Compounding interest
1. The idea
Interest can be added to an amount once a year or more often. Each time it is added, it starts to earn interest itself, so for the same quoted yearly rate, the more often interest is added, the more we end with. With interest added \(m\) times a year, each addition is at the yearly rate divided by \(m\). In the limit, with interest added at every moment, called continuous compounding, the growth factor becomes \(e^{Rn}\), the form used in the lessons on forwards:
\[ \begin{aligned} \text{value} &= A \left(1 + \frac{R}{m}\right)^{mn},\\ \text{value with continuous compounding} &= A\, e^{Rn}, \end{aligned} \]
where \(A\) is the amount invested, \(R\) the yearly rate as a decimal, \(m\) the number of times a year interest is added and \(n\) the number of years.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. Compounding on a chart
In the question, interest is added four times a year, at a quarter of the yearly rate each time: 6%/4 = 1.5%. Over 3 years that makes 12 additions: \(\$1{,}000 \times (1 + 0.06/4)^{12} = \$1{,}195.62\), against $1,180 with simple interest, where interest is never added to the amount.
Now move the number of years. The gap between the two lines is interest earned on interest.
4. One worked example, then your turn
Now continuous compounding with hints, one step at a time.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer with interest added a set number of times a year and one with continuous compounding. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: