3 pillar basel ii jmib_19_01.pdf

Journal of Money, Investment and Banking

ISSN 1450-288X Issue 19 (2011)

© EuroJournals Publishing, Inc. 2011

http://www.eurojournals.com/JMIB.htm

The Three Pillars of Basel II: A Diffusion Model with

Jumps of Banking Regulation

Basel, M. Al-Eideh

Quantitative Methods & Information Systems, College of Business Administration

Kuwait University, P. O. Box 5486, Safat 13055, Kuwait

E-mail: [email protected]

Abstract

In this research we try to consider the problem of analyzing the interaction between

the three pillars of the new Basel Accord (Basel II): capital adequacy requirements,

centralized supervision and market discipline. A cash-flows diffusion model with jumps of

commercial banks' behavior is developed where interaction between these three instruments

can be analyzed. The conditions under which market discipline can reduce the minimum

capital requirements needed to prevent moral hazard as well as the issues of regulatory

forbearance and procyclicality are also discussed.

Keywords: Cash-Flows Diffusion Model, Pillars of Basel II, Banking Regulations.

JEL Cassification Codes: E58; G21

1. Introduction Basel II is an international agreement that sets minimum requirements for the capital reserves held by

creditor basks. The Basel Committee on Banking Supervision (Basel Committee) was established in

1974, by the Bank of International Settlements (BIS), an international organization founded in Basel

(Switzerland) in 1930 to serve as a bank for central banks. The Basel Committee has gradually

developed common international standards of banking supervision that are supposed to be

implemented through national legislation, although the Committee has no international means of

enforcement. The main instruments developed by the Basel Committee are: The Basel Committee's

Concordant, The Basel Committee's 25 Core principles for Effective Banking Supervision, and most

importantly, The Basel Capital Accords.

In 1988, the supervisors of the Basel Committee agreed on how many financial reserves banks

must put aside when providing loans. For banks, it is costly to leave capital idle and they prefer to keep

as little capital in reserve as possible. But this temptation could threaten financial stability if the loans

are not entirely paid back, or when depositors suddenly collect their money. Thus, the use of reserve

requirements is one of the most important tools of prudential supervision, since it helps to prevent

excessive lending by banks and thus reduces the risk of bankruptcies (Cf. Berger et al. (1995), Jackson

and Perraudin (2000)). The basic principles of the Basel Capital Accord of 1988 (Basel I) are as

follows:

Principle 1: Banks must put aside 8 % of the amount of a loan in reserve when there is a 100 %

risk associated with that loan, as defined by the Basel Accord framework's risk assessment. If assessed

risk is lower, reserve requirements can be lowered accordingly.

Journal of Money, Investment and Banking - Issue 19 (2010) 6

Principle 2: Banks must make assessments of the potential for loan default for government,

bank and corporate borrowers. For instance, a bank that gives a loan to a government in an OECD

country does not have to put any money in reserve since according to the Basel principles, the risks of

non-repayment are none. In contrast, Basel I stipulates banks have to put aside 8% of all loans

provided to corporations.

Principle 3: Banks that give loans to other banks have to distinguish between short-term (up to

12 months) and long term loans. According to Basel I, the risk of providing short term loans to banks is

much less (only 20% risk) than providing long term loans (up to 100% risk for developing country

borrowers).

By the late 1990s, the Basel Committee began discussions to update Basel I. A revision was

needed to reflect current risk measurement techniques that were more sophisticated than the 1988

approach. Big banks were already increasingly using their own more detailed risk assessment

mechanisms, and the 1988 standard approach thus increasingly was regarded as a burden. In addition, a

new framework was needed to account for developments in financial markets, such as the common

practice of credit risk transfer. Finally, with Asia financial crises in the late 1990s, the need for better

financial supervision became very clear, further spurring the revision of the Capital Accord.

By May 2004, the Basel member countries reached a consensus on a new agreement, Basel II,

to replace the Basel Capital Accord of 1988 (Basel I) in the beginning of 2007. More than 30,000

financial institutions worldwide expected to adopt compliance with Basel II. This agreement aimed in

improving and supporting the risk management to promotes ensures the stability of the banking and

financial system, in order to maintain competitiveness and access to low–cost capital. This new

agreement differs from old one in some major respects. More specifically, it allows the use of internal

banking model systems to assess the riskiness of their portfolios and to determine their required capital

cushion. This applies to the credit and operational risks and decentralizes the regulatory capital

adequacy requirements for banks. Also, it acknowledges the importance of complementary

mechanisms to safeguard against bank failures, by explicitly building on two additional pillars next to

the traditional focus on minimum bank capital (E-L von Thadden (2004)). The new Basel Capital

Accord (Basel II) is basically based on the following three pillars:

Pillar 1: (Capital Adequacy Requirement). New risk assessment mechanisms and resulting

capital requirements: New methods are introduced to measure credit risk, the risk of non-payment

associated with bank lending. Different approaches are introduces such as the standardized approach

which measures the risks of a borrower by using private or public rating agencies that assess borrower

solvency, the internal rate based approach that allows a bank to use its own risk estimation systems as

long as they comply with certain criteria and information disclosure requirements, and the

securitization framework provided by Basel II and it aims to mitigate credit risk. Also, capital

requirements are introduced for operational risk, the risk associated with the internal processes of the

bank. Basel II offers three different approaches, varying in complexity, to assess these risks.

Pillar 2: (Supervisory Review). Changes in the supervisory processes: Banking supervisors get

more power and scope to intervene and monitor risk assessment systems of banks and banking supervisors

of the home and host countries of banks are required to make concrete plans to improve cooperation and

information exchange, and decrease the burden of banks to implement supervisory requirements.

Pillar 3: (Market Discipline). Market discipline through better disclosure of information by

banks: Banks have to publicize more differentiated data. The assumption is that when data indicate bad

banking behaviour, e.g. too many risky loans, the bank's clients and investors will react and put

pressure on the bank to correct the situation.

The articulation between these three pillars is far from being clear. Note that the recourse to

market discipline is rightly justified by common sense arguments about the increasing complexity of

banking activities, and the impossibility for centralized supervision to monitor in detail these activities.

Thus, as a complement to centralized supervision, it is important to encourage monitoring of banks by

professional investors and financial analysts. It is also suggested that the commercial banks some how

try to maintain economic capital way above the regulatory minimum or by intervention of supervisors.

7 Journal of Money, Investment and Banking - Issue 19 (2010)

Contradictorily, to insist so much on the need to enable early supervisory intervention if capital does

not provide a sufficient buffer against risk and to remain silent on the threshold and form of

intervention, while putting so much effort on the design of risk weights.

The reason for that is most of the theoretical models on banks capital regulations relies on static

models where the capital requirements are used to curb banks' incentives for excessive risk-taking and

where the choice of risk weights is fundamental. These models are failed to capture the intertemporal

effects as suggested by Hellwig (1998). Fore examples, in case of binding, the capital requirements can

only have an impact on banks' behaviour. Practically, capital requirements are binding for a very small

minority of banks and it may effects the behaviour of other banks. Also, in case of intertemporal

effects are taken into account, Blum (1999) suggested that the impact of more stringent capital

requirements may sometimes be surprising. Due to transitory efforts, modeling cost is obviously

additional complexity. In order to minimize this complexity, we will assume a stationary liability

structure, and rule out those transitory efforts. For simplicity, we will focus on one type of assets that

allowing deriving a Markov model of banks' behaviour with the value of the bank's assets as its state

variable (or, up to a monotonic transformation, the bank's capital ratio).

The objective of this research is to consider the problem of analyzing the interaction between

the three pillars of the new Basel Accord (Basel II): capital adequacy requirements, centralized

supervision and market discipline. We develop a diffusion model with jumps of commercial banks'

behavior where interaction between these three instruments (pillars) can be analyzed. Also, we

discussed the conditions under which market discipline can reduce the minimum capital requirements

needed to prevent moral hazard as well as the issues of regulatory forbearance and procyclicality.

This paper follows the approach of Decamps et al. (2004) and Dewatripont and Tirole (1994)

where the capital requirements should be viewed as intervention thresholds for banking supervisors

that should act as representatives of depositors' interest rather than complex schemes designed to curb

banks' asset allocation. Therefore, our model builds on a series of models that have adapted continuous

time models as the one used in the corporate finance to analyze the impact of the liability structure of

firms on their choices of investment and on their overall performance. Also, extension is made by

incorporating features that are essentials to capture the specificities of commercial banks.

Also, we consider that the banks can have the ability to finance investments with a positive net

present value. We omit the effect of introducing taxes and liquidation cost as usual according to

Modigliani and Miller (1958) theorem. Moreover, profitability of the investments requires costly

monitoring by the bank. Absent the incentives for the banker to monitor, the net present value of the

investments become negative. This incentive creates the needs for the regulator to act on behalf of

depositors to limit banks' leverage and to impose closure well before the net present value of bank's

assets become negative.

The reasons for the Modigliani and Miller theorem to be not valid in our model is due to the

value of the bank is indeed affected by closure decisions and by moral hazard which is captured by

assuming that bankers can shirk, which is shown that this occurs when the value of the bank's assets

falls below a certain threshold. In this case we derive the minimum capital requirement that is needed

to ensure this and is also needed to avoid moral hazard, but closure is not credible ex-post given that

the continuation value of the bank is positive. It is shown that the minimum capital requirement is also

reduced if the market discipline through compulsory subordinated debt and the possibility of regulatory

forbearance are introduced.

2. Literature Review We will not review the literature in details since there are enormous literature on the Basel Accord and

its relation to with the credit crunch. Good discussions can be found in Berger and Udell (1994),

Thakor (1996), Jackson et al. (1999), and Santos (2000). Also, Furlong and Keeley (1990), Kim and

Santomero (1988), Koehn and Santomero (1980), and Rochet (1992) have discussed the distortion of


banks' assets allocation that could be generated by the wedge between market assessment of asset risks

and its regulatory counterpart in Basel I. Bernanke and Lown (1991), Thakor (1996), Jackson et al.

(1999) have empirically tried to relate the theoretical arguments to the spectacular substitution of

commercial and industrial loans by investment in government securities in US banks in early 1990s,

shortly after the implementation of the Basel Accord and the Federal Deposit Insurance Corporation

Improvement Act (FDICA). Also, they established a positive correlation between bank capital and

commercial lending; causality can only be examined in a dynamic framework. More specifically, Peek

and Rosengren (1995) found that the increase in supervisory monitoring had also a significant impact

on bank lending decisions, even after controlling for bank capital ratios. Blum and Hellwig (1995) also

analyzed the macroeconomic implications of bank capital regulation.

Blum (1999) analyzed the consequences of more stringent capital requirements in a dynamic

framework and showed that these capital requirements may induce increasing risk taking by banks who

anticipate having difficulty meeting these capital requirements in future.

Hancock et al. (1995) studied the dynamic response to shocks in the capital of US banks using a

Vector Auto Regressive model and showed that US banks seem to adjust their capital ratios must faster

than they adjust their loans portfolios. Also, extension of this line research is done by Furfine (2001)

by building a structural dynamic model of banks behaviour calibrated on data obtained from some

large US banks on the period 1990-1997. He suggested that the credit crunch can not be explained by

demand effects but rather by raising capital requirements or by increasing regulatory monitoring and

also used his calibrated model to simulate the effects of Basel II and the suggested that its

implementation would not provoke a second credit crunch since that average risk weights on good

quality commercial loans will decrease when Basel II is implemented.

According to our objectives which summarizes on designing a tractable dynamic diffusion

model with jumps of banking behaviour where the articulation between the three pillars of Basel II can

be analyzed. This model mainly builds on the analysis of a simple continuous time model of

commercial banks’ behaviour of Decamps et al. (2004), and on the analysis of the impact of debt

maturity on asset substitution and firm value in the corporate finance models of Leland and Toft

(1996), Leland (1998), and Ericsson (2000), and on the analysis of the impact of solvency regulations

and supervision intensity on the behaviour of commercial banks in the banking models of Merton

(1974), (1977), Fries et al. (1997), Bhattacharya et al. (2002), and Milne and Whalley (2001). Also,

different authors studied banking behavior from different point of their views. For example: Mella-

Barral and Perraudin (1997), Anderson and Sundaresan (1996), Mella-Barral (1999), Calem and Rob

(1996), Froot and Stein (1998), etc.

Decamps et al. (2004) developed a simple continuous time model of commercial banks'

behaviour where interaction between these three pillars of Basel II can be analyzed. They showed that

market discipline can be used to reduce the closure threshold especially if there is a risk of regulatory

forbearance and also re-examined the traditional view on the supervisory role.

Leland and Toft (1996) obtained a closed form for the value of debt and equity and showed,

using numerical simulation, that risk shifting disappears for low maturity in conformity with the

intuition that short term debt allows to discipline managers. They investigated the optimal capital

structure which balances the tax benefits and the bankruptcy costs coming with debt. The optimal

capital structure is also investigated by Leland (1998) and Ericsson (2000) concerning with the asset

substitution problem where firm can modify the volatility of its assets' value. They showed how the

choice of assets' volatility influenced by the liability structure. Both considered the perpetual debt but

the constant renewal rate which serves as a disciplining instrument is introduced by Ericsson (2000).

Mella-Barral and Perraudin (1997) discussed the consequences of the capital structure on an

abandonment decision and obtained an underinvestment result from the fact that equity holders have to

inject new cash in the firm to keep it as an ongoing concern. As an elaboration to this aspect Anderson

and Sundaresan (1996) as well as Mella-Barral (1999) have studied the impact of possible

renegotiation between equity holders and debt holders allowing for the possibility of strategic default.


Merton (1977) studied the behaviour of commercial banks using a diffusion model. He actually

computed the fair pricing of deposit insurance in a context where supervisors can perform costly

audits. An extension of Merton's framework is done by Fries et al. (1997) whom they have introduced

a withdrawal risk on deposits by studying the impact of the regulatory policy of bank closures on the

fair pricing of deposit insurance. Note that the optimal closure rule has to trade-off between monitoring

costs and costs of bankruptcy. Thus the regulator may want to let the bank continue even when equity

holders have decided to close it under certain circumstances.

Bhattacharya et al. (2002) derived the closure rules that can be contingent on the level of risk

chosen by the bank and examined the complementarities between two policy instruments of bank

regulator, the level of capital requirements and the intensity of supervision.

Calem and Rob (1996) suggested a dynamic discrete time model of portfolio choice, and

analyzed the impact of capital based premia under the perfect audit hypothesis. They showed that a

tightening in capital requirement may lead to an increase in the risk of the portfolios chosen by banks,

and similarly, capital based premia may sometimes induce excessive risk taking by banks and this

never happen when capital requirements are stringent enough.

Froot and Stein (1998) suggested a model to the buffer role of bank capital so as to absorb

liquidity risks and determined the capital structure that maximizes the bank's value when there are no

audits nor deposit insurance.

Milne and Whalley (2001) developed a model where banks can issue subsidized deposits

without limit in order to finance their liquidity needs and studied the articulation between two

regulatory instruments, the intensity of costly auditing and the level of capital requirements, allowing

for the possibility of banks recapitalization. They also showed that banks' optimal strategy is to hold an

additional amount of capital above the regulatory minimum that used as a buffer which reduces the

impact of solvency requirements against future solvency shocks.

After this introduction, and a brief review of the enormous literatures related to the Capital

Accord of 1988 (Basel I) and the new Capital Basel Accord (Basel II), we move on the description of

our model.

3. The Cash-Flows Diffusion Model Diffusion processes have been used to model physical, biological, economic and social problems.

Examples include molecular motions of enumerable particles subject to interactions, security price

fluctuations in a perfect market, neurophysiological activity with disturbances, variations in population

growth, changes in number of species subject to competition, and gene distributions in evolutionary

development (Brockwell et al., 1983; Brockwell, 1985; Sorensen, 1988).

Consider the cash-flows generated by the bank's assets be modeled by a diffusion process

{ tX }; ,0≥t in which the diffusion coefficient G

σ and the drift coefficient Gµ are both proportional to

cash-flows tX , assuming all agents are risk neutral with a discount rateGr µ> where G stands for

"Good" technology. The diffusion process is interrupted by downward jumps occurring at a constant

rate c and having magnitudes with distribution function ( )⋅H . Then { tX }; ,0≥t is a Markov process

with the state space ),0[ ∞=S and can be regarded as the solution of the differential equation:

t G t G t t ttdX X dt X dW X dZµ σ −= + − (3.1)

where }{ tW a Wiener is process and }{ tZ is the compound Poisson process:

∑=

=tN

i

it YZ1

(3.2)

Here }{ tN is a Poisson process with mean rate c, where c is the downward jump rate, and

,....,,, 321 YYY is independent and identically distributed random variables with distribution

function ( )⋅H , with mean ( )1YEm = and variance ( )1

2 YVar=κ . Note that the moments of tZ can be


determined from the random sums formulas, and are ( ) cmtZE t = and ( ) ( )tmcZVar t

22 += κ , (cf.

Taylor and Karlin (1984), pp. 55, 201).

Note that the most genuine economic role is only satisfied if the bank monitors its assets with

fixed monitoring cost per unit of time which is equivalent to a continuous monetary outflow rb (in case

of variable monitoring cost it can also subtracted from Gµ ) where b is the present value of the cost of

monitoring the bank's assets forever.

Now consider the case of "Bad" technology, abbreviated by B , in case of the absence of

monitoring, the cash-flows diffusion model of bank's assets in equation (3.1) is then can be rewritten in

the form:

t B t B t t ttdX X dt X dW X dZµ σ −= + − (3.3)

where the diffusion coefficient G

σ and the drift coefficient Gµ are given by: 2 2 2 2

B G B Bσ σ σ σ≡ + ∆ ≥ and

B G G Gµ µ µ µ≡ − ∆ ≤ . Also for technical reasons, we assume that ( )2 / 2G G Gσ µ µ= + .

Suppose that the bank is closed. This means that the bank's assets are liquidated for a value

proportional to the current cash flows tXλ where λ is the coefficient of liquidation and is assumed to

satisfy

( )( )

( )( )2

22

2

22 2121

GGBB r

mmc

rr

mmc

r µ

κ

µλ

µ

κ

µ −

−++

−<<

−

−++

− (3.4)

As a result the "Bad" technology is always dominated by closure given that 00 xX = :

( )( ) 02

0

22

0

0

2y technologBad""

0x

r

xmmc

r

xdtXeE

BB

t

rt

X λµ

κ

µ<

−

−++

−=

∫∞+

−

Also, the net present of value of a bank who continuously monitors its assets, the "Good"

technology, is

( )( )

( )b

r

xmmc

r

xdtrbXeE

GG

t

rt

X −−

−++

−=

−∫

∞+

−

2

0

22

0

0

2y technologGood""

0 µ

κ

µ

so the "Good" technology dominates closure whenever is not too small assuming 1

G

Gr

nm

=-

and

1B

Br

nm

=-

:

( )2 2 2

0 02G G

x c m m b xν κ ν λ + + − − > (3.5)

or equivalently;

( )0 2 2 22G G

bx

c m mν κ ν λ>

+ + − − (3.6)

where

( )2 2 22G G

c m mν κ ν λ+ + − > (3.7)

while we denote by analogy

( )2 2 22B B

c m mν κ ν λ+ + − < (3.8)


Note that since 0c > then (3.7) is satisfied if and only if 211 1

2m k³ + - or

211 1

2m k£ - - for all 2 2k £ . But if 2 21 1

1 1 1 12 2

mk k- - < < + - is satisfied for all 2 2k £

then 2 2 2 0m mk + - < in (3.7).

Now assuming the banks continue forever, then the economic surpluses generated by the good

technology is positive when x is larger than the net present value NPV threshold

( )2 2 22G G

bNPV

c m mν κ ν λ=

+ + − −

this means that

( )2 2 22 0G G

c m m x bν κ ν λ + + − − − > (3.9)

when (3.7) is satisfied, while the economic surpluses generated by the bad technology is always

negative. Regarding the above discussion (3.9) could be negative for some downward jump rate c ,

actually for all 2 2k < and m between 211 1

2k- - and 21

1 12

k+ - . More specifically,

( )2 2 22 0G G

c m m x bν κ ν λ + + − − − < (3.10)

occurs if and only if

( )

( )2 2 22

G

G

b xc

m m

ν λ

κ ν

− −>

+ − (3.11)

which implies that this case of good technology will be a bad technology if c in equation (3.11)

is satisfied. On other word, the economic surpluses becomes negative.

Note that the following figure 3.1 represent the above cases of the surpluses generated by the

good (G) and the bad (B) technologies.

Figure 3.1: Economic surpluses generated by the good (G) and the bad (B) technologies.

G

x

B


Now assume the bank always monitors the value of its assets ( )GV x with good technology

using the liquidation threshold Lx which below Lx the bank is closed is then given by

( ) ( )0

L

Lrrt

G x t LV x E e x rb dt e x

τ

τ λ−−

= − + ∫ (3.12)

where Lτ is a random variable represent the stopping time at t Lx x= given that 0x x= . Using Karlin

and Taylor (1981) equation (3.12) can be written as

( ) ( )

( ){ }

2 2 2

1

2 2 2

2

2

G

G G G

a

G G L

L

V x c m m x b

xb c m m x

x

ν κ ν

ν κ ν λ

−

= + + − −

+ − + + − − ⋅

(3.13)

where

( ) ( )2

2 2 2 2

2 2 2 2 2

2 21 1 21

2 2

G GG

G G G G G

c m m c m m ra

κ κµ µ

σ σ σ σ σ

+ − + − = − + + − − + >

(3.14)

Thus the continuation value of the bank is equal to the net present value perpetual continuation

( )( )2 2 22G Gc m m x bν κ ν + + − − plus the option value associated to the irreversible closure decision at

threshold L

x . Therefore, the first-best closure threshold of the bank is the value of the cash flow

Lx that maximizes the option value associated to the irreversible closure decision is given by

( )2 2 2

1

2

GFB

GG G

abx

ac m mν κ ν λ

−= ⋅

+ + − − (3.15)

Note that the first-best closure threshold FB

x is smaller than the NPV threshold.

The main feature of commercial banking is deposit finance which mostly according to banks

liabilities consists of insure deposits with a volume normalized to one. Now in the absence of public

intervention which always consists of either liquidity assistant by the Central Bank or the closure by

the banking supervision authorities, liquidation of the bank occurs when the cash flows x received

from its assets are insufficient to repay the interest r on deposits. Then in this case the liquidation

threshold is given by

Lx r= (3.16)

As a consequence, the book value of the bank equity which equals to the book value of assets

minus the nominal value of deposits is positive. i.e.

( )2 2 22 1G G

c m m rν κ ν + + − > (3.17)

but liquidation does not permit payment of all deposits:

1rλ < (3.18)

Note that equation (3.17) captures the fact that, in the absence of liquidity assistance by the

central bank which will be discussed later in section 6, the solvent banks may be liquid which

guarantees that optimal capital requirements are positive. Also, equation (3.18) ensures that deposits

are risky.

Consequently, the present value PV of deposits is then given by

( ) ( )1

1 1

Ga

G L

L

xD x x

xλ

−

= − −

(3.19)

which leads to the market equity ( ) ( ) ( )G G GE x V x D x= - or equivalently


( ) ( )

( )( )

2 2 2

1

2 2 2

2 1

1 2

G

G G G

a

G G L

L

E x c m m x b

xb c m m x

x

ν κ ν

ν κ ν

−

= + + − − −

+ + − + + −

(3.20)

Note that deposits are risky because 1L

xl < which implies that the value PV of deposits

( )GD x is less than their nominal value 1, the difference corresponding to the liability of the Deposit

Insurance Fund which is covered by an insurance premium ( )01 D x- paid initially by the bank. But if

the bank ceases to monitor its assets, then the value of equity becomes

( ) ( )

( )( )

2 2 2

1

2 2 2

2 1

1 2

B

B B B

a

B B L

L

E x c m m x

xc m m x

x

ν κ ν

ν κ ν

−

= + + − −

+ − + + −

(3.21)

where

( ) ( )2

2 2 2 2

2 2 2 2 2

2 21 1 2

2 2

B BB

B B B B B

c m m c m m ra

κ κµ µ

σ σ σ σ σ

+ − + − = − + + − − +

(3.22)

Note that comparing the two equities of bad and good technologies in the above two formulas

equations (3.20) and (3.21). we conclude that ( ) ( )B GE x E x> for some values of x in some open

interval ( ),L Sx x where S

x is the point of intersection where ( )BE x becomes smaller than that of

( )GE x .

4. The Solvency Requirements Note that the basic reason for imposing a capital requirement in our model when

( ) ( )G L B LE x E x′ ′< there exist a region [ ],L Sx x where the bank chooses the bad technology (shirks) in

the absence of external intervention which reduces social welfare and ultimately provokes failure, the

cost being borne by the Deposit Insurance Fund (DIF). To avoid shirking, banking authorities which

sometimes represented by the Central Bank, a Financial Services Authority or by the DIF itself set a

regulatory closure threshold R

x below the bank is closed. Practically, this closure threshold can be

implemented by a minimal capital requirement. Thus the book value of equity is equal to the book

value of assets ( )2 2 22G G

c m m xν κ ν + + − minus the nominal value of deposits, which normalized to

1. Therefore, the solvency ratio r of the bank is given by

( )( )

2 2 2

2 2 2

2 1

2

G G

G G

c m m x

c m m x

ν κ νρ

ν κ ν

+ + − − = + + −

(4.1)

Note that ρ is an increasing function of x . If we denote the minimum capital ratio by R

ρ then

for every R

x x≥ implies that R

ρ ρ≥ which R

ρ finally defined by

( )( )

2 2 2

2 2 2

2 1

2

G G R

R

G G R

c m m x

c m m x

ν κ νρ

ν κ ν

+ + − − = + + −

(4.2)

Assume the monitoring cost b is not too small. Thus a solvency regulation is needed to prevent

insufficiently capitalized banks from shirking. Now following the same lines in the proof of


Proposition 2 in appendix A of Decamps et al. (2004), it is easily shown that the second best closure

threshold associated with the optimal ratio is the smallest value R

x of the liquidation threshold with

absence of shirking is given by

( )

( ) ( )2 2 2 2 2 2

1

2 2

G G B

R

G G G B B B

a b a ax

a c m m a c m mν κ ν ν κ ν

− + −=

+ + − − + + −

(4.3)

Regulation is needed whenever R

x r> which is equivalent to *b b> where

( ) ( )( ) ( )2 2 2 2 2 2

*2 2

1

G G G B B B G B

G

r a c m m a c m m a ab

a

ν κ ν ν κ ν + + − − + + − + − =

− (4.4)

Note that, when regulation is needed R Lx x r> = , then the implied solvency ratio Rr is

positive because it has assumed that ( )2 2 22 1G Gc m m rν κ ν + + − > which means that banks become

illiquid before they become insolvent especially in the absence of public intervention. Also, note that

for very large values of b , the liquidation value of the bank Rxl becomes greater than the nominal

value of deposits normalized to 1 and then deposit becomes riskless. In this case the incentives of

banks’ stockholders are not distorted by the limited liability option, they optimally decide to close the

bank when x hits the first best threshold FBx and the moral hazard constraint does not bind.

5. Market Disciplines Market discipline can be useful because of its possibility to produce additional information that the

regulator can exploit which is usually referred to indirect market discipline. For example, consider a

setup of Merton (1978) or Bhattacharya et al. (2002) where tx is only observed through costly and

imperfect auditing. As a result there is a positive probability that the bank will continue to operate in

the region [ ],L Rx x because of undetected by banking supervisors. If shirking is to be deterred more

stringent capital requirement or higher Rx has to be imposed to account for imperfect auditing. Note

that requiring the bank to issue a security such as subordinated debt whose payoff is conditional on

tx and is traded on financial markets would indirectly reveal to the value of tx and dispense the

regulator from costly auditing.

When tx is publicly observed, the supervisors can have recourse to a second form of market

discipline which sometimes known as direct market discipline which works by modifying structure of

banks. This is the idea behind the subordinated debt proposal which our model allows us to analyze

formally (cf. Calomiris (1998), Evanoff and Wall (2000)). In this case the banks are required to issue a

certain volume s of subordinated debt renewed with a certain frequency p . Note that both s and p are

policy variables of the regulator. Now keeping constant the total volume of outside finance, then the

volume of insured deposits becomes 1d s= - . We assume as in Ericsson (2000) that the subordinated

debt has an infinite maturity and is renewed according to a Poisson distribution with intensity p . Thus

the average time to maturity of subordinated debt is given by0

1ptpte dt

p

∞− =∫ .

Now we will consider the situation in which the regulator can commit to a closure threshold

Rx . We focus on the where Rx dλ < , so that deposits are risky while sub-debt holders and

stockholders are expropriated in case of closure. Now if we denote ( )GV x and ( )BV x are the value of

bank’s assets for Good and Bad technologies, ( )GD x and ( )BD x are the present value PV of insured

deposits for Good and Bad technologies, ( )GE x and ( )BE x are the value of equity for Good and Bad


technologies, and ( )GS x and ( )BS x are the market value of subordinated debt for Good and Bad

technologies.

For the case when the bank monitors its assets, then using the same adaptation used in the

previous section 3, it is easily shown that

( ) ( )

( ){ }

2 2 2

1

2 2 2

2

2

G

G G G

a

G G R

R

V x c m m x b

xb c m m x

x

ν κ ν

ν κ ν λ

−

= + + − −

+ − + + − − ⋅

(5.1)

and

( ) ( )1 Ga

G R

R

xD x d d x

xλ

−

= − −

(5.2)

where Ga is defined in equation (3.14).

Let p be the instantaneous probability that the subordinated debt is rapid at face value s , and it

has to be refinanced at price ( )GS x . i.e. ( )GS x is the solution of the following partial differential

equation

( ) ( )( ) ( ) ( )2 21

2G G G G G G

rS x sr p s S x xS x x S xµ σ′ ′′= + − + + (5.3)

Note that ( ) 0G RS x = . Thus ( )GS x is then given by

( )( )1

1

Ga p

G

R

xS x s

x

− = −

(5.4)

where

( )( ) ( )

22 2 2 2

2 2 2 2 2

2 21 12

2 2

G GG

G G G G G

c m m c m m r pa p

κ κµ µ

σ σ σ σ σ

+ − + − + = − + + − − +

(5.5)

Therefore, the first effect of direct market discipline is noticed from the exponent ( )1 Ga p-

which decreases when p increases. Thus ( )GS x increases in p . The value of the market equity

( ) ( ) ( ) ( )G G G GE x V x D x S x= - - becomes

( ) ( )

( )( )

2 2 2

1 1

2 2 2

2 1

2 +

G G

G G G

a a

G G R

R R

E x c m m x b

x xb d c m m x s

x x

ν κ ν

ν κ ν

− −

= + + − − −

+ + − + + −

(5.6)

In case of no market discipline when 0p = , the liability of the DIF is reduced when a fraction

s of insured deposits is replaced by subordinated debt. But this is exactly offset by the default premium

demanded by subordinated debtholders. Also, the value of the equity is reduced when s is increased

while keeping 1s d+ = because ( )G Ga p a≥ . Thus the bank will only issue subordinated debt if it is

imposed by the regulator or if it reduces the capital requirement.

Now if the bank shirking, the case of bad technology, the value of the equity ( )BE x using the

same adaptation as in section 3 too, it is easily shown


( ) ( )

( )( )( )

2 2 2

1 1

2 2 2

2 1

2 +

B G

B B B

a a p

B B R

R R

E x c m m x

x xd c m m x s

x x

ν κ ν

ν κ ν

− −

= + + − −

+ − + + −

(5.7)

where

( )( ) ( )

22 2 2 2

2 2 2 2 2

2 21 12

2 2

B BB

B B B B B

c m m c m m r pa p

κ κµ µ

σ σ σ σ σ

+ − + − + = − + + − − +

(5.8)

The necessary condition for shirking to be eliminated is 0∆ ≥ where

( ) ( )R G R B Rx E x E x′ ′ ∆ = − . A simple computation gives:

( ) ( ) ( )( ) ( )2 2 22 1 1R G R G G G R G G

x E x a c m m x a d b s a pν κ ν ′ = + + − − − + − −

and

( ) ( ) ( ) ( )2 2 22 1 1R B R B B B R B B

x E x a c m m x a d s a pν κ ν ′ = + + − − − − −

Thus

( ) ( )( )

( ) ( ) ( ) ( )

2 2 2 22

1

G G B B R G B R

G B G G B

a a x c m m x

a a d a b s a p a p

ν ν κ ν ν∆ = − + + − −

− − + − + −

(5.9)

Let ( )Rx p be the minimum value R

x that satisfies the inequality ( ) 0Rx∆ ≥ . Thus, the

minimum solvency ratio ( )Rx p that prevents bank shirking becomes

( )( ) ( ) ( ) ( )

( ) ( )( )2 2 2 2

1

2

G B G G B

R

G G B B G B

a a d a b s a p a px p

a a c m mν ν κ ν ν

− + − + − =− + + − −

(5.10)

or equivalently;

( ) ( )( ) ( ) ( )

( ) ( )( )2 2 2 20

2

G B G B

R R

G G B B G B

a p a p a ax p x s

a a c m mν ν κ ν ν

− − − = +− + + − −

(5.11)

Note that if the difference between the variances of the bad and good technologies is positive,

then the minimum solvency ratio R

x is a concave function of p which has a minimum value at *p ,

where 1

p is the average time to maturity of subordinated debt. But, if the difference between the

variances of the bad and good technologies is zero, then the minimum solvency ratio R

x is a

decreasing function of p . Also, the market discipline reduces the need for regulatory bank closures for

small values of p and the difference between the variances of the bad and good technologies.

Consequently, the increase in the frequency of renewal of subordinated debt will affect the shirking to

become more costly for bankers. Also, for fixed values of assets and deposits, the changes in the value

of equity come from changes of subordinated debt. But, for large difference between the variances of

the bad and good technologies, *p will be negative, then the introduction to subordinated debt is

counterproductive since it forces the regulator to increase the minimum capital requirement.

6. Supervisory Action In this section we will study how the efficiency of market discipline is affected by the attitude of

supervisory authorities. Note that the second pillar of Basel 2 is the supervisory review. The committee

views supervisory review as a critical compliment to minimum capital requirements and market


discipline (cf. Basel Committee, 2001, p. 30). Indeed, banking authorities are very often subject to

political pressure for supporting banks in disasters. In our model means providing public funds to the

banks who hit the thresholdR

x . Under the irreversibility assumptions, it is always suboptimal to let

banks go below this threshold. When net fiscal costs are not too high, the closure can be dominated by

continuation. Note that whenever the bank hits the boundary R

x x= the governments intervene and

considers the possibility of recapitalizing the bank up to R

x x+ ∆ with public funds. Thus the new assets

value ,g BOV reflects that future government intervention is anticipated. Note that BO refers to the Bail-

Out operation and { },g G B∈ for technology. Thus,

( ) ( ){ }, ,0

maxg BO R g BO Rx

V x V x x xγ∆ ≥

= + ∆ − ∆ (6.1)

In general, ( ),g BOV x can be written in the form

( ) ( ) 12 2 2

, 2 + ga

g BO g g g gV x c m m x b xν κ ν θ

− = + + − − (6.2)

Note that

;

0 ;g

b g Bb

g G

==

=.

This means that the government injects the minimum amount needed to stay above the critical

shirking level R

x which can be interpreted as liquidity assistance. Note that R

x becomes a reflecting

barrier with boundary condition: ( ),g BO RV x γ′ = . Following the same proof in Appendix B of Decamps

et al. (2004), then ( ),g BOV x for good technology G can be written as

( ) ( )( )2 2 2

12 2 2

,

22

1G G

G G a a

G BO G G R

G

c m mV x c m m x b x x

a

γ ν κ νν κ ν −

− − + − = + + − − − −

(6.3)

The liquidity assistant implies that ( ),G BO RV x is different than zero. If ( )G RV x γ′ < then the

cost of public funds is so high that the government prefers to close the banks. But for positive

( ),G BO RV x and ( )G RV x γ′ > , then it guarantees that the bailout is socially preferable to close.

In case of sub-debtholders and equityholders are wiped out, the decision to rescue the bank only

affects the deposit insurance fund which becomes the residual claimant of the assets value of the bank.

Note that the differential equation and boundary conditions that characterize the value of sub-debt and

equity are the same as before and thus g

S and g

E where { },g G B∈ are unchanged. Therefore, market

discipline is compatible with public liquidity assistance, provided that subordinated debtholders lose

their stake if the bank is rescued. But in case of sub-debtholders and equityholders are fully insured

when the bank hits the critical threshold R

x such that ( )g RS x s= where { },g G B∈ , the sub-debt

becomes riskless and its values equal to s . Also, the sub-debt term vanishes from the differential

equation that characterizes the equity g

E where { },g G B∈ , which implies that market discipline

becomes completely ineffective. This finally illustrates that the market discipline is indeed a useful

complement to the two other pillars of Basel II, the supervision and capital requirements.

Suppose now the bank has issued at least one security such as equity, certificates of deposits, or

subordinated debt, that are traded on a secondary market. Thus the price of security in our model is a

one to one function of the state variable x and inverting this function, the value of the bank’s cash

flows to condition its intervention policy can be inferred by the regulator. Therefore, the role bank

supervisors have to be re-examined their adoption of a gradual intervention policy instead of a constant

intensity of audit across all banks. For example, the regulator can set two thresholds R

x and I

x (with

R Ix x< ) which refers to closure and inspection thresholds respectively. Note that if

Ix x< the bank


is inspected, the technology chosen by the bank is revealed, and it is closed if the bank has chosen the

bad technology, g B= .

Under this regulatory policy, the value of equity when the technology is “good” is given by

( ) ( ) 12 2 22 1 Ga

G G G GE x c m m x b g xν κ ν − = + + − − − + (6.4)

where

( )( ) 12 2 21 2 Ga

G G G R Rg b c m m x xν κ ν − = + − + + −

But the value of equity when the bank is shirking becomes

( )( ) 12 2 22 1 for

0 otherwise

Ba

B B B I

B

c m m x g x x xE x

ν κ ν − + + − − + ≥ =

(6.5)

where

( )( ) 12 2 21 2 Ba

B B B I Ig c m m x xν κ ν − = − + + −

Note that G

a and B

a are both defined in equations (3.14) and (3.22) respectively. Also, note that

for a given value of I

x and a function B

E there is a minimum value of R

x such that G

E remains above

the value of B

E which is known as the value of tangent of two curves.

7. Conclusion In conclusion, this paper developed a cash-flows diffusion model with jumps of commercial banks'

behavior, where the three pillars of Basel II: capital adequacy requirements, centralized supervision

and market discipline, can be analyzed. More specifically, the pillar 1 of capital adequacy requirement

is interpreted as a closure threshold rather than an indirect mean of influencing banks’ asset allocation,

the pillar 2 of centralized supervision is re-examined the traditional view of on the supervisory role,

and the pillar 3 of market discipline is showed that it can be used to reduce this closure threshold when

there is a risk of regulatory forbearance.

It suggests that according to reliable signals given by market prices of the securities issued by

banks which are known as the indirect market disciplines, the supervisors can modulate the intensity of

their interventions from a simple audit to the closure of the bank. But the direct market discipline can

be effective if banking supervisors are protected from political interference. Also, indirect market

discipline cannot be used under all circumstances since market prices become erratic during crises

periods.

Acknowledgement This work was supported by Kuwait University, Research Grant No. [IQ 02/07].

References [1] Anderson, R. and Sundaresan, S. (1996). Design and Valuation of Debt Contracts. Rev. Finan.

Stud., Vol. 9, 37-68.

[2] Basel Committee (2001). Overview of the new Basel Capital Accord. Consultative document.

BCBS, Basel, Switzerland.

[3] Berger, A. N., Herring, R. J., and Szego, G. (1995). The role of capital in financial institutions.

J. Banking Finance, Vol. 19, 393-430.

[4] Berger, A. N and Udell, G. F. (1994). Did risk-based capital allocate bank credit and cause a

credit crunch in the US? J. Money, Credit, Banking, Vol. 26, 585-628.


[5] Bernanke, B. and Lown, C. (1991). The credit crunch. Brookings Papers on Economic Activity,

Vol. 2, 205-247.

[6] Bhattacharya, S., Plank, M., Strobl, G., and Zechner, J. (2002). Bank capital regulation with

random audits. J. Econ. Dynam. Control, Vol. 26, 1301-1321.

[7] Blum, J. (1999). Do capital adequacy requirements reduce risks in banking? J. Banking

Finance, Vol. 23, 755-771.

[8] Blum, J. and Hellwig, M. (1995). The macroeconomic implications of capital adequacy

requirements for banks. Europ. Econ. Rev., Vol.39, 739-749.

[9] Brockwell, P. J. (1985) The Extinction of a Birth, Death and Catastrophe Process and of a

Related Diffusion Models. Advanced Appl. Prob. 17, 42-52.

[10] Brockwell, P. J., Gani, J. and Resnick, S. I. (1983) Catastrophe Processes with Continuous

State-Space. Australian J. Stat. 25: 208-226.

[11] Calem, P. S. and Rob, R. (1996). The impact of capital-based regulation on bank risk-taking: a

dynamic model. Federal Reserve Board, Finance and Economics discussion series, 96-12,

February.

[12] Calomiris, C. W. (1998). Blueprints for a new global financial architecture. Available from

www.house.gov/jec/imf/blueprint.htm.

[13] Decamps, J-P., Rochet, J-C., and Roger, B. (2004). Three pillars of Basel II: optimizing the

mix. J. Finan. Intermediate, Vol.13, 132-155.

[14] Dewatripont, M. and Tirole, J. (1994). The Prudential Regulation of Banks. MIT Press,

Cambridge, USA.

[15] Ericsson, J. (2000). Asset substitution, debt pricing, optimal leverage and maturity. J. Finance,

Vol. 21, No. 2, 39-70.

[16] Erns-Ludwig von Thadden (2004). Bank capital adequacy regulation under the new Basel

Accord. J. Finan. Intermediate, Vol. 13, 90-95.

[17] Evanoff, D. D and Wall, L. D. (2000). Subordinated debt and bank capital reform. Federal

Reserve Bank of Chicago, WP-2000-07.

[18] Fries, S., Mella-Barral, P. and Perraudin, W. (1997). Optimal bank reorganization and the fair

pricing of deposit guarantee. J. Banking Finance, Vol. 21, 441-468.

[19] Froot, K. and Stein, J. (1998). A new approach to capital budgeting for financial institutions. J.

Finan. Econ., Vol. 47, 55-82.

[20] Furfine, C. (2001). Bank portfolio allocation: The impact of capital requirements, regulatory

monitoring, and economic conditions. J. Finan. Services Res., Vol. 20, 147-188.

[21] Furlong, F. and Keeley, N. (1990). A re-examination of mean-variance analysis of bank capital

regulation. J. Banking Finance, 69-84.

[22] Hancock, D., Laing, A. J., and Wilcox, J. A. (1995). Bank capital shocks: dynamic effects and

securities, loans, and capital. J. Banking Finance, Vol. 19, 661-677.

[23] Hellwig, M. (1998). Banks, markets, and the allocation of risk. J. Inst. Theoretical Econ. Vol.

154, 328-345.

[24] Jackson, P., and Perraudin, W. (2000). Regulatory implications of credit risk modeling. J.

Banking Finance, Vol. 24, 1-14.

[25] Jackson, P., Furfine, C., Groeneveld, H., Hancock, D., Jones, D., Perraudin, W., Redecki, L.,

and Yoneyama, N. (1999). Capital requirements and bank behaviour: the impact of the Basel

Accord. Working paper 1. Basel Committee on Bank Supervision.

[26] Karlin, S. and Taylor, H. (1981). A Second Course in Stochastic Processes. Academic Press,

New York.

[27] Kim, D. and Santomero, A. (1988). Risk in banking and capital regulation. J. Finance, Vol. 43,

1219-1233.

[28] Koehn, M. and Santomero, A. (1980). Regulation of bank capital and portfolio risk. J. Finance,

Vol. 35, 1235-1244.


[29] Leland, H. (1998). Agency cost, risk management and capital structure. J. Finance, Vol. 53,

1213-1243.

[30] Leland, H. and Toft, K. B. (1996). Optimal capital structure, endogenous bankruptcy, and the

term structure of credit spreads. J. Finance, Vol. 51, 987-1019.

[31] Mella-Barral P. (1999). The dynamics of default and debt reorganization. Rev. Finan. Stud.,

Vol. 12, No. 3, 535-579.

[32] Mella-Barral P. and Perraudin, W. (1997). Strategic debt service. J. Finance, Vol. 2, 531-556.

[33] Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. J.

Finance, Vol. 29, 449-469.

[34] Merton, R. C. (1977). An analytic derivation of the cost of deposit insurance and loan

guarantees: an application of modern option pricing theory. J. Banking Finance, Vol. 1, 3-11.

[35] Merton, R. C. (1978). On the cost of deposit insurance when there are surveillance costs. J.

Bus., Vol. 51, 439-452.

[36] Milne, A. and Whalley, A. E. (2001). Bank capital regulation and incentives for risk-taking.

Discussion paper. City University Business School, London, UK.

[37] Modigliani, F. and Miller, M. (1958). The cost of capital, corporate finance and the theory of

investment. Amer. Econ. Rev. Vol. 48, 261-297.

[38] Peek, J. and Rosengren, E. (1995). Bank capital regulation and the credit crunch. J. Banking

Finance, Vol. 19, 679-692.

[39] Rochet, J. C. (1992). Capital requirements and the behaviour of commercial banks. Europ.

Econ. Rev. Vol. 43, 981-990.

[40] Santos, J. (2000). Bank capital regulation in contemporary banking theory: a review on the

literature. BIS Working Paper No. 30, Basel, Switzerland.

[41] Sorensen, M. (1988) Likelihood Methods for Diffusions with Jumps. Research Report No. 159,

Department of Theoretical Statistics, Aarhus University.

[42] Taylor, H. M. and Karlin, S. (1984). An introduction to stochastic Modeling. Academic Press

Inc. London.

[43] Thakor, A. V. (1996). Capital requirements, monetary policy, and aggregate bank lending. J.

Finance, Vol. 51, 279-324.