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

Assume that the cash payments, are made at fixed periodic intervals (say annually), starting a year from now. If the growth rate is constant, then I can write,

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 . So that a dollar years from now is only worth dollars to me now.

I am now ready to find the present value of my growing perpetuity.

This is sometimes called the dividend discount or Gordon Growth model. To sum the infinite geometric series, has to be true.

Otherwise the series diverges and blows up in my face!

Suppose = $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?

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