Portfolio optimization: An application of risk-based loan pricing

Abstract

This research addresses the need for more effective risk-based loan pricing, motivated by the limitations of traditional fixed-rate models which inadequately account for borrower risk, acceptance probability, and portfolio interactions. The study is particularly relevant for financial institutions engaged in unsecured consumer lending, providing a structured approach to enhance pricing accuracy, profitability, and portfolio risk control. The research tackles the suboptimal pricing that arises when conventional models disregard both borrower behavior and marginal risk contribution. An optimized loan pricing framework was developed by integrating borrower acceptance probability and marginal risk metrics into a mixed-integer nonlinear programming (MINLP) model. The study built upon the foundational work of Chun and Lejeune (2020), where it employed value at risk (VaR), scenario-based simulations, and borrower acceptance models (linear, exponential, logistic). Computational challenges were addressed using sample average approximation (SAA), McCormick linearization, and concavification techniques. While their model integrates borrower acceptance and marginal risk, it has practical limitations. To address these, the study proposes: 1. A modified present value function to improve differentiability and numerical stability. 2. A simulated portfolio framework using real-world loan data to evaluate the pricing model under realistic conditions. 3. Model evaluation against a fixed-price scheme, demonstrating improved profitability and efficiency. Empirical simulations demonstrated that incorporating marginal risk and borrower response improved average profitability by approximately 67% compared to current pricing schemes. However, key limitations were identified. The MINLP problem’s nonconvex nature caused Couenne1 to frequently fail to find global solutions, while BONMIN2 provided only local optima. Optimized rates were, on average, 22% higher than market levels, raising concerns over practical borrower acceptance. Model stability improved with increased scenario sampling but worsened with larger existing loan portfolios. The study offers a practical framework for risk-based pricing but highlights the need for further methodological improvements, including solver enhancements, regulatory constraints, and dynamic multi-period portfolio considerations to contribute to practical and scalable implementation.