I'm having a trouble calculating (annualized from daily performance) Sharpe ratio, even though I've read some related posts here.
Say I have a daily performance, for example: $$[1.15, 1.2, 0.7]$$ means that after 3 trading days my cumulative (for, say 1 dollar investment) wealth is: cum_wealth=
$1*1.15*1.2*0.7=0.966$ which states that I actually lost money. When I calculate the Sharpe ratio I follow these steps:
- Subtract $1.0$ to get percentage: $X=[0.15, 0.2, -0.3]$
- Calculate sample expectance: $\hat{E}=(0.15+0.2-0.3)/3=0.0166$
- Calculate sample standard deviation: $\hat{\sigma}=0.224$
- Assuming the risk-free rate is 0%. Divide and annualize:
Sharpe
=$E/\sigma * \sqrt{252}=1.176$
I've constructed the example above specifically to show that I get positive Sharpe ratio when actually the portfolio loses money, which is counter-logic (if the math is correct)... What am I missing? Regards