📅 Observed Annualization
Annualizing a risk figure requires knowing how many periods a year contains, and that number can be measured from the observed data instead of being assumed in advance. LibreFolio measures it: the factor used to scale a per-period figure to an annual one is counted from the series that was actually analysed, never taken from a market convention fixed beforehand.
🔢 Formula
Two quantities are read off the series before any figure is annualized — the calendar span it covers, and the number of returns observed inside that span:
where:
- \(N\) = number of period returns actually used
- \(d_{baseline}\) = date of the first valuation (it opens the series and therefore produces no return of its own)
- \(d_{last}\) = date of the last return
- \(D\) = calendar days spanned, weekends and closures included
The factor \(f\) is then applied wherever a per-period figure has to be expressed on an annual basis. For volatility:
It is a count, not a convention
\(f\) has a literal reading: observations per year, counted. A series with \(N\) returns spread over \(D\) calendar days was observed at a rate of \(N/D\) per day, hence \(365N/D\) per year. No assumption is made about the instrument, its market calendar or its asset class — the rate is whatever the data turns out to be. The one piece of calendar knowledge involved decides what counts as an observation, never how many there should be: a stored price that only repeats the previous close on a weekend or a market holiday is a carry, not a quote (see below).
If the span collapses — a single valuation date, so \(D \leq 0\) — no factor exists and no annualized figure is produced. The alternative would be to invent one.
💡 Interpretation
The table below works the formula out for a few shapes of series. It is arithmetic, not the output of a LibreFolio run:
| Series | Returns \(N\) | Calendar days \(D\) | \(f = 365N/D\) | \(\sqrt{f}\) |
|---|---|---|---|---|
| Instrument quoted on trading days only, full year | 252 | 365 | 252.0 | 15.87 |
| Instrument trading every day (crypto), full year | 365 | 365 | 365.0 | 19.10 |
| Weekly-priced fund, full year | 52 | 365 | 52.0 | 7.21 |
| Instrument quoted on trading days, started mid-period | 126 | 183 | 251.3 | 15.85 |
| Daily series with gaps | 180 | 365 | 180.0 | 13.42 |
Four readings follow from those rows.
📈 √252 Is Recovered, Not Imposed
An instrument quoted only while its market is open contributes roughly 252 returns across a full calendar year, so \(f = 252 \times 365 / 365 = 252\) and the familiar \(\sqrt{252}\) comes out of the measurement. The conventional number is a result here, not an input — which is precisely why nothing is lost by refusing to hardcode it.
It comes out even when the price source delivers a row for every calendar day. Some sources repeat the last close on the days the market is closed — justETF, for one, repeats Friday's close on Saturday and Sunday. Counted as quotes, those rows would add a zero return that no market produced for every closed day, and push \(f\) towards 365. They are not counted: a stored price dated on a Saturday, a Sunday or a weekday market holiday whose close repeats exactly the close of the row before it is a carry — the last quote held over, not a new one — and its date adds no observation to the series. A price that moved on a closed day stays a quote, as a crypto-asset's weekend prices do, and so does an unchanged price on an ordinary weekday, such as a flat day of a bond. The instrument keeps its trading days, and \(f\) stays near 252. The full rule, with the exchanges whose holidays count, is under Data Quality.
A portfolio series is read on its holdings' quote days
That row describes an instrument's own quoted prices; the return series of a portfolio reaches it by another route. The portfolio's time-weighted return is computed one calendar day at a time, weekends and holidays included, carrying the last known price forward where nothing was quoted. The risk analysis reads it only on its observation days — the days on which at least one of the positions held at the end of the analysed period had a quote of its own — and chain-links the return of every other day into the next observation day, so the cumulative return is exactly what it was: only the sampling changes. A portfolio of stocks and funds quoted on trading days therefore contributes roughly 252 returns a year, like the first row, although its history has a point for every calendar day; one that holds an instrument quoted every calendar day, such as a crypto-asset, is observed every day, like the second. The factor follows the series the metric is computed on, which is exactly why it is measured rather than assumed.
🪙 A 24/7 Instrument Gives ≈ √365
An instrument that trades every calendar day is observed 365 times a year, so \(f \approx 365\). A hardcoded \(\sqrt{252}\) applied to it would understate its annualized volatility by a factor of:
The error is not a rounding detail: it is systematic, and it always points in the reassuring direction.
📐 The Factor Measures a Rate, Not a Length
Row four is a daily-priced stock observed for roughly half a year: \(f\) still lands near 252, because numerator and denominator shrink together. A shorter window does not shrink the factor — it only makes it noisier (see the limitations below).
🕳️ Gaps Lower It, Honestly
Holidays, missing prices, an asset that started mid-period: whatever was observed is what gets counted. A series with gaps is annualized as the sparse series it is, rather than as the dense series it was assumed to be.
📏 Coverage
Alongside the factor, a second quantity is published with the result: of the days on which the series could have been observed, the share on which it was.
where \(Q\) is the number of those days. Coverage says how densely the series was sampled, as a fraction between 0 and 1. It is not a second annualization factor: \(f\) says at what rate the series was observed, \(c\) says how many of its possible observations it kept. Which days count in \(Q\) depends on how the series is built:
- An instrument series — and any series built from the prices of several assets — runs on a joint calendar. Every date in the requested window on which at least one of the assets has a quote of its own is a candidate, and a candidate becomes an observation only if every asset can be valued on it, a price carried forward from an earlier date included. \(Q\) counts the candidates, the baseline aside. For an instrument quoted only while its market is open, the candidates are the market's own calendar, so its closed days cost nothing: a carry is not a quote, and adds no candidate.
- A portfolio series is its time-weighted return read on its observation days (see above), with its first point kept as the baseline. \(Q\) counts the observation days after the baseline, up to the last return. The portfolio history has a point for every calendar day, so for a portfolio held without interruption every observation day carries one, and coverage sits at the top of its range by construction. Only if none of the holdings has a quote of its own in the window is every calendar day kept, and \(Q\) is then the span \(D\).
What coverage does not say is whether the prices behind those observations were real. On a joint calendar, a date on which only some of the assets were quoted is still an observation, the others entering with a price carried forward; on a portfolio's observation day, every holding not quoted that day enters at its last known value. So a high coverage is not a certificate and a low one is not a defect: both describe how many of the possible observations the series kept, not the quality of what is inside them. That second question — were these prices quoted, or carried forward? — is the job of Data Quality.
Both figures travel with the result
Every risk result carries the inputs of its own annualization in its metadata — the number of observations, the calendar days spanned, the factor and the coverage. A published annual figure can therefore be recomputed by hand from the same inputs that produced it.
⚠️ Limitations
A short span makes the factor unstable
The denominator is the observed calendar span. Over a few weeks \(D\) is small, so one missing or one extra observation moves \(f\) appreciably, and \(\sqrt{f}\) with it. The annualized figure inherits that instability: it is being extrapolated from a window far shorter than the year it claims to describe.
Measuring the factor does not repair the assumption
Scaling by \(\sqrt{f}\) follows from variance adding across independent periods (see Volatility). If returns are autocorrelated — trends, volatility clustering, mean reversion — the scaled figure is biased regardless of how accurately \(f\) was counted. Measuring the factor removes a wrong constant; it does not remove the independence hypothesis underneath it.
Two figures annualized with different factors are also only comparable if the sampling behind them is comparable. A weekly-priced fund and a 24/7 instrument are both annualized correctly, and are still being observed in very different ways — which is what the observation count and the coverage are there to make visible.
🔗 Related
- 📊 Volatility — the figure most often expressed on an annual basis
- 🧪 Data Quality — what a low coverage means for the trustworthiness of a result
- 📐 Sharpe Ratio — a risk-adjusted ratio that also carries an annual scale
- 📊 Sortino Ratio — the downside-only variant, same scaling question