Use Arithmetic Returns for Multi-Period Sharpe Ratios

Summary

The document asks how to calculate an ex-ante Sharpe ratio from simulated return paths: from compounded total returns across paths, or by calculating a Sharpe ratio along each path and averaging those ratios. The response emphasizes that the conventional Sharpe ratio uses arithmetic returns rather than compounded geometric returns.

It describes a common practice of calculating the average periodic return above the risk-free rate and dividing by the standard deviation of periodic returns, often using monthly observations and annualizing the result. Some comparisons omit the risk-free adjustment and report a risk-return statistic instead. The answer is brief and does not specify a complete estimator for simulated paths, explain how path dependence should be handled, or discuss uncertainty in the estimates. Its guidance therefore clarifies the return convention but does not fully resolve every choice in the question’s multi-period simulation setup.

Key ideas

• The conventional Sharpe ratio is based on arithmetic returns rather than compounded geometric returns.
• A common calculation compares average periodic excess return with the standard deviation of periodic returns.
• Monthly returns are often used, with the ratio then annualized.
• Some strategy comparisons omit the risk-free rate and report a related risk-return statistic.
• The response does not fully prescribe how to aggregate simulated path-dependent Sharpe ratios.

Tags

# Calculating Ex-ante Sharpe Ratio in multi-period setting


# 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?

## Answer by Chris (score 2, accepted)

https://quant.stackexchange.com/a/45128

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.