Modelling Compound Interest

We are quite familiar with compound interest and the associated calculations from work we've studied previously.

Here we'll take a look at various way to model and represent compound interest so that we may clearly examine what is happening over time.

We can model compound interest in two main ways:

• Using a recurrence relation
• Using a table of value or spreadsheet

Modelling compound interest using a recurrence relation

Using a recurrence relation allows us to progressively analyse the effect of compounding our investment at each time period. In simple situations like those we'll examine here, we could also use the general rule for a sequence.

example 1

Let's take the simple example of investing $\$1000$$1000 at $10%$10% interest compounded annually.

What will the first few years of our investment look like?

End of Year 1	$1000\times1.10=\$1100$1000×1.10=$1100
End of Year 2	$1100\times1.10=\$1210$1100×1.10=$1210
End of Year 3	$1210\times1.10=\$1331$1210×1.10=$1331

We can clearly see that the balance at the end of each year is 1$10%$10% of the previous year.

Expressing this as a recurrence relation where $B_{n+1}$Bn+1​ is the balance at the end of year $n$n, we have:

$B_{n+1}=1.1B_n;B_0=1000$Bn+1​=1.1Bn​;B0​=1000

Notice the use of $B_0$B0​ here to indicate our initial investment. If instead we wanted to use $B_1$B1​, we need to be mindful that $B_1$B1​represents the balance at the end of year 1 and so we could write our recurrence relation as:

$B_{n+1}=1.1B_n;B_1=1100$Bn+1​=1.1Bn​;B1​=1100

I prefer to use $B_0$B0​ since writing it this way doesn't require me to do any calculations!

Recall that this recurrence relation represents a geometric progression, so we could also write a general or explicit rule for this scenario.

We could have $T_n=1100\times1.1^{n-1}$Tn​=1100×1.1n−1 and notice we have to use $T_1=1100$T1​=1100 in this case.

Or we could use $T_n=1000\times1.1^n$Tn​=1000×1.1n

I encourage you to be flexible with how you write and interpret your recursive and general rules - it will make your progress with this work so much easier!

 

Modelling Compound Interest Using a Spreadsheet

Let's explore this interactive compound interest spreadsheet.

You can change the amount invested (the blue cell) to any value you'd like to invest.

You can change the annual interest rate (the green cell) to any value. 

You can change the number of compounding periods (the pink cell) to quarterly ($4$4), monthly ($12$12), weekly ($52$52) or perhaps daily ($365$365).

What happens as you increase the number of compounding periods?

What happens as you increase the annual interest rate?

How has the value in cell C10 been calculated?

How has the value in D12 been calculated?

When you can answer these questions you're ready for the worked examples below.

 

 

Worked Examples

example 1

example 2