A Linear Ordinary Differential Equation Model of Assets and Loan Liquidity Management of Credit Unions: A Two-state Dynamic Model
Cynthia Ama Mensah *
Department of Mathematics, Statistics and Actuarial Science, Takoradi Technical University, Sekondi Takoradi, P. O. Box 256, Ghana and Department of Mathematical Sciences, University of Mines and Technology, Tarkwa, P. O. Box 237, Ghana.
Lewis Brew
Department of Mathematical Sciences, University of Mines and Technology, Tarkwa, P. O. Box 237, Ghana.
Monica Veronica Crankson
Department of Mathematical Sciences, University of Mines and Technology, Tarkwa, P. O. Box 237, Ghana.
*Author to whom correspondence should be addressed.
Abstract
This study proposes a closed-system linear ordinary differential equation (ODE) model to analyse and predict the dynamics of liquid assets and outstanding loans in credit union liquidity risk management. Using 40 quarterly observations (2015–2024) obtained from the Central Finance Facility (CFF) of the Ghana Co-operative Credit Unions Association (CUA), the study formulates a two-state deterministic model governed by the asset-to-loan transition rate α, the loan-to-asset recovery rate β, and the loan default rate δ. Closed-form solutions are derived using the Laplace transform technique, and discrete quarterly prediction equations are obtained using a first-order Taylor expansion. Stability analysis confirms that the system is asymptotically stable when α, β, and δ are positive. Model performance is evaluated using root mean square error (RMSE), mean absolute percentage error (MAPE), and the Scatter Index (SI). The optimal asset prediction model corresponds to α = 0.0869 (MAPE = 14.27%), and the optimal loan model to ω = 0.91 (RMSE = 225,548.63). Additionally, three machine-learning models—Support Vector Regression (SVR), XGBoost, and Random Forest—are employed to generate eight-quarter-ahead forecasts (2025–2026). SVR achieves the best out-of-sample performance for forecasting both assets (MAPE = 2.92%) and loans (MAPE = 3.70%). The study concludes that the ODE-based closed liquidity model provides an analytically rigorous and practically interpretable framework for credit union liquidity planning, and that SVR-based forecasting offers a useful complement for short-term projections.
Keywords: Credit unions, liquid assets, loan dynamics, ordinary differential equations, laplace transform, stability analysis, liquidity risk management, support vector regression, machine-learning forecasting