Converting a Treasury Bill Yield into a Daily CAPM Risk-Free Return
Summary
The discussion explains how to convert a quoted one-month Treasury bill yield into a daily return proxy for CAPM. The proposed conversion compounds the annualized yield over a daily calendar-year fraction, after converting a percentage quote to a decimal. This is presented as a practical approximation rather than a precise match to every instrument convention.
The reply also clarifies timing and investment assumptions. In a CAPM return for period t, the rate should ideally be known at the prior period, and a one-period investable rate would better match the decision horizon. Using a Treasury bill as a daily proxy implies selling it after a day, so its return is not literally risk-free over that holding period. An overnight interbank rate may fit the horizon more closely, while day-count conventions and other details remain. The answer says these complications are often neglected in practice; it does not compare proxy performance empirically.
Key ideas
- Convert a percentage annual yield to a decimal before deriving a daily compounded return.
- The rate used to explain period-t returns should ideally be observable when the investment decision is made.
- A one-month Treasury bill yield is not the same as an investable overnight rate.
- Treating a Treasury bill as a one-day risk-free investment assumes it can be sold the next day without uncertainty.
- Day-count conventions complicate precise conversion, though the answer describes the approximation as commonly used.
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# Proxy for daily risk-free return in CAPM
# Proxy for daily risk-free return in CAPM
Say I am estimating the following Capital Asset Pricing Model:
$$R_t = R^f_t - \beta(R^m_t - R^f_t)$$
where $R^f_t$ is the risk-free return, and $R^m_t$ is the return for some market index, say the S&P 500.
A common proxy for $R^f_t$ (for instance, see Fama and French (2004)) is the daily one-month yield on a Treasury bill. This data is easily available here.
Now, since those yields $Y$ are for holding it for 1 month, would I be correct in assuming that the proxy for the daily risk-free rate would be:
$$\widehat{R^f_t} = \sqrt[30]{Y_t}$$
Is this correct? If so, what assumptions am I implicitly making by generating the proxy for $R^f_t$ in such a manner (i.e. what am I assuming about investors?)?
## Answer by Igor Pozdeev (score 1, accepted)
https://quant.stackexchange.com/a/39692
Interest rates are usually reported in percent per year, so you should rather do
$$(1 + Y_t/100)^{1/365} - 1,$$
but there are a million of complications, most of which can be safely neglected. Think about what CAPM is doing: an investor at time $t-1$ is choosing between the risk-free asset and a stock to reap the payoff at time $t$. Hence, since the risk-free return at time $t$ is actually determined at time $t-1$, the risk-free rate in the formula should ideally be that of time $t-1$. Then, it should ideally be the actually investable 1-period rate, and if you are opting for the Treasuries, you are implicitly counting on selling it the next day, and so it's magically not risk-free anymore. A better rate for that purpose is the interbank overnight rate. Finally, day count conventions are a mess. But like I said, for all the practical purposes it is safe to convert the 1-month T-Bill rate to the daily return. Everyone does that anyway.
Also, see this question (link).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.