Why Risk-Free Leverage Leaves the Sharpe Ratio Unchanged

Summary

The document explains why borrowing at the risk-free rate to scale a risky portfolio increases its expected excess return and volatility by the same factor. The Sharpe ratio uses excess return, so the leveraged portfolio’s numerator is the leveraged portfolio return minus the risk-free rate: this simplifies to the original excess return multiplied by leverage. Its standard deviation is multiplied by that same amount, leaving the ratio unchanged.

The explanation corrects a calculation that subtracts borrowing costs from the portfolio return but then fails to subtract the risk-free rate from the leveraged return when forming the Sharpe ratio. The result assumes borrowing and lending occur at the same risk-free rate, and that leverage scales the risky exposure and volatility proportionally. Actual financing costs, constraints, or nonlinear effects can break those assumptions. The document offers an algebraic explanation rather than empirical evidence.

Key ideas

• The Sharpe ratio compares expected portfolio return above the risk-free rate with portfolio volatility.
• Leveraging by a factor scales both excess return and volatility by that factor under the stated assumptions.
• Subtract the risk-free rate from the leveraged portfolio return when calculating its Sharpe ratio.
• Borrowing and lending at the same risk-free rate is an assumption behind the invariance result.

Tags

# Why does the Sharpe ratio not change when the strategy is leveraged?


# Why does the Sharpe ratio not change when the strategy is leveraged?












It has been correctly stated that the Sharpe ratio of a strategy does not change when it is leveraged. I understand Eric's point that leveraging by $n$ multiplies both the return $x$ and volatility $\sigma$ by $n$. I also understand that we fund the leverage at risk free rate, and hence subtract $r(n-1)$ from the return. However, I cannot understand why this would not change the Sharpe ratio (sr) since $$sr = \frac{nx - (n-1)r}{n\sigma} = \frac{n(x-r)}{n\sigma} + \frac{r}{n\sigma} = \frac{(x-r)}{\sigma} + \frac{r}{n\sigma}$$ The sr is different, where am I going wrong?

## Answer by amsh (score 3)

https://quant.stackexchange.com/a/22651

Sharpe ratio = $\frac{r_p - r_f}{\sigma_p}$, where:

- $r_p$ is the expected portfolio return

- $\sigma_p$ is the portfolio's standard deviation

- $r_f$ is the risk free rate.

When you leverage '$n$' times:

- The leveraged portfolio return is $n r_p - (n-1) r_f$ (subtracting the cost of borrowing the money)

- The standard deviation increases to $n\sigma$

Hence:

"Leveraged Sharpe ratio" = $\frac{n r_p - (n-1) r_f - r_f}{n\sigma}=\frac{n(r_p-r_f)}{n\sigma}$ = $\frac{r_p - r_f}{\sigma_p}$ = Sharpe ratio.

Given any risky asset, one can generate an infinite expected return at the cost of added risk (by leveraging the investment). The Sharpe ratio mitigates "false advertising".

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.