Do capital adequacy requirements reduce risks in banking?

Do capital adequacy requirements reduce risks

in banking?

J

urg Blum

Institut zur Erforschung der wirtschaftlichen Entwicklung, University of Freiburg, D-79085 Freiburg, Germany

Received 4 February 1998; accepted 14 July 1998

Abstract

In a dynamic framework it is shown that capital adequacy rules may increase a bank's riskiness. In addition to the standard negative e€ect of rents on risk attitudes of banks a further intertemporal e€ect has to be considered. The intuition behind the result is that under binding capital requirements an additional unit of equity tomorrow is more valuable to a bank. If raising equity is excessively costly, the only possibility to increase equity tomorrow is to increase risk today. Ó 1999 Elsevier Science B.V. All rights reserved.

JEL classi®cation:G21; G28

Keywords:Capital adequacy rules; Banking regulation; Risk taking

1. Introduction

There is an ongoing debate about the e€ects of capital adequacy rules on banks' risk taking behavior. This paper introduces a new argument which has been neglected so far. It is shown that in a dynamic setting a new intertemporal e€ect can arise which leads to an increase in risk. The key insight is that under

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binding capital requirements an additional unit of equity tomorrow is more valuable to a bank. If raising equity is excessively costly, the only possibility to increase equity tomorrow is to increase risk today.

E€ects of capital adequacy rules on banks' behavior have been analyzed before. The literature most closely related to this paper deals with conse-quences of capital rules for banks' asset risk.1 _{For value-maximizing banks,} Furlong and Keeley (1989) demonstrate that capital requirements reduce risk-taking incentives, while Flannery (1989) concludes that higher risk risk-taking may be induced. In a mean±variance framework, Koehn and Santomero (1980, 1988), and Rochet (1992) show that improperly chosen risk weights may in-crease the riskiness of banks. Other authors argue that capital requirements reduce monitoring incentives, which reduces the quality of banks' portfolios (Besanko and Kanatas, 1993; Boot and Greenbaum, 1993). Dewatripont and Tirole (1995) view capital rules as a means to eciently allocate control rights between di€erent groups of claimholders, thereby indirectly in¯uencing the bank managers' incentives. Finally, Gehrig (1995) points out that capital re-quirements in¯uence the nature of strategic competition among banks. While each of the above articles emphasizes a di€erent aspect of capital adequacy requirements, they all have in common that they concentrate on static e€ects. In this context the present paper illustrates the importance of taking into ac-count the dynamic perspective of the banking business.

In the following the decision problem of a single bank is analyzed, both when it is regulated and when it is not regulated. The resulting optimal choices are compared with the ®rst-best solution of the model. Due to limited liability an unregulated bank has a tendency to take `excessive risks', i.e., risks higher than ®rst best. If the bank only faces a binding capital rule in the ®rst period, tightening the requirement decreases these risks. If capital requirements are implemented in the second period, however, banks may increase asset risk in period 1. This is true because tightening the regulation has two e€ects. First, a tighter restriction lowers the expected pro®ts of the bank. If pro®ts are lower, the bank has less to lose in the event of bankruptcy. Therefore, increasing risk, and hence the probability of default, is less costly for the bank the stronger the restriction.

Second, changes in the regulation a€ect the marginal return on risk. In the present model this marginal return on risk may be raised and therefore may reinforce the ®rst e€ect, which leads to an overall increase in risk. The reason is the fact that under a binding regulation equity tomorrow is more valuable to the bank. In a regime of binding capital requirements the amount that can be invested in the risky but pro®table asset is restricted to a multiple of the value

1_{See also the special issue of theJournal of Banking and Financeabout `The Role of Capital in}

of equity. This implies that an additional unit of equity leads to an additional investment larger than one unit in the risky asset. Due to this `leverage e€ect' equity is more valuable to a regulated bank. A bank facing binding capital rules therefore has a higher incentive to increase equity tomorrow. However, if a bank ®nds it prohibitively costly to raise additional equity in the capital market or is completely unable to do so, the only way to increase the amount of equity tomorrow is to increase risk today.2

2. The model

The model has the following simple time structure. At timetˆ0 a bank can invest its available funds. After one period at timetˆ1 returns are realized. If the bank does not default, another investment can be undertaken. Again after one period the ®nal returns are realized at time tˆ2, and all parties are compensated.

I assume that bank managers are risk neutral and act perfectly in the interest of shareholders. This implies that they maximize the expected value of equity. The bank is ®nanced by equity and deposits. While the initial stock of equity W0is exogenously given, the supply of depositsD0attˆ0 can be chosen by the bank. After one period, the bank has to pay the costs ofC…D0†(withC0;C00>0 and C…0† ˆ0). This cost function can be thought of as stemming from an incomplete competition framework. If banks are horizontally di€erentiated, they each enjoy a local monopoly. If they want to attract more deposits, they have to raise interest rates to capture a greater market share. Doing so the bank not only incurs the costs of these marginal deposits, but also raises the costs on all infra-marginal deposits. Hence the costs of deposits are rising at an in-creasing rate.

All depositors are protected by deposit insurance. Since deposits are fully insured, depositors always get their money back, no matter how risky the bank. Therefore, their demand of deposits is independent of the bank's riskiness. The assumption of universal risk neutrality enables us to separate risk e€ects due to risk choice from risk e€ects due to risk aversion of depositors, managers, and regulators.

Attˆ0 the bank has two investment opportunities: A safe asset with (gross) rate of returnRf P1 and a risky portfolio. The risk±return structure of the

2_{This leverage e€ect due to the rigid link between equity and the volume of loans induced by}

capital adequacy rules may also amplify macroeconomic ¯uctuations. If negative shocks to aggregate demand reduce the ability of ®rms to service their debts to banks, this reduction in debt service lowers bank equity. Because of the capital requirements, this in turn reduces bank lending and industry investment. See Blum and Hellwig (1995).

portfolio can be in¯uenced by the bank. In accordance with ®nance theory, at least in a certain range there is supposed to be a positive trade-o€ between risk and expected return. To avoid corner solutions with in®nite risk, I assume that after some point a further increase in risk leads to a decrease in expected re-turn. Speci®cally, I assume the following two-point distribution of the gross rate of returnR~;with the lower realization normalized to zero:

~

RˆX with probability p…X†;

~

Rˆ0 with probability 1ÿp…X†;

forXPRf;withp0…X†<0;p00…X†60 andp…Rf† ˆ1:These assumptions imply thatE‰R~jXŠ ˆp…X†X is strictly concave. In order for the expected return to be increasing inX atRf;I further assume thatp0…Rf†>ÿ1=Rf:The unique level of risk that maximizes expected return is denoted byX_. 3

Obviously, the safe asset is (weakly) dominated by the risky portfolio. Thus, since all the funds are invested in the risky portfolio, the probability of default is 1ÿp…X†for every givenX.

If the available funds attˆ1 are not sucient to cover the costsC…D0†, the bank defaults and all the available funds (if any) are transferred to the deposit insurance fund which pays depositors their contracted returns. Due to limited liability bank owners cannot be forced to pay any additional amount to cover unful®lled claims.

At timetˆ1 the model has the same structure as attˆ0. While the costs of deposits D1 are again C…D1† (with C0;C00>0 and C…0† ˆ0), the amount of equityW1 is determined by the initial amount W0 at tˆ0 plus the pro®ts or losses of the ®rst period. For convenience, however, I only consider a reduced form of a risky asset attˆ1. To simplify the exposition, I abstract from the uncertainty in the second period and replace the random variable in period 2 with its expected valueR>Rf. The model could easily be generalized to allow

for richer structures in period 2. While possibly more realistic, a true replica-tion of the ®rst-period structure in period 2, however, would render the model analytically intractable. The main consequence of the present approach is that the incentive for asset substitution in period 2 is neglected. But since we are interested in the choice of risk attˆ0, doing so does not qualitatively a€ect the results of the paper.

3_{The assumptions on the convexity of the cost function of deposits and the concavity of the}

return distribution are very similar to Gennotte and Pyle (1991). They also need two curvature assumptions to enable comparative statics and to guarantee interior solutions. They also make an assumption about the concavity of the return distribution. But since they have a completely elastic supply of deposits at the risk-free rate, they introduce a `cost function of loan initiation' which is increasing and convex in the level of investment and in their risk index.

2.1. First best

Absent any bankruptcy costs, a risk-neutral social planner chooses that level of risk that maximizes expected returns,X_{, i.e.,X satis®es}4

p0_{X†X‡p…X† ˆ0: …1†}

It is important to note that in this model it is socially ecient to incur a positive amount of risk. A perfectly safe investment policy, i.e.,X ˆRf;isnot

optimal. An increase in risk implies on the one hand a higher probability of default and on the other hand a higher return given no default. Up toX _the

net e€ect of such an increase in risk on the expected return is positive. By choosing a completely safe investment, these additional gains would be fore-gone.5

2.2. The unregulated bank

An unregulated bank solves max

X;D0;D1 p…X†f…

~

W1‡D1†RÿC…D1†g ‡ …1ÿp…X†† maxf0;ÿC…D0†g; whereW~1ˆX…W0‡D0† ÿC…D0†is the value of equity attˆ1 in case of suc-cess. Due to limited liability the value of equity in case of failure is zero and the bank cannot continue operation. Hence, the second term is zero. Inserting, we get

max

X;D0;D1 p…X†f‰X…W0‡D0† ÿC…D0† ‡D1Š

RÿC…D1†g: The optimality conditions are

p0_{X^†X^‡p…X^† ÿp}0_X^† C…D^0† ^ D0‡W0 ÿD^1RÿC…D^1† R…D^0‡W0† " # ˆ0; …2† ^ XÿC0_D^ 0† ˆ0; …3† RÿC0_D^ 1† ˆ0: …4†

The higher the (expected) pro®ts the bank makes by issuing deposits in period 2,D^1RÿC…D^1†, the lower the risk taken in period 1. The reason for this is the fact that in case of failure (and therefore bankruptcy) the bank not only has a

4_{The ®rst-best level of a variableZis denoted byZ. The (optimal) levels of an unregulated and} a regulated bank are denoted byZ^andZr_{;respectively.}

5_{In fact, most banking systems encourage banks taking risky positions. For example, in most}

industrialized countries the banking system as a whole does perform a positive amount of (risky) maturity transformation. See also Goodman and Santomero (1986).

payo€ of zero, but it also loses these pro®ts in period 2. If these future pro®ts, or `rents', are high, it is optimal for the bank to reduce risk to increase the probability of getting these rents at the expense of a lower value of equity at

tˆ1 in case of success.6_{If these rents are high enough, it is even possible that a} bank chooses a lower level of risk than ®rst best. In that case, capital adequacy requirements clearly would not be an appropriate instrument to remedy the situation. To avoid this possibility, I assume that the future rent is not too high, i.e.,RC …D^0†>D^1RÿC…D^1†. Given this assumption, attˆ0 the bank chooses a level of risk which is higher than ®rst best, X^ >X_{. This follows from a} comparison of Eqs. (1) and (2), and the concavity of the expected return function.

2.3. The regulated bank

In order to analyze the impact of capital rules on risk taking, I ®rst look at the case where only in the ®rst period a requirement is binding. Then I consider a bank that faces a requirement only in the second period. It turns out that the e€ect of capital adequacy rules on risk taking incentives can be totally di€erent in the two scenarios. While in the ®rst case capital adequacy rules lower the risk, in the second case they actually lead to anincreasein risk. Finally, the case where a bank is facing a binding requirement in both periods is discussed.

Basically, capital adequacy requirements have two e€ects on risk taking incentives. First, they in¯uence the marginal costs of taking risk. The tighter the requirements, the lower the pro®ts in case of success and the less a bank has to lose if it defaults. Since the marginal costs of taking risk are these pro®ts times the decrease in the probability of success, a higher requirement tends to increase risk.

Second, capital rules a€ect the marginal return of taking risk. Here the actual e€ect depends crucially on the regime we are in. If the regulation is only binding in the ®rst period, marginal returns of taking risk are reduced. In-creasing risk raises the rate of return in case of success. The gain from such an increase is proportional to the amount invested at this rate of return. Under a binding requirement the amount that can be invested is a multiple of the given value of equity. The tighter the regulation, the lower this multiple. Therefore, a stricter regulation today tends to reduce risk.

In contrast, a binding regulation in the second period increases the marginal return of taking risk in the ®rst period. An additional unit of equity in period 2 allows additional investment in the risky but pro®table asset of a multiple of this unit, as opposed to a one-for-one relationship in the unregulated case.

6_{See also Boot and Greenbaum (1993). There, banks have a lower incentive to monitor projects}

While the amount of equity today is exogenously given, the amount of equity tomorrow is not ®xed, but can actually be in¯uenced through the investment decision today. By increasing risk today the bank has a higher amount of equity available tomorrow in case of success. Therefore, the introduction of a capital requirement for tomorrow induces a higher risk today.

2.3.1. Binding requirement in the ®rst period

According to the Basel Accord of 1988 the value of equity has to be at least as high as a given fraction of all risk-weighted assets. 7 _{In this model a} so-called `Cooke ratio'c0in the ®rst period implies that a bank can invest no more thanW0=c0k0W0in the risky portfolio.8If the capital adequacy requirement is binding, as much as possible is invested in the risky asset,k0W0: For this, W0…k0ÿ1†deposits are necessary. Beyond that the bank can issue further de-positsD~0, as long as these funds are invested in the safe asset. The total amount of deposits is therefore Dr 0ˆmaxf…k0ÿ1†W0;C0ÿ1…Rf†g6D^0 …5† or Dr 0ˆ …k0ÿ1†W0‡D~0:

The problem the bank has to solve is max

X;D~0;D1

p…X†‰…W~1‡D1†RÿC…D1†Š; D~0P0;

where againW~1 is the value of equity attˆ1 in case of success, ~

W1ˆk0W0X ‡D~0Rf ÿC…Dr0†: The ®rst-order conditions are

p0_X†f…W~ 1‡D1†RÿC…D1†g ‡p…X†k0W0Rˆ0; …6† RÿC0_{D1† ˆ0; Rf ÿC}0_Dr 0†60; ~ D0‰RfÿC0…Dr0†Š ˆ0; D~0P0:

The ®rst term in Eq. (6) represents the marginal cost of increasing risk. The expression in curly brackets is the pro®t of the bank in case of success. In-creasing risk slightly lowers the probability of success byp0_{X†;and therefore} lowers the expected pro®t. The second term is the marginal return on risk. A

7_{For more details on the Basel Accord, see Dewatripont and Tirole (1995).}

8_{Note that these capital rules do not imply a maximum debt-to-equity ratio. Rather the leverage}

is indirectly determined through the banks' behavior. Also observe that in contrast to most models in the literature (and in accordance with reality) the capital requirements are not necessarily binding.

higher risk implies a higher rate of return on the total amount of funds invested in the risky asset attˆ0; k0W0. This higher return in case of success in turn translates into a higher amount of equity attˆ1, which can be reinvested in the risky asset with returnR. Hence, increasing risk by one unit increases the pro®ts given no default byk0W0R. At the optimal level of risk, marginal costs are equal to the marginal return.

We now want to determine the impact of an increase in the capital re-quirement on the risk taking behavior of the regulated bank. In order to do this, totally di€erentiating Eq. (6) and rearranging we get

dX dk0ˆ ÿ p0_X†W 0R X ÿC0…Dr0† ‡p…X†W0R p00_X†f…W~_{1‡D1†RÿC…D1†g ‡2p}0_X†k0W0R:

While the denominator is clearly negative, the numerator is not obviously of a given sign. This ambiguity re¯ects the two opposing e€ects of a tightening of the capital rule. The ®rst e€ect is the in¯uence of the Cooke ratio on the marginal cost of risk,p0_X†W

0R X ÿC0…Dr0†

. Since a regulated bank faces a binding restriction, the highest possible payo€ is lower than the maximal payo€ of an unregulated bank. The tighter the restriction, the lower this payo€, and increasing the probability of default by increasing risk becomes less costly, since a lower payo€ implies that a bank has less to lose in the event of default. The second e€ect is the impact of a changing requirement on the marginal return on risk,p…X†W0R. A tighter restriction reduces the amount that can be invested in the risky asset. Since the return from taking risk is proportional to this amount, a higher Cooke ratio reduces the marginal return from taking risk.

At the point where the capital regulation just becomes binding, by de®nition the risk of a regulated bank and the risk of an unregulated bank are exactly the same. From Eq. (3) we also know that at that pointX ˆC0_Dr

0†. Therefore, if the capital rule just becomes binding, a further tightening reduces the level of risk, dX=dk0>0. It turns out that the return e€ect always dominates the cost e€ect, i.e., increasing the requirement always reduces the level of risk.

Proposition 1.If the bank faces a binding capital adequacy requirement in the ®rst period, an increase in the requirement reduces the level of risk,dX=dc0<0:

The proof is in the appendix.

2.3.2. Binding requirement in the second period

If in contrast to Section 2.3.1 a bank faces a binding Cooke ratioc1 in the second period, the amount of funds that can be invested in the risky asset at

tˆ1 is restricted to W1=c1k1W1: Again, in addition to this the bank can invest as much money as desired in the safe asset,D~1:

The bank maximizes max

X;D0;D~1

p…X†‰k1W~1R‡D~1Rf ÿC…Dr1†Š; D~1P0;

where againW~1 is the value of equity attˆ1 in case of success, ~

W1ˆ …W0‡D0†X ÿC…D0†;

and the total amount of deposits in period 2 is given by Dr

1ˆ …k1ÿ1†W~1‡D~1: The ®rst-order conditions are

p0_X†fk1W~_1R‡D~_{1Rf ÿC…D}r 1†g ‡p…X†…W0‡D0†fk1Rÿ …k1ÿ1†C0…Dr1†g ˆ0; …7† X ÿC0_D 0† ˆ0; Rf ÿC0…Dr1†60; ~ D1‰RfÿC0…Dr1†Š ˆ0; D~1P0:

As before, the ®rst term in Eq. (7) represents the marginal cost of increasing risk and the second term the marginal return on risk. While the interpretation of the cost term is the same as in the previous section, the marginal return on risk is now di€erent. The key insight of the following argument is the fact that under a binding capital requirement themarginal value of equity is higher than in the absence of any regulation. Without regulation, an additional unit of equity attˆ1 is invested in the risky asset and generates an additional return of R. If a bank is facing a binding capital rule, an additional unit of equity allows an investment ofk1in the risky asset and generates a return ofk1R. Since k1>1, the bank has to increase the supply of deposits by…k1ÿ1†to ®nance the di€erence between the possible investment of sizek1 and the available unit of equity. These deposits cost…k1ÿ1†C0…Dr1†. SinceRPC0…Dr1†if the requirement is binding, the marginal value of equity at tˆ1 is higher in a regime of a binding capital rule:

k1Rÿ …k1ÿ1†C0…Dr1†PR:

To analyze the impact of an increase in the capital requirement on the risk taking behavior of the regulated bank, we totally di€erentiate and rearrange Eq. (7) to get dX dk1ˆ ÿ p 0_X†W~ 1 RÿC0…Dr1† n o ‡p…X†…D0‡W0† RÿC0…Dr1† n ÿH1 o h i p00_{X† k} 1W~1R‡D~1Rf ÿC…Dr1† n o ‡2p0_X†…D 0‡W0† h . k1Rÿ …k1ÿ1†C0…Dr1† n o ÿp…X†H2 i ; …8†

where H1ˆ …k1ÿ1†W~1C00…Dr1† if D~1ˆ0; H1ˆ0 if D~1>0 and H2ˆ …k1ÿ1†2…D0‡W0†2C00…Dr1† if D~1ˆ0; H2ˆ0 if D~1>0:

As before, the denominator is negative. But now it is possible that the nu-merator is also negative, i.e., it is possible that riskincreases if the capital re-quirement is tightened, dX=dk1<0. While as before a decrease ink1lowers the marginal cost of taking risk, it may also raise the marginal return on risk. As long as the marginal return increases or as long as it decreases less than the marginal cost falls, the level of risk will increase.9

At the point where the capital requirement just becomes binding, we know from Eq. (4) that RˆC0_Dr

1†: At that point, the numerator of Eq. (8) is therefore ÿp…X†…D0‡W0†H1<0; and risk unambiguously increases if the regulation is tightened. Ifc1is raised further, risk will still increase. Eventually, the sign of Eq. (8) will change and risk will fall again10_{. However, risk can}

never be reduced below the unregulated levelX^:This can be seen if we consider the tightest possible regulation,k1ˆ1:Comparing the ®rst-order condition

p0_Xr_†Xr_‡p…Xr_{† ÿp}0_Xr_† C…Dr0† Dr 0‡W0ÿ Dr 1Rf ÿC…Dr1† R…Dr 0‡W0† " # ˆ0 …9†

with Eq. (2), it is clear thatXr_>X^_{;since D}r

1Rf ÿC…Dr1†<D^1RÿC…D^1†:The relationship between the Cooke ratioc1 and the level of riskXris summarized in the following proposition (see also Fig. 1).

Proposition 2.When the capital requirement in the second period ®rst becomes binding atc1;tightening the requirement raises the level of risk,dX=dc1>0.If

the requirement is further increased, risk eventually falls again, but never below the level of an unregulated bank,Xr_{>X for all c}^

1>c1.

9_{Within an options framework, Flannery (1989) reaches a similar conclusion. He states that `an}

increase in the maximum permissible leverage [M] reduces portfolio risk. This occurs because each dollar of equity is more valuable the higher isM' (p. 249). In his model this is true because an increase in risk has two opposing e€ects. On the one hand it increases the option value of equity. On the other hand it reduces (expected) permissible leverage, which is smaller thanM. IfMis reduced, the former e€ect becomes more important and risk is increased.

10_{Using Eq. (9), it is straightforward to show that the denominator of Eq. (8) is positive at}

It is interesting to note that even if a bank only expects a future capital requirement to increase, this may lead to perverse e€ects on the risk choice today. In that case a future regulation, aimed at making the banking sector safer tomorrow, will actually make banking riskier today.

2.3.3. Binding requirement in both periods

If a bank is both subject to a binding requirement today and tomorrow, the net e€ect on risk is ambiguous. For instance, if a uniform capital requirement today and tomorrow is introduced, it is not clear whether the allocation is actually improved or if it is made worse.

Denoting the binding capital requirement in both periods by k, the bank's problem becomes

max X;D~0;D~1

p…X†‰kW~1R‡D~1Rf ÿC…Dr1†Š; D~0;D~1P0; whereDr

0 andDr1 are de®ned as before, andW~1ˆkW0X‡D~0RfÿC…Dr0†. The ®rst-order condition regarding the level of risk is now

p0_X†‰kW~

1R‡D~1Rf ÿC…Dr1†Š ‡p…X†kW0‰kRÿ …kÿ1†C0…Dr1†Š ˆ0: …10† To derive the e€ect of a change in the regulation on the level of risk, we again totally di€erentiate Eq. (10) and get

dX dk ˆ ÿp 0_{X† RÿC}0_Dr 1† ~ W1‡kW0…X ÿC0…Dr0†† n o ‡C0_Dr 1†W0…X ÿC0…Dr0†† h i D ÿp…X†W0‰2kRÿ …2kÿ1†C0…Dr1† ÿH3Š D ; …11†

whereD<0 is the left-hand side of Eq. (10) di€erentiated with respect toX, and

H3ˆk…kÿ1†C00…Dr1†fW~1‡ …kÿ1†W0…XÿC0…Dr0††g if D~1ˆ0;

H3ˆ0 if D~1>0:

The numerator of Eq. (11) is not necessarily of a single sign, i.e., the relation between the Cooke ratio and the level of risk need not be monotonic. As in the previously considered cases, the marginal cost e€ect always induces a higher level of risk. Whether the marginal return e€ect leads to an increase in risk depends on the parameters of the model. Interestingly, the less stringent the regulation, i.e., the lower the Cooke ratio, the more likely it is that a tightening of the regulation increases risk. This is true because the `leverage e€ect' 1=cis higher the lowerc. One unit of equity allows an investment of 1=cunits in the risky asset. The higher the leverage, the more valuable an additional unit of equity and, therefore, incurring a higher level of risk is more pro®table for the bank. A similar reasoning leads to the observation that risk is more likely to be increased the higher the initial amount of equityW0. Of course, if the amount of equity is very high, the regulation is not binding and the above reasoning does not apply. The problem of perverse incentive e€ects, therefore, is most pro-nounced for banks for which the requirement is just binding in period 2.

Finally, a tightening of the regulation is more likely to lead to an increase in risk the more convex the cost of deposits functionC…:†. A binding requirement reduces the amount of deposits a bank can issue, and hence lowers the mar-ginal costs of deposits. This means that issuing more deposits becomes more attractive to the bank. And since a bank may only issue more deposits if it has more equity available, it has a higher incentive to raise the level of risk in order to have a better equity position in case of success. The more convex the cost function, the stronger this incentive. Again, this e€ect is most pronounced for banks for which the requirement is just binding in the second period.

Eq. (11) can be somewhat simpli®ed, if we consider that level ofk, where the regulation is strictly binding in period one and just binding in period two.11_In that caseRˆC0_Dr

1†and Eq. (11) becomes

11_{The converse case is not possible since the level of equity is always higher in the second period}

dX dk ˆ ÿ

p0_X†RW

0…X ÿC0…Dr0†† ‡p…X†W0‰RÿH3Š

D :

Since the ®rst term in the numerator is negative, a sucient condition for risk to increase if the regulation is tightened is R<H3. As discussed above, this condition is more likely to be satis®ed the more convex the cost functionC…:†, the higherk, and the higher the initial amount of equityW0.

If the regulation is tightened further, risk will eventually decrease. Even a level of risk lower than ®rst best can be attained. To see this, we inspect the ®rst-order condition with respect to risk ifkˆ1:

p0_Xr_†Xr_‡p…Xr_{† ‡p}0_Xr_† R…D~0Rf ÿC…D~0†† ‡D~1Rf ÿC…D~1†

RW0

" #

ˆ0:

Since the expression in square brackets is positive and the expected return function is concave, it follows that the level of risk with maximum regulation is lower than ®rst best,Xr_<X.

To summarize, if a bank is subject to a capital regulation in both periods, risk initially falls as the requirement c is raised from a level where it is just binding in the ®rst period,c. As soon as the regulation also binds in the second period at ~c, a further tightening may raise the level of risk, even above the unregulated level. Eventually, risk will fall until it reaches a level lower than ®rst best at the maximal regulation. One possible example of the relationship between the level of risk and the Cooke ratiocis illustrated in Fig. 2.

3. Discussion

This paper has shown that in a dynamic model with incentives for asset substitution, capital adequacy requirements may actually increase risks. If regulators are mainly concerned about reducing the insolvency risk of banks, introducing capital rules, therefore, may not be such a good idea after all.12 One of the e€ects of such a regulation is the reduction of a bank's pro®ts. If future pro®ts are lower, a bank has a smaller incentive to avoid default. In addition to this, the `leverage e€ect' of capital rules raises the value of equity to the bank. For every dollar of equity, more than one dollar can be invested in the pro®table but risky asset. In order to raise the amount of equity tomorrow it may be optimal for a bank to increase risk today.

While the e€ects illustrated in this paper are potentially important, it has to be kept in mind that the present model is only an example rather than a general theory. For instance, one could analyze more general return distribution. Unfortunately, due to the vagueness of the concept of `risk', the generalization of the model to arbitrary return distributions is not possible. In order to be able to make meaningful statements, one necessarily has to restrict the class of distributions under consideration. The distribution used in this model has two important properties: (i) an increase in risk leads to a higher probability of default, and (ii) the conditional expected return given no default rises as risk is increased. Whatever de®nition of risk one wants to use, as long as these two properties are satis®ed, the results of this paper hold. For instance, one class of distributions that satis®es these conditions are distributions with the same mean that can be ordered according to the concept of `increasing risk' by Rothschild and Stiglitz (1970).13

Another consequence of the assumed two-point distribution is that changes in the amount of equity and deposits do not a€ect the default probability di-rectly. If we consider continuous return distributions, increasing the amount of equity, e.g., lowers the probability of bankruptcy ceteris paribus. While the calculations would be greatly complicated through the inclusion of those marginal e€ects, no additional insight would be gained. The qualitative results would still hold.

The results of this paper are a reminder that one has to be careful when assessing the e€ectiveness of capital adequacy rules. The present model illus-trates that by arguing in a purely static framework, important dynamic e€ects

12_{Alternatively, if regulation is to be implemented, it should be done quickly in order to reduce}

the scope of risk taking in the interim phase.

13_{Actually, it is possible that a so-called `mean-preserving spread' does neither increase the}

probability of default nor the expected return given no default. However, such an increase in risk can never decrease these two measures.

of the kind described here may be neglected. It is quite possible that the actual e€ects of the regulation are contrary to the ones intended.

4. Appendix A. Proof of Proposition 1

Rearranging Eq. (6) p0_{X†X‡p…X† ÿp}0_X† C…Dr0† ÿD~0Rf k0W0 ÿ D1RÿC…D1† Rk0W0 " # ˆ0

and denoting the expression in square brackets byAwe get p0_X†X‡p…X†

ÿp0_X†Aˆ0: Total di€erentiation yields

dX dAˆ

p0_X†

p00_{X†…XÿA† ‡2p}0_X†:

Because ofp0_{:†<0 andp}00_{:†60, this is positive ifXPA. To verify whether} XPAis satis®ed, simply note that in case of no default a bank always makes positive pro®t,Xk0W0‡D~0RfÿC…Dr0†>0. Therefore, risk falls ifAdecreases.

In the case of D~0ˆ0 we have

oA ok0ˆ o ok0 C…Dr 0† k0W0 ÿ D1RÿC…D1† Rk0W0 ! ˆC0…W…k0ÿ1††k0_kW₂0ÿC…W0…k0ÿ1†† 0W0 ‡ D1RÿC…D1† Rk2 0W0 >0; sinceC0_W 0…k0ÿ1††k0W0>C0…W0…k0ÿ1††…k0ÿ1†W0>C…W0…k0ÿ1††.

In the case ofD~0>0; Dr0ˆD~0‡W0…k0ÿ1†is constant (sinceC0…Dr0† ˆRf)

and oA ok0ˆ o ok0 C…Dr 0† ÿD~0Rf k0W0 ÿ D1RÿC…D1† Rk0W0 ! ˆRfk0W02ÿ …C…Dr0† ÿRf…Dr0ÿW0…k0ÿ1†††W0 k2 0W02 ‡ D1RÿC…D1† Rk2 0W0 : Since the last term is positive, this expression is certainly positive if

Rearranging, we get

Rf…Dr0‡W0† ÿC…Dr0†>0: SinceRf ˆC0…Dr0†, we can write this as

Dr

0C0…Dr0† ‡RfW0ÿC…Dr0†>0:

Because of the strict convexity of C…:† and C…0† ˆ0, this inequality always holds.

To summarize, Aincreases if k0 increases for all D~0P0. Therefore, since k0 ˆ1=c0, risk decreases ifc0 increases.

Acknowledgements

Without implicating them, I would like to thank Thomas Gehrig, Niklaus Muller, Georg Noldeke, Mark de Snaijer, Markus Staub, seminar participants at the University of Basel, and an anonymous referee for helpful comments and discussions. Especially Martin Hellwig deserves credit for his continued en-couragement and support of this work. Financial support of the WWZ Forderverein at the University of Basel and the Schweizerischer Nationalfonds is gratefully acknowledged.

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