Another type of interest is called compound interest. This interest is calculated using the principal amount and the interest from previous periods. Think of it as accumulating interest on interest. Is more interest earned or due using compound interest rather than simple interest? Yes, because the interest is being compounded. The higher the number of compounding periods, the greater the amount of the compound interest. Like simple interest, compound interest also has a formula.
Let's Watch
In the video below, an instructor will explain and demonstrate the use of compound interest. As you watch the video, pay close attention to what each letter in the formula represents.
Today, we're going to tackle a big, nasty-looking equation for finding compound interest – but don't fret! We're going to go through the equation step by step, variable by variable, so that the lesson is as digestible as possible. One we've got the formula down, it'll be smooth sailing.
So, here's the equation we're working with for compound interest:
\( A = P (1 + \frac{r}{n})^{nt} \)
- A stands for the amount you will have at the end of the timeframe.
- P stands for the principle or the amount that you initially invest.
- r is the rate that the bank gives you as a percent; be sure to convert to a decimal when you plug it in! For instance, a 3% rate is plugged in as .03, and an 11% rate is plugged in as .11.
- n represents the number of times per year that the principal is compounded.
- and t represents the time you are allowing the money to earn interest in years.
So, let's takes this equation for a spin. How much money would you end up with if you invested $2,000 at 6% interest for 4 years compounded semi-annually (twice a year)? At first, this question looks daunting. We're going to break it down just like we did the equation.
For our principal – our P value – we're going to plug in our 2,000 dollars. For our rate, we're going to plug in .06. For our n value, we're going to plug in 2. And for our t, we're going to plug in 4.
\( A = 2000 (1 + \frac{.06}{2})^{2\cdot4} \)
.06 divided by 2 equals .03, 1 plus .03 equals 1.03, and 2 times 4 equals 8, so we can reduce our equation to:
\( A = 2000 (1.03)^8 \)
1.03 to the eighth power is 1.26677008138, which we will round to 1.267, and 2000 times 1.267 equals $2534. So our solution is that if you invest $2,000 at a 6% rate for four years compounded semi-annually, you will end up with $2534.
A = $2534
See? It's not quite as complicated as it looks. The key to performing this kind of operation is to memorize the formula and know where to plug information into the correct variable. From there, it's simply a matter of simplifying for your solution. See you next time!
Question
What is the formula for compound interest? What does each letter in the formula represent?
The formula for compound interest is: \( A = P\left( 1 + \frac{r}{n} \right)^{nt} \). \( A \) represents the future value of the loan or investment, which is the total amount you will have paid or earned (including the principal) at the end of the term; \( P \) represents the principal or initial amount; \( r \) represents the interest rate (as a decimal); \( n \) represents the number of times per year that the interest is compounded; and \( t \) represents the time period of the loan or investment, in years.
