Annualisation Conventions and Their Pitfalls

Two funds pitch you on the same day. Both trade market-neutral, both report a Sharpe ratio computed from daily returns of 0.10. Fund A's tear sheet says its annualised Sharpe is 1.91. Fund B's says 1.59. On paper Fund A looks 20% better. It isn't — Fund A trades crypto and multiplied by the square root of 365, because crypto markets never close. Fund B trades equities and multiplied by the square root of 252, the count of US trading days. Identical skill, identical daily number, a 20% gap manufactured entirely by which calendar you picked.

That is the easy mistake, the one you can catch by asking "252 or 365?" The expensive mistake is quieter: annualising as if daily returns were independent, when they usually aren't.

Where the formula breaks

The standard move scales a daily Sharpe ratio by $\sqrt{N}$ to get an annual one, where $N$ is the number of trading periods in a year. That scaling is only correct if returns from one day to the next are uncorrelated — each day is a fresh coin flip. A strategy that holds positions for several days, marks them with any lag, or trades on a signal that itself decays slowly will show positive autocorrelation in its daily P&L. Positive autocorrelation means the "252 independent days" you're assuming are really something closer to 200 independent draws wearing daily clothes, and the naive $\sqrt{N}$ rule overstates how much the Sharpe ratio improves by averaging.

Lo's correction replaces $\sqrt{N}$ with a smaller multiplier once lag-1 autocorrelation $\rho$ is accounted for:

$$\text{multiplier} \;\approx\; \sqrt{N}\,\sqrt{\frac{1-\rho}{1+\rho}}.$$

In words: still scale by the square root of the number of periods, then shrink that by a factor driven by how strongly today's return predicts tomorrow's.

Worked example

A trend strategy rebalances weekly but marks daily, so consecutive daily returns carry $\rho = 0.15$ — a realistic figure for a slower, higher-turnover-lag book. Daily Sharpe is 0.10, $N = 252$.

Naive annualisation: $0.10 \times \sqrt{252} = 1.59$.

Corrected: multiplier is $\sqrt{252} \times \sqrt{0.85/1.15} = 15.87 \times 0.860 = 13.65$, giving an annual Sharpe of $0.10 \times 13.65 = 1.37$.

Convention	Annual Sharpe	Gap vs. correct
Naive, 252 days, no autocorrelation adjustment	1.59	+16%
Naive, 365 days (wrong calendar for this desk)	1.91	+39%
Lo-corrected, $\rho = 0.15$	1.37	—

Two convention choices — day count and independence — stack into a 39% overstatement on a strategy that, corrected, is a perfectly respectable but unremarkable 1.37.

A second, unrelated trap sits in total-return annualisation rather than Sharpe: compounding. A strategy averaging 2% per month arithmetically is not "24% a year." Compounded, $(1.02)^{12} - 1 = 26.8\%$. At low volatility the gap is cosmetic; at high volatility, where monthly returns swing between +8% and −6%, arithmetic averaging and geometric compounding can disagree by several percentage points, and only the compounded figure is what actually landed in the account.

The fix is procedural, not statistical: state the day count and the autocorrelation-adjusted multiplier next to every annualised number, and refuse to rank two strategies on Sharpe ratios computed under different conventions. See The Standard Error of the Sharpe Ratio for the companion problem of how uncertain that Sharpe estimate is in the first place.