Present Value of a Growing Annuity
A growing annuity is a series of payments that rises at a constant rate each period but stops after a fixed number of periods, like a salary with annual raises. Its formula is a growing perpetuity with the payments after the end date subtracted.
Growing annuity explained simply
A growing annuity is a series of payments that grows at a constant rate but stops after a set number of periods: PV = C₁ ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))ⁿ]. In the example, $1,000 growing 3% a year for 5 years at 8% is worth about $4,220.35.
What this growing annuity lesson covers
- What a growing annuity is and where it shows up
- The present value of a growing annuity formula
- Why it equals a growing perpetuity minus a delayed one
- How to check your answer by discounting each payment
Growing annuity formula, explained
- C₁
- first payment, at the end of period 1
- r
- discount rate per period
- g
- growth rate of payments per period (r ≠ g)
- n
- number of payments
How to value a growing annuity step by step (worked example)
You will receive $1,000 at the end of next year, growing 3% a year, for 5 years in total. The discount rate is 8%. What is it worth today?
- Inputs: C₁ = $1,000, r = 0.08, g = 0.03, n = 5.
- Growing perpetuity part: $1,000 ÷ (0.08 − 0.03) = $20,000.
- Shrink factor: 1 − (1.03 ÷ 1.08)⁵ ≈ 1 − 0.788982 = 0.211018.
- PV ≈ $20,000 × 0.211018 ≈ $4,220.35.
Answer: About $4,220.35 today (rounded to the cent).
Annuity vs growing annuity vs growing perpetuity
| Annuity | Growing annuity | Growing perpetuity | |
|---|---|---|---|
| Payments | Equal | Grow at g | Grow at g |
| Ends? | Yes | Yes | Never |
| Formula | C × [1 − (1 + r)⁻ⁿ] ÷ r | C₁ ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))ⁿ] | C₁ ÷ (r − g) |
Harder growing annuity questions
What if the growth rate equals the discount rate?
The formula divides by zero, but the answer is simple: each payment's growth exactly cancels its discounting, so PV = n × C₁ ÷ (1 + r). With g = r = 8%: 5 × 1,000 ÷ 1.08 ≈ $4,629.63.
Can g be higher than r in a growing annuity?
Yes. Unlike a growing perpetuity, a growing annuity ends, so the value stays finite. The formula still works; both brackets turn negative and the result is positive.
Common growing annuity questions
What is the formula for the present value of a growing annuity?
PV = C₁ ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))ⁿ], where C₁ is the first payment, r is the discount rate, g is the growth rate and n is the number of payments. Payments are assumed at the end of each period.
What is the difference between a growing annuity and a growing perpetuity?
Both have payments that grow at a constant rate, but a growing annuity stops after n payments while a growing perpetuity goes on forever. The growing annuity formula is the perpetuity value times a factor that removes the payments after period n.
Can the growth rate be higher than the discount rate in a growing annuity?
Yes. Unlike a growing perpetuity, a growing annuity has a finite number of payments, so the formula still works when g is greater than r. It only breaks when g equals r; then PV = n × C₁ ÷ (1 + r).
What is a graduated annuity?
A graduated or increasing annuity is another name for a growing annuity: payments that rise by a fixed percentage each period for a set term. Salary-linked pensions and rent with annual increases are common examples.