 Hello, and welcome to this session. This is Professor Farhad, and this session would look at the future value of an annuity, specifically ordinary annuity. This topic is covered in an introductory financial accounting course, the CPA exam as well as intermediate accounting. As always, I would like to remind you to connect with me only then if you haven't done so. YouTube is where you would need to subscribe. I have 1,600 plus accounting, auditing, finance, and tax lectures. This is a list of all my courses. If you like my lectures, please like them, share them, subscribe, put them in playlists. If they benefit you, it means they might benefit other people and connect with me on Instagram. On my website, farhadlectures.com, you will find additional resources such as PowerPoint slides, true, false, multiple choice, additional practice exercises that's going to help you supplement your accounting education and do well on your CPA exam. So if you're looking to improve your accounting performance, check out my website. A prerequisite for this session is the future value of a single amount. That's very helpful. I have the link in the description. It will help you tremendously understand the future value of an annuity. So in the prior session, we looked at the future value of a single amount. And basically, what does that mean? It means if we have $100 today, this is the present value. This is the present amount. And we want to know how much it's going to be worth into the future. We multiply the present value by 1 plus the interest rate raised to the end power, the period. So if we are talking about 10%, we'll take 100 times 1 plus 0.1 raised to the first power, and that's going to give us $110. I always mention why do I include 1? Why don't I take 100 times 0.1? That's equal to $10, and you have to add the 100 to the original amount. That's why you put 1 plus the interest rate. So this is the formula. Or you can go to the tables and equal to 1, I equal to 10%, and take 100 multiplied by the factor. And let's go to the factor to the future value of a single amount. So the future value of one of a single amount, 10%, one period, 1.1. So if we'll take 100 times the factor 1.1, it's going to give us $110. So this is what we learned in the previous session about the future value of a single amount. Now what we need to find out is not the future value of a single amount. We need to find out, and this is more realistic, where if you are investing, you invest 100 every month or 1,000 every month or 1,000 every year. Basically, if you put money away in your 401k or if you are purchasing an investment, you put the same amount of money at a regular interval. This is called an annuity. Again, if we don't know what annuity is, it's the same payment at the same interval. So every month, every quarter, semi-annually or annually, you'll invest the same amount. So what we need to find out is how much all these investments are worth into the future. So the future value of an annuity is the accumulated value of each annuity payment with interest as of the date of the final payment. To illustrate, assuming a 15% interest rate, the first $100 here, the first $100 that we invest, it's going to sit with us for two periods. Therefore, we'll take 100 times 1.15 raised to the second power. Simply put, we treated this $100 using this formula, this $100. The second $100 notice, it's going to be with us for only one period. So the second $100, 100 times 1 plus 0.15 raised to the first power. And the last $100, we don't really earn any interest on it. So it's 100 times 1 plus 0.15 raised to the zero power. So if we add them all up, they will add up to $347.25. Now, is there a shortcut to do this? Because sometime we might have many periods. And the answer is yes. We do have future value table where we could look up n equal to 3, i equal to 15. Now, unfortunately, unfortunately, we don't have the interest rate for the future value annuity at i equal to 15. But I can find what i equal to 15. What I can do is I can add up because I do have the future value of a single amount at 15. And I have 2 at 15 is 1.3225. One period at 15 is 1.5, 1.1500. And the last payment is exactly 1. Now, if I add them up, this is 5, 2, 7, 4, 3. So the factor, the future value annuity factor is 3.4725. If I take 100 times 3.4725, I will get $347.25. So simply put, rather than treating those $300 separate payments as separate payments, what I can do, I can use my future value annuity table. So there's a future value annuity table just like there's a present value annuity table to compute this all at once. So we can use the future value annuity table to find the future value of any amount giving an interest rate and a period of time. So simply put, if we take the payment, the payment times the factor, the factor from the table, the factor is based on n and i, depending on what the factor is, we'll find the future value of an annuity. Simply put, if we are giving the payment and the future value of an annuity, we can find the factor and from the factor we can either find n or i, just like what we did in the prior session. But let's work an example to show you how it works. Let's assume A expect to invest $1000 annually for 40 years. So we have n equal to 40 to yield an accumulated value of 154,762. So what do we have here is we have the future value of the annuity 154,762. And the payment is 1000. On the last date of the investment, for this to occur, what interest rate we must earn. So simply put, they're asking us what interest rate we must earn. So we should take the $1000 times the factor. We don't know what the factor is. We know in the factor we know n equal to 40. And that's going to give us 154,762. We can find the factor. How can I find the factor? If I take 154,762 and I divide this by 1000, that's going to give me the factor. And that's not the answer. The factor is just the factor. From the factor I can back into the answer. So the factor is 154.762. This is the factor. Now from the factor I can go to the future value annuity table n equal to 40. So let's go to the, make sure you are looking in the right table, future value of an annuity n equal to 40. And I'm looking one of the closest to 154.154.762. And the factor is here. It means I need to earn 6%. So I need to earn 6%. So I was able to find my interest rate. Let's take a look at this example. Steffi there expect to invest $10,000 annually that will earn 8%. So we have I equal to 8%. We have the payment equal to 10,000. How many annual investment? So we're looking at n must their makes to accumulate 303,243. So simply put, we have the payment as 10,000. So the payment 10,000 times the factor, which is we don't know the factor, equal to 303,243. Now we can find the factor. What's the factor equal to the factor equal to 303,243 divided by 1000. So we find the factor is 30.32, 30.3243. Now this is the factor and we know from the factor we are dealing with an I equal to 8%. So let's go to the future value annuity table. I equal to 8% right here. We know I equal to 8%. And we need to go across to find the closest thing to 30.324 right here. And if we go across, we need n equal to 16. We need 16 periods to find that. So that's good. Let's take a look at number 16. K. Mellon plans to to have woodheld. Let me just erase this plan to have woodheld $50 from her monthly paycheck. Be careful. It's a monthly and deposited in the savings account that earned 12% annually. So notice the deposit is monthly, but the earning is annually. So we have to make an adjustment. And they tell us here it's compounded monthly. If Mellon continue with her plan for two and a half years, how much will accumulate in the account on the date of the last deposit. So let me show you on a graph what this looks like. So this way it's it's very important that you can imagine this on a graph. So on a graph we have something like this. The whole period is 2.5 years, but it's monthly. So how many how many months do we have in a year and a month in a year? How many months do we have 12 months? So 12 plus 12 plus six, because it's two and a half years. That's 24. That's 30. So simply put, she is going to make 30 payments. So this is one, two, three, four, five, six, seven, eight, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. So she's going to make in 30 payments of $50 each. How much would that investment be worth at the end if she can earn 12% this is remember this is annually, but the deposit is monthly. So we know n equal to 30. What is i equal to? We have to be careful. We have, it's compounded monthly. We have to convert the 12% divided by 12 because we have 12 months in a year, i equal to 1%. So when we go to the table, we're going to take the $50 times the factor n equal to 30, i equal to 1%. So it's right here. So we have i equal to 1%. And we're going to go all the way to 30. And the factor is 30.7849. 30.7849. If we take 50 multiplied by that amount, she will have $1,394.25. So the value of this investment is $1,739.25. So that's the answer. Let's take a look at one more example. Let's take a look at this example. The future value of an annuity of an amount plus an annuity. Okay, start company decides to establish a fund that will be that will use 10 years from now to replace an aging production facility. And that's very common where the company put money away to replace their property, plant, and equipment. The company will make $100,000 initial contribution. So first they're going to make $100,000, one single sum to the fund and plans to make quarterly contribution of 50. So this is, so this is a single amount. So the $100,000 is only one time. And this is an annuity, the $50,000 is an annuity beginning three months from now. Now the fund earns 12%. So we are dealing with 12% compounded quarterly. So that's important. How much it's being compounded is extremely, extremely important. So what's the value of the fund? Well, we have two things to do. We have to compute the future value of the single amount, $100,000. This 12% is annually, we're going to take 12% divided by 4. So I equal to 3% and n equal to 10 times 4 and equal to 40. So we're going to be dealing in this example, I equal to 3 and equal to 40. Okay, now, first let's find the future value of the single amount, future value of one, n equal to 40. So n equal to 40, I equal to 3%, and the answer is 3.2620. So we're going to take the $100,000 multiplied by 3.2620. And this amount on its own, this $100,000 will grow up to be $326,200. Now we have to find the $50,000 annuity investment. So we're going to have to multiply it by the factor. We have to multiply it by the factor again, n equal to 40, I equal to 3. n equal to 40. And the factor is 75.4013. 75.4013. And that's equal to $3,770,065. Those two amounts together will amount to $4,096,265. So this is the future value of the amount of this investment. The $100,000 alone will grow up to be $326, and that $50,000 quarterly payment will grow to $3,770,065. Together they will equal to $4,096,265. I hope you were able to kind of follow in this session the future value, but don't worry, I will work in other few examples that are little bit more comprehensive that will include both. If you have any questions about this session, please let me know. And as always, I would like to remind you to visit my website for additional resources like this session, share it. 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