Converting Annual Treasury Bill Rates into Daily Risk-Free Returns
Summary
The document considers how to estimate daily risk-free returns for calculating excess returns on individual stocks and a broad equity market. It discusses using a three-month Treasury bill yield as a proxy, and raises a key data question: quoted yields may not be effective annual rates, so the day-count and compounding convention must be checked before converting them to daily returns.
One answer recommends using the risk-free series supplied with the Kenneth French data library, noting that its rates are annual and can be converted for use with log returns. Another answer derives a daily rate from a 90-day rate under a constant-rate assumption, using compounding across the period. These are practical suggestions, not a definitive comparison of data sources or conventions. The document does not establish how Yahoo Finance or Federal Reserve series quote their yields, and its annual-to-daily formula assumes a particular compounding basis and day count.
Key ideas
- Three-month Treasury bill yields can serve as a proxy for a risk-free return in basic excess-return calculations.
- Verify the yield quotation and day-count convention before converting an annual rate to a daily rate.
- The Kenneth French data library is suggested as a convenient source for a risk-free series used in research.
- Deriving a daily rate from a 90-day rate assumes the rate remains constant over the period.
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Full text
# How to calculate daily risk free interest rates
# How to calculate daily risk free interest rates
I'm working on an assignment in which I need to calculate excess returns for six stocks plus the S&P 500. I have computed daily logarithmic returns for every stock and for the market, I now need to calculate the risk free interest rate in order to be able to compute the excess return for every stock and the market.
The interest rate on three months T-Bills is a good proxy for the risk-free rate of return, but I have a lot of doubts on how to use data provided by Yahoo! Finance in order to compute the daily risk-free. Here are my assumptions and procedures:
- I use the 13 weeks treasury bill (ticker: ^IRX) historical quotes provided by Yahoo! Finance;
- Under the assumption that on Yahoo! Finance bond yields are quoted as Effective Annual Rate (EAR), the daily risk-free interest rate at time $t$ ($r_{f,t}^{daily}$) is computed as:
$$r_{f,t}^{daily}=(1+r_t)^{1/365}-1$$
where $r_t$ is the EAR rate at time $t$ provided by Yahoo.
Once computations are done, the excess return of stock $i$ at time $t$ is defined as:
$$\text{Excess Return}=r_{i,t}-r_{f,t}^{daily}$$
Questions:
- Is this procedure correct? Note that I do not need it to be exceptionally precise, it is just a basic exercise, but I would like it to be at least conceptually correct.
- The Federal Reserve also provides data for three months T-Bills. Are these rates also provided as EAR?
## Answer by phdstudent (score 4)
https://quant.stackexchange.com/a/22124
Just use the what most finance research papers use, i.e. the risk-free rate from the Kenneth French data library.
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
The rates are annual. So if you want log returns just take the log of $1+r^f_t$ and divide by 365.
## Answer by Andrew (score 0)
https://quant.stackexchange.com/a/44585
I think we can find daily risk free using following equation $$1 + r_{90} = (1 + r_1)^{90}$$ This follows from the fact there are no arbitrage opportunity. Here we assume that $r_1$ in the following periods will stay the same, that is non random. Thus, doing simple algebra we get $$r_1 = (1+r_{90})^{\frac{1}{90}} - 1$$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.