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Loan interest rates under risk-based capital requirements: The impact of banking market structure
Economic Modelling 32 (2013) 602–607
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Loan interest rates under risk-based capital requirements: The impact of banking market structure Inês Drumond a, b, 1, José Jorge a,⁎ a b
Faculdade de Economia, Universidade do Porto, CEF.UP, Portugal 2 DG-ECFIN, European Commission, Belgium
a r t i c l e
i n f o
Article history: Accepted 5 February 2013 Available online xxxx JEL classification: G21 L11
a b s t r a c t This paper analyzes how the effects of the introduction of risk-based bank capital requirements on bank loan rates depend on the market structure of the banking industry. We show that, when granting loans to borrowers under Basel II or Basel III capital requirements, banks with market power internalize an additional cost, in terms of regulatory capital, associated with the increase of borrowers' risk of default. As a result, the intermediation margin on bank loans increases with the changeover from non-risk to risk-based capital requirements, thereby making lending more expensive. © 2013 Elsevier B.V. All rights reserved.
Keywords: Risk-based capital requirements Basel accord Imperfect competition
1. Introduction The introduction of risk-based bank capital requirements, under both Basel II and Basel III, should be welcomed as it improves the adequacy of capital held by banks to the risk of their asset portfolios. Under these rules, the level of capital that a bank has to hold against a given exposure became a positive function of the credit risk of that exposure. 3 These developments in banking regulation have motivated additional research in this area. Nevertheless, the analysis of the effects of the implementation of risk-based capital requirements on bank loan rates under different banking industry structures has been overlooked. Conventional economic theory on pricing under imperfect competition states that prices are set as a markup over marginal costs. By developing a partial equilibrium model with oligopolistic banks, this paper shows that the Basel II and Basel III bank capital regulation adds another link to the conventional relationship between prices and quantities: a ⁎ Corresponding author at: Rua Dr. Roberto Frias, Porto, Portugal. Tel.: +351 225 571 100; fax: +351 225 505 050. E-mail addresses: [email protected] (I. Drumond), [email protected] (J. Jorge). 1 The views expressed in this paper are those of the authors and should not be attributed to the European Commission. 2 Centre for Economics and Finance at the University of Porto (CEF.UP) is funded by Programa Compete (European Regional Development Fund) and Fundação para a Ciência e a Tecnologia; project reference: PEst- C/EGE/UI4105/2011. José Jorge gratefully acknowledges financial support from Programa Compete (European Regional Development Fund) and Fundação para a Ciência e a Tecnologia; project reference: PTDC/ IIMECO/4895/2012. 3 Under the Basel Accord of 1988 (Basel I) only the borrowers’ institutional category was taken into account. 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.02.017
bank is aware that extending a new loan to a borrower increases the borrower's leverage and, thus, his risk of default; accordingly, that same bank is also aware that it will have to raise more costly bank capital under risk-based capital requirements and will account for this effect when setting loan interest rates. As a result, the intermediation margin on bank loans increases with risk-based capital requirements. We also show that borrowers with a level of risk such that the interest rate on their loans would not adjust with the changeover from non-risk to risk-based capital requirements under perfect competition, face a higher cost of funds with an oligopolistic banking system. 1.1. The implications of banking market structure The theoretical literature has identified two major effects of banking market structure on bank behavior. On the one hand, the fewer the number of banks, the larger is their market power and the smaller the total quantity of credit available to entrepreneurs. On the other hand, the fewer the number of banks, the higher the incentive to produce information, and therefore the larger the proportion of funds allocated to screened, high quality entrepreneurs.4 The empirical evidence suggests the existence of multiple effects of banking market structure for loan behavior and it does not provide definite answers regarding the sign and size of the information production effects. Yet, the market power approach is broadly validated in most studies. 4 Many theories stressing the role of production of information continue to rely on the existence of market power effects in the banking sector. For example, Gorton and He (2008), present a model in which banks collude in setting interest rates while competing in the production of information.
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Since the focus of this paper is not on the microfoundations of bank lending, we do not model explicitly the bank-firm relationship and the information channel and we choose the standard industrial organization (IO) approach to evaluate the Basel Accords. The new link presented in this paper exists whenever a bank with market power extends a loan to a firm and, therefore, the results are still valid in more general setups. The links and results presented in this paper are relevant for economic activity as long as two conditions are met. First, bank credit supply must affect economic activity; multiple studies support this assumption showing that banks play a specific role in the economy, as it is also evident in the recent financial crisis. Second, credit supply is inversely related to capital requirements (the so-called “bank capital channel”). Using the Call Reports, Hancock et al. (1995) apply a VAR methodology and find a strong relationship between bank capital and loan growth. The authors report that, after a shock to capital, larger banks adjust each component of their portfolio faster than smaller banks. Kishan and Opiela (2000) study the relationship between bank capital and monetary policy in the United States. They find that undercapitalized banks have the largest response of loans to monetary policy shocks, but the smallest response of time deposits, indicating that small, poorly capitalized banks are unable to raise alternative funds to sustain lending levels when monetary policy tightens. Other studies on European countries also corroborate these findings (see, for example, Jiménez et al., 2007, for the Spanish case). Studies estimating the medium-term impact of Basel III implementation on GDP growth show that economic output is mainly affected by an increase in bank lending spreads, as banks pass a rise in funding costs, due to higher capital requirements, to their customers (see, for example, Slovik and Cournède, 2011). Still, these studies do not account for the impact of regulation on intermediation margins and thus underestimate the full impact of new capital rules. Moreover, the Basel Committee is considering a surcharge on big systemically important financial institutions, forcing them to hold extra capital on top of the global minimums set last year. To the extent that many of these institutions have market power, the final impact of additional capital needs depends on the links described in this paper. The traditional approach to competition in financial intermediation derives largely from applying standard IO economics to the banking industry. Mandelman (2011) provides empirical evidence that economic expansions attract competitors to the banking industry and, as a result, established banks react by lowering interest rates, while during recessions few competitors enter and incumbents are able to sustain high profit margins. Countercyclical bank markups create a bank-supply channel that propagates and amplifies shocks to the economy and contribute to macroeconomic volatility. Dynamic macroeconomic models with sticky interest rates apply results in IO to understand the pass-through of monetary policy, namely Scharler (2008) considers financial intermediaries which operate in a fully competitive environment, and Hülsewig et al. (2009) and Güntner (2011) assume that banks extend loans to firms in an environment of monopolistic competition. Verona et al. (forthcoming) use a monopolistically competitive shadow banking system to explain countercyclical bank markups and to investigate whether monetary policy was responsible for the US boom-bust cycle of the 2000s. Van den Heuvel (2006) analyzed the impact of regulatory capital requirements when the bank faces a downward sloping demand curve for its loanable funds, thus laying the groundwork for a bank capital channel (Drumond, 2009, undertakes a survey of the existing literature on the bank capital channel). Gerali et al. (2010) analyze a DSGE model in which wholesale banks behave competitively and retail banks are monopolistic competitors, and in which banks have a target for their capital-to-asset ratio which the authors use as a shortcut for studying the implications and costs of regulatory capital constraints. Other authors introduced explicitly these constraints in dynamic
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macroeconomic models with a competitive banking sector, namely Jorge (2009) and Liu and Seeiso (2012). Yet, the literature is missing an integrated analysis of the pricing of loans under different banking structures and risk-based capital requirements. Our work fills this gap, thus providing the microeconomic foundations for dynamic macroeconomic models which aim at fully understanding the aggregate impact of bank capital regulation. In the next section we study the implementation of risk-based capital requirements under perfect competition, Bertrand competition and the Cournot oligopoly. We propose a proof of the existence and uniqueness of the Bertrand equilibrium which we believe to be of independent interest. Section 3 presents the empirical evidence on banking market structure, and the final section concludes with the implications for regulatory policy. 2. The model Our model economy consists of a set of M homogeneous banks and a set of J perfectly competitive firms, which borrow from those banks to buy physical capital. Both types of agents are risk neutral and the model is static.5 Banks raise funds from depositors and bank capital holders to finance the loans granted to firms. Let Dm and Sm be, respectively, the total amount of deposits and bank capital held by bank m. The gross return on deposits is given by RD, and RS represents the opportunity cost for the equity capital investment. We consider that the market for bank deposits is perfectly competitive, with RD identical across banks. The cost of funds RS is also identical for all banks.6 We further assume that the opportunity cost of bank capital is higher than the cost of bank deposits — there is a sizable theoretical and empirical literature to support this assumption (see, for instance, Stein, 1998, Gorton and Winton, 2000, and Bolton and Freixas, 2006). Let Ljm represent loans granted by bank m to firm j and Lj = M ∑m = 1 Ljm be the total amount of loans granted by all banks in the economy to firm j. Expected marginal productivity of physical capital is decreasing so that, the more a firm borrows in a given period, the lower the average expected return on borrowed funds. Under very general conditions, firm j's demand curve for loanable funds is downward sloping. Formally, we represent the demand function for loans by firm j by Lj = Lj(Rj) with dLj/dRj b 0 and d 2Lj/dRj2 ≤ 0. Given these regularity conditions, the inverse-demand function for funds by firm j is given by Rj = Rj (Lj) with dRj/dLj b 0 and d 2Rj/dLj2 ≤ 0, and represents the firm's willingness-to-pay for an amount of loans Lj. We assume that firms' risk of default on loans depends positively on firms' leverage ratio. We also consider that firms cannot issue equity — a reasonable hypothesis for small firms in the short run, which is the case we are interested in — implying that their net worth is fixed. Without loss of generality, we assume that firms' net worth is identical across firms and is standardized to one. Under these assumptions the level of risk of firm j depends only on Lj which, in turn, equals its leverage ratio. The minimum amount of regulatory capital that each bank has to raise under Basel II and III depends on the estimated credit risk of its portfolio. Let α(Lj) be the credit risk weight assigned to firm j and used to compute bank capital requirements.7 Under Basel I, bank capital requirements are not sensitive to loan risk and α(Lj) equals 1 for all Lj. We consider that under Basel II and III, α(Lj) is continuous with dα/dLj > 0 if Lj > 0, and α(Lj) = 0 otherwise. We further assume that dα/dLj + d2α/dLj2 > 0 for Lj > 0, which amounts to saying that function α cannot be too concave. There is one single value L⁎ such that α(L⁎) = 1. Note that a bank
5 Instead of the static model, we could have considered a dynamic setup but, for a preliminary regulatory analysis, intertemporal incentives to maintain a collusive arrangement may not be of first-order importance. 6 For our purposes, considering a different setup would not bring additional insights. 7 Under Basel II and III, function α(Lj) is determined either by the regulator, under the Standardized Approach, or by the bank, under the Internal Rating Based Approach.
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which has granted a loan to a firm with Lj = L⁎ is required to hold the same amount of regulatory capital under both regulatory regimes (risk-based or non risk-based capital requirements). In this context, the profit function of an individual bank m is given by
∫Rjm Ljm dϒ−RD Dm −RS Sm with
∫Ljm dϒ≡Dm þ Sm
0:08∫α Lj Ljm dϒ ≤ Sm ; where Rjm is the rate that bank m sets on a loan to firm j and Υ is the cumulative distribution of firms in the economy. The second constraint represents the regulatory capital requirement and holds with equality since, as mentioned before, the cost of raising bank capital is higher than the cost of raising deposits.8 The version of the Basel II Accord published in June 2004 proposed minimum capital requirements equal to 8% of risk-weighted assets and, later, the Basel III Accord revised these requirements and introduced additional capital buffers. For our purposes it is sufficient to have a constant ratio of a bank’s equity capital to its total risk-weighted assets and we maintain the 8% threshold. The profit function may be rewritten as h
h
i
∫ Rjm − 1−0:08α Lj
i D S R þ 0:08α Lj R Ljm dϒ:
We solve for the equilibrium under several market structures. To make the problem interesting, we assume that a monopolistic bank makes a strictly positive profit on each loan Lj. 2.1. Perfect competition in the banking industry The pricing of loans under perfect competition guarantees that interest rates equal the marginal cost of funds for the bank, and banks' profits are null. Proposition 1. Let the banking industry be characterized by perfect competition. The equilibrium loan rate to firm j is given by Rj = (1 − 0.08)RD + 0.08RS under Basel I, and Rj = [1 − 0.08α(Lj)] RD + 0.08α(Lj) RS under risk-based capital requirements. Proof. In a perfectly competitive market, banks are profit-maximizing price takers and all banks set the same loan rate to firm j, so that we define Rj = Rjm for all m. Each bank m chooses the loan size Ljm that maximizes its profit with firm j, solving h h i i D S max ∫ Rj − 1−0:08α Lj R þ 0:08α Lj R Ljm dϒ fLjm g with the first order conditions equal to h i D S S D dα L Rj ¼ 1−0:08α Lj R þ 0:08α Lj R þ 0:08 R −R dLj jm
∀j
8 Our model does not seek to explain the existence of capital requirements and we take them as exogenous. Some authors have argued that the relevant constraint on bank capital is economic rather than regulatory. Indeed, banks tend to hold a significant amount of capital above regulatory requirements (the so-called "capital buffers"). The reasons for holding capital buffers are manifold, e.g. as a result of market discipline or to avoid the dilution costs associated with having to issue new equity at short notice. Our results hold if we re-examine the role of banking market structure from the perspective of economic capital.
and the second order conditions equal to dα d2 α S D þ −0:08 R −R dLj dL2j
! ≤0
∀j:
Perfect competition drives profits to zero, so that Rj = [1 − 0.08α(Lj)] RD + 0.08α(Lj)RS. Under Basel I, α(Lj) = 1 and we obtain the result in the Proposition. Under Basel II and Basel III, the first order conditions are satisfied if and only if the loan size Ljm is extremely small. Therefore we obtain the result in the Proposition with an atomistic banking market structure. ■ We may thus conclude that the effects associated with the introduction of risk-based capital requirements on the cost of funds depend on the leverage of firm j. Firms with Lj > L⁎ are in a less favorable situation while firms with Lj b L⁎ are better off. Most importantly, firms with Lj = L⁎ are not affected by the transition to Basel II or III. 2.2. Imperfect competition in the banking industry The banking industry is one of the most heavily regulated industries in the economy and barriers to entry, like bank charters, are common. In this context, we now consider a small number of banks in an oligopolistic market structure. 2.2.1. Bertrand equilibrium With identical banks and constant marginal cost of finance, each bank offers a rate equal to the marginal cost of funds. Under Basel I, the credit risk weights are constant and, therefore, the marginal cost of finance is constant and it is easy to determine the loan rates. Yet, under Basel II and Basel III the marginal cost of funds is increasing in the total amount of loans Lj obtained by the firm. There are well-known existence and uniqueness problems when the marginal cost is not constant, and Dastidar (1995) has characterized the equilibria when the firm's cost function is increasing and convex in the total amount produced. But our framework differs from Dastidar (1995) in one important aspect. In Dastidar (1995), when a firm undercuts its rivals' price it captures and supplies the whole market, implying a substantial increase in production costs. In our framework, the increase in market share following a rate undercut implies a minor increase in marginal cost. This is because the marginal cost of funding a bank loan depends on the total amount of loans Lj obtained by the firm, and not so much on the size of the loan Ljm offered by the bank. A small price undercut raises Ljm substantially, but has a small impact on Lj so that the final result is a minor increase in the cost of finance. To better understand this, consider the example of two banks which offer the same loan rate (above their cost of funding) to a firm, and assume they have the same market share. When one bank undercuts the price slightly, it captures the market share of the other bank so that it supplies a loan twice as big as before. But the marginal increase in the cost of funding is small, because the total demand for loans by the firm does not change significantly. This contrasts with the traditional convex cost function in which doubling the production entails a substantial increase in marginal costs. This feature makes the structure of the game under Basel II and Basel III similar to the case of Bertrand competition with constant marginal costs and, unsurprisingly, the solution to both frameworks is identical. The next Proposition establishes that, under Bertrand competition, we obtain the same results as under perfect competition. Proposition 2. Let the banking industry be characterized by Bertrand competition. The equilibrium loan rate to firm j is given by Rj = (1 − 0.08) RD + 0.08RS under Basel I, and Rj = [1 − 0.08α (Lj)] RD +0.08α (Lj) RS under risk-based capital requirements.
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Proof. Under Basel I, the marginal cost of finance for bank m is equal to (1 − 0.08) RD + 0.08 RS, so that we have Bertrand competition with constant marginal costs. Hence, the loan rate equals the marginal cost of funding for the bank. Under Basel II and Basel III, the marginal cost of funds is increasing in the total amount of loans obtained by the firm Lj. Each bank m chooses the rate Rjm that maximizes its profit ∫ [Rjm − ([1 − 0.08α(Lj)] RD + 0.08α(Lj)RS)]Ljmdϒ knowing that there is price competition. The problem does not yield the usual smoothness conditions because undercutting loan rates has a discrete impact on profits. We focus on the loans granted to firm j and, for simplicity, we omit subscript j. Hence Ljm, Lj, Rjm and Rj become Lm, L, Rm and R. Let (R1, R2,…, Rm,…,RM) be the vector of rates offered by the banks to firm j. Bank m faces the following demand curve Dm ðR1 ; R2 ; …; Rm ; …; RM Þ 8 0 if Rm > Rn for some n ≠ m > <1 Rm ≤ Rn ; ∀n and m is the number ¼ LðRm Þ if >m of banks which set their rate equal to Rm : LðRm Þ if Rm b Rn ; ∀n ≠ m: The above means that if bank m sets a rate below all other banks then bank m gets the whole market, while if several banks charge the same rate they split the market equally. Define π m ðR; m Þ ¼ h i R− ½1−0:08α ðLÞRD þ 0:08α ðLÞRS L=m and let π(R) = πm (R, 1). We interpret function π m ðR; m Þ as the individual bank profit when m banks split the market equally, and function π(R) as the profit when one single bank supplies the firm. All banks set their rates simultaneously. When (R1, R2,…, Rm,…, Rm) is chosen, the profit of bank m is denoted by Em (R1, R2,…, Rm,…, Rm) and equals Em ðR1 ; R2 ; …; Rm ; …; RM Þ 8 0 if Rm > Rn for some n ≠ m > < Rm ≤ Rn ; ∀n and m is the number ¼ πm ðRm ; m Þ if of banks which set their rate equal to Rm > : π ðRm Þ if Rm b Rn ; ∀n ≠ m:
2
exists because
dL2j
Lemma. All banks offering the same rate R is an equilibrium if and only if πm R; M ≥π R . Proof. Upward deviations are not profitable as the bank looses market share and profit becomes null. In order to have a Bertrand equilibrium, downward deviations cannot be profitable either. Hence, πm R; M ≥ vm R for all m, where vm R ≡supfR′ bR g π R′ . Since function π(.) is ■ continuous then vm R ¼ maxR′ ∈½0;R π R′ ¼ π R . If there is an equilibrium in which all banks offer the same loan rate, then the loan rate must be equal to R0. This follows from the previous lemma because πm(R0,M) ≥ π(R0) = Mπm(R0,M) is equivalent to πm(R0, M) = 0. It follows that there is a unique equilibrium in which all banks offer a rate R0. Since π(R0) = 0 and L (R0) > 0, we obtain ■ R0 = [1 − 0.08α(L)] RD + 0.08α(L) RS. Under Basel II and Basel III there is no equilibrium in which banks set a loan rate above their marginal cost of funding. To see this, consider the case in which all banks set the same rate higher than their marginal cost. Two effects occur when one bank undercuts the loan rate slightly and unilaterally. First, it increases its market share substantially (the size of the loan to a firm doubles when there are two banks), thereby raising its profits. Second, there is a decrease in its unit profit because the rate charged decreases and the marginal cost increases. With a very small decrease in the loan rate, the reduction in the unit profit of the bank can be made arbitrarily small because the marginal cost does not change significantly. Hence the bank can make the second effect (almost) equal to zero, making a unilateral deviation profitable. 9 2.3. Cournot equilibrium The most interesting case occurs under the Cournot oligopoly structure. The market rate Rj is set at a level such that demand equals the total amount of loans supplied by banks, so that Rj = Rjm for all m. Consider a symmetric equilibrium in which all banks split the market equally, and let εL = −
^ be the minimum of such rates. the profit of the monopolist, let R Without loss of generality, we consider the case in which all banks offer the same rate. There is no equilibrium in which banks set a loan ^ because undercutting the loan rate would be profitable. rate above R ^ such that π(R 0) = 0. Lemma. There is one single rate R0 ≤R π ðRÞ . LðRÞ
be the elasticity of demand for loans.
Rj −ð1−0:08ÞRD −0:08RS 1 ¼ Rj Mε L
^ > 0, Since L R
then π(R 0) > 0. The unit profit function is continuous and strictly in ^ > 0 and πu(R D) ≤ 0, then there is a unique loan creasing. Since πu R
ð1Þ
under Basel I, and
≤0). If there is more than one rate that maximizes
Proof. Define the unit profit function as πu ðRÞ ¼
dLj Rj dRj Lj
Proposition 3. Let the banking industry be characterized by a Cournot equilibrium. The equilibrium loan rate to firm j satisfies
The vector (R1⁎, R2⁎,…, Rm⁎,…, Rm⁎) is a Nash equilibrium if Em(R1⁎, R2⁎,…, Rm⁎,…, Rm⁎) ≥ (R1, R2,…, Rm,…, Rm) for all Rm ≠ Rm⁎, ∀m. We have assumed that a monopolistic bank makes strictly positive ^ > 0 for which π R ^ > 0 (this rate profit, so that there is a finite rate R d Rj
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1 ¼ Mε L
h i S D dα Rj − 1−0:08α Lj RD −0:08α Lj RS −0:08 R −R Ljm dLj
Rj
ð2Þ
under risk-based capital requirements. Proof. Each bank m chooses the loan Ljm that maximizes its profit with firm j, knowing that there is competition in quantities. Bank m solves h h i i D S max ∫ Rj Lj − 1−0:08α Lj R þ 0:08α Lj R Ljm dϒ fLjm g
rate R 0 such that πu(R 0) = 0. Hence, the loan rate R 0 is the unique ^ such that π(R 0) = πu(R 0) · L(R 0) = 0. rate below R
■
The key condition for the existence of an equilibrium is that banks cannot have incentives to change loan rates, and the next lemma formalizes this condition.
9 Dastidar (1995) finds equilibria with prices higher than the marginal cost because price undercuts may not be profitable. An arbitrarily small price undercut leads to a big increase in production, which raises the marginal costs substantially. The reduction in unit profit may offset the increase in market share, rendering the unilateral deviation unattractive.
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with the first order conditions are equal to h i D S S D dα L Rj Lj − 1−0:08α Lj R −0:08α Lj R −0:08 R −R dLj jm dRj ¼− L ∀j dLj jm because
dLj dLjm
¼ 1. After dividing both terms by Rj, we obtain expression
(2) because M ¼
Lj . Ljm
Under Basel I, α(Lj) = 1 and we obtain expres-
sion (1). The second order conditions of bank m are equal to ! dα d2 α dRj d2 Rj S D b 0 ∀j: þ 2 −0:08 R −R þ dLj dLj dL2j dLj ■ Banks set the loan rate as a markup over marginal cost, and the equations in Proposition 3 show that the Cournot oligopoly structure adds a new channel through which risk-based capital requirements affect the rate Rj. This channel does not exist under Basel I and is captured by the term 0.08dα/dLj(RS − RD)Ljm. In order to gain some intuition, consider a firm that borrowed a total amount of L⁎. In this case, the bank is required to hold the same amount of regulatory capital under both regimes (risk-based and non risk-based capital requirements) as α(L⁎) = 1. Yet, Eqs. (1) and (2) show that the intermediation margin on loans increases with the changeover from Basel I to risk-based capital requirements: each time firm j gets a new loan (e.g. Ljm from bank m) its leverage and, consequently, its risk of default increase, thereby increasing the credit risk weight assigned to that firm and requiring bank m to hold more bank capital (an effect captured by the term 0.08dα/dLj); when pricing the new loan, bank m takes this effect into consideration as well as the higher marginal cost of issuing bank capital when compared to deposits (as the term RS − RD is positive). That is, under Basel II and Basel III, banks internalize the effect of extending new loans on their cost of finance, an outcome which is absent under Basel I. The following corollary formalizes these results. Corollary 1. Let the banking industry be characterized by a Cournot oligopoly. Firms which obtain an amount of bank loans equal to L ∗ under Basel I, face an increase in the cost of funds with the changeover to risk-based capital requirements. Proof. Let the optimal loan of bank m to firm j under Basel I be equal L dLj M
dR to L⁎/M, so that Rj −½1−0:08RD −0:08RS þ j
¼ 0. The first order
conditions under Basel II and III are given by h i dRj dα S D S D L −0:08 R −R Ljm Rj − 1−0:08α Lj R −0:08α Lj R þ dLj jm dLj ¼ 0 ∀j: If we substitute Ljm by L⁎/M in the left hand side of the previous equation we obtain a negative value. The second order conditions, given by d Rj dRj dα S d2 α S D D L þ2 −0:16 R −R −0:08 2 R −R Ljm b 0 2 jm dLj dLj dLj dLj 2
∀j;
guarantee that the left hand side is decreasing in the loan Ljm. Hence the value of the loan that solves the bank’s optimization problem under Basel II and III is such that Ljm b L⁎/M and, since Rj is a decreasing function in ■ Lj, the cost of funds increases under Basel II and Basel III. By focusing on the case in which firms' leverage and risk of default are such that the amount of capital that banks must set aside for regulatory purposes remains the same under both regulatory regimes
(risk-based and non risk-based capital requirements), the above corollary proves that, under a Cournot oligopoly, banks charge higher interest rates under Basel II and Basel III. Two other cases remain to be analyzed. When Lj > L⁎ under Basel I, it is straightforward to show that the cost of funds for the firm increases under Basel II and III because α(Lj) > α(L⁎) = 1. When Lj b L⁎ under Basel I, the effect of a shift to risk-based capital requirements over the cost of funds is unclear. On the one hand, α(Lj) b α(L⁎) = 1, which requires the bank to hold less bank capital, thereby decreasing the cost of bank loans. On the other hand, the intermediation margin increases as can be demonstrated by comparing Eqs. (1) and (2). In the previous corollary we compared the loan rate for different Basel Accords. We now compare the equilibrium intermediation margin on bank loans. Corollary 2. Let the banking industry be characterized by a Cournot oligopoly. The intermediation margin on loans increases with the changeover from Basel I to risk-based capital requirements. Proof. The intermediation margins under Basel I and the new Basel Rj Rj dα and þ 0:08 RS −RD Ljm , respectively. For Accords equal MεL
MεL
dLj
firms with the same (strictly positive) leverage, the intermediation margin increases. The increase of the intermediation margin makes lending more expensive under the new Basel Accords. The overall effect depends on the risk of the firms because less risky loans require less bank capital which may lead to lower loan rates. Yet, there is a tendency towards an increase in the cost of funding under the new Basel Accords. 2.4. Differences in pricing under risk-based capital requirements Under the Cournot oligopoly, banks internalize the effect of extending new loans on their cost of finance while, under perfect competition and Bertrand competition, banks fail to internalize the effect of their actions on the credit risk weights and on their marginal cost of funds. We may thus conclude that the structure of the banking industry matters when assessing the effects of Basel II and, more recently, Basel III. In the Cournot oligopoly, when bank m extends a new loan to firm j, it imposes a negative externality on every other bank as it forces its peers to hold more capital. Bank m partially internalizes the negative externality as the cost of the externality, measured by 0.08dα/ dLj(R S − R D), is multiplied by Ljm = Lj/M. A monopolist bank would fully internalize the externality, as Ljm = Lj, while under perfect competition, as Ljm converges to zero, banks fail to acknowledge the externality loss. In Bertrand equilibrium, banks fail to internalize the negative effects of price undercutting. This is because undercutting loan rates has a large positive impact on revenues and little impact on costs, so that banks have incentives to undercut rates unilaterally as long as profits are positive. 2.5. Demand for loanable funds We have maintained the demand for bank loans constant under non-risk and risk-based capital requirements. To the extent that the firm’s demand for loanable funds accounts for the effect of leverage over funding costs, the demand curve shifts with the move from non-risk to risk-based capital requirements. The demand for bank loans is likely to become more inelastic because credit becomes more expensive for highly leveraged firms and cheaper for firms with stronger credit quality. The focus of this paper is on bank behavior and, therefore, we do not model explicitly the firm’s problem. Still, the assessment of risk-based
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capital requirements would benefit from a thorough analysis of the microfoundations of bank loan demand.
structure when defining the risk weights used to compute bank capital requirements should be taken into account.
3. Empirical evidence on banking market structure
References
In the previous section we showed that, under a Cournot-type oligopoly, the intermediation margin rises with the introduction of risk-based capital requirements, while under perfect competition or under a Bertrand-type oligopoly there is no such effect. Since the market structure in banking matters for the economic consequences of bank capital regulation, it is important to characterize loan markets in those countries where banks are subject to risk-based capital requirements. Much of the empirical literature uses the Monti-Klein model to assess the bank market structure (see Klein, 1971, and Monti, 1972). Since the Monti-Klein model can easily be interpreted as a model of Cournot competition between a finite number M of banks, most studies estimate the impact of proxies for market power – like the number of banks, the degree of concentration in the banking sector, or the market share – on intermediation margins, while controlling for differences in efficiency. Using US data, Hannan (1991) shows that concentration is associated with higher loan rates, while Berger (1995) finds that market share is positively related to profitability. A number of studies confirm the existence of market power in bank loan markets by explicitly estimating the first order conditions of the Cournot model of bank pricing. For instance, Corvoisier and Gropp (2002) find that increased concentration leads to collusion and higher interest margins on bank loans in the euro area, while Marrouch and Turk-Ariss (2012), using longitudinal data on banks for 103 non-OECD countries, find that markups are a key determinant in loan pricing. All in all, the empirical literature on credit market competition favors the market power theories and the Cournot model, ascertaining the importance of market structure for bank capital regulation. 4. Concluding remarks This paper adds to the research agenda on risk-based capital requirements by highlighting the effects that the structure of the banking industry has on the setting of loan interest rates. In particular, we have shown that, in bank loan markets characterized by a Cournot oligopoly, the intermediation margins increase with the changeover from nonrisk to risk-based capital requirements, while margins remain constant in competitive markets. Consequently, our analysis suggests the hypothesis that the overall impact of risk-based capital requirements on loan interest rates depends on the distribution of risk and leverage across firms and on the market structure of the banking sector. The effects of the implementation of risk-based capital requirements on the borrowers' cost of funds in a country with a more competitive banking structure may be significantly different from the effects in a country with an oligopolistic structure, and the consideration of the market
Berger, A., 1995. The profit-structure relationship in banking—tests of market power and efficient-structure hypothesis. Journal of Money, Credit, and Banking 27, 404–431. Bolton, P., Freixas, X., 2006. Corporate finance and the monetary transmission mechanism. Review of Financial Studies 19, 829–870. Corvoisier, S., Gropp, R., 2002. Bank concentration and retail interest rates. Journal of Banking and Finance 26, 2155–2189. Dastidar, K., 1995. On the existence of pure strategy Bertrand equilibrium. Economic Theory 5, 19–32. Drumond, I., 2009. Bank capital requirements, business cycle fluctuations and the basel accords: a synthesis. Journal of Economic Surveys 23, 798–830. Gerali, A., Neri, S., Sessa, L., Signoretti, F.M., 2010. Credit and banking in a DSGE model of the Euro area. Journal of Money, Credit, and Banking 42, 107–141. Gorton, G., He, P., 2008. Bank credit cycles. Review of Economic Studies 75, 1181–1214. Gorton, G., Winton, A., 2000. Liquidity Provision, Bank Capital, and the Macroeconomy. (Available at SSRN: http://ssrn.com/abstract=253849). Güntner, J.H.F., 2011. Competition among banks and the pass-through of monetary policy. Economic Modelling 28, 1891–1901. Hancock, D., Laing, A., Wilcox, J., 1995. Bank capital shocks: dynamic effects on securities, loans, and capital. Journal of Banking and Finance 19, 661–667. Hannan, T., 1991. Bank commercial loan markets and the role of market structure: evidence from surveys of commercial lending. Journal of Banking and Finance 15, 133–149. Hülsewig, O., Mayer, E., Wollmershäuser, T., 2009. Bank behaviour, incomplete interest rate pass-through, and the cost channel of monetary policy transmission. Economic Modelling 26, 1310–1327. Jiménez, G., Ongena, S., Peydró, J.-L., Saurina, J., 2007. Hazardous times for monetary policy: what do twenty-three million bank loans say about the effects of monetary policy on credit risk-taking? CEPR Discussion Papers, p. 6514. Jorge, J., 2009. Why do bank loans react with a delay to shifts in interest rates? A bank capital explanation. Economic Modelling 26, 799–806. Kishan, R., Opiela, T., 2000. Bank size, bank capital and the bank lending channel. Journal of Money, Credit, and Banking 32, 121–141. Klein, M., 1971. A theory of the banking firm. Journal of Money, Credit, and Banking 3, 205–218. Liu, G., Seeiso, N.E., 2012. Basel II procyclicality: the case of South Africa. Economic Modelling 29, 848–857. Mandelman, F., 2011. Business cycles and the role of imperfect competition in the banking system. International Finance 14, 103–133. Marrouch, W., Turk-Ariss, R., 2012. Bank pricing under oligopsony–oligopoly: evidence from 103 developing countries. Bank of Finland's Institute for Economies in Transition Discussion Papers, pp. 1–2012. Monti, M., 1972. Deposit, credit, and interest rate determination under alternative bank objectives. In: Szego, G.P., Shell, K. (Eds.), Mathematical Methods in Investment and Finance. North-Holland, Amsterdam. Scharler, J., 2008. Do bank-based financial systems reduce macroeconomic volatility by smoothing interest rates? Journal of Macroeconomics 30, 1207–1221. Slovik, P., Cournède, B., 2011. Macroeconomic impact of Basel III. OECD Economics Department Working Papers, p. 844. Stein, J., 1998. An adverse selection model of bank asset and liability management with implications for the transmission of monetary policy. The RAND Journal of Economics 29, 466–486. Van den Heuvel, S., 2006. The bank capital channel of monetary policy. Society for Economic Dynamics 2006 Meeting Papers, p. 512. Verona, F., Martins, M., Drumond I., forthcoming. (Un)anticipated monetary policy in a DGSE model with a shadow banking system, International Journal of Central Banking.
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Loan interest rates under risk-based capital requirements: The impact of banking market structure Inês Drumond a, b, 1, José Jorge a,⁎ a b
Faculdade de Economia, Universidade do Porto, CEF.UP, Portugal 2 DG-ECFIN, European Commission, Belgium
a r t i c l e
i n f o
Article history: Accepted 5 February 2013 Available online xxxx JEL classification: G21 L11
a b s t r a c t This paper analyzes how the effects of the introduction of risk-based bank capital requirements on bank loan rates depend on the market structure of the banking industry. We show that, when granting loans to borrowers under Basel II or Basel III capital requirements, banks with market power internalize an additional cost, in terms of regulatory capital, associated with the increase of borrowers' risk of default. As a result, the intermediation margin on bank loans increases with the changeover from non-risk to risk-based capital requirements, thereby making lending more expensive. © 2013 Elsevier B.V. All rights reserved.
Keywords: Risk-based capital requirements Basel accord Imperfect competition
1. Introduction The introduction of risk-based bank capital requirements, under both Basel II and Basel III, should be welcomed as it improves the adequacy of capital held by banks to the risk of their asset portfolios. Under these rules, the level of capital that a bank has to hold against a given exposure became a positive function of the credit risk of that exposure. 3 These developments in banking regulation have motivated additional research in this area. Nevertheless, the analysis of the effects of the implementation of risk-based capital requirements on bank loan rates under different banking industry structures has been overlooked. Conventional economic theory on pricing under imperfect competition states that prices are set as a markup over marginal costs. By developing a partial equilibrium model with oligopolistic banks, this paper shows that the Basel II and Basel III bank capital regulation adds another link to the conventional relationship between prices and quantities: a ⁎ Corresponding author at: Rua Dr. Roberto Frias, Porto, Portugal. Tel.: +351 225 571 100; fax: +351 225 505 050. E-mail addresses: [email protected] (I. Drumond), [email protected] (J. Jorge). 1 The views expressed in this paper are those of the authors and should not be attributed to the European Commission. 2 Centre for Economics and Finance at the University of Porto (CEF.UP) is funded by Programa Compete (European Regional Development Fund) and Fundação para a Ciência e a Tecnologia; project reference: PEst- C/EGE/UI4105/2011. José Jorge gratefully acknowledges financial support from Programa Compete (European Regional Development Fund) and Fundação para a Ciência e a Tecnologia; project reference: PTDC/ IIMECO/4895/2012. 3 Under the Basel Accord of 1988 (Basel I) only the borrowers’ institutional category was taken into account. 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.02.017
bank is aware that extending a new loan to a borrower increases the borrower's leverage and, thus, his risk of default; accordingly, that same bank is also aware that it will have to raise more costly bank capital under risk-based capital requirements and will account for this effect when setting loan interest rates. As a result, the intermediation margin on bank loans increases with risk-based capital requirements. We also show that borrowers with a level of risk such that the interest rate on their loans would not adjust with the changeover from non-risk to risk-based capital requirements under perfect competition, face a higher cost of funds with an oligopolistic banking system. 1.1. The implications of banking market structure The theoretical literature has identified two major effects of banking market structure on bank behavior. On the one hand, the fewer the number of banks, the larger is their market power and the smaller the total quantity of credit available to entrepreneurs. On the other hand, the fewer the number of banks, the higher the incentive to produce information, and therefore the larger the proportion of funds allocated to screened, high quality entrepreneurs.4 The empirical evidence suggests the existence of multiple effects of banking market structure for loan behavior and it does not provide definite answers regarding the sign and size of the information production effects. Yet, the market power approach is broadly validated in most studies. 4 Many theories stressing the role of production of information continue to rely on the existence of market power effects in the banking sector. For example, Gorton and He (2008), present a model in which banks collude in setting interest rates while competing in the production of information.
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Since the focus of this paper is not on the microfoundations of bank lending, we do not model explicitly the bank-firm relationship and the information channel and we choose the standard industrial organization (IO) approach to evaluate the Basel Accords. The new link presented in this paper exists whenever a bank with market power extends a loan to a firm and, therefore, the results are still valid in more general setups. The links and results presented in this paper are relevant for economic activity as long as two conditions are met. First, bank credit supply must affect economic activity; multiple studies support this assumption showing that banks play a specific role in the economy, as it is also evident in the recent financial crisis. Second, credit supply is inversely related to capital requirements (the so-called “bank capital channel”). Using the Call Reports, Hancock et al. (1995) apply a VAR methodology and find a strong relationship between bank capital and loan growth. The authors report that, after a shock to capital, larger banks adjust each component of their portfolio faster than smaller banks. Kishan and Opiela (2000) study the relationship between bank capital and monetary policy in the United States. They find that undercapitalized banks have the largest response of loans to monetary policy shocks, but the smallest response of time deposits, indicating that small, poorly capitalized banks are unable to raise alternative funds to sustain lending levels when monetary policy tightens. Other studies on European countries also corroborate these findings (see, for example, Jiménez et al., 2007, for the Spanish case). Studies estimating the medium-term impact of Basel III implementation on GDP growth show that economic output is mainly affected by an increase in bank lending spreads, as banks pass a rise in funding costs, due to higher capital requirements, to their customers (see, for example, Slovik and Cournède, 2011). Still, these studies do not account for the impact of regulation on intermediation margins and thus underestimate the full impact of new capital rules. Moreover, the Basel Committee is considering a surcharge on big systemically important financial institutions, forcing them to hold extra capital on top of the global minimums set last year. To the extent that many of these institutions have market power, the final impact of additional capital needs depends on the links described in this paper. The traditional approach to competition in financial intermediation derives largely from applying standard IO economics to the banking industry. Mandelman (2011) provides empirical evidence that economic expansions attract competitors to the banking industry and, as a result, established banks react by lowering interest rates, while during recessions few competitors enter and incumbents are able to sustain high profit margins. Countercyclical bank markups create a bank-supply channel that propagates and amplifies shocks to the economy and contribute to macroeconomic volatility. Dynamic macroeconomic models with sticky interest rates apply results in IO to understand the pass-through of monetary policy, namely Scharler (2008) considers financial intermediaries which operate in a fully competitive environment, and Hülsewig et al. (2009) and Güntner (2011) assume that banks extend loans to firms in an environment of monopolistic competition. Verona et al. (forthcoming) use a monopolistically competitive shadow banking system to explain countercyclical bank markups and to investigate whether monetary policy was responsible for the US boom-bust cycle of the 2000s. Van den Heuvel (2006) analyzed the impact of regulatory capital requirements when the bank faces a downward sloping demand curve for its loanable funds, thus laying the groundwork for a bank capital channel (Drumond, 2009, undertakes a survey of the existing literature on the bank capital channel). Gerali et al. (2010) analyze a DSGE model in which wholesale banks behave competitively and retail banks are monopolistic competitors, and in which banks have a target for their capital-to-asset ratio which the authors use as a shortcut for studying the implications and costs of regulatory capital constraints. Other authors introduced explicitly these constraints in dynamic
603
macroeconomic models with a competitive banking sector, namely Jorge (2009) and Liu and Seeiso (2012). Yet, the literature is missing an integrated analysis of the pricing of loans under different banking structures and risk-based capital requirements. Our work fills this gap, thus providing the microeconomic foundations for dynamic macroeconomic models which aim at fully understanding the aggregate impact of bank capital regulation. In the next section we study the implementation of risk-based capital requirements under perfect competition, Bertrand competition and the Cournot oligopoly. We propose a proof of the existence and uniqueness of the Bertrand equilibrium which we believe to be of independent interest. Section 3 presents the empirical evidence on banking market structure, and the final section concludes with the implications for regulatory policy. 2. The model Our model economy consists of a set of M homogeneous banks and a set of J perfectly competitive firms, which borrow from those banks to buy physical capital. Both types of agents are risk neutral and the model is static.5 Banks raise funds from depositors and bank capital holders to finance the loans granted to firms. Let Dm and Sm be, respectively, the total amount of deposits and bank capital held by bank m. The gross return on deposits is given by RD, and RS represents the opportunity cost for the equity capital investment. We consider that the market for bank deposits is perfectly competitive, with RD identical across banks. The cost of funds RS is also identical for all banks.6 We further assume that the opportunity cost of bank capital is higher than the cost of bank deposits — there is a sizable theoretical and empirical literature to support this assumption (see, for instance, Stein, 1998, Gorton and Winton, 2000, and Bolton and Freixas, 2006). Let Ljm represent loans granted by bank m to firm j and Lj = M ∑m = 1 Ljm be the total amount of loans granted by all banks in the economy to firm j. Expected marginal productivity of physical capital is decreasing so that, the more a firm borrows in a given period, the lower the average expected return on borrowed funds. Under very general conditions, firm j's demand curve for loanable funds is downward sloping. Formally, we represent the demand function for loans by firm j by Lj = Lj(Rj) with dLj/dRj b 0 and d 2Lj/dRj2 ≤ 0. Given these regularity conditions, the inverse-demand function for funds by firm j is given by Rj = Rj (Lj) with dRj/dLj b 0 and d 2Rj/dLj2 ≤ 0, and represents the firm's willingness-to-pay for an amount of loans Lj. We assume that firms' risk of default on loans depends positively on firms' leverage ratio. We also consider that firms cannot issue equity — a reasonable hypothesis for small firms in the short run, which is the case we are interested in — implying that their net worth is fixed. Without loss of generality, we assume that firms' net worth is identical across firms and is standardized to one. Under these assumptions the level of risk of firm j depends only on Lj which, in turn, equals its leverage ratio. The minimum amount of regulatory capital that each bank has to raise under Basel II and III depends on the estimated credit risk of its portfolio. Let α(Lj) be the credit risk weight assigned to firm j and used to compute bank capital requirements.7 Under Basel I, bank capital requirements are not sensitive to loan risk and α(Lj) equals 1 for all Lj. We consider that under Basel II and III, α(Lj) is continuous with dα/dLj > 0 if Lj > 0, and α(Lj) = 0 otherwise. We further assume that dα/dLj + d2α/dLj2 > 0 for Lj > 0, which amounts to saying that function α cannot be too concave. There is one single value L⁎ such that α(L⁎) = 1. Note that a bank
5 Instead of the static model, we could have considered a dynamic setup but, for a preliminary regulatory analysis, intertemporal incentives to maintain a collusive arrangement may not be of first-order importance. 6 For our purposes, considering a different setup would not bring additional insights. 7 Under Basel II and III, function α(Lj) is determined either by the regulator, under the Standardized Approach, or by the bank, under the Internal Rating Based Approach.
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which has granted a loan to a firm with Lj = L⁎ is required to hold the same amount of regulatory capital under both regulatory regimes (risk-based or non risk-based capital requirements). In this context, the profit function of an individual bank m is given by
∫Rjm Ljm dϒ−RD Dm −RS Sm with
∫Ljm dϒ≡Dm þ Sm
0:08∫α Lj Ljm dϒ ≤ Sm ; where Rjm is the rate that bank m sets on a loan to firm j and Υ is the cumulative distribution of firms in the economy. The second constraint represents the regulatory capital requirement and holds with equality since, as mentioned before, the cost of raising bank capital is higher than the cost of raising deposits.8 The version of the Basel II Accord published in June 2004 proposed minimum capital requirements equal to 8% of risk-weighted assets and, later, the Basel III Accord revised these requirements and introduced additional capital buffers. For our purposes it is sufficient to have a constant ratio of a bank’s equity capital to its total risk-weighted assets and we maintain the 8% threshold. The profit function may be rewritten as h
h
i
∫ Rjm − 1−0:08α Lj
i D S R þ 0:08α Lj R Ljm dϒ:
We solve for the equilibrium under several market structures. To make the problem interesting, we assume that a monopolistic bank makes a strictly positive profit on each loan Lj. 2.1. Perfect competition in the banking industry The pricing of loans under perfect competition guarantees that interest rates equal the marginal cost of funds for the bank, and banks' profits are null. Proposition 1. Let the banking industry be characterized by perfect competition. The equilibrium loan rate to firm j is given by Rj = (1 − 0.08)RD + 0.08RS under Basel I, and Rj = [1 − 0.08α(Lj)] RD + 0.08α(Lj) RS under risk-based capital requirements. Proof. In a perfectly competitive market, banks are profit-maximizing price takers and all banks set the same loan rate to firm j, so that we define Rj = Rjm for all m. Each bank m chooses the loan size Ljm that maximizes its profit with firm j, solving h h i i D S max ∫ Rj − 1−0:08α Lj R þ 0:08α Lj R Ljm dϒ fLjm g with the first order conditions equal to h i D S S D dα L Rj ¼ 1−0:08α Lj R þ 0:08α Lj R þ 0:08 R −R dLj jm
∀j
8 Our model does not seek to explain the existence of capital requirements and we take them as exogenous. Some authors have argued that the relevant constraint on bank capital is economic rather than regulatory. Indeed, banks tend to hold a significant amount of capital above regulatory requirements (the so-called "capital buffers"). The reasons for holding capital buffers are manifold, e.g. as a result of market discipline or to avoid the dilution costs associated with having to issue new equity at short notice. Our results hold if we re-examine the role of banking market structure from the perspective of economic capital.
and the second order conditions equal to dα d2 α S D þ −0:08 R −R dLj dL2j
! ≤0
∀j:
Perfect competition drives profits to zero, so that Rj = [1 − 0.08α(Lj)] RD + 0.08α(Lj)RS. Under Basel I, α(Lj) = 1 and we obtain the result in the Proposition. Under Basel II and Basel III, the first order conditions are satisfied if and only if the loan size Ljm is extremely small. Therefore we obtain the result in the Proposition with an atomistic banking market structure. ■ We may thus conclude that the effects associated with the introduction of risk-based capital requirements on the cost of funds depend on the leverage of firm j. Firms with Lj > L⁎ are in a less favorable situation while firms with Lj b L⁎ are better off. Most importantly, firms with Lj = L⁎ are not affected by the transition to Basel II or III. 2.2. Imperfect competition in the banking industry The banking industry is one of the most heavily regulated industries in the economy and barriers to entry, like bank charters, are common. In this context, we now consider a small number of banks in an oligopolistic market structure. 2.2.1. Bertrand equilibrium With identical banks and constant marginal cost of finance, each bank offers a rate equal to the marginal cost of funds. Under Basel I, the credit risk weights are constant and, therefore, the marginal cost of finance is constant and it is easy to determine the loan rates. Yet, under Basel II and Basel III the marginal cost of funds is increasing in the total amount of loans Lj obtained by the firm. There are well-known existence and uniqueness problems when the marginal cost is not constant, and Dastidar (1995) has characterized the equilibria when the firm's cost function is increasing and convex in the total amount produced. But our framework differs from Dastidar (1995) in one important aspect. In Dastidar (1995), when a firm undercuts its rivals' price it captures and supplies the whole market, implying a substantial increase in production costs. In our framework, the increase in market share following a rate undercut implies a minor increase in marginal cost. This is because the marginal cost of funding a bank loan depends on the total amount of loans Lj obtained by the firm, and not so much on the size of the loan Ljm offered by the bank. A small price undercut raises Ljm substantially, but has a small impact on Lj so that the final result is a minor increase in the cost of finance. To better understand this, consider the example of two banks which offer the same loan rate (above their cost of funding) to a firm, and assume they have the same market share. When one bank undercuts the price slightly, it captures the market share of the other bank so that it supplies a loan twice as big as before. But the marginal increase in the cost of funding is small, because the total demand for loans by the firm does not change significantly. This contrasts with the traditional convex cost function in which doubling the production entails a substantial increase in marginal costs. This feature makes the structure of the game under Basel II and Basel III similar to the case of Bertrand competition with constant marginal costs and, unsurprisingly, the solution to both frameworks is identical. The next Proposition establishes that, under Bertrand competition, we obtain the same results as under perfect competition. Proposition 2. Let the banking industry be characterized by Bertrand competition. The equilibrium loan rate to firm j is given by Rj = (1 − 0.08) RD + 0.08RS under Basel I, and Rj = [1 − 0.08α (Lj)] RD +0.08α (Lj) RS under risk-based capital requirements.
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Proof. Under Basel I, the marginal cost of finance for bank m is equal to (1 − 0.08) RD + 0.08 RS, so that we have Bertrand competition with constant marginal costs. Hence, the loan rate equals the marginal cost of funding for the bank. Under Basel II and Basel III, the marginal cost of funds is increasing in the total amount of loans obtained by the firm Lj. Each bank m chooses the rate Rjm that maximizes its profit ∫ [Rjm − ([1 − 0.08α(Lj)] RD + 0.08α(Lj)RS)]Ljmdϒ knowing that there is price competition. The problem does not yield the usual smoothness conditions because undercutting loan rates has a discrete impact on profits. We focus on the loans granted to firm j and, for simplicity, we omit subscript j. Hence Ljm, Lj, Rjm and Rj become Lm, L, Rm and R. Let (R1, R2,…, Rm,…,RM) be the vector of rates offered by the banks to firm j. Bank m faces the following demand curve Dm ðR1 ; R2 ; …; Rm ; …; RM Þ 8 0 if Rm > Rn for some n ≠ m > <1 Rm ≤ Rn ; ∀n and m is the number ¼ LðRm Þ if >m of banks which set their rate equal to Rm : LðRm Þ if Rm b Rn ; ∀n ≠ m: The above means that if bank m sets a rate below all other banks then bank m gets the whole market, while if several banks charge the same rate they split the market equally. Define π m ðR; m Þ ¼ h i R− ½1−0:08α ðLÞRD þ 0:08α ðLÞRS L=m and let π(R) = πm (R, 1). We interpret function π m ðR; m Þ as the individual bank profit when m banks split the market equally, and function π(R) as the profit when one single bank supplies the firm. All banks set their rates simultaneously. When (R1, R2,…, Rm,…, Rm) is chosen, the profit of bank m is denoted by Em (R1, R2,…, Rm,…, Rm) and equals Em ðR1 ; R2 ; …; Rm ; …; RM Þ 8 0 if Rm > Rn for some n ≠ m > < Rm ≤ Rn ; ∀n and m is the number ¼ πm ðRm ; m Þ if of banks which set their rate equal to Rm > : π ðRm Þ if Rm b Rn ; ∀n ≠ m:
2
exists because
dL2j
Lemma. All banks offering the same rate R is an equilibrium if and only if πm R; M ≥π R . Proof. Upward deviations are not profitable as the bank looses market share and profit becomes null. In order to have a Bertrand equilibrium, downward deviations cannot be profitable either. Hence, πm R; M ≥ vm R for all m, where vm R ≡supfR′ bR g π R′ . Since function π(.) is ■ continuous then vm R ¼ maxR′ ∈½0;R π R′ ¼ π R . If there is an equilibrium in which all banks offer the same loan rate, then the loan rate must be equal to R0. This follows from the previous lemma because πm(R0,M) ≥ π(R0) = Mπm(R0,M) is equivalent to πm(R0, M) = 0. It follows that there is a unique equilibrium in which all banks offer a rate R0. Since π(R0) = 0 and L (R0) > 0, we obtain ■ R0 = [1 − 0.08α(L)] RD + 0.08α(L) RS. Under Basel II and Basel III there is no equilibrium in which banks set a loan rate above their marginal cost of funding. To see this, consider the case in which all banks set the same rate higher than their marginal cost. Two effects occur when one bank undercuts the loan rate slightly and unilaterally. First, it increases its market share substantially (the size of the loan to a firm doubles when there are two banks), thereby raising its profits. Second, there is a decrease in its unit profit because the rate charged decreases and the marginal cost increases. With a very small decrease in the loan rate, the reduction in the unit profit of the bank can be made arbitrarily small because the marginal cost does not change significantly. Hence the bank can make the second effect (almost) equal to zero, making a unilateral deviation profitable. 9 2.3. Cournot equilibrium The most interesting case occurs under the Cournot oligopoly structure. The market rate Rj is set at a level such that demand equals the total amount of loans supplied by banks, so that Rj = Rjm for all m. Consider a symmetric equilibrium in which all banks split the market equally, and let εL = −
^ be the minimum of such rates. the profit of the monopolist, let R Without loss of generality, we consider the case in which all banks offer the same rate. There is no equilibrium in which banks set a loan ^ because undercutting the loan rate would be profitable. rate above R ^ such that π(R 0) = 0. Lemma. There is one single rate R0 ≤R π ðRÞ . LðRÞ
be the elasticity of demand for loans.
Rj −ð1−0:08ÞRD −0:08RS 1 ¼ Rj Mε L
^ > 0, Since L R
then π(R 0) > 0. The unit profit function is continuous and strictly in ^ > 0 and πu(R D) ≤ 0, then there is a unique loan creasing. Since πu R
ð1Þ
under Basel I, and
≤0). If there is more than one rate that maximizes
Proof. Define the unit profit function as πu ðRÞ ¼
dLj Rj dRj Lj
Proposition 3. Let the banking industry be characterized by a Cournot equilibrium. The equilibrium loan rate to firm j satisfies
The vector (R1⁎, R2⁎,…, Rm⁎,…, Rm⁎) is a Nash equilibrium if Em(R1⁎, R2⁎,…, Rm⁎,…, Rm⁎) ≥ (R1, R2,…, Rm,…, Rm) for all Rm ≠ Rm⁎, ∀m. We have assumed that a monopolistic bank makes strictly positive ^ > 0 for which π R ^ > 0 (this rate profit, so that there is a finite rate R d Rj
605
1 ¼ Mε L
h i S D dα Rj − 1−0:08α Lj RD −0:08α Lj RS −0:08 R −R Ljm dLj
Rj
ð2Þ
under risk-based capital requirements. Proof. Each bank m chooses the loan Ljm that maximizes its profit with firm j, knowing that there is competition in quantities. Bank m solves h h i i D S max ∫ Rj Lj − 1−0:08α Lj R þ 0:08α Lj R Ljm dϒ fLjm g
rate R 0 such that πu(R 0) = 0. Hence, the loan rate R 0 is the unique ^ such that π(R 0) = πu(R 0) · L(R 0) = 0. rate below R
■
The key condition for the existence of an equilibrium is that banks cannot have incentives to change loan rates, and the next lemma formalizes this condition.
9 Dastidar (1995) finds equilibria with prices higher than the marginal cost because price undercuts may not be profitable. An arbitrarily small price undercut leads to a big increase in production, which raises the marginal costs substantially. The reduction in unit profit may offset the increase in market share, rendering the unilateral deviation unattractive.
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with the first order conditions are equal to h i D S S D dα L Rj Lj − 1−0:08α Lj R −0:08α Lj R −0:08 R −R dLj jm dRj ¼− L ∀j dLj jm because
dLj dLjm
¼ 1. After dividing both terms by Rj, we obtain expression
(2) because M ¼
Lj . Ljm
Under Basel I, α(Lj) = 1 and we obtain expres-
sion (1). The second order conditions of bank m are equal to ! dα d2 α dRj d2 Rj S D b 0 ∀j: þ 2 −0:08 R −R þ dLj dLj dL2j dLj ■ Banks set the loan rate as a markup over marginal cost, and the equations in Proposition 3 show that the Cournot oligopoly structure adds a new channel through which risk-based capital requirements affect the rate Rj. This channel does not exist under Basel I and is captured by the term 0.08dα/dLj(RS − RD)Ljm. In order to gain some intuition, consider a firm that borrowed a total amount of L⁎. In this case, the bank is required to hold the same amount of regulatory capital under both regimes (risk-based and non risk-based capital requirements) as α(L⁎) = 1. Yet, Eqs. (1) and (2) show that the intermediation margin on loans increases with the changeover from Basel I to risk-based capital requirements: each time firm j gets a new loan (e.g. Ljm from bank m) its leverage and, consequently, its risk of default increase, thereby increasing the credit risk weight assigned to that firm and requiring bank m to hold more bank capital (an effect captured by the term 0.08dα/dLj); when pricing the new loan, bank m takes this effect into consideration as well as the higher marginal cost of issuing bank capital when compared to deposits (as the term RS − RD is positive). That is, under Basel II and Basel III, banks internalize the effect of extending new loans on their cost of finance, an outcome which is absent under Basel I. The following corollary formalizes these results. Corollary 1. Let the banking industry be characterized by a Cournot oligopoly. Firms which obtain an amount of bank loans equal to L ∗ under Basel I, face an increase in the cost of funds with the changeover to risk-based capital requirements. Proof. Let the optimal loan of bank m to firm j under Basel I be equal L dLj M
dR to L⁎/M, so that Rj −½1−0:08RD −0:08RS þ j
¼ 0. The first order
conditions under Basel II and III are given by h i dRj dα S D S D L −0:08 R −R Ljm Rj − 1−0:08α Lj R −0:08α Lj R þ dLj jm dLj ¼ 0 ∀j: If we substitute Ljm by L⁎/M in the left hand side of the previous equation we obtain a negative value. The second order conditions, given by d Rj dRj dα S d2 α S D D L þ2 −0:16 R −R −0:08 2 R −R Ljm b 0 2 jm dLj dLj dLj dLj 2
∀j;
guarantee that the left hand side is decreasing in the loan Ljm. Hence the value of the loan that solves the bank’s optimization problem under Basel II and III is such that Ljm b L⁎/M and, since Rj is a decreasing function in ■ Lj, the cost of funds increases under Basel II and Basel III. By focusing on the case in which firms' leverage and risk of default are such that the amount of capital that banks must set aside for regulatory purposes remains the same under both regulatory regimes
(risk-based and non risk-based capital requirements), the above corollary proves that, under a Cournot oligopoly, banks charge higher interest rates under Basel II and Basel III. Two other cases remain to be analyzed. When Lj > L⁎ under Basel I, it is straightforward to show that the cost of funds for the firm increases under Basel II and III because α(Lj) > α(L⁎) = 1. When Lj b L⁎ under Basel I, the effect of a shift to risk-based capital requirements over the cost of funds is unclear. On the one hand, α(Lj) b α(L⁎) = 1, which requires the bank to hold less bank capital, thereby decreasing the cost of bank loans. On the other hand, the intermediation margin increases as can be demonstrated by comparing Eqs. (1) and (2). In the previous corollary we compared the loan rate for different Basel Accords. We now compare the equilibrium intermediation margin on bank loans. Corollary 2. Let the banking industry be characterized by a Cournot oligopoly. The intermediation margin on loans increases with the changeover from Basel I to risk-based capital requirements. Proof. The intermediation margins under Basel I and the new Basel Rj Rj dα and þ 0:08 RS −RD Ljm , respectively. For Accords equal MεL
MεL
dLj
firms with the same (strictly positive) leverage, the intermediation margin increases. The increase of the intermediation margin makes lending more expensive under the new Basel Accords. The overall effect depends on the risk of the firms because less risky loans require less bank capital which may lead to lower loan rates. Yet, there is a tendency towards an increase in the cost of funding under the new Basel Accords. 2.4. Differences in pricing under risk-based capital requirements Under the Cournot oligopoly, banks internalize the effect of extending new loans on their cost of finance while, under perfect competition and Bertrand competition, banks fail to internalize the effect of their actions on the credit risk weights and on their marginal cost of funds. We may thus conclude that the structure of the banking industry matters when assessing the effects of Basel II and, more recently, Basel III. In the Cournot oligopoly, when bank m extends a new loan to firm j, it imposes a negative externality on every other bank as it forces its peers to hold more capital. Bank m partially internalizes the negative externality as the cost of the externality, measured by 0.08dα/ dLj(R S − R D), is multiplied by Ljm = Lj/M. A monopolist bank would fully internalize the externality, as Ljm = Lj, while under perfect competition, as Ljm converges to zero, banks fail to acknowledge the externality loss. In Bertrand equilibrium, banks fail to internalize the negative effects of price undercutting. This is because undercutting loan rates has a large positive impact on revenues and little impact on costs, so that banks have incentives to undercut rates unilaterally as long as profits are positive. 2.5. Demand for loanable funds We have maintained the demand for bank loans constant under non-risk and risk-based capital requirements. To the extent that the firm’s demand for loanable funds accounts for the effect of leverage over funding costs, the demand curve shifts with the move from non-risk to risk-based capital requirements. The demand for bank loans is likely to become more inelastic because credit becomes more expensive for highly leveraged firms and cheaper for firms with stronger credit quality. The focus of this paper is on bank behavior and, therefore, we do not model explicitly the firm’s problem. Still, the assessment of risk-based
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capital requirements would benefit from a thorough analysis of the microfoundations of bank loan demand.
structure when defining the risk weights used to compute bank capital requirements should be taken into account.
3. Empirical evidence on banking market structure
References
In the previous section we showed that, under a Cournot-type oligopoly, the intermediation margin rises with the introduction of risk-based capital requirements, while under perfect competition or under a Bertrand-type oligopoly there is no such effect. Since the market structure in banking matters for the economic consequences of bank capital regulation, it is important to characterize loan markets in those countries where banks are subject to risk-based capital requirements. Much of the empirical literature uses the Monti-Klein model to assess the bank market structure (see Klein, 1971, and Monti, 1972). Since the Monti-Klein model can easily be interpreted as a model of Cournot competition between a finite number M of banks, most studies estimate the impact of proxies for market power – like the number of banks, the degree of concentration in the banking sector, or the market share – on intermediation margins, while controlling for differences in efficiency. Using US data, Hannan (1991) shows that concentration is associated with higher loan rates, while Berger (1995) finds that market share is positively related to profitability. A number of studies confirm the existence of market power in bank loan markets by explicitly estimating the first order conditions of the Cournot model of bank pricing. For instance, Corvoisier and Gropp (2002) find that increased concentration leads to collusion and higher interest margins on bank loans in the euro area, while Marrouch and Turk-Ariss (2012), using longitudinal data on banks for 103 non-OECD countries, find that markups are a key determinant in loan pricing. All in all, the empirical literature on credit market competition favors the market power theories and the Cournot model, ascertaining the importance of market structure for bank capital regulation. 4. Concluding remarks This paper adds to the research agenda on risk-based capital requirements by highlighting the effects that the structure of the banking industry has on the setting of loan interest rates. In particular, we have shown that, in bank loan markets characterized by a Cournot oligopoly, the intermediation margins increase with the changeover from nonrisk to risk-based capital requirements, while margins remain constant in competitive markets. Consequently, our analysis suggests the hypothesis that the overall impact of risk-based capital requirements on loan interest rates depends on the distribution of risk and leverage across firms and on the market structure of the banking sector. The effects of the implementation of risk-based capital requirements on the borrowers' cost of funds in a country with a more competitive banking structure may be significantly different from the effects in a country with an oligopolistic structure, and the consideration of the market
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