Most of us have either a checking or savings account that pays some form of interest. Most
banks use a formula called compounded interest to calculate the interest they pay you each month.
This video will address how you use compounded interest to solve interest problems.
Let's take a look at the formula first. Here it is: It looks like if a principal P
is invested at an interest rate I
expressed as a decimal and compounded N times a year in T years, it will grow to the amount A.
To reword this with maybe some simpler language: P is the starting amount or the initial investment.
I is the interest rate that must be expressed as a decimal. Remember to convert a percentage to a
decimal; we must remove the percentage symbol and divide by 100. N is the number of compounds per
year. So if it's compounded quarterly, N would be 4, since there are four quarters in a year.
Compounded monthly would be 12, and so on. T is time,
but it must be in years. So, of course, if two years, T would be 2.
Let's say they told you the time was 18 months. 18 months would be a year and a half, or 1.5 years.
Lastly, A is the amount after the given time.
Let's take a look at a couple of quick examples.
Suppose that you invest $1,000 at 8% interest compounded quarterly. How
much is in the account at the end of three years? Okay, let's set this up.
What would our principal be? Again, the principal is the
starting amount or the initial investment, so that's $1,000
times the quantity one plus I,
the interest rate as a decimal. Eight percent as a decimal would be 0.08.
Now this is compounded quarterly, so N would be 4,
and then we're raising this to the power of N times T. Well, we already said N was 4;
T is time in years, so 3. Now let's take a look at this for a moment. This may seem kind of odd,
but essentially what we're doing is we're getting common units. What I mean by that is this:
If we take an 8% annual interest rate and divide it by 4, we're essentially getting a quarterly
interest rate. If you multiply 4 times 3, that would be the number of quarters in three years.
So essentially what we're doing here is we're putting everything in quarters.
Okay, let's go ahead and go to the calculator and simplify this expression
on the right side. So we're going to enter it in pretty much just as we see it:
1,000 times the quantity one plus 0.08 divided by 4,
and that's going to be raised to the 4 times 3 power. I'll put 4 times 3 in parentheses.
Of course, I could just put 12, and that's pretty much all we have to do. This gives us
A, or the amount after 3 years, which is $1,268.24.
It's a pretty straightforward example of compounded interest. Now, for the next example,
what I want to do is compare what is going to happen to this balance if we change just one
condition. We're going to change how often it's compounded. Instead of compounded quarterly,
let's take a look at what we are going to do if it's compounded
daily. Same starting amount, same interest rate, same amount of time, and we'll see the difference.
Okay, so let's write out our formula.
The amount after three years equals the principal, still $1,000,
times the quantity one plus the interest rate as a decimal.
We're supposed to divide this by N. If it's compounded daily,
we assume there are 365 days in a year.
We raise this to the N times T power: 365 days times 3.
Again, we have a daily rate here, and we have the number of days in 3 years here.
Let's go back to our calculator and determine this amount.
You might be thinking which will give you more money. One nice thing about the graphing
calculator is instead of retyping everything that I just did, if I hit second enter,
it brings back the last expression. Except now what I can do is go back and edit anything I want.
So the only change is from changing this 4 to 365. Now 4 is a single digit, so I can overwrite the
4 with the 3. Now I have to insert the 65. Second delete is the insert and get the 65.
Of course, I could just delete everything and retype it, but I'm trying to save a little bit
of work here. I'll do the same thing with this 4. I'll overwrite the 4 with the 3
and then I'll insert the 65. And if we compare these amounts,
this amount is $1,271.22. As you can see from the previous problem, we had $1,268, so the amount has
increased. Hopefully, that makes sense, because if you're being paid interest on a daily basis
instead of a monthly basis, the more money you get paid more often, the better off you would be.
Now, this may not seem like a big difference, but of course,
if you're dealing in millions and billions or even trillions of dollars, it would add up.
Okay, that's pretty much how you deal with compounded interest.
I hope that helps, and I'll leave you with a thought.