Measuring banks’ liquidity risk: An option-pricing approach

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Measuring banks’ liquidity risk: An option-pricing approach Jinqing Zhang , Liang He , Yunbi An PII: DOI: Reference:

S0378-4266(19)30277-8 https://doi.org/10.1016/j.jbankfin.2019.105703 JBF 105703

To appear in:

Journal of Banking and Finance

Received date: Accepted date:

5 August 2018 19 November 2019

Please cite this article as: Jinqing Zhang , Liang He , Yunbi An , Measuring banks’ liquidity risk: An option-pricing approach, Journal of Banking and Finance (2019), doi: https://doi.org/10.1016/j.jbankfin.2019.105703

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Highlights  Within the framework of global game theory, we establish a boundary condition for bank runs  We show that there exists a unique Nash equilibrium for bank runs in our model  We obtain a formula for bank equity value with both bank run risk and insolvency risk  A bank’s optimal liquidity ratio is derived by maximizing its equity value  A deviation from a bank’s optimal liquidity ratio is a robust proxy for its liquidity risk

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Measuring banks’ liquidity risk: An option-pricing approach Jinqing Zhang Institute of Financial Studies Fudan University Shanghai, China 200433 Email: [email protected] Liang He Institute of Financial Studies Fudan University Shanghai, China 200433 Email: [email protected] Yunbi An1 Odette School of Business University of Windsor Windsor, Ontario, Canada N9B 3P4 Email: [email protected] Phone: 1-519-2533000-3133

Abstract This paper proposes a new approach to evaluating banks’ liquidity needs, which is not only wellgrounded theoretically, but is also easy to apply practically. Within the framework of a global game with imperfect information, we first establish a boundary condition for bank runs and show that there exists a unique Nash equilibrium for bank runs. Using the option-pricing approach, we then obtain a closed-form formula for the value of bank equity with both run risk and insolvency risk. Finally, a bank’s optimal liquidity ratio is derived by maximizing the value of bank equity. Using data on Chinese listed banks, we show that the deviation of the actual liquidity ratio from the optimal liquidity ratio in a bank represents a robust proxy for its liquidity risk. An increased liquidity shortfall leads to worsening liquidity problems, and this is particularly pronounced when the liquidity shortfall is high. Keywords: Bank runs; liquidity risk; insolvency risk; liquidity ratio; option pricing

1

Corresponding author.

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1. Introduction It is extremely important for commercial banks to maintain adequate liquidity to withstand the various stress events they will face. Banks with inadequate liquidity are inherently fragile, as they might not be able to cover unexpected cash outflows, leading to fire sales of assets or panic-based bank runs and ultimately causing a huge loss of assets or even bankruptcy. Such liquidity risk is particularly pronounced under the circumstances of economic and financial turbulence, since in this case liquidity varies substantially, and could dry up quickly. The global financial crisis of 2007 to 2009 highlights the vulnerability of the financial system to liquidity shocks, and demonstrates that bank failure is detrimental to the interests of shareholders, the stability of the banking system, and the real economy. Determining the optimal liquidity needs for banks is not only of interest to academics, but also of paramount importance for regulators and market investors. In both practice and research, one widely used method is to summarize and describe banks’ liquidity with a range of financial metrics. For instance, Rose (2002) summarizes the metrics that bank managers usually use to estimate liquidity needs, including the sources and uses of funds, the structure of funds, liquidity indicators, and market signals. Another example is the new liquidity regulation introduced in the Basel III Accord, which is based primarily on the two regulatory ratios aimed at reducing banks’ liquidity transformation: the liquidity coverage ratio (LCR) and the net stable funding ratio (NSFR). The problem with this method is that it could be difficult to collect the required data due to the incomplete disclosure of relevant information by banks, and it is almost impossible to update these liquidity indicators in a timely fashion due to relatively low disclosure frequency. In addition, while this approach, such as the LCR and NSFR rules, presents principles and guidelines for measuring and managing banks’ liquidity risk, it is not able to identify and analyze

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the incidence of a stress scenario and the way in which relevant variables change under such a scenario. For this reason, some researchers turn to indirect measures to proxy bank liquidity, such as transaction deposits (Gatev et al., 2007), the rate of changes in the aggregate balance sheet (Adrian and Shin, 2008), liquidity creation (Berger and Bouwman, 2009), and banks’ bids in central banks’ refinancing operation auctions (Fecht et al., 2011; Drehmann and Nikolaou, 2013). While these proxies can be used to characterize and understand the overall liquidity position and liquidity risk in a bank, they are not model-based and therefore lack a theoretical justification. Some previous studies including Calomiris et al. (2015), Diamond and Kashyap (2016), and Hugonnier and Morellec (2017) develop theoretical methods for determining liquidity requirements, but they focus primarily on how the requirements for minimum liquidity shift banks’ risk-taking behavior and on the policy implications of such requirements. Acharya et al. (2016) propose a transparent and practically applicable method for identifying a lower bound on the amount of cash that banks should be required to hold in order to avoid systematic crises. However, their method provides little insight into the risk arising from the uncertainty of depositors’ behaviors or coordination risk, as well as the interactions between liquidity risk and credit risk in the determination of liquidity needs. This paper proposes a new and theoretically well-grounded approach to evaluating a bank’s liquidity needs that accounts for risks of both bank runs and insolvency as well as their interactions. Since the incidence of bank runs is jointly determined by the bank’s operational nature and depositors’ coordination risk, we first establish a boundary condition for bank runs using Morris and Shin’s (2004) approach of modelling creditors’ coordination risk in debt refinancing. In this model, without common knowledge of fundamentals, a depositor’s premature

4

withdrawal decision is based on his/her own observed signals and the pre-emptive actions of those depositors with an information advantage. Within the framework of a global game theory with imperfect information (Carlsson and van Damme, 1993; Morris and Shin, 1998), we then show that there exists a unique Nash equilibrium for bank runs provided that private information is sufficiently precise. A bank run represents the ultimate manifestation of liquidity risk of the bank. It can lead to an immediate failure of the bank, even if it is still fundamentally solvent. The risks of bank runs and insolvency interact with each other and jointly determine bank stability, thus affecting the value of bank equity. Given this notion, we then extend Merton’s (1974) model by incorporating liquidity risk into the modelling framework. With both bank run risk and insolvency risk, bank equity can be viewed as a modified call-on-call option, where the boundary condition for bank runs constitutes the exercise threshold for the first call. This allows us to apply Geske’s (1979) option-pricing techniques to obtain a closed-form formula for the value of bank equity with liquidity risk. Finally, a forward-looking representative bank, recognizing the incidence of bank runs and insolvency, will choose to hold a level of liquidity in order to maximize shareholders’ value. In such a model, the optimal liquidity represents a bank’s voluntary choice in the best interest of shareholders, reflecting the trade-offs among risk, return, and liquidity. This is in contrast with the requirements for the LCR and NSFR as per the Basel III Accord, which are designed to be regulatory and rather ad hoc. Banks must necessarily hold a buffer of sufficient high-quality liquid assets to maintain the market’s confidence in liquidity and solvency, thereby insuring against the risk of bank runs. However, excessive liquidity not only hurts banks’ profitability, but also induces banks’ greater risk-taking behavior, exacerbating liquidity problems (Wagner, 2007;

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Acharya and Naqvi, 2012). A key insight from our value-based approach is that excessive or inadequate liquidity is inherently penalized by a reduction in equity value. Using data on Chinese listed banks, we show that the deviation of the actual liquidity ratio from the optimal liquidity ratio derived from our model can significantly explain the timeseries and cross-sectional changes in various aspects of liquidity and credit for banks. An increased liquidity gap leads to worsening liquidity problems in subsequent periods, such as diminished deposits and liquid assets, a higher implicit deposit rate, and a reduced asset expansion rate. Moreover, banks with a high liquidity shortfall tend to have a higher Merton’s default probability, a higher degree of leverage, as well as lower capital adequacy and Altman Zscores. Finally, equity returns can be explained by variations in the liquidity gap, and therefore, the liquidity gap represents an important risk factor in bank equity pricing. Our research contributes to the body of literature on bank runs by deriving an explicit boundary condition for bank runs. Diamond and Dybvig (1983) develop a model of demanddeposit contracts in which the relationship among depositors is depicted as a coordination game with two Nash equilibria: a run equilibrium and a no-run equilibrium. They show that the suspension of convertibility of deposits and the presence of deposit insurance can prevent bank runs, but do not provide a tool to predict whether or not the run equilibrium will occur. Following Diamond and Dybvig’s (1983) model, a number of subsequent studies examine the existence and properties of equilibria for bank runs and their policy implications (Postlewaite and Vives, 1987; Chang and Velasco, 2001; Caballero and Krishnamurthy, 2008). Given the existence of multiple equilibria, these studies are not able to determine the boundary condition for bank runs. Adopting a global game approach (Carlsson and van Damme, 1993; Morris and Shin, 1998), Morris and Shin (2004) and Goldstein and Pauzner (2005) develop a modified form

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of Diamond and Dybvig’s (1983) model in which fundamentals of the economy uniquely determine whether a bank run occurs, while depositors receive noisy private signals regarding the realization of fundamentals. In a model without common knowledge of fundamentals, these authors show how the unique equilibrium for bank runs is determined as a result of coordination failures. In a dynamic setting, He and Xiong (2012) further enrich the method for identifying the condition for runs by creditors in a financial firm by introducing uncertainties about creditors’ future decisions on rolling over their debt contracts. A challenge to the application of these models is that some parameters and expressions require the specification of depositors’ utilities, and thus are hard to estimate and compute. In contrast, we take a rather simple and implementable approach. As investors’ risk preferences are irrelevant in pricing options, we extend Morris and Shin’s (2004) model and derive a boundary condition for bank runs using the option-pricing approach. This approach is easy to apply in practice, as the boundary condition is identified only by information that is readily available, such as stock returns and publicly available accounting data. Our research also adds to the literature on valuing bank equity and measuring bank liquidity needs with the option-pricing approach. The pioneering work of Merton (1974) provides a solid theoretical framework for pricing equity in an enterprise. This method can be used not only to value bank equity, but also to estimate risk measures such as default risk, credit spread, and distance-to-default (Leland, 1994; Vassalou and Xing, 2004; Covitz and Downing, 2007; Duffie et al., 2007; Bharath and Shumway, 2008; Friewald et al., 2014). However, a direct application of Merton’s (1974) model can lead to systematically biased estimates of bank risk and equity value, since the model does not take into account insolvency arising from liquidity shortage. By incorporating liquidity risk into Merton’s (1974) framework, our model not only

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helps us better understand how bank equity is related to both liquidity and credit risks as well as their interactions, but also allows us to estimate the optimal liquidity level from the shareholders’ perspective. The remainder of this paper is organized as follows. Section 2 describes the basic model framework. Section 3 derives the boundary condition for bank runs and bank equity value, and then characterizes properties of the optimal liquidity ratio in a bank. Section 4 provides a numerical analysis of the effectiveness of our proposed liquidity measure using data on Chinese listed banks, while Section 5 concludes the paper. 2. The model We assume that the market is perfect and competitive. All securities in the market are divisible, and security trading is continuous. There are no transaction costs or taxes. There are no restrictions on short selling, and the full use of short-selling proceeds is allowed. The risk-free rate of interest is , which is constant and observable. Borrowing and lending are possible at the risk-free rate. Consider a representative bank, whose assets are worth . Total bank assets

are classified into liquid assets

and equity is worth

and illiquid assets

at time

, and thus

. Liquid assets are those assets that can be easily and immediately converted into cash at little or no loss of value (BCBS, 2013). Since liquid assets are typically risk free, their value process can be described as follows: .

(1)

Illiquid assets, however, generally have an uncertain future value. Following Black and Scholes (1973) and Merton (1974), we assume that the value of illiquid assets follows a geometric Brownian motion with constant drift

and volatility

8

:

( where

)

,

(2)

is a standard Brownian motion. Thus, the stochastic process for the value of total assets

is given as: ,( where

)

-

(

)

,

(3)

represents the ratio of liquid assets to total assets, referred to as the liquidity

ratio in this paper. Since banks finance long-term loans by short-term deposits, their liquidity risk arises primarily from the uncertainty of depositors’ withdrawal needs. This suggests that the liquidity ratio should be adjusted frequently in accordance with the bank’s debt maturities. We assume in this paper that the optimal liquidity ratio is determined at date 0, and remains unchanged until date . We denote

,

(

)

, and

We further assume that bank debt has a face value of

(

)

.

at time with maturity date .

The annualized return on bank debt is . In other words, for one dollar of deposits at date 0, a creditor will receive

dollars at time

if the bank remains solvent. Importantly,

creditors/depositors can withdraw their deposits at any time before maturity, which opens up the possibility of depositors running. We assume that if a deposit is withdrawn before it matures, then the depositor can get back only the original amount of deposit plus the interest accrued at the risk-free rate . According to Merton (1974) and Geske (1979), bank equity

satisfies the following

partial differential equation: .

(4)

In our model, a bank may fail because of either a bank run or insolvency. Thus, the value of bank equity is solved from Equation (4) subject to two boundary conditions. The first corresponds to the incidence of a bank run, which can be triggered at an exogenous time 9

(

).2 At

, a depositor’s running-on-the-bank decision is made based on the observed

public and private signals. The second corresponds to insolvency arising from inadequate capital when debt becomes due. In this case, the bank is unable to fulfill its obligations and becomes insolvent if

. Accordingly, the boundary conditions for the stochastic process of bank

equity can be specified as follows: , where

(5)

satisfies: ,

(6)

,

(7)

Similar to Merton’s (1974) model without consideration of liquidity risk, in our model bank equity is also regarded as a derivative instrument on bank assets and thus satisfies the same partial differential equation. Unlike Merton (1974), our model imposes an additional boundary condition, which is the condition for bank runs described in Equation (5). 3. Boundary condition for bank runs and determination of liquidity ratio 3.1 Boundary condition for bank runs In this section, following Morris and Shin’s (2004) approach of modelling creditors’ coordination risk in debt refinancing, we establish the threshold that triggers a bank run. Assume there is a continuum [0,1] of homogeneous depositors. At time signal regarding the value of bank assets

, all depositors receive a public

. It is important to note that the readily available

2

To simplify our analysis, we treat as a constant throughout the paper. However, our model can also be extended to the case in which is a random variable. In this case, the equity value is obtained by integrating our equity value expression with respect to the cumulative distribution function of under the risk-neutral probability measure.

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funds to meet short-term obligations are always less than the total assets at time

, due to the

lack of liquidity of some assets such as loans and advances. Illiquid assets cannot be borrowed or sold against for the full value they could generate. To avoid the potential losses that may arise from a fire sale of illiquid assets (Diamond and Rajan, 2005; 2011), banks often use illiquid assets as collateral to gain access to cash through repurchase agreements (Adrian and Shin, 2008; 2010; Fostel and Geanakoplos, 2008; Gorton and Metrick, 2009; 2012). With a repo haircut of h, banks can borrow solely (1 – h) dollars per dollar value of the collateralized assets (Gorton and Metrick, 2009; 2012; Shleifer and Vishny, 2010). Thus, the total cash flows that can be generated to meet liquidity needs at ( where

)(

)

are given as: ,

(8)

. Given the model’s assumptions, the prior distribution of

that depositors can infer

from the public signal is normal and given as follows: ( At

⁄ ,

, depositor

).

(9)

- also obtains his/her private signal

about

as follows:

, where

(10)

represents the noise in the private information with distribution

independent of

(

(

⁄ ), and

is

). After receiving the private signal, depositor immediately decides

whether or not to withdraw depositors. Previous literature typically models bank runs as a simultaneous move game, assuming that depositors receive signals on fundamentals and make withdrawal decisions simultaneously (Diamond and Dybvig, 1983; Goldstein and Pauzner, 2005). In this case, if funds are insufficient to meet the withdrawal requests from depositors, total funds are equally distributed among all depositors. However, as Brunnermeier (2001) points out, this 11

assumption does not reflect the fact that in reality, withdrawals by depositors occur sequentially. Moreover, this assumption clearly contradicts the property of global strategic complementarities that a depositor’s incentive to take an action (withdrawing deposits early or waiting until maturity) increases with the number of other depositors who take the same action. In fact, depositors observe information in sequence, and those with an information advantage make decisions earlier and their actions affect the payoff of other depositors (Yorulmazer, 2003; Gu, 2011). In this paper, we also assume that depositors receive their private signals in sequence. In addition, before obtaining

and making the running-on-the-bank

decision, depositor can observe the withdrawal history of others, and update his/her prior belief accordingly. Hence, depositor makes his/her withdrawal decision based on the public signal, the withdrawal history of others, as well as his/her own private signal We now turn to a representative depositor’s payoffs at time choices: withdrawing or waiting.

. under the two alternative

Since depositors are continuous in [0, 1], one single

depositor’s withdrawal decision has little impact on his/her bank’s ability to repay debts. Thus, with the practice of servicing depositors on a first-come, first-serve basis in the bank industry, a depositor gets paid in full if he/she chooses to withdraw his/her capital before his/her bank runs out of resources (Diamond and Rajan, 2001). Therefore, for each dollar of deposit, at time

the

depositor’s payoff if he/she withdraws the deposits is given by: .

(11)

This is consistent with the model assumption in Yorulmazer (2003) and Gu (2011), and is particularly suitable for our analysis, since we are interested in identifying the boundary condition for bank runs, which is determined by the actions of those depositors with an information advantage. 12

On the other hand, if a depositor does not withdraw (a no-run decision) at time

, his/her

payoff is contingent on the incidence of either a bank run or insolvency. If there is a run on the bank at time

, then the bank fails, which means that the depositor receives nothing. If there is , Equations (5) to (7) show that the bank’s equity value becomes

no run at time (

), where (

value

, a strike of

) is the price of a call on bank assets with initial

and time to maturity of

, the market value per dollar of deposits equals [

priority for repayments, then at time (

)]

. If the fraction of depositors who choose to withdraw at time

is n, then

. Thus, for each dollar of deposit, the depositor’s

a bank runs out of funds if payoff at time

. Suppose all bank debts have the same

if he/she leaves the deposits in can be summarized as: (

)

[ ,

)]⁄

(

(12)

It is noteworthy that a depositor’s observation of the withdrawal history of others can be very noisy, given banks’ limited information disclosure. Moreover, the specific amounts of deposits withdrawn are not observable. As a response to the actions of those who have already made their withdrawal decisions, depositor

updates his/her prior distribution of

in

Equation (9). The updated prior distribution is assumed as follows: (

⁄ ).

(13)

After receiving his/her private signal, using Bayesian rules, this depositor’s posterior distribution of variance

(

[

is normal with mean

]

(

)⁄(

) and

).

Now let us consider the optimal strategy for the depositor. To this end, we first solve for the model equilibrium under the assumption that all depositors follow a “threshold strategy”, in 13

which a creditor chooses to withdraw his/her deposits if his/her posterior belief about is lower than a threshold

, or

,

. In other words, a creditor will choose to withdraw deposits if and

only if: )⁄(

(

)

.

(14)

We will then demonstrate in Theorem 1 that this “threshold strategy” is the only strategy that depositors could apply in any equilibrium. Let

be the point below which the bank runs out of funds and is unable to repay

deposits. At time

, the proportion of creditors who withdraw deposits, which is (

proportion of depositors with

being lower than

, or

), is the

(

being lower than

)⁄

.

( ), by the definition of , we have:

As

*

∫

(

)⁄

+

.

(15)

From Equation (15), we can solve for the bank’s failure point

as a function of

. Only when

is greater than , a bank run is avoided. Since the posterior distribution of with mean |

)

(

and variance √

0

(

for depositor i at the switching point

is (

), the probability density function of )(

)

1. Note that

is normal

, according to Equation (8).

Based on Equation (12), the depositor’s expected payoff from leaving deposits in is given by: ∞

(

|

∫

∞

)

,

(

∫

(

)-√

)

(

0

| (

)(

) )

1

.

(16)

According to Morris and Shin (2004) and Angeletos and Werning (2006), whether or not a depositor claims early withdrawal depends on the benefits arising from withdrawing compared 14

with the benefits from waiting. When

is at the switching point

, depositor i should be

indifferent between the two choices, or (

|

)

.

(17)

Substituting Equations (11) and (16) into (17) yields the indifference condition for payoff: ∫

√

,

(

)-

0

(

)(

)

1

.

(18)

To obtain the boundary condition for bank runs , we need to solve Equations (15) and (18) simultaneously. The problem is that the values of

and

in these two equations and their

distributions depend on the sizes and sequences of private signals, and cannot be determined solely by the distribution of private signals. To understand this point, we assume that there are N depositors in total, and the sizes of their private signals satisfy consider two sequences of the arrivals of private signals: the sequence of well as the sequence of

. We now , as

. Apparently, for a particular depositor, the observed

number and proportion of withdrawals in the former case are always higher than those in the latter case. Thus, after observing the history of withdrawals, a depositor is more pessimistic in the former case. This means that the values of

and

and their distribution characteristics are

quite different in the two cases. Gu (2011) reveals that a sudden increase in the withdrawal rate leads to considerable bias in the prior beliefs about bank fundamentals among those depositors who remain undecided. In this case, the private information those depositors subsequently receive is not decisive, leading to inefficient runs. However, the values and sequence of private signals are unknown ex ante, rendering the values of and

and

undetermined. As can be seen from Equations (15) and (18), the solutions for

depend on the values of

and

and their distributions. Thus, it is difficult to solve

for the boundary condition ex ante. To simplify the analysis, we assume that depositors’ private 15

information is sufficiently precise (i.e., the values of

and

). As we will see below, under this assumption,

no longer impact the model solutions, and the boundary condition can be

identified by the distribution of private signals without knowing the values and sequence of private signals. This assumption is also adopted in many prior studies. For example, Goldstein and Pauzner (2005) use this assumption to simplify their model, and obtain a closed-form solution for the boundary condition of bank runs. In their study on the role of information in crises, Angeletos and Werning (2006) point out that the uniqueness of an equilibrium is ensured, if and only if private information becomes sufficiently precise. The practical implication of the assumed high precision of depositors’ private information is clear. Given the noise in the public signal and in the signal on other depositors’ actions, the private information that motivates a particular depositor to take an action is generally much more precise. When ∫

√

, Equation (18) is equivalent to the following equation: ,

(

)-

Equation (19) implies that when

0

(

)

1

.

(19)

, the indifference condition for payoff is identical for

all depositors. Thus, the switching point is also the same for all depositors, which is denoted as . When point

, n equals the probability that , the private signal

is lower than

(

)⁄

. At the

that depositor receives is normally distributed with mean

and variance ⁄ . Thus, Equation (15) becomes: .√ (

)/.

(20)

Using Equations (19) and (20), we can now solve for the two parameters of interest, and

, and therefore obtain a depositor’s running-on-the-bank equilibrium when all depositors

16

employ the same threshold switching strategy. Within Morris and Shin’s (2004) global game framework, we prove in Theorem 1 that the equilibrium is unique in the sense that the threshold is the only strategy that survives the iterated deletion of dominated

switching strategy around strategies.

Theorem 1. When depositors’ private information is sufficiently precise ( exists a unique equilibrium. In this equilibrium, if

and

solve Equations (19) and (20), then a

depositor claims early withdrawal provided that his/her posterior belief he/she chooses to wait (a switching strategy around

,

[

solves

; otherwise,

). (

In particular, as long as inequality true, when

), there

(

) )]

, and

holds .

Proof: See Appendix A. (

Note that the condition

)

generally holds true

in practice. 3 If otherwise, then according to the proof in Appendix, we have , meaning that a bank run occurs even if the bank still has sufficient funds to repay all its debts. When

, based on the boundary condition in Theorem 1, it is clear that there is no

bank run if and only if ,

(

)(

, where )-

,

is solved from the following equation:

(

)-

Thus, Equation (21) determines the asset value at time

3

This condition holds true in our empirical analysis.

17

.

(21)

below which a bank run is triggered.

To better understand the bank run triggering point

, we compare it with the run

with no noise in the depositors’ private information.

triggering point

can be solved from

the following equation (Mella-Barral and Perraudin, 1997): ( Since , ( )

) (

)(

)-

.

(22)

⁄

, we must have

as well as (

(

))⁄

. This suggests that the coordination risk arising from

uncertainties of other depositors’ behavior (Morris and Shin, 2004; He and Xiong, 2012) raises the probability of bank runs, thereby boosting the bank run triggering level. Note that the boundary condition in our model can be decomposed into the product of two parts. The first part is that the available cash flow at time words, ,

(

)-

)(

compared with

. In other

, which is the same as the condition set by Acharya et

al. (2016). The other part is Equation (22), the boundary condition under perfect information. Thus, the boundary condition of Equation (21) combines the two simplified cases, and therefore can better explain the incidence of bank runs. 3.2 Valuing bank equity with liquidity risk We show that a bank avoids a bank run if and only if

holds true. Since call

prices are monotonically increasing in the underlying asset price, inequality

is

equivalent to the following inequality: (

)

(

)

.

(23)

Thus, Equation (5) can be rewritten as follows: {

(

)

(24)

18

Equation (24) prompts us to value bank equity with bank run risk using the option pricing approach. More specifically, Equations (4) to (7) along with Equation (24) indicate that the payoff to bank shareholders can be viewed as follows. At time receive a call on bank assets with a maturity of

, shareholders have the right to

and a strike of

if the call value is

greater than , and receive nothing if otherwise. If shareholders are required to pay

, then shareholders’ payoff looks exactly like the payoff from a call on a call, in

the call at time

which the former call has a maturity date date

to receive

and a strike of

and a strike of

and the latter one has a maturity

. According to Geske (1979), the value of such a compound option is

given as follows: (

√ (

) )

√

(

),

(25)

where (

⁄

) (

⁄ )

√

√ (

⁄

) (

⁄ )

(26)

√ .

√

(

,

(27)

) is the bivariate cumulative normal distribution function with

integral limits and

(

as the correlation coefficient.

)

.

and

and

as the upper

satisfy Equations

(21) and (23), respectively. Since shareholders are not subject to the payment of

to receive the latter call at time

the value of bank equity at date 0 with liquidity risk is as follows: (

√

)

(

3.3 Optimal liquidity ratio and its properties

19

√

).

(28)

,

Recognizing the incidences of bank runs and insolvency, at time 0, a representative bank chooses a liquidity ratio to balance the benefits of maintaining sufficient liquidity holdings against the costs. More specifically, a bank chooses its optimal liquidity ratio

by maximizing

equity value: ,

where

,

-

(29)

is given in Equation (28). The optimal liquidity ratio derived from this model is comprehensive, as the model

captures not only the liquidity risk due to bank runs but also the insolvency risk. A bank with no run risk can become insolvent at the end. On the other hand, a bank may fail as a result of bank runs even if it might still be solvent at time T. These two types of risks are closely connected and jointly contribute to bank failure. The optimal liquidity ratio in Equation (29) reflects the bank’s optimal decision on liquid asset holdings, fully recognizing both sources of risk. This is in sharp contrast with other widely used liquidity methods (e.g., Acharya et al., 2016) that intend to measure a bank’s ability to solely meet its liquidity needs in the future. While there is no closed-form solution to Problem (29), we can analyze the mechanism through which liquidity ratio impacts bank equity first-order derivative of (

with respect to liquidity ratio can be decomposed as:

√ (

. From Equation (B.5) in Appendix B, the

⁄ )

( )√

)

(

)(

(

⁄

)

√

√

)

( √

)

.

The first and second terms of the above equation measure the effect of the liquidity ratio on the volatility of bank assets, which in turn impacts bank equity value. This effect is generally negative around the optimal liquidity ratio

. More specifically, the first term represents the

20

effect of liquidity ratio via asset volatility on the value of ComOpt given in Equation (25), which is negative. The second term measures the effect of liquidity ratio via asset volatility on the expected payoff of receiving the latter call in the call-on-call option, which is arguably trivial compared to the former effect. The third term measures the effect of the liquidity ratio on the liquidation value of bank assets, which in turn impacts the boundary condition for bank runs and bank equity value. Namely, as the liquidity ratio rises, the bank run boundary declines, which in turn positively impacts bank equity value. The ultimate effect of the liquidity ratio on bank equity depends on the magnitudes of these two effects. The optimal liquidity ratio is achieved when the two effects are balanced, or ⁄

. Now, let us consider whether bank runs are still possible even if a bank chooses to

maximize its equity value. Note that given our model assumptions, as long as chance that total assets drop to zero at time

. Therefore, if and only if

, there is a

, the risk of bank

runs can be completely eliminated. A bank can also choose the specific quantity of liquid assets, rather than the liquid asset ratio, to maximize its equity value. To completely avoid bank runs, the following must hold: .

(30)

In either case, to completely avoid bank runs, a bank must keep an extremely high liquidity ratio, which contradicts the objective of value maximization. Now we consider whether or not

can be the optimal solution to our model. It is

important to note that in Merton’s (1974) model without consideration of liquidity risk, banks will choose

to maximize equity value. This is sensible, as the model ignores the risk of

21

bank runs and also the effect of haircut on liquidating illiquid assets, which are exactly the reasons why banks hold liquid assets. This simply means that

is underestimated under the

traditional option pricing framework (Black and Scholes, 1973; Merton, 1974; Ronn and Verma, 1986). In fact, in our model with liquidity risk, it is still possible in theory that bank equity value is maximized at

. This is because the volatility of assets is assumed to be constant in our

model, whereas in reality the volatility can be changed by a shift in the loan structure. The following proposition shows that when the liquidity haircut

is sufficiently high, then

Proposition 1. When the liquidity haircut is sufficiently high (i.e., ⁄

at

. Thus,

.

is close to 1), then

must hold true.

Proof: See Appendix B. Previous studies show that the haircut level the haircut

in a repo is countercyclical, meaning that

is expected to be high (low) when the economy is weak (strong) (Adrian and Shin,

2009; Brunnermeier and Pedersen, 2009; Gorton and Metrick, 2009). In particular, during a period of financial turbulence, it is difficult for banks to seek short-term financing through repos, leading to an extremely high haircut level. Proposition 1 says that it is optimal for banks to maintain certain liquid assets when facing liquidity risk. Due to the complexity of the expression of bank equity value with liquidity risk, it is difficult to directly compute the first-order derivatives of interest. Thus, we illustrate the sensitivity of

with respect to the parameters of

to its determinants using a numerical example.

For this exercise, we set the benchmark values of parameters as ,

,

, and

.

22

,

,

,

Figure 1 plots

as a function of various parameters. It shows that

is negatively related to

the market value of total assets. This is because an increase in assets strengthens a bank’s ability to meet its obligations, which in turn reduces the possibility of bank failure of either type and also reduces the bank’s incentive to hold more liquid assets. Second, volatility of illiquid assets

increases with the

. A higher volatility of illiquid assets is associated with greater

uncertainties of the future value of bank assets. Thus, the bank tends to build up its liquid buffer to hedge against the increased liquidity risk if illiquid assets are particularly volatile. Third, as the book value of total debt

rises, the value of

rises. This is not surprising, as an increase in

represents greater obligations that the bank must meet at a later date, and thus the bank needs to maintain greater liquidity to withstand the possible stress events that could lead to a bank failure. Finally, the value of

rises as the haircut level

increases. Increased

translates into

reduced market liquidity, boosting the probability of bank runs. Therefore, the bank needs to increase the proportion of liquid assets to mitigate the effect of a high

on liquidity risk.

4. Empirical analysis In this section, we apply our model to estimate optimal liquidity ratios using real data on Chinese banks. Then, we examine the effectiveness of the liquidity gap in a bank, which is the difference between the optimal and actual liquidity ratios, in measuring and assessing the bank’s liquidity risk. 4.1 Sample and parameter estimation 4.1.1 Data In our analysis, we use data on Chinese commercial banks based on the following considerations. First, compared with the financial markets in developed countries, Chinese financial markets are still relatively immature. In developed markets, investors’ flight-to-quality 23

behavior can lead to substantial deposit inflows into the banking system during periods of financial turbulence (Gatev and Strahan, 2006), thereby biasing empirical results. However, in China, the lack of liquidity in the bond and commercial paper markets and the presence of stringent restrictions on investments inevitably inhibit the ability of Chinese investors to “flight to quality.” Thus, Chinese data is particularly suitable for the analysis of liquidity issues for banks, as the potential data problems associated with substantial deposit inflows during crisis are mitigated. Second, Chinese banks are primarily involved in the traditional business of deposittaking and loan-making, and bank profits are derived mainly from the spreads between lending and deposit rates. Other businesses such as asset securitization and loan commitment are small in terms of their size, share, and scope. As a result, the possibility of bank runs represents a major and important source of liquidity risk. Third, the deposit insurance program in China was initiated in 2015, and deposit insurance is capped at RMB 500,000. There is no deposit insurance during most of our sample period, and in theory, the insurance program in place from 2015 cannot prevent bank runs from happening in Chinese banks. We focus on 16 Chinese commercial banks that have a relatively long post-IPO period, and have relatively complete financial and accounting information available. The sample period extends from the end of 2006 to the third quarter of 2016. All the data including market interest rates, bank stock prices, and accounting data are obtained from the database iFinD. 4.1.2 Parameter specification and estimation To solve for the optimal liquidity ratio in our model and analyze its practical relevance in liquidity risk management, we need to specify and estimate the model parameters. Table 1 provides a summary of the specifications and calculations of various model parameters.

24

First, the interest rate on demand deposits is used as a proxy for the risk-free rate, because it is the risk-free rate that depositors can obtain in China. The yields on government bonds are not used to proxy the risk-free rate, as there is a liquidity premium in the yields given various restrictions on the trading of government bonds in China. Regarding the estimate for debt face value at time , previous studies assume that the face value of debt increases at the risk-free rate (Merton, 1977; Ronn and Verma, 1986). However, this assumption is not suitable for our analysis, since the incidence of bank runs and debt face value are highly correlated. While using heterogeneous estimates of debt returns in various banks may yield more accurate results, this complicates the model computation. Further, it may cause endogenous problems, as in the analysis the implicit deposit rate is used as one of the indicators for liquidity. Given that bank debt grows at a relatively steady rate, we set its return rate R at 3%, which is close to the average annual benchmark deposit rate in our sample period. In both academia and practice, it is common to consider T as a time horizon of one year (Charitou et al., 2013).4 Further, we set

equal to 0.25 years, which corresponds to the end of a

quarter, due to the following reasons. First, it is practically difficult to estimate

ex ante

without knowing depositors’ specific private information. Second, while depositors can obtain private information and make running-on-the-bank decisions at any time, they have particularly strong incentives to do so at the time when substantial public information is released. This is because obtaining and analyzing information entails costs and efforts. We set

as 0.25 years, as

this time of year typically corresponds to the auditing time, and audit reports are arguably the most important source of public information. Third, it is also around these dates when banks are

4

In theory, the best estimate of T is the duration of deposits. However, Charitou et al. (2013) show that this does not necessarily yield better results than the case of using the one-year horizon. In addition, it is generally difficult to obtain sufficient data to estimate the duration of deposits.

25

generally subject to reviews by regulatory authorities and face relatively tight liquidity constraints, 5 making it difficult for banks to meet short-term obligations through inter-bank lending/borrowing, and thus bank runs are more likely to happen. According to the model assumptions, we need stock prices and their volatilities at the beginning of each auditing period (one quarter). For this purpose, we use the closing price on the first trading day of each quarter, and the equity value at date 0 is obtained by multiplying the stock price by the number of shares outstanding in a bank. In addition, we employ the GJR(1,1) model to estimate the return volatility on each trading day. To obtain the volatility

for an

auditing period, we take the average of daily volatilities of the last month in the previous quarter and then annualize it by multiplying by √

. An alternative method is to match accounting data

with the market data on the audit report publishing dates in order to reflect the process of investors retrieving information from accounting and auditing reports. The problem with this method is that the stock price may contain not only the information about a bank’s performance in the last auditing period, but may also reflect the information from the end of the last auditing period until the publication date, creating endogenous concerns. For the liquidity haircut level, Gorton and Metrick (2009; 2012) and Shleifer and Vishny (2010) find that it is close to 20%. However, illiquid assets in different banks may have varying haircuts. Compared to US banks, Chinese commercial banks (especially those listed banks in our sample) are particularly important to the entire financial system of the country, and thus enjoy greater direct supports from governments. Accordingly, illiquid assets in these banks are subject to lower haircut in repos. In this paper, we set the haircut level at 15% for all banks. It is 5

For example, US banks must submit CALL reports, as per the Federal Financial Institutions Examination Council’s (FFIEC) requests. Chinese banks need to meet the quarterly requirements for review set by the macroprudential assessment (MPA) system.

26

noteworthy that although the value of

depends on the haircut level, the relative levels of

for

various banks are not affected by the haircut level used in the empirical analysis. Finally, solving Problem (29) requires estimates of asset value (

illiquid assets

), where

ratio at time 0, respectively. Since

and

and the volatility of

represent asset volatility and liquidity

is publically available, we only need to estimate

and

.

Merton (1974) proposes a numerical method to estimate these parameters. However, Crosbie and Bohn (2003) and Vassalou and Xing (2004) point out that high leverage can lead to biases in Merton’s (1974) estimates. Given that banks are highly leveraged, this problem can be further exacerbated. Therefore, we adopt Bharath and Shumway’s (2008) method to estimate banks’ asset volatilities. To this end, we first use the average debt volatility in the sample period to proxy debt volatility

. Namely,

. Setting debt volatility equal to a constant

can increase the robustness of the estimates of asset volatilities, mitigating the effects of substantial fluctuations in stock prices. Finally, we calculate the asset volatility as follows: . By plugging

,

(31)

, and other parameters into Equation (28), we can solve for

finally use the Newton Raphson algorithm to solve for

and

. We

in Problem (29).

4.2 Liquidity gap and subsequent variations in liquidity variables 4.2.1 Variables and regression model Given the optimal liquidity ratio obtained from our model, we can gauge a bank’s liquidity risk by comparing it with its actual liquidity ratio. Liquidity concerns arise if a bank fails to achieve the optimal level, and the greater the liquidity shortfall, the higher the liquidity risk. A proxy for liquidity gap is therefore the direct difference between the optimal and actual liquidity ratios. However, one may argue that as the actual liquidity ratio is a direct part of the 27

proxy, it is empirically difficult to distinguish between the effects of the liquidity measure and actual liquidity ratio. Additionally, as pointed out by Rose (2002), liquidity managers usually focus on changes in liquidity indicators rather than on the levels of these indicators. Since the optimal liquidity needs are derived from a theoretical standpoint, it is more sensible to compare the theoretical and actual liquidity ratios in a time-series and cross-sectional sense. Thus, we regress banks’ optimal liquidity ratios on the actual liquidity ratios, and use the residuals from the regression, referred to as the orthogonalized liquidity gap (OLG), to proxy banks’ liquidity risk. A higher OLG in a bank implies greater liquidity problems in the future. To examine this issue, we run the following model with fixed effects, using panel data:

, where

(32)

is a variable indicating a bank’s liquidity from a particular angle. We consider a

number of liquidity indicators from both the asset and liability sides. These variables include quarterly growth rate of deposits (GRD), quarterly growth rate of liquid assets (GRLA), implicit deposit rate (IDR), quarterly growth rate of loans (GRL), quarterly growth rate of assets (GRA), and loan-to-deposit ratio (LTD). The quarterly growth rate of a particular variable is calculated as the change in the variable in one quarter, standardized by the book value of assets at the beginning of the quarter (Cornett et al., 2011; Acharya and Mora, 2015). Following Acharya and Mora (2015), IDR is estimated as quarterly interest expenses divided by total deposits in the prior quarter. We measure credit origination using the loan-to-deposit ratio because Chinese listed banks are subject to strict regulations and supervision, and the loan-to-deposit ratio is one of the key measures used by Chinese regulators to assess a bank’s risk exposure.

28

We also include some control variables (Controls) to control for the effects of other characteristics at the bank level, including the liquid asset ratio (LAR), capital adequacy ratio (CAR),6 size as measured by the logarithm of total assets (Size), return on assets (ROA), and the ratio of nonperforming loans to total loans (NPL).7 We run the regressions with standard errors clustered at the bank level to account for potential serial correlations among bank variables. Table 2 reports the summary statistics of the regression variables. We see that the estimated optimal liquidity ratios range from 0 to 0.6293. The mean of these estimates is 0.4714, which is higher than the mean of actual liquidity ratios 0.2660. This suggests that these banks generally maintain a relatively low liquidity ratio compared with the optimal level. One possible explanation for this observation is that our model does not account for the effects of moral hazard as a result of government assistance on banks’ liquidity holdings. When a bank experiences financial difficulties, in order to avoid spillover disturbances to the stability of the financial system, the government is likely to revive the affected bank either by direct infusion of funds, or by a temporary reprieve from closure (Ronn and Verma, 1986). Such emergency liquidity support from the government adversely impacts banks’ incentives to maintain sufficient liquidity (Ratnovski, 2009). We note that due to the IPO effects or impacts by the macroeconomic conditions, some of the OLG estimates have extremely low values. To avoid the impact of extreme values on our regression results, we winsorize the 10 lowest values of OLG estimates. 4.2.2 Regression results

6

Capital adequacy ratio = (Tier I capital + Tier II capital)/risk weighted assets. Tier I comprises ordinary share capital, audited revenue reserves, future tax benefits, and intangible assets. Tier II comprises unaudited retained earnings, general provisions for bad debts, revaluation reserves, and perpetual subordinated debt. It also includes perpetual cumulative preferred shares and subordinated debt. 7 Non-performing loans include substandard loans, questionable loans, and lost loans.

29

Table 3 reports the results of the regressions in which the explained variables are various liquidity variables in banks. The key coefficients of interest are those on OLG reported in the first row. The results show that OLG indeed has an explanatory power for the variations in various aspects of liquidity for banks. More specifically, Columns 1 and 2 show that an increased OLG leads to diminished deposits and liquid assets in the subsequent period. The results in Column 3 show that banks choose to attract deposits by raising the deposit rate in an effort to alleviate the negative effects of reductions in deposits and liquidity. The results in Column 3 are more significant than those in Columns 1 and 2. This is primarily due to the fact that, to date, since a long-term systematic liquidity crisis has never occurred in the Chinese banking industry, an increase in deposit rates can help mitigate the impacts of deposit reductions. However, as shown in Columns 1 and 2, the effect of an increase in interest rates is insufficient to completely offset the reduction in deposits and liquid assets. The results in Columns 4 and 5 demonstrate how banks actively manage their assets according to the level of liquidity shortage OLG. Total assets and loan-to-deposit ratio are both expected to decrease as OLG increases. These results are in line with the empirical finding that greater liquidity risk exposure leads to downsizing and reduced lending of the affected banks (Cornett et al., 2011). The predictive powers of OLG are not only all statistically significant at the 5% level, but also of great economic relevance. For example, a 0.1 increase in OLG is associated with a 0.960 percentage point decline in deposit growth and a 1.519 percentage point decline in liquid asset growth, which are economically significant compared to the average growth rate of 3.02 percentage point and 1.43 percentage point, respectively. Regarding the control variables, we find that these liquidity measures are negatively related to the one-period lagged liquidity ratio, as the coefficients on the LAR for regression

30

models 1, 2, and 4 are negative and significant. This suggests that banks with less liquid assets tend to build up their liquidity buffer in the subsequent period in order to maintain a relatively stable liquidity ratio. Given the way in which OLG is defined, this also suggests that the power of OLG for explaining changes in subsequent liquidity is not driven solely by the explanatory power of LAR, as their coefficients would have opposite signs if otherwise. The results also show that an increase in bank size leads to a decline in the quarterly growth rates of deposits, liquid assets, total assets, and loan-to-deposit ratios, while leading to a higher implicit deposit rate, indicating a positive relationship between liquidity risk and bank size. This is in line with previous findings that despite the potential effects of the lender of last resort (LOLR) associated with bank size during crisis periods (Goodhart, 1999; Stern and Feldman, 2004; Goodhart and Huang, 2005), large banks are exposed to larger liquidity risk in normal time periods (Gatev and Strahan, 2006; Cornett et al., 2011). The coefficients on CAR and NPL are not significant, as they both measure only one particular aspect of credit risk and OLG already captures certain information about credit risk. The coefficients on ROA indicate that more profitable banks tend to have a better liquidity position. Collectively, these findings confirm that an increase in OLG predicts worsening liquidity problems in a bank. Thus, OLG is practically an effective measure of the liquidity risk faced by a bank. However, it is noteworthy that as our method to derive the OLG measure is inspired by Merton’s (1974) option-pricing method to measure the credit default probability (MDP), OLG should contain certain information about default probability. Nevertheless, it is primarily the concern about the quality of assets and insolvency that triggers bank runs (Rochet and Vives, 2004; Goldstein and Pauzner, 2005). To rule out the alternative explanation that the predictive

31

power of OLG for the changes in liquidity variables are solely due to MDP, we include MDP in Regression (32). Following Bharath and Shumway (2008), the variable MDP is estimated as follows: (

.

/ ( √

where

)

),

(33) .

The regression results are reported in Table 4. As expected, the coefficients on OLG remain statistically and economically significant, indicating that OLG can indeed capture a bank’s exposure to liquidity risk after controlling for the effect of credit risk. We note that the estimated coefficients on MDP are insignificant in most of the models. This indicates that Merton’s default probability has poor explanatory power for expected changes in liquidity, further confirming that OLG can better capture liquidity risk compared with the traditional measure of MDP. The effects of other control variables are all in line with those observed in Table 3. While our previous results indicate that the liquidity gap in a bank is related to various liquidity variables, the relationship may not be linear. If the liquidity gap is so large that the bank is close to the boundary of bank runs, then even a small reduction in liquid assets can cause a significant liquidity concern, resulting in a particularly strong relationship between the liquidity gap and variations in various liquidity indicators. On the other hand, a negative or low liquidity gap means that banks have sufficient or excess liquidity. In this case, banks may choose to slightly lower their liquid assets to achieve greater profitability without causing serious liquidity concerns. This leads to a less pronounced relationship between the liquidity gap and variations in various liquidity variables. Therefore, we expect that the greater the liquidity shortage in a bank, the higher its marginal effect on the bank’s subsequent liquidity. In other words, various liquidity 32

measures are more sensitive to OLG when the value of OLG is high. To examine this, we include the squared OLG (OLG2) as an explanatory variable in our models, and re-run the regressions. The results are reported in Table 5. The estimated coefficients on the variable OLG in Table 5 are now more significant or have a larger magnitude than those in Table 4. For example, the coefficient of OLG in the regression where the quarterly growth rate of deposits is the explained variable is now -0.2754 and significant at the 1% level. Thus, after controlling for the nonlinear effect, OLG is better able to predict a bank’s future liquidity problems. We note that the estimated coefficients of OLG2 all have the same sign as those of OLG and are all statistically significant at the 10% level, confirming that in general OLG has even a larger effect on a bank’s liquidity when the bank’s liquidity shortage is great. It is noteworthy that the coefficients on MDP are now more significant than those in Table 4, and the effect of MDP is the opposite of that of OLG in all models. As noted by Imbierowicz and Rauch (2014), the interactions between liquidity risk and credit risk may vary with the liquidity level. Thus, including the squared OLG in the regression helps improve the explanatory power of MDP. These results demonstrate that an increase in credit risk in a bank can lead to an increase in the bank’s deposits and liquid assets. This is because during our sample period, one regulatory policy in the Chinese banking industry is to calculate and adjust reserve requirements based primarily on a bank’s credit risk level. Banks are required to boost total reserves when facing relatively high credit risk, which results in more deposits and liquid assets. 4.2.3 Liquidity gap and bank equity value The derivation of the optimal liquidity ratio in our model implies that a deviation from the optimal liquidity ratio will lead to a reduced equity value. In addition, since investors are more concerned about liquidity shortfall, the negative effect of inadequate liquidity on equity

33

value tends to be larger than that of excess liquidity. For this reason, both OLG and OLG2 are expected to have an explanatory power for bank equity returns. It is important to note that financial reports are released three to four months after the date when transactions take place. To avoid look-ahead biases, when estimating the optimal liquidity ratio, in this exercise, we match accounting data with market data on the audit report publishing dates. In particular, we use the closing stock prices on the date prior to the publication date of the financial reports. Note that we adopt the GJR(1,1) model to estimate the volatility of daily stock prices. Since the estimated volatilities vary substantially, we first average the daily volatilities for the month prior to the publication date of financial reports, and then annualize the average by multiplying by √

.

To examine the way in which the liquidity gap OLG is related to bank equity returns, we run the following regression:

,

(34)

where the explained variable CARi,t is the stock return for bank i cumulated from the first trading day following the report release date, adjusted for market returns. 8 For this exercise, monthly, quarterly, and six-month CARs are considered. In addition to OLG and OLG2, other explanatory variables include the liquid asset ratio (LAR), Merton’s default probability (MDP), stock volatility (Sigma), returns on assets (ROA), logarithm of assets (Size), and book-to-market ratio (BM). All these explanatory variables are lagged one period. Bank fixed effect and time fixed effect are controlled. 8

We use CSI 300 index to calculate the market returns.

34

The regression results are reported in Table 6. We find that the estimated coefficient on OLG is negative and significant at the 1% level, regardless of the time period used for calculating the CARs. This is consistent with our expectation that the liquidity gap negatively impacts banks’ stock returns. The results also show that the estimated coefficients on OLG2 are all negative and significant at the 1% level, indicating that excess liquidity also reduces expected stock returns. We note that volatility is an important determinant of OLG, and idiosyncratic volatility is negatively related to expected returns (Merton, 1987; Barberis et al., 2001; Malkiel and Xu, 2002; Ang et al., 2006; Ewens et al., 2013; Stambaugh et al., 2015). Thus, the observed negative relationship between OLG and cumulative returns can be driven by the negative effect of volatility on returns. However, in Table 6 we show that this negative relationship is not completely driven by the negative effect of volatility, as the volatility effect is controlled in our regressions. 4.3 Robustness tests 4.3.1 Alternative measures of liquidity gap To examine whether our empirical results are robust with regard to how the liquidity gap is calculated, in this section we use the direct difference between the optimal and actual liquidity ratios as an alternative measure of liquidity risk. Specifically, the liquidity gap for a bank at a particular time is given as follows: .

35

(35)

We re-run our regressions to examine whether LG has an explanatory power for various liquidity indicators in a bank and for the stock returns. Table 7 reports the results, 9 which are largely consistent with the previous findings. However, the significance is slightly reduced. This result suggests that we can directly compare a bank’s liquidity ratio with its optimal liquidity ratio to gauge its liquidity risk. 4.3.2 Probability of bank runs and liquidity Our method also enables us to calculate the probability of bank runs under the riskneutral probability measure, which is given as follows: (

(

⁄

) ( √

⁄ )

).

(36)

Bank runs are the major source of liquidity risk in our analysis, and BRP is an important determinant of a bank’s optimal liquidity ratio. Thus, BRP is expected to have explanatory power for variations in liquidity. In theory, it is more accurate to estimate the real probability of bank runs. However, estimating the real probability needs expected returns on bank assets, which in themselves are difficult to estimate. It is important to note that the optimal liquidity ratio reflects not only the run risk, but also the insolvency risk and equity maximization, and therefore it contains more information and can better describe the entire risk profile of a bank than can the bank run probability alone. To examine whether BRP can predict changes in a bank’s liquidity, we replace the variable OLG with the BRP and MDP variables in Model (32), and re-run the regressions. The results reported in Table 8 are similar to those in Table 4, corroborating that banks with a higher probability of runs are more likely to see a reduction in their deposits and liquid assets and see an 9

To examine the potential problem of multicollinearity, we calculate the variance inflation factor (VIF) of the ordinary least squares regression result of each regression. All VIFs are significantly less than 10, indicating that multicollinearity is not a concern in our analysis.

36

increase in the implicit deposit rate in subsequent periods. On the asset side, these banks also manage their assets and loan expansions in a more prudent manner. Note that the difference between BRP and MDP is that they correspond to different boundary conditions. The BRP variable is derived based on the boundary condition for bank runs, while the MDP variable is calculated based on the boundary condition for insolvency. This explains why the effects of BRP on liquidity are the opposite of those of MDP. A high BRP can lead to a reduction in liquidity. On the contrary, the results in Table 8 indicate that a high MDP can lead to an increase in deposits and liquid assets. This is because banks with high credit risk are required to raise their deposits and liquid assets by regulatory authorities, while BRP represents the liquidity risk that has not attracted much attention from authorities during our sample period. We also notice that the coefficients on MDP in Table 8 are much more significant than those in Table 4. This is primarily due to the fact that the calculation of OLG requires the identification of both liquidity and insolvency boundaries, while BRP considers only the former. 4.3.3 Credit risk and liquidity risk When deriving the optimal liquidity ratio, we not only consider the default risk that is caused by bank runs, but also take insolvency risk into account. Liquidity risk and credit risk are closely connected and jointly contribute to bank failure (Diamond and Rajan, 2005; Imbierowicz and Rauch, 2014; Acharya and Mora, 2015). In most cases, a large lack of liquidity can lead to a substantial deterioration of credit quality and even bank insolvency. Ericsson and Renault (2006) show that any increase in illiquidity should be accompanied by an increase in credit risk proxies. However, Wagner (2007) and Acharya and Naqvi (2012) point out that due to agency problems, excess liquidity may also encourage excessive risk taking by banks, leading to a higher level of

37

credit risk. Thus, we expect that OLG can explain the future credit risk level, but the explanatory power is more significant when OLG is high. We estimate the following model to examine whether the liquidity gap is able to forecast changes in credit risk.

.

(37)

For this exercise, credit risk is measured by MDP, capital adequacy, Altman Z-score, 10 and leverage, while Controls include LAR, Size, and ROA. The results are reported in Table 9. We note that banks with a high liquidity shortfall tend to have a higher Merton’s default probability and a higher degree of leverage, as well as lower capital adequacy and Altman Z-scores. The results confirm that a liquidity shortfall in a bank can subsequently aggravate credit risk. In addition, the coefficients on the squared liquidity gap are positive and significant in Columns 1 and 2, indicating that liquidity risk is likely to lead to greater credit risk when the liquidity gap is particularly high. 5. Conclusions Within the framework of a global game with imperfect information, this paper establishes a boundary condition for bank runs, and shows that there exists a unique bank run equilibrium provided that depositors’ private information is sufficiently precise. Recognizing that a bank faces the risks of both runs and insolvency, we next evaluate bank equity using Merton’s optionpricing approach. A bank’s optimal liquidity needs can be derived by maximizing its equity

10

Altman Z-Score = 1.2 × working capital / total assets + 1.4 × retained earnings/total assets+ 3.3 × earnings before interest and tax/total assets + 0.6 × market value of equity/total liabilities + 1.0 × sales/total assets.

38

value. Both the liquidity ratio obtained from our model and the liquidity coverage ratio (LCR) (BCBS, 2013) address the same issue faced by banks, but from different perspectives. The LCR represents a regulatory requirement in order to mitigate banks’ liquidity risk. In contrast, the optimal liquidity ratio in this paper is derived from the perspective of shareholders, representing a bank’s best choice that reflects the trade-offs among return, insolvency risk, and liquidity risk. This paper provides a tool to assess banks’ liquidity risk for their regulatory authorities, managers, and outside investors. The model is practically implementable, as the calculation requires only publicly available data. Using data on Chinese listed banks, we show that the gap between a bank’s optimal and actual liquidity ratios represents a robust and reliable measure of various aspects of liquidity risk. A liquidity shortfall implies serious liquidity and credit issues for banks. This is particularly pronounced when the liquidity shortfall is high. In addition, a bank’s liquidity gap also plays an important role in explaining its expected stock returns.

39

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44

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45

Appendix A. Proof of Theorem 1 This proof proceeds in two steps. In the first step, we prove that when

is sufficiently large,

the solutions to Equations (19) and (20) exist and are unique. To this end, we denote: 𝑈( )

∫

(

)

,

√

(

)-

(

.

)

/

,

(A.1)

where ( ) is the implicit function inferred from Equation (20). To prove that Equations (19) and (20) have unique solutions, we need to prove that 𝑈 ( ) is strictly increasing in

.

From Equation (20), we have: 𝛷

√

𝜓

.

/.

(A.2)

On the other hand, Equation (A.1) can be rewritten as follows: 𝑈( )

∫

)√

(

𝜋⁄

(

∫ (

(

(

)

)

)

(

√ 𝜋⁄

) (

)

) (A.3)

We now simplify the terms of AA and BB on the right-hand side of Equation (A.3). ∫

(

)√

⁄

.

𝛽

/1

)

𝛽

.√ 0 ( )

/0

. Letting

0

(

.

/1/1.

(A.4)

and substituting Equation (A.2) into Equation (A.4), and we have: 0

.√ ( ( )

0

.

(

(

)/1 )

)/1 46

(

)

(A.5)

For the term BB= ∫

(

(

)

)

√

(

.

⁄

)

/

, we note that it

resembles the expected payoff from a call-on-call option, and thus we can use Geske’s (1979) compound option pricing formula to calculate it. More specifically, consider an asset Q with initial value

at time 0. The asset value follows the following stochastic process: .

Now consider the following call-on-call option. On the first exercise date (

of this compound option is entitled to pay the first strike price (

)

, the holder ), and

receive a call option. This call option gives the holder the right to buy the underlying asset Q for the second strike price

on the second exercise date

. Apparently, the

compound option will be exercised on the first exercise date only if the value of the second call (

(

) is higher than the first strike price ( (

)

)

), or equivalently

.

According to the assumed stochastic process of distribution of

at time (

, the risk-neutral probability

is given as:

(

⁄ )

For simplicity, denote

). (

⁄ )

as

. Now, we calculate the compound

option price in the case that the holder does not pay the first strike price to receive the second call. Using risk-neutral pricing, the price of this modified call-on-call is: ∫

(

))

𝜓(

∫

(

)

∫

(

)

∫

(

)

(

) (

( (

𝑦

𝑦

) )

.

√

.

√

47

𝑚)

(𝑦 𝑚)

(𝑦 𝑚)

.

√

)

(

(

√

,

)

/

/ (

(𝑚

) )-

/

.

(A.6)

Compared the expression of (A.6) with BB, we see that if ⁄ ( above

⁄ )

two

, then

equations (

⁄

and (A.6) are identical. Solving the

simultaneously

)⁄

as well as

⁄

gives:

and

. Thus, based on Geske’s (1979) compound option pricing

formula, we have: (

)

√

(

)

√

(

)

and

as

where 𝜓(

.

)

⁄ )

/ (

√ ,

√ (

) (

⁄ )

√ .

√

In addition,

(

) is the bivariate cumulative normal distribution function with

the upper integral limits and When

as the correlation coefficient. ⁄

,

and ( ))

√ (

. Moreover, ( ))

√ (

(

(

)

).

Thus, (

√

)

𝜓(

(

(

( )

)

.

(

) / (

.

/ (

⁄ )(

)

⁄ )(

)

√ ⁄ )(

)

√

)

).

Similarly, (

√

)

𝜓(

(

(

( )

(

)

.

) / ( √

48

.

/ ( √

⁄ )(

)

).

)

Based on Equation (A.7), it follows that when (

)

*

(

(

)

)

(

)

(

(

(

𝜓(

𝜓(

)

)

)

)

(

⁄ )(

) )

√

)+

).

(A.8)

Equations (A.5) and (A.8) imply that when 𝑈( )

⁄ )(

√ (

𝜓(

,

𝜓(

[

)

,

(

)

(

)].

(A.9)

Taking the first-order derivative of Equation (A.9) gives: 𝜓(

)

𝑑𝑈( 𝑑

)

[

(

𝜓(

)]

Based on Equation (A.10), we know that if

>0, then

)

,

𝑑𝑈( 𝑑

( )-. )

(A.10)

. To prove that

we take the first-order derivative of both sides of Equation (20) with respect to

>0,

and obtain the

following: (

)

.√

Solving for

/

√

( ))/.

.√ (

(A.11)

from Equation (A.11) gives the following: √

⁄(

(

)

Equation (A.12) implies that

)

√ 𝑑𝑈( 𝑑

)

.

(A.12)

. As a result, 𝑈 ( ) is strictly increasing in

Moreover, it can be inferred from (A.9) that

𝑈( )

according to Equation (20), we also have 49

(

. Since 𝑈( )

[

. )

( in

(

)]

)

, the solution to 𝑈 ( )

(

)(

. (

Based on Equation (A.2), denote (√

other words,

which cannot hold true. It follows that

)

). If

√

𝛷

/

. In

⟹

, then

⟺

𝜓

.

⟺

, ⟺

. On the other hand, according to the monotonicity of 𝑈( ), ), if and only if 𝑈(

(equivalently, (

) )

. Thus,

) (

if and only if

. Thus, the second part of the theorem is proved. Namely, as long as inequality (

)

[

solves

. Since 𝑈( ) is continuous

exists. Consequently, the solutions to Equations (19) and

(20) exist and are unique when

𝑑

)

(

holds true, when )]

, then

and

.

In the second step, following Morris and Shin (2004), we prove that the switching strategy around

is the unique equilibrium strategy that survives the iterated deletion of dominated

strategies. ̂) as depositor i’s expected payoff from leaving deposits in conditional on

Denote (

,

when all other depositors are using the switching strategy around the point ̂ . It is given as follows: (

̂)

∫

,

(̂ )

(

)-

√

⁄

.

(

)

/

,

(A.13)

where (̂ )

𝛷 (√ . ̂

( ̂)/).

(A.14) 50

̂) satisfies the

According to Morris and Shin (2004), we now need to prove that ( following properties. ̂ ) is strictly increasing in

a) Monotonicity. (

, and is strictly decreasing in ̂.

Similar to Equation (A.12), we have: (̂ )

⁄(

√

̂

)

√

.

Thus, ̂)

𝑢( ξ̂

̂

(̂ )

[

On the other hand, if ( ̂) ̂)

∫

(̂ )

∫ If ( ̂)

,

)(

, ,

)( 𝑢(

, it is obvious that

b). Full range: For any ̂, ( When (

̂)

(

)

̂)

̂)

(

) ⁄

√ )

.

⁄

√ ( √

) ⁄

(

(

(

)

(

.

(

)

)

.

)

)

. (A.15)

/

/ )

/

/ )

/

.

. Thus, (

̂) is strictly increasing in

, as

, and ( ̂)

, there must be a bank run. Thus, ( ,

(

. .

⁄

(

)-

) ⁄

√

√

)-

(

(̂ )

(

)-

( ,

(̂ )

( (̂ )

(

⁄

√

,

(

(̂ )

(̂ )

∫ ∫

,

∫

)]

̂) is strictly decreasing in ̂.

It follows that (

𝑢(

(̂ )

(

)-

c). Continuity: u is continuous.

51

(

̂)

.

, as

holds true. When )

.

. ,

The rest of the proof is similar to the proof in the Appendix of Morris and Shin (2004). satisfies the properties of full range and continuity, we can define the sequence of real

Since

numbers {

} as the solutions to the following equations:

(

)

(

)

, , for

From the monotonicity of

. as well as 𝑈 ( )

(

)

hold true. Thus, {

that for any j, inequalities

bounded sequence. Given that the solution to 𝑈( ) we have

, it is easy to prove } is a monotonic and

is unique as proved in the first step,

.

Similarly, we can define the sequence {

𝑈

} as the solutions to the following

equations: (

𝑈

(

𝑈

) 𝑈

,

)

We can also prove that

, for 𝑈

.

𝑈

𝑈

.

Next, we prove that for any given j, depositor i’s optimal strategy must be given as follows: g

𝜔( )

,

Let 𝑈(

𝜔 ) represent depositor i’s payoff from continuing the deposits conditional on

g

𝑈

when all other depositors’ strategy is 𝜔 . When

(A.16)

, since depositor i’s payoff U is

maximized when all other depositors are not running on the bank irrespective of the signal or 𝑈(

𝜔 )

(

), we have:

52

(

𝜔 )

𝑈(

)

(

)

.

(A.17)

Equation (A.17) implies that withdrawing is the dominant strategy when

. 𝑈

Similarly, we can prove that waiting till maturity is the dominant strategy when we prove that Strategy (A.16) holds for

.

For the inductive step, suppose that Strategy (A.16) holds for all strategies that satisfy Strategy (A.16) as

. We denote the set of , depositor i’s

. Among all strategies from

payoff is maximized when all other depositors adopt the switching strategy around 𝑈(

(

)

𝜔 𝑈(

). When )

𝜔

(

)

𝑈

. Namely,

, from u’s monotonicity, we have the following: (

)

.

(A.18)

Inequality (A.18) implies that withdrawing is the dominant strategy when the dominant strategy when

. Thus,

, and waiting is

. Thus, Strategy (A.16) holds for

. Consequently,

for any j, Strategy (A.16) holds. In addition, since 𝑈 ( )

around

)

, and 𝑈( )

when

(

)

, it constitutes a Nash equilibrium if every depositor adopts the switching

when strategy around

(

𝑈

. Note that

. As a result, the switching strategy

is the unique equilibrium strategy that survives the iterated deletion of dominated

strategies. Appendix B. Proof of Proposition 1 Taking the first-order derivative of Equation (28) with respect to yields: (

√

)

(

)

√

.

(

)

(

)

(B.1)

53

Based on Equation (23), we can rewrite (

as an implicit function of

and

, or

). Following Geske (1979), we have: (

𝑚

√

( (

)

)

(

𝑚

( )√

)

)

𝛷(

)

.

(B.2)

On the other hand, (

)

(

(

)

(

)

) (

) ( (

⁄

)

√

)

√

(B.3) )

. The implicit function theorem implies that:

√

(

(

.

√

Denote (

)(

)

(𝑑(

√

)[

(𝑑(

))

.

))]

(B.4)

Plugging Equations (B.2) and (B.3) into Equation (B.1), we have: (

√ (

( where

⁄ )

( )√

)

)(

(

⁄

)

√

(

)

√

)

√

,

(B.5)

is given in Equation (B.4) When | Thus, as |

, we have (

.Thus, as

) ( (

)[

)(

) √

, (𝑑)

(𝑑(

))]

.

(B.6)

, we have the following: ( √

)

.

(B.7)

54

0.55 0.5 0.45 0.4 0.35 700 702 704 706 708 710 712 714 716 718 720 722 724 726 728 730 732 734 736 738 740

Optimal Liquidity Ratio

Figure 1. Sensitivity of the optimal liquidity ratio to various factors

0.55 0.5 0.45 0.4 0.35 0.095 0.099 0.103 0.107 0.111 0.115 0.119 0.123 0.127 0.131 0.135 0.139 0.143 0.147 0.151 0.155 0.159 0.163 0.167 0.171 0.175

Optimal Liquidity Ratio

Total Assets Value (Billion Yuan) (a) Sensitivity of the optimal liquidity ratio to changes in total assets

Volatility of Illiquid Assets 0.55 0.5 0.45 0.4 0.35 590 592 594 596 598 600 602 604 606 608 610 612 614 616 618 620 622 624 626 628 630

Optimal Liquidity Ratio

(b) Sensitivity of the optimal liquidity ratio to changes in the volatility of illiquid assets

Total Liability Book Value at time 0 (Billion Yuan)

0.55 0.5 0.45 0.4

0.35 0.13 0.132 0.134 0.136 0.138 0.14 0.142 0.144 0.146 0.148 0.15 0.152 0.154 0.156 0.158 0.16 0.162 0.164 0.166 0.168 0.17

Optimal Liquidity Ratio

(c) Sensitivity of the optimal liquidity ratio to changes in the book value of total liabilities

Haircut (d) Sensitivity of the optimal liquidity ratio to changes in the haircut level

55

Table 1. Specifications and calculations of model parameters Parameters Tr T C0 r R B0 BT E0 σE h

Specifications and calculations 0.25 years 1 year Cash and equivalents plus due from central bank plus due from banks and other financial institutions plus lendings to banks and other financial institutions plus trading securities Interest rate on demand deposits 3% Total debt at time 0 B0eRT Stock price times the number of shares outstanding Estimated daily volatility times √ 15%

56

Table 2. Summary statistics of major variables Variable Mean 0.4714 Optimal liquidity ratio 0.0000 OLG 0.2660 LAR 0.2470 Bank run probability Quarterly growth rate of 0.0302 deposits Quarterly growth rate of 0.0143 liquid assets Quarterly growth rate of 0.0494 assets 0.6892 Loan-to-deposit ratio 0.1207 CAR 0.0125 NPL 3.0667 Size 0.0068 ROA

Std. 0.1130 0.0922 0.0712 0.1399 0.0325

Min 0.0000 -0.6460 0.0899 0.0000 -0.0433

Q1 0.4483 -0.0187 0.2168 0.1345 0.0085

Median 0.4997 0.0193 0.2564 0.2573 0.0251

Q3 0.5361 0.0505 0.3104 0.3577 0.0473

Max 0.6293 0.1522 0.5000 0.5436 0.2344

0.0401

-0.1436

-0.0096

0.0099

0.0339

0.1909

0.0499

-0.1111

0.0172

0.0391

0.0748

0.4168

0.0780 0.0242

0.4129 0.0371

0.6359 0.1092

0.7016 0.1189

0.7422 0.1310

0.8676 0.3067

0.0081 1.2176 0.0031

0.0035 0.4687 0.0009

0.0081 2.1054 0.0039

0.0102 2.9761 0.0067

0.0147 3.9757 0.0092

0.0831 5.6140 0.0155

This table reports the summary statistics of various variables in the regression models under consideration. The explained variable OLG is the orthogonalized liquidity gap, while LAR represents the liquid asset ratio at a bank. CAR is the capital adequacy ratio. NPL is the ratio of nonperforming loans to total loans. Size is the logarithm of assets. ROA is returns on assets. Q1 and Q3 represent the 25th percentile and 75th percentile, respectively.

57

Table 3. Liquidity gap and variations in liquidity

OLG LAR CAR Size NPL ROA Constant Observations R-square

(1) Quarterly growth rate of deposits -0.0960** (0.0429) -0.1033** (0.0395) -0.1434* (0.0708) -0.0065 (0.0072) -0.1198 (0.1989) 2.9664* (1.4407) 0.0479 (0.0292) 536 0.4722

(2) (3) Quarterly growth rate Implicit deposit of liquid assets rate -0.1519** 0.0126*** (0.0565) (0.0027) -0.2839*** 0.0115*** (0.0657) (0.0037) -0.1869 0.0034 (0.1667) (0.0079) -0.0214*** 0.0027*** (0.0070) (0.0004) 0.0736 0.0578** (0.2254) (0.0217) 4.0519* -0.3333*** (2.0719) (0.1046) 0.0704* -0.0029 (0.0353) (0.0020) 536 536 0.2940 0.7261

(4) Quarterly growth rate of assets -0.2464*** (0.0765) -0.3532*** (0.0890) -0.2877 (0.2021) -0.0316*** (0.0080) 0.1814 (0.3276) 4.9844*** (1.3085) 0.1667*** (0.0491) 536 0.3058

(5) Loan-to-deposit ratio -0.4053*** (0.1123) -0.1769* (0.0871) -0.0787 (0.2512) -0.1807*** (0.0159) -1.1057 (0.6739) 1.1707 (3.0405) 1.2423*** (0.0456) 536 0.5012

This table presents the results of regressions with various liquidity variables as the explained variable. The explained variable OLG is the orthogonalized liquidity gap, while LAR represents the liquid asset ratio at bank. CAR is the capital adequacy ratio. Size is the logarithm of assets. NPL is the ratio of nonperforming loans to total loans. ROA is returns on assets. The t-statistics are in parentheses. ***, **, and * denote 1%, 5%, and 10% significance, respectively.

58

Table 4. Liquidity shortfall and Merton’s default probability

OLG MDP LAR CAR Size NPL ROA Constant Observations R-square

(1) Quarterly growth rate of deposits -0.1086** (0.0430) 0.1061 (0.0736) -0.1110*** (0.0364) -0.1365* (0.0709) -0.0062 (0.0081) -0.2640 (0.2065) 3.1132** (1.4188) 0.1007** (0.0384) 529 0.4759

(2) Quarterly growth rate of liquid assets -0.1782*** (0.0572) 0.1804* (0.0895) -0.2968*** (0.0646) -0.1780 (0.1649) -0.0196** (0.0088) -0.2325 (0.2195) 4.3622** (2.0041) 0.1658*** (0.0430) 529 0.2911

(3) Implicit deposit rate 0.0124*** (0.0029) 0.0007 (0.0041) 0.0115*** (0.0037) 0.0030 (0.0079) 0.0024*** (0.0004) 0.0682*** (0.0222) -0.3402*** (0.0990) -0.0051** (0.0024) 529 0.7311

(4) Quarterly growth rate of assets -0.2329*** (0.0725) 0.0553 (0.1334) -0.3460*** (0.0971) -0.2829 (0.2046) -0.0318*** (0.0092) -0.0407 (0.3530) 5.3134*** (1.2926) 0.2886*** (0.0572) 529 0.3050

(5) Loan-to-deposit ratio -0.3817*** (0.1268) -0.1127 (0.1075) -0.1649 (0.0989) -0.0926 (0.2522) -0.1819*** (0.0169) -1.0043 (0.8260) 1.4891 (3.0904) 1.3149*** (0.0454) 529 0.4915

This table presents the results of regressions with various liquidity variables as the explained variable. The explained variable OLG stands for the orthogonalized liquidity gap, MDP stands for the Merton’s default probability, while LAR stands for the liquid asset ratio in a bank. CAR is the capital adequacy ratio. Size is the logarithm of assets. NPL is the ratio of nonperforming loans to total loans. ROA is returns on assets. The t-statistics are in parentheses. ***, **, and * denote 1%, 5%, and 10% significance, respectively.

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Table 5. Nonlinear impact of liquidity shortfall on liquidity risk

OLG OLG2 MDP LAR CAR Size NPL ROA Constant Observations R-square

(1) Quarterly growth rate of deposits -0.2754*** (0.0804) -0.5770** (0.2628) 0.2461*** (0.0712) -0.2190*** (0.0584) -0.1175 (0.0832) -0.0126 (0.0084) -0.3033 (0.1784) 2.4407* (1.2926) 0.1266*** (0.0414) 529 0.4857

(2) Quarterly growth rate of liquid assets -0.4752*** (0.1383) -1.0271*** (0.3455) 0.4294*** (0.1157) -0.4890*** (0.1069) -0.1442 (0.1674) -0.0308*** (0.0075) -0.3024* (0.1681) 3.1652 (2.1878) 0.2119*** (0.0471) 529 0.3127

(3) Implicit deposit rate 0.0206*** (0.0034) 0.0282*** (0.0082) -0.0061 (0.0036) 0.0168*** (0.0039) 0.0020 (0.0072) 0.0027*** (0.0005) 0.0701*** (0.0215) -0.3073*** (0.0931) -0.0063** (0.0025) 529 0.7362

(4) Quarterly growth rate of assets -0.4594*** (0.1547) -0.7834* (0.3862) 0.2453* (0.1309) -0.4926*** (0.1312) -0.2571 (0.2111) -0.0404*** (0.0088) -0.0940 (0.3144) 4.4004** (1.5837) 0.3238*** (0.0650) 529 0.3133

(5) Loan-to-deposit ratio -0.6464*** (0.1883) -0.9155* (0.4468) 0.1093 (0.1325) -0.3362** (0.1295) -0.0626 (0.2546) -0.1919*** (0.0191) -1.0667 (0.8048) 0.4221 (2.7925) 1.3560*** (0.0533) 529 0.5023

This table presents the results of regressions with various liquidity measures as the explained variable. The explained variable OLG stands for the orthogonalized liquidity gap, MDP stands for the Merton’s default probability, while LAR stands for the liquid asset ratio in a bank. CAR is the capital adequacy ratio. Size is the logarithm of assets. NPL is the ratio of nonperforming loans to total loans. ROA is returns on assets. The t-statistics are in parentheses. ***, **, and * denote 1%, 5%, and 10% significance, respectively.

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Table 6. Liquidity shortfall and equity returns

OLG

(1) One-month CARs -1.3700***

(2) One-quarter CARs -2.1258***

(3) Half-year CARs -1.5008***

(0.3132)

(0.3000)

(0.3010)

***

OLG2 LAR

-3.1319

-4.4691

-3.0075***

(0.7059)

(0.6895)

(0.9406)

-0.6676***

-1.1467***

-0.8308***

(0.1662)

(0.1818)

(0.2815)

2.4880 MDP

***

**

4.5274

(0.8742)

(0.9899)

**

Sigma ROA

***

BM

(0.9406)

-1.0286

-1.1844***

(0.1940)

(0.2449)

(0.2019)

-4.6363

-7.7291

5.1090

(2.9134)

(6.3108)

(9.7734)

-0.1413

-0.1433

-0.1677**

(0.0229)

(0.0415)

(0.0616)

-0.0297

0.0218

0.1164

(0.0430)

(0.0570)

0.4348 Constant

4.2221***

-0.5350

***

Size

***

***

***

0.6285

***

(0.0822) 0.7474***

(0.0932)

(0.1952)

(0.2535)

Observations

586

586

570

R-square

0.5511

0.5071

0.5158

This table presents the results of regressions with cumulative stock returns as the explained variable. The explained variable OLG stands for the orthogonalized liquidity gap, MDP stands for the Merton’s default probability, while LAR stands for the liquid asset ratio in a bank. ROA is returns on assets. Size is the logarithm of assets. BM stands for the book-to-market value ratio. The t-statistics are in parentheses. ***, **, and * denote 1%, 5%, and 10% significance, respectively.

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Table 7. Liquidity gap, and liquidity and equity return forecasts Panel A：Liquidity (1) Quarterly growth rate of deposits -0.0651 LG (0.0458) -0.3856** 2 LG (0.1529) 0.2090*** MDP (0.0660) -0.2045*** LAR (0.0518) -0.1069 CAR (0.0795) -0.0106 Size (0.0086) -0.3244 NPL (0.1995) 2.8663** ROA (1.2831) 0.1516*** Constant (0.0495) Observations 529 R-square 0.4848 Panel B：Equity returns

LG LG2 LAR MDP Sigma ROA Size BM Constant Observations R-square

(2) Quarterly growth rate of liquid assets -0.1532** (0.0597) -0.5403*** (0.1605) 0.3406*** (0.0961) -0.4534*** (0.0942) -0.1714 (0.1681) -0.0285*** (0.0074) -0.2711 (0.1933) 3.6349* (2.0529) 0.2634*** (0.0569) 529 0.3056

(3) Implicit deposit rate 0.0103*** (0.0023) 0.0206*** (0.0041) -0.0044 (0.0038) 0.0171*** (0.0039) 0.0028 (0.0079) 0.0027*** (0.0005) 0.0693*** (0.0219) -0.3133*** (0.0961) -0.0095*** (0.0027) 529 0.7353

(4) Quarterly growth rate of assets -0.1940** (0.0680) -0.5299** (0.1868) 0.2012 (0.1292) -0.4941*** (0.1276) -0.2760 (0.2084) -0.0395*** (0.0088) -0.0766 (0.3229) 4.6263*** (1.3964) 0.3908*** (0.0806) 529 0.3129

(5) Loan-to-deposit ratio -0.3321*** (0.1038) -0.5702* (0.2684) 0.0310 (0.1246) -0.3280** (0.1208) -0.1044 (0.2642) -0.1898*** (0.0183) -1.0124 (0.8198) 0.5819 (2.9703) 1.4529*** (0.0599) 529 0.4995

(1)

(2)

(3)

One-month CARs

One-quarter CARs

Half-year CARs

-0.2307* (0.1295) -2.3839*** (0.4906) -0.6966*** (0.1842) 2.3767** (0.8317) -0.5176** (0.1845) -3.3282 (2.9069) -0.1345*** (0.0259) -0.0223 (0.0453) 0.5744*** (0.1277) 586 0.5573

-0.5768*** (0.1797) -2.9478*** (0.5129) -1.1261*** (0.1881) 4.2273*** (0.9963) -0.9781*** (0.2458) -6.3602 (6.2773) -0.1367*** (0.0439) 0.0280 (0.0597) 0.8591*** (0.2307) 586 0.4948

-0.4032* (0.2184) -2.2647*** (0.4620) -0.8657*** (0.2745) 4.0922*** (0.9302) -1.1634*** (0.2001) 6.1558 (9.7318) -0.1625** (0.0638) 0.1214 (0.0853) 0.9248*** (0.2839) 570 0.5154

Panel A of this table presents the results of regressions with various liquidity measures as the explained variable, while Panel B reports the results of regressions with cumulative stock returns as the explained variable. The explained variable LG is the difference between the optimal and actual liquidity ratios in a bank, MDP stands for the Merton’s default probability, while LAR stands for the liquid asset ratio. CAR is the capital adequacy ratio. NPL is the ratio of nonperforming loans to total loans, Size is the logarithm of assets, and BM stands for the book-to-market value ratio. ROA is returns on assets. The t-statistics are in parentheses. ***, **, and * denote 1%, 5%, and 10% significance, respectively.

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Table 8. Bank run probability and liquidity

BRP MDP LAR CAR Size NPL ROA Constant Observations R-square

(1) Quarterly growth rate of deposits -0.0989** (0.0418) 0.2235** (0.0985) -0.1191** (0.0445) -0.0700 (0.0727) -0.0097 (0.0077) -0.3792* (0.1911) 2.9487** (1.3272) 0.1130*** (0.0329) 528 0.4789

(2) Quarterly growth rate of liquid assets -0.1953*** (0.0363) 0.4281*** (0.1196) -0.3330*** (0.0562) -0.0769 (0.1439) -0.0293** (0.0117) -0.4033* (0.1969) 3.5967 (2.0978) 0.2020*** (0.0462) 528 0.3015

(3) Implicit deposit rate 0.0083** (0.0029) -0.0069 (0.0060) 0.0101*** (0.0032) -0.0042 (0.0076) 0.0025*** (0.0005) 0.0783*** (0.0220) -0.3385** (0.1223) -0.0051* (0.0025) 528 0.7268

(4) Quarterly growth rate of assets -0.1842*** (0.0487) 0.2543* (0.1350) -0.3423*** (0.0787) -0.1429 (0.1726) -0.0361*** (0.0078) -0.2657 (0.3148) 5.1580** (2.0162) 0.3024*** (0.0488) 528 0.3064

(5) Loan-to-deposit ratio -0.4869*** (0.1120) 0.5303** (0.1959) -0.2894** (0.1189) 0.1038 (0.2566) -0.2112*** (0.0238) -1.3190** (0.5898) -1.2598 (2.9646) 1.4269*** (0.0850) 528 0.5425

This table presents the results of regressions with various liquidity measures as the explained variable. The explained variable BRP is the bank run probability, MDP stands for the Merton’s default probability, while LAR stands for the liquid asset ratio in a bank. CAR is the capital adequacy ratio. Size is the logarithm of assets. NPL is the ratio of nonperforming loans to total loans. ROA is returns on assets. The t-statistics are in parentheses. ***, **, and * denote 1%, 5%, and 10% significance, respectively.

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Table 9. Credit risk and liquidity risk

OLG OLG2 LAR Size ROA Constant Observations R-square

(1) Merton default probabilityt+1 0.2534*** (0.0352) 0.4172*** (0.1076) 0.1460*** (0.0336) -0.0062 (0.0074) -1.7204* (0.8597) 0.0707*** (0.0213) 577 0.6333

(2)

(3) Capital adequacyt+1

Leveraget+1 12.6399*** (2.8346) 35.0911*** (5.7636) 2.2022 (2.9836) -1.5514** (0.6837) -429.1511*** (95.5072) 11.6256*** (2.0697) 577 0.9064

-0.1899*** (0.0609) 0.2348 (0.2375) -0.0842 (0.0547) -0.0180 (0.0127) 3.3289 (2.1865) 0.1398*** (0.0403) 576 0.4096

(4) Altman Z-scoret+1 -0.3496*** (0.0455) 0.2483 (0.1946) -0.1762*** (0.0230) -0.0241*** (0.0057) 5.3738** (1.8949) 0.1877*** (0.0230) 577 0.8603

This table presents the results of regressions with various credit risk measures as the explained variable. The explained variable OLG stands for the orthogonalized liquidity gap, while LAR stands for the liquid asset ratio. Size is the logarithm of assets. ROA is returns on assets. The t-statistics are in parentheses. ***, **, and * denote 1%, 5%, and 10% significance, respectively.

64