# 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. – user42108 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! : ) – user42108 Mar 1 at 21:17

## 1 Answer

Look at the right side of the equation first. Earning can be treated as accumulated results for a relatively long period, which won't change too much if risk free rate increases. However, increasing risk free rate is gengerately accompanied by decreasing from capital market revaluation. For example, assume risk free is 0%, and a stock pays 5% annual dividend. If your saving account suddenly provides 5% annual return, will you take extra risk to purchase that stock? No. We won't consider buying that stock until the stock price drops, for instance a half, which means 10% annual dividend. That is an explanation for increasing risk free rate and increasing earning yield. For a rigorous mathematical argument, we may check a stock pricing model, such as Gordon Growth Model or other cashflow discount models. Risk free rate increasing leads to required capital return increasing, which finally hurts equity price.

Then we go back to the left side of the equation. How will the risk premium move if risk free rate increases? I think the answer lies beneath the risk appetite. When Federal increases rates. The whole market leverage ratio theoretically decreases as we borrow less money. The economy looks healthier, thus investor will require less risk premium.