We consider a lender (bank) who determines the optimal loan price (interest rates) to offer to prospective borrowers under uncertain risk and borrowers' response. A borrower may or may not accept the loan at the price offered, and in the presence of default risk, both the principal loaned and the interest income become uncertain. We present a risk-based loan pricing optimization model, which explicitly takes into account marginal risk contribution, portfolio risk, and borrower's acceptance probability. Marginal risk assesses the amount a prospective loan would contribute to the bank's loan portfolio risk by capturing the interrelationship between a prospective loan and the existing loans in the portfolio and is evaluated with respect to the Value-at-Risk and Conditional-Value-at-Risk risk measures. We examine the properties and computational difficulties of the associated formulations. Then, we design a concavifiability reformulation method that transforms the nonlinear objective function of the loan pricing problems and permits to derive equivalent mixed-integer nonlinear reformulations with convex continuous relaxations. We discuss managerial implications of the proposed model and test the computational tractability of the proposed solution approach.
Working Paper under submission.