 We'll come to the session in which we would look at the future value of an annuity. In the prior session, we looked at the future value of a single amount. If you are not comfortable with the future value of a single amount, I strongly suggest you go back and review the prior session, because if you don't understand the single amount, you won't be able to understand the future value of an annuity. This topic is covered in your accounting courses, finance courses, and of course the CPA exam. Whether you are an accounting student or a CPA candidate, I strongly suggest you take a look at my website, farhatlectures.com. I don't replace your CPA review course, whatever that CPA review course happens to be. I'm a useful addition to your CPA review course. I can help you understand your CPA review course better, which in turn will help you do better on the exam. Your risk is one month of subscription. You can give me a try, you like it, you see it's helping you, you keep it. If not, you cancel. Your potential gain is passing the exam. I have resources for other accounting courses, intermediate, advanced, auditing, governmental, so on and so forth. And my supplemental CPA review courses are aligned with Wiley, Becker, Roger, and Gleam. So it's very easy to go back and forth between your course and my resources. And I do have all the previously AI CPA released questions, and they are all contained detailed solution. Please connect with me on LinkedIn. If you haven't done so, take a look at my LinkedIn recommendation like this recording. Share it with others. Connect with me on Instagram, Facebook, Twitter, and Reddit. Let's talk about the future value of an annuity. As I mentioned, the future value of a single payment is a prerequisite session. It's very important that you understand this topic. Specifically, you need to understand the simple versus the compound interest, how the simple versus the compound interest work. So in the prior session, we looked at the future value of a single payment. Now we're going to be studying the future value of an annuity. So the question is, what is an annuity? Well, an annuity is a periodic payment or receipt of the same amount at the same interval. Well, it looks something like this. You have a time period, you know, whether it's monthly, yearly, let's assume yearly, year one, year two, year three, year four, year five, year six. And you have either making the same payment or receiving the same amount of money at those intervals. This is what an annuity looks like. For example, if you pay for a car, you make the same payment every month. If you have a mortgage, you make the same payment every month. That is an annuity. Now we have two types of different annuities. But the point is it's the same amount of money, either paid or receipts at the same time interval. Those payments are called rental because you pay rent, you know, at the beginning of every month, just rental, just to make you remember. We have two types of future value of an annuity. We have ordinary annuity and we have annuity due. And you need to know the difference between two. Once we understand ordinary annuity, you'll be able to understand annuity due. So ordinary annuity is when the payment or receipts start at the end of the period. Simply it would look something like this. So this is today. One, two, three, four. Those are the period. Nothing happened at point zero. For example, you have to make the first payment one month from now, the second payment, second month, third month and fourth month. Annuity due is totally different. In annuity due, you make four payments or four receipts starting today. Okay, so this is the annuity due, the green one. The annuity due will earn more interest because if you are making deposit, your deposit start, your first deposit start to earn money now. Versus an ordinary annuity, you don't make the first deposit until one year from now. So there is a difference between the two and we'll discuss those later. Starting with future value of an ordinary annuity. The best way to illustrate this is to work an example. Assume we invest $10,000 at the end of each year for the next five years and that money is invested at 10%. Let's take a look at it from a pictorial point of view. We're going to deposit 10K, 10K, 10K, 10K and 10K. So this is zero. This is one, two, three, four, three, one, two, zero, one, two, three, four, five. And we're going to be making those deposits. How much would our future value be worth? Well, I can compute this. Let me show you how I'll compute this. Well, if you remember what we learned in the prior session, when we looked at the future value of an amount and we said, well, let's start with this first payment, this $10,000. Well, this $10,000, it's going to earn one, two, three, four. So I'm going to go to this. I'm going to say we're earning 5%. And so the first $10,000, the first $10,000, it's going to earn 10 times 1.2155. The second $10,000, I'm going to highlight the second $10,000. And it's going to earn, let's see, let me change the color here. It's going to earn one, two, three periods. It's going to earn three periods. Therefore, we're going to multiply the second $10,000 by 1.1576. So the first $10,000, this $10,000 here will have one, two, three, four, four periods compounded at four periods. So the factor is 1.12155. The second $10,000 will be compounded for three periods, 1.15762. So if we take 10,000 times the factor, 10,000 times the factor, this third $1,000, it's going to earn only two periods. It's going to be times 1.105. And the fourth payment, it's going to only earn one period, 1.05, and the last $10,000 will not earn anything. Therefore, times one. Therefore, what I did is I computed the future value of this annuity and it happens to be $55,256. Now, what I did too, I computed all these, I added all these factors and they add up to 5.5265. So if I add all the factors, take the payment times the 5.5256, I can get to the 55,256. How much, this is how much my investment is worth. Now, we call this factor 5.526 future value annuity factor. Now I can, you can compute this future value annuity factor by taking 1 plus interest rate raised to the n minus 1 divided by the interest rate. So if you take 1 plus .05 raised to the fifth power minus 1, all divided by I, which is .05, you will get 5.5256. You'll get the factor. All what you have to do is take the factor and multiply it by the payment. Take the factor, which is the, we call it the future value annuity factor multiplied by 10,000 and you will find your answer. And it's as easy as that. Well, it's not as easy because you have to compute the future value annuity factor. That's the bad news. The good news is you really don't have to do that. Why? Because we do have a future value annuity table and this is the future value annuity table. Simply put, in this example here, it's five years at five periods. So we go, this is future value annuity table. This is a different table than the future value of a single amount. Here we are dealing with the future value of a single amount. Really, the annuity is a multiple single amounts, but if they are all the same, we have a table for that. So what we do is we come to here and we say, okay, the interest rate is five, the periods are five and the factor is 5.5256. Notice 5.5256 is right here. Therefore, you can find the future value of an annuity using the table. Well, let's find this if we can find this future value of this annuity. Adam will deposit $75,000 at the end of each six month period for the next three years. Notice Adam is making the payment twice a year. So this is not a yearly problem. This is a semi-annual problem. Three years, it means n, the periods equal to six. The rate of return on this investment is 10%. Well, 10% that's annually, since this is a semi-annual problem, we're going to have to divide this by two, interest rate equal to 5%. So to find the future value of this annuity, we're going to take the payment of $75,000, multiply it by the future value factor of the ordinary annuity, n equal to six, i equal to 5%. I'm going to come to the table here, n equal to six is here, i equal to five is here. They meet at 6.8019. So I'm going to take 75,000 times 6.8019. And by doing so, I'm going to take 75,000 times 6.8019. My future value is $510,142.50. I found the future value of my annuity. Notice here, I'm not using annual, I'm using semi-annual periods, so be careful. So this is how you find the future value of any annuity. Now you can give me a payment for how long, what interest rate, and I can find the future value of an annuity. Let's talk about the future value of an annuity due. Remember, what's the difference between ordinary annuity and annuity due? In an ordinary annuity, when Adam started to make those payments, one, two, three, four, five, six. One, two, three, four, five, six. The first 75,000 took place here. If it was, in an annuity due, the first 75,000 will take place here. Two, three, four, five, six. So Adam will earn more money because the first payment was made today. Started to earn, so we have this extra interest that we are going to earn. So this is the annuity due. The periodic rent occurring at the beginning of the period, not at the end. So basically we're going to earn an additional year of interest, and I just showed you on the graph for Adam where it's earned earlier. Now, in some textbook, they give you the future value of an annuity due table. If they give you the table, perfect. That's excellent. Use the table. Make sure you are looking at the table that says future value of an annuity due. Are we all okay with this? In most textbooks, they don't give you the stable. They expect you to know how to compute the future value of an annuity due. And how do you compute the future value of an annuity due? You will take the future value of an ordinary annuity. You multiply it by one plus the interest rate, and this is how you convert an ordinary annuity to an annuity due factor. Now, it's very important to understand when am I dealing with an annuity due, when the problems state the first payment is made today. And this is going to, you're going to see this when you are dealing with leases. When you have leases, usually the first payment is due immediately. So you are dealing with the future value of an annuity due. But all you have to do now is understand what the future value of an annuity due later on you will apply it. So let's see what is the future value of eight periodic payment of $800 starting today invested at 6%. Well, if we go back here, and this is what we're looking at, 1, 2, 3, 4, 5, 6, 7. So this is going to be 800, 800, 1, 2, 3, 4, 5, 6, 7, 8. And 800, each x is 800. Okay, so each x is 800. Notice it's starting at 0.1, it's starting at 0.0. 1, 2, 3, 4, 5, 6, 7. Okay, those are the eight payments. This is what we're looking at. What we have to do is we have to take 800 times the future value of an annuity ad annuity due. Now, we don't have the table for the future value of an annuity due. We're going to look up the table, the future value of an ordinary annuity. Eight payments, eight payments, let me change the colors, eight payments at 8%. And the factor is, I'm sorry, not 8%, 6%, 6%, right here, 6%. And the factor is 9.8975. Well, we're going to take 9.8975, not 7, 9.8975. So this should be 75. Let me do this and make sure I get this right. 9.89745 times 1.06. And that's going to be 10. So this is going to be 10.4912. I'm going to take 10.4912, 1297 to be more specific, multiplied by 800. And I should get 8393, 8393. So I just mistyped this should be 45. It should be 8393. If we go back to the Adam problem and we convert this annuity into an annuity due. So notice the ordinary annuity gave Adam 510,000. If Adam started to make the first deposit a year earlier, starting earlier, here's what's going to happen. We are dealing with a 5% investment and six periods. So it's going to be this one here, 6. It's going to be 6.8019 times 1.05. And yes, 5% 6 payment. I'm going to take 6.8019 times 1.05. And that's going to give us 7.1419. And we're going to multiply this by 75,000. And that's going to give Adam 535,649 dollars. And obviously it's more than 510,000. Why? Because Adam made a payment earlier starting the payment at the blue right here. Those are the blue payments and the blue payments will give you more future return because you earned more of the money, more money because you started earlier, started one period earlier. Sometime what's going to happen in the future value of an annuity is you are expected to find the rent payment. The rent payment is the periodic payment. Something to the fact of assume you need a down payment to buy a house and you need 14,005 years from now to buy a house. Assume you can find an investment that earns 8% some way annually. So here we go. You know you need 14,000. You know it's going to be 5 years but it's compounded some way annually. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So you know you're going to be earning 8% divide by 2 equal to 4% because it is annually. So the question is what should be your payment? So what payment do you have to make to make it equal to 14,000? Now remember that's easy. 14,000 to come up with the 14,000 you're going to take the payment times the future value of an ordinary annuity. The future value of an ordinary annuity. We know the future value. We don't know the payment but we know we have n equal to 10. We have n equal to 10, right? n equal to 10, 5 years semi-annually and i equal to 4%. We can find the future value of an ordinary annuity. We can find this part here. Whoops. We can find this part here. Okay, let's find it from the table. Well, from the table if we look at i equal to 4, n equal to 10, the factor is 12,0061. So now what we do is we say, okay, it's the 14,000 must equal. We don't know what the payment is. We don't know what the payment is. This is what we're looking for. But the payment times 12,0061 will give us 14,000. Now we can basically solve the payment by dividing both sides by 12,0061. And let's do that. So if I take 14,000 divided by 12,0061. And that's going to give me a payment of approximately 1166. Simply put, I have to deposit 1166 for the next 10 payments, a 10 payments, which is semi-annually every six months. And I will have and that money will earn 8%. And I will have in total 14,000. Well, the best way to show you this is to actually show you that it works. It's very important that you see the proof at least once. So you are convinced this thing will work. And here's the proof. The first six month you invest 1166. It's going to grow at 6%. Grow to be 1216. This amount plus 1166. It's going to give me at the beginning of the next six month, 2379. This amount will grow at 6%. It's going to grow to be 2474. Then this amount will add to it 1166. And it's going to be 3640. Then this amount will grow. And after 10, 6 months, you're going to have 14,000. You can do the math yourself if you'd like to. So simply put, you start with 1166. You let it grow at 4%. Six months later, you will add to it 1166. Then you let it grow. Then six months later, you will add 1166. So one and so forth until you get to the 10th payment. Sometime you might have to find the number of payments. In other words, the number of periodic rent, number of period and the rent. Assume your goal is to have 120,000 by making $9,615 deposit at the end of each year. So you want to have 120,000. You can afford to make this investment on a yearly basis. You can earn 8% on your money compounded annually. Keep it simple. How many deposits do you need to make? So how many periodic rent payment do you need to make? So here's what we have. We know we need 120,000. We know we can make the deposit, each deposit 9,615. We know i equal to 8%. The only thing is we don't know how many times do we need to make this payment. Well, easy. Here's what we can do. We can take 120,000. Simply put, if you really want to look at it from a mathematical perspective, it's 120. The future value equal to the periodic payment, which we know the periodic payment, 9,615 times a factor, future value factor, annuity factor. Now, we have everything except the factor. Well, we have to find what the factor is. Well, if we divide both sides by 9,615, so if we take 120 divided by 9,615, we're going to find the future value factor. And let's do that. Let's take 120,000 divided by 9,615. The factor is 12.48. Now, I know I'm going to be working with 8% here, and I'm going to go down until I find the closest thing to 12.48. 12.48 is right here, and it seems I need to make nine payments. So now, n equal to 9, I need to make nine payments. So if you would like to prove it to yourself. What do I mean by prove it to yourself? Say, if I deposit 9,615, let it grow for one year at 9%, get this number, then add to it 9,615, let it grow at 9%, and if you, I'm sorry, let it grow at, I apologize, not 9%, 8%, 8%, and do this for nine times, you should be able to get 120,000. Again, it's very important at the beginning to show the proof to yourself. You want to be convinced. You want to be convinced mathematically how this worked, because once you are convinced, that's it. Once you have it down, you're good to go. Now, what we're going to be doing next is computing the present value of an annuity. From an accounting perspective, I'm just going to set the ground now. The present value of an annuity is more important. The present value concept is more important for us than the future value. For one thing, the future value is easier. Two, the future value is mostly used in a finance course. When you're computing the future value of your 401K of your investment, the present value is what's really needed for accounting students. And this is what we will focus on in the next session. At the end of this recording, again, I'm going to remind you that to visit my website, farhatlectures.com, I can help you do better in your accounting courses. 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