# Exposition of Growth in a Perpetuity

Something that's bugged me since I've ever learned anything about finance: Philosophically(?) speaking, why does growth subtract from a perpetuity's return?

I know the mathematical explanation, but it's so counter-intuitive.

Why should a perpetuity with 0% expected growth be valued at the same proportion as one with high growth when holding the natural (risk free) rate constant?

Maybe if one uses the stock market as a transposition, you see that P/Es are higher with higher expected growth companies, reducing the net rate, explaining the $r - g$ in the denominator, but what about a natural rate of $X$ and a growth rate $g > X$? So, we're supposed to get paid for the privilege of holding this asset now?

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Take the dividend discount model for example. In your example the dividend growth rate $g$ would be greater than the investors required return $r$. So the required return should definitely be larger than the dividend growth rate. Finally, you should end up with the price of the stock after all. :-) – vanguard2k Nov 9 '12 at 9:26

## 1 Answer

I think the formula you refer to is

$$PV=\frac{C}{r-g}$$

If that's the case, then you do not subtract growth, the minus sign has an advantage on the present value.

The initial formula $PV=\frac{C}{r}$ assumes no evolution in $C$, but the other one assumes the that the payment will grow in time hence yes, you get paid for that.

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