Annuity present value explained
By Jude Wallis
The present value of an annuity is what a run of future payments is worth today. Ten annual payments of $1,000 discounted at 6 percent come to $7,360.09, because the payment arriving in year ten is worth far less now than the one arriving next year.
Present value
$7,360.09
Ordinary annuity: payments at the end of each period.
- Present value
- $7,360.09
- Payment
- $1,000.00
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In short
- Ten $1,000 payments at 6 percent are worth $7,360.09 today, well under the sum of the payments.
- Each payment is divided by one plus the period rate, raised to the number of periods until it arrives.
- Frequency matters: $500 a month for five years at 6 percent is worth $25,862.78.
- A higher discount rate lowers the value, because waiting is being priced more heavily.
- This assumes payments at the end of each period. Payments at the start are worth more.
Discounting is dividing, once for every period of waiting
A payment due next year is divided by one plus the rate. One due in two years is divided twice. By year ten at 6 percent, a $1,000 payment is worth a little over half its face amount today, and the ten discounted amounts add to $7,360.09.
That is the whole calculation. The formula usually written as an annuity factor is a shortcut for adding ten divisions, and the shortcut is worth knowing only after the divisions are.
The rate is a judgement, not a fact
Payment size and timing are usually observable. The discount rate is not: it is the return you could get elsewhere on money of similar risk, and it carries every opinion in the valuation. Raise it and the present value falls; lower it and the same payments look worth more.
So when two people disagree about what a pension buyout, a settlement offer or a lease is worth, they are almost always disagreeing about the rate rather than the arithmetic. Present value covers the single-payment case that this one is built from.
Frequency changes the answer
Monthly payments are not just an annual payment split twelve ways. Each arrives sooner than the annual equivalent would, so less discounting is applied, and the period rate is the annual rate divided by the number of periods. $500 a month for five years at 6 percent is worth $25,862.78 today.
Getting the period wrong is the most common error in this calculation. Using an annual rate with monthly payments overstates the discounting badly, and the answer will look plausible while being wrong by a wide margin.
Where it is used
Anywhere a stream is exchanged for a lump sum: pension buyouts, structured settlements, lease liabilities, valuing the fixed leg of anything. Compare it with the future value of the same stream in annuity present against future value, and with the endless case in how a perpetuity is priced. The annuity present value calculator takes payment, rate, frequency and term as four separate inputs for exactly the reasons above. This is educational material, not financial advice.
Worked examples
Ten payments of \$1,000 at 6 percent
An annuity pays $1,000 at the end of each year for ten years. At a 6 percent discount rate, what is it worth today?
- Divide each payment by 1.06 raised to the number of years until it arrives.
- Add the ten discounted values, which come to $7,360.09.
$7,360.09 today for ten future payments of $1,000.
\$500 a month for five years
A monthly annuity pays $500 for five years, discounted at 6 percent a year.
- The period rate is 6 percent divided by 12, and there are 60 payments.
- Discount each of the 60 payments and add them: $25,862.78.
$25,862.78. Sixty payments of $500 are worth well under their face total, but far more than an equivalent annual stream would be.
Common questions
Why is the present value below the sum of the payments?
Because money arriving later is worth less today. Discounting prices the wait, payment by payment.
What if payments come at the start of each period?
Multiply the result by one plus the period rate. That variant is an annuity due.
Which rate should I use?
The return available on money of similar risk. It is the input that carries the judgement in the whole calculation.
Is this financial advice?
No. It is educational material about valuing a stream of payments.
Keep reading
This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.