4. FINANCIAL FRAGILITY
4.2 B ANK R UNS AND C ONTAGIONS
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prudent and risky, this statement can still provide some explanations for the improvement for the increase of buffer which lead to the higher banking stability.
4.2 Bank Runs and Contagions
In this section, we discuss the conditions for the bank runs and contagions to happen, and how our model can effectively lower the possibility of these outcomes.
Firstly, we reconsider the condition that the bank would be bankrupt. For example, in region A, the bank has units of the short asset. The fraction of early consumers is in state , so in order to pay each early consumer units of consumption, the bank will have to get units of consumption by liquidating some of the long asset. This is the buffer. Bank A will fall into bankruptcy if the excess liquidity demands is greater than the buffer plus the net value of the interbank deposits of Bank A. We show the condition in equation (4.6).
(4.6) It’s important to talk about how the role of the interbank deposits to affect the bankruptcy condition. Let’s imagine that once Bank A tries to withdraw its deposits in other bank (Bank B), Bank B will observe such action and withdraw their deposit in Bank C as well to protect the value of their deposit. Because they think the excess liquidity demand in region A might spread to other regions and turn into a crisis, and they would like to withdraw the deposit now to get at least rather than liquidate the long asset (remember that the pecking order assume the bank will liquidate the deposits in other regions before liquidate the long asset). Finally, all banks will withdraw their deposit at date 1.
However, the value of the deposit in Bank B is different from the deposit in Bank A, making Bank A receiving some benefits from the interbank deposits. As the
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liquidation value , but because of Bank A’s liquidity problem, bank A gets some supports from its interbank deposits to deal with the excess liquidity demand. In Allen and Gale’s article, they think the liquidation value is determined simultaneously and must be the same for all the regions. Therefore, the mutual withdrawals simply cancel out each other. We have a different opinion about this and try to apply the interbank effect into equation (4.6) to affect the bankruptcy condition. To sum up, if is large enough and the inequality (4.6) is satisfied, then we can infer that the bank is bankrupt. We call this inequality as the
“bankruptcy condition”. Note that, there also exist a situation when is small enough and the inequality is not satisfied, bank A is insolvent but not bankruptcy.
However, the late consumers in region A are worse off because the premature liquidation of the long asset at date 1 prevent the bank from paying to depositors at date 2.
The second issue is about the contagion, or spillover effect, of the original liquidity shock in region A to other regions. Once the Bank A is bankrupt, there will be a spillover effect to region D. A deposit in region D is worth and a deposit in region A is worth , so banks in region D suffer a loss when cross holdings of deposits are liquidated. The liquidation of region D’s long assets will cause a loss to banks in region C, and this time the accumulated spillover effect large enough that region C too will be bankrupt. As we go from region to region the spillover gets larger and larger, because more regions are in bankruptcy and more losses have accumulated from liquidating the long asset. So once region D goes bankrupt, all the regions go bankrupt. Here we list the “spillover condition”
. (4.7)
The term is the amount promised to the banks in region C, and is the upper bound on the value of deposits in region A when it is liquidated (see equation
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(4.3)). Hence the left-hand side of the condition is the difference between liability and the upper bound on assets in the interbank deposit market for region D. If this exceeds region D’s buffer, the spillover will force banks in region D to be bankrupt.
Here we show that the both of the two effects will be reduced under the introduction of capital requirement. In proposition 2 we state the result of the bankruptcy problem, and in proposition 3 we turn to the spillover effect.
Proposition 2
Supposed the capital requirement is introduced, there will be less possibility for the bankruptcy to happen, that is,
We leave the complete proof in the Appendix B. From equation (4.6), we have the bankruptcy condition . Based on equation, we can calculate the minimum liquidity shock, , that will trigger the bankruptcy in region A. By comparing after and before the introduction of capital requirement, we can check whether the possibility of bankruptcy is getting lower.
That is, we want to testify whether the with the capital requirement, where represents the minimum liquidity shock that will trigger the bankruptcy after the capital requirement is introduced, and represents the one without capital requirement. The results are as below.
(4.8) We reasonably assume that for sufficient large return of the long asset. Obviously, if , the denominator will be smaller and the numerator
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will be larger, which makes the value of the ratio increase. However, we are still uncertain about the sign of the above solution. In order to figure out the net effect of the capital requirement, we take the partial derivative of .
(4.9)
This result is clear. As we raise the capital from the very beginning, the minimum level of liquidity shock that can trigger the bankruptcy in region A becomes larger.
We conclude that, after the regulation, there is lower probability of the bankruptcy in region A, which enhances the financial stability here in our model. This result actually meets our ex ante expectation.
In the next section, we discuss the spillover effect, which is described as the regional liquidity shock in region A to spread out to other regions (such as region D) and lead to the systematic liquidity risk in Allen and Gale’s model. There is a important question that whether the required capital can effectively prevent the spillover from happening, and we analyze this issue based on the condition (4.7).
Proposition 3
Supposed the capital requirement is introduced, the spillover effect will be reduced, that is, .
We remain the complete proof in the Appendix C too. In order to discuss whether the spillover effect has been successfully reduced after the capital requirement is introduced, we defined an indicator “SE”, which defined as the difference between the interbank liability and the bank buffer, for bank D from equation (4.7). If the spillover effect has been reduced, we can reasonably expect SE will
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decrease or even become negative to represent the improvement of the possibility of contagion. We show the result as below.
(4.10)
We reasonably assume that the term
for sufficiently large R to make
the difference positive. Here, is still the sufficient condition for the inequality to hold.
This result simply told us again that the cost of capital plays an important role in our model. Remember that in chapter 3.1 we conclude that if , the consumption in state 1 and 2 both decrease as the required capital increases; at the same time, the short asset investment y decreases while the long asset investment x increases. These properties made the buffer increase while the interbank liability decrease, which lead to lower possibility for the spillover to happen. The key here is that the interbank liability. Bank D owns to the Bank C, while it also has the asset of for the liquidation value of its deposit in Bank A. After the introduction of capital regulation, the liability for to Bank C decrease while the liquidation value might even increase for a sufficiently large r, pushing the entire interbank liability down.
Therefore, we conclude that after the capital regulation, there is less possibility for the financial contagions to happen.