Financial Intermediation

1
Financial Intermediation

• Lecture 7
• Debt Contracts and Credit Rationing

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Contents

• In this lecture we switch from symmetric to
  asymmetric information
• Information asymmetry adverse selection,
  probably the most well-known problem example
  from an insurance perspective
• Incentive Compatible insurance contracts
• Standard debt contracts based on verification
  problems
• Credit rationing the well-known Stiglitz-Weiss
  model

3
Information asymmetry

• Ex ante adverse selection problem of price
  increases increases riskiness of the pool of
  applicants
• During moral hazard. Change of behavior
• Ex post costly state verification

4
Adverse selection in insurance markets market
failure

• Risk-neutral insurer provides insurance
  contracts at a price ? and coverage d to
  applicants who are indistinguishable from one
  another
• Applicants (with wealth W) face a particular loss
  L
• One group, with size ?, has a high probability ?H
  on the loss. One group (size 1 - ?) has a low
  probability ?L
• Two states in state 1 loss occurs
• The fact that insurance can be bought at a price
  ? is equivalent to trading state-contingent
  consumption contracts at a relative price ?/(1-?)

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State-contingent outcomes

• Consumption (x1,x2) x1 W L d - ?d, x2 W
  - ?d
• Insurers profit (y1,y2) y1 -d ?d W L -
  x1, y2 ?d W - x2
• So y2 (? - 1)/? y1 (see the relative price (? -
  1)/? of trading state-contingent consumption) and
• x2 (W - ?L)/(1 - ?) - ?/(1 - ?) x1 these
  combinations of consumption are offered by the
  insurance company at a given premium ?
• The insurer cannot distinguish between the types
  of consumers and bases the insurance supply
  decision on an estimate ? of the loss probability

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Feasible consumption sets
x2
A is initial endowment point
Consumption
450
W
-?/(1-?)
A
Higher iso-profit lines to the origin
Note that ? lt ?
Iso-profit line through endowment point profits
are 0 see hereafter
-?/(1-?)
0
x1
W - L
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Expected profits for the insurer

• So ? can be seen as the probability of state 1,
  the case of a loss. Profit of the insurer is
  equal to G ? y1 (1 - ?) y2
• Iso-profit lines x2 (W - ? L - G)/(1 - ?)
  ?/(1 - ?) x1
• What is the quantity of insurance D offered (note
  that D differs from the demand for coverage d)?
  Maximize ? (? - 1) D (1 - ?)
  ?D (? - ?) D
• So for ? ?, any D is OK (typically not the case
  in the previous figure), but for ? lt ?, D 0
  (iso-profit lines closer to the origin) and for ?
  gt ? there is no solution
• In equilibrium the price of insurance ? must
  equal the probability of a loss as assessed by
  the insurer ?
• Note that in the previous figure ? lt ? holds, so
  D 0

8
We turn to the consumer group probabilities

• Consumer maximizes expected utility
  ?i u(W L (1 - ?)di) (1 - ?i) u(W - ?di)
  with respect to di, i H, L
• The budget restriction ?x1 (1 - ?)x2 W - ?L
  (use the expressions for x1 and x2 and solve for
  d)
• The equilibrium premium rate, based on a weighted
  average, is in between of the high and low ? ?H
  gt ? ?H (1 - ?) ?L gt ?L
• All high risk clients want full insurance, while
  low risk clients want (limited) insurance (this
  is what we will show hereafter)

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Utility and profit lines
450
x2
Higher utility
Full insurance on the 450-line
x2 lt x1 is not in the interest of the insurer
Higher profits
Iso-profit-line
Indifference-curve
0
x1
10
Insurance pool
Partial insurance
x2
450
? ?L
W
Low-risk consumers have a stronger preference
for x2
? ?
? ?H
Consumers want full insurance d L for ? lt ?
x1
0
W - L
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With the Figure

• Note that the indifference curve for H-consumers
  are steeper than the curve of a low-risk
  consumer
• At ? ? all high risk people want insurance,
  since ? lt ?H. The premium does not exceed the
  loss likelihood. Low risk consumers will not
  insure fully, but can do it partially
• The insurer probably makes a loss on high-risk
  applicants, maybe compensated by profits on
  low-risk clients
• Low-risk clients maybe insure incompletely at a
  price ? between ? and ?H

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Consequences

• There are too many assets of the lowest quality
  traded
• It is better to offer two contracts for both
  classes
• We need self-selection of customers
• Need to think about contract design what type of
  contract to be offered to high- and low-risk
  clients?

13
Optimal insurance contract

• The contract of course must maximize the
  objectives of the insurer,
• given that allocations are feasible,
• and must be individually rational, and
• must be incentive compatible there should be no
  incentives for any agent to misrepresent her
  private information

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Optimal contract under symmetric information
x2
A optimal this solution varies per client
W
A
-?/(1 - ?) -?/(1 - ?)
0
x1
W - L
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Revelation principle

• Any IC-contract can be obtained through a
  truth-telling contract
• Direct mechanism static Bayesian game in which
  each players single action is to submit a claim
  about his or her type.
• If telling the truth is a Nash equilibrium a
  direct mechanism is called incentive compatible
• Revelation principle a Nash equilibrium of any
  Bayesian game can be represented by an Incentive
  Compatible direct mechanism

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Optimal contract design with asymmetric
information

• We have to choose consumption sets for high risk
  (x1H, x2H) and low risk clients (x1L, x2L)
  to maximize expected profits
• ? ?H (W L - x1H) (1 - ?H)(W - x2H)
  (1 - ?) ?L(W L - x1L) (1 - ?L)(W -
  x2L)
• Subject to both individual rationality (IR) and
  Incentive compatibility (IC) constraints for both
  high and low risk clients

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IR and IC constraints

• ?H u(x1H) (1 - ?H) u(x2H) ? ?H u(W - L) (1 -
  ?H) u(W) Individual Rationality for H, and for L
• ?L u(x1L) (1 - ?L) u(x2L) ? ?Lu(W - L) (1 -
  ?L) u(W)
• ?H u(x1H) (1 - ?H) u(x2H) ? ?H u(x1L) (1 -
  ?H) u(x2L) Incentive Compatibility for H, and for
  L
• ?L u(x1L) (1 - ?L) u(x2L) ? ?L u(x1H) (1 -
  ?L) u(x2H)
• There are larger or equal signs, since
  indifference satisfies individual rationality and
  incentive compatibility

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How to explain rationality and incentive
compatibility?

• Graphical illustration for the allocations
• We have to look at utility and at profit lines
• What combinations are feasible?
• In any case is x2 lt x1 not feasible it is not in
  the interest of the insurer. x1 is the risky
  case!

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Step 1 rationality
x2
A to B expected profit of the insurer increases
we go to profit lines closer to the origin
A
B
Iso-profit line
Indifference curve
x1
0
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Intuition for Incentive Compatibility
x2
Potential area for low
Better for both
IRL
Worse for both
Potential area for high
IRH
0
x1
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Step 2 Incentive compatibility
For a given high-risk contract B, a low-risk
contract A must be above the indifference
curve of the low-risk through B ICL Idem for
the low-risk B above ICH
x2
A
B
ICL
ICH
x1
0
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Step 3 Given the low risk contract A optimal
high risk case
x2
E
A low risk contract C high risk contract
Implication high-risk contract will be
fully insured!
C
A
Range ABDCE gives more profitable contracts than
C, given A
D
Note that the IC-constraints are not binding at
first
B
ICH
0
x1
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Step 4 Given the high-risk C what is the
optimal low risk case?
x2
Given C, all contracts EBA are rational to
low-risk consumers and more profitable to the
insurer
E
W
A
EA contracts provided at the same price
B
ICL
C
ICH
0
x1
W - L
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Finally feasible contract
x2
Example of feasible contract full insurance for
high and partial for low risk
(x1L, x2L)
W
ICL
(x1H, x2H)
ICH
0
x1
W - L
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A Standard Debt Contract definitions

• Firms have investment projects, which require one
  unit of capital (no internal wealth) and produce
  a state-dependent return f(s), for all s ? S. We
  assume that f(s) is monotonically increasing
• p(s) is the probability of state s occurring
• Probability of an event E is P(E) ?s?E p(s)
• Lenders have an outside opportunity cost I 1
  i per unit of capital lending rate must exceed
  I. Monitoring a state requires a fixed cost ?.

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A lending contract

• A contract (?,?) is a pair of functions, such
  that
• ?(s) determines the state-dependent pay-off
• ?(s) ?0,1 specifies the states where monitoring
  takes place.
• S0(?) s ? S ?(s) 0 and S1(?) S - S0(?)
• We assume 0 ? ?(s) ? f(s) - ?.?(s) firms have
  no capital of their own and there are no other
  funds available to repay the loan. Costs of
  monitoring are paid by the firm

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Monitoring

• The firm reports a state sr. The lender can
  monitor the report. After monitoring, the true
  state will be established. The firm has an
  incentive to report a low sr
• A contract is incentive compatible if for all
  unmonitored states sr ? S0(?) there is no
  incentive to make false reports f(s) - ?(s) ?
  f(s) - ?(sr). If so, we get
• A contract (?,?) is incentive compatible if and
  only if ?(.) R is constant on S0(?), and R ?
  ?(s) for all s in S1(?)

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An Incentive Compatible Contract
?
R
I
s
0
Monitoring
No monitoring
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Optimal loan contracts (1)

• Must be Incentive Compatible
• Minimizes on the costs the costs of monitoring
• Firm returns ?(?,?) Ef - R.P(S0(?)) -
  ?s?S1(?) p(s) ?(s)
• Lenders return ?(?,?) ?s?S1(?) p(s) ?(s)
  R.P(S0(?)) - ? P(S1(?))
• Total returns ?(?,?) ?(?,?) Ef -
  ?P(S1(?)) so saving on monitoring costs is in
  the interest of all!

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Optimal loan contracts (2)

• A contract (?,?) that maximizes ?(?,?) subject to
  the following constraints
• ?(?,?) ? I individual rationality must be
  binding. Otherwise the R could be lowered,
  increasing firm profits
• f(s) - ? ? ?(s) ? 0 for all s ? S1(?)
• f(s) ? R ? 0 for all s ? S0(?)

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Optimal loan contracts (3)

• An optimal loan contract has the following form
• ?(s) R for s ? S0(?) and ?(s) f(s) - ? for
  s ? S1(?)
• ?(s) 0 for f(s) ? R and ?(s) 1 for f(s) lt R
• So it optimizes the needs of the borrower and the
  demands by the lender under minimal costs
  (restricting the monitoring zone)

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An optimal loan contract
f(s)
?
f(s) - ?
R
I
s
0
Monitoring
No monitoring
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Credit rationing

• Typical loan contract a fixed repayment with a
  bankruptcy provision in case of default
• This type of contract induces borrowers to take
  more risk than lenders want to
• This asymmetry in incentives may lead to credit
  rationing

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Notation

• f(s) is the state-dependent return function
• R is the repayment
• ?(s, R) maxf(s) - R,0 is the state-dependent
  return for the entrepreneur
• ?(s, R) minR, f(s) is the state-dependent
  repayment to the lender
• ?(s, R) ?(s, R) f(s) no monitoring costs

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Lender returns
Note that the return schedule is a concave
function
?
fA(s)
fB(s)
R
?(s,R)
?B
Lender prefers project B over A
?A
s
s0
0
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Borrower returns
Note that this is a convex function
?
fA(s)
fB(s)
?A(s,R)
Borrower prefers A over B
R
?B(s,R)
s
s0
0
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Conflict of interest

• Lenders are risk averse, borrowers risk seeking
• If lenders cannot observe the riskiness of the
  loan applicants, they might refrain from using
  the price mechanism adverse selection might be a
  consequence
• Illustration in a model (see Stiglitz-Weiss,
  1981)

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A credit rationing model (1)

• A bank faces a group of N borrowers two types
  (1 and 2), to be distinguished by their projects
• Two states H (high-) and L (low-risk) each with
  a probability 0.5
• (xtH,xtL) is the state-contingent returns of
  applicants of type t, t 1, 2
• Type 2 agents project return forms a
  mean-preserving spread of type 1s returns
  x2L lt x1L lt x1H lt x2H and
  ?1 (x1L
  x1H)/2 ?2 (x2L x2H)/2

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A credit rationing model (2)

• Each loan applicant has no funds and wants to
  borrow one unit
• The bank knows that half of the applicants is of
  type 2, but cannot distinguish the types
• The bank can raise funds by deposits at a rate I
  1 i following a supply schedule L(I) aI,
  agt0
• There is full competition in the banking sector,
  so expected lender profits E?(.,R)I
• In case of bankruptcy the bank can seize some
  value C from the customer

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A credit rationing model (3)

• Expected return schedule for a loan applicant of
  type t is Emax(xts - R, -C)
• For xtL C ? R 0.5 xtH xtL - R
• For xtH C gt R gt xtL C 0.5 xtH - R - C
• For R gt xtH C -C
• Define Rt Emax(s, Rt, C)0, so Rt xtH - C.
  This is the marginal applicant. R2 will be larger
  than R1 riskier applicants stay longer in the
  market

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A credit rationing model (4)
E?t
Returns for a loan applicant
(xtH xtL)/2
(xtH - xtL)/2 - C
Rt
xtH C
0
R
xtL C
-C
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A credit rationing model (5)

• Expected return schedule for the bank for type
  t is Emin(R, xts C)
• For xtL C ? R R
• For xtH C gt R gt xtL C 0.5 xtL R C
• For R gt xtH C 0.5 xtH xtL C

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A credit rationing model (6)
E?t
Returns to the lender
(xtHxtL)/2 C
xtL C
xtHC
0
xtLC
R
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Rationing at work
Returns get discontinuous through applicants
leaving the market
Low risk exit
E?t
x1H C
(x1Hx1L)/2 C
x1L C
x2L C
High risk exit
x2H C
0
x2L C
x1L C
R1
R2
R
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Lender behaviour

• It might not be in the interest of the bank to
  increase the interest rate
• The pool of applicants changes and high-risk
  applicants stay in the market
• Profits decrease and lead to sub-optimal results
• The bank goes to R1 and some of the applicants
  are denied credit