Macroprudential policy with capital buffers


BIS Working Papers

No 771


Macroprudential policy with capital buffers

by Josef Schroth


Monetary and Economic Department

February 2019


JEL classification: E13, E32, E44

Keywords: Financial frictions, Financial intermediation, Regulation, Counter-cyclical capital requirements, Market discipline, Access to funding


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© Bank for International Settlements 2017. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated.

ISSN 1020-0959 (print)

ISSN 1682-7678 (online)


Macroprudential policy with capital buffers

by Josef Schroth



This paper studies optimal bank capital requirements in a model of endogenous bank funding conditions. I find that requirements should be higher during good times such that a macroprudential “buffer” is provided. However, whether banks can use buffers to maintain lending during a financial crisis depends on the capital requirement during the subsequent recovery. The reason is that a high requirement during the recovery lowers bank shareholder value during the crisis and thus creates funding-market pressure to use buffers for deleveraging rather than for maintaining lending. Therefore, buffers are useful if banks are not required to rebuild them quickly.

JEL classification: E13, E32, E44

Keywords: Financial frictions, Financial intermediation, Regulation, Counter-cyclical capital requirements, Market discipline, Access to funding

1. Introduction

The recent financial crises in the United States and the European Union exposed tax­payers to potential losses from bank failures and significantly disrupted financial inter­mediation. A natural question arises from these experiences: Should regulators require banks to hold more capital and, if so, in what form? Higher minimum requirements reduce losses to stakeholders in case of bank failures but may constrain intermediation when it is most scarce—during financial crises. On the other hand, regulatory capi­tal buffers do not constrain intermediation provided the bank is willing to constrain payouts instead.

This paper develops a model of optimal bank capital and derives implications for bank capital regulation. There is only one type of bank debt in the model, such that capital takes the form of equity, and only one bank asset, loans to firms. There are two key frictions. First, bank equity is costly in the sense that bank shareholders demand a higher return than holders of bank debt. Second, holders of bank debt are wary of potential bank moral hazard in the sense that they require that bank shareholder value is not too low relative to the size of the bank balance sheet.

Together, the two frictions create a challenging risk-management problem for the bank. On the one hand, when bank equity is too low, then banks are debt-funding constrained because of fear of bank moral hazard, and lending margins are high be­cause of decreasing returns to scale at the firm level. On the other hand, banks hold costly equity—as a provision in case of low loan repayments—only if there is a strictly positive probability that they actually become funding-constrained. The two frictions therefore imply that banks lose access to the market for debt finance occasionally, at which point there is a credit crunch in the economy.

I study optimal capital regulation in the model by comparing the competitive- equilibrium allocation with the constrained-efficient allocation. Specifically, I interpret the difference between these two allocations as due to optimal capital regulation. There are two general-equilibrium channels that can be exploited in a constrained-efficient allocation. First, it is feasible, in the sense of satisfying bank participation constraints, to require banks to hold more costly equity during times of high loan repayments as long as overall bank profitability is somewhat raised. Second, it is possible to allow low bank equity, and high bank leverage, during times of low loan repayments as long as bank profitability is significantly raised temporarily for a while in a way that satisfies the bank debt funding constraint. In that sense, regulation trades off small and permanent against large but temporary distortions when taking measures to stabilize loan supply over time.

The main result of the paper is that optimal regulation requires banks to hold more equity when loan supply is high, but also allows banks to hold very little equity when loan supply is low. It is crucial that banks are also allowed to rebuild equity slowly after loan supply has been low—otherwise, loan supply would become very low when banks have very little equity. The reason is that if banks were to anticipate that they would have to rebuild costly equity quickly, then they would have lower shareholder value and, because of increased moral hazard concerns, reduced access to debt funding when equity is low. In other words, regulation must take into account that it cannot stimulate loan supply when bank equity is low by setting an equity requirement that is lower than the one implicitly imposed by the market for debt funding. Optimal regulation also requires banks to increase loan supply somewhat more slowly during a recovery from a credit crunch than they otherwise would. The resulting temporarily higher lending margins further raise bank dividend payout ratios during the time loan supply recovers, and further improves banks' access to debt funding during the time when loan supply is low. In that sense, when banks are offered a stake in the recovery from a credit crunch—through temporarily less onerous regulation and higher profit margins—then the credit crunch is less severe.

Three main policy implications can be derived from the analysis. First, minimum capital requirements should be as low as possible while still discouraging moral haz- ard. Second, any additional capital that a regulator wishes banks to hold should take the form of "capital buffers." The difference between a minimum requirement and a capital buffer is that banks are not forced to reduce the size of their balance sheet when they breach the latter. Specifically, banks may reduce equity payouts instead of deleveraging. Third, capital buffers that banks build up during good times are most effective in stabilizing lending during a financial crisis when banks are allowed to rebuild them slowly—while maintaining a high equity payout ratio—during the recovery from a financial crisis.

These policy implications can be compared with recent changes in recommenda­tions for bank regulation under the Basel Accord (Basel Committee on Banking Su­pervision, 2010) denoted "Basel III." First, the analysis suggests that market-imposed capital requirements are lower during financial crises for given bank borrower de­fault rates. Adherence to rigid microprudential capital requirements at all times may therefore not be optimal. In practice, giving banks some discretion in calculating risk-weighted assets during times of crisis can be justified for this reason—since bank margins are high when aggregate bank equity is low (for evidence and theory on "regulatory forbearance," see Huizinga and Laeven, 2012; Repullo, 2013; Repullo and Suarez, 2013). Second, there should be a buffer on top of market-imposed capital re­quirements—augmenting voluntary bank loan loss provisioning—that can be used to stabilize lending when bank equity is low. However, no equity payouts are allowed when this buffer is being used. This buffer resembles the capital conservation buffer under Basel III. Third, there should be an additional buffer that can be used when the first one is depleted. It can be used for lending. It can also be used for dividend payouts—or, equivalently, it is "released"—but only once the first buffer has been re­built. This additional buffer resembles the countercyclical capital buffer (CCyB) under Basel III. In my model, it is crucial that the regulator raises bank future profitability temporarily during the time when banks use the additional buffer to pay out divi­dends. The reason is that otherwise dividend payouts in the face of low bank equity would threaten bank solvency. In practice, bank profitability could be supported by recapitalizations financed by taxes on bank lending.

1.1. Related literature

Existing empirical work finds that financial crises are costly in terms of forgone output and generally lead to policy interventions that aim at restoring credit supply (Laeven and Valencia, 2013; Bernanke, 2018). Indeed, existing theoretical work suggests that interventions can improve welfare significantly because of the pivotal role that finan­cial intermediaries play (Bebchuk and Goldstein, 2011; Philippon and Schnabl, 2013; Sandri and Valencia, 2013; Schroth, 2016). However, theoretical work also stresses the importance of ex ante measures, such as capital buffers, which reduce the need to rely on ex post policy intervention (Lorenzoni, 2008; Martinez-Miera and Suarez, 2012; Be- genau, 2014; Clerc et al., 2015). The literature that trades off ex post intervention and ex ante measures relates bank access to funding to the liquidation value of bank assets during a default (e.g., Jeanne and Korinek, 2013). For example, during the 2007-2008 US financial crisis, there was a run in the market for secured bank funding when concerns about bank solvency suddenly emerged. This paper contributes to this lit­erature by relating the bank's decision to default to its future prospects. The approach is motivated by the fact that a defaulting bank loses its charter value, and the charter value depends (positively) on the bank's future prospects. Novel implications for bank regulation follow from this approach.

2. Model

This section describes an infinite horizon economy in discrete time with time periods t = 0,1,2,— There are aggregate productivity shocks st £ S = (sL, sH} C R++, where Pr(st = sL) = p in each period t = 1,2,— The initial state is given as s0. Define the sets St = S x St-1 for t = 1,2,... where S0 = (s0}. Let st denote the history of productivity shocks up to period t and the initial state, with s0 = s0, and define the probability measure nt on St. Denote conditional probabilities by nt (st+T |st) for any t and т = 1,2,    There is a measure one of identical short-lived firms producing a consumption good and investing using external funds obtained from a measure one of identical banks. Finally, risk-neutral households, also of measure one, value consumption and are endowed with one unit of labor each, which they supply inelastically. Households and banks trade one-period non-contingent bonds with each other.

2.1 Firms

Firms live for one period. They have access to a production technology that turns k units of the consumption good in period t and l units of labor in period t + 1 into F(k, l; st+1) = st+1kal1-a + (1 — 5)k units of the consumption good in period t + 1, where aggregate productivity st+1 is realized at the beginning of period t + 1, before l is chosen, and where 5 £ (0,1) is the depreciation rate. It is assumed that firms cannot sell bonds and do not have any internal funds such that they must borrow from a bank to fund capital investment. A firm that is born at the end of period t in state st produces in period t + 1 and maximizes its expected profit subject to solvency in each state of the world:

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wages wt+1 (st+1) and bank lending returns Rt+1 (st+1), the optimal firm labor input and capital investment choices are characterized as follows:

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It is assumed that firm profits accrue to households. Note that profits are zero for any realization of st+1 because of constant returns to scale.

2.2 Households

Households are risk-neutral and value consumption. They are endowed with one unit of labor in period t = 1,2,... and w0 units of the consumption good in period zero. Households discount future consumption using the subjective discount factor в < 1. Note that households are willing to trade the one-period non-contingent bond in finite quantity as long as its price is equal to в, implying a gross return of 1/в in an equilibrium of the model.

2.3 Banks

Banks are risk-neutral and value dividends dt(st). Let lt+1 (st) denote bank lending to firms in period t = 0,1,2,_____ Both dividend and lending choices are constrained to be non-negative. Banks face budget constraints as follows:

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where bt+1(st) denotes a bank's purchase of one-period non-contingent bonds at price в for t = 0,1,2,... and a0 > 0 denotes given initial bank equity. For t = 1,2,... define bank equity as at(ss) = Rt(ss)lt(st-1) + bt(st-1).

Banks discount future dividends using the discount factor в < 1. However, it is assumed that a bank enters an "accident" state at the beginning of each period with constant probability 1 — y/ > 0. When banks experience an accident, they pay all eq­uity—i.e., all loan repayments net of debt—into an "accident fund" and exit the econ­omy. The fund immediately distributes the collected equity among a measure 1 - Y/в of new banks that enter the economy. The assumption captures corporate governance problems and implies that banks effectively discount dividends using the lower dis­count factor y < в such that the value of a bank is Vo = Lf=o Y Ls es* dt(st)nt(st).

However, banks will generally not set dividends as high as possible because the timing of bank dividends determines a bank's incentive to engage in moral hazard, which in turn affects the bank's access to external funding. In particular, bank creditors (i.e., households) are willing to provide external funding to a bank as long as the bank values the future dividends it expects to enjoy more than a fraction в e (0,1) of its lending. The reason is that bank assets 1*+1 (s*) are not liquid and diminish by fraction в unless monitored by a bank. A bank could thus extract e£t+1(st) by defaulting and threatening creditors not to monitor their assets. This consideration is captured by the following no-default constraint that needs to be satisfied in every period t = 0,1,2,... in which the bank wishes to make use of external funding:

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The problem of a bank is thus to choose lending and bonds to maximize its value at date zero subject to (4), (5), (6) and dividend non-negativity. Let в* nt(s*)tft(s*) be the multiplier on the no-default condition (6) in period t, when lt+i(st) is chosen. It determines the change in the value of the bank's internal funds (equity) when the bankloses access to external funding—i.e., when the bank is constrained and cannot sell ad­ditional bonds. Let the value of internal funds be et nt(st )\t(st), i.e., the multiplier on the bank budget constraints. Then the first-order condition for bank lending lt+1 (s*) can be written as follows:

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Equation (7) says that banks are profitable, after adjusting their income for its riskiness, only at times when they lose access to external funding. The reason is that banks are competitive and would immediately compete away any risk-adjusted profit margin if their creditors would allow them to increase leverage. The model thus predicts that lending spreads are elevated during financial crises (Muir, 2017).

Let etnt(st)yt(st) denote the multiplier on dividend non-negativity. Then the first- order condition for dividends can be written as

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where sT denote sub-histories of st. The assumption that banks are more impatient than other participants in the bond market, i.e., y < в, implies that (6) will occasionally bind, as Proposition 1 shows.

Proposition 1. The bank no-default condition binds occasionally.

Proof. The first-order condition for bank bond holdings implies that the return on equity At converges almost surely. Hence, if the bank no-default constraint (6) were not binding occasionally, then tft = 0 almost surely and thus At — щ ^ 0 almost surely. But then dividends are zero almost surely, implying that (6) is in fact binding almost surely.

Bank equity is valuable because it can be used to relax the bank no-default con­straint and allow the bank to lend more and to attract more external funding at exactly those times when bank lending is profitable. Each bank is aware that low realizations of the aggregate shock lower equity of all other banks and increase the probability that other banks will lose access to external funding in the current or some future period. For this reason, each bank regards lending income as risky and extends lending only up to the point where their risk-adjusted profitability drops to zero. Banks thus en­gage in loan loss provisioning as a result of the "last bank standing effect" (Perotti and Suarez, 2002).

Equations (8) and (9) reveal that the bank's risk-management problem has both a forward-looking and a backward-looking component. On the one hand, internal funds (equity) in the current period can be used to reduce leverage. Lower leverage reduces the probability of losing access to market funding and being forced to cut dividends, potentially to zero, in future periods. On the other hand, internal funds in the current period can be used to pay dividends and thus increase access to market funding in all preceding periods through relaxing market-imposed no-default constraints. The model in this paper thus gives an example of how financial intermediaries evaluate risk differently compared with the representative household (He and Krishnamurthy, 2013; Adrian, Etula, and Muir, 2014). This difference in risk perception plays a crucial

2.4 Competitive equilibrium

Spot markets for labor, contingent bank loans, and one-period non-contingent bonds open in every period t. In every period t, the wage wt clears the labor market and the returns on loans Rt+1 clear the market for lending. Definition 1, below, characterizes a competitive equilibrium in terms of lending returns, wages, and bank actions.

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2.5 Deterministic steady state

Suppose sL = sH = 1 such that the economy does not experience any stochastic fluctuations. Define first-best lending as follows:

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Lemma 1 shows that banks provide less than the first-best amount of lending in steady state. The reason is that banks view equity as costly relative to external funding. The required return on bank lending is given by Rce = 1/в + e(e — yI/^y. This return is higher than the return on external funding, 1/e, but lower than the required return on

Lemma 1. Steady-state lending in the deterministic case in competitive equilibrium is given as follows:

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Proof. Banks pay strictly positive dividends in a steady state of the competitive equi­librium such that щ = 0. It follows from equations (8) and (9) that ipt =in a steady state. Further, banks are always borrowing-constrained due to their relative impatience. The amount of steady state lending in a competitive equilibrium then follows from equation (7).

3.   Macroprudential capital requirements

This section analyzes the model analytically. I will first discuss how the no-default con­straint (6) can be interpreted as a (market-imposed) bank capital requirement. Then, I will present first-order conditions that characterize the second-best allocation and imply a rationale for macroprudential capital regulation in the model economy.

3.1 Bank no-default constraint and capital requirements

Note that the first term in nt is the present value of pure profits where the bank's own discount factor is used rather than the bond market discount factor в- Since y < в, this term is lower for

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given lending returns than it is when bank profits are discounted using bond prices. The second term reflects the fact that usage of external funding, bt+T (st+T-1) < 0, is a way for the bank to increase its value. That is, there is a benefit for the bank from front-loading dividends and from back-loading debt repayments as a result of impatience. When bank budget constraints are used to substitute out dividends, the value of a bank at time t can be expressed as Vt(s*) = at(s*) + nt(s*). The no-default constraint (6) can then be reformulated as follows:

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With bank capital defined as expected equity, (14) gives a capital requirement that depends on the expected present value of bank future profits. This capital requirement is microprudential in the sense that its purpose is to guarantee the solvency of the bank only. For example, if the value of the bank does not exceed its equity, then permissible leverage is given by 1/в. If the bank is expected to have positive future profits, then itis allowed to have higher leverage because future profits serve as "skin in the game."

It is important to note that microprudential capital requirements are low in this economy, in the sense that banks often hold capital (equity) well above the require­ment stipulated by equation (14), implying that equation (14) will bind only occasion­ally. The reason is that banks seek to protect their charter value; that is, they risk-adjust income from lending to avoid low equity (and binding capital requirements) in states where the return on lending is high (loan loss provisioning). In that sense, market- imposed capital requirements already induce prudent behavior to some extent. Sec­tion 3.2 asks whether this extent is sufficient or whether additional macroprudential capital regulation is necessary.

The remainder of the paper studies second-best capital requirements, i.e., changes in the allocation of bank equity and lending that increase lending to firms weighted by the marginal product of capital subject to the market-imposed capital requirement (14). I interpret the difference between the second-best allocation and the competitive- equilibrium allocation—in which banks are only constrained by microprudential reg­ulation implied by (14)—as resulting from macroprudential capital regulation. I then discuss how macroprudential regulatory tools used in practice might be combined to­ward implementing the optimal macroprudential regulation, or second-best allocation, that is identified in the paper.

3.2 Optimal capital regulation

The capital requirement (14) gives rise to a pecuniary externality, in the sense of Green- wald and Stiglitz (1986). Hence, an exclusive reliance on loan loss provisioning moti­vated by market discipline may leave some inefficiencies in this economy unaddressed. The reason is that future asset prices, i.e., future lending returns [Rt+т(st+T)}T=1,2,..., enter (14) through expected future profits given by equation (13) at each point in time t = 0,1,2, A constrained social planner can therefore affect capital require­ ments, and thus permissible bank leverage, by affecting these future asset prices (as in Schroth, 2016). This paper focuses on how a constrained social planner can stabilize aggregate lending in the economy over time, by exploiting the pecuniary externality, and thus improve upon self-interested (competitive) individual provisioning by banks.

Because banks consider equity to be costly, relative to external funding, it is neces­sary to impose bank participation constraints in periods t = 0,1,2,...:

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Condition (15) ensures that banks prefer continuing to be banks rather than liquidating their assets. Note that the bank participation constraint is equivalent to nt (st) > 0. Condition (15) requires that the future profits that banks expect to earn are non­negative. The condition would never bind in a competitive equilibrium because banks are free to reduce lending and increase dividends. To see why it might be binding in a constrained-efficient—or second-best—allocation, consider the case in which bank lending is first best and bank debt is zero, such that the first term in nt is negative while the second is zero. A second-best allocation will thus allow for bank leverage or bank rents or both to discourage banks from liquidating themselves (recall also the discussion in Section 2.5).

Definition 2. The second-best allocation is given by sequences of dividends {Dt (st )}st eSt, t>0, bank bond holdings {Bt+1(st)}steSt, t>0 and bank lending {Kt+1 (st)}steSt, t>0 such that social welfare

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is maximized subject to initial bank equity a0, prices for labor and loans

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In a second-best allocation, the no-default constraint can be relaxed by increasing future profits. However, while an increase in future bank profits mitigates a severe credit crunch, it also creates socially costly distortions in future bank lending.

3.3 Analysis of the second-best allocation

Before continuing to the numerical part of the paper, first-order conditions that the second-best allocation must satisfy are discussed. Let ftnt(st)ft(st) be the multi­plier on the bank no-default constraint, ftnt(st)\t(st) be the multiplier on the bankbudget constraint, p1 nt(st)qt (st) be the multiplier on the participation constraint, and вnt(s{)^t(s{) the multiplier on dividend non-negativity. The first-order conditions for bonds and dividends can be combined as follows:

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where the terms sT denote sub-histories of sl. Equation (16) shows that the return on bank equity is forward-looking as well as backward-looking. The constrained planner values current equity more if it is more likely that equity will be scarce in future pe­riods, as indicated by binding dividend non-negativity constraints in the next period. However, the constrained planner also internalizes how higher equity in the current period can be used to increase the current dividend and thus relaxes bank no-default and participation constraints in all previous periods. Note that this intuition is almost the same as that in Section 2.3 (the bank participation constraint is ignored in Sec­tion 2.3 because it is satisfied by definition in competitive equilibrium). The difference is that the constrained planner is not impatient with respect to dividends and thus tends to value bank equity more highly. However, the bank participation constraint keeps the planner from back-loading dividends too much and from building up too much equity. The reason is that higher levels of equity necessitate higher rents from bank lending since the planner must deliver the return on bank equity 1/y.

The first-order condition for bank lending reveals that the second-best allocation may feature an excess risk premium on bank lending even if banks have further access to external funding, i.e., even if the no-default constraint does not bind such that

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That the second term on the left-hand side of equation (17) is non-negative can be seen by writing the first-order condition for dividends as follows:

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which is non-negative for all st+1. Excess returns enjoyed by banks therefore have a forward- and a backward-looking component. On the one hand, when the dividend non-negativity constraint is binding in the following period, then bank returns in­crease, which increases bank equity in that period and makes dividend non-negativity constraints bind less. On the other hand, when no-default or participation constraints have been binding in the past, bank returns increase, relaxing those constraints by increasing banks' ability to increase dividends in subsequent periods.

In summary, the intuition is as follows. When the no-default constraint binds, lending is severely reduced in the economy and excess lending returns shoot up. As a result, the value of bank internal funds increases, and this increase is long-lived by equation (17). This in turn leads to higher excess returns over a number of periods, increasing expected bank future profits immediately. The result is that the no-default constraint is being relaxed such that lending returns shoot up by less, at the social cost of somewhat higher lending returns over a number of future periods. That is, in a second-best allocation, the scarcity of bank lending is smoothed out over time.

3.4 Deterministic steady state

Analyzing the second-best allocation in deterministic steady state reveals that the trade-offs faced by the constrained planner are dynamic rather than static. Indeed, Lemma 2 shows that the second-best allocation is identical to the competitive-equilibrium allocation in steady state of the deterministic economy.

Lemma 2. Steady-state lending in the deterministic case in the second best is the same as in competitive equilibrium.

Proof. The bank participation constraint does not bind strictly because there is no bene­fit from having precautionary equity buffers in the deterministic case. Then multipliers are constant and satisfy f = (A — 1)(в - j)/j by equation (16). The second-best amount of bank lending can then be obtained from equation (17) as a function of the value of bank equity as follows:

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For the bank participation constraint to hold, bank lending must be lower than KCE because the bank participation constraint binds weakly in a deterministic steady state of the competitive equilibrium. Hence, it must be the case that KSB (A) < KCE. Because j < в, the effect of initially scarce bank equity on the steady-state value of bank equity decays geometrically by equation (16). The value of bank equity in steady state therefore depends only on the multiplier on the no-default constraint in steady state.

As a result, KSB (A) = KCE and

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4. Numerical analysis

This section analyzes the model numerically. Table 1 summarizes the choices of model parameter values used in this section. The choice of consumer discount factor в im­plies an annual risk-free interest rate of around 6 percent. The depreciation rate and capital income share are set to 12 percent and 35 percent, respectively. The choice for в implies a market-imposed capital requirement of 10 percent in normal times, when bank future profits are zero. The parameters characterizing the productivity shock process are chosen to allow the model to generate large crises. The economy experi­ences a "financial crisis" in period t if the bank "lending gap"—the difference between first-best lending KBB and actual lending—is 5 percent or higher. The parameter y that determines the bank's relative cost of equity is set such that the economy spends around 6 percent of the time in a financial crisis (see Figure 1).

In Section 3 it was shown that when the bank no-default condition binds in a second-best allocation, then lending returns are elevated for some time. Elevated fu­ture lending returns increase bank future profits and relax the no-default condition in the current period. The economic impact of a credit crunch, during which banks are forced to reduce lending due to insufficient access to external funding, is therefor

Image 9cgl

mitigated. However, granting future profits to banks creates economic distortions such that a second-best allocation would also require banks to hold more equity on aver­age. The idea is to limit use of an increase in future profits to the most severe credit crunches. As a result, banks are asked to increase their loan loss provisioning and can withstand more adverse shocks before the economy enters a credit crunch. On the rare occasions when the economy does enter a credit crunch despite higher provisioning, lending is stabilized by increasing bank future profits.

Figure 2 compares the second-best allocation with the competitive-equilibrium al­location for a given sequence of shocks. In particular, the high shock occurs between successive occurrences of the low shock for 1,2, and 4 times, respectively. Note that the high shock occurs sufficiently many times before each occurrence of low shocks for the economy to reach normal times, during which bank future profits are zero. Fig­ure 2(a) shows that banks in competitive equilibrium hold a voluntary capital buffer of 2.75 percent. A constrained planner would require an additional buffer of 3 percent such that bank capital in the second-best allocation during normal times reaches 15.75 percent.

Bank lending during normal times is lower in the second-best allocation than it is in competitive equilibrium, as can be seen in Figure 2(b). The reason is that banks must be compensated with a higher expected return on lending when they are re­quired to hold additional capital buffers because of the relatively higher cost of capital compared with external funding. However, bank lending is stabilized significantly in the second-best allocation compared with the competitive equilibrium. The reason is that the constrained planner can increase bank future profits at relatively low cost to offset decreases in bank equity whenever low-productivity shocks occur. The plan­ner can deliver future profits at low cost to banks because the planner smooths out the associated economic distortions over time. That is, in contrast to the competitive equilibrium, the second-best allocation delivers bank future profits by increasing long­term lending returns somewhat rather than increasing short-term lending returns a lot. Bank lending thus drops by less during financial crises, but it also recovers more slowly during the time banks are allowed to earn their future profits.

Figure 3 shows that expected excess returns are positive in normal times to com­pensate banks for the cost of capital buffers. Lending returns are more smoothed out in the second-best allocation; returns shoot up by less since financial crises are much less severe, but they stay elevated for longer to deliver increases in bank future profits more cheaply. During a financial crisis, lending returns shoot up sharply in compet­itive equilibrium, but banks are still forced to deleverage and reduce their reliance on external funding drastically. In contrast, banks increase their reliance on external funding during a financial crisis in the second-best allocation.

Because bank capital is costly, 7 < в, the extent to which the constrained planner is able to backload dividends to relax market-imposed no-default conditions during a financial crisis is limited. As a result, the dividend payout ratio increases temporarily following financial crises. Banks resume paying dividends earlier in the second-best al­location, compared with the competitive equilibrium, such that, in particular, rebuild­ing equity buffers takes a backseat to resuming dividend payouts in the aftermath of financial crises.

The constrained planner allows banks to have higher leverage during financial crises. On the one hand, the planner can satisfy market-imposed no-default condi­tions more easily by increasing bank future profits at relatively small cost. On the other hand, the planner is less averse to risk during crises, compared with banks in the competitive equilibrium, since any potential future equity scarcity can be partially

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offset by upward adjustments in future profits and since lending returns increase less during crises. Even though the planner prescribes additional equity buffers in normal times, the planner perceives equity to be relatively less scarce during times of financial crisis and is consequently less protective of it. Thus, while the second-best allocation features lower reliance of banks on external funding in normal times, the planner al­lows banks to aggressively replace lost equity with external funding during times of financial crisis.

Figure 4 shows the economy for a particular random draw of productivity shocks. The competitive equilibrium features a severe credit crunch during years 85-90. This credit crunch is much less severe in the second-best allocation. However, the economy takes a much longer time to recover from it. The constrained planner uses bank eq­uity more aggressively to maintain lending when bank earnings are low because of low-productivity shocks. The future profits that the planner must promise banks be­come large, and with them so does the dividend payout ratio. Subsequent low shocks deplete bank equity at periods when it has not yet had time to be rebuilt such that the planner has to adjust promised bank future profits upward repeatedly. As a re­sult, bank margins remain elevated—and bank lending remains depressed—for many years.

The second-best allocation is characterized by macroprudential capital regulation that avoids sharp reductions in bank lending and economic activity but, at the same time, can lead to a very persistent decline in lending and economic activity. One crucial assumption in my analysis is that the constrained planner can honor its promise to deliver bank future profits.

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4.1 Policy implications

The second-best allocation shows that there is a net benefit from requiring banks to hold additional equity. Such capital buffers are "always on" in the sense that banks should build them up gradually in good times while paying out dividends at the same time. High non-linearities as well as this gradualism imply that it is too late to turn on the capital buffer once the economy experiences financial stress in the form of losses on bank balance sheets.

Banks should be allowed to use capital buffers during credit crunches—for lending to firms and, eventually, for dividend payments. In case the economy is still in a credit crunch by the time capital buffers are exhausted, bank future rents can be increased to continue to stabilize lending. Bank future rents should be provided by distributing economic distortions over multiple periods, which has the side effect of slowing down the recovery from credit crunches. Credit crunches are much less severe in the second-best allocation than in competitive equilibrium such that the net effect on welfare is positive.

4.2 Relationship to regulatory practice

In practice, there is a broad consensus among regulators that banks should hold more capital on average (e.g., Fender and Lewrick, 2016). At the same time regulators have identified a number of "externalities" that determine what they consider the desired level of capital over time. Examples of such externalities are temporarily high levels of debt that might expose borrowers to sudden changes in asset prices, or exchange  rates, together with associated feedback effects. Another externality might be low bank provisioning stemming from a temporary scope to use accounting discretion that might expose the economy to sudden bank deleveraging. Determining the desired level of bank capital likely requires regulator judgment because not all externalities might be equally important at a given time. Indeed, while the Savings and Loan crisis was clearly related to insufficient recognition of risk by banks, the recent financial crisis was related to excessive borrowing as well as banks' hidden exposures.

In this paper, I focus on a different externality that is related to the price of bank equity. This externality is always important because it arises ex post, conditional on a financial crisis occurring. Specifically, promising banks future profits during a financial crisis lowers concerns about bank moral hazard, and thereby reduces pressure on banks to deleverage. The analysis in this paper suggests that regulation should take seriously its ability to affect bank profitability ex post in a way that complements the existing regulatory focus on addressing risk buildup ex ante.

4.3 Is a capital buffer harsh on banks?

Requiring banks to accumulate an additional capital buffer imposes costs on the econ­omy since bank equity is costly, y < в, and since the bank participation constraint (15) states that a bank cannot be forced to continue operating when its value falls short of its equity. An increased level of equity lowers the profits banks earn from leverage and makes it necessary for a constrained planner to compensate banks with profits from higher lending returns. In other words, the planner must allow banks to earn a higher return on assets such that banks can achieve their required return on equity with re­duced leverage. A planner thus trades off the benefit from increased resilience against the cost of more distorted lending returns when considering the size of a bank's capital buffer. Since bank dividends enter the planner welfare criterion stated in definition 2, the planner chooses a positive capital buffer.

Figures 5 and 6 compare the laissez-faire competitive equilibrium with the second- best allocation for the case in which bank dividends do not enter the planner welfare criterion at all. For example, a constrained planner may value bank dividends less in the case where foreigners enjoy some of these dividends. Figure 5 shows that a constrained planner that does not value bank dividends at all would ask banks to hold less equity than they do in the competitive equilibrium. In fact, the planner chooses bank lending above the first-best level during normal times. The reason is that the planner prefers that bank cash flows during normal times support wages rather than dividends—even at the cost of imposing losses on banks, lower bank equity and, compared with the case in which the planner values dividends directly, higher volatility of bank lending.

Leverage is higher in the second-best allocation, but severe credit crunches can be avoided by increasing bank future profits whenever banks experience low lending re­turns (Figures 5(b) and 5(d)). The bank participation constraint is satisfied—despite incurring losses in expectation during normal times—by anticipated temporary in­creases in profits that are large and frequent. Banks are not profitable during normal times but break even overall, since the planner treats them favorably during times of financial crisis.

Intuitively, when the planner does not value bank dividends, a high level of bank equity has a social cost that is excessive because of the bank participation constraint (15). As a result of market incompleteness, high realized lending returns lead to bank equity that is too high such that a planner prescribes lending above the first-best level, as well as negative expected lending returns, to achieve the desired lower level of equity. However, the planner still uses bank future profits to stabilize bank lending over time.

A constrained planner that does not value bank dividends would require banks to hold less equity and to lend excessively in good times. Such a planner would not see any reason to impose capital buffers even if credit-to-GDP measures are elevated—in fact, high credit-to-GDP becomes a policy implication. In contrast, a planner that val­ues bank dividends requires banks to hold more equity and somewhat restrict lending in good times. In that sense, a capital buffer is not harsh on banks because it is imposed by the planner that values bank well-being directly. A planner would always—whether valuing dividends or not—stabilize bank lending during credit crunches by adjusting future bank profits upward.

4.4 Tighter-than-necessary microprudential capital requirements

Figures 7 and 8 compare competitive equilibrium and second-best allocation for the case in which bank future profits do not enter the bank no-default constraint. This case can be interpreted as a microprudential regulator imposing a bank no-default condition that is tighter than the no-default constraint (6) imposed by market partic­ipants. The condition is then tighter than necessary to prevent default (Kehoe and Levine, 1993; Alvarez and Jermann, 2000). The additional tightness is ad hoc and not derived from macroprudential concerns. The second-best allocation can then be inter.

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Both individual loan loss provisioning and additional capital buffers are now higher. However, the constrained planner does not raise bank future profits to alleviate finan­cial crises. The reason is that for a given tight microprudential bank no-default con­straint, there is no scope for the macroprudential regulator to support bank lending in times of financial crisis. The following section studies regulatory capital buffers—capital requirements that are increasing in a bank's current-period bank dividend payout.

5. Regulatory capital buffers

The model can be used to study the properties of regulatory capital buffers (see Basel Committee on Banking Supervision, 2010, or "Basel III") in a dynamic economy. Specifically, in Section 5.1 the size of the regulatory capital buffer is a function of endogenous variables, and the focus is on how buffers affect transitional dynamics around financial crises. Section 5.2 uses a simpler capital buffer rule based on exoge­nous indicators and presents comparative statics with respect to the size of the buffer.

The benchmark model is extended by adding another possible value for productiv­ity shocks as follows:

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Further, productivity is now persistent, and the transition matrix Ps = Prob(s^+1|sf) is:

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I introduce an exogenous regulatory minimum capital requirement

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with capital defined as y Est+1 n(st+1 |st)at+1 (st+1) and set d1 = 0.06. I further intro­duce the following exogenous regulatory dividend payout restriction:

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The payout restriction (19) limits dividends to a fraction of "eligible earnings" which are simply defined as capital in excess of the minimum requirement. In practice, it is possible to define eligible earnings as a flow variable instead of a stock variable, but the latter may be more robust to bank accounting choices. Note that (19) implies the minimum requirement (18) because of the assumption that dividends must be non-negative.

5.1 Case of endogenous indicators

Under Basel III the size of the CCyB is a function of macroeconomic variables. In this section, I set Tt(st) as a function of the ratio of bank lending to expected GDP in a way that equates the size of the capital buffer to the capital buffer in the second-best allocation when bank lending to expected GDP is high. Alternatively, I could have taken two indicators: bank lending to GDP and growth of bank lending. Note that neither of these two indicators is sufficient by itself, but bank lending to expected GDP is. Figure 9 shows that the effect of the capital buffer is small but significant—bank

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lending drops by two percentage points less when the buffer had been fully built up before the financial crisis.

Figure 10 compares the competitive equilibrium with the regulatory buffer and minimum requirement to a second best in which the regulatory minimum requirement is imposed as well. The main difference is that in second best, buffers are about the same in normal times and boom times. There is no gain from varying the size of the buffer over time—unless a financial crisis occurs. During the recovery from a financial crisis, buffers should be rebuilt gradually.

5.2 Comparative statics with exogenous indicators

Suppose the buffer rule is simple in the sense that Tt (s*) = т(st) depends on st only. Because banks never pay out dividends in the lowest state sl, the payout restriction can be set arbitrarily when st = sl; fix it as t(sl) = 0. Figures 11 and 12 show welfare and the cost of bank credit, respectively, as a function of (t(sm),t(sh)). A function of bank lending to expected GDP or a decreasing function of bank lending to GDP as long as growth of bank lending is not too high.

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lower т makes the payout restriction (19) tighter. The lowest possible values for t(sm) and t(sh)—"tightness" of 1 in Figures 11 and 12—are the ones that make the bank participation constraint (15) bind.

Regulation asks banks to retain some of their windfall profits during times of eco­nomic boom. The idea is to use them to absorb losses in case the boom is followed by a bust. The role of buffers in normal times is to preserve some of the buffer built up during the boom and avoid that it is immediately paid out as soon as the boom ends. The same intuition also applies to the second-best buffers shown in Figure 10.

5.3 Buffer cyclicality

Figure 10 shows that the buffer a constrained planner would prefer is not necessarily higher in normal times than in boom times. In that sense there is nothing special about boom times when thinking about optimal buffer size. Indeed, Figure 13 shows that in the case in which there is no exogenous regulatory minimum requirement on

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top of the market-imposed equity requirement, the optimal buffer is actually higher in normal times compared with boom times. The reason is that bank losses are still far away, in expectation, during a boom. Paradoxically, therefore, the rationale of higher buffers in boom times might be to provision for binding regulatory constraints rather than actual bank losses.

6. Conclusion

Banks may lose access to external funding on occasion. This can create a socially costly credit crunch in the economy during which banks are forced to reduce their lending activity. This paper studies constrained-efficient capital regulation that aims to prevent and mitigate such credit crunches and derives two implications for macroprudential policy. First, additional capital buffers should be imposed ex ante. Because of strong non-linearities present in the model, such buffers should be always activated. Sec­ond, capital requirements and buffers should be reduced ex post during severe credit crunches. Bank default at increased levels of leverage is avoided by granting higher future profits to banks. A macroprudential regulator would affect bank profitability dynamically to smooth out the scarcity of bank lending over financial cycles.

The main policy implication is that capital buffers should be large eventually—however, banks should be given sufficient time to rebuild them during recoveries from finan­cial crises. The idea is to prevent treating banks harshly in the immediate aftermath of a crisis to avoid adding to concerns about bank moral hazard during crises. The pressure on banks to return external funding during a crisis is then lessened such that they can use buffers to maintain lending as much as possible. The CCyB under Basel III is time-varying and therefore can potentially be designed to take into account these implications.


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