What it means
An annuity, in finance, is simply a stream of equal payments made at regular intervals. These could be monthly savings, yearly deposits or quarterly contributions into a fund.
The future value looks forward to the end date and adds up everything the payments have grown into. Each payment earns interest for a different length of time.
The first deposit has the longest to grow, and the last deposit, in the most common version, earns nothing because it arrives at the very end. The formula combines all those growth periods in one calculation.
There are two main versions. An ordinary annuity pays at the end of each period, while an annuity due pays at the start of each period, which gives every payment one extra period of interest.
The future value of an annuity due is the ordinary result multiplied by one plus the interest rate for the period. Finance teams use the idea to plan for known future needs.
A company may set aside equal amounts each year to replace equipment, repay a bond at maturity or fund a pension promise, and the calculation shows whether the planned payments will reach the target. Individuals use it for goals such as a house deposit or education fund.
The nuance is that the result depends heavily on the assumed rate of return and on matching the rate to the payment period. A yearly rate has to be converted if payments are monthly, and the rate is an assumption rather than a promise, so it is wise to test several scenarios.
In practice
Real-world examples.
Example
A manufacturer plans to replace a machine in 6 years and sets aside $20,000 at the end of each year in a fund earning 5%. The finance team uses the formula to see whether the fund will reach the expected price. They find it comfortably does, so the plan is approved.
Example
A young professional saves $300 a month in a retirement account assuming a steady monthly return. She uses the future value formula to see how her balance could look after 30 years. The result shows her how much of the final amount comes from growth rather than her own deposits.
Example
A school board sets up a sinking fund to repay a $1,000,000 bond at maturity. It contributes equal annual amounts and uses the future value of an annuity to work out the size of the payment needed. The board reviews the assumed return every year.
Formula
Calculation
Future value of an ordinary annuity = P x [((1 + r)^n - 1) / r]
where P is the payment per period, r is the interest rate per period and n is the number of periods.
Suppose a company deposits $5,000 at the end of each year for 5 years into an account earning 4% a year. First, (1.04)^5 = 1.2166529. Then (1.2166529 - 1) / 0.04 = 0.2166529 / 0.04 = 5.4163225. Future value = 5,000 x 5.4163225 = $27,081.61. The company paid in 5 x 5,000 = $25,000, so interest earned is $2,081.61. If the deposits were made at the start of each year instead, the future value would be 27,081.61 x 1.04 = $28,164.88.Case study
Seen in the real world.
Brightwater Dairy is an illustrative, fictional company that needed to replace its bottling line in 8 years, with an expected cost of $800,000. The finance manager wanted to know how much to save each year without borrowing.
She assumed a 5% annual return and used the future value of an annuity formula in reverse, dividing the target by the annuity factor. With a factor of about 9.549 for 8 years at 5%, the yearly deposit came to roughly $83,800.
In this illustrative case, the board approved the plan but asked for a second scenario at 3%, which raised the yearly deposit to about $90,000. Seeing both figures helped the directors understand how much the plan depended on the assumed return.
Watch out
Common mistakes.
- Mixing up the rate and the period, for example using an annual rate with monthly payments without converting it.
- Using the ordinary annuity formula for payments made at the start of each period, which understates the result.
- Treating the assumed return as certain, when real returns vary from year to year.
Questions
People also ask.
What is the difference between future value of an annuity and future value of a lump sum?
A lump sum is a single payment growing over time, while an annuity is a series of equal payments each growing for a different length of time.
How do I handle monthly payments?
Divide the annual rate by 12 and use the total number of months as n, so that the rate and the period match.
Can the formula be used to find the payment needed?
Yes, rearrange it so that the payment equals the target divided by the annuity factor, which is the part in square brackets.
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