Development Microeconomics Tutorial SS 2006 Johannes Metzler Credit Ray Ch.4 Problem n9, Chapter 4. Consider a monopolist lender who lends to borrowers on a repeated basis. the loans are informal and are not backed up by written contracts. The lender has no way to recover a loan if the borrower chooses to default. The lender, however, threatens to cut off credit in the future to any defaulting borrower. Borrowers discount the next period s earnings by a discount factor of 0.5. Borrowers use the loan in cultivation. Cultivation can be done using one of two techniques. The first requires initial working capital of $00 and produces net output worth $300. The second requires $500 of working capital and yields net output of $,000. Find the amount of loan the lender will advance to each borrower every period in order to maximize his own profits. How much is the scheduled repayment and the implicit interest rate? What are these profits? How much does the borrower earn every period from the deal? Introduce a new factor into this scenario. Suppose the lender can keep some of the borrower s assets (like jewelry) as collateral, which he will seize in the case of default. The present value of the asset to the borrower is $300. Recalculate the optimal loan, repayment amount, implicit interest and profits in this case. Compare the two cases and summarize the effect of collateralization on the other terms of the loans. Does it increase or decrease the welfare of the borrower and the lender? Let s see which is the interest rate that lender can charge in the two cases to maximize his surplus. The maximum surplus available in the economy is given by the difference between the final output and the initial investment. Can the lender demand all this amount back as repayment? Maybe - if he wants to be sure that the borrower will not choose to default, he has to set the interest rate such that the borrower will be indifferent between paying back the loan or voluntarily default. We have, in other words, to take into account the outside option for the borrower, that is the payoff that he could get if he decide to break the contract and not give back the money. We know that if he defaults today, he will not be able to borrow anymore in the future, and we also know that he discount future earnings by a factor equal to 0.5. The interest rate on the loan that leaves him indifferent between defaulting and paying back the money is then the one that equalizes the present value of all his future earnings and his outside option. That is:
2 T [f(l) - L(+ i)] + δ [f(l) - L(+ i)] + δ [f(l) - l(+ i)] +... + δ [f(l) - L(+ i)] = f(l) () The left hand side of the equation shows the present value of the future earnings, while the right hand side represent the outside option for the borrower: if he decide to default, he will pocket the whole output produced and then he will get nothing from the next period on. This is where you have to change your notes from class!! I dropped the term [f(l) L(+i)] in the beginning (in my equation the first term was already multiplied/discounted by 0.5). My explanation was, that the lender invests now and earns the proceeds from his investment (e.g. harvest) in one period from now, and so the first cash inflow f(l) and outflow -L(+i) should already be discounted once (i.e., multiplied by 0.5 ). However, to be logically consistent, the right-hand side cash flow of the outside option (taking the output and defaulting/running: f(l) should also be discounted.) However, the decision to pay back vs. to default is taken by the borrower after one period/year when he has just finished the project and keeps the output f(l) in his hands. So from that point of view and point in time, if he runs, he gets f(l) without future possibilities to borrow (right-hand side of ()). If he wants to keep borrowing and thus pays back the original loan, he earns the difference between output and full repayment, f(l)-l(+i) both terms are not discounted. This is the same as assuming (as some of you interpreted from the wording of the question) that everything happens instantly: lending, production, output and paying back/defaulting would happen in an infinitesimally short time period (=now). So with a discount factor of δ and thus a discount rate r=/ δ,we sum from 0 (not ) to infinity, i.e., [f(l) - L( [f(l) - L( + i)]* t= 0 ( ) + r t = f(l) + i)] + [f(l) - L(+ i)]* t= ( ) + r t = f(l) (2) or (3) Applying the perpetuity formula to the second term on the left-hand side of (3) we get [f(l) - L(+ i)] [f(l) - L(+ i)] + = f(l) r 0.5f(L) - L For δ =0.5 (equivalent r=) this gives i =, the maximum rate that the L lender can charge on the loan without fearing the (voluntary) default of the Perpetuity: a constant stream of identical cash flows until infinity. The formula for determining the present value of a perpetuity is as follows: 2
borrower. In fact this is exactly what I had incorrectly calculated in class for the case of δ=2/3, r=/2. This yields i max =0.5 that means that the maximum interest rate that can be charged when the lender finances the first project is 50%. Which is the interest rate that he can charge if he wants to finance the second project? Using the same procedure, we get that imax2 = 0. So, if the lender finance the second project he cannot charge interests and the borrower will pocket all the surplus: a single cent more asked on the loan, and the borrower will run away with the money. The only feasible alternative for the lender is then to finance the first project. However, if he finance the first project asking for an interest rate of the 50%, from the next period on, the borrower will not need anymore credit from the lender. He will in fact receive $00, will get $300 from his investment, he will pay back the principal plus the interests (that is $50) keeping then $50 for him and, from the next period on, he will not need credit to continue his activity: he can adopt the same agricultural technique using his savings. That is probably something that is good for the economy as a whole, but that the lender would maybe dislike: he will have a profit of $50 in the first period and then nothing from the next period on. What happens if the lender can rely upon a collateral that the borrower Evaluates at $300? The line of reasoning goes as before. We have, however, to change the participation constraint for the borrower, given that now the outside option for the borrower is different: he can pocket the whole output, but he will loose $300. The new participation constraint is: [f(l) - L( 2 T + i)] + δ [f(l) - L(+ i)] + δ [f(l) - l(+ i)] +... + δ [f(l) - L(+ i)] = f(l) - 300 Again, the first term on the left-hand side is not discounted. Solving for i in the two cases, we get that: imax = 200% and that imax2 = 30%. As before, the lender will prefer to finance project one: the profits he gets from the deal are higher than financing project 2 (L i > L2 i2). Furthermore, in this case the borrower will always need credit given that the lender will extract all the surplus from him: the profits for the borrower are zero (f(l) L(+i) = 300 00(+2) = 0) while the lender will have a profit of $200 every period. What can be concluded on the effect of the collateralization on the welfare of the two agents? For sure the collateralization increases the welfare of the lender and decreases the welfare of the borrower. However, collateralization can help in creating a credit market that otherwise would be more difficult to start. 3
Using an alternative case of δ =2/3 as I did in class yields the following (check this at home!): Without collateral: imax = = 00% and imax2 = /3 o Note that the bargaining position of the lender is increased due to the lower discounting factor of the borrower (he can now charge 00% interest instead of 50% for project ) since the option to default becomes less valuable to the borrower. With collateral: imax = 3 = 200% and imax2 = 0.53 = 53% o Again, the lender s bargaining position is increased with collateral. 4
Problem n0, Chapter 4. SelfHelp is a newly formed credit cooperative which receives partial financing from government banks. Members of SelfHelp can deposit savings with the cooperative and they can also turn to SelfHelp for a loan if they need one. If a borrower defaults on a SelfHelp loan he is punished (which is equivalent to a loss of monetary value F) and excluded from future dealings with SelfHelp. The value (to the borrower) of these future dealing is some number S. However, there is no telling whether SelfHelp will survive in the future; denote the probability of survival by p. a) If each member is risk-neutral, what is the expected value of dealing with SelfHelp in the future? If SelfHelp survive, the borrower can deal with the institution a certain amount of money, and these dealings have a value S. However, if the institution does not survive, no dealings will take place and then there is no value for the borrower in having dealings with the dead institution. The expected value of future dealings is then given by: p S + ( p) 0 b) If a member has an outstanding loan of L and needs to repay it (along with interests at rate r), write down the value of his net gain from default. If a member has a loan and decide not to repay the loan, he can pocket the whole output (f(l)) generated by his investment saving the costs of repayment (L( + r)); he will incur in the monetary loss F and also he will loose all the future dealings with SelfHelp if she survives and this loss is evaluated by the borrower ps. All in all, then the net gain to the borrower D from defaulting (outside option) is given by D = f(l) F ps c) Now suppose that the probability of SelfHelp s survival depends on the percentage of borrowers who repay as well as the quantum of the government assistance should a high rate of default occur. Draw a graph with the repayment rate on one axis and the survival probability on the other, and show how this graph shifts with the amount of government assistance. 5
We can imagine the probability of survival of SelfHelp to be s-shaped, as in figure : for low repayment rates and given level of government assistance (set, for example at G0), the probability of survival of the institution is low. As the repayment rate increase, the probability of survival in the next period approaches. If the government increases his assistance in case of high rates of default (repayment rates below the threshold R), the line is shifted upward. In other words, for each level of the repayment rate below R, the probability of survival increases. d) Use parts (b) and (c) to show that in general there can be two outcomes or equilibria for the same parameters: (i) there is no (voluntary or strategic) default and SelfHelp survives with little or no government assistance or (ii) there are high rates of default and SelfHelp survives with low probability despite the government assistance. From part b) we can see that the probability of survival increases the net gain from strategic default, because it reduces the expected value of the future dealings with the institution. However the survival probability is not exogenously given but rather depends on the overall rate of repayment. That means that the behavior of each borrower, modifying the total rate of repayment, has an influence on the behaviour of the other borrowers. In other words, if a borrower think or expect that the prevailing rate of return will be low (maybe because he knows that in a certain area the harvest was particularly bad, so that many farmers will involuntarily default), his expected p will be low and consequently also the expected value of the future dealings with SelfHelp will be low, convincing then the borrower to voluntarily default. If the majority of 6
the borrowers has this kind of negative expectation on the global rate of repayment, the majority of them will default (in addition to the ones that will involuntarily default): the actual rate of return will be low and consequently SelfHelp will probably die despite the government help. On the other hand, if the majority of the borrowers expect a high rate of repayment, we will observe a low rate of strategical default, a high rate of repayment and the survival of SelfHelp even without the government help e) Show that a credible promise by the government to bail out SelfHelp in times of trouble can lead to a unique equilibrium in which all borrowers repay and little government assistance is actually required. Now, if the government can credible convince the population that in case of trouble he will not abandon SelfHelp, but he will rescue her, the expectations on the probability of survival are modified. The main point here is that the expected probability of survival that each borrower has in mind when he has to decide if defaulting or not, becomes less and less dependent on the global rate of repayment. In the extreme case where the government assures that he will save SelfHelp anyhow, even if the repayment rate is zero, the probability of survival raises to, irrespectively of the total repayment rate. If the future value of the dealings S is high enough, and if the probability of survival is very high even if many borrowers will default, the net gain of a strategic default can be quite low: all the borrowers that are able to do so, will repay the loans and actually few government help is needed. f) Using part (b), show how the survival probability of SelfHelp can affect the size of the loans it can make to the borrowers. To avoid strategic default the following condition has to hold: f(l) L( + r) f(l) F ps That can be turned into: F + ps L( + r) loss from default cost of repayment For a given interest rate, if the probability of survival is low, also the loss from a strategic default is low, and the only way to keep the previous condition is keeping L small. A small loan, however, could be not optimal from a social point of view, because often adopting new technologies (that could improve the rate of growth of the whole economy, or that could help small farmers to escape poverty) requires big investments. So a credible promise to rescue SelfHelp could be a good solution: from the one hand side, that will prevent strategic default and from the other hand, will help SelfHelp in giving bigger loans. 7