9.02 Compound interest

When interest is always calculated only on the principal or initial amount, it is called simple interest.  The amount of interest, when calculating simple interest, remains constant or fixed. However, most of the time when banks and financial institutions calculate interest, they are using compound interest.

Compound interest is calculated at the end of each compounding period, which is typically a day, month, quarter, or year. At the end of each compounding period, the total amount (principal plus interest) from previous compounding periods is used to calculate the new quantity of interest. We multiply the total amount by the interest rate and then add it to the total.

 

Worked example

Example 1

Suppose $\$500$$500 is invested in a compound interest account with an interest rate of $10%$10% p.a. compounded annually (that is, with a compounding period of one year) for $3$3 years. Then after one year, the interest is calculated:

Interest$=500\times10%=500\times0.1=\$50$=500×10%=500×0.1=$50

This interest is then added to the account

Balance after $1$1 year$=500+50=\$550$=500+50=$550.

After the second year interest is calculated again, but this time the interest rate is applied to the balance from the previous year:

Interest	$=550\times0.1=\$55$=550×0.1=$55
Balance after $2$2 years	$=550+55=\$605$=550+55=$605

 

Finally, after the third year we have

Interest	$=605\times0.1=\$60.50$=605×0.1=$60.50
Balance after $3$3 years	$=605+60.5=\$665.50$=605+60.5=$665.50

 

Hence, the interest is calculated on the previous year's amount including the interest.

 

Practice question

Question 1

 

The compound interest formula - annual compounding

Notice that in the above example, at the end of each compounding period there is a two step process: calculate the interest and then add it to the account balance. We could have combined these two steps as follows:

Balance after $1$1 year $=500+500\times0.1=500\times(1+0.1)=550$=500+500×0.1=500×(1+0.1)=550

This suggests a rule: 

New balance $=$= Previous balance $\times(1+0.1)$×(1+0.1)

In other words, we can find the balance at the end of each year by repeatedly multiplying by $(1+0.1)$(1+0.1)

Balance after $1$1 years	$=500\times(1+0.1)$=500×(1+0.1)
Balance after $2$2 years	$=500\times(1+0.1)\times(1+0.1)=500\times(1+0.1)^2$=500×(1+0.1)×(1+0.1)=500×(1+0.1)2
Balance after $3$3 years	$=500\times(1+0.1)\times(1+0.1)\times(1+0.1)=500\times(1+0.1)^3$=500×(1+0.1)×(1+0.1)×(1+0.1)=500×(1+0.1)3

This leads us to the compound interest formula. 

 

Practice questions

Question 2

 

Question 3

 

Compound interest formula - other compounding periods

What do we do if the interest is being compounded more frequently; perhaps daily, weekly, monthly, quarterly or semi-annually?

 

Notice that since $n$n is the number of compounding periods in a year, $\frac{r}{n}$rn​ is the interest rate per compounding period, and $nt$nt is the total number of compounding periods.

 

Worked example

Suppose $\$500$$500 is invested in a compound interest account with an interest rate of $10%$10% p.a. compounded semi-annually (that is, with a compounding period of $6$6 months) for $3$3 years.

Since the interest is being compounded semi-annually, the number of compounding periods in a year is $n=2.$n=2. The interest rate is $10%$10% per year and so the interest rate per compounding period as a decimal is $\frac{r}{n}=\frac{0.01}{2}=0.005$rn​=0.012​=0.005 . Moreover, in $3$3 years, there are a total of $nt=2\times3=6$nt=2×3=6 compounding periods. Now we can substitute into the formula:

$A$A	$=$=	$P\left(1+\frac{r}{n}\right)^{nt}$P(1+rn​)nt
	$=$=	$500\times\left(1+\frac{0.01}{2}\right)^{2\times3}$500×(1+0.012​)2×3
	$=$=	$500\times\left(1+0.005\right)^6$500×(1+0.005)6
	$\approx$≈	$\$670.05$$670.05

For comparison, if the $\$500$$500 is invested in a compound interest account with an interest rate of $10%$10% p.a. compounded annually for $3$3 years, then

$A$A	$=$=	$P\left(1+r\right)^t$P(1+r)t
	$=$=	$500\times\left(1+0.1\right)^3$500×(1+0.1)3
	$\approx$≈	$\$665.50$$665.50

Compounding more frequently produces more interest!

 

Practice questions

Question 4

Question 5

Question 6

 

Modelling compound interest using a spreadsheet

It is common to use tables to view the progression of an investment or loan. In the first investigation of this chapter we explore further how to use spreadsheets to automate calculating the values for such a table. This can be particularly useful when there are frequent compounds and/or a long investment/loan term. Using a spreadsheet we can quickly explore the impact of changes such as interest rates, compounding periods, or regular payments.

Exploration

Let's explore this interactive compound interest spreadsheet.

Created with Geogebra

Click on the coloured cells to explore the effect of:

• Changing the amount invested (the blue cell).
• Changing the annual interest rate (the green cell). 
• Changing the number of compounding periods (the pink cell) to quarterly ($4$4), monthly ($12$12), weekly ($52$52) or perhaps daily ($365$365).

Questions

1. What happens as you increase the number of compounding periods? Is the impact on interest earned significant?
2. What happens as you increase the annual interest rate? 
3. How has the value in cell C10 been calculated?
4. How has the value in D12 been calculated?

 

Practice question

Question 7