Lo's Autocorrelation-Adjusted Sharpe

The familiar rule for annualizing a Sharpe ratio — multiply the monthly Sharpe by $\sqrt{12}$ — is derived from an assumption that's easy to forget is even there: that returns in one period are statistically independent of returns in the next. Many strategies violate this. Illiquid assets that are priced with stale marks, strategies that hold positions across multiple measurement periods, or strategies whose returns are smoothed by infrequent valuation all produce positively autocorrelated returns — this month's return is correlated with last month's. Andrew Lo showed that when that's true, the standard $\sqrt{12}$ annualization is wrong, and typically overstates the true annualized Sharpe ratio.

Why autocorrelation inflates the naive Sharpe

Positive autocorrelation means a strategy's monthly returns look smoother than its true underlying volatility, because each observed return is partly explained by the previous one rather than being a fresh, independent draw. The standard deviation computed from those smoothed monthly numbers understates the strategy's real variability, and understating volatility in the Sharpe ratio's denominator directly overstates the resulting Sharpe. This is a well-known issue with strategies holding illiquid or infrequently-marked assets — private credit, some real estate strategies, or thinly-traded instruments — where smooth-looking return series can be an artifact of stale pricing rather than genuinely low risk.

Lo's correction

Lo derived an adjustment factor that replaces the naive $\sqrt{q}$ scaling (for $q$ periods per year) with a factor that accounts for the return series' autocorrelation structure. For a series with first-order autocorrelation $\rho$, the annualization factor becomes approximately:

$$\eta(q) = \frac{q}{\sqrt{q + 2\sum_{k=1}^{q-1}(q-k)\rho_k}} ,$$

replacing the naive $\sqrt{q}$. In plain English: instead of just scaling by the square root of the number of periods, the correction shrinks that scaling factor based on how much each period's return echoes the previous ones — the more positive the autocorrelation, the more the naive annualization overstates the true annualized Sharpe, and the correction pulls the multiplier down to compensate.

Worked example: a smoothed monthly series

A strategy shows a monthly Sharpe ratio of 0.30 with first-order monthly autocorrelation $\rho_1 = 0.4$ (ignoring higher-order terms for simplicity). The naive annualization: $0.30 \times \sqrt{12} \approx 1.04$. Using Lo's framework, positive autocorrelation of this size typically shrinks the effective annualization multiplier meaningfully below $\sqrt{12}$ — in illustrative examples from Lo's original paper, a monthly autocorrelation around 0.3–0.4 can cut the annualized Sharpe estimate by roughly 20–30% relative to the naive calculation, bringing the 1.04 estimate down toward the 0.75–0.85 range. The exact number depends on the full autocorrelation structure, but the direction is consistent: positive autocorrelation always pulls the corrected annualized Sharpe below the naive one.

What this means in practice

Whenever a strategy's return series shows meaningful serial correlation — common for illiquid or infrequently-marked positions, and also common in monthly hedge fund reporting more broadly — the naive $\sqrt{12}$ annualization should be treated with suspicion, and Lo's adjustment (or an equivalent correction) should be applied before comparing the annualized Sharpe against strategies with genuinely independent returns. Skipping this check is a well-documented way that illiquid strategies end up looking artificially attractive next to liquid ones on a Sharpe-ratio basis.