Sharpe Ratios With Autocorrelated Returns

You compute a daily Sharpe ratio for a strategy and multiply it by $\sqrt{252}$ to get an annual number, because that's the rule everyone learns. That rule assumes each day's return is statistically independent of the one before it — an assumption that's roughly fine for a liquid large-cap equity strategy and badly wrong for a slow, illiquid, or trend-following one, where today's return is correlated with yesterday's. Get the assumption wrong and the "annualized Sharpe" you report can be overstated or understated by a large margin, purely from a scaling error that has nothing to do with the strategy's actual quality.

Where the $\sqrt{n}$ rule comes from — and where it breaks

The standard annualization scales the mean return by $n$ (periods per year) and the standard deviation by $\sqrt{n}$, so Sharpe scales by $\sqrt{n}$. That $\sqrt{n}$ comes from how variance adds up: if daily returns are independent, the variance of an $n$-day sum is just $n$ times the daily variance. If returns are positively autocorrelated — a good day tends to follow another good day, as in a slow-moving trend strategy — the multi-day variance grows faster than $n$ times daily variance, because the days echo each other instead of adding independent information. The plain $\sqrt{n}$ rule then understates true multi-day variance, overstating the annualized Sharpe. Negatively autocorrelated returns (mean-reverting, as in some market-making strategies) work the other way: variance grows slower, and the naive rule understates the true annualized Sharpe.

A cleaner way to say it: with autocorrelation $\rho_1, \rho_2, \ldots$ at each lag, the correct scaling factor replacing plain $\sqrt{n}$ is

$$\sqrt{n + 2\sum_{k=1}^{n-1}(n-k)\rho_k} .$$

In plain English, that's the ordinary $n$ term plus a correction that adds up every pairwise lagged correlation, weighted by how many day-pairs sit that far apart. When all the $\rho_k$ are zero this collapses back to the familiar $\sqrt{n}$; when they're positive it grows faster and shrinks the properly-annualized Sharpe relative to naive scaling; when negative, it shrinks and the properly-annualized Sharpe grows relative to naive scaling.

Worked example. A trend-following strategy has a daily Sharpe ratio of 0.10 and a first-lag daily autocorrelation of $\rho_1 = 0.15$ (higher lags roughly zero). Naive annualization gives $0.10 \times \sqrt{252} \approx 1.59$. Using just the first-lag correction term, the adjusted scaling factor is approximately $\sqrt{252 + 2 \times 251 \times 0.15} \approx \sqrt{327} \approx 18.1$ versus the naive $\sqrt{252} \approx 15.9$ — about 14% larger. But that larger scaling factor divides into the corrected Sharpe calculation such that the properly annualized Sharpe comes out lower, around $1.39$ rather than $1.59$, since the true multi-day variance was bigger than the naive rule assumed. Reporting 1.59 instead of 1.39 overstates the strategy's risk-adjusted return by roughly 15%, purely from ignoring a modest amount of autocorrelation.

What this means in practice

The practical takeaway isn't to memorize the correction formula — it's to check whether returns show meaningful autocorrelation before trusting a naive annualized Sharpe, especially for strategies that hold positions for multiple days, use smoothed or stale prices, or trade illiquid instruments where price updates lag. A quick first-lag autocorrelation check is a cheap diagnostic: seeing $\rho_1$ noticeably away from zero signals that reported Sharpe ratios — yours or a manager's — need adjustment before being compared to a strategy with genuinely independent returns.