The repayment of a loan with equal payments and the minimization of the credit risk

A loan is typically repaid by equal payments at equal time intervals. Besides the amount of the first or the last payment may be incomplete, when the amount of the other payments are predetermined. In the nonlinear optimization model, which we propose in this paper, we assume that the creditor wants to construct the amortization schedule for repayment of a loan with minimal risk of default. For this purpose he must determine the optimal pair of the duration of the loan and the amount of the payments. On the base of the model considered, we propose a necessary and sufficient condition for optimality of this pair.