Liquidity Crunch in the Interbank Market: Is it Credit or Liquidity Risk, or Both?

Abstract

The interplay between liquidity and credit risks in the interbank market is analyzed. Banks are hit by idiosyncratic random liquidity shocks. The market may also be hit by bad news at a future date, implying the insolvency of some participants and creating a lemons problem; this may end up with a gridlock of the interbank market at that date. Anticipating such possible contingency, banks currently long of liquidity ask a liquidity premium for lending beyond a short maturity, as a compensation for the risk of being short of liquidity later and being forced to liquidate some illiquid assets. When such premium gets too high, banks currently short of liquidity prefer to borrow short term. The model is able to explain some stylized facts of the 2007–2009 liquidity crunch affecting the money market at the international level: (i) high spreads between interest rates at different maturities; (ii) “flight to overnight” in traded volumes; (iii) ineffectiveness of open market operations, leading the central banks to introduce some relevant innovations into their operational framework.

Appendix: The liquidity management problem

The crucial decision a bank has to take at t = 0 is whether to trade ST or LT in the interbank market. This choice affects her liquidity position and the actions she has to take in the next period. An analysis of this choice is necessary as a preliminary step, in order to understand how the banks’ payoffs shown below are derived, particularly in the proofs of Propositions 3 and 4.

Consider first a bank hit by a positive liquidity shock: she is long of liquidity at t = 0. Suppose she lends ST at this date:

• if the shock is permanent, at t = 1 she is still long of liquidity: she has to roll over her trade (lend again);

• if the shock is transitory, her liquidity position is balanced at t = 1: the deposit outflow (from 1 + x to 1) is funded by the incoming repayment of the interbank loan.

Now suppose that she lends LT at t = 0:

• if the shock is permanent, her position is balanced at t = 1: there is no inflow/outflow of liquidity at this date;

• if the shock is transitory, she is going to be short of liquidity at t = 1: she will have to fund the deposit outflow (from 1 + x to 1) by borrowing in the interbank market.

The same reasoning applies to a bank hit by a negative liquidity shock: she is short of liquidity at t = 0. Suppose she borrows ST at this date:

• if the shock is permanent, she is still short of liquidity at t = 1 : she has to roll over her trade (borrow again);

• if the shock is transitory, her liquidity position is balanced at t = 1: she repays her interbank debt by the deposit inflow (from 1 − x to 1).

Suppose instead that she borrows LT at t = 0:

• if the shock is permanent, her position is balanced at t = 1: there is no inflow/outflow of liquidity at this date;

• if the shock is transitory, she is going to be long of liquidity at t = 1: she will lend out the deposit inflow (from 1 − x to 1).

Table 1 summarizes the above discussion. It can be noted that if a bank trades LT at t = 0 and the shock turns out to be transitory, her liquidity position is reversed at t = 1 and she has to trade in a way opposite to what she did before. In particular, a bank initially hit by a positive liquidity shock turns out to be short of liquidity in the next period. To the contrary, if a bank trades ST at t = 0 and the shock is permanent, she has to roll over her trade at t = 1. Finally, if a bank trades ST at t = 0 and the liquidity shock is transitory, or alternatively if she trades LT and the shock is permanent, her liquidity position is balanced at t = 1: she does not need to take any action at this date.

Table 1 The liquidity management problem

Proof of Proposition 1

Interbank loans are riskless, so lenders’ reservation price is $\underline{R}_{h}=x$. As an alternative to borrowing, a short bank has to liquidate a share $\frac{x}{L_{G}}$ of her loan portfolio, giving up a return $\frac{x}{L_{G}}V_{G}=xl$, with l > 1; so $\overline{R}_{h}=xl$. Hence $\overline{R}_{h}>\underline{R}_{h}$: the feasibility condition is met. Due to competition among lenders, the equilibrium price is R _h  = x. At this price all banks short of liquidity borrow, since liquidating is more costly.□

Proof of Proposition 2

• (A) In a pooling equilibrium a lender in the interbank market is repaid with probability (1 − α), so lenders’ reservation price is given by $(1-\alpha)\underline{R}_{l}=x$. A good bank borrowing in the interbank market has a reservation price $\overline{R}_{l}=xl$. Hence $\overline{R}_{l}\geq \underline{R}_{l}$: the feasibility condition is met. A bad bank never repays an interbank debt, so the expected cost of borrowing in the interbank market is zero, while liquidating would cost $\frac{x}{L_{B} }V_{B}=xl$. Hence she is ready to (promise to) pay any price for borrowing and the feasibility condition is trivially met. Due to competition among lenders, the equilibrium price is $R_{l}=\frac{x}{(1-\alpha)}$. At this price all banks short of liquidity borrow, since liquidating is more costly.

• (B) A good bank short of liquidity has a reservation price $xl< \frac{x}{(1-\alpha)}$. Hence the pooling price—obtained in (A)—is not an equilibrium price, since the feasibility condition fails to hold for good banks. Banks long of liquidity are aware that only bad banks are ready to borrow (at any price) in the interbank market, so they do not lend.□

Proof of Proposition 3

ST market:

ST interbank loans are riskless, so lenders’ reservation price is $\underline{R}_{1}=x$. As an alternative to borrowing, a short bank has to liquidate a share $\frac{x}{L_{0}}$ of her loan portfolio, giving up a return $\frac{x}{L_{0}}V_{0}=xl$, with l > 1; so $\overline{R}_{1}=xl$. Hence $\overline{R}_{1}>\underline{R}_{1}$: the feasibility condition is met. Due to competition among lenders, the equilibrium price is R ₁ = x.

LT market:

LT interbank loans are repaid with probability π, so lenders’ reservation price is given by $\pi \underline{R}_{2}=x$ and borrowers’ reservation price is given by $\pi \overline{R}_{2}=xl$. Hence $\overline{R}_{2}>\underline{R}_{2}$: the feasibility condition is met. Due to competition among lenders, the equilibrium price is $R_{2}=\frac{x}{\pi }$.

At the equilibrium prices R ₁ and R ₂, banks short of liquidity borrow, since liquidating is more costly.

All banks are indifferent between trading ST and LT, since the expectation − as of t = 0 − of their final value is as follows.¹⁹ The expected value of a long bank lending ST is:²⁰

$$ p\left[ V_{0}+(1-k)R_{h}+k(1-\alpha)R_{l}-(1+x)\right] +(1-p)\left(V_{0}-1\right) =V_{0}-1 \label{longST} $$

(4)

The expected value of a long bank lending LT is:²¹

$$ \begin{array}{rll}&& p\left[ V_{0}+\pi R_{2}-(1+x)\right]\\ &&{\kern12pt}+(1-p)\left\{ V_{0}+\pi R_{2}-[(1-k)R_{h}+k(1-\alpha)R_{l}]-1\right\} =V_{0}-1 \label{longLT} \end{array} $$

(5)

The expected value of a short bank borrowing ST is:

$$ p\left\{ V_{0}-[(1-k)R_{h}+k(1-\alpha)R_{l}]-(1-x)\right\} +(1-p)(V_{0}-1)=V_{0}-1 \label{shortST} $$

(6)

The expected value of a short bank borrowing LT is:

$$ \begin{array}{rll} &&p\left[ V_{0}-\pi R_{2}-(1-x)\right]\nonumber\\ &&{\kern12pt}+(1-p)\left\{ V_{0}-\pi R_{2}+[(1-k)R_{h}+k(1-\alpha)R_{l}]-1\right\} =V_{0}-1 \label{shortLT} \end{array} $$

(7)

□

Proof of Proposition 4

ST market:

The same as in Case (A).

LT market:

LT interbank loans are repaid with probability π. Lenders’ reservation price is given by $\pi \underline{R}_{2}-x(1-p)k(l-1)=x$: the expected cost of funding a LT loan by liquidating at t = 1 must be subtracted from the expected return on such a loan. Borrowers’ reservation price is given by $\pi \overline{R}_{2}=xl$. It is easy to check that $\overline{R}_{2}>\underline{R}_{2}$: the feasibility condition is met. Due to competition among lenders, the equilibrium price is $R_{2}=\frac{x}{\pi } \left[ 1+(1-p)k(l-1)\right]$.

Banks long of liquidity are indifferent between lending ST and LT, since the expectation—as of t = 0—of their final value is as follows. The expected value if lending ST is:²²

$$ p\left[ V_{0}+x-(1+x)\right] +(1-p)\left(V_{0}-1\right) =V_{0}-1 \label{BlongST} $$

(8)

The expected value if lending LT is:²³

$$ p\left[ V_{0}+\pi R_{2}-(1+x)\right] +(1-p)\left\{ V_{0}+\pi R_{2}-\left[ (1-k)R_{h}+kxl\right] -1\right\} =V_{0}-1 \label{BlongLT} $$

(9)

□

At the equilibrium prices R ₁ and R ₂, banks short of liquidity borrow, since liquidating is more costly. They prefer to borrow either ST if $p<\frac{1}{2}$, or LT if $p>\frac{1}{2}$ (they are indifferent if $p=\frac{1 }{2}$), since the expectation—as of t = 0—of their final value is as follows. The expected value if borrowing ST is:

$$ p\left\{ V_{0}-\left[ (1-k)R_{h}+kxl\right] -(1-x)\right\} +(1-p)(V_{0}-1)=V_{0}-1-pkx(l-1) \label{BshortST} $$

(10)

The expected value if borrowing LT is:

$$ p\left[ V_{0}-\pi R_{2}-(1-x)\right] +(1-p)\left[ V_{0}-\pi R_{2}+x-1\right] =V_{0}-1-(1-p)kx(l-1) \label{BshortLT} $$

(11)