英文标题：
《On the implicit interest rate in the Yunus equation》
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作者：
Marc Diener (JAD), Pheakdei Mauk (JAD)
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最新提交年份：
2013
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英文摘要：
  In his book with Alan Jolis, Vers un monde sans pauvret\\\'e (1997) Yunus gives the example of a microcredit loan of 1000BDT reimbursed via 50 weekly settlements of 22BDT and correctly claims that this corresponds to the annual interest rate of 20%. But this is without taking into account that if the borrower has good reasons not to pay at one installment, she can postpone of one week all remaining settlements, under the same conditions, so without extra cost. This of course leads to a lower implicit interest rate. Introducing a simple geometric law model for the time between settlements, this turns the implicit interest rate into a random variable, whose laws is still unknown but for which we provide simulated empirical distribution density function. De ning by actuarial expected rate the real number r that satis es the expectation of the random Yunus equation, we compute this number as a function of the probability p of in-time installment. This allows in turn to compute the implicit probability p which is to the value of p corresponding to the observed 3% default rate, where in practice, \\default\" means \\more than four weeks delay\". The mathematical tool used is the probability generating function, the computer tool is the Scilab algebraic equation solver.
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中文摘要：
尤努斯在与艾伦·乔利斯合著的书《世界无国界报》（Vers un monde sans pauvret，1997）中举例说明了一笔1000孟加拉塔卡的小额信贷贷款，通过每周50笔22孟加拉塔卡的结算进行偿还，并正确地宣称这相当于20%的年利率。但这并没有考虑到，如果借款人有充分的理由不一次性付款，她可以在相同的条件下，将所有剩余的结算推迟一周，因此无需额外费用。这当然会导致较低的隐性利率。引入一个简单的几何定律模型来计算两次结算之间的时间，这将隐式利率转化为一个随机变量，其定律仍然未知，但我们提供了模拟的经验分布密度函数。通过精算期望率来定义满足随机尤努斯方程期望的实数r，我们计算这个数作为及时分期概率p的函数。这又允许计算隐式概率p，即与观察到的3%违约率对应的p值，在实践中，“违约”意味着超过四周的延迟。使用的数学工具是概率母函数，计算机工具是Scilab代数方程求解器。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：General Finance        一般财务
分类描述：Development of general quantitative methodologies with applications in finance
通用定量方法的发展及其在金融中的应用
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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