Schweser page 257 book 4 says: "Using the sustainable growth relation of g = ROE x b and observing that E1 = B0 = ROE, we can also derive the justified P/B from the gordon growth model as Justified P/B ratio = (ROE-g)/(r-g) Can anyone explain how they got this? It’s not a difficult formula to memorize, but I would just like to be able to understand it rather than just memorize it–and I’m not seeing how the numerator is obtained by having values for g and E1. Thank you.
Take the Single-Phase Residual Income model: V=BV0+[BV0(ROE-r)]/(r-g). (“V” is really price, although in the formula it is written as intrinsic value, or “V.”) Then divide both sides by BV0, to get P/B=1+(ROE-r)/(r-g). Then take the number “1” on the right hand side of the equation and rewrite it as (r-g)/(r-g), so that now you have P/B=(r-g)/(r-g)+(ROE-r)/(r-g). Now, you can smoosh the two fractions together because they are over the same denominator, r-g: P/B=(r-g+ROE-r)/(r-g). The r’s cancel out, so you get (-g+ROE)/(r-g), or (ROE-g)/(r-g). Does this help?
P/E1 = (1-b)/(r-g) now substitute E1 P/E1 = P / (B0 * ROE) = (1-b) / (r-g) now multiply both sides by ROE P/B0 = ROE(1-b) / (r-g) = (ROE - g) / (r-g)
P = D1 / (r - g) D1 = E1 * (1 - b) “the part of earnings that is paid out as a dividend” Therefore, P = E1 ( 1 - b) / (r - g) we know that E1 = ROE*B0 = net income / equity * equity (equity cancels out) P = (ROE*B0) (1 - b) / (r - g) P/B0 = ROE* (1 - b) / (r - g) P/B0 = (ROE - ROE*b) / (r - g) g = ROE*b “the return on equity that is retained by the firm” P/B0 = (ROE - g) / ( r - g)
u guys are great, thanks so much