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Annualizing the Sharpe Ratio and Its Sampling Distribution

Article Quant Q&A · Author: Carl
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Summary

The document poses a statistical question about annualized Sharpe ratios. It starts from the convention of multiplying a daily Sharpe ratio by the square root of the number of trading days in a year, under certain assumptions, and asks whether that scaling carries the daily ratio’s Student’s t distribution into the annualized figure. The author proposes a degrees-of-freedom adjustment based on the scaling factor and asks whether it is valid.

No answer, derivation, assumptions, or empirical evidence is included, so the suggested adjustment should be treated as a question rather than a result. The note highlights that annualization of a point estimate and the sampling distribution of that estimate are distinct issues. It does not discuss dependence between returns, estimation choices, or the conditions under which a t-based inference for Sharpe ratios applies.

Key ideas

  • The document asks how square-root annualization affects the sampling distribution of a Sharpe ratio.
  • It proposes a degrees-of-freedom adjustment but supplies no justification or answer.
  • Annualizing a Sharpe estimate does not by itself establish the distribution of that estimate.
  • Return dependence and other assumptions are not addressed.

Tags

StatisticsRisk managementBacktesting
Full text
# Did annual Sharpe ratio follows T distribution?


# Did annual Sharpe ratio follows T distribution?












Under some special condition, Sharpe ratio can be annualized by multiply $\sqrt{252}$, Since daily Sharpe ratio ($\frac{mean(r)}{std(r)}*\sqrt{T}$) follows student T distribution with degree of freedom T-1, then what about annual Sharpe ratio? I thought it should follows student T as well with degree of freedom $\frac{T}{\sqrt{252}} -1$ since: $$\text{annual SR}=\frac{mean(r)}{std(r)}*\sqrt{252}$$ so:$$\frac{mean(r)}{std(r)}*\sqrt{T} = \text{annual SR}*\frac{\sqrt{T}}{\sqrt{252}}$$ Is this correct?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.