5. Ordinary Annuity

We've already looked at the present value of a lump sum. Now we're gonna look at something called annuities. So in our last video, we looked at lump sums. And a lump sum was basically we had a line like this, and we had some amount of money that you got paid at the end, and the goal was to bring it back to find the present value, right? Well, annuities are a little bit different in that they are a series of payments. So you could have something like this where it's four years, and each year you get some amount of money. So this is just an investment vehicle where you get regular, ordinary payments.

The goal here is we're gonna try to find the present value. We're gonna try to find the present value of an annuity. Now, you could take each $9,000 payment and bring it back and add it up. So you could do this, but then once you start getting to something like, I mean, even four, if we had to do these four payments, that would be kind of a pain. But once you start getting into like 6, 10, 20, 30, 50, you're not gonna do it. It's not gonna happen. So in the same way that up here we used something called the present value of $1 or the future value of $1, depending, for annuities, we have something called the present value of an ordinary annuity or the future value of an ordinary annuity.

Now, real quick, just as an aside, this ordinary bit, what makes an annuity ordinary is that the payments come at the end of the period. So one year passes, and then you get the payment. A second year passes, and then you get the payment. That's what makes it ordinary. Ordinary just means payments at the end of the period. Let's look at an example, and I can walk you through what we're doing here. This example is another one of those would-you-rather questions. It's would you rather have $25,000 today or three payments of $9,000 at the end of each year, where the prevailing interest rate is 10%.

So would you rather have $25,000 today or get three payments of $9,000 at the end of each year? And let's assume that the market interest rate is 10%. And what this means, this market interest rate is 10%, that means if you got $25,000 today, you could earn 10% on it. So this schedule is gonna look like this. You're gonna have today, Year 1, Year 2, Year 3. Okay? And at the end of each of those years, you're gonna get $9,000.

Like I said in one of the other videos, you have to have all of your money in the same time to be able to compare it. So we can't really compare this $25,000 to this $27,000 'cause if you add all these up, it's $27,000, but we don't know which one is better. So we have to bring these $9,000 back to where the 25 is so that we can compare these nines with this 25. Now let me show you how to do it the long and painful way, and then I'll show you how to do it using the present value of an ordinary annuity.

So we could, because we know that these are all 10%, if we were invested, we could earn 10%. We could do it like this. $9,000 divided by 1.1 plus $9,000 divided by 1.1 for the second year, because it's compounded annually, so it's the second year. $9,000 divided by 1.1 for the third year. So this is Year 1, this is Year 2, and this is Year 3.

Now, that's effectively saying take this $9,000, bring it back, take this $9,000, bring it back, take this $9,000, bring it back. And what you'd get here is $8,182 plus $7,438 plus $6,762, and that would come out to $22,382. Now, doing it the long way, we've just discovered that the present value of an ordinary annuity at $9,000 for three years is $22,000. So the answer to the question is take the 25, 'cause 25 is more than 22. Now, what's the way that you want to actually do this on the test? Now, on the test, you wanna look up for 10% at three periods and find this number, this present value of an ordinary annuity number.

And in this case, it's 2.4869. And if you multiplied the $9,000 times 2.4869, turns out you get the exact same thing at $22,382. Now, you may be wondering, "Okay, but if this is all I have to do, why did you show me all this garbage up here?" Well, I showed you all of this because I want you to understand that this factor here, this 2.4869, is really a combination of these three factors, this 1.1, 1.1 to the second, 1.1 to the third.

So the present value of an ordinary annuity factor is just the combination of many present value of $1 factors. So instead of having to take $9,000 times the present value of $1 for one year and 10%, $9,000 times the present value of $1 for two years at 10%, and $9,000 times the present value of $1 for three years at 10%, and then add them all up, which is what we did here, that table gives you the right amount. It gives you the factor that you need to use to bring back all three of these payments at one time.

So you just take the payment amount times the present value of an ordinary annuity, and that gives you the present value of all three of those payments. Now, what you're gonna wanna look out for on the test is making sure that when you see a stream of payments like this, make sure that you use the ordinary annuity table. If you see a single payment like this, make sure you use the lump sum table or just the present value of a single dollar table. So you gotta make sure you're using the right table.

If it's multiple equal payments, that's an annuity. That's the definition of an annuity. An annuity is a series of equal payments, and an ordinary annuity is a series of equal payments made at the end of each period.