Choosing a Risk-Free Rate for Maximum Sharpe Portfolios

Summary

The document asks which risk-free rate to use when constructing a maximum Sharpe ratio portfolio from a rolling estimation window of monthly returns. It frames the problem within mean-variance portfolio theory, where the Sharpe ratio measures expected portfolio excess return relative to portfolio volatility. The answer says the theoretical risk-free rate is the rate at which investors can lend and borrow, an assumption that rarely holds exactly in practice.

As a practical proxy, it identifies the US three-month Treasury bill rate. For a more realistic borrowing constraint, it points to a construction associated with Black: form a mean-variance efficient portfolio uncorrelated with the market, then use its expected return as the shadow cost of borrowing in place of the risk-free rate. The document offers these choices conceptually but does not specify how to estimate the rate in an out-of-sample backtest or address implementation details such as aligning rate observations with monthly returns.

Key ideas

• The theoretical risk-free rate assumes investors can lend and borrow at the same rate.
• The US three-month Treasury bill rate is presented as a common industry proxy.
• Borrowing constraints can make a single risk-free rate an imperfect portfolio assumption.
• A market-uncorrelated efficient portfolio’s expected return can represent a shadow borrowing cost.

Tags

# Which riskfree rate to use for Maximum Sharpe Ratio Portfolio?


# Which riskfree rate to use for Maximum Sharpe Ratio Portfolio?












I am conducting out of sample backtests of the MV framework. But how exactly do I derive the Maximum Sharpe Ratio portfolio for this? The standard forumula of the Sharpe Ratio is given by:

$$\frac{(w r - r_f)}{\sqrt{w Σ w'}}$$

Lets say I have an estimation window of 60 monthly returns on whose basis I derive the optimal portfolio weights. Which risk free rate $r_f$ would I have to use here or would I have to use any at all?

I'd appreciate any help!

## Answer by Adam (score 2)

https://quant.stackexchange.com/a/51509

Within the context of portfolio theory, the risk-free rate is the interest rate at which investors may lend and borrow capital. This is generally not true in practice, but if you are willing to make this assumption then a common proxy used in industry is the US 3-Month T-Bill rates.

To impose a more realistic borrowing constraint, you could follow the approach in Black (1972) by constructing a mean-variance efficient portfolio uncorrelated with the market portfolio. Let's call the returns on this portfolio by $r_z$, then $\mathbb{E} r_z$ represents the shadow cost of borrowing and you can simply replace $r_f$ with $\mathbb{E} r_z$.

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.