The Impact of Recovery Value on Bank runs

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1 The Impact of Recovery Value on Bank runs Linda M. Schilling April 10, 2017 Abstract Recovery values after bank runs differ between countries while Basel III imposes uniform capital and liquidity regulation on banks in member countries of the Basel Committee on Banking Supervision. In a model where debt investors correctly anticipate recovery rates conditional on the event of a run, and take bank capital structure and coupon payments in the debt contract as given, changes in recovery rates change investors behavior and thus bank stability and welfare. This paper demonstrates that recovery values after runs impact socially optimal debt contracts and capital structures by changing incentives of uninsured debt investors to run on the bank. Capital regulation can be detrimental to bank stability, depending on recovery rate, since the socially optimal debt ratio may exceed the bank optimal debt ratio and the bank reacts to constraints on capital structure by altering debt contracts. To enforce particular stability levels, capital regulation has to be recovery rate specific and thus country-specific. This poses a challenge to transnational regulation under Basel III. Utrecht University and Bonn Graduate School of Economics, l.m.schilling@uu.nl. I thank Eugen Kovac, Gary Gorton, Joonhwi Joo, Kebin Ma, and Zhen Zhou for very insightful comments on the paper. 1

2 1 Motivation Country specific bankruptcy costs impact debt recovery rates through various channels such as allocating different sets of control rights to creditors, demanding different time periods the bank remains in bankruptcy, varying court-declared expenses, and so on. Bankruptcy laws, which determine country specific bankruptcy costs, differ from country to country. Chapter 11 of the U.S. bankruptcy code leaves control over firm s assets to some degree with management during debt renegotiations. Swedish bankruptcy law, in contrast, foresees a public auction where the firm is liquidated either piecewise or survives as a going concern and management and shareholders immediately lose their control rights. In France, the state imposes court-administered procedures in bankruptcy with the objective of maintaining employment. French bankruptcy courts are given control of the bankruptcy process and they are not mandated to sell firm assets to the highest bidder, see Davydenko and Franks (2008). Since bankruptcy costs depend on a country s legal framework, recovery values that determine payments to debt investors are country specific. Davydenko and Franks (2008) find median undiscounted recovery rates of 92% in the U.K., 67 % in Germany, and 56% in France. 1 Thorburn (2000) estimates recovery rates of Swedish firms as proportion of debt s face value 2 at a median of 25% for piecewise liquidation and 38% if the firm is auctioned in bankruptcy as going concerns. Analyzing U.S. firms, Bris et al. (2006) find mean recovery rates 3 of 1% for unsecured creditors of firms under Chapter 7 liquidations and 52% for unsecured creditors of firms under Chapter 11.,see also Franks and Torous (1994), Gupton and Carry (2000). Araten et al. (2004)). A further factor that influences debt recovery values after bank runs are interventions by central banks (lender of last resort). In the course of a run on a bank, in the Euro system Emergency Liquidity Assistance (ELA) can be granted by the national central bank to prevent a financial panic and contagion to other financial firms. While bankruptcy proceedings impose fixed costs that diminish recovery values, interventions by central banks increase the average recovery value to debt investors. The scale of intervention and recovery values may depend on liquidity mismatch faced by the bank. 4 In addition, liquidity of assets pledged as collateral to creditors plays an important role when liquidating firm assets. Recovery rates are higher the more liquid the collateral. Differences in debt recovery rates by country lead to an adaption of behavior by creditors ex ante. In an empirical study, Davydenko and Franks (2008) find that differences in creditors rights across countries cause banks to adapt their lending practices at loan 1 Recovery rates calculated as one less total loss over exposure at default. 2 Note that face values overstate market values. 3 Here, recovery rate is measured as fraction of initial claim which is distributed by the court in the case closure. 4 Here, liquidity mismatch is the gap between short-term debt outstanding and liquidity of the asset (Brunnermeier et al., 2013). 2

3 origination to companies in France, Germany and the UK. This paper analyzes how cross-country differences in recovery rates impact bank stability when uninsured debt investors correctly anticipate recovery values in case of default and can miscoordinate on runs. The recovery rate characterizes a wedge between the amount of liquidity the bank can raise using the asset as collateral to borrow in the money market and the amount of cash that is available for distribution to withdrawing debt investors in the course of the run. The bank funds withdrawals via collateralized borrowing to prevent liquidating assets and thus interruption of investment. If the amount of withdrawals exceeds the amount of cash available via pledging the asset as collateral (funding liquidity), the bank is forced to sell illiquid assets fast to satisfy claims. The amount of cash the bank can realize by selling assets (market liquidity) and funding liquidity differ in general, see Brunnermeier and Pedersen (2009). In a fast sale of illiquid assets total proceeds from sales undercut funding liquidity of the asset. In addition, a lender of last resort can get involved during the process of a run and provide liquidity assistance. We use the term recovery rate to denote the wedge between the liquidity threshold that triggers the run (asset s funding liquidity), and the amount of cash available for distribution to withdrawing debt investors after accounting for asset market values and liquidity assistance. More precisely, we define recovery rate as the fraction of asset (funding) liquidity available for distribution to withdrawing investors. Recovery rates and asset liquidity determine payments to debt investors during runs. Uninsured debt investors respond to changes in recovery rates in form of altered propensity to withdraw. The model allows for a unique equilibrium in which the probability of runs varies in recovery rates. When setting optimal contract coupons and capital structure, the bank takes into account how recovery rates interfere with the coordination game of debt investors. We examine different functional forms of recovery rate. Recovery rates may be a constant or non-constant fraction of asset liquidity. In the non-constant case, we allow recovery rates to depend on liquidity mismatch faced by the bank. By this we allow dependence of recovery rates on interventions of a lender of last resort where extent of an intervention may depend on liquidity mismatch of the bank of total short-term debt outstanding. In addition we parametrize each recovery rate function by a payoff parameter which allows to scale the wedge up or down within a specific recovery rate function. By numerical simulation we compare bank optimal to socially optimal capital structures and debt contracts across recovery rate functions. The social planner maximizes stability while the bank maximizes return on equity, given participation constraints. As main contribution of the paper we show, capital structures and contracts that implement the social optimum differ across recovery rates. When two banks invest in same assets and therefore face same asset liquidity, asset payoff risk and asset return, the 3

4 debt ratio δ Socially optimal capital structure (a) Socially optimal capital structure for three different recovery rate functions across payoff parameter b b debt ratio δ Comparison social versus bank optimal capital structure social optim debt ratio, const rec rate bank optim debt ratio, const rec rate (b) Comparison of Social Versus Bank Optimal Capital Structure b difference in model-implied optimal debt ratio can be up to 10 percentage points when the banks face different recovery rate function, see Figure (1a). The Modigliani Miller Theorem fails to hold in our model since changes in capital structure lead to changes in bank stability and therefore alterations in the total expected return on bank assets. 5 Contrasting bank optimal with socially optimal solutions, we show that for some recovery rate functions bank optimal debt ratio in fact may undercut the social optimal debt ratio meaning that observed lower debt ratios do not necessarily imply higher bank stability, see Figure (1b). This is, since the bank and the social planner have two channels to maximize return on equity respectively to maximize stability. The channels are debt ratio and contract coupons to debt investors. Higher contract coupons in the future induce debt investors to roll over, bank stability goes up but the contract becomes more expensive to the bank. Lower debt ratios can increase bank stability (see below), depending on recovery rate function, but decrease return per invested unit of equity should there be no run. Return on equity is zero in case of a run. To maximize return on equity, the bank has to find the right balance between maintaining stability and minimizing costs, and selects coupons and debt ratio accordingly. The bank s trade-off between costs and stability is affected by recovery rates since recovery rates determine payoffs to debt investors and thus likelihood of runs. In the scenario where social optimal debt ratio exceeds bank optimal debt ratio, the bank optimal contract coupon has to be lower than the coupon under the social optimum. 6 This case implies, when transitioning from the bank 5 Here, total expected return on bank assets does not equal return on the risky asset but takes into account that the asset needs to be liquidated in the incidence of a run and the set of states for which liquidation takes place alters in the debt ratio. 6 This is since otherwise stability under the bank optimal solution would be higher than stability under the social solution, a contradiction. 4

5 optimal to social solution, the stability benefits of an increase in coupon may outweigh the stability drawbacks of the accompanying increase in debt ratio. This scenario can emerge depending on recovery rate function and depending on payoff parameter of recovery rate. Last, we consider the effect of capital regulation on bank optimal solutions under varying recovery rate functions. If capital regulation excludes the bank optimal debt ratio, the bank is forced to adjust and sets a new coupon and a lower debt ratio to maximize return on equity subject to the constraint on capital structure and participation by debt investors. Capital regulation is socially desirable, if the stability level implied by the bank optimal solution under regulation approaches the social stability level. We show, welfare may increase under capital regulation depending on recovery rate function. If capital regulation demands the bank to set a lower debt ratio, in a first effect stability can improve and utility to debt investors increases, meaning that their participation constraint becomes non-binding. The bank appreciates enhanced stability but a lower debt ratio implicates a decline in return per invested unit of equity. To keep return on equity on a level closer to the unregulated bank optimal solution, the bank uses the second channel to optimize and lowers coupons to save costs on the contract. Debt ratio and contract coupon move together. The lowering of coupons has the side effect to decrease stability and utility to debt investors. The stability level implied by the bank optimal solution under regulation approaches the social stability level if stability gains through enforced decrease in debt ratio outweigh the stability loss via a decrease in coupons. For some recovery rate function, capital regulation has exactly this described, desired effect. We can however give a counterexample, the constant recovery rate function, for which the opposite is true. Here, stability implied by the bank optimal solution under regulation can be lower than stability level implied by the bank optimal solution of the unregulated bank since the decrease in coupons harms stability more than the enforced decrease in debt ratios. This result again highlights that capital regulation needs to be recovery-rate specific. The point of this paper is not to calculate optimal capital structures and contracts for varying recovery rates but to demonstrate that bank and social optimal capital structures and contracts differ across recovery value functions due to changed behavior of uninsured debt investors. We deliver several counterexamples to illustrate that without knowledge of a recovery value function the (non)-optimality of capital structure and the benefits of capital regulation cannot be verified. A unified, transnational regulatory framework not necessarily enforces equal stability levels across countries since countries differ in recovery value. 5

6 Related Literature This paper adds to the growing strand of literature on stability of financial intermediators against debt runs. Diamond and Dybvig (1983) analyze coordination behavior of depositors who share consumption risk by entering in deposit contracts with a bank. Risk-sharing among depositors leads to a liquidity mismatch between bank assets and liabilities and results in proneness of the bank to panic runs. The Diamond and Dybvig (1983) model yields multiple equilibria. In the good no run equilibrium welfare is higher than in the competitive outcome, while in the run equilibrium, the result is reversed. Postlewaite and Vives (1987) analyze a two player incomplete information game which yields a unique equilibrium with strictly positive probability of a bank run. As opposed to Diamond and Dybvig (1983), the model presented here features a unique equilibrium, and in contrast to Postlewaite and Vives (1987) there is uncertainty about the fundamental value of bank s assets. Chari and Jagannathan (1988) and Jacklin and Bhattacharya (1988) model information-based runs by introducing informative signals on the bank s asset returns. In contrast to these papers, our model features equilibrium uniqueness. To obtain a unique equilibrium, this paper employs technique from global games theory (Carlsson and Van Damme, 1993; Morris and Shin, 1998, 2001). The models closest to ours are Goldstein and Pauzner (2005), Morris and Shin (2009); Vives (2014) and Rochet and Vives (2004). The papers by Morris and Shin (2009); Rochet and Vives (2004); Vives (2014) study the impact of capital structure and asset liquidity on coordination behavior of debt investors in a global game in the context of collateralized funding or delegated decision making. Goldstein and Pauzner (2005) embed the Diamond and Dybvig model in a global game and show analytically that risk-sharing through deposit contracts is ex ante optimal although risk-sharing increases the probability of runs. Rochet and Vives (2004) derive policy recommendations for bank regulation and for interventions of the Lender of Last Resort in a model where solvent banks can be illiquid with strictly positive probability. Morris and Shin (2009) analyze how illiquidity risk and insolvency risk vary with balance sheet composition. Vives (2014) relates precision of public information, short-term debt ratio, competition for funds and fire-sale penalties to the degree of strategic complementarity of investors actions and thus fragility. König et al. (2014) analyze bank optimal portfolio choice under endogenous illiquidity risk due to runs. Our paper differs from these papers by introducing a new element, recovery rate functions, to the coordination problem of debt investors. Variations in recovery rates alter payoffs conditional on runs and therefore impact coordination behavior of debt investors, bank stability and bank and socially optimal contracts and capital structures ex ante. As opposed to Goldstein and Pauzner (2005) and Diamond and Dybvig (1983) in our model, debt investors have no insurance motive since they are risk-neutral and not subject to 6

7 exogenous liquidity shocks. In contrast, debt investors here invest in the bank to gain access to an asset which promises higher returns than storage. Consumption risk and risk-aversion can easily be incorporated in our model. We further depart from Goldstein and Pauzner (2005), by considering variations in asset liquidity and allowing equity financing of the bank. While Morris and Shin (2009), Rochet and Vives (2004); Vives (2014) and König et al. (2014) allow asset liquidation value to depend on the random state, in our model liquidation value is exogenous and deterministic. Morris and Shin (2009), Rochet and Vives (2004); Vives (2014) allow the bank to invest parts of its funds in a liquid asset while in our set-up the bank always invests in the risky, illiquid asset. The examples for recovery value functions analyzed in this paper are borrowed from related papers, we for instance analyze constant recovery rates as in Morris and Shin (2009) and recovery rates that imply payoffs which are independent of asset liquidity, as in Rochet and Vives (2004). Our paper further adds to the strand of literature which analyzes welfare consequences of financial regulation or government intervention in form of guarantees, bail outs or liquidity assistance. In a related model, Allen et al. (2015a) and Allen et al. (2015b) analyze the impact of different forms of government guarantees on the probability of banking crises. van der Heuvel (2016) analyzes social costs and benefits of liquidity and capital regulation in a general equilibrium growth model where banks create liquidity through financing illiquid loans with liquid deposits but are subject to Moral Hazard due to deposit insurance. Gorton and Winton (2016) analyze the impact of bank capital regulation in a model where more capital reduces the risk of bank failure but increases exposure of investors to liquidity shocks by forcing them to hold less demandable deposits and more information sensitive equity. To the best of our knowledge there is only one other paper that considers the impact of variation in recovery rates on debt roll over decisions in a setting with endogenous liquidity risk. In a dynamic game, He and Xiong (2012) analyze roll over decisions of debt investors who face a time-varying fundamental. Due to a staggered debt structure, creditors roll over decisions are made asynchronous. Since the amount of debt maturing in a short time period is small, there is no coordination problem between creditors maturing at the same time, in contrast to our model, but debt runs are motivated by fears that future maturing debt might not be rolled over leading creditors to run ahead of others. In fact, the model predictions of He and Xiong (2012) in the special case of a single debt investor are similar to ours, sufficiently low debt recovery rates lead the single investor to always roll over, while in our model lower debt recovery rates decrease bank run risk. But the result in He and Xiong (2012) flips as they introduce a staggered debt structure, smaller liquidation recovery rates amplify the rat race between creditors, the run threshold increases or vice versa, higher recovery rates lead to a more stable firm. 7

8 The difference in results is due to the fact that in our model, lower recovery rates lead to lower payoffs to debt investors conditional on the decision to withdraw and the occurrence of a run. The incentive to withdraw thus decreases as recovery rates go down. In He and Xiong (2012), the payoff to withdrawing is safe and not affected by recovery rates since the amount of maturing debt in a short time period is small. Instead, lower recovery rates lead to a lower expected continuation value when rolling over. The effects of recovery values on bank stability in this model are therefore reversed to the results in He and Xiong. 2 The Model There are two periods, three points in time 0, 1, 2, an one good (money). We assume no discounting between periods. There is a financial intermediary (bank), and two kinds of agents: a continuum of short-term debt investors [0, 1] of measure one, and a single equity investor. Both kinds of agents are risk-neutral and live for two periods. 7 At time zero, debt investors are symmetric and born each endowed with one unit of the good. Debt investors can consume in either period while the equity investor can only consume after period 2. At time zero, the equity investor is endowed with measure 1 units of the good. Debt investors and equity investors finance the bank s investment in a risky asset. Agents are born either as equity or debt investor, agents may not split their endowments to finance the bank in both ways. 8 Equity investors are passive and non-strategic. Investment and Collateralized Borrowing There exists a storage technology and an illiquid, risky, long-term investment opportunity T (risky asset). Debt investors and equity investors have no access to asset T, only to storage. Both types of investors gain access to T indirectly through investing in the bank. Investment in storage yields the initial investment for sure in every period. For every unit invested in period zero, the risky asset pays a return H with likelihood θ 0 and pays zero with probability 1 θ in period two, where the asset s random payoff probability θ (0, 1), or the economy s fundamental value (random state) is described in the information structure below. The expected asset return exceeds the return from storage E[θ]H > 1 (1) 7 That is, there is no exogenous liquidity shock in period one which prevents some agents to consume in period two. Thus agents face no consumption risk as in (Diamond and Dybvig, 1983) and (Goldstein and Pauzner, 2005). In particular, for tractability reasons we assume that there are no fundamental differences between the types of agents. Potential exogenous liquidity shocks and thus consumption risk to debt investors a la Diamond and Dybvig is straight forward to incorporate in the model and we expect the results of the paper to hold. Risk-aversion can easily be incorporated as well but results might change. 8 This assumption is for tractability reasons. 8

9 In period one, the risky asset generates no cash flows. To raise cash in period one, fractions of the asset can be pledged as collateral to borrow money from a third party in the money market via a repurchase agreement (repo). A repo transaction has two parties, the borrower (here the bank) and a lender. The lender lends cash to the borrower. To reduce the risk of the transaction to the lender, the borrower posts a collateral (asset) which goes into physical possession of the lender. Borrower and lender agree on that the collateral is repurchased by the borrower at the borrowed amount at a prespecified date. On top, the borrower pays interest i, the repo rate. If the collateral accrues interest during maturity of the repo, and the borrower can repay, accrued interest goes to the borrower. 9 If the borrower cannot repay, she defaults on the repo and the lender in the repurchase agreement may sell the collateral at market price. 10 Let fraction l (0, 1] be the exogenous amount of cash that can be raised by pledging one unit of the asset as collateral to a third party in a repo transaction (funding liquidity). 11 The bank is an investment expert. She has access to investing in risky asset T. To invest, the bank raises overall one unit in funds from equity and debt investors. To raise debt, she offers a contract to debt investors (explained below) and offers the residual value from investment to equity investors. The bank is in perfect competition for equity and makes zero profit. The bank s objective is to maximize shareholder value, she strategically selects her capital structure and interest payments (coupons) if offers in the debt contract: The bank sets debt ratio δ (0, 1), the fraction of its funds financed by uninsured shortterm debt, and equity ratio 1 δ, with the objective to maximize return per invested unit of equity (return on equity) to those equity investors it admits, subject to the outside option storage to both types of investors. Since the bank s balance sheet size is normalized to one, debt ratio δ equals the overall measure of outstanding short-term debt and the bank s capital structure is directly described by debt ratio δ. The bank only utilizes measure one unit of funds out of the aggregate endowment (measure two) of the economy. We make this assumption such that the bank optimizes over capital structure in an unconstrained way, she can select any debt ratio in (0, 1] and thus any equity ratio since the supply of equity and debt is sufficient. 12 We abstract from competition among equity or debt investors to finance the bank. 9 Note that this leads to a major distinction in pay-off structure compared to the case where the borrower has no access to the money market and has to sell parts of the asset to raise cash. Sold parts of the asset do not accrue interest to the previous owner even if the asset is bought back. 10 See Brunnermeier and Pedersen (2009). 11 Note, l is not the true value of the collateral in period 1 but the fraction of the value participants in the money market are willing to pay to accept the asset as collateral (overcollateralization), see also Dang et al. (2013). Fraction 1 l is called the haircut and corresponds to a safety margin to the lender. 12 When equity is very scarce, that is supply of equity in the economy is only measure τ << 1, we expect the results of the paper to change. 9

10 The bank invests collected funds of one unit in risky asset T. The bank cannot split her funds to also invest in storage. 13 In period one, the bank may pledge (fractions of) the asset as collateral to borrow cash from a third party in a repurchase agreement at repo rate i to satisfy potential debt claims by uninsured debt investors. If the bank can repay the counterparty of the repo in period two, she regains control of the asset and collects interest on the entire investment including the pledged fraction of the asset. Debt contract In order to attract funding by debt investors, the bank promises to pay interest on deposited funds. Every debt contract is characterized by two coupon payments, a period one (short-term) coupon of 1 unit and a period two (long-term) coupon of k (1, H) units. Henceforth, write (1, k) for the contract. At time zero, a debt investor can invest her unit of endowment in contract (1, k). The contract is liquid in the sense that only after period one, a debt investor decides on her action whether to choose the short-term coupon and ( withdraw ) her investment, by this earning coupon 1 or to roll over her investment and earn coupon k after period two. As a consequence, at time zero the amount of cash claimed by debt investors at the interim stage is not known to the bank. If a debt investor decides to withdraw, we also say that she runs on the bank. Debt investors cannot demand a fraction of their investment. 14 The parameter k (1, H) can be seen as an implicit forward interest payment which the bank pays to investors for leaving funds invested for another period. 15 The contract (1, k) and asset return probability θ are such that the expected payoff from rolling over exceeds the payoff from withdrawing E[θ]k > 1, (2) this constraint further implies that expected period two payoff from the contract exceeds utility from storage. We assume that, 1+i > k, that is collateralized borrowing via a repo is more expensive to the bank than borrowing from uninsured debt investors who roll over. The assumption can be justified noting that if the other party of the repo was unwilling to lend to the bank, the bank would default due to illiquidity. This gives all the bargaining power to the other party of the repo and implies a high repo rate i. In absence of this assumption, the bank had no incentive to raise funds via the debt contract in the first place and would jeopardize bank stability knowing that refinancing using the repo is cheaper given that she can borrow enough cash from the counterparty of the repo to prevent a run By this assumption we abstract from moral hazard on side of the bank. Incorporating moral hazard in the model would be very interesting but is beyond the scope of this paper. 14 This assumption is for tractability reasons. 15 The assumption k > 1 is necessary, otherwise withdrawing early was a dominant action. 16 For our results at the limit, when noise vanishes, the assumption 1 + i > k becomes irrelevant and results go through also for k > 1 + i, see below. 10

11 Further assume H > 1 + i = max(1 + i, k). That is, return of the risky asset exceeds gross repo rate and long-term coupon k. Endogenous Liquidation At period 1, each out of measure δ of debt investors has a claim on a coupon payment of one unit upon deciding to withdraw. The maximum measure of cash withdrawals the bank might face is thus δ. By seniority of debt, the bank is obliged to make the coupon payments under the premise of solvency. Denote by n [0, 1] the endogenous, ex ante random equilibrium proportion of debt investors who decide to withdraw after period 1 (aggregate action). Given the contract (1, k) and the measure of short-term debt funds δ (0, 1) collected by the bank, after period 1 the bank needs to pay out measure δn in cash to withdrawing investors. The bank finances withdrawals nδ via collateralized borrowing. The bank pledges fraction nδ/l of the asset as collateral to a lender in a repurchase agreement and obtains in return measure nδ in cash. The bank agrees to repay nδ(1 + i) to the lender after period two to gain back control of the pledged fraction of the asset. A run on the bank occurs, if after period one the measure of short-term funds claimed by withdrawing investors exceeds the amount that can be borrowed using the asset as collateral. That is, if n [0, 1] realizes such that nδ 1 > l or nδ > l (3) In case of a run, we assume that the bank does not enter in a repo but immediately defaults and is liquidated, see below. We relax this assumption in a later section where we allow for sequential liquidation. If funding liquidity l is sufficiently high for a given capital structure δ and contract (1, k), the occurrence of a run can be excluded ex ante. Since the proportion of investors who run on the bank cannot exceed one, runs can be excluded if δ l. We call such a bank run-proof. If instead a run cannot be excluded ex ante, if δ > l, the bank is run-prone. Liquidity Mismatch liquidity to total outstanding short-term debt Define the bank s liquidity ratio as the ratio of bank s asset ξ = l δ (4) The liquidity ratio is an indicator of the bank s liquidity mismatch. The larger asset liquidity l and the lower overall outstanding short-term debt, the higher the bank s liquidity ratio and the lower the bank s liquidity mismatch, that is the gap between potential cash withdrawals and cash available through selling the asset. Debt ratio δ enters the model only via liquidity ratio. In this model liquidity mismatch 11

12 (inverse of liquidity ratio) and capital structure of the bank are thus directly connected. Capital regulation will have a direct impact on liquidity mismatch of the bank since the bank cannot influence asset liquidity l. A run-prone bank has a debt ratio above asset liquidity δ (l, 1], l (0, 1] and thus a liquidity ratio below one, ξ (0, 1). Runs occur, if the proportion of withdrawing debt investors exceeds the bank s liquidity ratio n > ξ. Bankruptcy costs and Recovery value In the incidence of a run, n (ξ, 1], the bank cannot borrow enough cash to honour all debt claims. The bank thus defaults and its assets are liquidated fast. This process is costly. We model costs for fast unwinding of assets (bankruptcy costs) as a fraction of asset liquidity l. Let r b (ξ) = r(ξ, b), ξ (0, 1) (5) the recovery rate function, that is the fraction of asset liquidity l recovered after a default of the bank. We assume that recovery rates are strictly positive r(ξ, b) (0, 1), and continuously differentiable in parameter b and liquidity ratio ξ. We assume that recovery rates are monotonically increasing in b but may be increasing or decreasing in liquidity ratio. Here, parameter b describes the cost-efficiency of unwinding bank assets. The larger b, the lower costs and the higher the recovery rate. We will refer to b as the payoff parameter of the recovery rate function. The dependence of the recovery rate on liquidity ratio 17 ξ, a measure of the bank s liquidity mismatch, is justified by potential interventions by a lender of last resort. A lender of last resort may grant liquidity assistance if the bank is illiquid but solvent. Liquidity assistance in the course of a run increases the average value recovered to debt investors. A key indicator of a bank s liquidity risk is its liquidity mismatch, the gap between total short-term debt outstanding and liquidation value of assets, see Basel III (2013) on Liquidity Coverage Ratio LCR. When modeling recovery rates as function of liquidity ratio, the inverse of liquidity mismatch, we allow the lender of last resort to finetune its intervention depending on liquidity mismatch of the bank. For now, we let the recovery rate function general but we will look at concrete examples in later sections such as recovery rates that are independent of or linearly increasing in liquidity ratio: r c (b, ξ) = b r l (b, ξ) = bξ r i (b, ξ) = b ξ resulting in recovery value lr b (ξ) = bl resulting in recovery value lr b (ξ) = lbξ resulting in recovery value lr b (ξ) = bδ 17 The results remain when allowing recovery rate to depend on total short-term debt outstanding δ. 12

13 In addition, this general form of recovery value allows us to match recovery value functions previously observed in models of the related literature. 18 We assume that the bank s recovery rate function is common knowledge to all investors. Payoffs Debt investors choose actions only at the interim period and payoffs depend on the realization of the aggregate action, the proportion of debt investors who withdraw n. In the event of a run (n > ξ), after bankruptcy proceedings and asset sales, total proceeds l r b (ξ) are recovered (recovery value) and distributed to measure δ of debt investors. The proceeds are insufficient to cover debt investors original claims. 19 Each withdrawing investor has a claim on one unit and nδ is the measure of investors who withdraw. Altogether, they have a claim on measure nδ of funds while only measure lr(ξ) of cash is available for distribution to all debt investors after applying the bankruptcy cost for unwinding bank assets. withdrawing investor receives the share We make the assumption that in case of a run each lr b (ξ) δ = ξr b (ξ) (6) That is, withdrawing investors share the proceeds with all investors of measure δ, not only with withdrawing investors. In a later section we show that our results are robust to this assumption. Investors who roll over receive zero in case of a run. In a later section we show that results remain under (partial) desposit insurance. If the bank stays liquid in period one, if n ξ, all withdrawing investors receive the promised coupon payment of one unit and the game proceeds to period two. In period two, the return of the asset realizes as either H with probability θ or zero with probability 1 θ. 20 In case of payoff zero, remaining debt investors receive zero and the bank defaults on the repo. When the asset pays off H, the bank repays the counterparty of the repo, obtains back possession of the pledged fraction of the asset, receives accrued interest on the asset and pays off debt investors who rolled over. 21 Payoffs Debt Investors We assign the following payoffs to debt investors: 18 Constant recovery rates such as r c (b, ξ) = b were for instance modeled in Morris and Shin (2009) and He and Xiong (2012) while recovery rates resulting in recovery values independent of asset liquidity, such as r i (b, ξ), were modeled in Rochet and Vives (2004) under the assumption of delegated decision making. 19 Since a run occurred, we have lr < l < δn For instance, a loan is paid back including interest H or the borrower defaults completely. 21 She can do so, since the bank s net return is H δ n(1 + i) δ(1 n)k > 0 where she repays δn(1 + i) to the counterparty of the repo to obtain back possession of the pledged fraction of the asset and repays δ(1 n)k to remaining debt investors. We have H δ n(1 + i) δ(1 n)k > 0 since H > 1 + i = max(k, 1 + i). 13

14 Event/ Action Withdraw Roll-over { no Run k, p = θ 1 n [0, ξ] 0, p = 1 θ Run n (ξ, 1] ξ r b (ξ) 0 Table 1: In the event of no run, withdrawing investors obtain contract coupon 1 and investors who roll over obtain contract coupon k > 1 if the asset pays, with probability θ. In the event of a run, investors who roll over receive zero. Investors who withdraw share the recovery value, i.e. asset liquidity l multiplied with recovery rate r( ), with all other debt investors δ, lr/δ = ξr. Debt investor s utility difference between withdrawing in period 2 versus withdrawing early in period 1 is given by v(θ, n) = { θ k 1 ξ r b (ξ),if n ξ (no run),if n > ξ (run) For given contract (1, k), payoffs to debt investors are determined by funding liquidity l and debt ratio δ only through liquidity ratio ξ. Payoffs Equity Holders, return on equity (ROE) (7) The equity investor receives the residual value of investment after paying off all debt investors. She gets zero in case of a run. In case of no run she receives a return per invested unit (return on equity) of ( ROE = max 0, θ ) H δn(1 + i) δk(1 n) 1 δ where n is the endogenous equilibrium proportion of debt investors withdrawing from the bank, δn(1 + i) is the repayment of principal and interest to the counterparty of the repo, δk(1 n) is the payment to debt investors who rolled over, asset return is H with probability θ and 1 δ units of equity were invested. 22 The equity investor is passive. In equilibrium, debt investors aggregate action n which decides on the occurrence of a run and ROE, is determined by debt ratio δ and contract k. When setting δ and k, the bank takes the coordination problem of debt investors at the interim period as given to maximize expected return on equity to equity investors at the ex ante stage. Participation Debt investors and equity investors participate if expected utility from the contract respectively expected ROE is higher than utility from investing in storage. 22 ROE does not contain a repayment to withdrawing debt investors since the non occurrence of a run after period one implies that the bank could borrow enough cash from the counterparty of the repo to repay all withdrawing debt investors. The counterparty of the repo needs to be repaid instead. (8) 14

15 Information Structure Here, I follow Goldstein and Pauzner (2005). At time zero, the unobservable state θ U[0, 1] realizes and determines the return probability θ of the asset. Debt investors share a common prior about state θ. After period one, debt investors observe private, noisy and asymmetric signals about the state and hence the asset return probability θ i = θ + ε i, i [0, δ] where ε i are iid random noise terms, independent of θ and distributed according to U[ ε, +ε]. From the signal structure we see, signals convey information not only about the random asset return probability ut also about other investors signals. We assume, there exist states which yield dominant actions (dominance regions). The existence of dominance regions is crucial to obtain an equilibrium selection, see (Morris and Shin, 2001). There are states θ and θ such that if θ < θ, withdrawing is a dominant action whereas if θ > θ rolling over is the dominant action to debt investors. We refer to [0, θ] as the lower dominance region and call [θ, 1] the upper dominance region. The bound θ depends on the specific contract (1, k) and is given as the realization of θ such that 23 1 = θ k, i.e. θ = 1 k (9) For very high states θ θ, we impose that the asset earns return H already in period 1 and with certainty. To make this possible, we would need to impose an asset return probability function p(θ) = min( 1 θ, 1). This function equals one for θ θ and θ otherwise 1 θ. Note that as the bound to the upper dominance region θ approaches one, θ the function p(θ) converges uniformly to the identity p(θ) = θ. The bound θ is a constant independent of the debt contract and the asset. All calculations go through for the function p(θ) = min( 1 θ, 1) but to ease the analysis, we henceforth will assume that θ θ is close to one and we analyze instead the asset return probability function p(θ) = θ for θ U(0, 1), see for instance Allen et al. (2015a). The coordination problem vanishes for state realizations in the upper dominance region. When the asset pays off return H after period one already, the bank can always repay all withdrawing debt investors, H > 1 > δn for all n [0, 1] and debt ratios δ (0, 1). To ensure that debt investors may receive signals from which they can infer that the state has realized in either of the dominance regions, we assume that noise ε is sufficiently small such that θ(r, k) > 2ε and θ < 1 2ε hold. In particular, the bounds to the dominance regions are independent of debt ratio, asset liquidity and liquidity ratio. 23 Payoff k is the maximum payoff debt investors who roll over can obtain. By design of the contract, if θ realizes below θ, even in the absence of a run the expected payoff to rolling over is smaller than the payoff to withdrawing for every n [0, 1]. 15

16 Timing of debt investor s game At time zero, the state θ and payoff probability θ realize unobservably. Debt and equity investors invest. After period one, debt investors observe noisy, private signals and subsequently choose actions. The aggregate action n (proportion of withdrawing debt investors) realizes which determines whether the bank defaults due to a run or whether it stays liquid. In case of a run, all debt investors receive payoffs according to chosen actions and the game ends. If the bank stays liquid, the game proceeds to period 2 after paying debt investors who decide to withdraw. In period 2, the success of the risky investment is determined. In case of success, the counterparty of the repo and remaining debt investors are repaid, the extra proceeds go to equity investors. In case the asset does not pay, debt investors who rolled over and equity investors receive zero. Further the bank defaults on the repo. t " t # t $ θ and p θ realize. Investment occurs. Private signals θ ' realize. Actions whether to run are chosen. Asset returns realize. 3 Equilibrium: Coordination Game In this section we only consider the coordination game played by debt investors for every contract (1, k) and capital structure δ the bank might set. The bank takes asset liquidity l as given. For a debt ratio δ < l the coordination problem vanishes and the equilibrium becomes trivial, debt investors withdraw if they observe signals from which they infer that the state is in the lower dominance region, otherwise they roll over. The remaining section analyzes the case where asset liquidity is not sufficient to cover withdrawals by all debt investors, l < δ, and thus debt investors face a coordination problem. Investor s strategies are functions of the observed private signal into the action space. Given signal θ i, an investor s posterior belief about state θ, which gives the asset s return probability, is distributed uniformly on U[θ i ε, θ i + ε]. Signals and thus beliefs are correlated, for every state realization θ, investors observe signals in the range [θ ε, θ +ε]. The equilibrium concept is Bayes Nash. Proposition 3.1 (Existence and Uniqueness). The coordination game played by debt investors has a unique equilibrium. In equilibrium, all players play a symmetric threshold strategy around threshold signal θ. Denote by θ [θ ε, θ + ε] the equilibrium trigger signal. If an investor observes a signal θ i < θ she withdraws, if she observes a signal θ i > θ she rolls over debt. In case 16

17 θ i = θ she is indifferent and we assume she rolls over. The trigger θ is a function of contract k, capital structure δ, asset liquidity l and recovery rate function r b ( ). By nature of a symmetric trigger equilibrium played by a continuum of debt investors, the endogenous proportion of debt investors who withdraw, aggregate action n, is a deterministic function of the random state and the equilibrium trigger signal. Let n(θ, θ ) indicate the endogenous equilibrium proportion of investors demanding early withdrawal in period 1 when the true state is θ and the trigger is θ. The function n(θ, θ ) is given by the proportion 24 of investors who observe a signal below the trigger θ when the true state is θ, µ(θ i < θ θ). By the uniform distribution of the error term ε i, we have 1 + θ θ if θ [θ ε, θ + ε] 2 2ε n(θ, θ ) = 1 if θ θ ε 0 if θ θ + ε. In Figure (2), we have plotted the proportion of withdrawing investors as a function of the state for fixed equilibrium trigger θ. (10) Given a state realization θ, all investors observe signals in the range [θ ε, θ + ε]. Thus, for state realizations below θ ε, all investors obtain signals smaller than the trigger and hence withdraw, n = 1. Vice versa, for all state realizations above θ + ε, all investors observe signals larger than the trigger and hence roll over, n = 0. For state realizations in [θ ε, θ + ε] some investors observe signals above the trigger and roll over, while others observe signals below the trigger and withdraw. By the uniform distribution of the error term ε i, function n is linearly decreasing in state realization θ with slope 1, analogous to Goldstein and Pauzner 2ε (2005). To understand the importance of the equilibrium trigger signal, Figure 3 depicts how the size of equilibrium trigger influences the range of states for which runs occur. As signals to debt investors become precise ε 0, signals reveal close to perfectly the true state θ and runs occur for state and signal realizations below the trigger. The larger the trigger, the larger the range of signals for which investors withdraw and hence the larger the ex ante probability of a run. Bank stability improves, as the trigger decreases. 3.1 Comparative statics To derive the comparative statics of the equilibrium trigger, we use the payoff difference equation (PIE) which implicitly defines the equilibrium trigger θ [θ ε, θ + ε] [0, 1] as the signal at which a debt investor is indifferent between rolling over and withdrawing. Since payoffs are dependent on whether a run occurs or not, and given the trigger signal θ the posterior belief on state θ is uniform on [θ ε, θ +ε] the payoff difference equation 24 As the continuum of debt investors has measure δ, the proportion of debt investors observing signals below the trigger differs from its measure by factor δ. 17

18 n θ, θ 1 ξ = l δ θ θ ε θ, θ θ + ε θ 1 θ Lower dominance region Range of states for potential miscoordination Upper dominance region Figure 2: Proportion of debt investors who withdraw as a function of the state. Note that while the bounds to the dominance regions, θ and θ, and the critical state are states, the trigger θ is a signal. 0 θ θ θ 1 θ Panic Runs Runs No Run Figure 3: As noise vanishes, runs occur for state realizations below the trigger. The larger the trigger, the higher the ex ante probability of runs and the lower bank stability is given as 0 = 1 2ε θ +ε θ ε (kθ 1) 1 {n(θ,θ ) ξ} ξ r(ξ) 1 {n(θ,θ )>ξ}dθ (11) If the state realizes such that the endogenous proportion of withdrawing agents equals ξ, n(θ, θ ) = ξ, the bank is on the edge of a run. Define the critical state [θ ε, θ +ε] as the state at which the bank is at the edge of a default n(, θ ) = ξ (12) For state realizations below, n realizes such that a bank run occurs, n(θ, θ ) > ξ while for state realizations above the critical state no runs take place n(θ, θ ) < ξ. Substituting for n, we can rewrite the PIE as 0 = ξ 0 (kθ(n, θ ) 1) dn ξ ξ r(ξ) dn (13) 18

19 v θ, n kθ 1 ξr ξ ξ 1 n Figure 4: Payoff difference function v(θ, n) from equation (7) plotted for fixed θ as function of the endogenous proportion of withdrawing debt investors n. where θ(n, θ ) = θ + ε(1 2n) is the inverse of the function n(θ, θ ). We can simplify the PIE to derive an explicit formula for the equilibrium trigger: 0 = k ξ 0 (θ + ε(1 2n)) dn ξ ξ r(ξ) (1 ξ) (14) = ξ (kθ + kε(1 ξ) 1 r(ξ)(1 ξ)) (15) (16) The explicit formula for the trigger is found by solving that is, 0 = kθ + kε(1 ξ) 1 r(ξ)(1 ξ) (17) Lemma 3.1. The equilibrium trigger of the coordination game is explicitly given as [ ] 1 θ = min (1 + r(ξ) (1 ξ)) ε(1 ξ), θ k (18) where θ is the bound to the upper dominance region. As noise vanishes ε 0, the trigger takes the from lim ε 0 θ = min( 1 (1 + r(ξ) (1 ξ)), θ) (19) k Independently of the specified recovery value function r( ), we observe that as δ l or ξ 1, the trigger approaches the bound to the lower dominance region θ = 1/k: lim ε 0,ξ 1 θ = 1 k (20) 19

20 The trigger is below the bound to the upper dominance region θ (0, 1) if the parameters of the game are such that 1 (1 + r(ξ) (1 ξ)) < θ or k r(ξ) (1 ξ) < kθ 1 (21) Here we already see, since recovery value is a fraction r(ξ) (0, 1] and liquidity ratio is in (0, 1], the trigger will be below the upper dominance region only if k is sufficiently large. If instead r(ξ) (1 ξ) > kθ 1, the trigger stays constant at the bound to the upper dominance region θ Proposition Comparative statics The next proposition analyzes coordination behavior (the trigger) as a function of the contract k, liquidity ratio ξ and debt ratio δ. Proposition 3.2. The trigger θ monotonically decreases and is convex in coupon k. The trigger decreases in liquidity ratio at some point ξ (0, 1] if and only if the recovery value function r(ξ) satisfies r(ξ) (1 ξ) < r(ξ) (22) ξ If the inequality holds for all ξ (0, 1], the trigger monotonically decreases in liquidity ratio and thus monotonically increases in debt ratio δ. If recovery value function r(ξ) is concave in ξ, the trigger is concave in ξ too. The trigger is concave in δ if the trigger is concave and decreasing in ξ. The larger the trigger the higher the ex ante probability of runs and the lower bank stability. For exogenous asset liquidity l, liquidity ratio ξ is a sufficient statistic for debt ratio δ. Here, we have thus provided a sufficient condition on recovery rates for when bank stability deteriorates or improves in debt ratio. The result that a bank s (endogenous) liquidity risk due to runs can decrease in overall exposure towards short-term debt was first provided in Schilling (2016). The comparative statics in k are intuitive since the larger k, the larger the relative payoff from rolling over versus withdrawing. Therefore, a debt investor who is indifferent at a signal before an increase in k, will tend towards rolling over after the increase in k. For the comparative statics in ξ (in δ), the impact of an increase is ambiguous. When recovery value ξr(ξ) increases in liquidity ratio, payoffs from withdrawing versus rolling over conditional on a run go up which increases the incentive to withdraw. Thus, the trigger increases and stability goes down in a first effect ( payoff effect ). Changing liquidity ratio however also alters the critical proportion of withdrawing investors the 20