Annuities are a type of investment, but instead of contributing further deposits to our investment over time, instead we usually withdraw money from our annuity to use as income.
This means that eventually we will run out of money and the value of our annuity will be worth $\$0$$0.
We will examine modelling this type of annuity soon, but for now we'll begin by examining how we might grow an annuity, that is, grow an investment. We will build on familiar concepts involving compound interest.
Some useful formulae are given here to help you calculate present and future values of an annuity.
Future Value
$FV=a\left(\frac{\left(1+r\right)^n-1}{r}\right)$FV=a((1+r)n−1r)
Present Value
$PV=a\left(\frac{\left(1+r\right)^n-1}{r\left(1+r\right)^n}\right)$PV=a((1+r)n−1r(1+r)n)
where $a$a is the contribution per period paid at the end of the period,
$r$r is the interest rate per compounding period,
and $n$n is the number of compounding periods.
Notice that the Present Value formula has the Future Value formula contained within it. If we know the future value we can thus write the Present Value formula as:
$PV=\frac{FV}{\left(1+r\right)^n}$PV=FV(1+r)n
Worked Examples
example 1
Maria invests $\$6000$$6000 at the beginning of each year for $3$3 years in a savings account that pays $5%$5% p.a. with interest compounded semi-annually.
What is the value to which her first deposit will grow at the end of the $3$3 years correct to the nearest cent?
What is the value to which her second deposit will grow correct to the nearest cent?
What is the value to which her third deposit will grow correct to the nearest cent?
What is the future value of the annuity correct to the nearest cent?
How much interest will Maria earn on this annuity?
example 2
Find the contribution Mae needs to deposit into her savings account every 3 months, which pays $12%$12% p.a. with interest compounded quarterly, in order to reach her savings goal of $\$42576$$42576 in $3$3 years. Give your answer correct to the nearest dollar.
| Table of future value interest factors | |||||
| Interest rate per 3 month | |||||
| Periods | $2%$2% | $2.5%$2.5% | $3%$3% | $3.5%$3.5% | $4%$4% |
| $10$10 | $10.9497$10.9497 | $11.2034$11.2034 | $11.4639$11.4639 | $11.7314$11.7314 | $12.0061$12.0061 |
| $11$11 | $12.1687$12.1687 | $12.4835$12.4835 | $12.8078$12.8078 | $13.1420$13.1420 | $13.4864$13.4864 |
| $12$12 | $13.4121$13.4121 | $13.7956$13.7956 | $14.1920$14.1920 | $14.6020$14.6020 | $15.0258$15.0258 |
| $13$13 | $14.6803$14.6803 | $15.1404$15.1404 | $15.6178$15.6178 | $16.1130$16.1130 | $16.6268$16.6268 |
| $14$14 | $15.9739$15.9739 | $16.5190$16.5190 | $17.0863$17.0863 | $17.6770$17.6770 | $18.2919$18.2919 |
| $15$15 | $17.2934$17.2934 | $17.9319$17.9319 | $18.5989$18.5989 | $19.2957$19.2957 | $20.0236$20.0236 |
example 3
Judy wants to go on a long overseas trip in $7$7 years time and calculates that she will need at least $\$17300.42$$17300.42 to pay for it.
The bank Judy intends to use offers her an account which earns $12%$12% p.a. where interest is compounded monthly.
If she deposits one lump sum today and doesn't make any further contributions, what is the least amount Judy would need to deposit today so that she reaches her goal in time for her trip?
Judy realises that she might not have enough money to deposit today, so she decides to instead make regular, equal contributions every compounding period.
What is the least amount Judy would need to deposit each compounding period so that she reaches her goal in time for her trip?