Understanding Walter's Dividend Policy Model

Question

I'm trying to understand the justification for the mathematical formulation of the Walter model (1956), which provides an equation for the price of a stock based on present value of dividends and reinvestment of retained earnings. The equation is given by

$P = \frac{D}{K} + \dfrac{\frac{R(E-D)}{K}}{K}$,

where $P$ is the price, $D$ is the dividend, $R$ is the return on additional investment, $E$ is earnings, and $K$ is the cost of equity (i.e. required return).

I understand that $D/K$ reflects the present value of an infinite perpetuity. I also understand that $R(E-D)$ is the income obtained from reinvestment of retained earnings.

What I do not understand is why we end up with $K^2$ in the denominator of the second term. It seems to me that $R(E-D)/K$ is the present value of a stream of reinvestment income paid out to shareholders - this what I thought we wanted.

Dividing by $K$ suggests that shareholders are paid a stream of present values. This is specifically what I do not understand.

Answer 1

Say you have a company that has a 25% payout rate on EPS 1, and can invest retained earnings at say 10%. So every year, it has DPS 0.25 and retained 0.75

The D/K part is the NPV of receiving the 0.25 dividend in perpetuity, discounted by K.

Every year the 0.75 is then retained and invested. At 10% returns, that is 0.075 in perpetuity for that year's vintage, which has an NPV of 0.75/K.

But this 0.75/K is just a single year figure.

The company retains and invests the same 0.75 at 10% forever = worth 0.75/K every year. This is worth 0.75/K every year. In perpetuity, this is worth (0.75/K)/K, which is where you get your square.