School of Economics, Mathematics and Statistics
Birkbeck College
MSc Finance, MSc Economics
Topics in Finance
Lecture 6: Topics in Banking
1
Bank Runs
The Liquidity Insurance Model
• One good economy, with three periods, t = 0, 1, 2. Continuum of
ex-ante identical individuals, each endowed with one unit of good in
period 0, with which they finance consumption (C1 , C2 ) in periods 1
and 2.
• At t = 1, a fraction π1 of the population finds out that they want to
consume early: the utility function of this ‘type-1’ individual is u(C1 ).
The rest find out that they would like to consume late: the utility
function for ‘type-2’ is ρu(C2 ). Here ρ < 1 is the discount factor. In
ex-ante terms, the consumer’s expected utility can be written as
U (.) = π1 u(C1 ) + π2 ρu(C2 ),
where πi is the probability of being of type-i. The function u(.) is
increasing and concave.
• There are two technologies for transforming goods. Simple storage
returns one unit for every unit stored. An investment technology that
returns R > 1 after two periods or L < 1 after one period. Thus there
is a penalty for premature liquidation.
Autarky: No trade
Each individual chooses investment I independently of each other. Then
C1 (I) = 1 − I + LI ≤ 1,
while
C2 (I) = 1 − I + RI ≤ R,
with one of these inequalities being strict.
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Financial Markets
Suppose consumers can trade in period 1. Assume there is a riskless bond:
the price of the bond in period 1 is p and it pays one unit of good in period
2. Now, by investing I in period 0, the consumer gets
C1 (I) = 1 − I + pRI
which is possible by selling RI bonds. If she needs to consume late, she can
buy bonds with her liquid assets: she has 1 − I which will allow her to buy
1−I
p bonds: each bond gives her one unit of consumption in period 2.
C2 (I) =
1−I
+ RI
p
Market clearing in the bond market requires that p = R1 . Evaluating C1 and
C2 at this value, we get C1M = 1 and C2M = R, which Pareto dominates the
autarky allocation.
Pareto Optimal Allocation
We can find the Pareto optimal allocation by
max π1 u(C1 ) + ρπ2 u(C2 )
subject to the aggregate budget constraint π1 C1 + π2 CR2 = 1. The optimal
allocation (C1∗ , C2∗ ) satisfies the first-order condition,
u0 (C1∗ ) = ρRu0 (C2∗ ).
Fractional Reserve Banking and Its Stability
It may be possible to achieve the Pareto optimal outcome through a financial
intermediary. Suppose everyone deposits their endowment with a bank that
promises to pay C1∗ at time t = 1 to all type-1 individuals and C2∗ at time
t = 2 to all type-2 individuals. To meet these payments, the bank keeps
π1 C1∗ – a fraction of the deposits – in storage and invests the rest in the
long-term illiquid project. This could be interpreted as demand deposit
system based on fractional reserve banking.
Is this fractional reserve system stable? If the bank cannot observe the
consumers’ type, the answer depends on the behaviour and expectations of
individuals, especially the type-2 individuals. We consider two cases.
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Consider the case where C1∗ > C2∗ . (You can check, from the above
optimality condition, that this obtains whenever ρR < 1). Here type-2
individuals will want to withdraw C1∗ at 1 and store it themselves for consumption in period 2, as this improves on getting C2∗ from the bank in
period 2. In other words, type-2 individuals will pretend they are type 1.
In other words, everyone will want to withdraw their deposits in period 1.
As (C1∗ , C2∗ ) were set on the assumption that only a fraction π1 will withdraw early, the bank cannot meet these demands. In other words, fractional
reserve banking cannot work for this case.
Next, consider the case where C1∗ ≤ C2∗ . (This obtains if ρR ≥ 1). Here
there are two sub-possibilities.
• One, if every type-2 individual expects the bank to honour its commitments, they will be willing to wait. Only type-1 individuals will
withdraw so the proportion of population withdrawing is π1 . As long
as the bank holds π1 C1∗ in liquid assets, it will meet demand for withdrawals.
• Two, suppose one type-2 individual worries that other type-2 individuals may withdraw early. If so, the bank will be forced to liquidate its
long-term assets to meet this demand. In extreme cases, it may not
be able to meet the entire demand for withdrawals and will fail. If so,
it makes sense for this individual to withdraw too.
Thus there are two Nash equilibria: one in which all type-2 consumers
are willing to wait till period 2, and the second in which all depositors, type
1 and type 2, withdraw at t = 1. This second case can be described as a
bank run. Note that a bank run is inefficient.
Reading: see Freixas and Rochet, pages 191-197.
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2
The Lender-Borrower Relationship
2.1
Optimal Risk Sharing with Symmetric Information
• At date 0, a borrower borrows L from a lender to invest in a project.
The project is risky: it has a random return ỹ in period 1. Both lender
and borrow are risk-averse and care for consumption only in period 1.
Their utility functions are given by uL and uB , respectively.
• Suppose the return of the project is observable to both borrower and
lender (i.e., information is symmetric). The lender and borrower can
then set up a contract describing how the random return ỹ is to be
shared between them. Consider a repayment function R(y) that describes how much the lender receives when the realization of return is
y. If so, the borrower gets to keep y − R(y). It is reasonable to require
that
0 ≤ R(y) ≤ y
• Optimal risk sharing involves solving the following problem: maximize
the expected utility of the borrower subject to repaying the lender
enough that he is willing to lend. In other words, choose a repayment
function R(y) to
max EuB (ỹ − R(ỹ))
such that EuL (R(ỹ)) ≥ UL0
0 ≤ R(y) ≤ y,
where UL0 is the lender’s reservation utility.
• When the limited liability conditions do not bind, the optimal contract
involves optimal risk sharing
u0B (y − R(y))
= µ,
u0L (R(y))
where µ is a constant. Taking the log of this and differentiating we
get
u00B
u00L
0
(y
−
R(y))(1
−
R
(y))
−
(R(y))R0 (y) = 0
u0B
u0L
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• If we write
u00B
(x) = IB (x)
u0B
u00
− 0L (x) = IL (x)
uL
−
the previous relation can be re-written as
R0 (y) =
IB (y − R(y))
IB (y − R(y)) + IL (R(y))
Note that this result holds only if the limited liability constraints do
not bind. This implies that the sensitivity of the repayment (i.e., the
derivative of R(y)) is higher when the borrower is more risk averse
than the lender. This theoretical desideratum does not fit easily with
what we know of banking. In practice banks are typically not as risk
averse as borrowers, yet the risk in a typical bank loan is borne by the
borrower. In fact, the standard debt contract requires the borrower to
repay a fixed amount regardless of ỹ. So that we have R(y) = R for
all y. More accurately, once we factor in limited liability, the standard
debt contract takes the form
R(y) = min[R, y].
• In sum, commonly-observed debt contracts do not match up with optimal risk-sharing arrangements in the presence of symmetric information. One way to proceed is to allow the possibility that information
is not symmetric. In particular, we assume that the borrower can
observe ỹ but the lender cannot.
Reading: Freixas and Rochet, pages 92-94.
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2.2
Efficient Incentive-Compatible Contracts with Asymmetric Information
Suppose information is not symmetric: the borrower can observe ỹ readily,
but the lender can observe that value only through a costly audit. The lender
would like to minimise the need for costly audits, so would like to devise
contracts in which borrowers find it in their interest to report y truthfully.
Such a contract has the following structure.
1. The borrower reports a value yb for the return.
2. A repayment function R(b
y ) : specifies the repayment from borrower to
lender when borrower reports return to be yb.
3. An auditing rule: identifies a set S of reports (region of values of yb)
for which the lender audits the reported.
4. Penalty or reward function, P (y, yb), that specifies additional transfers
in case of an audit. We normalize payments so that the reward for
truthful reporting is zero.
An optimal contract must provide incentives to the borrower to be truthful, and be efficient in the sense of minimizing audit costs. How do we
understand what an optimal contract looks like?
• If penalty can be large, then it does not make sense for the borrower
to be untruthful when reporting anything in the audit region.
• Within the no-audit region, the payment R(b
y ) must be invariant: otherwise, borrower would always want to pick the value that is the lowest.
Let R be this constant value.
• R must be larger that all possible repayments in the audit region. If
not, there will be some value in the audit region where the borrower
is better off by reporting a value in the no-audit region.
In sum, a debt contract is incentive compatible if and only if there
exists a constant R such that
R(y) = R for all y ∈
/S
≤ R for all y ∈ S
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• Assume the borrower and lender are risk neutral. An incentive compatible contract is efficient if and only if it maximizes expected repayment for a given probability of audit. Such a contract must recover the
maximum amount in the event of an audit, and given limited liability
that maximum is just y. In other words, an audit takes place whenever
the repayment is less than R, and
R(y) = R for all y ∈
/S
= y for all y ∈ S
This can be interpreted as a standard debt contract.
Thus, a standard debt contract can be considered as an efficient incentivecompatible contract in a model of ‘costly state-verification’.
Reading: Freixas and Rochet, pages 95-96.
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2.3
Incentives to Repay: The Threat of Termination
If audits are not possible, the incentives for repayment may come from the
threat of termination.
• Borrower has a technology that converts loan L into ỹ. The technology
can be used repeatedly and the return in various periods is independently identically distributed (iid). Assume that ỹ can take one of two
values: yH with probability pH and or yL with probability pL = 1−pH .
Assume that yL < L, but that the expected value E(ỹ) > L.
• In a one-short relationship there would be no lending as the lender can
always pretend that ỹ = yL , causing the bank to lose money.
• A two-period relationship may involve some lending, if the bank can
commit to renew lending in the second period only if the lender repays
the agreed amount R at the end of period 1. Note that this repayment
will happen only if ỹ1 = yH . At the period of period 2, we would be
in the same situation as the one-shot relationship, so that the lender
will repay only yL . The bank will definitely lose money in period 2
but may still lend if expected profits from lending in the first period
are high enough. Ignoring any discounting, the expected profits of the
bank are
π=
−L
|{z}
initial outflow
+
p y +p R
| L L {z H }
+
exp. repayment in period 1
pH (−L + yL )
|
{z
}
exp. return from period 2 lending
which can be written as
π = −L + yL + pH (R − L)
• The bank would be willing to lend if π ≥ 0, which requires that
pH R ≥ L − yL + pH L.
(1)
• The borrower would be willing to repay in period 1 whenever she is
better off doing so:
−R + pH yH + pL yL
|{z}
|
{z
}
repayment
−yL
|{z}
exp. return in period 2
repayment in period 2
≥
−yL
|{z}
payment in period 1
or, equivalently, if expected return on the project
E(ỹ) = pH yH + pL yL ≥ R.
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(2)
• We combine the two requirements (1) and (2). Multiplying (2) by pH
we get
pH E(ỹ) ≥ pH R,
which can be combined with (1) to get
pH E(ỹ) ≥ L − yL + pH L,
or simplifying
pH [E(ỹ) − L] ≥ L − yL .
Reading: Freixas and Rochet, pages 101-102.
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2.4
Sovereign Debt
• Sovereign country makes an investment in a technology f (L), funded
by a foreign loan L contracted at interest rate r. The one-period profit
of the country is
π = f (L) − (1 + r)L
The country’s demand for capital LD is given by the condition that
the marginal rate of return on capital equal the price of that capital
f 0 (LD ) = 1 + r
• A sovereign country can default on its loan at any time. Assume that a
defaulting country will forever be excluded from capital markets: can
never borrow again. Assume that a defaulting country cannot store
any investment from one period to the next.
• Opportunity cost of default is the value of lost profits: assuming an
infinite horizon and a discount factor β < 1, this is given by
V (L) =
∞
X
β t [f (L) − (1 + r)L] =
t=1
β
[f (L) − (1 + r)L]
1−β
• For a country to have the incentive to repay, the repayment has to be
less than the opportunity cost of defaulting:
(1 + r)L ≤ V (L)
or
(1 + r)L < βf (L)
b such that
• Assuming decreasing returns to scale, there exists some L
the inequality holds for L ≤ L.
• Whether the optimal loan LD is feasible depends on whether β is large
enough.
Reading: Freixas and Rochet, pages 102-104.
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2.5
Collateral
• Agents borrow from a lender to invest. The investment can fail (ỹ = 0)
with probability θ or succeed (ỹ = y). The probability of failure differs
across agents. There are two types, low-risk types (θ = θL ) and high
risk types (θ = θH ), with θH > θL . Borrowers know their own types.
The lender cannot observe types but knows their proportions in the
population. All agents are assumed to be risk-neutral.
• A loan contract is given (R, C) where R is repayment and C is collateral. If the project succeeds, the borrower repays R and keeps y − R.
If the project fails, the lender liquidate the collateral. If so, the borrower loses C but the lender gets only a fraction δC, with δ < 1. The
‘efficiency cost’ of liquidation is (1 − δ)C.
• Consider an arbitrary contract (Rk , C k ). Its payoff to the lender is
(1 − θk )Rk + θk C k δ.
The payoff to the borrower is
(1 − θk )(y − Rk ) − θk C k .
The borrower will accept this contract as long as the payoff exceeds
his reservation utility U k . Assume, for simplicity that all borrowers
have the same reservation utility U, and that all bargaining power lies
with the lender who will drive each borrower to the point where the
borrower is indifferent between the contract and his outside option U.
(1 − θk )(y − Rk ) − θk C k = U
If the lender knows the borrower’s type, it is efficient to have contracts
with no collateral, as collateral costs θk C k to the borrower and yields
only θk C k δ to the lender. In this case, the optimal contract will have
(1 − θk )(y − Rk ) = U,
U
or Rk = y − (1−θ
Thus, if the lender could distinguish between
k) .
the two types, it would offer each a contract with no collateral, but
different repayment levels
RH
RL
U
(1 − θH )
U
= y−
(1 − θL )
= y−
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and C H = C L = 0. Note that as θL < θH , we have RL < RH : a
contract that the low risk-type finds just acceptable will have a lower
repayment requirement than a contract for the high-risk types.
• What happens if you cannot observe the types and must offer a common contract to each? Consider, first, a contract with no collateral.
High risk types will accept any contract that has R ≤ RH but low risk
types will accept contracts only with R ≤ RL . Recall that RL < RH .
For a common contract to be acceptable to both types, the highest
possible R that can be charged is RL . In other words, the high risk
types will derive some ‘informational rent’.
• It may be better to offer two contracts, one with (high interest rate,
zero collateral); and the second with (lower interest rate, some collateral). The idea is that high risk types will select the former contract
and the low risk types will choose the latter. In particular, consider
the contracts (RH , 0) and (R, C) where RH is as define above and R
and C are to be determined. For high-risk types to prefer (RH , 0) to
(R, C), it must be
(1 − θH )(y − RH ) ≥ (1 − θH )(y − R) − θH C.
(3)
For low risk types to prefer (R, C) to (RH , 0), we must have
(1 − θL )(y − R) − θL C ≥ (1 − θL )(y − RH ).
(4)
In addition, the contracts should be acceptable to the two types
(1 − θH )(y − RH ) ≥ U.
(5)
(1 − θL )(y − R) − θL C ≥ U.
(6)
Are there values of R and C that satisfy these conditions? Note first,
that by construction RH is such that (RH , 0) is just acceptable to the highrisk types so is (5) satisfied. Next, recall that, by construction, RL is such
that (RL , 0) provides utility U to the low-risk type. Given that RH > RL ,
the contract (RH , 0) must provide less than U to the low-risk types. Thus
any contract (R, C) that satisfies (6) must automatically satisfy (4).Thus
the essential relations are (3) and (6).
Reading: Freixas and Rochet, pages 119-121.
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