An ordinary annuity is a level-payment cash-flow stream paid at each period end and valued using present- or future-value factors.
An ordinary annuity is a series of equal payments made at the end of equally spaced periods. The term describes a cash-flow timing pattern used in finance; it does not necessarily mean an insurance annuity product.
For example, five $1,000 payments made at the end of five consecutive years form an ordinary annuity. The same payments made at the beginning of each year form an annuity due.
If the valuation date is time 0, an ordinary annuity with (n) annual payments places its cash flows at times 1 through (n). There is no payment at time 0.
| Time | 0 | 1 | 2 | … | n |
|---|---|---|---|---|---|
| Payment | $0 | PMT | PMT | … | PMT |
This convention matters because moving every payment one period earlier changes both present value and future value.
The present value of an ordinary annuity is:
Where:
The formula discounts each future payment back to time 0 and adds the results. A higher discount rate generally produces a lower present value, all else equal.
The future value immediately after the final payment is:
The first payment compounds for (n-1) periods, while the final payment is made on the future-value date and does not receive a full period of growth.
When (r=0), both formulas reduce to the undiscounted total of (PMT \times n).
Assume a contract pays $1,000 at the end of each year for five years. Use a 5% annual discount rate.
Under the stated assumptions, $4,329.48 today is financially equivalent to the five end-of-year payments when discounted at 5%.
The stream has a total nominal payment amount of $5,000, but its future value is higher because the earlier payments compound before the end of year five. These values are mathematical results based on a constant rate, not a promise that a real account or investment will earn 5%.
| Question | Ordinary annuity | Annuity due |
|---|---|---|
| When is each payment made? | End of period | Beginning of period |
| Is there a payment at time 0? | No | Yes |
| Which has the higher value at a positive rate? | Lower | Higher |
| How are values related? | Base formula | Ordinary value multiplied by (1+r) |
| Common timing example | Loan payment after a month of use | Rent paid before the month begins |
At a positive rate, an otherwise identical annuity due has a higher value because every payment is shifted one period earlier. At a zero rate, the two streams have the same value because timing creates no interest effect.
The pattern can appear in several settings:
These examples are not automatically identical. A loan payment may include principal and interest, a bond also has a principal repayment, and a retirement or insurance contract may have fees, guarantees, contingencies, or changing payments.
An insurance annuity is a contract. “Ordinary annuity” only identifies when a level cash-flow stream is paid. A contract may use beginning-of-period payments, changing payments, life-contingent payments, or other structures.
The formula’s (r) is a periodic rate. Converting a quoted annual rate depends on whether it is nominal, effective, or subject to another contract convention.
Cost-of-living adjustments, skipped payments, mortality contingencies, variable returns, and contract options require a more detailed model.
Inflation can reduce what fixed future payments buy. A nominal annuity calculation does not automatically provide a real, inflation-adjusted value.
The U.S. Securities and Exchange Commission’s Investor.gov annuity guide defines an annuity product as a contract with an insurance company and notes that costs, risks, and features vary by contract. Those product characteristics are separate from the ordinary-annuity formula. Review actual contract documents rather than inferring guarantees, liquidity, fees, or tax treatment from the cash-flow label.
This article is educational and does not provide personalized investment, lending, tax, insurance, or retirement advice.