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Risk-Free Rates, Earnings Yields, and Equity Risk Premiums

Article Quant Q&A · Author: K.K.
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Summary

The document examines whether a rising risk-free rate necessarily raises the earnings yield, defined in the discussion as earnings divided by price. It begins from an earnings-based relation between earnings yield, the equity risk premium, and the risk-free rate, and points out that the direction of change cannot be inferred from the two components moving in opposite directions without knowing their relative magnitudes.

The answer offers an economic intuition: if earnings change little while higher rates make competing safe returns more attractive, equity prices may fall, lifting earnings yield. It suggests using a dividend growth or discounted cash flow model to formalize how higher required returns can affect prices. The risk premium's response is discussed separately through risk appetite and leverage, but these claims are not derived rigorously or supported with data. Thus, the document does not establish that a higher risk-free rate alone guarantees a higher earnings yield or lower equity premium.

Key ideas

  • The identity relating earnings yield, the risk-free rate, and the equity risk premium alone does not determine how earnings yield changes.
  • A higher required return can lower equity prices and raise earnings yield if earnings remain relatively stable.
  • Discounted cash flow models can formalize how changes in required returns affect equity valuation.
  • The proposed link from higher rates to a lower equity risk premium depends on assumptions about risk appetite and leverage.

Tags

EquitiesUS marketsRisk management
Full text
# Equity risk premium and the earnings yield


# 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!

## Answer by xing gao (score 1, accepted)

https://quant.stackexchange.com/a/61479

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.