As we've seen, 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$$0.
Here we'll focus on different ways to model and analyse an annuity and we'll be relying on some of the functionality of our calculators.
Modelling an Annuity with a recurrence relation
example
Tahlia receives an inheritance of $\$250000$$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$$2000 to help pay for living expenses.
(a) Calculate the value of her annuity at the end of the first month.
Think: To calculate the value at the end of the first month, we will first need to add the interest owed to the investment and then subtract the withdrawal.
Do:
Value=$250000\left(1+\frac{7.2}{12\times100}\right)-2000$250000(1+7.212×100)−2000
Value= $\$249500$$249500
(b) Write a recursive rule that gives the value of the annuity, $V_n$Vn, at the end of $n$n months.
Think: Using what we've established in part (a) we can now write our recursive rule.
Do:
$V_n=1.006V_{n-1}-2000;V_0=250000$Vn=1.006Vn−1−2000;V0=250000
Note the use of $V_0$V0, the initial value of the annuity, as opposed to $V_1$V1. If I wanted to I could also use $V_1=\$249500$V1=$249500
(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:
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?
Think: If Tahlia never wants her annuity to run out, then she should only withdraw the amount of interest earned so that the value of her investment forever remains at $\$250000$$250000.
Do:
Interest=$0.006\times250000$0.006×250000
Interest=$\$1500$$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.
Modelling an Annuity with the CAS financial application
example
Jordan wins $\$75000$$75000 on an instant win ticket and decides to invest in an annuity which earns $8%$8% per annum compounded quarterly. Each quarter after he has accrued interest he withdraws $\$3000$$3000 to spend on regular fishing trips.
(a) If the investment continues in this way, how many fishing trips can Jordan afford to go on from the proceeds of his annuity?
Think: Using the financial facility on our calculator we know we are interested in calculating the value of $N$N.
Do:
I is our annual interest rate of $8%$8%.
PV is the present value of our investment and it is $-75000$−75000 because from Jordan's point of view he has given his money to the annuity
PMT is the payment Jordan receives of $\$3000$$3000 each quarter, and this is a positive value since the money returns to him
FV is the future value of our investment and Jordan wants to know when this amount will be $0$0.
P/Y is the number of payments or withdrawals made each year, which is $4$4
C/Y is the number of compounding periods each year, which is also $4$4
Solving we get:
So Jordan can afford to take $35$35 fishing trips.
(b) Jordan was hoping to afford $4$4 fishing trips a year for the next $10$10 years. How much should he withdraw each quarter to allow for this?
Think: This time we wish to solve for the PMT.
Do:
Solving we get:
So Jordan can only withdraw $\$2741.68$$2741.68 each quarter if he wants to achieve his goal.
Worked Examples
example 1
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?
Example 2
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.