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Microprudential Regulation in a Dynamic
Model of Banking
Gianni De Nicolò
International Monetary Fund and CESifo
Andrea Gamba
Warwick Business School, Finance Group
Marcella Lucchetta
University “Ca’ Foscari” of Venice, Department of Economics
This paper studies the quantitative impact of microprudential bank regulations on bank
lending and value metrics of efficiency and welfare in a dynamic model of banks that
are financed by debt and equity, undertake maturity transformation, are exposed to
credit and liquidity risks, and face financing frictions. We show that (1) there exists an
inverted U-shaped relationship between bank lending, welfare, and capital requirements,
(2) liquidity requirements unambiguously reduce lending, efficiency, and welfare, and
(3) resolution policies contingent on observed capital, such as prompt corrective action,
dominate in efficiency and welfare terms (noncontingent) capital and liquidity requirements.
(JEL G21, G28, G33)
The 2007–2009 financial crisis has been a catalyst for significant bank
regulation reforms, as the precrisis microprudential regulatory framework has
been judged inadequate to cope with large financial shocks. The new Basel III
framework envisions a raise in bank capital requirements and the introduction
of new liquidity requirements. At the same time, there is an active debate
regarding how to make prompt corrective action (PCA) policies and related
bank closure rules more effective in reducing the costs of government’s bank
For comments and suggestions, we thank without implications: the Editor, an anonymous referee, our discussants
Pete Kyle, Kai Du, Hayne Leland, Falco Fecht, Gyongyi Loranth, Wolf Wagner, Jean-Charles Rochet, Sudipto
Battacharya, Charles Calomiris, Stijn Claessens, Peter DeMarzo, Pablo D’Erasmo, Douglas Gale, Pedro Gete,
Gerhard Illing, Javier Suarez, Goetz van Peter, as well as the participants of the Financial Stability Conference
at Tilburg University (2011), the Conference on Stability and Risk Control in Banking, Insurance and Financial
Markets (2011) at the University of Venice, the ZEW/Bundesbank conference on Basel III and Beyond at the
Bundesbank Center in Eltville, the 12th Symposium on Finance, Banking, and Insurance at Karlsruhe Institute
of Technology (2011), the FED Cleveland Conference on Capital Requirements for Financial Firms (2012),
the FDIC/JFSR 12th Annual Bank Research Conference (2012), the AEA Meeting in Philadelphia (2014), and
seminar participants at Aarhus University, the International Monetary Fund, Warwick Business School, EIEF,
and the Federal Reserve Board. The views expressed in this paper are those of the authors and do not necessarily
represent those of the IMF or IMF policy. Send correspondence to Gianni De Nicolò, International Monetary
Fund, 700 19th Street NW, Washington, DC 20431, USA; telephone (202) 623-8200. E-mail: gdenicolo@imf.org.
© The Author 2014. Published by Oxford University Press on behalf of The Society for Financial Studies.
All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com.
doi:10.1093/rfs/hhu022
RFS Advance Access published April 1, 2014
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The Review of Financial Studies / v 0 n 0 2014
bailouts under deposit insurance.
1
Our study contributes to a sparse literature
analyzing microprudential bank regulation in the context of dynamic models
of banking.
2
To our knowledge, there is no study offering a joint assessment of
capital regulation, liquidity requirements, and PCA policies in a dynamic model
of banks which perform a well-defined intermediation function and deposits are
insured. A static or finite horizon model of banks can only partially capture how
banks make their policy decisions subject to the true shadow costs associated
with regulatory requirements. Formulating such a dynamic model is the main
contribution of this paper.
We study a partial equilibrium model in the tradition of the valuation
approach pioneered by Merton (1977) and Kareken and Wallace (1978), and
design it consistently with standard corporate finance setups adapted to the
specifics of banks, as advocated by Flannery (2012). The economy is driven
by a macroeconomic (systematic) risk factor, and financial markets are in
equilibrium. The banking system is composed of banks exposed to a systematic
risk factor and an idiosyncratic risk component. Investors’ preferences are
represented by a stochastic discount factor that is parameterized as in Jones
and Tuzel (2013), and it delivers countercyclical risk premia. With this pricing
kernel, we evaluate banks’ securities and derive measures of bank efficiency
and welfare.
Three features characterize our model. First, we analyze banks that
dynamically transform short-term liabilities into longer-term partially illiquid
assets whose returns are uncertain. This feature is consistent with banks’special
role in liquidity transformation, as banks in our model can be viewed as a
dynamic infinite horizon version of the intermediaries studied by Diamond and
Dybvig (1983) and Allen and Gale (2007). Second, we consider banks whose
deposits are insured. Deposit insurance is introduced because a key asserted
role of capital regulation is the abatement of the excessive bank risk taking
arising from moral hazard under partial or total insurance of its liabilities.
Thus, there is a potential role for capital requirements, and their effectiveness
in abating banks’ probability of default can be assessed. Third, banks can
be in financial distress. Such distress can be viewed as arising from market
incompleteness of the type analyzed by Allen and Gale (2004) as well as from
asymmetric information frictions that make equity issuance costly and make
banks unable to raise uncollateralized debt. These assumptions are meant to
capture an environment in which liquidity requirements may in principle have
1
The new Basel III framework is detailed in Basel III: A global regulatory framework for more resilient banks
and banking systems , Bank for International Settlements, Basel, June 2011. For a discussion of PCA reforms, see
Government Accountability Office, Bank Regulation: Modified Prompt Corrective Action Framework Would
Improve Effectiveness, Washington DC, June 2011.
2
Calem and Rob (1999) models banks of fixed size that do not issue equity. Bhattacharya et al. (2002) and Peura
and Keppo (2006) consider capital regulation only with bank closure rules implemented under banks’ random
audits. Estrella (2004) considers bank regulatory restrictions on value at risk, whereas Elizalde and Repullo
(2007) and Zhu (2008) consider capital regulation under simple bank closure rules.
2
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Microprudential Regulation in a Dynamic Model of Banking
a role in minimizing the risk that illiquid, but solvent, banks become insolvent,
and face increased costs to raise funding in case of financial distress.
We assess the impact of microprudential bank regulation in terms of value
metrics of bank efficiency and welfare. The first metric is enterprise value,
which is interpreted as the efficiency with which the bank carries out its maturity
transformation function. The second metric, called “social value,” proxies
the contribution to welfare associated with banking activities, as measured
by the discounted expected value of banks to all bank stakeholders and the
government. In the sequel, we refer to this metric interchangeably as social
value or welfare.
The optimal polices and metrics of efficiency and welfare of unregulated
banks are our benchmarks. Relative to these benchmarks, we compare policies
and value metrics of efficiency and welfare of banks subject to (1) capital
requirements resembling risk-based Basel II-type capital requirements, (2)
capital requirements and liquidity requirements resembling Basel-III liquidity
coverage ratios, and (3) a PCA provision that implements capital requirements
through restrictions on banks’payouts and a bank closure rule, both contingent
on observed levels of capital. We assess the impact of these bank regulations
quantitatively by simulating the model of a banking industry composed of banks
exposed to the same risks under a set of calibrated parameters, with regulatory
parameters mimicking current regulations. Because banks are exposed to the
same risks, we evaluate the impact of these regulations on a representative bank.
The analysis of bank optimal policies along the business cycle yields two
main results. First, under a mild capital requirement and the PCA, bank lending,
debt, and capital ratios are positively correlated with the systematic risk factor
and the correlations are lower than the ones exhibited by unregulated banks. On
the other hand, liquidity ratios are more negatively correlated with systematic
risk than in the unregulated case. In particular, these results suggest that risk-
based capital regulation does not necessarily enhance the procyclicality of bank
lending. Second, when liquidity requirements are added to capital requirements,
the positive correlation between capital ratios and the systematic risk factor
increases, especially in upturns. This happens because liquidity requirements
force banks to use retained earnings to build up liquidity buffers rather than
invest in lending. As a result, capital ratios become inflated in an upturn, but
they are not different from those of banks subject to capital regulation only in
a downturn. In other words, capital regulation and the PCA provide banks with
incentives to create liquidity buffers in downturns. With mandatory liquidity
requirements, however, banks are forced to build these buffers during upturns
as well, thereby depressing lending.
The steady-state (or unconditional) analysis of the model delivers three
key results. First, there exists an inverted U-shaped relationship between
bank lending and the stringency of capital requirements. Such a relationship
translates into an inverted U-shaped relationship between welfare and the
stringency of capital requirements. When capital requirements are mild, banks
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The Review of Financial Studies / v 0 n 0 2014
find it optimal to fulfill them through an increase in lending. This increase allows
banks to build up capital through increased revenues and retained earnings.
Relative to unregulated banks, the quantitative increase in bank lending due to
mild capital requirement is a notable 15% in our calibration. Banks’ probability
of default is also lower than that of their unregulated counterparts, indicating
that capital requirements are successful in reducing banks’ failures. However,
if capital requirements become too stringent, then banks find it optimal to
fulfill them through a reduction of lending, because lending exhibits decreasing
returns, and building equity through increased revenues and retained earnings
becomes too costly. These novel findings suggest the existence of optimal levels
of bank-specific regulatory capital under deposit insurance.
Second, liquidity requirements reduce bank lending, efficiency, and welfare,
with these reductions increasing monotonically with their stringency. This
occurs because liquidity requirements severely hamper banks’ maturity
transformation, forcing banks to use retained earnings to increase bond holdings
or reduce indebtedness, rather than investing them in lending. Moreover, when
liquidity requirements are added to capital requirements, they also destroy
the efficiency and welfare benefits of mild capital requirements, because bank
lending, efficiency, and social values are reduced relative to the bank subject to
capital regulation only. Quantitatively, the declines in bank lending and value
metrics of efficiency and welfare associated with liquidity requirements are
large, namely, in the order of 20%–25% in our calibration.
Third, a resolution procedure contingent on observed bank capitalization,
such as the PCA, dominates both capital and liquidity requirements in efficiency
and welfare terms. Recall that deposit insurance introduces incompleteness in
deposit contracts, as deposit payments are not contingent on the realization of
states of nature. This is inefficient, as these contingencies may be instrumental
in attaining optimality in banking environments similar to ours (see, e.g.,
De Nicolò 1996; Allen and Gale 1998). Noncontingent capital and liquidity
requirements are insufficient for, or even detrimental to, attaining optimality.
By contrast, the PCA introduces contingencies based on observed equity. These
contingencies substitute for the missing contingencies in deposit payments due
to deposit insurance. Thus, resolution procedures, such as the PCA, appear a
necessary tool to achieve optimality of bank regulation.
1. The Model
Time is discrete, the horizon is infinite, and a systematic (macroeconomic) risk
drives risk premia in a financial market equilibrium. The systematic risk is
denoted by u and follows an autoregressive process
u
t
= κ
u
u
t−1
+σ
u
ε
u
t
, (1)
where ε
u
t
is i.i.d. with a truncated standard normal distribution, κ
u
is the
autocorrelation parameter such that |κ
u
|<1, and σ
u
is the conditional standard
deviation.
4
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Microprudential Regulation in a Dynamic Model of Banking
Following Jones and Tuzel (2013), the preferences of the representative
investor are summarized by a one-period stochastic discount factor. Given the
transition to state u
t+1
from the current state, u
t
, the stochastic discount factor is
M(u
t
,u
t+1
)=βe
−g
t
ε
u
t+1
−
1
2
g
2
t
σ
2
u
, (2)
where the state-dependent coefficient of risk-aversion is defined as g
t
= g(u
t
)=
exp(γ
1
+γ
2
u), with 0<β<1, γ
1
>0 and γ
2
<1. Using this pricing kernel, we
will later on evaluate banks’ securities.
3
There is a finite set of heterogeneous banks indexed by j. Bank managers
maximize shareholders’ value, so there are no managerial agency conflicts.
Banks receive a random stream of short-term insured deposits, can issue risk-
free short-term collateralized debt, and invest in longer term risky productive
assets and short-term bonds. Thus, banks are exposed to credit and liquidity
risks. As detailed below, these risks are assumed to be affine functions
of a systematic risk factor and a bank idiosyncratic risk factor. The bank
idiosyncratic risk factor v
j
follows an autoregressive process
v
j
t
= κ
v
v
j
t−1
+σ
v
ε
j
t
, (3)
where ε
j
t
is i.i.d. with truncated standard normal distribution, and |κ
v
|<1.
We assume that ε
u
t+1
is independent of ε
j
t+1
for all j ’s and that the latter is
independent across banks. The random vector s
t
=(u
t
,v
t
) evolves according
to a stationary and monotone Markov transition function Q(s
t+1
|s
t
) jointly
defined by Equations (1) and (3). We denote S the state space of s, where S
is compact.
4
In the remainder of this section, we drop the superscript j ,aswe
will be solving a representative bank problem.
1.1 Bank’s balance sheet
On the asset side, a bank can invest in a liquid, one-period bond (a Treasury
bill), which yields a constant rate r
f
, and in a portfolio of risky assets, called
loans. We denote with B
t
the face value of the risk-free bond, and with L
t
≥0
the nominal value of the stock of loans outstanding in period t (i.e., in the time
interval (t −1,t]). Note that because B
t
can have unrestricted sign, we assume
it is the net position in bonds, thereby allowing the bank to simultaneously
borrow or lend at the rate r
f
. Similarly to Zhu (2008), we make the following
assumptions.
3
Other partial equilibrium approaches based on a reduced-form stochastic discount factor are Berk, Green, and
Naik (1999) and Zhang (2005). Differently from other functional forms of stochastic discount factors, the
one assumed by Jones and Tuzel (2013) has the convenient property of having a state-independent risk-free
discount factor: in our case, the gross yield of a risk-free zero coupon bond is
1/E
t
M
t+1
, where E
t
M
t+1
=
βe
−
1
2
g
2
t
σ
2
u
·E
t
e
−g
t
ε
u
t+1
= β
.
4
As detailed in Appendix C, the support of each state variable u and v is a compact set discretized using
Rouwenhorst’(1995) approach.
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The Review of Financial Studies / v 0 n 0 2014
Assumption 1 (Revenue function). The total revenue from loan investment
is given by Z
t
π(L
t
), where π(L
t
) satisfies π(0)=0, π>0, π
>0, and π
<0.
This assumption is empirically supported, as there is evidence of decreasing
return to scale of bank investments.
5
In our model loans can be viewed as
including traditional loans and as risky securities. Z
t
= Z(u
t
,v
t
) is a random
credit shock realized on loans in the same time period, which captures variations
in banks’ total revenues as determined by the systematic and idiosyncratic
shocks. Note that the choice variables B
t
and L
t
are set at the beginning of the
period, whereas Z
t
is realized only at the end of the period.
The maturity of deposits is set to one period. Bank maturity transformation
is introduced with the following:
Assumption 2. Aconstant proportion δ ∈ (0,1/2) of the existing stock of loans
at t, L
t
, becomes due at t +1.
The parameter δ<1/2 indexes the average maturity of the existing stock
of loans, which is 1/δ −1 > 1.
6
Thus, the bank is engaging in maturity
transformation of short-term liabilities into longer-term investments, as in
Diamond and Dybvig (1983). Under Assumption 2, the law of motion of L
t
is
L
t
= L
t−1
(1−δ)+I
t
, (4)
where I
t
is the investment in new loans if it is positive, or the amount of cash
obtained by liquidating loans if it is negative.
Convex asymmetric loan adjustment costs as in the Q-theory of investment
(see, e.g., Abel and Eberly 1994) are introduced to capture banks’ information
production costs about credit quality
Assumption 3 (Loan adjustment costs). The adjustment cost function for
loans is quadratic:
m(I
t
)=|I
t
|
2
χ
{
I
t
>0
}
·m
+
+χ
{
I
t
<0
}
·m
−
, (5)
where χ
{
A
}
is the indicator of event A, m
+
>0 and m
−
>0 are the unit cost
parameters, and costs are deducted from profits.
According to Assumption 3, a bank incurs screening and monitoring per-unit
costs m
+
when it increases lending, and per-unit liquidation costs m
−
when
loans are reduced. If m
−
>m
+
, then there is costly reversibility, because a bank
5
See for instance, Berger et al. (2005), Carter and McNulty (2005), and Cole, Goldberg, and White (2004).
6
The (weighted) average maturity of existing loans at date t , assuming the bank neither default nor makes any
adjustments on the current investment in loans, is
∞
s=0
s
δL
t+s
L
t
=
1
δ
−1, as the residual loans outstanding at date
t +s, s ≥ 0,isL
t+s
= L
t
(1− δ)
s
.
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Microprudential Regulation in a Dynamic Model of Banking
would face higher costs to liquidate investments rather than for expanding them.
This assumption is consistent with costs of breaking bank relationships higher
than those associated with an expansion of lending to old as well as to new
customers.
7
The magnitude of these loan adjustment costs will be pinned down
in our calibration.
On the liability side, the bank receives a random amount of one-period
deposits D
t+1
= D(u
t
,v
t
) at the beginning of period t +1, and this amount
remains outstanding during the period. The interest rate on deposits is r
d
≤r
f
,
where the difference between the rate on bonds and the remuneration of deposits
captures implicit costs of payment services associated with deposits. The
change D
t+1
−D
t
is an exogenous liquidity shock driven by the realizations
of systematic and idiosyncratic shocks.
Deposits are insured according to the following
Assumption 4 (Deposit insurance). The deposit insurance agency insures all
deposits. In the event that a bank defaults on deposits and on the related interest
payments, depositors are paid interest and principal by the deposit insurance
agency, which absorbs the relevant loss.
Under this assumption, with no change in the model, the depositor can
be viewed as the deposit insurance agency itself, whose claims are risky,
while deposits are effectively risk-free from depositors’ standpoint. Thus, the
difference between the ex ante yield on deposits and the rate on bonds can be
viewed as including a subsidy that the agency provides to the bank, as the cost
of this insurance is not charged to either banks or depositors.
To fund operations, a bank can issue one-period bonds and equity. Following
Hennessy and Whited (2005), we assume that a bank is constrained to issue
fully collateralized bonds, so that the bond yield is the rate r
f
. We denote
B
t
<0 the notional amount of the bond issued at t −1 and outstanding until t.
The collateral constraint is described below.
To summarize, at t −1 (i.e., at the beginning of period t), after the investment
and financing decisions have been made, a bank’s balance sheet equation is
L
t
+B
t
= D
t
+K
t
, (6)
where K denotes the ex ante book value of equity, or bank capital. Note that
B>0 denotes a positive risk-free investment (net of issued bonds), whereas
B<0 denotes the face value of issued bond (net of risk-free investment).
1.2 Bank cash flow
Once Z
t
and D
t+1
are realized at t, the current state (before a decision is made)
is summarized by the vector x
t
=(L
t
,B
t
,D
t
,u
t
,v
t
), as a bank enters date t with
7
For the costs associated with breaking up bank relationships as forgone monopoly rents due to holdup problems,
see Rajan (1992) and Sharpe (1990). A portion of liquidation costs
m
−
could also be viewed as capturing fire
sales costs arising from financial distress (see, e.g., Acharya, Shin, and Yorulmazer 2011; Hanson, Kashyap, and
Stein 2011).
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The Review of Financial Studies / v 0 n 0 2014
loans, bonds, and deposits in amounts L
t
, B
t
, and D
t
, respectively. Prior to
investment, financing, and cash distribution decisions, the total internal cash
available to a bank is
W
t
= W (x
t
)=y
t
−T (y
t
)+B
t
+δL
t
+(D
t+1
−D
t
). (7)
Equation (7) says that total internal cash W
t
equals earnings before taxes (EBT),
y
t
= y(x
t
)=π(L
t
)Z
t
+r
f
B
t
−r
d
D
t
, (8)
minus corporate taxes T (y
t
), plus the principal of one-year investment in bond
maturing at t, B
t
>0 (or alternatively the amount of maturing one-year debt,
B
t
<0), plus the repayment of maturing loans δL
t
, plus the net change in
deposits, D
t+1
−D
t
.
Consistently with current dynamic models of a nonfinancial firm (see,
e.g., Hennessy and Whited 2007), corporate taxation is introduced with the
following:
Assumption 5 (Corporate taxation). Corporate taxes are paid according to
the following convex function of EBT:
T (y
t
)=τ
+
max
{
y
t
,0
}
+τ
−
min{y
t
,0}, (9)
where τ
−
and τ
+
,0≤τ
−
≤τ
+
<1, are the marginal corporate tax rates in case
of negative and positive EBT, respectively.
The assumption τ
−
≤τ
+
is standard in the literature, as it captures a reduced
tax benefit from loss carryforward or carrybacks.
Given the available cash W
t
as defined in Equation (7) and the residual loans,
L
t
(1−δ), a bank chooses the new level of investment in loans, L
t+1
and the
amount of risk-free bonds B
t+1
(purchased if positive, issued if negative). As
a result, Equation (6) applies to B
t+1
, L
t+1
, and D
t+1
, and both L
t+1
and B
t+1
remain constant until the next decision date, t +1.
These choices may differ according to whether or not a bank is in financial
distress. If total internal cash W
t
is positive, it can be retained or paid to
shareholders. If W
t
is negative, a bank is in financial distress, because absent
any action, it would be unable to honor part, or all, of its obligations towards the
tax authority, depositors, or bondholders. When in financial distress, a bank can
finance the shortfall W
t
by liquidating loans, by issuing bonds (B
t+1
<0), or by
injecting equity capital. Overcoming this shortage of liquidity is expensive
because all these transactions generate either explicit or implicit costs. In
liquidating loans, a bank incurs the downward adjustment cost defined by
Equation (5), bond issuance is subject to a collateral restriction that can limit
bank debt capacity, and underwriting costs are paid when seasoned equity is
offered. We now present these latter two restrictions on the financial channels
of a bank.
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Microprudential Regulation in a Dynamic Model of Banking
Bond issuance is constrained by the following:
Assumption 6 (Collateral constraint). If B
t
<0, the amount of bond issued
by the bank must be fully collateralized. In particular, the constraint is
L
t
−m(−L
t
(1−δ))+π (L
t
)Z
d
−T (y
min
t
)−r
d
)D
t
+B
t
(1+r
f
)+D
d
−D
t
≥0,
(10)
where Z
d
is the worst possible credit shock (i.e., the lower bound of the support
of Z), D
d
is the worst-case scenario flow of deposits, and y
min
t
= π(L
t
)Z
d
+
r
f
B
t
−r
d
D
t
is the EBT in the worst-case end-of-period scenario for current L
t
,
B
t
and D
t
.
The constraint in Equation (10) reads as follows: the end-of-period amount
B
t
(1+r
f
) that the bank has to repay must not be higher than the after-tax
operating income, π (L
t
)Z
d
−r
d
D
t
−T (y
min
t
) in the worst-case scenario, plus
the total available cash obtained by liquidating the loans, L
t
−m(−L
t
(1−δ)),
plus the flow of new deposits in the worst-case scenario, D
d
, net of the claim
of current depositors, D
t
. Available cash would then be the sum of the loans
that will become due, L
t
δ, plus the amount that can be obtained by a forced
liquidation of the loans, L
t
(1−δ) net of the adjustment cost m(−L
t
(1−δ)), as
per Equation (5).
8
An obvious implication of this constraint is that a bank’s
indebtedness will be always bounded above.
We denote with (D
t
) the feasible set for a bank when the current deposit is
D
t
, defined as the set of (L
t
,B
t
) such that condition (10) is satisfied if B
t
<0,
with no restrictions being imposed when B
t
≥0:
(D
t
)=
{
(L
t
,B
t
)|
L
t
−m(−L
t
(1−δ))+D
d
+π(L
t
)Z
d
(1−τ
min
t
)
1+r
d
(1−τ
min
t
)
+B
t
1+r
f
(1−τ
min
t
)
1+r
d
(1−τ
min
t
)
≥D
t
,
B
t
<0
}
∪
{
B
t
≥0
}
, (11)
where τ
min
t
= τ
+
if y
min
t
>0 and τ
min
t
= τ
−
if y
min
t
<0.
Similarly to Cooley and Quadrini (2001), we assume that issuing equity is
costly according to the following assumption.
Assumption 7 (Equity issuance costs). A bank raises capital by issuing
seasoned shares incurring a proportional cost λ>0 on the value of new equity
issued.
Observe that when banks are not in financial distress, total costs of equity
issuance will be just a proportion of the total amount of equity issued. When
8
By assumption 9 introduced below, the support for deposits and credit shock processes is compact, implying that
the collateral constraint is well defined.
9
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The Review of Financial Studies / v 0 n 0 2014
|||
(L
t
,B
t
)
Z
t−1
D
t
→ Z
t
D
t+1
(L
t+1
,B
t+1
)
→ Z
t+1
D
t+2
(L
t+2
,B
t+2
)
...
t − 1
t
t +1
...
Figure 1
Bank’s dynamic
Evolution of the state variables (credit shock,
Z, and deposits, D) and of the bank’s control variables (cash and
liquid investments,
B, and loans, L), assuming the bank is solvent at each date.
banks are in financial distress, however, the cost of equity issuance will be
increased by an additional amount that the bank has to raise, owing to loan
liquidation costs. Therefore, the more severe the financial distress, the larger
the transaction cost incurred to raise equity financing.
9
As a result of the choice of (L
t+1
,B
t+1
), the residual cash flow to shareholders
at date t is
U
t
= U (x
t
,L
t+1
,B
t+1
)=W
t
−B
t+1
−L
t+1
+L
t
(1−δ)−m(I
t+1
). (12)
If U
t
is positive, it is distributed to shareholders (as either dividends or stock
repurchases). If U
t
is negative, it equals the amount of newly issued equity
inclusive of the higher cost due to underwriting fees. Hence, the actual cash
flow to equity holders is
e
t
= e(x
t
,L
t+1
,B
t+1
)= max{U
t
,0}+min{U
t
,0}(1+λ). (13)
Figure 1 depicts the evolution of the state variables and the related bank’s
decisions when the bank is solvent.
Lastly, bank’s insolvency occurs according to the following:
Assumption 8 (Insolvency). In the case of default, bank shareholders exercise
the limited liability option (i.e., equity value is zero), and the assets are
transferred to the deposit insurance agency, net of bankruptcy costs in
proportion η>0 of the size of a bank, proxied by the face value of deposits, D
t
.
Right after default a bank is reorganized as a new entity endowed with deposits
D
t+1
and new capital K
t+1
= D
u
−D
t+1
≥0, where D
u
is the upper bound of
deposit process. The restructured bank invests initially only in risk-free bonds,
B
t+1
= D
u
, so that L
t+1
= 0. The capital injected by the government in the new
bank is financed with general tax proceeds.
9
Note that even though we make the simplifying assumption that the costs of debt and equity issuance costs
are independent, thus assuming some segmentation of equity and debt markets, the shadow costs associated
with these two forms of financing are not necessarily independent, because the shadow cost of debt will be
determined by the extent to which banks have spare debt capacity, whereas, as observed, total equity issuance
costs are increasing in the degree of financial distress.
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Microprudential Regulation in a Dynamic Model of Banking
Assumption 8 embeds three features. First, because default is irreversible, a
new bank financed with initial public capital is formed to replace a defaulted
bank in order to preserve intermediation services. Second, when the government
intervenes and sets up a new bank, it does not incur any underwriting cost,
because no new shares are issued in the open market. Lastly, the government
is assumed to be able to finance any recapitalization of an individual bank with
general tax proceeds.
10
The mapping of systematic and idiosyncratic risk factors onto credit and
liquidity risks is defined by the following:
Assumption 9. The random vector X
t
=(Z
t
,logD
t+1
) is an affine function of
the state variable s
t
=(u
t
,v
t
):
X
t
= μ+Ns
t
, (14)
for given two-dimensional vector μ and two-times-two nonsingular
matrix N.
In particular, this assumption translates into the following law of motion for
(Z
t
,logD
t+1
):
Z
t
=(1−κ
Z
)Z +κ
Z
Z
t−1
+ξ
Z
t
logD
t+1
=(1−κ
D
)logD +κ
D
logD
t
+ξ
D
t
.
(15)
In the above equations, κ
Z
is the persistence parameter, σ
Z
is the conditional
volatility, and
Z is the long-term average of the credit shock; κ
D
is the
persistence parameter for the deposit process; σ
D
is the conditional volatility;
and
D is the long-term level of deposits. The error terms (ξ
Z
t
,ξ
D
t
) are related
to (ε
u
t
,ε
t
) and have correlation coefficient ρ. The parameters of the process
(Z
t
,D
t+1
) are estimated on data. In Appendix B we detail how the parameters
of the process (u
t
,v
t
) are related to the parameters of the process (Z
t
,D
t+1
).
10
Observe that no default can occur if B
t
<0 owing to the collateral constraint. From Equation (12), the cash flow
to shareholders is
L
t
−m(I
t+1
)+π(L
t
)Z
t
−T (y
t
)+B
t
(1+ r
f
)−D
t
(1+ r
d
)+D
t+1
−B
t+1
−L
t+1
.
The collateral constraint implies
L
t
−m(−L
t
(1− δ))+π (L
t
)Z
d
−T (y
min
t
)+B
t
(1+ r
f
)−D
t
(1+ r
d
)+D
d
≥0.
The part in brackets in the cash flow expression is nonnegative. Hence, the cash flow to equity holders can be
negative only if
L
t+1
is significantly higher than −B
t+1
. However, if the bank defaults, such a policy will be
undone (and set at levels
B
t+1
= D
u
, L
t+1
=0, as specified later). Therefore, shareholders will get a positive value
by avoiding default. Therefore, in the model, default of a nonregulated bank may occur only when
B
t
≥0.
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1.3 The unregulated bank program and the valuation of securities
Let E denote the market value of bank’s equity. Given the state, x
t
=
(L
t
,B
t
,D
t
,u
t
,v
t
), bank’s equity value is the result of the following program
E(x
t
)= max
{(L
i+1
,B
i+1
)∈(D
i+1
),i=t,...,T }
E
t
⎡
⎣
T
i=t
e(x
i
,L
i+1
,B
i+1
)
i
j=t
M(x
j−1
,x
j
)
⎤
⎦
,
(16)
where E
t
[·] is the expectation operator conditional on D
t
, on the state variables
at t,(u
t
,v
t
), and on the decision (L
t+1
,B
t+1
); M(x
t
,x
t+1
) is a discount factor
defined in Equation (2) (which only depends on the component u
t
of x
t
), such
that M(x
t−1
,x
t
)=1 att;(L
i+1
,B
i+1
) is the decision at date i, for i = t,..., and T
is the default date. Because the model is stationary and the Bellman equation
involves only two dates (the current, t, and the next one, t +1), we can drop the
time index t and use the notation without a prime for the current value of the
variables, and with a prime to denote the next-period value of the variables.
The value of equity satisfies the following Bellman equation
E(x)= max
0, max
(L
,B
)∈(D
)
e(x,L
,B
)+E
M(x,x
)E(x
)
. (17)
Compactness of the feasible set of a bank and standard properties of the value
function are described in Appendix A.
When a bank is solvent, the value of equity satisfies the following Bellman
equation:
E(x)= max
(L
,B
)∈(D
)
e(x,L
,B
)+E
M(x,x
)E(x
)
. (18)
We denote with (L
∗
(x),B
∗
(x)) the optimal policy when the bank is solvent.
When it is insolvent, shareholders exercise the limited liability option, which
puts a lower bound on E at zero. The default indicator function is denoted (x).
We solve Equation (17) to determine the value of equity and the optimal
policy, including the optimal default policy, , as functions of the current
state, x. We denote ϕ, the state transition function based on the optimal policy:
ϕ(x)=
⎛
⎝
L
∗
B
∗
D
⎞
⎠
(1−)+
⎛
⎝
0
D
u
D
⎞
⎠
. (19)
Equation (19) says that the new state is (L
∗
,B
∗
,D
) if the bank is solvent,
and (0,D
u
,D
) if the bank defaults. In case of default, a new bank is started
endowed with seed capital D
u
−D
and deposits, D
, and a cash balance D
u
, and
no loans. This new bank will revise its investment (together with the financing)
policy in the following decision dates.
The end-of-year cash flow from current deposits, D
t+1
, for a given realization
of the exogenous state variables, (Z
t+1
,D
t+2
), and on the related optimal policy,
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Microprudential Regulation in a Dynamic Model of Banking
is
f (x
t+1
|ϕ(x
t+1
))= D
t+1
(1+r
d
)(1−η(x
t+1
)). (20)
Hence, the ex ante fair value of newly issued deposits at t, from the viewpoint
of the deposit insurance agency (i.e, incorporating the risk of bank’s default),
is
F (x
t
)=E
t
[
M(x
t
,x
t+1
)f (x
t+1
|ϕ(x
t+1
))
]
= D
t+1
(1+r
d
)
(
1−ηP (x
t
)
)
, (21)
where P (x
t
)=E
t
[
M(x
t
,x
t+1
)(x
t+1
)
]
is the price of the relevant default
contingent claim. Dropping the dependence on the calendar date,
F (x)=D
(1+r
d
)
(
1−ηP (x)
)
. (22)
1.4 Value metrics of bank efficiency and welfare
A standard valuation concept is the market value of bank assets E(x)+f (x)−
B
−
, where f (x)=D(1+r
d
)(1−η(x)) from Equation (20), B
−
= min{B,0},
which includes the contribution of (cash-equivalent) one-period bonds B to
bank’s value, because bond investment helps reduce the potential costs triggered
by high cash flow volatility. However, this definition of market value does not
capture the role of banks as maturity transformers of liquid liabilities into
longer-term productive assets (loans). One of the key economic contributions
of banks identified in the literature is their role in efficiently intermediating
funds toward their best productive use (see, e.g., Diamond 1984; Boyd and
Prescott 1986). Banks play no such role if they just raise funds to acquire
risk-free (cash-equivalent) bonds. A suitable metric of bank efficiency is the
enterprise value of a bank, defined as EV (x)=
E(x)+f (x)−B
−
−B
+
, with
B
+
= max{B,0}. The enterprise value is thus the market value of equity plus
the value of deposits net of cash balances, or plus short-term debt, capturing
a bank’s ability to create “productive” intermediation.
11
Because B =B
+
+B
−
,
the enterprise value of the bank can be also written as EV (x)=E(x)+f (x)−B.
Our welfare metric of bank activities, called “social value,” is defined as
the sum of the values of banks’ activities to the government and to all banks’
stakeholders. In essence, this metric measures the contribution to welfare of
bank activities. The welfare metric is given by:
SV (x)=E(x)+D(1+r
d
)−B +G(x), (23)
where D(1+r
d
) is the book value of current deposits, and G(x) is the value of
the net payoff to the government, defined by the recursive equation
G(x)=(1−(x))
T (y
)+E[M(x,x
)G(x
)]
−(x)
ηD(1+r
d
)+K
(24)
with (x) denoting the default indicator at x. Equation (24) reads as
follows: so long as the bank is solvent ((x)= 0), taxes are collected, where
11
For the use of enterprise value as a metric of efficiency in the context of dynamic models of nonfinancial firms
see, for example, Gamba and Triantis (2008) and Bolton, Chen, and Wang (2011).
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E[M(x,x
)G(x
)] is the present discounted value of future tax proceeds. If the
bank is insolvent ((x)=1), then the government incurs direct bankruptcy costs
ηD, and injects new equity capital K
= D
u
−D
.
12
2. Bank Regulation
2.1 Capital requirement
In our model the capital ratio is the ratio of the book value of capital over the
book value of loans. Basel II-type capital regulation establishes a lower bound,
K
d
, on the book value of equity, set by the regulator as a function of a bank’s
risk exposure at the beginning of the period. In particular, this requirement
is a weighted average of risks associated with a bank’s exposure of assets of
different riskiness. Because a bank in our model has just one composite risky
asset, we set the weight applied to loans equal to 100%. Thus, the required
capital, K
d
, is at least a proportion, k, of loans outstanding at the beginning of
the period, L,orK
d
= kL. This requirement is equivalent to constraining net
worth to be positive ex ante. Given the definition of bank capital in Equation
(6), under the capital requirement, a bank’s feasible choice set is
(D)=
{
(L,B) | (1−k)L+B ≥ D
}
. (25)
When we compare the feasible choice set under the collateral constraint in
Equation (11) with the feasible set under the capital requirement, in general
neither (D)⊂(D) nor (D)⊂(D) in a proper sense.
The Bellman equation for the equity value of a currently solvent bank under
a capital requirement is given by Equation (18), the only difference being a
feasible set (D
)∩(D
) in place of (D
), because the bank is forced to
comply ex ante with the capital requirement. However, at the end of each
period, when the credit shock on existing loans Z
and the new deposit D
are realized, a bank may still face default risk if the innovations of the state
variables are particularly unfavorable.
2.2 Liquidity requirement
Current Basel III regulation introduces a mandatory liquidity coverage ratio as
a defense against the potential onset of severe liquidity stress. According to this
requirement, banks should hold a stock of high quality liquid assets such that
the ratio of this stock over the predicted net cash outflows over a 30-day time
period in the case of acute stress—as defined by the regulator—is not lower
than a certain percentage threshold.
12
Note that if the net payoff to the government is positive along a given path, the tax proceeds collected from a bank
in the past are sufficient to cover the recapitalization of a new bank. Otherwise, the shortfall of the government
will be covered by tax proceeds raised from other agents in the economy. In either case the government value
metrics of Equation (24) captures the net cost of banks to the government. Equivalently, the welfare metric is
the sum of enterprise value gross of bankruptcy costs, plus
G(x), which is net of bankruptcy costs incurred by
the government.
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Microprudential Regulation in a Dynamic Model of Banking
In our model, the stock of high quality liquid assets over the net cash outflows
over a bank’s planning period is given by the total cash available at the end
of the period over the total net cash flow in the worst-case scenario for both
credit and liquidity shocks. Formally, this liquidity coverage ratio should be
not lower than a level defined by the regulator, or
δL+Z
d
π(L)−T (y
min
)+B(1+r
f
)
D(1+r
d
)−D
d
≥. (26)
Hence, the feasible set for a bank complying with the liquidity requirement is
(D)=
(L,B) |
δL+D
d
+Z
d
π(L)(1−τ
min
)
(1+r
d
)−τ
min
r
d
+B
1+r
f
(1−τ
min
)
(1+r
d
)−τ
min
r
d
≥D
.
(27)
Thus, the liquidity ratio is the end-of-period total cash available in the worst-
case scenario over the end-of-period net cash outflows due to a variation in
deposits.
Figure 2 shows the constraints implied by the capital and the liquidity
requirements, as well as the collateral constraint for a specific choice of
parameters. It is apparent that when considered together, capital and liquidity
constraints may create considerable restrictions on a bank’s feasible choices.
2.3 Prompt corrective action (PCA)
An important objective of bank regulation is the minimization of losses of the
deposit insurance fund. This is achieved by PCA policies, which force banks to
5 10 15 20
L
-15
-10
-5
B
Liquidity
Capital
Collateral
Figure 2
Comparison of constraints
This figure presents the three feasible regions of
(L,B) defined by the collateral constraint, (D) in Equation
(11), the capital requirement,
(D) from Equation (25), and by the liquidity requirement, (D) from Equation
(27). The plot is based on the parameter values in Table 1, for a current
D =2.
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The Review of Financial Studies / v 0 n 0 2014
liquidate assets, suspend payouts contingent on prespecified observed levels of
capitalization, or close a distressed bank if capitalization is lower than a given
threshold. Correspondingly, we define a PCA rule contingent on observed ex
post bank capital. Formally, ex post bank capital in period t (as opposed to K,
which is the ex ante bank capital in period t), is
V = L+B −D+y −T (y)=K +y −T (y)
The regulator implements PCA according to the following:
Assumption 10 (PCA). If at time t the ex post bank capital satisfies the ex
ante capital ratio, that is V ≥ kL, no action is taken. If the ex-post bank capital
is such that 0<V <kL, the deposit insurer forces the distressed bank to restore
capital in the current period by replenishing the shortfall kL−V and to satisfy
the regulatory ratio in the next period. If the ex post bank capital is negative
(V ≤ 0), the bank is closed at time t and taken over by the deposit insurance
agency.
The PCA thus establishes that banks with a positive ex post capital, but lower
than satisfying regulatory ratio, are forced to finance the shortfall by liquidating
assets, raising collateralized debt, suspending payouts, or issuing new equity.
From Equation (12) we have
U = y −T (y)+K −K
−m(I )=V −m(I)−K
.
When V<kL, a bank is forced to finance the shortfall kL−V , as well as to set
the capital for the next period to the required level so that K
= V −m(I )−U ≥
kL
+(kL−V ). On the other hand, when the ex-post bank capital is negative, the
closure rule applies: a bank is expropriated from shareholders and is transferred
to the government. New capital K
= D
u
−D
is injected net of the positive value
of the bank’s capital, E(x)>0. In both cases, the government agency does not
incur bankruptcy costs (η = 0).
Note that the PCA actually implements a contingent enforcement of a capital
requirement. When triggered, such a requirement is more stringent than the ex
ante capital requirement for two reasons. First, the PCA implements a more
restrictive rule on ex ante book capital contingent on certain (low) realizations
of ex-post capital. Second, the risk of expropriation of shareholders based on
the realization of negative ex-post capital imposes a higher shadow cost on the
current capital requirement. For these reasons, we consider the PCA as a proper
regulation in itself, so that we can compare the impact of a contingent capital
requirement plus a closure rule, as implemented by the PCA just defined, with
(noncontingent) capital requirements.
13
13
In practice, however, PCA-type policies apply to banks that are already subject to other regulations, such as
capital requirements. In the sequel, we will also consider the impact of the PCA when implemented together
with capital and liquidity requirements.
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Microprudential Regulation in a Dynamic Model of Banking
For banks subject to the PCA, the Bellman equation of the solution of the
bank’s program when the bank is solvent (i.e., V
t
is positive) is as in Equation
(18). Under the PCA closure rule, the value of the net payoff to the government
is defined by the recursive equation
G(x)=(1−(x))
T (y
)+E[M(x,x
)G(x
)]
−(x)
K
−E(x)
, (28)
with (x) denoting the indicator of the event V<0atx.
As before, the government is assumed to be able to finance any
recapitalization shortfall of an individual bank with general tax proceeds.
However, if V ≤0 and collateralized debt is issued (B<0), bond holders,
old depositors, and the government are paid in full.
14
In this case, the
financing shortfall (net obligations minus available liquid funds) of the bank
is (1+r
d
)D −(1+r
f
)B +T (y)−(D
+π(L)Z). A portion of loan portfolio is
liquidated and the proceeds match (net of loan liquidation costs) the shortfall:
L−L
−m(L(1−δ)−L
)=(1+r
d
)D −(1+r
f
)B +T (y)−(D
+π(L)Z).
By solving this equation with respect to L
, a new level of loans, L
∗
,is
determined, which is positive because the bank satisfies the collateral constraint.
Clearly, in this reorganization procedure, an indirect cost is still incurred
because of loan liquidation costs, m(L(1−δ)−L
∗
).
3. Regulations in a Simplified Version of the Model
To illustrate some trade-offs on bank optimal policies implied by capital and
liquidity requirements under the assumption that banks are short lived, we
collapse our model to three periods. Now t +1 is the decision date, t +2 is the
final date, and the bank initial conditions are determined at t.
We make the following simplifying assumptions. The discount factor is
deterministic, so M(x
t
,x
t+1
)=β. There are no taxes and no adjustment costs,
and deposits are deterministic and constant (D
t
= D
t+1
= D
t+2
= D>0), δ =0,
β ≤(1+r
f
)
−1
, and r
d
= r
f
. Furthermore, we assume a simple two-point credit
shock distribution: Z
H
with probability p ∈(0,1), and Z
d
otherwise, where
Z
d
is such that Z
d
=
Z
L
L
t+1
π(L
t+1
)
, with Z
H
>0≥Z
L
≥−1. Under these assumptions,
the collateral constraint for B
t+1
<0, denoted with (C), the capital constraint,
14
This is because the collateral constraint in Equation (10)is
L+π(L)Z
d
+(1+r
f
)B −(1+r
d
)D −T (y
min
)≥ m(−L(1−δ))−D
d
,
the closure rule V ≤ 0 gives
L+π(L)Z +(1+r
f
)B −(1+r
d
)D −T (y) ≤0,
and the left-hand side of the second inequality is higher than the corresponding side of the first inequality.
Therefore, all stakeholders can be paid by liquidating the assets.
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denoted with (K), and the liquidity constraint, denoted with (L), are
B
t+1
≥
r
f
1+r
f
D −
1+Z
L
1+r
f
L
t+1
, (C)
B
t+1
≥D −(1−k)L
t+1
, (K)
B
t+1
≥
r
f
1+r
f
D −
Z
L
1+r
f
L
t+1
. (L)
Recall that by Equation (13), the cash flow to shareholders is U
t
= W
t
+L
t
−
B
t+1
−L
t+1
if U
t
>0, and U
t
(1+λ)ifU
t
<0, as the bank issues new equity at a
cost λ≥0.
The bank chooses (L
t+1
,B
t+1
) to maximize
e
t
+βE
t
[
e
t+1
]
=(W
t
+L
t
)(1+λ)−(1+λ−βp(1+r
f
))B
t+1
−(1+λ)L
t+1
+β
p
Z
H
π(L
t+1
)−(1+r
f
)D +L
t+1
+(1−p)max
0,(1+Z
L
)L
t+1
+(1+r
f
)B
t+1
−(1+r
f
)D)
. (29)
Because 1+λ>βp(1+r
f
), it is optimal to maximize debt (B
t+1
<0), because
profits are increasing in debt in the good state, whereas in the bad state losses
are bounded to be nonnegative by limited liability. This implies that at most
one of the constraints (C), (K), and (L) will be binding.
The unregulated bank maximizes Equation (29) subject to constraint (C).
Inserting (C) into Equation (29), the max{·} term in the third line of Equation
(29) vanishes. Therefore, the optimal loan level L
c
t+1
satisfies the (necessary
and sufficient) first-order condition
βpZ
H
π
(L
c
t+1
)=1+λ−βp −(1+λ−βp(1+r
f
))
1+Z
L
1+r
f
. (30)
Suppose now that the capital constraint (K) is tighter than (C); that is, (K) is
binding. The third line of Equation (29) becomes max{0,(1+Z
L
−(1+r
f
)(1−
k))L
t+1
}.
The optimal loan investment when (K) is binding, defined by L
k
t+1
, satisfies
βpZ
H
π
(L
k
t+1
)=1+λ−βp −(1+λ−βp(1+r
f
))(1−k). (31)
By comparing the right-hand sides of Equations (30) and (31), it is
straightforward to verify that L
k
t+1
>L
c
t+1
when (1+Z
L
)<(1+r
f
)(1−k), owing
to the strict concavity of function π.
Observe that the inequality (1+Z
L
)<(1+r
f
)(1−k) holds for relatively low
levels of k, but it is reversed for values of k close to one. Thus, there exists
a threshold value
ˆ
k such that L
k
t+1
<L
c
t+1
for all k>
ˆ
k. In other words, under
a sufficiently mild capital constraint (or k<
ˆ
k), lending is higher than in the
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Microprudential Regulation in a Dynamic Model of Banking
unregulated case. Thus, when (K) is binding, depending on parameters, lending
could be higher than in the unregulated case under mild capital requirements,
even though borrowing is lower (B
k
t+1
>B
c
t+1
holds when constraint (K) is more
stringent than (C)). This is because the capital requirement lowers the return of
holding cash relative to the expected return on loan investment. In sum, there
may exist parameter configurations such that the relationship between loans
and capital requirements is inverted U shaped. Interestingly, this result may
(but needs not) hold for any λ≥0.
Consider now the addition of a liquidity requirement to the capital
requirement and suppose that the liquidity constraint (L) is tighter than (K)
at the optimal choice L
k
t+1
; that is, (L) is binding. Replacing (L) in Equation
(29), the max{·} term turns into max{0,L
t+1
+(r
f
(−1)−1)D}.
If at the optimal solution L
t+1
+(r
f
(−1)−1)D ≤0, then L
t+1
satisfies
pZ
H
π
(L
t+1
)=r
f
−(1−p)Z
L
. (32)
Otherwise, L
t+1
satisfies
pZ
H
π
(L
t+1
)=r
f
−Z
L
. (33)
Comparing Equation (32) with Equation (30), it is easy to verify that the
right-hand side of Equation (30) is always strictly lower than that of Equation
(32). By strict concavity of the revenue function, this implies that L
t+1
<L
k
t+1
:
the liquidity constraint unambiguously reduces lending relative to the bank
subject to a (binding) mild capital constraint. Comparing Equation (30) with
Equation (33), the same result is obtained if p is close to one. Thus, there may
exist parameter configurations such that the liquidity constraint reduces lending
relative to the capital constraint.
The conclusions of this simplified version of the model may, or may not,
hold under complex dynamic trade-offs arising from the fully dynamic version
of our model, to which we now turn.
4. The Impact of Bank Regulation
In this section we illustrate the results of the calibration and simulation of the
model.
4.1 Calibration
Our calibration is based on three sets of parameters, summarized in Table 1.
The first set comprises parameters of the credit shock and deposits process. We
estimated the VAR in Equation (15) using U.S. yearly aggregate time series for
the period 1983–2009 for the entire universe of banks included in the Federal
Reserve Call Reports constructed by Corbae and D’Erasmo (2011). The shock
process was proxied by the return on bank investments before taxes, given by
the ratio of interest and noninterest revenues to total lagged assets. As can be
seen in Table 1, the shock process exhibits high persistence and the correlation
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The Review of Financial Studies / v 0 n 0 2014
Table 1
Base case model parameters
κ
Z
annual persistence of the credit shock 0.88
σ
Z
annual conditional SD of the credit shock 0.0139
Z unconditional average of the credit shock 0.0717
κ
D
annual persistence of the log of deposits 0.98
σ
D
annual conditional SD of the log of deposits 0.0209
D unconditional average of deposits $2
ρ correlation between log-deposit and credit shock −0.85
β
time discount factor 0.95
γ
1
constant price of risk parameter 3.22
γ
2
time varying price of risk parameter −15.30
r
f
annual rate on bonds 2.5%
r
d
annual rate on deposits 0%
τ
+
corporate tax rate for positive earnings 15%
τ
−
corporate tax rate for negative earnings 0%
δ annual percentage of reimbursed loan 20%
η bankruptcy costs 0.10
λ
flotation cost for equity 0.06
α
return to scale for loan investment 0.90
m
+
unit price for loan investment 0.04
m
−
unit price for loan fire sales 0.05
k
percentage of loans for capital regulation 4%
liquidity coverage ratio 20%
with the process of (log) deposit is negative. Estimates of the autocorrelation
process for (log) deposit produced estimates closed to unity, indicating the
possibility that such process has a unit root. To guarantee convergence of the
fixed point algorithm, we set this parameter equal to 0.98.
The second set of parameters is taken from previous research. The parameters
of the stochastic process of the macroeconomic risk factors u, κ
u
and σ
u
are
set equivalent, on an annual basis, to the quarterly values of 0.98 and 0.007
reported by Jones and Tuzel (2013). The parameters of the stochastic discount
factor, γ
1
=3.22 and γ
2
= −15.30, are also taken from Jones and Tuzel (2013).
The annual discount factor β is 0.95, equal to that used by Zhu (2008) and
Cooley and Quadrini (2001). The rate, r
f
, is set to 2.5% and the deposit rate,
r
d
, is set to zero. These values are consistent with the average effective cost of
funds documented by Corbae and D’Erasmo (2011). With regard to corporate
taxation, recall that the tax function is defined by the marginal tax rates, τ
+
and
τ
−
, for positive and negative income, respectively. Because we do not explicitly
consider dividend and capital gain taxation for shareholders, or interest taxation
for depositors and bond holders, the two marginal rates for corporate taxes need
to be considered net of the effect of personal taxes. For this reason, we choose
τ
+
= 15%, which is close to the values determined by Graham (2000) for the
marginal tax rate. The marginal tax rate for negative income is τ
−
= 0 to allow
for convexity in the corporate tax schedule.
Furthermore, the proportional bankruptcy cost is η =0.10. This is a value
close to the (structural) estimate of 0.104 based on data for U.S. nonfinancial
firms found by Hennessy and Whited (2007). Because this estimate is based
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Microprudential Regulation in a Dynamic Model of Banking
on nonfinancial firms, it can be viewed as a lower bound for bankruptcy costs
incurred in the financial sector. The annual percentage of reimbursed loan is
20%, so that the average maturity of outstanding loans is four years, in line
with the assumption made by Van den Heuvel (2009). The underwriting cost
for seasoned equity issuance is 6%, not far from the estimates provided by
Altinkilic and Hansen (2000) and Hennessy and Whited (2005).
The revenue function from loan investment is π (L)=L
α
as in Zhu (2008),
and our base case value for α is set equal to 0.90, which is a value in line
with that used in other studies. Lastly, we obtain m
+
=0.04 and m
−
=0.05 by
matching two moments from empirical data. The first moment is the average
ratio of bank credit over deposits, where bank credit includes loans and other
financial investments. From our dataset, this ratio is 1.271. The second moment
we match is bank’s book leverage, defined as deposits plus other financing
liabilities over loans and other financial investments. In the data, the average
book leverage is 0.89. The corresponding unconditional moments from a Monte
Carlo simulation of the model with the selected parameters are, respectively,
1.1153 and 0.9043. Hence, our calibration delivers per-unit loan liquidation
costs higher than per-unit costs of loan extensions (m
−
>m
+
).
The third set of parameters is based on regulatory prescriptions. These is
the ratio of capital to risk-weighted assets and the liquidity coverage ratio. The
benchmark capital ratio k is set equal to 4%, whereas the benchmark liquidity
ratio is set equal to = 20%.
The relationships between the credit and liquidity shocks affecting the
banking system, as well as the systematic (macroeconomic) risk factor and the
idiosyncratic factor, are derived in Appendix B and illustrated in Figure 3.As
expected, credit risk is positively correlated with the systematic risk factor,
as credit quality and loan demand increase in an upturn, and decline in a
downturn. On the other hand, liquidity risk, as captured by the dynamics
of insured deposits, is mildly negatively correlated with the systematic risk
factor. This pattern is consistent with an increase in savings in a downturn, as
insured deposits are a component of savings, and with the reallocation of agents’
portfolios towards safer assets. Lastly, the correlations between idiosyncratic
risk and credit and liquidity risks are fairly small, slightly negative for the
former, and almost null for the latter.
4.2 Bank policies along the business cycle
The correlations between optimal bank policies and the systematic (macroe-
conomic) risk factor, viewed as a proxy measure of the business cycle, are
summarized in Figure 4 and Tables 2–4. Five cases are considered: unregulated
banks, banks subject to capital requirements only, banks subject to both capital
and liquidity requirements, banks subject to the PCA only, and bank subject to
the PCA and capital requirements. Specifically, Figure 4 depicts lending and
debt policies for a given set of states centered on the steady-state (unconditional)
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The Review of Financial Studies / v 0 n 0 2014
−0.1
−0.05
0
0.05
0.1
−0.1
−0.05
0
0.05
0.1
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
idiosyncratic risk
systematic risk
Credit Shock
−0.1
−0.05
0
0.05
0.1
−0.1
−0.05
0
0.05
0.1
0.2
0.4
0.6
0.8
1
idiosyncratic risk
systematic risk
log−Deposit
Figure 3
Credit shock and deposit against systematic and idiosyncratic risk
The values are from the numerical solution of the model using nine points for
u and eleven points for v, based
on the estimated parameter values in Appendix B.
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Microprudential Regulation in a Dynamic Model of Banking
0.06 0.04 0.02 0 0.02 0.04 0.06
4
3.5
3
2.5
2
1.5
1
0.5
0
macroeconomic risk
B
*
Panel A: bond policy and macroeconomic risk
Non
Regulated
Capital Requirement
Capital and Liquidity Requirement
PCA
0.06 0.04 0.02 0 0.02 0.04 0.06
0.6
0.8
1
1.2
1.4
1.6
1.8
macroeconomic risk
L*/(L(1
))
Panel B: loan investment and macroeconomic risk
Non
Regulated
Capital Requirement
Capital and Liquidity Requirement
PCA
Figure 4
Bank’s policy
This figure illustrates the impact of regulatory restrictions on the bank’s policy related to loan investment
represented by the ratio
L
∗
/L(1− δ) and to short-term investment and financing with bonds, B
∗
, for the
nonregulated case, for the cases with capital constraint, with both capital and liquidity constraints altogether,
and the PCA case. These values are plotted against the macroeconomic risk factor,
u, in the upper panels and the
idiosyncratic risk factor,
v, in the remaining panels, and are obtained assuming that the bank is currently at the
steady state (so that the credit shock is 0.0717, and the deposits from the previous date are
D =2, respectively),
whereas
B =0, and L =4.7 so that the current bank capital (right before making the decision) is K =2.7.The
investment and financing policies are given by averaging out idiosyncratic risk when plotted against
u and
averaged across the systematic risk when plotted against
v. These results are based on the numerical solution of
the valuation problem in (18), as described in Appendix C, based on the parameter values in Table 1.
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0.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05
3
2.5
2
1.5
1
0.5
idiosyncratic risk
B
*
Panel C: bond policy and idiosyncratic risk
Non
Regulated
Capital Requirement
Capital and Liquidity Requirement
PCA
0.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
idiosyncratic risk
L*/(L(1 ))
Panel D: loan investment and idiosyncratic risk
Non
Regulated
Capital Requirement
Capital and Liquidity Requirement
PCA
Figure 4
Continued
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Microprudential Regulation in a Dynamic Model of Banking
median values.
15
Tables 2–4 are based on Monte Carlo simulation (the design
of the simulation exercise is detailed in Appendix C). In these tables we report
bank lending, capital ratios, and liquidity ratios for solvent banks sorted against
quartiles of total risk (the sum of systematic and idiosyncratic shocks, u+v). In
the top panel of each table the sorting of these variables is based on the whole
sample. The influence of the business cycle is obtained by sorting variables
conditional on the state of the economy being in a business cycle upturn (with
u=0.0352), or in a downturn (with u = −0.0352).
4.2.1 Unregulated banks. As shown in Figure 4 and Table 2, lending and
debt are positively correlated with the systematic shock. Recall that such a
shock translates into a positive credit shock, Z, and a negative liquidity shock,
because deposits are negatively correlated with the systematic shock. Thus, a
positive systematic shock prompts banks to increase lending, but also increases
their need of financing when deposits become relatively scarce, inducing them
to increase debt as well. Thus, a positive correlation between lending, debt, and
the systematic shock is a feature of banks’ optimal policies independently of
bank regulations.
As shown in Table 3, capital ratios are positively correlated with the
systematic shock, owing to lower deposits in an upturn (deposits are a negative
component of the ratio), whereas changes in loans and short-term debt almost
offset each other.
16
By contrast, as shown in Table 4, liquidity ratios are
negatively correlated with the systematic shock, because in a downturn debt
declines and deposits increase. The opposite holds in an upturn.
4.2.2 Capital requirements. Similarly to unregulated banks, banks under the
base capital requirement exhibit a positive correlation between the systematic
shock, lending, and debt (see Figure 4). However, banks invest more in loans
and choose less debt than their unregulated counterparts for all realizations of
the systematic shock. This implies that banks subject to a capital requirement
that we term “mild” find it optimal to increase loans at a rate proportionally
higher than the capital ratio coefficient (see Equation (25)), rather than by
reducing lending. In essence, a mild capital requirement reduces the rate of
return on short-term debt relative to the expected returns on loans, prompting a
higher investment in loans. In turn, higher revenues generated by higher lending
are employed in building capital. Interestingly, in a downturn banks subject to
a mild capital requirement reduce lending less than that of unregulated banks.
15
The analysis is centered at the steady states for both deposits (D =2) and credit shock (Z =0.0717), while choosing
B =0 to avoid the impact of current liquidity, and L =4.7, which is close to the unconditional median of L for
several versions of the model. As a result, bank’s capital is
K =2.7.
16
Note that (book) capital ratios may be negative for some realizations of systematic and idiosyncratic shocks.
Because there is no restriction on book capital and, as is standard in the literature, the (concave) loan function
has the book value of loans as an input, in the presence of debt a bank can be operating as long as the market
value of its capital is positive, even though its book capital can be negative.
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Table 2
Loan investment policy
Unconditional
e
u+v
0.99 1.01 1.02 1.06
Unregulated 1.12 1.01 0.87 0.73
Capital 1.18 1.13 1.00 0.91
Cap + Liq 0.92 0.81 0.76 0.63
PCA 1.23 1.13 1.03 0.86
PCA + Cap 1.18 1.13 1.01 0.87
Upturn
e
u+v
1.02 1.04 1.05 1.09
Unregulated 2.14 1.85 1.77 1.53
Capital 2.07 2.03 1.83 1.68
Cap + Liq 1.50 1.38 1.31 0.99
PCA 2.07 2.02 1.83 1.67
PCA + Cap 2.07 2.03 1.83 1.68
Downturn
e
u+v
0.95 0.97 0.98 1.02
Unregulated −0.04 −0.08 −0.08 −0.07
Capital 0.06 −0.03 −0.06 −0.07
Cap + Liq 0.07 −0.03 −0.02 −0.03
PCA 0.06 −0.05 −0.08 −0.09
PCA + Cap 0.06 −0.03 −0.05 −0.06
This table shows the simulated ratios L
∗
/((1− δ)L), where L
∗
is the optimal solution for a solvent bank, sorted
against quartiles of total risk,
e
u+v
, under the unregulated, the capital requirement, the capital and liquidity
requirement cases, the PCA case, and the one with PCA and capital requirement. We report the values from
the total sample (top panel), the values conditional on the economy being in an upturn, when the systematic
risk
u=0.0352 (middle panel), or in a downturn, when u =−0.0352 (bottom panel). These results are based on
the numerical solution of the valuation problem in Equation (18) and the simulation of the optimal solution, as
described in Appendix C, based on the parameter values in Table 1. The table presents the averages across the
economies of the time-series averages of the cross-sectional sortings.
This implies that risk-based capital regulation does not necessarily amplify the
contraction of lending in a downturn.
These mechanisms are reflected in the evolution of the capital and liquidity
ratios along the cycle. A mild capital requirement encourages banks to build-
up capital buffers in upturns to reduce the risk of costly loan liquidations
in downturns: this is witnessed by the lower correlation (to the point of
becoming almost zero) between capital ratios and the systematic shock relative
to unregulated banks (see Table 3). On the other hand (see Table 4), under
a mild capital requirement, liquidity ratios and the systematic shock become
more negatively correlated relative to unregulated banks. This occurs because
in a downturn banks use retained earnings and the financial resources freed by
the reduction in lending to strengthen their liquidity position. In other words, a
mild capital requirement contributes to strengthen banks’ liquidity position in
a downturn.
4.2.3 Liquidity requirements. The addition of a liquidity requirement to a
capital requirement changes bank optimal policies significantly. As shown in
Figure 4 and Table 2, banks’ lending and debt shrink for every realization of
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Microprudential Regulation in a Dynamic Model of Banking
Table 3
Capital ratios
Unconditional
e
u+v
0.99 1.01 1.02 1.06
Unregulated −5.75 −3.30 −1.10 −1.79
Capital 0.11 0.15 0.15 0.14
Cap and Liq 0.33 0.36 0.34 0.32
PCA 0.06 0.04 0.02 0.01
PCA + Cap 0.11 0.16 0.15 0.15
Upturn
e
u+v
1.02 1.04 1.05 1.09
Unregulated 0.18 0.14 0.12 0.06
Capital 0.19 0.18 0.15 0.13
Cap and Liq 0.62 0.60 0.59 0.53
PCA 0.19 0.19 0.15 0.13
PCA + Cap 0.19 0.18 0.15 0.13
Downturn
e
u+v
0.95 0.97 0.98 1.02
Unregulated −29.00 −23.49 −13.04 −14.29
Capital 0.06 0.20 0.19 0.19
Cap and Liq 0.07 0.21 0.20 0.19
PCA −0.07 −0.14 −0.17 −0.21
PCA + Cap 0.06 0.20 0.19 0.19
This table shows the simulated capital ratios (i.e., bank capital over loans, or K
∗
/L
∗
=(L
∗
+B
∗
−D
)/L
∗
, where
(L
∗
,B
∗
) is the optimal solution for a solvent bank and D
is the new possible level of deposits), sorted against
quartiles of total risk,
e
u+v
, under the unregulated, the capital requirement, the capital and liquidity requirement
cases, the PCA case, and the one with PCA and capital requirement. We report the values from the total sample
(top panel), the values conditional on the economy being in an upturn, when the systematic risk
u=0.0352 (middle
panel), or in a downturn, when
u= −0.0352 (bottom panel). These results are based on the numerical solution of
the valuation problem in Equation (18) and the simulation of the optimal solution, as described in Appendix C,
based on the parameter values in Table 1. The table presents the averages across the economies of the time-series
averages of the cross-sectional sortings.
the systematic shock, with the reduction in debt being the most dramatic. Thus,
the liquidity requirement turns out to be far more restrictive than the capital
requirement, forcing banks to reduce both debt and lending. Moreover, relative
to the case of banks subject to capital regulation only, the correlation between
lending and the systematic shock is drastically reduced.
The dominant tightness of the liquidity requirement is also reflected in the
evolution of capital ratios along the business cycle. As shown in Table 3,
capital ratios become inflated, being pushed up by a relatively large net
bond holding (the numerator of the ratio) and a lower investment in loans
(the denominator of the ratio). Note that the mechanism driving this result
is totally different from that induced by capital regulation: in that case, the
capital ratio is pushed up by retained earnings generated by higher revenues
from lending, while in this case, the capital ratio is mainly pushed up by a
significant reduction of lending. Therefore, under a liquidity requirement the
capital ratio and the systematic shock become significantly more positively
correlated. This increased correlation is the result of banks being forced to
reduce lending significantly in an upturn. In a downturn, however, capital ratios
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Table 4
Liquidity ratios
Unconditional
e
u+v
0.99 1.01 1.02 1.06
Unregulated −535.32 −270.79 −182.18 −129.10
Capital −534.41 −270.23 −181.53 −128.61
Cap and Liq 25.10 8.99 9.05 6.22
PCA −535.18 −270.67 −181.97 −128.52
PCA + Cap −534.41 −270.22 −181.51 −128.22
Upturn
e
u+v
1.02 1.04 1.05 1.09
Unregulated −26.32 −24.45 −23.35 −21.82
Capital −26.56 −25.43 −24.38 −23.01
Cap and Liq 0.91 0.75 0.91 0.70
PCA −26.77 −25.54 −24.45 −22.86
PCA + Cap −26.56 −25.42 −24.36 −22.88
Downturn
e
u+v
0.95 0.97 0.98 1.02
Unregulated −0.26 −0.12 −0.05 −0.01
Capital 2.59 3.01 3.20 3.34
Cap and Liq 3.04 3.40 3.42 3.52
PCA 2.53 2.98 3.15 3.31
PCA + Cap 2.58 3.01 3.21 3.39
This table shows the simulated liquidity ratios (i.e., end-of-period total cash available in the worst-case scenario
over the end-of-period net cash outflows due to a variation in deposits, or
(δL
∗
+π(L
∗
)Z
d
−T (y
min
)+B
∗
(1+
r
f
))/(D
(1+ r
d
)−D
d
), where (L
∗
,B
∗
) is the optimal solution for a solvent bank and D
is the new possible
level of deposits, sorted against quartiles of total risk,
e
u+v
, under the unregulated, the capital requirement, the
capital and liquidity requirement cases, the PCA case, and the one with PCA and capital requirement. We report
the values from the total sample (top panel), the values conditional on the economy being in an upturn, when
the systematic risk
u=0.0352 (middle panel), or in a downturn, when u=−0.0352 (bottom panel). These results
are based on the numerical solution of the valuation problem in Equation (18) and the simulation of the optimal
solution, as described in Appendix C, based on the parameter values in Table 1. The table presents the averages
across the economies of the time-series averages of the cross-sectional sortings.
are not significantly different from those attained by banks subject only to
capital regulation.
As shown in Table 4, liquidity ratios are significantly higher than the
prescribed level ( = 20%), because the (shadow) cost associated with the
liquidity constraint forces banks to hold precautionary cash to avoid hitting
that constraint. On the other hand, liquidity ratios and systematic shocks are
less negatively correlated, as witnessed by a comparison of the values under
an upturn and a downturn. As a result, in a downturn the increase in liquidity
holdings does not lead to liquidity ratios significantly different from those
attained by banks subject to capital regulation only.
4.2.4 PCA. Under the PCA, the correlations between lending, debt, capital,
and liquidity ratios are similar to those under capital regulation. However, as
shown in Table 3, the correlation of the capital ratio with the systematic shock
is higher than that under capital regulation, primarily owing to capital ratios
significantly lower in downturns. The lower capital ratios under the PCA in
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Microprudential Regulation in a Dynamic Model of Banking
a downturn indicate that banks will implement a lower reduction in lending,
while keeping their options open to either liquidate loans or issue equity in
the event that the realization of current earnings is unfavorable. In essence,
in a downturn the PCA appears to provide banks some flexibility, which is
unavailable under an (unconditional) ex ante capital requirement. In the next
section, we will show that the closure rule embedded in the PCA limits banks’
incentives to abuse such flexibility.
4.2.5 Summary. Our analysis of banks’ optimal policies along the business
cycle delivers two results. First, relative to unregulated banks, mild capital
requirements and the PCA reduce the correlation between lending, debt, and
capital ratio and macroeconomic shock, and makes the liquidity ratio more
negatively correlated with the macroeconomic risk, relative to unregulated
banks. Specifically, in an upturn banks increase lending at a rate consistent
with the buildup of capital buffers through retained earnings generated by
higher lending revenues, whereas in a downturn they reduce debt by building
up liquidity buffers. Perhaps not surprisingly, these results differ from those
obtained in static or semistatic models that do not allow banks to issue equity or
debt and to manage retained earnings. For example, procyclicality is enhanced
by capital requirements in the model of Repullo and Suarez (2013), who
consider short-lived banks that do not manage retained earnings, and are not
allowed to issue either debt or equity.
Second, the addition of liquidity requirements to capital requirements
reduces the procyclicality of lending and debt but also increases the
procyclicality of capital ratios. Yet capital buffers in downturns are not
significantly different from those resulting from banks subject only to capital
regulation. Thus, the reduction in lending procyclicality is skewed toward
upturns, significantly hampering lending.
4.3 Bank policies, efficiency, and welfare in the steady state
We now turn to the analysis of bank optimal policies and the metrics of
efficiency and welfare in the steady state obtained through Monte Carlo
simulation of the numerical solution of the bank’s optimal program (see
Appendix C for details). Table 5 presents statistics of policies, assets and
liabilities, welfare and efficiency metrics, and default frequency, when banks
are not regulated, when they are subject to capital requirements only, and when
they are subject to both capital and liquidity requirements. Table 6 presents a
comparison of the same statistics for banks under the PCA, and for banks subject
to the PCA superimposed on capital and liquidity requirements. In Table 7 we
report the same statistics under perturbed parameters of underwriting costs,
loan liquidation costs, and the degree of bank’s maturity transformation.
4.3.1 Capital requirements. Compared with the unregulated case, Table 5
shows that banks operating under the base case capital requirement (k = 4%)
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Table 5
The impact of bank regulation
Unreg. Capital Capital and Liquidity
k =4% k = 12% k =4% k =12% k =4%
= 20% = 20% = 50%
Loans (book) 4.41 5.08 4.96 3.71 3.75 3.71
Net bond holdings (book) −2.75 −2.30 −2.05 0.34 0.32 0.38
Bank capital (book) −0.32 0.80 0.92 2.07 2.09 2.12
Equity (mkt) 6.97 7.32 7.36 7.65 7.66 7.69
Deposits (mkt) 1.89 1.89 1.89 1.89 1.89 1.89
Enterprise value (mkt) 11.70 11.61 11.40 9.29 9.33 9.29
Government value (mkt) 0.82 0.97 0.97 0.90 0.90 0.91
Social value (mkt) 12.52 12.58 12.37 10.19 10.23 10.19
Default (%) 1.30 0.00 0.00 0.00 0.00 0.00
The table presents different dimensions of the bank, based either on book or on market values. The columns
represent different choices of parameters: the “Unregulated” case is obtained with the parameters in Table 1.
The case with capital constraint (“Capital”) has either
k =4%or k =12%. The case with both capital and liquidity
restrictions (“Capital & Liquidity”) is obtained for the base case parameters (
k =4% and = 20%), and two
alternative combinations, with
k =12% and = 20%, and with k =4% and =50%, respectively. These results
are based on the numerical solution of the valuation problem in Equation (18) and the simulation of the
optimal solution, as described in Appendix C, based on the parameter values in Table 1. The table presents the
averages across the simulated economies of the time-series averages of the cross-sectional averages (computed
on nondefaulted instances) of the different metrics.
Table 6
Prompt corrective action
Unreg. PCA Cap. PCA+ Cap. Cap.&Liq. PCA+
Cap.&Liq.
Loan (book) 4.41 5.12 5.08 5.03 3.71 3.72
Net bond holdings (book) −2.75 −2.38 −2.30 −2.25 0.34 0.34
Bank capital (book) −0.32 0.77 0.80 0.80 2.07 2.07
Equity (mkt) 6.97 7.46 7.32 7.30 7.65 7.65
Deposits (mkt)
1.89 1.88 1.89 1.89 1.89 1.89
Enterprise value (mkt) 11.70 11.81 11.61 11.53 9.29 9.30
Government value (mkt) 0.82 0.97 0.97 0.98 0.90 0.91
Social value (mkt) 12.52 12.78 12.58 12.50 10.19 10.20
Default (%) 1.30 3.71 0.00 0.00 0.00 0.00
PCA frequency (%) –
0.27 – 0.02 – 0
The table presents different dimensions of the bank, based either on book or on market values. The columns
offer a comparison among the three cases (unregulated bank, capital requirement, and capital plus liquidity
requirement) without and with the Prompt corrective action (PCA). These results are based on the numerical
solution of the valuation problem in Equation (18) and the simulation of the optimal solution, as described in
Appendix C, based on the parameter values in Table 1. The table presents the averages across the simulated
economies of the time-series averages of the cross-sectional averages (computed on nondefaulted instances) of
the different metrics.
lend more and have less debt. Remarkably, the steady-state percent increase
in lending relative to the unregulated case is a significant 15%. Because
deposits are not a control variable and follow the same exogenous process
of the unregulated case, the bank can fund these additional loans by reducing
payouts and increasing retained earnings and equity issuance. Specifically, from
Equations (12) and (13), given the choice of L
t+1
and B
t+1
, more earnings are
retained from W
t
, or shares are issued (incurring underwriting costs) if U
t
is
negative.
As a result of these optimal policies, banks also hold a higher capital ratio
than that prescribed by regulation. This is because the positive shadow price of
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Table 7
The role of equity issuance costs, adjustment costs and maturity transformation
Unreg. base
λ=0 λ = .2 m
−
= .08 δ = .1
Capital
Loans (book) 4.41 5.08 5.13 4.98 5.09 7.38
Net bond holdings (book) −2.75 −2.30 −2.32 −2.24 −2.31 −2.98
Bank capital (book) −0.32 0.80 0.83 0.75 0.80 2.41
Equity (mkt) 6.97 7.32 7.35 7.22 7.29 10.30
Deposits (mkt) 1.89 1.89 1.89 1.89 1.89 1.89
Enterprise value (mkt) 11.70 11.61 11.65 11.44 11.57 15.26
Government value (mkt) 0.82 0.97 0.99 0.95 0.97 1.37
Social value (mkt) 12.52 12.58 12.64 12.40 12.55 16.63
Capital and liquidity
Loans (book) 4.41 3.71 3.85 3.57 3.69 5.64
Net bond holdings (book) −2.75 0.34 0.31 0.38 0.35 0.64
Bank capital (book) −0.32 2.07 2.18 1.97 2.06 4.30
Equity (mkt) 6.97 7.65 7.80 7.49 7.64 10.86
Deposits (mkt) 1.89 1.89 1.89 1.89 1.89 1.89
Enterprise value (mkt) 11.70 9.29 9.47 9.09 9.27 12.20
Government value (mkt)
0.82 0.90 0.93 0.87 0.90 1.34
Social value (mkt)
12.52 10.19 10.40 9.96 10.17 13.54
The table shows two panels: on top the case of a bank with capital requirement and at the bottom the case of a
bank subject to both capital and liquidity restrictions. The table presents different dimensions of the bank, based
either on book or on market values. The columns represent different choices of parameters: the column denoted
“base” is the base case, with the parameters in Table 1. The others are obtained by changing only the parameter
used to denominate the column (e.g., in “
λ=0” all the parameters are at the base case value, but λ, which is set
to zero). These results are based on the numerical solution of the valuation problem in Equation (18) and the
simulation of the optimal solution, as described in Appendix C, based on the parameter values in Table 1.The
table presents the averages across the simulated economies of the time-series averages of the cross-sectional
averages (computed on nondefaulted instances) of the different metrics.
the capital constraint forces banks to manage their earnings and investments so
as to maintain a capital buffer that minimizes the risk that the constraint is hit. In
such an event, it would become too expensive to either liquidate loans or inject
new equity capital to comply with the regulatory restriction.
17
Importantly,
capital regulation results in a bank with a lower probability of default than in
the unregulated case. Thus, a capital requirement is unambiguously successful
in abating default risk under deposit insurance.
With regard to the efficiency and welfare metrics, a mild capital requirement
results in a small decrease in banks’ enterprise value (about 1%), which is offset
by an increase in government value. The higher government value stems from
higher tax receipts accruing from a larger taxable profit base, as well as from a
lower probability of bank default, which reduces expected bailout costs. As a
result, the welfare metrics is larger than that of the unregulated case.
However, the benefits of capital regulation can turn into welfare-reducing
costs if such regulation becomes too stringent. An increase in the capital
requirement (from k =4% to k = 12%) results in reductions of lending and
debt, as well as a reduction in efficiency and in welfare metrics. Recall that
17
These findings reflect the well-known result that constraints may not be binding on equilibrium paths (see, e.g.,
Ayagari 1995), and are consistent with the empirical evidence presented by Flannery (2005) and Flannery and
Kasturi (2008).
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banks satisfy a mild capital requirement by an increase in loans financed with a
combination of debt and retained earnings. But when this requirement becomes
too stringent, such a strategy becomes too costly. In such a case, payouts
need to be significantly reduced, because either raising new equity capital
generates equity issuance costs or reducing investment in unfavorable states
of nature creates loan liquidation costs. These costs compel banks to reduce
both lending and debt. Thus, there exists an inverted U-shaped relationship
between lending and welfare associated with capital requirements. This further
suggests the existence of optimal levels of regulatory capital as a function
of banks’ characteristics. In other words, under a mild capital requirement,
banks optimally choose to increase lending so as to generate higher revenues
supporting the building up of capital; if capital requirements are too high,
however, banks find it optimal to satisfy them by reducing lending. This latter
reduction also impacts negatively on welfare, as it reduces both the enterprise
and government values of bank activities.
4.3.2 Liquidity requirements. When liquidity requirements are added to
capital requirements, Table 5 shows that lending contracts dramatically relative
to the bank subject only to the base capital requirement (by about 26%). The
efficiency and welfare metrics declines significantly as well (by about 20%).
As noted earlier, the liquidity requirement generates overbloated banks’ book
capital ratios. Such high ratios may be viewed as an indication of safe, but very
inefficient, banks.
Furthermore, an increase in the capital requirement (from k =4%tok = 12%)
for the bank already subject to a liquidity requirement ( = 20%), implies only
small positive changes in lending and the efficiency and welfare metrics.
Similarly, an increase in the liquidity requirement (from = 20% to = 50%,
with constant k = 4%) implies relatively small changes in lending, efficiency,
and welfare. These results are due to the significant stringency of the liquidity
requirement, which makes banks’ optimal policies relatively insensitive to
changes in capital requirements.
4.3.3 PCA. As shown in Table 6, the PCA prompts banks to increase lending
relative to the unregulated case, as in the case of banks subject to capital
regulation only. Importantly, however, the PCA also achieves higher lending,
capital, and higher levels of the efficiency and welfare relative to banks
subject only to a capital requirement, even though default (in the form of
bank closure) is more frequent. Notably, our calibration implies that immediate
capital restoration when 0 <V <kL is triggered only 0.27% of the times, so
banks rarely incur this additional burden. This is because the PCA generates a
shadow cost of violation large enough to prompt banks to choose policies that
minimize the probability of these ex post violations. In essence, the PCA de
facto implements a contingent transfer of control of banks from the shareholders
to the deposit insurance authority that allows shareholders to get the upside
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potential of investment, while giving the controlling authority a way to mitigate
the welfare costs of liquidations and bailouts.
The superiority of the PCA over (noncontingent) capital and liquidity
requirements stems in part from the resolution of the difficulties introduced
by the absence of contingencies in deposit payments due to deposit insurance.
In setups similar to ours, when demandable deposits are either fully contingent
on the realization of the states, as in De Nicolò (1996), or when contingencies
are introduced, allowing deposit runs, as in Allen and Gale (1998), an optimal
banking allocation is achieved. Deposit insurance eliminates state contingency,
and noncontingent capital and liquidity requirements do not replace the lack of
contingencies of deposit payments. By contrast, the PCA is a state-contingent
intervention scheme that substitutes for the lack of contingencies in demandable
deposits, with these contingencies being induced now by a regulator acting as
a representative of depositor. Under deposit insurance, state contingencies are
moved to states of the world in which there is still equity value remaining.
It is interesting to assess the impact of the PCA depending on whether a
bank is hit by a systematic or idiosyncratic shock. In the sample generated
by our simulation, we find that the PCA is never triggered for any realization
of the idiosyncratic shock when the systematic shock is positive. Moreover,
conditional on a negative systematic shock, when the idiosyncratic shock
is positive, the PCA is triggered with a low frequency (0.08%), whereas it
occurs more frequently when such a shock is negative (0.19%). Therefore, the
PCA is triggered differently according to whether shocks are systematic or
idiosyncratic: bank closures and restructuring occur more often in the case of
an adverse systematic shock, whereas forced recapitalizations and restructuring
of payouts are more frequent in the case of an adverse idiosyncratic shock.
Finally, when the PCA is introduced in addition to banks subject to capital
requirements, banks are forced to hold a capital ratio higher than k in all states,
and this requirement is made more demanding by the PCA when V falls short
kL. Relative to the case of capital requirement only, the addition of the PCA
results in small reductions in lending, efficiency, and welfare metrics. Moreover,
the marginal effect of PCA is negligible when applied to banks that are already
subject to capital and liquidity requirements, owing primarily to the dominant
stringency of liquidity requirements, which nullify the benefits of the PCA.
4.3.4 The role of issuance costs, loan liquidation costs, and the degree
of maturity transformation. Here, we examine the impact of capital and
liquidity requirements under different parameters of underwriting costs, loan
liquidation costs, and the degree of bank’s maturity transformation. In Table 7
we consider: no issuance costs, as well as an increase of λ from the benchmark
value 0.06 to 0.2, an increase of loan liquidation costs, m
−
, from 0.05 to 0.08,
and a reduction of δ, the parameter gauging maturity transformation, from 20%
to 10%, indicating a longer loan maturity (from 4 years to 9 years).
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To what extent do underwriting costs contribute to determine the inverted
U-shaped relationship between capital requirements, lending, and welfare?
With no issuance costs, Table 7 shows that the inverted U-shaped relationship
between lending, welfare, and capital requirements is strengthened. This is
perhaps not surprising, because banks can cope with distress at a lower cost
and find it optimal to increase lending even under higher levels of capital
requirements. On the other hand, increasing issuance costs relative to the
benchmark generates a decline in lending, enterprise, and social values, but
this decline is relatively small. Note that these results hold for banks subject
to capital requirements only, as well as for banks subject to both capital and
liquidity requirements. Thus, the role of issuance costs in determining the costs
of more stringent capital requirements appears relatively less important than
the management of retained earnings.
18
To what extent do different levels of loan liquidation costs change the
impact of capital and liquidity requirements? When we increase the relevant
parameter (m
−
) in our simulation, lending, enterprise, and social values do
not change significantly, relative to the base case. This result reveals again the
key role of retaining earnings in supporting bank optimal choices: when facing
higher loan liquidation costs, a bank would respond by increasing lending and
retained earnings, with the latter increase aimed at minimizing the probability
of incurring large loan liquidation costs in the event of distress. However, the
addition of liquidity requirements to capital requirements still results in lower
lending and a worsening of the efficiency and welfare metrics, as in all previous
simulations. Therefore, the benefits of mild capital requirements may be even
more important when loan liquidation costs are high.
19
Lastly, the role of liquidity requirements in hampering the maturity
transformation function of banks is starkly illustrated by the case in which
banks have a longer loan maturity. Under capital requirements only, banks with
a larger maturity mismatch undertake a more intense maturity transformation,
as witnessed by higher levels of lending relative to banks with a milder maturity
mismatch. When liquidity requirements are added to capital requirements,
however, the reduction in lending, efficiency, and welfare is significantly greater
than that witnessed by banks whose loan maturity is shorter under capital
requirements only. Again, liquidity requirements are the more detrimental to
18
We run simulations treating equity issuance costs as undervaluation costs. In this case, although underpricing of
newly issued equity affects current shareholders, it benefits new shareholders because they can buy a share in the
bank’s capital for a lower price. Therefore, these costs are not deadweight costs but just a wealth transfer from
old to new shareholders, and are added to the welfare function. The qualitative results we obtain are essentially
the same as those just presented.
19
As a robustness check, we considered loan liquidation costs as fire sales costs, since these costs have been
identified as one of the key sources of systemic risk in the financial crisis of 2007–2009 (see, e.g., Kashyap,
Brener, and Goodhart 2011). The welfare criterion now includes these costs, because fire sales affect negatively
current shareholders but benefit investors who can buy assets at low fire sales prices. Although all other statistics
remain unchanged, the social values are higher than those obtained in Table 5, but still hump shaped.
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Microprudential Regulation in a Dynamic Model of Banking
lending, efficiency, and welfare, when a banks’ transformation of short-term
liabilities into longer-term assets is more intense.
4.3.5 Summary. Mild capital requirements induce banks to increase lending,
resulting in higher welfare relatively to unregulated banks, but these welfare
gains are destroyed when requirements become too strict. This result is
consistent with the finding of relatively high welfare costs associated with
capital regulation by Van den Heuvel (2008). The resulting inverted U-shaped
relationship between lending, welfare, and the level of capital requirements is
also robust to different levels of underwriting costs and loan liquidation costs.
When liquidity requirements are added to capital requirements, they undo
the benefits of mild capital requirements, because they prompt significant
reductions in lending, efficiency, and welfare. In essence, liquidity requirements
constrain banks’ maturity transformation function, forcing them to underinvest
in lending and overinvest in unproductive liquidity buffers.
Importantly, the PCA as a contingent resolution procedure dominates
(unconditional) capital and liquidity requirements in terms of lending,
efficiency, and welfare, because they provide stronger incentives for banks
to manage risk relative to requirements set ex ante. This is accomplished by
introducing contingencies based on observed equity, which effectively replace
those contingencies in deposit payments that might improve bank efficiency
and welfare, but that are ruled out by deposit insurance.
5. Concluding Remarks
In this paper we assess the impact of microprudential regulatory restrictions
in a dynamic model of banks under deposit insurance that are exposed to
credit and liquidity risks arising from systematic and idiosyncratic shocks, that
undertake maturity transformation, that invest in risky loans, that issue secured
debt and costly equity, and that may face financial distress. We found that capital
requirements do not appear to increase lending procyclicality and give banks
incentives to accumulate liquidity buffers in a downturn. In the steady state, we
uncovered an inverted U-shaped relationship between bank lending, welfare,
and regulatory capital ratios. By contrast, liquidity requirements significantly
reduce lending, bank efficiency, and welfare. Importantly, the implementation
of contingent capital requirements with a bank closure rule embedded in the
PCA dominates both noncontingent capital and liquidity requirements in terms
of lending, bank efficiency, and welfare.
Overall, these results support the argument byAdmati et al. (2011) that capital
requirements can be a substitute for liquidity requirements. Yet the results do not
support proposals of sharp increases in the stringency of capital requirements.
Rather, the dominance of the PCA over noncontingent regulatory prescriptions
supports the desirability of recent proposals to introduce contingencies in the
regulation of capital structure, such as those advanced in the pioneering work
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by Flannery (2005, 2009), and more recently by Hart and Zingales (2011).
Introducing resolution procedures with appropriate contingencies related to
observed levels of capital, such as the PCA, might be a necessary step in
moving toward optimal bank regulation.
Several extensions of our model seem worth pursuing in future work. First,
note that in our model bank risk choices arise from the dependence of the
volatility of cash flows on the level of lending, owing to the concavity of the loan
revenue function. A risk-shifting problem would arise if equity holders decided
to liquidate loan and distribute dividends, appropriating value from debt holders
and depositors. However, the collateral constraint prevents such a behavior.
Our setup therefore differs from those analyzed by Blum (1999) and Calem
and Rob (1999), in which a bank makes a choice of a loan revenue distribution.
Incentives for risk shifting might be induced by capital requirements when
banks are poorly capitalized and the costs of issuing equity are too high or
prohibitive, leading banks to “gamble for resurrection.” For example, in Calem
and Rob (1999) dynamic (infinite-horizon) model, bank risk taking increases
if capital requirements are sufficiently high. Yet in their model equity issuance
is ruled out and the size of a bank is fixed, so that banks cannot liquidate
assets. In a finite-horizon setting, Calomiris, Heider, and Horeova (2012) show
examples in which liquidity requirements might be a substitute for capital
requirements. However, this occurs because there is an exogenously given
choice of default in the final period. In this case, a positive bond investment (as a
“liquid” component of capital) that could be used to preserve the (continuation)
value of a bank is ruled out. Extending our model by allowing banks to choose
a loan revenue distribution might be useful to assess the robustness of the
foregoing conclusions. In any event, a contingent resolution policy, such as the
PCA, would likely eradicate at the source the potential risk-shifting problems
pointed out by these contributions.
Second, it might be interesting to explore the role of market-based
capital requirements, as our valuation framework makes it feasible to clearly
distinguish book from market values of equity in equilibrium. Third, our
model does not capture endogenous liquidity runs, because the high realization
of deposit withdrawals are exogenous. Thus, endogenous liquidity runs and
banks that actively seek out liquidity are ruled out. Extending our model
to include such features might make it feasible to address the issue raised
by De Nicolò, Favara, and Ratnoski (2012), regarding the extent to which
regulations such as capital and liquidity requirements may be complements or
substitutes.
Appendix A. Properties of the Unregulated Bank Program
Compactness of the feasible set of the bank can be shown as follows. Given the strict concavity
of π(L), there exists a level L
u
such that π(L
u
)Z
u
−rL
u
= 0, where r (which can be either r
f
or r
d
) is the cost of capital of the marginal dollar raised either through deposits or short-term
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Microprudential Regulation in a Dynamic Model of Banking
financing.
20
Thus, any investment L>L
u
would be unprofitable. This establishes an upper bound
on the feasible set of L, given by [0,L
u
] for some L
u
. With an upper bound on L, and because the
stochastic process D has compact support, the collateral constraint sets a lower bound B
d
(i.e., an
upper bound on bond issuance). Specifically, this is obtained by putting D
d
in place of D
t
and L
u
in place of L
t
in Equation (10).
Lastly, an upper bound on B can be obtained assuming that the proceeds from risk-free
investments made by the bank are taxed at a higher rate than the personal tax rate and that
floatation costs are positive. Specifically, assume that the current deposits D are all invested in
short-term bonds, B, with no investments in loans. To further increase the investment in bonds
of one dollar, the bank must raise equity capital. A shareholder thus incurs a cost 1+λ, where λ
is the floatation cost. This additional dollar is invested at the rate r
f
, so that at the end of the
year, the proceeds of this investment that can be distributed are (1+ r
f
(1−τ
+
)). Alternatively,
the shareholder can invest 1+λ in a risk-free bond, obtaining (1+λ)(1+r
f
). Because τ
+
≥0
and λ≥ 0, then (1+λ)(1+r
f
)≥(1+r
f
(1−τ
+
)), there is no incentive of the bank to have a cash
balance larger than D as long as either λ or τ
+
are strictly positive. The foregoing argument
is made for simplicity. If the shareholders are taxed on their investment proceeds at a rate τ
p
,
they obtain (1+ λ)(1+r
f
(1−τ
p
)) from their investment in the risk-free asset. If τ
p
≤τ
+
, then
(1+λ)(1+r
f
(1−τ
p
))>(1+r
f
(1−τ
+
)), and the bank has no incentive to increase the investment
in risk-free bonds beyond D. Moreover, if floatation costs associated with equity issuance are
strictly increasing in the amount issued, no assumption about differential tax rates are needed to
establish an upper bound on B. In conclusion, the feasible set of the bank can be assumed to be
[0,L
u
]×[B
d
,B
u
].
Furthermore, standard arguments establish the existence of a unique value function E(x)=
E(L,B,D,u,v) that satisfies Equation (18) and is continuous in all its arguments. The existence
and uniqueness of the value function E follow from the contraction mapping theorem (theorem 3.2
in Stokey and Lucas 1989). The continuity of E follow from the continuity of e and the monotonicity
of the Markov transition function of the process (u,v).
Appendix B. The Dynamics of Deposits and Credit Shocks
Introducing a more compact notation, the joint dynamics of the systematic and idiosyncratic risk
is described by equation
s
t
= Hs
t−1
+ζ
t
, (A1)
where
H =
κ
u
0
0 κ
v
,ζ
t
=
σ
u
ε
u
t
σ
v
ε
t
, E
t
ζ
t+1
·ζ
t+1
=
σ
2
u
0
0 σ
2
v
= T
and (ε
u
t
,ε
t
) are standard normal variates with truncated support. The dynamics of (Z
t
,logD
t+1
)in
(15)is
X
t
= X+KX
t−1
+ξ
t
, (A2)
where
X
t
=
Z
t
logD
t+1
,
X =
(1−κ
Z
)Z
(1−κ
D
)logD
,K=
κ
Z
0
0 κ
D
,ξ
t
=
ξ
Z
t
ξ
D
t
,
and
E
t
ξ
t+1
·ξ
t+1
=
σ
2
Z
ρσ
Z
σ
D
ρσ
Z
σ
D
σ
2
D
= .
20
Deposits and short-term bonds are the cheapest form of financing. If the same dollar were raised by issuing
equity, the cost would be higher, owing both to the higher cost of equity capital and to floatation costs. In this
case the upper bound would be even lower.
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The Review of Financial Studies / v 0 n 0 2014
In the bank’s model, to achieve the stochastic structure in Equation (A2), we have introduced
the transformation (14), X = μ+Ns, where μ=(μ
1
,μ
2
), and N is a matrix
N =
ν
Z,u
ν
Z,v
ν
D,u
ν
D,v
such that detN = 0. The bank’s program will be solved (and simulated) in the state space S , and
we will use the above affine transformation to calculate the values of the variables Z and D.
The parameters of the stochastic discount factor are taken from Jones and Tuzel (2013).
Therefore, we are left with eight unknown parameters describing the stochastic structure of the
model: μ
1
, μ
2
, κ
v
, σ
v
, ν
Z,u
, ν
D,u
, ν
Z,v
, ν
D,v
. From empirical data we calculate seven moment
conditions:
Z, logD, κ
Z
, σ
Z
, κ
D
, σ
D
, ρ, as reported in Table 1. In what follows, we derive the
equations that relate the unknown parameters to the moment conditions. From Equation (14), we
have s = N
−1
(
X −μ
)
. Replacing this in Equation (A1), after some manipulations we have
X
t
=
I −NHN
−1
μ+NHN
−1
X
t−1
+Nζ
t
.
This dynamics is equal to the dynamics in Equation (A2) if the following conditions are
simultaneously true: μ =
I −NHN
−1
−1
X, NHN
−1
= K, NTN
= . Given the second equation,
μ=
(
I −K
)
−1
X, so that the solution for μ =(μ
1
,μ
2
), is found in closed form relative to known
moments conditions. The second and third conditions set seven equations (three from NTN
= ,
given that the two matrices are symmetric, and four from NHN
−1
= K) in the remaining six
unknowns. We find a least square solution to this overidentified system of seven nonlinear
equations. Using the parameter values in Table 1 in the paper, with the additional parameters
for the macroeconomic risk, κ
u
=0.98, σ
u
=0.007, the robust solution obtained with a global
optimization routine is μ
1
=0.0717, μ
2
=0.6931, κ
v
=0.901992, σ
v
=0.009548, ν
Z,u
=1.660682,
ν
D,u
= −2.988127, ν
Z,v
= −0.798126, and ν
D,v
=0.044359. Although the model is overidentified,
the proposed calibration procedure does a good job at matching a number of moments of the process
X, from Equation (15), with moments of
ˆ
X = μ+ Ns, where s is the process in Equation (A1) with
the above estimated parameters. This is can be seen in Table A1.
To understand the relationship that this calibration generates between the variables (u,v) and
the variables (Z,D), we plot the values of Z and logD resulting from a numerical approximation of
the dynamic of (u,v) in Figure 3. The variable logD appears strongly countercyclical, and depends
only marginally on the idiosyncratic component, whereas the credit shocks Z are procyclical and
negatively correlated with idiosyncratic risk.
Table A1
Calibration
Original
X
ˆ
X
E[Z]0.0717 0.0718 0.0723
σ [Z]0.0293 0.0236 0.0429
E[logD]0.6931 0.6926 0.6920
σ [logD]0.0669 0.0471 0.0701
(1− κ
Z
)Z 0.0086 0.0129 0.0039
κ
Z
0.8800 0.8184 0.9357
σ
Z
0.0139 0.0135 0.0150
(1− κ
D
)logD 0.0347 0.0733 0.0374
κ
D
0.9500 0.8940 0.9450
σ
D
0.0209 0.0210 0.0226
ρ −0.8500 −0.7514 −0.8503
Results from the calibration procedure described in Appendix B. Column “Original” reports the value of the
moments from Table 1. In Column “
X” there is the mean value of the moments estimated from a simulation of
X. Column “
ˆ
X
” reports the mean value of the moments estimated from a simulation of s in Equation (A1), and
than using at each step of the simulated sample the transformation in Equation (14) to obtain
ˆ
X
.
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Microprudential Regulation in a Dynamic Model of Banking
Appendix C. Numerical Solution and Simulation of the Valuation
Problem
Given the dynamics of (u,v), we solve the program
E(x)=max
0, max
(L
,B
)∈A(D)
e(x,L
,B
)+E
M(x,x
)E(x
)
,
where function e(x,L,B), is defined in Equation (13), and A(D) is the case specific feasible
set defined differently for the unregulated and the regulated case. The solution of the Bellman
equation above is obtained numerically by a value iteration algorithm. The valuation model for
bank’s equity is a continuous decision and infinite-horizon Markov decision processes. The solution
method is based on successive approximations of the fixed point solution of the Bellman equation.
Numerically, we apply this method to an approximate discrete state-space and discrete decision
valuation operator.
21
The variables u and v are discretized using the numerical approach proposed by Rouwenhorst
(1995), as the stochastic process of systematic risk is quite persistent. The feasible interval for
loans, [0,L
u
], and for the face value of bonds, [B
d
,B
u
] (with B
d
<0<B
u
), is set so that they are
never binding for the equity maximizing program. We discretize [L
d
,L
u
], to obtain a grid of n
L
points
L=
L
j
= L
u
(1−δ)
j
|j =1,...,n
L
−1
∪
L
n
L
=0
such that, if the bank chooses inaction, the loan’s level is what remains after the portion δL has
been repaid. The interval [B
d
,B
u
] is discretized into n
B
equally-spaced values, making up the set
B. To keep the notation simple, we also denote x =(s,L,B) the generic element of the discretized
state.
For the set of parameters in Table 1,weuseL
u
= 18, B
d
= −7 and B
u
= 3. Given the properties
of the quadrature scheme, we solve the model using only five points for u and seven points for v.
However, we need to allow for many more points when discretizing the control variables, so we
choose n
L
= 29, and N
B
= 34. The tolerance for termination of the value function iteration is set
at 10
−5
.
Given the optimal solution, we can determine the optimal policy and the transition function
ϕ(x) in Equation (19) based on the arg-max of equity value at the discrete states x. The optimal
policy is used to generate fifty simulated economies, each characterized by a specific path of the
systematic shock, and comprising 2,000 independent paths/banks for the idiosyncratic shock, for
100 periods (years). In particular, given the simulated dynamics of the state variables (u,v)by
application of the recursive formula in Equation (15), we start from Z(0)=
Z and D(0)=D
d
. Then
setting a feasible initial choice L(0)= 0 and B(0)= D
u
(so that the initial bank capital is D
u
−D
d
),
we apply the transition function ϕ along each simulated path recursively. If a bank defaults at
a given step, then the current depositors receive the full value of their claim, while the deposit
insurance agency pays the bankruptcy cost. Afterward, a seed capital D
u
−D
d
is injected in the
bank. Together with deposit D
d
, the total amount D
u
is momentarily invested in bonds, B = D
u
,
whereas L =0. Then the “new” bank follows on the same path by applying the optimal policy. To
limit the dependence of our results on the initial conditions, we drop the first fifty steps.
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