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.
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.
PV-Annuity = PMT × [(1 − (1+r)⁻ⁿ) / r]
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.
FV-Annuity = PMT × [((1+r)ⁿ − 1) / r]
| 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 |
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.
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?