Rolling and Expanding Windows

A 20-day moving average needs to answer the same question every single day: "what happened over the last 20 days?" As one new day arrives, the oldest day falls off the back. That sliding behavior is a rolling window. An expanding window never drops anything — day 500's average is computed over all 500 days that came before it, not just the last 20.

Both show up constantly in quant code: rolling windows compute moving averages, rolling volatility, and rolling Sharpe ratios; expanding windows compute running statistics like a strategy's cumulative max drawdown or an out-of-sample model that should only ever "know" the past.

What each one computes

For a rolling window of size $N$, the value at time $t$ is a function of observations $t-N+1$ through $t$ only. Slide forward one step, and the window's leading edge drops the oldest point while gaining one new point. For an expanding window, the value at time $t$ is a function of every observation from the start of the series through $t$ — the window only ever grows.

Worked example: rolling mean by hand

Daily returns (in %) for a stock: 1, -2, 3, 0, 4, -1. Compute a 3-day rolling mean.

1. Day 3 (first full window): average of days 1-3 = $(1 - 2 + 3)/3 = 0.67$.
2. Day 4: window slides to days 2-4 = $(-2 + 3 + 0)/3 = 0.33$. Day 1 has dropped out entirely.
3. Day 5: window is days 3-5 = $(3 + 0 + 4)/3 = 2.33$.
4. Day 6: window is days 4-6 = $(0 + 4 - 1)/3 = 1.0$.

Compare to the expanding mean at the same points: day 4 would average all four days ($1-2+3+0)/4 = 0.5$, and day 6 would average all six ($1-2+3+0+4-1)/6 = 0.83$. The expanding value moves much less per step because each new point is diluted by a growing denominator — that damping is the whole difference in behavior between the two.

In pandas this is df["ret"].rolling(3).mean() versus df["ret"].expanding().mean(). Both return NaN until enough data exists to fill the first window — the rolling version needs its first $N-1$ rows to warm up, the expanding version only needs min_periods (default 1).

What this means in practice

Rolling windows are the right tool when you want a statistic to track current conditions — a 20-day realized volatility should reflect this month's market, not 2019's. Expanding windows are the right tool when leaking future information would be a bug: an expanding Sharpe ratio computed only from data available up to each date is a legitimate point-in-time metric, while a rolling Sharpe computed with a centered window secretly uses future returns.