10.07 Comparing simple and compound interest

In 5.01, we saw that if an investment accrues compound interest, it grows faster than an investment that earns simple interest. In general, we observed the following pattern:

The increase is the same each time step for simple interest, while the increase becomes steadily larger each step for compound interest.

What will these patterns look like in the long run? There are clues in the two formulas for calculating simple and compound interest:

$FV$FV	$=$=	$PV\left(1+rn\right)$PV(1+rn)	(simple interest)
$FV$FV	$=$=	$PV\left(1+r\right)^n$PV(1+r)n	(compound interest)

To help us explore these equations, let's use an interest rate of $5%$5% per annum, and an initial (present) value of $\$1000$$1000, in both kinds of cases. The equations become:

$FV$FV	$=$=	$1000\left(1+0.05n\right)=1000+50n$1000(1+0.05n)=1000+50n	(simple interest)
$FV$FV	$=$=	$1000\times1.05^n$1000×1.05n	(compound interest)

Note that when $n=1$n=1:

Simple interest             $FV$FV $=$= $1000+50n$1000+50n   $=$= $1000+50\times1$1000+50×1   $=$= $1000+50$1000+50   $=$= $\$1050$$1050

Compound interest           $FV$FV $=$= $1000\times1.05^n$1000×1.05n   $=$= $1000\times1.05^1$1000×1.051   $=$= $1000\times1.05$1000×1.05   $=$= $\$1050$$1050

After $1$1 period, the value of the investments will be the same.

And when $n=2$n=2:

Simple interest             $FV$FV $=$= $1000+50n$1000+50n   $=$= $1000+50\times2$1000+50×2   $=$= $1000+100$1000+100   $=$= $\$1100$$1100

Compound interest           $FV$FV $=$= $1000\times1.05^n$1000×1.05n   $=$= $1000\times1.05^2$1000×1.052   $=$= $1000\times1.1025$1000×1.1025   $=$= $\$1102.50$$1102.50

After the first time period, the balance starts to grow more quickly using compound interest.

So we have a linear equation and an exponential equation to model simple and compound interest.

To graph them, first remember that the subject of each equation, $FV$FV, is the future value of the investment, or how much money it will be worth in the future, so we can label the vertical axis "Money". The other variable $n$n represents the number of time periods that have passed, so we label the horizontal axis "Time".

Plotting the two functions $FV=1000+0.05n$FV=1000+0.05n and $FV=1000\times1.05^n$FV=1000×1.05n on the same set of axes gives us a smooth curve for each:

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

• 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) and both have the same value at the end of the first period.

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.

Practice question