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Theories of Bank Run Contagion

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There is little or no probability of bank insolvency, the type one depositors may prefer to follow the type two depositors and leave their deposit for longer period of time. Because they would like to take the advantage of the higher payoff on the bank’s assets and the expected higher return on investment compensate them for their impatience. So, when the probability of bank insolvency is low (based on the economic fundamentals), the impatient depositors may act as patient depositors and the bank will liquidated the funds at the end of maturity and divide the proceeds among the depositors equally.

But the bank faces serious liquidity crisis when the type two depositors pretends to act like type one and ask for early withdrawal. This situation occurs when the depositors receive signals that the economic fundamentals are bad and the performance of their bank is correlated with that of other insolvent banks from previous year. They become convinced that the probability of the bank insolvency is high and the type two depositors would prefer to make early withdrawal. Because in case of an insolvency, the bank might not differentiate the depositors based on their patience status and serve the liquidity demands on a first come, first served basis. As both patient and impatient depositors attempt to withdraw their deposits, the bank starts losing her reserve rapidly and leads to capitulation. The bank run in this case is caused by the factors which is not directly related to the bank’s financial activities and sometimes the bank faces run due to sudden liquidity shock even though she is not insolvent.

Bougheas (1999) has introduced OLG model instead of OLG framework because he wants to eliminate any discrimination between informed and uninformed depositors. So, there will be less chance of drawing any biased conclusion. The depositors in OLG model will only follow the changes in economic fundamentals and their perception about the correlation among banks’ returns. The researcher has drawn two major conclusions at the end of his theory. The first conclusion is that in an unregulated banking environment the bank run can be contagious only if the economic state is bad (recession). The probability of a bank run yielding a run on other solvent banks does not actually implies that the current banking system is unstable. But the second conclusion states that the run on a single bank may not cause a bank run contagion but certainly spread a negative signal about the macroeconomic stability. So, this negative signal might cause some financial turbulences to other banks like liquidity shortage, which can eventually lead to insolvency.

Panic- Based Bank Runs

Diamond and Dybvig (1983) (D&D model) presented the demand deposit contract exposes the bank to panic based bank run. Unfortunately, they did not provide us with precise mechanism to determine the possibility of bank run or the consequence of run on the aggregate economic welfare. Itay Goldstein and Ady Pauzner (2005) have done scientific research on a modified version of D&D model, where they also embrace the idea of economic fundamentals but not as determinants of bank run. The economic fundamentals are used as a mechanism to coordinates depositors beliefs on a certain outcome. They also consider the situation where the depositors acquire private signals and how those signals shape their perception about plausible bank run. Their modified D&D model also explore the underlying causality among optimum level of short term payment to the impatient depositors, risk sharing (demand- deposit contracts) and the bank run efficiency. Here, we are going to understand the theory of Panic based bank run, where the researchers (Goldstein & Pauzner, 2005) explain the welfare gain or loss due to bank run and according to them the run does not necessarily caused by unexplained feelings of depositors. They also propose some regulatory measures, which might be useful to eliminate or at least minimize the probability bank run and moral hazard problem regarding the deposit contracts. But we will untangle the regulatory intervention policies under different circumstances at concluding part of this scientific paper.

We are going to follow the scientific work of Goldstein & Pauzner (2005). Starting with a brief summary of traditional D&D model. The traditional Demand- Deposit model (D&D model) was proposed by Diamond & Dybvig (1983), where the researchers explained that the demand- deposit contracts offered by the banks, are very expedient to provide liquidity but at the same time these contracts expose banks to the possibility of panic-based run. In short sentence, the D&D contracts allow bank to create liquid claim on long term illiquid assets, but due to the maturity mismatch problem between the asset s(loans) and the liabilities (deposits) banks become vulnerable to the probability of panic-based bank run. The D&D model has two equilibriums. In case of Good equilibrium, only the impatient depositors (also short-term investors) claim for early deposit withdrawal. In order to satisfy their impatience, they are offered more than the liquidation value of the long-term asset, which results in welfare improving risk sharing situation. But in case of a bank run, Bad equilibrium happens. In Bad equilibrium, both impatient and patient depositors will ask for early deposits withdrawal, which eventually leads to a bank run (liquidity shock) and results in a welfare loss. But the problem with traditional D&D model is that it does not deliver us the required mechanism to anticipate which kind of equilibrium will happen or how to derive the probability of each equilibrium to occur.

There are several early scientific papers have been working on panic-based bank runs but from different perspectives. Chari and Jagannathan (1988) have presented Panic Based bank run which is caused by uninformed depositors. They showed that the uninformed agents make wrong interpretation of economic fundamentals (based on other agents’ run) and triggered panic attack. On the other hand, Peck and Shell (2003) have explored the idea of a more flexible deposit contracts, which may lead to complete elimination of bank run. According to Peck and Shell, under these contracts, the bank will be allowed to formulate the payment for the early withdrawers (impatient depositors) in order to stop the contagion. But due to moral hazard problem these contracts should not be allowed, we will discuss it later part of this paper. Goldstein and Pauzner (2005) have considered both of these scientific works and make necessary adjustments to analyse the interconnectedness of deposit contracts and the probability of bank runs.

The researchers also use economic fundamentals in their model, but not like Bougheas (1999) as a determinant of bank run. They use fundamental as an instrument to understand the depositors’ behaviours and the bank run is still panic based like in the traditional D&D model. In this model the economic fundamentals are stochastic and the depositors do not have communal knowledge of the fundamental status. They rather have slightly noisy private signals, which is more likely realistic enough. The researchers also shown that the bank run only happens when the economic fundamentals are below some certain critical value. But still the bank runs in this theory based on bad expectation. The depositors will undertake any actions (stay or run) based on her belief whether the others will do the same. So, a depositor can go for run simply because she presumes that the other would run too, even when the economic fundamentals are sufficiently strong.

Traditional D&D model explains that there are two equilibriums, in good equilibrium the bank increases the welfare and in bad equilibrium it decreases welfare. In order to resolve the difficulty regarding welfare and equilibrium, Goldstein and Pauzner (2005) follow the scientific works of Carlsson and van Damme (1993) and Morris and Shin (1998). In both of these papers (also in current theory) they showed that if the depositors receive noisy signals and formulate their actions based on it, this incident may lead to unique equilibrium. This unique equilibrium can be explained by the assumption of Global Strategic Complementarities. The Global strategic complementarities (GSC) stands for a situation where a depositor’s motivation to pursue early withdrawal is consistently increasing with the number of other depositors who pursue the same action. So according to GSC the motivation to run is highest not when all depositors go for it, rather when the aggregate demand for early withdrawals reach to the bankruptcy level. But beyond this level the incentives of go for early withdrawal may not increase with the number of other depositors who claim for early withdrawal. According to Goldstein and Pauzner (2005) once the bank exhausts the liquid reserve and due to asset fire sales leads to insolvency, the probability of getting any compensation from bank decreases drastically. This unique situation is called one sided strategic complementarities (OSC). OSC stands for a situation where the depositors would prefer to stay than go for run as long as the number of early withdrawers (depositors) is relatively small. So, this is how the researchers explain a uniqueness of depositors’ behaviour in panic-based run under modified D&D model (Goldstein & Pauzner, 2005).

The researchers also explore the relationship between the degree of risk sharing and the probability of bank run to find the optimum level of short term payment. They showed that a bank becomes more exposed to bank run when it offers a higher short-term payment. In order to have an efficient bank run to happen, the aggregate short-term payment has to be equal to the liquidation value of the long-term asset. Because in this particular situation the bank run occurs only when the expected long-term payoff is substantially low, the depositors can have better payoff if the long-term asset is liquidated early. On the other hand, if bank offers the short-term payoff more than the liquidation value of its long-term asset, it may face nonefficient bank run. Because if bank run happens, the bank will have to fire sale their long-term asset, even though their long-term return is substantially high. So, the ultimate question is how a bank formulates a deposit contract with optimum level of risk sharing to minimize the probability of bank run and increase the welfare? We will explore this question under two different situations where the depositors have no private signals and in other situation where the depositors acquire some private information.

Bank offers deposit contract impatient depositors with short term payments in order to satisfy their consumption needs. Then again, due to the high degree of risk aversion, the bank has to offer incentive compatible deposit contract to the impatient depositors and thus a wealth transfer from patient depositors to impatient one might be preferable. Even though this risk sharing might cause early liquidation of some of the long-term assets. But the types of depositors are private information, unknow to bank. So, the bank cannot formulate payments based on their types. So, they offer demand- deposit contracts to enable risk sharing, but in case of short term payment the bank has to face sequential constraint: equal amount to the depositors until its reserve is depleted. Two possible equilibriums are discussed by the researchers under this condition, if only the impatient depositors go for early withdrawal, they would receive their payment but there will be less payoffs for the patient depositors in future. But if both impatient and patient depositors go for early deposit run, there will be nothing left for future to gain. So, the agents would opt to demand early withdrawal and result in an unproductive equilibrium.

According to the traditional D&D model the optimum short-term payment is yielded under the assumption that the good equilibrium will be always preferred by the depositors. But evidently, the equilibrium is not always good and the risk sharing might not be always preferable. Moreover, it is hard to quantify the relationship between the quality of banking contracts and the probability of a bank run. This situation leads us to the discussion of unique equilibrium, where the depositors receive private signals.

Goldstein and Pauzner (2005) has modified the traditional D&D model by introducing private signals in the equilibrium. Under this modified model every depositor receives a private signal about the economic fundamentals and they formulate their actions based on this signal. An agent’s private signal can be regarded as her private assessment of the probability of long term payoffs. These private signals are idiosyncratic in nature and none of the depositors have any dominance over others regarding the quality of the information. Lack of completeness in the private signal makes it harder for a depositor to calculate her expected payoff, so she largely depends on her private signal to articulate her actions.

The private signals enable the depositors to formulate their decision based on their individual signal. If a depositor receives a higher signal, she will assume that the probability of achieving long term return on illiquid asset is also high. Therefore, she feels less motivated to demand early withdrawal. Furthermore, this private signal also serves as an indicator of other agents’ signals. If a depositor experience that other agents are also receiving high signal, she develops a belief that others will also not run and this leads to even less motivation to bank run. The researchers assume that there are two extreme cases of economic fundamentals. In both cases (extremely good and extremely bad), the agents actions are influenced largely by the status of economic fundamentals rather than the perceptions about other depositors’ actions. For example, in case extremely good fundamental region, the patient agents know that the long-term return from illiquid asset is almost certain to be achieved as well as the short-term payoffs. So, whatever her perception about others’ actions, she chooses not to run. Because there will be no need to liquidate the long-term asset to compensate the short-term payoffs and the long-term return is secured. Conversely, in case of lower dominance region, the fundamentals are very poor, that makes the Bank run very plausible. So, the patient agents realize that he expected return from waiting is significantly lower than that of short term withdrawal. So regardless of her anticipation of other depositors’ actions, her best response to this situation should be to go for early withdrawal.

Apart from these two extreme regions, there exists a vast intermediate dominance area. Unlike the extreme situations, the behaviour of depositors is not so predictable in this intermediate region. As we have already known that the private signals are incomplete and does not provide a complete knowledge about the other depositors’ signals. So, the depositor with private signal would most likely follow a unique equilibrium with a certain threshold. According to this modified D&D model, the patient agents choose to go for run only if they receive a private signal below threshold. Thus, in this intermediate region, the status of fundamentals act as a co ordination device, not as a determinant. The bank runs in this area are still based on bad expectation.

Goldstein and Pauzner (2005) has used a modified form of D&D model with slight adjustment like private signals, which leads them to unique equilibrium. In this unique equilibrium the depositors demand early deposit withdrawal only if the signal is below a certain threshold level. But the uniqueness of this equilibrium is that it does not follow the traditional idea of Global Strategic Complementarities. This unique equilibrium can be explained by one sided strategic complementarities that sometimes the motivation of run decreases with the increased number of early withdrawers (in case of bank insolvency). They also concluded that the bank cannot offer more than the optimum level of short term payment to the impatient depositors, because it might lead to bank run. So, it is not possible to acquire all of the available advantages of risk sharing.

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