# Equity risk premium and the earnings yield

I am trying to understand the relationship between the risk-free rate $$R_f$$ and the earnings yield of equities.

I have read that an increase in $$R_f$$ leads directly to a decrease in the equity risk premium $$ERP$$. I have also read that an increase in the risk-free rate leads to an increase in the earnings yield by way of a decreasing equity risk premium.

I am confused, and here's why. An earnings-based approach to the equity-risk premium leads to the formula $$ERP + R_f = \frac{E}{P}.$$ In order to conclude from this formula that $$\frac{E}{P}$$ is increasing in time, it is not enough to know only that $$ERP$$ is decreasing and that $$R_f$$ is increasing -- we need to know that $$ERP$$ is decreasing more slowly than $$R_f$$ is increasing.

My question is: Is there a fully rigorous mathematical argument that allows me to get directly from an increasing $$R_f$$ to an increasing $$\frac{E}{P}$$ without additional assumptions and which makes direct reference to a decreasing $$ERP$$? Basically: what am I missing?

I should maybe add that I am a topologist by training and that I am fairly new to mathematical finance.

Many thanks!

• How do you define the equity risk premium? ERP != E/P - RFR. Mar 1 at 19:37
• I’d define it as (expected return) - RFR. I have read that under certain assumptions, the expected return equals E/P. This is the only connection I could find between the three quantities, but I’d be happy to learn that there is something better out there.
– Tony
Mar 1 at 19:47
• My understanding is the ERP is the expected return on equities in excess of the RFR. "expected return" is presumably cap gains + distributions (e.g. divis) whereas EY is simply "earnings" (GAAP? non-GAAP?) divided by price. I don't see the connection between RFR, EP and ERP but perhaps someone can enlighten me! : ) Mar 1 at 21:17