3. PV of a Lump Sum

Let's talk about the present value of a lump sum. Now, I kind of already teased this in the video where we talked about what present value actually is, but let's go into a little bit more depth here. So the question is gonna be something like, is it better to have $1,000 now or $1,400 in three years? So it'll say three years, rate is, let's say, 10%, and would you rather have 1,400 in three or 1,000 now? So like I said in that introduction video, you can't compare 1,400 and 1,000 because they're at different times, and because you're able to earn 10%, you need to figure out what is this $1,000 going to grow to over three years, or what would it take to get to $1,400 over three years.

And so that's what we're saying when we say the present value of a lump sum. What we're gonna do is we're gonna take this $1,400 and drag it into the present, and then we can just compare the two numbers straight out. Now, on your test, you'll maybe have a chart that says 3 n, 10%, and the value is 0.7513. So what this is telling you is the present value of $1 over three periods at 10% is 0.7513. So if you were to bring this $1,400 back into the present, it would only be worth 75% of what it was three years from now.

So if we do that math and we say $1,400 times 0.7513 gives us $1,051. So this $1,400 brought into the present is worth $1,051. So right away that tells you, yeah, that's probably better because $1,051 is more than $1,000. Now, what if we wanted to drag this $1,000 into the future instead? Well, we don't have the chart right here, but what we can say is it's $1,000 times 1 plus the rate, so 1.1, to the number of periods. So what we're gonna do is we're gonna take the principal and multiply it by 1.1, which is 110%, and we're gonna do that three times.

So if we do that math and we say $1,000 times 1.1 to the third, we get $1,331. And again, as you would expect, the $1,400 is bigger. So the reason I did this both directions is to show you that on your test, no matter which one you get, if you get the present value of a dollar or if you get the future value of a dollar, you can go either direction. You can pull the $1,000 into the future, or you can pull the $1,400 from the future into the present. So in both cases, the $1,400 or the $1,051 is higher than the $1,000 or the $1,331.

So the answer to the question would be, you want the $1,400 in three years. Because if you got $1,000 now, the best you could do growing it at 10% would be to get to $1,331. You would have to get $1,051 now, and then if you grew that at 10%, you would get your $1,400. So when presented with a present value of a lump sum problem, go ahead and draw it out, and then based on whatever information you have, for example, here we had the present value chart. Well then draw a circle around it and pull the arrow back and say, "Okay, well, I'm pulling this $1,400 back into the present.

It's out in the future three years. I'm gonna pull it back into the present using this factor right here, this 0.7513 factor." And again, I wanna show you that if you had 1 divided by 1.1 to the third, there's your 0.7513 factor right there. So that's how these numbers are made. We said 1.1 times itself three times. Divide 1 by that number, you get your present value factor. So these will be on the test, but I just want you to know that's where that comes from.