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Group Survival Probability under Contagion in Microlending

Héctor Jasso-Fuentes, Alejandra Quintos, Xinta Yang

Abstract

In the context of micro-finance, a group of individuals undertake business projects that may interfere with one another. A contagious default happens if one person's project failure leads to the default of another group member. In this paper, we apply a probabilistic approach to analyze the impact of such contagion among investment group members. Firstly, a general formula is provided to compute the group survival probability with the presence of contagion effect. Then, special cases of this probability model are examined in detail. In particular, we show that if the investment group is homogeneous, defined in the paper, then including more members into the group will eventually lead to default with probability 1. This differs from the non-contagious scenario, where the default probability decreases monotonically with respect to the group size. Afterwards, we provide an upper bound of the optimal group size under the homogeneous setup; so, one can run a linear search within finite time to locate this optimizer.

Group Survival Probability under Contagion in Microlending

Abstract

In the context of micro-finance, a group of individuals undertake business projects that may interfere with one another. A contagious default happens if one person's project failure leads to the default of another group member. In this paper, we apply a probabilistic approach to analyze the impact of such contagion among investment group members. Firstly, a general formula is provided to compute the group survival probability with the presence of contagion effect. Then, special cases of this probability model are examined in detail. In particular, we show that if the investment group is homogeneous, defined in the paper, then including more members into the group will eventually lead to default with probability 1. This differs from the non-contagious scenario, where the default probability decreases monotonically with respect to the group size. Afterwards, we provide an upper bound of the optimal group size under the homogeneous setup; so, one can run a linear search within finite time to locate this optimizer.

Paper Structure

This paper contains 10 sections, 8 theorems, 51 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and Convention
  3. Model Setup
  4. Contagion Factor
  5. Main Results
  6. Special Cases for Constant Contagion
  7. Optimal Group Size under Constant Contagion and Homogeneity
  8. Declarations
  9. Funding
  10. Competing interests

Key Result

Lemma 1

Given an index set $I \subset \{1, 2, ..., n\}$, define the event $A_I = \{\eta_{i\in I} \le T, \eta_{j\in I^c} > T, \text{ all regular members default by } T\}$, where notation $\eta_{i\in I} \le T$ means $\eta_i \le T$ for all $i\in I$ as mentioned in the notation section above. We then have If $I = \{1, ..., n\}$, the right-side of equation:group-default-prob-with-natural-default-index-set-I-e

Figures (5)

  • Figure 1: Two examples of the function $q_{ij}(t)$
  • Figure 2: Probability of the group surviving as a function of group size $n$
  • Figure 3: Comparison between different upper bounds for the optimal group size
  • Figure 4: Another example of the group's survival probability with respect to $n$
  • Figure 5: Compare the true group survival probability with its approximation, in which $X_n$ is replaced by $nc_1$

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1: Immediate Constant Contagion
  • proof
  • Corollary 2: Contagion requires early default
  • proof
  • Theorem 2
  • proof
  • ...and 6 more