Risk Diversification through Multiple Group Membership in Microfinance Ratul Lahkar∗ and Viswanath Pingali† December 26, 2013
Abstract We consider group formation in the joint liability setting in microfinance. Joint liability imposes additional liability of having to repay for group partners should they fail to repay. Multiple group membership allows diversification of that risk, and therefore, is welfare enhancing for risk averse agents. Welfare enhancement occurs even when the total loan of an agent is unchanged. Therefore, multiple borrowing is not synonymous with overborrowing. Keywords: Microfinance, Joint Liability, Risk Diversification . JEL classification: D04; D81; G21.
∗
IFMR, 24, Kothari Road, Nungambakkam, Chennai, 600 034, India. email:
[email protected]. IIM Ahmedabad, Vastrapur, Ahmedabad, 380 015, India. email:
[email protected]. The author thanks Research and Publications Division, IIMA, for financial support. †
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1
Introduction
Evidence suggests that the expansion of microfinance institutions (MFIs) in developing countries has generated the phenomenon of multiple borrowing (McIntosh et al., 2005). This is the practice in which borrowers obtain credit from more than one MFI through membership in different groups. One explanation of multiple borrowing, popular in the media, is that following the proliferation of MFIs, clients are seeking to over-borrow beyond their requirements or capacity to repay.1 It has been feared that such indiscriminate borrowing may lead clients to a debt trap. Such concerns have been serious enough to trigger policy response from certain governments to curb multiple group membership.2 This popular explanation of multiple borrowing is clearly not satisfactory as it seems to assert that borrowers are irrational enough to borrow more than their needs simply because more credit is available. In fact, it is possible that multiple borrowing happens without increasing the total amount of loan. In that case, it cannot be interpreted as over-borrowing. Therefore, we need an alternative explanation of multiple borrowing. In this paper, we suggest risk diversification as such an explanation. Joint liability requires borrowers to carry the risk of their partners’ default. By becoming members of different groups and obtaining a fraction of her total loan requirement through each group, an agent is able to split up that risk into various small parts. This is welfare enhancing for risk averse agents, even without increasing the total amount of loans. Multiple borrowing, therefore, need not involve overborrowing. Instead, it may be a rational response by agents to the opportunity of diversifying risk that is presented by the expansion of microfinance. Of course, there may be costs to increased group membership as borrowers need to monitor a higher number of partners to ensure repayment. Borrowers decide on the optimal number of groups by comparing the higher benefit of more groups with the higher cost.3 Our paper contributes to the theoretical literature on the welfare implications of the expansion of microfinance. Its scope is narrow since it does not investigate the impact of this expansion on interest and default rates, which may significantly affect welfare.4 But it does present a novel explanation of multiple group formation and suggests that policy attempts to curb borrowing from multiple sources can reduce the capacity of borrowers to diversify risk. The rest of the paper is as follows. Section 2 describes our model. We present our main result in Section 3. Section 4 concludes with a few remarks on the possibility of empirical verification of our argument. The proof of the main result (Proposition 3.1) is in the appendix. 1
For example, see http://www.bbc.co.uk/news/world-south-asia-11997571. For example, the Government of India is proposing a law to limit a microfinance borrower to only one joint liability group and restricting the number of MFIs a borrower can approach to two. For full details of this proposed law, see, http://rbidocs.rbi.org.in/rdocs/PublicationReport/Pdfs/YHMR190111.pdf 3 Another explanation is that borrowers wish to substitute MFIs for other traditional sources (like moneylenders) without necessarily increasing their overall loan burden. However, since MFIs ration the amount of loan given to an individual, multiple borrowing is inevitable for such substitution. 4 See McIntosh and Wydick (2005) and Guha and Roy Chowdhury (2013) for the welfare implications of changes in interest and default rates as the number of MFIs increase. 2
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2
The Model
Consider a countably infinite set of agents. Each agent has an investment opportunity that requires $1 of financing. We fix the loan requirement to eliminate the need for over-borrowing. We go on to show that even with a fixed borrowing requirement, agents may seek multiple group membership as a rational measure to diversify risk. With probability p, which is identical for all agents, the investment succeeds and yields a gross return of Y > 1. With the residual probability, the investment fails and yields zero. Agents do not have any funds of their own and, instead, need to approach one or more MFIs to finance their investment. Since microfinance mostly operates on the principle of joint liability, we assume that agents can obtain a loan only if they approach a MFI as a group. Joint liability implies that each member of a group is liable not only for her own loan, but also the loans of all the other members in a group. Since MFIs cannot demand collateral to secure a loan, joint liability is intended as a mechanism to curb wilful default, i.e. default when an agent is capable of repaying. We assume that joint liability is able to curb wilful default entirely. Hence, whenever an agent’s investment succeeds, she repays not only her loan but also her share of the liability of any other member of her group whose investment fails. Borrowers are, of course, protected by limited liability. Therefore, they do not have to repay anything if their investment fails. Following standard practice in the theoretical microfinance literature (see, for example, Ghatak, 2000), we assume that a group consists of two members. However, we do not impose any limit on the number of groups that an agent can be a member of. Let us suppose that every agent is a member of n ∈ Z+ groups. An agent then secures n loans through membership of his n groups, each loan worth $ n1 .5 To ensure maximum risk diversification, we assume that each of the n groups an agent is a member of is distinct, i.e. the agent doesn’t have the same partner in any two groups. Therefore, every agent has a total of n partners for whom she is liable.6 We consider a competitive microfinance industry consisting of risk neutral firms. Let ρ > 1 be the cost a MFI incurs in lending out $1. It gives a total loan of $ n2 to each two–member group. The probability that the group fails to repay is the probability that both agents simultaneously fail in their investment. Therefore, each MFI charges gross interest rate R determined by the zero-profit condition 2 ρ 1 − (1 − p)2 R = 2 n n ρ ρ ⇒R= = . 2 (1 − (1 − p) ) p(2 − p)
(1)
All agents are risk-averse with a standard increasing and concave utility function on consumption u(c). We normalize u(0) to zero. If the investment succeeds, an agent consumes Y minus the 5 Usually, a MFI lends only once to an agent. Therefore, an agent may have to approach n distinct MFIs to obtain n loans. Whether the n lenders are distinct or not is not an important factor in our analysis. 6 The assumption of a countably infinite set of agents ensures that an agent never runs out of potential group partners.
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interest payment. Consumption in the event of investment failure is zero. Therefore, the gross utility obtained by an agent through membership of n distinct groups is Un
n X n n+1−k kR k = p (1 − p) u Y − R − n k k=0 n X n n−k kR k = p p (1 − p) u Y − R − . k n
(2)
k=0
If an agent’s investment is successful, then apart from the total of $1 she has to repay, she is also liable for an additional n1 for each of her group partners. The probability that k of her n partners default is nk pn+1−k (1 − p)k . She then needs to pay an additional amount of kR n along with her own interest liability of R.7 There may also be additional costs to group membership; for example, the higher monitoring required to ensure repayment if the number of partners increase. Let the cost of membership in n groups be gn , with g0 = 0. Therefore, the net utility from n group memberships is Bn = Un − gn .
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Main Result
We seek to show that Un increases with n. Let
ρ UI = pu Y − p
.
(3)
Equation (3) is the utility that an agent receives under individual liability if she can secure a collateral free loan of $1 at the break-even interest rate
ρ p.
The following proposition establishes
that as n increases, Un also increases and converges to UI . Intuitively, this happens because the expected interest paid under joint liability, irrespective of n, is also ρp . But conditional on success, the variance of the interest payment drops to zero, the same as under individual liability. The formal proof, which follows from the convergence of the binomial distribution to the normal distribution, is in the appendix. Proposition 3.1 Consider Un defined in (2) and UI as defined in (3). As n increases, Un increases, and further, Un → UI as n → ∞. Proposition 3.1 is the main result of this paper. It implies that the additional risk associated with joint liability tends towards zero as the number of groups an individual is a part of increases. Therefore, if there is no cost of group formation, an agent finds it rational to become a member of as many groups as possible. However, with gn positive and increasing, agents decide on the number of group memberships by considering the net benefit Bn = Un − gn . Each agent, therefore, becomes a member of n∗ ∈ Z+ 7
We are implicitly assuming Y > 2R to ensure an agent is able to repay if all her partners default.
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distinct groups, where n∗ < ∞ maximizes Un − gn .8 If n∗ > 1, we have multiple group membership.
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Conclusion
A necessary condition for our risk diversification hypothesis to be tenable is that loan sizes do not change with the increase in the number of MFIs. An analysis by Lahkar et al. (2013), using data from the Indian microfinance sector, shows that this is indeed the case. While the average loan size of a borrower has not changed, the number of loans taken have increased with an increase in the number of MFIs. This is, of course, not conclusive evidence in support of risk diversification as it is possible that borrowers are substituting from more traditional sources. But it does weaken the case for the over borrowing idea. We also note that risk diversification may not be possible by increasing the number of members in an existing group from, say, two to three. Group sizes are usually fixed by MFIs. Moreover, doing so just increases the additional liability of each member to $2 rather than diversifying the earlier liability of $1.
A
Appendix
Proof of Proposition 3.1. Given n, let Ck be the random variable Y − R −
kR n
. This is the
consumption level obtained by an agent when k of her partners fail to repay their loans. By (2), the variable k is binomially distributed with parameters n and (1 − p), i.e. k ∼ B(n, 1 − p). Hence, for any n, the expected consumption of the agent is kR ECk = pE Y −R− n 1 = p Y − R 1 + Ek n ρ 1 = p Y − 1 + n(1 − p) p(2 − p) n ρ = p Y − , p which is also the expected consumption under individual liability. To determine the declining variance in consumption, note that as n → ∞, B(n, (1 − p)) →d N (n(1 − p), np(1 − p)), the normal distribution with mean n(1 − p) and variance np(1 − p). It is also easy to verify that N (n(1 − p), np(1 − p)) →d δn(1−p) as n → ∞, where δn(1−p) is the Dirac distribution on n(1 − p). Hence, the variance in consumption approaches zero as n → ∞. Hence, Un increases as expected value stays constant and variance declines. To see the convergence in utility, denote by Φ(k) the distribution function of N (n(1−p), np(1−p)). The convergence in distribution and the boundedness of u Y − R − kR n by u(Y ) then implies that the expected value 8
The finiteness of n∗ follows from the concavity of Un and the convexity of gn .
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converges, or, Z lim Un = lim p
n→∞
n→∞
kR u Y −R− n
n(1 − p)R dΦ(k) = pu Y − R − n
ρ = pu Y − = UI , p
where the second equality follows from k →d δn(1−p) .
References [1] M. Ghatak, Screening by the Company You Keep: Joint Liability Lending and the Peer Selection Effect, The Economic Journal, 110 (2000), 601-631. [2] B. Guha, P. Roy Chowdhury, Microfinance Competition: Motivated Microlenders, Double Dipping and Default, J. Dev. Econ., 105 (2013), 86-102 [3] R. Lahkar, V. Pingali, S. Sadhu, Does Competition in the Microfinance Industry necessarily mean Over-borrowing? IIMA Working Paper Series No. WP2012-12-01 [4] C. McIntosh, A. de Janvry, E. Sadoulet, How Rising Competition among Microfinance Institutions affects Incumbent Lenders, The Economic Journal, 115 (2005) 987-1004. [5] C. McIntosh, B. Wydick, Competition and Microfinance, Journal of Development Economics, 78 (2005) 271-98.
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