Why Growth Raises the Present Value of a Growing Perpetuity

Summary

The document addresses the intuition behind the growing perpetuity formula, which values a stream of payments that increases over time. It contrasts this with the no-growth perpetuity formula and asks why growth appears subtracted from the discount rate in the denominator. The answer clarifies that, under the formula shown, subtracting growth increases present value relative to a constant payment stream.

The explanation is qualitative and brief: the no-growth case assumes payments stay fixed, while the growing case accounts for larger future payments. It does not derive the formula or explore its assumptions, such as the relationship between the discount rate and growth rate, nor does it analyze risk, changing rates, or practical valuation cases. The note is useful as a basic interpretation of the sign of growth in the perpetuity valuation expression.

Key ideas

• A no-growth perpetuity assumes its payment remains constant through time.
• A growing perpetuity reflects payments that increase over time.
• In the stated formula, subtracting growth from the discount rate raises present value.
• The answer offers intuition but does not derive the formula or explain its conditions of validity.

Tags

# Exposition of Growth in a Perpetuity


# 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?

## Answer by SRKX (score 2, accepted)

https://quant.stackexchange.com/a/4501

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.