Adverse Selection in Financial Markets
Reading. This piece covers Leland and Pyle (1977), as summarised in Freixas
and Rochet, Microeconomics of Banking, (pages 24-26 in the first edition or
pages 24-27 in the second edition) The second section is based on pages
118-121 in the first edition or pages 153-157 in the second edition.
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Introduction
Financial markets allow transfer of resources across time and across uncertain states of nature. Consider, for instance, an entrepreneur who has
access to an investment technology (“a project”), but no resources. A financial contract can allow the entrepreneur to acquire resources for investment in return for a promise to repay a later date. The repayment
obligation may be fixed (as in a debt contract) or variable. If the project is
risky (in the sense that the payoff could take high or low values), the financial contract can provide some degree of insurance: that is, the repayment
obligation can be so structured that the repayment obligation varies with
the payoff to the project. Further, in many cases, the repayment obligation
is constrained by the principle of limited liability: that the borrower does
not have to pay more than payout of the project.
Asymmetric information complicates financial contracts. Investment
projects can differ in their distribution of returns and the true distribution
may be known by the entrepreneur but not to the counter-party in the
financial transaction. This may make financial contracting harder.
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To see why, suppose projects differ in terms of the variability of returns. For any fixed repayment requirement an increase in the variability
of returns can raise the risk of default: this is, the risk that the payoff to
the project turns out to be less than the contracted repayment. If lenders
could assess the riskiness of the project, they can set repayment levels appropriately: in particular, riskier projects would have higher repayment
obligations to compensate for the greater risk of default. However, if the
riskiness of the project is not observable, high interest rates may well end
up attracting relatively risky projects. This is the standard problem of adverse selection.
Risk-averse entrepreneurs may prefer to raise resources by issuing equity rather than debt. Debt creates an inflexible repayment requirement
leading to substantial variability in net payoff. Equity allows them to share
the variability in payoffs, reducing the risk. In effect, they are in effect
selling the future uncertain return to the project for a fixed price. However, once again there is the problem of adverse selection. If entrepreneurs
know the quality of their project better than the potential investors, entrepreneurs with low quality projects would be the more inclined to sell
their projects than those who know their project to be good. We consider
various manifestations of this problem and mechanisms that can mitigate
it.
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Adverse Selection in Financial Markets
We model a competitive capital market with many risk-averse entrepreneurs
and risk-neutral investors. Each entrepreneur has a risky project. Given
the assumed pattern of risk preferences, it would be efficient to transfer all the risk from the entrepreneur to the investors. In particular, entrepreneurs would be happy to sell their project to risk-neutral investors
at a price that equals the expected return of the project: being risk averse,
they strictly prefer getting the expected value of the project to getting its
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random return. Investors, being risk neutral, are happy to buy the project
at a price that equals its expected return. This exchange leads to a Pareto
improvement.
Suppose, however, that the true quality of a project is known to its entrepreneur but not to potential investors. This model outlines how adverse
selection can then lead to inefficiency. Further, we show how to mitigate
this problem, entrepreneurs with good quality projects may prefer to partially self-finance the project to signal the superior quality of their project.
Assume each risky project requires investment of one unit, and provides a random net return R̃(θ, σ2 ), which is normally-distributed with
mean θ and variance σ2 . Assume that entrepreneurs have exponential
utility functions u(w) = −e−ρw , where ρ is the (constant) coefficient of
absolute risk aversion. Given that the return to a project is normallydistributed, entrepreneurs’ preferences can be described by mean-variance
utility functions: given wealth
w = W0 + R̃(θ, σ2 ),
(1)
where W0 is a constant and R̃(θ, σ2 ) is normally-distributed, each entrepreneur’s
expected utility from her random payoff is
1
u(W0 + θ − ρσ2 ).
2
(2)
If, on the other hand, she sells the project at some price P, her utility is
u(W0 + P).
(3)
An entrepreneur would be willing to sell her project if and only if u(W0 +
P) ≥ u(W0 + θ − 12 ρσ2 ), or equivalently, if
1
P ≥ θ − ρσ2 .
2
(4)
Investors are assumed to be risk neutral, so value any project by its
mean return θ. They would be willing to pay any price up to θ for buying
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the project. It would be Pareto improving for the risk-averse entrepreneurs
to sell, and for the investor to buy, the project at any price in the interval
[θ − 21 ρσ2 , θ ] . We assume for simplicity that all the bargaining power lies
with entrepreneurs, so the project sells at price P = θ.
Suppose now that projects differ: assume that the variance is the same
for all projects but mean return θ varies across projects. Let θi be the mean
return for entrepreneur i’s project. If θi is observed by both entrepreneur
and investors, prices will vary: we have Pi = θi . With symmetric information, better projects will sell at a higher price, but importantly all projects
will sell.
Consider, next, what happens with asymmetric information. Suppose
entrepreneur i knows θi , the expected return of her own project but investors cannot observe this value. If θi is not observable, all projects that
do sell must sell at the same price (call it P.) To identify the entrepreneurs
who will sell at this price, note that an entrepreneur who holds on to her
own project gets utility u(W0 + θi − 21 ρσ2 ). On the other hand, if she sells
her project, she gets utility u(W0 + P). She will sell if and only if
θi ≤ θ̂ ( P),
(5)
where
1
θ̂ ( P) = P + ρσ2
(6)
2
. In words, all entrepreneurs who know their project to have relatively
low expected return (θi < θ̂ ( P)) relative to the market price P will choose
to sell and those with good projects will not. This is the adverse selection
problem faced by investors.
Of course, once investors realize that only entrepreneurs with θi <
θ̂ ( P) sell at price P, they would be willing to offer any price. The actual
price must be such that investors expect to break even given the quality of
projects that are offered for sale.
P = E[θ |θ < θ̂ ( P)].
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(7)
The equilibrium with adverse selection is defined by a price P that allows investors to break even, and a θ̂ ( P) such that only those with expected return below θ̂ ( P) will sell their projects. In effect, both equations
(5) and (7) must hold for the sales that are actually carried out.
This equilibrium is generally inefficient. To see why consider, for example, the following special case. Suppose θ takes a low value θ1 with
probability π1 and a high value θ2 with probability π2 = 1 − π1 . Entrepreneurs with high θ are less likely to sell their projects. Would this
reluctance to sell undermine efficiency?
Efficiency requires that all projects be financed: this occurs only if P is
so high that θ̂ ( P) ≥ θ2 . That is
1
(8)
P + ρσ2 ≥ θ2
2
Of course, if both high type and low type sell at this outcome, so the competitive price of all projects must reflect the average expected output:
P = π1 θ1 + π2 θ2 .
(9)
1
π1 θ1 + π2 θ2 + ρσ2 ≥ θ2 ,
2
(10)
Together these imply
which can be written as
1 2
ρσ ≥ π1 (θ2 − θ1 ).
(11)
2
In words, the risk premium has to outweigh the adverse selection effect. If
it does not, the equilibrium outcome will be inefficient, in that those with
high return will not be able to sell their project. Only low-quality projects
will sell at a price θ1 .
Signaling Through Self-Financing
If condition (11) does not hold, we may still have a signaling equilibrium
in which entrepreneurs who know their project to be high return partly
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self-finance their project. Suppose entrepreneurs with high-return projects
(θ = θ2 ) self-finance some fraction α of their project. Entrepreneurs who
know their project to be low return (θ = θ1 < θ2 ) will be less inclined to
self finance. By choosing an α that is large enough, entrepreneurs with
good projects can distinguish themselves from owners of bad projects.
To formalise this, consider an outcome in which low-return projects
are identified by zero self-financing and sell at their expected return θ1 .
High-return projects sell at θ2 as long as they can persuade investors that
they are high return, by their willingness to self-finance a sufficiently high
proportion α. How high is sufficiently high?
The critical proportion must be so high that entrepreneurs with lowreturn projects are not tempted to mimic that. For any given level of selffinancing α, the ‘no-mimicking’ condition for low-return types is that
u(W0 + θ1 ) ≥ Eu[W0 + (1 − α)θ2 + α R̃(θ1 )].
(12)
The left side of this relation is the utility to low-return types of selling
the entire project (ie, zero self-financing) at price θ1 . Note that zero selffinancing reveals that the project is low return, so investors are willing to
pay only θ1 in this case.
The right side is the return from self-financing a proportion α of the
project and selling the rest. For the proportion they sell, (1 − α), they hope
to get the high price θ2 that high-return projects sell at. For the proportion
α they self-finance, they get a proportionate share of the random return of
the project. The mean return from this fractional holding to an investor
who knows her project to be low-return is αθ1 and its variance is α2 σ2 , so
that relation (12) amounts to
or
1
W0 + θ1 ≥ W0 + (1 − α)θ2 + αθ1 − ρσ2 α2 ,
2
(13)
α2
2( θ2 − θ1 )
≥
.
1−α
ρσ2
(14)
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Any α that satisfies this relation works. It generates an outcome in which
high-return projects are partially self-financed and the remaining fraction
sold at price θ2 ; low-return projects are sold sell entirely at price θ1 .
Any α that satisfies this condition works. There are a continuum of
such equilibria, differing in α. In fact the smallest value that achieves this
condition Pareto dominates other values.
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Adverse Selection in Credit Markets
Consider an entrepreneur who borrows one unit from a lender to invest in
a risky project and promises to repay R. The project has output ỹ, which is
assumed to take one of two values: ỹ = 0 (“failure”) or ỹ = y (“success”).
Let θ be the probability of failure, so that the expected return to the project
is (1 − θ )y. With limited liability, no repayment is made when the project
fails. If so, the expected return for the lender is
(1 − θ ) R,
(15)
while the expected return to the borrower is
(1 − θ )(y − R).
(16)
Suppose that there are two kinds of borrowers: those with projects that
are high-risk-and-high-return and those that are low-risk-and- low-return.
These projects can be denoted as (y H , θ H ) and (y L , θ L ), where θ H > θ L
(so that project H has higher risk of failure than project L), and y H > y L
(project H has higher payoff in the event of success).
Assume for simplicity that the expected return to the two projects is
the same: that is,
(1 − θ H ) y H = (1 − θ L ) y L .
(17)
If so, a risk-neutral entrepreneur investing in a project with own funds
would be indifferent between the two projects.
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We assume that the borrower knows his own type but the lender does
not. If the lender could observe the type of project, it would choose a
higher rate of interest for loans against the risky project. To fix ideas
with competitive lending and given risk-free interest rates R f , risk-neutral
lenders must expect to break even. The break-even interest rate Ri for
project of type i is given by
(1 − θ i ) R i = R f .
(18)
Clearly, R H > R L .
Suppose the lender cannot distinguish between the two types, so must
charge the same interest rate to all borrowers. Note that for any loan contract with positive R will be more attractive to entrepreneur with the highrisk projects rather than low-risk ones. To see why note that the expected
return to the entrepreneur is (1 − θ )(y − R) and
(1 − θ H )(y H − R) > (1 − θ L )(y L − R).
(19)
If lenders set interest rates based on average risk characteristics, only
borrowers who know their project to high risk will seek loans. This is
the familiar problem of adverse selection. If lenders cannot observe the
riskiness of the project directly, they are more likely to attract an adverse
selection of projects. They could set interest rates at R H , but if so, low risk
entrepreneurs will not be able to access capital markets: this inefficiency
is a direct consequence of informational imperfection.
Reservation utility
One way to mitigate the adverse selection is to use collateral as a screening
device.
Assume that borrowers also differ in their reservation utility U, which
measures the minimum expected payoff they need from the loan contract
in order to be willing to borrow. This can be viewed as the value of their
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alternative opportunities. Consider a loan contract with repayment requirement R. Given the reservation utility U, the highest repayment level
acceptable to the borrower must satisfy
(1 − θ )(y − R) ≥ U.
(20)
Given this participation constraint, the maximal value of R that a borrower
is willing to pay is
U
.
(21)
R = y−
1−θ
We assume that all the bargaining power lies with the lender so that it
will charge this maximal rate. The two types are assumed to differ in their
reservation utility, so that U L 6= U H
If the lender could observe θ i and U i , it would charge
RL = y −
UL
1 − θL
(22)
RH = y −
U
1 − θH
(23)
to the low-risk type
to the high risk-type.
We assume that low-risk borrowers have higher reservation utility than
high-risk borrowers, or that U L > U H . In fact, we make a stronger technical assumption:
UL
UH
>
.
(24)
1 − θL
1 − θH
Given our assumption, this implies that R L < R H , or that high-risk types
must repay a larger amount.
Assume that the population of borrowers contains equal numbers of
each type. If the lender cannot observe θ, it must charge a common R.
How should it set this value?
For any R L < R ≤ R H , the low-risk borrowers will choose not to
borrow, and the expected payoff from each borrower will be (1 − θ H ) R.
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The maximum payoff then obtains by choosing R = R H , with payoff
(1 − θ H ) R H .
On the other hand, if it sets R = R L , both types will borrow. As there
are equal proportions of each type, expected return per borrower will be
0.5(1 − θ H ) R L + 0.5(1 − θ L ) R L = (1 − θ ) R L ,
(25)
where θ is the average θ in the population.
Whether it is more profitable to charge R L and lend to all or charge
R H and lend selectively to high-risk types only depends on the parameter
values. We assume, for our purposes, that it is more profitable to lend to
all. At this outcome, high-risk types obtain an informational rent: they
borrow at R L even though they would be willing to pay R H > R L .
Signaling through Collateral
We see how contracts with collateral can improve on this outcome. Collateral C refers to some asset that the borrower hands over to the lender. A
loan contract is now given by ( 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 liquidates the collateral: we assume that in this
circumstance the borrower loses C but the lender gets only a fraction δC,
with δ < 1. The ‘efficiency cost’ of liquidation is (1 − δ)C.
A contract with collateral is given by ( Rk , C k ). The expected payoff to
the lender is now
(1 − θ k ) Rk + θ k C k δ,
(26)
and the expected payoff to the borrower is
(1 − θ k )(y − Rk ) − θ k C k .
(27)
The borrower will accept this contract as long as the payoff exceeds his
reservation utility U k . Assuming, as before, that all bargaining power lies
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with the lender, the contract that type k will be just willing to accept will
have
(1 − θ k )(y − Rk ) − θ k C k = U k .
(28)
Equivalently, with the lender having complete bargaining power
Rk = y −
θk
Uk
−
Ck .
1 − θk 1 − θk
(29)
Recall that if the lender can observe the borrower’s type, it is efficient
to have contracts with zero collateral. Use of collateral is inefficient as
it costs θ k C k to the borrower and yields only θ k C k δ to the lender. Thus,
with full-information, the contract for each type k will have C k = 0 and
k
.
R k = y − ( 1U
−θ k )
With asymmetric information, collateral can be used to separate the
two types. To anticipate the result, the lender would offer two contracts,
one with high interest rate and zero collateral intended for high-risk types;
and the second with lower interest rate and positive collateral intended for
low risk types. Provided the interest rate and collateral levels can be set
correctly, high risk types will indeed select the former contract and the low
risk types will select the latter.
In particular, consider the contracts ( R H , 0) and ( R, C ) where R H is as
defined above and R and C are to be determined. For high-risk types to
prefer ( R H , 0) to ( R, C ), it must be
(1 − θ H )(y − R H ) ≥ (1 − θ H )(y − R) − θ H C.
(30)
For low risk types to prefer ( R, C ) to ( R H , 0), we must have
(1 − θ L )(y − R) − θ L C ≥ (1 − θ L )(y − R H ).
(31)
In addition, the contracts should be acceptable to the two types
(1 − θ H )(y − R H ) ≥ U H .
(32)
(1 − θ L )(y − R) − θ L C ≥ U L .
(33)
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Are there values of R and C that satisfy these conditions? Note first,
that by construction R H is such that (R H , 0) is just acceptable to the highrisk types so is (32) satisfied.
Next, recall that, by construction, R L is such that (R L , 0) provides utility
U L to the low-risk type. Given that R H > R L , the contract ( R H , 0) must
provide less than U L to the low-risk types. We have
U L = (1 − θ L )(y − R L ) > (1 − θ L )(y − R H ).
(34)
Thus any contract ( R, C ) that satisfies (33) must automatically satisfy (31).
The essential relations are (30) and (33). Depending on the parameter values such a contact ( R, C ) can be found whenever these two conditions are
satisfied.
How does collateral separate the two types? Effectively, each type must
lose C in the event of default. Low risk types, who know their risk of
default to be low, do not mind this loss as much as high risk type.
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