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Escaping the Subprime Trap in Algorithmic Lending

Adam Bouyamourn, Alexander Williams Tolbert

Abstract

Disparities in lending to minority applicants persist even as algorithmic lending finds widespread adoption. We study the role of risk-management constraints, specifically Value-at-Risk ($\VaR$) and Expected Shortfall (ES), in inducing inequality in loan approval decisions, even among applicants who are equally creditworthy. Empirical research finds that disparities in the interest rates charged to minority groups can remain large even when loan applicants from different groups are equally creditworthy. We contribute an original analysis of 431,551 loan applications recorded under the Home Mortgage Disclosure Act, illustrating that disparities in data quality are associated with higher rates of loan denial and higher interest rate spreads for Black borrowers. We develop a formal model in which a mainstream bank (low-interest) is more sensitive to variance risk than a subprime bank (high-interest). If the mainstream bank has an inflated prior belief about the variance of the minority group, it may deny that group credit indefinitely, thus never learning the true risk of lending to that group, while the subprime lender serves this population at higher rates. We call this ``The Subprime Trap'': an equilibrium in which minority lenders can borrow only from high-cost lenders, even when they are as creditworthy as majority applicants. Finally, we show that a finite subsidy can help minority groups escape the trap: subsidies cover enough of the mainstream bank's downside risk so that it can afford to lend to, and thereby learn the true risk of lending to, the minority group. Once the mainstream bank has observed sufficiently many loans, its beliefs converge to the true underlying risk, it approves the applications of minority groups, and competition drives down the interest rates of subprime loans.

Escaping the Subprime Trap in Algorithmic Lending

Abstract

Disparities in lending to minority applicants persist even as algorithmic lending finds widespread adoption. We study the role of risk-management constraints, specifically Value-at-Risk () and Expected Shortfall (ES), in inducing inequality in loan approval decisions, even among applicants who are equally creditworthy. Empirical research finds that disparities in the interest rates charged to minority groups can remain large even when loan applicants from different groups are equally creditworthy. We contribute an original analysis of 431,551 loan applications recorded under the Home Mortgage Disclosure Act, illustrating that disparities in data quality are associated with higher rates of loan denial and higher interest rate spreads for Black borrowers. We develop a formal model in which a mainstream bank (low-interest) is more sensitive to variance risk than a subprime bank (high-interest). If the mainstream bank has an inflated prior belief about the variance of the minority group, it may deny that group credit indefinitely, thus never learning the true risk of lending to that group, while the subprime lender serves this population at higher rates. We call this ``The Subprime Trap'': an equilibrium in which minority lenders can borrow only from high-cost lenders, even when they are as creditworthy as majority applicants. Finally, we show that a finite subsidy can help minority groups escape the trap: subsidies cover enough of the mainstream bank's downside risk so that it can afford to lend to, and thereby learn the true risk of lending to, the minority group. Once the mainstream bank has observed sufficiently many loans, its beliefs converge to the true underlying risk, it approves the applications of minority groups, and competition drives down the interest rates of subprime loans.

Paper Structure

This paper contains 34 sections, 11 theorems, 71 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Literature
  4. Empirical Motivation
  5. Preliminaries and Model
  6. Loan Applicants
  7. Payoff distributions
  8. Observable Applicant Characteristics
  9. Lenders
  10. Bank payoff functions
  11. Risk Management Constraints
  12. Prior Bank Beliefs
  13. Differential Credit Records Drive Prior Variances
  14. Learning Through Lending
  15. Our Modeling Choices
  16. ...and 19 more sections

Key Result

Lemma 1

If bank $j$ lends to group $i$, it observes a return $\pi_{it}$ and then updates its posterior beliefs about the variance $\sigma_i^2$. Suppose that the bank observes a sequence of returns $\{\pi_1 ,\pi_2, \dots, \pi_M\}$. Then, it has posterior belief about the variance of group $i$: So that:

Theorems & Definitions (24)

  • Definition 1: Value-at-Risk ($\mathrm{VaR}$)
  • Definition 2: Bank's prior beliefs
  • Lemma 1: Belief Updating via Bayes Rule
  • Lemma 2: Unilateral Lending Thresholds
  • Lemma 3: Pooled Variance Threshold for Bank $L$
  • Lemma 4: Pooled Variance Threshold for Bank $H$
  • Corollary 1: Ordered variance thresholds
  • Theorem 1: Subprime Trap Equilibrium
  • proof
  • Remark 1: Purely Informational Failure
  • ...and 14 more