Competition and risk taking in a differentiated banking sector Martín Basurto Arriaga Tippie College of Business, University of Iowa Iowa City, IA 54-1994 Kaniṣka Dam Centro de Investigación y Docencia Económicas Carretera México-Toluca 3655, Mexico City, Mexico. Abstract We re-examine the relationship between the degree of deposit market competition and bank risk taking in a model where banks compete in differentiated deposit services. When banks invest their deposits directly, as has already been established in the extant literature, an increased degree of competition, measured either by greater degree of substitutability or by greater number of banks, induces the banks to take more risk in equilibrium. When banks invest their deposits in loans, and their borrowers choose the level of risk, the risk of bank failure depends on the number of banks in the economy, but is independent of the degree of substitutability in the deposit market. We further show that the relationship between risk and the number of banks may even be non-monotone. JEL classification: D40, D8, G1 Keywords: Bank competition; risk taking; loan contracts. 1. Introduction It is well-known that when banks are able to invest their deposits directly in risky projects, increased degree of deposit market competition induces banks to take more risk (e.g. Matutes and Vives, 1996, 000; Repullo, 004). A lower intermediation margin, implied by more competition in deposit market, incentivizes banks to take more risk as riskier projects, if successful, yield higher returns. This is due to the implied moral hazard problem since if a project fails the bank is not required to repay its depositors since it is protected by limited liability. 1 Boyd and De Nicoló (005), in their seminal paper, show that We owe thanks to Antonio Jiménez for helpful comments. Corresponding author. Email addresses: martinalberto-basurtoarriaga@uiowa.edu (Martín Basurto Arriaga), kaniska.dam@cide.edu (Kaniṣka Dam) 1 In a dynamic model of competition it is often argued that low charter value due to increased competition is the main reason for banks to get involved in high-risk investments (e.g. Hellmann et al., 000; Repullo, 004).
when banks are able to invest their deposits only in loans, the so-called positive relationship between competition and risk is reversed. In particular, they consider a banking sector where financial intermediaries compete in homogeneous deposit and loan services. Since in a lending relationship the borrowers face the moral hazard problems in choice of risk, incentive compatibility implies a positive association between the level of risk and loan rate. As greater competition induces lower rates in equilibrium, this in turn diminishes the risk of bank failure, which is called the risk-shifting effect of competition. We analyze a simple and tractable model of banking sector where banks compete in differentiated deposit services. Thus, both the degree of substitutability and the number of banks measure the degree of deposit market competition. When banks can directly invest their deposits, we find the usual positive relationship between competition and risk taking the greater the degree of substitutability or the greater the number of banks, higher is the risk taking in equilibrium. We next introduce a loan market where each bank lends its deposit to a borrower, and the probabilities of failure of the borrowers projects are correlated. We show that the equilibrium risk shifting depends on the number of banks, but does not depend on the degree of substitutability in the deposit market, and hence Boyd and De Nicoló s (005) result crucially depends on how the degree of competition is measured. The equilibrium risk shifting becomes independent of the number of banks if the borrowers projects are uncorrelated. We further show that, unlike Boyd and De Nicoló (005), the equilibrium risk shifting may even be non-monotonic in the number of banks in the economy. If the competition is in differentiated deposit and loan services, the degrees of substitutability of deposits and loans are also appropriate measures of competition in the respective markets, and the degree of substitutability in one market is not necessarily the same as that in the other. Dam et al. (014) show that in such a context the equilibrium risk depends on the level of competition of the loan market only. Martínez-Miera and Repullo (010) extends the model of Boyd and De Nicoló (005) by introducing correlation in loan defaults. They identify two countervailing effects: in addition to the usual riskshifting effect there is a margin effect which implies that lower loan rates caused by greater competition decrease banks revenues from performing loans, which would provide a buffer against loan losses. As a consequence, greater competition implies higher risk. Thus, there is in general a non-monotone relationship between competition and risk of bank failure. The main contribution of the present paper lies in providing a unified framework for deposit market competition where both the degree of substitutability and number of competitors measure the level of competition. Therefore, the behavior of equilibrium risk of bank failure with respect to competition crucially depends on the appropriate measure of competition.. The model Consider a banking sector consisting of two classes of risk neutral agents: a representative depositor and n banks. The economy lasts for 3 dates. At t = 0, banks simultaneously raise deposits by offering deposit rates r = (r 1,..., r n ). The supply of deposits with bank i is thus given by D i (r i, r i ), where r i = (r 1,..., r i 1, r i+1,..., r n ) with r i > 1 for all i. Deposit services offered by banks are differentiated with the degree of substitutability δ (0, 1). The inverse supply function of deposits of bank i is given Oligopolistic competition in differentiated banking products has been modeled earlier in the literature. Matutes and Vives (1996, 000) have used a linear-city model à la Hotelling, whereas Chiappori et al. (1995); Repullo (004); Dam et al. (014) have analyzed banking competition in a circular-city model à la Salop (1979).
by: 3 r i = 1 + D i + δd i, where D i = j i D j. Therefore, the supply of deposits with bank i is given by: D(r i, r i ) = (1 δ) + [1 + δ(n )]r i δr i (δ,n) where (δ, n) := (1 δ)[1 + δ(n 1)]. The degree of deposit market competition is measured by two parameters: the degree of substitutability δ and the number of banks n. At date 1, banks simultaneously choose their investment strategies. Each bank i has access to a continuum of constant-returns-to-scale risky technologies indexed by θ i [0, 1]. In particular, given an investment I, technology θ i yields: { (α + θ i )I with probability 1 θ i, f (I, θ i ) = (1) 0 with probability θ i, with α [1, 3]. The assumption that α 1 implies that a safe project viable, whereas α 3 does not rule out a risky project to be undertaken. Note that higher values of θ i imply greater per unit return, but at the same time, higher probability of failure, and hence a riskier project. 4 The choice of risk by each bank cannot be observed by the depositor, and hence banks face a potential moral hazard problem. Deposit contracts are subject to the banks limited liability meaning that in the case the project of a bank i fails it is not obliged to pay back the depositor. As a consequence, banks may opt for riskier project since it earns higher income with a positive probability. Such incentive problems of the banks will be crucial for the analysis of the deposit market competition where banks choose their own investment strategies. We assume, without loss of generality, that the deposits are fully insured by the central banking authority or by some insurer, e.g. the FDIC who charges a flat per unit insurance premium which is normalized to zero. Finally at t = 3, the project returns are realized and the depositor is paid off., 3. The deposit market equilibrium We analyze a symmetric subgame perfect Nash equilibrium (SPNE) of the deposit market game in which all banks choose the same levels of risk θ and the same deposit rates r. 5 At t = 1, given the supply of deposits D i (r i, r i ), each bank i solves max (1 θ i)(α + θ i r i )D i (r i, r) (M i ) {r i,θ i } subject to θ i = argmax ˆθ i { (1 ˆθ i )(α + ˆθ i r i )D i (r i, r) } = 1 (1 α + r i) θ(r i ). (ICB i ) by: 3 The following supply function may be derived from a quadratic indirect utility function of the depositor which is given n V = r i D i i=1 ( n i=1 D i + 1 n i=1d n i + δ D i D j ). i=1 j i 4 All our results hold good if we have assumed instead a decreasing-returns-to-scale technology (α + θ i )I γ with γ (0, 1], and a probability of success function π(θ i ) where π (θ i ) < 0 and π (θ i ) 0 for all θ i [0, θ] with π(0) = 1 and π( θ) = 0. 5 Since the banks are ex-ante identical, it is easy to show that there is a unique symmetric SPNE, and there are no asymmetric equilibria. 3
Constraint (ICB i ) is the incentive compatibility constraint of bank i which asserts that the bank will choose the risk level in order to maximize its expected payoff from deposit services given the deposit rates announced by all banks at the previous stage. The following proposition analyzes the behavior of the equilibrium deposit rate and risk taking with respect to the degree of deposit market competition. Proposition 1 Let r(δ, n) and θ(δ, n) be the deposit rate and risk shifting, respectively in the symmetric SPNE. (a) For a given number of banks n, the equilibrium deposit rate and risk level are monotonically increasing in the degree of substitutability δ. (b) For a given degree of substitutability δ, the equilibrium deposit rate and risk level are monotonically increasing in the number of banks n. The proofs of the above proposition and the other subsequent results are relegated to the appendix. Recall that we measure the degree of competition either by the degree of substitutability or by the number of banks. When competition increases, in order to attract more deposits each bank offers a higher deposit rate. As the intermediation margin m(δ, n) := α + θ(δ, n) r(δ, n) of each bank is lower with greater degree of competition, banks tend to take more risk in the expectation of maintaining a positive margin. Note that the intermediation margin is monotonically decreasing in both δ and n with lim δ 1 m(δ, n) = lim n m(δ, n) = 0 as δ 1 corresponds to perfect substitutability of deposit services, and n implies a perfectly competitive banking sector. The above results can be seen as a generalization of the results already established in the literature in the sense that risk taking in a banking sector is positively correlated with the degree of competition in the deposit market irrespective of how the degree of competition is measured. 4. Effect of loan contracts on risk taking Boyd and De Nicoló (005) show that when banks invest in loans, and their borrowers are the one who choose the riskiness of the projects, the well-known positive relationship between competition and risk taking is reversed. They consider an economy where banks compete in homogeneous deposit and loan services, and hence the number of banks n measures the degree of competition in both markets. They show that risk taking by banks decreases with the number of banks in the industry. In what follows we argue that the correlation between the equilibrium risk shifting and the intensity of the deposit market competition depends crucially on how one measures the degree of competition, and thus the conclusions of Boyd and De Nicoló (005) may not always hold good. In order to analyze the effect to loan contracts on risk taking, add two more dates to the initial timeline described in Section. After each bank i has its deposits D i in date 1, in the following date bank i lends up to D i at a loan rate ρ i to a risk neutral borrower/entrepreneur whom, without loss of generality, we refer to as entrepreneur i. 6 All entrepreneurs own the same technology as in (1), but now entrepreneur i faces a probability of failure π i (θ i,..., θ n ) which depends on the risk taking behavior of the rival banks. We assume the following functional form: π i (θ i,..., θ n ) = θ i + γθ i where θ i = θ j and γ [0, 1]. j i 6 Each bank i can also lend to a continuum of identical borrowers in which case the analysis would be similar. 4
The parameter γ measures the correlation of bank failures. We further assume that money market yields zero return so that each bank will invest all its deposits in loans. This is without loss of generality. At date 3, when the loan obligations are paid off, the bank pays back r i to the depositor. Both the borrowers and banks are protected by limited liability, i.e., if the project of the borrower of bank i fails neither the bank nor the depositor is repaid. Further, the entrepreneurs, without loss of generality, are assumed not to have direct access to money market. Given the constant returns to scale production technology thus each entrepreneur i will invest the entire loan obtain in the project, i.e., I i = D i. An optimal loan contract ρ i and risk level θ i for the bank-borrower relationship i solve the following maximization problem: max [1 π i(θ i, θ i )]ρ i D i D i (M i ) {θ i,ρ i } subject to θ i = argmax ˆθ i [1 π i (θ i, θ i )](α + ˆθ i ρ i )D i, (ICE i ) [1 π i (θ i, θ i )](α + θ i ρ i )D i 0. (IRE i ) A bank cannot observe the choice of risk by its borrower, and hence limited liability implies a moral hazard problem in risk taking. Borrower i will choose the risk level in order to maximize her expected utility which is asserted by the incentive compatibility constraint (ICE i ). Solving simultaneously the incentive constraints for n borrowers we get the optimal risk shifting of each borrower i as a function of the loan rates (ρ i, ρ i ), which is given by: θ i (ρ i, ρ i ) = ( γ)(1 α) + [ + (n )γ]ρ i γρ i ( γ)[ + (n 1)γ], (ICE i ) where ρ i = j i ρ j. Note that the optimal risk shifting of entrepreneur i is strictly increasing in ρ i since n, and strictly decreasing in ρ j for any j i. The constraint (IRE i ) is the individual rationality constraint of entrepreneur i whose outside option is normalized to zero. Given the incentive constraint (ICE i ), it is easy to show that (IRE i ) is automatically satisfied for each borrower i, and hence can be disregarded. The above maximization problem yields the optimal loan rate and risk level which are described in the following lemma. Proposition The equilibrium loan rates charged to and risk shifting chosen by the entrepreneurs are given by: ρ i = ρ(n) = θ i = θ(n) = 1 α + ρ(n) + (n 1)γ ( γ)[1 + α(1 + (n 1)γ)] 4 5γ + γ + γ(3 γ)n, () for all i = 1,..., n. (3) Moreover, the equilibrium loan rate and risk shifting depend on the number of banks, but are independent of the degree of substitutability in the deposit market. Recall that the degree of banking competition is measured by two parameters: δ and n. The above proposition contradicts Boyd and De Nicoló s (005) main finding in the following sense. If the intensity of competition is measured by the degree of substitutability in the deposit market, then risk taking in the banking sector is independent of the degree of competition. This is natural since, given the number of banks, the entrepreneurial choice of risk should depend only on the characteristics of the loan market, 5
and not on the fundamentals of the deposit market. On the other hand, since the probabilities of failure are correlated across borrowers, the equilibrium loan rate and risk shifting depend on the number of borrowers, and hence on the number of banks, which also measure the degree of deposit market competition. But such dependence is driven purely by the fact that γ > 0. If projects are not correlated, i.e., γ = 0, then we have ρ(n) = 1 + α, and θ(n) = 3 α. 4 In other words, the equilibrium loan rate and risk shifting both become independent of n. Now, the nature of dependence of the equilibrium risk shifting on n is in general ambiguous. To see this, differentiate () to get ρ (n) 0 if γ γ(α) 3 α. Note that γ(α) [0, 1] since 1 α 3. Now from (3) it follows that sign[θ (n)] = sign[ρ (n) γθ(n)]. Clearly, when γ γ(α), we have θ (n) < 0 which conforms to the finding of Boyd and De Nicoló (005). What happens when γ > γ(α). We will use an argument of continuity to show that for certain parametric configurations θ(n) is non-monotonic in n. Consider the case when γ = 1. Then, θ(n) = + n α (n + 1) = θ α (n + 3) (n) = (n + 1) 3. When n > 3, we have θ (n) < 0 since α 3. On the other hand, when n 3 and α > n + 3, we obtain θ (n) > 0. Since this property holds for γ = 1, by continuity it must hold for values of γ close to 1. Therefore, Proposition 3 Let θ(n) be the equilibrium risk shifting. (a) When γ γ(α), the equilibrium risk shifting is decreasing in the number of banks; (b) When γ 1 and n > 3, the equilibrium risk shifting is decreasing in the number of banks; (c) When γ 1 and n 3, the equilibrium risk shifting is increasing in the number of banks if α > n + 3. When the correlation of project failure is low enough the usual risk-shifting effect of Boyd and De Nicoló (005) implies that the equilibrium risk shifting decreases with the number of banks in the economy. From the expression of θ(n) it is easy to check that for high values of α the equilibrium risk shifting is increasing in γ. Therefore, when the correlation is almost perfect this effect dominates the usual risk-shifting effect, and hence θ (n) > 0. 5. Concluding remarks In this paper the main reason for introducing loan contracts is to show that the deposit and loan two markets are in practice dichotomous. Thus, the optimal behavior of the borrowers does not take into account the fundamentals of the deposit market. The equilibrium risk shifting depends on the number of banks only if the projects of the borrowers are correlated. This result should be interpreted carefully. We implicitly assume that there are more entrepreneurs with the need of funds to finance their projects 6
than banks. Thus, increasing the number of banks implies that more entrepreneurs are able to obtain financing. When projects are correlated, the dependence of the equilibrium risk shifting on the number of banks is actually driven by the correlation between the equilibrium risk shifting and the number of borrowers, which happens to equal to the number of banks in the present context. If there were m identical borrowers and n banks in the economy with m > n, and if each bank would finance m/n projects, then the equilibrium risk shifting would depend on m, the total number borrowers, and the number of banks would have been irrelevant in the context of risk taking. Appendix Proof of Proposition 1 Consider the maximization problem (M i ) of bank i. Substituting for θ = θ(r i ) into the bank s objective function, its expected profit at t = 0 reduces to: ( ) 1 + α ri Π i (r i, r) [1 θ(r i )][(α + θ(r i ) r i ]D i (r i, r) = D i (r i, r), where r is the deposit rate offered by each of the rival banks in a symmetric equilibrium. Therefore, each bank i chooses r i to maximize Π i (r i, r). The first-order condition of the above maximization problem yields the best reply function r i (r) of bank i which is given by: 1 [1 + α r i(r)][1 + δ(n )] = (1 δ) + [1 + δ(n )]r i (r) δ(n 1)r. Thus, putting r i = r = r(δ, n), we get r(δ, n) = The optimal risk level is thus given by: θ(δ, n) = 1 α + r(δ, n) Differentiating (4) with respect to δ we get r(δ, n) δ = (1 δ) + (1 + α)[1 + δ(n )]. (4) (1 δ) + [1 + δ(n )] = ( α)(1 δ) + [1 + δ(n )]. (5) (1 δ) + [1 + δ(n )] (r(δ, n) 1) + (n )(1 + α r(δ, n)) 3(1 δ) + (n 1)δ The above expression is strictly positive since δ (0, 1) and n. Since θ(δ, n) = θ(r(δ, n)), we have θ(δ, n) δ = θ r(δ, n) (r(δ, n)) = 1 r(δ, n) > 0. δ δ To prove part (b) of the proposition differentiate (4) with respect to n. This yields r(δ, n) n = δ[1 + α r(δ, n)] 3(1 δ) + (n 1)δ. (6) 7
Note that (4) can be written as r(δ, n) = 1 + α 1 + (1 δ) 1+δ(n ) < 1 + α. (7) Therefore, the expression in (6) is strictly positive. Now, similar to part (a) we have which completes the proof of part (b). θ(δ, n) δ = 1 r(δ, n) δ > 0, Proof of Proposition The first order condition of the borrower s maximization problem (ICE i ) is given by: θ i + γθ i = ( γ)θ i + γs θ = 1 α + ρ i, (8) where S θ n i=1 θ i. Summing the above equation over i, we get S θ = n(1 α) + n i=1 ρ i + (n 1)γ Substituting the above into (8) we obtain the expression for θ i (ρ i, ρ i ) given by (ICE i ). Define π i (ρ i, ρ i ) θ i (ρ i, ρ i ) + γ θ j (ρ i, ρ i ). j i. Note that π i = θ i θ j + (n )γ (n 1)γ + γ ρ i ρ i =. j i ρ i ( γ)[ + (n 1)γ] Bank i s maximization problem thus reduces to: max ρ i [1 π i (ρ i, ρ i )]ρ i D i D i. The first order condition of the above maximization problem is given by: 1 π i (ρ i, ρ i ) ρ i π i ρ i (ρ i, ρ i ) = 0. In the symmetric equilibrium, ρ i = ρ(n) for all i = 1,..., n. Substituting this into the above first order condition we get the expression for ρ(n) in Proposition. The expression for θ(n) follows from the incentive constraint (ICE i ) by substituting ρ i = ρ(n) and ρ i = (n 1)ρ(n). 8
Proof of Proposition 3 Differentiation of () with respect to n implies From (3) it follows that sign[ρ (n)] = sign[γ + α 3] = sign[γ γ(α)]. [ + (n 1)γ]θ (n) = ρ (n) γθ(n) = sign[θ (n)] = sign[ρ (n) γθ(n)]. Clearly for γ γ(α), we have ρ (n) < 0 which implies θ (n) < 0. Therefore, by continuity, θ (n) < 0 for values of γ in a small neighborhood of γ(α). Now we look for conditions so that there is γ ( γ(α), 1] for which θ (n) 0. Note that at γ = 1, we have θ(n) = + n α (n + 1) = θ α (n + 3) (n) = (n + 1) 3. So, θ (n) 0 if α (n + 3)/. Since α 3, this cannot hold if n > 3. Therefore, only for high values of α we have θ (n) 0. Since this is true for γ = 1 it is also true for γ close to 1. This completes the proof of the proposition. References Boyd, J. and G. De Nicoló (005), The theory of bank risk taking and competition revisited. The Journal of Finance, 60, 139 1343. Chiappori, P.-A., D. Pérez-Castrillo, and T. Verdier (1995), Spatial competition in the banking system: localization, cross subsidies and the regulation of deposit rates. European Economic Review, 39, 889 918. Dam, K., M. Escrihuela-Villar, and S. Sánchez-Pagés (014), On the relationship between market power and bank risk taking. Forthcoming in Journal of Economics. DOI 10.1007/s0071-013-0389-6. Hellmann, T., C. Murdock, and J. Stiglitz (000), Liberalization, moral hazard in banking, and prudential regulation: are capital requirements enough? The American Economic Review, 90, 147 165. Martínez-Miera, D. and R. Repullo (010), Does competition reduce the risk of bank failure? Review of Financial Studies, 3, 3638 3664. Matutes, C. and X. Vives (1996), Competition for deposits, fragility, and insurance. Journal of Financial Intermediation, 5, 184 16. Matutes, C. and X. Vives (000), Imperfect competition, risk taking, and regulation in banking. European Economic Review, 44, 1 34. Repullo, R. (004), Capital requirements, market power, and risk-taking in banking. Journal of Financial Intermediation, 13, 156 18. Salop, S. (1979), Monopolistic competition with outside goods. Bell Journal of Economics, 10, 141 156. 9