# 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:

$$\widehat{R^f_t} = \sqrt{Y_t}$$

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?)?

## 1 Answer

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).

• Thank you! I'm presuming that for the treasury website source I posted, the data reported are also the one year returns (therefore 365th root, not 30th root)? – user080517 May 8 '18 at 13:45
• @user080517 exactly. Fixed income rates are almost always reported per annum (and so they must be). – Igor Pozdeev May 8 '18 at 21:18