A Model of Simultaneous Borrowing and Saving. Under Catastrophic Risk

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1 A Model of Simultaneous Borrowing and Saving Under Catastrophic Risk Abstract This paper proposes a new model for individuals simultaneously borrowing and saving specifically when exposed to catastrophic risk. This risk leads to higher savings for the lowest income levels even when current consumption must be financed through borrowing at much higher interest rates. Empirically this is tested using data from Grameen bank in Bangladesh and comparing that to low income individuals in the United States where catastrophic risk exposure is less due to strong social safety nets and stronger financial markets following a disaster. This testing shows the model to hold and does explain partially the simultaneous borrowing and saving occurring in developing countries. 1

2 1 Introduction There has been a great deal of research on the borrowing and saving habits of individuals in developed countries. It is understood that individuals will use dis-savings, a drawdown of the savings, for consumption but at rates far lower than is suggested in neoclassical economic theory (Deaton, 1991). The two major lines of thought on this follow a behavioral format. The simultaneous borrowing and saving occurs due to a forced savings giving higher utility overall. The benefit may be in feigning poverty to reduce the claims on wealth by members of the extended family. Finally this simultaneous borrowing and saving may be Jonathan Morduch (2009) approaches the issue from the perspective that the threat of default on the loan has serious social consequences, building on earlier work by Karna Basu (2007), and that social consequence compels repayment of loans. This compulsion is greater than the internal drive to replenish savings which were spent down and so the negative spread is a payment to induce savings. The alternative proposed by Baland, Guirkinger, and Mali (2007) was that people with unobserved high savings use loans to signal financial distress falsely. The negative spread is therefore a way to state that financial constraints prevent loans or gifts to others that is not simply viewed as cheap talk. The alternative that I propose is that the threat of a large negative shock, or catastrophic shock, will threaten the ability to borrow in the future so current borrowing is used to a greater extent. The negative spread is a payment to transfer the risk of a catastrophe back to the lending institution. 2

3 The purpose of the model is to explain how a decision is made for borrowing and or lending is made by an individual. The model initially is for a microfinance or community banking which is in the same area and will suffer the catastrophe with the individual. The model is then expanded for more geographic dispersion to give some more stability to the microfinance institution. Savings are viewed first as a risk free vehicle and then an expansion is made in keeping with Basu where the savings has some risk. 2 Model 2.1 Assumptions The model is a two period model where an individual makes a banking decision in the first period. After the choice is made, nature will determine consumption in the second period. The individual is assumed to be a utility maximize that is risk averse. The choices available in the first period are to save or consume some portion of the initial endowment with the ability to also borrow, to temporally adjust later consumption to the first period. In the second period the savings are available for consumption after loans are repaid and all wealth is consumed. The individuals can choose to take wealth or loans and invest that in other business ventures which can be used for future consumption. All rates are considered to be gross rates (1 + r = R). To ensure the terminology is consistent the savings rate that one receives from savings is S. S is a risk free rate in the sense that savings in the bank is considered guaranteed to be available when needed for consumption. When there is a risk of the bank not honoring the obligation then the rate will be given the designator. In the model it is also possible that the savings are kept at home instead of at a bank and those savings earn no interest. The savings rate is lower than the interest rate of borrowing 3

4 and the interest rate of borrowing is L. If there is a possibility of the loan being forgiven then the lender will require a higher rate for that possibility and that higher rate is given by the designation. It is possible that the savings at home will reach a threshold necessary to make some minimum investment for a cottage industry (e.g. home looms, tea stands, or a rickshaw) and that investment will result in higher wealth available for consumption in period two. While the savings and this investment are occurring together, the return is different so the return on investment will be given by the designation P. All of these gross rates are assumed to be greater than one with. The choice variables for the individual will be then to determine how much to save and how much to borrow. Savings at home will be given the designation s. Savings at a banking institution, which will receive the interest rate given above, will be given the designation n. Savings which are sufficient for investment by exceeding the minimum threshold for starting a home business are given the designation i. Money which is borrowed from a banking institution will be given the designation b. The initial endowment of wealth in period one will be given the designation, ω, and that is not controlled by the individual. The risk some catastrophe could occur and destroy any savings at home and simultaneously any home based industry is given the designation δ. The risk that the bank will fail to honor the demand for repayment is given by the designation ε. The two risks are independent. 2.2 Initial (Base) Case In keeping with pervious literature, the case is initially explored with an agent with timeconsistent preferences. The individual will maximize utility in the following manner if there is an 4

5 ability to invest but there is no access to banking, or if the gross rate of return for savings is equal to one. If the investment is not feasible then the optimal savings choice would be to simply consume half of the endowment in the first period and consume half of the endowment in the second period. If the investment if feasible ( and with the restriction that ( ) since s is restricted to being nonnegative, then the individuals decision takes the form of the following. ( ( ( ) This would mean that the individual will invest in cottage industry without access to banking and so investment may be assumed to be worthwhile investment in the later models. This is only true though if the investment is affordable since s is nonnegative. With the possibility of investment and borrowing, the individual will have a slightly different utility maximization problem. The restriction on feasibility may be relaxed as the possibility of borrowing means that sufficient borrowing will occur to pay for the investment. The appropriate model in a case with banking is as follows. ( ( With this problem the individual has both savings at home and a loan which must be repaid with interest. The additional savings at home without interest would be there due to a desire for liquidity. 2.3 Safe Financial Case 5

6 If we have the case that individuals can invest in business ventures, borrow from a financial institution and save both at home, amount s, and in a bank, amount n, and receive interest in the latter case, our model is as follows. ( ( With risk free savings, individuals will not save at home and will decide to always save in the financial institution. Therefore the optimal value of s is zero. Since we know that S is less than L which is less than P we further can write the above maximization in two forms. If the cost of the investment is less than half of the initial endowment, ( ), then the investment would be made without the need to borrow. If the minimum investment is higher, then there would be borrowing sufficient to make the investment but no higher. The pair of functions would then be of the following form. ( ( ( ) ( ( ( ) 2.4 Risky Savings If one includes the possibility of savings being lost by the lending institution, the idea of borrowing and lending must be reexamined. There is some chance, 0 ε 1, that the savings are available in the next period. The two-period model can then be written in the following form with a higher rate on savings due to the added risk. ( ( ) ( ( 6

7 This model can now be thought of as a choice between investing in home savings with no risk, and investment in industry and a profit greater than the investment, and savings in a bank with a second period balance greater than the initial savings but with some risk. If the amount saved at home is greater than or equal to the required investment, i, all rational individuals will chose to invest. If there is some minimum threshold that must be reached though then that home savings may still occur, meaning that s < i is an upper bound for home savings. The borrowing which occurs can be for direct consumption in the first period or to invest in the profit making industry. The utility can be separated into two independent periods with period one being represented with the first term in the above and the second term being represented with one of the second terms depending on which state of the world is realized. The plot of the second period utility would be an increasing function of s since that is a simple transfer of wealth from period 1 to period 2. The plot of second period utility is not strictly increasing in n even though that is also savings since those savings are risky. The plot of second utility is strictly decreasing in the size of the loan b as that loan repayment with interest is required in that second period. 2.4 Catastrophic Risk Case The case could be made that the savings at home and the investment made are not truly risk free. The savings at home could be lost or stolen, for instance, and the investment in cottage industry could be destroyed in a fire or due to vandalism. The specific case that I explore here is to have a chance of the catastrophic event and the savings at home and the investment in industry are lost. To simplify the issue, I relax the condition of loss in the savings with the banking 7

8 institution. Because of this change I will call the chance of no loss occurring, δ, and the chance of loss will be (1-δ). The maximization problem then can be written as the following. ( ( ( ( With the above it is clear that with some chance of loss the chance for default exists if the rate of borrowing, L, is greater than the rate for savings, S. Normally the rates at which one can borrow is greater than the rate for savings. We know that if the loans that are taken are overcollateralized so that ns bl then there is no chance of default. 2.5 Catastrophic Risk Case with Loan Forgiveness If the chance exists that loss occurs and in such a case the loans would be forgiven then the model differs. The model then will be of the following form. ( ( ( ( Solving this would have the savings in the bank as a strategy to offset the risk of a catastrophic loss. Notice that in the above model, the risk of loss of bank savings has gone to zero. Adding the risk of loss will create a scenario where one could lose everything but with loan forgiveness in that case the model is not very different from above except that savings in the banking institution and savings at home coupled with investment at home will be in proportion to the ratio of δ/ε or the ratio of the relative risks. 2.6 Discounting Future Consumption So the next step is to evaluate the parameters of the above models with a discount of the future utility. This measure of discount is given by β < 1. This discounting could be for any 8

9 reason that future consumption is of less value than current consumption as a measure of the impatience. The models below will be under the same assumption regarding the gross rates of return as given below. With a time consistent framework the best response is the following, where the only choice is how much savings are necessary to induce an investment. It is possible that no loan will be taken if i is sufficiently small relative to the initial wealth, ω. ( ( ( ( ( ( ( ( ) ( The optimal time consistent model with no option for investment is given by in the following form. There is no reason to borrow without an investment opportunity which will pay for the increased amount due in the following period. Since there is the possibility epsilon that the savings in the bank is available we include that in mode with the chance that a disaster occurs and all home savings and any home industry is destroyed. ( ( ( ) ( ( We then can consider an individual that does not plan to invest. This individual will save some income for future consumption as above but we must also consider the savings at home as a function of the savings in the bank. In this case s(n) is simply the functional form of s given a value of n. ( ( ( ( ( ) ( ( ) 9

10 ( ( ( ( ( ) ( ( ) Now the next step is to evaluate the individual under the scenario where some higher loan is taken so that the individual is indifferent between saving at home and investing in a home project. This loan value, b*, is the one which makes the following true. In looking at the below we can see that b* is higher than b. ( ( ( ( ( ( ( ( The final step is to evaluate the scenario of having the individual in the situation where debt may be forgiven. We already have evaluated the decision to save at home in the face of risk of savings in a bank and the decision to take out loans when home savings are not sufficient to have investment. This builds on the early result of borrowing to save at home because it allows for increased consumption in the first period. So this is to evaluate how borrowing can be at a level, b**, where one is indifferent between having savings at home and savings at a bank. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 10

11 Therefore we can see that even though previous theory demonstrated that a rational agent would not choose to borrow and save if the rate L was greater than the rate S, it can be the case with the time inconsistent preferences that such will occur. 3 Comparative Statics 3.1 Example To evaluate the model for predictive values, I consider the above in equilibrium. To evaluate the time inconsistent preferences, each period is evaluated separately. To simplify the evaluation, the initial wealth, ω, is normalized to Implications 4 Empirical Analyses To test this model, I am comparing the simultaneous borrowing and savings rates of the poor in Bangladesh with the poor in the United States. The expectation is that there is a greater number of fully collateralized debt obligations held by the poor in Bangladesh as they are facing a risk that will remove their ability to use informal networks of borrowing from religious or family organizations when funds are needed most. The credit systems of the United States are geographically spread to minimize the risk of a single catastrophe impacting access and there is an existing social network to address immediate concerns (e.g. Federal Emergency Management Agency) so there is not that additional motive. 4.1 Data 11

12 The data for the savers in Bangladesh come from Grameen Bank. Grameen is a microfinance organization based in the capital of Dhaka but with branches across the entire country. In many areas, Grameen is the only banking facility for saving and a lower cost facility for borrowing (a more informal network of money lenders exists which very high interest rates). There is minimal underwriting for the borrowing of funds as the use of societal pressure and the mutual ownership of the bank by the borrowers compels high repayment rates. There is a potential issue of selection bias in that the data only includes the accounts held with the one bank and it is possible (but a violation of Grameen policy) to simultaneously borrow with other banks. This additional borrowing would only strengthen the case that borrowing acts as additional precautionary savings. The data for the United States comes from the Federal Reserve. That data is more robust in that the type of borrowing is defined by the use, so I can compare just consumption borrowing or total borrowing to see if there is a difference when mortgages and student loans are included. 4.2 Analysis form. The methodology used in the analysis of the data is a regression model of the following The vector B is the total borrowing by an individual. The alpha vector is the intercept term for each individual. The vector of betas is the relationship between current savings at the financial institution and the borrowing. The vector C is the demographic controls used for 12

13 matching the individuals in each sample group and the chi vector is the relative weight of those. There is an epsilon vector to represent noise. 5 Conclusion 6 Appendix 13

14 References Baland, Jean-Marie, Catherine Guirkinger, and Charlotte Mali Pretending to be Poor: Borrowing to Escape Forced Solidarity in Cameroon. (Mimeo) Basu, Karna A Behavioral Model of Simultaneous Borrowing and Saving. (Mimeo) Deaton, Angus Saving and liquidity Constraints. Econometrica 59, King, Amanda Swift and John T. King Golden Eggs versus Plastic Eggs: Hyperbolic Preferences and the Persistence of Debit. Journal of Economic Finance 35, Laibson, David Golden Eggs and Hyperbolic Discounting. The Quarterly Journal of Economics 1121 (2),