Hmm. I would say it has something to do with the Book Value being too low.

The P/B ratio is justified based on ROE, growth, and the required rate of return - this is the fair value of the multiple. It should be 2.

Calculating it from the fair value price (which is based the same fundamentals) then dividing it by $10 which was randomly plugged in gives a different result, 2.3.

I guess what this means is while it should be trading at 2, its overpriced trading at 2.3 - but since we calculated the price based on fundamentals … the book value is too low for it to be consistent with the fundamentals - as everything else is justified based on the gordon growth derivations and fundamentals.

Just another thought, considering that EPS is 2.5 and BV is 10, the ROE is 2.5% and therefore book value is consistent with fundamentals.

In that case are you sure the $1.00 dividend was the last paid dividend and not the next dividend to be paid? If it is the dividend for next period we wouldn’t grow it by the factor (1+g), and consequently, the two P/B ratios will be 2.

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

Yes - because P/B is the EPS at time T divided by Book value at T - 1, (ROE is the net income divided by beginning book value) and the question doesn’t really give you enough information to decipher which Book Value theyre talking about.