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Sharpe Ratio and Risk-Adjusted Metrics

Sharpe ratio
Sharpe=E[R]−rfσ\text{Sharpe} = \frac{E[R] - r_f}{\sigma}

Excess return per unit of volatility, which is precisely why leverage cannot improve it.

The standard way to compare strategies with different risk levels, since raw return alone rewards leverage rather than skill.

Scaling with time

Sharpeannual=Sharpedaily×252\text{Sharpe}_{\text{annual}} = \text{Sharpe}_{\text{daily}} \times \sqrt{252}

because excess return scales with TT while volatility scales with T\sqrt{T}. This is the same square-root rule everywhere else, and it means a Sharpe ratio is meaningless without its period.

A useful equivalence: the t-statistic of a return series is approximately its Sharpe ratio times years\sqrt{\text{years}}. So a strategy with Sharpe 1 needs about four years of data to reach a t-statistic of 2. This single relationship explains why track records are long and why short backtests prove nothing.

Key takeaway

t-statistic ≈\approx Sharpe ×years\times \sqrt{\text{years}}. A Sharpe of 1 over one year is not statistically distinguishable from zero, however good it looks.

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