4. PV of a Lump Sum

Let's do another present value of a lump sum problem. And here we're gonna do a couple of different things. So the problem will say something like, "In three years you'll need $100,000." So in three years you need $100,000. How much should you invest today if it earns 10%? So if you know you can earn 10%, and in three years you're gonna need $100,000, what do you invest today? So when you see a word problem like this, go ahead and draw it out. So you've got one period, two periods, and then at the end of the third you're gonna need $100,000.

So you can tell now that this is the present value of a lump sum problem. What you're trying to figure out is, if you're earning 10% a period, how much do you need to invest here to make it grow to $100,000? And this is important because the problem might not say, "Hey, heads up, I'm a present value of a lump sum problem." It might just say, "In three years you'll need $100,000. You can earn 10%." And you have to figure out, "Okay, this is a lump sum problem." Now, I wanna make this one a little bit more complicated, because this isn't super tricky.

So instead of saying earns 10%, I'm gonna say earns 10% and it's compounded semi-annually. So now, instead of having three periods, you really have six periods, because it's half a year, half a year, half a year, half a year, half a year, half a year, right? So now we have six periods. So what we need to do, whenever you see semi-annual, you need to be extremely careful and go back and make sure that you change both of these. 'Cause when it says semi-annual, you're no longer dealing with three periods at 10%. You're now dealing with six periods at 5%. So when you see semi-annual, make a huge deal about it, because you're gonna miss a problem if you don't.

You're gonna go forward and think, "Oh yeah, three years, 10%, no problem." But because it's compounded every half year, you have to double the number of periods and halve the interest rate. And this will be tricky, because on the test you'll see a table like this, and I'll draw it out for you real fast.

So on the problem, this will be the table you'll see, or you may have a whole sheet of present and future value, and so you're gonna be straight out of luck. So if on the test you see this, you may look and think, "Well, why is she giving me six periods? It's a three-period problem." Well, it's a three-period problem if you've messed it up. Because it says semi-annual, you need to make sure you're looking down here at the six, and you need to make sure you're looking at the five, because what you have to do is you have to do the i divided by 2 because it's semi-annual, which in this case gives us 5%.

So I can't stress enough how important it is that you make sure you mark it when it says semi-annually. And if it says monthly, you're not dividing by 2, then you're dividing by 12. So then you have three years times every month. You have 36 periods, and you have 10% divided by 12 months. You have a very low percentage rate. So make sure that you're checking in relation to a year, how is this compounded? Twice a year? Three times a year? Twelve times a year? Okay, I think you get the idea. So now we've decided we're at six periods for 5%.

So let's go over here. Six periods at 5% gives us 0.7462. So to drag this $100,000 all the way back to the present, we multiply times 0.7462. And I can just tell that's gonna be $74,620. So if I need $100,000 in three years compounded at 10% annually or 5% semi-annually, I need to invest $74,620. Because if you look, $74,620 times 1.05, which is our percentage rate, times six periods, gives us $99,997.93, which is gonna be $100,000, okay?

So you can just double-check by taking this present value and dragging it into the future using 1.05 to the sixth. And that is just kind of a gut check to make sure that you're doing this correctly. So what we said was we said that the present value, this number here that we're looking for, is equal to the future value times the present value of a dollar. And then we did it the other way, and we said the future value is equal to the present value times the future value of a dollar. So when you see a problem like this, make sure you know which one you're moving.

Here, we're moving the future value back, so what we need is the present value of a dollar, so you need to be looking at that chart. Now, if you're going the other way and you have the present value and you're trying to drag it into the future, you need the future value of a dollar, so make sure you're looking at that chart.