Annuity Formula
Reference for PV = PMT × (1-(1+r)^-n)/r and FV = PMT × ((1+r)^n-1)/r.
Covers ordinary and due annuities for loans, pensions, and retirement planning.
The Formulas
Present Value of Annuity: PV = PMT × [1 - (1+r)⁻ⁿ] / r
An annuity is a series of equal payments made at regular intervals. These formulas let you calculate either how much those payments are worth today (PV) or how much they will grow to in the future (FV).
Variables
| Symbol | Meaning |
|---|---|
| FV | Future value (total accumulated) |
| PV | Present value (lump sum equivalent today) |
| PMT | Payment amount per period |
| r | Interest rate per period |
| n | Total number of payment periods |
Example 1
You invest $500/month for 20 years at 7% annual return. What is the future value?
r = 0.07/12 = 0.005833, n = 20 × 12 = 240
FV = 500 × [(1.005833)²⁴⁰ - 1] / 0.005833
= 500 × [4.0387 - 1] / 0.005833
= $260,464 (you invested $120,000 — earned $140,464 in interest)
Example 2
What is a pension of $2,000/month for 25 years worth today at 5% discount rate?
r = 0.05/12 = 0.004167, n = 25 × 12 = 300
PV = 2000 × [1 - (1.004167)⁻³⁰⁰] / 0.004167
= $342,087 (the lump-sum equivalent today)
When to Use It
Use the annuity formula when:
- Planning retirement savings (how much will monthly contributions grow to?)
- Valuing a pension or structured settlement
- Calculating loan payments (PV annuity)
- Comparing lump-sum vs. periodic payment options
Key Notes
- Present value: PV = PMT × [1 − (1+r)^(−n)] / r: Discounts equal periodic payments back to today. Used to find the loan amount for a given payment, or to value a stream of future income. The bracketed factor is called the present value annuity factor (PVAF).
- Future value: FV = PMT × [(1+r)^n − 1] / r: Compounds equal periodic payments forward in time. Used for retirement savings projections — "If I contribute $500/month at 7%, what will I have in 30 years?"
- Ordinary annuity vs annuity-due: Ordinary annuity payments occur at the end of each period (most loans and savings products). Annuity-due payments occur at the beginning — multiply the ordinary annuity PV or FV by (1 + r) to convert.
- Perpetuity (infinite annuity): PV = PMT / r: An infinite stream of equal payments has a finite present value. Example: A $1,000/year perpetuity at 5% is worth PV = $20,000 today. Used to value preferred stocks and perpetual bonds (consols).
- Applications: Annuity formulas calculate mortgage payments, car loan payments, lease payments, pension valuations, lottery lump-sum vs annuity comparisons, and bond coupon present values.