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A new approach to rating scale definition with quantum-inspired optimization

Patrizio Spada, Laura Cappelli, Francesca Cibrario, Christian Mattia, Daniele Magnaldi, Matteo Argenton, Enrico Calore, Sebastiano Fabio Schifano, Concezio Bozzi, Davide Corbelletto

Abstract

In finance, assessing the creditworthiness of loan applicants requires lenders to cluster borrowers using rating scales. Financial institutions must define the scales in compliance with strict institutional constraints, resulting in solving a complex combinatorial constrained optimization problem. This contribution studies how to solve this problem using a Quadratic Unconstrained Binary Optimization (QUBO) model, a formulation suitable for quantum hardware. We validate this approach by testing the proposed formulation with classical heuristics. We then benchmark the results against a brute-force method to demonstrate consistent solution quality and highlight the framework's suitability for more complex scenarios.

A new approach to rating scale definition with quantum-inspired optimization

Abstract

In finance, assessing the creditworthiness of loan applicants requires lenders to cluster borrowers using rating scales. Financial institutions must define the scales in compliance with strict institutional constraints, resulting in solving a complex combinatorial constrained optimization problem. This contribution studies how to solve this problem using a Quadratic Unconstrained Binary Optimization (QUBO) model, a formulation suitable for quantum hardware. We validate this approach by testing the proposed formulation with classical heuristics. We then benchmark the results against a brute-force method to demonstrate consistent solution quality and highlight the framework's suitability for more complex scenarios.

Paper Structure

This paper contains 22 sections, 28 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Rating scale definition
  3. Rating scale definition and quantum computing
  4. A QUBO formulation for defining the rating scale
  5. QUBO models
  6. Constraints for the rating scale definition problem
  7. From constraints to QUBO penalties
  8. Logical constraint
  9. Monotonicity of the default rates
  10. Concentration
  11. Grade cardinality thresholds
  12. Code implementation and results
  13. A constrained implementation
  14. Building the QUBO formulation
  15. Assessing the monotonicity approximation
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Computational performance of the constrained algorithm.
  • Figure 2: Histograms of the cost function values associated with the confusion matrices reported in Tables \ref{['tab:confusion_matrices']}. The x-axis shows the cost function values computed for all binary strings. Bars are colored according to the corresponding confusion matrix category: true positives (TP), false positives (FP), true negatives (TN), and false negatives (FN).