Capital Adequacy and Liquidity in Banking Dynamics

Transcription

1 Capital Adequacy and Liquidity in Banking Dynamics Jin Cao Lorán Chollete October 9, 2014 Abstract We present a framework for modelling optimum capital adequacy in a dynamic banking context. We combine the (static) capital adequacy framework of (Repullo, 2013) with a dynamic banking model similar to that of (Corbae & D Erasmo, 2014), with the extra feature that the probability of systemic risk is endogenous. Unlike previous work, we examine frameworks to ameliorate bankruptcy using both capital adequacy and liquidity requirements. Since equity is costly, the social cost of regulation may be reduced if a regulatory capital requirement can be accompanied by other tools such as a liquidity buffer. Keywords: Risk-Taking; Capital Adequacy; Endogenous Systemic Risk; Liquidity Requirement; Banking Regulation JEL: E500, G210, G280 Cao is at Norges Bank, jin.cao@norges-bank.no. Chollete is at UiS Business School, loran.g.chollete@uis.no. The views expressed herein are those of the authors and do not necessarily reflect the views of Norges Bank. 1

2 1 Introduction What is the optimal capital adequacy ratio and how does it compare to liquidity requirements in terms of staving off systemic risk? In this paper we present a dynamic model for bank capital. In our framework, banks risk-taking incentives are endogenized while they optimize both their capital ratio and liquidity buffer, in order to maximize their long run cash flow. The key decision in our model is that each bank, after observing its profitability in a given period, decides how much risk to take in running projects. The bank thus has the incentive to do the following: Holding equity in order to absorb losses and avoid costly exit. The drawback of this option, however, is that raising equity is costly, so that banks prefer cheaper debt to equity. The bank s decision on equity holding does not only depend on its type, but also on its expectation of future shocks to project returns; Holding safe liquid assets, which serve as collateral for interbank borrowing when the bank is in distress. As in the equity case above, this decision depends on both the bank s current type and its expectation of future shocks. Setting a higher equity ratio and liquidity buffer reduce the bank s incentive in risk-taking, hence the likelihood of failure and system spillovers 1, generating a positive externality to the other banks. Individual banks do not internalize this externality and therefore the likelihood of bank failure is higher than the socially optimal level. Capital adequacy ratio and liquidity requirement are two important instruments in banking regulation, and mostly they have been studied separately in both banking theory and policy analysis. Capital requirement and liquidity regulation affect banks risk-taking incentive in different ways: holding more capital increases the banks skin-in-the-game and makes them more prudent in making investment decisions, while raising capital is costly for banks; holding more liquidity buffer makes banks more resilient under distresses, while it also implies higher opportunity cost in investing the high yield, risky assets. How these two instruments interact with each other, and whether one can substitute or complement to the other in regulatory design depend on the careful cost-benefit analysis, and are very much unexplored in previous research. To make up this gap, in this paper we examine a standard individual bank optimizing model, followed by a social planner s problem. By taking into account social costs, a regulator can set both optimal ratio of bank capital and liquidity buffer, which depend on bank types as well as macroeconomic indicators, and the banks risk-taking is lower than the laissez-faire solution. 1 In this paper we do not explicitly model the spillover; instead, there is a social cost when project failure occurs. 2

3 1.1 Literature review This paper partially builds on the literature on banks choice of leverage. This literature includes, for example, models based on moral hazard (e.g. (Allen, Carletti, & Marquez, 2011)) and models based on risk-shifting (e.g. (Repullo, 2013))). Our paper is closer to the latter stream of research. The majority of this research takes static setup, while in reality, banks are more likely to face dynamic decision problems. In a dynamic world, banks may have higher incentives to hold large capital buffers today, in order to stay in business and avoid costly exits in the future. Whether a bank indeed holds a large capital buffer depends on a variety factors, including the components of its asset portfolio, managers discount factors, expectation on future returns, and so on. From a regulator s point of view, banking regulation should account for banks incentives as well as the tradeoff between short term costs and long term gains. Consequently, a dynamic framework may lead to quite different model outcomes and policy implications. In light of these considerations, we adopt a dynamic framework in this paper to address these issues. Our framework also builds on the dynamic banking approach developed by (Corbae & D Erasmo, 2013). Unlike that literature, we focus on banks decision problems and abstract away from business cycle issues. The dynamic banking approach allows us to carry out a number of innovations we model more realistic balance sheets of banks with richer components, we closely examine bank decisions on funding sources and investment portfolios, and we study the roles of liquidity buffer and bank equity in constraining banks risk-taking behavior in an uniform modelling framework. We add to the dynamic banking literature by, instead of fixing the capital adequacy ratio exogenously by regulatory policy, allowing banks to choose their leverage instead. This innovation enables us to quantify the impact of capital requirements in restricting banks risk-taking, and derive an optimal regulatory capital ratio as a function of banks asset position and loan returns. Thus, our framework may shed some light on the current debate on capital adequacy ratio and countercyclical capital buffers. 1.2 Summary and contributions We develop a dynamic banking model, in which banks raise equity and deposits to support their investments in risky loans and safe government security, and have access to repo market when they are in distress. The banks choices on risk-taking, leverage, and liquidity buffer are all endogenized, and they differ from the social planner solution where the negative externality of bank failure is taken into account. Our paper contributes to the current debate on determination of the optimal capital ratio, and whether it should be countercyclical. We go a step further by exploring a potential companion to capital regulation. Specifically, since equity is costly the social cost of regulation may be reduced if a regulatory capital requirement can be accompanied by other tools, a liquidity buffer in this model. 3

4 2 Liquidity, Bank Capital and Optimal Risk-Taking Consider an economy that extends to 2 periods, t = 1, 2. There are two types of agents in the economy: A unit mass of investors, each of whom receives one unit of endowment at period 1. They invest the endowment in one of the banks at the beginning of period 1, and are repaid at the end of 1; A continuum of risk-neutral banks, who issue deposit contracts and equity to investors at period 1. Deposits are repaid at the end of the same period, with interest rate r D. Deposits are guaranteed by a complete deposit insurance so that r D can be normalized to a constant number, and investors are indifferent in choosing deposit contracts from different banks. Each bank is distinguished by its type θ, which is publicly observable and uniformly distributed, θ U[0, 1]. Then a bank of type θ invests l 1 in risky projects of type p 1, which yield a stochastic payoff at the end of the period. The bank can also hold a government security A 1, which yields a rate of return r A in the end of the period. The payoff of the risky project is given by { αz 1 (2z 2 θ p 1 ) with probability p 1, R = 0 with probability 1 p 1 where z captures the idiosyncraticproductivity shock to the investment technology and α is the productivity of technology. The realized z in each period is a random draw from two values {z g, z b }, which follow a Markov process with transition matrix [ ] π 1 π π 1 π with π > 1. 2 The timing of events is as follows: At the beginning of period 1, each bank raises funds one unit from each investor; The bank, starting with an initial net security position a 1, chooses how much to invest in risky assets l 1, and how much to invest in government securities A 1, as well as the riskiness of projects, p 1. On the liability side, a share of the bank s funds e 1 is taken as equity and the rest as deposits. The bank s choice set is thus a quadruple (l 1, A 1, p 1, e 1 ); 4

5 At the end of period 1, the technological shock z 2 2 is revealed. The project returns are determined and the bank collects the yield; Now the bank may wish to exit, liquidating all its assets at a discount ξ; otherwise it goes to secured overnight inter-bank market (e.g. collateralized repo), lending to or borrowing from other banks at inter-bank rate r B ; In the beginning of period 2, the bank pays out dividends on the equity and repurchases the collateral. Bank capital is costly such that equity holders require an excess return δ 0. The bank s problem is to choose capital ratio e 1, liquidity buffer A 1 and type of project p 1 to maximize its cash flow at period 2, i.e., max e 1,A 1,p 1 [0,1] E [W 1 (e 1, A 1, z 2, θ)], s.t. a A 1 + l 1. The bank s cash flow W 1 is determined by its decision on exit in the end of period 1, i.e. { W 1 (e 1, A 1, z 2, θ) = max W x=0 1 (e 1, A 1, z 2, θ), W1 x=1 (e 1, A 1, z 2, θ) } x {0,1} in which the value of staying in the business W x=0 1 (e 1, A 1, z 2, θ) = max B 2 A 1 1+r B {D 1 (e 1, A 1, z 2, B 1 )}, as well as the value of exit, s.t. a 2 = A 1 (1 + r B )B 2 0, D 1 (e 1, A 1, z 2, B 2 ) = π x=0 1 + B 2 (1 + δ)e 1, W x=1 1 (e 1, A 1, z 2, θ) = max { π x=1 1, 0 }. The bank s decision problem can be solved by backward induction. Suppose that the bank does not exit and survives in period t = 2 W x=0 1 (e 1, A 1, z 2, θ) = p 1 αz 1 (2z 2 θ p 1 )l 1 ( 1 + r D) (1 e 1 ) + r a A cl2 1 + A r B (1 + δ)e 1. (1) 2 Note that all end-of-the-period 1 variables are subscripted by 2 in order to match the inter-period variables such as B 2. 5

6 Instead, when { W1 x=1 (e 1, A 1, z 2, θ) = max ξ [p 1 αz 1 (2z 2 θ p 1 )l 1 + (1 + r a )A 1 ] ( 1 + r D) (1 e 1 ) 1 } 2 cl2 1, 0 > W x=0 1 (e 1, A 1, z 2, θ) the bank exits in the end of period 1. Given that W x=0 1 (e 1, A 1, z 2, θ) > W x=1 1 (e 1, A 1, z 2, θ), the bank s optimal choice of project can be found by taking first order condition W1 x=0 p 1 = αz 1 (2z 2 θ p 1 ) p 1 αz 1 = 0, p 1 = E [z 2 ] θ. And the bank s decision on holding liquidity buffer is derived from W x=0 1 = p 1 αz 1 (2z 2 θ p 1 ) + r a + c(1 + a 1 A 1 ) + 1 A r = 0, { B 1 A 1 = max c(1 + r B ) + (1 + a 1) + ra c p 1αz 1 (2E [z 2 ] θ p 1 ) c { 1 = max c(1 + r B ) + (1 + a 1) + ra c αz } 1E 2 [z 2 ] θ 2, 0. c The bank s incentive on holding liquidity buffer is determined by The interest return r a paid on holding government security; The cash raised from interbank borrowing at the repo rate r B ; The cost saved from issuing risky loans; The opportunity cost in issuing loans, which is captured by αz 1E 2 [z 2 ]θ 2. When z c 1 = z g so that E [z 2 ] is high as well, the bank holds lower buffer and issue more loans to reduce }, 0 the opportunity cost, and vice versa. However, if the return from risky project is too high, the bank will hold no buffer and invest everything on the risky projects. The bank s incentive on holding capital comes from inequality (1). If the inequality is never binding for e 1 [0, 1], the bank prefers all-equity financing, or, e 1 = 1. Otherwise the bank has the incentive to increase e 1 to reduce the interest paid on deposit, while in this case the cost of equity or the dividend paid on equity gets higher, too. Thus in equilibrium the bank s choice on equity always makes (1) binding. 6

7 In the static setup, the option of exit does not play much role in the bank s decision making. To see this, note that as long as 1 1+r B 1 and ξ < 1, p 1 αz 1 (2z 2 θ p 1 )l 1 ( 1 + r D) (1 e 1 ) + r a A cl2 1 + A r B > ξ [p 1 αz 1 (2z 2 θ p 1 )l 1 + (1 + r a )A 1 ] ( 1 + r D) (1 e 1 ) 1 2 cl2 1, or, the value of staying in the business is always higher than exiting as long as the cash flow is non-negative, and the bank only chooses to exit if such value is negative, W x=0 1 < 0. That is, there is no strategic exit. There is no exit averse, either, in the static setup. Since the future value of the bank s staying in business is zero, it is indifferent for the bank between exit and stay when W x=0 1 < 0. However, the results will change in a dynamic setup, in which the bank has to compare its liquidation value (if it chooses to exit) and the present value of its cash flow for the entire future (if it chooses to stay). Therefore, the bank may choose strategic exit to avoid too low future cash flow, or scramble for survival to avoid costly exit if the future cash flow is high enough. In both cases, the bank s choice on its balance sheet differs from that in the static setup. 3 Liquidity and Bank Capital in Dynamics In this section, we explore the bank s decision making in a dynamic setup. We follow most of the settings in the static model, while the banks live now in an infinite horizon. 3.1 Agents, technology and timing Consider an economy that is infinitely lived, t = 0, 1,..., +. The economy possesses the following features: A unit mass of investors, each of whom receives one unit of endowment at period t. They invest the endowment in one of the banks at the beginning of period t, and are repaid at the end of t; A continuum of risk-neutral banks, who issue deposit contracts and equity to investors at period t. Deposits are repaid at the end of the same period, with interest rate rt D. Deposits are guaranteed by a complete deposit insurance so that rt D can be normalized to a constant number, and investors are indifferent in choosing deposit contracts from different banks. Each bank is distinguished by its type θ, which is publicly observable and uniformly distributed, θ U[0, 1]. Then a bank of type θ invests l t in risky projects of type p t, which yield a stochastic payoff at the end of the period. The bank 7

8 can also hold a government security A t, which yields a rate of return rt A the period. The payoff of the risky project is given by { αz t (2z t+1 θ p t ) with probability p t, R = 0 with probability 1 p t in the end of where z captures the idiosyncraticproductivity shock to the investment technology and α is the productivity of technology. The realized z t in each period is a random draw from two values {z g, z b }, which follow a Markov process with transition matrix [ ] π 1 π π 1 π with π > 1. 2 The timing of events is as follows: At the beginning of each period t, each bank raises funds one unit from each investor; The bank, starting with an initial net security position a t, chooses how much to invest in risky assets l t, and how much to invest in government securities A t, as well as the riskiness of projects, p t. On the liability side, a share of the bank s funds e t is taken as equity and the rest as deposits. The bank s choice set is thus a quadruple (l t, A t, p t, e t ); At the end of period t, the technological shock z 3 t+1 is revealed. The project returns are determined and the bank collects the yield; Now the bank may wish to exit, liquidating all its assets at a discount ξ; otherwise it goes to secured overnight inter-bank market (e.g. collateralized repo), lending to or borrowing from other banks at inter-bank rate rt B ; In the beginning of period t + 1, the bank pays out dividends on the equity and repurchases the collateral. Bank capital is costly such that equity holders require an excess return δ Bank s decision making The bank operates a balance sheet as shown in Figure 1. It raises funding from equity and deposits, and invests in risky loans to the firms as well as in safe assets like government securities. We first present the various components of the bank s activities, then set up and solve the bank s value function. 3 Note that all end-of-the-period t variables are subscripted by t + 1 in order to match the inter-period variables such as B t+1. 8

9 Assets Liabilities Private loans to firms Downward sloping demand Other assets, may be Government securities Reserves, etc. Deposits With deposit insurance Equity Costly Payment of dividends Regulatory requirement Figure 1: The bank s balance sheet Components of Bank Activities. We now discuss the different components of the bank s balance sheet. First of all, the bank s net security position a t implies the following feasibility constraint 4 a + 1 l + A. The end-of-period profit of the operating bank (x = 0) is given by π x=0 = pαz(2z θ p)l ( 1 + r D) (1 e) + r a A 1 2 cl2. (2) For an exiting bank (x = 1) with limited liability, its profit is { π x=1 = max ξ [pαz(2z θ p)l + (1 + r a )A t ] ( 1 + r D) (1 e) 1 } 2 cl2, 0. (3) Thus far, the equity ratio e arises from the bank s optimal decisions, but it may also be dictated by regulatory requirements. We return to this issue in a later Section 5. In the end of period t, the bank participate overnight inter-bank market, or, collateralized repo market borrow at the end of period t and repurchase at the beginning of period t + 1 by lending cash to other banks (B < 0) or borrowing from other banks (B > 0) at 4 In the rest of the paper, we remove subscripts t for all period t variables, while variables in t + 1 are denoted by primes. 5 Suppose the bank is subject to capital requirement, or, it must hold at all times at least a fraction ϕ of risk-weighted assets in CET-1 or, equivalently e = A + l (1 e) ϕ (l + ωa) }{{}, risk-weighted l(1 ϕ) + A(1 ωϕ) (1 e) 0. 9

10 inter-bank rate r B using high quality assets as collateral B A 1 + r B. The net assets that the bank brings to the next period is thus a = A ( 1 + r B) B 0. The bank pays out dividends at the end of each period with the following two assumptions. D = π + B (1 + δ)e t, First, the pecking-order assumption that SEO is prohibitively expensive (documented by (Dinger & Vallascas, 2014)). That is, the bank can only raise short-term debt to pay out non-negative dividends instead of violating the capital requirement 6. Second, the equity premium assumption that equity holders require a premium for their risky investments, as in (Repullo, 2013). In the beginning of each period, the bank of type θ gets deposits from investors, chooses its loan supply l, equity e and security holdings A in order to maximize its cash flows. In the end of each period, the bank can choose to enter collateralized repo market for inter-bank borrowing B, and decide whether to exit x. Bank s Value Function. We now have summarized the components necessary to define the bank s problem. The bank s decision problem in the beginning of t is to maximize its cash flow over its future life horizon: V (a, θ, z) = max e,a,p [0,1] βw (e, A, z, θ), (4) s.t. a + 1 A + l. The bank s cash flow W is determined by its decision on exit, i.e. { W (e, A, z, θ) = max W x=0 (e, A, z, θ), W x=1 (e, A, z, θ) } x {0,1} in which the value of staying in the business W x=0 (e, A, z, θ) = max {D (e, A, z, B ) + E [V (a, θ, z ) z]}, B A 1+r B s.t. a = A (1 + r B )B 0, 6 A similar setup is in (Corbae & D Erasmo, 2014) D (e, A, z, B ) = π x=0 + B (1 + δ)e, 10

11 as well as the value of exit, W x=1 (e, A, z, θ) = max { π x=1, 0 }. 3.3 Solution to the Bank s Problem The bank s decision problem is solved by numerical simulation. The simulation is based on parameter values listed in Table 1. Parameter Table 1: Parameter values used in simulation Value The salvage value of one unit asset in liquidation ξ 0.7 Bank manager s discount factor β 0.95 Deposit rate r D Interest rate paid on government security / bank s reserve r a 0 Minimum required return to equity * δ 0.1 Repo rate r B Transition probability, good good and bad good π 0.90 Productivity shock in the good state z g 1 Productivity shock in the bad state z b 0.95 Bank s type θ 0.9 Bank s cost parameter in loan issuance c Productivity of bank s investment technology α 1.35 * Source: Return on Equity for all U.S. Banks, Figures 2 to 4 present the solution to the bank s decision problem, as described in Section 4. Figure 2 illustrates the bank s decision on holding government security A, given its initial net security position a. In the good state z g, investment on government security is a dominated strategy, since it is very likely that z = z g. However, in the bad state z b, the return from the investment is low and there is a positive probability that z = z b. Therefore, the bank has a incentive to hold A > 0 in order to avoid a costly exit. However, the optimal level of A also depends on the bank s decision on equity holding, which also reduces the bank s likelihood of failure. Figure 3 depicts the bank s decision on issuing equity. It is sometimes claimed in academic research that the bank s leverage ratio is procyclical that is there is a lower equity ratio in booms and a higher equity ratio in busts. Surprisingly, our simulation results show that the leverage ratio in our model can be both procyclical and countercyclical. The reason for this result is that instead of holding equity, banks can also use a liquidity buffer A to cushion 11

12 0.6 Zg Zb A a Figure 2: The bank s investment on government security adverse shocks. Thus, whether to invest more in holding a liquidity buffer or equity depends on the bank s cost-benefit calculations. When the initial net security position a is low and the economy is in the bad state z b, the return from risky assets is low. Since the economy is already in the bad state, the likelihood of getting stuck in the bad state and being forced to exit in the next period is significantly high. Consequently, the optimal solution is to reduce deposits in order, to avoid costly repayment to depositors when the bank is dissolved. This scenario leads to a higher equity ratio. On the contrary, when the economy is in the good state z g and the return from the risky assets is high, the bank need not worry about the lower bound of dividend payment. In this situation, the optimal strategy is therefore to accept more cheap deposits to maximize the cash flow D. This scenario leads to a lower equity ratio, and for this case bank leverage is indeed procyclical. However, things will be changed if the initial net security position a is high. Higher net security position allows the bank to increase its investment in risky assets in the good state z g. Here, the optimal strategy to maximize cash flow D is to reduce interest payments on deposits, leading to a higher equity ratio. In this case, the bank s leverage is actually countercyclical. Figure 4 shows the bank s decision on its net security position a carried to the next period, conditional on the following variables: its initial net security position a, current state of the world z and next period state of the world z. The bank faces a tradeoff: it has an incentive to hold a larger a in order to weather future adverse shocks. However, it 12

13 1 Zg Zb e a Figure 3: The bank s holding of equity has the opposite incentive, to reduce a for higher borrowing B from interbank market, in order to ensure current cash flow. The figure shows it is indeed the case: a falls when the bank s situation worsens, and it has to hold least a when the bank is in the bad state for two periods in a row. 4 Optimal Banking Regulation Banks need regulation because their failure entails a social cost, which is not internalized by individual banks. Assume that the social cost from failing loans of a bank with type θ is kz zαθl, proportional to the value of the bank s loan. Suppose the regulator has the same information as the market, and its objective is to maximize social welfare using various regulatory instruments, taking into account the social cost of non-performing loans. The regulator s problem is similar to that of the bank in Section 3, except that the profit of an operating bank (x = 0) in the end of each period (2) becomes: π x=0 = [pαz(2z θ p) (1 p)kz zαθ] l ( 1 + r D) (1 e) + r a A 1 2 cl2. (5) And for an exiting bank (x = 1) with limited liability, its value (3) now becomes { π x=1 = max ξ [pαz(2z θ p)l (1 p)kz zαθl + (1 + r a )A t ] ( 1 + r D) (1 e) 1 } 2 cl2, 0. (6) 13

14 0.5 Zg->Zg Zg->Zb Zb->Zg Zb->Zb a a Figure 4: The bank s net security position in period t + 1 The solution to the regulator s problem gives the capital adequacy ratio and optimal liquidity buffer, as functions of the bank s initial asset position, bank s type, and the current shock to the return of the loans. Using the model, one can explore further the following questions: 1. Countercyclicality of capital buffer and bank s risk-taking behavior. Since bank s leverage, loan supply and risk-taking are all endogenized, this model is one of the first that can in principle discern whether a countercyclical capital buffer can contain the credit cycle and restrain banks excessive risk-taking; 2. Liquidity regulation and capital requirement. Liquidity regulation and capital requirement are two of the most powerful regulatory tools in banking regulation, and indeed are two pillars in Basel III. Nevertheless, academics and policymakers usually discuss them separately 7 There is sparse literature on whether they are complementary or substitutive and how regulator can set both tools simultaneously. This model is one of the pioneers that study the interaction between the two policies: raising capital adequacy ratio effectively curbs the bank s incentive in issuing risky loans, while equity is costly to banks since they need to pay high premium to shareholders; holding more liquidity buffer increases opportunity cost for the banks since they have to scale down the investments in risky loans, while it offers cushion for the banks in the recession, allowing them to pay off depositors and avoid costly exits, as well as to have more net security 7 One exception is (Cao & Illing, 2011). 14

15 for future investments. The optimal liquidity coverage ratio and capital adequacy ratio depend on the balance between these costs and benefits. Figure 5 shows the implied capital adequacy ratio, derived from the solution to the regulator s problem, and Figure 6 presents the implied regulatory liquidity requirement. Since the social cost arises from the bank s non-performing loans, the first priority for the regulator is to directly reduce bank s risk-taking. Thus the regulator must require the bank to invest in safer loans and reduce its exposure to risky investments. Under this requirement, the success probability of bank s investments is higher and the loan supply (holding of safe government security) is lower (higher) in the regulator s solution. Somewhat counterintuitively, the regulatory capital adequacy ratio is lower than the non-regulatory solution, as in Figure 5. This outcome is reasonable in our model banks do not need as much costly equity to further contain their risk-taking behavior, because loan riskiness and loan supply are both restricted. Furthermore, Figure 5 suggests that the regulatory capital adequacy ratio should be countercyclical. The regulatory capital ratio should be higher in good times (when z g is realized), restricting the bank s excessive risk-taking and containing the credit supply. In bad times (when z b is realized), the ratio should be lowered to avoid a credit crunch. 1 Regulator solution under Zg Regulator solution under Zb Bank solution under Zg Bank solution under Zb e a Figure 5: The regulatory capital adequacy ratio 15

16 1 Regulator solution under Zg Regulator solution under Zb Bank solution under Zg Bank solution under Zb A a Figure 6: The regulatory requirement on holding government security 5 Conclusions We present a framework for modelling the optimum capital adequacy in a dynamic model of banking. We combine the static capital adequacy framework of (Repullo, 2013) with a dynamic banking model similar to that of (Corbae & D Erasmo, 2014), with the extra feature that the probability of systemic risk is endogenous. Unlike previous work, we examine frameworks to ameliorate bankruptcy using both capital adequacy and liquidity requirements. Since equity is costly, the social cost of regulation may be reduced if a regulatory capital requirement can be accompanied by other tools such as a liquidity buffer. 16

17 0.5 Regulator solution under Zg->Zg Regulator solution under Zg->Zb Regulator solution under Zb->Zg Regulator solution under Zb->Zb a a Figure 7: The bank s net security position in period t + 1 under capital and liquidity regulation 17

18 References Allen, F., Carletti, E., & Marquez, R. (2011). Credit market competition and capital regulation. Review of Financial Studies, 24, Cao, J., & Illing, G. (2011). Endogenous exposure to systemic liquidity risk. International Journal of Central Banking, 7 (2), Corbae, D., & D Erasmo, P. (2013). A quantitative model of banking industry dynamics (Tech. Rep.). University of Wisconsin. Corbae, D., & D Erasmo, P. (2014). Capital requirements in a quantitative model of banking industry dynamics (Tech. Rep.). University of Wisconsin. Dinger, V., & Vallascas, F. (2014). Are banks less likely to issue equity when they are less capitalized? (Tech. Rep.). University of Osnabrueck. Repullo, R. (2013). Cyclical adjustment of capital requirements: A simple framework. Journal of Financial Intermediation, 22,