Daily vs Weekly Charts: How the Expected Range Scales

Put the same expected-range indicator on a daily and a weekly chart of one symbol and two questions arrive together. One is arithmetic: how much wider should the weekly range be? The other is empirical: the calibration figures were measured on daily closes, so do they carry to the weekly or monthly chart?

The arithmetic question has a famous rule of thumb attached, and the rule is where the confusion starts. This article separates the two questions and reports what the working paper found when it ran the construction natively on weekly and monthly bars.

The square-root-of-time rule, and what it assumes

The rule says volatility grows with the square root of the horizon: a 1% daily standard deviation becomes about 1% × √5 ≈ 2.2% over five trading days and about 4.6% over 21. Run out to a year, it is the √252 conversion behind most annualized figures, walked through in what is rolling volatility.

The rule is exact under one condition: each period's return is independent of the last, drawn from the same distribution, with constant volatility. Variances then add, and the standard deviation of a sum grows with the square root of the number of terms. That condition is a model, not an observation. Real returns show volatility clustering, where large moves follow large moves, as described in what is volatility clustering, and volatility shifts between regimes. When the condition fails, the rule is an approximation. For a unit conversion that is usually fine; for a band that claims to contain the next close at a stated rate, it is a claim that needs testing.

A return-space band does not scale. It re-measures

The construction behind the Behavioral Transform Model, described on the How It Works page, is a rolling window of recent returns, their mean and standard deviation, and a range projected from the prior close. Every quantity there is defined per bar. On a weekly chart the returns in the window are weekly, Friday close to Friday close; on a monthly chart, month-end to month-end. The weekly band is not the daily band multiplied by √5; it is a fresh measurement of how large weekly moves have recently been.

The construction has no timeframe of its own. The window counts bars, not calendar time, so the canonical 60-bar window that spans about a quarter on the daily chart spans about fourteen months on the weekly and about five years on the monthly. The same property lets the band run down to intraday bars, the subject of does an expected range work on intraday charts; this article covers the other direction. It follows from building the range from returns rather than price levels, the distinction set out in return space vs price space. In price terms the weekly band is wider than the daily one, but not because anything was scaled: weekly returns are larger, and the window measured them.

What was measured going up

The paper's headline figures are daily-bar figures: across 40 instruments in five asset classes, the inner band contained about 71% of next closes historically and the outer band about 94%; past behavior is not a guarantee of future results. An appendix on multi-horizon calibration then applied the construction natively at weekly and monthly frequencies to the same universe, holding the window at 60 bars.

Inner-band containment came out at about 71% on weekly bars and about 72% on monthly, all three frequencies within 0.8 percentage points of one another and all roughly four points above the 67.46% that a 60-bar window would produce on Gaussian returns; the outer band was within half a point of its daily figure at both. The paper's summary is the sentence worth keeping: the coverage stability is a property of the construction at a fixed estimation window, not of the bar frequency. These are historical measurements.

One thing changes as the bar lengthens. The spread of containment rates across instruments was about 2 percentage points on daily and weekly bars and about 4.7 points on monthly, because monthly cells rest on far fewer observations. The average held; the individual rows are noisier.

The practitioner window, and why the raw number drops

Holding the window at 60 bars is one way up a timeframe. A common alternative holds calendar time instead: 12 bars on the weekly chart and 6 on the monthly, so each window still spans roughly a quarter to half a year. The paper reports that configuration too, and the FAQ page summarises it: raw inner-band containment fell to about 66% weekly and about 62% monthly.

That looks like the band losing calibration as the window shrinks. It is not, for the same reason that 67.46% rather than the textbook 68.27% is the right daily benchmark, as how to backtest a band indicator explains: a standard deviation estimated from 6 observations is noisy, and a ±1σ envelope built on it covers less of the distribution. The correct expectation for a 12-bar window on Gaussian returns is about 64%, and for a 6-bar window about 60%. Against those targets the measured 66% and 62% sat one to two points above, and the outer band at both frequencies landed within half a point of its corrected target. The FAQ's phrasing is the honest one: still well calibrated against the correct null, a softer envelope in raw terms because its proper target is lower.

Neither configuration is prescribed anywhere in the published work; the two rows report what each measured. A 6-bar and a 60-bar monthly band are different objects with different targets, each scored against its own, and the How It Works page's general trade-off, shorter reacts faster and longer is steadier, holds here as everywhere.

What happened when the square-root rule was used instead

The paper also ran the version of its own band that the rule of thumb implies: project the daily volatility estimate forward by √N to cover an N-day horizon, rather than measuring returns at the longer frequency. That projection contained about 63% of outcomes at a 20-day horizon and about 53% at 60 days, well below the 71% to 72% that native measurement produced. The paper's explanation is the one above: the independence assumption breaks across the regime variation a longer horizon spans.

The scope is narrow: two specifications of Oisigma's own construction, compared on the paper's universe. It says nothing about any other tool and does not make the square-root rule wrong as arithmetic. It says that, for the job of drawing a calibrated range at a longer horizon, giving the model the actual returns at that frequency was the better-calibrated choice, historically. Past behavior is not a guarantee of future results.

Where the evidence stops

The weekly and monthly figures are universe averages, with monthly rows individually noisy. A daily-chart band remains a one-bar-ahead range; the published record does not report a multi-week-ahead rate drawn on a daily chart. The crisis-onset limit, where a rolling window lags a fast volatility spike, is described in when volatility bands fail on daily data only; nothing published reports it at weekly or monthly resolution.

And, as with every result on this site, the paper validates the range's calibration, not the profitability of any use of it. Keeping one consistent way of reading structure across markets and timeframes is one of the uses described on the How It Works page; whether any such use delivers value after costs is an open question the calibration work does not address, on any timeframe.

The takeaway

The square-root-of-time rule answers an arithmetic question under an assumption; a return-space band answers a data question by re-measuring at whatever frequency the chart supplies. With the window held fixed, weekly and monthly containment stayed within a point of the daily figure across the universe, historically. With the window shortened to match calendar time, the raw rate fell for a reason that is about the window, not the frequency. And when the daily estimate was scaled up by the rule instead of re-measured, containment fell well short. The range does not scale; it is measured again, and that is why the daily figures carry upward.

Frequently asked questions

Does the expected range repaint on a weekly chart while the week is still forming? The band drawn for the current bar is fixed from prior closes and does not move, and once a bar closes its range and markers are never revised, as the How It Works page states. What can change while a weekly bar is forming is the live abnormal-move marker, which updates as the current price moves and only settles at the weekly close. Nothing published scores a bar before it closes, so a mid-week reading is a work in progress rather than a result.

Do Bollinger Bands® work on weekly charts? Oisigma has not published a per-timeframe figure for Bollinger Bands®, so this site cannot verify a claim in either direction. The one published measurement is a daily-bar one: at the canonical (20, 2) settings, the band contained about 83% of next closes historically against the roughly 95% its ±2 standard deviations nominally imply, per the 97-year audit. Whether that gap widens, narrows, or holds on weekly bars is untested here.

Does a wider monthly range mean the market is more volatile? No. The monthly range is wider in price terms because monthly returns are larger in absolute size than daily returns, not because the market's condition has changed. A 5% move that would be unusual on a daily bar can be ordinary on a monthly bar, and the band on each chart is measuring size relative to that chart's own recent returns. Reading the monthly band as a statement about daily volatility, or the daily band as a statement about the month, is comparing quantities the construction never put on the same scale.

How many bars does the monthly figure rest on? Far fewer than the daily one. An instrument with 30 years of history contributes roughly 7,500 daily bars but only about 360 monthly bars, and an instrument without at least 61 monthly bars was excluded from the 60-bar monthly cell altogether. That is why the cross-asset spread of monthly containment rates was more than twice the daily spread, and why a single instrument's monthly figure should be read as an estimate with a wide interval around it, even though the universe average landed within a point of the daily figure, historically.

The quickest way to see re-measurement rather than scaling is to open the same symbol on a daily and a weekly chart with the band on both and watch each recalculate from its own returns. BTM draws a return-space expected range on any TradingView timeframe, with its calibration record and its limits published on the Proof page, and you can start a free 30-day trial to compare the two on the markets you already follow.