Profit-Based Lending Discrimination

• The paper introduces a principled profit-based measure that quantifies lending discrimination via differences in expected profit or ROI across protected groups.
• It employs empirical, synthetic, and counterfactual methodologies to estimate group-specific profits, identifying calibration errors and bias effects.
• The work integrates fairness constraints into credit modeling, revealing trade-offs between operational profitability and reducing discriminatory profit gaps.

A profit-based measure of lending discrimination provides a principled, economically interpretable framework for quantifying disparate impact in credit decisions through differences in expected profit or return on investment (ROI) across protected groups. This approach captures the realized financial implications of algorithmic or human lending decisions, extends beyond approval rate or error-based fairness measures, and offers a direct link between group-level model calibration, underwriting practices, and observed disparities in loan profitability.

1. Formal Framework and Key Definitions

Let $X$ denote the underwriting features for a loan applicant, $p(X)$ the true probability of default, and $\hat p(X)$ the credit model’s predicted probability of default, with a quoted interest rate (or APR) $r(X)$ and normalized principal. The expected profit per unit lent is given by

$$\pi\bigl(\hat p(X),\,r(X)\bigr) = (1-p(X))\,r(X) - p(X)$$

where $p(X)$ governs expected loss and $(1-p(X)) r(X)$ expected revenue. For each protected group $g$ (e.g., race, gender), define group-conditional average profit

$\pi_g = \mathbb{E}[\pi(\hat p(X), r(X)) | G = g]$

and the profit gap (“profit-based discrimination”): $\Delta_\pi(g, g') = \pi_g - \pi_{g'}$ A fair, risk-neutral allocation implies $\Delta_\pi(g, g') = 0$ for all $g, g'$ (Coots et al., 23 Dec 2025).

Analogous definitions are used in simplified binary lending settings: $$\pi(y, \hat y) = \begin{cases} +C & \text{if } \hat y = 1,\, y = 1 \ -B & \text{if } \hat y = 1,\, y = 0 \ -C & \text{if } \hat y = 0,\, y = 1 \ 0 & \text{if } \hat y = 0,\, y = 0 \end{cases}$$ where $C$ is the net return to a good loan and $B$ the loss given default. The group profit gap is

$$\Delta\pi = \mathbb{E}[\pi | A=\text{adv}] - \mathbb{E}[\pi | A=\text{disadv}]$$

as in (Kozodoi et al., 2021, Bansal, 8 May 2025).

2. Measurement Methodologies

To estimate group-specific profits in practice:

• Empirical IRR/Net Profit Approach: Aggregates actual cash flows for each loan and solves for the internal rate of return (IRR) or mean profit per group, typically applied to rich cohort data with observed repayments, APRs, and risk scores. Group labels may be observed or imputed probabilistically via Bayesian Improved Surname Geocoding (BISG) and name-based gender inference (Coots et al., 23 Dec 2025).
• Synthetic or Simulated Evaluation: On synthetic data, approval decisions $A_i$, outcomes $y_i$, and profits $\pi_i$ are simulated under various fairness constraints. Group-level sums and ROI are computed:

$$\mathrm{Profit}_G = \sum_{i \in G} A_i (y_i\,r\,L - (1-y_i)\,d\,L)$$

and

$$\Delta\mathrm{Profit} = \mathrm{Profit}_{G^c} - \mathrm{Profit}_G$$

with $d$ the default-loss proportion (Bansal, 8 May 2025).

• Structural Counterfactual Simulation: Human or algorithmic lending policies are imitated in structural models, and counterfactual “bias-free” regimes (removing preference or belief biases) are compared. The difference in aggregate expected profits, $$\Delta\Pi = \Pi_\text{unbiased} - \Pi_\text{biased}$$, identifies the profit penalty of bias (Hu et al., 2022).

3. Drivers of Profit Disparities: Model Calibration and Bias

Observed profit-based discrimination can arise from model calibration errors or explicit biases:

• Calibration Error: Let

$$\mathrm{CalErr}(g) = \mathbb{E}[\hat p(X) - p(X) | G=g]$$

Systematic underestimation ($\mathrm{CalErr} < 0$) of default risk for group $g$ lowers APRs and boosts $\pi_g$, yielding a negative $\Delta_\pi$ for that group; overestimation depresses $\pi_g$ (Coots et al., 23 Dec 2025).

• Preference- and Belief-Based Bias: Human evaluators may discount certain groups via explicit latent utility penalties (taste-based bias) or shift prior beliefs (belief-based bias), affecting $P(\text{approval}|x, g)$ and resulting realized profits (Hu et al., 2022). Counterfactual removal of these biases quantifies their profit impact.
• Indirect/Proxy Discrimination: Unobserved mediators correlated with both protected status and repayment risk can induce profit gaps in statistical models, as seen in P2P contexts (Shen et al., 2022).

4. Integration with Fairness Constraints and Optimization

Profit-parity constraints or penalties can be directly incorporated into credit model training: $$\min_f L_0(f) + \lambda |\Delta\pi(f)| \quad \text{or} \quad \max_f \mathbb{E}[\pi(f)] - \lambda |\Delta\pi(f)|$$ with $\lambda$ trading off total profit and parity. Alternatively, hard constraints $|\Delta\pi(f)| \leq \epsilon$ can be imposed (Kozodoi et al., 2021). Empirically, moderate reductions in profit gap ($\sim$20% of mean profit) can be achieved with minimal profit cost ($<$5%), while eliminating the gap entirely typically requires substantial profit sacrifice ($>$30%) (Kozodoi et al., 2021, Bansal, 8 May 2025).

A unified "profit-based discrimination index" $D^{(m)}$ may be formulated as

$$D^{(m)} = w_1 \frac{\Delta\mathrm{Profit}_{\mathrm{loss}}^{(m)}}{\mathrm{Profit}_{\mathrm{baseline}}} + w_2 \frac{|\Delta\mathrm{Profit}^{(m)}|}{\mathrm{Profit}_{\mathrm{baseline}}}$$

where $\Delta\mathrm{Profit}_{\mathrm{loss}}$ captures the economic cost of fairness constraints and $|\Delta\mathrm{Profit}|$ the residual group gap (Bansal, 8 May 2025).

5. Empirical Findings in Real and Simulated Lending Settings

Empirical application of profit-based discrimination has yielded the following key findings:

• Loans to men and Black applicants on a major U.S. fintech platform earned lower average IRR (e.g., 7.7% for Black vs. 8.5% for White, 8.3% for men vs. 9.1% for women). The estimated profit gap for both race and gender was $-0.8\%$ IRR, indicating favorable terms for these groups (Coots et al., 23 Dec 2025).
• Underlying source: The "blind" risk model systematically underestimated risk for Black borrowers and overestimated it for women by approximately 0.8% and 0.4% respectively. Calibration correction (explicitly including race/gender) eliminated the IRR gap but contravened U.S. disparate-treatment law (Coots et al., 23 Dec 2025).
• In synthetic simulations, fairness interventions (e.g., demographic parity, equal opportunity) reduced group profit disparities but often at the expense of overall profitability. The severity of this trade-off depends on the constraint and market parameters (interest rate, default loss) (Bansal, 8 May 2025).
• In micro-lending settings, human taste- and belief-based gender biases decreased firm profit, and machine learning models could mitigate these losses by neutralizing such biases (Hu et al., 2022).
• In large-scale P2P platforms, female borrowers with comparable actual returns were significantly more likely to receive funding, yet taste-based discrimination (higher required return thresholds for women) persisted alongside rational statistical discrimination (Shen et al., 2022).

6. Extensions, Limitations, and Practical Guidance

• Extension to Other Lending Forms: The structural and empirical profit-discrimination frameworks are adaptable to different credit products (e.g., mortgage, credit card), provided sufficient observed or imputed repayment and approval data is available (Hu et al., 2022).
• Practical Use for Lenders: Regularly analyze group-specific profit, monitor the profit gap ($\Delta\pi$), and set thresholds for intervention. Consider pilot studies to tune $\lambda$ (the fairness–profit parameter) against real operational constraints (Kozodoi et al., 2021, Coots et al., 23 Dec 2025).
• Limitations: Existing models often assume uniform loan size and cost structure; lack of segment-level heterogeneity may obscure subgroup-specific effects. Many studies stop short of granular causal analysis or dynamic/longitudinal fairness impact (Moldovan, 2022, Coots et al., 23 Dec 2025).
• Legal and Regulatory Challenges: Directly correcting calibration through explicit use of protected-class information may eliminate profit-based discrimination but can conflict with disparate-treatment and fair-lending statutes (e.g., ECOA/FHA in the U.S.) (Coots et al., 23 Dec 2025).
• Robustness Issues: Accurate profit estimation requires rich outcome data and credible missingness assumptions in the presence of unfunded or censored loans (Shen et al., 2022).

7. Relationship to Standard Fairness Metrics

Profit-based discrimination metrics complement but do not supplant conventional statistical fairness measures (demographic parity, equal opportunity, predictive parity). In practice, profit differences are often assessed alongside traditional group-fairness metrics:

• Profit as a Performance, Not a Fairness Metric: Some studies treat profit as an axis to be balanced against pure fairness metrics (e.g., SPD, DI, AOD), not as a fairness metric per se (Moldovan, 2022).
• Unified Efficiency Frontiers: Plotting Pareto frontiers of (profit, $|\Delta\pi|$) allows transparent visualization of trade-offs between profitability and profit-based fairness (and, by extension, traditional fairness criteria) (Kozodoi et al., 2021, Bansal, 8 May 2025). A plausible implication is that the "efficient frontier" varies with credit product structure and market regime, necessitating context-specific tuning.

Summary Table: Core Definitions and Metrics

Metric Type	Mathematical Expression	Reference
Per-loan profit	$\pi(\hat p, r) = (1 - p) r - p$	(Coots et al., 23 Dec 2025)
Group profit	$\pi_g = \mathbb{E}[\pi(\hat p, r) | G = g]$	(Coots et al., 23 Dec 2025)
Profit gap	$\Delta_\pi(g, g') = \pi_g - \pi_{g'}$	(Coots et al., 23 Dec 2025)
Structural gap	$$\Delta\Pi = \Pi_\text{unbiased} - \Pi_\text{biased}$$	(Hu et al., 2022)
Unified index	$$D^{(m)} = w_1 \frac{\Delta\mathrm{Profit}_{\mathrm{loss}}}{\mathrm{Profit}_\mathrm{baseline}} + w_2 \frac{|\Delta\mathrm{Profit}|}{\mathrm{Profit}_\mathrm{baseline}}$$	(Bansal, 8 May 2025)

Profit-based measures provide a concrete, economically grounded basis for quantifying and managing lending discrimination, clarifying the interplay of model calibration, fairness interventions, and operational constraints. They are essential for both regulatory compliance audits and the design of equitable, profitable credit-scoring systems.