Compounded Interest — Transcript

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1. 0:01Most of us have either a checking or savings  account that pays some form of interest. Most
2. 0:07banks use a formula called compounded interest to  calculate the interest they pay you each month.
3. 0:13This video will address how you use compounded  interest to solve interest problems.
4. 0:21Let's take a look at the formula first.  Here it is: It looks like if a principal P
5. 0:25is invested at an interest rate I
6. 0:28expressed as a decimal and compounded N times a  year in T years, it will grow to the amount A.
7. 0:37To reword this with maybe some simpler language: P  is the starting amount or the initial investment.
8. 0:47I is the interest rate that must be expressed as  a decimal. Remember to convert a percentage to a
9. 0:52decimal; we must remove the percentage symbol and  divide by 100. N is the number of compounds per
10. 1:00year. So if it's compounded quarterly, N would  be 4, since there are four quarters in a year.
11. 1:08Compounded monthly would be  12, and so on. T is time,
12. 1:15but it must be in years. So, of  course, if two years, T would be 2.
13. 1:20Let's say they told you the time was 18 months. 18  months would be a year and a half, or 1.5 years.
14. 1:28Lastly, A is the amount after the given time.
15. 1:32Let's take a look at a couple of quick examples.
16. 1:36Suppose that you invest $1,000 at 8%  interest compounded quarterly. How
17. 1:44much is in the account at the end of  three years? Okay, let's set this up.
18. 1:54What would our principal be?  Again, the principal is the
19. 1:56starting amount or the initial  investment, so that's $1,000
20. 2:05times the quantity one plus I,
21. 2:08the interest rate as a decimal. Eight  percent as a decimal would be 0.08.
22. 2:18Now this is compounded quarterly, so N would be 4,
23. 2:23and then we're raising this to the power of  N times T. Well, we already said N was 4;
24. 2:29T is time in years, so 3. Now let's take a look  at this for a moment. This may seem kind of odd,
25. 2:36but essentially what we're doing is we're getting  common units. What I mean by that is this:
26. 2:41If we take an 8% annual interest rate and divide  it by 4, we're essentially getting a quarterly
27. 2:47interest rate. If you multiply 4 times 3, that  would be the number of quarters in three years.
28. 2:56So essentially what we're doing here is  we're putting everything in quarters.
29. 3:02Okay, let's go ahead and go to the  calculator and simplify this expression
30. 3:08on the right side. So we're going to  enter it in pretty much just as we see it:
31. 3:131,000 times the quantity  one plus 0.08 divided by 4,
32. 3:25and that's going to be raised to the 4 times  3 power. I'll put 4 times 3 in parentheses.
33. 3:33Of course, I could just put 12, and that's  pretty much all we have to do. This gives us
34. 3:40A, or the amount after 3  years, which is $1,268.24.
35. 4:03It's a pretty straightforward example of  compounded interest. Now, for the next example,
36. 4:08what I want to do is compare what is going to  happen to this balance if we change just one
37. 4:14condition. We're going to change how often it's  compounded. Instead of compounded quarterly,
38. 4:22let's take a look at what we are  going to do if it's compounded
39. 4:25daily. Same starting amount, same interest rate,  same amount of time, and we'll see the difference.
40. 4:32Okay, so let's write out our formula.
41. 4:36The amount after three years  equals the principal, still $1,000,
42. 4:44times the quantity one plus  the interest rate as a decimal.
43. 4:52We're supposed to divide this  by N. If it's compounded daily,
44. 4:55we assume there are 365 days in a year.
45. 5:02We raise this to the N times  T power: 365 days times 3.
46. 5:10Again, we have a daily rate here, and we  have the number of days in 3 years here.
47. 5:16Let's go back to our calculator  and determine this amount.
48. 5:21You might be thinking which will give you  more money. One nice thing about the graphing
49. 5:27calculator is instead of retyping everything  that I just did, if I hit second enter,
50. 5:34it brings back the last expression. Except now  what I can do is go back and edit anything I want.
51. 5:42So the only change is from changing this 4 to 365.  Now 4 is a single digit, so I can overwrite the
52. 5:494 with the 3. Now I have to insert the 65.  Second delete is the insert and get the 65.
53. 5:59Of course, I could just delete everything and  retype it, but I'm trying to save a little bit
54. 6:03of work here. I'll do the same thing with  this 4. I'll overwrite the 4 with the 3
55. 6:09and then I'll insert the 65.  And if we compare these amounts,
56. 6:17this amount is $1,271.22. As you can see from the  previous problem, we had $1,268, so the amount has
57. 6:32increased. Hopefully, that makes sense, because  if you're being paid interest on a daily basis
58. 6:37instead of a monthly basis, the more money you  get paid more often, the better off you would be.
59. 6:44Now, this may not seem like a  big difference, but of course,
60. 6:47if you're dealing in millions and billions or  even trillions of dollars, it would add up.
61. 6:56Okay, that's pretty much how you  deal with compounded interest.
62. 7:00I hope that helps, and I'll  leave you with a thought.