The role of asymmetric information

Transcription

1 LECTURE NOTES ON CREDIT MARKETS The role of asymmetric information Eliana La Ferrara Credit markets are typically a ected by asymmetric information problems i.e. one party is more informed than the other. Informational asymmetries give rise to the classical problems of "adverse selection" and "moral hazard". In particular: i. Adverse Selection: agents hold private information before the relationship starts. Information asymmetry ex ante (Hidden Information). ii. Moral Hazard: agents action is not veri able, or agents receive private information after the relationship has begun. Information asymmetry ex post (Hidden Action) Stiglitz-Weiss Model (1981) 1. Adverse Selection (Hidden Information) Suppose a lender faces many potential borrowers. Each borrower would like to borrow a xed amount of money (L) to nance a risky project with return Q. Higher returns are associated to riskier projects. The distribution of returns is F (Q; #); where greater # corresponds to greater borrower risk and # 2 [#, #]. The lender cannot ascertain the riskiness of a project (hidden information problem). Assumptions: -Borrowers have limited liability -Both lender and borrowers are risk neutral Problem: The lender sets the interest rate i in order to maximize his expected pro ts. Model: The borrower can repay the loan or not depending on the project realization. His pro ts B are given by: Given borrower s behavior, lender s pro ts L are given by: B (Q; i) = max fq (1 + i)l; Cg (1) L (Q; i) = min fq + C; (1 + i)lg (2) The borrower defaults on the loan if Q (1 + i)l < C, where C represents borrower s collateral. In case of default, the borrower does not repay the loan and pays to the lender the maximum he can pay back, i.e. Q + C. Starting from Q = (1 + i)l C; the borrower repays the loan, and borrower pro ts are an increasing function of project riskiness. Claim 1 : For a given interest rate, ^{, there exists a critical value ^# such that only individuals with # > ^# will borrow from the lender. Proof: From Figure 1 it is evident that borrower s pro ts are bounded below and are a convex function of the return Q of the project. Thus, borrower s expected pro ts increase with risk (" # =)" E B ) The value ^# solves E B = 0; therefore: Z 1 0 max fq (1 + ^{)L; Cg df (Q; ^#) = 0 (3) 1

2 Borrower s expected pro ts, E B ; are increasing in #, hence: EB > 0 if # > ^# E B < 0 if # < ^# Claim 2 : As the interest rate, ^{; increases, the critical value ^# increases. Proof : This follows from di erentiating eq. (3): d^# d^{ (3) (3) =@^# > 0 (3) =@^{ < 0, (3) =@^# > 0 according to Claim 1. When the interest rate increases, only riskier individuals will borrow, as they have higher expected returns. Claim 3 : Lender expected pro ts are a decreasing function of the riskiness of the loan. Proof : From gure 2, it is evident that lender s pro ts are bounded below and are a concave function of Q. Therefore lender s expected pro ts decrease with risk (" # =)# E L ): Conclusions If borrower type is unobservable to the lender, the interest rate becomes an indirect screening device. An increase in the interest rate has two e ects on lender s expected pro t, E L : - An increase in the interest rates raises lender s expected returns in case of repayment. For given L and repayment rate: " i )" (1 + i)l )" E L - An increase of the interest rate has an adverse selection e ect causing safe borrowers to quit the market, hence leading to a decrease in lender s expected return (# E L ) For this reason lender s expected pro t function is non-monotonic in the interest rate (see Figure3). In particular there will be a value i such that for i > i the expected pro t will be decreasing in i because the adverse selection e ect dominates the increase in (1 + i)l. This means that the lender will not raise the interest rate above the threshold i : Consequently if demand for credit exceeds supply at i = i the interest rate does not increase automatically to equilibrate demand and supply and credit is rationed (credit rationing). Notes a) Even with risk neutrality, limited liability of borrowers leads to: - preference for risk among borrowers - aversion to risk among lenders This is due to the fact that lenders bear all downside risk. b) The assumption that lenders do not know borrowers characteristics is not realistic in close knit rural communities. 2

3 2. Moral Hazard (Hidden Action) Suppose now that project realization depends on borrower s e ort which is unobservable to the lender (hidden action). Before signing the contract all borrowers are identical (symmetric information ex ante), but after the loan has been granted, each borrower can decide the e ort level to be devoted to project realization. Let L be the loan amount required to start a project and C the collateral required to access the loan (with C < L). The e ort level (e) chosen by borrowers a ects the probability of project success so that project returns Q with prob p(e) are given by: 0 with prob (1 p(e)) The cost of e ort is e and p() is a concave function of e that is p 0 (e) > 0 and p 00 (e) < 0: Assumptions: Both lender and borrower are risk neutral Borrower has limited liability i.e. in case of project failure he does not have any obligation, and he loses the collateral C: Model solution: 1. First Best: Suppose that the project has been self nanced by the farmer or equivalently that borrower s e ort is observable by the lender who can specify in the contract the optimal amount of e ort. In this rst best case the objective for the farmer is to chose the optimal e ort level to maximize expected project returns, net of e ort and project costs. Therefore the problem is: max [p(e) Q (1 p(e)) 0] e L e F.O.C. p 0 (^e F B ) = 1 Q (1) Equation (1) determines the optimal rst best e ort level i.e.the level of e ort which maximizes expected project returns. 2. Second Best: Suppose now that the farmer cannot self nance the project and borrows L or equivalently that the lender cannot monitor borrower s e ort: In case of project success, the borrower has to repay R (1 + i)l; whereas in case of project failure he loses the collateral C: The borrower solves: max [p(e)(q R) + (1 p(e))( C)] e e F.O.C. p 0 (^e SB ) = 1 Q + C R (2) Equation (2) gives the optimal level of e ort chosen by the borrower. Di erently from the rst best case in which the optimal level of e ort only depends (positively) on project returns in the case of success, the second best optimal e ort level also depends on the collateral (C) and on the total debt (R). 3

4 ^e SB (R; C + ) Remembering that the probability of success p(e) is a concave function (p 0 (e) > 0 ; p 00 (e) < 0) ; it follows that the level of e ort chosen by the borrower is a positive function of C > 0 ; and a negative function of < 0 Intuitively, a higher liability in case of project failure (i.e. a higher collateral) will induce the borrower to increase e ort, whereas a higher debt burden will induce less e ort by lowering borrower s net pro t in case of success. In order to compare rst and second best solutions, we need to analyze the lender s pro t (). Depending on credit market structure such pro t may be greater or equal to zero. If the credit market is competitive = 0; whereas a monopolistic market implies > 0: In any case, since lenders can quit the market with no cost, we must have that > 0: Lender s expected pro t are given by: p(e)r + [1 p(e)] C L > 0 Collecting p(e) : p(e)[r C] {z } >0 + [C L] > 0 {z } <0 By assumption we know that C < L: Hence [R C] must be greater than zero.to guarantee that > 0: Rewriting equations (1) and (2): p 0 (^e F B ) = 1 Q p 0 (^e SB ) = 1 Q + C R {z } <0 (1) (2) Remembering that p(e) is concave it s now clear that: e F B > e SB Implications: With ex post information asymmetry the borrower has incentives to under supply costly e ort. In particular, the farmer cares about project returns and has an incentive to put in higher e ort when he can fully retain gains from e ort, i.e.returns from higher success probability ( rst best). If instead the farmer needs to borrow money to nance the project or if the lender cannot force the borrower to provide a certain level of e ort, the debt burden will reduce borrower s payo in case of project success and therefore debt will lower borrower s e ort incentives (debt overhang). Note that the reduction of e ort w.r.t. the rst best is higher the higher is the debt burden (R) i.e. the higher the interest rate. This last implication further extend the credit rationing result to situations with ex post information asymmetries. 4

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