# Sharpe ratio 1 and probability to lose money

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 ?

• 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. Commented Mar 14 at 14:51

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, $$P\left(\sum_{i = 1}^4 R_i \leq 0\right) = P\left(Z \leq -2\right) \approx 2.28\%.$$