# Compare ex-post Sharpe ratio of multiple portfolios?

Say I want to compare the Sharpe ratio of two portfolios, is it necessary to look at the excess return? Or can you just compare their average return divided by their standard deviation?

So basically, since in both cases you subtract the same proxy for risk-free rate (of course under the assumption that both were held over the exact same time horizon), can you just leave that proxy out?

Naturally, the value of the Sharpe ratio is different from the value obtained by this shortcut, but I am only interested in the relative position of the sharp ratios between each other.

I assume that if with this shortcut portfolio A will do better than portfolio B, then the Sharpe ratio of portfolio A is also always higher than portfolio B. Since if that would not be the case then theoretically you could pick a proxy for the risk-free rate that gets the desired result?

• $\frac{r_1}{\sigma_1} > \frac{r_2}{\sigma_2}$ does not imply that $\frac{r_1-r_f}{\sigma_1} > \frac{r_2-r_f}{\sigma_2}$, so $r_f$ does matter. – Alex C Apr 5 at 23:27

## 2 Answers

You can compare the ratio ex-risk-free rate, but it's no longer technically a Sharpe ratio.

As others have noted, removing rf from the equation can give differing ranks (this is particularly true where r is close to rf), but for all intents (and simplicity's sake), making the comparisons excluding rf is a valid assessment of risk-adjusted return, often referred to simply as risk/return ratio or something similar.

Aside from simplicity, there are also theoretical arguments supporting doing it this way (eg, you're likely receiving close to the rf on margin and/or cash in your account, hence nullifying its subtraction; as an HF, we target absolute returns, making the rf reference not meaningful).

The return in Sharpe ratio has to be risk-adjusted return. If you use the absolute return, you ignore the return you could possibly get without baring any risk. Sharpe ratio is used to analyze how much greater a return the person is obtaining in relation to the level of additional risk taken to generate that return.

Return of asset minus Risk-free rate is the return you reward for taking the risk. The Standard Deviation you take is the risk you take.