Using a Time-Varying Risk-Free Return in the Sharpe Ratio

Summary

The document asks how to choose the risk-free rate when calculating a Sharpe ratio over a period in which daily annualized returns and their average and volatility are known. It questions whether using the average one-year Treasury yield over that period is consistent with the calculation.

The response frames the risk-free rate as the realized return from a risk-free investment over the same dates. It describes tracking a one-dollar investment by compounding each day’s posted annual rate over a daily fraction of a year, then using the investment’s cumulative gain as the Sharpe ratio’s risk-free return. This aligns the comparison period with the risky asset’s measurement period.

The example assumes daily rates are available and uses a particular day-count convention. The note does not discuss alternative compounding conventions, rate instruments, or how to handle uncertainty in rates, so those choices may need to be specified for other datasets.

Key ideas

• The Sharpe ratio compares risky-asset returns with the return available from a risk-free investment over the same period.
• A changing annual risk-free rate can be converted into a cumulative investment return by compounding across the observation dates.
• The resulting cumulative gain, rather than an unadjusted average annual yield, is the period-matched risk-free return.
• The example uses daily rates and a stated daily fraction of a year, so its compounding convention is an assumption.

Tags

# Risk-free rate in the Sharpe ratio


# Risk-free rate in the Sharpe ratio












Let's say I have data stemming from date $t_1$ to date $t_2$, I have the annualized return each day during those two dates. Therefore I know the expected return and the standard deviation of the returns between those two dates. But what should I use as risk-free rate in the calculation of the Sharpe ratio: $$ \frac{E(r)-r_f}{\sigma}$$ It seems like to be consistent, it would be natural to use the average 1-year risk-free rate from the Treasury yield curve during dates $t_1$ and $t_2$.

## Answer by user8948 (score 1)

https://quant.stackexchange.com/a/47481

The formula wants to compare the (expected) return of a risky asset with the (known) return of a risk-free alternative. You need therefore to think of $r_f$ as the return of a risk-free asset.

The easiest way to know you're doing it right is to reconstruct this asset on a spreadsheet. To consider a typical case: if you have per annum risk-free rates posted daily, the cumulative value of an initial investment of \$1 would be

$$s_0 = 1, \quad \quad s_{t+1}= s_t \times (1+i_{t+1})^{1/252}$$

where $i_t$ is the interest rate on day $t$. The return on the risk-free asset then would simply be $r_f = s_{t_2 - t_1} -1$.

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.