Interpreting Expected Stock Growth Relative to the Risk-Free Rate

Summary

The document asks whether the ratio of a stock’s expected growth rate to the risk-free rate has an established name or interpretation when the stock is modeled with geometric Brownian motion. It is primarily a question about terminology, not a proposed trading signal or valuation method.

A response connects the ratio to the Capital Asset Pricing Model. Under that model, expected stock return equals the risk-free rate plus beta times the market’s expected excess return; dividing by the risk-free rate expresses the ratio in terms of beta and the market-return-to-risk-free-rate ratio. This is an algebraic restatement of CAPM, not evidence that the ratio has a standard standalone name or predictive value. Interpretation also depends on the model’s assumptions and the risk-free rate being meaningful as a denominator; the post does not explore those limitations or provide empirical tests.

Key ideas

• The author asks whether expected stock growth divided by the risk-free rate has a standard name.
• The response uses CAPM to express the ratio through stock beta and the market’s excess return.
• The formula provides a model-based interpretation but does not establish a separate accepted metric.
• No empirical validation or trading application is presented.

Tags

# What is this ratio: expected returns on stock divided by risk free rate?


# What is this ratio: expected returns on stock divided by risk free rate?












So this ratio has come up in some work I'm doing and I can't seem to figure out if it is attested in the literature. Here's the setting:

Given a risk free rate $r(t)$ and a stock price which follows a geometric brownian motion, $\frac{dS(t)}{S(t)} = \mu_S(t)dt + \sigma_S(t)dW(t)$, what is the significance/definition/terminology for the ratio of expected growth in stock prices to the risk free rate, $\frac{\mu_S(t)}{r(t)}$? This is more of a question regarding vocabulary/interpretation, so any ideas would be greatly appreciated.

## Answer by hvedrung (score 2)

https://quant.stackexchange.com/a/20620

Maybe it makes sense to refer to CAPM theory: it is expected that stock return is proportional to market excess return and it beta $$ \mu_S(t) = r(t) + \beta_S\cdot(r_m(t)-r(t)).$$ Where $r_m(t)$ is market expected return and $\beta_S$ is stock beta.

Thus according to CAPM it's expected that $\frac{\mu_S(t)}{r(t)} = 1+\beta_S\cdot(\frac{r_m(t)}{r(t)}-1).$

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