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