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Alright. So the formulas we've been using so far, that future value equals present value times one plus r to the end or the other one where we rearrange that same formula. Those were for lump sums of money. That was for when we had a lump sum of money. Right? We were talking about, okay I'm trying to save $12,000 at a future point in time or I'm depositing $100 in the bank. Right? There's just one lump sum of money and it's not a stream of cash flows. Okay When we deal with time value of money, we also talk about a specific stream of cash flows called an annuity. Okay. We talk about annuities. So an annuity. Well these are payments of the same amount of money. So when I say the same amount of money it has to be $500 every time or $1000 every time there's a payment and it has to be a regular and I want to say equal intervals, so regular intervals and it's the same amount of money. So we're going to say something like, okay every year there's gonna be a $500 payment every year or every month there's gonna be a $500 payment. As long as it's equal intervals. It could be every day, a $500 payment whatever it is, It has to be the same amount of money at the equal intervals for it to be an annuity. Okay, most likely you're gonna see annuities that are gonna be interest payments and interest payments that you usually make annually. So you're gonna be making an annual interest payment or semiannual interest payments where you're making interest payments every six months rather than every 12 months. Okay. So finding the present value of an annuity. So when I say the present value of an annuity how much is this stream of payments? So it's not just, how much is this one lump sum of money worth today? No. How much is this stream of payments worth today or the future value? How much is this stream of payments worth at some future date? Well those formulas, they're beyond the scope of this class, luckily you're not going to have to use a formula when we're calculating the present value of an annuity, which is usually what we're gonna be doing is the present value of an annuity. We're gonna have tables, we're gonna have what's called a present value table or a future value table that's gonna give us some ratio that we use rather than have to do a whole formula. And it makes the calculation a lot easier. Okay, We're not gonna talk about the tables in this video, we'll talk about it in a future video. But what I wanna do is just get familiar with the topic of annuities and we're gonna do timelines here uh to get familiar with annuities. So let's start with this example here. The example is you have reached retirement and have earned a pension that will pay you $10,000 annually for the next five years, let's visualize this information on a timeline. Okay, so you've earned a five year pension that's gonna give you $10,000 each year for five years. So what's gonna happen is let's draw our timeline and we're gonna be right here at year zero right now, then there's year one year, two, year three, your four and your five. Right? So for the next five years you're going to get $10,000 payments. And generally when we talk about annuities, there's no payment right away. All these payments start one year from now. Okay. And that's called an ordinary annuity. There's other types of, But we're not going to get into those in this course. This course deals with ordinary annuities where the payments start one year from now. Okay, so that's exactly what we have here. We're going to get $10,000 annually for the next five years. So that means we're gonna get a cash flow of $10,000 here, a cash flow of $10,000 here, $10,000 here. So every year, $10,000 for five years. Right? So notice before when we were drawing our timelines, there was just one cash flow, we just drew, okay, we need this $12,000, 2 years from now or whatever the cash flow was, there was always just one cash flow that we wrote in and then we found out what that was worth at different point in time. So when we find the present value of an annuity we don't find the present value of each of these cash flows separately. Yes it's possible to discount each of these cash flows to today's date separately. But that would take a long time right? We would have to use our present value formula over and over again for each of these cash flows at different points in time. So what we do with the present value of an annuity we're gonna take all of the cash flows and we're gonna bring them all back using our Our table will learn how to do that and we're gonna find the present value of the annuity. So it's gonna be some amount that's worth the same amount as if you were to take $10,000 each year for five years. Okay so we're gonna find the present value of an annuity like that and it's gonna use the same kind of principles. We're gonna have some we're gonna have some n for the number of periods. So in this case would be five periods. We would have some are for our interest rate. But now instead of a future value or a present value. Well we're searching for a present value. Right what is that annuity worth today? We're gonna instead have the payment, whoops. B. A. P. M. T. The payment that's the annuity payment. Okay so those are the variables we'll we'll be working with once we get to the tables and stuff, but at this point I just want you to get familiar with what an annuity looks like. Notice how this follows the rules of an annuity that we're getting $10,000 each year for five years. So it's the same amount of money for five p Periods for equal periods, right? For each year. So annually, it's not okay, you're going to get $10,000 in a year and then $5,000 in two years and then $8,000 in three years. That wouldn't be an annuity. Okay. The annuity has to be the same amount of money each period for equal equal space in between each payment, which is usually a year like this. All right. So why don't you guys go ahead and try and build the timeline in the next problem?

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