Sharpe Ratios When the Risk-Free Rate Is Stochastic

Summary

The document asks whether the Sharpe ratio denominator should account for a time-varying risk-free rate, including its volatility and correlation with portfolio returns. The answer frames the ratio as return per unit of risk for a zero-cost position. Under that interpretation, an investor borrows at the risk-free rate before investing, so variation in the financing rate contributes to the risk of excess returns.

The response also notes that a foreign-exchange strategy’s Sharpe ratio implicitly reflects these considerations. It does not give a formula, derivation, or empirical comparison with the conventional calculation, so it serves as conceptual guidance rather than a complete prescription. The appropriate treatment depends on the return series and financing assumptions: when the funding rate varies, measuring only portfolio-return volatility may omit refinancing risk embedded in the excess-return position.

Key ideas

• A Sharpe ratio can be viewed as return per unit of risk on a zero-cost position.
• A variable borrowing rate can add refinancing risk to an investment financed at that rate.
• The risk of excess returns can therefore depend on movements in the risk-free rate and its relation to portfolio returns.
• Foreign-exchange strategy Sharpe ratios implicitly include these rate effects, according to the response.

Tags

# Sharpe Ratio with Stochastic Interest Rate?


# Sharpe Ratio with Stochastic Interest Rate?












All versions of the Sharpe ratio that I've seen seem to assume that the risk-free rate is constant, and the standard deviation of the excess return in the denominator simplifies to the standard deviation of the portfolio return. Is it appropriate to include instead the standard deviation of the risk-free rate and the correlation of the portfolio return and risk-free rate when calculating the standard deviation of the excess return in the denominator?

## Answer by Igor Pozdeev (score 2)

https://quant.stackexchange.com/a/41150

Sure! Sharpe ratio must be defined as the return per unit risk on a zero-cost position. The notion you are referring to achieves this by assuming borrowing at a risk-free rate before investing, so refinancing risks should matter.

On a side note, the Sharpe ratio of any ForEx strategy would implicitly have the stuff you mention accounted for.

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