9.05 Comparing simple and compound interest

Simple interest is calculated only on the principal (that is, the initial amount) so the amount of interest being added to a loan or investment remains constant or fixed. Compound interest is interest earned on the principal amount plus interest on the interest already earned. In general, we observed the following pattern:

With simple interest the balance will increase with a constant step up. Graphing the balance at the start of each period will form a straight line pattern. For compound interest instead of the value of your investment increasing in a straight line the steps become larger and larger, a graph of the balance will grow exponentially and look something like this:

Over time, simple interest will continue to grow by the same amount each year, while compound interest will grow faster and faster.

Notice:

• The simple interest graph is a straight line and the compound interest graph is a smooth curve.

• Both graphs are increasing.

• The simple interest line is increasing at a constant rate and the compound interest curve is increasing at an increasing rate.

• Both graphs have the same $y$y-intercept (present value or principal).

Knowing the basic shape of the curve that each type of investment makes will help us think about key points in the life cycle of an investment, and compare investment strategies.

 

Worked example

Consider a deposit of $\$1000$$1000 into an online account for $2$2 years that pays $5%$5% p.a. interest.

(a) Find the difference in interest earned in $2$2 years if the interest rate is calculated as a simple rate compared to a compound interest rate compounding annually.

Think: We can calculate the interest earned on a simple interest account using the simple interest formula, $I=P\times R\times T$I=P×R×T. We can compare this to the interest earned on a compound investment using the compound interest formula to find the future value of the investment and then subtract the principal investment to find the interest earned. The compound interest formula for annual compounding is: $A=P\left(1+r\right)^n$A=P(1+r)n, where $A$A is the final amount of money, $P$P is the principal, $r$r is the annual interest rate, $n$n is the number of years.

Do:

Simple interest:

$I$I	$=$=	$P\times R\times T$P×R×T
	$=$=	$\$1000\times0.05\times2$$1000×0.05×2
	$=$=	$\$100$$100

Compound interest:

$A$A	$=$=	$P\left(1+r\right)^n$P(1+r)n
	$=$=	$\$1000\left(1+0.05\right)^2$$1000(1+0.05)2
	$=$=	$\$1102.50$$1102.50

 

So the total compound interest earned over the two years would be $\$102.50$$102.50 which is $\$2.50$$2.50 more than what was earned with simple interest.

Reflect: Although a $\$2.50$$2.50 difference may not seem like much, think of how much the difference would have been if a million dollars was invested instead of a thousand, or if the investment was made for twenty years instead of two.

(b) Find the equation which describe the future value $FV$FV of the investment after $n$n years, if:

(i) Simple interest is paid annually on the account.

Think: The value of the investment starts at $\$1000$$1000 and increases by $5%$5% of the initial investment each year. This is a constant increase of $\$50$$50 per year.

Do: $FV=1000+50n$FV=1000+50n

Reflect: This is the equation of a line with a $y$y-intercept of $\$1000$$1000 and a gradient of $\$50$$50 per year.

(ii) Compound interest is paid annually on the account.

Think: The value of the investment starts at $\$1000$$1000 and increases by a multiplier of $1.05$1.05 each year.

Do: $FV=1000\left(1.05\right)^n$FV=1000(1.05)n

Reflect: This is the equation of an exponential curve with $y$y-intercept $\$1000$$1000 and increasing by $5%$5% each year. Knowing the equations of the two investments would also allow us to use graphing software to compare them visually.

 

The questions associated with this lesson will focus on comparing simple and compound interest investments. We will look further to compare loans in the following investigation.

Practice questions

Question 1

Question 2

Question 3