Capital Regulation, Strategic Behaviour of Banks and. Stability

Transcription

1 Capital Regulation, Strategic Behaviour of Banks and Stability Eva Schliephake Preliminary Version of March 1, Abstract This paper analyzes the impact of capital requirements on the competitive behavior of banks and the resulting implications for the stability of the banking sector. In order to prevent banks from taking excessive risk, minimum capital requirement regulation forces banks to refund a substantial amount of their investments with equity capital. This creates a buffer against loan losses but also increases the cost of funding. A resulting change in the loan interest rate can result in higher risk taking by the borrowers, which destabilizes the banking sector. This article argues that banks also strategically react to capital regulation. A certain capital structure may be costly to adapt in the short term. Therefore, banks are more likely to adapt their loan interest rate amounts than raising capital. This reduces competitive forces of loan interest rate competition. It is shown that the enhanced price setting power can reverse the effect capital requirements have on stability in Bertrand competition. By changing the competitive environment in the banking sector the capital requirement regulation can increase the riskiness of the banking sector even if the regulation would enhance stability under Bertrand competition. JEL classification: G21, K23, L13 Keywords: Capital Requirements, Risk Shifting, Endogenous Competition Corresponding author (Eva.Schliephake@ovgu.de), Scientific Assistant, Department of Economics and Management, Otto von Guericke University, Magdeburg. This research has been made possible by a grant (KL 1455/1 1) from the German Research Foundation (Deutsche Forschungsgemeinschaft) and by the Friedrich Naumann Stiftung für die Freiheit through grants from the Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung). 1

2 1 Introduction This paper analyses the effect of capital requirement regulation on the stability of banks in an imperfect loan market where few oligopolistic banks compete in loan interest rates and decide on optimal loan capacities. Since loans are assumed to be homogenous, price competition is fierce and banks undercut each other in loan rates until the interest payments equal the marginal refunding cost of the asset investment - the classical Bertrand result. Excessive risk taking occurs since the Bertrand banks do not internalize the downside risk of their loan investments, hence, loan interest rates are set even below the marginal expected refunding cost. As a result too many loans are supplied and banks take excessive risk compared to a fully liable competitors that fully internalize the probability of asset defaults. It is shown, that a minimum capital requirement that perfectly reflects the externalized downside risk can offset the excessive risk taking. However, regulating the capital structure also changes the competitive behaviour of the banks, i.e., the timing of strategic decisions. If banks prefer to adapt lending rather than the capital capital structure when they are confronted with a binding capital requirement, the regulation can change Bertrand competition into a two stage game where banks first choose optimal loan capacities and in the second stage loan interest rates are chosen to clear capacities. The idea that capital regulation creates a credible precommitment to loan interest rates and thereby reduces the intensity of interest rate competition was first discussed in Schliephake and Kirstein (2010). They develop a detailed analysis of the conditions under which capital requirement regulation changes the strategic interaction among oligopolistic banks from strategic complementarity in price setting to strategic substitutes in capacity choices. While showing that higher capital requirements reduce the incentives of banks to undercut in interest rates, their model neglects uncertainty and risk sensitivity of capital requirements. In contrast, this paper focuses on the impact of increased price setting power on the riskiness of the banking sector. The Cournotization effect, as we call the changed timing of strategic decisions due to capital regulation, results in loan interest rates above the Bertrand loan interest rates. This implies strictly positive profits for the 2

3 banks but also higher loan interest burdens to the borrowers. In order to analyze the effect of the Cournotization on the bank stability, it is necessary to establish the missing link between the literature on competition and stability as well as the capital regulation and stability literature. The literature the relationship between competition on stability can be roughly classified in two streams. The first stream argues that more competition erades stability because it reduces the charter value of the bank and therefore increases the incentives to take more risk as argued for example fby Matutes and Vives (2000), Hellmann et al. (2000), Repullo (2004) and Allen and Gale (2000). Moreover, Allen and Gale (2004) argue that reduction in the charter value of banks also decreases effort on monitoring, thereby further increasing the riskiness of the bank. However, these models take the investment risk of banks as exogenously given and just focus on the competition on the liability side of the bank. Boyd and De Nicolo (2005) (BdN) extend this framework by considering also for competition in the loan assetst and allowing borrowers to react to higher loan interest rates. Building on the seminal work of Stiglitz and Weiss (1981) their model assumes that not only the limited liability of banks gives rise to risk shifting, but also that risk of loan assets increases in the loan interest rate. Assuming perfect correlation of loan defaults, this extension actually reverses the conventional wisdom of higher competition leading to instability. If risk shifting takes place in the loan asset market, increased competition can actually reduce the probability of bank failure, since lower loan rates reduce loan asset risk. Recent work on the effect of competition on stability combines the both earlier approaches. Hakenes and Schnabel (2011) and Martinez-Miera and Repullo (2010) point out that the effect of competition on stability is ambiguous and depends on the correlation between loan defaults. Martinez-Miera and Repullo (2010) extend the BdN model and allow for imperfect correlation among the investment projects. They show that if not all loans default at the same time, the impact of competition on banking stability is generally non-monotonous. They show that, with imperfect correlation of loan default, an additional stability enhancing effect arises from reduced competition. Less aggres- 3

4 sive competition leads to higher loan rates, which increases the default risk of the loan (risk-shifting) but also leads to higher profits per non-defaulting loan. Similar contradictory are the results of the theoretical literature on the impact of minimum capital regulation on the stability of the banking sector. A bank refunding structure that contains substantial amounts of equity certainly has advantages considering the stability of the bank. Equity provides a buffer against unexpected shocks and hence reduces the states of nature in which a bank becomes illiquid or even insolvent. If banks would internalize all costs and benefits of bank capital, the banks private optimal capital structure would coincide with the socially optimal capital structure as it has been discussed by Gale (2003). Yet, banks do not fully internalize the benefits of equity since part of the risk is externalized to the safety net. Setting minimum capital requirements is a regulatory instrument of shifting the risks borne by the depositors or insured by the safety net back to the shareholders. Hence, moral hazard and the incentive of excessive risk taking is reduced as shown by Hellmann et al. (2000), Repullo (2004) and Allen et al. (2006). Furthermore, higher capital requirements reduce the risk of contagion among banks as pointed out by Allen and Carletti (2011). Generally, when capital is costly, an increase in capital requirement regulation has multiple effects that stabilize and destabilize the banking sector. Firstly, higher equity funding decreases the amount of deposit funding and therefore decreases states of nature in which a bank can fail (the buffer effect of equity that stabilizes the bank). Secondly, it increases the marginal cost of financing investments. These higher cost of funding decrease the banks profitability, which decreases the banks charter value, which gives incentives for higher risk taking (destabilizes the bank). At the same time, the higher cost of funding decreases the activity of the banks. Less activity means less lending, which results in increased loan interest rates. On the one hand, higher loan interest rate lead to higher earnings on non-defaulting loans, which can offset losses from defaulting loans (the margin effect that stabilizes the bank). On the other hand do higher loan rates reduce the earnings of borrowers, which induces risk shifting of project investments (the BdN effect that destabilizes the bank). The net effect of capital requirement regulation on banking stability depends on which effect of the induced higher equity funding prevails. 4

5 Empirical evidence for the relationship of capital requirements, competition and financial stability are similar ambigous. Carletti and Hartmann (2003) provide a good overview of the mixed empricial findings on the relationship between competition and stability, starting with Keeley (1990) who finds that the erosion in the US banks market power that resulted from deregulation caused an increase in the bank failure rates during the 1980 s. Similarly, Beck et al. (2006) provide evidence that more concentrated banking systems are less vulnerable to systemic risk because more concentrated banks tend to diversify their risks more. Schaeck et al. (2009) also come to the result that a more concentrated banking system is less fragile to systemic risk. Berger et al. (2009) find that though more competitive banking systems tend to take more risk but also compensate the higher risk with higher capital to asset ratios and are, thus, less fragile to systemic risk. Schaeck and Cihak (2011) find empirical evidence that a bank s capital structure is one of the channels through which competition may have an impact on the stability of the banking sector. However, there is little empirical evidence, that more stringent capital regulation actually improve the stability of a particular banking sector as pointed out by Barth et al. (2008). Building on the rather mixed theoretical prediction and empirical evidence, recent empirical work focuses on the role that the market and institutional environment plays in the determination of the relationship between competition and stability of the banking sector Beck et al. (2010) seize the theoretical suggestions on non-monotone relationships among competition and stability and try to identify the most prominent factors that determine the amplitude and direction of the relationship. Based on cross-country data they find that competition tends to be stability reducing the more stringent capital regulation is; the more restricted banking activities are; and the more homogenous the banking sector is. In praticular they find that more stringent capital regulation tends to have an amplyfying effect on the competition-stability relationship, regardless of the sign of the particular relationship. To the author s knowledge, there only exist two theoretical paper, that try to simultaneously analyze the that competition and capital regulation have on bank stability: Hakenes and Schnabel (2011) show that the ambiguous effect of competition bank s risk taking translates into ambiguous effects of capital requirement on the stability of the 5

6 banking sector. Though their model tries to capture the influence of correlation among loan defaults, the simplification of their model still assumes that either all loans default at the same time or no default at all occurs. Banks themselves can only influence the probability with which defaults occur. Hence, there is no positive marginal effect of higher profits from non-defaulting loans that can buffer losses from defaulting loans. Martinez- Miera (2009) analyzes the impact of capital requirement regulation on the bank failure under different market structures when loan defaults are imperfectly correlated. He argues that if the asset risk of the bank s loan portfolio is not perfectly correlated, capital requirements have ambiguous effects on the stability of a bank. In contrast do Martinez-Miera (2009), this paper does not take the competitive envirnonment as given but explicitly considers changes in the competitive structure due to strategic reaction of banks to the regulation. The Cournotization effect of capital requirement regulation then decreases the competitive forces, i.e., reduces the incentives to undercut competitors in loan interest rates. This leads to increased margins per loan, which again reinforces the two effects from increased loan interest rates: the margin effect, if loan defaults are not perfectly correlated and the risk shifting effect by borrowers. This paper shows that the net effect on banking stability depends on how much market power is gained by the Cournotization effect, which respectively depends on the number of oligopolistic competitors; the correlation of loan defaults and the intensity of the risk shifting if Cournotized banks pass the cost of higher capital requirement to borrowers or not. This in turn determines if capital requirement regulation of oligopolistic bank markets enhances or erodes the stability of the sector, when it simultaneously reduces the incentives to undercut in loan interest rates. The paper is organized as follows: In section 2 the basic model of oligopolistic loan interest rate competition and capital regulation is set up. Section 3 illustrates the Cournotization effect. In section 4 the model is extended to borrower risk shifting. Based on a single risk factor model the effects of Cournotization on the probability of bank failure is discussed with perfect correlation of loan defaults. Then the results are generalized to imperfect correlated defaults. Section 5 concludes. 6

7 2 Model Setup Consider a single-period model of n banking firms. In the beginning of the period each bank has access to deposit finance (D) at a constant cost (r D 0). Deposits are insured at a flat insurance premium, normalized to zero without loss of generality. Hence, the supply of deposits to bank is independent of the riskiness of the bank s asset investment. In alliance with the current regulatory system, this paper takes the existence of the fixedrate deposit insurance as given (explicit or implicit by the bail out policy). Each bank is run by a bank manager, who can acquire equity K from shareholders, which have an alternative (equally risky) investment opportunity with return r K = r D + c. This fixed opportunity cost reflect the higher cost of capital compared to the deposit insured lending. This assumption is not undisputed in the literature. In particular, Admati et al. (2010) elaborate the weaknesses of the assumption that bank capital costly to the society. Here, it is however only assumed that raising equity is relatively more costly to the specific bank, which can be interpreted as a consequence of the deposit insurance system. The insured depositors do not expect a risk premium while the liable equity investors do. Another interpretation is that higher opportunity cost compared to deposit funding reflects the additional benefits that deposits create to the depositors. The role of the bank as an financial intermediary is therefore welfare enhancing. Unnecessarily high capital requirement regulation would erode the bank s role of as a financial intermediary offering depositing services i.e. a capital to asset ration of 1 would not allow at all for financial intermediation in the sence of providing deposit services. Hence, equity is assumed to be costly to the bank and pure equity funding in our simple model setup would be inefficient in the absence of bank Moral Hazard. The banks invest their funds into standard debt contract loans L, which they attract by setting a loan rate r. The demand for loans is assumed to be inversely related to the offered loan interest rate Bank i faces the following demand when competing in prices. 7

8 L i = L(r i ) 1 m L( i) 0 if r i < min(r i ) if r i = min(r i ) if r i > min(r i ) Where min(r i ) denotes the infimum of the loan interest rates set by the competitors and m captures the subset of the n banks that actually choose the infimum loan interest rate. Borrowers that receive the loan, invest it into risky projects that give a return of y (1 + r D ) with probability (1 p) and that fail with default probability p, i.e., giving a return of zero to the borrower, who therefore fails to repay her loan. Projects cannot be liquidated, hence in case of failure the bank receives nothing, too. This simplifying assumption is not crucial and does not change any results as long as the liquidation value of the projects is below the equity capital share, the project is financed with. If the liquidation value is below the equity, the bank is perfectly safe and will never go bankrupt. We exclude this trivial case by setting the liquidation value equal to zero. Hence, banks always fail when all of their projects default unless they are fully equity financed. Therefore, the bank returns are min(r, y) with probability (1 p) and zero otherwise. Assume for the moment that project returns are perfectly correlated. 1 The balance sheet constraint is given with L = K + D (1) A (risk sensitive) minimum capital requirement is defined as the requirement to refund a specific proportion of assets (of a specific risk type) with equity βl K. Where β is the risk sensitive capital to asset ratio the regulator sets with β (p) < 0, lower risk of default (higher success probabilities) requires less capital. Since the opportunity cost of capital is above the cost of deposit funding, in equilibrium the constraint will always be binding: βl = K (2) 1 Later we will see that this is a quite strong assumption and correlation plays a crucial role for the relationship between capital requirement regulation, competition and stability of the banking sector. 8

9 . The capital requirement, therefore, constrains the balance sheet: D = (1 β) L. 2.1 How Capital Requirements Correct Risk Taking Incentives In this simple linear model, where equity is more costly than deposit fundingthe banks raise no equity K = 0 in the absence of capital regulation, thereby fully exploiting the subsidy on deposits from the deposit insurance. The subsidy is reflected in the cost p D that is externalized to the deposit insurance system. A bank hence has the following profit function: (1 p) ((1 + r)l(r) (1 + r D )D) In equilibrium, the balance sheet constraint 1 will be binding and reduces to L = D. The pricing competition in loan market drives down the loan price among banks to zero expected profits. Π = (1 p) L((1 + r)) ((1 + r) (1 + r D )) = 0 r = r D The expected loan default costs are externalized to the deposit insurance system i.e. prices are set too low, since they do not reflect the overall risk of the loan portfolio. A fully liable bank manager instead would price the loan with respect to the full risk: 0 = (1 p) L(r)(1 + r) (1 + r D ) hence, without a safety net, the loan price would be (1 + r) = (1 + r D) (1 p) In order to reduce the negative externalities of the safety net, a regulator forces the banks to refund its loan assets partially by costly equity. With binding capital requirements, the zero profit condition becomes 9

10 Π = (1 p) ((1 + r)l(r) (1 + r D )D) r K K = 0 (3) Under the assumptions made, the zero profit condition can be rewritten with the help of 1 and 2 to: Π = [(1 p) ((1 + r) (1 + r D )(1 β)) (1 + r K ) β] L((1 + r)) = 0. (4) With Bertrand competition, banks again undercut prices until they make zero profits resulting in equilibrium loan interest rate of. ( ) (1 + rk ) r = r D + β (1 p) (1 + r D) (5) A regulator that wants banks to set loan interest rates as if they were fully liable, wants to force banks to set (1+r) = (1+r D) p. Substituting this in 4 and solving for β gives β (p) = p(1 + r D ) (1 + r K ) (1 p) (1 + r D ) With β (p) p = deposit rate. (r D r K )(1+r D ) ((1+r K ) (1 p) (1+r D )) 2 < 0 for any cost of capital greater than the With perfect information and contingent contracting it would be possible to induce banks to act as if they were fully liable. The optimal risk sensitive capital requirement that offsets bank risk shifting is β (p) = p(1+r D ) (1+r K ) (1 p) (1+r D ). requirement, the Bertrand price in equilibrium would be (1 + r) = (1+r D) (1 p). Facing such a capital If the banks can raise unlimited equity at any time and deposits can be attracted at constant rates, the introduction of a carefully designed capital requirement implements the full liability behaviour of banks in perfect (or unconstraint Bertrand) competition. Competitive prices reflect the portfolio asset risk r = (1+r D) (1 p) the form of too many risky loans) is mitigated L(r) < L(r D ). and excessive risk taking (in 10

11 3 Cournotization of Bertrand Banks In line with the theoretical and empirical findings it is assumed that bank managers avoid to increase equity but adapt their asset portfolios when facing an regulatory equity shortage. 2 Assuming that there are prohibitive costs of recapitalizing immediately changes the sequence of decisions made and influences the competitive environment. The Bertrand competition among banks becomes a two stage decision making process, where in a first stage, the bank has to define the capital structure, in the second stage competition in loan interest rates takes place. t=1 According to minimum capital requirement regulation, banks choose optimal K i, i = 1...n, i j with K = n i K i. t=2 After observing the opponents K i bank i chooses optimal r i (r i, K) For simplicity we assume that capital can only be raised in t = 1 (the cost of immediate recapitalization is prohibitively high) 3. Note that the capital decision is sunk in stage 2. Hence, the (marginal) cost of capital in stage two is zero. Instead of influencing the marginal cost of investing in loan assets in the second stage, the regulatory minimum capital requirement sets an upper bound on the banks ability to supply loans (and thereby a lower bound to equilibrium loan interest rates as shown below). L i (r i (r i ), r i ) K i β(p) Let r ( ) be the inverse demand function that is decreasing and concave in the loan quantities supplied, i.e., r(0) > r D, r ( ) < 0 and r ( ) 0. The optimal prices chosen in the second stage therefore depend on the amount of capital raised by each bank in the first stage. 2 Anecdotical evidence, that capital constrained banks adapt assets rather than liabilities can be found during the recent financial crisis, where many banks faced difficulties in replacing lost capital in a timely fashion. Calomiris and Herring (2012) provide evidence that despite the urgent need to replace lost capital in 2008, financial institutions preferred to wait. They argue that stock prices were so low that the issuance of significant amounts of capital in order to cover the large losses incurred would have implied substantial dilution of stockholders including existing management. These observations suggest that bank managers try to avoid an immediate increase in capital in order to satisfy market demand and prefer to reduce the demand for loans by increasing the loan rate. 3 Schliephake and Kirstein (2010) allow for recapitalization in stage two and show that if recapitalization is costly enough, the capital raised in the first stage becomes a binding constraint of the second stage. 11

12 ( ) K r i (r i, K) r min β(p) In other words, for given capital it would never be profitable to undercut r min since this would imply a demand above the capacity, i.e. a demand that can not be served due to the regulatory restrictions; implying lower profits. Consider first the trivial case, where all banks have raised sufficient capital to serve the loan market demand at the Bertrand price with externalized downside risk. Formally, this means that the capital raised by banks in the first stage exceeds K β(p) > L(r D) The second stage pricing decision would just be the same as in the unregulated case. The fierce price competition is not constrained by the first stage capital decision. Hence, the non-profit condition would be 0 = p L(r) ((1 + r) (1 + r D )) r = r D Consider now the case, where the raised amount of capital is sufficiently small. With sufficiently small, it is assumed that the capacity to provide loans to borrowers from the first stage capital decision bindingly constraints the price competition L(r D ) > K β(p) and the capacities are so low, that (under any rationing rule) the remaining demand whenever r i < r min i is below the monopoly (optimal) loan output L M i ( K) > Ki β(p) L M i ( K)) is the monopoly loan amount in the residual market. for i, j, where 12

13 Lemma 1. (From Tirole (1988)) For sufficiently small capacities to lend that are set in the first stage, the second stage loan rate competition yields a unique Nash Equilibrium that is independent of any rationing rule, namely a loan interest rate that just clears ( ) capacities: r K β(p) Proof. If the infimum of the loan interest rates set by the competitors equals the capacity clearing interest rate, undercutting the opponent s price can never be profitable. Consider contrariwise the case where r i < r( K). Since recapitalization is assumed not to be possible in stage two, price undercutting would only lead to excess demand for loans, which cannot be served due to the binding minimum capital requirement. Undercutting in loan rates is not profitable, since both banks already lend their whole capacity to borrowers. Furthermore, a price increase is not profitable, since profits are assumed to be strictly convex in loan quantities and the capacity is assumed to be smaller than the residual demand monopoly quantity. For any r( K) < r i bank i receives the residual demand, after the m other banks loan applicants up to their capacity. The assumption L M i ( K) > Ki β(p) implies by definition Π(LM i ) > Π( Kj β(p) ). The inverse demand function ) (K i, K j )) < r. Since the resulting profit maximizing loan quantity implies r(l M j ( Kj β(p) in the residual market is higher than the small capacity, the respective profit maximizing loan rate must be lower than the constrained optimal loan rate that clears capacity. Hence, overbidding can never be profitable (Figure 1 illustrates this point) The underlying assumption that capacities are chosen sufficiently small in the first stage seem to be quite restrictive. One sufficient condition for low capacities would be very high cost of capital. Yet, the seminal work of Kreps and Scheinkman (1983) shows that for concave inverse demand function and efficient rationing of the residual demand, installing sufficiently low capacities is the unique sub-game perfect equilibrium of the two stage game regardless of the investment cost of capacity (cost of capital in this model). 4 Since, this model focuses on the effects of a change in the competitive structure induced 4 Davidson and Deneckere (1986) show that this result is not robust against different rationing rules. With alternative rationing rules, competitors find it optimal to build up capacities that are not sufficiently low but below the demand for selling the product at marginal cost. For capacities that are not sufficiently small sub-game perfect strategies only exist in mixed strategies. However, regardless the specific rationing rule, the separation of decisions into capacity buildup and price competition leads to reduced incentives to undercut in prices and, therefore, positive profits. 13

14 Figure 1: The overbidding competitor s profit with sufficiently low capacities. by capital requirement regulation on the riskiness of banks, it is assumed for simplicity, that borrowers are rationed according to the efficient rationing rule and accordingly that the result of Kreps and Scheinkman (1983) can be applied. Anticipating the second stage capacity clearing equilibrium, banks choose equity in the first stage that maximises stage objective function: max K i Π i = (1 p) ((1 + r)(l i + L j ) L i (1 + r D )D i ) (1 + r K )K i (6) s.t. L i = K i + D i 14

15 K i = β(p)l i This is the classical Cournot competition objective function, where (1+r K ) -the cost of capital - can be interpreted as the marginal cost of investing into loan capacity. Defining k(p) := 1 β(p), the first order condition is: ( ) Π i r( K) = (1 p) k K i + p ( (1 + r( K i K K)) k (1 + r D )(k 1) ) (1 + r K ) = 0. i The symmetric Cournot (Nash) Equilibrium is characterized by values of K i = K that simultaneously satisfy the system of first order conditions for all banks i = 1...n 5 : Where K := n K and h( K) = (r( K) [ r D ) + β h ( K) K + n h(k) = 0 (7) (1 + r D ) (1+r K) (1 p) ]. The first term reflects market power rents that result from the strategic commitment to Cournot capacities in the first stage. The term captures the effect of a decreasing demand that is taken into consideration when capacities are build up in the first stage. The second term reflects each banks expected payoff per unit of loans. The first order condition can be rewritten as r ( K) ( [ K n + (r( K) r D ) + β (1 + r D ) (1 + r ]) K) = 0 (1 p) Lemma 2. The Cournotization effect reflects an increase in market power and can be represented as a decrease n, i.e. n β < 0. Proof. If the number of competitors approach infinity, the market power term vanishes and the sub-game perfect outcome approaches the one stage Bertrand equilibrium outcome lim n r( K) ( ( )) = r D + β (1+rK ) (1 p) (1 + r D ) The effect of capital regulation on the strategic interaction can, hence, be approximated by an increase in the number of 5 The closed form can be written as r ( K) ( K k n + (r( K) ) r D ) k + (1 + r D ) (1+r K ) = 0 p 15

16 banks from infinity to some finite number n <. The lower the number of banks in the oligopolistic market, the lower the competitive forces in the two stage game and the higher the marginal profits compared to the Bertrand equilibrium. Lemma 3. If equity can not be changed in the short run, a binding minimum capital requirement that allows banks to strategically commit to loan capacities, ceteris paribus, leads to an underinvestment in risky loans compared to the intended Bertrand result. In equilibrium, loan interest are above regulated Bertrand interest rates and loan quantities are below regulated Bertrand quantities. Proof. Total differentiation of the first order condition 7 gives dk dn = h( K) h ( K) + (1 + n) h ( K) > 0 Clearly, since r ( K) < 0 it must be true that h ( K) = r ( K) < 0. Similar, h ( K) 0 results in r ( K) 0. Hence, the denominator is negative. Hence, the Cournotization effect that is approximated by a decrease in n leads to lower overall loan capacities than in the Bertrand outcome. Formally, for n < the Cournot Nash equilibrium loan quantity is lower L C (n) < L B, equilibrium loan interest rates are higher R C > R B and strictly positive profits are above zero Π C > Π B = 0. Proposition 1. An increase in capital requirements leads to lower investments in equity (loan capacity). This results in higher equilibrium loan interest rates compared to the Bertrand result. Proof. Differentiating the first order condition 7 and using the Proposition 2 gives d K n (β)(h( K)) [ ] + n(β) (1 + r D ) (1+r K) dβ = p h ( K) + (1 + n) h ( K) < 0 but then r ( K) < 0 implies that dr dβ = r ( K) dk dβ > 0. 16

17 Higher capital regulation reduces the equilibrium loan supply because capital is costly (classical cost effect) and because it increases market power, reflected in decreasing n which decreases the equilibrium loan supply even further. The capital requirement regulation that was designed to reduce the overinvestment in risky assets in order to internalize the risk leads to an underinvestment in risky assets. Other than inducing banks to invest in the optimal amount of assets the capital requirement enhances banks profits on the cost of borrowers that have to pay higher loan rates than intended by the regulation. If minimum capital requirement ratios do not take into account this Cournotization effect but solely the default risk of loan investments, risk sensitive capital requirements do not achieve the intended outcome of a regulator (the internalization of the downside risk). This is not new. Often, optimal capital requirement regulation is described to be a trade-off between financial stability and efficiency because it is meant to enhance stability at the cost of a reduction in lending. So what is left to be discussed is the question if the Cournotization effect of capital requirement regulation enhances or erodes the stability of the banking sector. 4 Cournotization and Risk-Shifting The previous discussion concentrated on the changes in competitive behaviour of banks when capital requirement regulation is tightened, while the risk taking behaviour of borrowers was assumed to be exogenously given. The results do not necessarily hold in a arguably more realistic environment, where not only banks react to changing cost of investments but also borrowers. Following the seminal work of Stiglitz and Weiss (1981) the optimal risk choice of borrowers varies with loan interest rates. Since not only banks, but also borrowers are limited liable and, hence, protected against downside risk of investments, higher loan interest rates, reduce the profitability of borrowers investment projects, which gives incentives to search for higher yields at the cost of higher riskiness of the project. The model is therefore, extended to the optimal responses of borrowers to differing loan rates resulting from tightening the capital requirement regulation. 17

18 The intuition is that the individual default probabilities of projects is partly controlled by the borrowers decision to control risk. This could either reflect a certain costly effort that borrowers spend to enhance the success of their projects or by the unobservable choice of the particular project the borrower invests in. The less profitable projects become that are financed by bank loans, the less effort borrowers are willing to spend and the lower are the success rates of their projects. This is in line with the idea of risk-shifting introduced by Boyd and De Nicolo (2005). However, the distribution of defaults in the actual portfolio held by a bank, which determines the default probability of a bank, does not only depend on the individual default probabilities but also on the correlation between the defaults of each loan. In other words, not only the probability of loan repayment of each loan is important for the stability of a bank but also how many loans default at the same time. 4.1 Borrower Risk Shifting and the Aggregate Loan Failure Rate The basic setup in this section follows Martinez-Miera and Repullo (2010). Consider a continuum of penniless entrepreneurs captured with i, who have access to risky projects of fixed size, normalized to 1. The entrepreneurs can spend effort on an individual alternative (e.g. employment) to obtain a utility level b[0, B]. The reservation utility is continuously distributed on [0, B] with the cumulative distribution function G(b). Let G(u) denote the measure of entrepreneurs that can obtain an alternative utility less or equal to u. In case of success, projects yield a risky return α(p j ) and zero otherwise. The component p i is the endogenously chosen probability of default and reflects the costly effort an entrepreneur spends on the project to enhance expected output. 6 Hence, a riskier project 6 Boyd et al. (2009) explicitly model the optimal effort choice of entrepreneurs: The projects yield an output of ỹ + z. The total return component y is random and distributed with the density function f(y) and the cumulative density F (y) on the closed interval [0, A], which is known by the bank and the borrowers. The component z is endogenous and reflects the costly effort the borrower is willing to spend on the project to enhance output. The effort cost is c(z) a strictly increasing, twice differentiable convex cost function.for a given contracted loan rate a borrower, hence, defaults whenever y y r z. Knowing the loan rate offered the entrepreneur chooses his optimal effort in order to maximize his expected profit: max z max z A + z r A y A y differentiation yields z R L(r) = (y + z r)f(y)dy c(z) Integrating by parts yields the objective function: F (y)dy c(z), resulting in the first order condition: 1 F (r z) = c (z). Total F (r z) F (r z) c (z) < 0. Higher loan rates imply less optimal effort, which 18

19 (less costly effort) yields to a higher success return to the borrower, i.e. α(p i ) is assumed to be positive, concave and increasing in p i. As inmartinez-miera and Repullo (2010) it is assumed here that α(0) < α (0) in order to get interior solutions. As before, the bank offers a standard debt contract with limited liability of borrowers: In case of project success with probability (1 p i ), the bank receives the contracted loan interest r and in case of default with probability p i the bank receives nothing, since the project s liquidation value is assumed to be zero. For any given loan rate, the entrepreneur maximizes his payoff: max u(r) = (1 p j )(α(p j ) r) (8) pi s.t. u(r) u i The first order condition is characterized with: (1 p j ) α (p j ) α(p j ) + r = 0 Which implicitly defines a unique default choice p j (r).7 Using the envelop theorem it can be shown that u(r) r = (1 p (r)) < 0. For any optimal effort choice, an increase in the loan rate decreases borrowers uility. L(r) denote the total loan demand, which exactly equals L(r) = G(u(r)). For any given loan rate r a measure of G(u(r)) obtains an alterntive utility less or equal to u(r) and, therefore, demands a loan.since G > 0 it is straightforward that the total demand for loans is decreasing in the loan interest rate. Total differentiation of the first order condition gives Let dp dr = 1 (1 p) α (p) 2α (p) > 0 An increase in loan interest rates increases the probability of default of the loan. This translates into higher risk. 7 The assumption α(0) > α (0) secures a unique interior solution for any loan interest rate in the intervall α(0) α (0) < r < α(1). 19

20 is what we call the BdN effect.. Since entrepreneurs are homogenous in their objective function, except for the exogenous reservation utility, all entrepreneurs will choose the same default probability p j (r) = p (r), {j u j u(r)} or opt for their outside option. A bank that is lending to L i (r) borrowers, faces individual loan defaults of p (r) in the portfolio. However, only if all project default at the same time (perfect correlation of defaults), the bank s portfolio risk of default is also equal to p (r). The very restrictive assumption of perfect correlation simplifies the analysis but disregards positive effects of increased price setting power. If not all loans default at the same time, a higher return on non-defaulting loans can offset more losses from defaulting loans. This positive effect on bank stability has been neglected in the analysis of Boyd and De Nicolo (2005). Martinez-Miera and Repullo (2010) explicitly allow for imperfect correlation and call the stability enhancing effect of market power the margin effect. To model imperfect correlation, they use a single risk factor model of Vasicek (2002). The Vasicek single factor model provides closed form loss rate distribution for large homogenous portfolio, which depends only on two parameters, a loan default rate and a factor ρ [0, 1] that measures the correlation among defaults. Since the single risk factor model is also used as the base model for the current capital requirement regulation in the Basel Accords, the risk model seems to be an appropriate choice in order to analyse the impact of the Cournotisation effect on the banking stability. Let the project s outcome be driven by the realization of a latent variable y j that is the sum of a deterministic term decreasing in p j, and two independent standard normally distributed risk factors: a systematic risk factor Z and an idiosyncratic risk factor ε j : y j = φ 1 (p j ) + ρ Z + 1 ρ ε j Where φ( ) refers to the cumulative distribution function of a standard normal random variable and φ 1 its inverse. If ρ = 1 all project defaults are perfectly correlated and depend only on the observed state of nature. If ρ = 0 there is no systematic risk and project defaults are perfectly uncorrelated. The project fails, whenever y j < 0. Hence, the 20

21 individual probability of failure is P [y j < 0] = [ ρ Z + 1 ρ εj < φ 1 (p j ) ]. Since the sum of two standard normal distributed variables itself is also a standard normally distributed variable this is equal to P [y j < 0] = φ [ φ 1 (p j ) ] = p j. The probability that the latent variable is negative equals to the individual probability of failure. Lemma 4. For a given optimal choice of p (r) and exogenous correlation among defaults ρ, the probability of observing a certain aggregate loan failure rate smaller or equal to x [0, 1] is ( F (x; p(r( K))) 1 ρ φ 1 (x) φ 1 ) (p) = φ ρ Proof. See Appendix An increase in the the optimal choice of default probabilities leads to a first-order dominance shift in the distribution of the failure rate, i.e. F p < 0. When all projects fail independently (ρ 0) there is no systematic but only idiosyncratic risk in the portfolio. By the law of large numbers the probability that the fraction of failures in the investment portfolio is equal to p is then equal to unity. Hence, provided that the return of (1 p) projects exceeds the liabilities to depositors, the perfectly diversified bank would never fail and always repay its liabilities. If the portfolio s project defaults are perfectly correlated (ρ 1) the whole portfolio fails with probability p and with probability 1 p no project fails giving the same return structure as if the bank invested only in one project. Hence, with probability (1 p) the bank fails and does not repay its depositors The Equilibrium Anticipating the risk shifting effect of increased loan rates on the aggregate portfolio default, the banks have to build up optimal loan capacities by raising optimal levels of equity. Similar to the above analysis, the banks take into account their limited liability in case of bank default. However, if project defaults are not perfectly correlated, the project s probability of default is not equal to the bank s probability of default. In particular, the first stage objective function can now be written as: 8 Note that if a bank could freely choose the correlation parameter ρ in this setup it will prefer to perfectly correlate the defaults of its portfolio in order to exploit the option value of the deposit insurance i.e. externalise (1 p) of the deposit refunding cost to the deposit insurance. 21

22 K i /β [max [ (1 x) ( (1 + r( K))) (1 + r D ) (1 β) ), 0 ] β(1 r K ) ] (9) A bank goes bankrupt whenever the revenue from non-defaulting loans can not compensate the liabilities to its depositors, i.e.: (1 ˆx) (1 + r( K)) (1 + r D ) (1 β) = 0 Which gives a critical default rate ˆx where the bank fails whenever more defaults occur than the critical rate. ˆx( K) = (r( K) r D ) + β(1 + r D ) 1 + r( K) (10) Note that ˆx β > 0, ˆx r D < 0 and ˆx ( K) = r ( K)(1 β)(1+r D ) (1+r( K)) 2 < 0 since r ( K) < 0. The critical failure rate above which banks fail, increases in capital requirements, decreases in r D and decreases in loans supplied, since the loan interest rate is decreasing in total loans supplied. The bank s objective function therefore can be written as: Π(K i, K i ) = K i 1 β. with ˆˆx Partial integration yields: 9 0 ( (r( K) rd ) + β(1 + r D ) x (1 + r( K)))dF (x, p(r( K)) ) (1 + r K ) ( ) 1 Π(K i, K i ) = K i β h( K, β) (1 + r K ) 9 Integration ( by parts yields [(r K ]ˆx i/β rd ) + β(1 + r D ) x (1 + r))f (x, p(r( K)) 0 ˆx 0 F (x, p(r( K) ) ( 1)(1 + r)dx β r K ( which reduces to K ˆx i/β 0 F (x, p(r( K) ) (1 + r)dx β r K. 22

23 ˆx( ˆ K,β) h( K, β) := F (x, p(r( K))) (1 + r( K))dx 0 Note that h( K, β) can be interpreted as a modified inverse demand function, that gives the expected revenue of a loan per capacity investment. It therefore takes not only into account the decreasing inverse demand but also risk shifting and accordingly higher default rates. Since the inverse demand as well as the risk shifting effect are decreasing and concave it is not necessarily the case that also the modified demand is decreasing and concave in K: K,β) + ˆx( 0 h ( K) := F (ˆx, p(r( K))) (1 + r) ˆx ( K) [ F (x, p(r( K))) F + (1 + r( K)) p p (r( K)) ] r ( K) dx We already know that ˆx ( K) < 0, hence, the first term is unambiguously negative: higher loan capacities reduce loan interest rates which decreases the bankruptcy rate of a bank. The sign of the term in the integral is, however, ambiguous. An increase in capacity decreases equilibrium loan interest rates in the second stage, which has direct effect on the return of non-defaulting loans (F (x, p(r( K))) is positive) but a reduction in the loan interest rate also decreases the loan default probability (risk shifting, p (r( K)) > 0) Which leads to a first-order stochastic dominance shift in the distribution of the failure rate ( F p < 0). In order to get interior solutions we make from here on the rather strong assumption that h ( K, β) < 0 and h ( K, β). With this assumption we can define the unique symmetric Nash Equilibrium with the first order condition: h ( K) K + n (h( K) β (1 + r K ) ) = 0 The question remains if an increase in capital requirement that also has a collusive effect leads to higher or lower aggregate lending capacities, when the banks also take into account the resulting risk shifting of their collusive behaviour. 23

24 4.3 The Impact of Capital Requirements on Banking Stability The probability of bank failure is the probability, with which the losses of simultaneously defaulting loans are bigger than the gain from the non-defaulting loans: q(r( K), β) = P [x > ˆx] = ( 1 F (ˆx(β, r( K)); p(r( K)) ) ( = φ φ 1 (p(r( K))) ) (11) 1 ρ φ 1 (ˆx(β,r( K))) ρ An increase of capital requirement effects this bank failure probability in three ways: dq(r( K(β)), β) dβ = q β ( K(β), β) }{{} Buffer( ) + q r ( K(β), β) }{{} dr dβ }{{} 0 Risk Shifting(+)+Margin( ) Cournotization(±) Higher equity has a direct buffer effect, which reduces the bank s probability of failure. It also has an indirect effect through the change in the loan interest rate. The increase in loan leads to a risk shifting effect where borrowers choose higher default risk as an reaction to higher interest rates and a margin effect which implies that higher interest rates return higher margins on non-defaulting loans and therefore stabilize the banks if the loan defaults are not perfectly correlated. For a detailed derivation of this result see the Appendix (2) and Martinez-Miera (2009). It becomes clear that the impact of capital requirement regulation on the stability of the banking system depends on on the one hand on the relative size of the risk shifting effect and the margin effect, which depends on the correlation of project failures and on the effect it has on interest rate setting. Lemma 5. (from Martinez-Miera (2009)) In Bertrand competition, an increase in capital requirements increases the equilibrium loan interest rate: dr dβ > 0. Proof. A symmetric Bertrand equilibrium is characterized by ˆβ := K L(ˆr) (each unit of loan is exactly financed by β equity capital) and an equilibrium interest rate r = ˆr such that Π(ˆr, ˆβ) 0 and r < ˆr Π(r, ˆβ) > 0. The Bertrand price undercutting argument then implies that Π(ˆr, ˆβ) = 0. Equity financing is more costly that deposit financing. 24

25 Therefore, for a capital structure β < K L(r) it must hold for any loan interest rate that Π(r, ˆβ) K > Π(r, L(r) ). Hence, banks will never finance their loans with more equity than required by regulation. Similar, an increase in the minimum capital requirement ˆβ < β implies that Π(ˆr, ˆβ) > Π(ˆr, β). But Π(ˆr, ˆβ) = 0 implies Π(ˆr, β) < 0, therefore, ˆr cannot be the equilibrium interest rate (no banking industry can exist with negative profits). The definition of a Bertrand equilibrium interest rate requires r < ˆr Π(r, ˆβ) > 0. Since Π(ˆr, ˆβ) > Π(ˆr, β), a fortiori, r < ˆr Π(r, β) > 0 implying that r > ˆr Proposition 2. If banks can recapitalize without any cost, increasing capital requirements leads to an increase in the probability of bank failure when loan defaults are perfectly correlated. With imperfect correlation the impact is ambiguous. Proof. The buffer effect of equity funding is: ( ) ( 1 ρ φ 1 (p) 1 ρ φ 1 (ˆx) φ ρ ρ ) ( dφ 1 (ˆx) dˆx ) 1 + rd 1 + r( K) < 0 The effect of a change in the loan interest rate on the bank stability can be summarized as: ( ) ( 1 ρ φ φ 1 (p) 1 ρ φ 1 (ˆx) ρ q r ( K(β), β) = ) (( dφ 1 (p) dp ) p (r) 1 ρ ( ) ) dφ 1 (ˆx) dˆx ˆx (r) It is easy to see that the correlation between bank failures determines which effect prevails. With perfect correlation (ρ 1) the margin effect as well as the buffer effect disappear. The only effect that remains is the risk shifting effect. If (ρ 1) we obtain: dq(r( K(β)), β) dβ ( ) ( 1 φ 1 (p) 1 ρ φ 1 ) ( (ˆx) dφ 1 ) (p) = ρ φ p (r) ρ dp }{{} >0 dr dβ Whenever dr dq(r( K(β)),β) dβ > 0, dβ > 0: the bank s probability of failure increases in capital 25