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Suppose someone is doing some daily trading and tells you their annualised sharpe ratios for the following years:

2004: 0.7
2005: 1.2
2006: 1.1
2007: -0.2

Is it possible to get the annualised Sharpe ratio for the period 2004-2007? And if so, how?

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    $\begingroup$ Consider the simplest case of 2 periods, $r=0$ and the returns over the two periods being independent. You can then write that the log-return over the full period $x$ is $x_1+x_2$. Then, you have that the overall sharpe ratio: $s = \frac{\mu_1+\mu_2}{\sqrt{\sigma_1^2+\sigma_2^2}} = \frac{\mu_1+\mu_2}{\sqrt{s_1^2 \mu_1^2 + s_2^2\mu_2^2}}$. Meaning even in that simple case you cannot compute $s$ from the knowledge of only $s_1$ and $s_2$. $\endgroup$ – Quantuple Aug 7 '17 at 9:37
  • $\begingroup$ @Quantuple This should really be an answer. $\endgroup$ – sashkello Aug 8 '17 at 2:02
  • $\begingroup$ @sashkello Duly noted. $\endgroup$ – Quantuple Aug 8 '17 at 7:26
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No, in general you can't combine Sharpe ratios in this way.

In the special case that the volatility is the same in each year, and the returns aren't too pathological, then the aggregate Sharpe ratio will be close to the average of the Sharpe ratios for each year, i.e. in your case it would be (0.7 + 1.2 + 1.1 - 0.2) / 4 = 0.7.

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Consider the simplest case possible: 2 periods with independent log-returns and zero risk-free rate.

You can then write that the log-return over the full period $x$ is the sum of the individual log-returns over each sub-period: $x = x_1+x_2$. Also, the overall sharpe ratio is by definition $$ s = \frac{\Bbb{E}[x]}{\sqrt{\text{Var}[x]}} $$ Under our assumptions it can further be expressed as $$s = \frac{\mu_1+\mu_2}{\sqrt{\sigma_1^2+\sigma_2^2}} = \frac{\mu_1+\mu_2}{\sqrt{s_1^2 \mu_1^2 + s_2^2\mu_2^2}}$$ showing that, even in the simplest case possible, you cannot compute $s$ from the sole knowledge of the individual Sharpe ratios $s_1$ and $s_2$.

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