# Gordon Growth Model

Consider a growing perpetuity: a stream of growing cash payments that goes on forever.

Assume that the cash payments, ${c_1, c_2, c_3, ..., c_{\infty}}$ are made at fixed periodic intervals (say annually), starting a year from now. If the growth rate ${g}$ is constant, then I can write, $\displaystyle c_i = c_1 (1 + g)^{i-1}$

However, one dollar in my hand today is worth more than one dollar a year from now.

That is, I discount future cash payments.

Suppose that the discount rate is ${d}$. So that a dollar ${i}$ years from now is only worth ${1/(1+d)^i}$ dollars to me now.

I am now ready to find the present value of my growing perpetuity. $\displaystyle \begin{array}{rcl} PV & = & \dfrac{c_1}{(1+d)} + \dfrac{c_2}{(1+d)^2} + \dfrac{c_3}{(1+d)^3} + ...\\ & = & \dfrac{c_1}{1+g} \sum\limits_{i=1}^{\infty} \left(\dfrac{1+g}{1+d}\right)^i \\ & = & \dfrac{c_1}{1+g} \dfrac{\left(\dfrac{1+g}{1+d}\right)}{1-\left(\dfrac{1+g}{1+d}\right)}\\ & = & \dfrac{c_1}{1+g} \dfrac{1+g}{d-g}\\ & = & \dfrac{c_1}{d-g} \end{array}$

This is sometimes called the dividend discount or Gordon Growth model. To sum the infinite geometric series, ${g < d}$ has to be true.

Otherwise the series diverges and blows up in my face!

Suppose ${c_1}$ = $1, and the discount rate is 10%. Let us estimate the PV of the cash flow for different growth rates. For a perpetuity (or immortal stock) with these stable characteristics -a cash stream that starts paying$1 from next year and grows indefinitely at 5% – I should willing to pay $20. In some ways, the discount rate is my opportunity cost. I demand a d = 10% return! If I can buy this security at$20 (a forward P/E ratio of 20), then I will earn my required rate of return (=discount rate).

What happens if I demand a higher rate of return?

Say I demand d = 20% like Warren Buffett! Well, then I have find some sucker who will sell me this stuff at $6.67. And hope I am not one of these suckers. Let me go back to d = 10%, and see what I should be willing to pay for this perpetuity if it grows a tad below the discount rate – say 9.99%. Holy guacamole! With these growth characteristics (which are nearly impossible to get in real life – perhaps the subject of a later post), I should be willing to pay$10,000 for this stuff!

That is a forward PE of 10,000. And I heard somebody crying that Facebook was expensive at a 100 P/E?