Why the Risk-Free Rate Can Change Sharpe Ratio Rankings
Summary
The document explains why subtracting the risk-free rate matters when comparing Sharpe ratios. Since the rate is subtracted from each fund’s return before dividing by its volatility, it changes each fund’s numerator relative to its own risk. The ranking can therefore differ from a comparison that uses raw returns, even if the same risk-free rate applies to both funds.
A numerical example shows Fund A ranked higher when the risk-free rate is 1%, while Fund B ranks higher when the rate is zero. The discussion also links excess returns to the Sharpe ratio’s leverage invariance: scaling an excess-return position scales both expected return and volatility, leaving the ratio unchanged. Using raw returns does not preserve that property. The examples are illustrative; the document does not address estimation choices, changing rates, or other practical limits of Sharpe comparisons.
Key ideas
- Subtracting the risk-free rate can change which of two funds has the higher Sharpe ratio.
- The same risk-free rate does not affect funds equally after each return is scaled by its own volatility.
- Using excess returns makes the Sharpe ratio invariant to scaling a position in excess returns.
- The numerical comparison illustrates the ranking effect but does not discuss estimation uncertainty.
Tags
Full text
# Sharpe Ratio, risk free rate
# Sharpe Ratio, risk free rate
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!
## Answer by msitt (score 4, accepted)
https://quant.stackexchange.com/a/33650
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.
## Answer by Matthew Gunn (score 1)
https://quant.stackexchange.com/a/33668
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.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.