The Sharpe Ratio

Sharpe ratio measures risk-adjusted return; annualizing it by $\sqrt{\text{periods}}$ relies on an i.i.d. returns assumption.

1. The idea

The Sharpe ratio measures excess return per unit of volatility, letting you compare strategies with different risk levels on equal footing. A higher Sharpe indicates more consistent risk-adjusted performance, not necessarily higher raw returns.

Because returns are usually measured per period (daily, monthly), Sharpe is often annualized by multiplying by $N$ where $N$ is the number of periods per year. This scaling comes from the fact that under i.i.d. returns, variance scales linearly with time while mean scales linearly too, so the ratio (mean/std) scales with $N$ .

This $N$ scaling breaks down if returns are autocorrelated (e.g., momentum or mean-reversion), since then variance doesn't simply add across periods — a key caveat interviewers like to probe.

2. Why it matters

Sharpe ratio appears constantly in trading interviews as a risk-adjusted performance metric, and the annualization formula (plus its i.i.d. assumption) is a frequent follow-up question.

3. Core mathematics

Sharpe ratio (per period)

Sharpe = \frac{E [ R ] - R _{f}}{σ ( R )}

$R$ = period return, $R_{f}$ = risk-free rate

Annualization

Sharpe_{annual} = Sharpe_{period} \times N

$N$ periods per year, e.g. 252 for trading days

4. How to recognize it

Think of this when…

Asks to compare two strategies with different volatilities
Mentions annualizing a daily or monthly Sharpe ratio
Tests whether returns need to be independent for the $N$ rule

5. Worked example

Worked example

A strategy has a daily mean return of 0.05% and daily standard deviation of 1%, with negligible risk-free rate. Assuming i.i.d. daily returns and 252 trading days per year, find the annualized Sharpe ratio.

1
Compute daily Sharpe ratio.
$Sharpe_{daily} = \frac{0.0005}{0.01} = 0.05$
2
Annualize by multiplying by $252$ .
$Sharpe_{annual} = 0.05 \times 252 \approx 0.05 \times 15.87$
Answer: Annualized Sharpe ≈ 0.79.

6. Common mistakes

Multiplying the Sharpe ratio by $N$ instead of $N$
Annualizing with $N$ when returns are known to be autocorrelated, overstating or understating true risk-adjusted performance
Ignoring the risk-free rate subtraction when it's material

7. Quick check

Annualizing a monthly Sharpe ratio uses a multiplier of:

The $N$ annualization rule assumes:

8. Practice

3 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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