Theory of Probability &Its Applications, Volume 66, Issue 2, Page 326-335, January 2021.
We propose and study a simple model for microcredit using two sums, with a random number of terms, of identically distributed random variables, the number of terms being Poisson distributed; the first sum accounts for the payments---the payables---made to the collective vault by the participants, and the second sum, subtracted from the first, accounts for the loans received by the participants, the receivables. Under a global independence hypothesis, we define, by mean of moment generating functions, an easily implementable condition for the sustainability of the collective vault. That is, if, for all the participants and at any time, on average, the sum of the loans is strictly less than the sum of the payments to the collective vault, then the probability of the collective vault failing can be made arbitrarily small, provided the loans only start to be accepted after a sufficiently large delay. We present numerical illustrations of collective vaults for exponential and chi-squared distributed random terms. For the practical management of such a collective vault it may be advisable to have loan granting rules that destroy the independence of the random terms. We present a first simulation study that shows the effect of such a loan granting rule, that removes the independence hypothesis on maintaining the stability of the collective vault.

中文翻译：

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小额信贷合作银行计划的随机模型

概率论及其应用，第 66 卷，第 2 期，第 326-335 页，2021 年 1 月。
我们提出并研究了一个简单的小额信贷模型，它使用两个总和，具有随机项数，同分布的随机变量，项数为泊松分布；第一笔款项是参与者向集体金库支付的款项——应付款项，第二笔款项减去第一笔款项后，是参与者收到的贷款，即应收账款。在全球独立假设下，我们通过矩生成函数定义了一个易于实现的集体金库可持续性条件。也就是说，如果对于所有参与者，在任何时间，平均而言，贷款的总和严格小于支付给集体金库的总和，那么集体金库失败的概率可以任意小，前提是贷款只有在足够长的延迟后才开始被接受。我们展示了指数和卡方分布随机项的集体金库的数值说明。对于此类集体金库的实际管理，可能建议制定破坏随机条款独立性的贷款授予规则。我们提出了第一个模拟研究，显示了这种贷款发放规则的影响，它消除了维持集体金库稳定性的独立性假设。