Optimal Group Size in Microlending
OOptimal Group Size in Microlending
Philip Protter ∗ Statistics DepartmentColumbia UniversityNew York, NY, 10027ORCID [email protected]
Alejandra Quintos †‡ Statistics DepartmentColumbia UniversityNew York, NY, 10027ORCID [email protected]
July 30, 2020
ABSTRACT
Microlending, where a bank lends to a small group of people without credit histories, beganwith the Grameen Bank in Bangladesh, and is widely seen as the creation of MuhammadYunus, who received the Nobel Peace Prize in recognition of his largely successful efforts.Since that time the model of microlending has received a fair amount of academic attention.One of the issues not yet addressed in detail, however, is the issue of the size of the group.(Some attention has nevertheless been paid, see the appropriate references in this paper.)Instead of a game theory approach, we take a mathematical approach to the issue of anoptimal group size, where the goal is to minimize the probability of default of the group. Todo this, one has to create a model with interacting forces, and to make precise the hypothesesof the model. (In previous work we have addressed the issue of what is a fair rate of interestto charge for such a loan.) We show that the original choice of Muhammad Yunus, of agroup size of five people, is, under the right (and, we believe, reasonable) hypotheses, eitherclose to optimal, or even at times exactly optimal, i.e., the optimal group size is indeed fivepeople. ∗ Supported in part by NSF grant DMS-1612758 † Supported in part by the Fulbright-Garc´ıa Robles Program ‡ Corresponding author a r X i v : . [ q -f i n . M F ] J u l INTRODUCTION
Microfinance analyzes the lending mechanisms for people without access to traditional creditsystems because of their low income, lack of collateral, or credit history. There are manylending mechanisms studied in the literature, but we will focus on group lending, which wasintroduced by Muhammad Yunus of the Grameen bank in Bangledesh. He won the NobelPeace Prize (2006) for his efforts. In this mechanism, a loan is made to all members of thegroup for a fixed period of time (often less than a year).One of the crucial components of group lending is known as contingent renewal, which is apenalty that eliminates or reduces access to future loans to all members in a group if any ofthem defaults. This is to say, the default of at least one individual provokes the default ofthe group. Note that by default of a member we mean that she stops paying her share ofthe loan. The group lending mechanism is thought to be useful because it induces: (i) peerselection of members in a group since they are better informed than the lender about otherpotential borrowers, (ii) peer pressure to help enforce payments by other members in thegroup, and (iii) peer monitoring between the group members, but particularly by a leader, toensure continued performance on the loan. In this paper, we will focus on modeling the lastelement of the previous list. See the articles F. Diener et al. 2009, M. Diener and Mauk 2007,and M. Diener and Santos 2016, for a study of the consequences of implementing variouspenalties for default.The existing academic literature primarily focuses on understanding why the group lendingmechanism is successful in reducing defaults. Both static and dynamic models have beenanalyzed (see Stiglitz 1990; Varian 1990; Conlin 1999; Morduch 1999; Chowdhury 2005,Chowdhury 2007; Tedeschi 2006). A related and somewhat unexplored issue is to determinean optimal group size. We define optimal size as the one that maximizes the probability ofno default of the group. Obviously, there are other important factors in the calculation ofthe default of a group, including – especially – the personalities of the members of the group,local traditions, a cultural sense of responsibility, and the like. We choose to focus, in thiswork, exclusively on the size.The problem of an optimal group size has been analyzed by economists in the past (seeArmend´ariz and Morduch 2010; Gin´e et al. 2010; Ahlin 2015, Ahlin 2017). Most of them takean intuitive, experimental and, often verbal approach. In contrast, ours is more probabilisticin nature, and thereby quantifiable. A first approach to what we present here is the articleof Jarrow and Protter 2019. We also note that the problem of group size has been tackledusing tools from Game Theory (see Rezaei, Dasu, and Ahmadi 2017).An outline for this paper is as follows. Section II presents the model and our main theorem,while Section III outlines an interpretation for it. Section IV shows an example and discussesits intuitiveness, and Section V concludes. 2
I THE MODEL
Caveats:
We begin with a few caveats. Different scenarios of Microlending have beenconsidered in the academic literature. For example, there is the issue of the size of the loanaffecting the group in group lending, and in particular its size (with larger loans leading tosomewhat larger groups, see Rezaei, Dasu, and Ahmadi 2017). In this paper, the loan size isfixed, and we will not consider its influence on the group size. Some researchers take as theirpoint of view the maximization of the profits of the lending institution (see, e.g., Bourjadeand Schindele 2012). This runs counter to the spirit embraced in Yunus 1998, and again,while a valid consideration, it is not our concern in this paper. We are concerned only withminimizing the possibility of default on the loan; admittedly this is related to maximizingthe profits of the lender, along with the interest rates charged (see Jarrow and Protter 2019).Finally, we mention that we do not discuss the transaction costs of banks, and what the effectof group lending is on them. We also implicitly assume that the members of a given groupform a fairly homogeneous collection of people (see Devereux and Fishe 2007, later echoed inBourjade and Schindele 2012.) This homogeneity assumption is reflected in our assumptionof identical distributions, within a group of a given size, allowing the distributions to changewith the group size.We now introduce the notation for our model. Let N i be the event of no default of member i in a group of size k ( k ∈ Z + , k ≥ N k be the event that the group of size k does notdefault and, ϕ ( k ) := 1 − P k ( N ), i.e., the probability of default of member 1 in a group ofsize k . Recall that, as explained in the introduction, the group lending mechanism impliesthat if at least one member of the group defaults, then the whole group defaults. This iswhat we call default of a group .We make the following assumptions:1. For fixed size k , the group members are independent and identically distributed.2. The probability of no default of one person depends on the size of the group. We makethis explicit by writing: P k ( N ).3. P k ( N ) >
0. Otherwise the problem is trivial as the members will default for sure.We are interested in finding an optimal group size, that is, finding the number of people k ∗ that maximizes the probability of no default of the group. Using our assumptions, alongwith our definition of default of a group, this translates into maximizing: P ( N k ) = P (cid:32) k (cid:92) i =1 N i (cid:33) = [ P k ( N )] k = (1 − ϕ ( k )) k (1)For a moment, suppose that ϕ ( · ) is constant in k , hence as ϕ ( · ) <
1, (1 − ϕ ( · )) k decreasesas k increases. So, in order to have a maxima in (1), it makes sense to require that 1 − ϕ ( k )increases with k , which means that ϕ ( k ) needs to decrease with k . The question is then, atwhat speed? This motivates the following theorem.3 heorem 1. Let ϕ ( x ) = f ( x ) , for all x ∈ R + If:1. f ( x ) > for all x ≥ f ( x ) ∈ C f (cid:48) ( x ) > for all x ≥
4. There exist a, b ∈ R ( a < b, a ≥ such that either:i) f ( a ) − af (cid:48) ( a ) = and f ( b ) − bf (cid:48) ( b ) = 1 andii) f (cid:48)(cid:48) ( x ) < for all x ∈ ( a, b ) oriii) f ( a ) − af (cid:48) ( a ) = 1 and f ( b ) − bf (cid:48) ( b ) = andiv) f (cid:48)(cid:48) ( x ) > for all x ∈ ( a, b ) Then (1 − ϕ ( x )) x has a unique maximizer x ∗ in ( a, b ) . Moreover, if a , b are unique, then x ∗ is the unique maximizer.Proof. Let S ( x ) := (cid:80) ∞ n =0 (cid:0) n +1 (cid:1) (cid:0) n +2 (cid:1) (cid:16) f ( x ) (cid:17) n Note: • S ( x ) is a decreasing function in x . • S ( x ) ∈ (cid:0) , (cid:1) , for all x ∈ R because:12 <
12 + ∞ (cid:88) n =1 (cid:18) n + 1 (cid:19) (cid:18) n + 2 (cid:19) (cid:18) f ( x ) (cid:19) n = S ( x ) < ∞ (cid:88) n =0 (cid:18) n + 1 (cid:19) (cid:18) n + 2 (cid:19) = 1Set h ( x ) := f ( x ) − xf (cid:48) ( x ) and note that in ( a, b ), h ( x ) is a monotone function because ofcondition (4ii) or (4iv). More explicitly: h (cid:48) ( x ) = f (cid:48) ( x ) − [ f (cid:48) ( x ) + xf (cid:48)(cid:48) ( x )] = − xf (cid:48)(cid:48) ( x ) > <
0) for all x ∈ ( a, b )Moreover, the monotonicity of h ( x ) and condition (4i) or (4iii) imply < h ( x ) < x ∈ ( a, b )In this way for all x ∈ ( a, b ): • Both S ( x ) and h ( x ) are continuous and monotone4 S ( x ) is bounded between (cid:0) , (cid:1) . This actually holds for all x ∈ R + • h ( x ) increases from to 1 (or decreases from 1 to )Then there exists a unique x ∗ ∈ ( a, b ) such that h ( x ∗ ) = S ( x ∗ ) (2)We shall see that this x ∗ is actually the unique maximizer. Thanks to equation (2), we have: f ( x ∗ ) − x ∗ f (cid:48) ( x ∗ ) = ∞ (cid:88) n =0 (cid:18) n + 1 (cid:19) (cid:18) n + 2 (cid:19) (cid:18) f ( x ∗ ) (cid:19) n ⇐⇒ ∞ (cid:88) n =0 (cid:18) n + 1 (cid:19) (cid:18) n + 2 (cid:19) (cid:18) f ( x ∗ ) (cid:19) n + x ∗ f (cid:48) ( x ∗ ) − f ( x ∗ )= ∞ (cid:88) n =0 (cid:18) n + 1 (cid:19) (cid:18) f ( x ∗ ) (cid:19) n − ∞ (cid:88) n =0 (cid:18) n + 2 (cid:19) (cid:18) f ( x ∗ ) (cid:19) n + x ∗ f (cid:48) ( x ∗ ) − f ( x ∗ )= ∞ (cid:88) n =1 (cid:18) n (cid:19) (cid:18) f ( x ∗ ) (cid:19) n − − ∞ (cid:88) n =2 (cid:18) n (cid:19) (cid:18) f ( x ∗ ) (cid:19) n − − f ( x ∗ ) + x ∗ f (cid:48) ( x ∗ )= ∞ (cid:88) n =1 (cid:18) n (cid:19) (cid:18) f ( x ∗ ) (cid:19) n − − ∞ (cid:88) n =1 (cid:18) n (cid:19) (cid:18) f ( x ∗ ) (cid:19) n − + x ∗ f (cid:48) ( x ∗ )= ∞ (cid:88) n =1 (cid:18) n (cid:19) (cid:18) f ( x ∗ ) (cid:19) n +1 − ∞ (cid:88) n =1 (cid:18) n (cid:19) (cid:18) f ( x ∗ ) (cid:19) n + x ∗ f (cid:48) ( x ∗ )( f ( x ∗ )) (3)Now, recall we want to find a maxima for (1 − ϕ ( x )) x . This is equivalent to maximizing U ( x ) := x ln (1 − ϕ ( x )).Note U (cid:48) ( x ) = ln (1 − ϕ ( x )) − x − ϕ ( x ) ϕ (cid:48) ( x )It suffices to find x ∗ (the maximizer) such that U (cid:48) ( x ∗ ) = 0, which is equivalent to g ( x ∗ ) = 0where g ( x ) := [1 − ϕ ( x )] ln (1 − ϕ ( x )) − xϕ (cid:48) ( x )Recall: ln(1 − y ) = − (cid:80) ∞ n =1 y n n , if | y | <
1. Then: g ( x ) = [1 − ϕ ( x )] (cid:34) − ∞ (cid:88) n =1 ϕ n ( x ) n (cid:35) − xϕ (cid:48) ( x )= ∞ (cid:88) n =1 ϕ n +1 ( x ) n − ∞ (cid:88) n =1 ϕ n ( x ) n − xϕ (cid:48) ( x )= ∞ (cid:88) n =1 (cid:18) n (cid:19) (cid:18) f ( x ) (cid:19) n +1 − ∞ (cid:88) n =1 (cid:18) n (cid:19) (cid:18) f ( x ) (cid:19) n + x f (cid:48) ( x )( f ( x )) Finally, it is easy to see that this last line and (3) imply g ( x ∗ ) = 0 and we can conclude x ∗ isthe unique maximizer in ( a, b ). If a, b are unique, it is clear that the maximizer is unique.5 emark. The conditions (4i) and (4iii), which may seem mysterious at first glance, areinspired by Taylor’s Theorem, from calculus.One can argue that, in the previous theorem, we heavily used the continuity of f ( x ) and thefact that x ∈ R + and, as we are optimizing with respect to the number of people, we shouldhave taken k ∈ Z , but we can always round x ∗ to the closest integer to get k ∗ . III INTERPRETATION OF THE THEOREM
Recall formula (1) P ( N k ) = P (cid:16)(cid:84) ki =1 N i (cid:17) = [ P k ( N )] k = (1 − ϕ ( k )) k . As we briefly discussedin section (II), because of our independence and identical distribution assumptions, thereare 2 interacting forces affecting P ( N k ). On the one hand, P ( N k ) = (1 − ϕ ( k )) k decreases as k increases because 0 < ϕ ( k ) <
1. On the other hand, we set a fortiori ϕ ( k ) to decrease as k increases with the hope to find a maximizer k ∗ . Lending to a group has advantages overlending to an individual, but as the size of the group increases, the advantages diminish andtend to zero. There should, therefore, be some happy (and optimal) compromise of a groupsize being big, but not too big!There are two opposing forces here. As the group size increases, the responsibility forperforming one’s tasks becomes dispersed, increasing the likelihood that one or more memberof the group may default. Typically, there will be a leader or primary organizer, the forcebehind the loan, and she will need to ride herd on the other members, keeping them in line,if need be. The larger the group, the more diffused her efforts will be, and therefore theless effective. In our model, this is captured by ϕ ( k ) as different functions give different“peer pressure levels/intensities” and hence different k ∗ (optimal group size). On the otherhand, as the group size contracts, each person becomes more important, making it harderto recover from a mistake, or a temporary period of misfortune. In the limit case of only oneborrower, lenders in Ghana (for example) have found that, there being no peer pressure atall, the borrower has a serious probability of simply absconding with the money. Mathematically, as the group size increases, because of our assumptions, there are moreindependent chances of failure as it is riskier to have k + 1 possible defaults than k . Thiscauses P ( N k ) to decrease as k increases. Note that the effect of this force is free of the choiceof ϕ ( k ).Now, the issue is to find the right speed of decay of ϕ ( k ). This is addressed by our theorem.It is important to note that our theorem is useful because not all ϕ ( k ) work. One can check,for example, that the seemingly natural choice of ϕ ( k ) = k does not have a finite maximizergreater or equal to 2. Personal conversation of the first author in Accra, Ghana, August 22, 2018; with Prof. Dr. OlivierMenouken Pamen, of AIMS, Ghana V EXAMPLE
In this section, we provide a function that satisfies our theorem and whose maximizer isclose to 5, i.e. x ∗ ≈
5. As explained in
Banker To The Poor (1998), this is the group sizeproposed by Muhammad Yunus. Let us consider the following choice of the function f : f ( x ) = x α + (ln x ) β (4)This example captures two different forces at play. The part of f ( x ) given by x α representsthe leader’s ability to influence the group’s performance, while the component (ln x ) β repre-sents the quality of the group at play. We chose ln x , which has a distinctly slower growthrate than x , for quality of the group because we think that this is less relevant than theleader’s ability.Let us consider α = p and, for simplicity, its reciprocal, i.e., β = 1 /p , for all p ∈ (cid:2) , (cid:3) . Theexponents of x and of ln x are chosen in this way because we want to have opposite forcesfor the interaction of the leader and the group. That is, the less effective the leader is, thebetter group quality we need.The cases p = and p = 1 are relevant as they lead to x ∗ = 5 .
13 and x ∗ = 4 .
62 respectively.Therefore, in the extreme cases, i.e., either an excellent leader or a high-quality group, theoptimal group size is 5.We wish to note that, although f ( x ) = x p , for all p ∈ (1 / ,
1) works, we believe this functiondoes not capture the complexity of the situation we are trying to model. For this function,when p is close to 1, eg. p = 0 . x ∗ = 503 .
45. This should notbe surprising as x . is close to x , which as previously discussed does not have a finitemaximizer. Moreover, when p is close to 1 /
2, e.g. p = 0 . x ∗ = 1 . Condition (3): f (cid:48) ( x ) > x ≥ Proof of (3): f (cid:48) ( x ) = px p − + p (cid:0) x (cid:1) (ln x ) p − As x ≥
2, it is clear f (cid:48) ( x ) > Condition (4i):
There exist a and b such that f ( a ) − af (cid:48) ( a ) = and f ( b ) − bf (cid:48) ( b ) = 1 Proof of (4i):
Set h p ( x ) := f ( x ) − xf (cid:48) ( x ) = x p +(ln x ) p − px p − p (ln x ) p − = (1 − p ) x p +(ln x ) p − (cid:16) ln x − p (cid:17) Claim: h p ( x ) is increasing in x Proof of claim: ∂∂x h p ( x ) = (1 − p ) px p − + (cid:16) p − (cid:17) (ln x ) p − (cid:104) ln x − p (cid:105) x + x (ln x ) p −
7t is clear (1 − p ) px p − >
0. So, it suffices to show (cid:18) p − (cid:19) (ln x ) p − (cid:20) ln x − p (cid:21) x + 1 x (ln x ) p − ≥ x ≥ e . As ln x + 1 ≥ ≤ p ≤
2, it follows that ln x + 1 ≥ p p (cid:18) ln x + 1 − p (cid:19) ≥ p (cid:18) ln x − p (cid:19) − (cid:18) ln x − p (cid:19) + ln x ≥ (cid:18) p − (cid:19) (cid:18) ln x − p (cid:19) + ln x ≥ h p ( x ) is increasing for x ≥ e Claim: h p ( e ) = (1 − p ) e p + 1 − p is concave in p and hence there exists a local maxima,namely p ∗ . (Recall ≤ p ≤ Proof of claim: ∂∂p h p ( e ) = − e p + (1 − p ) e p + p = − pe p + p ∂ ∂p h p ( e ) = − e p − pe p − p < ⇒ h p ( e ) is concaveNow, to find the maxima, we set the derivative equal to 0, i.e. ∂∂p h p ( e ) = 00 = − pe p + 1 p p e p pe p Set u = p , we need to solve ue u = , which we do by using the product logarithm.Hence u = W (cid:0) (cid:1) and thus p ∗ = 3 W (cid:0) (cid:1) ≈ . ⇒ h p ( e ) | p =0 . ≈ . < Claim: h p ( e ) < for all p ∈ (cid:2) , (cid:3) Proof of claim:
As we have shown that h p ( e ) is concave in p and that h p ∗ ( e ) ≈ . < , it follows that h p ( e ) < for all p ∈ (cid:0) , (cid:1) We only need to check the endpoints, h p ( e ) | p = ≈ − . h p ( e ) | p =1 = 0Hence h p ( e ) < for all p ∈ (cid:2) , (cid:3) . This, along with h p ( x ) being continuous, increasingin x ≥ e and lim x →∞ h p ( x ) = ∞ , imply that there exists a such that h p ( a ) = and that a ≥ e ≈ . ⇒ a ≥ laim: h p ( e ) = (1 − p ) e p + 2 p − (cid:16) − p (cid:17) is concave in p Proof of claim: ∂∂p h p ( e ) = − e p + 2(1 − p ) e p + (cid:16) p − p (cid:17) (ln 2)2 p − + (cid:16) p (cid:17) p − ∂ ∂p h p ( e ) = − e p + 4(1 − p ) e p − p p + 2 p (cid:16) − p (cid:17) ln 2 p − p ln 2 p + 2 p − (cid:16) − p (cid:17) (ln 2) p Now we show that ∂ ∂p h p ( e ) <
01. It is clear that − e p + 4(1 − p ) e p <
02. As ln 2 < − p ≤ (cid:18) − p (cid:19) ln 2 < − (cid:18) − p (cid:19) ln 2 < − p p + 2 p (cid:18) − p (cid:19) ln 2 p <
03. Similarly (cid:18) − p (cid:19) ln 22 < − (cid:18) − p (cid:19) ln 22 < p ln 2 p + 2 p − (cid:18) − p (cid:19) (ln 2) p < Claim: h p ( e ) ≥ p ∈ (cid:2) , (cid:3) Proof of claim:
As we have shown that h p ( e ) is concave in p , it suffices to show h p ( e ) | p = ≥ h p ( e ) | p =1 ≥ h p ( e ) | p = = e > e > h p ( e ) | p =1 = 1Hence h p ( e ) ≥ p ∈ (cid:2) , (cid:3) . This, along with h p ( x ) being continuous and increasingin x implies that there exists b such that h p ( b ) = 1 and that b ≤ e ≈ . ⇒ b ≤ ondition (4ii): f (cid:48)(cid:48) ( x ) < x ∈ ( a, b ) for some a, b > e ≤ a < b Proof of (4ii): f (cid:48)(cid:48) ( x ) = p ( p − x p − + p (cid:104)(cid:16) − pp (cid:17) (cid:0) x (cid:1) (ln x ) p − − (cid:0) x (cid:1) (ln x ) p − (cid:105) = p ( p − x p − + p (cid:0) x (cid:1) (ln x ) p − (cid:104) − pp − ln x (cid:105) p ( p − x p − ≤ p ≤ ≤ p ≤ x + 1 ≥ x ≥ e , we get p ≤ ln x + 1 . Hence, − pp − ln x ≤
0, which implies the 2nd term is non-positive for all x ≥ e Hence for f ( x ) = x p + [ln x ] p , using our theorem, we can claim that the maximizer x ∗ ∈ [3 ,
7] for all p ∈ (cid:2) , (cid:3) . It is worth noticing that for this particular f ( x ), we can obtain anarrower interval in the following way:To find the maximizer x ∗ of (1 − ϕ ( x )) x , we need to set the derivative equal to 0, which, asnoted in the proof of the theorem, is equivalent to solving [1 − ϕ ( x )] ln (1 − ϕ ( x )) − xϕ (cid:48) ( x ) =0. Using ϕ ( x ) = f ( x ) = x p +[ln x ] /p , let us define: H ( x, p ) : = [1 − ϕ ( x )] ln (1 − ϕ ( x )) − xϕ (cid:48) ( x )= (cid:32) − x p + [ln x ] p (cid:33) ln (cid:32) − x p + [ln x ] p (cid:33) + p x p ln x + (ln x ) p p ln x (cid:104) x p + (ln x ) p (cid:105) Then, after fixing p , we need to find x such that H ( x, p ) = 0Note H ( x, p ) is decreasing in x for all p ∈ (cid:2) , (cid:3) . Hence, for fixed p ∈ (cid:2) , (cid:3) and for all x ≤ .
48, we have H ( x, p ) ≥ H (3 . , p ), which impliesmin p H ( x, p ) ≥ min p H (3 . , p )By calculations, we know:min p H (3 . , p ) = H (3 . , . ≈ . > x ≤ .
48 and for all p ∈ (cid:2) , (cid:3) H ( x, p ) ≥ min p H ( x, p ) ≥ min p H (3 . , p ) > H ( x, p ) = 0 does not have a solution when x ≤ .
48 and p ∈ (cid:2) , (cid:3) . So, x ∗ must be in the interval (3 . , ∞ ).As H ( x, p ) is decreasing in x , H ( x, p ) ≤ H (5 . , p ) for all p ∈ (cid:2) , (cid:3) and for all x ≥ . H (5 . , p ) ≤ max p H (5 . , p ) = H (5 . , ≈ − . <
0. Joining these 2 facts, weget for all x ≥ . p ∈ (cid:2) , (cid:3) : H ( x, p ) ≤ H (5 . , p ) ≤ max p H (5 . , p ) < H ( x, p ) = 0 does not have a solution when x ≥ . p ∈ (cid:2) , (cid:3) . So, x ∗ must be in (0 , . x ∗ ∈ (3 . , .
4) for all p ∈ (cid:2) , (cid:3) By numerical calculations, we can see that when p ∈ [0 . , . p ∈ [0 . , x ∗ ∈ [4 . , . V CONCLUSIONS
This paper provides a theoretical model for the determination of the optimal number ofpeople in a group loan. As these loans are intended for low-income borrowers with littleor no collateral, and with no credit history, one of the starting points to maximize therepayment rate is to determine the best possible size of the group. An empirical study ofthe proposed model awaits subsequent research.11
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