An annuity is a series of equal cash flows, , at equal time intervals for a fixed number of periods. Mortgages, car-loan payments, bond coupons, lease payments, and equal-instalment savings plans are all annuities.
Assumptions: payments equal in size, equally spaced in time, each one at the end of its period (the ordinary annuity convention). An annuity due puts payments at the beginning of each period instead, same idea shifted by one period.
Four standard equivalences relate an annuity to a present value at time zero or a future value at time :
- accumulates the annuity to a future lump sum.
- gives the equal deposit needed to accumulate to (sinking fund).
- is the present value of the stream.
- is the equal payment that amortises over periods (capital recovery).
The factors come from compounding; Compound interest factor has the explicit formulas.
A worked instance: you want to save up $50,000 for a down-payment in 5 years, and the savings account pays 4% effective annual interest. How much do you need to deposit at the end of each year?
So roughly $9,231/year for 5 years builds up to $50,000 at 4%.
A second instance: a $300,000 mortgage at 5% over 25 years. What’s the annual payment?
(Real Canadian mortgages compound semi-annually but are paid monthly, so the actual conversion is a bit more involved. See Nominal vs effective interest rate for period mismatches.)
Capitalized value is the present-worth of an annuity that runs forever, a perpetual annuity. Take the limit as in and you get
So $1,000/year forever at 5% is worth $20,000 today. This applies to long-lived assets where the gap between “very long” and “forever” is negligible: civic infrastructure, endowments, perpetual bonds.
For non-uniform series, see Arithmetic gradient series and Geometric gradient series.