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.

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    $\begingroup$ 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 '19 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 '19 at 16:43
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    $\begingroup$ 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 '19 at 15:52
  • $\begingroup$ @AlexC Okay, thanks! $\endgroup$ – sane Oct 10 '19 at 15:58

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