Odds Ratios and Relative Risk

A credit model finds that borrowers with a low credit score default 20% of the time, versus 5% for borrowers with a high credit score. A headline reads "low-score borrowers are 4 times as likely to default" — that's the relative risk (0.20/0.05 = 4). But the model's own coefficients, from a logistic regression, are naturally expressed as an odds ratio, and if you plug in the same numbers you get a very different-sounding 4.75. Both numbers are correct, computed from the identical underlying probabilities — they're just answering subtly different questions, and confusing the two is one of the most common misinterpretations of model output in credit risk and classification work.

An analogy: probability of rain vs. betting odds on rain

If there's a 20% chance of rain, a weather forecaster states the probability directly: 1 in 5. A bookmaker instead quotes odds: for every $1 bet on rain, you'd win $4 if it rains (20% probability implies odds of 1:4, "4 to 1 against"). These aren't the same number — probability and odds are related but different scales, and the gap grows the further probability sits from 0 or 50%. Relative risk compares two probabilities directly, the way the forecaster does; the odds ratio compares two sets of betting odds, the way the bookmaker does — so ratios of odds diverge from ratios of probabilities whenever the underlying probabilities aren't small.

The mechanics, one piece at a time

For an event with probability $p$, the odds are $p/(1-p)$ — the ratio of "happens" to "doesn't happen." Given two groups with event probabilities $p_1$ and $p_2$, the relative risk is simply

$$RR = \frac{p_1}{p_2},$$

a direct ratio of probabilities. The odds ratio is instead the ratio of each group's odds:

$$OR = \frac{p_1/(1-p_1)}{p_2/(1-p_2)}.$$

In plain English: relative risk asks "how many times more likely is the event," while the odds ratio asks "how many times greater are the betting odds on the event." When both $p_1$ and $p_2$ are small (a rare event, like default in a healthy loan book), $1-p_1$ and $1-p_2$ are both close to 1, so $OR \approx RR$ — the two measures nearly agree. But as either probability grows away from zero, $(1-p)$ shrinks noticeably below 1 and the two measures diverge, with the odds ratio always exaggerating the relative-risk figure whenever both probabilities exceed 50%, or understating it in a mirrored way for probabilities both below 50% but not small.

Worked example 1: the credit default numbers, both ways

$p_1 = 0.20$ (low-score group), $p_2 = 0.05$ (high-score group). Relative risk: $RR = 0.20/0.05 = 4.0$ — low-score borrowers are 4 times as likely to default. Odds: low-score odds $= 0.20/0.80 = 0.25$; high-score odds $= 0.05/0.95 \approx 0.0526$. Odds ratio: $OR = 0.25/0.0526 \approx 4.75$. The odds ratio (4.75) overstates the relative risk (4.0) by about 19% here — not enormous, but enough that reporting "4.75 times more likely to default," when the logistic regression coefficient's exponentiated value is handed over uncritically as if it were relative risk, is simply incorrect and overstates the true risk multiple.

Worked example 2: where the gap becomes severe

Now suppose $p_1 = 0.60$ and $p_2 = 0.30$ — a common event, like a stock beating its sector in a quarter under two signal regimes. Relative risk: $RR = 0.60/0.30 = 2.0$. Odds: $0.60/0.40 = 1.5$ and $0.30/0.70 \approx 0.4286$. Odds ratio: $OR = 1.5/0.4286 \approx 3.5$ — 75% larger than the true relative risk. This is exactly the trap with logistic regression output on common events: a fitted "3.5x" odds ratio sounds far more dramatic than the true doubling of probability it corresponds to, and reporting it as "3.5 times more likely" is a meaningful, misleading exaggeration.

The logistic curve below is exactly the function a credit or default model fits, and its coefficients exponentiate into odds ratios — watch how the curve's steepest, most probability-sensitive region sits away from the extremes, which is where odds and probability diverge most.

What this means in practice

Logistic regression, the default tool for credit scoring, default prediction, and binary classification generally, produces coefficients that exponentiate directly into odds ratios — not relative risk. Whenever the event being modeled isn't rare (defaults in a stressed portfolio, up-days for a stock, any outcome occurring more than roughly 10-15% of the time), reporting a logistic regression's odds ratio as if it were "X times more likely" materially overstates the true probability multiple, and this error compounds when odds ratios from different baseline probabilities are compared to each other as if they meant the same thing.