Slides 9 Debraj Ray Columbia, Fall 2013 Credit Markets Collateral Adverse selection Moral hazard Enforcement 0-0
Credit Markets
Formal sector:State or commercial banks. Require collateral, business plan
Usually lend to registered firms or small businesses Usually charge the lowest rates of interest
Often give directed credit to “priority sectors”: (exports, agriculture, small business)
Informal sector:
Moneylender, trader, landlord, shopkeeper . . . Better information
Better enforcement capabilities (multi-market) Specific abilities to take certain types of collateral Can price out distortions via multi-market contact
Quasi-formal organizations:
Grameen Bank, microfinance NGOs
Sometimes use group liability to avoid default Rigid repayment schedules
Working capital rather than fixed capital loans
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Some Stylized Facts
What follows is all interrelated, of course. Collateral matters: the richer get better deals. Relationship between collateral and interest rates
The form of collateral matters too:
Landlords lend to tenants and small-holders.
Traders lend to farmers producing the same crop they trade. Occupation-based lending: Bangladeshi immigrants.
Collateral or limited liability interacts with strategic incentives: a↵ects who seeks credit (adverse selection)
a↵ects how credit is deployed (moral hazard) a↵ects repayment incentives (strategic default)
a↵ects lender-borrower combinations (segmentation) a↵ects design: group lending and peer monitoring
a↵ects loan types: e.g., working capital vs. fixed capital
generates credit rationing: interest rates don’t clear the market.
0-4 Example.
Projects A and B, startup cost 100,000.
Rates of return 15% and 20% (revenues 115,000 and 120,000). Bank rate of interest 10%
Perfect coincidence of interest between bank and borrower. A0: 230,000 or 0 equal prob, expected return unchanged. Under limited liability, bank strictly prefers Project B. But the borrower strictly prefers Project A0! (Why?)
Two extensions of the example:
The bank attracts type A0 borrowers (adverse selection)
Risk Premia
How about charging higher interest rates to compensate? Say p = prob. repay, i = interest rate, r = opportunity cost: Then p(1+i) = 1+r, or
i = 1+r
p 1.
The problem is that i a↵ects the repayment probability! Aside: In fact, sometimes a deliberate ploy by lender. Lender valuation of collateral V`, borrower valuation Vb.
Borrower prefers to repay if
L(1+i)< Vb+F
where L = loan size, i = interest rate, F = default cost.
0-6 Aside, contd.
Lender wants his money back if L(1+i)> V`
Thus loan repayment in the interest of both parties if Vb+F > V`.
But if
Vb+F < V`.
interest rate may be adjusted to facilitate collateral seizure. (Note: analysis works best for inelastic loans.)
Adverse Selection Over Projects
Stiglitz and Weiss (1981)
Borrowers di↵er in “riskiness”. cdf of returns R given by F(R, ✓)
Assume same mean: R RdF(R, ✓) = R RdF(R, ✓0) for all ✓, ✓0.
Larger ✓’s more risky in the sense of strict SOSD: Z y 0 F(R, ✓)dR > Z y 0 F(R, ✓0)dR for all y in interior of support, whenever ✓ > ✓0.
Startup cost of project: B, borrow at rate r, collateral C. Limited liability in loan repayment; repay if
R+C (1+r)B.
0-8
So borrower’s return if project pays o↵ R is given by ⇡(R, r)⌘ max{R (1+r)B, C},
and lender’s return is given by
⇢(R, r) ⌘ min{R+C, B(1+r)}.
Expected payo↵ to borrower of type ✓ is therefore ˆ⇡(✓, r) =
Z
⇡(R, r)dF(R, ✓),
And expected payo↵ to lender from borrower of type ✓ is ˆ⇢(✓, r) =
Z
⇢(R, r)dF(R, ✓). ˆ⇢ decreasing in ✓, but ˆ⇡ increasing in ✓.
R
P
ayof
fs
C
R
P
ayof
fs
C
Expected payo↵ to borrower increases in ✓, opposite for lender.
0-10
These relationships with ✓ have the following implication: Define a threshold ✓(r) by the condition
ˆ⇡(✓(r), r) ⌘ 0. Then ✓(r) is increasing in r and the set
{✓|✓ ✓(r)} will enter the market.
If riskiness not observed, borrower quality will degrade in r. Lender profit rate with interest rate r:
d(r)⌘ 1
B[1 G(✓(r))]
Z 1
✓(r) ˆ⇢(
✓, r)dG(✓) 1
r L ende r P ayof f 0 r*
With free entry and exit of banks, d(r) is also deposit rate. Deposit supply ˆS(d), but S(r) ⌘ ˆS(d(r)) nonmonotonic. Demand curve for loans D(r) downward-sloping (why?)
Equilibrium: A set of interest rates such that no bank wants to deviate from charging them, with positive deposits at each rate.
0-12 Notes on equilibrium:
If more than one r, depositors indi↵erent, borrowers prefer low-est r.
Lowest eq r cannot have S(r) > D(r), and if S(r) = D(r), it must be only rate.
Otherwise there are unsatisfied depositors, and a deviant bank can take them at lower rate and make profits.
Credit rationing: D(r)> S(r) at the lowest equilibrium rate. Theorem.
(a) Suppose S(r⇤) = D(r⇤) for some r and S(r) < S(r⇤) for all r < r⇤. Then there is no equilibrium with credit rationing.
(b) Suppose that for every r⇤ with D(r⇤) = S(r⇤), there is r < r⇤
Proof of Part (a).
S(r⇤) =D(r⇤) for some r⇤, S(r)< S(r⇤) for all r < r⇤.
Then only equilibrium is at r⇤.
0-14
If any eq r below r⇤, then every other eq r0 above r⇤ to maintain
depositor indi↵erence.
Deviant bank: o↵ers r⇤, and some d 2 (d(r), d(r⇤)), takes all the
rationed borrowers from r, makes positive profit.
If every eq r no less than r⇤, let r1 be lowest of them.
By earlier note, D(r1) S(r1) and if r1 = r⇤, then unique.
So suppose that r1 > r⇤. Then S(r1) D(r1) < D(r⇤) =S(r⇤).
So S(r)> S(r⇤) and consequently, d(r)> d(r⇤).
Deviant bank charges r = r⇤, o↵ers d0 2 (d(r⇤), d(r)), gets all depositors and borrowers, makes positive profits.
Proof of Part (b).
For every r⇤ with D(r⇤) = S(r⇤), there is r < r⇤ with S(r)> S(r⇤).
D r S r r S, D D r S r r S, D D r S r r S, D
Then equilibrium must involve credit rationing.
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Suppose not, then single-r equilibrium at r⇤ with S(r⇤) =D(r⇤).
We know that S(r) > S(r⇤) for some r < r⇤.
So d(r)> d(r⇤).
Deviant bank o↵ers r and d 2(d(r⇤), d(r)), makes profits. Remains to demonstrate equilibrium with credit rationing. Let r⇤⇤ be the smallest maximizer of d(r).
If D(r⇤⇤)> S(r⇤⇤), then done: set r = r⇤⇤.
Otherwise, D(r⇤⇤)< S(r⇤⇤) (equality violates part (b)). In this case, let r⇤ be Walrasian equilibrium below it.
By assumption, there is r1 < r⇤ with S(r1) > S(r⇤).
Let r2 be the smallest value of r > r1 such that d(r2) =d(r1).
Such r2 exists because r1 is a local but not global maximizer. r r** r* D(r) S(r) r1 r2 S(r2) =S(r1)> S(r⇤) = D(r⇤)> D(r2), so excess supply at r2. 0-18 Complete the proof:
Take the equilibrium set to be {r1, r2}.
If all funds go to r1, excess demand, if all to r2 excess supply.
Using depositor indi↵erence, find allocation to exactly match demand and supply.
Check that this is an equilibrium with credit rationing. (No one can deviate, not even to r⇤⇤.)
Moral Hazard in Project Choice
Now we allow borrowers to choose from di↵erent projects. Projects indexed by ✓.
Return Q(✓) with prob p(✓), 0 with prob 1 p(✓).
Arrange such that Q(✓) increasing, wlog p(✓) decreasing. Each project requires the same loan size of B.
Borrower with collateral C, facing r, chooses ✓ to max p(✓) [Q(✓) B(1+r)] [1 p(✓)]C,
where Q(✓) exceeds B(1+r), otherwise no borrowing. Determines riskiness ✓(C, r) as a function of C and r.
0-20 Proposition:
✓(C, r) is decreasing in C and increasing in r.
Collateral induces safety; interest rates induce risk.
Proof. Define Z = B(1+r) C. Borrower picks ✓ to maximize p(✓)[Q(✓) Z].
Let Z1 > Z2. Let ✓1 and ✓2 be the two maxima (assume unique):
Then p(✓1)[Q(✓1) Z1] > p(✓2)[Q(✓2) Z1],
while p(✓2)[Q(✓2) Z2] > p(✓1)[Q(✓1) Z2].
Adding these two inequalities, we can conclude that
[p(✓1) p(✓2)] (Z1 Z2)< 0.
Credit rationing:
Entirely possible that for low values of C, max
r p(✓(C, r))(1+r) < 1+ ¯r
where ¯r is opportunity cost of funds. Collateral and interest:
Under competitive lending,
p(✓(C, r))B(1+r) + [1 p(✓(C, r))]C = B[1+ ¯r].
LHS as function of r. Nonmonotone, with end-point conditions.
0-22 B(1+r)_ p(θ(C,r)B(1+r) + [1- p(θ(C,r)]C r r _ C ↑ r*↓
Equilibrium rate of interest falls as collateral goes up. Argument does not work for monopoly lending.
Moral Hazard and the Debt Overhang
Costly e↵ort can influence success probabilities. Project requires startup of B.
Output is either Q (good) or 0 (bad).
Probability of good output is p(e), where e = agent e↵ort. If agent is self-financed, choose e to maximize
p(e)Q e L.
Assume unique choice e⇤; described by the first-order condition
p0(e⇤) = 1
Q
This is the efficient, or first-best level of e↵ort.
0-24 Now consider a debt-financed agent.
R = (1+r)B is total debt, C < B is collateral.
Now the e↵ort choice of a borrower facing a debt R given by max
e p(e)(Q R) [1 p(e)]C e
Optimal choice ˆe(R, w) is defined by the first-order condition: p0(e) = 1
Q+C R.
ˆe(R, C) < e⇤.
ˆe(R, C) is decreasing in R and increasing in C. This is the debt overhang.
Equilibrium debt and e↵ort in the credit market
e R
Lender's Iso-Payoff Curves
Borrower's Incentive Constraint
E1 E2 E3
Proposition. Equilibria with higher lender profits involve higher interest rates, but lower levels of e↵ort and social surplus.
Note: interest rate ceiling even at E3.
0-26
Competition: E↵ect of an increase in collateral. Evaluate at old level of lender payo↵
e R
Lender's Iso-Payoff Curve
Borrower's Incentive Constraint
E
E'
Monopoly: E↵ect of an increase in collateral.
Lender maximizes in each case, subject to incentive constraint.
e R
Lender's Iso-Payoff Curve
Borrower's Incentive Constraint
E E'
(Generally) R and e will both go up.
0-28
Strategic Default
Focus in this section: outside options, variable loan size Q = F(L)
Borrow L repeatedly (working capital) Stationary contract (L, R).
Fundamentalincentive constraint: If repay, get [F(L) R]/(1
).
If default, get F(L) today and v per period starting tomorrow. F(L) R v.
Combine enforcement constraint and lender iso-payo↵.
L
R
Lender's Iso-Payoff Line R - (1+r)L = constant Self-Enforcement Constraint
!v
!F(L) - R - !v = 0
0-30
Second-best: maximize borrower payo↵ over feasible set:
L R
!v
Borrower's Iso-Payoff Line F(L) - R = constant
L R
!v
Borrower's Iso-Payoff Line F(L) - R = constant
(A) IC Not Binding (A) IC Binding
E
E
In the first case, self-enforcement constraint not binding. Focus on second case, where it is.
Proposition. Equilibria with higher lender profits involve higher interest rates, but lower levels of e↵ort and social surplus.
Use diagram to prove higher interest rates.
L
R
Lender's Iso-Payoff Line R - (1+r)L = constant Self-Enforcement Constraint E2 E1
E3
!v
0-32
E↵ect of lowering the outside option v. Depends on lender power
L R E !v E' L R E !v E'
Information and Equilibrium in Credit Markets
Where does the outside option v come from?
Expected value conditional on termination of relationship. The reason matters, how much is known to others.
Thus, v may also incorporate punishment following default. Say competitive lending market. Recall, borrrowers maximize
F(L) (1+r)L subject to the incentive constraint
F(L) 1+rL v. Let w(v) be the maximized value.
0-34
Say that post-default, borrower approaches lenders sequentially. A lender uncovers past default with probability p.
If so, the lender refuses loan, the borrower waits one period, finds new lender.
Continues until new lender agrees to lend.
By symmetry, new contract will have value w(v). So in general equilibrium v must be given by
v = p v+ (1 p)w = 1 p
1 pw. Then we can write v = (1 ⇢)w, where
⇢ ⌘ p(1 )
1 p
Notes on the scarring factor:
If p is close to one, so is the scarring factor ⇢. On the other hand, ⇢ ! 0 as converges to one.
(But wait for a final verdict on the net e↵ect of patience.) Theorem. Define
⇢⇤ ⌘ (1+r)(1 )ˆL/
F(ˆL) (1+r)ˆL 2 (0, 1),
where ˆL is the maximizer of F(L) 1+rL.
Then a competitive equilibrium exists if and only if ⇢ ⇢⇤. Implications for theories of social capital.
Information and development.
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Proof. Necessity. Start with equilibrium loan L.
Inspect maximization problem to see that L ˆL (why?). If, contrary to the proposition, we have ⇢ < ⇢⇤, then
v = (1 ⇢)[F(L) (1+r)L] = [F(L) (1+r)L] ⇢[F(L) (1+r)L] > [F(L) (1+r)L] ⇢⇤[F(L) (1+r)L] = [F(L) (1+r)L] (1+r)(1 )/ F(ˆL)/ ˆL (1+r)[F(L) (1+r)L] [F(L) (1+r)L] (1+r)(1 )/ F(L)/L (1+r)[F(L) (1+r)L] = [F(L) (1+r)L] (1+r)(1 )L/ = F(L) 1+rL,
where the weak inequality in above string above uses L ˆL. But this contradicts the enforcement constraint.
Proof, contd. Sufficiency. Suppose that ⇢ ⇢⇤. Define ˆv ⌘ (1 ⇢)[F(ˆL) (1+r)ˆL].
At v = ˆv, the borrower’s max problem has a solution, because:
F(ˆL) 1+r ˆL = F(ˆL) (1+r)ˆL (1+r)(1 ) ˆL = (1 ⇢⇤)[F(ˆL) (1+r)ˆL]
(1 ⇢)[F(ˆL) (1+r)ˆL] = ˆv.
At this point, (1 ⇢)w(ˆv) (1 ⇢)[F(ˆL) (1+r)ˆL] = ˆv. On the other hand, there is ¯v ˆv such that
F(ˆL) 1+r ˆL = ¯v.
That is, ˆL is the only feasible solution when v = ¯v.
At this point, (1 ⇢)w(¯v) = (1 ⇢)[F(ˆL) (1+r)ˆL] = ˆv ¯v. w(v) is continuous, so there is v⇤ such that (1 ⇢)w(v⇤) =v⇤.
0-38
Interlinked Contracts
Market segmentation:
Landlord lends to tenant, the trader to farmers, etc. Reasons for interlinkage:
Assurance. Fixed costs need to be covered in trading. So “tie up” farmers by giving them loans in exchange for the promise of output sales to him.
Enforcement. Using double-threats; e.g., remove tenancy con-tract as well as future loan concon-tracts if default on the loan concon-tract. Nonmarketable Collateral. In interlinked relationships, easier to accept non marketable collateral. (Note” doesn’t explain if the contract per se is interlinked.)
Summary
Fundamental to credit markets is the problem oflimited liability. Limited liability a↵ected by the ability to post collateral.
This has three channels of influence:
Via adverse selection: project mix becomes excessively risky Via moral hazard: borrowers put in too little e↵ort to repay Via strategic default: borrowers take outside options
In all cases, lender power is correlated with inefficient outcomes. Lender power more likely when there is segmentation.
