2. Simple vs Compound

Let's talk about simple versus compound interest. And I think it was Einstein that said there's no force more powerful on Earth than compound interest, and you'll see why here in a second. On simple interest, it's, well, simple. If you invest $1,000 at 9%, so you're earning 9% interest, after one year you will have earned $90. A year later, you're gonna earn another $90. If you keep it invested for three years, you're gonna earn another $90. Because simple interest is just the principal times the rate. So for simple, it's principal only.

So each year you keep it invested, you earn 9% of principal. Now, on the other hand, compound is where things get fun. After the first year, you earn $90. But after the second year, you earn 9% on that first 90 that you earned. So now you earn $98. You see where this is going?

On the third year, you're earning 9% on this first 90 and on this second 98. So you earn somewhere around $107. So you end up earning $295. And as you can see, this just keeps increasing so that you're earning more and more, faster and faster. And this right here, this schedule right here is why you want to put as much money in retirement as early as possible. The most money you can put in your retirement accounts when you get out of college, put all of it in there. Because in the end, you're gonna earn way more off of your earned interest than you are off of your principal.

So the earliest you can start putting money into retirement, do it, because over 30, 40 years, compound interest just goes bananas, and you're gonna be very well off and very happy that you did that right when you got out of college. Okay, enough life advice. So this is the way that you do it manually. Now, what happens when you see a problem that says over 10 years, 20 years, 30 years? Well, then you've got a problem because you don't wanna do this, especially on this side, you don't wanna do this 30 times. So over here, what did we actually do? Well, we said $1,000 times .09 times 3.

Isn't that what we effectively did? Because $1,000 times .09 will give us 90. We added 90 three times, which is the same as multiplying by three. So over on the simple side, it's just principal times rate times period. And you could see that as P times I times N, or P times little i times n, something like that. Principal times rate times period. And that will give you your $270 in interest. Now, the harder question is what did we do on this side? Well, we said it was $1,000 times 1.09 gave us $1,090, times 1.09 gave us $1,188, times 1.09 gave us $1,295, minus our $1,000 gives us the $295 in interest.

So over here it's much more complicated. So what we functionally did was we said it's $1,000 times 1.09 times 1.09 times 1.09, and that will give us our $1,295, because this gets us to $1,090, this gets us to $1,188, and this gets us to our $1,295. So going even further, we say it's $1,000 times 1.09 to the third. Now, if we take 1.09 to the third and solve that out, 1.09 raised to the third power, we get 1.295.

So 1.295 times $1,000 is going to give us that $1,295. And the reason I went through all of that was because I wanted to show you this number right here. This 1.295. This 1.295 is the future value of $1 at three periods for a rate of 9%.

So if you were to look up the future value of $1, go down to the 3 row and over to the 9% row, you would find 1.295. Because this right here, all this garbage we just did, that is how those future value and present value tables came into existence. That's how those are calculated. They're not just made up, and it's not some complicated process even. It's just saying, "Okay, well, at 9% for three periods, you just do 1.09, 1.09, 1.09 three times and you end up with 1.295."

Now on your test and on all your homework and stuff, you're gonna have the future value and the present value tables, I would assume. So you're not gonna need to work all this out on the test, especially because on your test, I don't think you have calculators capable of doing exponents. So much beyond three, four, or five, and it's gonna be a huge pain anyway. But I did want you to see that the future value and the present value tables are calculated in a really straightforward and really understandable way, and something that you can figure out all on your own. Because you may have one of them on there, just like this one that we just did, where it's pretty simple.

It's only three years at 9%, and you may not have the future value of a dollar. And so what you have to do is you have to say, "Okay, well, it's $1,000. It's my principal times my rate three times." So over here, I gave you a nice little formula, P times I times N. On the compound side, it's principal times 1 plus the rate raised to the period. So it's $1,000 times 1 plus the rate of .09 raised to three periods, or $1,000 times 1.09 three times. So what we've done now is we've said the present value is $1,000, and after one, two, three years, the future value is $1,295, 'cause we said it's something like $1,090, $1,188, $1,295.

So what you just did, if you were following along, is you just calculated the future value of $1 for n = 3, i = 9%. 'Cause the future value of $1 is... Well, let's just put it over here. The future value of $1 is 1.295. $1 becomes $1.295, so $1,000 must become $1,295. And now, just for giggles, let's see if we can go from future back to present. Well, we know that to get from present to future, we multiplied by 1.09 three times. Now, what if we wanted to know how do we go back to $1?

Well, you've got the three periods, and to go from the first period, you have to divide by 1.09. Second period, you divide by 1.09. And third period, you divide by 1.09. Let's see if I'm being honest here. 1.295 divided by 1.09 divided by 1.09 divided by 1.09. Pretty close. I'm gonna go ahead and call that one. So when the rate is 9% and the period is three years, what you're doing is you're dividing the 1.95 by 1.09 three times, and that gets you all the way back to a dollar.

So what we basically just did was we took the future, $1.295, divided it by 1.09 to the third and said, "We get $1." Well, obviously, because 1.09 to the third is 1.295. So if we flipped it and said 1.295 times 1 over 1.09 to the third, which is the same thing as dividing it by, right? So these two right here are the exact same, if you'll remember your algebra. 1.295 divided by 1.09 to the third is the same as 1.295 times 1 over 1.09 to the third.

That still equals a dollar. But what is this value right here? What does that reduce down to? Well, it's simply 1 divided by 1.295. So you get 0.7704. So this value is 0.7704. Okay? So this is the present value of $1 for three periods at 9%. So if you were to look up in an actual chart at three periods, 9%, you would find present value is 0.7704. Let's try that out. Let's see if I'm telling the truth. $1,295 times 0.7704: $997.60.

We're gonna call that close enough and say that the present value would be $1,000. So if you wanted to bring the $1,000 forward, you use 1.295. If you wanna bring the $1,295 back, you use .7704. This stuff truly is a little bit tricky to understand. It's kinda hard to understand the future value at some period for some rate. Why are we doing all of this stuff? But what I wanted to show you was starting simply, starting simply by saying $1,000 at 9% for three years. We ended up with $295 in interest, or $1,295 total.

Right? And the way that we did that was first, we did it manually. $1,000 times 9% is $90. $1,090 times 9% is $98. $1,188 times 9% is $107. All of that adds up to $1,295. Then we did it a little bit better by saying, instead of calculating just the interest, we calculate the whole amount 'cause we're gonna use that again. So we said $1,000, 1.09, $1,090. 1.09, $1,188. 1.09, $1,295. Got it. Then I said, "Well, why don't we just drop out these middle ones?" Because that's effectively what we're doing.

You can say $1,000 times 1.09, 1.09, 1.09, and you get the $1,295. Then even simplify it a little further, whenever you see these three right here, you should say, "Oh, well, that's just 1.09 to the third." So you can do your principal times 1.09 to the third, and this is just a number. You can solve for 1.09 to the third, and you get 1.295. So if you did $1,000 times 1.295, you get $1,295, and that's way easier than doing all this, right?

Now, I told you this is the future value of $1, and you will be able to look this up in your book for most of them, but should it not be there in your book, you should feel comfortable and confident that you know how to figure it out yourself. 'Cause we said the future value of $1 at three periods, this is right here, three periods, and 9%, the number that we used, is 1.295. And then we took it a step further and we said, "Okay, well, that's the future value of a dollar at n = 3 and i = 9%.

What's the present value of a dollar at n = 3, i = 9%?" And what we decided was, instead of just multiplying it by 1.09 to the third, which really is 1.295, the inverse of that would just be 1 over 1.295, which turns out to be .7704. And now I'm wondering if I did that math wrong. 1 divided by 1.295. Of course, I did. .7722.

Man, even with a calculator. .7722. So this should actually be 0.7722, which means that should also be .7722. So if you'd done the math correctly, you would just say 1 over 1.295, and that gives you your present value of $1. Now, if you wanna see something crazy, let's take the present value of $1, .7722, times the future value of $1. Let's see if I can do this one right.

And that gives you $1. How fun is that? So what this little formula right here actually tells you is if you had 77 cents in the present and brought it to the future, you'd have a dollar. If you had a dollar twenty-nine in the future and brought it to the past, you'd have a dollar. That's not super useful, but it is a cool little math trick, and it does prove the point that the present value of a dollar times the future value of a dollar gives you a dollar.