This page will explain how to derive justified P/B ratio formula:

$\displaystyle \text{justified P/B ratio} = \frac{ \text{ROE} - g }{r-g}$ ,

where:
ROE = return on equity
r = required rate on equity
g = expected growth rate in dividend and earnings

Derivation

From Gordon Growth Model, we know

$\displaystyle \text{justified stock price} = V _ 0 = \frac{ D _ 1 }{r - g}$ ... ①

Using $B_0 = \text {Current book value of equity}$ , we can express the justified P/B ratio as

$\displaystyle \text{Justified P/B} = \frac{V _ 0}{B_0}$

Since next year's earning $E_1$ can be calculated as

$\displaystyle E_1 = B_0 \times \text{ROE}$

$\displaystyle \rightarrow B_0 = \frac{E_1}{\text{ROE}}$ ... ②

①÷② gives

$\displaystyle \frac{V _ 0}{B _ 0} = \frac{ D _ 1}{r - g} \times \frac{\text{ROE}}{ E_1}$

$\displaystyle \qquad = \frac{ D _ 1}{E_1} \times \frac{\text{ROE}}{ r - g }$

$\displaystyle \qquad = (1-b) \times \frac{\text{ROE}}{ r - g }$

$\displaystyle \qquad = \frac{\text{ROE} - b \times \text{ROE}}{ r - g }$ ... ③

where b = retention rate.

Since sustainable growth rate can be calculated as $g = b \times \text{ROE}$ , we get

$\displaystyle ③= \frac{\text{ROE} - g}{ r - g }$

babaye's notes