# Sharpe Ratio, risk free rate [closed]

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!

## 2 Answers

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.