Dividend Payouts and Rollover Crises

Abstract

The dividend policies of banks received much attention in the wake of the 2007–2008 financial crisis. The U.S. banking sector maintained large dividend payouts throughout 2007 and 2008, even as losses were increasing rapidly (Acharya, Shin, and Gujral 2009), so that aggregate dividends paid by U.S. banks in 2008 exceeded their aggregate earnings by about 30% (Floyd, Li, and Skinner 2015). One explanation for banks’ dividend policies during the early stages of the financial crisis is that they reflected a form of moral hazard. Scharfstein and Stein (2008) argue that banks engaged in risk shifting through their dividend policies. Another explanation focuses on a signaling role of dividends in response to rollover risk. Acharya et al. (2011) suggest that U.S. banks were worried that cutting dividends could induce a run by their short-term creditors.

At first glance these two views seem incompatible. After all, if high dividend payouts are associated with bank insiders engaging in risk shifting, then they are also bad news about the bank’s ability to survive a run. Consequently, paying dividends in the middle of a rollover crisis should exacerbate rather than soften a run by short-term creditors. Second, and related to the first point, while the role of dividends as a (positive) signal of future profitability has been well established (e.g., Bhattacharya 1979), the question of the role of dividends as a signal of available liquidity in the midst of a rollover crisis has received far less attention. This is worrying, because a proper understanding of that role is central for the design of dividend regulation policies that can improve financial stability.

In this paper, we examine theoretically the role of dividends when banks are subject to coordination-based rollover crises or runs. We show that higher dividends can (in equilibrium) be interpreted as good news about available liquidity even if banks can (and in some cases do) use them to engage in risk shifting. Moreover, the interaction between signaling and coordination reverses the signaling incentives relative to the established view of dividend signaling: the lower-quality types overpay dividends in an attempt to mimic higher-quality types, rather than higher-quality types overpaying dividends in order to separate from lower types. Finally, when dividends signal available liquidity, banks’ dividend choices fail to internalize an informational externality that operates through the inferences and actions of lenders. In that case a cap on dividends that forces surviving banks to pool on a common dividend level improves financial stability.

We consider a bank that is financed by a continuum of short-term lenders that face a coordination problem when rolling over maturing debt.¹ If a sufficient share of lenders refuse to roll over (run, for short), then the bank does not have enough liquidity to repay all lenders and is forced to fail. In that case, an individual lender is better off running than rolling over. At an initial stage, prior to the rollover episode, the bank (owner) chooses a dividend payout. It derives a positive payoff from consuming the dividends but also incurs a reduction in the liquidity available to meet the demands of running lenders and in the value of bank assets, given successful rollover. Therefore, a bank that expects an unsuccessful rollover has an incentive to pay out all available liquidity in dividends. Additionally, and consistent with the existing dividend signaling literature, we assume that (conditional on surviving the rollover episode) the marginal cost of liquidity is lower for higher-quality banks.

We introduce dividend signaling to this environment by assuming that the lenders observe the bank’s dividends and make their rollover decisions based on the inferences they draw about the bank’s type and its available liquidity. Therefore, the dividend choice of the bank acts as an endogenous signal about the bank’s probability of surviving the run. To ensure equilibrium uniqueness, we also assume that lenders observe the dividend with small idiosyncratic noise.² With rollover risk, the ability to pay dividends means that some banks that can survive the rollover instead choose to pay out all available liquidity as dividends. We call this direct negative effect on survival the resilience effect of dividends. However, dividends also convey information, so they indirectly affect the incentives of short-term lenders to roll over their debt. We call this second indirect effect on survival the signaling effect.

Whether the signaling effect reinforces or counters the resilience effect depends on whether higher dividends are good or bad news about the bank’s type and its available liquidity. Nevertheless, how lenders interpret a higher dividend signal may be endogenous to the behavior of banks, thus, giving rise to multiple equilibria (Angeletos, Hellwig, and Pavan 2006). We show conditions under which higher dividends are interpreted as good news and, moreover, no equilibria exist in which higher dividends are interpreted as bad news. Our conditions are a simple strengthening of the assumptions on the existence of dominance regions familiar from global games. Specifically, if a sufficiently large share of low-quality banks always fail and pay no dividends, then it is possible for dividends to signal good news about survival in equilibrium. Intuitively, even if some failing banks are paying higher dividends than some surviving banks, if lenders’ priors are such that, “on average,” failing banks are associated with limited liquidity and a low dividend payout, then higher dividends can be interpreted as good news in equilibrium.

Under these conditions, if lenders observe dividend signals with sufficiently small noise, then the signaling effect dominates the resilience effect and paying dividends can actually decrease the total liquidity outflow that the bank experiences. Intuitively, the high signal precision means that the dividend choice of the bank affects the inference and actions of many lenders, so that a reduction in the liquidity outflow due to a lower dividend payout is dominated by the increase in the run size. One implication of facing such a trade-off is that banks would never choose a dividend that leaves them on the downward sloping part of the liquidity outflow schedule. Therefore, banks choose to distort their dividends up to a level where the total liquidity outflow is again increasing in dividends. At that level any additional benefits associated with dividend signaling are small and so a large set of (surviving) banks choose similar dividend payouts. Therefore, the equilibrium outcome of a strong signaling effect is a dividend policy that features both higher dividend payouts and a lower sensitivity of dividends to the bank type compared to the dividend policy absent rollover.

Unlike dividend signaling about future profitability, in the case with dividend signaling about available liquidity and rollover risk the banks that distort their dividends the most in equilibrium are low type (surviving) banks rather than high types. The intuition for this reversal in signaling incentives is simple: In the presence of a coordination-based rollover episode, the lower-quality (surviving) banks, which are more exposed to the rollover, have stronger incentives to signal that they have sufficient available liquidity. In contrast when signaling about future profitability, it is higher types that have stronger incentives to distort their dividends up and separate from lower types.

As lenders observe bank dividends with smaller and smaller noise, the strengthening of the signaling effect relative to the resilience effect lowers the bank failure cutoff – the value of fundamentals at which a bank is indifferent between failing and surviving. We show this interesting feature of the signaling effect in the limiting case in which the noise vanishes and lenders are almost perfectly coordinated. In that case for a bank that is subject to a run, even a marginal increase in dividends ends up inducing all lenders to choose to roll over.³

Despite this intriguing effect of dividend signaling on the failure cutoff, we show that from the point of view of a regulator with preferences for minimizing the bank failure cutoff, the banks’ equilibrium dividend payouts can be inefficiently high during a rollover episode. There are two sources of inefficiencies: the risk shifting that banks may engage in, which mechanically raises the failure cutoff, and an informational externality that banks fail to take into account when choosing their dividends. We show that the optimal dividend policy in a rollover episode consists of a common dividend payout for all surviving banks. This effective dividend cap pools all surviving banks together, which decreases the dividend signal cutoff at which a lender is indifferent between running and rolling over. At the same time more failing banks are forced to pay zero dividends under the optimal dividend policy. Intuitively, the regulator wants failing and surviving banks to pay sufficiently distinct dividends, so that, given the noise in lender observations, the lenders can identify failing from surviving banks more easily. Therefore, with dividend signaling, a binding cap on dividends is a more effective macroprudential tool compared to a complete dividend restriction (Goodhart et al. 2010).

Finally, we discuss the empirical relevance of our theory. We test two salient implications of the signaling effect, namely that it leads to a dividend policy that features both higher dividend payouts and a lower sensitivity of dividends to fundamentals compared to the dividend policy absent rollover. We also provide a test to identify the signaling effect from the resilience effect, leveraging on the observation that for banks with relatively high fundamentals, higher exposure to rollover risk moves dividends in opposite directions under the two effects. We document two novel facts consistent with our model. Consistent with the first implication, we show that surviving banks which were more reliant on short-term funding prior to the 2007-2008 financial crisis paid higher dividends during the crisis. Also, consistent with the second implication, we show that, across industries, dividend payouts are less variable in industries that are more reliant on short-term funding.

1. Related Literature

Our paper is related to the growing literature on bank dividend payouts, particularly during a financial crisis, and the optimal policy response to those (Acharya, Le, and Shin 2016; Floyd, Li, and Skinner 2015; Hirtle 2014; Cziraki, Laux, and Loranth 2016). Acharya, Le, and Shin (2016) study a model of bank dividend payouts, in which risk shifting by the bank equity holders due to a possible low future franchise value influences bank dividend payouts. When banks are linked through an inter-bank market, this risk shifting interacts with an additional dividend externality that may trigger a systemic crisis. Our modeling approach complements this important framework by studying the informational role of dividends when banks are exposed to a coordination-based run. We argue that with dividend signaling there is an additional informational externality that banks fail to internalize in addition to the risk-shifting inefficiencies. However, rather than arising from direct spillovers via bank linkages, in our model, the informational externality arises through the inference of lenders and their rollover decisions.

The informational role of dividends relates our model to the seminal paper on dividend signaling about future profitability of Bhattacharya (1979) and a large subsequent literature (Miller and Rock 1985; John and Williams 1985; Hausch and Seward 1993; Guttman, Kadan, and Kandel 2010; Baker, Mendel, and Wurgler 2016). Bhattacharya (1979) argues that with asymmetric information about future profitability, if the marginal cost of paying dividends is decreasing in the firm’s type (so there is single crossing), then dividends can serve as a signal to outside investors that separates higher from lower profitability firms. In contrast, we show that with coordination-based runs the signaling incentives are completely reversed. Specifically, it is the lower-quality (surviving) banks that have the stronger signaling incentives. Therefore, despite single crossing, the interaction of the coordination-based run and signaling in our framework pushes the banks that signal through dividends toward pooling rather than toward separation in equilibrium.

The reduced sensitivity of dividends to fundamentals that results from these dividend signaling incentives relates the paper to the partial pooling result of Guttman, Kadan, and Kandel (2010). However, there are several important differences. Conceptually, the reduced sensitivity of dividends in our framework is the unique outcome of the signaling incentives imposed by the underlying coordination game and is not driven by out-of-equilibrium beliefs. Consequently, our model makes new testable predictions about when dividends should be expected to be less sensitive to fundamentals. Finally, the informational externality that we uncover in our framework is a unique feature of the interaction between dividend signaling and coordination.

Our paper is related to the large literature on global games of regime change (e.g., Carlsson and van Damme 1993; Morris and Shin 1998) and, particularly, to global game models of bank runs (Goldstein and Pauzner 2005; Rochet and Vives 2004) and rollover crises (Morris and Shin 2004).⁴ We contribute to this important literature by analyzing how banks use their dividend payouts to manage the rollover crisis. In addition, while most of these models assume an exogenous information structure for lenders or an exogenous resilience level for banks, both the information structure of lenders and the resilience level of banks are endogenous in our model. This endogenous information structure relates our paper to a growing literature on information acquisition in global games (He and Manela 2016; Szkup and Trevino 2015; Yang 2015; Ahnert and Kakhbod 2017) and also to papers studying the effects of information quality and transparency on stability (Iachan and Nenov 2015; Moreno and Takalo 2016; Ahnert and Martinez-Miera 2019).

Our paper is particularly related to models of signaling in global games. Angeletos, Hellwig, and Pavan (2006) and Angeletos and Pavan (2013) consider a regime-change game in which a policy maker can take a costly policy action to influence the cost of attacking. They show that there exist multiple equilibria, depending on how the policy action is interpreted. For example, there always exists an “inactive-policy” equilibrium in which agents ignore the policy action when choosing to attack and the policy maker anticipates this and does not intervene. There also exists an “active-policy” equilibrium in which only intermediate types choose to intervene. In contrast, there is a unique equilibrium interpretation of dividend signals in our framework. One reason for this is that the bank enjoys a direct utility from paying dividends, which rules out any “inactive-policy” type equilibria. In addition, the dividend action has no direct effect on the cost of attacking and, rather, affects the lenders’ actions only through their inferences. Finally, because dividends have a large impact on lenders’ inferences for all bank types, all surviving banks signal through dividends in equilibrium.

Edmond (2013) studies a model of regime-change in which the regime can engage in costly manipulation of the private information of agents considering staging a revolution. In equilibrium, agents try to infer the true type of the regime given the signals they observe. His framework features a unique equilibrium. As in Edmond (2013), our economy also admits a unique equilibrium despite the signaling effect of the bank’s actions and the endogenous information structure that arises. However, while he studies how a regime engages in costly manipulation of agents’ private signals about its type (i.e., propaganda), we study how a bank optimally chooses its dividend policy when faced with a coordination-based run. Also, in our framework, the direct effect of paying out dividends is to weaken the ability of the bank to survive the rollover episode. In contrast, in Edmond (2013) the regime’s action cannot be destabilizing.

Goldstein and Huang (2016a) study how a regime can increase the probability of survival by committing to abandoning the status quo for some fundamentals. However, the information transmission that takes place in their model, and which ends up stabilizing the regime, is more in the spirit of the Bayesian persuasion literature (Kamenica and Gentzkow 2011) rather than through sending a costly signal.⁵

Finally, our focus on the link between dividend signaling and financial stability relates our paper to the recent literature on the effects of information disclosure on financial stability (for instance, from stress-testing as in Bouvard, Chaigneau, and de Motta (2015), Faria-e Castro, Martinez, and Philippon (2016), and Goldstein and Leitner (2018) or credit ratings as in Goldstein and Huang (2016b) and Holden, Natvik, and Vigier (2018)). In contrast to many of these papers, we focus on information generated by one of the parties in the rollover game, which maximizes its own payoff, rather than a third party (i.e., a regulator) who has an explicit objective to improve financial stability.

2. Model

Consider an economy with three periods, |$t\in\{0,1,2\}.$| There is a bank with an exogenously given balance sheet. At |$t=0$| the bank (owner) chooses how much to pay in dividends from the bank’s total available liquidity. The bank has a continuum of short-term creditors who make rollover decisions on their debt at |$t=1$|⁠. The bank uses its total remaining available liquidity (net of the |$t=0$| dividend payout) to repay creditors who refuse to roll over. The bank fails if it cannot pay all creditors that refuse to roll over. At |$t=2$| creditors obtain payoffs based on their actions and the outcome of the rollover episode, and the bank owner obtains any remaining equity value. Figure 1 illustrates a summary of the timing of events. We now provide the details for this environment.

Figure 1

Timing of events

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2.1 The bank

2.1.1 Assets

At the beginning of |$t=0$|⁠, the bank holds a portfolio of assets that deliver payoffs at |$t=2$|⁠. At |$t=0$| and |$t=1$| the bank can convert part of these assets into liquidity by selling or borrowing against them as collateral. We call the maximum liquidity the bank can obtain in this way, its total available liquidity and denote it by |$\overline{\ell}$|⁠. The bank uses liquidity to make a dividend payment |$d$| at |$t=0$| and to meet redemptions by short-term creditors at |$t=1$|⁠. We let |$l$| equal the sum of the dividend payment and the redemptions by short-term lenders that the bank chooses to meet. Therefore, |$l\le\overline{\ell}$| is the total liquidity outflow from the bank at the end of |$t=1$|⁠.

Assumption B1.|$\frac{v_{\theta}\left(\theta,l\right)}{-v_{l}\left(\theta,l\right)}$| is strictly decreasing in |$l$|⁠.

We briefly discuss some of our assumptions on |$v$| and their implications for the analysis and for the empirical relevance of the model. The assumption |$v_{l\theta}>0$| is a single crossing condition that is often made in signaling models (e.g., Bhattacharya 1979). In standard signaling environments this assumption implies a positive link between dividends and future profitability, consistent with empirical evidence (see, e.g., Nissim and Ziv 2001; Cziraki, Laux, and Loranth 2016). While this condition is only sufficient for our results, as we discuss in Section 3.2.5, we maintain it in most of the analysis to facilitate comparison with the existing dividend signaling literature. To provide additional intuition for the single crossing condition and the portfolio restrictions that it implies, in the Online Appendix we present one possible microfoundation for |$v$| that gives rise to this condition (and the other conditions on |$v$|⁠). Our example is a continuous and “smoothed” version of the asset portfolios commonly assumed in banking models, where assets are grouped into discrete asset classes based on their liquidity. Finally, Assumption B1 ensures that a bank’s incentives to fully liquidate its portfolio and pay it out as dividends at |$t=0$| are decreasing in |$\theta$|⁠.⁶

2.1.2 Liabilities and bank payoff

2.2 The lenders

At |$t=1$|⁠, after dividends are paid out, the lenders decide whether to roll over their debt to |$t=2$| or refuse to roll over (or run, for short). We assume that the lenders have a uniform prior about |$\theta$| over |$\left[-K,K\right]$| for |$K>0$|⁠. Lenders observe additional information, which we detail in Section 3. Because the information will be heterogeneous across agents, we will denote the probability with respect to lender |$i$|’s information set by |$\Pr{}_{i}\left\{ .\right\} $|⁠.

2.3 Dominance regions

We assume that there exist lower and upper dominance regions.

• Lower dominance region: There exists a |$\underline{\theta}>-K$|⁠, such that for |$\theta<\underline{\theta}$|⁠, |$\overline{\ell}\left(\theta\right)=0$|⁠.

• Upper dominance region: There exists a |$\overline{\theta}<K$|⁠, such that for |$\theta>\overline{\theta}$|⁠, |$\overline{\ell}\left(\theta\right)>1$|⁠, and |$\lambda\bar{\ell}(\theta)<v(\theta,1)$|⁠.

• Multiplicity region: For |$\theta\in\left(\underline{\theta},\overline{\theta}\right)$|⁠, |$\overline{\ell}\left(\theta\right)\in\left(0,1\right)$|⁠.

Therefore, banks with very weak fundamentals are insolvent and fail with probability one for any |$A\in\left[0,1\right]$|⁠. Conversely, banks with very strong fundamentals can meet all demands for withdrawals. Furthermore, it is never optimal for such banks to liquidate all their assets at |$t=0$|⁠.¹¹ In between, whether or not a bank can survive depends on whether lenders coordinate on running or rolling over. If all lenders run, then the bank cannot survive, and if all lenders roll over, the bank can survive. The equilibrium concepts that we work with are standard and are included in the Online Appendix.

2.4 No-run benchmark

To highlight the interaction between dividends and rollover risk, we first characterize the dividend payout of a bank that does not face a run. We assume that

Assumption B2.|$\lambda\ge-v_{l}\left(\underline{\theta},0\right)$|⁠,

so that even banks with low values of |$\theta$| find it optimal to pay dividends. We view this particular case as the empirically relevant one in light of our discussion of the risk-shifting incentives by banks in the Introduction.

Proposition 1 shows that, in the absence of runs, banks with higher portfolio quality pay higher dividends. This outcome is a direct implication of the single crossing condition, |$v_{l\theta}>0$|⁠.

3. Equilibrium Analysis

We characterize equilibrium outcomes under two different information structures. First, we consider the case when lenders do not observe the dividend choice of the bank and instead observe an exogenous private signal about the bank’s type. Then, we introduce dividend signaling by assuming that lenders observe the bank’s dividend and make inferences about the bank’s type based on that information and their prior beliefs.

3.1 Exogenous information and the resilience effect

Figure 2 illustrates the optimal dividend policy in this equilibrium. To gain some intuition, note first that with exogenous lender signals the size of the run does not depend on the dividend payout of the bank. Moreover, as in other global games models (Morris and Shin 2003), the strategic uncertainty resulting from dispersed private information determines a run size of |$A\left(\theta_{f},\hat{\theta}\right)=p$| for a bank at the equilibrium failure threshold, |$\theta_{f}$|⁠. Such a bank is indifferent between paying out |$\overline{\ell}\left(\theta_{f}\right)$| and failing or paying out |$d^{*}\left(\theta_{f}\right)$|⁠, enduring a run of size |$p$|⁠, and surviving. Therefore, the total liquidity outflow for a bank at the failure threshold is |$d^{*}+p$|⁠. In this case, higher dividend payouts always increase the liquidity outflow of the bank. Contrast this with the case when banks do not pay dividends. In that case the failure threshold is not determined by whether the bank chooses to survive but by whether survival is feasible given the run size |$p$|⁠.

Figure 2

Exogenous information and the no-run benchmark

This figure shows the equilibrium dividend policies under the no-run benchmark (dashed line) and the exogenous information case (solid line).

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Because the liquidity outflow for the bank at the failure threshold is only |$p$| in that case, and a higher liquidity outflow can be met only by a bank with higher type, paying dividends clearly leads to more banks failing. We call this the resilience effect associated with dividend payouts.

Finally, as Figure 2 illustrates, compared to the no run benchmark, in the exogenous information case a bank that chooses to survive distorts its dividend payout below the no-run dividend level, |$d_{nr}\left(\theta\right)$| determined by Equation (7). Specifically, |$d^{*}\left(\theta\right)=\max\left\{ 0,d_{nr}\left(\theta\right)-A\left(\theta,\hat{\theta}\right)\right\} $|⁠. Therefore, for a bank that chooses to survive, payouts to running lenders crowd out dividend payouts.

3.2 Endogenous information and the signaling effect

Next, we introduce dividend signaling by assuming that lenders observe the bank’s dividend and make inferences about the bank’s type based on that information and their prior beliefs. We assume that dividends are observed by lenders with idiosyncratic noise. Formally, each lender observes a private signal about dividends, |$d_{i}=d\left(\theta\right)+\eta_{i}^{d}$|⁠, with |$\eta_{i}^{d}\sim_{i.i.d.}N\left(0,\alpha^{-1}\right)$|⁠, where |$d\left(\theta\right)$| is the dividend choice of a bank with type |$\theta$|⁠, and |$\alpha$| denotes the dividend signal precision. We make this assumption to abstract away from the equilibrium multiplicity arising because of common certainty, resulting from the observation of a public signal (as in Woodford 2002, Myatt and Wallace 2014, Kolbin 2015, Gaballo 2016, or Angeletos and Lian 2018). In addition to its technical role, we can interpret the private noise in dividend observations as a reduced form for limited attention by lenders.¹³

3.2.1 Dividend payouts with signaling

When lenders follow a monotone strategy |$\hat{d}$|⁠, a bank reduces the run it is facing by paying higher dividends. We call this the signaling effect. It is useful to define |$d_{\min}$| as the minimizer of |$l\left(d\right)$|⁠.¹⁵ We will consider economies with |$\hat{d}\in\left(0,1\right)$| and |$\alpha$| sufficiently large, so that |$d_{\min}>0$|⁠.¹⁶ In that case, as Figure 3 shows, the liquidity outflow that a bank experiences is decreasing for some values of |$d$|⁠, so that the signaling effect dominates the resilience effect.

Figure 3

Liquidity outflow as a function of dividends paid

This figures shows the total liquidity outflow |$d+A\left(d,\hat{d}\right)$|⁠, for a given |$\hat{d}$|⁠, as a function of the dividend |$d$|⁠. In this case, |$\alpha$| is sufficiently high so that the liquidity outflow is decreasing in a region around |$\hat{d}$|⁠.

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We focus on this case, because it highlights how signaling affects the equilibrium dividend policies of banks. Given the interpretation of |$\alpha$| as a parameter that determines the degree of lender attention toward publicly available information, we also believe that it is empirically plausible in many relevant cases, for example, when banks have sophisticated institutional lenders. Proposition 3 characterizes the solution to the bank problem in that setting.

As in the exogenous information case, a bank chooses whether to fail or survive by comparing the payoff from paying out |$\overline{\ell}\left(\theta\right)$| and failing the rollover episode, against the payoff from paying a dividend |$d^{*}$| and surviving a run of size |$A\left(d^{*}\left(\theta_{f}\right),\hat{d}\right)$|⁠. A bank at the failure threshold is then exactly indifferent between failing (and paying out |$\overline{\ell}\left(\theta_{f}\right)$|⁠) and surviving (and paying out |$d^{*}\left(\theta_{f}\right)$|⁠).

The dividend payout of surviving banks is given by Equation (15). The left-hand side corresponds to the marginal benefit from paying out one more dollar of dividends, whereas the right-hand side corresponds to the marginal cost. Because of the signaling effect, the (effective) marginal cost of paying dividends is lower compared to both the no run case (cf. Equation (7)) or the case with exogenous information (cf. Equation (11)). Intuitively, paying out one more dollar in dividends leads to a total liquidity outflow of only |$1+A_{d}<1$|⁠. The lower effective marginal cost implies that a bank has incentives to distort its dividend payout above the no-run dividend level, |$d_{nr}\left(\theta\right)$|⁠. On the other hand, as in the exogenous information case, the resilience effect tends to push toward the bank setting a dividend payout below the no-run level, due to crowding out by the run. When the signaling effect is strong, the former effect dominates, and the bank increases its dividend above |$d_{nr}\left(\theta\right)$|⁠.¹⁷

The signaling effect is particularly strong for banks with type |$\theta$|⁠, such that |$d_{nr}\left(\theta\right)\le d_{\min}$|⁠. For these banks, increasing the dividend above |$d_{nr}\left(\theta\right)$| is associated with a lower liquidity outflow. Banks then optimally choose a dividend of at least |$d_{\min}$|⁠. In addition, the marginal impact on the size of the run given by |$A_{d}$|⁠, decreases strongly in |$d$| around |$d_{\min}$|⁠. Intuitively, because lenders care about whether the bank fails or survives the rollover episode (rather than the specific bank type), the dividend payouts of banks with higher fundamentals are already interpreted by most lenders as strong evidence that the bank will survive the rollover episode, so any upward distortion in dividends has only a small effect on lenders’ inference. Therefore, all banks that distort their dividends above the no-run level choose payouts above but close to |$d_{\min}$|⁠. In summary, a strong signaling effect induces (surviving) banks with different types to choose similar dividend payouts.

3.2.2 Lenders’ inference and actions

Unlike the exogenous information case, how lenders interpret a higher dividend signal – whether as good or bad news about bank survival – can now be endogenous to the behavior of banks. On the other hand, the banks’ behavior depends on how lenders interpret higher dividends. Therefore, it can easily be the case that there are multiple equilibria as in Angeletos, Hellwig, and Pavan (2006). Moreover, from Equation (16) and as illustrated in Figure 4, even when higher dividends are good news about survival and lead to fewer lenders running, the dividend policy of banks is nonmonotone and some failing bank types choose to pay higher dividends than some surviving banks. Therefore, a lender’s posterior belief about the bank failing given dividend signal |$d_{i}$|⁠, |$\Pr(\theta<\theta_{f}|d_{i})$|⁠, may not always decrease with |$d_{i}$|⁠. Put differently, a higher dividend signal may not always be good news about bank survival.

Figure 4

Equilibrium dividend policies for low (left panel) and high (right panel) dividend signal precisions

This figure shows the equilibrium dividend policy in the no-run benchmark (dashed lines) and the dividend signaling case (solid lines). In the left panel, |$\alpha=400$|⁠; in the right panel, |$\alpha=1,000,000$|⁠. See the Online Appendix for additional details.

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Intuitively, if a lender with signal |$d_{i}$| expects a lower dividend from a bank that fails compared to a bank that survives, then a lender with a marginally higher signal is more optimistic about the bank surviving. To apply Lemma 1, notice that by Proposition 3, |$d^{*}\left(\theta\right)>d_{\min}$|⁠, so |$E_{D,i}\left[d^{*}\left(\theta\right)\right]>d_{\min}$|⁠. Therefore, higher dividends are always good news about bank survival if |$E_{N,i}\left[\overline{\ell}\left(\theta\right)\right]<d_{\min}$|⁠. One sufficient condition for this inequality is that |$K$| is sufficiently large, so that the lower dominance region (in which |$\overline{\ell}\left(\theta\right)=0$|⁠) is large. In that case lenders expect most failing bank to have no available liquidity, including for paying dividends. We use this observation to characterize the lenders’ actions.

3.2.3 Equilibrium characterization

Turning to equilibrium characterization, Proposition 9 in the appendix combines the results from Propositions 3 and 4, and characterizes equilibria in monotone strategies for this economy with the property that higher dividends are good news about bank survival. Furthermore, it shows conditions under which, if the monotone strategy equilibrium is unique, it is the unique equilibrium of this economy. To show equilibrium uniqueness, one has to show that in any equilibrium higher dividends can only be interpreted as good news.¹⁸ A sufficient condition for this is that the dominance regions (parametrized by |$K$|⁠) are large and there is single crossing, |$v_{l\theta}>0$|⁠, so that the dividend payouts of surviving banks are increasing in |$\theta$|⁠. Intuitively, because banks with very low |$\theta$| always pay no dividends and fail, and banks with very high |$\theta$| always pay dividends and survive, when these types are sufficiently prevalent, a higher private signal is always interpreted as good news about survival, regardless of the actions of lenders with intermediate signals or the dividend policies of banks in the multiplicity region.

Figure 4 illustrates the equilibrium dividend policies for one particular example. The figure plots both the dividend policy with endogenous information (solid line), and the dividend policy in the no-run case (dashed line) for banks in the multiplicity region. Banks below the failure threshold pay out all available liquidity as dividends. In contrast to the exogenous information case, surviving banks pay a dividend that is higher than their no-run dividend. Furthermore, the dividend payouts of surviving banks are more similar and vary less with |$\theta$| relative to the no-run case. As the signal precision is increased and the signaling effect is strengthened, the dividend payouts of banks close to the failure threshold become even more similar (and even less sensitive to |$\theta$|⁠).

Hence, dividend signaling in the presence of rollover risk implies a stronger upward distortion in dividends for (surviving) banks with low types. This is intuitive – in the presence of rollover risk banks signal available liquidity and the ability to survive a rollover episode. The lower-quality banks which are more exposed to rollover thus signal “more.” This is opposite to the established view of signaling future profitability (Bhattacharya 1979), where the incentives to signal high future profitability induce higher types to signal more and separate from lower types. In the Online Appendix, we consider a modification of our model that illustrates this outcome. We use that modified environment to point out three notable differences with our model. First, signaling about future profitability implies no dividend distortion for the lowest bank type. In contrast, signaling about available liquidity means that the distortion is largest for the lowest (surviving) bank type. Second, signaling about future profitability implies dividend distortions for all types, including the highest types. In contrast, signaling about available liquidity (combined with single crossing) means that the distortion becomes arbitrarily small with type, because both the equilibrium run size (⁠|$A\left(d,\hat{d}\right)$|⁠) and the marginal effect of higher dividends on the run size (⁠|$A_{d}\left(d,\hat{d}\right)$|⁠) go to zero as |$d$| increases. Third, signaling about future profitability may imply that dividend distortions are actually increasing in type.

3.2.4 A limit result

Next, we show the following stark result for the limiting case when lender signals become arbitrarily precise.

To build some intuition for this result, note that for any |$d>\hat{d}$|⁠, a higher value of |$\alpha$| ends up decreasing the liquidity outflow, |$l\left(d\right)=d+A\left(d,\hat{d}\right)$|⁠. Intuitively, when |$\alpha$| is larger, lenders are more coordinated and the same dividend choice influences the actions of more lenders. Put differently, less noise in the observation of dividends strengthens the signaling effect. As |$\alpha\to\infty$|⁠, lenders become almost perfectly coordinated and so |$A\to0$|⁠, for |$d>\hat{d}$|⁠. Therefore, even a marginal increase in dividends induces (almost) all lenders to choose to roll over. Consequently, any surviving bank will face no run, including a bank at the failure cutoff |$\theta_{f}$|⁠, and the liquidity outflow will only equal the dividend payout itself. This, however, can only be consistent with indifference between survival and failure if |$\theta_{f}=\theta^{nr}$| – the “failure” cutoff when there is no run on the bank. Figure 4 illustrates the strengthening of the signaling effect for finite values of |$\alpha$|⁠. More precise dividend signals reduce the failure cutoff and bring it closer to the no-run cutoff. Additionally, |$d_{\min}$| decreases, which further reduces the cost associated with signaling.¹⁹ One interesting implication of the limiting case, (also suggested by Figure 4), is that when lenders observe arbitrarily precise signals, |$d^{*\prime}\left(\theta_{f}\right)\to0$|⁠. Therefore, in equilibrium, banks that are close to the failure cutoff (approximately) pool on their dividend payouts.²⁰

3.2.5 Dividend signaling without single crossing

We extend our analysis to cases in which the single crossing condition, |$v_{l\theta}>0$|⁠, does not hold. The single crossing condition holds in many realistic settings linking bank type to the underlying portfolio. Nevertheless, it is violated if, for example, a high |$\theta$| bank has less cash than a low |$\theta$| bank but has more (or higher-quality) illiquid assets. Similarly, if banks have to liquidate assets at a common price that is independent of the underlying asset quality, then |$v_{l\theta}<0$|⁠, as well.

Equilibrium characterization changes little without the single crossing condition.²¹ Specifically, Proposition 3 holds independently of the single crossing condition. Intuitively, even if the single crossing condition is reversed, so |$v_{l\theta}<0$|⁠, all surviving banks choose to pay a dividend above |$d_{\min}$|⁠, the dividend payout that minimizes the total liquidity outflow. Given Proposition 3, the single crossing condition is not relevant for the lenders’ inference, either, because it is still the case that |$d^{*}\left(\theta\right)>d_{\min}$| and so having a sufficiently large share of very low types that fail and pay no dividends means that higher dividends are still interpreted as good news about survival in equilibrium. Therefore, Proposition 4 continues to hold, as well.

Where the single crossing condition matters is for equilibrium uniqueness. Specifically, it is no longer clear that in any equilibrium higher dividends can only be interpreted as good news. However, one can show equilibrium uniqueness under different conditions than the single crossing condition. For example, as long as there are sufficiently many banks in the upper dominance region that pay a positive dividend (that is also bounded away from zero), Proposition 9 will still hold as well.

In terms of equilibrium outcomes, without single crossing, the equilibrium dividend payouts of surviving banks need not be increasing in |$\theta$|⁠. Instead, the dividend payouts of these banks could be nonmonotone or even decreasing in |$\theta$|⁠. Moreover, it will no longer be the case that the lowest surviving types distort their dividend payout the most relative to the no-run case. Instead, it is high surviving types, which would prefer to pay a low dividend in the absence of a rollover episode, that may have to distort up their dividend payout the most.

4. Policy Implications

The possibility that banks can influence the coordination-based run they face via their dividend choices has important implications for dividend regulation aimed at improving financial stability during a rollover crisis. In particular, suppose that a regulator cares about minimizing the set of banks failing due to a coordination-based run (i.e., minimizing the failure threshold |$\theta_{f}$|⁠). In this section, we characterize the dividend policy, denoted by |$d^{P}\left(\theta\right)$|⁠, that achieves this. We will call this policy the optimal dividend policy for concreteness.

We assume that the regulator knows |$\theta$|⁠, so that there is no asymmetric information friction between the regulator and the bank(s). We also assume that the regulator respects the information sets of lenders and so cannot directly communicate any information to the lenders. The first assumption ensures that we derive a benchmark optimal dividend policy akin to “first-best” optimal policies in economies with asymmetric information. The second assumption is standard in the literature on optimal policy in games with dispersed information(Angeletos and Pavan 2007). Later on, we discuss the implications of relaxing each of these assumptions.

Second, by Equation (20), the bank’s dividend policy influences the marginal lender with dividend signal cutoff |$\hat{d}$| and, hence, the size of the run, |$A$|⁠, that other bank types experience. This second effect operates through the lenders’ inference. We will call the first source of inefficiency a “risk shifting externality,” whereas the second an “informational externality.” We have the following characterization result.²²

The most striking difference between the optimal dividend policy, |$d^{P}\left(\theta\right)$|⁠, and the equilibrium dividend policy |$d\left(\theta\right)$| from (16) is that the dividend payout for surviving banks is capped at |$d_{\min}^{P}$| and, moreover, that cap is binding for all surviving banks. In contrast, |$d\left(\theta\right)$| is increasing in |$\theta$| (due to single crossing). Intuitively, under the equilibrium dividend policy, a lender observing the marginal dividend signal, |$\hat{d}$|⁠, assigns a relatively low probability that the bank has a very high type, as higher types pay much higher dividends than |$\hat{d}$|⁠. Suppose that all surviving types are mandated to pay the same dividends at some level close to |$\hat{d}$|⁠. Because all of these banks survive the run, observing a dividend of |$\hat{d}$| becomes stronger evidence in favor of survival, and the lender observing |$\hat{d}$| becomes strictly better off rolling over – the marginal lender becomes a lender with a lower dividend signal.

Another important feature of the optimal dividend policy is that more failing banks are forced to pay zero dividends compared to the equilibrium dividend policy. Intuitively, the regulator can lower |$\hat{d}$| in two ways – by mandating that all surviving banks pool on paying the same dividend or, due to the noise in lender dividend signals, by ensuring that failing and surviving banks pay as distinct dividends as possible.

Proposition 6 provides a partial characterization of the optimal dividend policy, because |$d^{P}\left(\theta\right)\in\left\{ 0,\overline{\ell}\left(\theta\right)\right\} $| in the set |$\theta\in\left[\overline{\ell}^{-1}\left(\hat{d}^{P}\right),\overline{\ell}^{-1}\left(\hat{d}^{P}+C_{0}+C_{1}\right)\right)$|⁠. However, that set is small for large values of |$\alpha$|⁠, because both |$C_{0}$| and |$C_{1}$| tend to 0 as |$\alpha\to\infty$|⁠. Therefore, as lender signals become perfectly precise, we can characterize the optimal dividend policy (almost) fully. In that case we can also determine the smallest failure cutoff |$\theta_{f}^{P}$| and the associated dividend cutoff |$\hat{d}^{P}$|⁠.

This result is intuitive in light of the regulator’s objective. The regulator does not put any weight on the bank’s payoffs but only cares about minimizing the failure cutoff, so he chooses a dividend policy to achieve the smallest feasible failure cutoff. In the limit, as |$\alpha\to\infty$| and lenders become perfectly coordinated, the smallest feasible failure cutoff is at the lower dominance threshold, so |$\theta_{f}^{P}\to\underline{\theta}$|⁠. In contrast, with no dividend regulation, by Proposition 5, the failure threshold approaches |$\theta^{nr}>\underline{\theta}$|⁠. Moreover, the distance between the two thresholds increases with |$\lambda$|⁠.

4.1 Discussion

The two key features of the dividend policy from (22) are the full dividend restriction for low failing types and the binding dividend cap for high surviving types. This structure is somewhat different from macroprudential dividend regulation measures commonly proposed in the literature, which call for a restriction of dividend payouts of all banks (Goodhart et al. 2010). The reason for the difference is the dividend signaling effect, which a regulator can also utilize when stabilizing the financial system. Below, we will examine the robustness of these features of the optimal dividend policy to a set of alternative modeling assumptions.

4.1.1 Asymmetric information

In the presence of asymmetric information between the regulator and the bank, the dividend policy in (22) may not be incentive compatible for some bank types for the following reasons. First, the policy (22) may contain nonmonotonicities. For example, if |$d^{P}\left(\theta_{1}\right)=\overline{\ell}\left(\theta_{1}\right)>d^{P}\left(\theta_{2}\right)$|⁠, for |$\theta_{1}<\theta_{2}$|⁠, then with asymmetric information, a bank with type |$\theta_{2}$| may be better off paying |$d^{P}\left(\theta_{1}\right)$|⁠. Second, the policy mandates a full dividend restriction for some bank types, for which paying a higher dividend may be feasible. In the Online Appendix we argue that the region of fundamentals where these issues arise is small for high values of signal precision and show that in that case a two-dividend menu, given by |$d\in\left\{ 0,\overline{d}\right\} $|⁠, for an appropriately chosen value of |$\overline{d}$|⁠, approximates well the optimal policy |$d^{P}\left(\theta\right)$|⁠.

4.1.2 Regulator communication

In the presence of dividend signaling, a regulator who observes |$\theta$| uses the dividend policy to (indirectly) communicate information to lenders about the bank’s ability to survive a run. Abstracting away from the small nonmonotonicity region, the dividend policy (22) then looks like a binary disclosure rule that pools bank types into two groups – the failing banks and the surviving banks. This is similar to the disclosure rules analyzed in Goldstein and Leitner (2018) in the context of stress testing, though in a different economic environment. In our setting, rather than directly disclosing the results of the stress test via a binary scoring rule, the regulator communicates the results of the stress test by either allowing the bank to pay a dividend or not. Therefore, interpreted through the lens of stress testing and information disclosure, the dividend policy we derive implies a link between stress test results and dividend payouts.

One can use the insight that the regulator communicates with lenders via the bank’s dividend policy to also understand direct disclosures by the regulator in the context of our model. In the Online Appendix we argue that optimal dividend policy (22) is qualitatively unchanged even with direct (but noisy) regulator disclosure. In that case the regulator can use both the dividend and its own direct disclosure to improve the overall precision of the lenders’ information.

4.1.3 Alternative policy objective

The assumption that the regulator only cares about minimizing bank failure is appropriate when thinking about policies that promote financial stability but is generally quite stark, because it disregards any direct benefits from paying dividends. Therefore, in the Online Appendix we consider an alternative notion of optimality, namely, maximizing the expected bank payoff prior to the realization of the fundamental |$\theta$|⁠. A bank that can commit to this optimal dividend policy will internalize the informational externality. We show that when |$\lambda$| is small, this optimal dividend policy also features a binding cap on dividends for surviving banks and a zero dividend payoff for low types.

5. Testable Implications and Empirical Relevance

Our model predicts that higher rollover risk leads to higher average dividend payouts. We capture higher rollover risk by increasing the lenders’ incentive to run (⁠|$p\uparrow$|⁠) or the bank’s reliance on short-term debt (⁠|$b\uparrow$|⁠).²³

Intuitively, as |$p$| is increased, the marginal lender needs to be more optimistic about bank survival to be indifferent between running and rolling over. As a consequence, the new marginal lender becomes an agent that observes a higher dividend signal – |$\hat{d}$| goes up. Because |$d_{\min}$| – the lower bound on dividend payouts for surviving banks, defined in Section 3.2.1 – is increasing in |$\hat{d}$|⁠, it follows that dividends also increase. A change in the reliance on short-term debt, |$b$|⁠, has an analogous effect. We illustrate the comparative statics for |$p$| and |$b$| from Proposition 8 in Figure 5, away from the limiting case of perfect signal precision.

Figure 5

Dividend policy comparative statics with respect to |$p$| (left panel) and |$b$| (right panel)

This figure shows the optimal dividend policy with dividend signaling for different values of |$p$| and |$b$|⁠. The left panel considers two different values of |$p$| (⁠|$p=0.5$| for the solid line, and |$p=0.7$| for the dashed line), and the right panel shows the equilibrium for two different values of |$b$| (⁠|$b=0.5$| for the solid line, and |$b=0.5$| for the dashed line). In both panels, the black dotted line is the optimal dividend under the no-run benchmark.

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A second important prediction of our model, which we discussed in Section 3.2.1 in the context of Proposition 3, is that the upward distortions in dividends due to signaling lowers the sensitivity of dividends to fundamentals, because banks with different types choose similar dividend payouts in equilibrium. This prediction is illustrated in Figure 5, where we also plot the dividend policy in the no run case.

We document two novel empirical facts and relate them to our main testable implications.²⁴ We first investigate cross-sectional variation in the evolution of dividend payouts during the 2007 financial crisis. We test the first prediction (cf. Proposition 8) – that an increase in rollover risk increases the average dividend payout – by investigating whether banks that were more reliant on short-term funding and, therefore, all else equal, were subject to more rollover risk, had higher dividend payouts as the crisis progressed. We rank U.S. banks according to the share of liabilities in short-term debt in 2006 and examine the behavior of dividend payouts for banks in the first and fourth quartiles as the financial crisis was unfolding.

As shown in Panel (a) of Figure 6, both groups have similar trends in dividend growth before 2007. However, after 2007 their dividend payments diverge sharply. Banks that relied relatively less on short-term debt decreased their dividend payouts starting in 2007. In contrast, banks that relied relatively more on short-term debt stayed on their pre-2007 dividend growth trend during 2007 and 2008.

Figure 6

Yearly nominal dividend payments for large U.S. banks with different reliance on short-term debt

The figure shows the evolution of dividend payouts for U.S banks. In each panel, we split banks into two groups, based on their degree of short-term funding. We follow Hirtle (2014) and focus on large banks, defined as bank holding companies with more than |${\$}$|500 million in assets as of Q12006. Bank holding companies are then grouped into four quartiles based on their short-term debt relative to total liabilities in 2006. The figure compares the dividend payments of the 1st and the 4th quartiles. Short-term debt is defined as the sum of repurchase agreements, time deposits above |${\$}$|100,000, and federal funds. Left panel: All banks. Right panel: Only banks with average RoA in 2009 above the median. Source: Y-9C reports of bank holding companies.

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While this pattern is consistent with our model, it does not allow us to identify whether the signaling or the resilience effect dominates during the crisis. Specifically, if the resilience effect dominates, it is possible for dividend payouts to go up with higher exposure to rollover risk, driven by an increase in dividends by failing banks that ends up outweighing any decline in dividends by surviving banks due to the crowd-out effect.²⁵ To distinguish between these two effects, we use the observation that among the set of surviving banks, which are banks with relatively high fundamentals, higher exposure to rollover risk implies that dividend payouts move in opposite directions under the two effects. Specifically, if the signaling effect dominates, these banks would increase dividends, while if the resilience effect dominates, they would decrease dividends.

To implement this test empirically, we use a proxy for the bank fundamentals and repeat the analysis for a subset of relatively strong banks. Specifically, we consider a subsample of banks with average Return on Assets (RoA) in 2009 above the median.²⁶ Panel (b) of Figure 6 plots the evolution of dividends for this subsample of banks. The pattern is very similar to the one in the left panel. Overall, we conclude that Figure 6 is consistent with a strong signaling effect.

Next, we test the second prediction (cf. Proposition 3) – exposure to rollover risk decreases the sensitivity of dividends to fundamentals. In Figure 7, we plot the standard deviation of dividend growth against the average share of short-term debt relative to assets for different U.S. industries. Therefore, variation in the share of short-term debt captures variation in the exposure to rollover risk, while variation in the standard deviation of dividend growth captures variation about the sensitivity of dividends to fundamentals in a sector. These two quantities are negatively related, suggesting that dividends are less variable in industries that rely more on short-term funding. This is consistent with dividend payouts being less sensitive to fundamentals in industries where rollover risk is higher.

Figure 7

Industry-level standard deviation of dividend growth and short-term debt reliance

The figure shows a binned scatter plot of industry-level short-term debt share (short-term debt as share of total assets) (x-axis) plotted against the standard deviation of that industry’s dividend growth. Short-term debt is defined as current liabilities. The industry-level correspond to 4-digit SIC code, excluding financials (SIC codes 6000–6999) and utilities (SIC codes 4900–4949). Sources: CRSP and Compustat.

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6. Why Do Banks Signal through Dividends?

Our analysis focuses on dividend payouts as a signal of a bank’s available liquidity. This is motivated by the recent financial crisis and the possible use of dividend signaling to manage coordination-based runs. However, banks also take other actions in times of financial stress to signal “balance sheet strength” that seem to worsen their liquidity positions. For example, Duffie (2010) provides a description of a hypothetical dealer bank’s actions in response to financial stress. He notes that the bank “... takes actions that worsen its liquidity position in a rational gamble to signal its strength and protect its franchise value. [The bank] wishes to reduce the flight of its clients, creditors, and counterparties.” Such actions include compensating clients for losses on investments arranged by the bank or continuing with over-the-counter (OTC) derivative trades that reduce available liquidity. Another related action, which is more relevant for classical bank runs, is the decision of a bank that faces a run by depositors to not suspend convertibility of deposits into cash immediately but instead to service withdrawing depositors. Although these actions are not our proximate motivation, our theoretical framework can be interpreted more broadly and used to analyze their signaling effects as well.

A bank can take other possible actions to convey information about its liquidity position and balance sheet. One alternative is information disclosure in the form of updated balance sheet data or other forms of bank health statements. A key challenge with such disclosures, however, is that such statements may not be credible. As an example, Lehman Brothers boasted a Tier 1 capital ratio of 11 % – well above the regulatory minimum – 5 days before its bankruptcy (The Economist 2010). Another costly alternative to dividend payments is share repurchases. Without any difference in signaling, share repurchases look similar to dividends from the bank’s perspective by constituting an outflow of liquidity. Empirically, however, the signaling content of share repurchases is less pronounced (Hirtle 2014). One possible explanation is that share repurchases are most often discretionary and conducted in the open market. Given stock market volatility, it may be hard for market participants to infer from market prices whether a bank has discontinued a share repurchase program to save on liquidity.

7. Concluding Comments

U.S. banks paid large amounts in dividends during the financial crisis despite mounting losses. In our paper we study a framework that incorporates two distinct views of the underlying reasons for this behavior – risk shifting and signaling. We show that both of these increase the dividend payouts of lower-quality banks when there is a coordination-based run. First, more banks choose to liquidate early because of the rollover crisis. Second, banks that choose to survive pay higher dividends than is optimal for them in the absence of a rollover crisis, because lenders interpret higher dividends as good news about the bank’s available liquidity. Therefore, through the lens of our model, both forces lead to banks with low asset quality paying high dividends during a financial crisis.

The signaling effect and related reluctance of banks to cut dividends due to an increased risk of a rollover crisis may also explain why banks were reluctant to issue new equity (essentially a negative dividend) during the financial crisis (Bigio 2012). Examining the implications of new equity issuance in an environment with rollover crises is a potentially important extension of our framework that we leave for future research.

Acknowledgments

We thank Itay Goldstein (the editor) and two anonymous referees for suggestions that greatly improved the paper. We also thank Toni Ahnert, Felipe Iachan, Gisle Natvik, Øyvind Nilsen, and Tuomas Takalo and seminar participants at BI Norwegian Business School, the 30th Stony Brook International Conference on Game Theory, FIRS 2019, SNDE 2017, and EEA 2017 for valuable comments and suggestions. This paper should not be reported as representing the views of Norges Bank. The views expressed are those of the authors and do not necessarily reflect those of Norges Bank. Supplementary data can be found on The Review of Financial Studies web site.

Appendix

A.1 Details on the Dividend Signaling Equilibrium Characterization

We first derive a condition under which in any monotone strategy equilibrium of this economy, the cutoff |$\hat{d}\in\left(0,1\right)$|⁠. To this end, we characterize the optimal dividend policy of a bank that faces a run of |$A=1$| regardless of its fundamentals |$\theta$|⁠.

By Lemma 1 and the discussion after it, it follows that for any |$\alpha$|⁠, there is a sufficiently large |$K$|⁠, such that the left-hand side of (A3) is decreasing in |$\hat{d}_{\max}$| and so (A3) can have at most one solution. Furthermore, the left-hand side of that expression can be made arbitrarily large (arbitrarily close to 0) for sufficiently small (large) values of |$\hat{d}_{\max}$| and so there exists a solution. We can now state condition B3 which ensures that in any monotone equilibrium, |$\hat{d}<1$|⁠.

Assumption B3.|$\hat{d}_{\max}<1$|⁠.

Notice that |$\hat{d}_{\max}$| is the signal of a marginal lender who is indifferent between running and rolling over if all other lenders run regardless of their signal, and hence banks suffer a run of |$A=1$| regardless of |$\theta$|⁠. Therefore, this is the lender cutoff in the most pessimistic possible case, when other lenders run regardless of their signals and only banks in the upper dominance region survive. As we show in the proof of Proposition 9 below, this condition then ensures that in any monotone strategy equilibrium, |$\hat{d}<\hat{d}_{\max}<1$|⁠.

Assumption B4.|$\hat{d}_{\min}>0$|⁠.

Therefore, |$\hat{d}_{\min}$| is the signal of a marginal lender who is indifferent between running and rolling over if all other lenders roll over regardless of their signal and hence the bank experiences no run for any |$\theta$|⁠. Therefore, this is the lender cutoff in the most optimistic possible case, when other lenders do not run regardless of their signal and only banks with |$\theta<\theta^{*}$| fail. As we show in the proof of Proposition 9 , this condition then ensures that in any monotone strategy equilibrium, |$\hat{d}>\hat{d}_{\min}>0$|⁠.

Finally, to show equilibrium uniqueness, we additionally have to show that there cannot exist equilibria in which higher dividends are interpreted as bad news about survival. Single crossing, |$v_{l\theta}>0$|⁠, is a sufficient condition for this together with |$K$| being sufficiently large, as we will show next.

An equilibrium in monotone strategies consists of a lender cutoff |$\hat{d}$|⁠, a bank cutoff |$\theta_{f}$| and a bank dividend policy |$d\left(\theta\right)$|⁠.

A.2. Omitted Proofs

Footnotes

References