Time Value of Money Part V
Compound Interest Calculation – The Card at Every Birthday Party
When I was young, my grandparents started a college account for me. Instead of giving me a present or a gift card for my birthday, they would always give me a card letting me know the amount that they had contributed to my college account. They also told us that if we would contribute to the account, they would match whatever we saved with our own money. As a young kid, it was tough to round up some of my hard earned money to put it into an account where I felt like I would never see it again. In the end, I made some contributions that were graciously matched by my grandparents. As I got older, I was about two years out from college and wanted to estimate how much I’d have in the account for my first year of college. To do this, I needed to work through a compound interest calculation that we’ll cover below.
Let’s say that I had $5000 in the savings account and wanted to know what it would be worth two years in the future. We’ll also say that the interest rate that I was able to get was 4%. Now up to this point, we’ve only used time frames of one year. So how do we figure out how much that we’d have in two years? We’ll need to apply the concept of compounding interest. To do this, let’s work our way through this compound interest calculation year by year.
Since we have $5000 now, we can calculate how much we’ll have in one year at a 4% interest rate. What we are calculating in this section is the future value of the $5000 in one year. By multiplying $5000 by 1 plus the interest rate, we get:
$5000 x (1 + 4%) = $5000 x (1 + 0.04) = $5000 x (1.04) = $5200
Since we started with $5000 and ended up with $5200 in one year, we got $200 interest in the first year. So to figure how much more interest we’d get in the second year, we can just add another $200 and get $5400, right? Well, not exactly. The one piece that we’re missing is that we actually get to earn interest in the second year on the $200 interest that we made in the first year. Because of this, we’ve actually got to multiply the $5200 we had at the end of the first year by 1 plus the interest rate again to find out how much we’ll have after two years.
$5200 x (1 + 4%) = $5200 x (1 + 0.04) = $5200 x (1.04) = $5408
Note that by getting extra interest in year two on the interest we made in year one, we actually got $8 more than we would have had if it was not compounded.
I Feel the Power From the Compound Interest Calculation
Since we can see how to solve the problem by breaking it into two steps, let’s try it again by putting the whole thing into one compound interest calculation. Since in the first year we multiplied $5000 by 1.04 (which equaled $5200) and then multiplied that value by 1.04 again in year two, we can write it out like this:
$5000 x 1.04 x 1.04 = $5408
So we’ve combined the two year’s of equations into one pretty short compound interest calculation. Is there any way that we can make it even easier? Well there’s another trick that we’ll use in the future so we can apply it to a number of examples. Remember back in math class how they would show a number that’s multiplied by itself? Let’s take 1.04 from the previous example below:
1.04 x 1.04 = 1.04 ^ 2
The little upward pointing arrow is spoken as “to the power” when describing it. So for this example, we’d say that we have 1.04 to the 2nd power. Essentially, this is just a math way of saying that you take 1.04 multiplied by itself. So for this problem, we can simplify it down to:
$5000 x 1.04 ^2 = $5408
Key Takeaway: When working to find the future value of an amount further out than one time period (in this case longer than one year out), we get interest on the previous interest which we have to account for. This is referred to as compounding and we’ll be using the compound interest calculation in a lot of future topics.
In the next small business financial management lesson, we’ll go through a couple more examples on how we can use the equations in this case on a number of different problems.