What is the present value of a growing annuity formula?

PV = P1 ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))n], with P1 the first payment, r the discount rate, g the growth rate and n the number of payments.

What if the growth rate equals the discount rate?

The formula divides by zero, so use PV = n × P1 ÷ (1 + r). The calculator does this automatically.

Can g be larger than r?

Yes. Unlike a growing perpetuity, a growing annuity has a finite number of payments, so its value is finite for any g.

When is the first payment?

One period from now. For a first payment today, multiply the result by (1 + r).

Present Value of Growing Annuity Formula

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Present Value of a Growing Annuity Calculator

A growing annuity is a fixed number of payments that increase by a constant percentage each period, such as a salary with annual raises, rent with an escalation clause or a pension indexed to inflation. Its present value is the growing perpetuity P₁ ÷ (r − g) minus the part of that perpetuity that starts after the last payment, which gives P₁ ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))ⁿ].

Formula

PV = P₁r − g × [1 − (1 + g)(1 + r)ⁿ]

Where

PV: present value of all n payments
P₁: first payment, at the end of period 1
r: interest (discount) rate per period
g: growth rate of the payment per period
n: number of payments

Calculator

First payment (P₁)

Rate per period (r) (%)

Growth rate (g) (%)

Number of payments (n)

Result and step-by-step solution

Present value of the growing annuity

12,250.04

Payments grow from 1,000.00 to 1,753.51 over 20 periods.

Convert the rates
r = 0.08, g = 0.03, n = 20
Growth-to-discount ratio
1 + g1 + r = 1.031.08 = 0.953704, raised to n: 0.953704²⁰ = 0.387498
Apply the formula
PV = 1,000.000.05 × (1 − 0.387498) = 12,250.04

How to use the formula

The ratio (1 + g) ÷ (1 + r) is the key quantity: it is how much each payment’s present value changes from one period to the next. When g equals r it is exactly 1, every payment has the same present value P₁ ÷ (1 + r), and the calculator switches to the special-case formula n × P₁ ÷ (1 + r).

In the example, 20 annual payments starting at 1,000 and growing 3% a year, discounted at 8%, are worth 12,250.04. A flat 1,000 a year for 20 years would be worth only 9,818.15, so the raises add about 25% to the value. For the balance these payments would build instead, see the future value of a growing annuity.

Frequently asked questions

What is the present value of a growing annuity formula?: PV = P₁ ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))ⁿ], with P₁ the first payment, r the discount rate, g the growth rate and n the number of payments.
What if the growth rate equals the discount rate?: The formula divides by zero, so use PV = n × P₁ ÷ (1 + r). The calculator does this automatically.
Can g be larger than r?: Yes. Unlike a growing perpetuity, a growing annuity has a finite number of payments, so its value is finite for any g.
When is the first payment?: One period from now. For a first payment today, multiply the result by (1 + r).

Last reviewed: September 26, 2026. Calculations run in your browser and were checked against numpy-financial, SciPy and published spreadsheet examples.