# Risk-Free Rate In CAPM

Let's start out with the CAPM equation itself:

$E(R_i) = R_{f1} + \beta_{i}(E(R_m) - R_{f2})$

Are there cases where one should choose a different $R_{f1}$ and $R_{f2}$ (Risk Free Rates Of Interest) or do they always have to be equal?

And what if I had to opt for the Swiss government bonds that now have a negative yield (-0.21%)? Should I just take the mean value over the last 10 years and use this mean value as my $R_{f1}$ & $R_{f2}$? And does it matter what kind of bond yield data (daily, weekly, monthly, yearly) I choose for the "last 10 years"?

• To give a partial answer, from my understanding of the model, only a single risk free rate should be used as they are related. The model is saying the expected return $E(R_i)$ is equal to the risk free rate + the market risk premium the security demands which is the market return, minus the risk free rate times $\beta$ (systematic risk) – AfterWorkGuinness Oct 21 '15 at 1:21
• Well, from my point of view it would definitely make sense to use the same risk-free rate, but maybe there's some form of risk-free rate optimisation one could do in order to assimilate CAPM to extraordinary situations, such as the negative government bond yield. – Lumberjack88 Oct 21 '15 at 1:30
• They should be the same, if no risk free rate is available, you should use Black's zero-beta CAPM. – zsljulius Oct 21 '15 at 1:39
• In DCF framework, $R_f$ should correspond to modelling horizon, so if you're modelling long term equity it should be 10 or 30 Yr government bond. Beware, $R_f$ in CAPM should be the same as the $R_f$ used to calculate equity risk premium, not two as you showed in your formula. You may find more by looking at Ibbotson's ERP tables. – Sergey Bushmanov Oct 21 '15 at 11:05

The CAPM equation is an straight line and risk free rate ($$Rf$$) is your intercept. At least in standard version of the CAPM is not possible to have more than 1 risk free rate, in fact in similar case the highest should always preferred in investment point of view. In any case the standard assumption speak about only one risk free rate. In CML framework sometimes is shown the possibility to lend and borrowed at two different rate but I never seen this possibility in CAPM setting.
About the negative $$Rf$$: is not a problem in mathematical point of view. Maybe this situation affect personal risk aversion but this is another story.
About the data: actually choice of $$Rf$$ is not obvious. Is possible to choose short term or long term bond but also deposit rate or other. The time horizon and estimation technique are your choices. From "micro" point of view It depend from actual investment possibility and assumptions. From "macro" point of view you have to make more relevant assumptions. Note that for $$Rm$$ the same is true.