Annuities
What this domain tests
An annuity is a series of equal payments made at regular intervals. The exam tests the future value of an ordinary annuity (sinking funds, retirement savings) and the present value (loan payouts, lease valuation, installment payoffs).
Core rules
- Ordinary annuity: payments at the end of each period. Annuity due: payments at the beginning of each period.
- Future value: FV = PMT × [((1 + i)^n − 1) ÷ i].
- Present value: PV = PMT × [(1 − (1 + i)^(−n)) ÷ i].
- i is the rate per period and n is the total number of payments - convert them exactly as in compound interest.
- An annuity due equals the ordinary annuity value multiplied by (1 + i), because every payment earns one extra period of interest.
- A sinking fund is a future-value problem: what periodic deposit reaches a stated target.
- Total interest earned = future value − (payment × number of payments).
Worked example
$500 is deposited at the end of each quarter for 3 years at 8% compounded quarterly. Find the accumulated value. i = 0.08 ÷ 4 = 0.02; n = 3 × 4 = 12 payments. (1.02)^12 = 1.268242; minus 1 = 0.268242; ÷ 0.02 = 13.412089. FV = 500 × 13.412089 = $6,706.04. Total deposited = 500 × 12 = $6,000, so interest earned is about $706.04.
Traps that cost marks
- Counting years instead of payments for n.
- Using the annuity-due formula when deposits are made at period end, which overstates the fund.
- Confusing PV with FV: money received later is discounted, not accumulated.
- Dividing the annual rate by 12 when payments are quarterly.
- Assuming the last payment earns interest - under an ordinary annuity it earns none.
Where it appears
Retirement contributions, lease payment schedules, mortgage payment amounts, and bond interest streams are all annuities, so identifying the payment timing is the first step in every one of these problems.