Annuity math is built on the time value of money — the principle that a dollar today is worth more than a dollar in the future. There are two primary scenarios: calculating how much a sum grows (future value), and calculating how much regular income a sum can generate (payout/present value).
The Four Core Annuity Formulas
FV = PV × (1 + r)^n
PV = present value, r = interest rate per period, n = number of periods
→ Use full payout calculator to see monthly income from this amount.
FV = PV×(1+r)^n + PMT × [((1+r)^n − 1) / r] × (1 + r×type)
type = 0 for ordinary annuity (end of period), type = 1 for annuity due (beginning)
PMT = PV × [r(1 + r)^n] / [(1 + r)^n − 1]
This is the "present value of an annuity" solved for PMT. Reduces balance to exactly zero at term end.
PV = PMT × [(1 − (1 + r)^−n) / r]
Used to value a stream of future payments in today's dollars — the inverse of the payout formula.
→ Compare immediate annuity pricing in the immediate annuity calculator
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Worked Examples
Example 1: Future Value of a Lump Sum
$100,000 at 5% annually for 20 years:
FV = $100,000 × (1.05)²⁰ = $100,000 × 2.6533 = $265,330
Interest earned: $165,330 — more than the original principal.
Example 2: Monthly Income from a Balance
$200,000 at 5% annually over 20 years (monthly payments):
r = 5%/12 = 0.4167%, n = 240 months
PMT = $200,000 × [0.004167 × (1.004167)²⁴⁰] / [(1.004167)²⁴⁰ − 1] = $1,320/month
Example 3: Future Value with Monthly Contributions
$50,000 lump sum + $500/month for 25 years at 5.5%:
r = 5.5%/12 = 0.4583%, n = 300 months
FV ≈ $50,000 × (1.004583)³⁰⁰ + $500 × [((1.004583)³⁰⁰ − 1)/0.004583] ≈ ~$410,000
Excel Annuity Functions
Excel includes built-in time-value-of-money functions that implement the same formulas:
| Function | Formula | Example |
|---|---|---|
| FV | FV(rate, nper, pmt, pv) | =FV(5%/12, 240, -500, -50000) |
| PV | PV(rate, nper, pmt) | =PV(5%/12, 240, 1320) |
| PMT | PMT(rate, nper, pv) | =PMT(5%/12, 240, -200000) |
| NPER | NPER(rate, pmt, pv) | =NPER(5%/12, -1320, 200000) |
| RATE | RATE(nper, pmt, pv) | =RATE(240, -1320, 200000)*12 |
Note: In Excel PMT functions, outflows (contributions, present value) are entered as negative numbers.
Ordinary Annuity vs. Annuity Due
An ordinary annuity makes payments at the end of each period. Most mortgages, retirement accounts, and investment products work this way. An annuity due makes payments at the start — rent and insurance premiums are classic examples.
The annuity due is worth slightly more because each payment gets one additional compounding period. On a $1,000/month contribution over 30 years at 6%, an annuity due yields approximately $47,000 more in future value than an ordinary annuity. Our main annuity calculator lets you toggle between both types.
The Most Common Calculation Mistakes
Mistake 1: Mismatching rate period with payment period. If payments are monthly, the rate must be the monthly rate (annual ÷ 12) and n must be in months (years × 12). Using annual rates with monthly periods overstates growth dramatically.
Mistake 2: Confusing accumulation rate with payout rate. When buying an insurance annuity, the rate quoted for growth (e.g., 5.5%) differs from the payout rate (e.g., 5.0%) used to calculate monthly income. Always ask your insurer for both numbers separately. See our annuity payout calculator for the distribution-phase formula.
Mistake 3: Ignoring compounding frequency. Monthly compounding yields more than annual compounding at the same stated rate. A 5% rate compounded monthly has an effective annual yield (APY) of 5.116%. Always compare APY, not nominal rate, across products.
Frequently Asked Questions
What is the formula to calculate annuity future value?
What is the difference between an ordinary annuity and annuity due?
How do I calculate monthly annuity payments?
Can I calculate an annuity in Excel?
What is the effective annual rate (APY) of an annuity?
Common Calculation Scenarios
See real worked numbers for the most common annuity planning questions — each page applies these formulas to a specific amount and age with full rate comparison tables.