Annualising the Sharpe Ratio

A backtest reports a daily Sharpe ratio of 0.05. That number alone means almost nothing to a portfolio manager, who thinks in years, not days. Converting it isn't as simple as scaling up a return figure, because a Sharpe ratio is a ratio of two things that scale differently over time — and mixing that up is one of the most common ways a strategy's real edge gets misquoted.

Why the square root, not the number itself

$$\text{SR}_{\text{annual}} = \text{SR}_{\text{period}} \times \sqrt{N} .$$

In words: $N$ is the number of periods in a year — 252 for daily returns, 12 for monthly. A return's mean scales linearly with time (twelve months of return simply add up), but its standard deviation scales with the square root of time, because independent risks partially cancel rather than stack directly. Since the Sharpe ratio is mean divided by standard deviation, the linear numerator and the square-root denominator combine so the whole ratio scales by $\sqrt{N}$ — not $N$.

Set the exponent to $0.5$ and this is exactly the curve annualization follows: Sharpe grows with the square root of the holding period, flattening out as $N$ gets large, never in a straight line.

Worked example 1 — daily to annual

A strategy's daily Sharpe ratio is 0.05. With 252 trading days in a year, $N = 252$ and $\sqrt{252} \approx 15.87$. The annualized Sharpe is $0.05 \times 15.87 \approx 0.79$ — a solid, believable figure. Multiplying by 252 directly instead of $\sqrt{252}$ would have produced a Sharpe of 12.6, an obviously absurd number that no real strategy sustains, and a dead giveaway that whoever produced it annualized the return correctly but forgot volatility scales differently.

Worked example 2 — monthly to annual, and the naive mistake

A monthly-rebalanced fund reports a monthly Sharpe of 0.15. With $N = 12$, $\sqrt{12} \approx 3.46$, giving an annual Sharpe of $0.15 \times 3.46 \approx 0.52$. A marketing deck that instead multiplies by 12 outright reports a Sharpe of 1.80 — more than three times too high, and exactly the kind of error that makes an unremarkable strategy look like a rare find.

What this means in practice

The $\sqrt{N}$ rule assumes returns are independent and identically distributed period to period — no autocorrelation. Strategies that trade illiquid assets, or that smooth their own reported returns (common in some hedge fund reporting), tend to show positive serial correlation, which inflates the naive $\sqrt{N}$-annualized Sharpe above its true value; Lo's adjustment corrects for this by discounting the scaling factor based on how correlated consecutive returns actually are.