On the implicit interest rate in the Yunus equation

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.