Simple and compound interest

What is interest?

Let’s start with a few simple definitions.

If you put $10,000 into an investment that offers 2% interest, you will earn 2% of $10,000 in some time period. The time period could be a month, a year, or it could be anything depending on the investment. The amount of interest earned is added to the initial principal amount to calculate the total at the end of the period. In this case, the principal is $10,000, meaning that we started with $10,000. We can calculate the interest as 2% of 10,000, i.e. $0.02\times 10000=200$, so the interest earned is $200.

$$\begin{array}{rl}\text{(interest rate)(principal)}& =\text{interest\hspace{0.33em}earned}\\ .02(10,000)& =200\end{array}$$

$$\begin{array}{rl}\text{(interest earned) + principal}& =\text{total\hspace{0.33em}account}\\ 200+10,000& =10,200\end{array}$$

For this investment, after one time period of interest, you would have $10,200.

The example above involves just one time period. What would happen if you earned 2% interest for two separate time periods? Depending on whether the investment pays simple interest or compound interest, the result will be different.

Simple interest equation

Simple interest is calculated when the interest is earned only on the principal. Although the total account grows over time, only the original principal is used to calculate the interest earned. This means that the interest earned per year doesn’t change, and the total account value grows linearly.

Let’s try a simple interest question.

If you put $15,000 into an account that earns simple interest at a rate of 1% per 4 months, how much money will be in the account after 3 years?

To solve this, all we need to do is plug the numbers into the equation.

$$\begin{array}{rl}\text{Total}& =P(1+rt/100)\\ & =15000(1+1(9)/100)\\ & =15000(1+9/100)\\ & =15000(1.09)\\ & =16350\end{array}$$

There is a “gotcha” here to be aware of. The question asks how much money will be in the account after $3$ years, but the period is every $4$ months, so we need to find how many periods are in total to use for our $t$ variable. We have $3\ast 12=36$ months of total time, divided into $36/4=9$ periods of interest.

Compound interest equation

Compound interest is calculated a little differently from simple interest. Compound interest adds interest earned to the principal, and uses that as the base for the next interest payment. As the account value grows over time, so does the amount of new interest earned in every period. This can multiply into higher and higher gains per year, hence the word compound.

Let’s try a compound interest question.

How much total interest would you earn in 4 years from a $1,000 investment that pays 3% annual interest?

$$\begin{array}{rl}\text{Total}& =P(1+r/100{)}^t\\ & =1000(1+3/100{)}^4\\ & =1000(1.03{)}^4\\ & =1000(1.12551)\\ & =1125.51\end{array}$$

Since the total after 4 years is $1,125.51, the amount of interest earned is just the total minus the principal: $1125.51-1000=125.51$.

Note that the question doesn’t explicitly mention the word compound. Typically questions will specifically call out whether you’re dealing with simple or compound interest, but if it isn’t mentioned, compound interest is the standard in the real world, so it’s likely that’s the correct one to use.

Simple interest vs. compound interest

If you’re wondering, a multi-period compound interest account will always earn more money than a simple interest account if both account interest rates, time periods, and principals are equal.

For example, take a look at the following outcomes with $1,000 principal, 5% interest, and a 10-year time horizon.

• Simple interest, annual: $1,500
• Compound interest, annual: $1,628.89
• Compound interest, semi-annual: $1,638.62
• Compound interest, quarterly: $1,643.62
• Compound interest, monthly: $1,647.01
• Compound interest, daily: $1,648.66

The more frequently your interest compounds, the more money you’ll make!