Tools · Return frequency
Returns compound. Volatility takes the square root.
Converting a figure between daily, monthly and annual is two different operations depending on whether it is a return or a volatility, and each carries an assumption. This does both and states what it assumed.
Converter
This page is showing a worked example. The table below converts a 1.0% monthly return and 4.0% monthly volatility to every other frequency. Enable JavaScript to enter your own; the table and method on this page do not need it.
- Annualised return
- 12.68%
- compounded
- Annualised volatility
- 13.86%
- scaled by √periods
- Periods a year
- 12
- as selected
| Frequency | Periods a year | Return | Volatility |
|---|---|---|---|
| Daily | 252 | 0.047% | 0.87% |
| Weekly | 52 | 0.230% | 1.92% |
| Monthly | 12 | 1.000% | 4.00% |
| Quarterly | 4 | 3.030% | 6.93% |
| Annual | 1 | 12.683% | 13.86% |
Method
Two different operations.
Applying the wrong one is among the most common errors in a performance table, and it is not a small error: a 1.0% monthly return is 12.68% a year compounded, not 12 times 1.0%.
Returns
annual = (1 + r)n − 1
Each period's gain earns the next period's gain, so returns compound rather than add. Going the other way, the per-period rate is the n-th root: (1 + annual)1/n − 1. Multiplying a monthly return by 12 always overstates the year.
Volatility
annual σ = σ × √n
Variance adds across independent periods, and volatility is the square root of variance, so volatility scales by the square root of the number of periods. Monthly volatility becomes annual by multiplying by √12, or about 3.46.
Limits
The assumption inside the square root.
- Independence. √n scaling holds when periods are independent. Where returns are autocorrelated — trending or mean-reverting — it is wrong in a predictable direction: positive autocorrelation makes the true annual volatility higher than the scaled figure, negative autocorrelation lower.
- The trading-day count. 252 is a convention for daily data, not a fact. Different markets and different providers use different counts, and a figure annualised with one and compared against a figure annualised with another is not a like-for-like comparison.
- Constant volatility. The conversion assumes the level of volatility does not change through the period being scaled.
- Annualising short records. Converting a three-month return to an annual rate produces a number, not an expectation. This website does not publish an annualised return until a record reaches twelve months, for that reason.
This converter is provided for general information only and is directed to wholesale and professional investors. It is not personal advice, and it is not an offer, invitation or recommendation to acquire any financial product. Investing involves risk, including the possible loss of capital. Past performance is not a reliable indicator of future performance.
Definitions
Terms on this page.
Each links to the glossary entry, which states the convention the term assumes as well as what it means.
- Annualised return — compounding, not multiplying.
- Volatility — scaled by the square root of time.
- Autocorrelation — what breaks the square-root rule.
- Cumulative return — the total the conversions are consistent with.
Questions
Common questions.
How do I annualise a monthly return?
Compound it: (1 + monthly)12 − 1. A 1.0% monthly return annualises to 12.68%, not to 12%. Multiplying by 12 ignores the compounding and understates the figure.
Why is volatility multiplied by the square root of 12?
Because variance, not volatility, adds across independent periods. Twelve months of variance is 12σ², and the volatility of that is σ√12. It is the same reason a longer measurement window does not increase uncertainty in proportion to its length.
Should I use 252 or 365 days?
252 for returns measured on trading days, which is what a price series gives you. 365 applies to figures that accrue on calendar days, such as interest. Mixing the two is a common source of small, persistent errors.
Can I annualise a return from three months of data?
Arithmetically yes, meaningfully no. The conversion assumes the rate continues, which three months of data gives no basis for. Short-period figures are better quoted as what they are.