Contagious Failure

In the previous section we have shown that liquidity shocks in statesS₁andS₂create incentives for banks to form connections by exchanging interbank deposits at date 0. However, interbank deposit exchange works well only if there is no excess demand for liquidity in the financial system as a whole. If there are non-insurable idiosyncratic shock in statesS~₁,...,S~_n, the cross holdings of interbank deposits can cause problems of contagion. That is, the shock may affect initially only one institution and then

spread to other bank(s) through interbank deposit exchange. In order to evaluate the contagion risk we introduce the concept of loss given default (LDG) which is the loss of value a bank incurs on its interbank deposits when its counterpart fails at date 1. We illustrate the process of contagious failure using examples 1 and 3.

In example 1, Bank A connects with Bank B and the amount of interbank deposit exchange is . Suppose that BankAsuffers from idiosyncratic shock at date 1, it then faces an excessive liquidity demand of. Bank A’s liquidity demand is thus(q), whereas other banks face a liquidity demand ofq only. BankAhas liquid asset which is worth(q), and it can use amount of liquid assets in the form of Bank B’s claims and liquidate amount of its interbank claims on Bank B. However, since Bank B also has the liquidity need of q, it needs to liquidate the interbank deposit from BankA33. We then have:

           q q ) (

where the right-hand side is Bank A’s liquidity demand, which is less than the liquidity supply in the left-hand side. Bank A cannot meet the liquidity need and the long asset fails. Given that interbank deposits are subordinated debt relative to consumers’ deposits, Bank A has to satisfy consumers’ deposit withdraw of(1q) before meeting the interbank deposit withdrawal. Since(1q) , Bank B then suffers from a loss given default of . To meet the liquidity deficit, Bank B has to liquidate an amount of the long asset that equals . Liquidating the illiquid asset prematurely, however, incurs the penalty rater1. The maximum amount of a long

33 _{Note that bank finds it optimal to liquidate the interbank deposits before liquidating the long asset,} because the return of interbank deposit is one at date 1 and 2, whereas the return of long asset israt date 1 andRat date 2. SinceRr_{; the pecking order of asset liquidation is liquidating short asset first,} liquidating interbank deposit second and liquidating long asset last, see also Allen and Gale (2000).

67 asset that can be liquidated without causing bankruptcy is that Bank B has to meet consumers’ deposit withdrawals(1q)at date 2. We then have:

        R q R r CB ( 1)(1 ) (4)

whereCBdenotes Banks’ capital buffer which depends on the rate of early liquidation of long assetr, the amount of consumer’s deposit withdraw at date 2 and the return of a long asset at date 2. If the capital buffer is less than the loss given default, i.e.

 

CB , Bank B fails due to contagion. In general, bank i suffers from contagious failure as long as

i

LGD

CB  (5)

With regard to example 3, if BankA suffers from idiosyncratic shock at date 1, by the same argument, Bank A cannot meet the liquidity need and the long asset fails. BankBthen suffers from a loss given default of . Hence, to meet its liquidity deficit, BankB has to liquidate its interbank deposit2from Bank C. However, Bank Calso needs to liquidate the interbank deposit of2from BankB.We then have:

 

) 2 2

(q  q

where the left-hand side is Bank B’s liquidity supply and the right-hand side its liquidity demand. Bank B faces an excess liquidity demand of . Again, since the capital buffer is less than the loss given default, BankBfails. It has to liquidate all its long assets at date 1. BankB’s liquidity supply consist of early liquidation of its long assets at date 1r(1q), short assets(q), and interbank deposits,2, from BankC. Its liquidity demand is 1 from consumer deposit withdrawal and2from Bank C’s interbank deposit withdraw. Ifris small, we then have:

1 ) ( ) 1 ( q  q  r (6)

Under this condition, Bank C suffers from a loss given default of2. Bank C’s liquidity supply is the sum of own short asset(q), interbank deposit , and its interbank deposit in BankD,subtracting the loss given default2.

  

) 2

(q   

The total liquidity supply for Bank C is then(q), which consists of its liquidity demandqfrom consumers’ deposits withdraw and from Bank D’s interbank deposit withdraw. BankChas to liquidate an amount of long asset equal to2. SinceCB , BankCthen fails. Its total liquidity supply is then

) ( ) 1 ( q  q r

which is less than 1 given condition 6. Bank Dthus suffers from a loss given default of , which also fails given that the same argument as described above applies.