when comparing the Sharpe Ratio (SR) of two different funds, does it make a difference, whether I use excess returs (returns - risk free rate) or returns (without dedcuting the risk free rate, assuming the risk free rate is always 0%) in the numerator? Since I am subtracting the same risk free rate from the returns of the two funds, the result (eg. Fund A has a higher SR than Fund B) should be the same, irrespective of calculating with excess returns or returns?!

many thanks in advance!


No, this is not the same. For example, consider the scenario $$ \begin{align*} r_A &= 10\% \quad\quad \sigma_A = 10\% \\ r_B &= 1.5\% \quad\quad \sigma_B = 1\% \\ \end{align*} $$ If $r_f=1\%$, $$ \text{SR}_A=0.90 \quad\quad \text{SR}_B=0.50 $$ then $A$ has the higher sharpe.

Now if $r_f=0\%$, $$ \text{SR}_A=1.00 \quad\quad \text{SR}_B=1.50 $$ then $B$ has the higher Sharpe.


A motivating idea for the Sharpe Ratio is that the measure is invariant to leverage. Let's say we lever up $\alpha$ on the excess return $r^A - r^f$ to have the excess return $r^x = \alpha \left(r^A - r^f \right) $

Trivially, the Sharpe Ratio is unchanged:

\begin{align*} \mathit{SR} &= \frac{\operatorname{E}[\alpha \left( r^A - r^f \right) ]}{\operatorname{Stdev}\left( \alpha \left( r^A - r^f \right) \right) } = \frac{\operatorname{E}[ r^A - r^f ]}{\operatorname{Stdev}\left( r^A - r^f \right) } \end{align*}

On the other hand if you you have returns rather than excess returns, this doesn't work. Let return $r = (1 + \alpha) r^A - \alpha r^f$. Observe the measure would not be invariant to leverage.

$$ ?? = \frac{\operatorname{E}[ (1 + \alpha) r^A - \alpha r^f]}{\operatorname{Stdev}\left( (1 + \alpha) r^A - \alpha r^f]\right)} \neq \frac{\operatorname{E}[ r^A ]}{\operatorname{Stdev}\left( r^A \right)}$$

An excess return (or zero cost portfolio return) is the return on a portfolio that is equally long and short. The difference between two returns is an excess return.


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