Asymmetric information in finance

Asymmetric information in finance

Financial contract

Is a contract agreed upon by a borrower and a lender.

The contract is signed only if both parties’ participation constraint is fulfilled:

Participation constraint:

expected return from the contract ≥ expected return from best alternative course of action (opportunity cost)

Example:

An entrepreneur (borrower) is considering the following investment project:

Risky Investment at time t: I = 100

Cash Flow at time t + 1: CFs = 300 CFf = 0

Probabilities: α = 0.7 1 – α = 0.3

Expected value EV(I) = α CFs + (1 – α) CFf = 210

Loan contract L = 100

Lender’s opportunity cost: forgone opportunity to invest L = 100 in bonds at ‘sure’

interest rate r = 10%. That is:

Lender’s opportunity cost = (1 + r)L = 110

Limited liability: contractual clause that, in case the investment fails, the borrower is not forced to pay the capital and interest owed by using personal assets. Under this clause, the financial contract is risky for the bank. In what follows we assume limited liability.

Participation constraint

Assumption 1: borrower and lender are risk neutral

Assumption 2: the borrower does not have any other investment opportunity.

Under assumption 1,

lender’s participation constraint is:

the risk-adjusted interest rate r^L fixed by the bank must be sufficient to yield expected revenue from loan L ≥ opportunity cost (1 + r)L

Bank’s expected revenue from L:

α(1 + rL)L + (1 −α)min [(1 + rL)L, CFf] ≥ (1 + r)L because CFf = 0 this yields:

α(1 + rL)L ≥ (1 + r)L

Lender’s participation constraint is just fulfilled at risk-adjusted interest rate rL such that: α(1 + rL)L = (1 + r)L

By fixing the risk adjusted interest rate, the lender is indifferent between the risky contract and the sure asset.

In the example: ^1+rL=1+r

α =1+0.1

0.7 =1.57 r_L=0.57

Remark: whenever min CFf < (1 + r)L, we have ^rL>r

Borrower expected profit:

E π =α[^CFs−(1+r_L)L] = EV ^{−α (1+rL)L} = 210 – 0.7[1.57(100)] = 100.1 ^≈ 100 Under assumptions 1, 2, we have:

Borrower’s participation constraint is ^{E π ≥ 0} . This is fulfilled.

Forms of asymmetric information in financial contracts:

Hidden information: (pre-contractual opportunism)

The borrower overstates the true success probability α of the project, in the attempt of obtaining a lower risk adjusted interest rate r^L.

Hidden action: (post-contractual opportunism)

- the borrower mis-reports the true ex-post cash flow, in the attempt of avoiding the payment of ^(1+rL)L

- the borrower is investing the money received by the lender on a riskier investment project than was agreed with the bank (moral hazard).

Hidden information: adverse selection

Suppose there are two types A, B of productive investment projects carried out by type A and B entrepreneurs.

- The frequency pA, pB of types A and B in the population is common information

- B is riskier than A project cost: LA = LB = L - EVA = αA CFs,A + (1 – αA) CFf,A = αA CFs,A CFf,A = 0

- EVB = αB CFs,B + (1 – αB) CFf,B = αB CFs,B CFf,B = 0 Assume: EVA = EVB and CFs,B > CFs,A and αA > αB

Risk adjusted interest rate on loan:

1+r_{L, A} = ^1+r_α

A

<

B

=

The expected payment to the lender is the same ^(1+r)L , under both types.

Type B entrepreneurs hide their type to the bank, in the attempt of getting a lower interest charge. The bank is indeed unable to distinguish type A and B.

The bank charges the same risk adjusted interest factor (1 + rL) to every borrower.

(1 + r^L) is so fixed that:

(1 + r) = pA αA (1+r_L) + pB αB (1+r_L)

(1 + r) = ^(1+rL) ( pA αA + pB αB )

1+r_L = _p ^1+r

Aα_A+p_Bα_B

=

s

<

As a result of hidden information, low-risk A entrepreneurs and high-risk B entrepreneurs are charged the same interest rate ^rL .

Moreover, in this example, we have that the low-risk and the high-risk projects have the same expected value: EVA = EVB

This implies that if A and B pay the same interest rate ^rL , the risky project B yields a higher expected profit than the safer project A:

E π_A=α_A[^CFs , A−(1+r_L)L] = EVA −α_A(1+r_L)L E π_B=α_B[^CFs , B−(1+r_L)L] = EVB −α_B(1+r_L)L

EVA = EVB and ^αA > ^αB imply ^{E π}A < ^{E π}B at any rL

Adverse selection: At interest rate ^rL > ^{rL, A} type A entrepreneurs have an incentive to change their type and switch to type B project.

The outcome is that type A projects disappear from the market, that is there is an endogenous change in the frequencies of types A and B:

p_A=0 , → ^pB=1 and ^1+rL=1+r_{L ,B} .

Hidden action: moral hazard

rL = risk adjusted interest rate fixed by the bank

l = project with low EV (expected cash flow) αl = success probability of l h = project with high EV (expected cash flow) αh = success probability of h L = cost of project l and h

CFh,s = cash flow if success CFh,f = 0 = cash flow if failure CFl,s = cash flow if success CFl,f = 0 = cash flow if failure CFl,s > CFh,s and αh > αl such that EVh > EVl

Expected cash flow: EVh = αh CFh,s + (1 − αh)0

>

Because EVh

>

Expected profit

EП_h=EV_h−α_h(1+r_L)L EП_l=EV_l−α_l(1+r_L)L

Borrower’s incentive compatibility constraint:

EП_h≥ EП_l that is: ^EVh−α_h(1+r_L)L ≥ EV_l−α_l(1+r_L)L

Can be written as:

α

EV_l≥(¿ ¿h−α_l)(1+r_L)L EV_h−¿

This implies

α

(¿¿h−α_l)L ≥(1+r_L) (1+r_Lmax)=EV_h−EV_l

¿

That is: ^{rLmax≥ rL}

Incentive compatible interest factor

Eπ

EVh Eπh

EVl

Eπl

(1 + rL)

(1 + rL)max

incentive compatible range of rL

Contract Feasibility:

Contract is feasible if it meets the participation constraints of both parties, and the borrower’s incentive-compatibility constraint. In the range rL < rL,max , only h projects are financed, and the bank’s participation constraint is rL > ^1+rL,h = ^1+r_α

h .

Thus the contract is feasible in the interest rate range ^1+r_α

h

<

Effect of a credit constraint D on the incentive-compatible interest rate Credit constraint: bank unwilling to finance the whole project cost L borrower is forced to contributing to the project with own funds D

Expected profit with credit constraint: D > 0

EП_h=EV_h−D−α_h(^1+rL)^(L−D)

EП_l=EV_l−D−α_l(^1+rL)^(L−D)

The incentive-compatibility constraint that makes the low-risk project h more attractive than the high-risk project l is

EП_h≥ EП_l ,

that is: ^EVh−αh(^1+rL)^{(L−D)≥ EV}l−α_l(1+r_L)(L−D) EV_h−EV_l≥(α_h−α_l)(1+r_L)(L−D)

This can be written as:

D(^αh−α_l)^+EVh−EV_l≥(α_h−α_l)(^1+rL)^L

(^1+rL, max∨D> 0) =

α

(¿¿h−α_l)(L−D) EV_h−EV_l

¿

≥(^{1+ r}L)

Conclusion:

The upper-bound ^{rL, max} to the incentive-compatible risk-adjusted interest rate rL is an increasing function of the credit constraint D. By imposing a higher credit

constraint D, the bank is free to fix a higher incentive compatible interest rate than it would be the case otherwise.

Exercise:

Two projects: A, B Cost: LA = LB = 100

Cash flow if s: CFs,A = 600 CVS,B = 880 αA,S = 0.8 αB,S = 0.5 Cash flow if s: CFf,A = 0 CVf,B = 0

The ‘safe’ interest rate is r = 0.5

What is the maximum incentive compatible interest rate rL? Is the bank prepared to lend money L at such a rate?

Solution:

EV_A=600∗0.8=480 EV_B=880∗0.5=440

rL, max is the max. value of rL such that:

EП_A≥ EП_B that is: EV_A−α_A(1+r_L)L ≥ EV_B−α_B(1+r_L)L Can be written as:

α

EV_B≥(¿¿A−α_B)(1+r_L)L EV_A−¿

This implies

α

(¿¿A−α_B)L ≥(1+r_L) (1+r_Lmax)=EV_A−EV_B

¿

(^1+rLmax)⁼_0.3∗L^{40 =1.33}

That is, the maximum incentive compatible interest rate is ^rLmax = 0.33

But, is the bank prepared to lend at an interest rate rL≤0.33 ? Too see this, we must check the bank’s participation constraint:

we call rL,Athe lowest interest rate at which the bank is prepared to lend money L on project A.

Participation constraint:

αA(1 + rL,A)L = (1 + r)L ^{1+rL, A} = 1+r

α_A =1 . 05

0 . 8 =1. 3125

Remark :

if safe interest rate r = 0.1 then ^1+rL, A = 1+r α_A =1.1

0.8=1.375

because:

participation constrained r_{L, A} = 0.375 > 0.333 = r_{L, max} = max incentive compatible rate, the loan contract L = 100 is not signed.

The bank will then propose a different loan contract, such that the borrower has to participate with wealth D to project finance, and the bank contributes L−D .

This will raise rL, max according to:

(^1+rL, max) =

α

(¿¿A−α_B)(L−D) EV_A−EV_B

¿

= _0.3(100−D)⁴⁰

If D = 20, and safe interest rate r = 0.1, then rL, max = 0.666 > rL, A and the contract meets the participation and the incentive-compatibility constraints if r_{L, A} = 0.375 < r_L < 0.666.