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Econometric Research and Special Studies Department
Banking competition, risk, and regulation
W. Bolt and A. F. Tieman
Research Memorandum WO&E no. 647
March 2001
De Nederlandsche Bank
BANKING COMPETITION, RISK, AND REGULATION
W. Bolt and A. F. Tieman E-mail addresses: (corresponding author) [email protected], Ph: +31-20-5242916; [email protected]; Fax:
+31-20-5242529. The authors thank Peter Vlaar for critical comments and helpful discussions.
March 2001
Research Memorandum WO&E no. 647/0102
De Nederlandsche Bank NV
Econometric Research and
Special Studies Department
P.O. Box 98
1000 AB AMSTERDAM
The Netherlands
ABSTRACT
Banking competition, risk, and regulation
W. Bolt and A. F. Tieman
In a dynamic framework banks compete for customers by setting lending conditions for the loans they
supply, taking into account the capital adequacy requirements posed by the regulator. By easing its lending conditions a bank faces a tradeoff between attracting more demand for loans, thus making higher
per-period profits, and a deterioration of the quality of its loan portfolio, thus a higher risk of failure. Our
main results state that more stringent capital adequacy requirements lead commercial banks to set more
stringent loan conditions to their customers, and we show that increased competition in the banking industry leads banks to behave more risky. In this model we also look at risk-adjusted capital requirements
and show that risk-based regulation is effective. We extend the basic model to have banks choose both
their lending conditions and the level of bank capital. In this extended model it turns out that it may be
beneficial for a bank to hold more equity than prescribed by the regulator, even though equity is more
expensive than attracting deposits. We show that the same conclusions with respect to the effectiveness
of regulation hold as in the standard model.
Keywords: Banking competition, risk profile, failure rate, capital requirements
JEL Codes: E44, G28, L16
SAMENVATTING
Bank competitie, risico en regulering
W. Bolt en A. F. Tieman
In een dynamisch model concurreren banken op basis van de verstrekkingscriteria op leningen die zij
aanbieden, rekening houdend met de eisen opgelegd door de toezichthouder. De bank moet hierbij een
afweging maken: aan de ene kant vergroten soepelere condities de vraag naar leningen, hetgeen de kortetermijn winst doet stijgen, maar aan de andere kant verslechtert de kwaliteit van de leningenportefeuille,
waardoor het faillissementsrisico toeneemt. Onze belangrijkste resultaten stellen dat banken minder risico nemen wanneer de kapitaalvereisten strenger zijn en dat toegenomen competitie leidt tot risicovoller
bankgedrag. Ook laten we zien dat risico-gewogen kapitaalvereisten een effectief middel zijn voor regulering. Als uitbreiding van het model bestuderen we het geval waarin de bank niet alleen haar risico
kiest, maar ook haar optimale niveau van eigen vermogen. De analyse toont aan dat optimaal gedrag van
banken ertoe kan leiden dat zij meer eigen vermogen aanhouden dan strikt vereist door de reguleerder,
ofschoon financiering met eigen vermogen duurder is dan het aantrekken van deposito’s. Dezelfde conclusies met betrekking tot de effectiviteit van regulering kunnen worden getrokken als in het standaard
model.
Trefwoorden: Bank competitie, risicoprofiel, faillissementskans, kapitaalvereisten
JEL Codes: E44, G28, L16
1
INTRODUCTION
While banking competition has been the subject of extensive research in the field of microeconomics, a
large part of this literature has primarily focused on the liabilities’ side of the bank’s balance sheet. Recent research has stressed the importance of competition for deposits among banks and its consequences
for a bank’s risk taking behaviour, financial regulation and financial stability as a whole. Matutes and
Vives (2000) analyse a model of imperfect competition and show that, irrespective of the existence of
deposit insurance, banks post too high deposit rates whenever social failure costs are large. In their view,
effective regulation should encompass deposit rate regulation and asset restrictions to improve welfare.
Cordella and Yeyati (1998) find that lower entry costs foster competition in deposit rates and reduce
banks’ incentives to limit risk exposure. In a recent article, Hellmann, Murdock and Stiglitz (2000) show
in a dynamic model of moral hazard that competition can undermine prudent bank behaviour. In terms
of optimal regulation, they find that a combination of capital requirement and deposit rate controls can
achieve Pareto-efficient outcomes 1 .
Relatively little attention has been devoted to competition in terms of bank assets. In this paper we
present a model of loan competition between banks. We focus on competition in terms of lending conditions, i.e. a bank with less stringent lending conditions and easier acceptance criteria attracts more
demand. However, by relaxing these lending conditions too much, the quality of the loan portfolio may
be negatively affected, causing higher failure rates. Hence, in our model, competition for loans induces
a trade-off for the bank between boosting market share in the short run and securing continuation of its
operations in the long run 2 .
As a leading example we consider the Dutch market for mortgage lendings. Over the last few years this
market has been characterized by an easing of lending conditions and a sharp increase in the demand
for loans, mainly as a consequence of increased competition, see The Nederlandsche Bank (2000). The
increase in mortgage lendings is partly due to the sharp increase of the prices of houses used as collateral
for the mortgages. But especially the easing of the mortgage lending conditions, induced by increased
competition on the market for mortgage lendings, has had large effects on the demand for mortgage
loans. Lending conditions were eased by e.g. providing loans against second and temporary income,
raising the acceptable mortgage debt service/income ratio, and increasing the loan maximum relative to
1
For a spatial model of competition for depositors see Freixas and Rochet (1997), see Battacharya, Boot and
Thakor (1998) for a survey on these topics.
2 Niinimäki (2000) develops a model of ‘double’ competition, i.e. competition for loans and deposits, and shows
that deposit insurance has no effect on risk-taking with respect to loan competition. However, this analysis concerns a so-called entrepreneurial bank without equity finance, so it cannot assess the regulatory aspects of imposing
capital requirements and its consequences for risk-taking and banks’ failure rates. In contrast, Vlaar (2000) considers the interaction between regulation and banking competition, where banks compete as Cournot duopolists on
the market for loans. He finds that higher capital requirements impose a higher burden on the inefficient bank in
terms of profits, but that these requirements may be strongly welfare improving.
the forced-sale value of the property. According to a study of De Haas, Houben, Kakes, and Korthorst
(2000) performed by the Dutch central bank, an average Dutch household was able to have an 86%
higher maximum mortgage in 1998 compared to 1994. The higher lending capacity by itself has had a
upward effect on housing prices. This development has resulted in an increased vulnerability of certain
categories of Dutch home-owners with respect to a possible decline in housing prices. As a consequence,
it also had a detrimental effect on the risk profile of the commercial banks’ loan portfolios. This article
studies these consequences of increased competition for bank vulnerability.
In our model, banks react strategically to competitive forces in the market for loans. Loosely stated,
competition drives profits down since margins are lower, because of stronger incentives to take on more
risk in terms of setting easier lending conditions, thus inducing higher failure rates. So, competition can
undermine prudent banking and that is where banking regulation and prudential supervision come into
play. To promote the safety and soundness of the banking system, banks need to comply to regulatory
requirements. More specifically, in our study the main objective of banking regulation is to limit the
probability of bank failure. To achieve his goal we assume that the regulator can only use a capital
adequacy requirement (CAR) as its instrument. Hence, in setting their lending conditions, banks must
take the capital adequacy ratio imposed by the regulator into account. Given this imposed CAR, a
bank’s management seeks to maximize the discounted stream of future profits of a bank. Our analysis
shows that, given some regularity condition on market structure, it is indeed the case that higher CARs
lead to less risk taking both by tightening the lending conditions and by banks having larger reserves.
Consequently higher CARs lead to to lower failure rates.
Capital adequacy requirements have always played an important role in banking regulation, as perhaps
best exemplified by the 1988 Capital Accord introduced by the Basel Committee on Banking Supervision. Other things being equal, higher capital ratios lower the probability of bankruptcy since bank capital
acts as a buffer for unexpected low returns on its loan portfolio. However, since issuing equity is likely to
be more costly than attracting liabilities, holding too much capital might inflict on a bank’s charter value,
which may in fact lead to more risk-taking. In this sense, a ‘one-size-fits-all’ capital ratio, as prescribed
in the 1988 Capital Accord, may induce the wrong incentives. Incentive-based reasoning has triggered a
change in the landscape of banking regulation, best proof of which is the recently issued proposal for a
New Basel Capital Accord. This new framework intends to align regulatory capital requirements more
closely with underlying risks, and to provide banks with a menu of options for the assessment of capital
adequacy, enhancing incentives for a better risk management. Intuitively, better incentives are provided
when capital adequacy is contingent on a bank’s individual risk taking behaviour. In this spirit the New
Accord proposes that banks with a higher risk appetite will find their capital requirement increasing,
and vice versa. We implement regulation by imposing a contingent regulatory rule in our model. Our
analysis confirms that implementing a contingent rule is very effective in meeting the regulator’s target
of achieving a maximum admissible failure rate.
Looking at actual capital ratios one often sees that banks hold more capital than prescribed, indicating
that the imposed CAR is not a binding restriction. This may be partly motivated by the buffer function
of bank capital. Sudden losses realized on a bank’s loan portfolio will then not immediately lead to
regulatory action. Indeed, an extended version of our model in which banks endogenously choose their
level of equity shows that a bank will choose a relatively high level of bank capital and tighter lending
conditions. It allows the bank to make a better cost-benefit tradeoff, which results in higher charter values
and lower failure rates. Optimal regulation may now take the form of a contingent rule with respect to
equity: more risk appetite must be countered by more equity.
The article is outlined as follows. In the next section we describe the model setup. In Section 3 we first
analyze the basic model with a uniform returns distribution analytically in Section 3.1, and look at the
effect of optimal regulation by a contingent rule in this framework in Section 3.2. Section 3.3 presents
further numerical analyses of the basic model with a more realistic beta returns distribution. In Section 4
we extend the basic model to incorporate the banks with an endogenous choice of equity. The analysis of
this model is again performed numerically using a beta returns distribution. Finally Section 5 concludes.
2
THE BASIC MODEL
Consider competition between two banks, labelled i = 1; 2, which both operate for an infinite number
of periods labelled t
=
1; 2; : : : . Bank managers are risk neutral. To abstract from agency problems
between bank managers and owners, we assume that bank managers are entitled to a fixed percentage of
the bank’s profits as long as the bank does not fail and have the same discount rate as the shareholders3 .
A bank’s management chooses its optimal bank lending policies by stipulating its lending conditions (or
acceptance criteria) for granting loans, which in turn determine the riskiness of the bank’s loan portfolio.
Intuitively, the idea is that less stringent lending conditions cause riskier portfolios, which degenerate
the quality of these portfolios, inducing higher failure rates and ultimately leading to more vulnerable
banks. In our model, these lending conditions are captured by a single index parameter αi 0; i = 1; 2.
A higher value of the parameter αi corresponds to an easing of the lending conditions and is assumed
to decrease the average of the returns distribution and increase the variance of the loan portfolio. As an
opposing force, by relaxing the lending conditions it becomes easier for consumers to take out loans, i.e.
their overall loan capacity increases, yielding an increase in the demand for loans. In this way, a bank
may be able to sustain its (current) profits in the loans market. Formally, bank i faces the following linear
demand curve
Li (αi ; α j ) = L + l1 αi
l2 α j ;
with l1 ; l2 > 0; i; j = 1; 2; j 6= i:
(1)
Thus, the loans of the two banks may be regarded as (imperfect) substitutes, where the parameter l2
reflects the degree of substitution between the loans of the two banks. By imposing demand curves
which depend only on lending conditions, the nature of competition among the two banks in this model
is completely due to competition in the lending conditions, and not in the lending rates 4 . In fact, the
model can be regarded as Bertrand competition with loan conditions as strategic variables.
All potential loans are (imperfectly) diversified and share the same expected return per unit of loanable
funds. The interest percentage charged on all loans is equal, but the probability that a loan is not repaid
or is only partially repaid varies with αi . More specifically, similar to the specification in Matutes and
Vives (2000), denote by R̃i the stochastic return per unit of supplied loans by bank i. The random variable
R̃i is distributed according to a distribution function Fi which is twice continuously differentiable with
positive density on the interval [γi ; γi ], with γi 0, i = 1; 2. Thus the worst that can happen to a bank is
that none of its loans are repaid and the return on loans is 0. By choosing its lending conditions parameter
3
Alternatively, another interpretation would be to say that the bank managers are the shareholders of the bank, see
Chu (1999).
4 This aspect captures the Dutch mortgage market relatively well, where we do not see so much competition
in mortgage rates, but rather in the acceptance criteria and lending conditions of these supplied mortgages, see
De Haas, Houben, Kakes, and Korthorst (2000).
αi , bank i influences the distribution of returns from its loan portfolio. This influence takes the form of a
reduction in the average return of the loans portfolio and an increase in its variance. Formally, we denote
the corresponding density function by fi
=
f (R̃i ; αi ), where αi
2 [α α] is the aforementioned lending
;
conditions index, allowing for the possibility that α = ∞. Higher values of αi are associated with lower
expected returns E R̃i
= Re (αi )
and higher variances of returns var(R̃(αi )). That is, dRe (αi )=dαi < 0 and
dvar(R̃(αi ))=dαi > 0. The average return for α = 0 (’minimal risk’) is denoted Re (0) = R̄.
Our main focus is on banking competition in the loans market. To simplify, we abstract from considerations at the liabilities side of the banks’ balance sheet by assuming that at the beginning of every period
t, both banks have unlimited access to an inelastic supply of deposits. Deposits at bank i are labelled Di .
Deposits are short and will be repaid at the end of the period at a fixed deposit rate rd . Moreover, we neglect any possible liquidity problems on the part of the bank by assuming that deposits are fully insured
in a government run fixed rate deposit insurance scheme, see e.g. Van den Heuvel (1999) or Hellman,
Murdock, and Stiglitz (2000) 5 . Moreover, both banks supply all loans demanded by the public. Apart
from loans Li , and deposits Di , both banks need to issue equity Ei , i = 1; 2, in order to be able to deal with
possible bad returns on their loan portfolio. Since equity bears a higher risk than deposits do, risk averse
equity holders demand an equity premium of ρ > 0 on top of the risk free rate rd 6 . A higher ratio of
equity versus (risky) loans on a bank’s balance sheet means that it has a lower probability of bankruptcy.
The banking industry regulator prescribes a minimum equity-to-loans ratio to the commercial banks by
setting a capital adequacy ratio of 100k percent, 0 < k < 1. Note that this k corresponds to the CAR
actually discussed in the Basel Accords. In the standard version of the model we assume that no bank
will raise more equity than necessary for adhering to the regulator’s requirements, since equity is more
costly than deposits. This assumption results in
Ei = kLi ;
i = 1; 2;
(2)
where the equality sign follows from the binding nature of the capital adequacy requirements. In Section 4 we posit an extended model in which the amount of equity is endogenously determined by the
bank management.
The standard setup above leads bank i to make a (stochastic) profit (or loss). Note that the maximum loss
the equity holders of a bank may suffer never exceeds the amount of equity initially raised by the bank,
because of limited liability. Here, we focus on excess profits of the bank, i.e. profits which remain after
repayment of deposits with interest and equity including compensation demanded by the shareholders,
5
For convenience, we assume that the paid insurance premium is zero, see also Van den Heuvel (1999).
In Hellman, Murdock, and Stiglitz (2000) the rate of return on equity is determined endogenously as the equilibrium rate that clears the market for bank equity capital. It is shown that in equilibrium ρ > 0. Alternatively, Gorton
and Winton (1997) also derive that bank capital is costly endogenously in a general equilibrium model.
6
see e.g. Vlaar (2000) for a similar definition. This profit function looks like
n
πi (αi ; α j ) = max R̃i Li (αi ; α j )
n
= max
(R̃i
r d Di
(r + ρ)Ei
d
[r + ρk])Li (αi ; α j )
d
;
;
(r + ρ)Ei
d
o
o
(r + ρ)kLi (αi ; α j )
d
;
i; j = 1; 2; j 6= i;
(3)
where we used the binding nature of the capital adequacy requirements and the initial balance sheet
equality Di
= (1
k)Li . A bank which realizes positive expected excess profits is able to meet (on
average) the return demanded by the equityholders and will thus be able to raise equity on the capital
market in the future.
Both banks are subject to prudential regulation by the same regulating body. At the end of each period,
the regulator inspects the balance sheet of both banks. If the return on its loan portfolio is not sufficient to
repay all depositors, that is, whenever equity becomes negative, the regulator closes down the bank and
its management is dismissed. Utility for the management after dismissal is set to 0. After the dismissal
of the management and the declared bankruptcy of the bank, a new bank identical to the old one comes
into operation with a new management which takes the place of the bankrupted bank. The probability of
such a bankruptcy is equal to
θi (αi ; α j ) = Pr[bankruptcy ] = Pr[πi <
(r + ρ)Ei ] = Pr
d
h
R̃i Li (αi ; α j ) < rd (1
k)Li (αi ; α j )
i
;
(4)
which, since Li (αi ; α j ) cancels out, translates into
θi (αi ) = Fi rd (1
Z
k)
r d (1 k)
=
∞
f (x; αi )dx;
(5)
so that the bankruptcy probability for bank i is independent of αj . We furthermore assume that when
a bank is not bankrupt at the end of period t, it continues operations into period t + 1 with the same
initial balance sheet as in period t. Thus, we implicitly assume excess profits to be paid out as extra
dividends and negative excess profits to be compensated by shareholders supplying additional equity and
thus recapitalizing their bank. In effect, this stationarity assumption enables us to analyze the model in a
static framework.
We are now able to express expected excess profits by
πei (αi ; α j ) = E πi (αi ; α j ) =
Z
θi (αi )(rd + ρ)kLi (αi ; α j ) +
∞
r d (1 k)
f (x; αi )(x
[r + ρk])Li (αi ; α j )dx;
d
i; j = 1; 2; j 6= i:
(6)
A bank’s management is maximizing its discounted expected excess profits. Given the absence of agency
problems and given that the management receives no payoffs after being dismissed, it maximizes the
following value function over αi
∞
Vi (αi ; α j ) = ∑
t =0
1
θi (αi )
rd + ρ
t
πei (αi ; α j ) =
πei (αi ; α j )
1
1 θi (αi )
r d +ρ
:
(7)
Often in the literature, Vi is referred to as the charter value of bank i, i = 1; 2. From the charter value one
clearly sees the three distinct effects a change in the risk index αi has for bank i. Consider an increase in
αi . First, this results in increased demand at bank i. Second, the profit per unit of loan demanded at bank
i decreases. These two effects combined lead to the conclusion that for a generic returns distribution, a
change in αi has an ambiguous effect on the per-period profits of bank i. Third, an increase in αi leads to
a higher default probability of the bank and thereby reduces its charter value. For notational convenience
we label the above discount rate by δi (αi ) = [1
θi (αi )]=[rd + ρ]. The discount rate consists of two
components. First, since the profits πi accrue to the equity holders, the time preference rate for equity
1=(rd + ρ). Second, the continuation probability of the flow of profits is 1
θi (αi ) . Multiplying these
two components yields the discount rate δi (αi ).
The timing of events during a time period is illustrated in Figure 1. At the beginning of period t both
banks choose their lending conditions αi , i = 1; 2 simultaneously. These lending conditions determine
the demand for loans at each bank. At the same time instant, in order to fund these loans, both banks attract the necessary deposits and raise equity. After loans are made, deposits attracted, and equity issued,
the stochastic return on the loans is realized. Subsequently, at the end of period t, the regulator assesses
whether the bank has positive equity. If not, the bank is declared bankrupt, settles its debts through the
deposit insurance scheme and ceases operations. A new identical bank takes its place. In case of positive
equity, the bank repays its depositors the amount of deposits plus interest, and equity is paid out including compensation and (possibly negative) excess dividends to the shareholders. The bank continues its
operations, and the game proceeds to the next period t + 1.
Figure 1
The timing of events.
Simultaneous
choice of αi .
t
Realization of
return on loans R i .
Demand for loans L i (αi ; α j ).
Acquisition of deposits D i .
Issuance of equity E i .
Payout of
(excess) dividend.
Inspection by
regulator.
Possible failure.
t +1
3
STRATEGIC ANALYSIS OF THE BASIC MODEL
Since the choice of the lending conditions parameter αi of one bank influences the profits of the other
bank, our model of bank behavior presents a strategic decision-making problem for both banks. Given
α j , bank i’s manager maximizes the discounted sum of profits over αi . This yields bank i’s reaction
function, i = 1; 2, to bank j’s behaviour, j = 1; 2; j 6= i. Formally, from bank i’s first order condition we
have
∂Vi
∂αi
=
[1
δi (αi )]
∂πei (αi ;α j )
∂αi
πei (αi ; α j ) ∂[1
δi (αi )]2
[1
δi (αi )]
∂αi
= 0;
(8)
which can be rewritten as
επ (αi ; α j ) = εδ (αi );
(9)
where επ (αi ; α j ) = (∂πei =∂αi )(αi =πei ) denotes the elasticity of profits with respect to αi , and εδ (αi ) =
(∂ ( 1
δi )=∂αi )(αi =(1
δi )) the ‘elasticity’ of the discount factor with respect to αi . In fact, equation (9)
presents a marginal cost-benefit tradeoff: the gains of taking up more risk in terms of increasing current
profits are just outweighed by the future losses expressed in terms of a lower discount rate, that is, a
higher failure rate. In principle, solving (9) leads to bank i’s reaction function αRi (α j ), i = 1; 2. In
a symmetric equilibrium one must have αi
=
(symmetric) equilibrium value α .
3.1
α j , so that επ (α ; α )
= εδ (α ),
implicitly defines the
A Uniform Returns Distribution
For a general returns distribution, an explicit analytical expression for α cannot be found. Therefore,
we perform the analysis for a uniform distribution of returns. Consider a returns distribution for bank i
with x 2 [A
aαi ; B], A; a > 0, A < B, 0 αi Aa , which depends linearly on the loans condition index
αi , i.e. the distribution function is f (x; αi ) =
1
B A+aαi ,
when x 2 [A
aαi ; B] and 0 elsewhere. A bank
which takes minimal risk, i.e sets αi = 0, has an expected return of E (R̃i ) = R̄ =
A+B
2 .
For this uniform
distribution the charter value of bank i is equal to
Vi (αi ; α j ) =
8
C2 αi
>
L(αi ; α j ) CC13 +
;
>
+C4 αi
<
>
>
:
when F (rd (1
k); αi ) 2 [0; 1]
(10)
L(αi ; α j ) C6 +CC5 7 αi ;
when F (rd (1
k); αi ) = 0
with
C1
=
(r + ρ)(B + ((1
C2
=
2ak(rd + ρ)2 ;
C3
=
2
C4
=
2a(rd + ρ);
C5
=
(B
C6
=
2(B
C7
=
2a(rd + ρ
d
2
k)rd )2 + 2Ak(rd + ρ)
k)rd + B(rd + ρ
(1
(1
A(rd + ρ))
1)
k)rd )(rd + ρ) B
A)(rd + ρ
2B(rd + kρ));
;
rd
k(rd + 2ρ)
;
1);
1):
Solving the first order condition for αi yields the reaction funtion αRi (α j ). Subsequently solving α j
=
αRj (αRi (α j )) yields the symmetric equilibrium value α as a function of the exogeneous variables of the
model. The explicit analytical expression for α can be found in the appendix.
Figure 2
The two reaction curves and the line α1 = α2
α2
7
α1 = α2
6
5
α
4
α1 (α2 )
3
α2 (α1 )
2
1
1
2
3
4
5
6
7
α1
Figure 2 presents the reaction curves αRi (α j ), i; j = 1; 2, j 6= i of the two commercial banks for a uniform
returns distribution on [0:7
1
5 αi ; 1:7],
demand function Li (αi ; α j ) = 2 + 5αi
a return on deposits of 1:05, an equity premium of 0:1 and a
2α j . The CAR is set to k = 8%, the current minimum capital
requirement as prescibed by the Basle Accord. The intersection of the reaction curves is a graphical
representation of α , which analytical calculations show to be equal to 4:1. Note that because of the
symmetry of the problem the intersection point of the reaction curves lies on the line α1 = α2 , which is
also drawn in the figure.
Interest goes out to the comparative statics of α with respect to the capital adequacy ratio k and increased
competition, described by an increase in l2 7 . In Figure 3 we show the dependence of α on the capital
adequacy ratio k for the parameters mentioned above. The figure clearly show the general features of the
comparative statics on k. For the given set of parameter values, an increase in the capital adequacy ratio
leads to less risk taking on the part of the banks which set more stringent loans conditions. Since less
risk taking is equivalent to imposing tougher conditions on loans supply, an increase in the CAR leads to
a decline in the total demand for loans.
Observation 3.1. The higher the fixed capital adequacy ratio, the less risk commercial banks take on.
Figure 3
The dependence of α on k
α
6
5
4
3
2
1
0:05
0:1
0:15
0:2
0:25
0:3
k
The result in Observation 3.1 stems from marginal considerations. In order to clarify the different
marginal effects, we consider all these effects separately for the uniform returns distribution under consideration 8 . When the CAR k increases, this has two distinct effects on the charter value Vi . First, the
probability of failure of the bank decreases, leading to an increase in the charter value. Also, the marginal
benefits of further increasing k will diminish. Second, the profit per unit of demanded loan decreases,
leading to lower per-period profits and lower marginal costs of further increasing k. As to a change in
the risk index parameter α, similar reasoning applies. A lower value of α has three distinct effects on the
charter value. First, it leads to a lower failure rate, thereby increasing the marginal failure rate and thus
decreasing the marginal costs of increasing α in terms of the charter value. Second, it causes demand to
7 The comparative statics features described below are general in the sense that it occurs for all values in the wide
range of parameter values we looked at.
8 By doing so we neglect all crossterms. This enables a more clear presentation of the different effects.
fall, thereby decreasing per-period profits. Since demand is linear in α, the fall in demand by itself does
not lead to a change in the marginal revenues. Third, the revenues per unit of loan will increase, thereby
increasing per-period profits. For a uniform returns distribution, revenues per unit of loan are also linear
in α, yielding, as before, that marginal revenues are not affected by a change in α 9 . In the case of a
uniform returns distribution, the marginal revenues of the multiplication of both terms in the expression
for expected per-period profits is linear in α. Consequently marginal revenues of increasing α slightly
are lower for lower values of α.
Now, we consider the influence of increasing k on α through the failure rate and the per-period profit
channels separately. From the above it follows that increasing k leads to lower marginal benefits when
considering the bank’s failure rate. Since bank management always equates marginal benefits to marginal
costs, this means that the marginal costs of changes in the failure rate have to decrease as well. Again
from the above, it directly follows that lower marginal costs are achieved by lowering α, yielding support
for the comparative statics result in Observation 3.1. Considering the per-period profits of the bank, a
higher k leads to lower marginal costs. Equating lower marginal benefits in terms of α to these lower
marginal costs in terms of k, leads to a lower α. Thus, the per-period profit channel considerations
work in the same direction as the failure rate channel considerations. Together these channels yield the
comparative statics result in Observation 3.1.
Comparative statics results of α on l2 are shown in Figure 4. From the figure, we see that more intense
competition leads to more risk-taking by the banks. This is caused by the more fierce competition
for market share as a consequence of the increased competition. Since more risky portfolios are the
consequence of looser conditions on loans supply, it means that the total demand for loans increases
when competition becomes more intense.
Observation 3.2. The more intense banking competition, the more risk commercial banks take on.
Seen from a technical point of view, Observation 3.2 follows from standard Bertrand price competition
considerations. Increased competition harms per-period profits and has no direct effect on the probability
of bankruptcy. In order to partially compensate for the profit decrease, the banks will increase the risk
in their portfolio, thereby ceteris paribus increasing demand. Of course, through this channel of loans
portfolios bearing more risks, indirectly the failure rate will increase.
The loosening of lending conditions in an environment with more competition is clearly in accordance
with empirical findings in the Dutch mortgage market, already described in Section 1. Indeed increased
9
Note that this is no longer true when we consider a beta returns distribution furtheron in the paper. This allows
for the possibility that for a beta returns distribution, Observation 3.1 is refuted for certain parameter values.
Figure 4
The dependence of α on l2
α
9
8
7
6
5
4
3
2
1
2
3
4
5
l2
competition has led to increased vulnerability of the commercial banks’ balance sheets. Recognizing
these developments, the Dutch central bank has, in its capacity as banking industry supervisor, put forward several policy measures which should result in lower vulnerability of the commercial banks. Most
of the proposed measures come down to a more adequate risk weighting of the supplied mortgage loans.
In the model above, more adequate risk weighting means making the capital adequacy ratio for bank
i dependent on the riskiness αi of this bank’s loans portfolio. Below, we develop such a risk-adjusted
capital adequacy framework.
3.2
Risk-Adjusted Capital Requirements
We assume that the regulator’s goal is a stable financial system. To achieve the goal, it sets a maximum
admissible probability of bankruptcy Pmax 2 [0; 1] for commercial banks as a target. Its instrument now
becomes a contingent rule, by which it specifies how the CAR depends on the risk a commercial bank
takes and Pmax . Thus, k(αi ; Pmax ) is then a parameter in the function θi which gives the probability of
failure. Thus the optimal k(αi ; Pmax ) is the solution to θi (αi ; k(αi ; Pmax )) Pmax which for a uniform
returns distribution as specified above yields
k(αi ; Pmax ) 1
1+B
(1
Pmax)(B
1 + rd
A + aαi)
:
Observe that the returns distribution we consider implies that B > A
forward that
∂k(αi ;Pmax )
∂Pmax
<
(11)
aαi . Therefore, it follows straight-
0, that is, when the maximum admissible probability of bankruptcy is decreased,
the banks will be required to hold more equity in relation to their level of risky loans. From the expres-
sion for k(αi ; Pmax ) it also follows directly that
∂k(αi ;Pmax )
∂αi
>
0, confirming that indeed a regulator would
like to see a bank with a more risky portfolio hold more equity.
Observation 3.3. The optimal risk-adjusted capital adequacy ratio is higher when a commercial bank
takes on more risk.
Observation 3.4. The optimal risk-adjusted capital adequacy ratio is higher for a lower admissible
probability of failure.
The regulator now communicates to the commercial banks that it will set the CARs according to k(αi ; Pmax ).
The banks will thus optimize their charter value, given these risk-weighted CARs. For the set of parameter values chosen above, the restriction on k(αi ; Pmax ) will be binding for all reasonable (i.e. small) values
of Pmax . Filling out this binding restriction in the charter value of the banks and subsequent computations
results in linear reaction curves, and an elegant expression for the equilibrium solution
α =
l1 A
(B r d )r d
B Pmax r
aL
l1 + 1 Pmax + r +Pmax r +2 ρ
a (2 l1
l2 )
:
(12)
The comparative statics for α in the case of risk-weighted regulation show that α is increasing in Pmax ,
i.e. when the regulator sets a lower admissible probability of bankruptcy, the banks will react by taking
less risk and vice versa. Thus, indeed the risk weighting of the CARs has the desired effect that the risks
in the banking system will diminish. At the same time, because the banks are placing more demanding
conditions on their supply of loans, the total loans market demand declines.
Observation 3.5. A lower admissible probability of failure leads to less risk taking by the commercial
banks.
3.3
A Beta Returns Distribution
One of the main advantages of choosing a uniform distribution to model the stochastic returns on the
bank’s loan portfolio is its simple mathematical form and analytic tractability. However, its drawback
lies in the restrictive nature of its density function, which is constant on the support of the distribution.
Thus, shifting the expectation of the distribution necessarily implies altering the distribution’s support,
which was done in the previous section.
A more natural and flexible distribution to model the stochastic returns is a beta distribution. A beta
distribution has the appealing feature that a change in its parameters yields a shift in the probability
mass, while its support remains constant. Thus, we can specify that returns will always be within certain
bounds, while at the same time manipulating the average returns and higher moments of the distribution
as a function of the loan condition parameter αi . The price to be paid of using a beta distribution is in
loosing the ability to derive analytical expressions. Therefore, we present a numerical exercise to grasp
the robustness of the results presented in the previous two subsections.
Let R̃i = R̄(1
b
>
X̃i) where X̃i follows a beta distribution on the interval [0; 1] with parameters (a(αi ); b),
0. We simply posit a(α)
E (R̃i ) =
R̄b
b+aαi ,
=
aα, a
>
0. We then have that the support of R̃i is [0; R̄], and that
which is decreasing in αi , indicating that the quality of the loan portfolio is decreasing as
more risk in terms of a loosening of the lending conditions is taken 10 . Given a fixed parameter b, larger
values of αi shift the probability mass towards the region of low returns, i.e. towards zero. Figure 5
shows how the probability mass shifts on [0;R̄] for R̄ = 1:2 and for different values of αi : higher values
of αi put more mass on the lower returns region. Further, no risk corresponds to αi = 0 in which case
E (R̃i ) = R̄. The probability of bankruptcy is 0 for αi = 0 and increases in αi , to approach 1 as αi
Figure 5
! ∞.
Illustration of beta density functions for different αi with a = 0:25, b = 5 and R̄ = 1:2
f (R)
5
αi = 1:5
4
3
αi = 6
2
αi = 12
1
R
0:2
0:4
0:6
0:8
1
1:2
As before, given a fixed α j , a higher αi induces on the one hand a higher probability of default but on
the other hand enables bank i to attract more loans at a lower profit per unit of loans, thus possibly enhancing its current profits. Bank i has to weigh these two opposing forces optimally. Table 1 presents the
symmetric Nash equilibria αi = αj = α for different values of the capital ratio, and shows corresponding equilibrium probabilities of default, volumes of loans and expected charter values. This numerical
10
The variance of the returns is increasing for relevant values of α, for large α though the variance starts decreasing.
example closely follows the one presented in Section 3.1 in terms of parameter values: we put deposit
interest rate rd at 1:05, average return on loans without risk R̄ at 1:2, equity premium ρ at 0:1, demand for
loans at Li (αi ; α j ) = 2 + 5αi
2α j and the beta density parameters a = 0:25 and b = 5. The table shows
that a higher capital ratio induces lower risk taking, i.e. a lower α in equilibrium. As a consequence the
bankruptcy probabilities drop, a result which is stated in Observation 3.6 below. These findings confirm
our results of Section 3.1 and point to robustness with respect to the choice of probability distribution for
the returns. We remark that in the particular example in Table 1, for a capital adequacy ratio k = 0:08,
the probability of bankruptcy is around 9%. The resulting density for α
=
1:31 closely compares to
the density drawn most to the right in Figure 5. Note that in the table the charter value of the banks
increases when the CAR becomes more stringent. This suggests that when banks could freely choose
the level of their equity-to-loans ratio, they would choose a ratio higher than k, thereby rendering the
capital adequacy constraint non-binding. In such a setting the banks of course have to take account of
the fact that, in contrast to the case with a fixed binding CAR, their competitors can always choose to
strategically lower his capital ratio if this yields more profit, i.e. that a non-binding CAR does not act
as a credible precommitment device to the banks. We extend the current model to incorporate such an
unconstrained choice of equity in Section 4.
Observation 3.6. Higher fixed capital adequacy ratios lead to lower failure probabilities of the commercial banks.
Table 1
Symmetric Nash equilibria for different capital adequacy ratios k
k
α
θi (α )
Li (α ; α )
0:05
0:08
0:10
0:15
0:20
1.38
1.31
1.27
1.21
1.16
0.121
0.091
0.076
0.050
0.032
6:15
5:92
5:81
5:63
5:49
Vi
2:19
2:28
2:32
2:36
2:35
Explanatory Note: Parameter values are a = 0:25, b = 5, R̄ = 1:2, r d
= 1:05, ρ = 0:1, L = 2, l 1 = 5, l2 = 2.
Note, however, that Observation 3.6 -compared to Observation 3.1 in the previous section- is stated in
terms of failure rates and does not say that increased fixed capital adequacy ratios lead to less risk taking
on the part of the banks. In the beta returns case, a further qualification is needed, which we illustrate
graphically. As already argued in the previous section, the equilibrium value of the risk parameter α
stems from marginal considerations, as captured by equation (9). The intersection of the elasticity curves
επ (α; α) and εδ (α) gives the Nash equilibrium. Figure 6 shows these curves for capital ratio k = 0:08
(solid lines) and k = 0:20 (dashed lines). We see that, on the one hand, the ‘per-period profit elasticity’
line shifts to the left for the increased capital ratio, since a higher k leads to lower per-period profits,
and on the other hand, the ‘failure rate elasticity’ line shifts to the right indicating that probability of
failure decreases as k rises. The net effect of a shift in k from 0:08 to 0:20 is a lower value of α , as also
indicated by Table 1.
Figure 6
Illustration of the Nash equilibrium for k = 0:08 (solid) k = 0:20 (dashed)
επ (α; α)jk=0:08
εδ (α)jk=0:08
0:5
0:4
0:3
0:2
εδ (α)jk=0:20
επ (α; α)jk=0:20
0:1
α
1:1
1:2
1:3
1:4
1:5
0:1
0:2
The location of the ‘per-period profit elasticity’ lines in Figure 6 also depends on market characteristics
such as initial market demand measured by L, and the rate l1
l2 at which demand grows when α
increases, while the location of the ‘failure rate elasticity’ line does not depend on L, l1 , or l2 . A higher
initial market demand results in lower values of the per-period profit elasticity, since a relatively large
autonomous part makes the per-period profits less responsive to changes in α, inducing less need for risk
taking. Hence, for a given capital adequacy ratio k, the equilibrium level of α decreases with a larger
initial market size L. Also, a higher value of L causes the ‘per-period profit elasticity’ line to be flatter.
Therefore, for a large L the inverse relation between α and k may be reversed: the intersection of a
flatter leftward-shifted ‘per-period profit elasticity’ line with the rightward-shifted ‘failure rate elasticity’
line may yield an intersection point shifted to the right, i.e. the equilibrium value of α may increase for
larger k.
The next table depicts this result. Table 2 shows that if the initial size of the market is big (L = 4)
compared to the equilibrium volume (e.g. Li (α ; α ) = 6:2 at k = 0:08), risk taking actually increases
for higher CARs. However, failure rates will continue to go down. We conclude that with for general
returns distributions, Observation 3.1 only holds for some regularity condition on market structure, a
result which the regulator needs to take into account. In the beta returns case this condition says that the
value of L=(l1
l2 ) has to be small enough for Observation 3.1 to carry over.
As in the previous section, we have also checked the sensitivity of the results to more intense competition,
Symmetric Nash equilibria for a large initial size of the market L = 4
Table 2
k
α
θi (α )
Li (α ; α )
0:05
0:08
0:10
0:15
0:20
0.70
0.75
0.77
0.80
0.81
0.057
0.048
0.043
0.031
0.021
6:09
6:24
6:31
6:40
6:42
Vi
3:79
3:80
3:80
3:78
3:72
Explanatory Note: Other parameter values are a = 0:25, b = 5, R̄ = 1:2, r d
= 1:05, ρ = 0:1, l 1 = 5, l2 = 2.
as measured by an increase in l2 . Table 3 shows the results for a fixed CAR of k = 0:08 in terms of optimal
α ’s, and corresponding failure rates, volume of loans and charter values. From the table we see that the
analytical results found for a uniform returns distribution are confirmed in the beta returns distribution
case: More fierce competition leads to the commercial banks to take on more risk, which results in higher
failure rates and lower charter values.
Observation 3.7. The more intense banking competition, the more risk commercial banks take on, the
higher failure rates, and the lower charter values.
Symmetric Nash equilibria for different values of l2 at k = 0:08
Table 3
l2
α
θi (α )
Li (α ; α )
1
2
3
4
1.02
1.31
1.76
2.48
0.069
0.091
0.130
0.194
6:09
5:92
5:52
4:48
Vi
2:97
2:28
1:43
0:57
Explanatory Note: Other parameter values are a = 0:25, b = 5, R̄ = 1:2, r d
= 1:05, ρ = 0:1,
L = 2, l 1 = 5.
We continue by analysing optimal regulation as in Section 3.2. Hence we assume that the regulator fixes
a maximum bankruptcy probability Pmax as its objective and enforces a capital adequacy ratio k(αi ; Pmax )
on the banks, which is contingent on a bank’s risk profile, such that θi (αi ; k(αi ; Pmax )) Pmax . In calculating an optimal strategy the banks will take this contingent rule into account. The resulting equilibrium
α determines an equilibrium capital ratio k = k(α ; Pmax ).
Table 4
Symmetric Nash equilibria in the presence of risk-adjusted regulation
Pmax
α
k(α ; Pmax )
Li (α ; α )
0:01
0:03
0:05
0.92
1.08
1.18
0.31
0.20
0.15
4:75
5:24
5:55
Vi
2:30
2:42
2:39
Explanatory Note: Parameter values are a = 0:25, b = 5, R̄ = 1:2, r d
Table 4 gives the results for Pmax
=
= 1:05, ρ = 0:1, L = 2, l 1 = 5, l2 = 2.
1%, 3% and 5%. It turns out that the imposed contingent rule is
effective in reducing the bankruptcy probability to the imposed level. It results in higher required capital
ratios and induces less risk taking. For instance, to get the probability of bankruptcy down to 3% the
induced capital adequacy ratio rises to 20%, while a failure rate of 1% corresponds to a CAR of 31% in
this example. Thus we observe qualitatively the same behavior as in Observation 3.5: A lower admissible
probability of failure leads to less risk taking by the commercial banks.
4
AN EXTENTION: ENDOGENOUS EQUITY
In the basic model, we assumed that banks will never hold more equity than the minimum amount prescribed by the regulator, since raising equity is more costly than attracting deposits. However, bank management weights the marginal benefits of taking on more risk by easing its lending conditions (i.e. creating higher per period profits) against the marginal costs (i.e. allowing a higher probability of bankruptcy).
In other words, while equity is more expensive to a bank than deposits, in a dynamic context, holding
more equity also prolongs the expected lifetime of the bank in operation, thus yielding a more extended
flow of profits. Hence, it might very well be that bank management chooses to hold more than the minimal prescribed amount of equity. This observation is in accordance with the real world observation that
many banks hold more equity than the minimal amount prescribed according to the CARs.
Below, we investigate a model in which we drop the assumption Ei = kLi . Such a model presents bank i
with an optimization problem over both lending conditions and equity, i.e. over the two variables αi and
Ei , subject to the condition that Ei =Li kmin , with kmin now being the minimum capital requirement imposed by the regulator 11 . It results in expressions for per period profits and the probability of bankruptcy
that are different from those presented before. Using the balance sheet equality Li = Ei + Di , per period
profits amount to
n
π(αi ; α j ; Ei ) = max R̃i Li (αi ; α j )
n
= max
rd )Li
(R̃i
r d Di
(r + ρ)Ei
ρEi ;
(r + ρ)Ei
d
d
(r + ρ)Ei
d
;
o
o
(13)
:
The probability of bankruptcy of bank i will now, through its demand L(αi ; α j ), also depend on α j , as
follows.
θi (αi ; α j ) = Pr R̃i < r
d
1
Ei
Li (αi ; α j )
=F
r
d
1
Ei
Li (αi ; α j )
(14)
Analogous to the analysis of the basic model, combining the bankruptcy probability with the per period
profit formula yields expected per period profits πei (αi ; α j ) and bank i’s charter value Vi (αi ; α j ).
We analyze this model for the beta returns distribution as specified in Section 3.3, with the same parameOne may alternatively say that the bank is simultaneously optimizing over α i and its capital ratio Ei =Li , see also
Vlaar (2000). Note however that this ratio also depends on α j via the demand for loans.
11
ter values as before: a = 0:25, b = 5, R̄ = 1:2, rd = 1:05, ρ = 0:1, L = 2, l1 = 5, l2 = 2. Figure 7 plots the
(symmetric) charter value for different values of α = αi = α j and equity-to-loans ratio Ei =Li . One may
=L = 12; 5% (with a failure
verify that the symmetric maximum in Figure 7 is attained for α̃ = 0:65 and Eg
rate of 2.9% and charter value 2.6), however this maximum does not correspond to the symmetric Nash
equilibrium, since players cannot coordinate on these equilibrium actions. Given that bank i chooses
αi
=
=L, bank j is better off by deviating from these actions. Numerical optimization
α̃ and Ei =Li = Eg
yields the symmetric Nash equilibrium for α
=
1:24 and (E =L) = 12:1%, which in this example is
well above the eight percent, as currently prescribed in the Basel Accord. This equilibrium ratio of equity
over loans yields an equilibrium level of equity E = 0:70 relative to an equilibrium volume of loans
of 5.7. Consequently the bankruptcy probability drops down to 6.3%. Indeed we conclude that banks,
when unconstrained in their choice of equity, may choose to hold more equity than prescribed to them
by the regulator in order to lower their probability of failure. The volume of the loans supplied slightly
decreases, from 5.9 in the case of fixed regulation at k = 8% to 5.7 now 12 .
Observation 4.1. When given the choice, commercial banks may choose to hold more equity than the
minimum equity level required by the regulator.
Although not shown here, comparative statics results on l2 are qualitatively the same as in the previous
subsection. Increasing l2 as a measure of increased competition causes banks to take on more risk and
less capital to secure their market share, as in Observation 3.2. As a result, failure rates increase. Since
the capital adequacy ratio k is no longer binding for small k, performing comparative statics analysis on
the CAR is not useful.
In contrast to Section 3 where the imposed CAR was always binding, here in this extended model with
endogenous equity the imposed CAR loses some of its bite, since it now acts as a minimum capital
requirement. As long as it is not binding -that is, in the above example as long as kmin 12; 1%- it does
not impinge on the equilibrium outcome. As already analysed in the previous section, in order to better
align capital requirements with underlying risks, the regulator may impose a contingent rule. In this
extended model this would mean setting a contingent rule on equity, in the sense that more risk must be
countered by more capital. Such a rule is equivalent to the contingent rule on k in the fixed equity case of
Section 3. Let the ‘contingent equity’ rule be denoted by E (αi ; α j ), such that θ(αi ; α j ; E (αi ; α j )) Pmax .
Calculations show that reducing failure rates in this way is effective. For e.g. Pmax = 3% -which is strictly
smaller than the bankruptcy probability of 6.3% in the example above-, in equilibrium a bank takes on
less risk and holds more equity than in the unconstrained case without a contingent rule, i.e. α = 1:08
12 We were unable to derive analytical solutions for the model with endogenous equity and a uniform returns
distribution. The numerical results for such a model are qualitatively same as the results obtained here for a beta
returns distribution.
Figure 7
Charter values for the endogenous equity model
2:6
Vi
2:4
2:2
0:2
2:0
Ei =Li
0:25
0:5
0:1
0:75
α
1
1:25
and (E =L)
=
0
20%. The effect on the volume of the loans market is contractionary, decreasing from
5.7 tot 5.2. These findings exactly correspond to the second row of Table 4 of Section 3. In summary,
we find that imposing a contingent rule for equity is effective in reducing the failure rates in the banking
sector.
Observation 4.2. In a setting with an endogenous choice of equity, imposing an optimal risk-adjusted
minimum level of equity reduces commercial banks’ failure rates.
5
CONCLUSIONS
We have modelled a framework in which commercial banks compete for loans by setting loans conditions. The model enables us to investigate the effects of regulation of and competition among the banks.
We analyse the model analytically for a uniform returns distribution and numerically for a beta returns
distribution. The most important conclusions to be drawn are that increased competition will lead to more
risk taking by the banks. The mechanism here is that more competition leads ceteris paribus to lower perperiod profits, which lowers the costs of bankruptcy to the bank management and thus makes them more
prone to seek more risk in order to increase demand. We therefore conclude that increased competition
in the banking industry warrants increased attention of the financial sector regulatory authority.
Setting a higher fixed capital adequacy ratio enables the regulator to reduce the probability that commercial banks go bankrupt. Such regulation is more effective when the risk-demand elasticity is low.
When risk-demand elasticities are higher, we found that, although the failure probability of commercial
banks still decreases when regulation becomes tougher, the banks partially compensate for the tougher
regulation by taking on more risks in their loans portfolios.
As an alternative to fixed capital adequacy ratios, a regulator can implement risk-weighted regulation,
as is the tendency reflected in the New Basel Accord on capital adequacy. Our analysis shows that riskweighted regulation is effective: the capital adequacy requirements increase in the risk taken by the bank
and decrease in the maximal failure probability deemed acceptable by the regulator. We conclude that
risk-adjusted capital adequacy requirements are thus a useful tool for the regulatory authority to control
the increased vulnerability of the banking system as a consequence of, among other factors, the increased
competition.
We have extended the basic model by specifying a model in which bank management maximizes profits
over both the loans conditions index and equity. We show that management may choose to hold more
equity than required by the regulator, even though raising equity is more expensive than attracting deposits. The rationale here is that although additional equity lowers per period profits, it prolongs the
expected lifetime of the bank and thus the expected length of the stream of future profits. In such a
setting, risk-adjusted capital adequacy requirements are useful when the regulator wants to reduce the
failure probabilities below their equilibrium values.
A
THE EXPLICIT EXPRESSION FOR α IN THE UNIFORM CASE
Some algebraic manipulations yield
α =
1
4 a2 k l1 (rd + ρ)
2
a rd + ρ
4 A k l1 ρ
a2 (B
(1
l2 ) rd
2 B k (2 l1
1
k)) l1 + 2 (1
(1
k ) r ) (B
k)2 l22 r
(1
2k
4 l1 (a L + (A
8 k l1 (a L + A (l1
2 B k ( 2 l1 + l2 )
2
r
d
1
k) r) rd + ρ
(1
k) k l2 + (1
+ l2 r + 2 k l1 ρ
B2 l22 + rd 2 k
(1
B2 l2 + rd 4 k (A
2
+
+
k)) l1 ) + 4 (A
l2 )) ρ +
l22 r + 4 k l1 (l1
k)2 l2 r
l2 ) ρ
(1
k)) l1 l2 + (1
1=2 :
k) l22
+
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