I have built a return process $\{x_t, t = 1,\dots,T\}$ for an asset. Suppose I have generated $K$ sample paths $\{x_t^j, t=1,\dots,T\}, j=1,\dots,K$. I think of two ways to compute the Sharpe ratio.

The first is based on the total return over the whole time period, $\frac{\frac{\sum_{j=1}^K\prod_{t=1}^T (1+x_t^j)}{K}-\prod_{t=1}^T (1+r_{ft})}{\sigma(\prod_{t=1}^T (1+x_t^j))}$. ($r_{ft}$ is the risk free process).

The second is path-dependent. For each individual sample path $j$, I can compute a path-dependent Sharpe Ratio $s_j$. Then I take an average of $s_j$. I can even compute the standard deviation.

Which one is correct?


Sharpe ratio is calculated using arithmetic returns, not geometric return.

It's most often calculated using monthly returns, taking an average less the risk-free rate and SD, and then usually annualizing using the square of 12. It's also not uncommon to omit the rf portion, and calculate a risk/return type stat and compare strategies that way.


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