I think ^ you mean 2.5 / 10 = 25%… Those derivations you use should get the same answer. In addition - the point of different ratios is to have different “tools” to be able to compute an accurate price. IE, you would use P/B in certain instances and P/E in another. It makes sense to compare the justified P/B with the market P/B, ie trading high or low or vs comps.

P = D0 (1+g) / r-g = E1*(1-b) / (r - g)

Divide both sides by book value

P/B = E1 / B (1 - b) / (r-g)

But, E1/B is equal to ROE

So, P/B = ROE (1 - b) / (r-g)

=> P/B = ROE - ROE*b / (rig)

Using definition of growth, ROE * b = g

So, substituting

P/B = ROE - g / (r-g)

So, to your point, assuming a constant payout ratio of 1 / 2.5 = 40% => b = 60% => g = .6 * .25 => g = 15%

So, now we’ve got .25 - .15 / .2 - .15 = 2. This is the justified ratio. Implied price is then BV * 2 = $20. If P/B is an appropriate indicator of value, comparing it to the market price is useful