Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

The Safer, the Riskier:
A Model of Financial Instability and Bank Leverage
Ryo Kato and Takayuki Tsuruga

Discussion Paper No. E-10-014

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

February, 2011 (revised March, 2015)

The Safer, the Riskier: A Model of Financial Instability
and Bank Leverage∗
Ryo Kato† and Takayuki Tsuruga‡
March 16, 2015

Abstract
We examine the role of bank leverage to explain why the 2007-08 financial crisis unfolded
at a time when the economy appears to be less fragile to crisis risks. To this end, we extend
the model introduced by Diamond and Rajan (2012) to a variant where the probability offinancial crises varies endogenously. In our model, aggregate liquidity shock plays a key rolein precipitating a crisis because high liquidity demand in a highly leveraged banking system
is likely to expose the economy to greater crisis risks. We consider an example of a “safe”
environment where liquidity demand tends to be low on average. Using numerical analysis, weshow that the “safer” environment could incentivize banks to raise their leverage, resulting in abanking system that is more vulnerable to liquidity shocks.
JEL Classification: E3, G01, G21
Keywords: Bank run; Financial crisis; Maturity mismatch∗
We thank two anonymous referees, Robert De Young, Koichi Futagami, Katsuya Ue, Dan Sasaki, Kenta Toyofuku,
Noriyuki Yanagawa and seminar and conference participants at the Bank of Japan, Kobe University, Kyoto University,
University of Tokyo, the Annual Meeting of the Japan Society of Monetary Economics, the Annual Meeting of theJapanese Economic Association, and the Third International Symposium in Computational Economics and Financefor helpful comments. We are also indebted to Daisuke Oyama and other members of the Banking Theory Study
Group for their helpful discussion. Takayuki Tsuruga gratefully acknowledges the financial supports of the Grantin-aid for Scientific Research, Japan Securities Scholarship Foundation, and Nomura Foundation for Social Science.
Views expressed in this paper are those of the authors and do not necessarily reflect the official views of the Bank ofJapan.
†
Bank of Japan; e-mail: ryou.katou@boj.or.jp‡
Graduate School of Economics, Kyoto University and the Centre for Applied Macroeconomic Analysis ; e-mail:tsuruga@econ.kyoto-u.ac.jp1

1

Introduction

A consensus on the 2007-08 crisis is that, with hindsight, the banking sector had been exposed tohigh risks of insolvency before the crisis took place. Conversely, there was a widely-held perception
in the run-up to the crisis that the banks were placed in a “safe” environment.1
Relatedly, a notable fact is that funding liquidity for banks, was quite affluent before the crisis.
Gorton and Metrick (2012 a,b) point out that the market for repurchase agreements (repo), which
provides short-term funding for banks and other financial institutions, rapidly grew in the periods
prior to the 2007-08 crisis. In the meantime, prices of funding liquidity remained at a quite lowlevel. For example, the TED spread, an indicator of funding liquidity, declined to historically low
levels in the early 2000s and leveled off until 2006.2 Figure 1 plots the TED spread, together withnet repo funding to banks and broker-dealers from 1998:Q1 to 2008:Q4. The combination of thelow prices of funding liquidity and expanding short-term funding markets implies that demand forfunding liquidity by banks was low relative to the supply. In the presence of such abundant liquidity
or relatively low demands for liquidity, banks can feel safe, standing a distance away from risks offinancial crises. A key question is why the financial crisis unfolded at a time when the banking
sector was considered surrounded by such a “safe” environment.
This paper aims to explain how a “safer” environment can increase the probability of bank runs.
A key to understand this “the safer, the riskier” case is the banks’ endogenous risk-taking. We
show that the banks’ risk-taking with higher leverage offsets, or even dominates, the exogenously
improved environment in terms of the bank run probability. In particular, a “safe” environment,
represented by a low demand for funding liquidity, incentivizes banks to raise their leverage. The
increased leverage can result in a higher risk of bank runs.
Our model is based on the framework of bank runs developed by Diamond and Rajan (2001,2012). In Diamond and Rajan (2001), banks’ commitment to repaying demandable deposits works
as a disciplinary device for banks to raise funds. While the maturity transformation promotes
1

For example, Reinhart and Rogoff (2009, p.208) noted that the 2007-08 financial crisis came as a surprise fromthe viewpoint of investors because “the financial meltdown of the late 2000s was a bolt from the blue, a ‘six-sigma’
event.”
2
Other similar indicators of funding liquidity, such as LIBOR-OIS spread and repo rates remained at low levels
as well.

2

financial intermediation, such funding via demandable liabilities exposes banks to some risks ofbank runs. Using a simplified framework of Diamond and Rajan (2001), Diamond and Rajan (2012,
hereafter DR) demonstrated how and why low interest rate policies deployed by a central bank raisebanks’ leverage and endangers the financial stability. In this paper, we use the DR’s framework fora different purpose, that is, to answer the above-mentioned key question. To assess the effect of“safe” environment for banks, two distributions for liquidity (preference) shocks are compared: the“risky” and “safe” distributions in the sense that the depositors’ demands for liquidity are high or
low, on average. Our finding that “the safer, the riskier” implies that the bank run probability canincrease even in the absence of any policy intervention, which is in a sharp contrast to the main
focus of DR.
While we model banks funded by demand deposits following the classic literature of banking, thebanks in our paper can refer to broader financial intermediaries that raise funds via short-term debts(e.g., a repo) and invest them in longer-term assets, by maturity transformation. (See Diamondand Rajan, 2001.) The lower price for funding liquidity in the pre-crisis periods can be translated
into a model of classic banking where fewer depositors (i.e., suppliers of short-term funding forbanks) come to banks to withdraw their deposit. In the same spirit, Gorton and Metrick (2012b)
point out that the “run on repo” which happened in the 2007-08 financial crisis was a systemicbank run. In line with this view, we focus on the aggregate liquidity shock that affects depositors
as the liquidity suppliers for banks. As a result, we can assume that “bank runs” and “financial
crises” are interchangeable in our very stylized model.
In comparison with the early studies discussing the mechanism of financial crises, this paper
relies neither on contagion nor externalities.3 In a similar spirit of DR, there are a number ofstudies arguing that the growing expectations of bank bailouts or the low interest rate policy by
the central banks might be responsible for the crisis.4 We do not claim that all these factors did
3

The role of contagion for understanding crises is emphasized by Allen and Gale (2000), Dasgupta (2004), Allen
and Carletti (2006), and Acharya and Yorulmazer (2008), Allen, Babus, and Carletti (2012), and Castiglionesi,
Feriozzi, and Lorenzoni (2012). Lorenzoni (2008) and Jeanne and Korinek (2010) develop models in which financiallyconstrained borrowers take on more risks due to pecuniary externalities. Stein (2012) and Gersbach and Rochet(2012) discuss pecuniary externalities in models with banks.
4
Examples include Farhi and Tirole (2012), Jiménez, Ongena, Peydró and Saurina (2014), and Maddaroni and
Peydró (2011). The low interest rate policy is closely related to the risk-taking channel of monetary policy. Angeloni,
Faia, and Lo Duca (2014) introduce demandable deposits as a disciplinary device in Diamond and Rajan (2001)3

not play critical roles for the 2007-08 crisis. Rather, this paper provides an example where thecrisis probability rises even without these factors such as policy interventions and externalities.
The rest of the paper is organized as follows. Section 2 introduces the model. In Section 3, we
discuss the main numerical results with some robustness checks. Section 4 concludes.

2

The Model

2.1

Setup

We consider a variation of the economy developed by DR in which the bankers are intermediating
the funds from households to entrepreneurs via maturity transformation. Most of the assumptions
are maintained in line with the original DR model except for the households’ preference and theflow of their income. In DR, households’ utility function is given by U (C1 , C2 ) = log (C1 ) +log (C2 ), where Ct is consumption at date t. Also, DR assume that the random shock arises fromthe uncertainty over expectations on future income and consider finite discrete aggregate states.
By contrast, we eliminate uncertainty with respect to households’ income while incorporating amore straightforward random shock regarding liquidity preference into our model. Specifically,
households’ utility function is given by U (C1 , C2 ) = θ log (C1 ) + (1 − θ) log (C2 ), where θ is acontinuous random variable with a support θ ∈ (0, 1). Here, θ can be interpreted as a “liquidity
shock,” which indicates how much liquidity is needed at date-1 consumption. Seemingly, the utility
function takes the same form as the expected utility in Allen and Gale (1998). However, weemphasize that there is neither an early consumer nor a late consumer in this economy. Ourmodel includes only single type of households, who are subject to perfectly correlated liquidity
shocks across households. In our model, θ is the only source of the aggregate uncertainty, which
precipitates a crisis in this economy. The utility function provides the advantage that we can focus
on aggregate uncertainty and an endogenously changing crisis probability in a straightforwardmanner.
Following DR, we assume three types of agents: (i) households, (ii) entrepreneurs, and (iii)
into their dynamic stochastic general equilibrium model and study the monetary transmission in the model withendogenous probability of bank runs.

4

bankers. As assumed by DR, while the households are risk averse, the entrepreneurs and bankers
are risk-neutral.
The economy lasts for three dates (t = 0, 1, 2). At date 0, households are born with a unitof a good. By assumption, no household consumes at date 0. Rather, they deposit all the date-0
endowments into banks. Bankers compete to offer the most attractive promised deposit payment D
to households (per unit of endowment deposited). Then, bankers lend the households’ endowment
to entrepreneurs. Each entrepreneur invests a unit of the good to launch a long-term project at
date 0. These transactions are settled before the realization of the liquidity shock.
At date 1, the liquidity shock θ is realized. Households determine the date-1 withdrawal w1 tosmooth out their consumption, given the realized θ (and their fixed endowment at dates 1 and 2).
Turning to entrepreneur’s project, each of the projects yields a random output Ỹ2 at its completionat date 2. Outcomes of projects follow a uniform distribution with a support [0, Ȳ2 ]. In thismodel, there is no aggregate uncertainty in Ỹ2 , and thus the financial stability entirely relies on theaggregate uncertainty in θ. If a project is prematurely liquidated, the project produces X1 (< 1)at date 1. If each banker needs to liquidate all projects to meet a high liquidity demand (i.e., fullwithdrawal of the deposit), a crisis takes place at date 1. Otherwise, at date 2, households consume
the rest of deposits together with the date-2 endowment.
In the following sections, we first describe the agent’s decisions after the realization of θ andthen the bankers’ choice of D before the realization of θ.

2.2

Demand for liquidity

A household chooses its withdrawal w1 , given deposit face value D, the one-period gross interest
rate r12 (from date 1 to 2), and the liquidity shock θ. The interest rate r12 represents the price forliquidity, which equates the demand (withdrawal) with the supply (liquidated projects) of liquidity.
Throughout this paper, we focus on an economy in which storage technology is not available tohouseholds and bankers.55

In the appendix, which is available upon request, we present the model with storage and show that our main
results are essentially the same as in the model without storage.

5Given that a crisis is not taking place, the households’ maximization problem is given bymax θ log C1 + (1 − θ) log C2 ,w1s.t. C1 = e1 + w1

(1)
(2)

C2 = e2 + r12 (D − w1 ) ,(3)

where et is the household’s endowment at date t. Here, θ determines the need for liquidity foreach date. If θ is low, households’ deposits are likely to be fully repaid by bankers over the two
periods, which means that the households smooth out their consumption. If θ exceeds a thresholdvalue, however, the households’ deposits are not fully repaid. Then, a crisis (i.e., a run on theentire banking system) takes place and each household receives X1 at date 1 and nothing at date2 from the bankers. Thus, the households fail to smooth out their consumption and end up with
C1 = e1 + X1 and C2 = e2 . Note that, based on the “business cycle view” as argued by Gorton(1988) and Allen and Gale (1998, 2007), we implicitly assume an information structure in whichcrises are precipitated as a Nash equilibrium when bankers are revealed to be insolvent.6
When the households can smooth out their consumption, the intertemporal first-order condition
for consumption [θ/ (1 − θ)] (C1 /C2 )−1 = r12 is satisfied. Due to the budget constraints (2) and(3), the withdrawal can be written asµw1 = θ

¶e2+ D − (1 − θ) e1 .
r12

(4)

It is convenient to define the households’ lifetime income in normal times by m:

m = e1 + D +

e2
.
r12

(5)

The log-utility implies that consumption in normal times is proportional to m, namely, C1 = θm
6Other types of Nash equilibriums, including a coordinated bank holiday, could exist depending on the information
structure. A quick fix to exclude such equilibriums is to assume a belief that, while all households are in facthomogeneous, an infinitesimally small number of households may have different preferences from others. Allen andGale (1998, 2007) discuss this in more detail.

6and C2 = (1 − θ) r12 m.

2.3

Banks assets and supply of liquidity

Entrepreneurs and bankers in our model replicate those in DR. Each banker is a relationship lenderthat has obtained special knowledge of the entrepreneurs’ business, and this knowledge ensures
the banker’s collection skill to acquire a fraction γ Ỹ2 (< Ỹ2 ) of the output from the entrepreneurs.
The collection skill is assumed to be not transferable to other lenders. Following DR, we denote
the realization of Ỹ2 by Y2 and assume that Y2 becomes known at date 1. As in DR, we assume
that each banker lends to enough entrepreneurs. As a result, all the symmetric bankers share an
identical portfolio. Let the bankers’ assets be A (r12 ). Then, A (r12 ) can be expressed as
1
A (r12 ) =Ȳ2

Z Y2 (r12 )0

1
X1 dY2 +
Ȳ2

Z Ȳ2

γY2
dY2 ,
Y2 (r12 ) r12(6)

where the first term of the equation indicates the supply of liquidity (i.e., the values of liquidated
projects), while the second term represents completed projects evaluated at t = 1. In (6), Y2 (r12 )denotes the cut-off level of return on projects satisfying Y2 (r12 ) = r12 X1 /γ. The cut-off level ofreturn on projects can be understood from the bankers’ liquidation decision: if they liquidate aproject to meet households’ liquidity demand, they would obtain X1 at date 1. Conversely, if they
let the project continue, the present value of the continued project is γY2 /r12 . Taking r12 as given,bankers’ liquidation decision is made by comparing X1 with γY2 /r12 . This comparison determines
the cut-off level of return on projects. Furthermore, it can be easily shown that A0 (r12 ) < 0 for the
economically meaningful range of r12 .7
Bankers become insolvent if the solvency condition D ≤ A (r12 ) is violated. In this case, crises
are precipitated: the bankers liquidate all of the entrepreneurs’ projects, repay X1 to households,∗ , which satisfies the solvency
and lose all their assets. We then define the threshold interest rate r12

condition with equality:∗
D = A (r12),

7

The proof is available upon request in the separate appendix.

7(7)

∗ strictly decreases with D since A0 (·) ≤ 0.8 In other words, a higher level of D requires
where r12
∗ , which can be written as r ∗ = A−1 (D) ≡ r ∗ (D),a lower level of the threshold interest rate r1212
12

for the bankers to be solvent. Note that there is a clear distinction between the threshold interest∗ and the price of liquidity, r . While the former is solely determined by D in (7) at daterate, r12
12

0, the latter reflects the supply and demand in the liquidity market at date 1.

2.4

Market equilibrium

In normal times, the following liquidity market clearing condition holds:µθ¶(X1 )2 r12e2+ D − (1 − θ) e1 =
,
r12
γ Ȳ2

(8)

which has two roots for r12 but gives only one positive r12 . The left-hand side of the equation
points to liquidity demand (4), while the right-hand side indicates supply from project liquidation
shown in (6).
∗ is solely determined by D in (7), the threshold value of
Whereas the threshold interest rate r12∗ together with (8). Let θ ∗ be the threshold value ofθ that precipitates crises is determined by r12∗ = r ∗ (D), we haveθ that precipitates crises if and only if θ > θ∗ . Evaluating r12 in (8) at r1212¢¡∗ / γ Ȳ
(X1 )2 r12
2 + e1.
θ =∗e1 + D + e2 /r12∗

(9)

∗ = r ∗ (D) is strictly decreasing in D, θ ∗ is also strictly decreasing inwhich indicates that, since r1212

D. When we emphasize this relationship between θ∗ and D, we express θ∗ as θ∗ (D) and express its
first derivative as θ∗0 (D). We also note that, because a larger liquidity shock increases households’
withdrawal, a smaller θ∗ , by definition, points to a higher crisis probability. Denoting π as theprobability of the financial crises, π can be expressed as π (θ∗ ) = 1 − F (θ∗ ), where F (θ) is the
cumulative distribution function of θ.

8∗∗0Note that r12(D) ≡ A−1 (D) and, by the inverse function theorem, r12(D) = 1/A0 < 0.

8

2.5

Bankers’ Choice of Leverage

Diamond and Rajan (2001) argued that demand deposits D serve as a commitment device forbankers. Demand deposits, like other short-term funding vehicles, can compensate for the lack oftransferability of the bankers’ collection skill to others (e.g., households) and thus promote liquidity
creation. In line with this argument, the bankers in our model need to determine the face valueof deposits before observing θ. As a result of competition, the bankers make a competitive offer
of deposits for households. The competitive offer maximizes the household welfare taking the
distribution of θ as given.9 Here, given θ, the choice of D has a one-to-one relationship with thebank leverage. The bank leverage in our model can be defined as D/ [A (r12 ) − D] and is determined
once D is chosen. Therefore, in our model, the optimal choice of D and the optimal choice of bankleverage can be treated interchangeably.
Formally, the bankers choose D to maximize
Z θ∗ (D)0{θ log (θm) + (1 − θ) log [(1 − θ) r12 m]} dF (θ)
Z 1[θ log (e1 + X1 ) + (1 − θ) log (e2 )] dF (θ) ,+θ ∗ (D)(10)∗ = r ∗ (D) from (7). Here the first term of (10) corresponds tosubject to (5), (8), (9), and r1212

the utility from consumption under no crisis, while the second term points to the utility fromconsumption under a crisis. The integral is taken over θ ∈ (0, θ∗ ] for the first term, because anyθ that is lower than or equal to the threshold value does not precipitate crises. In contrast, thesecond term indicates that bankers recognize that consumption smoothing is impossible for θ > θ∗ .
The first-order condition for D is given by·¸¾½µ ∗ ∗ ¶
∗ m∗
(1 − θ∗ ) r12
θ m∗∗+ (1 − θ ) log
π 0 (θ∗ ) θ∗0 (D)θ log
e1 + X1e2¶¸
Z θ∗ · µ1e2 ∂r121 − θ ∂r12=
1− 2+
dF (θ) ,m
r12 ∂Dr12 ∂D0

9(11)

In the model, the bankers in fact are maximizing their own profits by household welfare maximization. See Allenand Gale (1998) and DR for more details on the bankers’ optimization problem.

9∗ . By the definition of π (θ) , π 0 (θ ∗ ) is equal to −f (θ ∗ ), where f (θ ∗ )where m∗ = e1 + D + e2 /r12

denotes the probability density function evaluated at θ = θ∗ . The partial derivative ∂r12 /∂D canbe implicitly defined by the liquidity market clearing condition (8).
In choosing the optimal D, bankers strike the right balance between the marginal benefit andcost of increasing D on behalf of the households. The right-hand side of (11) can be interpreted as
the marginal benefit of increasing D through changes in households’ lifetime income and interest
rate. Intuitively, a higher D allows households to receive higher income from their deposits and toenjoy more consumption at both dates. Hence, as far as θ ≤ θ∗ , households obtain higher returns
from increasing D.
The left-hand side of (11) represents the marginal cost of increasing D. The terms in the curly
brackets indicate the loss of the utility due to a crisis. The term outside the curly brackets assesses
the marginal changes in the crisis probability in response to an increase in D. Hence, putting them
all together, we can interpret the left-hand side as the marginal cost of increasing D.

3

Simulating the Model

3.1

Calibration

To evaluate impacts of changes in environment surrounding the banking sector, we numericallysolve the model. For simulating the model, we need to determine five parameters: the maximum
productivity of entrepreneurs’ projects (Ȳ2 ), the value of liquidated project (X1 ), the bankers’
collection skill (γ), and households’ endowment at dates 1 and 2 (e1 , e2 ). For the distribution of θ,
we assume that θ is generated from the beta distribution.
In terms of assigning values for parameters, we borrow parameter values from DR wheneverfeasible. We take Ȳ2 = 3.5, X1 = 0.95 and γ = 0.90 from DR. In parameterizing households’
endowment, we do not borrow parameter values from DR since our model differs from DR in termsof the assumption on the aggregate uncertainty. For this reason, we calibrate e1 and e2 to ensurethat two targets are matched with the U.S. data. The first target is the bank capital ratio of 10.3

10

percent. The targeted bank capital ratio is taken from the U.S. data in 2007.10 The second target
is the crisis probability of 4.76 percent. We take the target crisis probability from Laeven andValencia (2013) who report that there were two banking crises in the U.S. between 1970 and 2011.
Given their finding, the frequency of banking crises in the U.S. is 4.76 percent per year (i.e., 2/42
= 0.0476). We employ this value as a proxy for the crisis probability in the U.S. banking sector.11
Moment matching of the bank capital ratio and the crisis probability results in e1 = 0.305 ande2 = 0.405. The calibrated values of et give rise to a plausible pattern in the economy before thecrisis. Namely, households’ income is expected to grow in the run-up to the crisis, which was also
observed in the U.S. economy.
The remaining parameters required to solve the model are those for the distribution of θ. We
set benchmark parameters for the beta distribution to ensure that µθ = 0.5 and σ θ = 0.05, whereµθ and σ θ denote the mean and the standard deviation of θ, respectively.12 The liquidity preference
shocks are, by nature, unobservable and there is no solid empirical counterpart on these parameters.
Hence, we perform extensive robustness analysis for these parameters.
We numerically compute the equilibrium by solving the system of nonlinear equations. The∗ , θ ∗ , m, and m∗ . The
equations in the system are (8) and (11) together with the definitions of r12

first column of Table 1 shows the computation results under the benchmark calibration. Bankers
set the level of the deposit face value D at 1.052 and the resulting θ∗ is 0.584.
Figure 2 plots the households’ expected utility over a variety of deposit face values D. The
figure also articulates the sub-components of the utility. The smooth bell shape of the utility canbe understood as the weighted average of the two sub-components, (i) the expected utility in theabsence of a run E (U |no run) and (ii) the expected utility under a run E (U |run). In the figure,the probability of a crisis is represented by the ratio of the distance along the vertical axis between
the solid and the upper dashed lines to that between the upper and lower dashed lines.

10

The data source is from the World Development Indicators published by the World Bank.
We also used the frequency of banking crises in North America reported by Reinhart and Rogoff (2013) for
robustness analysis. While our calibrated values of et differ slightly from the benchmark calibration, the results are
qualitatively unaltered.
12
In parameterizing µθ , we follow DR who assumed the equal weight on the date-1 utility and the date-2 utility.
11

11

3.2

Results

The experiment that we perform here investigates changes in the distribution of the underlying
shock θ. Table 1 compares the crisis probabilities for a few cases where we change µθ while keepingσ θ unchanged. Recall that the probability of a crisis was targeted at 4.76 percent in the initial“risky” distribution with µθ = 0.50 (Case 1 in Table 1). Now, suppose that µθ declines to 0.35,
which corresponds to a “safer” environment in our model.13
With the lower mean of 0.35 in the “safer” distribution, the probability of a crisis declines from4.76 to 0.0004 percent (Case 2 in Table 1), if bankers keep their leverage unchanged at the level
under Case 1. Thus, a decrease in µθ implies that bankers find that crises are precipitated byextremely small upper tail risks (i.e., a risk of a large θ). Put differently, they recognize that thefundamentals are safe.
This is not the end of the story, however. Table 1 also reports that, when µθ declines, the bankleverage increases (i.e., D = 1.21 in Case 3). In our model, bankers have a strong incentive to raise
D when they face a smaller upper tail risk. Though a higher leverage gives rise to higher returns tohouseholds, it also increases the risk of bankers’ insolvency. As a result, the high leverage elevatesthe crisis probability to 5.77 percent. Therefore, it is not always true that the “safer” the economy,
the more secure the banking system.
Figure 3 shows how bank leverage affects the crisis probability through θ∗ . If bankers do notreact to the change in the distribution of θ, θ∗ remains unchanged at θ∗ = θ∗R (= 0.58). In the safer
distribution in Cases 2 and 3, this θ∗R implies a crisis probability of nearly zero. However, if bankers
react to the changes in fundamentals correctly, θ∗ decreases from θ∗R to θ∗S (= 0.43), giving rise toa higher crisis probability (region A in Figure 3).
What policy-relevant lesson can we learn from our analysis? To assess crisis risks in the banking
sector, it is essential to take into account the bankers’ own behaviors and responses to the environments, together with the environments surrounding them. Otherwise, risk assessment, particularly
when assuming the bankers’ behavior held unchanged, can give rise to an over-optimistic conclu13
The value of the lower µθ = 0.35 can be inferred by the actual decline in the TED spread observed in the early
2000s. (See Figure 1.) We confirmed that the size of the decline in the TED spread is comparable to that in thecorresponding price of funding liquidity in our model, E (r12 |no run).

12

sion (a too low crisis probability estimate, for example). To accurately estimate crisis probabilities,risks must be assessed based on the detailed knowledge about the banks’ risk-taking, including
their endogenous reactions to changes in environments surrounding them.

3.3

Robustness

Because our simulation exercise has provided only one example of the “the safer, the riskier” case,one may naturally ask whether this result is robust to changes in parameters. We thus examine
the parameter ranges where “the safer, the riskier” is the case. Figure 4 plots π and D against µθ ,while other parameters (except for σ θ ) are kept constant. The upper panel shows that the crisis
probabilities are downward-sloping for a wide range of µθ under various σ θ . This downward-sloping
portion of the curve indicates that a “safe” environment represented by a low µθ raises the crisis
probability if µθ is less than about 0.50.
In the lower panel of Figure 4, we observe that the curve for D shifts upward as σ θ decreases,
implying that a decrease in the volatility of liquidity preference shock fuels bankers’ risk-taking.
Using their dynamic stochastic general equilibrium model, Gertler, Kiyotaki, and Queralto (2012)
also find that banks issue more short-term debt in the economy calibrated with a smaller volatilityof a shock.14 Therefore, regarding banks’ risk-taking, the upward shifts in the curve are consistent
with their result. On the other hand, our model further enables us to evaluate whether a decrease involatility of shock elevates the crisis probability. Our simulation results in the upper panel suggest
that, unlike the case of µθ , a decrease in volatility lowers the crisis probability because it dominates
the effect of banks’ risk-taking in terms of crisis probability.
Figure 5 also investigates whether the crisis probability remains decreasing in µθ even if wechange other calibrated parameters e1 , Ȳ2 , γ, and X1 . Overall, the panels in the figure suggestthat the curves for the crisis probabilities are downward-sloping if µθ is sufficiently low. In thisrobustness analyses, we choose the parameter ranges to ensure that our robustness analyses satisfy
the following three criterions: (i) D < 2.5 so that the face value of demand deposit should not be
unreasonably high; (ii) the expected bank capital ratio is strictly higher than 3 percent, which is
14

In Gertler, Kiyotaki, and Queralto (2012), a shock to the aggregate capital in the economy is assumed rather
than a shock to liquidity preference.

13

substantially low, relative to the bank capital ratio targeted for the benchmark simulation; (iii)
0.025 < π < 0.135, with which we limit equilibrium crisis probabilities to a reasonable range,
compared to the empirical studies.

4

Concluding remarks

We argued that the banking system can be incentivized to take on more risks by “safer” environments and thus can expose the economy to a higher crisis probability. In our experiment, we
focused on the liquidity preference that had relatively abated as reflected in the data and translated
it into a decline in the mean of liquidity preference shocks. Our model shows that the banks’ risktaking reacting to the improved environment often dominates the outright effect of the improved
environment itself, in terms of the crisis probability. Our numerical simulations may provide an
explanation why the 2007-08 financial crisis unfolded amid an economic environment favorable tothe banking system. Beyond understanding the recent crisis, our main result in this paper suggeststhat risk assessment by policymakers should be made based not solely on environment surroundingbanks, but on the banks’ risk-taking together.
In terms of the normative implication, we note that Allen and Gale’s (2003) argument canbe applied to the present model. In other words, the laissez-faire banking sector in our model
achieves the constrained-efficient allocation of risks as well as resources. As long as the planneruses the same contracting technology as the banking sector, policy interventions aiming at crisis
prevention are likely to undermine welfare.15 To examine how to reduce the crisis probability
in a welfare-improving manner, the introduction of a welfare-relevant pecuniary externality intoa richer model is useful in considering policy implications. For the purpose, Kato and Tsuruga(2013) develop a model with pecuniary externalities by extending the model introduced in thispaper. Furthermore, other potentially important factors, such as contagion, imperfect information,and irrational over-optimism may amplify crises and enhance the importance of policy interventions
for crisis prevention.
15

On the policy perspectives, DR consider a variety of policy measures, assuming that the social planner has
technologies superior to the banking sector (e.g., state-contingent bank bailouts).

14

References[1] Acharya, V. V., and T. Yorulmazer (2008), “Information Contagion and Bank Herding,” Journal of Money, Credit and Banking, 40 (1), pp. 215-231.
[2] Allen, F., A. Babus, and E. Carletti (2006), “Asset Commonality, Debt Maturity and Systemic
Risk,” Journal of Financial Economics, 104 (3), pp. 519-534.
[3] Allen, F., and E. Carletti (2006), “Credit Risk Transfer and Contagion,” Journal of Monetary
Economics, 53 (1), pp. 89-111.
[4] Allen, F., and D. Gale (1998), “The Optimal Financial Crises,” Journal of Finance, 53 (4),pp. 1245-1284.
[5] Allen, F., and D. Gale (2000), “Financial Contagion,” Journal of Political Economy, 108 (1),pp. 1-33.
[6] Allen, F., and D. Gale (2003), “Capital Adequacy Regulation: In Search of a Rationale,” in R.
Arnott, B. Greenwald, R. Kanbur, and B. Nalebuff eds., Economics for an Imperfect World:
Essays in Honor of Joseph Stiglitz, MIT Press.
[7] Allen, F., and D. Gale (2007), Understanding Financial Crises, Oxford University Press.
[8] Angeloni, I., E. Faia, and M. Lo Duca (2014), “Monetary Policy and Risk Taking,” unpublishedmanuscript.
[9] Castiglionesi, F., F. Ferriozzi, and G. Lorenzoni (2012), “Financial Integration and Liquidity
Crises,” unpublished manuscript.
[10] Dasgupta, A. (2004), “Financial Contagion through Capital Connections: A Model of theOrigin and Spread of Bank Panics,” Journal of the European Economic Association, 2 (6), pp.
1049-1084.
[11] Diamond, D. W., and R. Rajan (2001), “Liquidity Risk, Liquidity Creation, and Financial
Fragility: A Theory of Banking,” Journal of Political Economy, 109 (2), pp. 287-327.
[12] Diamond, D. W., and R. Rajan (2012), “Illiquid Banks, Financial Stability, and Interest Rate
Policy,” Journal of Political Economy, 120 (3), pp. 552-591.
[13] Farhi E., and J. Tirole (2012), “Collective Moral Hazard, Maturity Mismatch, and Systemic
Bailouts,” American Economic Review, 102 (1), pp. 60-93.
[14] Gersbach, H., and J.-C. Rochet (2012), “Aggregate Investment Externalities and Macroprudential Regulation,” Journal of Money, Credit, and Banking, 44 (2), pp. 73-109.

15[15] Gertler, M., N. Kiyotaki, and A. Queralto (2012), “Financial Crises, Bank Risk Exposure andGovernment Financial Policy,” Journal of Monetary Economics, 59, Supplement, pp. S17-S34.
[16] Gorton G. (1988), “Banking Panics and Business Cycles,” Oxford Economic Papers, 40 (4),pp. 751-781.
[17] Gorton G., and A. Metrick (2012a), “Who ran on repo?” NBER working paper, No. 18455.
[18] Gorton G., and A. Metrick (2012b), “Securitized Banking and the Run on Repo,” Journal of
Financial Economics, 104 (3), pp. 425-451.
[19] Jeanne, O., and A. Korinek (2010), “Managing Credit Booms and Busts: A Pigouvian Taxation
Approach,” NBER Working Paper No. 16377.
[20] Jiménez, G., S. Ongena, J. L. Peydró, and J. Saurina (2014), “Hazardous Times for Monetary
Policy: What Do 23 Million Loans Say about the Impact of Monetary Policy on Credit RiskTaking?” Econometrica, 82 (2), pp. 463-505.
[21] Kato, R., and T. Tsuruga (2013), “Bank Overleverage and Macroeconomic Fragility,” Kyoto
University Research Project Center Discussion Paper Series, E-12-002.
[22] Laeven, L., and F. Valencia (2013), “Systemic Banking Crises Database,” IMF Economic
Review, 61 (2), pp. 225-270.
[23] Lorenzoni, G. (2008), “Inefficient Credit Booms,” Review of Economic Studies, 75 (3), pp.
809-33.
[24] Maddaloni, A., and J. L. Peydró (2011), “Bank Risk-Taking, Securitization, Supervision, and
Low Interest Rates: Evidence from the Euro Area and U.S. Lending Standards,” Review of
Financial Studies, 24 (6), pp. 2121-2165.
[25] Reinhart, C., and K. Rogoff (2009), This Time is Different: Eight Centuries of Financial
Folly, Princeton University Press.
[26] Reinhart, C., and K. Rogoff (2013), “Banking Crises: An Equal Opportunity Menace,” Journalof Banking and Finance, 37 (11), pp. 4557-4573.
[27] Stein, J. (2012), “Monetary Policy as Financial-Stability Regulation,” Quarterly Journal of
Economics, 127 (1), pp. 57-95.

16

Table 1: Numerical simulation of crisis probabilities (π) for µθµθ
Crisis probabilityDθ∗

Case 1
0.500
4.760
1.052
0.584

Case 20.350
0.0004
1.052
0.584

Case 3
0.350
5.767
1.214
0.431

Note: The crisis probability is expressed in terms of percent. The crisis probabilities and θ∗ in Cases 1 and 3 arecomputed from the optimal leverage. The crisis probability and θ∗ in Case 2 are computed under the assumptionthat the leverage in Case 1 is kept unchanged even when µθ is reduced to 0.35.

17Figure 1: TED spread and net repo funding to banks and broker dealers

2000

Net repo funding (dashed)

TED spread (solid)

2

1

1000

0
1998

1999

2000

2001

2002

2003
2004
year

2005

2006

2007

2008

2009

Note: The solid line represents the TED spread (the spread between 3-month LIBOR based on US dollars and3-month Treasury Bill rate) from Federal Reserve Economic Data. The unit is percent. The dashed line shows net
repo funding to banks and broker dealers taken from the Financial Accounts of the United States deflated by theConsumer Price Index for All Urban Consumers (All Items). The unit of net repo funding is billions of 2005 U.S.
dollars.

18

Figure 2: Bankers’ leverage and households’ utility

0.1

E[U], E[U|No run], E[U|Run]

0.05

E[U]
E[U|No run]E[U|Run]

0
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
1.02

1.04

1.06

1.08

1.1
1.12
Deposits

1.14

1.16

1.18

1.2

Note: The solid line represents the utility level against the face value of deposits that is computed from (10). The
upward-sloping dashed line is the expected utility conditional on no bank run that is computed from the first lineof (10) divided by 1-π. The downward-sloping dashed line is the expected utility conditional on a bank run that iscomputed from the second line of (10) divided by π . The calibration is based on the assumption that a liquidity
shock follows a beta distribution with a mean of 0.50 and a standard deviation of 0.05.

19Figure 3: Comparisons for distributions for θ

8

7

Probability density

6

Case 1

Cases 2 and 3

5

4

3

2

1

A
0
0.15

0.2

0.25

0.3

0.35

0.4

*0.45

θS

0.5

0.55

0.6

θ*R

0.65

Note: The solid line represents the probability density function based on a beta distribution with a mean of 0.50 anda standard deviation of 0.05 (Case 1). The dashed line is the probability density function of a beta distribution with∗a smaller mean of 0.35 but with the same standard deviation (Cases 2 and 3). Here θRis the threshold value of aliquidity shock that precipitates a crisis under Cases 1 and 2, while θS∗ is the threshold value corresponding to Case
3. The region A in the figure corresponds to the crisis probability in Case 3.

20

Figure 4: Crisis probabilities (π) and the optimal level of the deposit face value (D) against µθCrisis probabilities against the mean of θσθ = 0.02

0.14

σθ = 0.05
Crises probability

0.12

σ = 0.08
θ

0.1
0.08
0.06
0.04
0.02
0.2

0.3

0.4
μθ

0.5

0.6

D against the mean of θ
2
1.9

D

1.8
1.7

σθ = 0.02

1.6

σθ = 0.05

1.5

σθ = 0.08

1.4
1.3
1.2
1.1
1
0.2

0.3

0.4
μθ

0.5

0.6

Note: The curve represents the crisis probabilities (the upper panel) and the optimal level of D (the lower panel)against the mean of θ. Each curve in both panels is plotted for σθ of 0.02, 0.05, and 0.08.

21

Figure 5: Robustness analysis

0.11

0.11

e1 = 0.20

0.1

e1 = 0.30

0.09

e1 = 0.40

0.08
0.07

0.07
0.06

0.05

0.04
0.4
μθ

0.5

Ybar2 = 7.0

0.08

0.05

0.3

Ybar2 = 4.5

0.09

0.06

0.2

Ybar2 = 2.0

0.1
Crisis probability

Crisis probability

0.12

0.6

0.2

0.3

0.4
μθ

0.11
Crisis probability

0.1
0.09
0.08
0.07
0.06

0.6

X1 = 0.60

0.12
Crisis probability

γ = 0.60
γ = 0.80
γ = 0.99

0.5

X = 0.80
1

X1 = 1.00

0.1
0.08
0.06

0.05
0.2

0.3

0.4
μ

0.5

0.6

0.2

θ

0.3

0.4
μ

0.5

0.6

θ

Note: The curve in each panel represents the crisis probabilities against the mean of θ. Each curve is plotted forvarious parameters in the model. The upper-left panel plots the crisis probability when e1 = 0.20, 0.30, and 0.40.
The upper-right panel plots the crisis probability when Ȳ2 = 2.0, 4.5, and 7.0. The lower-left panel plots the crisis
probability when γ =0.60, 0.80, and 0.99. The lower-right panel plots the crisis probability when X1 = 0.60, 0.80,
and 1.00.

22