7.05 Annuities
Annuities are investments in which we make regular deposits, or regular withdrawals.
Two key features of annuities are:
- The amount deposited or withdrawn each period remains constant for the duration of the annuity.
- The balance in the account accrues interest at the end of each passing period. These returns are fixed and do not fluctuate with share market or interest rate fluctuations.
Making regular contributions
Here is an example where regular, equal deposits are made into an investment account and interest is compounded at the end of each period.
At the beginning of each year for $3$3 years, $\$500$$500 is deposited into an investment account that earns $3%$3% per annum interest.
- The first $\$500$$500 accrues interest for $3$3 years.
- The second $\$500$$500 accrues interest for $2$2 years.
- The third $\$500$$500 accrues interest for $1$1 year.
We are interested in the total value of the investment after $3$3 years.
Notice that the funds in the account will increase quickly because of the regular contributions and the interest paid on the balance. We can determine the present or future value of a particular contribution to an annuity using the following formula:
$FV=PV(1+\frac{r}{100k})^{kn}$FV=PV(1+r100k)kn
- $PV$PV is the present value of a withdrawal or deposit
- $k$k is the number of compounding periods per year
- $n$n is the number of years
- $r$r is the annual interest rate, for example if rate is $6$6% p.a. then $r=6$r=6
- $FV$FV is the future value of the withdrawal or deposit amount (after $kn$kn compounding periods)
It is not a coincidence that this is the compound interest formula which we encountered earlier! In an annuity each equal, regular contribution earns compound interest, and we can use the formula on each contribution separately to find the future value of the annuity as a whole.
From the example above:
- the first contribution will have a value of $\$500\times1.03^3$$500×1.033
- the second will have a value of $\$500\times1.03^2$$500×1.032, and
- the last contribution will have a value of $\$500\times\left(1.03\right)$$500×(1.03)
by the end of the investment
Making regular withdrawals
In a reverse situation, we may start with a single sum investment from which regular, equal amounts are withdrawn. This is another type of annuity, and a common example is superannuation.
Say you retire with a superannuation fund of $\$500000$$500000 from which you withdraw $\$5000$$5000 at the end of each month. The interest compounds on the remaining balance at $3%$3% per annum.
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After $1$1 month, the initial $\$500000$$500000 will have earned interest of $0.25%$0.25%, and then the first $\$5000$$5000 will be withdrawn, so there will be $\$\left(500000\times1.0025-5000\right)$$(500000×1.0025−5000) left in the account.
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This balance will then earn a month’s interest before another $\$5000$$5000 is withdrawn, and so on.
The funds in the account will decrease and eventually the balance will be $\$0$$0 because the regular withdrawals are more than the interest we make on the balance. So in this case, we’d be interested in how long our superannuation will last us.
Practice questions
QUESTION 1
Which of the following are types of annuities?
An account in which you make regular contributions and the interest is paid at the end of each period.
AAn account in which you make contributions when you have spare money and the interest is paid at the beginning of each period.
BAn account from which you withdraw contributions that decrease as the balance decreases.
CAn account in which you make regular withdrawals and the interest is paid at the end of each period.
D
QUESTION 2
Amber made a single $\$20000$$20000 deposit into a savings account, with interest compounding yearly at $6.8%$6.8% p.a. Calculate the balance in the account after $5$5 years, correct to the nearest dollar.
QUESTION 3
Maria invests $\$4000$$4000 at the end of each year for $4$4 years in an investment account that pays $5%$5% per annum with interest compounded monthly.
What value will her first deposit have grown to at the end of the $4$4 years?
What value will her second deposit have grown to at the end of the $4$4 years?
What value will her third deposit have grown to at the end of the $4$4 years?
What is the future value of the annuity correct to the nearest cent?
QUESTION 4
Sandro has a superannuation fund of $\$600000$$600000. He withdraws $\$40000$$40000 at the beginning of each year as an income stream in his retirement. The interest compounds monthly on the remaining balance at $3%$3% per annum.
Calculate the balance in the account at the end of the first year, correct to the nearest dollar.
By how much did the balance decrease in the first year? Is this the same amount as his withdrawal? Explain your answer.