The annual Sharpe ratio hides an assumption about your returns

The standard way to annualise a monthly Sharpe ratio is to multiply it by the square root of twelve. Textbooks teach it, data providers use it, and our own calculator used it until this week.

That step is not a unit conversion. It falls out of a more general formula once you assume each month's return tells you nothing about the next, and for the assets where people care most about the Sharpe ratio, that assumption is the one least likely to hold.

What the convention says

Morningstar's methodology document states it without qualification:

"The annualized Sharpe Ratio is the product of the monthly Sharpe Ratio and the square root of twelve. This is equivalent to multiplying the numerator by 12 (to produce an arithmetic annualized excess return) and the denominator by the square root of 12."

That is exactly right as arithmetic. Annualising a mean return multiplies by 12; annualising a standard deviation multiplies by √12, if the monthly variances simply add. Twelve over the square root of twelve is the square root of twelve, and the ratio scales accordingly.

The conditional in that sentence is doing all the work, and it is not printed. We searched the whole document — 54,705 characters — for serial correlation, autocorrelation, independent, i.i.d. and smoothed. Each appears zero times.

The general case

Andrew Lo set out the correct scaling in 2002. Converting a periodic Sharpe ratio to an annual one over q periods is not multiplication by √q but by a factor that depends on how the returns are correlated with themselves:

η(q) = q ÷ √( q + 2 × Σ (q − k) ρk )

where ρ_k is the correlation between returns k periods apart. Set every ρ to zero — assume months are independent — and the sum vanishes:

η(12) = 12 ÷ √12 = 3.4641 = √12

⚠️ Read that identity the other way round. √12 is Lo's formula, evaluated at zero autocorrelation. The convention is therefore a special case doing duty as a general rule, which is a different complaint from being wrong.

How much it moves

Assume returns follow a simple first-order process, so ρ_k = ρ^k, and evaluate:

A fund with a monthly autocorrelation of 0.3, which is unremarkable for assets not marked to market daily, has its annualised Sharpe ratio overstated by roughly a third under the standard method.

The part most write-ups get wrong

The usual summary is that √12 flatters. That is true only for positively autocorrelated returns, and it is stated as though it were the whole story.

⚠️ Look at the bottom row. When returns are negatively autocorrelated — mean-reverting, which is a real pattern in some strategies — √12 understates the annualised Sharpe ratio, by 16.9% at ρ = −0.2. The convention has no fixed bias: the sign of its error is set by the data, and calling it "overstating" quietly assumes the case you are trying to detect.

Why it bites hardest where it matters most

Positive monthly autocorrelation is not exotic. It arises mechanically wherever holdings are valued infrequently or by appraisal rather than by trade: property funds, private credit, infrequently traded small caps, and anything whose monthly mark is partly a smoothed version of the previous one. Smoothing suppresses month-to-month variation, which lowers the measured standard deviation, which raises the Sharpe ratio — and then √12 raises it again.

Liquid, daily-marked equity portfolios tend to have autocorrelation close to zero, which is the case the convention was built for and where it is fine. The assets where a high Sharpe ratio is used as a selling point are disproportionately the assets where the assumption fails.

What this does not mean

It does not mean Morningstar is doing something improper. A methodology document is a statement of method, the convention is universal, and publishing a figure computed the same way as everyone else's is what makes it comparable. Applying Lo's correction unilaterally would produce a number that could not be compared with any other provider's.

It does not mean √12 should be abandoned. For liquid portfolios it is very close to right, and it has the considerable virtue of being computable from data everyone has.

What it means is that an annualised Sharpe ratio is a two-part claim, one part measured and one part assumed, and only the measured part is usually shown. If the autocorrelation of the underlying returns is not shown, you cannot tell which part is which.

What to do about it

If you are given an annualised Sharpe ratio, ask what period the underlying returns were measured over and whether the holdings are marked to market. Monthly returns on appraisal-valued assets is the combination to be careful with.

If you have the return series, compute the lag-1 autocorrelation before you annualise. That is one line in a spreadsheet, and it tells you immediately whether √12 is roughly right or off by a third.

And do not compare an annualised figure with a periodic one. Our Sharpe ratio calculator now asks how often your returns are measured, reports the periodic and annualised figures side by side, and prints the assumption in the working rather than hiding it.

What we changed on our own tool

The calculator did something worse than assume independence. It did not annualise at all. It computed a monthly Sharpe ratio from a monthly series and then graded it against thresholds that are annual conventions, including an explicit comparison with the long-run figure for a major equity index. Its own sample data scored 0.74, which the tool called acceptable. Annualised, the same series gives 2.57, which the tool's own scale calls excellent — a two-grade jump.

Its benchmark table had a second fault in the same family, subtracting a per-period risk-free rate from annual benchmark returns and inflating one index's quoted Sharpe ratio by 75%. Both are fixed, the arithmetic now lives in a tested module rather than in the page, and the annualisation step is labelled as an assumption on screen.

Where this comes from, and what will date it

The annualisation sentence is quoted from Morningstar's own published methodology paper on standard deviation and the Sharpe ratio, and the absence of any independence caveat is a count over that document's full extracted text rather than an impression. The scaling factor is from Andrew Lo's 2002 paper on the statistics of Sharpe ratios.

Every figure in the table above was computed here from that formula under a first-order autocorrelation assumption, not quoted from anywhere, and the zero-autocorrelation row reproducing √12 exactly is the check that the implementation is right.

⚠️ The table has two limits. The ρ^k decay is a modelling choice made for legibility; real return series do not follow it exactly, so treat the percentages as illustrating the shape of the error rather than as a lookup table. And we have quoted one provider's methodology because it publishes one — the convention is industry-wide, and singling out the document that is transparent enough to state its method plainly would be the wrong reading.

This will not date. The piece is an argument about what a formula assumes, made against a paper from 2002 and an identity that holds by construction. The situation would change with better disclosure. If providers published the autocorrelation alongside the ratio, the assumption would become visible, and this piece would be a description of how things used to be reported.