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.
What you will learn
- 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
The formula
- C₁
- payment at the end of period 1
- r
- discount rate per period
- g
- constant growth rate of payments (must be less than r)
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.
Common 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.