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


  • So Yeon Chun

    () (McDonough School of Business, Georgetown University, Washington, District of Columbia 20057; INSEAD, 77305 Fontainebleau, France)

  • Miguel A. Lejeune

    () (Department of Decision Sciences, George Washington University, Washington, District of Columbia 20052)


We consider a lender (bank) that 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 because of 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 concavifiability 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 multiloan 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.

Suggested Citation

  • So Yeon Chun & Miguel A. Lejeune, 2020. "Risk-Based Loan Pricing: Portfolio Optimization Approach with Marginal Risk Contribution," Management Science, INFORMS, vol. 66(8), pages 3735-3753, August.
  • Handle: RePEc:inm:ormnsc:v:66:y:2020:i:8:p:3735-3753
    DOI: 10.1287/mnsc.2019.3378

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    1. Bernd Engelmann & Ha Pham, 2020. "A Raroc Valuation Scheme for Loans and Its Application in Loan Origination," Risks, MDPI, Open Access Journal, vol. 8(2), pages 1-20, June.

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