Introduction
An annuity is a style of investment from which individuals usually withdraw a regular amount of funds. The withdrawal in many cases is repeated until the funds run out, that is the value of the annuity reaches \$0. A special case of annuities are perpetuities where rather than the funds dwindling the investment value remains constant.
We can solve annuity problems using the finance application of our calculators, or by modelling the problem with a recurrence relation.
Model annuities with the financial application
We can solve problems involving annuities using a financial application by setting the present value (PV) as the initial value invested and future value (FV) as the remaining funds at the end of the investment (often \$0). The present value should be entered as a negative to indicate depositing the money for investment and the payment (PMT) will be positive as this is returned to the investor.
Examples
Example 1
Victoria invests \$190\,000 at a rate of 12\% per annum compounded monthly.
Victoria wants to determine what her equal monthly withdrawal should be if she wants the investment to last 20 years.
Fill in the value for each of the following. Put an X next to the variable we wish to solve for.
| Variable | Value |
|---|---|
| N | ⬚ |
| I(\%) | ⬚\% |
| PV | ⬚ |
| PMT | ⬚ |
| FV | ⬚ |
| P/Y | ⬚ |
| C/Y | ⬚ |
Hence determine the amount of the monthly withdrawal. Give your answer to the nearest cent.
When using a financial application for an annuity:
N is the number of instalment periods, or how long the annuity lasts.
I\% is our annual interest rate.
PV is the present value of our investment. This should be represented by a negative value, as the investment represents money has set aside.
PMT is the amount withdrawal each time period.
FV is the future value of our investment which should be 0.
P/Y is the number of payments or withdrawals made each year.
C/Y is the number of compounding periods each year.
Model annuities with recurrence relations
This is similar to modelling an investment with regular payment with the payment in this case being withdrawn and hence, negative in the form shown below.
For a principal investment, P, at the compound interest rate of r per period and a payment d withdrawn per period, the sequence of the value of the investment over time forms a first order linear recurrence.
The recursive sequence which generates the value, V_{n}, of the investment at the end of each instalment period is: V_{n}=V_{n-1}\times (1+r)-d, \, \text{where }V_{0}=P
The recursive sequence which generates the value, V_{n}, of the investment at the beginning of each instalment period is: V_{n}=V_{n-1}\times (1+r)-d, \, \text{where }V_{1}=P
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 which gains interest each compounding period according to some rate r, and after this we then withdraw a payment of 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:
| \displaystyle V_{0} | \displaystyle = | \displaystyle P |
| \displaystyle V_{1} | \displaystyle = | \displaystyle P\times (1+r)-d |
| \displaystyle V_{2} | \displaystyle = | \displaystyle (P\times (1+r)-d)\times(1+r)-d |
| \displaystyle = | \displaystyle P\times (1+r)^{2}-d\times((1+r)+1) | |
| \displaystyle V_{3} | \displaystyle = | \displaystyle (P\times (1+r)^{2}-d)\times((1+r)+1))\times (1+r)-d |
| \displaystyle = | \displaystyle P\times (1+r)^{3}-d\times((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 ((1+r)^{n-1}+(1+r)^{n-2}+\ldots+(1+r)^{2}+(1+r)+1)
We can simplify this using the equivalence:x^{n-1}+x^{n-2}+\ldots+x^{2}+x+1=\dfrac{x^{n}-1}{x-1} by letting x=(1+r):(1+r)^{n-1}+(1+r)^{n-2}+\ldots+(1+r)^{2}+(1+r)+1=\dfrac{(1+r)^{n}-1}{(1+r)-1}
Now we can rewrite the general equation as: V_{n}=P\times (1+r)^{n}-d\times \dfrac{(1+r)^{n}-1}{1+r-1}which simplifies to the formula for annuities: V_{n}=P\times (1+r)^{n}-d\times\dfrac{(1+r)^{n}-1}{r}
where V_{n} is the annuity value after n compounding periods, P is the principal investment, r is the interest rate per compounding period, and d is the amount being withdrawn at the end of each period.
Examples
Example 2
Lachlan received an inheritance of \$100\,000. He invests the money at 8\% per annum with interest compounded annually at the end of the year. After the interest is paid, Lachlan withdraws \$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 in terms of A_{n - 1} that gives the value of the account after n years and an initial condition A_{0}.
What is the value of the investment at the end of year 10?
By the end of which year will the annuity have run out?
For a principal investment, P, at the compound interest rate of r per period and a payment d withdrawn per period, the sequence of the value of the investment over time forms a first order linear recurrence.
The recursive sequence which generates the value, V_{n}, of the investment at the end of each instalment period is:V_{n}=V_{n-1}\times (1+r)-d, \, \text{where }V_{0}=P
The recursive sequence which generates the value, V_{n}, of the investment at the beginning of each instalment period is:V_{n}=V_{n-1}\times (1+r)-d, \, \text{where }V_{1}=P
Explicit formula for annuities: