The Procyclical Eﬀects of Bank
Capital Regulation
Rafael Repullo
CEMFI and CEPR

Javier Suarez
CEMFI and CEPR

May 2009
Abstract
We assess the procyclical eﬀects of bank capital regulation in the context of a dynamic
equilibrium model of relationship lending in which banks are unable to access the equity markets every period. Importantly, banks anticipate that shocks to their earnings
as well as the cyclical position of the economy can impair their capacity to lend in the
future and, as a precaution, hold capital buﬀers. We find that these buﬀers are countercyclical under risk-insensitive capital requirements (e.g. Basel I) and procyclical underrisk-sensitive capital requirements (e.g. Basel II). Yet, under risk-sensitive capital requirements, the higher buﬀers maintained in expansions are insuﬃcient to prevent asignificant contraction in the supply of credit at the arrival of a recession. We show
that, in the context of Basel II, making the underlying confidence levels slightly countercyclical would strongly reduce the procyclical eﬀects on the supply of credit withoutcompromising banks’ long-run solvency targets.
Keywords: Banking regulation, Basel II, Business cycles, Capital requirements, Credit
crunch, Loan defaults, Relationship banking.
JEL Classification: G21, G28, E43

We would like to thank Sebastián Rondeau for his excellent research assistance and Matthias Bank, Jos
van Bommel, Thomas Gehrig, Claudio Michelacci, Oren Sussman, Dimitri Tsmocos, Lucy White, Andrew
Winton, and seminar audiences at the 2008 AFA Meetings, 2007 EFA Meetings, 2007 European, Far Eastern,
and Latin American Meetings of the Econometric Society, Bank of Portugal Conference on “Bank competition”, ECB Conference on “The implications of changes in banking and financing on the monetary policy
transmission mechanism”, ETH Zurich Conference on “Banking and the macroeconomy”, Bank of Spain,
CEMFI, the Federal Reserve Board, the New York Fed, the Richmond Fed, the European University Institute, and the Universities of Berlin (Humboldt), Oxford, Tilburg, and Zurich for their comments. Financial
support from the Spanish Ministry of Education (Grant SEJ2005-08875) is gratefully acknowledged. Address: CEMFI, Casado del Alisal 5, 28014 Madrid, Spain. Phone: 34-914290551. E-mail: repullo@cemfi.es,suarez@cemfi.es.

1Introduction

The financial crisis that started in the summer of 2007 has generalized the concern about
the procyclicality embedded in the financial system and, more specifically, in institutionaland regulatory aspects such as accounting rules, trading and risk-management rules, andbank capital requirements. All recent reviews of the challenges for the immediate reform offinancial regulation (e.g. Brunnermeier et al. (2009), FSF (2009)) stress the need to makethe financial system less procyclical. This paper focuses on providing a framework for thequalitative and quantitative analysis of the procyclical eﬀects of bank capital requirements.
We use the model to compare the cyclical behavior of the supply of credit under riskinsensitive (Basel I) and risk-sensitive (Basel II) capital requirements, and to assess potential
countercyclical corrections of the latter.
Even when capital requirements are essentially flat, like in Basel I, bank capital regulation
has the potential to be procyclical because bank profits tend to turn negative during recessions, impairing banks’ lending capacity. The problem is potentially much more importantwhen capital requirements are risk-sensitive, like in the Internal Ratings Based (IRB) approach of Basel II, since in this case, even if a bank’s total capital were not directly impaired,
the rise in risk associated with the arrival of a recession might lead to a severe contractionin the supply of credit.1
Two key conditions are necessary for these contractionary eﬀects to occur. First, somebanks must find it diﬃcult to respond to the higher capital requirements by issuing new
equity. Second, some of their borrowers must be unable to switch to other sources of finance.2
However, these conditions are not suﬃcient for the existence of significant procyclical eﬀects,
since banks anticipate that shocks to their earnings as well as the cyclical position of the1
In the IRB approach capital requirements are an increasing function of banks’ estimates of the probabilityof default (PD) and loss given default (LGD) of each loan, and these inputs are likely to rise in downturns. In
theory banks must feed the capital requirement formulas with “through-the-cycle” estimates of the relevant
PDs and LGDs, but a variety of reasons (short data samples, ratings migrations over the cycle, etc.) makethe actual estimates very likely to vary over the cycle. The extreme case in which this happens is when“point-in-time” estimates of PDs and LGDs are utilized.
2
These conditions have been noted by Blum and Hellwig (1995) and parallel the conditions in Kashyap,
Stein, and Wilcox (1993) for the existence of a bank lending channel in the transmission of monetary policy.

1economy can impair their capacity to lend in the future and, as a precaution, may holdcapital in excess of the regulatory requirements. The critical question is whether capital
buﬀers (that will endogenously respond to regulatory changes) will be suﬃcient to neutralize
the procyclicality added by the new requirements.
To address this issue we develop a tractable dynamic equilibrium model of relationshipbanking in which the business cycle is modeled as a two-state Markov switching process thataﬀects business failures and, thus, loan default probabilities. At every date a continuum of
entrepreneurs enters the market. They demand funds for two consecutive periods, giving riseto an overlapping generations structure. Consistent with the view that relationship banking
makes banks privately informed about their borrowers, we assume that (i) borrowers become
dependent on the banks with which they first start a lending relationship, and (ii) banks withongoing relationships have no access to the equity market. The first assumption captures thelock-in eﬀects caused by switching costs and the potential lemons problem faced by banks
when a borrower is switching from another bank.3 The second assumption captures theimplications of these informational asymmetries for the market for seasoned equity oﬀerings,
which can create prohibitive transaction and dilution costs for urgent recapitalizations. It
is consistent with the experience seen during the current crisis and with a view alreadyexpressed by the Basel Committee on Banking Supervision (2004, paragraph 757) whenapproving Basel II: “It may be costly for banks to raise additional capital, especially if thisneeds to be done quickly or at a time when market conditions are unfavorable.” It can also
be seen as a convenient reduced-form for the observed delays in banks’ recapitalizations.4
The combination of relationship lending and the inability of banks with ongoing relationships to access the equity market establishes a natural connection between the capital
shortages of some banks at a given date and the credit rationing of some borrowers at that
date. Yet, each cohort of new borrowers is assumed to be funded by banks that renew their3

See Boot (2000) for a survey of the relationship banking literature. Several papers explicitly analyze
the costs of switching lenders under asymmetric information (e.g., Sharpe, 1990) as well as the trade-oﬀs
behind the possible use of multiple lenders as a remedy to the resulting lock-in eﬀects (e.g., Detragiache etal., 2000). We are implicitly assuming that these alternative arrangements are prohibitively costly.
4
Barakova and Carey (2001) study the time to recovery of US banks that became undercapitalized in the
1984-1999 period, showing that they needed an average of 1.6 years to restore their capital positions.

2

lending relationships, have access to the equity market, and hence face no binding limits totheir lending capacity.
In order to isolate the potential cyclicality coming from the supply side of the loan market,
we abstract from demand-side cyclicality and aggregate feedback eﬀects that might mitigate
and exacerbate, respectively, the aggregate implications of the cyclicality in banks’ lending
capacity. The model, however, could serve as a building block of a more comprehensive
dynamic stochastic general equilibrium model in which, say, part of the production comesfrom entrepreneurial firms that require relationship bank finance.
We define equilibrium under the assumption of free entry into the banking sector, which
implies that the banks’ net present value at the dates in which they can issue equity mustbe zero. We characterize the equilibrium loan rates and banks’ capital decisions in eachstate of the economy, and derive a number of comparative statics results. Assuming that
equity financing is more costly than deposit financing, we show that capital requirements
increase equilibrium loan rates, but have an ambiguous eﬀect on capital holdings. On the
one hand, the higher prospects of ending up with insuﬃcient capital call for the holding oflarger buﬀers; on the other hand, higher capital requirements reduce the profitability of future
lending, and thus the bank’s interest in preserving its future lending capacity. Our analyticalexpressions suggest that the shape of the distributions of loan losses in diﬀerent states ofthe economy matter for determining which eﬀect dominates. Since the impact of capitalrequirements on the supply of second period loans is, therefore, analytically ambiguous, we
assess it numerically.
For the numerical analysis, we describe the distributions of loan losses according to thesingle risk factor model of Vasicek (2002), which provides the foundation for the IRB capitalrequirements of Basel II. Under this model, capital requirements have an exact value-atrisk interpretation: required capital is such that it can absorb the potential losses of a loan
portfolio over a one-year horizon with a probability (or confidence level) of 99.9%.5 Our
5As shown by Gordy (2003), the single risk factor model also has the feature that the contribution of agiven loan to value-at-risk is additive, that is, it depends on the loan’s own characteristics and not of thoseof the portfolio in which it is included.

3

analysis focuses on the comparison between the risk-insensitive regime of Basel I and therisk-sensitive regime of Basel II, as well as on the proposal and assessment of a specificcyclical correction of Basel II.
We find that when the value of the ongoing lending relationships is large enough and thecost of equity capital is not very large, banks optimally choose to keep capital buﬀers. Under
realistic parameterizations, Basel II leads banks to hold buﬀers that range from about 2%of assets in recessions to about 5% in expansions. The procyclicality of these buﬀers reflectsthat banks are concerned about the upsurge in capital requirements that takes place whenthe economy goes into a recession. We find, however, that these equilibrium buﬀers are
insuﬃcient to neutralize the eﬀects of the arrival of a recession, which may cause a verysignificant reduction in the supply of credit.6 Under the flat capital requirements of BaselI, the same economies would exhibit slightly countercyclical buﬀers and essentially no credit
crunch eﬀects.7 For the purposes of comparison, we also compute the equilibrium in alaissez-faire environment without capital regulation, finding that banks’ capital buﬀers (inthis world, pure “economic capital”) would be of around 5% and not very cyclical, and credit
rationing would be more cyclical (and on average higher) than under Basel I, but less cyclical(and on average lower) than under Basel II.
Our results also show that the probabilities of bank failure under the Basel II regime
are substantially lower than under the Basel I regime and, as one would expect, much lower(about 100 folds!) than in the laissez-faire benchmark. This suggests that the business cycleside-eﬀects of Basel II may have a payoﬀ in terms of the long-term solvency of the banking
system. It also suggests the possibility of ameliorating the procyclical impact of Basel II byintroducing some small adjustments in the IRB capital requirements. In this dimension, we
consider the possibility of modifying the cyclical profile of confidence levels in a way that
6Supervisors seem aware of this possibility. For instance, Greenspan (2002) claims that “The supervisoryleg of Basel II is being structured to supplement market pressures in urging banks to build capital considerably
over minimum levels in expansions as a buﬀer that can be drawn down in adversity and still maintain adequate
capital.”7
Some papers, starting with Bernanke and Lown (1991), point out that the introduction of Basel I caused
a credit crunch in the US during the months preceding the cyclical peak of 1990. But no credit crunch
episode has been detected after banks adjusted their capital holdings to the higher requirements.

4

keeps their long-term average at 99.9%, but lessens the target in those situations in which
credit rationing turns out to be the highest under the Basel II regime. We find that theseadjustments may achieve significant reductions in procyclicality without major costs in termsof banks’ long-term solvency.
The papers closest to ours are Estrella (2004), Peura and Keppo (2006), and Zhu (2008).
Estrella (2004) considers the dynamic optimization problem of a bank when its dividend
policy and equity raising processes are subject to quadratic adjustment costs, loan losses
follow a second-order autoregressive process, and bank failure is costly. The paper focuseson the comparison between the optimal capital decisions of the bank in the absence ofregulation and under a value-at-risk rule, concluding that they are very diﬀerent. Peura andKeppo (2006) consider a bank with an asset portfolio of exogenous size in the context ofa continuous-time model where raising equity takes time. A supervisor checks at random
intervals of time whether the bank complies with a minimum capital requirement. The paper
finds that the bank may hold capital buﬀers in order to reduce the risk of being closed forholding insuﬃcient capital when audited. Finally, Zhu (2008) adapts the model of Cooley andQuadrini (2001) to the analysis of banks with decreasing returns to scale, minimum capitalrequirements, and linear equity-issuance costs. Assuming ex-ante heterogeneity in banks’capital positions, the paper finds that for poorly-capitalized banks, risk-sensitive capitalrequirements increase safety without causing a major increase in procyclicality, whereas forwell-capitalized banks, the converse is true. Relative to these three papers, we simplify thedetails of the banks’ dynamic optimization problem and embed such problem in the context
of an equilibrium model of relationship banking with endogenous loan rates. Additionally,
we adopt the realistic loan default model of the IRB approach of Basel II and focus on theimplications for the dynamics of aggregate bank lending.
The rest of the paper is organized as follows. Section 2 presents the model. In Section 3
we analyze the capital decision of a representative bank. Section 4 defines the equilibrium
and provides the comparative statics of equilibrium loan rates and banks’ capital holdings.
In Section 5 we present the numerical results concerning the size and cyclical behaviorof capital holdings, capital buﬀers, credit rationing, and probabilities of bank failure in a5

number of parameterizations of the model. In Section 6 we examine adjustments of theBasel II framework that reduce its procyclical eﬀects. Section 7 discusses the robustness of
our results to changes in some of the key assumptions of the model. Section 8 concludes.
Appendix A contains the proofs of the analytical results, and Appendix B discusses the
choice of parameter values for the numerical analysis.

2

The Model

Consider a discrete time economy in which time is indexed by t = 0, 1, 2, ... The economy ispopulated by three classes of risk-neutral agents: entrepreneurs, banks, and investors.

2.1

Entrepreneurs

Entrepreneurs belong to overlapping generations formed by a continuum of measure one of exante identical and penniless individuals who remain active for up to two periods (three dates).
Entrepreneurs have the opportunity to undertake investment projects with the following
characteristics. The first period project of an entrepreneur born at date t requires a unit
investment at that date. At date t + 1 the project yields 1 + a if it is successful, and 1 − λif it fails, with a > 0 and 0 < λ < 1. The second period project of an entrepreneur born atdate t requires μ units of investment at date t + 1. The return at date t + 2 of this projectis independent of the return of the initial project, and equals (1 + a)μ if it is successful, and(1 − λ)μ if it fails, so parameter μ measures the scale of the second period project.
All projects operating from date t to date t + 1 have an identical probability of failure
pt . The outcomes of contemporaneous projects exhibit positive but imperfect correlation,
so their aggregate failure rate xt is a continuous random variable with support [0, 1] andcumulative distribution function (cdf) Ft (xt ) such thatZ 1
xt dFt (xt ).
pt = Et (xt ) =0

For simplicity, we consider the case in which the history of the economy up to date t
only aﬀects Ft (xt ) (and, thus, pt ) through an observable state variable st that can take two
6values, h and l, and follows a Markov chain with
qh = Pr (st = h | st−1 = h) and ql = Pr (st = h | st−1 = l) .
Moreover, we assume that the cdfs corresponding to the two states, Fh (·) and Fl (·), areranked in the sense of first-order stochastic dominance, so that the probabilities of businessfailure in each state satisfy
ph > pl .
Thus states h and l may be interpreted as recession (high business failure) and expansion(low business failure) states, respectively.

2.2

Banks

Banks are competitive intermediaries specialized in channeling funds from investors to entrepreneurs. Following the literature on relationship banking, we assume that the financing
of an entrepreneur in this economy relies on a sequence of one-period loans granted by thesingle bank from which the entrepreneur obtains his first loan. We also assume that setting
up the relationship with the entrepreneur makes the bank incur some cost c, to be subtracted
from first period revenue.8 Finally, for simplicity, we abstract from the possibility that partof the required second period investment μ is internally financed by the entrepreneur.9
Banks are funded with deposits and equity capital, both of which are raised from investors. To simplify the analysis we assume that deposits are fully insured (at a zero premium), and their supply is perfectly elastic at a risk-free rate that we normalize to zero.10
We also assume that investors require an excess return δ ≥ 0 on each unit of equity capital.
The cost of capital δ is intended to capture in a reduced-form manner distortions (such as
8This cost might include personnel, equipment and other operating costs associated with the screening
and monitoring functions emphasized in the literature on relationship banking.
9
This simplification is standard in relationship-banking models; see, for example, Sharpe (1990, p. 1072)
or von Thadden (2004, p. 14). Moreover, if entrepreneurs’ first-period profits are small relative to therequired second-period investment (as in our numerical analysis below), the quantitative eﬀects of relaxing
this assumption would be negligible.
10
In our numerical analysis the probability of bank failure is a small fraction of the 0.1% target of BaselII, so the fair deposit insurance premium would be negligible.

7

agency costs of equity or debt tax shields) that introduce a comparative disadvantage ofequity financing relative to deposit financing–in addition to deposit insurance.11
We introduce an important imperfection concerning banks’ equity financing: While banks
entering the market or renewing their portfolio of lending relationships can raise new equity
in an unrestricted manner, recapitalization is impossible for banks with ongoing lending
relationships. This assumption is intended to capture in a simple way the long delays or
prohibitive dilution costs that a bank with opaque assets in place might face when organizing
an urgent equity injection.12
Banks are managed in the interest of their shareholders, who are protected by limited
liability. Entry to the banking sector is free at all dates, but banks are subject to a capitalrequirement that obliges them to hold a capital-to-loans ratio of at least γ s on the loans madewhen the state of the economy is s. This formulation encompasses both the risk-insentitive(Basel I) and risk-sensitive (Basel II) type of regulation, as well as a laissez-faire environment
with zero capital requirements. In Basel I the capital requirement is (for corporate loans)a constant ratio γ l = γ h = 8%. Basel II aims at a better alignment of capital requirementswith the underlying banking risks, and consequently requires higher capital for riskier loans.
In our setup there is no cross-sectional heterogeneity among borrowers but the state of
the economy aﬀects the risk of the representative loan, so Basel II amounts to a capitalrequirement in the high default state, γ h , higher than the capital requirement in the low
default state, γ l .13
11

Further to the reasons for the extra cost of equity financing oﬀered by the corporate finance literature,
Holmström and Tirole (1997) and Diamond and Rajan (2000) provide agency-based explanations specificallyrelated to banks’ monitoring role.
12
These costs are most likely related to asymmetric information. Specifically, in a world in which banks
learned about their borrowers after starting a lending relationship (like in Sharpe, 1990) and borrower
quality were asymmetrically distributed across banks, the market for seasoned equity oﬀerings might beaﬀected by a lemons problem (like in Myers and Majluf, 1984). Thus, after a negative shock, banks withlending relationships of poorer quality would be more interested in issuing equity at any given price, which
explains why the prices at which new equity could be raised may be unattractive to banks with higher-quality
relationships and why, in suﬃciently adverse circumstances, the market for those SEOs may collapse.
13
The precise Basel II formula that relates the capital requirement γ s to the loans’ probability of default
ps will be described in Section 5.1. Although Basel II stipulates that estimates of the probability of default“must be a long-run average of one-year default rates” (Basel Committee on Banking Supervision, 2004,
paragraph 447), industry practices based on point-in-time rating systems, the dynamics of rating migrations,and composition eﬀects make the eﬀective capital charges on a representative loan portfolio very likely to8

To guarantee that the funding of investment projects is attractive to banks at all dates,
we assume that
(1 − ps )(1 + a) + ps (1 − λ) − c > (1 − γ s ) + γ s (1 + δ),(1)

for s = h, l. Thus, in all states of the economy, the expected return per unit of investment,
net of the setup cost c, is greater than the cost of funding it with 1 − γ s deposits and γ scapital.

3

Banks’ Capital Decision

Consistent with the assumption that banks with ongoing lending relationships may facecapital constraints, we assume that entrepreneurs born at date t obtain their first periodloans from unrestricted banks that can raise capital at this date. This allows us to analyze
the banking industry as if it were made of overlapping generations of banks that operate fortwo periods, specialize in loans to contemporaneous entrepreneurs, and cannot issue equity
at the interim date.14
In this economy, the supply of loans to the entrepreneurs who start up at date t might
be aﬀected by the recapitalization constraint faced by their banks at date t + 1. In fact,banks will be aware of this and, in order to better accommodate the eﬀect of negative
shocks to their first period income or possibly higher capital requirements in the case ofrisk-sensitive capital regulation, they may hold a buﬀer of equity capital on top of the first
period regulatory minimum.
To understand the financing problem faced by each generation of entrepreneurs in thiseconomy, consider a representative bank that lends to the measure one continuum of entrepreneurs starting up at date t, possibly refinances them at date t + 1, and gets liquidated atdate t + 2.15 Let s and s0 denote the states of the economy at dates t and t + 1, respectively.
be higher in recessions than in expansions. See, for example, Kashyap and Stein (2004), Catarineu-Rabell,Jackson, and Tsomocos (2005), Gordy and Howells (2006), and Saurina and Trucharte (2007).
14
Notice that a bank that can raise capital at date t is essentially identical to a new bank established atthat date.
15
It will become obvious that banks that can issue equity face constant returns relative to the size of theirloan portfolio.

9At date t the representative bank raises 1 − ks deposits and ks ≥ γ s capital, and invests
these funds in a unit portfolio of first period loans. The equilibrium interest rate on these
initial loans, denoted rs , will be determined endogenously, as explained below, but under our
perfect competition assumption the bank takes it as given. Since the supply of deposits isperfectly elastic at a zero interest rate, rs should be interpreted as the spread between initialloan rates and deposit rates.
At date t + 1 the bank gets 1 + rs from the fraction 1 − xt of performing loans (that is,
those extended to entrepreneurs whose projects are successful) and 1 − λ from the fraction
xt of defaulted loans, and incurs the setup cost c, so its assets are 1 + rs − xt (λ + rs ) − c,while its deposit liabilities are 1 − ks . Thus its capital at date t + 1 is
ks0 (xt ) = ks + rs − xt (λ + rs ) − c,(2)where xt is a random variable whose cdf conditional on the state of the economy at date t
is Fs (xt ). If ks0 (xt ) < 0 the bank fails, while if ks0 (xt ) ≥ 0 it can operate for a second period.

Using the definition of ks0 (xt ), it is immediate to show that bank failure occurs when thedefault rate xt exceeds the critical valuexbs =ks + rs − c.
λ + rs(3)

The entrepreneurs that start up at date t demand an amount μ of second period loans
at date t + 1.16 At this stage entrepreneurs are dependent on the bank, so their demand isinelastic as long as the loan interest rate does not exceed the success return of the projects
in the second investment period. Thus the second period loan rate will be a.
Since the bank cannot issue new equity at date t + 1, its maximum lending capacity isgiven by the ratio between its available capital ks0 (xt ) and the capital requirement γ s0 , which
depends on the state of the economy s0 at date t + 1. Thus, whenever ks0 (xt ) ≥ 0 there are

two cases to consider: the case with excess lending capacity, ks0 (xt ) ≥ γ s0 μ, and the casewith insuﬃcient lending capacity, ks0 (xt ) < γ s0 μ. Using the definition of ks0 (xt ) in (2), it is16

Note that this includes entrepreneurs that defaulted on their initial loans. This is because under ourassumptions such default does not reveal any information about the entrepreneurs’ second period projects.

10immediate to show that the latter case arises when the default rate xt exceeds the criticalvaluex
ess0 =

ks + rs − c − γ s0 μ,
λ + rs(4)

ess0 < xbs < 1, one
which is obviously smaller than x
bs , defined in (3). Thus, whenever 0 < x
can find three diﬀerent situations at date t + 1, depending on the realization of the default
rate: for xt ∈ [0, xess0 ], the representative bank has excess lending capacity; for xt ∈ (exss0 , xbs ],the bank has insuﬃcient lending capacity; and for xt ∈ (bxs , 1] the bank fails. We next derive

the expected continuation payoﬀs of the bank’s shareholders in each of the two cases where
the bank does not fail.
When there is excess lending capacity at date t + 1 the bank finances μ loans using(1 − γ s0 )μ deposits and γ s0 μ capital. Since ks0 (xt ) ≥ γ s0 μ, the bank pays a dividend of

ks0 (xt ) − γ s0 μ to its shareholders at date t + 1.17 At date t + 2 the bank gets 1 + a from thefraction 1−xt+1 of performing loans and 1−λ from the fraction xt+1 of defaulted loans, so its
assets are [1 + a − xt+1 (λ + a)]μ, while its deposit liabilities are (1 − γ s0 )μ. Thus shareholders’expected payoﬀ, conditional on the state of the economy at date t + 1, can be expressed as
μπ s0 , where
π s0 =Z 1
0

max {γ s0 + a − xt+1 (λ + a), 0} dFs0 (xt+1 )(5)is the expected gross equity return on a per-unit-of-loans basis. The value of shareholders’
stake in the bank at date t + 1, inclusive of the dividend ks0 (xt ) − γ s0 μ, can be written as
vss0 (xt ) = (βπ s0 − γ s0 )μ + ks0 (xt ),(6)

where β = 1/(1 + δ) is the shareholders’ discount factor implied by the cost of capital δ.
The first term in (6) measures the net present value contribution of the capital that remains
invested in the bank up to date t + 2. Assumption (1) guarantees that βπ s0 > γ s0 , so thatsuch contribution is positive.18
17

Since entrepreneurs born at date t + 1 borrow from banks that can raise equity at that date, the bankmay use the excess capital to either pay a dividend to its shareholders or to reduce the deposits to be raisedat this date. However, under deposit insurance and δ ≥ 0, the second alternative is strictly suboptimal.
R1
18
To see this, notice that πs0 > 0 [γ s0 + a − xt+1 (λ + a)] dFs0 (xt+1 ) = γ s0 + a − pt+1 (λ + a), but assumption(1) implies a − pt+1 (λ + a) > δγ s0 and hence π s0 > (1 + δ)γ s0 = γ s0 /β.

11When there is insuﬃcient lending capacity at date t+1 the bank finances ks0 (xt )/γ s0 loans
with [ks0 (xt )/γ s0 ] − ks0 (xt ) deposits and ks0 (xt ) capital. At date t + 2 shareholders’ expected

payoﬀ, conditional on the state of the economy at date t+1, can be expressed as π s0 ks0 (xt )/γ s0 ,where π s0 is the expected gross equity return on a per-unit-of-loans basis given by (5). When
there is insuﬃcient lending capacity at date t + 1 the bank pays no dividends at that dateand, hence, the value of shareholders’ stake in the bank is justvss0 (xt ) =βπ s0 0k (xt ).
γ s0 s(7)

As before, assumption (1) implies that βπs0 > γ s0 , and hence shareholders strictly benefit
from keeping ks0 (xt ) invested in the bank.
Putting together the two cases, as well as the case in which the bank fails, we can express
the market value of the bank at date t + 1, inclusive of dividends, as⎧(βπ s0 − γ s0 )μ + ks0 (xt ), if xt ≤ x
ess0 ,⎪⎪⎪⎨
βπ s0 0
vss0 (xt ) =ks (xt ),if x
ess0 < xt ≤ x
bs ,
⎪
γ
0
⎪
s
⎪
⎩
bs ,
0,
if xt > x(8)which is a continuous and piecewise linear function of xt .19 Going backward one period, thenet present value of the representative bank that in state s holds capital ks and charges aninterest rate rs on its unit of initial loans isvs (ks , rs ) = βEt [vss0 (xt )] − ks ,(9)where the operator Et (·) takes care of the fact that, at date t, vs is subject to the uncertainty
about both the state of the economy at date t + 1 (which aﬀects the second period capitalrequirement γ s0 and gross equity return πs0 ) and the default rate xt of initial loans (which
determines the capital ks0 (xt ) available at that date).
Taking as given the initial loan rate rs , the representative bank that first lends to a
generation of entrepreneurs in state s will choose its capital ks so as to maximize vs (ks , rs )19

Note that βπ s0 > γ s0 implies that if the bank does not fail at date t + 1 the market value of its equity
vss0 (xt ) is strictly greater than the accounting value ks0 (xt ).

12subject to the constraint ks ∈ [γ s , 1]. Since vs (ks , rs ) is continuous in ks , for any given interest
rate rs , the bank’s capital decision always has a solution. In Appendix A we show that thefunction vs (ks , rs ) is neither concave nor convex in ks , and that we may have interior solutions
or corner solutions with ks = γ s . When the solution is interior, there is a positive probability
that the bank has insuﬃcient lending capacity in the high default state s0 = h (and possibly
also in the low default state s0 = l), and there is a positive probability that the bank has
excess lending capacity in the low default state s0 = l (and possibly also in the high default
state s0 = h). The intuition for this result is as follows. If in the two possible states at datet + 1 the bank had a zero probability of finding itself with insuﬃcient lending capacity, thenit would have an incentive to reduce its capital at date t in order to reduce its funding costsat that date. On the other hand, if in the two possible states at date t + 1 the bank had azero probability of finding itself with excess lending capacity, then it would have an incentive
either to increase its capital at date t and thereby relax the capital constraint at date t + 1,
or to go to the corner ks = γ s .20

4Equilibrium

In the previous section we have characterized banks’ capital and lending decisions at thedates in which they can raise capital, as well as at the dates in which they cannot. This
analysis has taken as given the interest rate rs at the beginning of a lending relationshipin state s, with the continuation loan rate being the success return a of the second period
investment projects. In order to define an equilibrium, it only remains to describe how theinitial loan rate is determined.
Given our free entry assumption, in equilibrium the pricing of these loans must be suchthat the net present value of the bank is zero under the bank’s optimal capital decision.
Were it negative, no bank would extend loans. Were it positive, incumbent banks would
have an incentive to expand, and new banks would profit from entering the market. Hence
20

The possible preference for the corner ks = γ s is due to the fact that, in this case, the function vs (ks , rs )is (locally) either decreasing or convex in ks ; see Appendix A for the details.

13

in each state of the economy s = h, l we must havevs (ks∗ , rs∗ ) = 0,(10)ks∗ = arg max vs (ks , rs∗ ).

(11)

for
ks ∈[γ s ,1]Therefore we may define an equilibrium as a sequence of pairs {(kt , rt )} describing thecapital-to-loan ratio kt of the banks that can issue equity at date t and the interest rate rt
charged on their initial loans, such that each pair (kt , rt ) satisfies (10) and (11) for s = st ,where st is the state of the economy at date t.
The existence of an equilibrium is easy to establish. Diﬀerentiating (10) we havedvs
∂vs dk∗ ∂vs= ∗ s +,
drs
∂ks ∂rs ∂rs
where the first term is zero, by the envelope theorem, and the second is positive, becauseof the higher interest payments at date t + 1 (see Appendix A for details). So vs (ks∗ , rs ) is
continuous and monotonically increasing in rs . Moreover, for suﬃciently low interest rates
we have vs (ks∗ , rs ) < 0, while for rs = a assumption (1) implies vs (ks∗ , rs ) > 0. Hence weconclude that there is a unique rs∗ that satisfies vs (ks∗ , rs∗ ) = 0.21

4.1

Comparative statics

The structural parameters that describe the economy are the following: The success returna (which determines the interest rate of continuation loans), the loss given default λ, thescale of the second period projects μ, the cost of setting up a lending relationship c, the costof bank equity capital δ, the probabilities of transition from each state to the high default
state qh and ql , and the capital requirements γ h and γ l . To complete the description of
the economy, one must also specify the state-contingent cdfs of the default rate, Fh (xt ) andFl (xt ).
21
However, since the function vs (ks , rs ) is neither concave nor convex in ks , there may be multiple optimalvalues of ks corresponding to rs∗ .

14

Table 1 summarizes the comparative statics of the equilibrium interest rates on initialloans rs∗ , which are derived in Appendix A. The table shows the sign of the derivative drs∗ /dz
obtained by diﬀerentiating (10) with respect to each exogenous parameter z. The eﬀects
of the various parameters on rs∗ are inversely related to their impact on the profitability ofbanks’ lending activity. Other things equal, a and μ impact positively on the profitability ofcontinuation lending; λ aﬀects negatively the profitability of both initial lending (directly)
and continuation lending (directly and by reducing the availability of capital in the second
period); c has a similar negative eﬀect (with no direct eﬀect on the profitability of continuation loans); δ increases the cost of equity funding in both periods; γ h and γ l increase theburden of capital regulation in the corresponding initial state, as well as in the corresponding
continuation state (which will be h or l with probabilities qs and 1 − qs , respectively); finally,
qs decreases the profitability of continuation lending because, in the high default state h,loan losses are higher and the corresponding capital requirement γ h may also be higher.

Table 1. Comparative statics of the initial loan rate rs∗
z

=

drs∗
dz

a

λ

μ

c

δ

qs

γh

γl

−

+

−

+

+

+

+

+

Table 2 summarizes the comparative statics of the equilibrium initial capital ks∗ chosen
by the representative bank in an interior solution–obviously, when the solution is at thecorner ks∗ = γ s , marginal changes in parameters other than the capital requirement γ s do
not change ks∗ . As further explained in Appendix A, the recursive nature of the comparative
statics of the system given by (10) and (11) makes it convenient to decompose the eﬀects
of the change in any parameter z into a direct eﬀect (for constant rs∗ ) and a loan rate eﬀect(due to the change in rs∗ ):dks∗∂k ∗ ∂k ∗ dr∗= s + s s.
dz∂z
∂rs dz15

Loan rate eﬀects can be easily determined. Diﬀerentiating the first-order condition thatcharacterizes ks∗ in an interior solution gives∂ 2 vs ∂ks∗
∂ 2 vs+= 0.
∂ks2 ∂rs ∂ks ∂rs
The coeﬃcient of ∂ks∗ /∂rs is negative, by the second-order condition that characterizes ks∗
in an interior solution, and the second term is negative (see Appendix A). Hence ∂ks∗ /∂rs isnegative, which implies that the signs of loan rate eﬀects are the opposite to those in Table
1. Intuitively, the initial capital ks and the initial loan rate rs are substitutes in the role ofproviding the bank with suﬃcient capital for its continuation lending (see the definition of
ks0 (x) in (2)). In an interior solution, the marginal value of ks is decreasing in ks , and thus
also in rs , so a larger rs reduces the bank’s incentive to hold excess capital.

Table 2. Comparative statics of the initial capital ks∗(in an interior equilibrium)
z

=

a

λ

μ

c

δ

qs

γh

γl

∂ks∗
∂z

(direct eﬀect)

+

?

+

+

−

?

?

?

∂ks∗ drs∗∂rs dz(loan rate eﬀect)

+

−

+

−

−

−

−

−

dks∗
dz

(total eﬀect)

+

?

+

?

−

?

?

?

For the parameters a, μ, and δ, the direct and the loan rate eﬀects point in the same
direction, so the total eﬀect can be analytically signed. In essence, higher profitabilityof continuation lending (captured by a and μ) and lower costs of capital (captured by δ)
encourage banks to hold larger capital buﬀers in order to better self-insure against the default
shocks that threaten its continuation lending. For the setup cost c, the direct and the loan
16

rate eﬀects have unambiguous but opposite signs, so the total eﬀect is ambiguous. The
positive direct eﬀect comes from the fact that, by the definition of ks0 (x) in (2), c subtracts
to the bank’s continuation lending capacity exactly like ks adds to it, without aﬀecting the
profitability of such lending and hence the marginal gains from self-insuring against default
shocks. The direct eﬀects on ks of the parameters λ, qs , γ h , and γ l have ambiguous signs.
Increasing any of these parameters simultaneously reduces the profitability of continuation
lending and impairs the expected capital position of the bank when such lending has tobe made. The value of holding excess capital in the initial lending period falls, but theprospects of ending up with insuﬃcient capital increase. So the profitability of continuation
lending and the need for self-insurance against default shocks move in opposite directions.
The resulting ambiguity of the direct eﬀects extends to the total eﬀects.
The details of the analytical expressions suggest that the shape of the distributions ofdefault rates matter for the determination of these eﬀects, which eventually becomes a question to be elucidated either empirically or by numerically solving the model under realistic
parameterizations. In the rest of the paper, we resort to the second alternative, if only because the experience accumulated with Basel II capital requirements is still very short andthe corresponding data is heavily aﬀected by the unprecedented severity of the current crisis.

5

Numerical Results

To further explore the forces that aﬀect banks’ initial capital buﬀers as well as to assess theimplications for the dynamics of lending under diﬀerent regulations, we numerically solve themodel in a number of plausible scenarios. Importantly, in all scenarios we assume that thestate-contingent probability distributions of the default rate, described by the cdfs Fh (xt )and Fl (xt ), conform to the single risk factor model that underlies the capital requirements
associated with the IRB approach of Basel II.22 This means that we assess the implicationsof the new capital requirements under the assumption that the supervisor’s model of refer22

The single factor model is due to Vasicek (2002) and its use as a foundation for the capital requirementsof Basel II is due to Gordy (2003).

17

ence is correct.23 In line with the one-year ahead value-at-risk perspective of Basel II, theparameterization assumes that each model period corresponds to one calendar year.

5.1

The single risk factor model

Suppose that the project undertaken by entrepreneur i at date t fails if yit < 0, where yit isa latent random variable defined byyit = αt +p√ρt ut + 1 − ρt εit ,where αt is a parameter determined by the state of the economy at date t, ut is a single factorof systematic risk, εit is an idiosyncratic risk factor, and ρt ∈ (0, 1) is a state-contingent
parameter that determines the correlation among project failures. It is assumed that ut andεit are N(0, 1) random variables, independently distributed from each other and over time,
as well as, in the case of εit , across projects. Let Φ(·) denote the cdf of a standard normal
random variable. Conditional on the information available at date t, the probability of failureof project i is pt = Pr (yit < 0) = Φ(−αt ), since yit ∼ N(αt , 1), which implies αt = −Φ−1 (pt ).
With a continuum of projects, the aggregate failure rate xt is only a function of therealization of the single risk factor ut . Specifically, by the law of large numbers, xt coincides
with the probability of failure of a (representative) project i conditional on the information
available at t and the realization of ut :
¶
µ −1
√
³
´
p
Φ (pt ) − ρt ut√−1√
xt = gt (ut ) = Pr −Φ (pt ) + ρt ut + 1 − ρt εit < 0 | ut = Φ.
1 − ρtUsing the fact that ut ∼ N(0, 1), the cumulative distribution function of the aggregate failure
rate can be expressed as¡¢
Ft (xt ) = Pr (gt (ut ) ≤ xt ) = Pr ut ≤ gt−1 (xt ) = Φ¶µ√1 − ρt Φ−1 (xt ) − Φ−1 (pt ).
√ρt

In Basel II the correlation parameter ρt is assumed to be a decreasing function of thestate-contingent probability of default pt . Hence we postulate the following state-contingent
23

Of course, the model could be similarly solved under alternative specifications of the relevant cdfs, but
in that case the requirements set under the regulatory formula described below would not have the direct
value-at-risk interpretation implied by our parameterization.

18

probability distributions of the default rate¶µ√1 − ρs Φ−1 (x) − Φ−1 (ps ),Fs (x) = Φ
√
ρs

(12)

where, as stipulated by Basel II for corporate loans, the correlation parameter ρs is decreasingin the probability of default ps according to the formula¶µ1 − e−50ps.
ρs = 0.12 2 −1 − e−50(13)

In the IRB approach of Basel II, capital must cover the one-year ahead losses due to loan
defaults with a probability of 99.9%. Hence the capital requirement in state s is given by
γ s = λFs−1 (0.999), where Fs−1 (0.999) is the 99.9% quantile of the distribution of the default
rate. Using (12), the Basel II capital requirement becomes¶µ −1√
Φ (ps ) + ρs Φ−1 (0.999)√,
γ s = λΦ1 − ρs(14)

where ρs is given by (13). This is the formula for corporate exposures of a one-year maturity
that appears in Basel Committee on Banking Supervision (2004, paragraph 272).24 It should
be noted that Basel II establishes that the expected losses, λps , should be covered withgeneral loan-loss provisions, while the remaining charge, λ(γ s − ps ), should be covered withcapital. However, from the perspective of our analysis, provisions are just another form ofequity capital, so the distinction between the expected and unexpected components of loan
losses is immaterial.

5.2

Benchmark scenarios

Table 3 shows the set of parameter values that define the three benchmark scenarios considered in our numerical analysis. The scenarios diﬀer in the volatility of the state-contingent
probabilities of default pl and ph . We briefly comment on them here, relegating further discussion in the light of available data on the US banking sector to Appendix B. Note that
because of our normalization of the risk-free rate to zero, all interest rates and rates of returnin the parameterization should be interpreted as spreads over the risk-free rate.
24

The Basel II formula incorporates an adjustment factor that is increasing in the maturity of the exposure,and equals one for a maturity of one year.

19Table 3. Parameter values in the benchmark scenarios
A. Common parameters
a
μ
λ
c
δ
qh
ql
0.04 1.00 0.45 0.03 0.04 0.64 0.20

B. Probability of default (PD) scenarios
Scenarios

Benchmark PDs
ps (%)

Basel II requirements
γ s (%)

Low
volatility

s=l
s=h

1.20
2.91

6.87
10.01

Mediumvolatility

s=l
s=h

1.10
3.26

6.60
10.51

High
volatility

s=l
s=h

1.00
3.62

6.31
11.00

The three PD scenarios are defined so as to keep the expected capital charge under
Basel II equal to 8%, which is the capital requirement under Basel I. Appendix B
discusses the choice of parameter values in the light of available data on US banks.

Panel A of Table 3 contains the parameters that are common to the three scenarios. The
value of the success return a = 0.04 implies that the interest rate of continuation loans is
4%.25 The loss given default (LGD) parameter λ = 0.45 is taken from the Basel II “foundation IRB” formula for unsecured corporate exposures.26 The scale of the second period
projects μ = 1 provides a neutral starting point–fine-tuning this parameter would require
some empirical estimation of the growth rate of loan exposures along a typical corporate
lending relationship or, alternatively, of the asset growth rate in a representative business
25

The success return could be higher, as long as the part that can be pledged to the bank without destroying
the entrepreneur’s incentives is set at 4%. See Holmström and Tirole (1997) for a discussion of the conceptof pledgeable return.
26
In the “advanced IRB” approach banks are allowed to use their own internal models to estimate λ.

20

financed by banks. The cost of setting up a lending relationship c = 0.03 is chosen so as
yield realistic initial loan rates. The cost of bank capital δ = 0.04 is intended to capture
the tax disadvantages of equity financing relative to deposit financing. The probabilities oftransition to the high default state, ql = 0.20 and qh = 0.64, imply expected durations of 5
years for the low default state and 2.8 years for the high default state, which we calibrate
according to the observed behavior of the charge-oﬀ ratio of FDIC-insured commercial banks
in the US during the period 1970-2004.27
Panel B of Table 3 shows our choices for the probabilities of default (PDs) in each state,
pl and ph , and the corresponding Basel II capital requirements, γ l and γ h , implied by (14).
In each scenario we have chosen the PDs such that the long-run average capital requirement
under Basel II (given the underlying unconditional probabilities of visiting each state) is 8%,
as under the risk-insensitive Basel I regulation.28 The idea is to allow for a comparison of thecyclical eﬀects of Basel I and Basel II that is not aﬀected by a change in the long-run averagelevel of the capital requirements. The three scenarios only diﬀer in the magnitude of thecross-state variation in the PDs–and all of them are within a range that can be consideredempirically plausible.
It is important to note that we are assuming that the LGD parameter λ does no depend on
the state s. Altman et al. (2005), using data on bond defaults, find that LGDs are positively
correlated with default rates. In the context of our model, however, cyclical variation in
LGDs has a similar eﬀect on banks’ profits and Basel II capital requirements as an increasein the cyclical variation in PDs. Hence, the eﬀects of our conservative assumption could beoﬀset by placing more weight on the results corresponding to the scenarios with more volatile
PDs.
27

Expected durations are computed as ql + 2(1 − ql )ql + 3(1 − ql )2 ql + ... = 1/ql = 5 for state l, and(1 − qh ) + 2qh (1 − qh ) + 3qh2 (1 − qh ) + ... = 1/(1 − qh ) ' 2.8 for state h. It should be noted that, in linewith the empirical findings of Bruche and González-Aguado (2006), in our parameterization credit cycle
downturns last longer than typical business cycle downturns.
28
The unconditional probabilities of the low and the high default state, denoted φl and φh , can be obtainedby solving the system of equations ql φl +qh φh = φh and φl + φh = 1, which gives φl = (1−qh )/(1−qh +ql ) '0.64 and φh = ql /(1 − qh + ql ) ' 0.36.

21

5.3

Capital buﬀers and procyclicality

Table 4 shows initial loan rates rs∗ , initial capital ks∗ , and the implied capital buﬀers ∆∗s =ks∗ −γ s , for s = h, l, in each of the scenarios described in Table 3 and under the two regulatory
frameworks that we want to compare: Basel I, with a flat capital requirements of 8%, andBasel II, with the requirements given by (14). As a reference, we also include the results ina laissez-faire environment without capital requirements (γ h = γ l = 0).
Table 4. Initial loan rates, capital, and capital buﬀers(all variables in %)Scenarios

rs∗Basel I
ks∗
∆∗s

rs∗

Basel II
ks∗
∆∗s

rs∗

Laissez-faire
ks∗
∆∗s

Low
volatility

s=l
s=h

1.2
2.4

11.0
11.2

3.0
3.2

1.2
2.5

11.9
12.2

5.0
2.2

0.7
1.8

5.2
5.4

5.2
5.4

Mediumvolatility

s=l
s=h

1.2
2.7

11.0
11.2

3.0
3.2

1.2
2.8

11.7
12.5

5.1
1.9

0.6
2.1

5.1
5.3

5.1
5.3

High
volatility

s=l
s=h

1.1
3.0

10.9
11.1

2.9
3.1

1.1
3.1

10.7
12.6

4.3
1.6

0.5
2.3

5.0
5.2

5.0
5.2

The parameters that define each of the three scenarios and the associated Basel II capitalrequirements are described in Table 3. The Basel I capital requirement is always 8%.

The results show that initial loan rates are always higher in the high default state,reflecting the need to compensate banks for both a higher PD and a lower prospective
profitability of continuation lending (since the high default state h is more likely to occur
after state h than after state l). These rates are very similar in the two Basel frameworks,
confirming previous results from static models predicting that the loan pricing implicationsof Basel II will be small.29 Basel II slightly increases loan rates in the high default state, andinduces no significant change in loan rates in the low default state. These eﬀects may be
explained by the fact that Basel II significantly increases banks’ capital in the high default
state, but has a smaller impact on capital in the low default state.
29

See Repullo and Suarez (2004).

22

The results also show that, in order to preserve their future lending capacity, banks holdsizeable capital buﬀers. Under Basel I, the cyclical variation in PDs has a rather small impact
on capital decisions, although excess capital tends to be larger in the high default state (where
loan losses can be expected to cause a larger reduction in future lending capacity) than in
the low default state.30 Under Basel II the cross-state variability in PDs visibly translates
into greater variability of both total capital and capital buﬀers. Interestingly, the cyclicalpattern of the buﬀers gets reversed, from slightly countercyclical in Basel II to strongly
procyclical in Basel II. The main reason for this reversal is that, under Basel II, banks in the
low default state l anticipate that if the economy switches to the high default state h thecapital requirement will increase from γ l to γ h . This jump in capital requirements implies
a reduction in their lending capacity so, to preserve continuation lending, they have an
incentive to hold larger precautionary capital buﬀers than under Basel I, where the capitalrequirement stays at 8%. Symmetrically, under Basel II, banks in the high default state hanticipate that if the economy switches to the low default state l the capital requirement
will decrease from γ h to γ l , so they have an incentive to hold smaller capital buﬀers thanunder Basel I. The numerical results for the three scenarios show that the first eﬀect (higher
buﬀers in state l) turns out to be more important than the second eﬀect (lower buﬀers instate h), which implies that the move from Basel I to Basel II will increase the long-run
average level of the capital buﬀers (computed with the unconditional probabilities of visiting
each state).31
As for the laissez-faire environment, the results in Table 4 confirm that, under our parameterization, the “economic capital” chosen by banks starting lending relationships iswell-below the regulatory capital of any of the two Basel frameworks, but significantly different from zero (and very similar across states), which reflects banks’ interest in preserving
their valuable lending during the second period.32 The lower initial interest rates in both30This is consistent with the existing evidence about the behavior of capital buﬀers under Basel I–
including Ayuso et al. (2004) with Spanish data, Lindquist (2004) with Norwegian data, and Bikker andMetzemakers (2004) with data from 29 OECD countries–and raises doubts about the interpretation thatsuch evidence reflects banks’ myopia.
31
The increase in the medium volatility scenario is of 0.88 percentage points.
32
Elizalde and Repullo (2007) discuss the concept of economic capital (and its relationship with the23

states (relative to the Basel frameworks) reflect the savings on the costs of equity financing
due to the use of less capital in the two lending periods.
Table 5 compares the cyclical behavior of credit rationing under Basel I and Basel II,
as well as in the laissez-faire environment. Lending in any given period is made up of
initial loans, whose quantity is always one, and continuation loans, whose quantity varies
with the lending capacity of the banks that are unable to issue equity in that period. We
denote by credit rationing the expected percentage of continuation projects that cannot be
undertaken because of banks’ insuﬃcient lending capacity. Table 5 shows credit rationing
in state s0 = l, h when it is reached from state s = l, h according to any of the four possible
sequences (s, s0 ). In our simple model, investment and hence expected gross output (thereturns from the funded investment projects) are linearly related to total credit, given thestate of the economy, so we can use credit rationing as a summary statistic of aggregate
economic activity.
In Basel I (as well as in the laissez-faire environment) credit rationing does not depend onwhether the arrival state s0 is a high or a low default state, since the capital requirement isconstant (at 8% or 0%, respectively). Rationing only depends on the profits realized during
the previous period, which determine the capital available for continuation lending. The
distribution of this random variable depends on the state s of the economy in the previous
period. This explains why the figures for Basel I (and the laissez-faire environment) in Table
5 only vary with s in each scenario, and are smaller for s = l than for s = h.
regulatory capital of Basel II) in a model where banks are concerned about the loss of charter values when
they become insolvent.

24

Table 5. Credit rationing(all variables in %)Scenarios
Low volatility(s, s0 ) = (l, l)(s, s0 ) = (l, h)(s, s0 ) = (h, h)(s, s0 ) = (h, l)Unconditional
Medium volatility(s, s0 ) = (l, l)(s, s0 ) = (l, h)(s, s0 ) = (h, h)(s, s0 ) = (h, l)Unconditional
High volatility(s, s0 ) = (l, l)(s, s0 ) = (l, h)(s, s0 ) = (h, h)(s, s0 ) = (h, l)Unconditional

Credit rationing in state s0
Basel I
Basel II
Laissez-faire
1.4
1.4
2.5
2.5
1.8

0.3
4.9
3.8
0.7
1.7

2.2
2.2
4.5
4.5
3.0

1.4
1.4
2.7
2.7
1.9

0.3
10.7
4.5
0.6
2.6

2.1
2.1
5.2
5.2
3.2

1.3
1.3
3.0
3.0
1.9

0.4
24.4
5.3
0.5
4.6

2.0
2.0
6.1
6.1
3.5

The parameters that define each of the scenarios and the associated Basel II capitalrequirements are described in Table 3. The Basel I capital requirement is 8%. Credit
rationing is the expected percentage of continuation projects that cannot be undertaken
because of banks’ insuﬃcient lending capacity. The rows show credit rationing in states’ when it is reached from state s according to the sequence (s,s’) in the first column.
Rows labeled ‘unconditional’ show weighted averages based on the unconditional
probabilities of each state.

Under Basel II, the impact of bank profits is also present, but the overall eﬀects on
credit rationing are dominated by the cross-state variation of the capital requirements, andits endogenous eﬀects on capital buﬀers. Thus the sequences with (s, s0 ) = (l, h), and thenthose with (s, s0 ) = (h, h), systematically exhibit the largest credit rationing. Intuitively, inthe low default state l the transition to the high default state h is less likely than continuing25

in h after being in h; additionally, in state l the required capital is lower than in state h. For
both reasons, banks hold less capital in s = l than in s = h (see Table 4). But then if theeconomy ends up in s0 = h, the combination of a lower capitalization in the previous period
and a higher current requirement explains the very sizable contractions in lending capacity
shown in Table 5.33 In particular, for the medium volatility scenario, when the economy
goes from the low to the high default state (and despite of the fact that banks hold a capital
buﬀer of 5.1% in the low default state) an average of 10.7% of the continuation projects arerationed, a figure that goes up to 24.4% in the high volatility scenario.
Thus Basel II implies significantly larger cyclical variation in credit rationing (and consequently in investment and output) than Basel I. Its incidence on the average level of credit
rationing, shown in the rows labeled ‘unconditional’ in Table 5, depends on the volatility of
PDs along the cycle. For the medium volatility scenario, the extra cost of Basel II in termsof long-run average credit rationing amounts to about 0.7% of the potential continuation
investment.
The behavior of credit rationing in the laissez-faire environment is very much an amplified
version of what is observed under Basel I, with levels of rationing that are between 50% and100% larger. In relation with Basel II, however, the comparison depends on the volatilityof PDs along the cycle: except in the low volatility scenario, the laissez-faire exhibits lower
cyclicality and, in the high volatility scenario, it even exhibits lower unconditional expectedcredit rationing.

5.4

Banks’ solvency

We next compare the various regulatory regimes in terms of banks’ solvency. Table 6 reports
the probability of failure of the representative bank for each of the scenarios described inTable 3 and each of the possible states of the economy. These probabilities are diﬀerent
for banks making initial loans (that in state s start with capital ks , earn interest rs on
performing loans, and pay the cost c of starting up their lending relationships) and banks
33

Interestingly, for the sequences with s0 = l (which entail the lowest credit rationing under Basel II), theeﬀect of bank profits becomes visible again, producing lower rationing in the (l, l) sequence than in the (h, l)sequence.

26

making continuation loans (that in state s start with capital γ s , earn interest a on performing
loans, and do not pay c). Unlike in the results on credit rationing, these probabilities are
purely forward-looking (i.e., they do not depend on the state of the economy in the previous
period) and hence we only report their conditional-on-s and unconditional values.
Table 6 shows that the probabilities of bank failure are much more uniform across states
under the risk-sensitive capital requirements of Basel II than under the constant 8% capitalrequirement of Basel I. Conditional on the state of the economy, the link between the level ofthe requirements and the level of solvency of second period banks is direct (since these banks
hold no capital buﬀers and the net interest income earned on performing loans is the same
in both states), so not surprisingly Basel II implies a significant improvement in solvency
in the high default state h and a reduction in solvency in the low default state l, with theunconditional eﬀect being clearly positive. For first period banks there are additional eﬀects
coming from the endogenous capital buﬀers and loan interest rates. Our results in Table
4 show that the latter eﬀects are very small, so solvency is inversely related to the total
holdings of capital. Hence Basel II increases the solvency of first period banks in state h,and (in the low and medium volatility scenario) it also increases their solvency in state l,despite imposing lower requirements than the 8% of Basel I. The unconditional eﬀect of BaselII on the solvency of first period banks is positive in the three scenarios. All in all, Basel II
roughly halves the probabilities of bank failure associated with Basel I, and makes the riskof failure more evenly distributed over time. This suggests that the risk-sensitive capitalrequirements of Basel II have a payoﬀ in terms of the long-term solvency of the banking
system.

27

Table 6. Banks’ solvency(all variables in %)Scenarios
Low volatility
1st-period banks:s=l
s=h
Unconditional
2nd-period banks: s = l
s=hUnconditional
Medium volatility
1st-period banks: s = l
s=hUnconditional
2nd-period banks: s = l
s=hUnconditional
High volatility
1st-period banks: s = l
s=hUnconditional
2nd-period banks: s = l
s=hUnconditional

Probability of bank failure
Basel I
Basel II
Laissez-faire
0.025
0.094
0.050
0.008
0.054
0.024

0.016
0.051
0.028
0.014
0.018
0.015

2.185
4.492
3.013
1.023
5.721
2.710

0.022
0.115
0.056
0.006
0.074
0.030

0.014
0.054
0.029
0.014
0.019
0.015

2.080
5.210
3.203
0.867
7.195
3.139

0.019
0.140
0.063
0.005
0.099
0.039

0.023
0.059
0.036
0.013
0.019
0.015

1.968
6.126
3.461
0.723
8.895
3.657

The parameters that define each of the scenarios and the associated Basel II capitalrequirements are described in Table 3. The Basel I capital requirement is 8%. Rows
labeled ‘unconditional’ show weighted averages based on the unconditional probabilitiesof each state.

It is worth noting that the probabilities of bank failure are very small under both
regulations–unconditionally, they range between 0.024% and 0.063% under Basel I, andbetween 0.015% and 0.036% under Basel II. Interestingly, the combination of capital buﬀers
and net interest income earned on performing loans makes the latter much lower than the
0.1% implied by the 99.9% confidence level of Basel II. This combination also explains the28

fact that the probabilities of bank failure under the laissez-faire environment are not very
high–unconditionally, they range between 2.710% and 3.657%.

6

Policy analysis

Our previous results show that the risk-sensitive Basel II regime induces much more cyclicality in the supply of bank credit than the risk-insensitive Basel I regime. Specifically, under
its original formulation, the Basel II regime tends to produce a particularly strong reduction
in banks’ lending capacity (and a rise in credit rationing) when the economy goes into a
recession. The results also indicate that bank solvency is higher under Basel II than under
Basel I.
These results suggest that the comparison of Basel I and Basel II in welfare terms is nottrivial and will crucially depend on the (structural or reduced-form) imputation of a social
cost to bank failures.34 Although the model could be extended to perform such a welfare
analysis, the discussion in this section will focus on less normative point: we will show that
it is possible to reduce the procyclical eﬀects of Basel II without fully renouncing to risksensitiveness or to the targeted long-term confidence. The adjustments that we consider
consist of modifying the cyclical profile of the confidence levels set by the regulator in such a
way that keeps their long-term average at 99.9%–the current level in the IRB approach ofBasel II–but lessens the target in those states (or sequences of states) where credit rationing
turned out to be the highest under the Basel II regime.
Table 7 shows the results of two specific policy experiments of this kind. Both are
performed under the parameterization of the medium volatility scenario described in Table
3. Policy 1 reduces the confidence level in the high default state h to 99.8% and increases
the confidence level in the low default state l to η > 99.9% so as to maintain the long-run
average at 99.9%. Thus η solves:
η × φl + 0.998 × φh = 0.999,
34

Repullo and Suarez (2004) perform this type of welfare analysis in a static setup where procyclicality is
not a concern, but capital requirements imply a deadweigh loss due to the extra cost of equity financing.

29where φl and φh are, respectively, the unconditional probabilities of the low and the highdefault state.35 Such a small adjustment causes a relevant change in capital requirements(from 6.6% to 7.9% for γ l and from 10.5% to 9.3% for γ h ), modifying banks’ optimal buﬀers(which become less procyclical), and smoothing the cyclicality of credit rationing. As shownin Panel A of Table 7, credit rationing in the sequences (l, h) and (h, h) falls from 10.7% and4.5%, respectively, to less than 4% in both sequences. Unconditionally, it falls from 2.6% to
1.9%, which is its unconditional value under Basel I. Interestingly, although the probabilities
of bank failure in the high default state h obviously increase, they remain lower than 0.08%
in all cells, and their unconditional average only increases from 0.029% to 0.040% for first
period banks, and from 0.015% to 0.017% for second period banks.
In Policy 2 we confine the reduced 99.8% confidence level to periods where state h occurs
after state l. The objective is to reduce the credit rationing detected in the second periodof sequences with (s, s0 ) = (l, h). The capital requirement when h occurs after h is left
unchanged, while the confidence level applied to the second period of the sequences (l, l)and (h, l) is increased so as to to keep the long-run average confidence level at 99.9%. By
construction, Policy 2 makes smaller adjustments in the Basel II capital requirements, so it
will be less eﬀective than Policy 1 in terms of smoothing credit rationing, but it will also beless significant in terms of its implications for banks’ solvency. The results in Table 7 showthat credit rationing in the second period of the (l, h) sequence gets substantially reduced,
but not as much as with Policy 1, while the unconditional probabilities of bank failure arealmost unchanged relative to those of Basel II.
All in all, our policy experiments show the feasibility of achieving significant gains interms of credit rationing without major costs in terms of banks’ long-term solvency. Thiscan be achieved with cyclical adjustments that preserve the value-at-risk foundation of theBasel II requirements. The choice between Policy 1 and Policy 2 (or the fine tuning of
their details) should eventually depend on the trade-oﬀ between the gains in terms of asmoother and lower credit rationing, and the losses in terms of less smooth and slightly
higher probabilities of bank failure.
35

See Footnote 28 for the expressions of φl and φh in terms of the transition probabilities ql and qh .

30

Table 7. Procyclicality correction(all variables in %)A. Credit rationing

0(s, s ) =Basel I
Basel II
Policy 1
Policy 2(l, l)
1.4
0.3
0.8
0.5

Credit rationing in state s0(l, h)(h, h)(h, l)
1.4
2.7
2.7
10.7
4.5
0.6
3.7
3.6
1.6
4.4
4.4
0.6

Unconditional
1.9
2.6
1.9
1.9

B. Banks’ solvency(s, s0 ) =Basel I
Basel II
Policy 1
Policy 20(s, s ) =Basel I
Basel II
Policy 1
Policy 2

Probability of failure of 1st-period banks in state s0(l, l)(l, h)(h, h)(h, l)
Unconditional
0.022
0.115
0.115
0.022
0.056
0.014
0.054
0.054
0.014
0.029
0.017
0.079
0.079
0.017
0.040
0.019
0.054
0.054
0.019
0.031
Probability of failure of 2nd-period banks in state s0(l, l)(l, h)(h, h)(h, l)
Unconditional
0.006
0.074
0.074
0.006
0.030
0.014
0.019
0.019
0.014
0.015
0.007
0.035
0.035
0.007
0.017
0.011
0.035
0.019
0.014
0.016

The parameters and associated Basel II capital requirements are those of the medium volatility scenario
described in Table 3. The Basel I capital requirement is 8%. Policy 1 reduces the confidence levelof the Basel II formula to 99.8% in state h and increases it in state l so as to keep the unconditional
average at 99.9%. Policy 2 sets the confidence level at 99.8% only when state h occurs after state l,compensating it when state l occurs so as to keep the unconditional average at 99.9%. Columns
labeled ‘unconditional’ show weighted averages based on the unconditional probabilities of each state.

7

Discussion

In this section we discuss some simplifying features of our model, including the distributionof banks’ market power in the first and second period of the lending relationships, the use of31

short-term loan contracts, and the assumption that banks with ongoing relationships have noaccess to the equity market. We consider the possible eﬀects of relaxing these assumptions,noting that in some cases our conclusions about the procyclicality of Basel II would be
strengthened.

7.1

Competition and market power

In our model banks are perfectly competitive in the market for first period loans and act
as monopolists in the market for second period loans. This is justified by the fact thatborrowers get locked-in after one period due to some (unmodeled) asymmetric information
problem (e.g., one that makes borrowers searching for a new bank after one period look like“rejected” by the initial bank, which leads to a standard lemons problem in the refinancing
market). We have opted for an all-or-nothing modeling of the lock-in eﬀect for several
reasons. The most obvious one is tractability: if the market for continuation loans weremore competitive, the degree of eﬀective competition and the resulting loan rates would
vary with the degree of credit rationing. Even in the polar case of perfect competition in
the market for second period loans, over the range of poor realizations of the risk factor thatlead to rationing, banks would be able to appropriate scarcity rents from their (non-rationed)borrowers.36 Relative to the current modeling, the more competitive market for second periodloans would entail lower continuation rents for banks, and hence lower incentives for themto keep capital buﬀers. This uncovers a second justification for our assumption: it yields a
conservative estimate of the procyclicality induced by capital regulation.

7.2

Short-term loan contracts

We have described the relationship between entrepreneurs and banks as instrumented by asequence of one-period loans. One might wonder whether focusing on short-term contracts
builds in the “imperfection” that drives our main results. The answer is yes and no. With
long-term contracts, or more generally contracts that diﬀer from the sequence of one-period
36

In the context of the single risk factor model of Basel II, either all banks or no bank are capital constrainedin the market for second period loans, so the second period loan rate would be either the monopoly rate aor some break-even rate that makes continuation lending a zero net present value investment for the banks.

32loans on which we focus, there might be room for improving over the credit allocation
outcomes obtained in our analysis. For example, for given capital buﬀers, setting higher
loan rates in the first period and lower loan rates in the second would reduce the incidenceof credit rationing, since banks would have more capital to support their second period
lending. But in the context of our model, long-term contracts pose important commitment
and enforcement problems. In particular, they would have to specify the loan rates in thefirst and the second period, as well as the rationing scheme to be used in those cases where
the bank ends up with insuﬃcient lending capacity (since otherwise the bank might try torenegotiate the terms of the second period loans by threatening entrepreneurs to put themin the pool of rationed borrowers). This means that default rates in the first period would
have to be verifiable, and banks would have to be restricted in their ability to pay dividends.
They might also have to commit to maintain a capital buﬀer in the first period. The reason
for this is that when competing for borrowers in the first period, banks internalize the wholesurplus that the underlying investment projects generate over the two periods, but once therelationship starts, banks only internalize the return of their current and future lending. If
the second period loan rate is lower than a, then banks do not take into account part ofthe continuation surplus when deciding how much capital to hold–recall that first period
capital buﬀers are held for the purpose of reducing the expected credit rationing in the second
period. All in all, the previous considerations suggest that there are serious commitment
and enforcement problems that limit the possible improvements in the allocation of credit
that long-term contracts might bring about.

7.3

Imperfect access to the equity market

The assumption that banks with ongoing relationships have no access to the equity market
is obviously crucial for our results. With perfect, frictionless access to the equity market
in the interim period, there would be no credit rationing among second period borrowers,
except in the rare event that their bank fails (and it is not recapitalized by its shareholders).
Banks in such a context would most likely hold no buﬀers, at least under the high capitalrequirements of Basel I or Basel II. Given the ample evidence of equity-issuance frictions,33

the key question is whether the specificities of our approach–that ties these frictions to theinformational asymmetries associated with relationship lending–drive the results.
A more general way of capturing equity-issuance frictions would be to assume that accesscan occur with some (exogenous) probability υ < 1. Changes in parameter υ could then be
used to evaluate the “marginal” eﬀects of the friction on capital buﬀers and credit rationing.
One could also explore situations in which υ is contingent on the state s0 of the economy at
the interim date. This extension would probably confirm the results obtained in the current
setup and might even reinforce our conclusions about the procyclicality of the risk-sensitive
capital requirements of Basel II.37
It should also be noted that the really strong assumption that we are making is that banks
can frictionless access the equity market when they renew their stock of lending relationships.
This assumption is instrumental to achieving a tractable OLG structure in the context of
an infinite horizon problem that otherwise would be characterized by longer memory andmore complex dynamics (namely, credit rationing might also aﬀect first period loans andthe pricing of these loans might depend on the full history of the economy). A modelingalternative would be to assume a structure similar to the one in the popular Calvo (1983)
model of staggered price setting, i.e., that in each period a fraction of the banks can issue
new equity. In this context one would have to discuss the allocation of the newly born
entrepreneurs to the existing banks. Would they demand loans to the recapitalizing banks
only? If not, how would the pricing of the new loans be determined?

7.4

Other extensions

The framework used in this paper could also be extended in a number of other directions.
First, we could consider lending relationships that extend over more than two periods. If
relationships last for T periods and banks cannot raise equity for the whole length of therelationship, the qualitative results should be very similar to ours. Such a model would, ofcourse, yield richer dynamics, as the eﬀect of a shock would propagate over several periods.
37

If banks can with some probability access the equity market in the interim period, they would have lower
incentives to keep capital buﬀers, so depending of parameter values, the incidence of credit rationing might
be higher.

34

Second, we could consider alternative distributions Fs (xt ) of the default rate xt , keeping
the specification of the Basel II capital requirements as in (14), but allowing for thicker
tails in the loan loss distribution. Third, we could incorporate cyclically-varying demand.
One easy way of doing this would be to let the scale μ of second period projects varywith the state s of the economy. A bigger challenge would be to introduce some “downward
sloping” aggregate demand for loans, which could be done by assuming that entrepreneurs areheterogeneous in their opportunity cost of becoming active in the first period. The alreadyexisting variability in projects’ success probabilities would tend to produce (procyclical)
variability in the demand for loans. Further variability might be introduced by replacing
the current success return a for some as . Finally, one could allow for feedback eﬀects fromconstrained to unconstrained entrepreneurs by letting at = a(It ) instead of a, where a(It ) is
an increasing (and possibly concave) function of It , the aggregate investment at date t. This
will capture demand externalities or technological complementarities similar to those studied
in endogenous growth theory. Analyzing these alternative models is beyond the scope of thispaper.

8

Concluding Remarks

In many early supervisory and industry reports on the implications of Basel II, it was standard to first recognize the potential cyclical eﬀects of the risk-sensitive capital requirements
and then qualify that, given than most banks held capital in excess of the regulatory minima,
the practical incidence of the procyclicality problem was likely to be small if not negligible.
In light of the disastrous experience accumulated during the current crisis, the temptation
now is to presume that the buﬀers voluntarily chosen by the banks will never be “suﬃcient”
to prevent the procyclicality of bank lending and a more radical regulatory reform is needed.
While many of the writings defending the early views do not have the extension or thetechnical nature required to elaborate on the foundations of their claims, others unveil two
related misconceptions. The first misconception is that the holding of capital buﬀers meansthat capital requirements are “not binding.” Under a purely static perspective this would

35

be tautologically true. In a convex optimization problem, it would also be true that smallchanges in the level of the requirements would not alter the optimal capital holdings. In a
dynamic problem, however, this need not be the case: banks may hold capital buﬀers in agiven period because they wish to reduce the risk of facing a statically “binding” requirementin the future. Perhaps these precautions make future requirements “not binding” when
the time comes, but clearly their presence alters banks’ capital decisions and the wholedevelopment of future events. So observing that banks hold capital buﬀers does not meanthat capital requirements do not matter.
A second, related misconception is to accept that the cyclical behavior of capital buﬀers
observed under some particular regulatory regime (either Basel I or Basel II) can be somehow
extrapolated, without much qualification, to any other regulatory regime. The results inthis paper suggest that the behavior of buﬀers can radically diﬀer across regulatory regimes,
which in part might explain why many observers changed their views on the suﬃciency or
insuﬃciency of capital buﬀers as the current crisis unfolded. In other words, if buﬀers areendogenously aﬀected by the prevailing bank capital regulation (even if they appear not to“bind”), reduced-form predictions of what would occur under diﬀerent regulatory regimes
do not resist the Lucas’ critique.
Our model provides a tractable framework in which it is possible to evaluate the cyclicaleﬀects of bank capital regulation without incurring in these misconceptions. To keep theanalysis as transparent as possible, we have simplified on a number of dimensions. For
example, we abstract from demand side fluctuations and aggregate feedback eﬀects thatmight mitigate and exacerbate, respectively, the supply-side eﬀects that we identify. As wehave discussed above, one could take our model as a building block for a fuller dynamic
general equilibrium model with a production sector partly composed of entrepreneurial firmsthat rely on relationship bank lending. One could also think about extensions that generalize
our modeling of the frictions related to banks’ access to equity financing. Our contribution,from this perspective, is to show that the interaction of relationship lending (which makessome borrowers dependent on the lending capacity of the specific bank with which they
establish a relationship) with frictions in banks’ access to equity markets (which makes some36

banks’ lending capacity a function of their historically determined capital positions and thecapital requirements imposed by regulation) has the potential to cause significant cyclicalswings in the supply of credit.
Under realistic parameterizations, the model produces capital buﬀers and equilibriumloan rates whose levels and cyclicality in the Basel I regulatory environment are in linewith those observed in the data. The same parameterizations when applied to the Basel II
environment suggest that its risk-sensitive requirements imply a substantial increase in theprocyclicality of bank lending. Specifically, despite banks taking precautions and holding
larger buﬀers during expansions in order to have a reserve of capital for the next recession(when capital requirements rise), the arrival of recessions is normally associated with asizeable credit crunch, as capital-constrained banks are induced to ration credit to some of
their dependent borrowers.
Having a model that explicitly accounts for the endogenous determination of capital
buﬀers and equilibrium loan rates is important for a rigorous analysis of the type of countercyclical adjustments that recent proposals on regulatory reform advocate. As an illustration, we have shown that small cyclical adjustments in the confidence level of Basel II
would substantially reduce the incidence of credit rationing over the business cycle withoutcompromising the long-run solvency targets implied in the original regulation.

37

Appendix
A

Proofs of analytical results

Solutions to the representative bank’s capital decision Using the definition of vss0 (xt )in (8), the net present value vs (ks , rs ) of the representative bank that in state s holds capital
ks and charges an interest rate rs on its unit of initial loans may be written asvs (ks , rs ) = qs vsh (ks , rs ) + (1 − qs )vsl (ks , rs ),

(15)

where
vss0 (ks , rs ) = β

"Z
x
hss0
0

βπ s0
[(βπ s0 − γ s0 )μ + ks0 (x)] dFs (x) +
γ s0

Z xes

x
hss0

#

ks0 (x) dFs (x) − ks .

(16)

By the definitions (4) and (3) of x
ess0 and x
bs , the function vss0 (ks , rs ) has the following
properties:

1. For ks ≤ c − rs we have x
ess0 < xbs ≤ 0, so∂vss0= −1 < 0.
∂ks

ess0 ≤ 0 < x
bs , so
2. For c − rs < ks ≤ c − rs + γ s0 μ we have x
β 2 π s0
∂vss0
=
Fs (b
xs ) − 1 ≶ 0, and
∂ks
γ s0

xs )
∂ 2 vss0
β 2 π s0 Fs0 (b
=
> 0.
2
∂ks
γ s0 (λ + rs )ess0 < 1, so
3. For c − rs + γ s0 μ < ks < c + λ + γ s0 μ we have 0 < x

∂vss0
β
=
[βπ s0 Fs (bxs ) − (βπ s0 − γ s0 )Fs (exss0 )] − 1 ≶ 0,
∂ks
γ s0

and
β
∂ 2 vss0[βπs0 Fs0 (b=xs ) − (βπs0 − γ s0 )Fs0 (exss0 )] ≶ 0.
2∂ks
γ s0 (λ + rs )4. For c + λ + γ s0 μ ≤ ks we have 1 ≤ x
ess0 < xbs , so∂vss0= β − 1 < 0.
∂ks38

Hence the function vss0 (ks , rs ) is linearly decreasing or strictly convex for ks ≤ c − rs + γ s0 μ,
linearly decreasing for ks ≥ c + λ + γ s0 μ, and may be increasing or decreasing, and concave
or convex for c − rs + γ s0 μ < ks < c + λ + γ s0 μ. Introducing the constraint ks ∈ [γ s , 1] (andassuming that parameter values are such that c + λ + γ s0 μ < 1) it follows that the problem
maxks ∈[γ s ,1] vss0 (ks , rs ) has either a corner solution with ks = γ s , or an interior solution withks ∈ (c − rs + γ s0 μ, c + λ + γ s0 μ). In the latter case we have 0 < x
ess0 < 1, so there is apositive probability Fs (exss0 ) that the bank has excess lending capacity in state s0 , and apositive probability 1 − Fs (exss0 ) that the bank has insuﬃcient lending capacity in state s0 .

Since γ l ≤ γ h implies c − rs + γ l μ ≤ c − rs + γ h μ and c + λ + γ l μ ≤ c + λ + γ h μ, we conclude
that the problem maxks ∈[γ s ,1] [qs vsh (ks , rs ) + (1 − qs )vsl (ks , rs )] has either a corner solution
with ks = γ s , or an interior solution with ks ∈ (c − rs + γ l μ, c + λ + γ h μ). In the latter case,there must be a positive probability that the bank has insuﬃcient lending capacity in states0 = h (and possibly also in state s0 = l), and a positive probability that the bank has excess
lending capacity in state s0 = l (and possibly also in state s0 = h).
Comparative statics of the initial loan rate The sign of drs∗ /dz for z = a, λ, μ, c, δ, qs , γ h , γ lcan be obtained by total diﬀerentiation of (10):∂vs dks∗ ∂vs drs∗ ∂vs++= 0.
∂ks dz∂rs dz∂z(17)

When ks∗ is interior, the first-order condition for a maximum that follows from (11) gives∂vs /∂ks |(ks∗ ,rs∗ ) = 0, so the first term in (17) vanishes. Moreover when ks∗ is interior it must

be the case that 0 < x
ess0 < 1 for at least one state s0 , so diﬀerentiating (15) and (16) we
have

∂vs
∂vsh
∂vsl
= qs
+ (1 − qs )
> 0,
∂rs
∂rs
∂rs

since
∂vss0
=β
∂rs

"Z
x
hss0
0

βπ s0
(1 − x) dFs (x) +
γ s0

Z xes

x
hss0

(1 − x) dFs (x) ≥ 0,with strict inequality for at least one state s0 . Hence we are left with:
¶−1
µ
drs∗
∂vs
∂vs
=−
.
dz
∂rs
∂z
39

#

(18)

Similarly, in a corner solution with ks∗ = γ s we have dks∗ /dz = 0 for all z 6= γ s , in whichcase the first term in (17) also vanishes and (18) obtains again. Finally, for z = γ s , we havedks∗ /dγ s = 1 and, thus,

¶−1 µ
¶
µ
drs∗
∂vs ∂vs
∂vs
,
=−
+
dγ s
∂rs
∂γ s ∂ks

where ∂vs /∂ks |(ks∗ ,rs∗ ) ≤ 0, since otherwise fixing ks∗ = γ s would not be optimal. With
these expressions in mind, the results in Table 1 can be immediately related to the (selfexplanatory) signs of the partial derivatives of vs (ks∗ , rs∗ ) that we summarize in Table A1 (andwhose detailed expressions we omit, for brevity).
Table A1. Eﬀects on the net present value of the bank
z
∂vs
∂z

=

rs

a

λ

μ

c

δ

qs

γh

γl

+

+

−

+

−

−

−

−

−

Comparative statics of the initial capital When the optimal initial capital in state s
is at the corner ks∗ = γ s , with ∂vs /∂ks |(ks∗ ,rs∗ ) < 0, marginal changes in any parameter otherthan γ s will have no impact on ks∗ , while obviously dks∗ /dγ s = 1. Thus, in what follows wefocus on the more interesting interior solution case.38
The sign of dks∗ /dz for z = a, λ, μ, c, δ, qs , γ h , γ l can be obtained by total diﬀerentiationof the first-order condition ∂vs /∂ks = 0 that characterizes an interior equilibrium:∂ 2 vs dks∗
∂ 2 vs drs∗
∂ 2 vs++= 0.
∂ks2 dz∂ks ∂rs dz∂ks ∂z(19)

By the second-order condition we have ∂ 2 vs /∂ks2 < 0, which givesµ 2 ¶−1 µ 2
¶
dks∗
∂ vs
∂ vs
∂ 2 vs drs∗
=−
+
.
dz
∂ks2
∂ks ∂z ∂ks ∂rs dz
38

The case with ks∗ = γ s and ∂vs /∂ks |(ks∗ ,rs∗ ) = 0 is a mixture of both cases since, depending on the sign ofthe eﬀect of the marginal variation in a parameter, the optimal decision might shift from being at the corner
to being interior. A similar complexity may occur if the change in a parameter breaks some underlying
indiﬀerence between an interior and a corner solution (or between two interior solutions). We will omit thediscussion of these cases, for simplicity.

40

Hence the sign of dks∗ /dz coincides with the sign of the second term in brackets, which has
two components: the direct eﬀect of z on ks∗ (for constant rs∗ ) and the loan rate eﬀect (dueto the eﬀect of z on rs∗ ). The signs of the direct eﬀects shown in the first row of Table 2
coincide with the signs of the cross derivatives ∂ 2 vs /∂ks ∂z summarized in Table A2 (whose
detailed expressions we omit, for brevity).
Table A2. Eﬀects on the marginal value of capitalz∂ 2 vs∂ks ∂z

=

rs

a

λ

μ

c

δ

qs

γh

γl

−

+

?

+

−

−

?

?

?

The signs of the loan rate eﬀects shown in the second row of Table 2 can be simply obtained
from the results summarized on Table 1 and the fact that by diﬀerentiating (15) and (16)one can show that∂ 2 vs∂ 2 vsh∂ 2 vsl= qs+ (1 − qs )< 0,
∂ks ∂rs∂ks ∂rs∂ks ∂rswhere∂ 2 vss0
β=bs )Fs0 (bxs ) − (βπ s0 − γ s0 )(1 − x
ess0 )Fs0 (exss0 )].
[βπ s0 (1 − x∂ks ∂rs
γ s0 (λ + rs )To check this notice that the second-order condition ∂ 2 vs /∂ks2 < 0 implies¸∙πh
βπ h − γ h 0πl
βπ l − γ l 0< qsxs ) qs + (1 − qs )Fs (exsh ) + (1 − qs )Fs (exsl ).
βfs (bγhγlγhγless0 , together with the fact that γ l ≤ γ h ,Hence using the definitions (3) and (4) of x
bs and x
we have 1 − x
bs < 1 − x
esl ≤ 1 − xesh , so we conclude that¸∙πl
βπ l —γ lπh
βπ h —γ h
< qs
β(1−b
xs )fs (bxs ) qs + (1—qs )(1−exsh )Fs0 (exsh )+(1—qs )(1−exsl )Fs0 (exsl ),γhγlγhγlwhich after some reordering proves the result.

41

B

Discussion of parameter values

Interest rate on continuation loans: a = 0.04. The interest rates on banks’ marginal
lending and borrowing activities are not available in standard statistical sources. A common approach is to proxy them with implicit average rates computed from accounting figures. According to the FDIC Statistics on Banking for the years 2004 to 2007 (available at
http://www2.fdic.gov/SDI/SOB/), Total interest income of all US commercial banks represents, on average, 5.74% of Earning assets, while Total interest expense represents 2.32% ofTotal liabilities. This yields an average net interest margin of 3.42%. Yet Service charges
on deposit accounts are 0.55% of Total deposits, which implies that deposit-funded activities
yield an average intermediation margin of 3.97%. This number is very close to our assumed
4%. (See Figure 1 for quarterly data on the net interest margin of US banks over a longer
period.)Cost of setting up a lending relationship: c = 0.03. This is a rather conservative
estimate of the importance of intermediation costs. According to the FDIC Statistics on
Banking for the years 2004 to 2007, Total non-interest expense of all US commercial banks
42

represented an average of 3.97% of Total assets.
Cost of bank capital: δ = 0.04. Based on the estimates of Graham (2000) for non-financial
corporations, an annual discount rate of 4% is a rather conservative estimate for the tax
disadvantage of equity financing. To see this, consider the standard measure of the marginal
tax shield of debt financing, net of personal taxes: MTS = [(1−τ i )−(1−τ c )(1−τ e )]/(1−τ i ),where τ i , τ c , and τ e are the marginal tax rates on personal interest income, corporate income,and personal equity income, respectively. As in Hennessy and Whited (2007), set τ i = 0.29
and consider τ c = 0.40 as an upper bound to τ c (based on the combination of the top
statutory federal rate and the average state rate as reported by Graham, 2000). Suppose,
conservatively, that τ e = 0, so that the marginal investor manages to make its equity income
fully exempt from personal taxation. Then we get MTS ' 0.04 for τ c = 0.32, where this lastchoice is consistent with US data. In particular, for US commercial banks over the period
2004-2007, Applicable income taxes represented, on average, 31.7% of Pre-tax net operating
income. As in Hennessy and Whited (2007), this number can be seen as the expected tax
rate in a situation in which the representative bank earns positive taxable income and hence
faces an eﬀective corporate tax rate of τ c with a probability of 80%, while it faces an eﬀective
zero tax rate with a probability of 20%.
Probabilities of transition to the high default state: ql = 0.20 and qh = 0.64. In our
Markov switching setup, the expected durations of states l and h are 1/ql and 1/(1 − qh ),respectively. We calibrate these durations using data from the FDIC Historical Statistics on
Banking (available at http://www2.fdic.gov/hsob/index.asp). Specifically, we compute theannual ratio of Net loan and lease charge-oﬀs to Gross loans and leases for FDIC-insured
commercial banks over the period 1969-2004, and we detrend the series using the standard
HP-filter for annual data. The resulting series includes 20 below-average observations in
4 complete low default phases (implying an average duration of 20/4 = 5 years) and 14
above-average observations in 5 complete high default phases (implying an average durationof 14/5 ' 2.8 years). The observations corresponding to 1969 and 2004 belong to censored
below-average phases that are not taken into account. The imputed expected durations are43

in line with Koopman et al. (2005), that identify a stochastic cycle in US business failure
rates with a period of between 8 and 11 years.
PD scenarios: pl ∈ [1.00, 1.30] and ph ∈ [2.88, 3.63]. In the Special Report “Commercial
Banks in 1999” (available at http://www.philadelphiafed.org/files/bb/bbspecial.pdf), theFederal Reserve Bank of Philadelphia oﬀers data on the experience of US commercial banks
during the full cycle of the 1990s. Following the 1990-1991 recession, the aggregate ratio of
Non-performing loans to Total loans was slightly above 3% in years 1990-1993, declined toslightly above 2% in 1993, and remained below 1.5% (with a downward trend) for the restof the decade. It is also possible to check the realism of our PD scenarios by looking at theratio of Loan losses to Total loans, whose quarterly evolution over recent years appears in
Figure 2. Notice that under our assumption about the value of the LGD parameter, λ = 0.45
(which we take from the “foundation IRB” approach of Basel II), the average default rate
behind the series in Figure 2 should be 1/0.45 ' 2.22 times the ratio depicted there, which
again suggests the realism of our choice of PDs slightly above 1% in low default states andaround 3% in high default states.

44

References
Ayuso, J., D. Pérez, and J. Saurina (2004), “Are Capital Buﬀers Procyclical? Evidence
from Spanish Panel Data,” Journal of Financial Intermediation, 13, 249-264.
Altman, E., B. Brady, A. Resti, and A. Sironi (2005), “The Link between Default andRecovery Rates: Theory, Empirical Evidence, and Implications,” Journal of Business,
78, 2203-2227.
Barakova, I., and M. Carey (2001), “How Quickly Do Troubled U.S. Banks Recapitalize?
With Implications for Portfolio VaR Credit Loss Horizons,” Federal Reserve Board.
Basel Committee on Banking Supervision (1988), International Convergence of Capital
Measurement and Capital Standards, Basel: Bank for International Settlements.
Basel Committee on Banking Supervision (2004), International Convergence of Capital
Measurement and Capital Standards. A Revised Framework, Basel: Bank for International Settlements.
Bernanke, B., and C. Lown (1991), “The Credit Crunch,” Brookings Papers on Economic
Activity, 2, 205-239.
Bikker, J., and P. Metzemakers (2004), “Is Bank Capital Procyclical? A Cross-Country
Analysis,” De Nederlandsche Bank WP No. 009/2004.
Blum J., and M. Hellwig (1995), “The Macroeconomic Implications of Capital Adequacy
Requirements for Banks ,” European Economic Review, 39, 739-749.
Boot, A. (2000), “Relationship Banking: What Do We Know?,” Journal of Financial Intermediation, 9, 7-25.
Bruche, M., and C. González-Aguado (2006), “Recovery Rates, Default Probabilities andthe Business Cycle,” CEMFI Working Paper No. 0612.

45

Brunnermeier, Markus, Andrew Crockett, Charles Goodhart, Avi Persaud, and Hyun Shin(2009), “The Fundamental Principles of Financial Regulation”, Geneva Reports on theWorld Economy, 11, preliminary draft.
Calvo, G. (1983), “Staggered Prices in a Utility-Maximizing Framework,” Journal of Monetary Economics, 12, 383-398.
Catarineu-Rabell, E., P. Jackson, and D. Tsomocos (2005), “Procyclicality and the New
Basel Accord — Banks Choice of Loan Rating System,” Economic Theory, 26, 537-557.
Cooley, T., and V. Quadrini (2001), “Financial Markets and Firm Dynamics,” American
Economic Review, 91, 1286-1310.
Detragiache, E., P. Garella, and L. Guiso (2000) “Multiple versus Single Banking Relationships: Theory and Evidence,” Journal of Finance, 55, 1133-1161.
Diamond, D. W., and R. G. Rajan (2000), “A Theory of Bank Capital,” Journal of Finance,
55, 2431-2465.
Elizalde, A., and R. Repullo (2007), “Economic and Regulatory Capital in Banking: Whatis the Diﬀerence?,” International Journal of Central Banking, 3, 87-117.
Estrella, A. (2004), “The Cyclical Behavior of Optimal Bank Capital,” Journal of Banking
and Finance, 28, 1469-1498.
FSF (2009), Report of the Financial Stability Forum on Addressing Procyclicality in the Financial System, Financial Stability Forum, April. Available at http://www.fsforum.org/
publications/r_0904a.pdf.
Gordy, M. (2003), “A Risk-Factor Model Foundation for Ratings-Based Bank Capital
Rules,” Journal of Financial Intermediation, 12, 199-232.
Gordy, M., and B. Howells (2006), “Procyclicality in Basel II: Can We Treat the Disease
Without Killing the Patient?,” Journal of Financial Intermediation, 15, 395-417.
46

Graham, J. (2000), “How Big Are the Tax Benefits of Debt?,” Journal of Finance, 55,
1901-1941.
Greenspan, A. (2002), “Cyclicality and Banking Regulation,” Remarks at the Conference on
Bank Structure and Competition, Federal Reserve Bank of Chicago, Chicago, IL, May
10. Available at http://www.federalreserve.gov/BoardDocs/Speeches/2002/20020510/.
Hennessy, C., and T. Whited (2007), “How Costly Is External Financing? Evidence froma Structural Estimation,” Journal of Finance, 62, 1705-1745.
Holmström, B., and J. Tirole (1997), “Financial Intermediation, Loanable Funds, and theReal Sector,” Quarterly Journal of Economics, 112, 663-691.
Kashyap, A., and J. Stein (2004), “Cyclical Implications of the Basel II Capital Standards,”
Federal Reserve Bank of Chicago Economic Perspectives, 1st Quarter, 18-31.
Kashyap, A., J. Stein, and D. Wilcox (1993), “Monetary Policy and Credit Conditions:
Evidence form the Composition of External Finance,” American Economic Review,
83, 151-195.
Koopman, S. J., A. Lucas, and P. Klaassen (2005), “Empirical Credit Cycles and CapitalBuﬀer Formation,” Journal of Banking and Finance, 29, 3159-3179.
Lindquist, K.-G. (2004), “Bank’s Buﬀer Capital: How Important Is Risk?,” Journal ofInternational Money and Finance, 23, 493-513.
Myers, S., and N. Majluf (1984), ”Corporate Financing and Investment Decisions whenFirms Have Information that Investors Do Not Have,” Journal of Financial Economics,
13, 187-221.
Peura, S., and J. Keppo (2006), “Optimal Bank Capital with Costly Recapitalization,”
Journal of Business, 79, 2163-2201.
Repullo, R., and J. Suarez (2004), “Loan Pricing under Basel Capital Requirements,” Journal of Financial Intermediation, 13, 496-521.
47

Saurina, J., and C. Trucharte (2007), “An Assessment of Basel II Procyclicality in Mortgage
Portfolios,” Journal of Financial Services Research, 32, 81-101.
Sharpe, S. (1990), “Asymmetric Information, Bank Lending, and Implicit Contracts: A
Stylized Model of Customer Relationships,” Journal of Finance, 45, 1069-1087.
Vasicek, O. (2002), “Loan Portfolio Value,” Risk, 15, December, 160-162.
Von Thadden, E.-L. (2004), “Asymmetric Information, Bank Lending and Implicit Contracts: The Winner’s Curse,” Finance Research Letters, 1, 11-23.
Zhu, H. (2008), “Capital Regulation and Banks’ Financial Decisions,” International Journalof Central Banking, 4, 165-211.

48