The Impact of Regulator Leniency on Stress Tests and Bank Lending, Study notes of Finance

Download The Impact of Regulator Leniency on Stress Tests and Bank Lending and more Study notes Finance in PDF only on Docsity! Stress Testing and Bank Lending∗ Joel Shapiro†and Jing Zeng‡ University of Oxford and Frankfurt School of Finance and Management May 2019 Abstract Bank stress tests are a major form of regulatory oversight. Banks respond to the strictness of the tests by changing their lending behavior. As regulators care about bank lending, this affects the design of the tests and creates a feedback loop. We demonstrate that stress tests may be (1) lenient, in order to encourage lending in the future, or (2) tough, in order to reduce the risk of costly bank defaults. There may be multiple equilibria. Regulators may strategically delay stress tests. We also analyze bottom-up stress tests and banking supervision exams. Keywords: Bank regulation, stress tests, bank lending JEL Codes: G21, G28 ∗We thank Matthieu Chavaz, Paul Schempp, Eva Schliephake, Anatoli Segura, Sergio Vicente, Daniel Weagley and the audiences at Amsterdam, ESSEC, Exeter, Loughborough, Lugano, Zurich, the Bundes- bank ”Future of Financial Intermediation” workshop, the Barcelona GSE FIR workshop, the EBC Network conference, and the MoFiR Workshop on Banking for helpful comments. We also thank Daniel Quigley for excellent research assistance. †Säıd Business School, University of Oxford, Park End Street, Oxford OX1 1HP. Email: Joel.Shapiro@sbs.ox.ac.uk ‡Frankfurt School of Finance and Management, J.Zeng@fs.de. 0 1 Introduction Stress tests are a new policy tool for bank regulators that were first used in the recent financial crisis and have become regular exercises subsequent to the crisis. They are assessments of a bank’s ability to withstand adverse shocks, and are generally accompanied by requirements intended to boost the capital of those banks who have been found to be at risk. Naturally, bank behavior reacts to stress testing exercises. Acharya, Berger, and Roman (2018) find that all banks that underwent the U.S. SCAP and CCAR tests reduced their risk by raising loan spreads and decreasing their commercial real estate credit and credit card loans activity.1 Regulators should take into account the reaction of banks when conducting the tests. One might posit that if regulators want to boost lending, they might make stress tests more lenient. Indeed, in the case of bank ratings, Agarwal et al. (2014) show that state level banking regulators rate banks more leniently than federal regulators due to concerns over the local economy and this may lead to more bank failures. In this paper, we study the feedback effect between stress testing and bank lending. Banks may take excess risk or not lend enough to the real economy. Regulators react with either a lenient or tough approach. We demonstrate that this behavior may be self-fulfilling and result in coordination failures. A regulator may prefer to conduct an uninformative stress test or to strategically delay the test. We show that when capital is more available or the bank is more systemic, stress tests will be more informative. We compare stress tests to banking supervision exams, and show that the latter’s relative informativeness depends on lending externalities. In the model, there are two sequential stress testing exercises. For simplicity, there is one bank that is tested in both exercises. Each period, the bank decides whether to make a risky loan or to invest in a risk free asset. The regulator can observe the quality of the risky loan, and may require the bank to raise capital (which we denote as “failing” the stress test) 1Connolly (2017) and Calem, Correa, and Lee (2017) have similar findings. regulator allows the bank to perform the test (as in Europe). We find that results may be more or less informative depending on the value the regulator places on lending vs. costly defaults. In the model, uncertainty about the regulator’s preferences plays a key role. Given that (i) increased lending may come with risk to the economy and (ii) bank distress may have systemic consequences, there is ample motivation to keep this information/intention private. This uncertainty may also arise from the political process. Decisionmaking may be opaque, bureaucratic, or tied up in legislative bargaining.5 Meanwhile, governments with a mandate to stimulate the economy may respond to lobbying from various interest groups or upcoming elections.6 There is little direct evidence of regulators behaving strategically during disclosure exer- cises, but much indirect evidence. The variance in stress test results to date seem to support the idea of regulatory discretion.7 Beyond Agarwal et. al. (2014) cited above, Bird et al. (2015) show U.S. stress tests were lenient towards large banks and strict with poorly capi- talized banks, affecting bank equity issuance and payout policy. The recent Libor scandal revealed that Paul Tucker, deputy governor of the Bank of England, made a statement to Barclays’ CEO that was interpreted as a suggestion that the bank lower its Libor submis- sions.8 Hoshi and Kashyap (2010) and Skinner (2008) discuss accounting rule changes that the government of Japan used to improve the appearance of its financial institutions during the country’s crisis.9 5Shapiro and Skeie (2015) provide examples of related uncertainty around bailouts during the financial crisis. 6Thakor (2014) discusses the political economy of banking. 7The 2009 U.S. SCAP was widely perceived as a success (Goldstein and Sapra, 2014), with subsequent U.S. tests retaining credibility. European stress tests have varied in perceived quality (Schuermann, 2014) with the early versions so unsuccessful that Ireland and Spain hired independent private firms to conduct stress tests on their banks. 8The CEO of Barclays wrote notes at the time on his conversation with Tucker, who reportedly said, “It did not always need to be the case that [Barclays] appeared as high as [Barclays has] recently.” This quote and a report on what happened appear in the Financial Times (B. Masters, G. Parker, and K. Burgess, Diamond Lets Loose Over Libor, Financial Times, July 3, 2012). 9Nevertheless, stress tests do contain significant information that is valued by markets (Flannery, Hirtle, and Kovner (2017) demonstrate this and survey recent evidence). 4 Theoretical Literature There are a few papers on reputation management by a regulator. Morrison and White (2013) argue that a regulator may choose to forbear when it knows that a bank is in distress, because liquidating the bank may lead to a poor reputation about the ability of the regulator to screen and trigger contagion in the banking system.10 Boot and Thakor (1993) also find that bank closure policy may be inefficient due to reputation management by the regulator, but this is due to the regulator being self-interested rather than being worried about social welfare consequences as in Morrison and White (2013). Shapiro and Skeie (2015) show that a regulator may use bailouts to stave off depositor runs and forbearance to stave off excess risk taking by banks. They define the type of the regulator as the regulator’s cost of funding. We also have potential contagion through reputation as in these papers, but model the bank’s lending decision and how it interacts with the choice of the regulator to force banks to raise capital. Furthermore, we define the regulator’s type as whether it lenient (uninformative) or strategic. There are several recent theoretical papers on regulatory disclosure and stress tests. Goldstein and Sapra (2014) survey the stress test and disclosure literature to describe the costs and benefits of information provision. Prescott (2008) argues that more information disclosure by a bank regulator decreases the amount of information that the regulator can gather on banks. Bouvard, Chaigneau, and de Motta (2015) show that transparency is better in bad times and opacity is better in good times. Goldstein and Leitner (2018) find a similar result in a very different model where the regulator is concerned about risk sharing (the Hirshleifer effect) between banks. Williams (2017) looks at bank portfolio choice and liquidity in this context. Orlov, Zryumov, and Skrzypacz (2018) show that the optimal stress test will test banks sequentially. Faria-e-Castro, Martinez, and Philippon (2016) demonstrate that stress tests will be more informative when the regulator has a strong fiscal position (to stop runs). In contrast to these papers, in our model reputational incentives drive the regulator’s 10Morrison and White (2005) also model reputation as the ability of the regulator to screen, but do not consider the effect of ex-post learning about the type of the regulator. 5 choices. In addition, we incorporate capital requirements as a key element of stress testing and focus on banks’ endogenous choice of risk. We also don’t allow the regulator to commit to a disclosure rule (as all of the papers except for Prescott (2008) and Bouvard, Chaigneau, and de Motta (2015) do). Our paper identifies the regulator’s reputation concern as a source of fragility in the banking sector. In a different context, Ordonez (2013, 2017) show banks’ reputation concerns, which provides discipline to keep banks from taking excessive risk, can lead to fragility and a crisis of confidence in the market. 2 The model We consider a model with three types of risk-neutral agents: the regulator, the bank and a capital provider. The model has two periods t ∈ {1, 2} and the regulator conducts a stress test for the bank in each period. We assume that the regulator has a discount factor δ ≥ 0 for the payoffs from the second period bank, where δ may be larger than 1 (as, e.g., in Laffont and Tirole, 1993). The discount factor captures the relative importance of the future of the banking sector for the regulator. For simplicity, we do not allow for discounting within a period. We now provide a very basic timeline of each period, and then proceed in the following subsections to discuss each aspect in detail. In each period t, where t = {1, 2}, there are three stages: 1. Bank investment choice; 2. Stress test and (possible) recapitalization; 3. Payoffs realize. 6 then the expected value of a good loan is lower than than the capital provider’s outside option. This also implies that recapitalization is infeasible for the bad loan. Second, if the opportunity cost of capital is low, then the expected value of a bad loan is higher than the capital providers’ outside option. This also implies that recapitalization is feasible for the good loan. We assume that the capital provider has some bargaining power β due to the scarcity of capital, enabling it to capture a fraction of the expected surplus of the bank. Raising capital thus results in a (private) dilution cost for the bank’s owners. The banking literature generally views equity capital raising as costly for banks (for a discussion, see Diamond (2017)). We model this cost as dilution due to the bargaining power of a capital provider, which fits our scenario of a public requirement by a regulator, though other mechanisms that impose a cost on the bank when trying to shore up capital would also work.16 We make the following assumption on the effect of recapitalization: Assumption 3. [α + γ(1− α)(1− d)] (R− 1) < R0 − 1. This assumption implies that the bank’s owners may not find it worthwhile to originate the risky loan, because the expected payoff may be lower than that from investing in the safe asset (R0− 1). This is because, if the bank originates a risky loan, and the loan is good (with probability α), the bank’s owners receive at most the residual payoff of R − 1 after repaying the debtholders; if the loan is bad, however, the bank’s owners may not receive the value of the loan because it may have to raise capital (with probability 1− γ). 2.4 Regulatory preferences We now examine the externalities from risky lending that affect social welfare and, hence, the regulator’s preferences. 16For example, the bank may be forced to sell assets at fire-sale prices. This is a loss in value for the bank. And those who are purchasing the assets are distorting their investment decisions, as in our model. Hanson, Kashyap, and Stein (2011) discuss this effect and review the literature on fire sales. 9 There are two social costs of risky lending. The first is the cost to society of a bank default at stage 3. Specifically, if a bank operates without being recapitalized and the borrower repays 0 at stage 3, the bank defaults and a social cost to society D is incurred. The cost of bank default may represent cost of financing the deposit insurance payout,17 the loss of value from future intermediation the bank may perform, the cost to resolve the bank, or the cost of contagion. The second social cost of risky lending is the capital provider’s opportunity cost; the alternative investment that goes unfunded when the capital provider recapitalizes the bank. This is only incurred if ρ = ρL. We make the following assumption about the social costs of risky lending. Assumption 4. dD > ρL − 1 > 0. This assumption states that a strategic regulator finds that it is beneficial to recapitalize a bank whose risky loan is known to be bad, but not a bank whose risky loan is known to be good. Finally, we add one more potential externality, which we call the social benefit of risky lending: if the bank originates a risky loan at stage 1, it generates a positive externality equal to B. Broadly, increased credit is positively associated with economic growth and income for the poor (both across countries and U.S. states, see Demirgüç-Kunt and Levine, 2018).18 2.5 Regulator reputation The regulator can be one of two types, a strategic type or a lenient type. The strategic type trades off the social benefits and costs associated with recapitalization when deciding whether to fail a bank. The lenient type is behavioral and always passes the bank. A lenient type could also be considered an uninformative type, as its test does not screen banks. Agents 17The deposit insurance payout would be costly if (i) deposit insurance wasn’t fairly priced, or (ii) there is a cost (e.g., political) of using the deposit insurance fund. 18Moskowitz & Garmaise (2006) provide causal evidence the social effects of credit allocation such as reduced crime. 10 may view this type as not conducting “serious” stress test exercises. The behavior of the lenient type regulator can be microfounded by a high social net benefit of risky lending.19 In subsection 7.1.2, we demonstrate that our qualitative results still hold if we replace the behavioral lenient type with a behavioral tough type who always fails banks and recapitalizes them. The regulator knows its own type, but during the stress testing of bank t (where t = {1, 2}), the owners of the bank and the capital provider are uncertain about the regulator’s type. These agents have a belief that, with probability 1 − zt, the regulator is strategic. With probability zt, the regulator is believed to be a lenient type. In our model, z1 is the probability that nature chooses the regulator to be a lenient type. The term z2 is the updated belief that the regulator is a lenient type after the first period stress test. 2.6 Summary of timing The regulator conducts stress testing of the bank in first period, and then in the second period if the bank has not defaulted in the first period. If the bank defaults in the first period, the bank is closed down and does not continue into the second period. At the beginning of the second period, the beliefs about the type of the regulator will be updated depending on the result of bank’s stress test and the realized payoff of the bank in the first period. The timing is illustrated in Figure 1. We assume that the probabilities that the risky loan opportunity is good in the second period is independent of whether the risky loan opportunity is good in the first period, and that the type of the regulator is independent from the quality of the banks’ risky loans. Furthermore, the regulator’s type remains the same in both periods. We use the equilibrium concept of Perfect Bayesian equilibrium. 19Specifically, if the lenient type regulator has a low cost of bank default D′ or a large benefit from risky lending B′, then passing the bank with certainty is indeed the unique equilibrium strategy. 11 + (1− α)(1− z2)(1− γ)(1− φ)(1− d)R︸ ︷︷ ︸ fail and recapitalized ≥ R0 − 1. (3) The bank originates a risky loan if and only if the expected payoff to the bank’s owners is higher when it originates a risky loan (represented by the left hand side of Eq. 3) than when it invests in the safe investment (represented by the right hand side of Eq. 3). Notice that the expected payoff to the bank’s owners when it originates a risky loan consists of two terms. First, if the bank does not raise capital and there is no default, it receives the net payoff R − 1 at stage 4. This is the case if the loan is good, if the loan is bad and the regulator is lenient so that the bank passes the stress test (and the loan doesn’t default), or if the loan is bad and the bank fails the stress test but recapitalization is infeasible (and the loan doesn’t default). Second, if the bank fails the stress test and is recapitalized, which is the case if the loan is bad and the regulator is strategic, the bank’s owners face dilution during recapitalization and thus their payoff is only the retained share φ of the bank’s equity. The equity is priced after the stress test and reflects the equilibrium choice of the regulator in the second period. Proposition 1. In the second period, there exists a unique equilibrium. There exists a unique threshold z∗2 < 1, such that in equilibrium the bank originates a risky loan if and only if z2 ≥ z∗2, given by ∆(z∗2) = 0, where: ∆(z2) ≡ [α + (1− α)(1− d)] (R− 1)− (R0 − 1)︸ ︷︷ ︸ profit differential without recapitalization − (1− α)(1− z2)(1− γ) (ρL − (1− d) + β [(1− d)R− ρL])︸ ︷︷ ︸ dilution cost of recapitalization . (4) If the bank extends a risky loan at stage 2, the lenient type regulator passes the bank with certainty and the strategic regulator passes the bank with certainty if and only if the bank’s loan is good. Moreover, there exists β̄ < 1, such that z∗2 > 0 if and only if β > β̄. 14 0 1 0.4 0.5 0.6 0.7 0.8 z∗2 z2 Equilibrium project choice Risky loan Safe investment Risky loan Safe investment Figure 2: The expected payoff to the bank’s owners. The parameters used in this plot are: R0 = 1.5, R = 2, α = 0.3, d = 0.4, ρL = 1.1, γ = 0.2, β = 0.25 and D = 0.5. This implies that z∗2 = 0.25. This proposition states that the bank’s incentive to originate a risky loan takes into account two factors. On the one hand, the bank benefits from originating the risky loan because it produces a higher expected profit than the safe investment (the profit differential term in ∆(z2)). On the other hand, the bank faces a dilution cost whenever it is required to raise capital, because the capital provider extracts rents (the dilution cost in ∆(z2)). When recapitalized, which occurs with probability (1−α)(1−z2)(1−γ), the bank’s cost of funding increases from 1 − d to ρL + β [(1− d)R− ρL]. Here, 1 − d represents the bank’s cost of repaying depositors, taking into account the deposit insurance, and ρL +β [(1− d)R− ρL] is how much the bank must pay the capital provider. Since the bank only faces the possibility of failing the stress test and thus having to raise capital if it extends a risky loan, the bank only originates the risky loan if the gains from higher NPV outweighs the potential dilution cost of recapitalization. Importantly, the bank originates a risky loan only if the regulator’s reputation of being the lenient type is sufficiently high, i.e. z2 ≥ z∗2 , as illustrated in Figure 2. This is because the lenient type regulator does not require the bank to raise capital even if the bank’s risky loan is bad (Table 1), reducing the expected dilution cost to the bank. Since the regulator’s reputation z2 determines the bank’s investment decision in equilib- 15 rium, we now turn to understanding how the regulator’s reputation affects surplus. Let UR and U0 denote the strategic regulator’s expected surplus from the bank in the second period when the bank originates a risky loan and invests in the safe asset, respectively. We can express the expected surplus as follows: UR = [α + (1− α)(1− d)]R− 1 +X, (5) U0 = R0 − 1, (6) where X represents the net social costs of risky lending, given by: X ≡ B − (1− α) [γdD − (1− γ)(ρL − 1)] . (7) When the bank extends a risky loan, the strategic regulator internalizes the net social benefits of risky lending, consisting of the positive externality of bank lending B as well as the social costs of a potential bank default. Conditional on a bad loan (with probability 1−α), the expected social costs of a potential bank default include the expected cost of bank default dD if recapitalization is infeasible (with probability γ) and the forgone net return from the capital providers’ alternative investment ρL−1 when the bank is recapitalized (with probability 1 − γ). Notice that X encapsulates all of the externalities from risky lending; the following analysis will only use X rather than the individual components. It then follows from Proposition 1 that the strategic regulator’s expected surplus, for a given reputation z2, denoted by U(z2), is given by U(z2) =  UR, if z2 > z∗2 , U0, if z2 < z∗2 , λUR + (1− λ)U0 for some λ ∈ [0, 1], if z2 = z∗2 , (8) where we have taken into account that, if z2 = z∗2 , the bank is indifferent between originating 16 type, given that the bank fails the stress test in the first period. The first period bank surplus effect is positive if the first bank’s risky loan is good as there is no risk of default. This effect is negative if the risky loan is bad, given Assumption 4. The reputation effect depends on the regulator’s posterior reputation after it grades the first period bank and the payoffs are realized. Given that the lenient type regulator passes the bank, if the strategic regulator fails the bank in the first stress test, it is revealed to be strategic (zf2 = 0); the bank will then realize that it will be recapitalized in the second period if its risky investment is of bad quality. In contrast, if the strategic regulator passes the bank in the first stress test, it is pooled with the lenient type regulator who also passes the bank. In equilibrium, the posterior probability that the regulator is the lenient type, given that the bank passes the first stress test and then realizes a payoff of R is given by zR2 (πg, πb) = [α + (1− α)(1− d)]z1 [α + (1− α)(1− d)]z1 + [απg + (1− α)(1− d)πb](1− z1) (10) where πg and πb denote the strategic regulator’s probability of passing the bank in the first stress test, given that the bank’s risky loan is good and bad, respectively. As a result, passing or failing the bank in the first period may lead to different investment decisions by the bank in the second period, and hence has a reputation effect. The following lemma establishes the set of possible equilibrium stress testing strategies, which narrows down our analysis. Lemma 1. In any equilibrium, the stress testing strategy of the strategic regulator is either: Informative: it passes the first bank if and only if the bank’s risky loan is good; Lenient: it passes the bank with certainty if the bank’s risky loan is good, and passes the bank with positive probability π∗b > 0 if the loan is bad; or Tough: it passes the bank with probability π∗g < 1 if the bank’s risky loan is good, and fails the bank with certainty if the loan if bad. 19 This lemma follows from the fact that, in any equilibrium, the strategic regulator faces strictly greater incentives to pass a bank with a good risky loan than a bank with a bad risky loan. This can be seen in Eq. 9. Passing a bank with a bad risky loan results in a possible costly default, while passing a bank with a good risky loan does not have this possibility. A bank default generates two costs. First, it generates a social cost of default D in the first period. Second, it leads to a loss of expected surplus U(z2) in the second period. Lemma 1 thus represents all possible equilibrium strategies of the strategic regulator in the first period: every possibility where the probability with which the strategic regulator passes a bank with a good risky loan is weakly larger than the probability with which it passes a bank with a bad risky loan. We now show that for intermediate levels of the net social benefits of risky lending X, there is a unique equilibrium in which the strategic regulator’s stress testing strategy in the first period is identical to its strategy in the second period. Proposition 2. There exist cutoffs X and X, with X < X, such that for X ∈ [X,X], there exists a unique equilibrium in the first period in which the stress testing strategy of the strategic regulator in the first period is identical to that in the second period described in Proposition 1, and is fully informative. For intermediate levels of the net social externality of bank lending (X ∈ [X,X]), the expected surplus for the strategic regulator is not too sensitive to the bank’s investment decision in the second period. That is, for intermediate X, the social values of the risky project and the safe asset are close, so the bank’s investment choice in the second period does not affect the regulator’s surplus much and the regulator can choose its static optimum stress testing strategy in the first period. Proposition 2 shows that, in this case, the equilibrium is unique and is fully informative. In the following sections, we will show that the other two types of equilibria described in Lemma 1 can arise if the net social benefits of risky lending X is either low or high, and depend on the regulator’s reputation building incentives. 20 4.1 High net social benefits of risky lending X > X If the strategic regulator fails the first period bank and recapitalizes it, the regulator reveals the fact that it is strategic. The bank then faces a strong incentive to invest in the safe investment in the second period, in order to avoid failing the stress test. If the strategic regulator passes the bank in the first period, however, it pools with the lenient regulator, increasing the incentive for the bank to engage in risky lending in the second period. If the benefit of lending by the bank in the second period is sufficiently large, the regulator may want to pass the bank in the first period, even when its risky loan is bad, in order to gain a reputation of leniency. In the following proposition, we demonstrate that for high X, there is still an equilibrium in which the strategic regulator’s stress testing strategy in the first period is identical to its strategy in the second period, but reputation building incentives to encourage lending by the second period bank can lead to another equilibrium for the bank’s stress test. Proposition 3. For high net social externalities of bank default X > X, there exists an equilibrium in the first bank’s stress test that is either fully informative or lenient (as described in Lemma 1). There exist δ̄b, δb ∈ R+ ∪ {∞}, with δ̄g ≥ δb, such that • the fully informative equilibrium exists if and only if δ ≤ δ̄b; and • a lenient equilibrium exists if and only if δ ≥ δb. Moverover, δb 6= ∞ if and only if β ≥ β̄ and z1 > z̄1, and δ̄b 6= ∞ if and only if β > β̄ and z1 > z̄1, where β̄ is defined in Proposition 1. In particular, there exists δb, such that the equilibrium is unique unless δ = δb or β = β. Proposition 3 shows that, for certain parameters, the equilibrium stress testing strategy of the regulator in the first period is the same as its strategy in the second period, and is illustrated in Table 1. However, this proposition also shows that the strategic regulator’s reputation building incentives to encourage lending by the bank in the second period can 21 Strategic regulator Lenient regulator q1 = g Pass with probability π∗g < 1 Pass q1 = b Fail Pass Table 3: Equilibrium stress testing in the first period when the strategic regulator wants to build reputation to reduce excessive risk-taking by the second bank in the second period. Proposition 4 shows that, for certain parameters, the equilibrium stress testing strategy of the regulator in the first period is the same as its strategy in the second period, and is illustrated in Table 1. However, this proposition also shows that the strategic regulator’s reputation building incentives to reduce excessive risk taking by the bank in the second period can lead to an equilibrium with a tough stress test in the first period. Table 3 depicts the stress testing in the tough equilibrium in the first period described in Proposition 4. In the fully informative equilibrium, the strategic regulator passes the bank with a good risky loan in the first period, which maximizes the expected surplus from the bank. Failing the bank in this case would possibly result in a costly recapitalization of the bank with no benefit, since the good loan will not default. However, in the tough equilibrium, by failing this bank, the strategic regulator is able to reveal its willingness to recapitalize a bank, and thus reduce the bank’s incentive to engage in excessive risk taking in the second period. Proposition 4 identifies two further necessary and sufficient conditions for an equilibrium with reputation building to reduce excessive risk-taking to exist. First, the private cost of capital (β) must be sufficiently high. This makes it possible to make the bank refrain from taking excessive risk. Second, the reputation concern (δ) of the regulator must be sufficiently high, so that the regulator’s reputational benefit outweighs the short-term efficiency loss when recapitalizing the bank with a good risky loan. Proposition 4 also indicates that the informative and tough equilibria coexist when δ ∈ (δg, δ̄g). This is due to a strategic complementarity between the regulator’s first period stress test and the bank’s belief updating process in the second period. We discuss this in more detail in the next section. U.S. stress tests have generally been regarded as much more strict than European ones. 24 First, the Federal Reserve performs the stress test itself on data provided by the banks (and does not provide the model to the banks), whereas in Europe, it has been the case that the banks themselves perform the test. Second, the U.S. stress tests have regularly been accompanied by Asset Quality Reviews, whereas this has been infrequent for European stress tests. Third, one of the most feared elements of the U.S. stress tests has been the fact that there is a qualitative element that can (and has been) used to fail banks.20 In line with our results above, the fact that U.S. stress tests have been institutionalized as occurring on a yearly basis implies that reputation concerns are important. Furthermore, a swifter recovery from the crisis means that capital raising for banks was likely to be easier in the U.S. 5 Discussion Having characterized the equilibria of the model, we discuss the implications of the model in this section. First, we examine the reasons for equilibrium multiplicity. Second, we point to the possibility that stress tests may be strategically delayed. Third, we consider how stress tests may vary with the availability of capital. Finally, we explore the implications of the model for stress test design when banks are systemic. 5.1 Multiplicity Proposition 4 imply that there exist parameter values for which the fully informative equilib- rium coexists with a tough equilibrium. This is because the regulator’s reputation concern is self-fulfilling. Specifically, the strategic regulator’s stress testing strategy and the bank’s belief updating process are strategic complements when the net social benefits of risky lending are low (X < X). Here, the bank realizes that the strategic regulator’s surplus is lower when the 20The qualitative element for domestic banks was removed for domestic banks in March 2019 (see “US financial regulators relax Obama-era rules,” by Kiran Stacey and Sam Fleming, Financial Times, March 7, 2019). 25 bank makes the risky loan and therefore when the strategic regulator is perceived to be the lenient type. If the bank conjectures that the strategic regulator adopts a tougher stress test strategy (lower πg), the bank infers that the regulator who passes the bank in the first period is more likely to be lenient (higher zR2 ). Consequently, the bank increases its risk- taking in the second period after a pass result in the first period, resulting in even lower expected surplus for the strategic regulator. In turn, this further decreases the net gain for the strategic regulator from passing the bank in the first period, justifying a tougher testing strategy. It is indeed this strategic complementarity that leads to equilibrium multiplicity. By contrast, the regulator’s stress testing strategy and the bank’s belief updating process are strategic substitutes when the net social benefit of risky lending is high (X > X). Here, the bank realizes that the strategic regulator’s surplus is larger when the bank extends the risky loan and therefore when the strategic regulator is perceived to be the lenient type. If the bank conjectures that the strategic regulator adopts a lenient stress testing strategy (higher πb), the bank infers that the regulator who passed the bank in the first period is more likely to be the strategic type (lower zR2 ). Consequently, the bank may refrain from originating a risky loan in the second period after a pass result in the first period, resulting in lower expected surplus for the strategic regulator. In turn, this reduces the net gain for the strategic regulator from passing the bank in the first period.21 Because of this strategic substitutability, the type of equilibrium multiplicity does not arise.22 21While formally, multiplicity arises only when X is high, the driver for this is the assumption that the behavioural type is lenient. The strategic complementarity flips (i.e., there is co-existence of informative and lenient equilibria) when we change the behavioral type to being tough. We describe this in Section 7.1.2. 22Note that Proposition 3 implies that there exists multiplicity in the knife-edge cases when δ = δb and when β = β̄, but for different reasons than the strategic complementarity discussed above. In both cases, the fully informative equilibrium coexists with a lenient equilibrium. If δ = δb, then in both types of equilibria, the bank invests in the risky loan in the second period if and only if it passes the stress test in the first period. Such multiplicity stems from the fact that the bank’s investment decision in the second period follows a threshold strategy. Therefore a range of stress testing strategies (in terms of the mixing probability πb) leads to posterior beliefs held by the bank that are consistent with the same investment strategy in the second period, implying the same reputation effect on the regulator’s stress testing incentives that justifies the mixed strategies. If β = β̄, then in any equilibrium, after the failure of the first bank, the posterior reputation of the regulator is zf2 = 0, such that the bank is indifferent between investments in the second period. As a result, risky lending by the bank in the second period after failing the stress test in the first period justifies a fully informative equilibrium, while safe investment by the bank in the second period after failing the stress test in the first period justifies a lenient equilibrium. 26 tion building incentives to incentivize lending (X > X). This is because when consider- ing whether to pass a bad bank in the first period, the strategic regulator trades off the cost/benefit of recapitalizing the bank in the first period against the regulator’s cost/benefit of affecting the bank’s investment decision in the second period. While the reputation effect depends only on the bank’s updated belief about the regulator’s type, the cost of passing a bad bank in the first period is larger if the cost of a potential bank failure in the first period is larger. The strategic regulator’s stress testing strategy when facing reputation building incentives to curb excessive risk taking (X < X) is unaffected, since in this case the strategic regulator’s main focus is whether to pass or fail a good bank, which does not run the risk of default. In both U.S. and Europe there have been ongoing debates since the inception of stress tests about how large/systemic a bank must be in order to be included in the stress testing exercise. To the extent that larger and more systemic banks have higher expected cost of default, our model predicts that they should be subject to (weakly) more informative tests. 6 The bank and the regulator both learn the asset quality In this section, we consider a stress test where the signal about the quality qt of the bank’s risky loan observed by the regulator during the stress test is also observed by the bank. This could be the case because: • The stress test only uncovers the private information the bank already has about its loan quality.23 This is indeed the case for banking supervision examinations. These exams are conducted on a regular basis by collecting information and assessing the health of banks on multiple dimensions and have real effects.24 They do not use 23For example, in Walther and White (2018) the regulator and the bank both observe the bank’s asset value, while creditors do not. They consider the effectiveness of bail-ins in this scenario. 24Agarwal et. al. (2014) demonstrate real effects of exams: leniency leads to more bank failures, a 29 information from the entire banking system to assess the position of each bank (which can be the source of the regulator’s private information in the baseline model). In the United States, this has historically been conducted using the CAMEL rating system, though in recent years variations on this rating system have been implemented;25 or • The stress test produces/uncovers new information but regulators share that informa- tion with the bank. This second case resembles the European stress test exercises, which use a “bottom-up” approach where the regulator provides the model and basic parameters to banks, who perform the test themselves.26 In contrast, the U.S. uses a “top-down” approach where the regulators does the test themselves and do not provide all of the information about the model or results.27 As in the baseline model, the equilibrium in the second period for given belief z2 held by the bank is as described in Lemma 1. Unlike in the baseline model, here the bank in the second period forms posterior beliefs z2 = zR2,q1 (z2 = zf2,q1) about the probability that the regulator is lenient given the bank passes (fails) the stress test in the first period and the loan quality in the first period is q1. Therefore in the first period, taking the bank’s posterior beliefs described above as given, the incentives of the strategic regulator to pass the bank is characterized by Gq1(z R 2,q1 , zf2,q1), where Gq1(·) is defined by Eq. 9. In equilibrium, as in the baseline model, the posterior belief that the regulator is lenient given that the bank fails the stress test in the first period is zf2,q1 = 0, since only a strategic higher proportion of banks unable to repay TARP money in the crisis, and a larger discount on assets of banks liquidated by the FDIC. Hirtle, Kovner, and Plosser (2018) demonstrate real effects of more banking supervision effort (measured by hours). 25The RFI/C(D) system was recently supplanted by the LFI system for large financial institutions. See (https://www.davispolk.com/files/2018-11- 06 federal reserve finalizes new supervisory ratings system for large financial institutions.pdf). 26Note that we do not model the inherent moral hazard problem when a bank is permitted to do its own stress test. 27See Baudino et al (2018) and Niepmann and Stebunovs (2018) for a discussion of top-down vs. bot- tom up approaches. The U.S. has recently made more information available about its test after com- plaints about opacity by banks (https://uk.reuters.com/article/uk-usa-fed-stresstests/fed-gives-u-s-banks- more-stress-test-information-unveils-2019-scenarios-idUKKCN1PU2GE). 30 regulator fails a bank. In addition, the posterior belief of the bank that the regulator is lenient given that it had a loan of quality q1 and passed the first stress test is given by: zR2,q1(πq1) = z1 z1 + (1− z1)πq1 . (11) We can now compare the results in this case to those in the baseline model and examine the effect of bank information on the equilibrium stress testing strategy of the regulator. Proposition 5. The equilibria when the bank has information about the risky loan’s quality qt are characterized as follows. • For intermediate levels of net social benefits of risky lending X ∈ [X,X], there exists a unique informative equilibrium (as described in Proposition 2). • For high net social benefits of risky lending X > X, there exists an equilibrium for the first bank’s stress test, which is either fully informative or lenient. The parameter space in terms of (β, z1, δ) for which the fully informative equilibrium exists is strictly smaller than in the baseline model, and that for which the lenient equilibrium exists is strictly larger than in the baseline model. • For low net social benefits of risky lending X < X, there exists an equilibrium for the first bank’s stress test, which is either fully informative or tough. The parameter space in terms of (β, z1, δ) for which the fully informative equilibrium exists is strictly larger than in the baseline model, and that for which the tough equilibrium exists is identical to the baseline model. Proposition 5 provides two key insights. First, when the net social benefit of risky lending is high (X > X), the bank having knowledge of the risky loan’s quality exacerbates the strategic regulator’s reputation concerns, resulting in a less informative stress test. The reason is as follows. Since the strategic regulator is more likely to pass a bank with a good loan than a bank with a bad loan, after a pass on the first stress test, the bank’s posterior 31 the behavioral regulator is lenient are also the only types of equilibria when the behavioral regulator is tough. Proposition 7. Consider the model with a tough type regulator. In the second period, there exists a unique equilibrium in which, if the bank extends a risky loan, the strategic regulator passes the bank with certainty if and only if the bank’s loan is good. An equilibrium exists in the first period, and the possible equilibria are as described in Lemma 1. • For X ∈ [X,X], there exists a unique equilibrium in which the stress testing strategy of the strategic regulator in the first period is identical to that in the second period. • For X > X, the equilibrium is either fully informative or lenient. There can exist a unique equilibrium, or a fully informative equilibrium can coexist with a lenient equi- librium. • For X < X, the equilibrium is either fully informative or tough. There can exist a unique equilibrium, or a fully informative equilibrium can coexist with a tough equilib- rium. In particular, there exists δg, such that the equilibrium is unique unless δ = δg or β = β̄. Notice that, in contrast to the baseline model, when the behavioral regulator is a tough type, equilibrium multiplicity that arises due to the regulator’s self-fulfilling reputation con- cern exists only with a lenient equilibrium, but not with a tough equilibrium. This is because, in this case, the strategic regulator’s stress testing strategy and the bank’s belief updating process are strategic complements only when the net social benefit of risky lending is high (X > X). Here, the bank realizes that the strategic regulator’s surplus is larger when the bank originates the risky loan and the regulator is perceived to be the lenient type. If the bank conjectures that the strategic regulator adopts a more lenient stress test strategy (higher πb), the bank infers that the regulator who passes the bank in the first period is 34 more likely to be the strategic regulator (who would be relatively lenient). Consequently, the bank is more likely to originate a risky loan in the second period after a pass result in the first period, resulting in higher expected surplus for the strategic regulator. This further increases the net gain for the strategic regulator from passing the bank in the first period, justifying a more lenient testing strategy.28 7.2 No deposit insurance In this section, we no longer assume that the bank is funded by fully insured deposits. Instead, we assume that, at stage 1 of each period, the bank raises one unit of funds from- competitive debtholders maturing at stage 1.5 with an exogenous repayment of 1.29 Then bank then chooses whether to invest in the safe investment or originate a risky loan. At stage 1.5, the bank must rollover its debt. After observing the bank’s investment decision, the debtholders require a promised repayment of R̃t ∈ [1, R] at stage 3. Accordingly, should the bank fail the stress test at stage 2, it is required to raise R̃t unit of capital in order to eliminate bank default. Notice that Assumption 1 implies that the bank is always solvent, and thus financing at stage 1 and refinancing at stage 1.5 are always feasible. In order to account for the potentially higher amount of recapitalization, we modify Assumptions 2 and 4, respectively, as follows: Assumption 5. R < ρH and (1− d)R ≥ RmaxρL, where Rmax ≡ 1 α+(1−α)(1−d) . Assumption 6. dD > Rmax(ρL − 1) > 0. The maximum promised repayment Rmax is derived given debtholders’ belief that the bank is not going to be recapitalized. We can now solve the model by backward induction and show that the main results of the baseline model remain valid. 28Like in the baseline model, there exists multiplicity when the net social benefit of risky lending is low X < X due to the threshold nature of the bank’s investment decision in the second period. 29While the promised repayment can be greater than 1, the renegotiation-proof repayment is equal to 1 assuming that the bank’s liquidation value is equal to 1. 35 7.2.1 Stress testing in the second period We begin by characterizing the equilibrium in the second period. If the bank makes a safe investment at stage 1, it is clear that the bank will not default and therefore requires no capital at stage 2. The bank rolls over its debt at stage 1.5 with a promised repayment of R̃2 = 1. If the bank makes a risky investment at stage 1, Assumption 6 ensures that the strategic regulator passes the bank if and only if the loan is good, as depicted in Table 1. At stage 2, given the stress test result, the bank raises R̃2 unit of capital if it fails the stress test. Analogous to Eq. 2, the equity given to the capital providers is a fraction φ(R̃2), determined by φ(R̃2)(1− d)R = R̃2ρL + β [ (1− d)R− R̃2ρL ] . Anticipating the stress testing strategy of the regulator at stage 2, we can now proceed back to the point when the bank rolls over its maturing debt at stage 1.5 and determine the promised repayment R̃2(z2) to debtholders: [α + (1− α) (1− [z2 + (1− z2)γ] d)] R̃2(z2) = 1. Eq. 13 takes into account that debtholders will be repaid if the loan doesn’t default. There is no default if the is of good quality (with probability α). If the loan is of bad quality (with probability 1−α), defaults with probability d if (i) the regulator is lenient (with probability z2) or (ii) the regulator is strategic (with probability 1−z2) but recapitalization is unfeasible (with probability γ). We can now analyze the bank’s investment decision at stage 1. At stage 1, given the promised repayment R̃2 to debtholders and the fraction of equity φ it will need to sell to capital providers in exchange for capital, the bank originates a risky loan if and only if [α + (1− α) [z2 + (1− z2)γ] (1− d)] ( R− R̃2 ) 36 baseline model. Moreover, another type of equilibrium multiplicity can arise for high net social benefits of risky lending (X > X) due to endogenous social cost of recapitalizing the bank. Specifically, if the investors conjecture that the strategic regulator adopts a more lenient stress test strategy (higher πb), they require a higher promised repayment R̃1 since they expect higher probability of default. Consequently, the regulator would have to require the bank to raise a higher amount of capital (equal to R̃1) to eliminate a potential bank default, increasing the social cost of recapitalizing the bank. In turn, this raises the net gain for the strategic regulator from passing the bank, justifying a more lenient stress testing strategy. 8 Conclusion Stress tests have been incorporated recently into the regulatory toolkit. The tests provide assessments of bank risk in adverse scenarios. Regulators respond to negative information by requiring banks to raise capital. However, regulators have incentives to be tough by asking even some safe banks to raise capital or to be lenient by allowing some risky banks to get by without raising capital. These incentives are driven by the weight the regulator places on lending in the economy versus stability. Banks respond to the leniency of the stress test by altering their lending policies. We demonstrate that in equilibrium, regulators may be tough and discourage lending or lenient and encourage lending. These equilibria can be self-fulfilling and the regulator may get trapped in one of them, leading to a loss of surplus. Banking supervision exams will lead to similar results but be less informative. It would be of great interest to study regulators’ reputation and effects on the real economy when stress tests deal with multiple banks in a macroprudential setting. 39 References [1] Acharya, Viral, Allen Berger, and Raluca Roman. 2018. Lending Implications of U.S. Bank Stress Tests: Costs or Benefits?. Journal of Financial Intermediation. 34: 58-90. [2] Agarwal, S., D. Lucca, A. Seru, and F. Trebbi. 2014. Inconsistent Regulators: Evidence from Banking, Quarterly Journal of Economics, 129(2), pp.889- 938. [3] Baudino, P., Goetschmann, R., Henry, J., Taniguchi, K. and W. Zhu. 2018.Stress- testing banks – a comparative analysis. Bank of International Settlements, FSI Insights on policy implementation 12. [4] Bird, A., S. A. Karolyi, T. G. Ruchti, and A. Sudbury. 2015. Bank Regulator Bias and the Efficacy of Stress Test Disclosures. Mimeo. [5] Boot, A. W. A., and A. V. Thakor. 1993. Self-interested bank regulation. American Economic Review 83:206–11. [6] Bouvard, M., P. Chaigneau, and A. de Motta. 2015. Transparency in the financial system: Rollover Risk and Crises. Journal of Finance, 70, 1805–1837. [7] Calem, P., Correa, R., Lee, S.J. 2017. Prudential Policies and Their Impact on Credit in the United States Working Paper. [8] Cho, I.-K., and D. M. Kreps. 1987. Signaling games and stable equilibria. Quarterly Journal of Economics 102:179–221. [9] Connolly, M.F. 2017. The Real Effects of Stress Testing in a Financial Crisis: Evidence from the SCAP. Working Paper. [10] Demirgüç-Kunt, A and R Levine (2018): Finance and growth, Cheltenham: Edward Elgar Publishing. 40 [11] Diamond, W. 2017. Safety Transformation and the Structure of the Financial System. Working Paper. [12] Eisenbach, T. M., Lucca, D. O., and R. M. Townsend, 2017, The economics of bank supervision, Working paper Federal Reserve Bank of New York. [13] Faria-e-Castro, M., Martinez, J., Philippon, T., 2016. Runs versus Lemons: Information Disclosure and Fiscal Capacity. Review of Economic Studies, forthcoming. [14] Flannery, M., B. Hirtle, and A. Kovner. 2017. Evaluating the information in the federal reserve stress test. Journal of Financial Intermediation 29, 1–18. [15] Garmaise, M., and Moskowitz T. 2006. Bank mergers and crime: the real and social effects of credit market competition. J. Finance 61(2):495–538. [16] Goldstein, I., and Y. Leitner. 2018. Stress tests and information disclosure. Journal of Economic Theory, forthcoming. [17] Goldstein, I. and H. Sapra. 2014. “Should Banks’ Stress Test Results be Disclosed? An Analysis of the Costs and Benefits.” Foundations and Trends in Finance. Volume 8, Number 1. [18] Hanson, S., Kashyap, A., and Stein, J., 2011. A macroprudential approach to financial regulation. Journal of Economic Perspectives 25, 3–28. [19] Hirtle, B., Kovner, A., and M. Plosser. 2018. The Impact of Supervision on Bank Performance. Mimeo. [20] Hoshi, T., and A. Kashyap. 2010. Will the U.S. bank recapitalization succeed? Eight lessons from Japan. Journal of Financial Economics 97:398–417. [21] Laffont, J.-J. and J. Tirole. 1993. Theory of incentives in procurement and regulation. Massachusetts: MIT Press. 41 9 Proofs 9.1 Proof of Proposition 1 ∆(z2) given by Eq. 4 is obtained by substituting Eq. 2 into Eq. 3 to eliminate φ and rearranging. Notice that ∆(z2) is strictly increasing in z∗2 . Moreover, ∆(1) > 0 as implied by Assump- tion 1, and ∆(z2) → −∞ as z2 → −∞. Therefore a unique z∗2 as defined by ∆(z∗2) = 0 exists, where z∗2 < 1. We now derive a condition for z∗2 > 0. This is the case if and only if ∆(0) = [α + γ(1− α)(1− d)] (R− 1)− (R0 − 1) − (1− α)(1− γ)(1− β) [(1− d)R− ρL] < 0. (16) Notice the above expression is strictly decreasing in β. Moreover, Assumption 3 implies that ∆(0) < 0 for β = 1. Therefore there exists a unique β̄ < 1, such that z∗2 > 0 if and only if β > β̄, where β̄ is defined by [α + γ(1− α)(1− d)] (R− 1)− (R0 − 1)− (1− α)(1− γ)(1− β̄) [(1− d)R− ρL] = 0. (17) 9.2 Proof of Proposition 2 Let X be defined such that UR = U0, i.e., [α + (1− α)(1− d)]R− 1 +X = R0 − 1. (18) 44 Let X be defined such that (1− d)UR = [(1− d) + d(1− γ)]U0, i.e., (1− d) ( [α + (1− α)(1− d)]R− 1 +X ) = [(1− d) + d(1− γ)] (R0 − 1). (19) It is straightforward to show that X > X. We now consider the case where X ∈ [X,X]. Notice that X ≥ X and the fact that zR2 (πg, πb) ≥ zf2 = 0 for all (πg, πb) implies that UR ≥ U(zR2 (πg, πb)) ≥ U(zf2 ) ≥ U0. It follows that: Gg(z R 2 (πg, πb), 0) = (1− γ)R̃1(πb)(ρL − 1) + δ [ U(zR2 (πg, πb))− U(zf2 ) ] ≥ (1− γ)R̃1(πb)(ρL − 1) > 0. (20) Moreover, this also implies that: Gb(z R 2 (πg, πb), z f 2 ) = (1− γ) [ R̃1(πb)(ρL − 1)− dD ] + δ [ (1− d)U(zR2 (πg, πb))− [(1− d) + d(1− γ)]U(zf2 ) ] ≤ (1− γ) [ R̃1(πb)(ρL − 1)− dD ] + δ [(1− d)UR − [(1− d) + d(1− γ)]U0] < 0, (21) where the second inequality follows because of Assumption 4 and X ≤ X. Since Gg(z R 2 (πg, πb), 0) > 0 > Gb(z R 2 (πg, πb), z f 2 ) for all (πg, πb), in this case there exists a unique equilibrium in which the strategic regulator passes the bank in the first period if and only if the risky loan is good. 9.3 Proof of Proposition 3 Since X > X > X, we have Gg(z R 2 (πg, πb), 0) > 0 for all (πg, πb) as shown in the proof of Proposition 2. Therefore πg = 1 in any equilibrium. 45 Before we proceed, given πg = 1, we establish some properties of Gb(z R 2 (1, πb), z f 2 ), where we have zR2 (1, πb) > zf2 = 0. • If β = β̄, where β̄ is defined in Proposition 1, then Eq. 8 implies that U(zR2 (1, πb)) = UR for all πb ∈ [0, 1] and U(zf2 ) takes a continuum of values in [U0, UR]. In turn, this implies that Gb(z R 2 (1, πb), z f 2 ) takes a continuum of values between (1− γ) [(ρL − 1)− dD] < 0 and (1− γ) [(ρL − 1)− dD] + δ [(1− d)UR − [(1− d) + d(1− γ)]U0]. • If β 6= β̄, then z∗2 > 0 and let us define π̂b such that zR2 (1, π̂b) = [α + (1− α)(1− d)] z1 [α + (1− α)(1− d)] z1 + [α + (1− α)(1− d)π̂b] (1− z1) = z∗2 . (22) Notice that zR2 (1, πb) is strictly decreasing in πb, therefore Gb(z R 2 (1, πb), 0) is continuous and decreasing in πb. Specifically, Gb(z R 2 (1, πb), 0) is equal to (1− γ) [(ρL − 1)− dD] + δ [(1− d)UR − [(1− d) + d(1− γ)]U0] for all πb < π̂b, is equal to (1−γ) [(ρL − 1)− dD] < 0 for all πb > π̂b, and takes a continuum of values in between at πb = π̂b. We can now characterize the two possible types of equilibrium: a fully informative equi- librium with πb = 0 and a lenient equilibrium with πb > 0. First, an equilibrium with πb = 0 exists if and only if Gb(z R 2 (1, 0), 0) ≤ 0. Notice that Gb(z R 2 (1, 0), 0) > 0 implies that U(zR2 (1, 0)) > U(zf2 ), which in turn implies that Gb(z R 2 (1, 0), 0) is strictly increasing in δ and that zR2 (1, 0) ≥ z∗2 ≥ 0. Therefore there exists δ̄b(β, z1) ∈ R+∪{∞}, such that an equilibrium with πb = 0 exists if and only if δ ≤ δ̄b(β, z1), where δ̄g(z1) 6=∞ if and only if zR2 (1, 0) > z∗2 > 0. More specifically, δ̄b(β, z1) is defined by δ̄b(β, z1) =  δb, if β > β̄ and z1 > z̄1, ∞, otherwise. (23) where β > β̄ ensures that z∗2 > 0 by Proposition 1 and z̄1 is defined such that zR2 (1, 0) = z∗2 , 46 definition of z̄1 given by Eq. 26, and δg is defined by (1− γ)(ρL − 1) + δg [UR − U0] = 0. (29) Second, an equilibrium with πg < 1 exists if and only if Gg(z R 2 (πg, 0), 0) ≤ 0 at some πg < 1. Since we have shown above that Gg(R̃1(0), zR2 (πg, 0), 0) is increasing in πg, an equilibrium with πg < 1 exists if and only if Gg(R̃1(0), zR2 (0, 0), 0) = Gg(R̃1(0), 1, 0) ≤ 0. Notice that Gg(R̃1(0), 1, 0) ≤ 0 implies that Gg(R̃1(0), 1, 0) is strictly decreasing in δ and that 1 ≥ z∗2 ≥ 0. Therefore there exists δg(z1) ∈ R+ ∪ {∞}, such that an equilibrium with πg < 1 exists if and only if δ ≥ δg(β, z1), where δg(z1) 6=∞ if and only if 1 ≥ z∗2 ≥ 0. More specifically, δg(β, z1) is defined by δg(β, z1) =  δg, if β ≥ β̄, ∞, otherwise, (30) where β ≥ β̄ ensures that z∗2 ≥ 0 by Proposition 1, and δg(z1) is defined by Eq. 31. 1 > z∗2 is satisfied by Proposition 1. To summarize, an equilibrium with πg = 1 exists if and only if δ ≤ δ̄g(β, z1), whereas an equilibrium with πg < 1 exists if and only if δ ≥ δg(β, z1). Finally, it is immediate from Eqs. 30 and 32 that δ̄g(β, z1) ≥ δg(β, z1), with strict inequality if and only if β ≥ β̄ and z1 ≤ z̄1. 9.5 Proof of Corollary 1 Using similar logic as the proof of Proposition 3, an equilibrium with πb = 1 exists if and only if Gb(z R 2 (1, 1), 0) = Gb(z1, 0) ≥ 0. Notice that Gb(z1, 0) ≥ 0 implies that Gb(z1, 0) is strictly increasing in δ and that z1 ≥ z∗2 ≥ 0. an equilibrium in which πb exists if and only if z1 ≥ z∗2 and δ ≥ δb. This is condition is equivalent to z1 ≥ z∗2 and δ ≥ δ̄b, since z1 ≥ z∗2 implies that z1 > z̄1 and thus δ̄b = δb. 49 9.6 Proof of Corollary 2 This corollary follows immediately from Propositions 3–4, the implicit function theorem, and the observation that Gg(·) is decreasing in γ1, and Gb(·) is increasing in γ1. 9.7 Proof of Corollary 3 This corollary follows immediately from Propositions 3–4, the implicit function theorem, and the observation that Gg(·) is independent of D1, and Gb(·) is decreasing in D1. 9.8 Proof of Proposition 5 We prove this proposition by analyzing the three regions of X separately. • X ∈ [X,X]. The proof is identical to the proof of Proposition 2. • X > X. The characterization of the equilibrium follows the logic of the proof of Proposition 3. Recall that zR2,g(πg) > zf2,g = 0 for all πg ∈ [0, 1]. We then have that Gg(z R 2,g(πg), z f 2,g) > 0 for all πg and therefore πg = 1 in any equilibrium. The equilibrium is thus either fully informative or lenient. A fully informative equilibrium (i.e. one with πb = 0) exists if and only if Gb(z R 2,b(πb), 0) ≤ 0, whereas a lenient equilibrium (i.e. one with πb > 0) exists if and only if Gb(z R 2,b(πb), 0) ≥ 0 for some πb > 0. We first show that, the parameter space for which the fully informative equilibrium exists is (weakly) smaller than in the baseline model, and that for which the lenient equilibrium exists is (weakly) larger than in the baseline model. This follows because, for any πb ∈ [0, 1], Gb(z R 2,b(πb), 0) ≥ Gb(z R 2 (1, πb), 0), since Gb(z R 2 , 0) is increasing in zR2 and zR2,b(πb) ≥ zR2 (1, πb). We now show that the above statement holds with strict inequality by demonstrating that there exist parameter values for which the fully informative equilibrium is the unique equilibrium in the baseline model but does not exist in the model in which the 50 bank observes qt. Specifically, suppose β > β̄, z1 = z̄1 − ε, where ε > 0, and δ > δb, where δb is defined by Eq. 27. In this case, z1 < z̄1 implies that δ̄b(β, z1) = δb(β, z1) = ∞ as shown in the proof of Proposition 3, therefore the unique equilibrium in the baseline model is fully informative. Recall that z̄1 is defined such that zR2 (1, 0) = z∗2 . Therefore for ε sufficiently small, z1 < z̄1 implies that ∆(0) ≤ ∆(zR2 (1, 0)) < 0 < ∆(zR2,b(0)). This implies that U(zR2,b(0)) = UR and U(zf2,b) = 0, and therefore G(zR2,b(0), 0) > 0 as implied by δ > δb. Therefore a fully informative equilibrium does not exist when the bank observes qt, whereas by continuity of Gb(z R 2,b(πb), 0) in πb, a lenient equilibrium does exist. • X < X. The characterization of the equilibrium follows the logic of the proof of Proposition 4. We have that Gb(z R 2,b(πb), z f 2,b) < 0 for all πb and therefore πb = 0 in any equilibrium. The equilibrium is thus either fully informative or tough. A fully informative equilibrium (i.e. one with πg = 1) exists if and only if Gg(z R 2,g(1), 0) ≥ 0, whereas a tough equilibrium (i.e. one with πg < 1) exists if and only if Gg(z R 2,g(πg), 0 ≤ 0 for some πg < 1. We first show that the parameter space for which the fully informative equilibrium exists is (weakly) larger than in the baseline model, and that for which the tough equilibrium exists is (weakly) smaller than in the baseline model. This follows because, for any πg ∈ [0, 1], Gg(z R 2,g(πg), 0) ≥ Gg(z R 2 (πg, 0), 0), since Gg(z R 2 , 0) is decreasing in zR2 and zR2,g(πg) < zR2 (πg, 0). Following similar logic as for the case where X > X, we can show that the parameter space for which the fully informative equilibrium exists is strictly larger than in the baseline model. Specifically, suppose β > β̄, z1 = z̄1 − ε, where ε > 0, and δ > δg, where δg is defined by Eq. 31. For ε sufficiently small, the unique equilibrium in the baseline model is tough, whereas a fully informative equilibrium exist in the model in which the bank observes qt. 51 exists β < β̄, such that s∗2 > 0 if and only if β > β. Proof. Eq. 35 is obtained by substituting Eq. 33 into Eq. 34 to eliminate φT . Moreover, notice that ∆T (s2) is increasing in s2, and ∆T (1) = ∆(0). Therefore s∗2 < 1 if and only if β < β̄, since β̄ is defined such that ∆(0) = 0 at β = β̄. Finally, s∗2 > 0 if and only if ∆T (0) < 0. This is the case if and only if β > β, where β is defined by γ [α + (1− α)(1− d)] (R− 1)− (R0 − 1) + (1− γ)(1− β) ([α + (1− α)(1− d)]R− ρL) = 0. (34) We have that β < β̄ because ∆T (s2) is increasing in s2 and decreasing in β. It then follows that the strategic regulator’s expected surplus from the bank in the second period, for a given reputation s2, denoted by UT (s2), is given by UT (s2) =  UR, if s2 > s∗2, U0, if s2 < s∗2, λUR + (1− λ)U0 for some λ ∈ [0, 1], if s2 = s∗2. (35) We now move to analyzing the equilibrium stress test of the regulator for the bank in the first period, given the equilibrium in the second period. The incentives of the strategic regulator to pass the first period bank is given by GT g (sp2, s R 2 ) = (1− γ)(ρL − 1) + δ [ UT (sp2)− UT (sR2 ) ] , GT b (sp2, s R 2 , s 0 2) = (1− γ) [(ρL − 1)− dD] + δ [ (1− d)UT (sp2)− (1− d)UT (sR2 )− d(1− γ)UT (s0 2) ] . (36) Analogous to Eq. 9, the first term in Eq. 38 represents the net gain in terms of the expected surplus from the bank in the first period and the second term represents the reputation 54 concern in terms of the expected surplus from the bank in the second period. In contrast to the baseline setup, passing the first period bank reveals that the regulator is strategic, i.e. sp2 = 1. Subsequently, the bank continues to the second period with probability 1 if its risky loan is good, or with probability 1 − d if its risky loan is bad. The term sR2 (s0 2) is the posterior belief held by the market about the probability that the regulator is strategic, given that the first period bank fails the stress test and that the realized payoff is R (0). Since the bank recapitalizes with probability 1 − γ after failing the stress test, it continues to the second period with probability 1 if its payoff is R and with probability 1 − γ if its payoff is 0. In equilibrium, the posterior probabilities are given by sR2 (πg, πb) = [α(1− πg) + (1− α)(1− d)(1− πb)] s1 [α(1− πg) + (1− α)(1− d)(1− πb)] s1 + [α + (1− α)(1− d)] (1− s1) , s0 2(πb) = (1− πb)s1 (1− πb)s1 + (1− s1) . (37) In particular, if the first bank fails the stress test, the market updates its belief to sR2 , s 0 2 ≤ s1 if the bank realizes a payoff of R and 0, respectively, reflecting the fact that the strategic regulator is less likely to fail a bank than the tough regulator. Moreover, we have sR2 < s0 2, since the strategic regulator is more likely to fail a bad bank than a good bank. We can now prove this proposition by considering the three regions of X separately. • X ∈ [X,X]. Recall that sp2 = 1 ≥ s0 2(πb) ≥ sR2 (πg, πb) for all (πg, πb). Therefore X ≥ X implies that UR ≥ UT (sp2) ≥ UT (s0 2(πb)) ≥ UT (sR2 (πg, πb)) ≥ U0 for all (πg, πb). It follows that GT g (sp2, s R 2 (πg, πb)) = (1− γ)(ρL − 1) + δ [ UT (sp2)− UT (sR2 (πg, πb)) ] ≥ (1− γ)(ρL − 1) > 0. (38) 55 Moreover, we have GT b (sp2, s R 2 (πg, πb), s 0 2(πb)) = (1− γ) [(ρL − 1)− dD] + δ [ (1− d)UT (sp2)− (1− d)UT (sR2 (πg, πb))− d(1− γ)UT (s0 2(πb)) ] ≤ (1− γ) [(ρL − 1)− dD] + δ [(1− d)UR − [(1− d) + d(1− γ)]U0] < 0, (39) where the last inequality follows from X ≤ X. Since GT g (sp2, s R 2 (πg, πb)) ≥ 0 ≥ GT b (sp2, s R 2 (πg, πb), s 0 2(πb)) for all (πg, πb), there exists a unique equilibrium in which the regulator passes the bank in the first period if and only if the risky loan is good. • X > X. As shown above, this implies that GT g (R̃T 1 (πb), s p 2, s R 2 (πg, πb)) > 0 for all (πg, πb). Therefore πg = 1 in any equilibrium. Given πg = 1, we first establish some properties of GT b (1, sR2 (1, πb), s 0 2(πb)). Notice that sR2 (1, πb) and s0 2(πb) defined by Eq. 39 are strictly decreasing in πb, implying that GT b (1, sR2 (1, πb), s 0 2(πb)) is increasing in πb. We can now characterize the two possible types of equilibrium: a fully informative equilibrium with πb = 0 and a lenient equilibrium with πb > 0. First, an equilibrium with πb = 0 exists if and only if GT b (1, sR2 (1, 0), s0 2(0)) = (1− γ) [(ρL − 1)− dD] + δ [ (1− d)UT (1)− (1− d)UT (sR2 (1, 0))− d(1− γ)UT (s1) ] ≤ 0 (40) Notice that GT b (1, sR2 (1, 0), s1) ≤ 0 implies that GT b (1, sR2 (1, 0), s1) is strictly increasing in δ and that 1 ≥ s∗2 ≥ sR2 (1, 0). Therefore there exists δ̄ T b ∈ R+ ∪ {∞}, such that a 56 librium with πg < 1 exists if and only if limπg→1G T g (1, sR2 (πg, 0)) ≤ 0. Notice that limπg→1G T g (1, sR2 (πg, 0)) ≤ 0 implies that UT (1) < limπg→1 U T (sR2 (πg, 0)), which in turn implies that limπg→1G T g (1, sR2 (πg, 0)) is strictly decreasing in δ and that 1 ≥ s∗2 > sR2 (1, 0). Therefore there exists δTg ∈ R+ ∪ {∞}, such that the an equilibrium with πg < 1 exists if and only if δ ≥ δTg , where δ̄ T g 6= ∞ if and only if 1 ≥ s∗2 > sR2 (1, 0). More specifically, δTg is defined by δTg =  δg, if β ∈ (β, β̄] and s1 < s̄2, ∞, otherwise, (47) where β ∈ (β, β̄] ensures that 1 ≥ s∗2 > 0 and s < s̄2 ensures that s∗2 > sR2 (1, 0). 9.11 Proof of Lemma 2 After substituting Eq. 12 into Eq. 14 to eliminate φ, we have that the bank originates a risky loan if and only if ∆̃(z2) ≥ 0, where ∆̃(z2) = [α + (1− α)(1− d)]R−R0 − (1− α)(1− z2)(1− γ) ( R̃2(z2)(ρL − 1) + β [ (1− d)R− R̃2(z2)ρL ]) . (48) We first show that ∆̃(z2) is strictly increasing in z2. The derivative with respect to z2 is: ∂∆̃(z2) ∂z2 (1− α)(1− γ) = ( R̃2(z2)(ρL − 1) + β [ (1− d)R− R̃2(z2)ρL ]) − (1− z2) ∂R̃2(z2) ∂z2 [(1− β)ρL − 1] , (49) where: ∂R̃2(z2) ∂z2 = [ R̃2(z2) ]2 (1− α)(1− γ)d > 0. (50) 59 Consider two cases. First, if (1 − β)ρL − 1 ≤ 0, both terms in Eq. 51 are positive and therefore ∂∆(z2) ∂z2 > 0 for all z2. Second, if (1 − β)ρL − 1 > 0, then using Eq. 52, we can rewrite Eq. 51 as ∂∆̃(z2) ∂z2 (1− α)(1− γ) = β(1− d)R + [(1− β)ρL − 1] R̃2(z2) [ 1− (1− z2)(1− α)(1− γ)dR̃2(z2) ] . The definition of R̃2(z2) given by Eq. 13 implies that 1− (1− z2)(1− α)(1− γ)dR̃2(z2) > 0 and thus ∂∆̃(z2) ∂z2 > 0 for all z2. Moreover, ∆̃(1) > 0 by Assumption 1 and ∆̃(z1) → −∞ as z2 → −∞. Therefore there exists a unique z̃∗2 < 1, defined by ∆̃(z̃∗2) = 0, such that ∆̃(z2) ≥ 0 and thus the bank originates a risky loan if and only if z2 > z̃∗2 . We now derive a condition for z̃∗2 > 0. This is the case if and only if ∆̃(0) = [α + (1− α)(1− d)]R−R0 − (1− α)(1− γ) ( R̃2(0)(ρL − 1) + β [ (1− d)R− R̃2(0)ρL ]) < 0. (51) Notice the above expression is strictly decreasing in β. Moreover, it must be the case that ∆̃(0) < 0 for β = 1. To see this, notice that, at β = 1, we have ∆̃(0) = [α + (1− α)(1− d)]R−R0 − (1− α)(1− γ) [ (1− d)R− R̃2(0) ] < 0 ⇔ [α + (1− α)γ(1− d)]R−R0 + (1− α)(1− γ)R̃2(0) < 0 ⇔ [α + (1− α)γ(1− d)] ( R− R̃2(0) ) − (R0 − 1) < 0, (52) which is implied by Part (ii) of Assumption 3. Therefore there exists a unique β̃ < 1, such that z∗2 > 0 if and only if β > β̃, where β̃ is defined by [α + (1− α)(1− d)]R−R0− (1−α)(1−γ) ( R̃2(0)(ρL − 1) + β̃ [ (1− d)R− R̃2(0)ρL ]) = 0. (53) 60 9.12 Proof of Proposition We prove this proposition by analyzing the three regions of X separately. • X ∈ [X,X]. The proof is identical to the proof of Proposition 2. • X > X. The characterization of the equilibrium follows the logic of the proof of Proposition 3. Recall that zR2 (πg, πb) > zf2 = 0 for all (πg, πb). We then have that G̃g(R̃2(πb), z R 2 (πg, πb), z f 2 ) > 0 for all πg and therefore πg = 1 in any equilibrium. The equilibrium is thus either fully informative or lenient. • X < X. The characterization of the equilibrium follows the logic of the proof of Proposition 4. We have that G̃b(R̃2(πb), z R 2 (πg, πb), z f 2 ) < 0 for all πb and therefore πb = 0 in any equilibrium. The equilibrium is thus either fully informative or tough. 61