Annuity present vs future value
By Jude Wallis
Present value discounts a stream back to today; future value compounds it forward to the end. Ten annual payments of $1,000 at 6 percent are worth $7,360.09 now and $13,180.79 on the last payment date. Neither figure is the plain sum of the ten $1,000 payments.
| Present value of an annuity | Future value of an annuity | |
|---|---|---|
| Valuation date | Today, before any payment has been made. | The date of the final payment. |
| Direction of the arithmetic | Divide by one plus the rate, once per period of waiting. | Multiply by one plus the rate, once per period of growth. |
| On this annuity | $7,360.09. | $13,180.79. |
| Cash actually paid | Ten payments of $1,000, adding to more than the present value. | The same ten payments of $1,000, adding to less than the future value. |
| The question it answers | What would I pay today for this income. | What will I have if I keep paying this in. |
| Where it is used | Pricing a pension, a lease, or a settlement offer. | Sizing a savings plan or a sinking fund. |
On this page
One rate, two directions
The two figures are the same annuity seen from opposite ends of the same ten years. Discounting each of the ten $1,000 payments back to today gives $7,360.09. Compounding each of them forward to the final payment date gives $13,180.79. Grow the present value at 6 percent for ten years and you land on the future value; the two are never independent facts.
Both sit either side of the plain sum of ten $1,000 payments. The present value is lower because money arriving later is worth less now. The future value is higher because early payments have had years to earn. The rate is doing all of the work in both directions.
Which one the question wants
If someone offers to buy your income stream, the present value is the price to argue about. If you are trying to reach a target, the future value is the one to check, and the sinking fund calculator runs it backwards: the deposit that reaches $13,180.79 in ten years at 6 percent is exactly the $1,000 annual payment, which is what the second worked example below verifies.
Getting this the wrong way round is expensive rather than academic. Valuing a ten-year stream at $13,180.79 when the question was what to pay today overstates it by nearly 80 percent. Present value is the anchor concept, and how annuity future value works covers the forward direction.
The assumptions inside both
Each figure assumes payments arrive on schedule, at the end of each period, and that one rate applies throughout. Move the payments to the start of the period and both numbers rise by a factor of one plus the rate, which is the whole difference an ordinary annuity against an annuity due turns on. The annuity present value calculator and the future value annuity calculator take the timing as an input rather than a footnote. This is educational material, not financial advice.
Worked examples
\$1,000 a year for ten years, valued today
An annuity pays $1,000 at the end of each year for ten years, and the discount rate is 6 percent. What is it worth now?
- Discount each payment by 1.06 raised to the number of years until it arrives.
- Add the ten discounted payments: $7,360.09.
The present value is $7,360.09, well under the sum of the ten $1,000 payments the annuity makes.
The deposit that reaches \$13,180.79
Run the same annuity forward instead. What annual deposit at 6 percent reaches $13,180.79 after ten years?
- Each deposit compounds at 6 percent for the years remaining after it is made.
- Solve for the deposit that makes the ten compounded amounts total $13,180.79, and it is $1,000.
$1,000 a year, which confirms that $13,180.79 is the future value of the same annuity whose present value is $7,360.09.
Common questions
Why is the present value less than the payments add up to?
Because $1,000 in ten years is worth less than $1,000 today. Discounting prices that wait.
Do the two use the same rate?
In this comparison yes, which is what makes them two views of one annuity. A different rate makes them different problems.
What if payments arrive at the start of each period?
Both values rise by one plus the period rate. That variant is called an annuity due.
Is this financial advice?
No. It is educational material about valuing a payment stream from either end.
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.