An annuity is a style of investment from which individuals usually withdraw a regular amount of funds until the value of the annuity is $\$0$$0.
We can solve annuity problems using the finance application of our calculators, or by modelling the problem with a recurrence relation.
Modelling an annuity with the calculator financial application
Select the brand of calculator you use below to work through an example involving annuities.
Casio Classpad
How to use the CASIO Classpad to complete the following tasks involving annuities using the inbuilt financial solver.
Jordan invests $\$75000$$75000 in an annuity which earns $8%$8% per annum compounded quarterly. At the end of each quarter he withdraws $\$3000$$3000 to spend on a quarterly fishing trip.
If the investment continues in this way, how many fishing trips can Jordan afford to go on from the proceeds of his annuity?
Jordan was hoping to afford $4$4 fishing trips a year for the next $10$10 years. To achieve this he will reduce the amount he withdraws for each trip. How much can he afford to withdraw each quarter?
TI Nspire
How to use the TI Nspire to complete the following tasks involving annuities using the inbuilt financial solver.
Jordan invests $\$75000$$75000 in an annuity which earns $8%$8% per annum compounded quarterly. At the end of each quarter he withdraws $\$3000$$3000 to spend on a quarterly fishing trip.
If the investment continues in this way, how many fishing trips can Jordan afford to go on from the proceeds of his annuity?
Jordan was hoping to afford $4$4 fishing trips a year for the next $10$10 years. To achieve this he will reduce the amount he withdraws for each trip. How much can he afford to withdraw each quarter?
Modelling an annuity with a recurrence relation
This is similar to modelling an investment with regular payment with the payment in this case being withdrawn and hence, negative in the form shown below.
For a principal investment, $P$P, at the compound interest rate of $r$r per period and a payment $d$d withdrawn per period, the sequence of the value of the investment over time forms a first order linear recurrence.
The sequence which generates the value, $V_n$Vn, of the investment at the end of each instalment period is:
- Recursive sequence:
$V_n=V_{n-1}\times(1+r)-d$Vn=Vn−1×(1+r)−d, where $V_0=P$V0=P
- Recursive sequence:
$V_n=V_{n-1}\times(1+r)-d$Vn=Vn−1×(1+r)−d, where $V_1=P$V1=P
Worked example
example 1
Tahlia receives an inheritance of $\$250000$$250000 and decides to invest the entire amount in an annuity earning $7.2%$7.2% per annum compounded monthly. At the end of each month, once the interest has been paid into her account, Tahlia withdraws $\$2000$$2000 to help pay for living expenses.
(a) Calculate the value of her annuity at the end of the first month.
Think: To calculate the value at the end of the first month, we will first need to add the interest owed to the investment and then subtract the withdrawal.
Do:
| $\text{Value }$Value | $=$= | $250000\left(1+\frac{7.2}{12\times100}\right)-2000$250000(1+7.212×100)−2000 |
| $=$= | $\$249500$$249500 |
(b) Write a recursive rule that gives the value of the annuity, $V_n$Vn, at the end of $n$n months.
Think: Our initial value, at the start of the annuity is $250000$250000. Remember the annual rate of $7.2$7.2% must be divided by $12$12 to get a monthly rate of $0.6$0.6%, or $0.006$0.006.
Do:
$V_n=1.006V_{n-1}-2000$Vn=1.006Vn−1−2000; $V_0=250000$V0=250000
(c) Use the sequence facility of your calculator to determine during which year and month Tahlia's annuity will end.
Think: To do this we will need to use our recursive rule and then scroll through our table of values until we find the first time the value of the annuity turns to a negative value.
Do: For an example showing how to create a table of values for a sequence inspect the first examples given in a previous lesson. Scrolling through your created table you should see the following:
| $n$n | $V_n$Vn |
|---|---|
| $229$229 | $\$5422.07$$5422.07 |
| $230$230 | $\$3454.60$$3454.60 |
| $231$231 | $\$1475.33$$1475.33 |
| $232$232 | $-\$515.82$−$515.82 |
We can see the annuity first holds a negative value during month $232$232. Therefore it will take $19.33$19.33 years. So this occurs after $19$19 years and $4$4 months.
(d) How much should Tahlia withdraw each month if she wishes her annuity to last indefinitely (become a perpetuity)?
Think: If Tahlia never wants her annuity to run out, then she should only withdraw the amount of interest earned in the first month, so that the value of her investment forever remains at $\$250000$$250000.
Do:
| $\text{Interest }$Interest | $=$= | $0.006\times250000$0.006×250000 |
| $=$= | $\$1500$$1500 |
(e) If Tahlia could find a higher interest rate for her investment, but kept the withdrawal amount the same, would the annuity end sooner or later?
Think: What does it mean if the interest rate was higher? It means she would earn more from her annuity each month.
Do: Therefore, if Tahlia earns more but withdraws the same amount, her annuity will last longer.
Reflect: The recursive rule could also be written as $V_{n+1}=1.006V_n-2000$Vn+1=1.006Vn−2000; $V_0=250000$V0=250000.
Always read the question carefully to know which notation to use.Practice questions
QUESTION 1
Victoria invests $\$190000$$190000 at a rate of $12%$12% per annum compounded monthly.
We will use the financial solver on our CAS calculator to determine what Victoria's equal monthly withdrawal should be if she wants the investment to last $20$20 years.
Fill in the value for each of the following. Type an $X$X next to the variable we wish to solve for.
$N$N $\editable{}$ $I$I$%$% $\left(\editable{}\right)%$()% $PV$PV $\editable{}$ $PMT$PMT $\editable{}$ $FV$FV $\editable{}$ $P$P$/$/$Y$Y $\editable{}$ $C$C$/$/$Y$Y $\editable{}$ Hence determine the amount of the monthly withdrawal.
Give your answer to the nearest cent.
QUESTION 2
Lachlan received an inheritance of $\$100000$$100000. He invests the money at $8%$8% per annum with interest compounded annually at the end of the year. After the interest is paid, Lachlan withdraws $\$9000$$9000 and the amount remaining in the account is invested for another year.
How much is in the account at the end of the first year?
Write a recursive rule for $A_n$An in terms of $A_{n-1}$An−1 that gives the value of the account after $n$n years and an initial condition $A_0$A0.
Write both parts on the same line separated by a comma.
What is the value of the investment at the end of year $10$10?
Round your answer to the nearest cent.
By the end of which year will the annuity have run out?
Question 3
Emma sells her business and with the profit of $\$150000$$150000 sets up an annuity. She will pay herself $\$3500$$3500 monthly from her annuity which earns interest of $7.6%$7.6% per annum compounded quarterly.
Write a recursive rule that gives the value of her annuity, $A_n$An, at the end of quarter $n$n.
Write both parts of the rule (including the initial value $A_0$A0) on the same line, separated by a comma.
Use your calculator to determine the balance of her annuity after $3$3 years, correct to the nearest cent.
Use the previous part to determine after how many years and quarters the annuity will close.
The annuity will close after $\editable{}$ year(s) and $\editable{}$ quarter(s).
If interest was calculated monthly rather than quarterly, what effect, if any, would this have on the life of her annuity?
No effect
AThe annuity would last longer
BThe annuity would not last as long
C
Calculating annuities without technology
If we want to calculate annuities without technology, we will need to find an annuity formula that perfectly describes what is happening to the annuity at every step.
We know that an annuity starts with a principal investment $P$P which gains interest each compounding period according to some rate $r$r, and after this we then withdraw a payment of $d$d. Then, we repeat this until the annuity's value becomes zero.
Let's start by representing this information in an equation for the first few compounding periods:
| $V_0$V0 | $=$= | $P$P |
| $V_1$V1 | $=$= | $P\times\left(1+r\right)-d$P×(1+r)−d |
| $V_2$V2 | $=$= | $\left(P\times\left(1+r\right)-d\right)\times\left(1+r\right)-d$(P×(1+r)−d)×(1+r)−d |
| $=$= | $P\times\left(1+r\right)^2-d\times\left(\left(1+r\right)+1\right)$P×(1+r)2−d×((1+r)+1) | |
| $V_3$V3 | $=$= | $\left(P\times\left(1+r\right)^2-d\times\left(\left(1+r\right)+1\right)\right)\times\left(1+r\right)-d$(P×(1+r)2−d×((1+r)+1))×(1+r)−d |
| $=$= | $P\times\left(1+r\right)^3-d\times\left(\left(1+r\right)^2+\left(1+r\right)+1\right)$P×(1+r)3−d×((1+r)2+(1+r)+1) |
Because the sequence is recursive, we begin to see a pattern emerge after expanding and simplifying the result of each compounding period. In fact, we can generalise these equations to get:
$V_n=P\times(1+r)^n-d\times\left((1+r)^{n-1}+(1+r)^{n-2}+\ldots+(1+r)^2+(1+r)+1\right)$Vn=P×(1+r)n−d×((1+r)n−1+(1+r)n−2+…+(1+r)2+(1+r)+1)
However, this looks quite messy, so we also want to simplify this equation by applying a useful equivalence.
For any sequence of powers we have the equivalence:
$x^{n-1}+x^{n-2}+\ldots+x^2+x+1=\frac{x^n-1}{x-1}$xn−1+xn−2+…+x2+x+1=xn−1x−1
Test this equivalence for yourself by checking that:
$\left(x^{n-1}+x^{n-2}+\ldots+x^2+x+1\right)\times(x-1)=x^n-1$(xn−1+xn−2+…+x2+x+1)×(x−1)=xn−1
Using this equivalence, we can rewrite the general equation as:
$V_n=P\times\left(1+r\right)^n-d\times\frac{\left(1+r\right)^n-1}{1+r-1}$Vn=P×(1+r)n−d×(1+r)n−11+r−1
which simplifies to:
$V_n=P\times\left(1+r\right)^n-d\times\frac{\left(1+r\right)^n-1}{r}$Vn=P×(1+r)n−d×(1+r)n−1r
where $V_n$Vn is the annuity value after $n$n compounding periods, $P$P is the principal investment, $r$r is the interest rate per compounding period, and $d$d is the amount being withdrawn at the end of each period.