A Linear Ordinary Differential Equation Model of Assets and Loan Liquidity Management of Credit Unions: A Two-state Dynamic Model

Determining Insolvency Regions in Banks: A Stochastic Dynamic Approach Integrating Liquidity and Credit Risk

We develop a continuous-time structural dynamic model to determine the exact insolvency regions of banks arising from the non-linear interaction between liquidity and credit risk. While existing literature predominantly treats these risks in isolation or via reduced-form specifications, we explicitly model the feedback loop where funding shocks and regulatory constraints force balance-sheet adjustments that can lead to endogenous insolvency. By incorporating Basel III regulatory requirements (LCR and NSFR) into a stochastic optimal control framework, we solve for the exact insolvency boundary using the Hamilton-Jacobi-Bellman (HJB) equation. To bridge the gap between theoretical complexity and supervisory practice, we derive and validate a surrogate analytical approximation function that allows for real-time monitoring. Calibrated using granular balance-sheet data from the Iranian banking sector, our model reveals significant non-linear threshold effects: the joint occurrence of liquidity stress and credit portfolio defaults disproportionately accelerates the transition toward insolvency compared to their individual effects. The proposed surrogate function offers supervisors a computationally efficient tool for stress testing and early warning systems. Our findings provide novel insights into financial frictions in emerging markets and offer a rigorous framework for integrated risk management.

Deterministic and Stochastic Modeling of Deposit–Loan Dynamics with Optimal Regulatory Control

Banks must balance deposit stability, loan expansion, and regulatory compliance while operating under liquidity constraints and financial risks. This study presents a mathematical model to examine the dynamics of bank deposits and loans under the influence of liquidity mechanisms and regulatory policies. The model proceeds in three stages: a deterministic nonlinear model, a dynamic optimal control model, and a stochastic model. Under the deterministic model, deposit withdrawals are liquidity-dependent, leading to a feedback mechanism in which liquidity improves deposit stability while financing loan growth. The theoretical results demonstrate the model’s positive and bounded solutions and show the existence and local stability of equilibria. Several parameters are based on regulatory policies or calibrated from Indonesian banking data, while the unknown parameters are estimated using the particle swarm optimization (PSO) algorithm. The results show that the proposed model is capable of fitting and predicting the data and has slightly lower mean absolute percentage errors for in-sample and out-of-sample compared with the benchmark model, and achieves comparable directional forecasting performance based on the index of directionality. Sensitivity analysis shows that the capital adequacy ratio supports lending, whereas an increased reserve requirement limits lending. An optimal control approach is developed by considering the reserve and capital requirements as time-varying policy variables. By applying Pontryagin’s maximum principle, we establish the necessary conditions for optimality. Numerical experiments demonstrate that the optimal control regulation enhances financial ratios, particularly the loan-to-deposit and liquidity ratios, at a reasonable cost. Finally, the stochastic model accounts for random variations in withdrawals and credit risks. Simulation-based observations reveal that although the system becomes more volatile, the mean dynamics are close to the deterministic case. Our framework offers a data-based and analytically tractable approach for studying the dynamics of banking variables and the effects of regulatory policies. The proposed model provides a mathematical tool for assessing the long-term effects of regulatory policies on banking performance and can assist bank managers and regulators in designing strategies that balance lending activity and liquidity resilience.

The Decoupling of Credit Growth and Operational Risk: An Integrative EPVw Approach under Clean Surplus Valuation

This study investigates the dynamic, non-linear relationship between private sector credit expansion and operational risk within financial institutions. Utilizing 2025 monthly loan data, the research proposes an integrative quantitative framework coupling Standard Score (Z-score) Probability Density Functions with the Weighted Expected Present Value (EPVw) model, incorporating a 0.995644 discount factor. Empirical results reveal a critical decoupling between credit volume and risk magnitude. While the loan portfolio expanded continuously from 6,742 billion in January to a peak of 7,318 billion in December, calculated risk (EPVw) followed a parabolic, bell-shaped distribution. Maximum risk exposure converged mid-year (June–August), driven by the highest statistical probability density as loan values hovered around the annual mean (μ = 6,997 billion). Conversely, despite peak volumes in December, year-end capital risk collapsed to its lowest point (377 billion) due to an extreme right-tail distribution shift (Z = 2.02) minimizing probability density weights (0.051). These findings challenge traditional linear risk assumptions, proving volatility concentrates within normal baselines rather than volume peaks. This study provides a vital framework for asset-liability management, ensuring capital reserves are optimally calibrated against market extremes, aligned with Clean Surplus Relation principles.

Cointegration and Error Correction Analysis of Non-Performing Loans and Return on Equity of Deposit Money Banks in Nigeria (2000-2025)

This research investigated the impact of non-performing loan ratio (NPLR), loan loss provision ratio (LLPR), and capital adequacy ratio (CAR) on the financial performance of Deposit Money Banks in Nigeria from 2000 to 2024, using return on equity (ROE) as a performance indicator. Employing an ex-post facto research design, secondary data were collected from the annual reports of selected banks. The analysis utilized the Augmented Dickey-Fuller (ADF) unit root test, Johansen cointegration test, and Error Correction Model (ECM) to evaluate both short- and long term relationships among the variables. Findings indicate the existence of a long-run equilibrium relationship among the variables, as evidenced by the error correction term ECM (-1), which recorded a coefficient of -0.964289 and a significant p-value of 0.0066. This implies that approximately 96.43% of short-run disequilibrium is corrected annually. In the short run, NPLR (coefficient = 0.200765, p = 0.7856), LLPR (coefficient = 0.508688, p = 0.3621), and CAR (coefficient = 0.260466, p = 0.9269) exerted positive but statistically insignificant effects on ROE. The study concludes that although NPLR, LLPR, and CAR do not significantly influence bank performance in the short run, a stable long-run equilibrium relationship exists among the variables. The study therefore recommends strengthening loan monitoring and recovery practices, improving loan loss provisioning policies, and maintaining adequate capital levels to support sustainable profitability and financial resilience.

Exchange-Rate Volatility and Financial Stability in the Banking Sector: Distributional Evidence from G7 and High-Income European Economies

The present study examined how volatility in exchange rates shapes banking-sector financial stability across the G7 and six high-income European countries, consisting of 13 developed economies. The study analyses the time period from 2000 to 2023. To measure volatility, the present study employed the GARCH(1,1) conditional variance of monthly real effective exchange rates. Stability is measured through the following two supporting indicators: Bank Z-score (solvency) and Non-Performing Loan (NPL) ratio (credit quality). Our analysis combines the Fully Modified OLS and two-step System GMM for analysing long-run and dynamic effects. To assess distributional heterogeneity, Method of Moments Quantile Regression (MMQR) is employed, while Dumitrescu–Hurlin tests are used for examining causality. The results show that volatility in exchange rates significantly reduces bank solvency and elevates credit risk. These effects are highly uneven: the adverse impact falls on the most fragile banking systems—those in the lower quantiles of the Z-score distribution and the upper quantiles of the NPL distribution. Causality runs unidirectionally, moving from volatility to instability. Institutional quality, which is proxied by the rule of law and regulatory quality, is seen to significantly decrease the credit-risk channel but not the solvency channel. Our findings provide implications for developed economies and support targeted, fragility-sensitive macro-prudential policy.

Influence Structure Capital and Risk Credit to Profitability in Banking Companies Listed on the Indonesia Stock Exchange (IDX) Period (2019-2023)