Converting Annual Risk-Free Rates to Weekly Rates

Summary

The document explains how to convert an annual risk-free rate to a weekly rate for use in calculations such as the Sharpe ratio. The correct conversion depends on the rate convention. With continuously compounded returns, annual and weekly log rates scale linearly with time, so the annual rate is divided by the number of weeks in a year.

For rates compounded at discrete intervals, the weekly rate must instead be chosen so that compounding it over the year reproduces the annual gross return. The document gives the corresponding compounding relationship rather than a numerical example. It does not recommend a particular risk-free benchmark or proxy, despite that being part of the original question; it addresses only the frequency conversion. Users therefore need to select a suitable rate series and align its convention and observation period with the returns used in their analysis.

Key ideas

• Rate conversion depends on whether returns use continuous or discrete compounding.
• For continuously compounded rates, the weekly rate is the annual rate divided by the weeks per year.
• For discrete compounding, choose a weekly rate whose repeated compounding matches the annual gross return.
• The document does not identify a benchmark to use as the risk-free proxy.

Tags

# Answer by Stéphane (score 2, accepted)


# What should I use as a proxy for the risk-free rate? And how do I transform the yearly risk-free rate to the weekly risk-free rate?












In order to calculate the Sharpe Ratio, I need the risk-free rate. What is the most convenient proxy for the risk-free rate? And how do I transform the yearly risk free rate to a weekly risk free rate?

Best, memecon

## Answer by Stéphane (score 2, accepted)

https://quant.stackexchange.com/a/51437

A lot of work in asset pricing relies on continuously compounded rate of returns in which case if $r_{fw}$ is the weekly rate of return, then the gross return over a year would be given by $\exp(r_{fy}) = \exp(r_{fw} \times 52)$ which implies $r_{fw} = r_{fy}/52$.

However, suppose you have to work with interest rates that compound at discrete point in time, then you could work by using $(1+r_{fy}) = (1 + r_{fw})^{52}$. This asks what would be the weekly compounded rate that gives you exactly the right yearly rate.

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.