Systemic Risk and the Optimal Seniority Structure of Banking Liabilities

The paper argues that systemic risk must be taken into account when designing optimal bankruptcy procedures in general, and priority rules in particular. Allowing for endogenous formation of links in the interbank market we show that the optimal policy depends on the distribution of shocks and the severity of …re sales.


Introduction
An important issue that the design of any bankruptcy procedure must resolve is the allocation of priority rights among the various claimants of the corresponding entity's assets. By de…nition the value of the assets of a bankrupt entity is below the value of its liabilities and, therefore, a rule is needed for allocating those assets to the various claimants. Often the allocation of priority gives rise to a complex hierarchy among the creditors having at the top holders of secured debt and right down at the bottom the providers of equity. There is an extensive literature in …nancial economics 1 that studies the optimal design of bankruptcy procedures. The main aim of the design is to make investment attractive to creditors by shifting su¢cient risk on those agents who have control over decision-making. In a setting with multiple stages of …nancing by multiple creditors the design of priority rights takes the form of a hierarchical structure.
A policy area that has attracted a lot of attention, especially in the aftermath of the 2008 crisis, is the design of priority rules for banks. What is striking is the variety of rules applied around the globe (Lenihan, 2012;Wood, 2011). Many countries in response to the crisis have either only recently introduced or are in the process of introducing depositor preference rules. These include Greece, Portugal, Hungary, Latvia and Romania that have to implement such rules as part of the conditions that they need to meet in order to participate in EU/IMF programmes. In the UK the Vickers report recommends the introduction of a depositor preference rule (ICB Report 2011).The arguments supporting the rules in place are mainly about the incentives that these rules provide to depositors and other creditors to monitor the activities of banks. The lower a creditor is on the priority list, the less likely is that she will receive (at least full) compensation in the case of bankruptcy and therefore the stronger are her incentives to ensure that the borrower does not take excessive risks. While nobody disagrees with the validity of the last statement there is substantial variation in opinions about which is the most suitable party to perform the monitoring service.
In this paper, we argue that systemic risk is another aspect of banking that must be taken into account when designing bankruptcy procedures. In general, the design of optimal priority rules is focused on the welfare of a bank's creditors, namely, its depositors, other debt holders and equity owners. However, when systemic risk is a concern, the design must also take into consideration the welfare of third parties and, in particular, all those who provided funds to the rest of the banking system. The degree to which a bank's credit providers will be a¤ected when the bank becomes insolvent it depends on their position on the priority ladder. When among those creditors that are not getting paid in full are other banks, as long as these banks stay solvent the losses will be absorbed by equity holders. However, when these banks become insolvent the losses will be absorbed by their creditors and this process will continue till either the system clears or all banks become bankrupt. The total systemic losses will depend on the value of assets that can be recovered when banks become insolvent and thus go into liquidation. The sale of assets in depressed markets, '…re sales' as they are known in the literature, further deteriorates balance sheets and thus enhances the fragility of the …nancial system. 2 It is straightforward to show that if the value of assets is fully recovered under liquidation, that is the market value is equal to the book value, then the seniority structure does not matter. This is because the total losses are limited to the value of initial losses. 3 With this in mind, …re sales are going to be important in our work.
Recent work on systemic risk, models contagion in banking following a network approach. 4 One important …nding of this literature is that the structure of the network matters. Thus far, this structure is exogenously given. This can be …ne as long as the seniority structure of liabilities is …xed. 5 However, our aim is to compare the welfare implications of two alternative priority structures, namely, depositor seniority and bank seniority, and to do so we need to take into consideration the impact of seniority structure on the formation of the interbank network. Again, it is straightforward to show that if the network structure of the intertbank market could not be a¤ected by the choice of the seniority structure then the optimal option would be to allocate seniority to banks. The reason is that as more of the losses are absorbed by depositors the lower will be the losses absorbed by the interbank market and thus the lower the risk of further insolvencies. 6 However, this is no longer true when the network structure is endogenous. The problem is that allowing for endogenous network formation complicates signi…cantly the analysis and therefore we will ignore the incentives for monitoring that each seniority structure o¤ers. However, this is an important issue and any related policy debate needs to include it. We are going to o¤er our thoughts on this issue in the concluding section of the paper.
We will analyze a model of the banking system where banks …nance their investments by two types of borrowing, namely, retail deposits and loans from other banks. Given that our main goal is to understand how alternative priority policies might a¤ect the network structure we focus on the optimal lending policy of a bank that has some excess liquidity. The main disadvantage of adopting a partial equilibrium approach is that by de…nition the network structure that we consider does not satisfy conventional equilibrium solutions. However, existing analytical models of systemic risk have predominantly focused on networks with symmetric structures. 7 Although, it is simple to demonstrate that symmetric 2 In traditional models of bank panics the fear of liquidation costs are what drives depositor runs (see Diamond and Dybvid, 1983;Allen and Gale, 2000). See Shleifer and Vishny (2010) for a review of the related literature. 3 The only thing a¤ected by the seniority structure is the distribution of losses between depositors and bank owners. However, the seniority structure should minimize total losses leaving their distribution to other policy instruments. 4 For reviews of the literature see Allen and Babus (2009) and Bougheas and Kirman (2015). 5 In the literature so far the analysis is carried out under the supposition that in the case of bankruptcy depositors have priority. See, for example, Acemoglu et al. (2015). 6 We are intentionally ignoring concerns about the impact of the priority structure on the likelihood of liquidity runs that can lead to insolvencies and hence systemic risk. The reason is that there are other instruments, such as deposit insurance, that are more appropriate to deal with such concerns (see, Diamond and Dybvig, 1983). 7 See for example, Acemoglu et al. (2015) and Allen and Gale (2000).
structures can be stable, symmetry is not a characteristic feature of actual …nancial networks. 8 The main advantage of our approach is that it allows us to compare all the options that a bank could have when considering forming a new link. We show that as long as the network structure is …xed our results are consistent with those in the existing literature. However, we will also be able to go a step further by examining what happens when the policy regime can a¤ect the formation of links.
The bank in our model has three lending options: the …rst one is to o¤er the loan to a bank that has zero net obligations to the rest of the banking system, the second option is to o¤er the loan to a bank that is a net borrower, and the third option is to o¤er the loan to a net lender. We derive the lender bank's pro…t maximizing choice under both priority rules and then we derive the corresponding social optimum choice. Given that priority rules only matter when a bank fails, our results will be sensitive to (a) the size of the shock and (b) the probability distribution of shocks across the banking system. We carry out the analysis under two alternative scenaria regarding the distribution of shocks. In one case, we assume that each bank is hit by a shock with the same probability but bigger banks are hit by proportionally bigger shocks, while in the other case we assume that the probability that a bank is hit by a shock is proportional to its size but the size of the shocks are the same for each bank. The former scenario corresponds to the case where a bank's various asset returns are strongly correlated (for example, when the portfolio is dominated by regional mortgages) in which case a shock would a¤ect the balance sheet uniformly. The latter scenario corresponds to the case where bank portfolios are well diversi…ed in which case a shock would only a¤ect a fraction of the asset side of a bank's balance sheet. 9 Gabaix (2011) has extensively studied and compared such shocks and analyze their implications for the distribution of …rm size. In this paper, we examine how such shocks can a¤ect a small network of banks.
There are two main results. In general, under bank seniority the pro…t maximizing network structure is also the one that maximizes social welfare. The intuition of this result is straightfoward. Having the depositors absorb as much of the losses as possible allows banks to ful…ll their obligations against each other thus minimizing systemic risk. However, there is a very important exception. When the joint likelihood of (a) extreme high initial losses, (b) catastrophic …re sales, (c) low pro…tability, and (d) lack of asset diversi…cation, is very high 8 See the review article by Bougheas and Kirman (2015) for examples. 9 Correlation here refers to the asset returns of a bank's portfolio. Our intention is to study how the priority structure of banking libilities a¤ects the transmission of shocks from one bank to another and thus we are ignoring correlations of asset returns across the banking system where multiple banks fail simultaneously. As long as the correlation of returns across the banking system is not perfect our main argument is still valid although the results become weaker. Of course, as Acharya (2009) has demonstrated, any design of new policies needs to consider the incentives that these policies o¤er to banks to have their portfolios correlated and thus enhancing systemic risk. The argument is that if banks expect that it is more likely that they will be bailed out during systemic events then they have an incentive to increase correlation. the pro…t maximizing network structure under bank seniority is not optimal. Although the likelihood that all four conditions are met is low when they are all met systemic losses can be catastrophic. These four conditions provide a fair characterization of the state of the US banking system before the 2008 …nancial crisis. Taking the above results together we …nd that the optimal policy would, ceteris paribus, depend on the likelihood of such events. In the next section of the paper we review some other work in the literature that has identi…ed other factors that must be considered when designing optimal bankruptcy policies.
We organize our work as follows. After a brief review of the related theoretical literature in Section 2 we describe our model and in Section 3 we present our results. In the …nal section we conclude. All derivations are provided in the Appendix.
Related Literature Responding to the 1980s Savings and Loans crisis, the US Congress enacted the 1991 Federal Deposit Insurance Corporation Improvement Act followed by the 1993 Depositor Preference Act. The introduction of these Acts motivated a long debate among …nancial economists, legal scholars and policymakers that is reviewed in Bougheas and Kirman (2016). Here we restrict our attention to some theoretical developments that are more closely related to this present work.
Some experts argue that non-depositor priority rights provide strong incentives to depositors to discipline the banks. Actually, Calomiris and Kahn (1991) have suggested that by its very nature demandable debt, which allows depositors to withdraw their funds at will, o¤ers the required market discipline device. However, this argument is weakened when we allow for the possibility that uninformed depositors of nevertheless safe banks might also panic thus generating a wide-spread panic throughout the banking system.
Others believe that banks and other creditors are more suitable monitors. For example, Rochet and Tirole (1996) argue that interbank exposures generated through transactions in the interbank market provide strong incentives for banks to monitor other banks and therefore interbank loans should be junior to deposits. Kaufman (2014) has challenged this claim by arguing that how e¤ective the banks are as monitors depends on their beliefs about the likelihood that the government would intervene in times of crises in which case banks would consider transactions in the interbank market as bearing low risk. Birchler (2000) also supports depositor preference on the grounds that banks have an informational advantage relative to a large number of small depositors. Moreover, he argues that raising funds by o¤ering to depositors a standardized product with priority rights is a more e¢cient than having each depositor sign a bilateral contract with a bank. Thus, the introduction of a priority list reduces the amount of resources devoted to socially ine¢cient information gathering and such an arrangement seems to be ideal for banks that raise funds from a large number of uninformed investors.
In contrast to the above mentioned studies, Freixas et al. (2004) o¤er a mixed view. In their model banks provide two services, namely, screening and monitoring. By screening potential applicants they improve the pool of loans that they o¤er while by monitoring …rms that have been o¤ered loans banks ensure that these …rms perform well. The optimal seniority structure depends on which of the two moral hazard problems associated with the two services is more pressing.
Most of the work on the seniority status of bank loans has focused on the interbank market where such loans are not secured. However, on the liability side of their balance sheet banks have other claims by …nancial institutions that are secured and therefore have top priority. Bolton and Oehmke (2015) analyze the seniority status of some types of derivatives and conclude that while these claims provide risk sharing opportunities, their position on the top of the priority ladder can lead to ine¢ciencies as it transfers risk to other bank creditors such as depositors. 10

The Banking Network
In order to assess the welfare implications of each of the two alternative priority rights allocation policies we need …rst to understand how each of these options would a¤ect the structure of the interbank network which in turn would depend on the pro…t maximizing decisions of each bank in the network. The dynamic formation of the interbank market network is a di¢cult problem that has recently attracted some attention (Babus, 2015;Cohen-Cole et al., 2010 ). This work takes the priority ladder of banking liabilities as exogenously given. Our work is further complicated by the dependence of the network structure on the priority ladder that requires making very complex welfare comparisons. With that in mind we will examine a very simple banking network that will allow us to make such comparisons and then we will consider its relevance for more realistic environments.
There are two types of risk-neutral agents: bank owners and depositors. There are four banks which, for mnemonic reasons, we denote as  , , , . We keep bank balance sheets very simple. On the asset side bank  has customer loans,   , and may have loans o¤ered to other banks. Let    ( 6 = ) denote loans from bank  to bank . On the liability side bank  has customer deposits,   , may have deposits from other banks and equity   . Let    ( 6 = ) denote deposits in bank  from bank . Balance sheets must satisfy the constraints and the interbank market must satisfy the constraints The net interest rate on consumer loans is equal to   1, the interest rate on deposits is equal to 0 and interbank interest rate is also equal to 0. 11 The initial balance sheets of the four banks are as follows: Each bank has funded one unit of loans with its own deposits. In addition, bank  had an extra unit of deposits that it loaned to bank  that used it to …nance an extra unit of loans. Thus, the balance sheets of banks  ,  and  are constructed so that they capture the three possible cases of net interbank exposures. Bank 's (neutral) net exposure is zero, bank  is a net lender and bank  is a net borrower. 12 Suppose that bank  (decision-maker) obtains an extra unit of deposits that is willing to loan to another bank. All other three banks can fund an extra unit of consumer loans. In what follows we provide answers to the following four questions: ² Assuming that depositors have priority, to which other bank will bank  o¤er the loan to maximize its pro…ts?
² Assuming that depositors have priority, which bank should receive the loan so that social welfare is maximized?
² Assuming that banks have priority, to which other bank will bank  o¤er the loan to maximize its pro…ts?
² Assuming that banks have priority, which bank should receive the loan so that social welfare is maximized?
The answers to the four questions only matter when there is a banking crisis and in particular when a bank other than  goes into liquidation. Thus, we assume that one of the other three banks has to write o¤ some of its assets. The answers will also depend on many other modeling choices such as the likelihood and size of shocks and the expectations of bank  about future changes in each bank's balance sheets including changes in the interbank network. We will begin by analyzing a benchmark case that provides simple and intuitive answers. Later we will discuss how our results might be a¤ected if we move to more complex variations of our model. Our benchmark model satis…es the following restrictions: Assumption 1 (Myopic Expectations) After bank  o¤ers the loan to one of the other three banks it does not expect any further changes in any of the balance sheets. 13 Assumption 2 (Catastrophic Fire Sales) When any bank, other than the bank initially hit by a shock, goes bankrupt (systemic losses) the value of customer loans on the its bank's balance sheet is completely wiped out.
The …rst assumption is made for analytical convenience. It is hard to see how we can make any progress within our framework by considering any alternative. Given that decisions matter only in case of a systemic event any forward looking solutions would depend on the likelihood of such events after each possible choice that banks could make in the future. Having said that, a bank would always face one of the three choices that we are considering in this work. Our assumption would not be too restrictive as long as banks consider the indebtedness status (net borrower or net lender) of a potential borrower as more important than the size of net indebtedness.
The second assumption is only imposed to minimize the number of cases that we need to consider but it will become clear that our main conclusions do not depend on it.
Clearly, the answers to the four questions will also depend on the beliefs that bank  has about the size distribution of the initial shock. Suppose that bank  is the one in ‡icted by the initial shock. Then, let   denote bank 's liquidation value of customer loans. Thus, the size of the shock is given by   ¡   . We will compare the answers to the four questions under two alternative scenaria related to the distribution of shocks across the banking system. 14 Proportional Shocks The probability that any of the banks  or  or  becomes insolvent is equal to 1 3 . Shocks are proportional to the value of the in ‡icted bank's customer loans (     =  8). Identical Shocks   ¡  =   ¡  6 1, for every bank  or ; The probability that a bank becomes insolvent is proportional to the value of its customer loans.
The above two scenaria regarding the distribution of shocks across the banking system capture two polar cases. With proportional shocks the underlying 13 The alternative assumption would be to have farsighted banks. The literature on farsighted networks is very young. There is some progress made in proving existence results but not much on characterizing the solutions (see, for example, Dutta et al., 2005). 14 For the moment we do not impose any restrictions on the joint distribution of  and    assumption is that scale does not lead to diversi…cation. Put di¤erently, as a bank grows (in terms of customer loans) it replicates its existing portfolio (extreme specialization). Thus, as a bank's customer loans grow in size so does the bank's exposure to the risks associated with the particular sector …nanced by the bank. Under the assumption that each sector is equally likely to be in ‡icted by a negative shock and thus the corresponding …rms become unable to meet their obligations with their bank, shocks are proportional. In contrast, the implication of the assumption of identical shocks is extreme diversi…cation. Thus, as a bank doubles in size (again, in terms of customer loans) it also doubles the number of sectors it …nances. Under the same supposition as above, that is each sector is equally likely to be in ‡icted by a negative shock, now all shocks have the same size (assuming each sector has similar …nancial needs) but as a bank doubles the number of sectors that it …nances it also doubles the probability that it will be impacted by a shock.

Results
In this section, we describe and o¤er some intuitive explanations of our main results. The detailed derivations can be found in the Appendix. The solution of the model proceeds in two steps. Firstly, we derive for each priority case and for each option that bank  has for o¤ering the loan, bank 's pro…ts and social welfare. The latter is de…ned as total bank pro…ts plus total available deposits after any bank resolution. 15 This exercise is completed for all admissible values of net interest on loans, , and liquidation values,   . These derivations are very straightforward but tedious and appear in the Appendix under the subsection heading 'Preliminary Derivations'. Secondly, we use the calculations from the …rst step to derive bank 's pro…t maximizing choice and also the loan o¤er that maximizes welfare. The analysis is completed by choosing the priority rights policy that would maximize welfare conditional on bank 's pro…t maximizing choice. This step is repeated for each of the two restrictions on the distribution of shocks across the banking system.
The following two Propositions summarize the results.
Proposition 1 (Proportional Shocks): (a) Under depositor seniority, for any values of  and , it is never optimal for bank  to o¤er the loan to bank .
(b) Under bank seniority, for any values of  and , o¤ering the loan to bank  weakly dominates the alternative two options. 15 To simplify our analysis we have assumed linear utility. The absence of any curvature matters only for one important case where we carefully discuss the various trade-o¤s. We have also assumed equal weighting between equityholders and depositors. Our model does not distinguish between bank managers and equityholders and as we have argued above there is no a priori for the social planner to favor depositors over equityholders. If one of the aims is to protect depositors in order to guarantee adequate liquidity then this can be achieved by alternative policy instruments such as deposit insurance.
(c) O¤ering the loan to bank  maximizes welfare for any values of  and  except the worst case scenario of very high initial losses and very low pro…tability.
It is clear that, keeping the structure of the network …xed, bank 's expected pro…ts are higher under bank priority for any of the three loan o¤er options as more of the losses are absorbed by depositors.
Parts (a) and (b) are important as they demonstrate that the expected pro…t maximizing choice under bank priority is not the same as under depositor priority. The result can be best understood by considering bank 's expected pro…ts when it o¤ers the loan to bank  which has already o¤ered a loan to bank . Under depositor priority bank  is potentially exposed to failures of either bank  or bank . In contrast, under bank priority there is a bu¤er of deposits at bank  protecting bank .
To understand part (c) of the Proposition we observe that as long as the network structure is …xed bank seniority maximizes welfare. (Social welfare is at least as high under bank seniority for all values of the shocks.) This is because having depositors absorb the losses prevents the spread of the crisis to other banks. Ignoring for the moment the worst case scenario (low  and low ), we also …nd that o¤ering the loan to bank  is also the social welfare maximizing case. Form the point of view of social welfare we care about both depositors and equityholders. Given that bank  has a higher value of deposits, under bank seniority, o¤ering the loan to this bank reduces the likelihood that the crisis spreads. Certainly, this would mean that depositors su¤er most of the losses. But as aggregate losses are low this is only a distributional issue.
The above argument is not true for the worst case scenario (low  and low ). In that case o¤ering the loan to bank  does not maximize welfare. When the shocks are large it is better for the network not to be too connected (Acemoglu et al., 2015a; Cabrales et al. 2014). Low connectivity reduces contagion. However, from the point of view of bank  o¤ering the loan to bank  is never dominated by the other two choices. The reason is that bank  does not take into account the e¤ect of its choice on depositors. This creates a con ‡ict between the equilibrium pro…t-maximizing choice and the one that maximizes social welfare. Of course, from an ex ante point of view everything depends on the relative likelihood of these extreme events (fat tails).
Thus, the pro…t maximizing choice in not necessarily the same as the social welfare maximizing choice when the structure of the network is a¤ected by the allocation of priority rights. As we have already argued that is not the case when the network structure is …xed. Next, we consider the alternative structure for the distribution of shocks.

Proposition 2 (Identical Shocks):
(a) Under depositor seniority the optimal choice of bank  would be either to o¤er the loan to bank  or bank  depending on the distribution of shocks.
(b) Under bank seniority bank  will be indi¤erent across the three choices.
(c) Welfare is maximized by o¤ering the loan either to bank  or bank .
This structure for the distribution of shocks would arise when banks have well diversi…ed portfolios. Banks with bigger balance sheets would be in ‡icted by shocks more often, however, the size of the shocks would not depend on the size of the balance sheets. With depositor seniority, the optimal choice of bank , would depend on the exact speci…cation of the distribution of shocks, and it would be either to o¤er the loan to bank  or to bank , but never to bank . This is because by o¤ering the loan to bank , bank  is exposed to any shocks in ‡icting either bank  or bank  (indirectly). In contrast, under bank seniority given that the size of the shocks are relatively small, and thus the losses are absorbed by depositors, expected pro…ts are identical under all three choices.
Social welfare is maximized by avoiding o¤ering the loan to bank . This is because in the latter case there is a high concentration of loans in bank  and thus when this bank fails the potential for losses is higher. Thus, when shocks are relatively small bank seniority induces networks that maximize welfare.
Considering the two Propositions together we draw the following conclusions.
Corollary 1 (a) The structure of the network is not invariant to the policy regime.
(b) The pro…t maximizing choice in not necessarily the same as the social welfare maximizing choice when the structure of the network is a¤ected by the policy regime.
We regard the two alternative restrictions that we have imposed on the distribution of shocks across the banking system as two polar cases of a much broader space of such distributions. Considering together the results of Propositions 2 and 3 we …nd that under bank seniority bank  would o¤er the loan to bank . In general, this would also be the social welfare maximizing choice. However, this would not be the case if the joint likelihood of (a) extreme high initial losses, (b) catastrophic …re sales, (c) low pro…tability, and (d) lack of asset diversi…cation, is very high. These four conditions provide a fair characterization of the status of the US banking system before the 2008 …nancial crisis. Our results suggest that while under most circumstances bank seniority would o¤er more protection against systemic risk, the higher connectivity that it encourages, might exacerbate the systemic consequences of extreme events.
It is important to keep in mind that our analysis has neglected other important factors that must be considered when designing bankruptcy rules. The issues are very complex to be considered in one uni…ed model. However, by focusing on just a single aspect we hope that we have identi…ed some of the trade-o¤s that need to be considered when considering such a complicated design problem.

Conclusion
Responding to the 2008 …nancial crisis policy-makers around the world have introduced, or are in the process of introducing, new bank resolution procedures, knows as 'bail-ins' that would require a failing bank's creditors to bear the costs of restoring it back to health. In USA the relevant legislation has been introduced through the Dodd-Frank Act while in the European Union such measures where enacted through the European Stability Mechanism and the Bank Recovery and Resolution Directive. The purpose of these new policy measures is to shift the burden of bank failures away from taxpayers (bail-outs) and onto unsecured creditors. When resolution takes place under a bail-in procedure, creditors instead of directly receiving the proceeds from the liquidation of the failing bank's assets as it would happen under normal bankruptcy procedures would now have their claims converted into equity. But still the exact value of their equity holdings will depend on their place in the priority ladder. Avgouleas and Goodhart (2015) o¤er a critical review of these policies and discuss how each of three groups of creditors, namely, depositors, other …nancial institutions and bondholders, is going to bear the burden. By focusing on issues related to systemic risk our work contributes in this discussion given that these policies were designed to eliminate the systemic consequences of failures of large …nancial institutions.
We have not delivered an unconditional optimal policy recommendation. This is not too surprising given that we already know that while some types of network structures are better at protecting the system during mild episodes the same structures can prove catastrophic during extreme events (Acemoglu et al., In order to keep our analysis simple, we have completely ignored other reasons for supporting one or another priority rights policy. However, there is a substantial body of work that has examined the incentives that such policies o¤er to various types of creditors to monitor banks. What has been absent from this discussion so far are their potential systemic risk implications. Keeping the analysis tractable also meant that we had to impose a couple of strong restrictions on our model. The analysis in the paper was carried out under the assumption that …re sales are catastrophic. We have also carried out the analysis for the case of …re sales that are not catastrophic and it turns out that the general message does not change. As we have pointed out in the Introduction, in the absence of …re sales, priority rights are irrelevant for the social welfare implications of systemic risk. For values of …re sales in the intermediate range we get many more cases to consider making the presentation of the results cumbersome. However, given that we are interested in comparing choices ex ante much of the intermediate variation vanishes out. What is more worrisome is our assumption of myopic expectations. What it implies is that bankers do not expect any further changes in the network structure. Although, recently there has been some progress in the direction of endogenizing the formation of banking networks (Babus, 2015; Cohen-Cole et al. 2010), the complexity of the issues that we have attempted to address in this paper it does not allow us to use their methods. We chose our network structure as it captures the three alternative types of borrowers that a lending bank might meet (positive exposures, negative exposures, zero exposures). Any other lending bank would be facing similar options. Clearly, as the network structure gets more complicated a bank's decisions will not depend only on the net exposures of their potential lenders but also on the exposures further down the line.
In summary, our results have been derived from a very simple network structure under naive behavioral assumptions. Having said that we believe that the results are very intuitive and at the very least are a good starting point for addressing important policy issues.

Appendix
Throughout our analysis we assume that when a bank becomes insolvent its assets are distributed to its creditors according to priority rules and all parties at the same priority level share their allocated assets in proportion to their corresponding claims. Let ¦  denote bank 's pro…ts and   …nal withdrawals (consumption) by the depositors of bank . As a result of potential bank liquidations we have   6   .
We begin by deriving bank 's pro…ts, ¦  , and social welfare (post-bankruptcy customer deposits plus total bank pro…ts),  , for each of the two priority cases and for each of the three options of bank , for all admissible values of  and   . Then we compare bank 's pro…t maximizing choice with the social welfare maximizing choice for each seniority structure and for each of the two restrictions on the distribution of shocks across the banking system.

(a) Loan from Bank  to Bank 
The new balance sheets of banks  and  are given by: 16 We consider separately the three cases of initial insolvencies:

Case 1 Bank  goes bankrupt
In this case the shock only a¤ects bank  (direct hit) and its creditor bank . Depending on the value of   2 [0 2] we need to consider two cases: (a)   6 1. Depositor seniority implies that   =   . Thus, bank  will lose its deposits at bank  , go bankruptcy itself and, by Assumption 2,   = 0. Thus, we have ¦  = 0 and  = 3(1 + ) +   given that   = 2, ¦  = ,   = 1 and ¦  = 2.
In this case the depositors of bank  get all their deposits back,   = 1, and bank  recovers   ¡ 1 of its loan to bank  . There are two cases to consider depending on whether or not bank  remains solvent: (i)  +    2. In this case the bank gets liquidated as the value of its assets that is the sum of the value of its loans 1 +  plus the funds that managed to recover from bank  are less than the value of its liabilities which is equal to 2. The depositors of the bank will get   =   ¡ 1 and thus we have as in the previous case ¦  = 0 and  = 3(1 + ) +   but now the higher value of   is shared between the depositors of the two a¤ected banks.
(ii)  +   > 2. Now bank  is solvent and thus where for the derivation of welfare we notice that the depositors of all banks got their funds back and the sum of the pro…ts of the two una¤ected banks is equal to 3.

Case 2 Bank  goes bankrupt
The bankruptcy only a¤ects bank  whose assets include a loan of 1 unit to bank . Therefore, its depositors will receive   2 [0 1] plus a deposit of 1 unit at bank . Given that all banks other than  have not been a¤ected, we have:

Case 3 Bank  goes bankrupt
The bankruptcy will also a¤ect bank  that has o¤ered a loan to bank . Symmetry implies that the welfare results in this case are exactly the same as those derived from the case when bank  goes bankrupt. When bank  receives the new loan from bank  there is a symmetric network structure, namely, two banks o¤ering a loan and two banks receiving a loan. Moreover, banks  and  are the two banks receiving the loans. The only di¤erence is that in this case bank  is solvent. Thus, we have the following cases: (a)   6 1 The new balance sheets of banks  and  are given by: Given that other banks are not a¤ected, we have:

Case 2 Bank  goes bankrupt
We need to consider two cases depending of the value of   2 [0 2]: (a)    1. The depositors of bank  receive   plus a deposit of 1 unit at bank . Bank 's loan to bank  will not be repaid and therefore bank  will go bankrupt and, by Assumption 2,   = 0. Thus, In this case the depositors of bank  are fully compensated by receiving   plus 2 ¡   of deposits at bank . Bank  recovers   ¡ 1 of its loan to bank  and there are two cases to consider depending on whether or not bank  remains solvent. The analysis is exactly the same as in the case when bank  o¤ers the loan to bank  and the latter becomes insolvent. The only di¤erence is that we need to replace   with   . Thus,

Case 3 Bank  goes bankrupt
The bankruptcy will also a¤ect bank  that has o¤ered bank  a loan and potentially bank  that has o¤ered bank  a loan. Note that   2 [0 2].
(a)    1. The payo¤ of the depositors of bank  is given by   =   and bank 's loan is not repaid. There are two cases to consider depending on whether or not bank  remains solvent.
(i)   1 2 . Given that the assets of bank  are equal to 2(1 + ) and the liabilities are equal to 3, the inequality implies that bank  also goes bankrupt. The bankruptcy implies that that   = 0 which, in turn, implies that bank  goes bankrupt and thus   = 0. Only the isolated bank  survives.
Now the depositors of bank  are fully paid,   = 1 and bank  recovers   ¡ 1. Once more, there are two cases to consider depending or not bank  remains solvent. The only di¤erence with the above case is that now the assets of bank  are larger by   ¡ 1. Thus, now we have The new balance sheets of banks  and  are given by:

Case 1 Bank  goes bankrupt
This is similar to the case when bank  o¤ers the loan to bank . Given that other banks are not a¤ected, we have:

Case 2 Bank  goes bankrupt
The payo¤ to depositors of bank  is equal to   , where   2 [0 1], plus a deposit of 1 unit at bank . All other banks are not a¤ected.

Case 3 Bank  goes bankrupt
We need to consider the following two cases depending on the value of   2 [0 3].
(a)    1. The payo¤ of depositors of bank  is given by   =   and the loans to banks  and  are not repaid and thus both banks go bankrupt with only bank  surviving.
Now the depositors of bank  are fully paid,   =   = 1, and banks  and  each recover   ¡1 2 from their one 1 unit of loan to bank . What happens to the two banks depends on the value of   . Each of these two banks have assets equal to 1 +  + and liabilities equal to 2. Thus, we have two cases to consider: Bank  and  are solvent.

Bank Seniority
(a) Loan from Bank  to Bank  See Table 1 for the balance sheets of banks  and  and Table 2 for the balance sheets of banks  and .

Case 1 Bank  goes bankrupt
Depending on the value of   2 [0 2] we need to consider two cases: (a)    1. Given that banks have seniority   = 0 and bank  receives a payo¤ equal to   and depending on its value we need to consider two cases: (i)   1 ¡   . Bank  goes bankrupt and   =   .
and bank 's loan to bank  is fully repaid.

Case 2 Bank  goes bankrupt
Given that the only liabilities of bank  are customer deposits, the outcome is exactly the same as the case with depositor seniority.

Case 3 Bank  goes bankrupt
The bankruptcy will also a¤ect bank  that has o¤ered a loan to bank . Symmetry implies that the welfare results in this case are exactly the same as those derived from the case when bank  goes bankrupt. When bank  receives the new loan from bank  there is a symmetric network structure, namely, two banks o¤ering a loan and two banks receiving a loan. Moreover, banks  and  are the two banks receiving the loans. The only di¤erence is that in this case bank  is solvent. Thus, we have the following cases: Table 1 for the balance sheets of banks  and  and Table 2 for the balance sheets of banks  and .

Case 2 Bank  goes bankrupt
The creditor bank  has its loan repaid by obtaining 1 unit of deposits at bank . The payo¤ of depositors of bank  is equal to   =   .

Case 3 Bank  goes bankrupt
The bankruptcy will also a¤ect bank  that has o¤ered bank  a loan and potentially bank  that has o¤ered bank  a loan. Note that   2 [0 2].
(a)    1. The payo¤ of the depositors of bank  is given by   = 0 and bank  receives   . There are two cases to consider depending on whether or not bank  remains solvent.
(i)   1¡  2 . Given that the assets of bank  are equal to 2(1 + ) +   and the liabilities are equal to 3, the inequality implies that bank  also goes bankrupt. The bankruptcy implies that that   = 0 and bank  receives   .
However, the inequality implies that bank  with assets equal to 1 +  +   and liabilities equal to 2 also goes bankrupt, hence   =   .
Bank  is solvent and hence bank  is not a¤ected.
Bank 's loan is fully repaid and   =   ¡ 1. Table 1 for the balance sheets of banks  and  and Table 2 for the balance sheets of banks  and .

Case 1 Bank  goes bankrupt
As in the case when bank  o¤ers the loan to bank  the only bank a¤ected is bank  .

Case 2 Bank  goes bankrupt
The payo¤ to depositors of bank  is equal to   plus 1 unit of deposits at bank .

Case 3 Bank  goes bankrupt
Depending on the value of   2 [0 3] we need to consider two cases: (a)    2. In this case banks  and  each receive   2 and   = 0. What happens to banks  and  depends on the value of   . Each bank's assets are equal to 1 +  +   2 while the corresponding liabilities are equal to 2. Then, once more, we need to consider tow cases: (i)   1 ¡   2 . Both banks go bankrupt and   =   =   2 .  Table 1 shows the expected pro…ts of bank , [¦  ], for each of the 3 loan o¤er options and for all possible values of per unit of loan pro…ts, , and liquidation value per unit of loan, . The last column of the table indicates the optimal choice, that is the one that maximizes bank 's expected pro…ts, under the supposition that shocks per unit of loans and pro…ts per unit of loan can only take the value that corresponds to that particular row of the table.
[Please insert Table 1 about here] The restriction that shocks are proportional implies that, for example, the probability that a bank with 2 units of customer loans will be in ‡icted by a shock of size  is equal to the probability that a bank with 3 units of customer loans will be in ‡icted by a shock of size 3 2 . Then, in deriving the above table if, for example, bank  has 2 units of customer loans and thus   2 [0 2] we have set   = 2, where  2 [0 1], in the derivations of the last section.
Consider Table 1A. For the …rst three rows we have   1 3 and   1 2 . For the derivation of the …rst row we focus on sub-section 5.1.1.(a) that is when bank  o¤ers the loan to bank  . From case 1(a) we …nd that if bank  goes bankrupt ¦  = 0. Similarly, from case 2(a) we …nd that if bank  goes bankrupt ¦  =  and if bank  goes bankrupt ¦  = . Under the supposition of proportional shocks each bankruptcy event is equiprobable and thus we conclude that [¦  ] = 2 3 . Similarly, for the derivation of the second row we focus on sub-section 5.1.1.(b) that is when bank  o¤ers the loan to bank  and in particular cases 1, 2(a) and 3(a)(i) and for the derivation of the third row we focus on sub-section 5.1.1.(c) that is when bank  o¤ers the loan to bank  and in particular cases 1, 2 and 3(a). Comparing the three rows we …nd that under the supposition that liquidation values and pro…ts satisfy   1 3 and   1 2 bank  would be indi¤erent between o¤ering the loan to banks  and . For the next three rows of the table we now have  > 1 2 . The only di¤erence between this case and the one considered above is that when bank  o¤ers the loan to bank  we need to consider case 3(a)(ii) instead of case 3(a)(i). Now all three choices result in the same expected pro…ts.
Next, consider Table 1B. When 1 3 6   1 2 and bank  o¤ers the loan to bank , if the latter goes bankrupt the liquidation value of its assets is higher than its obligations to depositors. Therefore, its remaining assets will be equally distributed to its two creditor banks, namely,  and , and what happens to these banks it depends on the values of both  and . Given that   1 2 , as long as   3 4 , we have   1 ¡ 2 3  and thus the relevant case is 3(b)(i). The derivation of expected pro…ts follows exactly the same logic as the one used for the derivations of  Table 1C. We compare the …rst three rows of the table and the next three rows for the case when bank  o¤ers the loan to bank . All depends on whether bank  stays solvent in which case bank  also stays solvent (compare cases 3(b)(i) and 3(b)(ii)). Next, focusing on the middle six rows of the table we observe that there is a di¤erence between the …rst three rows of this group and the last three rows when bank  o¤ers the loan to bank . This is exactly the same case discussed in the previous paragraph and the results depend on whether or not bank  can remain solvent after the bankruptcy of bank . Finally, comparing the last six rows of the table we …nd that there is a di¤erence between the …rst three rows of this group and the last three rows when bank  o¤ers the loan to either bank  or bank . It all depends on whether or not bank  remains solvent after the bank that was o¤ered the loan went into bankruptcy. Table 2 shows the expected pro…ts of bank  for the case of bank seniority.

Bank Seniority
[Please insert Table 2 about here] Consider Table 2A. Focusing on the …rst three rows we observe that when the level of pro…ts  is very low, bank  always becomes insolvent when the bank to which it o¤ered the loan becomes insolvent. Thus, with probability 2 3 the bank remains solvent. Comparing the …rst three rows with the next three rows we …nd that the expected pro…ts of bank  increase when it o¤ers the loan to bank . Comparing cases 3(a)(i) and 3(a)(ii) we …nd that for su¢ciently high values of  bank  remains solvent after the bankruptcy of bank . Next, we compare the middle six rows and …nd that there is a di¤erence depending on whether or not bank  becomes insolvent when it o¤ers the loan to bank  and the latter becomes insolvent (cases 1(a)(i) and 1(a)(ii)). Moving to the last six rows we …nd that there is a di¤erence in expected pro…ts when bank  o¤ers the loan to bank . As  moves above the threshold that separates the last three rows from the three rows above them bank  remains solvent despite the bankruptcy of bank .
Di¤erences in the other two tables also capture how further increases in  and  a¤ect the expected pro…ts of bank  when it o¤ers the loan to bank  and the latter goes bankrupt. In Table 2B we have exactly the same results as for the case above. In contrast, in Table 2C,  is su¢ciently high for bank  to fully recover its loan to bank .

Social Welfare
Depositor Seniority For the case of depositor seniority and under the restriction that shocks are proportional, Table 3 shows the expected social welfare, [ ], for each of the 3 loan o¤er options and for all possible values of per unit of loan pro…ts, , and liquidation value per unit of loan, . The last column of the table indicates the optimal choice, that is the one that maximizes expected social welfare, under the supposition that shocks per unit of loans and pro…ts per unit of loan can only take the value that corresponds to that particular row of the table.
[Please insert Table 3 about here] Notice that the second term of all expressions is the same. The reason is that at any time there are 5 units of consumer loans in the books of banks  ,  and  and under proportional shocks 5 3  equals the expected liquidation value of these loans. Thus, from now on we focus on the …rst term. For each pair of values for  and  and for each possible loan o¤er by bank  we derive the …rst term by adding the welfare results for the corresponding three bankruptcy cases divide by 3 given that each of the three banks becomes insolvent with the same probability. The various cuto¤s are the same as the ones derived for the derivation of bank 's expected pro…ts under depositor seniority. As an example, consider the …rst entry of Table 3A. The loan is o¤ered to bank  and, therefore, we focus on Section 5.1.1.(a) and take the average of the welfare values of cases 1(a), 2 and 3(a). Notice that in this case   = 2,   =  and   = 2 and thus, as we pointed above, the average liquidation value is equal to 5 3 . As another example, consider the last entry of Table 3C. The loan is o¤ered to bank  and, therefore, we focus on Section 5.1.1.(c) and take the average of the welfare values of cases 1, 2 and 3(b)(ii). Notice that in this case   = ,   =  and   = 3 and thus, once more, the average liquidation value is equal to 5 3 .
Bank Seniority Table 4 below shows the expected social welfare for the case of bank seniority.
[Please insert Table 4 about here] The cut-o¤ values correspond to those of Table 2. For the derivations we have followed the same steps as for the case of depositor seniority but this time we used the results of Section 5.1.2.

Depositor vs Bank Seniority
Proof of Proposition 1 Parts (a) and (b): The proof follows from a direct comparison of Tables 1 and 2. When comparing Tables 1 and 2 we need to keep in mind that what matters is the value of expected pro…ts under each policy regime. However, it is clear that the exact probability weights do not matter. Table 1 shows that under depositor seniority it is never optimal for bank  to o¤er the loan to bank . For any values of  and  by o¤ering the loan to bank , bank  makes at least as high pro…ts as when o¤ering the loan to bank . Then, bank  optimal choice will be either to o¤er the loan to bank  or to bank  with the optimal choice depending on the distributions of  and . In contrast, under bank seniority, Table 2 shows that o¤ering the loan to bank  is the dominant choice for any values of  and .
Part (c): The proof follows from the proofs of parts (a) and (b) and the following observations: According to Table 4, bank seniority that provides incentives to bank  to o¤er the loan to bank , maximizes welfare in all cases but the top one which corresponds to the worst case scenario of very high initial losses and very low pro…tability. For this particular case, welfare would be maximized by the depositor seniority option (see Table 3) that provides incentives to bank  to o¤er the loan to either bank  or bank .

Comparing Bank Seniority to Depositor Seniority (Identical Shocks)
With identical shocks we have   ¡   =   ¡   6 1, for every bank  or .
Put di¤erently, given that the losses are restricted to be at most equal to 1 unit of consumer loans,  equals 1 minus these losses and is identical across banks.

The Optimal Choice of Bank 
Depositor Seniority For the case of depositor seniority and under the restriction that shocks are identical, Table 5 below shows the expected pro…ts of bank , [¦  ], for each of the 3 loan o¤er options and for all possible values of per unit of loan pro…ts, , and  values de…ned above. The last column of the table indicates the optimal choice, that is the one that maximizes bank 's expected pro…ts, under the supposition that shocks per unit of loans and pro…ts per unit of loan can only take the value that corresponds to that particular row of the table.
onsider the …rst row of the table that is derived using the results of Section 5.1.1.(a) Given that the losses cannot exceed one unit and banks  and  have two units of loans each, the relevant cases are 1(b)(i), 2 and 3(b)(i) . Thus, for the derivations we let   =   = 1 +  and   =  and the entry in the …rst column follows from  +    2 ,   1 ¡  and from   1¡ 2 )   1 ¡ . The total number of customer loan units on the books of banks  ,  and  is equal to 5 and, thus, the probability that bank  will go bankrupt is equal to 2 5 , the corresponding probability for bank  is equal to 1 5 and the corresponding probability for bank  is equal to 2 5 . Then, [¦  ] = 2 5 £ 0+ 1 5 £  + 2 5 £  = 3 5 . For the second row we use the results of Section 5.1.1.(b) and in particular cases 1, 2 and 3(b)(i). In this case we have   =   = 1 +  and   = . For the case 2(b)(i) the cut-o¤ value for  is the same as above but for case 3(b)(i) we have a new cut-o¤ value given by 1¡ 2 . Lastly, the probabilities that banks ,  and  become insolvent are equal to 1 5 , 2 5 and 2 5 , respectively. For the third row we use the results of Section 5.1.1.(c) and in particular cases 1, 2(b)(i) and 3(b)(i). In this case we have   =   =  and   = 2 + . For the case 3(b)(i) the cut-o¤ value for  is given by 2 +    2 ,   1 ¡ . Lastly, the probabilities that banks  ,  and  become insolvent are equal to 1 5 , 1 5 and 3 5 , respectively.
Following the same steps we have completed the table.
Bank Seniority Table 6 shows the expected pro…ts of bank  for the case of bank seniority. Bank seniority completely protects bank  form the insolvencies of any other bank when the size of the shocks are restricted to be less than 1.

Social Welfare
Depositor Seniority Table 7 shows the social welfare results for the depositor seniority case. For the derivation of the entries we follow exactly the same steps as those that we followed for the derivation of Table 5. The only di¤erence is that now we use the results for  instead of [¦  ].  Bank Seniority Table 8 below shows the expected social welfare for the case of bank seniority. In all 3 entries of the table the total welfare of depositors is equal to 5 + . There are 6 units of deposits in the banking network and each time a bank defaults its depositors lose 1 ¡ . Pro…ts are lower when bank  o¤ers the loan to bank  because of the concentration of loans in one bank.

Depositor vs Bank Seniority
Proof of Proposition 2 Parts (a) and (b): This follows directly from Tables  5 and 6.