In the textbook I read the following:

We can increase the present value of a share of common stock with a new investment only if $ROE > r$ , where $r$ is a discount rate (capitalization rate). If a new investment results in $ROE < r$, the price of the stock will decline even though earnings could be higher.

I have a trouble to understand the mathematical relationship bertween $ROE$ and $r$. In order to understand that I did the following operations. Since $$ROE=\frac{EPS}{P},$$ where $EPS$ is earnings per share, $P$ is stock price. From another hand, we have $$P=\frac{EPS}{r}+PVGO,$$ where $PVGO$ is present value of growth opportunities. Doing simple algebra the later yields: $$r=\frac{EPS}{P-PVGO}.$$ Does it mean that the difference between $ROE$ and $r$ is determined by $PVGO$? Please kindly explain the difference between those concepts.

  • 1
    $\begingroup$ ROE is an accounting rate of return on the existing business, defined as $\frac{EPS}{BVE}$ where BVE is Book Value of Equity. $r$ is the expected return on the stock (determined in the stock market). They coincide in the hypothetical case of a no growth, or steady state company, which has no opportunities to expand its business by reinvesting earnings (hence it distributes all its earnings), but such companies are rare and so generally r and ROE are different. $\endgroup$
    – Alex C
    Oct 9, 2019 at 15:15
  • $\begingroup$ @AlexC Thank you for very useful comment! So am i right that the difference between $ROE$ and $r$ is determined by $PVGO$? Is there any strict mathematical relationship between those two terms as I tried to derive in my posted question? $\endgroup$
    – sane
    Oct 9, 2019 at 16:43
  • 1
    $\begingroup$ I could make up a problem where $r$ and $ROE$ have two arbitrary values, so I don't see the point in trying to find a mathematical relationship between two such conceptually different things. $ROE$ is associated with the business, $r$ has to do with the stock, i.e. how the stock market values the business (and its associated growth opportunities). $\endgroup$
    – Alex C
    Oct 10, 2019 at 15:52
  • $\begingroup$ @AlexC Okay, thanks! $\endgroup$
    – sane
    Oct 10, 2019 at 15:58


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.