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 earn 2% of $10,000 over one time period. The time period could be a month, a year, or something else, depending on the investment.

First, calculate the interest earned:

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

So the interest earned is $200.

Then add that interest to the principal to get the total amount in the account at the end of the period:

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

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

That example uses just one time period. If you earn 2% interest for multiple time periods, the result depends on whether the investment pays simple interest or compound interest.

Simple interest equation

Simple interest is calculated when interest is earned only on the original principal. Even though the total account grows over time, the interest each period is always based on the same starting principal.

This means:

• The interest earned per period stays the same.
• The total account value grows linearly over time.

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, you plug values into the equation. The key step is making sure $t$ matches the given time period.

• The interest period is every 4 months.
• 3 years is $3\times 12=36$ months.
• The number of 4-month periods is $36/4=9$.

So $t=9$.

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

Compound interest equation

Compound interest is calculated differently from simple interest. With compound interest, each period’s interest is added to the account, and then the next period’s interest is calculated using this new (larger) amount.

This means:

• The amount of interest earned each period can increase over time.
• The total account value grows faster than linear growth.

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?

First, calculate the total amount after 4 years:

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

The total after 4 years is $1,125.51. The total interest earned is the total minus the principal:

$1125.51-1000=125.51$.

Note that the question doesn’t explicitly say compound. Many problems will tell you whether to use simple or compound interest. If it isn’t stated, compound interest is the standard in most real-world investments, so it’s often the correct assumption.

Simple interest vs. compound interest

A multi-period compound interest account will always earn more money than a simple interest account when the principal, interest rate, and total time are the same.

For example, here are the 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!