How is logistic regression used for default prediction and how do I interpret coefficients?

Question

AcadiFi · Accepted Answer

Logistic regression models the log-odds of default as a linear combination of predictors: ln(PD / (1 - PD)) = beta_0 + beta_1 X1 + ... + beta_k Xk. PD itself is recovered via PD = 1 / (1 + exp(-Z)) where Z is the linear predictor.

Suppose a scorecard for Bluefield Auto Finance has three variables after WoE transformation: Z = -2.10 + 0.85 WoE_FICO + 0.62 WoE_DTI + 0.41 WoE_LTV. A coefficient of 0.85 on WoE_FICO means a one-unit increase in the WoE (i.e., moving from a worse to a better FICO bin) multiplies the odds of being good by exp(0.85) = 2.34. Because WoE is ln(P(good)/P(bad)) scaled by bin, a positive coefficient on WoE indicates the variable is predictive in the expected direction.

For a specific borrower with WoE_FICO = 0.40, WoE_DTI = -0.25, WoE_LTV = 0.10: Z = -2.10 + 0.34 - 0.155 + 0.041 = -1.874. PD = 1 / (1 + exp(1.874)) = 13.3%.

```mermaid
flowchart LR
    A[Raw vars] --> B[WoE bins]
    B --> C[Logistic fit]
    C --> D[Log-odds Z]
    D --> E[PD = sigmoid]
```

FRM exam points: coefficients on raw (non-WoE) variables should be interpreted as log-odds changes per unit of the variable. Maximum likelihood fits, convergence via Newton-Raphson, and statistical significance via Wald tests are all fair game.

How is logistic regression used for default prediction and how do I interpret coefficients?

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