ECON7003 Money and Banking. Hugh Goodacre. Lectures 1-2. BANK RUNS Bank deposits and uncertain liquidity demand. The Diamond and Dybvig 1983 model, Spencer,

ECON7003 Money and Banking. Hugh Goodacre.Lectures 1-2.

BANK RUNS

Bank deposits and uncertain liquidity demand.

The Diamond and Dybvig 1983 model, Spencer, ch. 10 version.

1. Trading risk in a two-individual ‘society’.

2. The bank deposit contract.

Preview:

3. Measures to prevent bank runs.

Withdrawal of deposits on demand normally no problem, despite:

low deposit : asset ratio

high gearing in bank sector

(a) Scale economies:

→ withdrawal demands unlikely to be correlated.

For banking system as a whole, likely to be inversely correlated:

Debits-credits net out!

(b) Tradable money market instruments:

e.g. Certificates of Deposit (CDs).

To meet fluctuations in liquidity needs.

These advantages are basic to bank’s profit through intermediation:

i.e. Asset transformation:

Short-term / instantly withdrawable deposits → long-term / illiquid assets

‘Maturity transformation’

Small-size deposits → large-size assets:‘Size transformation’

Low-risk instrument, i.e. deposit, → high-risk.:‘Risk transformation’

In each case:Interest on asset > interest on liability → bank profit.

BUT:

Loss of confidence in bank

→ withdrawals not motivated by ‘genuine’ liquidity requirement / transactions motive.

May be contagious and → panic.

In panic, those at end of queue may not be paid in full:

Even if bank is solvent and all its assets are liquidated

Costs of liquidation

Loss of:

customer relationships

confidential information, etc.

i.e. Destruction of ‘informational capital’ / intangible assets.

Inevitably undervalued in ‘fire sale’ conditions.

→ Net value > 0 when functioning

may → < 0 if sold off hurriedly.

Asymmetric information problem facing bank:

Bank unable to distinguish between:• withdrawals for ‘genuine’ / transactions purposes• withdrawals through panic

→ cannot pay ‘in sequence’:

Gain time → avoid fire sale

→ liquidate assets at better price.

3-period model of bank runs and measures to prevent them.

Assumption:Bank liabilities all consist of deposits withdrawable on demand.

Each individual has a primary investment of 1 in period 0yields 1 if liquidated and consumed in period 1yields R > 1 if liquidated and consumed in period 2.

i.e. R ≡ 1 + r

Individuals are of 2 types:

Type 1s ‘die’ in period 1having first liquidated their investment and consumed its entire value.

Type 2s survive period 1 but ‘die’ in period 2having by that time liquidated their investment and consumed its entire value.

The overall proportion (p) of type 1s is publicly knownin period 0

i.e. There is no aggregate uncertainy.

but individuals do not find out which type they are until period 1, and this information is private.

i.e. There is individual uncertainy.

i.e. Requirement for liquidation of investment in period 1 drives the demand for liquidity.

‘Cost of early death’ is R – 1.

Because R > 1, type 2s optimally set C1 = 0.

Individual’s expected utility E [U] in period 0:

E [U] = p.U(C11 + C2

1) + (1 – p).U(C12 + C2

2)

Type 1s: Expectation of a constant is a constant →

E[C11] = C1

1 = 1

E[C21] = C2

1 = 0 Type 2s: Expectation that they optimise →

E[C12] = 0

E[C22] = R

→ Substituting:

E [U] = p.U(1 + 0) + (1 – p).U(0 + R)

→ E [U] = p.U(1) + (1 – p).U(R)

‘Society’ of two individuals where p = ½

Learning own type ≡ revelation of type of other !

i.e. Full ‘state verification’ / no informational asymmetry.

→ Socially optimal risk-sharing contract possible in period 0:Type 2 will pay fixed sum (π) to type 1 in period 1.

→ Individual 1 consumes C1 = 1 + π in period 1.

Individual 2 consumes C2 = R(1 – π) in period 2.

Only requirement: Mechanism for enforcing contract.

Deriving optimal scale of transfer (π):

We need to find the value of π which maximises total social utility (SU) ≡ U(C1) + U(C2)

Express period 2 budget constraint i.t.o. C1:

C1 = 1 + π

→ π = C1 - 1

Substituting into C2 = R(1 – π) we have:

C2 = R[1 – (C1 – 1)]

= R(2 – C1) = 2R – RC1

Substituting into expression for total social utility, we have:

SU = U(C1) + U(2R – RC1)

Differentiating SU and setting to zero to maximise, we have:

SU = U(C1) + U(2R – RC1)

→ dSU / dC1 = MU1 – R.MU2 = 0

→ MU1 / MU2 = R = 1 + r

i.e. MRS (in consumption) = MRT (through investment)

We define the values which solve these equations as:

C1*, C2*, and π*

C2

2 C1

2R ← Vertical intercept:Period 2 social budget constraint:

C2 = R(2 – C1)Solving for C1 = 0:

C2 = 2R

Horizontal intercept:Maximum possible consumption by both types (‘social’ consumption) is 2. ↓

Social budget line

C2

R

2 C1

1

2R

Allocation point under autarchy / no trading of risk

i.e. Social level of consumption under autarchy is:

1 + R

A

C2

2 C1

2R

450

450 line indicates complete absence of risk between ‘states’ / outcomes

C2

R

2 C1

1

2R

450

With trading in risk / contract to pay π, ‘social IC’ reaches tangency with BC at A'

A' is closer to the 450 line, indicating a reduction in risk

With no trading in risk, ‘social indifference curve’ cuts BC at A

A'

A

It is on a higher social IC curve, showing that trading in risk results in a socially preferable outcome to autarchy.

C2

C2*

2

A

C1

1

2R

450

A'Rπ*

π*

At A', individual 1 consumes C1*

due to receiving π*

At A', individual 2 consumes C2*

due to loss of Rπ*

C1*

R

Note: C2* > C1*

BUT:

Society of more than two individuals:

Information on own type remains private in period 1:

→ life expectancy and liquidity requirements no longer publicly revealed.

→ asymmetric information problem in designing contract for trading risk.

An intermediary / bank now offers a deposit contract capable of achieving same degree of insurance as in the two-individual case.

i.e. :

All type 1s will consume C1* = 1 + π in period 1.

All type 2s will consume C2* = R(1 – π) in period 2.

C2* > C1* → type 2s still have motive to set C1 = 0

BUT: Bank can only fulfil this contract if only type 1s withdraw their deposits in period 1.

i.e. for ‘genuine’ liquidity requirement.

Fragility of this result:

In period 1 liabilities > assets

→ bank relies on type 2s not withdrawing.

Period 1 liabilities > assets:

Recall the assumption: All the bank’s assets / funds are sourced from its depositors.

Let there be N depositors, then the funds available to the bank for distribution to depositors in period 1 are:

N.1 = N

The bank’s liabilities to depositors in period 1 are: N.C1*

And N.C1* > N !

Let p = ½

‘Good’ outcome period 1:

Type 2s will optimise by setting C12 = 0

Only type 1s withdraw deposits in period 1.

→ Liquidity demand in period 1 is:

pNC1* + (1 – p)N.0

= ½NC1* < N

i.e. Bank’s liabilities do not exceed its assets.All deposit withdrawal demands can be met.

‘Bad’ outcome period 1:

Type 2s fear a bank run / begin to withdraw deposits in period 1.

If all do so (‘bank panic’), type 2 liquidity demand in period 1 is:

(1-p).NC1* = ½NC1*.

→ Total liquidity demand:

½NC1* + ½NC1*

= NC1* > N

i.e. Bank’s assets insufficient to meet liabilities.Some depositors get 0.

Deposit : liability ratio of banks in period 1:

N : N.C1*

i.e. 1 : C1*

Assumption: No deposit insurance arrangements are in place.

Maximum proportion of depositors who can withdraw their deposits in period 1 in the presence of a run:

Deposits divided by liabilities:

N / NC*

i. e. deposits : liabilities ratio (1 : C1*) expressed as a fraction:

f = 1 / C1*

C1* > 1

→ f < 1

Fraction of depositors who get nothing through being last in the queue:

1 – f= 1 - 1 / C1*

= (C1* - 1) / C1*

We have: C1* = 1 + π

Substituting: 1 – f = (1 + π – 1) / C1* = π / C1*

i.e. Fraction who receive nothing is π / C1*

i.e. Intermediation / bank deposits offer solution to informational problems of trading in risk of early death.

BUT

That solution is not robust to fear of bank’s insolvency:

Such fear may → self-fulfilling prophecy / fear becomes general (‘panic’).

‘Sequential service constraint’ / bank cannot meet all withdrawal demands / ‘last in queue’ get nothing.

Expectations of run may → actual run, with no change in fundamentals.

Banks are inherently ‘fragile’.

If fear is contagious, may threaten whole banking system.

Preview: The ‘good’ and ‘bad’ outcomes will be defined as Nash equilibria.

Measures to prevent bank runs.

Influence expectations / provide confidence.

Make ‘good’ Nash equilibrium unique.

3 possible solutions:

Action by banks themselves:Suspend convertibility

Government actions:Government-backed deposit insuranceLender of last resort facility