Annualisation in investing: how monthly and daily figures are scaled to annual rates

Investment performance is reported on an annual basis by convention, but the underlying data is rarely annual. Monthly net asset values, daily price series, and even higher-frequency observations are the actual inputs—and converting them to annual figures requires specific conventions. The annualisation step is methodological glue that older articles assume and never explain, and it is the source of much confusion when figures from different sources do not match.

What annualisation is

Annualisation is the methodology of converting performance statistics computed at one frequency (monthly, weekly, daily) into the equivalent annual figure. The conversion differs by statistic: returns scale geometrically; volatilities scale by the square root of time; ratios that combine the two scale according to whichever convention is applied to the inputs.

The conventions are simple but not always intuitive, and disagreement about the right convention is the most common source of mismatched performance figures across different reports of the same underlying strategy.

How each statistic is annualised

Returns. Returns annualise geometrically. A monthly return of 1% does not annualise to 12% (the arithmetic sum) but to (1 + 0.01)¹² − 1 = 12.68% (the geometric compound). For a series of monthly returns, the annualised figure is the geometric mean: ((Π(1 + rᵢ))^(12/n)) − 1, where n is the number of months and the product is over the observed monthly returns. This is the same calculation as CAGR, expressed in annualised form.

Volatility. Volatility annualises by the square root of the number of periods per year, on the assumption that returns are independently distributed. Monthly volatility annualises by √12; daily volatility by √252 (using the trading-day convention, not calendar days). The square-root-of-time rule follows from the variance of a sum of independent random variables being the sum of the variances; standard deviation, the square root of variance, therefore scales by the square root of the number of periods.

The independence assumption is empirically imperfect. Daily equity returns show small but real autocorrelation that violates the assumption; the annualised volatility computed by √252 scaling is therefore typically a slight underestimate of the true annual volatility. For most practical purposes, the bias is small enough that the convention is preserved.

Sharpe ratio. The Sharpe ratio is excess return divided by volatility. The numerator scales with the period (monthly excess return × 12 is approximately monthly Sharpe × 12); the denominator scales with √period. The ratio therefore scales by √period: monthly Sharpe × √12 ≈ annualised Sharpe. This is the standard convention, and it explains why annualised Sharpe ratios for the same underlying strategy can look different at different reporting frequencies.

Drawdowns and maximum drawdown. Drawdowns are not annualised. The maximum drawdown is whatever the deepest peak-to-trough decline in the chosen sample is, regardless of period. Reporting the same maximum drawdown number annualised would lose its meaning—drawdown depth is the relevant statistic, not its frequency-equivalent.

What the evidence shows

The biggest practical consequence of annualisation conventions is for short-horizon backtests of strategies with non-normal return distributions. A six-month backtest of a strategy with daily returns can show an annualised Sharpe of 3 simply because the very small sample size and the square-root-of-time scaling of volatility produce an unrealistic ratio. A one-year backtest of the same strategy might show an annualised Sharpe of 1, and a multi-year backtest a Sharpe of 0.7. The strategy's true Sharpe is whatever the longest sample suggests; the short-sample annualised figure is an artefact.

The reverse pattern affects long-horizon volatility comparisons. A strategy with daily volatility of 0.6% annualises to 9.5% if returns are independent (0.6 × √252) but can be substantially higher if returns exhibit positive autocorrelation, as is common in less-liquid asset classes. The reported annualised volatility from monthly data is typically more accurate than from daily data for these strategies, simply because monthly returns are less affected by the autocorrelation that biases the daily-to-annual conversion.

The annualisation of performance ratios across different reporting frequencies is a recurrent source of confusion in the academic literature. Studies that report monthly Sharpe ratios cannot be directly compared with studies that report annualised Sharpe ratios without applying the √12 conversion, and the conversion is sometimes applied inconsistently in the literature itself.

Limitations and trade-offs

Annualisation assumes a stable underlying process. The square-root-of-time rule for volatility, the geometric chaining of returns, and the corresponding ratio conversions all assume that the statistical properties of the return-generating process are constant over the annualisation horizon. Regime shifts, structural breaks, and time-varying volatility all break the assumption to varying degrees.

For shorter-horizon performance evaluation, the annualisation step inflates the natural noise of the underlying estimates by the square-root-of-time multiplier. A monthly Sharpe estimate from 24 observations has approximately √24 = 4.9 standard errors of dispersion in its annualised form; the same Sharpe estimate from 240 observations has about 0.55. The annualised number looks the same in both cases, but its statistical reliability is very different.

The practical recommendation is to annualise consistently, report the underlying frequency, and treat all annualised figures from short samples with the discount the sample size implies. Probabilistic Sharpe ratios and confidence intervals make this discount explicit.

Annualisation in pfolio

pfolio reports annualised metrics—CAGR, annualised volatility, annualised Sharpe—as standard, computed via standard time-period scaling: geometric chaining for returns, square-root-of-time for volatility. The underlying period frequency depends on the asset's price data.

Related articles

• CAGR explained: what compound annual growth rate means for your portfolio
• Volatility in investing: how to measure and manage portfolio risk
• Geometric vs arithmetic mean return: when each measure is the right one to use
• Probabilistic Sharpe ratio: testing whether a Sharpe is statistically distinguishable from a target
• Sharpe ratio confidence intervals: how to quantify the noise around a Sharpe estimate
• Log returns vs simple returns: when each convention is appropriate