Calculating Ex-ante Sharpe Ratio in multi-period setting

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?

1 Answer

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.