishe creation of moral hazard incentives and the resulting loss of
arket discipline. Like all regulatory solutions to market failure,eposit insurance protections and bank resolution procedures areecond-best arrangements that result in incentive incompatibili-ies. Knowing (or suspecting) their deposits are protected from loss,nsured (and to a lesser extent, uninsured) depositors have littlencentive to monitor the financial condition of their banks, and havehe perverse incentive to make deposits at troubled banks payingbove-market interest rates. The deposit insurance put option givesanagers of troubled banks incentives to “gamble for resurrec-
ion” by paying above-market rates for deposits and investing thoseunds in risky loans. Extending deposit insurance protection to allank creditors in failed banks, or providing financial assistance toeep insolvent banks open, reduces market discipline further andxacerbates the risk-taking behaviors of both bank depositors andank managers.
.1. Regulator incentives
As a first principle, one might reasonably presume that gov-rnment deposit insurers strongly identify with their mission ofrotecting insured depositors and, when administratively possible,his culture can easily err on the side of protecting uninsured depos-tors and non-deposit creditors as well. Such predilections may bexacerbated when political and/or economic pressures arise to pre-ent illiquidity at all costs—for instance, during economic criseshen numerous large banks become insolvent. Whether or not
hese predilections rise to the level of serious principle-agent prob-ems is the subject of some debate (Kane, 1990; Mishkin, 1992).ane and Klingebiel (2004) weigh in with an especially cynicalssessment: Regulators exhibit a bias toward bailing out all deposi-ors because they do not want to be blamed (rightly or wrongly) forhe bank failure by disgruntled (unprotected) depositors. Lookingrom a different angle, Kane (1995) shows how existing legal andegulatory arrangements (including the prompt corrective actioneatures of the Federal Deposit Insurance Corporation Improve-
ent Act of 1991) create incentives for regulators to practiceorbearance.
Regardless of regulator motive, making uninsured depositorshole reduces deposit market discipline: it reinforces the incen-
ives for depositors to lend to risky banks, and it enhances the valuef the deposit insurance put option. Numerous proposals have been
1 Hoggarth et al. (2002) estimate that the economic costs of a systemic bank failurevent could run as high as 15–20% of a nation’s GDP.
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nd a partial “advance dividend payment” to uninsured deposi-ors, the amount of the latter based on a first approximation of thealue of the failed bank’s assets.2 Slight delays in paying deposi-ors may have positive market discipline effects by imposing costsn depositors who knowingly provide funds to risky banks; underhis line of thought, if authorities can credibly commit to such aractice, depositors will have incentives to monitor the banks andemand higher rates on funds they deposit in risky banks. A con-inuing line of policy proposals in this same vein is provided byaufman (2004), Mayes (2004), Kaufman and Eisenbeis (2005) andarrison et al. (2007).
.2. Resolution policy in practice
Fig. 1 briefly describes the main resolution techniques used byhe FDIC since its inception; some of these techniques have onlyecently become available to the FDIC via legal changes or improvedechnologies, while others are no longer available or have fallennto disuse over time.3 The various techniques are displayed in rankrder based on the degree to which they preserve liquidity. Becausehere is a roughly inverse relationship exists between preserving
arket liquidity and enhancing market discipline among theseechniques, Fig. 1 also illustrates the primary economic tradeoffliquidity versus discipline) facing policymakers. Finally, the fig-re also shows that borrower liquidity and depositor liquidity tendo be positively correlated across these techniques—for example,hen a receivership liquidates a failed bank’s assets gradually over
ime, some depositors will be denied full access to their (uninsured)unds, and some borrowers will be denied full access to their linesf credit, until enough assets are sold to cover these obligations.
Over the past three-quarters of a century, bank resolution prac-ices in the U.S. have swung between the extremes of Fig. 1pectrum. Prior to the establishment of the FDIC in 1933, bankailures were typically resolved in a manner analogous to chap-er 7 bankruptcies of non-bank corporations. This approach clearlyavored imposing discipline over preserving liquidity. Insolventanks were closed and a receiver was named to manage the resolu-ion, usually the state banking authority for state chartered banksr the Office of the Comptroller of the Currency (OCC) for nationalanks. The receiver was responsible for liquidating the assets ofhe failed bank and repaying the depositors and other creditors ofhe bank. This process could take many years, during which depos-tors and other liability holders of the failed bank lost access toheir funds.4 Borrowers also faced large costs, having to establishew credit relationships at other banks while still having to pay offny existing loans from the closed bank. A number of studies haveocused on the economic impact of bank failures during the 1920s
nd early 1930s; these studies find that bank failures by them-elves, even if unaccompanied by an extended financial panic, hadegative effects on the economy (e.g., Bernanke, 1983; Calomiris2 There is an important difference between an uninsured depositor dividendased on perfected asset estimates and the “advance dividend” to uninsured depos-
tors. The latter is based on a conservative estimate of the value of the failed bankn cases in which the resolution authority wishes to provide liquidity but lackshe time, information, or other resources necessary to complete a full insuranceetermination.3 Additional details concerning the resolution techniques listed in Fig. 1, as well
s a short history of their use in the U.S., are contained in a companion Appendix Ahat is available upon request from the authors.
4 Anari et al. (2005) showed that depositors and other creditors of national bankshat failed in 1929 received only 66.12% of their funds; only about 20% of this amount13.22 cents on the dollar) were returned during the first year, approximately doublehat amount was returned during the second year, and declining amounts wereeturned each year after that. The average liquidation period was about four years.
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al Stability 9 (2013) 612– 627
nd Mason, 2003; Kupiec and Ramirez, 2009; Ramirez and Shively,012).
The creation of FDIC and the introduction of federal depositnsurance greatly mitigated the immediate ill effects of bank fail-re. Insured depositors (and often uninsured creditors as well)ere made whole immediately when a failed bank was closed, and
epeated application of this practice defused incentives for bankuns. Moreover, most borrowers retained access to their credit linesuring FDIC failed bank resolutions. While over 10,000 banks hadailed during the 1920s alone, only a handful of banks failed annu-lly during the 1940s through the 1970s. But polices that favoredepositor liquidity to the exclusion of market discipline created
ax practices and moral hazard incentives. The extreme supervi-ory forbearance practices by regulatory authorities during the U.S.avings and loan crisis of the 1980s is the textbook example. Insol-ent thrift institutions were permitted to continue operating, and inhe most extreme cases of forbearance, authorities provided finan-ial assistance to thrifts without removing thrift management andithout a pledge of additional support from thrift owners.5 As Kane
1989, 1995) has argued, allowing these “zombie” thrifts to oper-te, virtually without any safeguards, permitted thrift managers toamble for resurrection by making risky loans with big financialpsides. Ultimately, these extreme practices cost $153 billion inesolution costs, with the U.S. taxpayers paying about $125 billionf this (Curry and Shibut, 2000).
Congress responded to the savings and loan crisis (and theave of commercial bank failures that followed) with legislationesigned to tilt failed bank policy away from blanket protec-ion of liquidity and toward imposing at least some discipline onninsured creditors and bank management. The Federal Deposit
nsurance Corporation Improvement Act (FDICIA) of 1991 con-trained the decision-making latitude of bank supervisors andegulators. Among other changes, FDICIA established a “promptorrective action” (PCA) regime that restricted the activities of trou-led institutions before they became insolvent; mandated that theDIC resolve failed banks in the “least costly” manner; increased therequency and regularity of safety and soundness examinations;nd took an initial step toward risk-based deposit insurance pri-ing. But these steps proved to be inadequate in the fall of 2008,hen the FDIC was powerless to resolve failing financial institu-
ions such as Lehman Brothers, Bear Stearns, Citigroup and Bank ofmerica, where were either too large, too complex, or were orga-ized as non-bank corporations beyond the legal reach of the FDIC.o avoid a massive reduction in economy-wide liquidity, the Trea-ury and the Federal Reserve now had no choice but to intervene,nd it required a series of extraordinary and costly steps to stabilizeredit markets, shore up the solvency of large financial institutions,nd prevent the systemic crisis from spreading even further.6
After the events in the fall of 2008, policy makers sought to cre-te a set of authorities that would allow an “orderly liquidation” ofystemically important bank and non-bank financial firms, while
5 The assistance came in the form of regulatory accounting adjustments thatllowed thrifts to carry nonperforming loans at artificially high values, thus inflatingheir accounting capital and making the thrift “book solvent.”
6 A partial list of actions taken during September and October of 2008 includes:he Federal Reserve Board created hundreds of billions of dollars in special lend-ng facilities and direct asset purchase programs, all aimed at providing increasingiquidity for suppliers of short-term credit. The Federal Reserve Bank of New Yorkrovided an $85 billion line of credit to American International Group (AIG), a large.S. insurance company that also experienced a liquidity shortage, due to large losses
n its sales of credit default swaps on subprime mortgage-backed securities. And the.S. Treasury injected $115 billion of equity into eight of the largest U.S. bankingompanies (Bank of America, Bank of New York Mellon, Citigroup, JPMorgan Chase,organ Stanley, State Street, Goldman Sachs, and Wells Fargo) under the Troubled
sset Repurchase Program (TARP).
R. DeYoung et al. / Journal of Financial Stability 9 (2013) 612– 627 615
Resolution TechniqueA shor t description of each
resolution technique
Most liquidity preserved
Least li quidity p reserv ed
Open bank assistanceCash or in-kind assistance provided to
bank, ban k o wne rs remain intact
Forbearance
All owi ng inso lvent o r undercapitalized bank to co ntinue to op erate, o ften with old man agemen t intact. No c ash or in-
kind assistan ce is p rovided.
Bridge bank
A tempo rary Nation al Ban k c reated with F DIC in con trol. Assets and mos t liabilities of failed bank transferred to
new bank. Old o wnership, ho lding compan y c red itors, and man agemen t
are severed fr om ban k.
Purchase and assumptionAcquirer of failed bank pu rch ases
designa ted assets fr om the failed b ank and assu mes the liab ilit ies.
Partial pa yout
Acquirers o f fail ed ban k may on ly wish to b id on a sub -set o f the failed
banks d epo sits. Re maining depo sit ors paid directl y b y F DIC.
Asset liqu idation
Failed b ank a ssets are li quida ted b y the FDIC o r it s de signe es. Uninsu red
Depositor co verage is limited to the proc eed s of the sale.
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appbcdtoocan embark on such sales and other actions without waiting for areorganization plan to be developed and approved by a bankruptcyjudge.7
Fig. 1. A list of failed bank resolution techniq
aintaining market liquidity and forcing stockholders and debtolders to bear the losses in event of failure. Under the Dodd-Frankall Street Reform and Consumer Protection Act of 2010, the FDIC
s named as the receiver of such failed financial companies, andhe Act provides the FDIC it with the necessary legal authority toesolve these firms. All assets, deposits, and other financial con-racts would transfer to a temporary “bridge financial company”nd the FDIC would provide the bridge company with the liquidityeeded to continue all essential functions and maintain its ongoingsset values (funding made available via an FDIC line of credit withhe U.S. Treasury). A new board directors would be appointed toun the temporary firm, which would hire new senior managers.he bridge structure would provide the FDIC with the time nec-ssary to assess the true values of the failed firm’s assets, and theDIC would advance dividend payments to non-insured creditorss these assessments allowed. To support this orderly liquidationrocess, all large complex financial institutions are required to fileesolution plans (so-called “living wills”) with regulators on a reg-lar basis. Ideally, the FDIC would sell the bridge company—nowmaller, less complex, and solvent—to private investors within ahort period of time.
. The technology of failed bank resolution
A number of factors constrain the ability of banking author-ties to efficiently resolve an insolvent financial firm—that is, toe-organize its ownership and management, re-allocate its assets,
nd at the same time preserve the liquidity of its depositors,reditors and borrowers. We characterize these constraints intohree categories: legal limits that prevent the authorities fromaking certain actions; asymmetric or missing information thatoc
order by the amount of liquidity preserved.
etards the resolution process; and economic spillovers due tohe shortcomings of the resolution process (systemic effects). Thenefficiencies imposed by these constraints may cause bankinguthorities to abandon attempts to impose discipline on failedanks and their customers and instead “bail out” these banks.hile in theory these constraints should be less binding under
he Dodd-Frank Act, the constraints remain important for at leastwo reasons: the FDIC’s new authorities under Dodd-Frank remainntested, and these new authorities are not generally available toanking authorities outside the U.S.
.1. Legal limitations
The FDIC’s special legal authority to take over insolvent banksnd act as receiver differs materially from the regular bankruptcyrocedures used to resolve insolvent non-banks. Bankruptcy lawsrotect the owners of insolvent firms from their creditors, andankruptcy proceedings typically take weeks or months to con-lude. In contrast, as receiver the FDIC steps in on behalf of theepositors and other creditors, and the owners never regain con-rol of the firm. The FDIC can act quickly—generally overnight orver a weekend—to provide depositors with access to all or mostf their funds, has broad discretion to sell the bank’s assets, and
7 Bliss and Kaufman (2011) argue that the U.S. bankruptcy practices could be tail-red in a way that lessens or eliminates its inefficiencies when applied to bankingompanies; these changes would arguably allow insolvent banks to be reorganized
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16 R. DeYoung et al. / Journal of F
The FDIC has been granted these special powers because depos-tor illiquidity resulting from a slow, bankruptcy-like resolutionrocess would cause disruptions to the payments system, financialarkets, and the real economy. In countries where bank regula-
ors lack these special powers, bankruptcy courts must be used toesolve insolvent banks, creating strong incentives for regulatorso subsidize financially troubled banks rather than closing them.8
ut even in the U.S. these powers have been limited. As discussedbove, the FDIC special resolution authority did not extend to thensolvent non-bank financial companies at the center of the 2008nancial crisis.
.2. Information problems
Immediately after taking control of an insolvent bank, the FDICegins the insurance determination process. The aim is to quicklyetermine how many of the bank’s deposits are insured, howany of the deposits are uninsured, and to whom these deposits
re owed. Insurance determination has historically been a manualrocess, working from depositor signature cards and other paperecords kept at banks’ branch offices. Over time, technologicalmprovements—some as prosaic as requiring banks keep depositecords electronically and in uniform format, and equipping mem-ers of the government resolution team with laptop computersnd wireless Internet—have automated this process and is usuallyompleted overnight.9
Insured depositors have full and immediate access to thensured portion of their funds, either at the acquiring bank (for pur-hase and assumption, or P&A, resolutions) or by some form of airect payout from the FDIC (for asset liquidation or partial payoutesolutions). Uninsured depositors do not have immediate and fullccess to their funds, and are issued receivership certificates thatepresent their claims. The percentage of their funds they will even-ually receive, and the delay in receiving those funds, is a function ofhe asset valuation process. Uninsured depositors usually receive apartial dividend” payment relatively quickly, once the FDIC has annitial estimate of the value of the failed bank’s assets. If the bank’sssets are complex (e.g., structured asset-backed securities), illiq-id (e.g., loans to small businesses), of questionable quality (e.g.,erivatives contracts with troubled counterparties), international
n scope, or otherwise difficult to value, then uninsured depositorsill receive smaller initial dividends and will have to wait longer for
ny additional dividend payments.10 This combination of imme-
iate access to partial funds, delays in receiving additional funds,nd a potential loss of some funds provides substantial liquidityor uninsured depositors while providing incentives under whichather than being closed or bailed out. These changes in the Bankruptcy Code, if suc-essful, would represent a new resolution technique for banks within the meaningf our model.8 An international survey conducted by the FDIC in 2000 found that failed banks
sed the regular corporate bankruptcy process in 12 of 18 advanced economies,ncluding Austria, Belgium, Germany, Spain, Sweden, Taiwan and the UK (Bennett,001), although the UK has since adopted legislation (Banking Act, 2009) creatingDIC-like bank resolution authority. A more recent World Bank study found thatnly 18% of 143 surveyed countries in 2008 had bank insolvency laws separate andifferent from regular bankruptcy laws (Marinc and Vlahu, 2011).9 For example, in 2008 the FDIC mandated that large banks establish electronic
ecords containing all of their deposit account information, with the goal of trans-orming insurance determinations from a slow and manually intensive process ton automated overnight process.10 Cumming and Eisenbeis (2010) propose that complex banking companies adopt
simpler and more transparent form of corporate organization that would arguablyake the asset valuation process faster and more accurate. In the same vein, require-ents that banks conduct derivatives transactions through separately capitalized
entral exchanges could speed the process by reducing the valuation problemsssociated with counterparty default risk.
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al Stability 9 (2013) 612– 627
epositors may better monitor bank risk-taking and potentially dis-ipline the bank by requiring higher interest rates or withdrawingheir funds.
Bank borrowers are also dealt a degree of discipline during thisrocess. Borrowers may temporarily lose access to the undrawnortions of their credit lines and/or a portion of any compensatingeposit balances. In a P&A resolution, borrowers retain access toheir existing credit lines in the short-run, but the acquiring bank
ay or may not renew the credit relationship. In a liquidation res-lution, the existing banking relationship dissolves, and borrowersncur the information costs necessary to establish entirely newnancial relationships with other banks.
.3. Systemic effects
The FDIC is able to impose delays and losses on uninsuredepositors at small banks without causing systemic economic andnancial disruptions. In contrast, large banks have hundreds ofhousands of depositors, borrowers and other counterparties in
arkets throughout the country; imposing delays or losses whenesolving a large insolvent bank can cause substantial financialnd economic disruption. As banks grow larger and/or more sys-emically important, the external costs of imposing discipline onepositors, borrowers and bank decision-makers can increase dra-atically.As the FDIC attempts to resolve increasingly larger and/or more
nancially complex failed banks, completing an overnight insur-nce determination and a relatively complete initial asset valuationecomes more difficult, due to both the sheer number of depositccounts to be administered and the large number and potentialomplexity of assets to be valued. Fig. 2 illustrates the potentialcope of this problem. Prior to 2008, the largest FDIC insuranceetermination was First City Houston in 1992 with 322,983 sep-rate deposit accounts; in sobering contrast, Bank of Americaurrently has over 60 million separate deposit accounts. The dis-uption of such a large number of liquidity arrangements—on bothides of the balance sheet—could have significant macro-economicffects.
. Modeling the impact of technology on bank resolutionolicy
We will now formalize much of the above discussion in a game-heoretic framework. The technology of failed bank resolution isentral to our model, and this marks an innovation to the existingheory literature on bank failure policy. The model allows us toemonstrate how technological limits impact the choices availableo the bank resolution authority, leading to increases in both bankailouts and bank risk-taking.
Our model contains many of the characteristics present in ear-ier theory models of failed bank resolution. Like much of therevious literature (e.g., Freixas, 1999; Goodhart and Huang, 2000;ordella and Yeyati, 2003), the regulator in our model faces aradeoff: it can close a failed bank, and by doing so impose mar-et discipline that reduces moral hazard incentives, or it can bailut the bank, and by doing so preserve market liquidity andvoid potential systemic harm to financial markets and the macro-conomy. The banks in our model can choose to run a “complex”usiness strategy that is both highly prone to failure and, in thease of failure, imposes large reductions in market liquidity; given
imited technologies for resolving failed banks, this can pose a too-omplex-to-resolve (TCTR) problem for the resolution authorityimilar to the TBTF problem present in most of the extant litera-ure (e.g., Ennis and Malek, 2005). Thus, as in a number of previous
R. DeYoung et al. / Journal of Financial Stability 9 (2013) 612– 627 617
Larg est Insured Institutions (as of Ju ne 201 0)
Domestic Depo sits
($ billion s)
Depos it Accou nts
(number)
Ban k o f Ameri ca NA $829 64,080,664
Wells Fargo & Company $719 92,432,109
Cit ibank $254 24,144,341
JPMo rgan Chase Bank NA $633 46,588,519
US Ban k, NA $169 12,395,340
Five Largest FDIC Insurance Determinations
Deposits
($ billion s)
Deposit Accounts
(number)
IndyMac Bank, F SB $28.5 301,878
First City Houston , NA $2.5 322,983
NetBank $2.3 191,194
ABN Financial NA $1.8 27,209
Silver State Bank $1.7 20,677
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Fig. 2. Five largest U.S. commercial banks
tudies (e.g., Mailath and Mester, 1994; Acharya and Yorulmazer,007), the regulator in our model faces a time inconsistency prob-
em which makes it difficult to credibly commit to a disciplinaryesolution policy. Moreover, our model demonstrates how polit-cal pressure, macro-economic conditions, or herding by banksxacerbates the regulator’s problem and leads to increased reg-latory forbearance (e.g., Acharya, 2001; Acharya and Yorulmazer,008a,b; Brown and Dinc, 2009). We solve our game in randomtrategies, but unlike other studies that use this equilibrium con-ept to suggest a policy of “constructive ambiguity” (e.g., Freixas,999; Goodhart and Huang, 2000; Gong and Jones, 2010), our usef random strategies is merely a convenient equilibrium device andot a policy prescription. In contrast to previous studies that exploithe differences between solvency-driven and liquidity-driven bankailures (e.g., Diamond and Rajan, 2002; Freixas et al., 2003; Freixasnd Parigi, 2010), we model failed banks as pure insolvencies andence are concerned only with bank resolution policy, not with
ender-of-last-resort policy.
.1. Game set-up
We construct a multi-period game between a government res-lution authority (RA) and a banking industry comprised of aon-trivially large number of identical investors who have toecide each period how to allocate their capital to a non-trivially
arge number of banks. The banks have access to two loan pro-uction processes: simple loan production and complex loanroduction. Simple loans are easy to value and the simple loan pro-uction process (e.g., originate and hold; core deposit funding; noff-balance sheet obligations) is transparent and easy to unwindn bankruptcy and hence generates relatively small social and/or
acroeconomic externalities upon bank failure. Complex loans areifficult to value and the complex loan production process (e.g.,riginate, securitize and sell; financial market rather than depositunding; off-balance sheet obligations) is opaque and difficult to
nwind in bankruptcy and hence generates larger failure external-ties. These two loan production processes are separable and bothxhibit diminishing returns; banks can mix these processes, and itill be useful in some cases to refer to a bank as “mostly complex”
tFcS
ve largest FDIC insurance determinations.
r “highly complex” depending on its loan mix. We will also usehe terms “investors” and “bank(s)” interchangeably below.
We emphasize that what we call “complexity” is not the sames “large size.” While both of these characteristics will make fail-re resolution more difficult, the former is more likely than the
atter to cause economy-wide externalities—i.e., one side of theentral liquidity-versus-discipline tension facing the RA in ourodel. The assets created and held by complex banks are more
ifficult to value (e.g., deeply subordinated tranches of loan secu-itizations, loan servicing contracts, venture capital investments),omplex banks are more likely to be interconnected with othernancial institutions and hence exposed to their risks (loans orssets sold with recourse, trading positions in financial deriva-ives contracts), and complex bank funding is more apt to runrepos, purchased fed funds, commercial paper) and hence encour-ge runs at similarly funded banks. And although large banks doend to be more complex than smaller banks, there is far from
one-to-one correspondence between these two characteristicsmong financial companies in the U.S. For example, MetLife is theixth largest financial holding company in the U.S. with assets ofround $800 billion, and under the provisions of the Dodd-Frankct it is classified as a systemically important financial institution
SIFI). Despite its large size, however, MetLife resembles the easy-o-unwind simple banks from our model. Its main liabilities arensurance contracts backed largely by fixed income assets that areasy-to-value and sell, and its commercial banking affiliate issuesnly about $10 billion of deposits, has zero trading assets, and itserivatives positions are limited to vanilla interest rate swaps, for-ards and futures (data from www.fdic.gov). Perhaps even more to
he point, the FDIC was able to resolve the failed $300 billion sav-ngs bank Washington Mutual (WAMU) in September 2008 withoutny macro-economic disruption and without any cost to the tax-ayer. The resolution imposed losses on WAMU’s shareholders andninsured creditors and transferred WAMU’s retail deposit fran-hise to JPMorgan Chase. This clean resolution of such a large bankas possible WAMUs assets (mainly mortgage loans) were easy
o value quickly. At the other extreme, the Federal Reserve andDIC have been charged with identifying which non-bank finan-ial firms with assets less than $50 billion should be classified asIFIs.
6 inancial Stability 9 (2013) 612– 627
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Banks issue deposits at the beginning of the period, invest thoseunds (along with the investors’ capital) in either simple loans LSr complex loans LC that mature at the end of the period, and usehe loan proceeds to pay back depositors. If a bank’s investmentroceeds are greater than its deposit liabilities, the resulting profitsre distributed to the investors who play the game again in the nexteriod. Otherwise, if a bank’s investment proceeds are too small toay off its depositors, then the bank becomes insolvent, its investors
eave the game, and an equal amount of new investors arrive at thetart of the next period.11
Loans default with probability �i (i = C,S) for complex and sim-le loans; each �i follows a two-part stochastic process consistingf a macroeconomic (systematic) shock felt by all banks and aank-specific (idiosyncratic) shock that is distributed indepen-ently across all banks. We place no constraints on the relativealues of �C and �S, and we allow complex and simple loanefaults to be uncorrelated. Banks follow an internal (i.e., non-egulatory) value-at-risk (VaR) capital policy that protects the bankgainst default in all states of nature except for the tail-risk eventn which both complex loans and simple loans default. Thus, aank fails with probability ϕ = �C�S and survives with probability
− ϕ = (1 − �C)(1 − �S) + (1 − �C)�S + �C(1 − �S).A failed bank generates a social externality, the costs of which
re truly external to the banks in our model and hence we need notodel them explicitly. Consistent with recent experience, we sim-
ly assume that the social externality increases in the amount ofC at the failed bank. Within the context of our model, it is naturalo consider an externality that manifests itself as macro-economiclliquidity. For example, if depositors at failed complex banks haveo wait to receive access to their funds, then fewer deposits wille available in the next period of the game, reducing the supply of
iquidity to all banks but more importantly reducing the amountf liquidity in the economy. Similarly, if assets at failed complexanks take time to be sold off (or if they can be sold off only at aiscount), increased investor uncertainty about the value of simi-
ar financial assets can reduce the amount of available liquidity innancial markets in general.
.2. Game without bank failure regulation
A game without a resolution authority (RA) lacks strategic inter-ction. Investors maximize profits each period as follows, as if theyere playing a one-period game:
maximize :LC ,LS
� = (1 − �C )FC (LC ) + (1 − �S)FS(LS)
subject to : L = LC + LS
(1)
here the Fi(Li) are increasing and concave profit functions foromplex and simple loans satisfying the standard conditionsi(0) = 0, F′
i(0) = +∞ and limLi→∞ F′i(Li) = 0. L is the fixed exogenous
emand for loans, which the bank can satisfy using any combina-
ion of the simple and complex production processes. Given thatoth production processes exhibit diminishing returns and satisfy11 The arrival of new investors is not technically necessary so long as the totalumber of solvent investors remains “non-trivially large.” We could just as easilyssume that the surviving banks proportionately absorb the deposits and remainingssets of the failed bank. Either assumption obviates modeling strategic interac-ions that occur as the industry becomes concentrated and market power developsEnrico Perotti and Javier Suarez, European Economic Review, 2002). Moreover, thisssumption simplifies the solving of the dynamic (repeated) version of the game. Inny case, entry is relatively easy in U.S. banking markets, either via new (de novo)ank charters, geographic expansion by existing banks, or expansion by non-banknancial services firms into banking product markets.
hsaobi(L
4
T
Fig. 3. Game without regulation.
he above standard conditions, the solution LC* to this problem is
nterior and given by the first order condition:
1 − �C )F ′C (LC ) − (1 − �S)F ′
S(L − LC ) = 0
here LC* is the amount of complex loans produced in a game with-
ut regulation. The solution is straightforward and is illustrated inig. 3.
Eq. (1) assumes additive separability between the profits gen-rated by simple and complex banking; by definition, this meanshat there are no cost or revenue synergies between the two bank-ng methods. While one can perhaps imagine synergies betweenhese two banking approaches, cost and revenue synergies acrossanking lines of business have been notoriously difficult to quantityBerger et al., 1999; Mester, 2008) and many banking researchersonclude that they limited at best. Nevertheless, in the face of eitherroduction or marketing synergies, each of the single-productrofit functions would surely maintain their fundamental, well-ehaved economic properties (increasing at a decreasing rate withutput); hence, synergies would only serve to increase the heightf the parabolic shape shown in Fig. 3 and consequently have noualitative effect on the equilibra in the one-period and infiniteorizon games described below. Similarly, we make no assump-ions in Eq. (1) about the relative costs, revenues or profits of simplend complex loans. Again, so long as the two portions of (1) are eachell-behaved profits functions (concave down in output), then the
oint profit function in Fig. 3 will have an interior maximum regard-ess of the relative expenses and revenues of simple and complexoans. For example, scale economies for complex loan productionrelative to smaller scale economies for simple loan production)ould simply skew the inverted-U shape in Fig. 3 toward the right
ut maintain an interior optimum.Because we assume that all investors are identical, all banks will
ave the same LC*. Some of these banks will fail—loan default is
tochastic and contains an idiosyncratic component—but as statedbove the failure probability is unrelated to LC
*. In the absencef bank failure regulation, these failed banks enter the regularankruptcy process, which (because it protects banks against cred-
tors for some period of time) fosters macro-economic illiquiditythe social externality) in amounts that are positively related toC
*.
.3. Bank failure regulation
We now introduce a bank failure resolution authority (RA).he resolution process works more quickly than bankruptcy
inancial Stability 9 (2013) 612– 627 619
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R. DeYoung et al. / Journal of F
roceedings, returns failed bank deposits to the banking system inime for the next round of the game, which mitigates the macroeco-omic liquidity shock. The RA’s technology set is finite, however,nd it lacks the ability to quickly resolve banks with large amountsf complex loans. Letting LT represent the limits of resolution tech-ology (i.e., the lowest level of failed bank complexity that the RA
s unable to resolve), the RA closes failed banks when LC < LT andails out failed banks when LC ≥ LT.12
In a failed bank closure, the RA seizes the insolvent bank andays off the depositors, using the bank’s (insufficient) investmentroceeds plus an insurance fund which is capitalized prior to thetart of the game.13 The bank’s owners receive zero profit distribu-ion (i.e., the bank fails with limited liability), are prohibited fromlaying the game again, and new investors enter the game at thetart of the next period. In a bailout, the RA pays off the deposi-ors (or, equivalently, gives the bank the funds necessary to pay offhe depositors) and pays investors an amount b large enough sohat they can play the game again in the next period (i.e., recapi-alizes the bank).14 Bailout in our model is similar in nature to thepen bank assistance policy discussed above and used sparingly inhe past by the FDIC, and is similar in spirit to the U.S. Treasury’sARP program and other ad hoc actions used to support insolventnd/or illiquid financial institutions during the recent financial cri-is. Bank closure in our model is similar to the depositor payoutolicy discussed above and used often by the FDIC.
Faced with the task of resolving an insolvent bank, the RAaximizes its own utility by selecting a method of resolution
hat maintains some non-negative amount of liquidity (LIQ) andmposes some non-negative amount of market discipline (DISC).
he RA’s preferences for LIQ and DISC need not be exactly consis-ent with social welfare—for example, as discussed above in Section, the regulator may identify closely with depositors, hence favor12 These two alternative resolution techniques, “closure” and “bailout,” are meanto represent the two ends of the liquidity-discipline spectrum illustrated in Fig. 1.imiting the RA’s tradeoff to just two techniques keeps the model tractable with noeal loss of generality.13 Imagine that the insurance fund is capitalized by charging each of the “non-rivially large number of investors” a small and identical entry fee at the beginningf the game. For the sake of simplicity, we characterize a flat-rate insurance pre-ium; conceptually, including a risk-based deposit premium would make little
ifference because in our model bank insolvency risk is by assumption unrelated toank complexity. Moreover, the central focus of our model is neither insolvency riskor systemic risk; rather, we focus on how innovations in bank failure resolutionechnology can improve regulators’ ability to resolve complex failed banks, and byoing so change bank incentives and result in an equilibrium with banks that are
ess complex and a regulator that is less likely to bail out complex banks shouldhey fail. Of course, one can imagine a complexity-based (as opposed to a flat-rate)nsurance premium large enough to deter banks from choosing to be complex inhe first place—however, such a solution requires that the RA monitor all banks (inur model, the RA takes no actions until a bank fails), can accurately detect (withoutype I and II errors) complexity at non-failed banks and in non-crisis situations, andan do this at a reasonable cost. We choose to avoid these complications to keepur model tractable, and further note that none of this matters unless in the end theA has both the technology and the political will to actually close complex failedanks, which is the main lesson from our model. Other studies have focused on
nnovations in deposit insurance pricing. For example, Acharya et al. (2010) explorehe relationships between systemic risk and deposit insurance pricing, and showhat charging higher premiums to large banks with correlated insolvency risks can
itigate systemic risk and minimize costs to the insurance fund.14 We make the simplifying assumption that the bailout payment b is unrelated tooan complexity LC . If complexity makes it difficult for the RA to value a bank’s assets,t should also make it impossible to determine the size of b needed to recapitalizehe bank. The RA could easily deal with this problem by including loan guaranteeslong with b. Issuing these guarantees would be costless to the RA in the short-un. As the value of the complex loans are revealed in the long-run, the RA coulday out extra capital above b to banks that incur larger than expected loan losses,nd fund these payments by collecting capital back from banks that incur smallerhan expected loan losses. This only requires the RA to be able to estimate b in annbiased fashion across multiple banks.
s
wotdc
L
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L
twi
Fig. 4. The RA’s problem.
omewhat more liquidity and somewhat less discipline than is wel-are maximizing—but in any case we specify both LIQ and DISC asconomic goods in the RA’s utility function. The RA is constrainedn its choices of LIQ and DISC by available resolution technologies T.
ith little loss of generality we specify this utility maximizationroblem as follows using a Cobb-Douglas utility function and a
inear technology constraint T:
maximizeLIQ,DISC
U = ˛LIQ ˇDISC� (2)
ubject to : T ≥ �LIQ + ωDISC (2a)
0 ≤ LIQ < 1 (2b)
0 ≤ DISC < 1 (2c)
here � and ω are, respectively, the “prices” of maintaining a unitf liquidity LIQ or imposing a unit of discipline DISC. To emphasizehe policy tradeoff between maintaining liquidity and imposingiscipline, we arbitrarily set � = 1 and then rewrite the technologyonstraint as:
IQ = T − ωDISC (2a’)
hich allows the straightforward interpretation of the slope ω ashe “liquidity price of discipline,” that is, the marginal rate at whichn efficient RA can transform one unit of sacrificed liquidity intodditional discipline. Consistent with the discussion above, ω willecline with efficiencies in insurance determination, asset valua-ion, and other activities necessary to resolve a failed bank, and ωill increase with the size and complexity of the failed bank being
esolved. The constraints (2b) and (2c) scale the problem so that theolution set lies within the unit square (see Fig. 4).15 Maximizing2) with respect to LIQ and DISC yields the RA’s optimal resolution
ethod:
IQ∗ = ˇT
+ �
15 The RA cannot maintain maximum liquidity (LIQ = 1) because failed bank resolu-ions require the bank to be shut down for a short period of time (typically over theeekend). The RA cannot impose maximum discipline (DISC = 1) because deposit
nsurance will shield some creditors from the costs of bank failure.
6 inancial Stability 9 (2013) 612– 627
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20 R. DeYoung et al. / Journal of F
ISC∗ = �T
ω( + �)
The results are economically intuitive. Discipline increases withhe RA’s strength of preference for discipline �, decreases with theA’s strength of preference for liquidity ˇ, and decreases with the
iquidity price of discipline ω. Liquidity increases with the RA’strength of preference for liquidity and decreases with the RA’strength of preference for discipline �. Both liquidity and disciplinencrease, proportionate with their strength of preferences and �,
ith efficiencies that relax the technology constraint T.The solution is illustrated in Fig. 4. The indifference curves
re drawn with a relatively shallow slope, indicating an RAith a strong preference for maintaining market liquidity and
weak preference for imposing market discipline ( > �).16 (Forur Cobb-Douglas utility function this policy tradeoff is given by�U(1/ˇ)DISC( − �/ˇ)/ˇ˛(1/ˇ) which shows a diminishing will-
ngness by the RA to substitute DISC for LIQ.) The four indifferenceurves correspond to ordered utility levels � where �4 > �3 > �2 > �1.he technology constraint T has linear slope ω (the liquidity pricef discipline) and runs between “bailout,” which generates highiquidity (depositors are fully paid off) and low discipline (investorslay again next period), and “close,” which generates lower levelsf liquidity (some depositors receive haircuts) but higher levels ofiscipline (investors are wiped out).
The RA clearly prefers insolvent banks that are mostly simple.hen a relatively simple bank fails, the RA has access to the more
fficient resolution technology Tsimple with a low liquidity cost ofiscipline, and will prefer closure (utility = �3) over bailout (util-
ty = �2). When a relatively complex bank fails, then the RA hasccess only to a less efficient resolution technology Tcomplex with aigh liquidity cost of discipline, and will prefer bailout (utility = �2)ver closure (utility = �1). To make the model easier to solve wenclude the “quiet life” utility �4 which the RA consumes whenhere are no bank failures in a given period. The arrow suggests aechnology expansion path: As the resolution technology improvesi.e., as LT increases), the RA will close, rather than bail out, increas-ngly complex banks.17
The investor maximization problem changes with the introduc-ion of the RA and the possibility of being bailed out. Investors now
aximize profits as follows:
axLC ,LS
{(1 − �C )FC (LC ) + (1 − �S)FS(L − LC ), if LC < LT
(1 − �C )FC (LC ) + (1 − �S)FS(L − LC ), ϕb, if LC ≥ LT
(3)
here banks that choose LC ≥ LT and hence are too complex toesolve (TCTR) gain access to the expected bailout subsidy ϕb.
ence, the solution to the game with bank failure regulation willepend on the technology LT that is available to the RA. There arehree cases. In the first case, illustrated in Fig. 6a, the resolution16 Preferences such as these are plausible in a number of scenarios: (a) the RAnd/or elected officials who influence the RA identify strongly with depositors andence prefer liquidity over discipline; (b) the RA and/or economic authorities that
nfluence the RA feel that an illiquidity shock would harm the macro-economy andence prefer liquidity over discipline; and/or (c) the RA and/or bank supervisors who
nfluence the RA wish to conceal the financial deterioration of a bank or the bankingystem, and the regulatory forbearance that results naturally generates liquidityather than discipline. As illustrated at length in Appendix A, the resolution choicesade by FDIC during the post-deposit insurance era have been largely consistentith a strong preference for liquidity over discipline.
17 Improvements in the insurance determination process, improved asset valua-ion techniques, and other efficiencies in bank resolution would relax the technologyonstraint by reducing the liquidity price of discipline; in contrast, greater bank sizer increased bank complexity would make the resolution process more difficult,ightening the constraint by increasing the liquidity price of discipline.
siLilb
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(a
Fig. 5. Game tree with payoffs.
echnology is very imperfect, such that LT ≤ LC* and does not con-
train investors’ choices. Investors choose LC = LC* and the bank gets
ailed out if it fails. In the second case, illustrated in Fig. 6b, the res-lution technology is only weakly imperfect, such that LC
* < L� ≤ LT,here L� is defined by the condition
�(LC > L∗C) = �(L∗
C)
(1 − �C )FC (L∗C) + (1 − �S)FS(L − L∗
C) = (1 − �C )FC (L�) + (1 − �S)FS(L − L�) + ϕb.
nvestors are once again unconstrained and choose LC = LC*, but in
his case the bank gets closed if it fails. In the third case, illustrated inig. 6c, the resolution technology is moderately imperfect, such thatC
* < LT < L� . In this case the investors are constrained by the reso-ution technology, choose LC = LT = LC
**, and the bank gets bailed outf it fails. Thus, we have the first main implication from our model:he presence of a resolution authority with imperfect resolutionechnology can increase the amount of complexity in the bankingystem (i.e., LC
** > LC*).
We now solve two versions of the game with regulation: a one-eriod game in which the RA’s response to a bank failure markshe end of the game, and an infinite horizon game in which banksan change their loan mixes after each RA response. In what fol-ows, we assume the most interesting case (illustrated in Fig. 6c)n which the RA’s resolution technology is moderately imperfect.onsistent with this case, we assume that “simple” banks chooseC
* and earn expected profits �S(LC*), and we assume that “com-
lex” banks choose LC** and earn expected profits �C(LC
**) = �C + ϕb,here �C is the non-governmental part of its expected profits.18 (Asiscussed above, it may be instructive to think about two same-ized banks, one of which chooses LC** and hence is a SIFI whosensolvency will create systemic external costs, and one that choosesC* and hence is a non-SIFI.) Because the resolution technologys moderately imperfect, it holds that �C(LC
**) > �S(LC*) > �C. The
ast inequality holds because LC* is the optimal solution without
ailouts.
.4. One-period game
The solution of the one period game is straightforward, and is
asy to see in the game tree illustrated in Fig. 5. The RA alwaysloses failed simple banks (node 4) because �3 > �2 and it alwaysails out failed complex banks (node 5) because �2 > �1. Banks will18 We note that each of the production options available to the bank in this set-upLC
* and LC**) include both simple loans and complex loans. This implies that banks
lready know how to produce both types of loans at the beginning of the game.
R. DeYoung et al. / Journal of Financi
Fig. 6. Game with regulation. Panel A: very imperfect resolution technology. Bankoptimizes at LC
* and failed banks are bailed out. Panel B: weakly imperfect resolutiontio
atpitot
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• The equilibrium path strategy Be: If st = NB, the bank chooses to besimple with certainty.
19 There is no inconsistency here: one can think of the RA playing the samerepeated game simultaneously with each bank in the industry. This merely requiresus to assume that the RA’s preference ordering (�4 > �3 > �2 > �1) is invariant to thenumber of banks that fail in any given period. Nevertheless, in our comparative sta-tics analysis below, we analyze the impact that political pressure or macro-economiccircumstances may have on the RA’s policy choices during times of multiple and/orlarge bank failures.
20 This ‘one-period memory’ makes sense for our problem. The history of bankfailure and resolution in the U.S. has been marked by discrete episodes of bank
echnology. Bank optimizes at L∗C
and failed banks are closed. Panel C: moderatelymperfect resolution technology. Bank optimizes at L∗∗
Cand failed banks are bailed
ut.
lways choose to be complex (node 2), due to the RA’s TCTR policyhat introduces the regulatory wedge b that makes expected com-lex profits �C(LC
**) exceed expected simple profits �S(LC*). Thus,
n the subgame perfect equilibrium of the one-period game, all ofhe banks are complex because this strategy promises unambigu-usly higher expected returns, and the RA will always choose to bailhem out should they become insolvent. Without a history of past
fbtpf
al Stability 9 (2013) 612– 627 621
ctions, or the promise of future actions, the RA will not close annsolvent complex bank, because it cannot consume the expecteduture benefits (i.e., fewer complex banks) that would derive frommposing discipline today.
.5. Infinite horizon game
In the one-period game, the RA’s current policy decisions cannotnfluence the industry’s future business model decisions. Specif-cally, the RA might choose to forfeit some short-run utility (bylosing insolvent complex banks and suffering reduced liquidityoday) in order to establish a credible reputation that makes thendustry less likely to choose complexity in the future. In ordero study this interplay between banks’ and RA’s actions we studyn infinitely repeated version of the one-period game describedbove. We add the assumption that all players weight future pay-ffs with positive discount factors: ı < 1 for the RA and < 1 for theanks. Although our objective is to characterize the strategic inter-lay between the RA and the banking industry, in what follows wenalyze a repeated version of the one-period game described aboveetween the RA and a single bank.19
In what follows, we derive the conditions that support an equi-ibrium in which a bank repeatedly chooses to be simple withertainty. We focus on Markov strategies in which the past influ-nces current decisions only through its effect on the state variablesFudenberg and Tirole, 1991, chapter 13; Maskin and Tirole, 2001).here are two possible states of the world at the beginning of eachtage of the game, defined by whether there was a bailout at time
− 1 (denoted by st = B) or there was no bailout at t − 1 (st = NB). Ift = NB, then the game repeats with the successful bank (i.e., whenhere was no bank failure at t − 1) or with the new replacementank (i.e., when there was a failure at t − 1 and the RA closed theank). Otherwise if st = B, the game repeats with the bailed outank.20
The desired equilibrium upon which we focus has only simpleanks that are always closed when they fail. The banks and the RAlay one set of strategies when the desired equilibrium obtains, and
different set of strategies when the desired equilibrium does notbtain. We consider the following profile of strategies for the RA:
The equilibrium path strategy RAe: The RA closes insolvent simplebanks with certainty.The off-equilibrium path strategy RAo: The RA closes insolventcomplex banks with probability q and bails them out with prob-ability 1 − q.
nd we consider the following profile of strategies for the bankingndustry:
ailures (e.g., the 1930s; the late 1980s and early 1990s; and the late 2000s) followedy clear policy shifts in response to those episodes (e.g., the creation of the FDIC andhe Glass-Steagall Act; the FDIC Improvement Act; and the Dodd bill). Enough timeassed between these events for both banks and regulators to ‘forget’ the past, andocus only on the new episode.
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The off-equilibrium path strategy Bo: If st = B, the bank chooses tobe simple with probability p and complex with probability 1 − p.
The following proposition provides the conditions under whichhis profile of strategies constitutes an equilibrium in which bankshoose the simple business model and the RA always closes failedanks:
Proposition: There exists a non-negative value ı- such that forny ı ∈ (ı-; 1) there also exists the following “disciplinary equilib-ium” in the infinitely repeated game:
The RA always closes a failed simple bank. Furthermore, the RAis indifferent in expected payoffs between closing or bailing outa failed complex bank and chooses closure with probability q∗ =1 − (1−(1−ϕ))
ϕ[�S+(1−(1−ϕ))b] (�S − �C ).When st = NB, banks always choose the simple business model.But when st = B, banks are indifferent in expected payoffs betweenthe simple and complex business models and choose the simplemodel with probability p∗ = 1 −
(1ı
− 1)
�2−�1ϕ(�3−�2) .
The proof of this proposition appears in Appendix A.The first important insight of the proposition is that, in order
o credibly establish a disciplinary mechanism that encourageshe bank to make mostly simple loans, the RA should randomizeetween closing and bailing out failed complex banks.21 The RAixes its response to complex bank failure proportionately (i.e.,
*) so that the banks are just indifferent between being complexnd simple—in other words, banks cannot increase their expectedrofits by deviating from the simple bank strategy. More explic-
tly, the RA is indifferent between bailing out and closing a failedomplex bank if and only if
1 + ı(1 − ϕ)�4 + ϕ�3
1 − ı= �2 + ı
(1 − ϕ)�4 + ϕ�3 − (1 − p)ϕ(�3 − �2)1 − ı
.
(4)
The left-hand side is the value of playing closure to the RA. Therst term is the immediate utility �1 from closing the failed complexank. The second term is the discounted utility arising from theeplacement bank choosing the mostly simple loan model in futureeriods, i.e., the future returns from imposing discipline today. Theight-hand side is the value of playing bailout to the RA. The firsterm is the immediate utility �2 from bailing out the complex bank.he second term is the discounted utility arising from the bailedut bank randomizing between the complex and simple lendingtrategies in future periods. It is instructive to rewrite this equationo compare the RA’s immediate utilities to its future utilities:
2 − �1 = ı
1 − ı(1 − p)ϕ(�3 − �2), (5)
here the left-hand side is the immediate utility from bailing outhe complex failed bank relative to closing it (illiquidity avoided),
nd the right-hand side is the expected future utility from clos-ng the complex failed bank relative to bailing it out (moral hazardvoided). When a bank chooses to be simple after a bailout withery high probability (p close to 1) then the future gain of today’slosure becomes negligible and the RA will prefer to bail out the21 Note that an RA threat to always close a failed complex bank is not a solutionnd outside our real world experience: if the RA could credibly commit to alwayslosing failed complex banks, then banks would never choose to be complex, andstablishing this credibility would be unimportant for the RA.
bR
p
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al Stability 9 (2013) 612– 627
anks.22 Similarly, when a bank chooses to be complex after aailout with very high probability (p close to 0) then the future gainf today’s closure becomes larger than the immediate gain from aailout. By this logic we obtain the RA’s best response function,hich stipulates that the RA closes failed complex banks if p < p*,
ails out such banks if p > p*, and is indifferent between these twoctions if p = p*.
The second important insight of the proposition is that theisciplinary equilibrium exists only if the discount factor ı is suf-ciently high—that is, only when the future matters for the RA. Aisciplinary equilibrium requires ı > ı-; otherwise, the RA preferslways bailing out failed complex banks, as the future utility conse-uences associated with this action will get deeply discounted. Bye-arranging the above expression for p* we can derive a boundaryondition for this cutoff threshold, ı- = 1/1 + (ϕ(�3 − �2)/(�2 − �1).
e can gain some intuition by rewriting the boundary conditions
�2 − �1)(
1ı
− 1)
< ϕ(�3 − �2). (6)
he disciplinary equilibrium requires that the expected marginaltility from avoiding moral hazard in the future (right-hand side)xceeds the marginal utility from avoiding illiquidity by bailing out
complex bank today (left-hand side) by a factor of (1/ı − 1).
.6. Comparative statics
Explicit expressions for the comparative static results (i.e., theartial derivatives of ı, q*, and p* with respect to model param-ters �1, �2, �3, �4, ϕ, �s, �c, b, and ı) are shown in Appendix.23 We report the signs of the comparative static tests here, alongith logical interpretations of these results that are consistent with
he structure of our model. The relevance of these findings for theurrent debate on failed bank resolution policy is discussed in theection that follows.
The threshold ı identifies the discount factor that separateshe disciplinary equilibrium from the non-disciplinary equilibrium.
hen ı > ı (i.e., the future is relatively important), the RA acceptslliquidity today in exchange for reducing moral hazard incen-ives in the future; when ı < ı (i.e., the future is unimportant),he RA accepts moral hazard incentives in the future in exchangeor reducing illiquidity today. Changes in the values of the modelarameters influence the sensitivity of this inter-temporal tradeoff.eteris paribus, increases in �1, �3 and ϕ make the disciplinary equi-
ibrium more attractive to the RA, and thus push ı lower. Highertility from closing failed complex banks (�1) makes bailouts rel-tively less attractive to the RA; higher utility from closing failedimple banks (�3) makes simple banks relatively more attractive tohe RA. A higher probability of the bank default state ϕ increases theA’s expected marginal utility from avoiding moral hazard and/orecreases its expected marginal utility of avoiding illiquidity (seehe discussion that accompanies Eq. (5) above). In contrast, anncrease in �2 makes the disciplinary equilibrium less attractiveo the RA, and thus pushes ı higher. Obviously, higher utility fromailing out failed banks (�2) makes discipline less attractive to the
A.The RA’s off-the-equilibrium-path behavior is given by q*, therobability that the RA will close a failed complex bank. Ceteris
22 This is why a RA announcement that it will always close failed complex bankss not credible.23 Although �s , �c and ϕ are functions, we treat them as parameters in the com-arative statics section in order to provide some additional intuition behind ouresults.
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aribus, increases in b and �c make complex lending more attrac-ive to banks by increasing its expected returns; an increase in therobability of bank failure ϕ makes complex lending more attrac-ive by creating greater possibilities for bailouts that extend thexpected life of the bank; and an increase in makes a longerxpected life more valuable to the bank. In response, the RAecomes more likely to impose discipline, thus increasing q*. Inontrast, an increase in �s makes simple lending more attractive toanks. In response, the RA becomes less likely to impose discipline,hus decreasing q*.
The bank’s off-the-equilibrium-path behavior is given by p*, therobability that the bank chooses to make mostly simple loans.eteris paribus, increases in �1 and �3 directly strengthen the RA’sreferences to establish discipline, making banks more likely at theargin to make simple loans; an increase in �2 obviously has the
pposite effect. A higher probability of bank failure ϕ increases theuture chances of illiquidity and moral hazard behavior while anncrease in ı makes the RA care more about these future statesf the world, both of which make the RA more likely to imposeiscipline and hence the bank becomes more likely to make simple
oans.Two of these results bear special attention: (1) An increase in
1 (holding �2 constant) is equivalent to a reduction in the liquid-ty price of imposing discipline on failed complex banks (see Fig. 4).hus, the comparative static results ∂ı/∂�1 < 0 and ∂p*/∂�1 > 0 indi-ate that more efficient failed bank resolution technologies aloneill make the disciplinary equilibrium more likely to obtain. An RA
hat can close a failed complex bank more efficiently—that is, pre-erving more liquidity and generating more discipline—will imposehe disciplinary equilibrium more often (∂ı/∂�1 < 0) and banks fac-ng this RA will have a higher probability of making simple loans∂p*/∂�1 > 0).24 (2) A decrease in the RA’s discount factor ı (holdingonstant the banks’ discount factor ) means that the RA increas-ngly values current liquidity and/or discipline at the expense ofuture liquidity and/or discipline. This is most likely to occur duringn economic downturn or financial crisis, when preserving cur-ent liquidity becomes relatively more important than preventinguture moral hazard incentives. Such a time revaluation could beriggered if, for example, multiple complex banks become insolventithin months or weeks, the threat of contagion increases due toerding or inter-connectedness, the government or central bankressures the RA to keep banks open, and/or the bank supervisoryuthority pressures the RA to forebear to cover up gross super-isory mistakes. Thus, in a world where bailouts are possible, theomparative static result ∂p*/∂ı > 0 indicates that systematic devel-pments or political events can create incentives for banks to makeomplex loans.
. Implications and conclusions
When a bank fails in the U.S., the FDIC is assigned as a receiver
or the insolvent bank and has special powers to take immedi-te and unilateral action to resolve the situation. These specialesolution powers yield potential macro-economic efficiencies:24 These comparative static results are based on a shock in the technology availableo the RA at the beginning of the repeated game, and assumes that the technologyemains static after that. Technically, a change in LT (or any of the other parameters)uring the game would invalidate our equilibrium solution. As discussed at length inection 3 above, resolution technology does change over time, often unpredictably,ith changes in laws and regulations and with advances in information technol-
gy. For a change in LT to materially influence either the banks’ or the RA’s actionsoday—and hence the comparative static results—then (a) the players would have toxpect the future change in LT and (b) because these expected changes would occurn the future, their effects would be discounted in the players’ decisions.
fipnhaficetncTFf
al Stability 9 (2013) 612– 627 623
epositors and line-of-credit customers can have immediate accesso their funds, thus avoiding illiquidity problems in the local,egional or nationwide economies in which failed bank operates.ut these protections make bank depositors and borrowers pas-ive counterparties, reducing banks’ exposure to market disciplinend encouraging bank managers to take greater insolvency risk.his policy tradeoff—which we refer to here as the liquidity pricef discipline—is the underlying motivation for much of the debatever how we should regulate financial institutions. We construct
theory model of failed bank resolution policy that is centered onhis tradeoff.
We model a repeated game between a utility maximizing bankesolution authority (RA) and profit maximizing banks that canenerate negative externalities should they fail. In our model,anks choose to be either complex or simple, where complex-
ty is unrelated to the probability that a bank fails, but makes aank difficult for regulators to unwind if it does fail. Thus, ourodel does not focus on the positive relationship between risk-
aking and the probability of bank failure; rather, it focuses onhe positive relationship between failed bank complexity and theropensity of regulators to bail out insolvent banks. In equilibrium,he degree of complexity chosen by banks and the propensity ofegulators to bailout failed banks are dependent on two exoge-ous conditions: (1) the technology set available to the RA and2) the political and economic pressures under which the RA mustperate.
We define resolution technology set as the tradeoff the RAust make between preserving liquidity for the customers of a
ailed bank (and by extension, liquidity in the macro-economy)nd imposing market discipline on the bank’s other stakeholders,.e., the liquidity cost of discipline. A positive (liquidity increasing)hock to the resolution technology set generates two importantesults. First, improved technology makes the RA more likely toursue a disciplinary resolution policy, closing failed banks ratherhan providing them with financial assistance. Second, an improve-
ent in the RA’s technology makes banks less likely to pursueomplex business strategies that make them difficult for the RAo efficiently resolve in the case of failure. We specify the politicalnd/or economic pressure facing the RA even more simply as theA’s value of time. The logic is straightforward: in an environment
n which bank closures create negative externalities that threatenhe current health of the macro-economy, policymakers will dis-ount the long-run consequences of bank bailouts (increased moralazard incentives) relative to the short-run social and political ben-fits of bank bailouts (preventing financial disruptions, avoidingank failures and the blame that goes with them). In our model,hen the RA increasingly discounts future consequences, banks
ecome more likely to pursue risky business strategies that makehem difficult to efficiently resolve.
We can use our model as a prism for viewing the policy solu-ions imposed on insolvent banking companies during the recentnancial crisis. In some cases, extant laws and regulations simplyrevented authorities from applying their existing resolution tech-ologies to insolvent financial institutions. For example, the FDICad the legal authority to resolve insolvent banks, but lacked thisuthority over parent bank holding companies (e.g., Citigroup, thenancial holding company that owns Citibank) or non-bank finan-ial firms (e.g., Bear Stearns, AIG). Outside the U.S., the FSA hadven less legal authority, so effectively policymakers in the UK hado choose between nationalizing Northern Rock or letting it enterormal commercial bankruptcy. In other cases, the size and/or
omplexity of insolvent financial firms (e.g., some of the initialARP recipients) simply outstripped the technological ability of theDIC to close them down without inducing large losses in liquidityor bank customers, for counterparties of bank customers exposed
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24 R. DeYoung et al. / Journal of F
o contagion-like effects, and for investors potentially exposed toncreased uncertainty in financial markets. Pressure was felt byecision makers to provide immediate assistance to large complexnancial firms to prevent further disruptions to financial marketsnd the macro-economy.
While it is generally agreed—though ultimately notrovable—that government assistance prevented a larger melt-own, the decisions to provide this assistance were made with
ittle thought of the longer term consequences for bank risk-aking.25 Absent the technological ability to close the veryargest insolvent banking companies, federal bank regulatorsncouraged/facilitated the purchase of these banks by otherarge banks (Wells Fargo purchased Wachovia in October 2008;ank of America purchased Merrill Lynch in September 2008),ctions that substantially increased the size and complexity ofhe surviving acquirers.26 Even if we experience continuousmprovements in resolution technology that allow regulatorso close increasingly larger banks, allowing the largest bankso grow even larger nullifies these technological gains. In our
odel, in which such scenarios correspond to a higher rate ofime discount for the RA, increased focus on the present relativeo the future (i.e., the discounting of future moral hazard conse-uences) makes banks more likely to choose high-risk businessodels.The seeming inconsistency of U.S. regulators’ treatment of Bear
tearns and Lehman Brothers—providing assistance for JP Morgan-hase to purchase the former investment bank, but several months
ater standing by and forcing the latter investment bank to enterankruptcy—can also be interpreted within the framework of ourodel. McDonald and Robinson (2009) argue that Lehman Brothers
EO Richard Fuld falsely believed that the United States govern-ent would save the company, and with this put option in hand
uld played a game of brinksmanship, engaging in unnecessarilyisky behavior and rejecting as too low private sector offers to pur-hase controlling interest in the firm (e.g., an $18 per share from theorea Development Bank in August 2008). Assuming this charac-
erization is an accurate one, the behavior of Lehman is consistentith a departure from the desired simple equilibrium strategy, and
he seeming reversal of regulatory policy is consistent with the ran-om strategy necessary to support the desired equilibrium in theuture.
The “orderly liquidation authority” provision in the Dodd-rank Act of 2010 represents a positive technological shock for.S. bank regulators. The ability to place systemically importantnancial companies (not just banks) into receivership provides anlternative to ad hoc policy interventions necessary to avoid theconomy-wide reductions in liquidity that could result if theserms gained corporate bankruptcy protections. The legislation alsoives regulators the authority to force a decrease in the operationalnd/or financial complexity of large systemic financial firms. In
he context of our model, this new authority expands the technol-gy set available to the RA, reduces the liquidity cost of discipline,akes the closure of large insolvent bank more likely, and reduces25 In a September 19, 2008 address, President George W. Bush said that TARPshould be enacted as soon as possible” because “our entire economy is in dan-er” and that “the risk of not acting would be far higher.” These remarks were madehortly after Federal Reserve Chairman Ben Bernanke and U.S. Treasury Secretaryenry Paulson lobbied the White House to provide large and immediate assistance
o financial markets and troubled financial firms.26 In September 2008, the FDIC seized and closed Washington Mutual Bank—athe time, the largest U.S. thrift institution with assets of about $300 billion—andold its assets and most of its liabilities to JPMorgan Chase. The parent company,
ashington Mutual Corporation, stripped of its major asset, declared bankruptcyhe following day.
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al Stability 9 (2013) 612– 627
he incentives for banks to choose high-risk business models. Theseheoretical effects, of course, are conditional on the RA’s rate of timeiscount (i.e., pressures to act in the short run) remaining low. Theuccess of the Dodd-Frank provisions may ultimately depend onhether regulators can invoke their new authority to close large,
omplex financial firms without succumbing to the inevitable polit-cal and economic pressures to do otherwise during a financialrisis.
At the other end of the policy spectrum lie proposals that wouldreatly, or even completely, reduce the probability of bank insol-ency. Indeed, a world in which banks do not fail is a world inhich bank failures do not need to be resolved. Various examples
re mentioned in Dodd-Frank, most notably: the co-called Volckerule that would eliminate proprietary trading at banks; “skin-in-he-game” provisions that would require banks to hold a portion ofisky assets they create and sell off; and contingent capital schemeshat would keep failed banks operating by converting the claimsf junior creditors to an ownership stake. Such policies are notagic bullets: while they may reduce the number of failed banks
hat regulators must resolved, they come with attendant costs ofheir own. Eliminating proprietary trading surely does not elim-nate credit risk, the underlying cause of bank insolvency duringhe financial crisis. Reductions in credit risk large enough to sub-tantially reduce bank failures can only be accomplished by greatlyurtailing lending (by definition, loans are risky), which in our eco-omic system is the central purpose of banks. Similarly, holdingack some portion of loan securitization tranches (the skin) mayhange bank incentives but reduces the total amount of risk inhe system only by reducing securitizations, which in the end allo-ates less capital to loans (the game). Finally, bail-ins leave in placehe fundamental deficiencies of failed banks—that is, the organiza-ional, behavioral, geographic, and product mix inefficiencies thataused poor bank performance in the first place—rather than real-ocating bank assets to more efficient and sounder financial firms.imply stated, we must not forget that bank failures, like bankrupt-ies of inefficient non-banking firms, are essential phenomena forfficient reallocation of society’s resources in the long run. This callsor a regulatory regime that not only expects and allows banks toail, but also improves our capacity for resolving those failed banks.
cknowledgements
The views expressed in this paper are those of the authors and doot necessarily reflect positions held by the Federal Reserve Bankf Kansas City or the Federal Deposit Insurance Corporation. Theuthors thank two anonymous referees; the journal editor Iftekharasan; Robert Bliss, Ken Jones, George Kaufman, Paul Kupiec, Roseushmeider, Emre Ergungor, Carlos Ramirez, Laurence Scialom,nd seminar participants at City University of London, Erasmusniversity, Federal Deposit Insurance Corporation, Federal Reserveank of Cleveland, Tilburg University, and University of Paris-anterre for their insights and suggestions.
ppendix A. Proof of the proposition
We derive conditions under which the strategies described inhe proposition constitute a subgame perfect equilibrium of ournfinitely repeated game. We use the concept of the Markov Per-ect Equilibrium, i.e., we derive conditions under which the Markovtrategies mentioned in the paper constitute a subgame perfect
quilibrium.The RA’s strategy on the equilibrium path (RAe). First, we showhat the RA always prefers to close failed “simple” banks. If the RAticks to its strategy RAe and closes the banks, the closure yields
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he immediate utility �3. Moreover, the closure in period t implieshat st+1 = NB and the “simple” strategy by the new investors inhe future. Thus, the RA receives from period t + 1 on the presentalue of future utility from closing the failed “simple” banks, whichs equal to PV(RAe) = ((1 − ϕ)�4 + ϕ�3)/(1 − ı). PV(RAe) takes intoccount that in each period the RA receives �4 with probability
− ϕ (the “simple” banks survives) or �3 with probability ϕ (theA closes the failed “simple” banks). Formally, PV(RA)e is derivedsing the following equation:
V(RAe) = (1 − ϕ)(�4 + ıPV(RAe)) + ϕ(�3 + ıPV(RAe)).
Hence the utility from closing the failed “simple” banks at period is:
3 + ı(1 − ϕ)�4 + ϕ�3
1 − ı.
Now consider the situation in which the RA makes a one-timeeviation and bails out the failed “simple” banks. It receives the
mmediate utility �2. However, the bailout results in st+1 = B, andhe bailed-out investors and the new investors (should the existingnvestors be banned from the banking banks in the future) play thetrategy Bo, i.e., they randomize between the “simple” and “com-lex” business strategy. This alters the present value of the RA’suture utility, which is denoted by PV(BOS). PV(BOS) is a solution tohe following equation:
V(BOS) = p[(1 − ϕ)�4 + ϕ�3 + ıPV(BOS)] + (1 − p)[(1 − ϕ)�4
+ ϕq�1 + ϕ(1 − q)�2 + ıPV(BOS)].
his equation takes into account that after a bailout both the RAnd the investors randomize between their pure strategies. Withrobability p the banks become “simple”. With probability 1 − phe banks become “complex”. The RA receives �4 with probability
− ϕ if the banks succeed. Otherwise, the RA receives either �1,hen it closes the failed “complex” banks with probability ϕq, or �2ith probability ϕ(1 − q) when it bails out such banks. After solving
he last equation with respect to PV(BOS) the utility from the RA’sne-time deviation is
2 + ı(1 − ϕ)�4 + ϕ�3 − (1 − p)ϕ[�3 − (q�1 + (1 − q)�2)]
1 − ı.
The RA sticks to its strategy of closing down the failed “simple”anks if and only if
3 + ı(1 − ϕ)�4 + ϕ�3
1 − ı≥ �2 + ı
(1 − ϕ)�4 + ϕ�3 − (1 − p)ϕ[�3 − (q�1 + (1 − q)�2)]1 − ı
This can be rewritten as
3 − �2 ≥ −(1 − p)ıϕ�3 − (q�1 + (1 − q)�2)
1 − ı.
This expression holds always because �3 > �2 > �1 implies thathe LHS of the last expression is positive and the RHS negative.ence, the RA will never deviate from RAe under the stated assump-
ions.The investors’ strategy on the equilibrium path (Be). Second, we
tudy the investors’ decision to choose the “simple” bank whenhere was no bailout in the previous period. Assume that st = NBobserve that in this state both the existing investors that suc-eeded in the period t − 1 and the new investors after closure ofhe old bank at t − 1 decide about its strategy for period t). Whenhe investors choose the “simple” bank, their payoff is:
s = �s + (1 − ϕ)�s + · · · + t−1(1 − ϕ)t�s + · · · = �s
1 − (1 − ϕ) ′
here the bank succeeds with probability 1 − ϕ and is closed withrobability ϕ.
w
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al Stability 9 (2013) 612– 627 625
If the investors deviate and choose the “complex” bank, this hashe following consequences for their payoff. When the “complex”ank succeeds, the investors return to choosing a “simple” bank inhe next period and its payoff is (�c + (1 − ϕ)Vs). When the bankails while being “complex”, the RA starts to play RAo. RAo prescribeshat a failed “complex” bank is closed with probability q (implyingt+1 = NB and the “simple” strategy by the new investors), or oth-rwise it is bailed out (which implies that the investors will playo from st+1 = B on). Denote the investors’ continuation payoff fromlaying Bo as VB0 . Then VB0 is given by the following equation:
B0 = p(�s + (1 − ϕ)VB0 ) + (1 − p)(�c + (1 − ϕ)VB0 )
+ (1 − p)ϕ(1 − q)(b + VB0 ),
here the bank is “simple” with probability p or “complex” withrobability (1 − p). In the latter case the failed “complex” bankeceives b and is allowed to continue with probability (1 − p)ϕ (−q).hen:
B0 = p�s + (1 − p)�c + (1 − p)ϕ(1 − q)b1 − (1 − ϕ + (1 − p)(1 − q)ϕ)
.
Hence the payoff from deviating from Be is:
�c + (1 − ϕ)Vs) + ϕ(1 − q)(b + VB0 ).
Because the last expression depends on p and q, we have to findixed strategies played by the both parties out of the equilibrium
ath in order to check under which conditions the investors do noteviate from Be.
The mixed strategy of the RA. Third, we have to find the off-quilibrium response of the RA to the failed “complex” banks. Ifhe RA plays a mixed strategy once it deals with the failed “com-lex” banks, it has to be indifferent between closure and bailout.he RA’s utility from closing the failed “complex” banks amounts tohe immediate utility �1 and the future continuation value ıPV(RAe)the closure implies st+1 = NB). The RA’s utility from bailing out theailed “complex” banks amounts to the immediate utility �2 andhe future continuation value ıPV(BOC). PV(BOC) is the RA’s utilityfter the investors play Bo following the bailout and is the solutiono the following equation:
V(BOC) = p[(1 − ϕ)�4 + ϕ�3 + ıPV(BOC)] + (1 − p)[(1 − ϕ)�4
+ ϕ�2 + ıPV(BOC)].
Solving out for PV(BOC) the RA’s indifference condition reads:
1 + ı(1 − ϕ)�4 + ϕ�3
1 − ı= �2 + ı
(1 − ϕ)�4 + ϕ�3 − (1 − p)ϕ(�3 − �2)1 − ı
r
ı
1 − ı(1 − p)ϕ(�3 − �2) = �2 − �1.
Both sides of the last equality are positive because of �3 > �2 > �1.The last equality describes the RA’s reaction to the investors’
ehavior. The RA’s payoff from bailouts is increasing in p: the higherhe probability p that the investors play “simple”, the higher thexpected utility of bailouts of the “complex” banks because theyccur less frequently. From this expression we can derive the RA’sest response given the strategy played by the investors:
close the “compelx” banks for p < p∗indifferent between closing and bailing out the "complex" banks for p = p∗bail out the "complex" banks for p > p∗
here
∗ = 1 −(
1ı
− 1)
�2 − �1
ϕ(�3 − �2).
626 R. DeYoung et al. / Journal of Financial Stability 9 (2013) 612– 627
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he mixed strategies of the investors. Fourth, we derive the off-quilibrium mixed strategy for the investors. After the investorsbserve st = B, they will randomize according to Bo, which requireshat the investors are indifferent between the “simple” and “com-lex” bank. Being “simple” delivers Vs to the investors. Denote asc the investors’ payoff from the “complex” bank. If the “complex”ank is successful, the investors’ expected profit is �c and the futureontinuation value is Vc with the probability 1 − ϕ. If the “com-lex” bank fails with probability ϕ, it is closed with probability q,ut with probability 1 − q it is allowed to continue and receives
+ Vc. Formally Vc comes from the following equation:
c = �c + (1 − ϕ)Vc + ϕ(1 − q)(b + Vc)
nd it is equal to
c = �c + ϕ(1 − q)b1 − (1 − qϕ)
.
The investors are indifferent between the “complex” or “sim-le” bank after a bailout, when Vs = Vc. Solving this equation for qelivers that the investors are indifferent for
∗ = 1 − (1 − (1 − ϕ))ϕ[�s + (1 − (1 − ϕ))b]
(�s − �c) ∈ (0; 1).
* < 1 follows because of �s > �c. q* > 0 followsecause of �s < �c + ϕb. Indeed, q* > 0 is equivalent tos − (ϕ)/(1 − (1 − ϕ))�S < �C + ϕb, which is weaker thans < �c + ϕb. The reaction function of the investors is as fol-
ows. Hence, the investors’ best response is to set up the “complex”ank for q < q*, the “simple” for q > q*, and they are indifferent for
= q*.Finding the mixed strategies. Fifth, we combine the best responses
f the RA and the investors to find the optimal off-equilibriumtrategies. q* is always between 0 and 1. p* is always lower than 1.he following two figures summarize the potential cases depend-ng on the parameters of the model (Fig. A1).
The solid line represents the best response of the RA, q(p), andhe dashed line the one of the investors, p(q). Now we will show thathe only case which supports an equilibrium, in which the investorshoose to the “simple” industry, is the case in which p* is between 0
nd 1. When p* ≤ 0, the RA always bails out the “complex” industry.his cannot lead to a desired equilibrium because the investorsould always choose to the “complex” industry and deviate frome.
line) and the investors (dashed line).
Now, we will check if for parameters such that p* > 0 the result-ng out-of-equilibrium mixed strategies can support the desiredquilibrium. First, p* > 0 holds for
>1
1 + (ϕ(�3 − �2)/(�2 − �1)= ı- ∈ (0; 1).
econd, in order to check whether the investors do not deviate frome, we insert p* and q* in the condition
s ≥ (�c + (1 − ϕ)Vs) + ϕ(1 − q)(b + VB0 ),
erived when checking the one-time deviation from Be.It turns out that the investors are indifferent between deviat-
ng or not. This is intuitive because the investors are indifferentetween the “simple” and “complex” industry out of the equi-
ibrium, so the same has to hold on the equilibrium path. Thisnalizes the proof of the claim in the proposition that the aboveentioned mixed strategies support the disciplinary equilibrium
or any ı ∈ (ı-; 1).�
ppendix B. Comparative static results
Comparative statics for ı-:
∂ı
∂ϕ= −(�3 − �2)/(�2 − �1)
(1 + (ϕ(�3 − �2))/(�2 − �1))2< 0
∂ı
∂�1= − ϕ(�3 − �2)/(�2 − �1)
(1 + (ϕ(�3 − �2))/(�2 − �1))2< 0
∂ı
∂�2= − ϕ(�3 − �2)/(�2 − �1)2
(1 + (ϕ(�3 − �2))/(�2 − �1))2< 0
∂ı
∂�3= − ϕ
(�2 − �1)(1 + (ϕ(�3 − �2))/(�2 − �1))2< 0
Comparative statics for q*:
∂q∗∂
= �s(�s − �c)
ϕ[�s + (1 − (1 − ϕ))b]2> 0
∂q∗∂b
= (1 − (1 − ϕ))2ϕ
ϕ[�s + (1 − (1 − ϕ))b]2> 0
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A
A
A
B
B
B
B
B
C
C
C
C
D
DE
F
F
F
FG
G
H
H
K
K
K
K
K
K
K
K
M
M
M
M
M
M
M
R. DeYoung et al. / Journal of F
To derive the next three results, we treat ϕ, �C, and �S as param-ters that freely vary rather than as functions. Doing so providesome additional intuition.
∂q∗∂�c
= 1 − (1 − ϕ)ϕ[�s + (1 − (1 − ϕ))b]
> 0
∂q∗∂�S
= −(1 − (1 − ϕ))(�C + (1 − (1 − ϕ))b)
ϕ[�s + (1 − (1 − ϕ))b]2< 0
∂q∗∂ϕ
= (�s − �c)((1 − )�s + (1 − (1 − ϕ))2b)
[ϕ[�s + (1 − (1 − ϕ))b]]2> 0
Comparative statics for p*
∂p∗∂ı
= 1ı2
�2 − �1
ϕ(�3 − �2)> 0
∂p∗∂�1
= (1/ı) − 1ϕ(�3 − �2)
> 0
∂p∗∂�2
=(
1ı
− 1)
�3 − �1
ϕ(�3 − �2)2< 0
∂p∗∂�3
=(
1ı
− 1)
�2 − �1
ϕ(�3 − �2)2> 0
Unlike ∂q */∂ϕ above, the following derivative ∂p */∂ϕ is a com-arative static result because ϕ = �C�S increases equally with bothC and �S and that we impose no restrictions on the relative mag-itudes of �C and �S.
∂p∗∂ϕ
=(
1ı
− 1)
�2 − �1
ϕ2(�3 − �2)> 0
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