I understand the application of the various RI methics, although the derivation of the single stage model is annoying me a little.

Where I am encountering the problem is the logical reasoning behind moving from the original RI equation. Starting with this equation:

P0/B0 = [(ROE - g)/(r-g)]

what are the logical behind the steps necessary to get to the below equation:

P0/B0 = 1 + [(ROE - r)/(r-g)]*B0

The reasoning is more important to me than the mathematical modification, as it is very simple to see that the two are logially equivallent by subbing in [(r-g)/(r-g)] where the 1 is, yet this derivation seems incredibly less straight forward than the justified P/B, P/E, P/S, etc.

The final equation seen is every easy to translate once you’ve gotten things into the base form, but I would still prefer to avoid any memorization if I can simply derive it every time:

V0 = B0 + [(ROE - r)/(r-g)]*B0

Any help would be greatly appreciated.