Risk-Based Loan Pricing: Portfolio Optimization Approach with Marginal Risk Contribution

Abstract

We consider a lender (bank) who determines the optimal loan price (interest rate) to offer to prospective borrowers under uncertain borrower response and default risk. A borrower may or may not accept the loan at the price offered, and both the principal loaned and the interest income become uncertain due to the risk of default. We present a risk-based loan pricing optimization framework that explicitly takes into account the marginal risk contribution, the portfolio risk, and a borrower's acceptance probability. Marginal risk assesses the incremental risk contribution of a prospective loan to the bank's overall portfolio risk by capturing the dependencies between the prospective loan and the existing portfolio, and is evaluated with respect to the Value-at-Risk and Conditional Value-at-Risk measures. We examine the properties and computational challenges of the formulations. We design a reformulation method based on the concaviability concept to transform the nonlinear objective functions and to derive equivalent mixed-integer nonlinear reformulations with convex continuous relaxations. We also extend the approach to the multi-loan pricing problems, which feature explicit loan selection decisions in addition to pricing decisions. We derive formulations with multiple loans that take the form of mixed-integer nonlinear problems with nonconvex continuous relaxations and develop a computationally efficient algorithmic method. We provide numerical evidence demonstrating the value of the proposed framework, test the computational tractability, and discuss managerial implications.

Keywords: pricing, revenue management, credit risk, marginal risk, Value-at-Risk, Conditional Value-at-Risk, willingness-to-pay, mixed-integer nonlinear stochastic programming