Calculating Ex-ante Sharpe Ratio in multi-period setting

Question

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?

Answer 1

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.