Present Value of a Growing Perpetuity
A growing perpetuity is a stream of payments that never ends and grows at a constant rate each period. Its present value is the first payment divided by the gap between the discount rate and the growth rate. It is the engine behind the dividend growth model and terminal values.
Growing perpetuity explained simply
A growing perpetuity pays forever and grows at a constant rate. Its value is the first payment divided by the gap between the discount rate and the growth rate: PV = C₁ ÷ (r − g). In the example, $600 growing 3% a year at an 8% rate is worth $12,000.
What this growing perpetuity lesson covers
- What a growing perpetuity is and how it differs from a plain perpetuity
- The formula PV = C₁ ÷ (r − g) and why it uses next period's payment
- Why the growth rate must be below the discount rate
- How it connects to the Gordon growth model and terminal value
Growing perpetuity formula, explained
- C₁
- payment at the end of period 1
- r
- discount rate per period
- g
- constant growth rate of payments (must be less than r)
How to value a growing perpetuity step by step (worked example)
An investment pays $600 at the end of next year, and the payment grows 3% every year forever. The discount rate is 8%. What is it worth today?
- Inputs: C₁ = $600, r = 0.08, g = 0.03.
- Find the gap: r − g = 0.08 − 0.03 = 0.05.
- Divide: PV = $600 ÷ 0.05 = $12,000.
- Without growth it would be $600 ÷ 0.08 = $7,500, so growth adds a lot of value.
Answer: $12,000 today.
Perpetuity vs growing perpetuity
| Perpetuity | Growing perpetuity | |
|---|---|---|
| Payments | Equal forever | Grow at rate g forever |
| Formula | C ÷ r | C₁ ÷ (r − g) |
| $600 at 8% (g = 3%) | $7,500 | $12,000 |
| Where you meet it | Preferred shares | Gordon growth model, terminal values |
What if the growth is negative? David's take
Textbooks almost always show g as positive. Try a negative one. My working hypothesis for AI-era products: start expensive, as long as you're cheaper than any competitor with something similar, then drop the price rapidly as copycats arrive, because AI makes reverse engineering easy. If those $600 payments shrank 10% a year instead of growing 3%, the stream would be worth $600 ÷ (0.08 + 0.10) ≈ $3,333 instead of $12,000. The formula handles it. The question is whether you're honest about which way g is going.
From David's concept: Nokia Arc Pricing Curve. This is David's opinion, alongside the standard textbook answer above.
Harder growing perpetuity questions
What if the discount rate is 7% instead of 8%?
$600 ÷ (0.07 − 0.03) = $15,000. A one-point change in r moves the value by 25%, because the r − g gap is small.
What if the first payment only arrives at the end of Year 3?
Value the perpetuity one period before the first payment, at Year 2: $600 ÷ 0.05 = $12,000. Then discount 2 years: $12,000 ÷ 1.08² ≈ $10,288.07.
Common growing perpetuity questions
What is the growing perpetuity formula?
PV = C₁ ÷ (r − g), where C₁ is the payment one period from now, r is the discount rate and g is the constant growth rate. It only works when r is greater than g.
Why must the growth rate be lower than the discount rate?
If g equals or exceeds r, each future payment is worth as much or more today than the one before, so the sum never settles and the value is infinite. Mathematically, r − g would be zero or negative.
Is the growing perpetuity the same as the Gordon growth model?
Yes, the Gordon growth (dividend discount) model is a growing perpetuity applied to dividends: share price = next year's dividend ÷ (cost of equity − dividend growth rate). It is also used for terminal value in DCF.
Should I use this year's payment or next year's in the formula?
Next year's. The formula needs C₁, the first payment to be received. If you are given the payment just made, C₀, multiply it by (1 + g) first.