8.7  Residual Income:  Constant Growth Model

Recall in chapter 5 we considered the simplest form of the basic dividend model, the constant dividend growth model.  In this model, d_t = d*(1+g)^t so the current dividend, d, grows for every at rate g.  Note that d_t is a forecast of the dividend at time t.  Mathematically this has a nice implied simplification which can be expressed as follows:

This simplifies to:

An immediate practical problem with this model was that it was applicable only to dividend paying stocks.  However, under clean surplus accounting a₁ = CI₁ – (BV₁ – BV₀), where CI₁ is comprehensive income for the next period, BV₁ is end of period book value per share and 0 denotes the present, we can interpret a₁ as the dividend paid out next period (d₁).  As a result, we can extend the basic constant growth model to all firms, irrespective of whether they pay dividends or not, by invoking the clean surplus accounting assumption. 

Under constant growth the change in the book value equals CI₁*RR (the retention ratio) = ROE*RR*BV₀ when ROE (Return on Equity is measured using Comprehensive Income).  In the beginning of chapter 3 we developed accounting or fundamental growth as ROE*RR = g.  Extending this definition to measuring ROE using Comprehensive Income the change in the book value from time 0 to time 1 equals BV₀*g which provides the constant dividend growth model extended to all firms by substituting for d₁ as follows:

The advantage of the latter form is that it is applicable to all stocks irrespective of whether or not they pay dividends.