Bank Competition, Stability, and Capital Regulation

Transcription

1 Bank Competition, Stability, and Capital Regulation Hendrik Hakenes Max Planck Institute, Bonn Isabel Schnabel University of Mainz, MPI Bonn, and CEPR August 24, 2007 Abstract: We analyze the effects of capital regulation on bank stability in a model where banks compete for loans and deposits, and where banks and firms are subject to a risk-shifting problem. We show that the risk-taking of banks and firms are complementary to each other; higher risk-taking by firms gives banks an incentive to choose higher risks. Moreover, we show that tighter capital requirements increase the risk of individual loans and may also increase a bank s probability of default because they relax the competition for loans. Due to these competitive effects, capital regulation may destabilize the banking sector. Keywords: Bank competition; loan market competition; capital regulation; riskshifting; banking stability. JEL-Classification: G21, G28. D43. We thank John Boyd, Gianni De Nicolò, Robert Hauswald, and Martin Hellwig for helpful suggestions. We also thank the participants of the Meeting of the European Finance Association in Ljubljana, as well as seminar participants at the Max Planck Institute in Bonn for comments. Financial support from Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged. Max Planck Institute for Research on Collective Goods, Kurt Schumacher Str. 10, Bonn, Germany. hakenes@coll.mpg.de Johannes Gutenberg-Universität Mainz, Mainz, Germany. isabel.schnabel@unimainz.de

2 I. Introduction The question of how competition affects banking stability has attracted a lot attention in recent years. It is a widely held view that there is a trade-off between competition and stability in banking. The argument goes that competition erodes banks profit margins and hence charter values, which in turn increases risk-taking incentives because banks have less to lose in an insolvency (see, e. g., Keeley, 1990). In an important paper, Boyd and De Nicolò (2005) have shown that this trade-off is not robust to the introduction of loan market competition. In their model, higher competition induces banks to lower their loan rates, which mitigates the borrowers moral hazard problem, and hence their risk-taking. Under this view, competition increases banking stability. 1 The differing results are driven by the way that banks risk-taking is modeled. In the first type of models, banks hold a portfolio of projects and directly determine the riskiness of these projects; given limited liability and deposit insurance, banks are subject to a moral hazard problem in choosing their risk (see, e. g., Keeley, 1990; Hellmann, Murdock, and Stiglitz, 2000; Allen and Gale, 2004; Repullo, 2004). In Boyd and De Nicolò s words, banks solve a portfolio problem. Boyd and De Nicolò (2005), on the other hand, assume that banks extend loans to entrepreneurs who themselves determine the risk of their projects. Given limited liability, the entrepreneurs are subject to a moral hazard problem, just as in the classical model by Stiglitz and Weiss (1981). In this case, banks solve an optimal contracting problem (see also the recent contributions by Boyd, De Nicolò, and Jalal, 2006; Martínez and Repullo, 2007). Our paper analyzes what the described results imply for the effectiveness of capital regulation in increasing the stability of the banking sector. For this purpose we develop a model that encompasses both approaches described above by allowing for both a portfolio and an optimal contracting problem. The optimal contracting problem looks exactly as in the paper by Boyd and De Nicolò (2005). We modify that model by adding a portfolio problem on the side of banks. More specifically, we allow the bank to choose the correlation of its loans. We make the simplest possible assumptions about this correlation choice, which allows us to obtain closed-form solutions for all of our results. 2 In addition, we introduce costly bank equity and 1 See, however, the paper by Martínez and Repullo (2007) who argue that competition can destabilize banks even in the presence of loan market competition because it reduces their capital buffers. 2 Boyd, De Nicolò, and Jalal (2006) also add a portfolio problem by allowing the bank to invest in a safe and a risky asset. 2

3 capital requirements. In a first step, we analyze the effect of higher competition on banks riskiness and characterize the conditions under which one obtains a trade-off between competition and stability. In a second step, we study the impact of stricter capital requirements on a bank s riskiness. In both parts, we consider the effect on the riskiness of individual loans, on the bank s correlation choice, and finally on the bank s probability of default. Our model yields a number of interesting results: As in Boyd and De Nicolò (2005), higher competition decreases the riskiness of individual loans. In contrast, the effect on the bank s correlation choice is ambiguous: In addition to the standard charter value effect described by Keeley (1990) and others, there is a countervailing effect that tends to mitigate a bank s risk-taking in the face of fiercer competition. The reason is that gambling is worthwhile for the bank only if the risk chosen by entrepreneurs is sufficiently large. In this sense, a bank s and its entrepreneurs risk-taking are complementary to each other. Therefore, we call this the risk complementarity effect. The overall impact on a bank s probability of default is ambiguous. We then show that capital regulation affects banks risk by changing the intensity of competition in the banking sector. Stricter capital requirements attenuate the competition for loans, implying higher loan rates, and hence higher risk-taking on the side of firms. Therefore, stricter regulation increases the probability of default of a single loan. Due to the risk complementarity described above, stricter capital requirements may also induce the bank to choose a higher correlation of loans. Because of these competitive effects, stricter capital requirements can lead to a higher probability of default of banks. Generally, conditions that yield a risk-decreasing effect of competition tend to yield a risk-increasing effect of capital regulation. Hence, if there is a trade-off between competition and stability, then capital regulation tends to have a positive effect on bank stability. If, however, an increase in competition stabilizes the banking system, then capital regulation tends to have perverse effects, and stricter capital requirements tend to increase the probability of bank failure. The paper proceeds as follows: In the following section, we describe the model setup. In Section III, we derive the equilibrium. Section IV deals with the effect of competition on banks risk-taking, whereas Section V analyzes the effect of capital regulation. Section VII concludes. 3

4 II. Model Setup Our setup follows closely the model presented by Boyd and De Nicolò (2005). Consider an economy with three dates, 0, 1, and 2. There are three types of agents: entrepreneurs, depositors, and banks. All agents are risk neutral. Entrepreneurs There is a continuum of entrepreneurs who have no own resources, but have access to risky projects of fixed size, normalized to 1. Projects have constant returns to scale and yield, per invested unit, S with probability p(s), and 0 otherwise. p(s) satisfies the following conditions: p(0) = 1, p( S) = 0 for some S >0, p (S) < 0, and p (S) 0 for all S [0, S]. Hence, as in the paper by Boyd and De Nicolò (2005), p(s) S is a strictly concave function of S that reaches a maximum at S,wherep (S ) S + p(s ) = 0. The entrepreneurs (date 1) choice of S is unobservable by the bank and the depositors. At date 2, the bank can only observe and verify at no cost whether the project has been successful or unsuccessful. By assumption, financing contracts are simple debt contracts. The aggregate demand for loans is represented by a downward-sloping inverse demand curve, denoted by r L ( L), where L denotes the aggregate loan volume in the economy. r L ( L) satisfies r L (0) > 0, r L ( L) < 0, and r L ( L) 0. Depositors The aggregate supply of deposits is represented by an upward-sloping inverse supply curve, denoted by r D ( D), where D denotes the aggregate deposit volume in the economy. Analogously to the loan market, r D ( D) satisfies r D (0) 0, r D ( D) > 0, and r D ( D) 0. Moreover, we assume that r D (0) <r L (0), which ensures the existence of an equilibrium. Deposits are insured, so that the deposit supply does not depend on the bank s risk. Banks pay a strictly positive, flat deposit insurance premium, α. Banks There are N banks with large initial resources. The volume of loans extended by bank j is denoted by L j. Loans are financed by deposits, D j, and equity, E j, hence L j = E j + D j. Aggregate deposits in the banking sector are equal to D = N j=1 D j, aggregate loans are L = N j=1 L j. Banks compete for deposits and loans in a Cournot fashion. In setting loan rates, banks take into account the best responses of entrepreneurs to their choice of the loan rate. Bankers demand an expected return of r E for providing equity. Equity is 4

5 expensive, i. e., r E >p(0) r L (0). 3 Hence, equity finance would be inefficient in the absence of a moral hazard problem, but it can help to prevent excessive risk-taking by banks. 4 Assume that the regulator imposes a minimum capital requirement, i. e., E j βl j,whereβ is the required capital ratio (e. g., 8%). 5 The introduction of capital requirements is one major deviation from the setup introduced by Boyd and De Nicolò (2005). The other major innovation concerns the bank s asset side: We assume that a bank can influence the correlation of its loans, ρ j [0, 1]. In contrast, Boyd and De Nicolò assume that the entrepreneurs projects are perfectly correlated. Allowing for a choice of correlation, we add a portfolio problem to the optimal contracting problem analyzed by Boyd and De Nicolò. Hence, our approach encompasses the models containing only a portfolio problem (as in Keeley, 1990; Allen and Gale, 2004), and those considering only an optimal contracting problem (as in Boyd and De Nicolò). More specifically, we assume that the projects financed by a bank have some natural correlation ρ 0. However, the bank can exert effort to increase or decrease the correlation. In doing so, the bank incurs a (non-monetary) cost C j,whichis proportional to the size of its portfolio, i. e., C j = C(L j,ρ j )=L j c(ρ j ). The cost function c(ρ j ) satisfies the following conditions: c(ρ 0 )=0andc (ρ 0 ) = 0 for some ρ 0 [0, 1], and c (ρ j ) > 0 for all ρ j [0, 1]. Hence, the cost function is a strictly convex function with a minimum at ρ 0 ; this implies that any deviation from the natural correlation is costly. We make the following simplifying assumption about the bank s correlation choice: For a given choice of ρ, all projects are perfectly correlated with probability ρ, and perfectly uncorrelated with probability 1 ρ. In both cases, the expected portfolio payoff of the projects is equal to p(s) S. But the probability of default is equal to 1 p(s) in the first case, and (due to the law of large numbers) zero in the second case. Hence, by setting ρ, the bank directly influences its default probability. Note that ρ has a natural interpretation as the correlation between any two projects in the portfolio: With probability ρ, the correlation between any two projects is 1, with probability 1 ρ, the correlation is 0. As a consequence, the ex-ante correlation between any two projects (and, hence, loans) is ρ 1+(1 ρ) 0=ρ. 3 This assumption is common in the literature (see, e. g., Hellmann, Murdock, and Stiglitz, 2000; Repullo, 2004). 4 In our model, the main purpose of capital regulation is to mitigate the banks moral hazard problem. Other rationales for capital regulation have been modelled by Morrison and White (2005). 5 Later we will also consider risk-adjusted capital requirements. 5

6 Figure 1: Time Structure t 0 Banks choose the volume of deposits, D j, and the volume of loans, L j Banks choose the correlation within their loan portfolios, ρ j t 1 Entrepreneurs choose the risk of their projects, S, andinvest t 2 Projects mature; repayment of deposits by banks or deposit insurance; banks are residual claimants Time Structure The time structure of the game is as follows (see Figure 1). At date 0, each bank j chooses the volume of deposits, D j, and the volume of loans, L j, subject to the constraint that it has to comply with the capital regulation, i. e., E j = L j D j βl j. The deposit rate is given by r D ( D) =r D ( j D j), and the loan rate by r L ( L) =r L ( j L j). Then, banks choose the correlation within their loan portfolios, ρ j, incurring costs c(ρ j ) for each unit of loans. At date 1, entrepreneurs invest the money from the bank in a project, choosing the risk parameter S. Finally, at date 2, the entrepreneurs projects mature. If a project is successful, the entrepreneur repays his loan (with interest) to the bank. If the aggregate return to the bank is sufficient, the bank repays its deposits, and retains the remaining returns. Otherwise, the deposit insurance enters into the bank s obligations, and the bank obtains nothing. III. Equilibrium We solve the model by backward induction. We will concentrate on symmetric equilibria, and we will drop the index j when there is no danger of confusion. At date 1, the entrepreneurs choose the project risk S in order to maximize expected profits for a given loan rate r L. An entrepreneur s expected return is p(s)(s r L ). The first-order condition is h(s) :=S + p(s) p (S) = r L. (1) Note that h(s) would be equal to 0 in the first-best solution. Since h (S) =2 p(s) p (S)/p (S) 2 > 0, S will be too high, compared to the first best, for any r L > 0. An increase in the loan rate induces an entrepreneur to choose higher risk (S) and moves him further away from the first-best solution. (1) defines an implicit function S(r L )=h 1 (r L ), with S (r L ) > 0. 6

7 Before the entrepreneurs choice of S, banks choose the correlation ρ j of the loans in their portfolio for given deposit, loan, and equity volumes. Bank j s expected profit is Π j = ρ j p(s) [ r L ( L) L j (r D ( D)+α) ] D j +(1 ρ j ) [ p(s) r L ( L) L j (r D ( D)+α) ] D j re E j L j c(ρ j ) = p(s) r L ( L) L j [ρ j p(s)+(1 ρ j )] (r D ( D)+α) D j r E E j L j c(ρ j ). (2) Here we make use of the fact that all firms will choose the same S in equilibrium. The first-order condition with respect to ρ j is Π j ρ j =(1 p(s)) (r D ( D)+α) D j L j c (ρ j )=0, (3) where S is a function of r L ( L) because banks anticipate the entrepreneurs risk choices. The first-best solution would have L j c (ρ j ) = 0, implying that ρ j = ρ 0. Since c (ρ j ) > 0, ρ j will be higher than the first best, ρ 0, for any p(s) < 1. Banks overspecialize due to limited liability and deposit insurance. Note that an increase in the deposit rate (or, alternatively, the deposit insurance premium) induces the bank to choose higher risk (ρ) and moves it further away from the first-best solution. (3) defines an implicit function ρ j ( D, D j, L, L j ). At the beginning of date 0, banks choose the profit-maximizing volumes of deposits, equity, and loans. Due to the balance sheet identity, L j = D j + E j ; hence, only two of the three variables can be chosen independently. Since equity is expensive, a profit-maximizing bank takes no more equity than is required by regulation. 6 As a consequence, we can express E j and D j as a function of L j : E j = βl j and D j = (1 β) L j. This implies that the bank s optimization problem from the viewpoint of date 0 can be written as a function of its loan volume, [ max L j p(s)rl ( L) (1 β) [ ρ j p(s)+(1 ρ j ) ] ( (r D (1 β) L) + α) βre c(ρ j ) ] L j s. t. S + p(s) p (S) = r L( L) ( and (1 p(s)) (r D (1 β) L) + α)(1 β) c (ρ j )=0. (4) The first restriction defines the entrepreneurs risk-taking, S, as a function of L; the second restriction defines the bank s correlation choice, ρ j, as a function of L. 6 In reality, capital requirements are often not binding. See Allen, Carletti, and Marquez (2006) for an analysis of this phenomenon. 7

8 IV. Competition and Bank Risk In this section, we analyze how an increase in competition affects the risk of banks in our setup. We first consider the effect of competition on the riskiness S of a single bank loan (i. e., on the entrepreneurs risk-taking), then the effect on a bank s correlation choice ρ, and finally on a bank s probability of default, PD = ρ (1 p(s)). First of all, note that, as N increases, the banking sector expands, i. e., the aggregate volume of loans L goes up. Lemma 1 Higher bank competition increases the size of the banking sector (i. e., the aggregate loan volume), d L/dN > 0. As a consequence, the loan rate r L ( L) decreases. Because D j =(1 β) L j, Lemma 1 implies that the aggregate deposit volume D also increases and the deposit rate r D ( D) increases. These results are typical for models à la Cournot; as new firms enter the market, the aggregate volume increases, and the price moves closer to the competitive outcome. According to (1), an decrease in the loan rate entails a decrease in the entrepreneurs risk-taking S. Proposition 1 Higher bank competition decreases the entrepreneurs risk-taking, ds/dn < 0. This is just a restatement of the main result by Boyd and De Nicolò. Higher competition for loans leads to a reduction in loan rates, which mitigates the entrepreneurs moral hazard problem. In the model by Boyd and De Nicolò, this result directly translates into lower bank risk. In our model, the effect of competition on bank risk also depends on the bank s correlation choice. The effect of competition on the bank s choice of ρ j can be derived from (3), which can be rewritten as ( ( ) (1 ) c (ρ) =(1 β) r D (1 β) L) + α p(s). (5) The bank trades off the marginal cost of a higher correlation (c (ρ)) against the marginal benefit, which is the saving (or externalization) of interest payments (including the deposit insurance premium) in the default state. An increase in bank competition affects the correlation choice of banks through the accompanying increase in the aggregate loan volume of the banking sector (see Lemma 1). Defining 8

9 Ψ:= ( r D ((1 β) L)+α )( 1 p(s( L)) ) and totally differentiating (5), we can derive the effect of competition on a bank s correlation choice, dρ dn = ρ L L N = 1 β Ψ L c (ρ) L N with Ψ L =(1 β) r D( D) ( 1 p(s) ) ( r D ( D)+α ) p (S) S (r L ) r L( L). (6) The sign of dρ/dn is identical to the sign of Ψ/ L. (6) shows that an increase in N affects a bank s correlation choice through two channels: First, it raises the deposit rate through its effect on the aggregate loan (and hence deposit) volume. As a result, the bank s margin shrinks, which induces the bank to choose a higher portfolio correlation ρ j, implying a higher portfolio risk. This effect is the standard charter value effect found in the literature on the tradeoff between competition and stability (as formulated, for example, by Allen and Gale, 2004). Second, it lowers the loan rate and, hence, the entrepreneurs risk-taking (Proposition 1), which translates into a lower default probability of the bank s loans. This makes gambling less attractive for the bank because default occurs less frequently. In this sense, the bank s and the entrepreneurs risk-taking are complementary to each other. Therefore, we call this effect the risk complementarity effect. The second effect reinforces the risk-decreasing effect of competition described by Boyd and De Nicolò. An increase in competition reduces the entrepreneurs risk-taking, which, in turn, dampens the bank s incentive to choose a high portfolio correlation. From (6), we can derive the effect of competition on a bank s correlation choice, as characterized in the following proposition. Proposition 2 Higher bank competition increases a bank s portfolio correlation, dρ/dn > 0, ifandonlyif Ψ/ L >0, i.e.if (1 β) r D ( D) r D ( D)+α p (S) 1 p(s) S (r L ) r L( L). (7) Hence, an increase in the number of banks tends to entail a higher correlation of a bank s loans if the deposit rate reacts strongly to a change in aggregate deposits, if the entrepreneurs success probability reacts weakly to a change in risk, if the entrepreneurs risk-taking reacts weakly to a change in the loan rate, and if the loan rate reacts weakly to a change in the aggregate loan volume. For high β, the left-hand-side of (7) converges towards zero, and more competition always decreases a bank s correlation choice. The reason is that for high β, banks refinance mostly through equity, so the elasticity of the deposit supply becomes irrelevant for the bank s risk choice. 9

10 Finally, let us discuss how an increase in competition affects a bank s overall risk, as measured by the bank s default probability PD = ρ (1 p(s)). Abank spd depends on two factors: the riskiness of its loans (1 p(s)), and the correlation of its loans ρ. Taking the derivative with respect to N yields d PD dn = dρ dn (1 p(s)) ρp (S) ds dn. (8) From Propositions 1 and 2, we know that the sign of the first term depends on condition (7), whereas the subtrahend is always positive implying that the second term is negative. We get the following proposition. Proposition 3 Higher bank competition increases a bank s probability of default, d PD/dN > 0, if and only if dρ dn >ρ p (S) 1 p(s) ds dn. (9) The right-hand-side of (9) is strictly positive. This implies that an increase in the number of banks will decrease a bank s probability of default in all cases in which banks choose a less correlated portfolio in reaction to fiercer competition. In such situations, the effects of competition on the riskiness of single loans and on the bank s portfolio correlation go in the same direction. If banks increase their portfolio correlation as N increases, the overall effect on the bank s PD is ambiguous. V. Capital Regulation and Bank Risk In this section, we analyze how capital regulation affects banks risk-taking in our setup. As before, we will consider the effect of capital regulation on the riskiness of a single bank loan, on a bank s correlation choice, and on a bank s probability of default. We will see that, in our setup, capital regulation affects banks risk not only by aligning the interests of the bank and its creditors, but also through competition. Consider an increase in the capital requirement β. This makes banking less attractive because it raises capital costs, which induces banks to choose lower deposit and loan volumes. Lemma 1 Stricter capital requirements decrease the size of the banking sector (i. e., the aggregate loan volume), d L/dβ < 0. 10

11 The decrease in the aggregate loan volume translates into an increase in the loan rate r L ( L). The volume for deposits decreases even more strongly than the loan volume. In addition to the shrinking size of the banking sector, a lower share is financed through deposits. Hence, the deposit rate r D ( D) decreases as well. Due to the increase in the loan rate, the entrepreneurs will choose higher risk. Proposition 1 ds/dβ > 0. Stricter capital requirements increase the entrepreneurs risk-taking, Strikingly, a tightening of capital regulation increases the risk of individual loans because it attenuates the competition for loans. This straight-forward result would also follow in the setup described by Boyd and De Nicolò (2005) and does not depend on our modeling of the bank s portfolio problem. Lower competition for loans leads to an increase in loan rates, which exacerbates the entrepreneurs moral hazard problem. As before, the effect on overall bank risk depends also on the bank s correlation choice. As in the above section, the bank s portfolio choice is governed by (5), which can be written as c (ρ) =(1 β) Ψ, where Ψ is defined as above. Ψ depends on β through r D and through L. Totally differentiating (5), we can derive the effect of capital regulation on a bank s correlation choice: dρ dβ = 1 [ ( Ψ Ψ+(1 β) L c (ρ) L β + Ψ )], (10) β where Ψ/ L is as in (6), L/ β < 0 according to Lemma (1 ), and Ψ/ β = (1 p(s)) r D ( D) L <0. A tighter capital regulation affects a bank s correlation choice through several channels: First, a stricter capital regulation (an increase in β) forces the banker to hold a higher equity share. As a consequence, the banker has a higher stake in the bank and has, therefore, more to lose in an insolvency, which restrains the bank s risk-taking. This channel is captured by the first term in brackets in (10), which is unambiguously negative. The sign of the second term in brackets is determined by the sign of Ψ/ L, which depends on whether (7) holds. There are again two countervailing effects: On the one hand, an increase in β decreases the deposit rate through its effect on the aggregate deposit volume D. Through the standard charter value effect, this lowers the bank s incentives to choose a high portfolio correlation. Hence, this effect goes in the same direction as the first effect: A stricter capital regulation induces banks to choose a less correlated (and hence less risky) portfolio. However, there is a third channel, which goes in the opposite direction. An increase in β 11

12 increases the loan rate and thereby the entrepreneurs risk-taking (Proposition 1 ), which increases the entrepreneurs probability of default (1 p(s)) and makes gambling more attractive. This is again the risk complementarity effect described above. Finally, an increase in β lowers the deposit volume, and hence the deposit rate, even for a constant loan volume, which reinforces the charter value effect described above. From (10), we can derive the effect of capital regulation on a bank s correlation choice, as characterized in the following proposition. Proposition 2 Stricter capital requirements increase a bank s portfolio correlation, dρ/dβ > 0, if and only if ( Ψ (1 β) L L β + Ψ ) > Ψ. (11) β Remember that Ψ is strictly positive. Hence, if Ψ/ L >0 (thus if condition (7) is satisfied), stricter capital requirements decrease a bank s risk-taking because L/ β < 0. Hence, (7) is a sufficient condition for capital requirements to have a disciplining effect on a bank s choice of portfolio correlation. In contrast, Ψ/ L <0 (i. e., condition (7) is not satisfied) is a necessary condition for stricter capital requirements inducing a bank to increase its portfolio correlation. However, it is not sufficient. The effect of β working through the loan rate (the third channel described above) has to be strong enough to overcompensate not only the second, but also the first channel. Note the parallelism between the effect of competition on risk-taking and the impact of capital regulation. If fiercer competition increases the bank s risk-taking, then a stricter capital regulation reduces it. Formulated the other way around, if an increase in β induces the bank to choose a higher correlation, then increased competition makes the bank choose a smaller correlation. These results are summarized in the following corollary. Corollary 1 A positive effect of bank competition on a bank s correlation is a sufficient condition for a negative effect of capital requirements on the bank s correlation, dρ/dn > 0 = dρ/dβ < 0. A positive effect of capital requirements on a bank s correlation is a sufficient condition for a negative effect of bank competition on the bank s correlation, dρ/dβ >0 = dρ/dn <0. Finally, we discuss how a tighter capital regulation affects a bank s default probability, ρ (1 p(s)). Taking the derivative with respect to β yields d PD dβ = dρ dβ (1 p(s)) ρp (S) ds dβ. (12) 12

13 From Propositions 1 and 2, we know that the sign of the first term depends on condition (7), whereas the subtrahend is always negative implying that the second term is positive. We get the following proposition: Proposition 3 Stricter capital requirements increase a bank s probability of default, d PD/dβ > 0, if and only if dρ dβ >ρ p (S) ds 1 p(s) dβ. (13) The proposition shows that tighter capital regulation may render a bank more risky. From Proposition 1, we know that a higher β makes every single loan in a bank s portfolio more risky. From Proposition 2, we know that a stricter capital regulation can also induce the bank to choose a more highly correlated portfolio. In such situations, it is obvious that the bank s PD must increase. However, even if a higher β makes the bank take a less correlated portfolio, the effect on the entrepreneurs risk-taking may dominate, leading to a higher probability of default for the bank. VI. Extensions A. Risk-adjusted capital requirements So far, we have considered only flat capital requirements. One interesting question is whether our results carry over to risk-adjusted capital requirements, as in the new Basel Capital Accord ( Basel II ). Note that the credit risk model underlying the Accord yields portfolio invariant capital requirements that depend only on the risk of a specific loan (namely, its probability of default, exposure at default, and loss given default), and not on the portfolio it is added to (see Basel Committee on Banking Supervision, 2005). The correlations to be used by banks are determined by the supervisors, and are therefore exogenously given for the banks. In our model, a risk-based capital requirement according to Basel II would refer to the risk of individual loans, S. Hence, assume that the capital requirement β(s) isa function of S, wheredβ(s)/ds > 0. What does this imply for a bank s risk-taking? It is clear that such a risk-based capital requirement will induce the bank to choose a higher deposit volume than flat capital requirements. A higher deposit volume relaxes competition for loans, decreases loan rates, reduces firms risk-taking, and thereby reduces the capital costs. 13

14 Risk-based capital requirements also affect the consequences of a tightening of capital regulation (or, analogously, of a decrease in the number of banks): As described above, tighter capital requirements (or a decrease in the number of banks) increase the risk of individual loans. With risk-based capital regulation, this implies even higher capital requirements, and we get a multiplier effect: The capital requirement β(s) rises because S rises, which attenuates the competition for loans, raises loan rates, increases firms risk-taking S, which raises capital requirements even further. Hence, the perverse effects of capital regulation would be reinforced. This result underlines the importance of truly risk-adjusted capital requirements that take into account a bank s actual portfolio correlation (ρ), and not only the riskiness of a single loan. (To be completed.) VII. Conclusion Our paper has shown that the discussion about the relationship between competition and stability in banking also has some bearing on the effectiveness of capital regulation in maintaining banking stability. A stabilizing effect of capital regulation tends to obtain in those situations where competition has a destabilizing effect, and vice versa. Given the prominence of capital requirements in today s regulation, the empirical relationship between competition and stability should thus be of great interest to policy makers. The existing empirical evidence on the presumed trade-off between competition and stability in banking is rather mixed. Following the seminal contribution by Keeley (1990), various studies have found evidence supporting or rejecting the existence of such a trade-off (see, e. g., Jiménez, Lopez, and Saurin, 2007 for supporting evidence, and Boyd, De Nicolò, and Jalal, 2006 and De Nicolò and Loukoianova, 2006 for evidence refuting the existence of a trade-off). These mixed findings are consistent with our model, which predicts different relationships between competition and stability, depending among other things on the elasticities of deposit supply and loan demand. In the light of our model, the empirical findings suggest that capital regulation may not be suited in all circumstances to prevent excessive risk-taking in banking. 14

15 References Allen, Franklin, Elena Carletti, and Robert Marquez, 2006, Credit Market Competition and Capital Regulation, Working paper. Allen, Franklin, and Douglas Gale, 2004, Competition and Financial Stability, Journal of Money, Credit, and Banking 36, Basel Committee on Banking Supervision, 2005, An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel. Boyd, John H., and Gianni De Nicolò, 2005, The Theory of Bank Risk Taking and Competition Revisited, Journal of Finance 60, Boyd, John H., Gianni De Nicolò, and Abu Al Jalal, 2006, Bank Risk Taking and Competition Revisited: New Theory and New Evidence, Working paper. De Nicolò, Gianni, and Elena Loukoianova, 2006, Bank Ownership, Market Structure, and Risk, Working paper. Hellmann, Thomas, Kevin C. Murdock, and Joseph E. Stiglitz, 2000, Liberalization, Moral Hazard in Banking, and Prudential Regulation: Are Capital Requirements Enough?, American Economic Review 90, Jiménez, Gabriel, Jose A. Lopez, and Jesús Saurin, 2007, How Does Competition Impact Bank Risk Taking?, Working paper. Keeley, Michael C., 1990, Deposit Insurance, Risk and Market Power in Banking, American Economic Review 80, Martínez, D., and Rafael Repullo, 2007, Does Competition Reduce the Risk of Bank Failure?, Working paper. Morrison, Alan D., and Lucy White, 2005, Crises and Capital Requirements in Banking, American Economic Review 95, Repullo, Rafael, 2004, Capital Requirements, Market Power, and Risk-Taking in Banking, Journal of Financial Intermediation 13, Stiglitz, Joseph E., and Andrew Weiss, 1981, Credit Rationing in Markets with Imperfect Competition, American Economic Review 71,