Annuities

An annuity is a series of equal payments made at regular intervals — rent, a loan instalment, a pension, a fixed subscription. Because every payment is the same, a single formula replaces discounting each one separately.

Present value of an annuity

For a payment PMT each period, rate i, over n periods:

PV = PMT * (1 - (1 + i)^-n) / i

You will receive 1,000 units at the end of each year for 5 years, discounted at 8%:

PV = 1000 * (1 - (1.08)^-5) / 0.08
PV = 1000 * (1 - 0.68058) / 0.08
PV = 1000 * 3.99271 = 3,992.71

So a 5-year stream of 1,000 per year is worth about 3,993 units today — not 5,000, because of discounting.

Future value of an annuity

If instead you save PMT each period and let it grow:

FV = PMT * ((1 + i)^n - 1) / i

Saving R$ 500 per month at 0.6% per month for 36 months:

FV = 500 * ((1.006)^36 - 1) / 0.006
FV = 500 * (1.24036 - 1) / 0.006
FV = 500 * 40.0594 = 20,029.69

Three years of R$ 500 deposits (R$ 18,000 paid in) grow to about R$ 20,030 — the extra R$ 2,030 is compound interest.

Why annuities matter

Loan payments, retirement planning and bond coupons are all annuities. Recognising the shape lets you price them in one step instead of summing dozens of discounted flows.