Insights

How to Compute Abnormal Returns

Alphanume Team · April 27, 2026

Benchmark models for event-study returns — picking the right one for your question.

Abnormal returns are the difference between actual returns and the returns predicted by a benchmark model. Computing them properly is the foundation of every credible event-study — including post-offering drift, lock-up expiration effects, and any other event-driven analysis. The mechanics are simple; the choice of benchmark model has real implications and is the place researchers most often introduce hidden assumptions.

The basic formula

For each security i and each date t in the event window:

AR(i,t) = R(i,t) − E[R(i,t)]

Where R(i,t) is the actual return and E[R(i,t)] is the return predicted by the benchmark model. The cumulative abnormal return across an event window of T days is the sum: CAR(i) = Σ AR(i,t) for t in event window.

Benchmark model choices

Several standard benchmarks, in increasing complexity:

1. Mean-adjusted returns. E[R(i,t)] = average return of the security over an estimation window prior to the event. Simple, but uncontrolled for market direction.

2. Market-adjusted returns. E[R(i,t)] = market return on date t (e.g., S&P 500 or CRSP value-weighted index). Removes broad market direction; assumes beta of 1.

3. Market model. E[R(i,t)] = α + β × R_market(t), where α and β are estimated from regression over an estimation window prior to the event. The standard academic approach for event studies.

4. Multi-factor models. Fama-French 3-factor (market, size, value) or 5-factor (adding profitability and investment) or Carhart 4-factor (adding momentum). More precise but estimation noise grows.

5. Matched-firm benchmark. E[R(i,t)] = return of a matched control firm or matched portfolio (by size, sector, and style). Particularly useful for long-horizon studies where factor-model precision degrades.

Which to use

Rule of thumb:

Short event windows (1–10 days): Market model is the standard. Multi-factor models add little.
Medium event windows (10–60 days): Market model still works; matched-firm approaches gain useful precision.
Long event windows (60–365 days): Matched-firm or matched-portfolio approaches are preferred. Factor-model estimates become unstable.

For dilution-event drift specifically — where the relevant windows are weeks to months — matched-firm approaches are commonly used.

The estimation window

For models requiring parameter estimation (market model, factor models), the estimation window matters:

Typical choice: 120–250 trading days ending a defined buffer (typically 21–30 days) before the event.
Buffer rationale: Avoid contamination from pre-event drift if the event is anticipated by the market.
Window length tradeoffs: Longer windows give more precise estimates but assume stability of parameters; shorter windows are more responsive but noisier.

Cumulative vs buy-and-hold abnormal returns

Two ways to aggregate abnormal returns over an event window:

CAR (cumulative abnormal return): Sum of daily abnormal returns. Standard for short windows. Easy to interpret as additive.

BHAR (buy-and-hold abnormal return): Compound the security's returns, compound the benchmark's returns, and take the difference. More appropriate for long windows because it reflects actual investor experience.

The CAR-BHAR divergence grows with the length of the event window. For 60+ day windows, BHAR is generally preferred.

Statistical testing

To determine whether observed abnormal returns are statistically distinguishable from zero:

Cross-sectional t-test: Standard approach for testing whether the mean CAR across an event sample is significantly nonzero.
Standardized CAR test: Divides each security's CAR by its event-window standard deviation. More robust to heteroskedasticity.
Bootstrap or permutation tests: Robust to non-normality and small samples.
Sign tests: Non-parametric; useful when the return distribution has heavy tails.

For dilution-event samples — which tend to have heavy tails due to occasional very-large moves — non-parametric tests often produce more credible inference than parametric tests.

Common errors

Confounding events. Other corporate actions (earnings, M&A) during the event window contaminate the signal. Either exclude these events or model them.
Selection bias in the estimation window. Picking estimation windows that produce favorable parameters is data mining.
Survivorship bias in the event sample. Excluding events whose subjects later delisted biases the result — see survivorship bias.
Look-ahead in the benchmark. Using factor-model loadings computed using post-event data is look-ahead — see look-ahead bias.
Currency or trading-day mismatches. For cross-market analyses, ensure dates and returns are aligned.