Slides 9 Debraj Ray Columbia, Fall 2013 Credit Markets Collateral Adverse selection Moral hazard Enforcement
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
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: affects who seeks credit (adverse selection) affects how credit is deployed (moral hazard) affects repayment incentives (strategic default) affects lender-borrower combinations (segmentation) affects design: group lending and peer monitoring affects loan types: e.g., working capital vs. fixed capital generates credit rationing: interest rates don t clear the market.
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. A : 230,000 or 0 equal prob, expected return unchanged. Under limited liability, bank strictly prefers Project B. But the borrower strictly prefers Project A! (Why?) Two extensions of the example: The bank attracts type A borrowers (adverse selection) The borrower diverts money to type A projects (moral hazard)
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 affects the repayment probability! Aside: In fact, sometimes a deliberate ploy by lender. Lender valuation of collateral V l, borrower valuation V b. Borrower prefers to repay if L(1 + i) < V b + F where L = loan size, i = interest rate, F = default cost.
Aside, contd. Lender wants his money back if L(1 + i) > V l Thus loan repayment in the interest of both parties if V b + F > V l. But if V b + F < V l. interest rate may be adjusted to facilitate collateral seizure. (Note: analysis works best for inelastic loans.) This aside makes very clear that i affects default.
Adverse Selection Over Projects Stiglitz and Weiss (1981) Borrowers differ in riskiness. cdf of returns R given by F (R, θ) Assume same mean: RdF (R, θ) = RdF (R, θ ) for all θ, θ. Larger θ s more risky in the sense of strict SOSD: y 0 F (R, θ)dr > y 0 F (R, θ )dr for all y in interior of support, whenever θ > θ. Startup cost of project: B, borrow at rate r, collateral C. Limited liability in loan repayment; repay if R + C (1 + r)b.
So borrower s return if project pays off R is given by π(r, r) max{r (1 + r)b, C}, and lender s return is given by ρ(r, r) min{r + C, B(1 + r)}. Expected payoff to borrower of type θ is therefore ˆπ(θ, r) = π(r, r)df (R, θ), And expected payoff to lender from borrower of type θ is ˆρ(θ, r) = ρ(r, r)df (R, θ). ˆρ decreasing in θ, but ˆπ increasing in θ.
Payoffs Payoffs C R C R Expected payoff to borrower increases in θ, opposite for lender.
These relationships with θ have the following implication: Define a threshold θ(r) by the condition ˆπ(θ(r), r) 0. Then θ(r) is increasing in r and the set {θ θ θ(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(θ(r))] θ(r) ˆρ(θ, r)dg(θ) 1 d(r) will typically be non-monotonic in r.
0Lender Payoff r* r With free entry and exit of banks, d(r) is also deposit rate. Deposit supply Ŝ(d), but S(r) Ŝ(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.
Notes on equilibrium: If more than one r, depositors indifferent, borrowers prefer lowest 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 with S(r) > S(r ). Then there is credit rationing in equilibrium.
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.
If any eq r below r, then every other eq r above r to maintain depositor indifference. Deviant bank: offers r, and some d (d(r), d(r )), takes all the rationed borrowers from r, makes positive profit. If every eq r no less than r, let r 1 be lowest of them. By earlier note, D(r 1 ) S(r 1 ) and if r 1 = r, then unique. So suppose that r 1 > r. Then S(r 1 ) D(r 1 ) < D(r ) = S(r ). So S(r) > S(r ) and consequently, d(r) > d(r ). Deviant bank charges r = r, offers d (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 ). r r D r D r S r S, D S r S, D r Then equilibrium must involve credit rationing.
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 offers r and d (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 r 1 < r with S(r 1 ) > S(r ). Choose r 1 as first conditional maximizer of S(r) below r.
Let r 2 be the smallest value of r > r 1 such that d(r 2 ) = d(r 1 ). Such r 2 exists because r 1 is a local but not global maximizer. r D(r) S(r) r ** r 2 r * r 1 S(r 2 ) = S(r 1 ) > S(r ) = D(r ) > D(r 2 ), so excess supply at r 2.
Complete the proof: Take the equilibrium set to be {r 1, r 2 }. If all funds go to r 1, excess demand, if all to r 2 excess supply. Using depositor indifference, 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 different projects. Projects indexed by θ. Return Q(θ) with prob p(θ), 0 with prob 1 p(θ). Arrange such that Q(θ) increasing, wlog p(θ) decreasing. Each project requires the same loan size of B. Borrower with collateral C, facing r, chooses θ to max p(θ) [Q(θ) B(1 + r)] [1 p(θ)] C, where Q(θ) exceeds B(1 + r), otherwise no borrowing. Determines riskiness θ(c, r) as a function of C and r.
Proposition: θ(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 θ to maximize p(θ)[q(θ) Z]. Let Z 1 > Z 2. Let θ 1 and θ 2 be the two maxima (assume unique): Then p(θ 1 )[Q(θ 1 ) Z 1 ] > p(θ 2 )[Q(θ 2 ) Z 1 ], while p(θ 2 )[Q(θ 2 ) Z 2 ] > p(θ 1 )[Q(θ 1 ) Z 2 ]. Adding these two inequalities, we can conclude that [p(θ 1 ) p(θ 2 )] (Z 1 Z 2 ) < 0. Therefore p(θ) is increasing in Z.
Credit rationing: Entirely possible that for low values of C, max r p(θ(c, r))(1 + r) < 1 + r where r is opportunity cost of funds. Collateral and interest: Under competitive lending, p(θ(c, r))b(1 + r) + [1 p(θ(c, r))]c = B[1 + r]. LHS as function of r. Nonmonotone, with end-point conditions.
C _ B(1+r) r* p(θ(c,r)b(1+r) + [1- p(θ(c,r)]c r _ r Equilibrium rate of interest falls as collateral goes up. Argument does not work for monopoly lending.
Moral Hazard and the Debt Overhang Costly effort 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 effort. If agent is self-financed, choose e to maximize p(e)q e L. Assume unique choice e ; described by the first-order condition p (e ) = 1 Q This is the efficient, or first-best level of effort.
Now consider a debt-financed agent. R = (1 + r)b is total debt, C < B is collateral. Now the effort choice of a borrower facing a debt R given by max e p(e)(q R) [1 p(e)]c e Optimal choice ê(r, w) is defined by the first-order condition: p (e) = 1 Q + C R. ê(r, C) < e. ê(r, C) is decreasing in R and increasing in C. This is the debt overhang. Lender return given by π = p(e)r + [1 p(e)]c B[1 + r].
Equilibrium debt and effort in the credit market R E 3 Lender's Iso-Payoff Curves E 2 E 1 Borrower's Incentive Constraint e Proposition. Equilibria with higher lender profits involve higher interest rates, but lower levels of effort and social surplus. Note: interest rate ceiling even at E 3.
Competition: Effect of an increase in collateral. Evaluate at old level of lender payoff R Lender's Iso-Payoff Curve E E' Borrower's Incentive Constraint e R comes down, e goes up.
Monopoly: Effect of an increase in collateral. Lender maximizes in each case, subject to incentive constraint. R Lender's Iso-Payoff Curve E E' Borrower's Incentive Constraint e (Generally) R and e will both go up.
Strategic Default Focus in this section: outside options, variable loan size Q = F (L) Borrow L repeatedly (working capital) Stationary contract (L, R). δ). Fundamental incentive constraint: If repay, get [F (L) R]/(1 If default, get F (L) today and v per period starting tomorrow. δf (L) R δv. Fundamental self-enforcement constraint.
Combine enforcement constraint and lender iso-payoff. R Self-Enforcement Constraint!F(L) - R -!v = 0 Lender's Iso-Payoff Line R - (1+r)L = constant L!v
Second-best: maximize borrower payoff over feasible set: R R E E Borrower's Iso-Payoff Line F(L) - R = constant Borrower's Iso-Payoff Line F(L) - R = constant L L!v (A) IC Not Binding!v (A) IC Binding 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 effort and social surplus. Use diagram to prove higher interest rates. R Self-Enforcement Constraint E 2 E 1 E 3 Lender's Iso-Payoff Line R - (1+r)L = constant!v L
Effect of lowering the outside option v. Depends on lender power R E' R E E' E L L!v!v Competition: interest rate, L ; monopoly: opposite effects
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 + r δ Let w(v) be the maximized value. L v.
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 = pδv + (1 p)w = 1 p 1 δp w. Then we can write v = (1 ρ)w, where ρ p(1 δ) 1 δp can be viewed as the scarring factor.
Notes on the scarring factor: If p is close to one, so is the scarring factor ρ. On the other hand, ρ 0 as δ converges to one. (But wait for a final verdict on the net effect of patience.) Theorem. Define ρ (1 + r)(1 δ) ˆL/δ F ( ˆL) (1 + r) ˆL (0, 1), where ˆL is the maximizer of F (L) 1+r δ L. Then a competitive equilibrium exists if and only if ρ ρ. Implications for theories of social capital. Information and development.
Proof. Necessity. Start with equilibrium loan L. Inspect maximization problem to see that L ˆL (why?). If, contrary to the proposition, we have ρ < ρ, then v = (1 ρ)[f (L) (1 + r)l] = [F (L) (1 + r)l] ρ[f (L) (1 + r)l] > [F (L) (1 + r)l] ρ [F (L) (1 + r)l] = [F (L) (1 + r)l] (1 + r)(1 δ)/δ [F (L) (1 + r)l] F ( ˆL)/ ˆL (1 + r) (1 + r)(1 δ)/δ [F (L) (1 + r)l] [F (L) (1 + r)l] F (L)/L (1 + r) = [F (L) (1 + r)l] (1 + r)(1 δ)l/δ = F (L) 1 + r L, δ where the weak inequality in above string above uses L ˆL. But this contradicts the enforcement constraint.
Proof, contd. Sufficiency. Suppose that ρ ρ. Define ˆv (1 ρ)[f ( ˆL) (1 + r) ˆL]. At v = ˆv, the borrower s max problem has a solution, because: F ( ˆL) 1 + r δ (1 + r)(1 δ) ˆL = F ( ˆL) (1 + r) ˆL δ = (1 ρ )[F ( ˆL) (1 + r) ˆL] (1 ρ)[f ( ˆL) (1 + r) ˆL] = ˆv. ˆL At this point, (1 ρ)w( ˆv) (1 ρ)[f ( ˆL) (1 + r) ˆL] = ˆv. On the other hand, there is v ˆv such that F ( ˆL) 1 + r δ ˆL = v. That is, ˆL is the only feasible solution when v = v. At this point, (1 ρ)w( v) = (1 ρ)[f ( ˆL) (1 + r) ˆL] = ˆv v. w(v) is continuous, so there is v such that (1 ρ)w(v ) = v.
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 contract as well as future loan contracts if default on the loan contract. Nonmarketable Collateral. In interlinked relationships, easier to accept non marketable collateral. (Note doesn t explain if the contract per se is interlinked.) Removal of Distortions. Multi-dimensional pricing.
Summary Fundamental to credit markets is the problem of limited liability. Limited liability affected 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 effort 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.