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Can subordinated debt constrain banks’ risk taking?
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Journal of Banking & Finance 32 (2008) 1110–1119 www.elsevier.com/locate/jbf
Can subordinated debt constrain banks’ risk taking? Jijun Niu
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Faculty of Business Administration, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada V5A 1S6 Received 4 July 2006; accepted 18 September 2007 Available online 1 October 2007
Abstract This paper presents a model in which requiring banks to issue a proper amount of subordinated debt can constrain their risk taking both before and after debt issuance. The main idea is that the prospect of issuing debt motivates banks to invest in safe assets before debt issuance; holding such assets then constrains their risk taking after debt issuance. The model helps understand the existing empirical findings, and offers a new testable prediction. It also suggests that: (1) regulators should set the amount of subordinated debt within a range; and (2) subordinated debt cannot entirely substitute for equity capital. Ó 2007 Elsevier B.V. All rights reserved. JEL classification: G21; G28 Keywords: Subordinated debt; Risk taking; Bank regulation; Market discipline
1. Introduction Many economists believe that requiring banks to issue some amount of subordinated debt can constrain their risk taking. The idea seems intuitive: if creditors charge riskier banks a higher interest rate, banks would think twice before taking excessive risk. This idea is referred to as direct market discipline.1 Empirical studies, however, seem to have produced conflicting findings. Some researchers examine cross-sectional data. They find that creditors indeed charge riskier banks a higher interest rate, and conclude that subordinated debt can constrain banks’ risk taking (see, e.g., Covitz et al., 2004; Morgan and Stiroh, 2001 and Sironi, 2003). Others look at time-series data. They find no change of the banks’ risk-taking behavior before and after debt issuances (see
Krishnan et al., 2005).2 When does risk reduction occur, then, if subordinated debt can constrain banks’ risk taking? To answer this question, we propose a theoretical model. The model studies how a bank chooses between two types of assets: safe or risky. A safe asset has a higher expected return, but a risky asset provides a higher return when it succeeds. The bank first chooses an asset to invest its existing funds. It then raises some new funds by issuing insured deposits and subordinated debt. After that the bank chooses another asset to invest its new funds. At the time of debt issuance creditors can observe which type of asset the bank has already invested in, but they cannot contract on the bank’s future asset choice. The bank pays a flat-rate deposit insurance premium, and defaults when both of its assets fail. We show that there exists a range of the amount of subordinated debt such that below this range, the bank invests
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Tel.: +1 778 782 4491; fax: +1 778 782 4920. E-mail address: [email protected] 1 When the market prices of bank debts help regulators and other private creditors detect risky banks, subordinated debt constrains the banks’ risk taking indirectly. This idea is referred to as indirect market discipline. See Flannery (2001) for an excellent discussion on the various aspects of market discipline. 0378-4266/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2007.09.020
2 Several researchers attempt to directly measure the effect of subordinated debt on banks’ risk-taking behavior. The findings are mixed. Bliss and Flannery (2000) find that bond price changes do not reliably influence subsequent bank behavior. Ashcraft (2006) documents that the presence of subordinated debt has a positive effect on the future outcomes of distressed banks.
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in two risky assets. This is because a flat-rate deposit insurance scheme provides an incentive for the bank to take excessive risk. Interestingly, when the amount of subordinated debt is within this range, the bank invests in two safe assets. In other words, subordinated debt constrains the bank’s risk taking both before and after debt issuance. The intuition is as follows. Because the creditors cannot charge the bank an interest rate based on its future asset choice, they have to anticipate it. They do so by observing which type of asset the bank has already invested in. A bank holding a safe asset has a higher opportunity cost of taking risk, and hence is more likely to invest in another safe asset after debt issuance. Accordingly, the creditors charge such a bank a low interest rate. This provides the bank with an incentive to invest in a safe asset before debt issuance; holding such an asset motivates the bank to invest in another safe asset after debt issuance. When the amount of subordinated debt is above this range, the bank again invests in two risky assets. This is because when the amount of debt is too large, even a bank holding a safe asset will invest in a risky asset after debt issuance, because doing so would reduce its expected cost of debt. But at the time of debt issuance rational creditors can anticipate the bank’s future asset choice, and charge it a high interest rate. Anticipating this, the bank invests in a risky asset even before debt issuance. The above analysis suggests that when subordinated debt constrains banks’ risk taking, researchers are expected to document three findings. First, banks reduce their risk before they issue debt. Second, creditors charge riskier banks a higher interest rate. And third, banks do not change their risk-taking behavior after they have issued debt. Our model thus suggests that the existing empirical findings are consistent with the idea that subordinated debt can constrain banks’ risk taking. It also suggests that future research needs to examine whether banks reduce their risk before they issue debt. There have been a number of proposals calling for increased use of subordinated debt;3 our model supports these proposals. It further suggests that regulators should set the amount of subordinated debt within a range. In the paper we discuss how the bounds of this range depend on a number of factors. In addition, our model suggests that subordinated debt cannot entirely substitute for equity capital to constrain banks’ risk taking. Our paper is most closely related to Blum (2002). Blum raises an important question: how can a bank credibly commit to choosing a given level of risk after debt issuance? He shows that if the bank cannot commit, then requiring it to issue subordinated debt would aggravate its risk-taking incentives. We propose that the bank can use its existing safe asset as a commitment device.
3 See BCBS (2003) or Board and Treasury (2000) for a comprehensive review of these proposals.
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Our paper is also related to Calomiris and Kahn (1991), Flannery (1994) and Niinima¨ki (2001). These papers explain how short-term debt can discipline banks. Their rationale is that by giving creditors the right to withdraw their funds at any time, banks are deterred from taking excessive risk. Our paper explains how long-term debt can also discipline banks.4 Boot and Schmeits (2000) show that the potential benefits of conglomeration depend on the effectiveness of market discipline. Our paper complements their results by showing that a proper amount of subordinated debt can improve market discipline. In a continuous-time setting, Decamps et al. (2004) and Rochet (2004) show that subordinated debt can constrain banks’ risk taking. They focus on examining the optimal mix of the three pillars of the New Basel Accord. By contrast, our paper helps understand the existing empirical findings and offers a new testable prediction. Our paper also explains why the amount of subordinated debt should be set within a range, and why subordinated debt cannot entirely substitute for equity capital. The paper proceeds as follows. Section 2 presents the baseline model. Section 3 examines the bank’s asset choices. Section 4 checks the robustness of the results, and Section 5 discusses policy implications. Section 6 concludes. 2. The baseline model 2.1. The sequence of events We consider a representative bank (a banking entrepreneur) that operates in a risk-neutral economy. The risk-free interest rate is normalized to zero. There is a bank regulator in this economy. The bank can invest in two types of assets: safe or risky. Either type of asset requires an initial investment of $1. A safe asset yields a return of RS when it succeeds, and 0 when it fails. The probability of failure is hS. A risky asset yields a return of RR when it succeeds, and 0 when it fails. The probability of failure is hR. We assume that 0 < hS < hR < 1; 1 < R S < RR ; 0 < D ð1 hS ÞRS ð1 hR ÞRR ; i.e., neither type of asset is riskless, a risky asset provides a higher return when it succeeds, but a safe asset has a higher expected return. There are four dates: 0, 1, 2, and 3. At date 0, the bank is endowed with k 2 [0, 1] amount of equity capital, and (1 k) amount of deposits. All the deposits are insured by the government.5 We assume that the insurance 4 Subordinated debts are usually issued with initial terms to maturity in excess of 10 years (see BCBS, 2003 and Board and Treasury, 2000). 5 See Blum (2002) for an analysis of the impact of deposit insurance in a similar model.
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premium cannot be made contingent on the bank’s risk. Following Blum (2002), the deposit insurance premium is set equal to zero.6 At this date, the bank chooses an asset to invest its existing funds. We label this asset as the bank’s existing asset. At date 1, the bank raises $1 by issuing insured deposits and subordinated debt. Both markets are competitive; hence depositors and subordinated creditors earn zero expected profit on their investments. We assume that creditors can observe which type of asset the bank has chosen at date 0. However, they cannot contract on the bank’s asset choice after debt issuance. At date 2, the bank chooses another asset to invest its new funds. We label this asset as the bank’s new asset. At date 3, all the deposits and subordinated debts mature, and the cash flows of both assets are realized. We assume that the bank’s total cash flows are observable and verifiable, and the cash flows of the two assets are independent. We also assume that the bank defaults only when both of its assets fail.7 This assumption gives the bank the benefit of diversification from investing in different assets, and is consistent with the key idea of Diamond (1984). When the bank defaults, it incurs bankruptcy costs of CB(CB < 1). Think of CB as including the loss of the bank’s charter and the damage to the banking entrepreneur’s reputation. 2.2. Discussion of key modeling assumptions The baseline model makes three key assumptions. The first is that the creditors can observe which type of asset the bank has chosen before debt issuance, i.e., its original risk choice. Thus far the empirical evidence is inconclusive on how accurately investors can observe the riskiness of a bank. For example, Morgan (2002) examines the pattern of disagreement between bond raters. He finds that banks are inherently more opaque than nonbanking firms. Ashcraft and Bleakley (2006) find that banks are able to exploit private information in order to manage earnings and the real information content of public disclosure. In contrast, Flannery et al. (2004) study both the market microstructure properties of banks’ equity prices and the analyst forecast errors of banks’ earnings. They find that banks are no more opaque than nonbanking firms. We will show in Section 4 that our main results hold as long as the creditors can imperfectly observe the bank’s original risk choice. There are several reasons to justify this assumption. First, a large academic literature documents 6 We could have assumed that the insurance premium is set equal to a positive constant such that the insurance provider breaks even. Depending on whether the insurance premium is paid ex ante or ex post, deposit insurance can have different impacts on the bank’s risk-taking incentives. But as long as the insurance premium cannot be made contingent on the bank’s risk, our main results will hold. 7 A similar assumption appears in Boot and Schmeits (2000). This assumption would be satisfied if the cash flows generated by the success of either asset are sufficient to pay off all the depositors and creditors of the bank.
that creditors can at least partially observe the riskiness of a bank, and charge riskier banks a higher interest rate (see, e.g., Covitz et al., 2004; Morgan and Stiroh, 2001 and Sironi, 2003). Second, creditors would mandate information disclosure as a requirement for investing. Indeed, Covitz and Harrison (2004) find that banks convey private information to the market during bond issuances. Finally, the New Basel Accord specifies a set of disclosure requirements that aims to encourage market discipline. These requirements would help creditors better assess the riskiness of a bank (see BCBS, 2004). The second key assumption is that the regulator cannot observe the bank’s original risk choice, or, even if she can observe it, she cannot shut the bank down based on her observation.8 This assumption is reasonable because it is difficult to imagine that in practice bank regulators would be allowed to shut a bank down based on soft information about the quality of its investments. The third key assumption is that once the bank has invested in an asset, it cannot reverse its investment decision.9 We will show that our main results continue to hold even when the bank can shift a fraction of its investment after debt issuance. There are reasons to believe that it is often costly for banks to quickly and significantly adjust the composition of their assets. To begin with, a fundamental function performed by banks is to collect demand deposits and make long-term capital investments that are somewhat irreversible (Diamond and Dybvig, 1983). Banks would incur substantial costs if they prematurely terminate such investments, and hence the composition of their assets changes only gradually. Indeed, Hancock et al. (1995) find that banks typically take several quarters, and in many instances a few years, to completely adjust the composition of their assets to new desired levels following shocks to their capital positions. Selling loans is an alternative for banks to adjust the composition of their assets. Doing so, however, is also costly. Banks usually obtain proprietary information about their borrowers in the lending process, and hence adverse selection is a concern for potential buyers of bank loans (see Gorton and Pennacchi, 1990). In addition, banks are relationship lenders (see, e.g., Diamond and Rajan, 2000). They develop valuable skills of collecting loans in the loan originating process, and such skills cannot be easily transferred to others. The loan originating banks cannot credibly commit to monitoring loans for others once these loans are sold. Therefore potential buyers are willing to buy loans only at a discount. 8
Berger et al. (2000) find that supervisory assessments and bond rating agency assessments complement one another, in the sense that each information set helps to forecast the other group’s assessments of bank conditions. They also find that supervisory assessments are less accurate than bond market assessment in predicting future changes in bank performance. 9 Myers and Rajan (1998) make a similar assumption. They assume that banks hold a mix of illiquid loans and liquid securities as assets. The illiquid loans enhance the banks’ capacity to issue debt.
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3. The bank’s asset choices In this section we examine the bank’s asset choices (1) when it issues only insured deposits, and (2) when it issues a combination of insured deposits and subordinated debt. We also examine the impact of equity capital on the bank’s asset choices. 3.1. Insured deposits Suppose that at date 1, the bank raises the entire $1 of needed funds by issuing insured deposits. Because the market for insured deposits is competitive, the face value of these deposits is $1, whether the bank invests in the safe assets or the risky assets. The bank chooses two assets to maximize its expected profit. It can invest in two safe assets, two risky assets, or one safe asset and one risky asset. Because we wish to emphasize the bank’s incentives to take excessive risk, we introduce the following parametric assumption. Assumption 1. The expected return of a safe asset exceeds that of a risky asset by an amount that is not too large D<
1 2 hR h2S ð2 k C B Þ: 2
When this assumption is not satisfied, the bank will always invest in the safe assets, and thus there is no need for regulation. We adopt the convention that when the bank is indifferent between a safe asset and a risky asset, it chooses the safe asset. We then have the following result. Proposition 1. When the bank issues only insured deposits, it invests in two risky assets.
zero expected profit. Not able to write contract on the bank’s future asset choice, they have to anticipate it. They do so by observing which type of asset the bank has already invested in. In this section we assume that the creditors can perfectly observe the type of the bank’s existing asset; later we relax this assumption. In equilibrium the creditors’ expectations must be correct. We restrict attention to pure strategy equilibria. Because safe assets have a higher expected return, the regulator wishes to motivate the bank to only invest in the safe assets. The bank chooses its two assets sequentially: one before and the other after debt issuance. We thus need the following parametric assumption. Assumption 2. The expected return of a safe asset exceeds that of a risky asset by an amount that is not too small:
3.2. Subordinated debt Now suppose that the regulator requires the bank to issue s (0 < s 6 1) amount of unsecured, subordinated debt, and (1 s) amount of insured deposits. When the bank defaults, the insured depositors will receive compensation from the government, but the subordinated creditors will lose their investments in the bank. Consequently, the subordinated creditors care about the bank’s risk, i.e., its asset choices. Because the market for subordinated debt is competitive, the creditors charge the bank an interest rate to make
1 2 hR h2S ð1 k C B Þ; 2 hS h2R h2S ð2 k C B Þ : DP 3 2hS þ ðhR þ hS Þ 1 h2S
ð2AÞ D P ð2BÞ
When this assumption is not satisfied, it is not possible to constrain the bank’s risk taking using subordinated debt alone. We can verify that neither inequality implies the other, and hence both are necessary. We can also verify that Assumptions 1 and 2 always define a range of possible values for D. The next proposition shows that when D is within this range, requiring the bank to issue a proper amount of subordinated debt motivates it to invest in two safe assets. Proposition 2. There are two parameters 2D ; s ¼ 2 k CB 2 hR h2S
Proof. See Appendix. h This result is easy to understand. A risky asset provides the bank with a higher return when it succeeds. Meanwhile, because of limited liability the bank does not need to compensate its depositors when it defaults. This convex payoff structure provides an incentive for the bank to take excessive risk. When the expected return of a safe asset exceeds that of a risky asset only by a small amount, the bank invests in two risky assets in order to take maximum risk.
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s ¼
½D hS ðhR hS Þð2 k C B Þ 1 h2S h3S ðhR hS Þ
;
where 0 < s 6 1 and s 6 s, such that when the regulator sets the amount of subordinated debt s within the range of ½s; s, a subgame perfect Nash equilibrium exists, in which: (i) at date 0, the bank invests its existing funds in a safe asset; (ii) at date 1, the creditors charge the bank an interest rate of h2S = 1 h2S . They would charge the bank a higher interest rate of h2R = 1 h2R had the bank invested its existing funds in a risky asset; (iii) at date 2, the bank invests its new funds in another safe asset; (iv) when s < s or s > s, no such equilibrium exists.
Proof. See Appendix. h To understand this result, consider how the amount of subordinated debt affects the bank’s risk-taking incentives before and after debt issuance. Suppose that the bank
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needs to issue a small amount of debt. At the time of debt issuance, if the creditors observe that the bank has invested in a safe asset, they will charge the bank a low interest rate, for two reasons. First, other things remaining the same, such a bank is less likely to default, because default only happens when both assets fail. Second, such a bank is more likely to invest in another safe asset. Indeed, if after debt issuance the bank invests in a risky asset and when this asset fails, the bank has to compensate its depositors and creditors using the cash flows of its existing asset, unless that asset fails as well. A safe asset is less likely to fail, and hence holding a safe asset increases the bank’s opportunity cost of taking risk. The creditors’ rules for setting interest rate provide an incentive for the bank to take low risk before debt issuance. When the amount of debt is above a lower bound, the incentive is strong enough for the bank to invest in a safe asset. Holding this asset then motivates the bank to invest in another safe asset after debt issuance. Once the debt has been issued, however, the bank’s incentives change. Now the larger the amount of debt, the stronger the bank’s incentives to take risk, because doing so would reduce its expected cost of debt. When the amount of debt is above an upper bound, even a bank holding a safe asset will invest in a risky asset. But rational creditors can anticipate the bank’s future asset choice, and would ask for a high interest rate. Anticipating this, the bank invests in a risky asset even before debt issuance. An alternative way to understand Proposition 2 is to think about the important question raised by Blum (2002): how can a bank credibly commit to choosing a given level of risk after debt issuance? Blum shows that if the bank cannot commit, then requiring it to issue subordinated debt would aggravate its risk-taking incentives. We propose that the bank can use its existing safe asset as a commitment device. The bank can promise the creditors that it will invest in a safe asset after debt issuance. Having already invested in a safe asset makes the bank’s promise credible. The bank’s commitment ability, however, is limited: a safe asset is still not riskless. Thus when the bank has to issue a very large amount of debt it is no longer able to commit. Proposition 2 implies that there can be no observable change of the banks’ risk-taking behavior before and after debt issuance, even when subordinated debt constrains their risk taking. Future research needs to examine whether banks reduce their risk before they issue debt. 3.3. Equity capital In recent years capital requirements have been the primary regulatory tools in bank regulations. In the New Basel Accord, capital requirements are designated as one of the three pillars underpinning prudential regulation, together with supervisory review process and market discipline.
Our model sheds some light on how subordinated debt and equity capital can work together to constrain banks’ risk taking. Proposition 3. An increase of equity capital reduces the lower bound on subordinated debt, and increases the upper bound on subordinated debt. Proof. ds/dk < 0. ds=dk > 0. h The intuition is as follows. Equity capital increases the bank’s opportunity cost of taking risk, because the bank will lose its equity capital when it defaults. As a result, when the bank holds a larger amount of equity capital, a smaller amount of subordinated debt is sufficient to motivate it to invest in the safe assets. Moreover, the bank will not invest in the risky assets even after it has issued a larger amount of subordinated debt. In practice, regulators typically permit a substitution of subordinated debt for equity capital. Proposition 3 implies that a small amount of subordinated debt can reduce the needs of equity capital. But when the bank has to issue a large amount of subordinated debt, more equity capital is called for to ensure that the bank does not take excessive risk. 4. Extensions This section extends the baseline model in two directions to check the robustness of the results. First, we consider the case where the creditors can only imperfectly observe the type of the bank’s existing asset. Second, we consider the case where the creditors can use bond covenants to partially restrict the bank’s future asset choice. We will show that our main results continue to hold in both cases. 4.1. Imperfect observation Following Boot and Schmeits (2000), we model imperfect observation as follows: at the time of debt issuance, with probability a (0 < a < 1) the creditors can observe the type of the bank’s existing asset, and with probability (1 a) they cannot observe it.10 The parameter a measures how accurately the creditors can assess the riskiness of the bank. Proposition 4. There are two parameters a ¼ max s 1 h2S þ h2S ; h2S ðhR hS ÞD ; ðhR þ hS Þ½D hS ðhR hS Þð2 k C B Þ sD ¼ s 1 h2S a h2S ; where h2S < a 6 1 and s < sD 6 1, such that: 10 Alternatively, we could assume that at the time of debt issuance, only a fraction a of creditors can observe the type of the bank’s existing asset, and the rest of the creditors cannot observe it. The results would be the same.
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(i) when a P a and the regulator sets the amount of subordinated debt within the range of ½sD ; s, a subgame perfect Nash equilibrium exists in which the bank invests in two safe assets; (ii) dsD/da < 0; (iii) when a < a, no such equilibrium exists.
Proof. The proof is a simple modification of the Proof of Proposition 2, and is available from the author upon request. h The intuition is as follows. Before the bank issues debt, it considers whether to invest in a safe asset or a risky asset. The bank weighs the moral hazard gains from taking excessive risk against the expected increase in its funding costs, which depend on how accurately the creditors can observe the bank’s original risk choice, and how large the amount of debt is. When the creditors cannot accurately observe the bank’s original risk choice, the regulator has to require the bank to issue a larger amount of debt in order to provide the bank with proper incentives. Doing so, however, aggravates the bank’s risk-taking incentives after debt issuance. Therefore, when the accuracy of the creditors’ observation falls below a critical level, using subordinated debt alone cannot motivate the bank to invest in the safe assets. 4.2. Bond covenants Thus far we have assumed that once the interest rate is determined, the creditors can no longer influence the bank’s risk taking. In practice, however, creditors may continue to influence the bank’s risk taking through bond covenants (see Goyal, 2005). To study how our results change when the creditors can partially restrict the bank’s risk taking through bond covenants, we modify the baseline model as follows: the bank will incur costs of CA if it invests in a risky asset after debt issuance. Think of CA as the penalty arising from the violation of the covenants. For expositional clarity, we assume that CA is not too large Assumption 3 CA <
1 2 hR h2S ð2 k C B Þ D: 2
We then have the following result. Proposition 5. There are two parameters sA and sA (both are defined in Appendix), where 0 < sA < 1 and sA < sA , such that: (i) when the regulator sets the amount of subordinated debt within the range of ½sA ; sA , a subgame perfect Nash equilibrium exists in which the bank invests in two safe assets; (ii) dsA/dCA < 0; (iii) dsA =dC A > 0.
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Proof. See Appendix. h The intuition is clear. A higher value of CA means that it is more costly for the bank to take risk; hence the lower bound on subordinated debt is decreased. Moreover, the bank is less likely to invest in a risky asset even after it has issued a large amount of subordinated debt; hence the upper bound on subordinated debt is increased. 5. Discussion The existing subordinated debt proposals show considerable disagreements on the required amount of debt (see Board and Treasury, 2000, pp. 45). Our model suggests that regulator needs to set s, the fraction of debt funded by subordinated debt versus insured deposits, within a range. We now use a simple example to illustrate how the lower and upper bounds of this range depend on a number of factors. The baseline parameters are as follows: RS = 1.1, hS = 0.1, RR = 1.35, hR = 0.3, k = 0.1 and CB = 0.7. The results are summarized in Table 1. Panel A illustrates that when the return of the safe asset decreases, the lower bound on subordinated debt increases, and the upper bound on subordinated debt decreases. Notice that when the return is sufficiently small, the lower bound exceeds the upper bound, implying that in this case subordinated debt alone cannot induce the bank to invest in the safe assets. The policy implication is immediate: subordinated debt may not be able to provide discipline when regulators need it most, i.e., during times of aggregate financial distress when returns of safe assets are low.11 During such times regulators need to rely more on other regulatory instruments, such as supervisory review. Panel B illustrates the results in Proposition 4. As the accuracy of the creditors’ observation decreases, the lower bound on subordinated debt increases. Panel C illustrates the results in Proposition 5. Bond covenants decrease the lower bound and increase the upper bound on subordinated debt. There are several other issues to discuss. First, the present model assumes that once the bank has invested in an asset, it cannot reverse its investment decision. What would happen if the bank can shift a fraction of its investment in the safe asset into the risky asset after debt issuance? It turns out that when the fraction is small, it is not optimal for the bank to do so. Therefore, the results in Proposition 2 continue to hold.12 Intuitively, a small shift would not change the bank’s probability of default and thus its expected cost of debt, but would reduce its expected investment return. More general, our results hold when it is costly for banks to quickly and significantly adjust the composition 11 Competition may also reduce safe returns. For some interesting analysis on bank competition and prudential regulation, see Hellmann et al. (2000) and Repullo (2004). 12 The proof is available from the author upon request.
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Table 1 Results of the numerical example Panel A: The resulting lower and upper bounds when RS changes 1.100 1.095 1.090 1.085 1.080 1.075 RS Lower bound (s) 0.075 0.187 0.300 0.412 0.525 0.637 Upper bound ðsÞ 103.950 81.675 59.400 37.125 14.850 7.425 Panel B: The resulting lower and upper bounds when a changes a 1.000 0.900 0.800 0.700 0.600 0.500 Lower bound (sD) 0.075 0.083 0.094 0.108 0.126 0.152 Upper bound ðsÞ 103.950 103.950 103.950 103.950 103.950 103.950 Panel C: The resulting lower and upper bounds when CA changes CA 0.000 0.001 0.002 0.003 0.004 0.005 0.075 0.062 0.050 0.037 0.025 0.012 Lower bound (sA) Upper bound ðsA Þ 103.950 108.900 113.850 118.800 123.750 128.700
of their assets. For banks that are mainly in the traditional lending business, we believe that this is a reasonable assumption, for the reasons discussed in Section 2. For banks that are mainly in the business such as security trading, this assumption is questionable, because such banks usually hold a large amount of highly tradable securities, and receive a large amount of trading revenues. However, for such banks there exist other mechanisms through which subordinated debt can constrain their risk taking. For example, such banks would have strong incentives to develop and maintain the reputation of being a low-risk counterparty in trading transactions. The yields of the subordinated debts of these banks can then serve as a valuable signal to both private investors and bank regulators. Second, in the present model the safe asset has a higher NPV than the risky asset, and subordinated debt helps produce the efficient outcome. One might ask: if the risky asset has a higher NPV, will subordinated debt prevent the bank from producing the efficient outcome? It turns out that this need not be a concern. The reason is the following. In equilibrium the creditors always break even, and thus subordinated debt does not affect the expected returns from investing in either type of asset. When the risky asset has a higher NPV, the bank will invest in it whether it is required to issue subordinated debt or not, because doing so would maximize both the investment returns and the wealth transfer from the deposit insurance provider. In other words, subordinated debt will not prevent the bank from producing the efficient outcome. Third, the present model does not consider the possibility of distress. What would happen if the bank becomes distressed when it has to issue subordinated debt? Information asymmetries are likely to become larger during distress, and thus the presence of subordinated debt may amplify the bank’s risk-taking incentives at this point. However, the empirical evidence seems to say otherwise, as Ashcraft (2006) documents that the mix of subordinated debt in bank capital has a beneficial impact on the future outcomes of distressed banks. Ashcraft speculates that this is because creditors are more likely to impose restrictive covenants on
distressed banks.13 Identifying the exact mechanism is an important topic for future research. Finally, if regulators impose a subordinated debt requirement on banks, would banks shift their balance sheets in a fashion such that the subordinated debt requirement would not affect their risk taking? This question is highly complex, and is inevitably part of the broader question of how much capital a bank should maintain. To address this question, we need a model with endogenously determined level of bank capital. We leave this question to future research. 6. Conclusion The ongoing regulatory debate and the seemingly conflicting empirical findings call for a theoretical model to organize our thoughts on how subordinated debt constrains banks’ risk taking. This paper proposes such a model. At the core of the model is a simple argument: requiring banks to issue subordinated debt motivates them to invest in safe assets before debt issuance; holding such assets then constrains their risk taking after debt issuance. As a result, researchers may observe no change of the banks’ risk-taking behavior before and after debt issuance, even though subordinated debt has constrained their risk taking. Future research needs to examine whether banks reduce their risk before they issue debt. The amount of subordinated debt affects banks’ risktaking incentives differently before and after debt issuances. Therefore, regulators should set the amount of debt within a range. We show that the bounds of this range depend on a number of factors, such as the return of the safe asset, the creditors’ ability to assess the riskiness of the bank, and their ability to use bond covenants. Because these factors differ across countries (see BCBS, 2003), our model suggests that the optimal amount of subordinated debt varies across countries. Franchise value (also known as charter value) also can reduce banks’ risk-taking incentives, because banks will lose their valuable charters when they default (see Keeley, 1990). Our model suggests that franchise value and subordinated debt can work together to constrain banks’ risk taking. One feature distinguishes our model from franchise value theory: while usually it is difficult for banks to increase their franchise values, they can be motivated to reduce their risk. Requiring banks to issue subordinated debt aims at providing them with such incentives. Acknowledgments I am grateful to Erwan Morellec and Elu von Thadden for their constant support and advice. I would like to thank 13 Goyal (2005) finds that charter values significantly affect the likelihood of restrictive covenants in bank debt contracts.
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Alex Edmans, Michel Habib, Enrique Schroth, Giorgio Szego (the editor), Lucy White, Yu Zhang, seminar participants at Simon Fraser University, the 2006 NFA meeting, and especially three anonymous referees for helpful comments and suggestions, and the University of Rochester for its hospitality. Financial support from NCCR FinRisk and the Swiss National Science Foundation is gratefully acknowledged.
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We solve the model by backward induction, and prove the proposition using a number of preliminary results. Claim 1. If I = S, then the bank will choose J = S if and only if s 6 s, where s is defined by ½D hS ðhR hS Þð2 k C B Þ 1 h2S s ¼ : ð3Þ h3S ðhR hS Þ Proof. When I = S, the subordinated creditors set B = BL. Now if the bank chooses J = S, its expected profit, denoted VSLS, is given by
Appendix Proof of Proposition 1. Recall that initially the bank is endowed with k amount of equity capital, and (1 k) amount of insured deposits. It then issues $1 of insured deposits so that the total amount of insured deposits becomes (2 k). When the bank defaults, it incurs bankruptcy costs of CB. The bank chooses its two assets by comparing its expected profit in the following three cases. Case 1: The bank invests in two safe assets. Its expected profit in this case is 2ð1 hS ÞRS 2 þ k þ h2S ð2 k C B Þ: Case 2: The bank invests in two risky assets. Its expected profit in this case is 2ð1 hR ÞRR 2 þ k þ h2R ð2 k C B Þ: Case 3: The bank invests in one safe asset and one risky asset. Its expected profit in this case is ð1 hS ÞRS þ ð1 hR ÞRR 2 þ k þ hS hR ð2 k C B Þ:
V SLS ¼ 2ð1 hS ÞRS 1 h2S ð2 k s þ BL Þ h2S C B : ð4Þ Alternatively, if the bank chooses J = R, its expected profit, denoted VSLR, is given by V SLR ¼ ð1 hS ÞRS þ ð1 hR ÞRR ð1 hS hR Þ ð2 k s þ BL Þ hS hR C B : It is easy to verify that VSLS P VSLR if and only if s 6 s. h Claim 2. If I = R, then the bank will choose J = R. Proof. Because I = R, the subordinated creditors set B = BH. Now if the bank chooses J = S, its expected profit, denoted VRHS, is given by V RHS ¼ ð1 hR ÞRR þ ð1 hS ÞRS ð1 hS hR Þ ð2 k s þ BH Þ hS hR C B :
It follows from Assumption 1 that the bank can maximize its expected profit by investing in two risky assets. Proof of Proposition 2. Let I denote the type of asset that the bank chooses to invest in before debt issuance: I = S if the bank invests in a safe asset, and I = R if the bank invests in a risky asset. Similarly, let J denote the type of asset that the bank chooses to invest in after debt issuance: J = S if the bank invests in a safe asset, and J = R if the bank invests in a risky asset. At date 1 the bank issues s amount of subordinated debt. Let B denote the face value of the debt. We assume that the subordinated creditors set B = BL when they observe that the bank has chosen I = S, and set B = BH when they observe that the bank has chosen I = R, where BL and BH are defined by s ; 1 h2S s : BH ¼ 1 h2R
BL ¼
ð1Þ ð2Þ
Note that setting B = BL is the same as charging the bank an interest rate of h2S = 1 h2S , and setting B = BH is the same as charging the bank an interest rate of h2R = 1 h2R .
Alternatively, if the bank chooses J = R, its expected profit, denoted VRHR, is given by V RHR ¼ 2ð1 hR ÞRR ð1 h2R Þð2 k s þ BH Þ h2R C B : ð5Þ We have V RHR V RHS ¼ h2R hS hR ð2 k s þ BH C B Þ D > h2R hS hR ð2 k C B Þ D 1 > h2R h2S ð2 k C B Þ D > 0; 2 where the first inequality is obtained because BH > s, and the third inequality follows from Assumption 1. Therefore, if I = R, the bank will choose J = R. h Claim 3. The bank will choose I = S and J = S if and only if s 2 ½s; s, where s is defined by s ¼ 2 k CB
2D : h2R h2S
We claim that 0 < s 6 1 and s 6 s.
ð6Þ
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Proof. If the bank chooses I = S, the subordinated creditors set B = BL. It then follows from Claim 1 that the bank will choose J = S if and only if s 6 s. In this case the expected profit of the bank is VSLS, where VSLS is given by Eq. (4). On the other hand, if the bank chooses I = R, the subordinated creditors set B = BH. It then follows from Claim 2 that the bank will choose J = R. In this case the expected profit of the bank is VRHR, where VRHR is given by Eq. (5). It is easy to verify that VSLS P VRHR if and only if s P s. h Assumption 1 implies that s > 0. Because s 2 (0, 1], to make it possible of having some s 2 ½s; s, we need to ensure two conditions. First, we need s 6 1. This condition is equivalent to Assumption 2A. Second, we need s 6 s. This condition is equivalent to Assumption 2B. Hence by Assumption 2, both conditions are satisfied. Claim 4. When s 2 ½s; s, the subordinated creditors make zero expected profit. Proof. When s 2 ½s; s and the bank has chosen I = S, the subordinated creditors set B = BL. Claim 1 implies that the bank will choose J = S. Therefore the bank will default with probability h2S . Eq. (1) then implies that the subordinated creditors make zero expected profit. On the other hand, when the bank has chosen I = R, the subordinated creditors set B = BH. Claim 2 implies that the bank will choose J = R. Therefore the bank will default with probability h2R . Eq. (2) then implies that the subordinated creditors make zero expected profit. Combining above results, we see that if and only if s 2 ½s; s, a subgame perfect Nash equilibrium exists in which the bank chooses I = S and J = S, and the subordinated creditors make zero expected profit. h Proof of Proposition 5. The proof is a modification of that of Proposition 2. We find that there is no change of the value of VSLS, while the value of VSLR is reduced by an amount of CA. Consequently VSLS P VSLR if and only if s 6 sA , where C A 1 h2S sA ¼ s þ 3 : hS ðhR hS Þ We then find that there is no change of the value of VRHS, while the value of VRHR is reduced by an amount of CA. It follows from Assumption 3 that VRHR > VRHS. Finally, we find that VSLS P VRHR if and only if s P sA, where CA sA ¼ s 2 hR h2S Observe that sA < s and sA > s. It is easy to verify that 0 < sA, dsA/dCA < 0, and dsA =dC A > 0. References Ashcraft, A., 2006. Does the market discipline banks? New evidence from the regulatory capital mix. Federal Reserve Bank of New York Staff Reports, no. 244.
Ashcraft, A., Bleakley, H., 2006. On the market discipline of informationally opaque firms: Evidence from bank borrowers in the federal funds market. Federal Reserve Bank of New York Staff Reports, no. 257. Basel Committee on Banking Supervision, 2003. Markets for bank subordinated debt and equity in Basel Committee member countries. Bank for International Settlements, Basel. Basel Committee on Banking Supervision, 2004. International convergence of capital measurement and capital standards (a revised framework). Bank for International Settlements, Basel. Berger, A., Davies, S., Flannery, M., 2000. Comparing market and supervisory assessments of bank performance: Who knows what when? Journal of Money, Credit and Banking 32, 641–667. Bliss, R., Flannery, M., 2000. Market discipline in the governance of U.S. bank holding companies: Monitoring vs. influencing. Europe Finance Review 6, 361–395. Blum, J., 2002. Subordinated debt, market discipline, and banks’ risk taking. Journal of Banking and Finance 26, 1427–1441. Board of Governors of the Federal Reserve System and United States Department of the Treasury, 2000. The feasibility and desirability of mandatory subordinated debt. Boot, A., Schmeits, A., 2000. Market discipline and incentive problems in conglomerate firms with applications to banking. Journal of Financial Intermediation 9, 240–273. Calomiris, C., Kahn, C., 1991. The role of demandable debt in structuring optimal banking arrangements. American Economic Review 81, 497– 513. Covitz, D., Harrison, P., 2004. Do banks time bond issuance to trigger disclosure, due diligence, and investor scrutiny? Journal of Financial Intermediation 13, 299–323. Covitz, D., Hancock, D., Kwast, M., 2004. A reconsideration of the risk sensitivity of U.S. banking organization subordinated debt spreads: A sample selection approach. Federal Reserve Bank of New York Economic Policy Review (September), 73–92. Decamps, J., Rochet, J.-C., Roger, B., 2004. The three pillars of Basel II: Optimizing the mix. Journal of Financial Intermediation 13, 132–155. Diamond, D., 1984. Financial intermediation and delegated monitoring. Review of Economic Studies 51, 393–414. Diamond, D., Dybvig, P., 1983. Bank runs, deposit insurance, and liquidity. Journal of Political Economy 91, 401–419. Diamond, D., Rajan, R., 2000. A theory of bank capital. Journal of Finance 55, 2431–2465. Flannery, M., 1994. Debt maturity and the deadweight cost of leverage: Optimally financing banking firm. American Economic Review 84, 320–331. Flannery, M., 2001. The faces of ‘‘market discipline’’. Journal of Financial Services Research 20, 107–119. Flannery, M., Kwan, S., Nimalendran, M., 2004. Market evidence on the opaqueness of banking firms’ assets. Journal of Financial Economics 71, 419–460. Gorton, G., Pennacchi, G., 1990. Financial intermediaries and liquidity creation. Journal of Finance 45, 49–71. Goyal, V., 2005. Market discipline of bank risk: Evidence from subordinated debt contracts. Journal of Financial Intermediation 14, 318–350. Hancock, D., Laing, A., James, W., 1995. Bank capital shocks: Dynamic effects on securities, loans, and capital. Journal of Banking and Finance 19, 661–677. Hellmann, T., Murdock, K., Stiglitz, J., 2000. Liberalization, moral hazard in banking, and prudential regulation: Are capital requirements enough? American Economic Review 90, 147–165. Keeley, M., 1990. Deposit insurance, risk, and market power in banking. American Economic Review 80, 1183–1200. Krishnan, C., Ritchken, P., Thomson, J., 2005. Monitoring and controlling bank risk: Does risky debt help? Journal of Finance 60, 343–378. Morgan, D., 2002. Rating banks: Risk and uncertainty in an opaque industry. American Economic Review 92, 874–888. Morgan, D., Stiroh, K., 2001. Market discipline of banks: The asset test. Journal of Financial Services Research 20, 195–208.
J. Niu / Journal of Banking & Finance 32 (2008) 1110–1119 Myers, S., Rajan, R., 1998. The paradox of liquidity. Quarterly Journal of Economics 113, 733–771. Niinima¨ki, J.-P., 2001. Intertemporal diversification in financial intermediation. Journal of Banking and Finance 25, 965–991. Repullo, R., 2004. Capital requirements, market power, and risk-taking in banking. Journal of Financial Intermediation 13, 156–182.
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Rochet, J.-C., 2004. Rebalancing the three pillars of Basel II. Federal Reserve Bank of New York Economic Policy Review (September), 7– 21. Sironi, A., 2003. Testing for market discipline in the European banking industry: Evidence from subordinated debt issues. Journal of Money, Credit and Banking 35, 443–472.
Journal of Banking & Finance 32 (2008) 1110–1119 www.elsevier.com/locate/jbf
Can subordinated debt constrain banks’ risk taking? Jijun Niu
*
Faculty of Business Administration, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada V5A 1S6 Received 4 July 2006; accepted 18 September 2007 Available online 1 October 2007
Abstract This paper presents a model in which requiring banks to issue a proper amount of subordinated debt can constrain their risk taking both before and after debt issuance. The main idea is that the prospect of issuing debt motivates banks to invest in safe assets before debt issuance; holding such assets then constrains their risk taking after debt issuance. The model helps understand the existing empirical findings, and offers a new testable prediction. It also suggests that: (1) regulators should set the amount of subordinated debt within a range; and (2) subordinated debt cannot entirely substitute for equity capital. Ó 2007 Elsevier B.V. All rights reserved. JEL classification: G21; G28 Keywords: Subordinated debt; Risk taking; Bank regulation; Market discipline
1. Introduction Many economists believe that requiring banks to issue some amount of subordinated debt can constrain their risk taking. The idea seems intuitive: if creditors charge riskier banks a higher interest rate, banks would think twice before taking excessive risk. This idea is referred to as direct market discipline.1 Empirical studies, however, seem to have produced conflicting findings. Some researchers examine cross-sectional data. They find that creditors indeed charge riskier banks a higher interest rate, and conclude that subordinated debt can constrain banks’ risk taking (see, e.g., Covitz et al., 2004; Morgan and Stiroh, 2001 and Sironi, 2003). Others look at time-series data. They find no change of the banks’ risk-taking behavior before and after debt issuances (see
Krishnan et al., 2005).2 When does risk reduction occur, then, if subordinated debt can constrain banks’ risk taking? To answer this question, we propose a theoretical model. The model studies how a bank chooses between two types of assets: safe or risky. A safe asset has a higher expected return, but a risky asset provides a higher return when it succeeds. The bank first chooses an asset to invest its existing funds. It then raises some new funds by issuing insured deposits and subordinated debt. After that the bank chooses another asset to invest its new funds. At the time of debt issuance creditors can observe which type of asset the bank has already invested in, but they cannot contract on the bank’s future asset choice. The bank pays a flat-rate deposit insurance premium, and defaults when both of its assets fail. We show that there exists a range of the amount of subordinated debt such that below this range, the bank invests
*
Tel.: +1 778 782 4491; fax: +1 778 782 4920. E-mail address: [email protected] 1 When the market prices of bank debts help regulators and other private creditors detect risky banks, subordinated debt constrains the banks’ risk taking indirectly. This idea is referred to as indirect market discipline. See Flannery (2001) for an excellent discussion on the various aspects of market discipline. 0378-4266/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2007.09.020
2 Several researchers attempt to directly measure the effect of subordinated debt on banks’ risk-taking behavior. The findings are mixed. Bliss and Flannery (2000) find that bond price changes do not reliably influence subsequent bank behavior. Ashcraft (2006) documents that the presence of subordinated debt has a positive effect on the future outcomes of distressed banks.
J. Niu / Journal of Banking & Finance 32 (2008) 1110–1119
in two risky assets. This is because a flat-rate deposit insurance scheme provides an incentive for the bank to take excessive risk. Interestingly, when the amount of subordinated debt is within this range, the bank invests in two safe assets. In other words, subordinated debt constrains the bank’s risk taking both before and after debt issuance. The intuition is as follows. Because the creditors cannot charge the bank an interest rate based on its future asset choice, they have to anticipate it. They do so by observing which type of asset the bank has already invested in. A bank holding a safe asset has a higher opportunity cost of taking risk, and hence is more likely to invest in another safe asset after debt issuance. Accordingly, the creditors charge such a bank a low interest rate. This provides the bank with an incentive to invest in a safe asset before debt issuance; holding such an asset motivates the bank to invest in another safe asset after debt issuance. When the amount of subordinated debt is above this range, the bank again invests in two risky assets. This is because when the amount of debt is too large, even a bank holding a safe asset will invest in a risky asset after debt issuance, because doing so would reduce its expected cost of debt. But at the time of debt issuance rational creditors can anticipate the bank’s future asset choice, and charge it a high interest rate. Anticipating this, the bank invests in a risky asset even before debt issuance. The above analysis suggests that when subordinated debt constrains banks’ risk taking, researchers are expected to document three findings. First, banks reduce their risk before they issue debt. Second, creditors charge riskier banks a higher interest rate. And third, banks do not change their risk-taking behavior after they have issued debt. Our model thus suggests that the existing empirical findings are consistent with the idea that subordinated debt can constrain banks’ risk taking. It also suggests that future research needs to examine whether banks reduce their risk before they issue debt. There have been a number of proposals calling for increased use of subordinated debt;3 our model supports these proposals. It further suggests that regulators should set the amount of subordinated debt within a range. In the paper we discuss how the bounds of this range depend on a number of factors. In addition, our model suggests that subordinated debt cannot entirely substitute for equity capital to constrain banks’ risk taking. Our paper is most closely related to Blum (2002). Blum raises an important question: how can a bank credibly commit to choosing a given level of risk after debt issuance? He shows that if the bank cannot commit, then requiring it to issue subordinated debt would aggravate its risk-taking incentives. We propose that the bank can use its existing safe asset as a commitment device.
3 See BCBS (2003) or Board and Treasury (2000) for a comprehensive review of these proposals.
1111
Our paper is also related to Calomiris and Kahn (1991), Flannery (1994) and Niinima¨ki (2001). These papers explain how short-term debt can discipline banks. Their rationale is that by giving creditors the right to withdraw their funds at any time, banks are deterred from taking excessive risk. Our paper explains how long-term debt can also discipline banks.4 Boot and Schmeits (2000) show that the potential benefits of conglomeration depend on the effectiveness of market discipline. Our paper complements their results by showing that a proper amount of subordinated debt can improve market discipline. In a continuous-time setting, Decamps et al. (2004) and Rochet (2004) show that subordinated debt can constrain banks’ risk taking. They focus on examining the optimal mix of the three pillars of the New Basel Accord. By contrast, our paper helps understand the existing empirical findings and offers a new testable prediction. Our paper also explains why the amount of subordinated debt should be set within a range, and why subordinated debt cannot entirely substitute for equity capital. The paper proceeds as follows. Section 2 presents the baseline model. Section 3 examines the bank’s asset choices. Section 4 checks the robustness of the results, and Section 5 discusses policy implications. Section 6 concludes. 2. The baseline model 2.1. The sequence of events We consider a representative bank (a banking entrepreneur) that operates in a risk-neutral economy. The risk-free interest rate is normalized to zero. There is a bank regulator in this economy. The bank can invest in two types of assets: safe or risky. Either type of asset requires an initial investment of $1. A safe asset yields a return of RS when it succeeds, and 0 when it fails. The probability of failure is hS. A risky asset yields a return of RR when it succeeds, and 0 when it fails. The probability of failure is hR. We assume that 0 < hS < hR < 1; 1 < R S < RR ; 0 < D ð1 hS ÞRS ð1 hR ÞRR ; i.e., neither type of asset is riskless, a risky asset provides a higher return when it succeeds, but a safe asset has a higher expected return. There are four dates: 0, 1, 2, and 3. At date 0, the bank is endowed with k 2 [0, 1] amount of equity capital, and (1 k) amount of deposits. All the deposits are insured by the government.5 We assume that the insurance 4 Subordinated debts are usually issued with initial terms to maturity in excess of 10 years (see BCBS, 2003 and Board and Treasury, 2000). 5 See Blum (2002) for an analysis of the impact of deposit insurance in a similar model.
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premium cannot be made contingent on the bank’s risk. Following Blum (2002), the deposit insurance premium is set equal to zero.6 At this date, the bank chooses an asset to invest its existing funds. We label this asset as the bank’s existing asset. At date 1, the bank raises $1 by issuing insured deposits and subordinated debt. Both markets are competitive; hence depositors and subordinated creditors earn zero expected profit on their investments. We assume that creditors can observe which type of asset the bank has chosen at date 0. However, they cannot contract on the bank’s asset choice after debt issuance. At date 2, the bank chooses another asset to invest its new funds. We label this asset as the bank’s new asset. At date 3, all the deposits and subordinated debts mature, and the cash flows of both assets are realized. We assume that the bank’s total cash flows are observable and verifiable, and the cash flows of the two assets are independent. We also assume that the bank defaults only when both of its assets fail.7 This assumption gives the bank the benefit of diversification from investing in different assets, and is consistent with the key idea of Diamond (1984). When the bank defaults, it incurs bankruptcy costs of CB(CB < 1). Think of CB as including the loss of the bank’s charter and the damage to the banking entrepreneur’s reputation. 2.2. Discussion of key modeling assumptions The baseline model makes three key assumptions. The first is that the creditors can observe which type of asset the bank has chosen before debt issuance, i.e., its original risk choice. Thus far the empirical evidence is inconclusive on how accurately investors can observe the riskiness of a bank. For example, Morgan (2002) examines the pattern of disagreement between bond raters. He finds that banks are inherently more opaque than nonbanking firms. Ashcraft and Bleakley (2006) find that banks are able to exploit private information in order to manage earnings and the real information content of public disclosure. In contrast, Flannery et al. (2004) study both the market microstructure properties of banks’ equity prices and the analyst forecast errors of banks’ earnings. They find that banks are no more opaque than nonbanking firms. We will show in Section 4 that our main results hold as long as the creditors can imperfectly observe the bank’s original risk choice. There are several reasons to justify this assumption. First, a large academic literature documents 6 We could have assumed that the insurance premium is set equal to a positive constant such that the insurance provider breaks even. Depending on whether the insurance premium is paid ex ante or ex post, deposit insurance can have different impacts on the bank’s risk-taking incentives. But as long as the insurance premium cannot be made contingent on the bank’s risk, our main results will hold. 7 A similar assumption appears in Boot and Schmeits (2000). This assumption would be satisfied if the cash flows generated by the success of either asset are sufficient to pay off all the depositors and creditors of the bank.
that creditors can at least partially observe the riskiness of a bank, and charge riskier banks a higher interest rate (see, e.g., Covitz et al., 2004; Morgan and Stiroh, 2001 and Sironi, 2003). Second, creditors would mandate information disclosure as a requirement for investing. Indeed, Covitz and Harrison (2004) find that banks convey private information to the market during bond issuances. Finally, the New Basel Accord specifies a set of disclosure requirements that aims to encourage market discipline. These requirements would help creditors better assess the riskiness of a bank (see BCBS, 2004). The second key assumption is that the regulator cannot observe the bank’s original risk choice, or, even if she can observe it, she cannot shut the bank down based on her observation.8 This assumption is reasonable because it is difficult to imagine that in practice bank regulators would be allowed to shut a bank down based on soft information about the quality of its investments. The third key assumption is that once the bank has invested in an asset, it cannot reverse its investment decision.9 We will show that our main results continue to hold even when the bank can shift a fraction of its investment after debt issuance. There are reasons to believe that it is often costly for banks to quickly and significantly adjust the composition of their assets. To begin with, a fundamental function performed by banks is to collect demand deposits and make long-term capital investments that are somewhat irreversible (Diamond and Dybvig, 1983). Banks would incur substantial costs if they prematurely terminate such investments, and hence the composition of their assets changes only gradually. Indeed, Hancock et al. (1995) find that banks typically take several quarters, and in many instances a few years, to completely adjust the composition of their assets to new desired levels following shocks to their capital positions. Selling loans is an alternative for banks to adjust the composition of their assets. Doing so, however, is also costly. Banks usually obtain proprietary information about their borrowers in the lending process, and hence adverse selection is a concern for potential buyers of bank loans (see Gorton and Pennacchi, 1990). In addition, banks are relationship lenders (see, e.g., Diamond and Rajan, 2000). They develop valuable skills of collecting loans in the loan originating process, and such skills cannot be easily transferred to others. The loan originating banks cannot credibly commit to monitoring loans for others once these loans are sold. Therefore potential buyers are willing to buy loans only at a discount. 8
Berger et al. (2000) find that supervisory assessments and bond rating agency assessments complement one another, in the sense that each information set helps to forecast the other group’s assessments of bank conditions. They also find that supervisory assessments are less accurate than bond market assessment in predicting future changes in bank performance. 9 Myers and Rajan (1998) make a similar assumption. They assume that banks hold a mix of illiquid loans and liquid securities as assets. The illiquid loans enhance the banks’ capacity to issue debt.
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3. The bank’s asset choices In this section we examine the bank’s asset choices (1) when it issues only insured deposits, and (2) when it issues a combination of insured deposits and subordinated debt. We also examine the impact of equity capital on the bank’s asset choices. 3.1. Insured deposits Suppose that at date 1, the bank raises the entire $1 of needed funds by issuing insured deposits. Because the market for insured deposits is competitive, the face value of these deposits is $1, whether the bank invests in the safe assets or the risky assets. The bank chooses two assets to maximize its expected profit. It can invest in two safe assets, two risky assets, or one safe asset and one risky asset. Because we wish to emphasize the bank’s incentives to take excessive risk, we introduce the following parametric assumption. Assumption 1. The expected return of a safe asset exceeds that of a risky asset by an amount that is not too large D<
1 2 hR h2S ð2 k C B Þ: 2
When this assumption is not satisfied, the bank will always invest in the safe assets, and thus there is no need for regulation. We adopt the convention that when the bank is indifferent between a safe asset and a risky asset, it chooses the safe asset. We then have the following result. Proposition 1. When the bank issues only insured deposits, it invests in two risky assets.
zero expected profit. Not able to write contract on the bank’s future asset choice, they have to anticipate it. They do so by observing which type of asset the bank has already invested in. In this section we assume that the creditors can perfectly observe the type of the bank’s existing asset; later we relax this assumption. In equilibrium the creditors’ expectations must be correct. We restrict attention to pure strategy equilibria. Because safe assets have a higher expected return, the regulator wishes to motivate the bank to only invest in the safe assets. The bank chooses its two assets sequentially: one before and the other after debt issuance. We thus need the following parametric assumption. Assumption 2. The expected return of a safe asset exceeds that of a risky asset by an amount that is not too small:
3.2. Subordinated debt Now suppose that the regulator requires the bank to issue s (0 < s 6 1) amount of unsecured, subordinated debt, and (1 s) amount of insured deposits. When the bank defaults, the insured depositors will receive compensation from the government, but the subordinated creditors will lose their investments in the bank. Consequently, the subordinated creditors care about the bank’s risk, i.e., its asset choices. Because the market for subordinated debt is competitive, the creditors charge the bank an interest rate to make
1 2 hR h2S ð1 k C B Þ; 2 hS h2R h2S ð2 k C B Þ : DP 3 2hS þ ðhR þ hS Þ 1 h2S
ð2AÞ D P ð2BÞ
When this assumption is not satisfied, it is not possible to constrain the bank’s risk taking using subordinated debt alone. We can verify that neither inequality implies the other, and hence both are necessary. We can also verify that Assumptions 1 and 2 always define a range of possible values for D. The next proposition shows that when D is within this range, requiring the bank to issue a proper amount of subordinated debt motivates it to invest in two safe assets. Proposition 2. There are two parameters 2D ; s ¼ 2 k CB 2 hR h2S
Proof. See Appendix. h This result is easy to understand. A risky asset provides the bank with a higher return when it succeeds. Meanwhile, because of limited liability the bank does not need to compensate its depositors when it defaults. This convex payoff structure provides an incentive for the bank to take excessive risk. When the expected return of a safe asset exceeds that of a risky asset only by a small amount, the bank invests in two risky assets in order to take maximum risk.
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s ¼
½D hS ðhR hS Þð2 k C B Þ 1 h2S h3S ðhR hS Þ
;
where 0 < s 6 1 and s 6 s, such that when the regulator sets the amount of subordinated debt s within the range of ½s; s, a subgame perfect Nash equilibrium exists, in which: (i) at date 0, the bank invests its existing funds in a safe asset; (ii) at date 1, the creditors charge the bank an interest rate of h2S = 1 h2S . They would charge the bank a higher interest rate of h2R = 1 h2R had the bank invested its existing funds in a risky asset; (iii) at date 2, the bank invests its new funds in another safe asset; (iv) when s < s or s > s, no such equilibrium exists.
Proof. See Appendix. h To understand this result, consider how the amount of subordinated debt affects the bank’s risk-taking incentives before and after debt issuance. Suppose that the bank
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needs to issue a small amount of debt. At the time of debt issuance, if the creditors observe that the bank has invested in a safe asset, they will charge the bank a low interest rate, for two reasons. First, other things remaining the same, such a bank is less likely to default, because default only happens when both assets fail. Second, such a bank is more likely to invest in another safe asset. Indeed, if after debt issuance the bank invests in a risky asset and when this asset fails, the bank has to compensate its depositors and creditors using the cash flows of its existing asset, unless that asset fails as well. A safe asset is less likely to fail, and hence holding a safe asset increases the bank’s opportunity cost of taking risk. The creditors’ rules for setting interest rate provide an incentive for the bank to take low risk before debt issuance. When the amount of debt is above a lower bound, the incentive is strong enough for the bank to invest in a safe asset. Holding this asset then motivates the bank to invest in another safe asset after debt issuance. Once the debt has been issued, however, the bank’s incentives change. Now the larger the amount of debt, the stronger the bank’s incentives to take risk, because doing so would reduce its expected cost of debt. When the amount of debt is above an upper bound, even a bank holding a safe asset will invest in a risky asset. But rational creditors can anticipate the bank’s future asset choice, and would ask for a high interest rate. Anticipating this, the bank invests in a risky asset even before debt issuance. An alternative way to understand Proposition 2 is to think about the important question raised by Blum (2002): how can a bank credibly commit to choosing a given level of risk after debt issuance? Blum shows that if the bank cannot commit, then requiring it to issue subordinated debt would aggravate its risk-taking incentives. We propose that the bank can use its existing safe asset as a commitment device. The bank can promise the creditors that it will invest in a safe asset after debt issuance. Having already invested in a safe asset makes the bank’s promise credible. The bank’s commitment ability, however, is limited: a safe asset is still not riskless. Thus when the bank has to issue a very large amount of debt it is no longer able to commit. Proposition 2 implies that there can be no observable change of the banks’ risk-taking behavior before and after debt issuance, even when subordinated debt constrains their risk taking. Future research needs to examine whether banks reduce their risk before they issue debt. 3.3. Equity capital In recent years capital requirements have been the primary regulatory tools in bank regulations. In the New Basel Accord, capital requirements are designated as one of the three pillars underpinning prudential regulation, together with supervisory review process and market discipline.
Our model sheds some light on how subordinated debt and equity capital can work together to constrain banks’ risk taking. Proposition 3. An increase of equity capital reduces the lower bound on subordinated debt, and increases the upper bound on subordinated debt. Proof. ds/dk < 0. ds=dk > 0. h The intuition is as follows. Equity capital increases the bank’s opportunity cost of taking risk, because the bank will lose its equity capital when it defaults. As a result, when the bank holds a larger amount of equity capital, a smaller amount of subordinated debt is sufficient to motivate it to invest in the safe assets. Moreover, the bank will not invest in the risky assets even after it has issued a larger amount of subordinated debt. In practice, regulators typically permit a substitution of subordinated debt for equity capital. Proposition 3 implies that a small amount of subordinated debt can reduce the needs of equity capital. But when the bank has to issue a large amount of subordinated debt, more equity capital is called for to ensure that the bank does not take excessive risk. 4. Extensions This section extends the baseline model in two directions to check the robustness of the results. First, we consider the case where the creditors can only imperfectly observe the type of the bank’s existing asset. Second, we consider the case where the creditors can use bond covenants to partially restrict the bank’s future asset choice. We will show that our main results continue to hold in both cases. 4.1. Imperfect observation Following Boot and Schmeits (2000), we model imperfect observation as follows: at the time of debt issuance, with probability a (0 < a < 1) the creditors can observe the type of the bank’s existing asset, and with probability (1 a) they cannot observe it.10 The parameter a measures how accurately the creditors can assess the riskiness of the bank. Proposition 4. There are two parameters a ¼ max s 1 h2S þ h2S ; h2S ðhR hS ÞD ; ðhR þ hS Þ½D hS ðhR hS Þð2 k C B Þ sD ¼ s 1 h2S a h2S ; where h2S < a 6 1 and s < sD 6 1, such that: 10 Alternatively, we could assume that at the time of debt issuance, only a fraction a of creditors can observe the type of the bank’s existing asset, and the rest of the creditors cannot observe it. The results would be the same.
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(i) when a P a and the regulator sets the amount of subordinated debt within the range of ½sD ; s, a subgame perfect Nash equilibrium exists in which the bank invests in two safe assets; (ii) dsD/da < 0; (iii) when a < a, no such equilibrium exists.
Proof. The proof is a simple modification of the Proof of Proposition 2, and is available from the author upon request. h The intuition is as follows. Before the bank issues debt, it considers whether to invest in a safe asset or a risky asset. The bank weighs the moral hazard gains from taking excessive risk against the expected increase in its funding costs, which depend on how accurately the creditors can observe the bank’s original risk choice, and how large the amount of debt is. When the creditors cannot accurately observe the bank’s original risk choice, the regulator has to require the bank to issue a larger amount of debt in order to provide the bank with proper incentives. Doing so, however, aggravates the bank’s risk-taking incentives after debt issuance. Therefore, when the accuracy of the creditors’ observation falls below a critical level, using subordinated debt alone cannot motivate the bank to invest in the safe assets. 4.2. Bond covenants Thus far we have assumed that once the interest rate is determined, the creditors can no longer influence the bank’s risk taking. In practice, however, creditors may continue to influence the bank’s risk taking through bond covenants (see Goyal, 2005). To study how our results change when the creditors can partially restrict the bank’s risk taking through bond covenants, we modify the baseline model as follows: the bank will incur costs of CA if it invests in a risky asset after debt issuance. Think of CA as the penalty arising from the violation of the covenants. For expositional clarity, we assume that CA is not too large Assumption 3 CA <
1 2 hR h2S ð2 k C B Þ D: 2
We then have the following result. Proposition 5. There are two parameters sA and sA (both are defined in Appendix), where 0 < sA < 1 and sA < sA , such that: (i) when the regulator sets the amount of subordinated debt within the range of ½sA ; sA , a subgame perfect Nash equilibrium exists in which the bank invests in two safe assets; (ii) dsA/dCA < 0; (iii) dsA =dC A > 0.
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Proof. See Appendix. h The intuition is clear. A higher value of CA means that it is more costly for the bank to take risk; hence the lower bound on subordinated debt is decreased. Moreover, the bank is less likely to invest in a risky asset even after it has issued a large amount of subordinated debt; hence the upper bound on subordinated debt is increased. 5. Discussion The existing subordinated debt proposals show considerable disagreements on the required amount of debt (see Board and Treasury, 2000, pp. 45). Our model suggests that regulator needs to set s, the fraction of debt funded by subordinated debt versus insured deposits, within a range. We now use a simple example to illustrate how the lower and upper bounds of this range depend on a number of factors. The baseline parameters are as follows: RS = 1.1, hS = 0.1, RR = 1.35, hR = 0.3, k = 0.1 and CB = 0.7. The results are summarized in Table 1. Panel A illustrates that when the return of the safe asset decreases, the lower bound on subordinated debt increases, and the upper bound on subordinated debt decreases. Notice that when the return is sufficiently small, the lower bound exceeds the upper bound, implying that in this case subordinated debt alone cannot induce the bank to invest in the safe assets. The policy implication is immediate: subordinated debt may not be able to provide discipline when regulators need it most, i.e., during times of aggregate financial distress when returns of safe assets are low.11 During such times regulators need to rely more on other regulatory instruments, such as supervisory review. Panel B illustrates the results in Proposition 4. As the accuracy of the creditors’ observation decreases, the lower bound on subordinated debt increases. Panel C illustrates the results in Proposition 5. Bond covenants decrease the lower bound and increase the upper bound on subordinated debt. There are several other issues to discuss. First, the present model assumes that once the bank has invested in an asset, it cannot reverse its investment decision. What would happen if the bank can shift a fraction of its investment in the safe asset into the risky asset after debt issuance? It turns out that when the fraction is small, it is not optimal for the bank to do so. Therefore, the results in Proposition 2 continue to hold.12 Intuitively, a small shift would not change the bank’s probability of default and thus its expected cost of debt, but would reduce its expected investment return. More general, our results hold when it is costly for banks to quickly and significantly adjust the composition 11 Competition may also reduce safe returns. For some interesting analysis on bank competition and prudential regulation, see Hellmann et al. (2000) and Repullo (2004). 12 The proof is available from the author upon request.
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Table 1 Results of the numerical example Panel A: The resulting lower and upper bounds when RS changes 1.100 1.095 1.090 1.085 1.080 1.075 RS Lower bound (s) 0.075 0.187 0.300 0.412 0.525 0.637 Upper bound ðsÞ 103.950 81.675 59.400 37.125 14.850 7.425 Panel B: The resulting lower and upper bounds when a changes a 1.000 0.900 0.800 0.700 0.600 0.500 Lower bound (sD) 0.075 0.083 0.094 0.108 0.126 0.152 Upper bound ðsÞ 103.950 103.950 103.950 103.950 103.950 103.950 Panel C: The resulting lower and upper bounds when CA changes CA 0.000 0.001 0.002 0.003 0.004 0.005 0.075 0.062 0.050 0.037 0.025 0.012 Lower bound (sA) Upper bound ðsA Þ 103.950 108.900 113.850 118.800 123.750 128.700
of their assets. For banks that are mainly in the traditional lending business, we believe that this is a reasonable assumption, for the reasons discussed in Section 2. For banks that are mainly in the business such as security trading, this assumption is questionable, because such banks usually hold a large amount of highly tradable securities, and receive a large amount of trading revenues. However, for such banks there exist other mechanisms through which subordinated debt can constrain their risk taking. For example, such banks would have strong incentives to develop and maintain the reputation of being a low-risk counterparty in trading transactions. The yields of the subordinated debts of these banks can then serve as a valuable signal to both private investors and bank regulators. Second, in the present model the safe asset has a higher NPV than the risky asset, and subordinated debt helps produce the efficient outcome. One might ask: if the risky asset has a higher NPV, will subordinated debt prevent the bank from producing the efficient outcome? It turns out that this need not be a concern. The reason is the following. In equilibrium the creditors always break even, and thus subordinated debt does not affect the expected returns from investing in either type of asset. When the risky asset has a higher NPV, the bank will invest in it whether it is required to issue subordinated debt or not, because doing so would maximize both the investment returns and the wealth transfer from the deposit insurance provider. In other words, subordinated debt will not prevent the bank from producing the efficient outcome. Third, the present model does not consider the possibility of distress. What would happen if the bank becomes distressed when it has to issue subordinated debt? Information asymmetries are likely to become larger during distress, and thus the presence of subordinated debt may amplify the bank’s risk-taking incentives at this point. However, the empirical evidence seems to say otherwise, as Ashcraft (2006) documents that the mix of subordinated debt in bank capital has a beneficial impact on the future outcomes of distressed banks. Ashcraft speculates that this is because creditors are more likely to impose restrictive covenants on
distressed banks.13 Identifying the exact mechanism is an important topic for future research. Finally, if regulators impose a subordinated debt requirement on banks, would banks shift their balance sheets in a fashion such that the subordinated debt requirement would not affect their risk taking? This question is highly complex, and is inevitably part of the broader question of how much capital a bank should maintain. To address this question, we need a model with endogenously determined level of bank capital. We leave this question to future research. 6. Conclusion The ongoing regulatory debate and the seemingly conflicting empirical findings call for a theoretical model to organize our thoughts on how subordinated debt constrains banks’ risk taking. This paper proposes such a model. At the core of the model is a simple argument: requiring banks to issue subordinated debt motivates them to invest in safe assets before debt issuance; holding such assets then constrains their risk taking after debt issuance. As a result, researchers may observe no change of the banks’ risk-taking behavior before and after debt issuance, even though subordinated debt has constrained their risk taking. Future research needs to examine whether banks reduce their risk before they issue debt. The amount of subordinated debt affects banks’ risktaking incentives differently before and after debt issuances. Therefore, regulators should set the amount of debt within a range. We show that the bounds of this range depend on a number of factors, such as the return of the safe asset, the creditors’ ability to assess the riskiness of the bank, and their ability to use bond covenants. Because these factors differ across countries (see BCBS, 2003), our model suggests that the optimal amount of subordinated debt varies across countries. Franchise value (also known as charter value) also can reduce banks’ risk-taking incentives, because banks will lose their valuable charters when they default (see Keeley, 1990). Our model suggests that franchise value and subordinated debt can work together to constrain banks’ risk taking. One feature distinguishes our model from franchise value theory: while usually it is difficult for banks to increase their franchise values, they can be motivated to reduce their risk. Requiring banks to issue subordinated debt aims at providing them with such incentives. Acknowledgments I am grateful to Erwan Morellec and Elu von Thadden for their constant support and advice. I would like to thank 13 Goyal (2005) finds that charter values significantly affect the likelihood of restrictive covenants in bank debt contracts.
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Alex Edmans, Michel Habib, Enrique Schroth, Giorgio Szego (the editor), Lucy White, Yu Zhang, seminar participants at Simon Fraser University, the 2006 NFA meeting, and especially three anonymous referees for helpful comments and suggestions, and the University of Rochester for its hospitality. Financial support from NCCR FinRisk and the Swiss National Science Foundation is gratefully acknowledged.
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We solve the model by backward induction, and prove the proposition using a number of preliminary results. Claim 1. If I = S, then the bank will choose J = S if and only if s 6 s, where s is defined by ½D hS ðhR hS Þð2 k C B Þ 1 h2S s ¼ : ð3Þ h3S ðhR hS Þ Proof. When I = S, the subordinated creditors set B = BL. Now if the bank chooses J = S, its expected profit, denoted VSLS, is given by
Appendix Proof of Proposition 1. Recall that initially the bank is endowed with k amount of equity capital, and (1 k) amount of insured deposits. It then issues $1 of insured deposits so that the total amount of insured deposits becomes (2 k). When the bank defaults, it incurs bankruptcy costs of CB. The bank chooses its two assets by comparing its expected profit in the following three cases. Case 1: The bank invests in two safe assets. Its expected profit in this case is 2ð1 hS ÞRS 2 þ k þ h2S ð2 k C B Þ: Case 2: The bank invests in two risky assets. Its expected profit in this case is 2ð1 hR ÞRR 2 þ k þ h2R ð2 k C B Þ: Case 3: The bank invests in one safe asset and one risky asset. Its expected profit in this case is ð1 hS ÞRS þ ð1 hR ÞRR 2 þ k þ hS hR ð2 k C B Þ:
V SLS ¼ 2ð1 hS ÞRS 1 h2S ð2 k s þ BL Þ h2S C B : ð4Þ Alternatively, if the bank chooses J = R, its expected profit, denoted VSLR, is given by V SLR ¼ ð1 hS ÞRS þ ð1 hR ÞRR ð1 hS hR Þ ð2 k s þ BL Þ hS hR C B : It is easy to verify that VSLS P VSLR if and only if s 6 s. h Claim 2. If I = R, then the bank will choose J = R. Proof. Because I = R, the subordinated creditors set B = BH. Now if the bank chooses J = S, its expected profit, denoted VRHS, is given by V RHS ¼ ð1 hR ÞRR þ ð1 hS ÞRS ð1 hS hR Þ ð2 k s þ BH Þ hS hR C B :
It follows from Assumption 1 that the bank can maximize its expected profit by investing in two risky assets. Proof of Proposition 2. Let I denote the type of asset that the bank chooses to invest in before debt issuance: I = S if the bank invests in a safe asset, and I = R if the bank invests in a risky asset. Similarly, let J denote the type of asset that the bank chooses to invest in after debt issuance: J = S if the bank invests in a safe asset, and J = R if the bank invests in a risky asset. At date 1 the bank issues s amount of subordinated debt. Let B denote the face value of the debt. We assume that the subordinated creditors set B = BL when they observe that the bank has chosen I = S, and set B = BH when they observe that the bank has chosen I = R, where BL and BH are defined by s ; 1 h2S s : BH ¼ 1 h2R
BL ¼
ð1Þ ð2Þ
Note that setting B = BL is the same as charging the bank an interest rate of h2S = 1 h2S , and setting B = BH is the same as charging the bank an interest rate of h2R = 1 h2R .
Alternatively, if the bank chooses J = R, its expected profit, denoted VRHR, is given by V RHR ¼ 2ð1 hR ÞRR ð1 h2R Þð2 k s þ BH Þ h2R C B : ð5Þ We have V RHR V RHS ¼ h2R hS hR ð2 k s þ BH C B Þ D > h2R hS hR ð2 k C B Þ D 1 > h2R h2S ð2 k C B Þ D > 0; 2 where the first inequality is obtained because BH > s, and the third inequality follows from Assumption 1. Therefore, if I = R, the bank will choose J = R. h Claim 3. The bank will choose I = S and J = S if and only if s 2 ½s; s, where s is defined by s ¼ 2 k CB
2D : h2R h2S
We claim that 0 < s 6 1 and s 6 s.
ð6Þ
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Proof. If the bank chooses I = S, the subordinated creditors set B = BL. It then follows from Claim 1 that the bank will choose J = S if and only if s 6 s. In this case the expected profit of the bank is VSLS, where VSLS is given by Eq. (4). On the other hand, if the bank chooses I = R, the subordinated creditors set B = BH. It then follows from Claim 2 that the bank will choose J = R. In this case the expected profit of the bank is VRHR, where VRHR is given by Eq. (5). It is easy to verify that VSLS P VRHR if and only if s P s. h Assumption 1 implies that s > 0. Because s 2 (0, 1], to make it possible of having some s 2 ½s; s, we need to ensure two conditions. First, we need s 6 1. This condition is equivalent to Assumption 2A. Second, we need s 6 s. This condition is equivalent to Assumption 2B. Hence by Assumption 2, both conditions are satisfied. Claim 4. When s 2 ½s; s, the subordinated creditors make zero expected profit. Proof. When s 2 ½s; s and the bank has chosen I = S, the subordinated creditors set B = BL. Claim 1 implies that the bank will choose J = S. Therefore the bank will default with probability h2S . Eq. (1) then implies that the subordinated creditors make zero expected profit. On the other hand, when the bank has chosen I = R, the subordinated creditors set B = BH. Claim 2 implies that the bank will choose J = R. Therefore the bank will default with probability h2R . Eq. (2) then implies that the subordinated creditors make zero expected profit. Combining above results, we see that if and only if s 2 ½s; s, a subgame perfect Nash equilibrium exists in which the bank chooses I = S and J = S, and the subordinated creditors make zero expected profit. h Proof of Proposition 5. The proof is a modification of that of Proposition 2. We find that there is no change of the value of VSLS, while the value of VSLR is reduced by an amount of CA. Consequently VSLS P VSLR if and only if s 6 sA , where C A 1 h2S sA ¼ s þ 3 : hS ðhR hS Þ We then find that there is no change of the value of VRHS, while the value of VRHR is reduced by an amount of CA. It follows from Assumption 3 that VRHR > VRHS. Finally, we find that VSLS P VRHR if and only if s P sA, where CA sA ¼ s 2 hR h2S Observe that sA < s and sA > s. It is easy to verify that 0 < sA, dsA/dCA < 0, and dsA =dC A > 0. References Ashcraft, A., 2006. Does the market discipline banks? New evidence from the regulatory capital mix. Federal Reserve Bank of New York Staff Reports, no. 244.
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