Single Stage RI Formula Derivation

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.

As stated in my above post:

I know how to make the modification. The question is WHY do we make the modifications? To give a parallel, in the case of the P/E formula, you simply take the formula for price and divide it by earnings. In the case of single stage RI models, I do not understand the reasoning, although I do understand the math.

The reason why you make the transformation is so that you can get to valuation. Think about it - P0/B0 by itself is a number… it represents nothing. Ever seen a stock trading at 9.5x on an index? P0/B0 is simply a multiple. The first stage model gives a dollar value of the firm, with a unit of dollars (or whatever currency you’re using).

Unit of P0 is , unit of B0 is . In multiple form, both units cancel out, and you’re left with a number (multiple). This is good for analysis, but not good from a valuation perpective.

Using the transformed formula, you have P0 on the left side in form. You bring B0 to the right hand side to value the firm... you also end up with on the right hand side. Mathematical integrity is preserved since you end up with the same unit on both sides.

I understand shifting it out of the multiple, but that still doesn’t explain the form change between the first two equivalent formulas. Your explanation doesn’t actually answer the question at hand at all.

Start with:

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

Then multiply by B0 to put things in price terms:

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

Critical Stage: What I am gathering from here is that they want to break apart the RI from the base value, where:

RI = (ROE - r) * B0

Yet we do not have things in the form (ROE - r) on the numerator, so we’ll have to adjust. Let x be the conversion factor necessary to accomplish this:

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

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

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

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

x = B0 * (g - r) / (r - g)

Subbing x back into the main pricing equation:

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

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

By substitution from the original conversion factor:

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

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

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

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

Here you can see that it is equivalent to the Single Stage RI formula:

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

tl;dr - The math works, but it is ludicriously annoying and would take too long on an exam. Is there a faster way to logically derive this?

Your critical stage is simply over analyzing the problem at hand. You don’t need to go through that much work: simply add and subtract “r” (refer to the last post in the other thread I pointed to) and rearrange the terms to end up with (ROE - r)*B0 in the numerator.

I guess I just need to be cognizant of the fact that RI requires (ROE - r) and modify my equation as early as possible. Thanks for the help.