Discrete-Time Dynamics of Deposit-Loan Volumes Model with Repayment Rate: Standard and Non-Standard Approaches

Abstract

In the banking system, the repayment rate of loans, which is influenced by interest rates and nonperforming loans, plays an important role in the bank’s cash flow. In this paper, we propose a discrete model of deposit–loan volumes by considering the repayment rate. The proposed model involves the standard forward Euler discretization and the non-standard finite difference (NSFD) scheme. The numerical schemes of both models are explicitly defined. Both models have three fixed points, i.e., the transaction-free point, the loan-free point, and the active-transaction point. The transaction-free fixed point is unstable, while the other two are locally asymptotically stable under certain conditions. The stability of the Euler model’s fixed point depends on the stepsize h. This indicates that the NSFD model is dynamically more consistent since it does not depend on h. Numerical simulations also confirm that the stability property of the NSFD model’s fixed points does not depend on h. Meanwhile, the stability of the fixed points of the Euler model depends on h. The simulations also show that the Euler model undergoes period-doubling and Neimark–Sacker bifurcations. This is indicated by changes in the stepsize that cause the convergence of the solution to shift into oscillations or even chaos. The chaotic condition is an undesired or even avoided situation in the banking sector. High and irregular fluctuations lead to the failure of policy control and liquidity projection. We also performed a case study using weekly loan data from September 2022 to March 2025 via parameter estimation. We use two performance metrics, i.e., the coefficient of determination (R2) and the root mean square error (RMSE). Both models produce realistic parameter values and provide a good fit to the data trend, as observed visually and from R2. Based on RMSE, the NSFD model performs better than the Euler model. Moreover, the larger the h, the better the performance. These results suggest the use of the NSFD model, which has better relevance and accuracy than the Euler model.