Annuities and savings Updated November 2026

Growing annuity

A growing annuity is a finite series of payments that grow at a constant rate g each period. Its present value is PV = C1 / (r - g) x (1 - ((1 + g) / (1 + r))^n), where C1 is the first payment. It is a growing perpetuity cut off after n payments.

Syntax

PV = C1 / (r - g) x (1 - ((1 + g) / (1 + r))^n)   (if r = g: PV = n x C1 / (1 + r))

Part	What it is
`C1`	First payment, one period from now
`r`	Discount rate per period
`g`	Growth rate of the payment per period
`n`	Number of payments

Examples

First payment 100, growing 3% a year for 5 years, r = 8%
PV = 100 / (0.08 - 0.03) x (1 - (1.03 / 1.08)^5)

Result: PV = 422.04

Rent of 1,000 next year, rising 2% a year for 10 years, r = 7%
PV = 1,000 / 0.05 x (1 - (1.02 / 1.07)^10)

Result: PV = 7,606.50

Check with g = 0: 100 for 3 years at 5%
PV = 100 / 0.05 x (1 - (1 / 1.05)^3)

Result: PV = 272.32 (same as the level annuity)

Common mistakes

Using the current payment instead of the first future payment C1; if the payment just made was 100 and growth is 3%, C1 is 103.
Dividing by zero when r equals g; use PV = n x C1 / (1 + r) in that case.
Mixing an annual growth rate with a monthly discount rate; both must be per the same period.

Practice Growing annuity

These drills check your answer on live data, free, no account needed.

Growing annuityA pension pays $13,000 at the end of the first year, and each later payment is 1% larger than the one before. There are 24 payments in all. At a discount rate of 7%, what is the pension worth today?
Growing perpetuityA rental property's net cash flow next year will be $8,000, and it is expected to grow 1% a year forever. At a required return of 9%, what is the property worth today?

Questions

What is the future value of a growing annuity?

FV = C1 / (r - g) x ((1 + r)^n - (1 + g)^n). For the first example: 100 / 0.05 x (1.08^5 - 1.03^5) = 620.11, which equals the PV of 422.04 compounded 5 years at 8%.

Can I do a growing annuity in Excel?

There is no built-in function. Either type the formula directly or list the growing payments in a column and use =NPV(r, range).

PerpetuityA perpetuity pays a fixed amount forever and is worth PV = C / r. The growing perpetuity formula C1 / (r - g), examples for preferred stock and dividends.
Present value of an annuityHow to find the present value of equal payments with PV = PMT x (1 - (1 + r)^-n) / r. Worked examples, Excel PV and why this is the loan formula in disguise.
Annuity formula (future value)The future value of an ordinary annuity is PMT x ((1 + r)^n - 1) / r. Examples for yearly and monthly deposits, the Excel FV function and common errors.
NPV (net present value)Net present value is the sum of discounted cash flows minus the initial investment. Examples at 10% and 8%, the accept rule, Excel NPV and the sign mistake.