Key point
An annuity is a stream of equal payments at equal intervals over a fixed horizon. "Equal" is the load-bearing word — if the payments grow over time you have a growing annuity (a more advanced topic) and if they continue forever you have a perpetuity (next module). The defining annuity examples in your life: a salary that pays the same nominal amount every period (until your next raise), Social Security that pays a fixed monthly check, a 30-year fixed-rate mortgage where you pay the same dollar amount every month for 360 months. Each of these is a textbook annuity.
Key point
Two flavors based on when payments hit. (1) Ordinary annuity: payments fall at the END of each period. Most loans, bonds, and salaries work this way — you receive your paycheck at the end of the pay period; your mortgage is due at the end of the month. (2) Annuity due: payments fall at the BEGINNING of each period. Most rent, subscriptions, and insurance premiums work this way — you pay the landlord on day 1, not day 30. The dollar difference between the two for the same nominal flows is exactly one period of interest: an annuity due is worth (1+r) more than an ordinary annuity, because every payment earns one extra period of compounding.
Formula
PV-Annuity = PMT × [(1 − (1+r)⁻ⁿ) / r]
Key point
The PV-annuity formula is just a collapsed geometric sum: PV = PMT/(1+r) + PMT/(1+r)² + PMT/(1+r)³ + ... + PMT/(1+r)ⁿ. Each term is the PV of one period's payment (using the discounting formula from pfvi-5b). Sum them up; the closed-form bracket is the algebra that compresses the sum. The bracket [(1 − (1+r)⁻ⁿ) / r] is called the "annuity factor" — once you know it for a given (r, n), every annuity at that rate and horizon is just PMT × factor. A 30-year mortgage at 6%/12 monthly = factor of 166.79. So a $300,000 mortgage requires PMT = $300,000 / 166.79 ≈ $1,798/month. That is the entire mortgage formula — re-3's payment formula, derived.
Formula
FV-Annuity = PMT × [((1+r)ⁿ − 1) / r]
Compare
| Scenario | Inputs | Annuity factor | Answer |
|---|---|---|---|
| Mortgage payment | $300K loan, 6% APR, 30 years (360 mo), r = 0.5%/mo | 166.79 | PMT = $300K ÷ 166.79 ≈ $1,799/month |
| Auto loan payment | $35K loan, 7% APR, 5 years (60 mo), r ≈ 0.583%/mo | 50.50 | PMT = $35K ÷ 50.50 ≈ $693/month |
| Pension PV | $40K/yr for 20 yrs at 5% (income replacement) | 12.46 | PV = $40K × 12.46 ≈ $498K (the lump sum equivalent) |
| Retirement at 25 | Save monthly to hit $1M at 65, 7% APR, 480 mo | FV factor 2,624 | PMT = $1M ÷ 2,624 ≈ $381/month |
| Retirement at 45 | Same goal, 20 years (240 mo), same return | FV factor 521 | PMT = $1M ÷ 521 ≈ $1,920/month — 5x the early-starter payment |
Key point
Loan amortization is the schedule that shows how each annuity payment splits between interest (on the remaining balance) and principal (paying down the loan). Early payments are mostly interest; late payments are mostly principal. On a 30-year mortgage at 6%, your first $1,799 monthly payment splits as $1,500 interest + $299 principal. By month 360, the same $1,799 splits as $9 interest + $1,790 principal. The total payment is constant; the mix flips. Most personal-finance pain ("why is so little going to principal?") becomes obvious once you see this is an arithmetic property of any amortizing annuity, not a bank trick.
Try it
Check-in
Check-in
Sit with the ideas.
You want $1,000,000 at age 65. You start contributing today at age 25 and earn 7% annually on every dollar you save. Roughly how much do you need to deposit at the end of every month to hit the goal?