I came across the following interview problem and I am looking for a possible solution. We have a strategy with risk free return 0 and sharpe ratio 1. What is the probability to lose money over four years ?
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$\begingroup$ I tried the following : if $r_i$ is the daily return, I am looking for $S = P(\sum_{~1000}r_i < 0)$ approximating the four years to 1.000 days. Sharpe ratio of 1 gives me that $E(r) = \sigma$. I am trying to use the central limit theorem yielding $\frac{S-nE(r)}{\sigma \sqrt{n}}$ is approximately $N(0, 1)$. I am stuck on how to use this. $\endgroup$– marc33Commented Mar 14 at 14:51
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1 Answer
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I would assume annual returns, because that's usually how the Sharpe ratio is stated. Let's assume returns are normally distributed and independent with mean $\mu > 0$. Then $R_i \sim{\mathcal{N}(\mu, \mu^2)}$ for $i = 1, 2, 3, 4$. Because of my independence assumption, $\sum_{i = 1}^4 R_i \sim{\mathcal{N}(4\mu, 4\mu^2)}$. The $Z$-score corresponding to 0 is $Z = \frac{0 - 4\mu}{2\mu} = -2$. Hence, $$ P\left(\sum_{i = 1}^4 R_i \leq 0\right) = P\left(Z \leq -2\right) \approx 2.28\%. $$
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$\begingroup$ Doesn't this give you a sharpe of 2, not 1? $\endgroup$ Commented Apr 4 at 23:43
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$\begingroup$ Over four years, but over one year the Sharpe ratio is 1. $\endgroup$ Commented Apr 8 at 13:55
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$\begingroup$ um but isnt the sharpe supposed to be 1? $\endgroup$ Commented Apr 9 at 19:20
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1$\begingroup$ Over one year it is 1. Sharpe ratios are typically annualized, so I'm assuming that when you say the Sharpe ratio is 1 you mean the Sharpe ratio over one year is 1. Within this construction, the Sharpe ratio scales up by sqrt(T), where T is the number of years the strategy has been run. In particular, note that over four years the Sharpe ratio is sqrt(4) = 2. In general, it's impossible to have a Sharpe ratio of 1 over every time period. $\endgroup$ Commented Apr 10 at 16:53