6.01 Modelling annuities recursively

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• The amount deposited or withdrawn each period remains constant for the duration of the annuity.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. 

 

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In a reverse situation, we may start with a single sum investment from which regular, equal amounts are withdrawn.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.

• 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the initial $\$500000$$500000 will have earned interest of $0.25%$0.25%, and then the first $\$5000$$5000 will be withdrawn, so there will be $\$\left(500000\times1.0025-5000\right)$$(500000×1.0025−5000) left in the account.

• This balance will then earn a month’s interest before another $\$5000$$5000 is withdrawn, and so on.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 question

QUESTION 1

When investing money we should try to optimise our financial gains by comparing different investment options. The best way to do this is to model each annuity and then determine its value well into the future.

In section 6.01 we saw that an annuity involves an initial investment followed by regular and equal deposits or withdrawals on which you earn compound interest. Because of these regular and equal amounts, we can model annuities using a recurrence relation.

Analysing our investment with a recurrence relation

To develop our first recursive model, let's look at the following scenario.

We'll invest $\$5000$$5000 at $6%$6% per annum interest compounded monthly. At the end of each month, after the interest has been applied, we'll make an additional deposit of $\$100$$100.

In order to develop a recursive model for this annuity, we first need to change our interest rate from annual to monthly by dividing by $12$12, so we get $0.5%$0.5% per month. At the end of each month, we increase the balance by $0.5%$0.5% ($\times1.005$×1.005) and then add $\$100$$100.

The balance at the end of the first few months may now be calculated as follows.

End of month 1	$1.005\times5000+100$1.005×5000+100	$=$=	$\$5125$$5125
End of month 2	$1.005\times5125+100$1.005×5125+100	$=$=	$\$5250.63$$5250.63
End of month 3	$1.005\times5250.63+100$1.005×5250.63+100	$=$=	$\$5376.88$$5376.88

We are dealing with money, so we have rounded each balance to $2$2 decimal places and then use the rounded amount in the following row.

Notice a recurring pattern:

$\text{Balance at the end of the month }=\text{balance at the beginning }\times1.005+100$Balance at the end of the month =balance at the beginning ×1.005+100

We call this a recurrence relation because the same relationship between the beginning and end balance repeats itself every time period.

It describes what is happening from month to month, and can be used to track how the annuity will grow.

To examine the balance in a few years time, we don't want to manually calculate the values in the table. Luckily, there are a couple of ways we can use these recurrence relations and technology to do these calculations for us.

 

Analysing our investment with a spreadsheet

We can model our investment using a spreadsheet, like the one in the applet below.

In this scenario, an initial amount of $\$5000$$5000 has been invested, and interest of $6%$6% is earned per annum. At the end of each period, additional deposits of $\$100$$100 are made that increase the closing balance.

 

At this point, we want to create a formula that relates the closing balance and the opening balance each time period, and then let the technology do the calculations for us for as many time periods as we like.

The percentage interest rate in $B2$B2 is an annual rate, so we need to divide this by the number of compounding periods to get the equivalent rate per period. A $6%$6% annual rate is equivalent to a $0.5%$0.5% monthly rate.

Column $B$B shows the opening balance each period and Column $E$E shows the closing balance of the previous period, so we have:

$B7=B1$B7=B1

$B8=E7$B8=E7

$B9=E8$B9=E8

and so on.

To calculate the interest in Column $C$C, we divide the value in cell $B2$B2 by the number of compounding periods in a year (in this case $12$12 months). The opening balance each time period that appears in column $B$B is increased by this new monthly rate.

$\text{C7 }=\frac{\text{B2 }}{12\times100}\times\text{B7 }$C7 =B2 12×100​×B7

$\text{C8 }=\frac{\text{B2 }}{12\times100}\times\text{B8 }$C8 =B2 12×100​×B8

and so on.

The closing balance each time periods is the result of taking the opening balance and adding the intereste earned and the deposit made.

$\text{E7 }=\text{B7 }+\text{C7 }+\text{D7 }$E7 =B7 +C7 +D7

$\text{E8 }=\text{B8 }+\text{C8 }+\text{D8 }$E8 =B8 +C8 +D8

and so on.

By creating a spreadsheet where each cell in column $E$E is defined recursively, we can generate the future value of the annuity after any number of time periods.

Note that here our extra deposits always match the number of compounding periods each year.

Feel free to change the original investment amount, the annual interest rate, how often the interest is compounded and the regular deposit amount in order to investigate the effect on the investment's final value.

Determining information from a model

Sometimes we may be presented with an incomplete model which requires us to determine the recurrence relation (or formula) underlying it.

The first $5$5 quarters of an annuity's life are modelled in the spreadsheet below.

To calculate the quarterly interest rate for this annuity, we need to use one of the interest rate values in the table. In Quarter $2$2, the opening balance of $\$10450$$10450 earns interest of $\$209$$209. Let $r$r be the quarterly interest rate.

$\$10450\times r=209$$10450×r=209, so $r=0.02$r=0.02.

The quarterly interest rate is $2%$2%, which is equivalent to $8%$8% per annum.

The balance at the end of the fifth quarter is contained in cell $E6$E6. It is the result of increasing the opening balance in $B6$B6 by $2%$2%, and adding a deposit of $\$250$$250.

$E6=1.02\times B6+D6$E6=1.02×B6+D6

To determine how much as originally invested, we need to write a formula here to solve for $X$X in the table. The opening balance earns interest of $2%$2%, and has a deposit of $\$250$$250 added to it to give us the closing balance.

(c) How much was originally invested?

$1.02X+250=10450$1.02X+250=10450

$X=10000$X=10000

So $\$10000$$10000 was originally invested.

We can write a recurrence relation that gives the balance of the investment at the end of each quarter.

$\text{Closing balance }=1.02\times\text{Beginning balance }+\$250$Closing balance =1.02×Beginning balance +$250

The initial balance is $\$10000$$10000.

When we combine the starting value of the annuity with an equation relating the balance at start and end of each month, we have a recursive model that allows us to find the closing balance for each period.

Once we have this relation, we can use a spreadsheet, calculator or other technology to generate a sequence which can tell us the size of our investment any time in the future.

Analysing our annuity with other technology

Any technology that is able to generate a sequence can be used to observe the value of our annuity well into the future. If we use a calculator, we must assign the following symbols to each part of our recurrence relation.

• Initial balance $\rightarrow$→ $a_0$a0​
• Starting balance at beginning of period $\rightarrow$→ $a_n$an​
• balance $\rightarrow$→ $a_{n+1}$an+1​

In this example, our initial investment amount is $\$5000$$5000, monthly interest rate is $r=0.5%$r=0.5%, and we are interested in the value of the annuity after $2$2 years (or $24$24 months).

After inputting $a_0$a0​ and the recursive formula for $a_{n+1}$an+1​, the calculator generates a sequence for us. This allows us to see our annuity will be worth $\$8179$$8179 after $24$24 months.

We can calculate the regular deposit amount that is made each month. On the display, we can see that the recursive relation is given by $a_{n+1}=1.005a_n+100$an+1​=1.005an​+100. It tells us we multiply the beginning balance each month ($a_n$an​) by $1.005$1.005, and add the deposit amount to get the closing balance $a_{n+1}$an+1​.

The regular monthly deposit is $\$100$$100.

Practice questions

Question 2

Question 3