Immediate (Ordinary) Annuity Calculator

Calculate Periodic Payments for an Ordinary Annuity

Ordinary Annuity Payment Formula

Formula

\[ PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}} \]

Where:

$PMT$ — periodic annuity payment (the amount paid at the end of each period)
$P$ — present value of the annuity (the lump sum or principal)
$r$ — interest rate per period, expressed as a decimal (e.g. 5% = 0.05)
$n$ — total number of payment periods

This formula gives the level payment that, discounted at rate $r$, exactly exhausts the present value $P$ over $n$ periods. Because each payment is assumed to occur at the end of its period, the first payment is discounted by one full period.

Ordinary Annuity – Calculation Example

Example: you have $10,000 today and want to withdraw it in equal annual payments over 10 years, with the fund earning 5% per year, compounded annually. Payments are taken at the end of each year.

\[ PMT = \frac{10{,}000 \times 0.05}{1 - (1 + 0.05)^{-10}} = \frac{500}{1 - 1.05^{-10}} = \frac{500}{0.386087} \approx 1{,}295.05 \]

Result: the annuity payment is $1,295.05 per year. Ten payments of this amount, each discounted at 5%, add back to exactly $10,000 in present value.

How to Use the Calculator

Enter the Present Value of Annuity — the lump sum you are converting into payments (or the loan amount you are repaying).
Enter the Annual Interest Rate as a percentage (e.g. 5 for 5%). If your payments are monthly, divide the annual rate by 12 and enter that, with periods counted in months.
Enter the Number of Periods — the total count of payments, in the same time unit as the interest rate.
Read the Annuity Payment — the fixed amount paid at the end of every period.

What Is an Immediate (Ordinary) Annuity?

An ordinary annuity is a series of equal payments made at regular intervals, with each payment occurring at the end of its period. It is called an immediate annuity because payments begin one period after the annuity is set up — for example, you hand over a lump sum to an insurance company today and receive your first payment at the end of the first month or year.

The end-of-period timing is what distinguishes an ordinary annuity from an annuity due, where payments fall at the beginning of each period. Because every ordinary-annuity payment is delayed by one period relative to an annuity due, its payments are worth slightly less in present-value terms, and the required payment for a given present value is correspondingly smaller.

The mathematics works in both directions. If you have a pool of money (a retirement fund, a lottery payout, a loan you are repaying), the formula tells you the equal payment that draws it down to exactly zero after $n$ periods while it earns interest at rate $r$. The same calculation underlies fixed-rate mortgages, car loans, and pension payouts, which is why the payment produced here is often identical to a lender's amortization payment.

Interpreting the result is straightforward: a higher interest rate or a longer term reduces the payment, because the balance earns more and has more time to be drawn down. A larger present value, naturally, increases the payment proportionally — doubling $P$ doubles $PMT$.

Assumptions, Limitations, and Practical Tips

Rate and period must match. The interest rate $r$ must be the rate per period, and $n$ must count the same periods. For monthly payments on a 6% annual loan, use $r = 0.06/12 = 0.005$ and $n$ in months. If you need this handled automatically for a monthly schedule, see the Monthly Annuity Payment calculator.
Payments are level. The formula assumes every payment is identical. If you expect the payment to start later or change over time, the Deferred Annuity Payment calculator covers delayed-payment cases.
The rate is fixed and compounding is discrete. Variable rates, continuous compounding (handled by the Continuous Compounding Present Value calculator), fees, taxes, and inflation are not modeled.
Rounding matters over long horizons. Real-world payments are rounded to cents, so the final balance may differ by a small residual; lenders adjust the last payment to close it out.
Zero interest is a special case. When $r = 0$, the formula degenerates (division by zero), and the payment is simply $PMT = P / n$. If you are comparing annuity payouts against other investment options, the Net Present Value calculator can help evaluate which stream is worth more today.

Frequently Asked Questions

How do I calculate an ordinary annuity payment by hand?

Use the formula $ PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}} $. First convert the annual rate to a rate per period (divide by 12 for monthly), then raise $1 + r$ to the power $-n$, subtract that from 1, and divide $P \cdot r$ by the result. For example, $50,000 at 0.5% per month over 60 months gives $ PMT = \frac{50{,}000 \times 0.005}{1 - 1.005^{-60}} \approx \$966.64 $.

What is the difference between an ordinary annuity and an annuity due?

In an ordinary annuity each payment arrives at the end of the period; in an annuity due each arrives at the beginning. Because annuity-due payments come one period earlier, they are worth more, so the same present value supports slightly smaller payments. To convert between them, multiply: $ PMT_{due} = PMT_{ordinary} \div (1 + r) $. If you receive payments at the start of each month, use the annuity-due version instead of this calculator.

How do I convert an annual interest rate for monthly payments?

Divide the annual nominal rate by 12 and express payments in months. A 6% annual rate becomes $ r = 0.06 / 12 = 0.005 $ per month, and a 30-year loan means $ n = 30 \times 12 = 360 $ periods. Do not use 6% with 360 periods — that mismatch is one of the most common calculation errors and produces wildly wrong payments.

What happens if the interest rate is 0%?

The formula divides by $r$, so it fails at exactly 0%. In that case the payment is simply $ PMT = P / n $: a $12,000 pool paid out over 24 months is $500 per month with no interest earned or charged. For very small rates close to zero, the formula still works, and the payment approaches $P/n$ as $r$ approaches zero.

Why is my payment different from what my lender or insurer quotes?

Real-world quotes include fees, insurance charges, taxes, profit margins, or rate spreads that this formula does not model. Lenders also round payments to cents and adjust the final payment, causing tiny differences. If the gap is large, check that the rate and period units match, that compounding frequency agrees, and whether the quoted product is an annuity due or includes extra charges. Treat this result as the pure mathematical payment, not a binding quote.

Is this the same payment a lender uses for a mortgage or car loan?

Yes. A standard fixed-rate amortizing loan is mathematically an ordinary annuity: the loan amount is the present value, and the level payment drawn from this formula pays it off exactly over $n$ periods. This is why the calculator's result typically matches a lender's principal-and-interest payment. However, real monthly bills may add escrow, taxes, or insurance on top of that base payment.

What counts as a good annuity payout rate?

For a lifetime immediate annuity, payouts of roughly 5–7% of the purchase price per year are common, depending on age and interest rates. The mathematical payout from this calculator can look higher for long terms because it returns principal plus interest and eventually ends, while a lifetime annuity pays as long as you live. Annuity terms vary significantly between providers, so compare quotes and consider consulting a financial advisor before committing funds.