What it means
The word "annuity" simply means a stream of equal payments over a set length of time. What makes it "ordinary" is the timing: the first payment lands one full period from now, not today.
This is the standard assumption in most finance formulas, which is why it is called ordinary. The reason timing matters is the time value of money, the idea that a dollar today is worth more than a dollar next year because today's dollar can be invested.
Payments at the end of a period spend an extra period unpaid, so their present value is lower than if they were paid at the start. The difference is real money when the amounts are large.
Businesses use the ordinary annuity formula to price and compare things that pay a steady amount. Typical uses include working out the value of a lease with fixed year-end payments, the price of a bond's coupon stream, or what a monthly loan repayment should be.
If you are lending, you are effectively buying an annuity from the borrower. The close cousin is the annuity due, where the payments come at the start of each period.
Rent is a good example because it is paid in advance, while a mortgage payment is typically in arrears and so closer to an ordinary annuity. Using the wrong version produces an answer that is off by one period of interest.
A practical shortcut is the annuity factor, which is simply the bracketed part of the formula worked out once for a given rate and number of periods. Tables and spreadsheet functions such as PV and FV store these factors, so most people never calculate them by hand.
What matters is knowing which timing assumption the spreadsheet is using, because the default is the end of the period. The same machinery explains why a longer term lowers the monthly repayment on a loan but raises the total interest paid.
Stretching the same amount over more periods spreads the payments out, so each one is smaller, yet more periods of interest are charged on the balance still outstanding.
In practice
Real-world examples.
Example
A logistics company signs a vehicle lease requiring $10,000 at the end of each of the next four years. The finance team discounts the payments at 8% and records a lease value of about $33,121. This is the amount a lender would advance against the contract today.
Example
A saver puts $5,000 into an investment account at the end of every year for 10 years. Because each deposit is made at year end, the first one earns no interest in its first year. The saver uses the future value formula to project the balance at retirement.
Example
A software firm buys a customer contract that pays $2,000 at the end of every month for three years. The buyer discounts those 36 payments at a monthly rate to decide the most it should pay for the contract.
Formula
Calculation
Present value of an ordinary annuity = payment x (1 - (1 + r)^-n) / r
Future value of an ordinary annuity = payment x ((1 + r)^n - 1) / r
Here r is the interest rate per period and n is the number of periods. An investment pays $10,000 at the end of each year for 4 years and the discount rate is 8%. First, 1.08^4 = 1.3605, so 1 / 1.3605 = 0.7350. The present value is 10,000 x (1 - 0.7350) / 0.08 = 10,000 x 3.3121 = $33,121. The future value of the same payments at the end of year 4 is 10,000 x (1.3605 - 1) / 0.08 = 10,000 x 4.5061 = $45,061.Case study
Seen in the real world.
Kestrel Print Works is a fictional printing business, and this is an illustrative story. The owner was offered a new press for $120,000 cash or $32,000 at the end of each year for four years.
The financial controller treated the instalment option as an ordinary annuity and discounted it at the company's borrowing rate of 8%. She found the present value of the four payments was 32,000 x 3.3121 = $105,987.
That was lower than the $120,000 cash price, so the instalment plan was cheaper in present value terms, as long as the business could earn or borrow at 8%. The illustrative lesson is that comparing the total of the payments ($128,000) with the cash price ($120,000) would have pointed to the wrong decision.
Watch out
Common mistakes.
- Using the ordinary annuity formula for payments made at the start of each period, which understates their present value.
- Mixing an annual interest rate with monthly payments, when the rate and the number of periods must use the same time unit.
- Comparing the simple total of payments with a cash price, ignoring the time value of money.
Questions
People also ask.
What is the difference between an ordinary annuity and an annuity due?
In an ordinary annuity payments fall at the end of each period, and in an annuity due they fall at the start, which makes the annuity due worth more.
Is a mortgage an ordinary annuity?
Usually yes, because repayments are made at the end of each month, although the exact terms depend on the loan contract.
Can an ordinary annuity have a different payment each period?
No, an annuity by definition has equal payments, and uneven payments are handled as separate cash flows.
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