《货币经济学》课程文献阅读：Bank capital, interbank contagion, and bailout policy（JBF,2013）

Journal of Banking Finance 37(2013)2765-2778 Contents lists available at SciVerse Science Direct Banking Financ ELSEVIER journalhomepagewww.elsevier.com/locate/jbf Bank capital, interbank contagion, and bailout policy Cross Mark Suhua Tian, Yunhong Yang Gaiyan Zhang.* 200433 China Peking University, 5 Yiheyuan Road, Beijing 100871, China College of Business Admin University of missouri at St Louis, One University Blvd. St Louis, Mo 63121, USA A INFO ABSTRACT This paper develops a theoretical framework in which asset linkages in a syndicated loan eceived 9 February 2011 althy bank when its partner bank fails. We investigate how capital constraints vailable online 13 April 2013 of the healthy bank to takeover or liquidate the exposure held jointly with the failing ba bank's ex ante optimal capital holding and possibility of contagion are affected by anticipa EL classification: policy, capital requirements and the joint exposure. We identify a range of factors tha weaken the possibility of contagion and bailout. Recapitalization with common stock rather than pre- ferred equity injection dilutes existing shareholder interests and gives the bank a greater incentive to L51 Id capital to cope with potential contagion. Increasing the minimum regulatory capital does not nec. ssarily reduce contagion, while the requirement of holding conservation capital buffer could increase the bank,'s resilience to avoid contagion e 2013 Elsevier B V. All rights reserved. pital holding egulatory capital requirement Liquidation 1 Introduction extent banks can absorb external shocks internally during a financial crisis. Improved understanding of this issue can help the authorities There is a longstanding and ongoing debate about whether gov- better balance the benefits of government bailout, in containing the ernment bailout is necessary during a financial crisis and, if so, in contagion of a financial crisis, from its substantial costs what form it should be provided Some believe that government In this paper, we develop a theoretical framework in which a bailout of banks will save banks and their projects, minimizing a healthy bank(Bank 1)can become infected when its partner bank domino effect in the financial system and the loss of employment:( Bank 2) in a joint exposure to a syndicated loan fails and defaults Bailing out Wall Street bankers is necessary to keep the Us economy on its share of loan. We analyze the impact of Bank 1s capital hold from crumbling even further and taking American workers down with ing and the size of its exposure on contagion or continuation of it "(Barack Obama, US president, 29 September 2008). nt exposures. Furthermore, we investigate how Bank 1's capital However, others believe that banks can self-adjust, finding a prior to the crisis and possibility of contagion are affected by antic new equilibrium without help from the government: " Bailout is ipated bailout and regulation policies and a number of important not necessary. The banking industry can handle this mess intemally factors related to Bank I's exposure. and does not need subsidies. "(Bert Ely, a leading expert on banking Our study employs the inventory theoretic framework of bank and finance in the washington policy community, 24 September capital, which advocates that banks maintain a buffer of capital in Therefore, the banks'ability to self-adjust plays a key role in gov- ernment bailout decisions. Given the potential drawbacks of govern es the federal budget deficit and may even drag the untry into a fiscal crisis. Hellmann et al (2000) cite a world Bank study showing ment bailout, it is important to understand whether and to what that the costs related to financial crises can reach 40 percent of GDP. During the 2008 nder the Troubled Asset Relief Program(TARP) European governments interven Corresponding author. Tel +1 314 516 6269: fax: +1 314 516 6600. ailaddresses:tiansuhua@fudan.edu.cn(STian)yhyang@gsm.pku.edu.cn(Y Dexia by Belgium, France, and Luxembourg(E150 billion). Hypo Real Estate Bank by Yang), zhangga@umsledu(G. Zhang) Germany(E50 billion), ING by Dutch government(E35 billion), and others 0378-4266s- see front matter o 2013 Elsevier B V. All rights reserved. http://dx.doiorg/10.1016j-jbankfin.2013.03.024

Bank capital, interbank contagion, and bailout policy Suhua Tian a , Yunhong Yang b , Gaiyan Zhang c,⇑ a School of Economics, Fudan University, 600 Guoquan Road, Shanghai 200433, China bGuanghua School of Management, Peking University, 5 Yiheyuan Road, Beijing 100871, China c College of Business Administration, University of Missouri at St. Louis, One University Blvd., St. Louis, MO 63121, USA article info Article history: Received 9 February 2011 Accepted 20 March 2013 Available online 13 April 2013 JEL classification: G21 E42 L51 Keywords: Interbank linkages Optimal capital holding Contagion Bailout policy Regulatory capital requirement Takeover Liquidation abstract This paper develops a theoretical framework in which asset linkages in a syndicated loan agreement can infect a healthy bank when its partner bank fails. We investigate how capital constraints affect the choice of the healthy bank to takeover or liquidate the exposure held jointly with the failing bank, and how the bank’s ex ante optimal capital holding and possibility of contagion are affected by anticipation of bail-out policy, capital requirements and the joint exposure. We identify a range of factors that strengthen or weaken the possibility of contagion and bailout. Recapitalization with common stock rather than pre￾ferred equity injection dilutes existing shareholder interests and gives the bank a greater incentive to hold capital to cope with potential contagion. Increasing the minimum regulatory capital does not nec￾essarily reduce contagion, while the requirement of holding conservation capital buffer could increase the bank’s resilience to avoid contagion. 2013 Elsevier B.V. All rights reserved. 1. Introduction There is a longstanding and ongoing debate about whether gov￾ernment bailout is necessary during a financial crisis and, if so, in what form it should be provided. Some believe that government bailout of banks will save banks and their projects, minimizing a domino effect in the financial system and the loss of employment: ‘‘Bailing out Wall Street bankers is necessary to keep the US economy from crumbling even further and taking American workers down with it.’’ (Barack Obama, US president, 29 September 2008). However, others believe that banks can self-adjust, finding a new equilibrium without help from the government: ‘‘Bailout is not necessary. The banking industry can handle this mess internally and does not need subsidies.’’ (Bert Ely, a leading expert on banking and finance in the Washington policy community, 24 September 2008). Therefore, the banks’ ability to self-adjust plays a key role in gov￾ernment bailout decisions. Given the potential drawbacks of govern￾ment bailout, it is important to understand whether and to what extent banks can absorb external shocks internally during a financial crisis. Improved understanding of this issue can help the authorities better balance the benefits of government bailout, in containing the contagion of a financial crisis, from its substantial costs.1 In this paper, we develop a theoretical framework in which a healthy bank (Bank 1) can become infected when its partner bank (Bank 2) in a joint exposure to a syndicated loan fails and defaults on its share of loan. We analyze the impact of Bank 1’s capital hold￾ing and the size of its exposure on contagion or continuation of joint exposures. Furthermore, we investigate how Bank 1’s capital prior to the crisis and possibility of contagion are affected by antic￾ipated bailout and regulation policies and a number of important factors related to Bank 1’s exposure. Our study employs the inventory theoretic framework of bank capital, which advocates that banks maintain a buffer of capital in 0378-4266/$ - see front matter 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2013.03.024 ⇑ Corresponding author. Tel.: +1 314 516 6269; fax: +1 314 516 6600. E-mail addresses: tiansuhua@fudan.edu.cn (S. Tian), yhyang@gsm.pku.edu.cn (Y. Yang), zhangga@umsl.edu (G. Zhang). 1 Government bailout increases the federal budget deficit and may even drag the country into a fiscal crisis. Hellmann et al. (2000) cite a World Bank study showing that the costs related to financial crises can reach 40 percent of GDP. During the 2008 global financial crisis, the US government spent $250 billion to recapitalize the banks under the Troubled Asset Relief Program (TARP). European governments intervened to rescue financial institutions, such as Fortis by the Benelux countries ($16 billion), Dexia by Belgium, France, and Luxembourg (€150 billion), Hypo Real Estate Bank by Germany (€50 billion), ING by Dutch government (€35 billion), and others. Journal of Banking & Finance 37 (2013) 2765–2778 Contents lists available at SciVerse ScienceDirect Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf

apital buffer(as in Basel) outside se the bank's resilience to avoid con-

excess of regulatory requirements to reduce future costs of illiquid￾ity and recapitalization.2 In our model, two banks jointly make a syn￾dicated loan for an indivisible project. When an external shock leads the partner bank to discontinue its business operations, Bank 1 has two options: (a) accepting the liquidation of the syndicated project and receiving a comparatively low liquidation value, or (b) taking over all of the interest of Bank 2 in the indivisible project. Bank 1 also anticipates that the government may inject common equity or pre￾ferred equity into it if Bank 2 becomes distressed. If Bank 1’s capital level after taking over or liquidating the distress loan is lower than the regulatory capital requirement, the bank will be liquidated with the loss of all future dividends payments to shareholders. Thus, the failure of Bank 2 forces Bank 1 into liquidation and contagion occurs. In our analysis, we first provide the basic accounting analysis using balance sheet developments to examine when continuation of the joint project is possible, when contagion may emerge, and when bailout is needed to prevent contagion. Then we extend the analysis using the technique of dynamic stochastic optimiza￾tion to investigate Bank 1’s value to shareholders when it takes over or liquidates the joint project, and its value to shareholders prior to the shock allowing for the possible bank actions after the crisis. Bank 1’s decision in the crisis is based on the relative values after taking over or liquidating the joint project. Then we charac￾terize the optimal ex-ante capital holding and compare it with the regulatory capital requirement to examine whether contagion happens and how much capital in the form of common stock or preferred stock must be provided when bailout is necessary. Our simulations show that contagion will not occur if the healthy bank properly anticipates Bank 2’s failure and increases its ex-ante optimal capital holding to accommodate the joint project that may fail. However, if Bank 1 seriously underestimates the probability of the shock, its capital level will be lower than the regulatory require￾ment for taking over or liquidating the project, triggering contagion. In addition, if it has a high fraction of its assets invested in the joint project, a low bargaining power over the project, an exposure smal￾ler than Bank 2’s exposure in the joint project, or a large loss of mar￾ket value of the project, its capital level is more likely to be lower than the required capital level to take over or liquidate the project. In sum, low capital ratios play a key role in promoting contagion and forcing liquidation. Interbank contagion can be minimized if the surviving banks are well capitalized and capable of making opti￾mal choices in response to potential external shocks. Our model provides several important policy implications. First, a higher anticipated probability of bailout will lead Bank 1 to hold less capital, reflecting the risk of moral hazard. Second, when the government injects funds in the form of common equity rather than preferred stock, it dilutes existing shareholder interests more and hence provides a stronger incentive for Bank 1 to hold more capital, reducing moral hazard. Third, increasing the minimum reg￾ulatory capital ratio per se may increase the possibility of conta￾gion if Bank 1’s increase of optimal capital buffer is not sufficient to match the increased capital requirement. Finally, the require￾ment of holding conservation capital buffer (as in Basel III) outside periods of stress could increase the bank’s resilience to avoid con￾tagion during the crisis. These results, collectively, provide theoret￾ical support for the global government efforts to promote robust supervision and regulation of financial firms and give new insight into how this task can be best undertaken.3 Three contributions of our analysis are noted. First, our study adds to the theoretical bank contagion literature by examining interbank contagion due to banks’ joint exposure to a common as￾set. In our model, contagion arises from uncertainties of banks’ as￾sets side, which differs from the common theoretical framework (such as bank-run models) for analyzing contagion from liabili￾ties-side risk due to maturity mismatch. In the seminal paper by Diamond and Dybvig (1983), bank-run is caused by a shift in depositors’ expectations due to some commonly observed factor such as a sunspot. In more realistic settings, Chari and Jagannathan (1988), Gorton (1985) rely on asymmetric information between the bank and its depositors on the true value of loans to induce bank runs, while Chen (1999) relies on Bayesian updating deposi￾tors who learn from interim bank failures that lead to bank runs. Allen and Gale (2000) propose that contagion arises because a liquidity shock in one region can spread throughout the economy due to interregional claims of one bank on other banks. While the above bank contagion literature has focused mainly on deposit withdrawals as a propagation mechanism, a distur￾bance on the lending side can propagate and infect the system. This possibility deserves more attention from the theoretical perspec￾tive. Honohan (1999) shows disturbances can be transmitted through lending decisions due to banks over-committing to risky lending. Our paper adds to this strand of studies by examining con￾tagion arising from lending-side risk, in particular, due to banks’ joint exposure to a syndicated loan. This is supported by empirical evidence in Ivashina and Scharfstein (2010), who find that banks co-syndicated with Lehman suffered more stresses of liquidity, indicating that Lehman’s failure put more of the funding burden on other members of the syndicate and exposed them to increased likelihood that more firms would draw on their credit lines. Although our model deals with potential contagion arising from exposure to a syndicated loan agreement, the implications can be ex￾tended to more general situations of interbank linkages, for example, exposure to a common asset market such as sub-prime mortgage backed securities, or a situation with direct counterparty exposure. The counterparty contagion hypothesis predicts that firms with close business or credit relationships with a distressed firm will suffer ad￾verse consequences from the financial troubles of the distressed firm (Davis and Lo, 2001; Jarrow and Yu, 2001).4 Given the complexity of interbank linkages, counterparty risk is even more worrisome for finan￾cial institutions. In the spirit of our model, whether other banks will fail in the wake of the collapse of a counterparty bank depends on whether their optimal capital holding before the shock exceeds the minimum 2 This strand of literature posits that banks treat their capital holding strategy as an inventory decision that allows them to be forward-looking by increasing their capital levels as necessary or adjusting their asset portfolios in response to any future breach of regulatory capital requirements. The buffer stock model of bank capital was first proposed by Baglioni and Cherubini (1994), later developed by Milne and Robertson (1996), Milne and Whalley (2001), Milne (2004), and in discrete time by Calem and Rob (1996). Peura and Keppo (2006) extend the continuous-time framework to take account of delays in raising capital. Milne and Robertson (1996) state that banks maintain extra capital in excess of minimum regulatory requirements in order to reduce the potential future costs of illiquidity and recapitalization. Milne (2002) further examines the implications of bank capital regulation as an incentive mechanism for portfolio choice. Milne (2004) argues that banks’ risk-taking incen￾tives depend on their capital buffer, not on the absolute level of capital. Our focus is different. We consider the bank’s optimal capital decision and interbank contagion using the inventory framework. 3 For example, the US Department of the Treasury states that ‘‘capital and liquidity requirements were simply too low. Regulators did not require firms to hold sufficient capital to cover trading assets, high-risk loans, and off-balance sheet commitments, or to hold increased capital during good times to prepare for bad times.’’ (Financial regulatory reform: a new foundation, 2010. See http://www.financialstability.gov/ docs/regs/FinalReport_web.pdf) 4 Empirically the counterparty contagion hypothesis is supported by Hertzel et al. (2008), Jorion and Zhang (2009), Brunnermeier (2009), Chakrabarty and Zhang (2012), Iyer and Peydro (2011), among others. As Helwege (2009) points out, government bailout is necessary if counterparty contagion is a major contagion channel for financial firms. The related interbank contagion literature relies on contractual dependency such as a bilateral swap agreement to induce contagion when one party is unable to honor the contract (e.g., Gorton and Metrick, 2012). Another interbank contagion channel is when fire-sale of illiquid assets by one bank depresses asset prices and prompts financial distress at other institutions (e.g., Shleifer and Vishny (1992), Allen and Gale (1994), Diamond and Rajan (2005), Brunnermeier (2009), Wagner (2011)). 2766 S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778

S. Tian et al Jourmal of Banking 8 Finance 37(2013)2765-2778 Investment =B+S Shock Fig. 1. Investment and cash flow for Project G capital requirement after the banks take action such as liquidating or able to bailing out failed banks because it induces banks to differentiate taking over the assets associated with the failed bank. theirrisks. Kashyapet al (2008) propose replacing capital requirements Second, using the inventory buffer model of bank capital to study by mandatory capital insurance policy so that banks are forced to hoard contagion allows us to model banks' precautionary risk management liquidity Chari and Kehoe(2010) show that regulation in the form of behaviors before crisis happens. Banks' optimal capital holding prior ex-ante restrictions on private contracts can increase welfare while to the crisis is endogenously determined. Within the inventory ex-post bailouts trigger a bad continuation equilibrium of the policy framework, the bank manages inventory reserves in order to cope game. Farhi and Tirole(2012)propose a model that banks choose tocor vith uncertain outcomes. If the bank has sufficient inventory re- relate their risk exposures in anticipation of imperfectly targeted gov serves to take over the joint assets of other banks, the failure of ernment intervention to distressed institutions. one bank does not necessarily lead to contagion. So when the risk Our study is closely related to Philippon and Schnabl(2013) of failure of other banks is properly understood, the possibility of who analyze public intervention choices(buying equity, purchas contagion in the inventory setup becomes relatively remote Govern- ing assets, and providing debt guarantees) to alleviate debt over ment bailout is not always necessary if a bank can internally cope hang among private firms. They find that with asymmetric with potential contagion arising from asset linkages. information between firms and the govern buying equity An alternative is the conventional approach in which a bank cap dominates the two other interventions. we also consider bailou ital is a continuously binding constraint, similar to a household bud with equity injection, but our study further shows that common et or a firm s feasible production set. with this approach, one banks stock bailout is preferable ex ante to preferred equity bailout be- takeover of another bank,s assets is impossible because this would cause it induces banks to target for a higher level of capital holding violate the binding capital requirement. Liquidation of a joint project and thus reduces the government bailout budget. is the only possible outcome. If the bank invests a large share of assets in the project and the loss ratio is high, the failure of one bank leads Acharya et al. (2010)that government support to surviving banks directly to the failure of its partner banks in a joint project. In order conditional on their liquid asset holdings increases banks' incen- to prevent such interbank contagion, it is necessary for the govern- tive to hold liquidity, and that support to failed banks or uncondi- ity of continuing the joint project without government intervention. their study stresses the role of banks'asset composition, our focus Hence contagion becomes excessively mechanical in the conven- is the role of banks' capital holdings in anticipation of common tional set-up, which is inherently biased towards government bailout. stock or preferred equity bailout. Third, our paper adds to the bailout literature by explicitly examin- The rest of the paper is organized as follows. We ing how government bailout policy (injection of common equity versus benchmark model setup in Section 2 In Section 3, we provide the ba preferred equity)affects banks' ex ante capital buffer and possibilities sic accounting analysis to examine when interbank contagion may interbank contagion, and how banks' capital holding prior to the cri- emerge due to a failure of a partner bank In Section 4 we derive the is, in turn, affects the level of government bailout. Earlier studies have solution for a banks optimal capital-asset ratio prior to the crisis for addressed whether, when, and how to bail out a bank. Our study com- dealing with a partner banks potential failure in anticipation of gov- plements the literature by providing a case for why a bailout is not al- ernment intervention. Section 5 shows simulation results for the rela- ways necessary to help a healthy bank survive contagion. tionships between a number of public policy and banks' investment Spurred by the recent financial crisis, there is a growing literature parameters and the level of ex-ante capital holding, possibility of con- on bank bailouts. Acharya and Yorulmazer(2008)point out that tagion, and government bailout amounts. Section 6 concludes granting liquidity to surviving banks to take over failed banks is prefer 2. The model related to earlier work stu g bank behavior under ca uirement constraints. Diamond and Rajan(2000) argue that the optimal bank In this section, we set up a framework to describe how a bank pital structure reflects a tradeoff between the effects of bank capital on liquidity determines its optimal capital-ass that banks eation, the expected costs of bank distress, and the default risk of borrowers. Bolton maintain a buffer of capital that exce set ratIo and Freixas(2006)posit that bank lending is constrained by capital adequacy in order to reduce the potential future costs of ity and recap- hubbard et al.(2002)find that banks that maintain more capital charge italization and the contagion effects of failure of its partner bank. nterest rate on loans. Jokipii and Milne(2008)show that capital buffers of the banks in the eu15 have a significant negative Co-movement with the cycle, which exacerbates the pro-cyclical impact of Basel cation as a capable monitor could distort bank closure policy. dreyfus assume that a banking group enter into a syndicated loan surance coverage is optimal. Rochet and Tirole(1996)derive the optimal prudential agreement to finance part of an investment, B, in an indivisible Pro- conducting undesired rescue opera- ject G Financing for the rest of the project, S, is obtained by issuing ons. Gale and vives(2002)argue that a bail out should be restricted ex ant equity or debt, or comes from other sourc oral hazard concerns. Gorton and Huang(2004 )show that the government bailout Ject G is bein rovides more effective liquidity than private investors. Diamond See the review of theoretical literature by philippon and Schnabl(2013),w that the market is perfect and that the financing metho al work on bailouts such as giannetti and simonov ailable for rest of the investment do not affect the cash flow of project g or the 011 Glasserman and Wang(2011). returns on investment B that the banking group receives

capital requirement after the banks take action such as liquidating or taking over the assets associated with the failed bank. Second, using the inventory buffer model of bank capital to study contagion allows us to model banks’ precautionary risk management behaviors before crisis happens. Banks’ optimal capital holding prior to the crisis is endogenously determined. Within the inventory framework, the bank manages inventory reserves in order to cope with uncertain outcomes. If the bank has sufficient inventory re￾serves to take over the joint assets of other banks, the failure of one bank does not necessarily lead to contagion. So when the risk of failure of other banks is properly understood, the possibility of contagion in the inventory setup becomes relatively remote. Govern￾ment bailout is not always necessary if a bank can internally cope with potential contagion arising from asset linkages.5 An alternative is the conventional approach in which a bank’ cap￾ital is a continuously binding constraint, similar to a household bud￾get or a firm’s feasible production set. With this approach, one bank’s takeover of another bank’s assets is impossible because this would violate the binding capital requirement. Liquidation of a joint project is the only possible outcome. If the bank invests a large share of assets in the project and the loss ratio is high, the failure of one bank leads directly to the failure of its partner banks in a joint project. In order to prevent such interbank contagion, it is necessary for the govern￾ment to inject equity in other banks. However, this omits any possibil￾ity of continuing the joint project without government intervention. Hence contagion becomes excessively mechanical in the conven￾tional set-up, which is inherently biased towards government bailout. Third, our paper adds to the bailout literature by explicitly examin￾ing how government bailout policy (injection of common equity versus preferred equity) affects banks’ ex ante capital buffer and possibilities of interbank contagion, and how banks’ capital holding prior to the cri￾sis, in turn, affects the level of government bailout. Earlier studies have addressed whether, when, and how to bail out a bank.6 Our study com￾plements the literature by providing a case for why a bailout is not al￾ways necessary to help a healthy bank survive contagion. Spurred by the recent financial crisis, there is a growing literature on bank bailouts.7 Acharya and Yorulmazer (2008) point out that granting liquidity to surviving banks to take over failed banks is prefer￾able to bailing out failed banks because it induces banks to differentiate their risks.Kashyap et al. (2008)propose replacing capital requirements by mandatory capital insurance policy so that banks are forced to hoard liquidity. Chari and Kehoe (2010) show that regulation in the form of ex-ante restrictions on private contracts can increase welfare while ex-post bailouts trigger a bad continuation equilibrium of the policy game. Farhi and Tirole (2012) propose amodel that banks choose to cor￾relate their risk exposures in anticipation of imperfectly targeted gov￾ernment intervention to distressed institutions. Our study is closely related to Philippon and Schnabl (2013), who analyze public intervention choices (buying equity, purchas￾ing assets, and providing debt guarantees) to alleviate debt over￾hang among private firms. They find that with asymmetric information between firms and the government, buying equity dominates the two other interventions. We also consider bailout with equity injection, but our study further shows that common stock bailout is preferable ex ante to preferred equity bailout be￾cause it induces banks to target for a higher level of capital holding and thus reduces the government bailout budget. Our results on bailout policy also complement the findings of Acharya et al. (2010) that government support to surviving banks conditional on their liquid asset holdings increases banks’ incen￾tive to hold liquidity, and that support to failed banks or uncondi￾tional support to surviving banks has the opposite effect. While their study stresses the role of banks’ asset composition, our focus is the role of banks’ capital holdings in anticipation of common stock or preferred equity bailout. The rest of the paper is organized as follows. We introduce our benchmark model setup in Section 2. In Section 3, we provide the ba￾sic accounting analysis to examine when interbank contagion may emerge due to a failure of a partner bank. In Section 4 we derive the solution for a bank’s optimal capital-asset ratio prior to the crisis for dealing with a partner bank’s potential failure in anticipation of gov￾ernment intervention. Section5 shows simulation results for the rela￾tionships between a number of public policy and banks’ investment parameters and the level of ex-ante capital holding, possibility of con￾tagion, and government bailout amounts. Section 6 concludes. 2. The model In this section, we set up a framework to describe how a bank determines its optimal capital-asset ratio, assuming that banks maintain a buffer of capital that exceeds the regulatory requirement in order to reduce the potential future costs of illiquidity and recap￾italization and the contagion effects of failure of its partner bank. 2.1. One project We assume that a banking group enter into a syndicated loan agreement to finance part of an investment, B, in an indivisible Pro￾ject G. Financing for the rest of the project, S, is obtained by issuing equity or debt, or comes from other sources.8 Project G is being Cash flow t = 0 t =∞ Shock t =T Investment =B+S Fig. 1. Investment and cash flow for Project G. 5 Our paper is also related to earlier work studying bank behavior under capital requirement constraints. Diamond and Rajan (2000) argue that the optimal bank capital structure reflects a tradeoff between the effects of bank capital on liquidity creation, the expected costs of bank distress, and the default risk of borrowers. Bolton and Freixas (2006) posit that bank lending is constrained by capital adequacy requirements. Hubbard et al. (2002) find that banks that maintain more capital charge a lower interest rate on loans. Jokipii and Milne (2008) show that capital buffers of the banks in the EU15 have a significant negative co-movement with the cycle, which exacerbates the pro-cyclical impact of Basel II. 6 For example, Boot and Thakor (1993) model that a desire for the regulator to acquire a reputation as a capable monitor could distort bank closure policy. Dreyfus et al. (1994) discuss whether the setting of ceilings on the amount of deposit insurance coverage is optimal. Rochet and Tirole (1996) derive the optimal prudential rules while protecting the central banks from conducting undesired rescue opera￾tions. Gale and Vives (2002) argue that a bail out should be restricted ex ante due to moral hazard concerns. Gorton and Huang (2004) show that the government bailout for banks in distress provides more effective liquidity than private investors. Diamond and Rajan (2005) propose a robust sequence of intervention. 7 See the review of theoretical literature by Philippon and Schnabl (2013), who also discuss several recent empirical work on bailouts such as Giannetti and Simonov (2011), Glasserman and Wang (2011). 8 Given that our main research objective is not designing the capital structure for Project G, we assume that the market is perfect and that the financing methods available for rest of the investment do not affect the cash flow of Project G or the returns on investment B that the banking group receives. S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778 2767

S Tian et aL /Joumal of Banking S Finance 37(2013)2765-2778 implemented in two phases At t=0, the banks invest in Project G. 2 denote the risk of investment I A and(1-l), and Z, and Z2 After that, the assets in place generate cash flow, which gives the are Brownian motions, with the correlation coefficient of @12. banks a return on their investment. Project G will repay the banks We assume that R1>R2 and o1>0 in full as long as the project is viable. However, a shock causes on Bank 1 chooses 0 to maximize the shareholders value. mea- bank in the banking group to go into distress and default on its share sured by the expected discounted value of future dividends f the loan at time t=t, which arrives according to a poisson pro- date its own share of the loan in Project G(in which cae the V(C)=maxe e/e"pt0, dt +e-PH(G)I ss. The other banks in the group has to decide whether to liqui project will be liquidated as well)or to take over the failed bank's where p represents both the discount factor(p>0)and, because loan in Project G Fig. 1 shows the timeline for the scenario deposits are unremunerated, the excess cost of equity relative to 22. Two banks bank debt. The first term in the brackets represents the cumulative discounted cash flow generated by the investment project before We assume that the banking group consists of two banks: Bank the shock occurs, and the second term in the brackets represents the discounted cash flow when the shock occurs. The specific form Bank 2. Bank i(i=1, 2)holds a fixed amount of non-tradable of H(Cr)depends on which action Bank 1 takes when the shock hap- alued at A at t=0. The capital of Bank i, denoted by C is the value of its equity. The bank has raised the difference be- pens. We discuss it in detail in Section 4. Regulators constantly tween assets and capital by issuing short-term deposits of D, -A,- compare the net worth c of Bank i with the minimum regulatory C, assuming an infinitely elastic supply of deposits fully insured If Ct. Regulators constantly audit the net worth of a bank. If the net worth of a bank is lower than the minimum regu- latory requirement, it has to be liquidated Its debt holders will (1)The total existing assets of the banks are fixed, and the banks then be repaid in full out of deposit insurance, but its shareholders an adjust only their dividend bayou (2) The banks are able to finance all cash flow needed instanta- will receive nothing. neously by taking out deposit insurance or absorbing more deposits at zero cost. 23. One shock Take Bank 1 as an example. At any time t, Bank 1 pays dividends At a random time t, a shock (the systemic crisis)arriving at a rate 0 subject to 0>0. Cash flow affects net worth C and hence according to a Poisson process causes Bank 2 to default on its share deposits D according to of the loan and require termination of the syndicated loan unless Bank 1 takes over the loan in its entirety. Bank 1 expects the inten- dC=AR:+(1-b)AR2-odt +O1lAdz1 + O2(1-h)Ad=-dd (1) sity of the shock to be o>yo. 3 Bank 1 has to decide whether to lig where r, and r2 denote the expected return of investment I,A and undate its own loan in Project G (in which case the project will be (1-11) in excess of the deposit interest rate, respectively. a, and liquidated as well)or to take over the failed banks loan in Project G Bank 1 also expects the government to offer an equity capital 9 We thank the referee for his her suggestion of introducing a jump process for the injection to Bank 1 in the form of preferred equity or common i We assume that government capital In our model sume that the capital structure decision is determined after injection will give Bank 1 the new desired capital level C, depen the initial investment decision is made. In other words, L Is ! h because we are dated. 4 If Bank 1 accepts the bailout, it will choose the optimal nously and the ing on whether the joint project has been taken over or liqui anks, endogenous selection of l, will make the calculation more complicated. The injection amount k to maximize its shareholders value after it takes takes the form of preferred equity, the shareholders of preferred choose the two-bank setting to examine the potential a potential role for one banks Our model setup, in which a bank holds two components of correlated as allout. Our model can be generalized As discussed later both the size of the distressed loan as a fraction of total assets and he correlation coefficient are important in determining the banks optimal capital B= A ng a random asset maturity wi process is that at any point before the shock, the expected remaining time-to-shock is For a fixed amount B, given that the lat of banks in the group, the always 1/o ower the fraction L, we can examine the In reality. in the event of a government bailout, the injected increases the banks capitalization to well above the mir en the number of banks in gh. When I is large, the loan its own interests and limit the possibility of failure. This assumpti to Project G is a large fraction of investment for bank i. This occurs when the number with our framework, where capital plays an invento the referee for of banks in the group is low pointing this out

implemented in two phases. At t = 0, the banks invest in Project G. After that, the assets in place generate cash flow, which gives the banks a return on their investment. Project G will repay the banks in full as long as the project is viable. However, a shock causes one bank in the banking group to go into distress and default on its share of the loan at time t = T, which arrives according to a Poisson pro￾cess.9 The other banks in the group has to decide whether to liqui￾date its own share of the loan in Project G (in which case the project will be liquidated as well) or to take over the failed bank’s loan in Project G. Fig. 1 shows the timeline for the scenario. 2.2. Two banks We assume that the banking group consists of two banks: Bank 1 and Bank 2. Bank i (i = 1, 2) holds a fixed amount of non-tradable assets valued at Ai at t = 0. The capital of Bank i, denoted by Ci, is the book value of its equity. The bank has raised the difference be￾tween assets and capital by issuing short-term deposits of Di = Ai - Ci, assuming an infinitely elastic supply of deposits fully insured by the regulator. We assume that the original asset allocation of Bank i has been optimally made. The total assets of Bank i can be divided into two components: liAi and (1 li)Ai, (0 6 li 6 1), where liAi is the amount lent by Bank i to Project G and (1 li)Ai is the amount invested in other pro￾jects.10 According to our assumptions, B ¼ P2 i¼1liAi. 11 Regulators constantly audit the net worth of a bank. If the net worth of a bank is lower than the minimum regulatory requirement, it has to be liquidated. Its debt holders will then be repaid in full out of deposit insurance, but its shareholders will receive nothing. We make the following assumptions in line with Milne (2002, 2004) to obtain an analytical solution: (1) The total existing assets of the banks are fixed, and the banks can adjust only their dividend payouts. (2) The banks are able to finance all cash flow needed instanta￾neously by taking out deposit insurance or absorbing more deposits at zero cost. Take Bank 1 as an example. At any time t, Bank 1 pays dividends at a rate h subject to h P 0. Cash flow affects net worth C and hence deposits D according to dC ¼ ½l1AR1 þ ð1 l1ÞAR2 hdt þ r1l1AdZ1 þ r2ð1 l1ÞAdZ2 ¼ dD ð1Þ where R1 and R2 denote the expected return of investment l1A and (1 l1)A in excess of the deposit interest rate, respectively, r1 and r2 denote the risk of investment l1A and (1 l1)A, and Z1 and Z2 are Brownian motions, with the correlation coefficient of x12. 12 We assume that R1 > R2 and r1 > r2. Bank 1 chooses h to maximize the shareholders’ value, mea￾sured by the expected discounted value of future dividends: VðCÞ ¼ maxh E Z T 0 eqt htdt þ eqTHðCT Þ ð2Þ where q represents both the discount factor (q > 0) and, because deposits are unremunerated, the excess cost of equity relative to bank debt. The first term in the brackets represents the cumulative discounted cash flow generated by the investment project before the shock occurs, and the second term in the brackets represents the discounted cash flow when the shock occurs. The specific form of H(CT) depends on which action Bank 1 takes when the shock hap￾pens. We discuss it in detail in Section 4. Regulators constantly compare the net worth C of Bank 1 with the minimum regulatory requirement C ¼ As, in which s is the required capital-asset ratio. If C w0.13 Bank 1 has to decide whether to liq￾uidate its own loan in Project G (in which case the project will be liquidated as well) or to take over the failed bank’s loan in Project G. Bank 1 also expects the government to offer an equity capital injection to Bank 1 in the form of preferred equity or common stock with the probability k. We assume that government capital injection will give Bank 1 the new desired capital level C⁄ , depend￾ing on whether the joint project has been taken over or liqui￾dated.14 If Bank 1 accepts the bailout, it will choose the optimal injection amount K to maximize its shareholders’ value after it takes over or liquidates Project G with the injected capital. If the bailout takes the form of preferred equity, the shareholders of preferred 9 We thank the referee for his/her suggestion of introducing a jump process for the shock. 10 In our model, we assume that the capital structure decision is determined after the initial investment decision is made. In other words, li is given exogenously and the bank determines its optimal capital-asset ratio based on the given li. Because we are interested mainly in the impact of an external shock on the optimal capital holding of banks, endogenous selection of li will make the calculation more complicated. The assumption that the bank’s original portfolio choice is independent of the capital structure is a possible limitation of our model. 11 For simplicity, we choose the two-bank setting to examine the potential interbank contagion issue. With one surviving bank, contagion is possible and there is a potential role for one bank’s takeover of the joint assets and for government bailout. Our model can be generalized to one failed bank and N-1 surviving banks, in which case: B ¼ XN i¼1 liAi For a fixed amount B, given that the larger the number of banks in the group, the lower the fraction li, we can examine the effect of the number of banks in the group by li. When li is small, the loan to Project G is a small fraction of investment for bank i. This occurs when the number of banks in the group is high. When li is large, the loan to Project G is a large fraction of investment for bank i. This occurs when the number of banks in the group is low. 12 Our model setup, in which a bank holds two components of correlated assets, departs from that of Milne (2002, 2004), who treats all banks’ assets homogeneously. As discussed later, both the size of the distressed loan as a fraction of total assets and the correlation coefficient are important in determining the bank’s optimal capital holding. 13 An important advantage of assuming a random asset maturity with a Poisson process is that at any point before the shock, the expected remaining time-to-shock is always 1//. 14 In reality, in the event of a government bailout, the injected capital increases the bank’s capitalization to well above the minimum regulatory requirement to protect its own interests and limit the possibility of failure. This assumption is also consistent with our framework, where capital plays an inventory role. We thank the referee for pointing this out. 2768 S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778

S. Tian et al Journal of Banking 8 Finance 37(2013)2765-2778 Symbols. ook value of Bank 1's equity ank I's deposits, which is equal to the difference between A and c ank 1's investment ratio in Project G ank 1's dividend payout rate(0> 0) R1. R The expected returns of investment I,A and investment(1-l) in excess of the deposit interest rate(r1> R2) Z1, lities of investment I,A and investment (1-1)A,d1>a bank debt The minimum regulatory capital level for Bank 1 before the shock The relative ratio of Bank 2s investment in Project G over Bank 1's investment in Project G apital level after the joint project has bee The optimal bailout amount for Bank 1 to maximize its shareholders'value after it takes over or liquidates Project G with the injected capital The ratio of mark-to-market value of Project G over the initial investment value The minimum regulatory capital level for Bank 1 to take over the loan The minimum regulatory capital level for Bank 1 to liquidate the loan ity will receive only the fixed dividend; they will not share in the bargaining power, x could be equal to 0 and the actual payou de gain should the bank recover. In contrast, since common be the greatest one, nl. we further take into account of"mar k shareholders will share in the upside potential, an injection market accounting", which could lead to a mark down in common equity will dilute existing shareholder interests. ation of the(impaired) joint project in the event of continuation A Table 1 summarizes and explains the notation used in the from 1 to y. The lowest possible valuation would be the price paid for the assets: the highest possible valuation is the original accounting value, so 1-xE <y<1 3.Bank's balance sheet development upon the shock When the crisis occurs, Bank 1 faces a choice between two comes, liquidation or continuation without government support. In this section we provide a preliminary accounting analysis of how parameter assumptions affect the possible balance sheet 1. If the joint project is continued, then the bank must inject addi- developments. We identify when continuation of the joint project tional cash into the wiring it to ra is possible, when contagion will happen and when bail out can be additional deposits ). The balance sheet now becomes as in Bal- sed to prevent liquidation of joint projects. Doing this first pro- ance sheet 2 beloy vides helpful intuition and makes the subsequent technical exposi- Asset y(1+n)l is the mark-to-market value of Project G after tion in Sections 4 and 5 easier to follo Bank 1 takes over the project, liability (1-xE(nl)is the addi- As described in Section 2. the initial balance sheet of Bank 1 can tional deposit raised by Bank 1, capital (1-yXl+n)l is the be formulated as. net change of capital level due to capital loss, arising from accounting mark down of taking over Project G offset by Bank Assets Is capital gain from its bargaining power, (nl)xE. 1- D Assets Liabilities y(1+n) Cr=1-Dr-[(1+n)(1-y)-nxl Balance sheet 1 1+[n-(1+n)(1-y)!1+[n-(1+n)1-y) When the shock happens, Bank 2 defaults on its share of the loan. If Bank 1 decides to liquidate its loan in Project G, l, Project G will be liquidated. We assume s is the loss-given-default ratio Balance Sheet 2 (LGD)of Project G. If Bank 1 decides to take over the failed bank's This implies that the capital ratio alters from G to T*=nl continuing the joint project will depend on the bargaining power We define1=51#+=哪-Cr=厘世#需 of bank 1 Now the capital ratio will fall(41 <0) if xEn<(1-Dr)n+ Dr(1+ The lowest possible price will be the recovery value from liqui- nX1-y), i.e, if the bargaining gain from its bargaining power dation(1-E)nl(because bank 2 can still get this amount by refus ing to sell the assets ). The other extreme is if Bank 1 must pay full accounting value for the loan(if Bank 2 has all the bargaining allow the bank to increase its capital (provided that this bargaining gain exceeds any project can then be written as x(1-4)+(1-x]nl=(1-xenl If x is equal to 1, Bank 1 has stronger bargaining power, the actual marked up the Barclays accounts as"negative good will because they paid much less payout could be the lowest one, n(1-5)l. If Bank 2 has a stronger for the assets than their accounting value. we thank the referee for pointing this out

equity will receive only the fixed dividend; they will not share in the upside gain should the bank recover. In contrast, since common stock shareholders will share in the upside potential, an injection of common equity will dilute existing shareholder interests. Table 1 summarizes and explains the notation used in the model. 3. Bank’s balance sheet development upon the shock In this section we provide a preliminary accounting analysis of how parameter assumptions affect the possible balance sheet developments. We identify when continuation of the joint project is possible, when contagion will happen and when bail out can be used to prevent liquidation of joint projects. Doing this first pro￾vides helpful intuition and makes the subsequent technical exposi￾tion in Sections 4 and 5 easier to follow. As described in Section 2, the initial balance sheet of Bank 1 can be formulated as: Assets Liabilities 1 l Dt l Ct = 1 Dt 1 1 Balance Sheet 1. When the shock happens, Bank 2 defaults on its share of the loan. If Bank 1 decides to liquidate its loan in Project G, l, Project G will be liquidated. We assume n is the loss-given-default ratio (LGD) of Project G. If Bank 1 decides to take over the failed bank’s loan in Project G, the amount paid for taking over the assets and continuing the joint project will depend on the bargaining power of Bank 1. The lowest possible price will be the recovery value from liqui￾dation (1 n)nl (because bank 2 can still get this amount by refus￾ing to sell the assets). The other extreme is if Bank 1 must pay full accounting value for the loan (if Bank 2 has all the bargaining power over the sale of the assets). Let x 2 [0,1] represent the bar￾gaining power of Bank 1, the price paid for the assets in the joint project can then be written as [x(1 n) + (1 x)]nl = (1 xn)nl. If x is equal to 1, Bank 1 has stronger bargaining power, the actual payout could be the lowest one, n(1 n)l. If Bank 2 has a stronger bargaining power, x could be equal to 0 and the actual payout will be the greatest one, nl. We further take into account of ‘‘mark to market accounting’’, which could lead to a mark down in the valu￾ation of the (impaired) joint project in the event of continuation from 1 to y. The lowest possible valuation would be the price paid for the assets; the highest possible valuation is the original accounting value, so 1 xn R2) r1, r2 The volatilities of investment l1A and investment (1 l1)A, r1 > r2 Z1, Z2 The Brownian motions of investment l1A and investment (1 l1)A x12 The correlation coefficient of Z1 and Z2 q The discount factor and the excess cost of equity relative to bank debt C The minimum regulatory capital level for Bank 1 before the shock occurs s The capital-asset ratio required by the regulator n The relative ratio of Bank 2’s investment in Project G over Bank 1’s investment in Project G / The intensity of the Poisson process for the arrival of a financial crisis k The probability of government bailout in the form of preferred equity or common stock C⁄ Bank 1’s new desired capital level after the joint project has been taken over or liquidated K The optimal bailout amount for Bank 1 to maximize its shareholders’ value after it takes over or liquidates Project G with the injected capital n The loss-given-default ratio (LGD) of Project G x The bargaining power of Bank 1 over Project G y The ratio of mark-to-market value of Project G over the initial investment value b C The minimum regulatory capital level for Bank 1 to take over the loan bb C The minimum regulatory capital level for Bank 1 to liquidate the loan 15 Possessing bargaining power in the acquisition of the joint assets (x > 0), may allow the bank to increase its capital (provided that this bargaining gain exceeds any accounting mark down of asset values); and it is possible that the increase in capital is so large that the capital ratio of the bank actually rises rather than falls after it acquires the distressed assets. An illustration is the acquisition by Barclays Group of the assets of Lehman Brothers North America in 2008, which were immediately marked up the Barclays accounts as ‘‘negative good will’’ because they paid much less for the assets than their accounting value. We thank the referee for pointing this out. S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778 2769

S Tian et aL /Joumal of Banking S Finance 37(2013)2765-2778 not sufficient to offset the fall in the capital ratio from the increase when the bank liquidates project G). These have simple closed form in the balance sheet and the mark down in the value of assets the analytical solutions of the general form Ajexpm C+ A2expm 2C bank will be unable to continue the joint project if These obtain because, post-crisis ly decision of the bank is i -(nmox-ym-lC for some target le- vel of buffer capital C. mi and m2 are constants determined by the Liabilities elevant equation of motion for the post-crisis evolution of C and Al 1-l D and A2 are constants of integration. The three free parameters A. A2 and c are determined by three boundary conditions applying at 1-l C. C and C. Appendix A states the equations of motion, the bound- 1-l ary conditions and the resulting closed form solutions. Balance sheet 3 4. 1. Parameter restriction Capital falls by Is and the capital ratio changes from 1 -Dr to --De-s The difference is 42=-(1-Dr)=-. This implies that As shown in the previous section, the regulatory capital re- the capital ratio will fall(42C, then Bank 1 will to liquidate Project G, which is in turn lower than the to take over Project G when the shoc be able to take over the project and survive without government assistance If C>C, then Bank 1 will be able to liquidate the joint (3)C<C<C, that is, the minimum regulatory capital require- project and survive without government assistance. But if both 如 than that C<C and c< c then there is contagion, and the failure of Bank 2 required to liquidate Project G, but lower he amount required to take over Project G when the occurs shown In the following two sections, we use stochastic dynamic program- ming technique to examine the banks choice between liquidation ++yu<5<i+i+my y(I-m holds, the capital required and continuation in the crisis with anticipation of bailout policy to take over Project G will always be lower than the capita We compare Bank 1's ex-post value to shareholders under different required to liquidate the project, i. e, C<C<C 17For exam- scenarios, which is then used to analyze the value function and cap- ple, if the regulatory capital ratio is 10%, x=0, n=l, andy= 1 italization decisions of the bank prior to the crisis. Then we conduct Condition(1)will hold as long as the loss-given-default ratio simulations to investigate the impact of anticipated shock intensity. of the bank loan is higher than 30%, which is supported by the public policy(bailout in the form of common stock or preferred empirical evidence. We therefore choose this plausible con- stock, and regulatory capital requirements)and parameter value (e.g, the exposure to the distressed loan) on Bank 1's capitalization dition and focus on the first case C<c< C, in the subse decisions, possibility of contagion, and government bailout amounts. quent analysis 4. Endogenous capital holding decision willingness to pay Bank 2 will be Using stochastic dynamic programming, we analyze the post- constrain s atively age com hed t the erior adqurnge miw pomie cish crisis value of Bank 1 to shareholders and the value prior to the cri- flow So Bank 1 is less willing to pay a high price to take over the distre sis. There are two possible post-crisis value functions: U(C)(for the thank the referee for suggesting us to consider the complexity of the actual payout by case when the bank takes over project G)and wc)(for the case Bank I to Bank 2 due to the bargaining process, the project's cash flow, and Bank I's Gupton et al., 2000) examine 181 bank loan defaults(mostly syndicated loans systemic crisis. and find that the mean bank-loan value in default is 69.5% vene to prevent bank failure or and 52.1% for Senior Unsecured loans. Therefore the loss-give ecovery rate), is 30.5% for Senior Secured bank loans and 47.9% as preferred shares In Section 4, we will solve for the bailout amounts that must be loans Bank loans usually have a higher recovery rate than other forms of debt. fitc Section 5, we use simulations to show ntagion or maximize shareholder's value. In (2005) historical recovery rates of Senior Unsecured bonds for 24 industries how the bailout amounts depend upon the over the period 2000 to 2004. The mean of average loss-given-def across industrie

not sufficient to offset the fall in the capital ratio from the increase in the balance sheet and the mark down in the value of assets. The bank will be unable to continue the joint project if CT½ð1þnÞð1yÞnxnl f1þ½nð1þnÞð1yÞ1g < s or if CT < ð1 þ ½n ð1 þ nÞð1 yÞlÞs ½nxn ð1 þ nÞð1 yÞl ¼ b C. 2. The joint project is liquidated, in which case with a loss-given￾default ratio of Project G, n, depositors are repaid the recovery from the liquidated loan (1 n)l and the balance sheet of the bank becomes that presented below as balance sheet 3. Assets Liabilities 1 l DT l(1 n) CT ¼ 1 DT ln 1 l 1 l Balance Sheet 3. Capital falls by ln and the capital ratio changes from 1 DT to 1DTln 1l . The difference is D2 ¼ 1DTln 1l ð1 DT Þ ¼ lðCTnÞ 1l . This implies that the capital ratio will fall (D2 < 0) provided the loss given default n is greater than the capital ratio before failure, 1 DT. There will be contagion if the fall in capital is large enough to push bank 1 into liquidation i.e. if CT þ D2 ¼ CT þ l CTn 1l < s or CT < ð1 lÞs þ nl ¼ bb C . From the above discussion, it is clear that the impact of the systemic crisis, and the choices available to the bank when such a crisis oc￾curs, will vary according to the amount of capital it holds at the time of the crisis, CT, the size of its exposure to the joint project rel￾ative to the bank’s total assets l, the relative exposure of the two banks to the joint project n, the loss ratio of the project after liqui￾dation n, the bank’s bargaining power over the impaired assets x, and the accounting treatment of jointly held assets y. There are two critical levels of capital b C and bb C . If C P b C, then Bank 1 will be able to take over the project and survive without government assistance. If C P bb C , then Bank 1 will be able to liquidate the joint project and survive without government assistance. But if both C < b C and C < bb C then there is contagion, and the failure of Bank 2 forces Bank 1 into liquidation without government assistance.16 In the following two sections, we use stochastic dynamic program￾ming technique to examine the bank’s choice between liquidation and continuation in the crisis with anticipation of bailout policy. We compare Bank 1’s ex-post value to shareholders under different scenarios, which is then used to analyze the value function and cap￾italization decisions of the bank prior to the crisis. Then we conduct simulations to investigate the impact of anticipated shock intensity, public policy (bailout in the form of common stock or preferred stock, and regulatory capital requirements) and parameter values (e.g., the exposure to the distressed loan) on Bank 1’s capitalization decisions, possibility of contagion, and government bailout amounts. 4. Endogenous capital holding decision Using stochastic dynamic programming, we analyze the post￾crisis value of Bank 1 to shareholders and the value prior to the cri￾sis. There are two possible post-crisis value functions: U(C) (for the case when the bank takes over project G) and W(C) (for the case when the bank liquidates project G). These have simple closed form analytical solutions of the general form A1expm1C + A2expm2C. These obtain because, post-crisis, the only decision of the bank is to pay or retain dividends and to continue in operation until, even￾tually, capital falls to the minimum regulatory required level b C and the bank must close. This is a standard problem of optimal balance sheet management, previously solved by Milne and Robertson (1996), Radner and Shepp (1996) and others. Optimal policy is bar￾rier control, paying no dividends if C < C⁄ , and otherwise to make sufficient dividend payments to maintain C P C⁄ for some target le￾vel of buffer capital C⁄ . m1 and m2 are constants determined by the relevant equation of motion for the post-crisis evolution of C and A1 and A2 are constants of integration. The three free parameters A1, A2 and C⁄ are determined by three boundary conditions applying at b C; bb C and C⁄ . Appendix A states the equations of motion, the bound￾ary conditions and the resulting closed form solutions. 4.1. Parameter restriction As shown in the previous section, the regulatory capital re￾quired for Bank 1 to take over the distressed loan is b C ¼ fð1 þ ½n ð1 þ nÞð1 yÞlgs þ ½ð1 þ nÞð1 yÞ nxnl; while the capital required for Bank 1 to liquidate the distressed loan is bb C ¼ ð1 lÞs þ nl. In comparison with the minimum capital requirement for Bank 1 before the shock occurs, i.e., C ¼ s, several possible relationships among C; b C, and bb C exist: (1) C < b C < bb C , that is, the regulatory capital requirement for Bank 1 before the shock is lower than that required to take over Project G, which is in turn lower than the amount required to liquidate Project G when the shock occurs. (2) C < bb C < b C, that is, the minimum regulatory capital require￾ment for Bank 1 before the shock is lower than that required to liquidate Project G, which is in turn lower than the amount required to take over Project G when the shock occurs. (3) bb C < C < b C, that is, the minimum regulatory capital require￾ment for Bank 1 before the shock is higher than that required to liquidate Project G, but lower than the amount required to take over Project G when the shock occurs. It can be easily shown that if ð1þnÞð1yþysÞ 1þnx < n < s x þ ð1þnÞð1yÞð1sÞ nx holds, the capital required to take over Project G will always be lower than the capital required to liquidate the project, i.e., C < b C < bb C . 17 For exam￾ple, if the regulatory capital ratio is 10%, x = 0, n = 1, and y = 1, Condition (1) will hold as long as the loss-given-default ratio of the bank loan is higher than 30%, which is supported by the empirical evidence.18 We therefore choose this plausible con￾dition and focus on the first case, C < b C < bb C , in the subse￾quent analysis. 16 So far we assume that the government takes no action to avert the systemic crisis. However, the government may choose to intervene to prevent bank failure or liquidation of assets by providing additional capital, either as common equity capital or as preferred shares. In Section 4, we will solve for the bailout amounts that must be provided by the authorities to avoid contagion or maximize shareholder’s value. In Section 5, we use simulations to show how the bailout amounts depend upon the parameters of the model. 17 The willingness to pay Bank 2 will be affected by capitalization of Bank 1. If Bank 1 has relatively low capital, the takeover will have a relatively small benefit to its own shareholders. The loss of value because of moving closer to minimum capital constraint is relatively large, compared to the benefit of acquiring a new positive cash flow. So Bank 1 is less willing to pay a high price to take over the distressed loan. We thank the referee for suggesting us to consider the complexity of the actual payout by Bank 1 to Bank 2 due to the bargaining process, the project’s cash flow, and Bank 1’s capitalization. 18 Gupton et al., 2000) examine 181 bank loan defaults (mostly syndicated loans) and find that the mean bank-loan value in default is 69.5% for Senior Secured loans and 52.1% for Senior Unsecured loans. Therefore the loss-given-default ratio (1- recovery rate), is 30.5% for Senior Secured bank loans and 47.9% for senior unsecured loans. Bank loans usually have a higher recovery rate than other forms of debt. Fitch (2005) reports historical recovery rates of Senior Unsecured bonds for 24 industries over the period 2000 to 2004. The mean of average loss-given-default ratio is 67% across industries. 2770 S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778

S. Tian et al Jourmal of Banking 8 Finance 37(2013)2765-2778 2771 4. 2. Bank 1's problem Under Condition(iv), an injection of common equity dilutes exis ing shareholder interests and hence provides a stronger incentive In anticipation of crisis, Bank 1's post-crisis value functions in or Bank 1 to hold more capital to cope with the failure of other different scenarios (liquidation of the joint project, continuation banks. The term-(1-A)ovo)reflects the expected effect on Bank without bailout, or continuation with bailout)will determine Bank I's continuation value, -vO), from the shock and no government 1's choice of whether to liquidate or continue the joint asset, and bailout, which occurs with probability(1-2)odt. hence its pre-crisis value function, VO), and its target level of cap- Finally, if CC, the level amount of capital injection that maximizes Bank 1's value function and change of Bank 1's value function can be continuously adjusted if it takes over (liquidates) Project G. Under Condition(iii), the by changing dividend policy in the neighborhood of C In Condition shareholders of preferred stock will receive only the fixed dividend (4)and (5, c" is the desired long-run or target level of capitalization and will not share in the upside gain should the bank recover. at which all earnings are paid out. At C any increment to capital is

4.2. Bank 1’s problem In anticipation of crisis, Bank 1’s post-crisis value functions in different scenarios (liquidation of the joint project, continuation without bailout, or continuation with bailout) will determine Bank 1’s choice of whether to liquidate or continue the joint asset, and hence its pre-crisis value function, V(C), and its target level of cap￾ital, C⁄ . The application of standard techniques shows that the ex￾ante value function V satisfies Hamilton–Jacobi–Bellman (HJB henceforth) differential equation in the following general form: qV ¼maxh hþ½lR1 þ ð1lÞR2 hVc þ1 2 r2 1l 2 þr2 2ð1lÞ 2 þ2lð1lÞr1r2x12 h iVcc ( )þKðCÞ ð3Þ where K(C) takes the following forms depending on the relation￾ship between C; C; b C, and bb C : / maxfWðC nlÞ VðCÞ;UðC ½ð1 þ nÞð1 yÞ nxnlÞ VðCÞg if b C b CC ½UðC ½ð1þnÞð1yÞ nxnlþK1Þ K1; max K2> b b C C ½WðC nlþK2Þ K2;0g VðCÞ 8 b CC C CþK1 UðC ½ð1þnÞð1yÞ nxnlþK1Þ  h i; max K2> b b C C C CþK2 WðC nlþK2Þ h i) VðCÞ 8 >>>>>>>: 9 >>>>= >>>>; ð1kÞ/VðCÞ if C C, the level and change of Bank 1’s value function can be continuously adjusted by changing dividend policy in the neighborhood of b C. In Condition (4) and (5), C⁄ is the desired long-run or target level of capitalization at which all earnings are paid out. At C⁄ any increment to capital is S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778 2771

S Tian et aL/Joumal of Banking S Finance 37(2013)2765-2778 diately as a dividend (referred to as barrier control). Val- Table 2 ues of capital holdings above c cannot be obtained because of the The impact of shock intensity on al capital holding, contagion and bailout continuous sample paths of the assumed diffusion The amounts i=0.5,l=0.3.n=0.8,x=0.1,y=098.t bank always wishes to retain a buffer of capital to reduce the ex ed cost of not meeting the regulatory capital requirement. 0.121720.121720.121720.12172 Therefore, the optimal policy is to pay dividends at as high a level as possible when C exceeds C, but otherwise to retain all earnings Condition (4)arises because control is instantaneous at the C m 0.1140260.1134920.113068 0.1405820.141071 003169200322255 boundary. Bank 1's value prior to the crisis is equal to the optimal 002778930005824490.005135380.00464646 capital level, C, when the capital is chosen optimally. Condition (5)is a consequence of an optimally selected C. Otherwise, the va- lue function could be increasing at C by a small shift of C in the direction that v<1 The impact of the anticipated government bailout probability on optimal capital holding, contagion and bailout amounts =0.5, 1=0.10, n=1, x=0.15, y=0.96. t=0.08,=0.50. roof. See Appendix E.口 In Fig. 2, we present pictures of the value function(MO), U(o) C 0.08786 and wo) to show the changes in the capital that follow different 01 0.1039640.1039190.1038 0.103826 actions(continuation, liquidation) based on the solution of the dy- 0.1039720.1039330.103893 namic models Eqs.(Al-2)and (Al-3)for U(C)and eqs.(Al-5)and 2594550002638220002683050.0027291000277646 (Al-6)for WC in Appendix A. The solid, dashed and dot-dashed Kcom000259191000263012000266922000270925000275025 lines represent VO U(O and w(C respectively. The figures in the four panels differ horizontally by parameter range and verti- cally by type of bailout Cpre( Com)is the optimal capital ratio se- when Bank 1 takes over Project G, C, with the optimal level of ex- lected by Bank 1 prior to the crisis with the anticipated preferred ante capital holding under bailout in the form of common stock stock (common stock) bailout. C is the required capital for Bank and preferred equity, Ccom and Cpre We also show the bailout I to take over Project G. The difference between C and Cpre(C amounts of common stock and preferred equity that are needed to and Com) is the minimum amount of capital that must be provided and A Shareholder's value for continuation of Project G, Kipe om, where Pre=C+[(1+n)(1-y)-nxsJl-Cpr by the government for Bank 1 to continue the project with the Km=C+((1+n)(1-y)-nxg-Coom and C is the new desired anticipated preferred stock(common stock) bailout. Panel(a) capital level C when Bank 1 takes over Project G C<akc<a and Panel (c)c<c<c<2 show the Below are the baseline parameter values case when Bank 1 has sufficient capital to take over Project G with- out bailout, while Panel(b)(t<ce <c<a) and Panel(d) 0.02 0.04 0.01 0. 02 0.05 0.05 C<Com <C<a show the case when bailout is ne Several observations can be made from Fig. 2. First, Bank 1's 5.1. The impact of shock intensity, bailout policy, and regulatory post-crisis value functions, U(C) and W(O), are non-linear functions capital requirement on contagion and bailout amounts of the capital ratio post crisis. Second, the shareholder's value U(o hen Bank 1 takes over Project G is always higher than M when Table 2 presents simulation results to illustrate the relationship between the anticipated shockintensity and Bank I's optimal capita Bank 1 liquidates Project G, no matter whether the firm receives holding, possibility of contagion, and bailout or not, or bailout takes the form of preferred nd bailout amounts to show the stock. The comparison of U(O and w(o) determines that Bank 1 economic magnitude, we assume that the total assets of Bank 1, A, will choose to continue Project G and the target level of capital are $100 billion. The capital holding required to take over Project to the crisis. Therefore, our subsequent analysis focuses on G, C, is $12. 172 billion. It is apparent that Com is always higher than case of continuation of the project. third the pre-crisis target or equal to Cpre. Contagion occurs for a wider range of values of o in 1, which is endogenously determined by anticipation of preferred stock bailout than common stock bailout solving Eq (3), depends on payoffs under these different scenarios. (we use numbers in bold to indicate contagion). For example, when p=0., Com is $.1071 billion, exceeding the required capital ra- 5. Bank optimal capital holding, interbank contagion, and tio to take over Project G, while Chre is s113068 billion, lower than C government bailout That is, Bank 1 is willing to set aside s2. 8 billion more for an antici Ited common stock bailout. Intuitively, if Bank 1 views a shock and Since we cannot obtain a closed-form solution for the optimal a common stock bailout as likely, it will keep more capital in order to apital holding for Bank 1, we use simulations to examine the im- Table 4 pact of a number of parameters on Bank 1s optimal capital he The impact of the regulatory capital ratio on optimal capital holding, contagion and and whether interbank contagion will emerge. These factors in- bailout amountsφ=0.5,=0.5,l=0.1,n=2,x=0.15.y=095,=050. clude exogenous variables and factors related to Bank 1's exposure 0.11 0.12 to Project G. For each case, we compare the required capital level for interested readers to investigate other cases. Please also refer our working paper ersion for more detailed simulation results and discussions

paid immediately as a dividend (referred to as barrier control). Val￾ues of capital holdings above C⁄ cannot be obtained because of the continuous sample paths of the assumed diffusion process. The bank always wishes to retain a buffer of capital to reduce the ex￾pected cost of not meeting the regulatory capital requirement. Therefore, the optimal policy is to pay dividends at as high a level as possible when C exceeds C⁄ , but otherwise to retain all earnings. Condition (4) arises because control is instantaneous at the C⁄ boundary. Bank 1’s value prior to the crisis is equal to the optimal capital level, C⁄ , when the capital is chosen optimally. Condition (5) is a consequence of an optimally selected C⁄ . Otherwise, the va￾lue function could be increasing at C⁄ by a small shift of C⁄ in the direction that Vc < 1. Proof. See Appendix B. h In Fig. 2, we present pictures of the value function (V(C), U(C) and W(C)) to show the changes in the capital that follow different actions (continuation, liquidation) based on the solution of the dy￾namic models Eqs. (A1-2) and (A1-3) for U(C) and Eqs. (A1-5) and (A1-6) for W(C) in Appendix A. The solid, dashed and dot-dashed lines represent V(C), U(C) and W(C) respectively. The figures in the four panels differ horizontally by parameter range and verti￾cally by type of bailout. C pre C com   is the optimal capital ratio se￾lected by Bank 1 prior to the crisis with the anticipated preferred stock (common stock) bailout. b C is the required capital for Bank 1 to take over Project G. The difference between b C and C preðb C and C com) is the minimum amount of capital that must be provided by the government for Bank 1 to continue the project with the anticipated preferred stock (common stock) bailout. Panel (a) C < b C < C pre < bb C  and Panel (c) C < b C < C com < bb C  show the case when Bank 1 has sufficient capital to take over Project G with￾out bailout, while Panel (b) C < C pre < b C < bb C  and Panel (d) C < C com < b C < bb C  show the case when bailout is necessary. Several observations can be made from Fig. 2. First, Bank 1’s post-crisis value functions, U(C) and W(C), are non-linear functions of the capital ratio post crisis. Second, the shareholder’s value U(C) when Bank 1 takes over Project G is always higher than W(C) when Bank 1 liquidates Project G, no matter whether the firm receives bailout or not, or bailout takes the form of preferred or common stock. The comparison of U(C) and W(C) determines that Bank 1 will choose to continue Project G and the target level of capital prior to the crisis. Therefore, our subsequent analysis focuses on the case of continuation of the project. Third, the pre-crisis target level of capital, C pre C com  , which is endogenously determined by solving Eq. (3), depends on payoffs under these different scenarios. 5. Bank optimal capital holding, interbank contagion, and government bailout Since we cannot obtain a closed-form solution for the optimal capital holding for Bank 1, we use simulations to examine the im￾pact of a number of parameters on Bank 1’s optimal capital holding and whether interbank contagion will emerge.19 These factors in￾clude exogenous variables and factors related to Bank 1’s exposure to Project G. For each case, we compare the required capital level when Bank 1 takes over Project G, b C, with the optimal level of ex￾ante capital holding under bailout in the form of common stock and preferred equity, C com and C pre. We also show the bailout amounts of common stock and preferred equity that are needed to maximize shareholder’s value for continuation of Project G, K pre and K com, where K pre ¼ C u þ ½ð1 þ nÞð1 yÞ nxnl C pre; K com ¼ C u þ ½ð1 þ nÞð1 yÞ nxnl C com and C u is the new desired capital level C⁄ when Bank 1 takes over Project G. Below are the baseline parameter values: r1 R1 r2 R2 x12 q A 0.02 0.04 0.01 0.02 0.05 0.05 1 5.1. The impact of shock intensity, bailout policy, and regulatory capital requirement on contagion and bailout amounts Table 2 presents simulation results to illustrate the relationship between the anticipated shock intensity and Bank 1’s optimal capital holding, possibility of contagion, and bailout amounts. To show the economic magnitude, we assume that the total assets of Bank 1, A, are $100 billion. The capital holding required to take over Project G, b C, is $12.172 billion. It is apparent that C com is always higher than or equal to C pre. Contagion occurs for a wider range of values of / in anticipation of preferred stock bailout than common stock bailout (we use numbers in bold to indicate contagion). For example, when / ¼ 0:9; C com is $14.1071 billion, exceeding the required capital ra￾tio to take over Project G, while C pre is $11.3068 billion, lower than b C. That is, Bank 1 is willing to set aside $2.8 billion more for an antici￾pated common stock bailout. Intuitively, if Bank 1 views a shock and a common stock bailout as likely, it will keep more capital in order to Table 3 The impact of the anticipated government bailout probability on optimal capital holding, contagion and bailout amounts / = 0.5, l = 0.10, n = 1, x = 0.15, y = 0.96, s = 0.08, n = 0.50. k 0.1 0.3 0.5 0.7 0.9 b C 0.08786 0.08786 0.08786 0.08786 0.08786 C pre 0.104008 0.103964 0.103919 0.103873 0.103826 C com 0.10401 0.103972 0.103933 0.103893 0.103852 K pre 0.00259455 0.00263822 0.00268305 0.0027291 0.00277646 K com 0.00259191 0.00263012 0.00266922 0.00270925 0.00275025 Table 2 The impact of shock intensity on optimal capital holding, contagion and bailout amounts k = 0.5, l = 0.3, n = 0.8, x = 0.1, y = 0.98, s = 0.10, n = 0.50. / 0.1 0.3 0.5 0.7 0.9 b C 0.12172 0.12172 0.12172 0.12172 0.12172 C pre 0.115931 0.114752 0.114026 0.113492 0.113068 C com 0.11694 0.117928 0.139893 0.140582 0.141071 K pre 0.0297867 0.0309655 0.031692 0.0322255 0.03265 K com 0.0287782 0.0277893 0.00582449 0.00513538 0.00464646 Table 4 The impact of the regulatory capital ratio on optimal capital holding, contagion and bailout amounts / = 0.5, k = 0.5, l = 0.1, n = 2, x = 0.15, y = 0.95, n = 0.50. s 0.08 0.09 0.10 0.11 0.12 b C 0.0948 0.10665 0.1185 0.13035 0.1422 C pre 0.11154 0.104859 0.114848 0.124838 0.134828 C com 0.111553 0.1234 0.117999 0.126985 0.136416 K pre 0.0034052 0.0219367 0.023797 0.0256574 0.0275179 K com 0.0033923 0.00339517 0.0206462 0.0235102 0.0259295 19 We used Mathematica to generate simulation results. Due to space constraint, we only report a subset of simulation results. However, the code is available upon request for interested readers to investigate other cases. Please also refer our working paper version for more detailed simulation results and discussions. 2772 S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778

S. Tian et al Jourmal of Banking 8 Finance 37(2013)2765-2778 2773 able 3 illustrates the impact of the anticipated probability of Bank Is bargaining power, x, and the optimal capital holding s =0.5, 2=0.5, 1=02, the government bailout. When the probability of bailout goes from n=1,y=1,t=0.10.÷=0.50. 0.1 to 0.9, Ce decreases by $18.2 million from $.4008 billion to S103826 billion, while Com decreases by a smaller amount of $15.8 million from $.401 billion to $.3852 billion. The higher the 117935 anticipated probability of the government bailout, the lower the ex ante capital holding Bank 1 will maintain, and the greater 000295751 amount of bailout will be needed, reflecting the moral hazard Next, we examine in Table 4 the impact on contagion and bailout amounts if the public policy on capital requirement is changed. The avoid contagion when the crisis materializes. This underscores the ecently finalized Basel Ill requires banks to hold 4.5% of common ontagion. To help Bank 1 to reach the new optimal capital level that 4%)of risk-weighted assets. Basel lll also introduces an additional maximizes its shareholder's value, the government needs to inject capital conservation buffer of 2.5%, which is designed to ensure that 3.265 billion(Kipre)and 0.46 billion(K om)in the case of preferred banks build up capital buffers outside periods of stress which can be drawn down as losses are incurred and to avoid breaches of mini- ital holdingfor an anticipated common equity bailout allows a lower mum capital requirements. our simulation shows that increasing level of government recapitalization. tagion, in fact, this could increase contagion. But imposing the con- Contagion is more likely to occur if Bank 1 underestimates servation buffer as Basel ll could help banks to increase resilience probability of crisis. For example, when o is 0.9(Com is 14. 1071 bil As shown in Table 4, when t is 0.08, both Ce and Com are great lion) but Bank 1 estimates the shock intensity to be 0.3, Com er than 0.08, no contagion will occur. However, if the authority (11.7928 billion) will be lower than C, thus contagion occurs. The government needs to inject an amount of 2.779 billion rather creases t(0.09, 0.10, 0.11 or 0.12), contagion could occur under the anticipated preferred stock bailout because C goes up by a higher than 0.465 billion if Bank 1 correctly estimated the shock intensity level than C:. Similarly, contagion will occur when increases to fo.9. An unexpected external shock is more likely to cause inter 0.10.0.11 or 0.12 under the bank contagion and large bailout requirements, as shown in recent in the last two rows, K and K"om increase with T, suggesting that financial crisis imply increasing minimum capital requirement is not a cure-all d=05=05k=01n=1x010y=096r=0.10.E=050 =05=05k01n=2x010y=096r=0.10.5=050 c=c;"c-即*n-)-m -g+n01-y)-m a C<c<c<c b)C<C<C<c =05=050.1,n=1x0.0y=096r=0.10=0.50 4=05=05k01n=2x0.10y=096r=010.=050 Equity Vale U(C-[Q+nXl-y)-mmD (cC<c<c d Fig. 2. Bank 1's pre-crisis and post-crisis value functions. Note: (a)and(b)shows Bank I's pre-crisis and post-crisis value functions with the anticipated preferred stock allout; (c)and (d) shows Bank 1's pre-crisis and post-crisis value functions with the anticipated preferred stock bailout

avoid contagion when the crisis materializes. This underscores the importance of keeping capital buffer in anticipation of interbank contagion. To help Bank 1 to reach the new optimal capital level that maximizes its shareholder’s value, the government needs to inject 3.265 billion K pre  and 0.46 billion K com   in the case of preferred and common stock bailout, respectively. Bank 1’s more ex ante cap￾ital holding for an anticipated common equity bailout allows a lower level of government recapitalization. Contagion is more likely to occur if Bank 1 underestimates probability of crisis. For example, when / is 0.9 (C com is 14.1071 bil￾lion) but Bank 1 estimates the shock intensity to be 0.3, C com (11.7928 billion) will be lower than b C, thus contagion occurs. The government needs to inject an amount of 2.779 billion rather than 0.465 billion if Bank 1 correctly estimated the shock intensity of 0.9. An unexpected external shock is more likely to cause inter￾bank contagion and large bailout requirements, as shown in recent financial crisis. Table 3 illustrates the impact of the anticipated probability of the government bailout. When the probability of bailout goes from 0.1 to 0.9, C pre decreases by $18.2 million from $10.4008 billion to $10.3826 billion, while C com decreases by a smaller amount of $15.8 million from $10.401 billion to $10.3852 billion. The higher the anticipated probability of the government bailout, the lower the ex ante capital holding Bank 1 will maintain, and the greater amount of bailout will be needed, reflecting the moral hazard problem. Next, we examine in Table 4 the impact on contagion and bailout amounts if the public policy on capital requirement is changed. The recently finalized Basel III requires banks to hold 4.5% of common equity (up from 2% in Basel II) and 6% of Tier I capital (up from 4%) of risk-weighted assets. Basel III also introduces an additional capital conservation buffer of 2.5%, which is designed to ensure that banks build up capital buffers outside periods of stress which can be drawn down as losses are incurred and to avoid breaches of mini￾mum capital requirements. Our simulation shows that increasing the absolute regulatory minimum will not necessarily reduce con￾tagion, in fact, this could increase contagion. But imposing the con￾servation buffer as Basel III could help banks to increase resilience. As shown in Table 4, when s is 0.08, both C pre and C com are great￾er than 0.08, no contagion will occur. However, if the authority in￾creases s (0.09, 0.10, 0.11 or 0.12), contagion could occur under the anticipated preferred stock bailout because b C goes up by a higher level than C pre. Similarly, contagion will occur when sincreases to 0.10, 0.11 or 0.12 under the anticipated common stock. As shown in the last two rows, K pre and K com increase with s, suggesting that simply increasing minimum capital requirement is not a cure-all Table 5 Bank 1’s bargaining power, x, and the optimal capital holding / = 0.5, k = 0.5, l = 0.2, n = 1, y = 1, s = 0.10, n = 0.50. x 0 0.05 0.10 0.15 0.20 b C 0.12 0.115 0.11 0.105 0.1 C pre 0.113809 0.113832 0.126471 0.12158 0.117935 C com 0.117594 0.131471 0.126476 0.121584 0.117935 K pre 0.027084 0.0220611 0.00442152 0.00431263 0.00295751 K com 0.023299 0.0044222 0.0044167 0.00430878 0.00295751 Fig. 2. Bank 1’s pre-crisis and post-crisis value functions. Note: (a) and (b) shows Bank 1’s pre-crisis and post-crisis value functions with the anticipated preferred stock bailout; (c) and (d) shows Bank 1’s pre-crisis and post-crisis value functions with the anticipated preferred stock bailout. S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778 2773

S Tian et aL /Joumal of Banking S Finance 37(2013)2765-2778 solution. Instead, it may increase the burden of the government of the conservation capital buffer, which increases its capitalbo5g when crisis happens. However, if Bank 1 holds an additional 中=0.1A=05n=12x=0.15y=096r=008=0.50 ng to 12.5% when t is 10%, the bank can take over Project G and p-12 ntagion since it is higher than C(0. 1185). Gover bailout is not necessary as Bank 1 can draw down its capital buffer to avoid loss 527 The impact of Bank 1s exposure to Project G on contagion and 002 Next, we show how a range of factors related to Bank 1's expo- ure to Project G affect Bank 1s optimal capital holding. of contagion and bailout amounts. Fig. 3 displays th Fig 4. The impact of Bank 1s investment ratio, I, on government bailout amounts. Itio, I, as I changes from0 to 1. when I is small enough(e.g, less than oi 5). Bank b will have suticient capital to avoid contagion K:. 小=05=05/=0Lx=01y=095r=0105=050 drops discretely. within this range, C is increasing in 4, indicating that Bank 1 holds more capital buffer if it invests more of its assets in Project G. However, C(the dot-dashed line) increases at a faster bos rate than Com or Cpre, indicating contagion when I is large The relationship between I and bailout amounts is shown in Fig 4. As I increases, bailout amount K increases but at different 002 rates in two intervals. The interval to the left of jump discontinuity corresponds to the case where Bank 1 can survive without govern- ment bailout as C >C. To maximize shareholder,s value, Bank 10.01 only needs a tiny amount of bailout. The right interval shows that the bailout amounts are considerably higher when C<C More er, Kcom is always lower than Pre. Intuitively, more capital buffer held by Bank 1 with the anticipated common equity helps the gov- ernment to save bailout budget. Fig. 5. The impact of two banks' relative investment ratio, n, on government bailout Next, if the relative investment ratio in Project g of Bank 2 over Bank 1, n, increases within a certain range(e. g, lower than or equal to 1), Bank 1's ex ante optimal capital holding increases as well and no contagion occurs. However, as n increases further( 2.5), i.e., Bank 2 holds a greater fraction of distressed loan than Bank 1, Bank 1's capital holding is lower than C and contagion hap- pens. Fig. 5 shows the relationship between bailout amounts and The bailout amounts are insignificant if Bank 2s investment ratio is smaller or close to Bank 1s investment ratio in Project G. However, when n becomes large enough, Kpre and Kcom jump up discretely. 03 Intuitively, it is more expensive for the government to help Bank 1 to take over the large fraction of loan held by Bank 2. 002 addition, we examine how Bank 1's bargaining power, x, af- fects contagion possibility and bailout amounts As shown ble 5. when x increases from 0.10 to 0.20. Bank hold lower p=0L=0.5n=12x=015y=0.9r=0085=050 098 Fig. 6. The impact of the mark-to-market value of Project G, y, on government =C or to th mates its bargaining power, contagion will occur. For example, if Bank 1 estimates its bargaining power to be 0. 2, its optima 0.12 holding ratio is 11.7935 billion. However, if its actual barg power is 0, the regulatory capital requirement is 12 billion, exceeds the banks capital holding ratio and contagion will happen. 0.10 There is a large drop of bailout amount, K, once x exceeds a certain level, and the decline occurs at a lower value of x for Kcom than Kore- Presumable, a bank with weak bargaining power will rely on more capital injection from the government. Finally, when Bank 1 underestimates the loss of the loan value g. 3. The impact of Bank 1's investment ratio on the optimal capital holding. due to marking-to-market, contagion could happen. The bailout

solution. Instead, it may increase the burden of the government when crisis happens. However, if Bank 1 holds an additional 2.5% of the conservation capital buffer, which increases its capital hold￾ing to 12.5% when s is 10%, the bank can take over Project G and avoid contagion since it is higher than b C (0.1185). Government bailout is not necessary as Bank 1 can draw down its capital buffer to avoid loss. 5.2. The impact of Bank 1’s exposure to Project G on contagion and bailout amounts Next, we show how a range of factors related to Bank 1’s expo￾sure to Project G affect Bank 1’s optimal capital holding, possibility of contagion and bailout amounts. Fig. 3 displays the relationship between C⁄ and the investment ratio, l, as l changes from 0 to 1. When l is small enough (e.g., less than 0.15), Bank 1 will have sufficient capital to avoid contagion ðC > b CÞ. Once l becomes large enough (e.g., greater than 0.2), C⁄ drops discretely. Within this range, C⁄ is increasing in l, indicating that Bank 1 holds more capital buffer if it invests more of its assets in Project G. However, b C (the dot-dashed line) increases at a faster rate than C com or C pre, indicating contagion when l is large. The relationship between l and bailout amounts is shown in Fig. 4. As l increases, bailout amount K⁄ increases but at different rates in two intervals. The interval to the left of jump discontinuity corresponds to the case where Bank 1 can survive without govern￾ment bailout as C > b C. To maximize shareholder’s value, Bank 1 only needs a tiny amount of bailout. The right interval shows that the bailout amounts are considerably higher when C < b C. More￾over, K com is always lower than K pre. Intuitively, more capital buffer held by Bank 1 with the anticipated common equity helps the gov￾ernment to save bailout budget. Next, if the relative investment ratio in Project G of Bank 2 over Bank 1, n, increases within a certain range (e.g., lower than or equal to 1), Bank 1’s ex ante optimal capital holding increases as well and no contagion occurs. However, as n increases further (n = 1.5, 2, or 2.5), i.e., Bank 2 holds a greater fraction of distressed loan than Bank 1, Bank 1’s capital holding is lower than b C and contagion hap￾pens. Fig. 5 shows the relationship between bailout amounts and n. The bailout amounts are insignificant if Bank 2’s investment ratio is smaller or close to Bank 1’s investment ratio in Project G. However, when n becomes large enough, K pre and K com jump up discretely. Intuitively, it is more expensive for the government to help Bank 1 to take over the large fraction of loan held by Bank 2. In addition, we examine how Bank 1’s bargaining power, x, af￾fects contagion possibility and bailout amounts. As shown in Ta￾ble 5, when x increases from 0.10 to 0.20, Bank 1 will hold lower amounts of capital prior to the crisis. If Bank 1 seriously overesti￾mates its bargaining power, contagion will occur. For example, if Bank 1 estimates its bargaining power to be 0.2, its optimal capital holding ratio is 11.7935 billion. However, if its actual bargaining power is 0, the regulatory capital requirement is 12 billion, which exceeds the bank’s capital holding ratio and contagion will happen. There is a large drop of bailout amount, K⁄ , once x exceeds a certain level, and the decline occurs at a lower value of x for K com than K pre. Presumable, a bank with weak bargaining power will rely on more capital injection from the government. Finally, when Bank 1 underestimates the loss of the loan value due to marking-to-market, contagion could happen. The bailout Fig. 6. The impact of the mark-to-market value of Project G, y, on government bailout amounts. Fig. 3. The impact of Bank 1’s investment ratio on the optimal capital holding. Fig. 5. The impact of two banks’ relative investment ratio, n, on government bailout amounts. Fig. 4. The impact of Bank 1’s investment ratio, l, on government bailout amounts. 2774 S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778