Justified P/B Ratio

CFA Level II Equities
#1

I need some guidance to better understand this equation conceptually.

Justified P/B ratio: (ROE - g )/(r - g)

Why do we deduct growth from ROE in the numerator?

Here is what I think through: If a firm earns a 25% ROE, and my required rate of return is 13%, but I assume a 4% growth factor, then the justified P/B multiple should be more like (.25)/(.13-.04) = 2.8x. I’m not sure why we would conceptually deduct growth from their ROE in the numerator.

#2

I agree that shown like this, it doesn’t make much sense. However you can think about P/B also this way:

P/B ratio = 1 + ((ROE-r)/(r-g))

Stated like this, it looks very much like the way you’d compute the value under the Residual Income Model (in which book value plays an important role), where:

V = BV + BV*(ROE-r)/(r-g)

Hope it helps linking the two together and makes a bit more sense this way!

#3

A few items to consider; First and foremost the Justified P/B ratio and the other multiples in this chapter are based on the Gordon Growth Model. Review the basics of that to clear up anything you need with that. GGM formulas:

P= D0(1+g)/(r-g)

P = D1 / (r - g)

P=E1 ( 1 - b) / (r - g) where E1=ROE

Remember that g= RR * ROE. This is the sustainable growth rate of a company. Look at it as how much a firm commits to reinvestment(RR) and the quality of reinvestment(ROE). This is the amount of growth the company can sustain with its current equity value. This growth factor is removed from both the numerator and denominator to establish a value now for the multiple.

If you need more help than this I will write it up a different way.

#4

great post Analyst_24. I struggled with this…thanks mate.

#5

Hi Analyst 24, could you tie in your comments above and relate that to PVGO? Also, for all assumed mature companies using GGM, is it assumed that ROE = r? Does this mean that the company is no growth, or grows at a constant rate? Also if ROE = r , does this mean the compAny is not adding value (in order for that to happen, ROE would be > r, in which case the company is successfully investing in + NPV projects, otherwise it would just pay out all the earnings as dividends)?

sorry for all the questions but I am just trying to tie in Corp finance, GGM and the RI Model together and I’m really struggling on the PVGO aspect and the ROE = r (as it relates to growth and value) concepts too

#6

I think I need more help on this. Simply subtracting the growth rate from numerator and denominator is weird. You could also divide by 1+ growth rate if you wanted to account for it. I don’t understand how you could just look at the formula and say “Hey! This is how it should be.” Can anyone help me view why is it the way it is? Much appreciated.

#7

Here goes:

P_0 = \frac{D_1}{r - g}
= \frac{E_1 × \left(1 - b\right)}{r - g}
= \frac{B_0 × ROE_1 × \left(1 - b\right)}{r - g}

So,

\frac{P_0}{B_0} = \frac{ROE_1 × \left(1 - b\right)}{r - g}
= \frac{ROE_1 - ROE_1 × b}{r - g}
= \frac{ROE_1 - g}{r - g}

The numerator – (ROE – g) – is the portion of ROE paid in dividends (i.e., not retained). You can think of this as a sort of Gordon growth model, where the numerator is the percentage of earnings retained, instead of the dividend paid out.

3 Likes
#8

S2000 magician

As always - a fantastic post!

#9

Thanks.

I hope that ankitgu (OP) finds it helpful.

#10

I think there is a mistake above. The numerator ROE - g is the part paid in dividends isn’t it ? as b is the retention rate so g = ROE * b is the part of ROE that is kept in the firm’s BS.

and ROE - g = ROE paid in dividends

am i right ?

#11

I believe that you are. Good catch. I’ve corrected it.