do not understand derivation of P/B formula

CFA Level II
#1

Hello,

I am confused by the derivation of the below P/B formula:

(ROE - g)/(r-g) = justified P/B ratio.

Can someone break down this calculation for me please?

Thanks,

#2

You start with Gordon growth, derive the justified leading P/E ratio, then move to the justified P/B ratio:

P0 = D1 / (rCE − g)

P0/E1 = (D1 /E1) / (rCE − g)

P0/E1 = (1 − b) / (rCE − g)

P0 / (B0 × ROE) = (1 − b) / (rCE − g)

P0/B0 = [ROE × (1 − b)] / (rCE − g)

P0/B0 = [ROE – (ROE × b)] / (rCE − g)

P0/B0 = (ROE – g) / (rCE − g)

#3

is it worth knowing that entire derivation, or should we just focus on the final formula for the exam?

#4

I’d focus on the formula, and its implications:

  • If ROE > rCE, P0 > B0; the company is adding value for the investors
  • If ROE < rCE, P0 < B0; the company is eroding value for the investors
#5

use tangible solves all your problems

#6

Except, perhaps, unintelligibility.

#7

s2000 Did a really good job of explaining it but if anyone’s brain works like mine I learned it better by thinking of it in two components (price and book value of equity).

I was solving a problem earlier that gave ROE, g, (1-b), r and recent dividend asking for justified P/B. I thought of it as price and book value of equity as separate components. Looking to get to BV of equity I knew I had ROE (which is NI or earnings/book value of equity). To get closer to the BV of equity I needed to factor out the earnings component. The earnings component can be derived by taking dividends/(1-b). Now you would take 1/ROE = BV of equity / earnings and multiply by the earnings you calculated from dividends/(1-b). You are left with the book value of equity being your B component of P/B.

Price using a single-stage GGM would be calculated by D1 / (r-g) giving you the P component.

Taking the P component (D1/(r-g)) and dividing by the B component (Earnings/ROE) gives you:

[(D1*ROE)/Earnings] / (r-g)

D1/Earnings is (1-b) leaving you with:

(1-b)(ROE)/(r-g) Which simplifies to [ROE - b*ROE]/(r-g)

Giving you: (ROE-g)/(r-g)

Last part of the derivation is essentially what s2000 put. Breaking it up in two pieces is definitely a little messier but IMO is more intuitive and doesn’t require you to remember a formula that wasn’t already used in L1.

#8

Thanks for this!!!

#9

My pleasure.