 Hello, and welcome to the session. This is Professor Farhad. In this session, we would look at the present value of an annuity. Specifically, we're going to be looking at an ordinary annuity. This topic is covered in financial accounting, introductory course, the CPA exam. The topic is also covered in intermediate accounting, a little bit more in depth if you're interested if this wasn't good enough for you. As always, connect with me on LinkedIn 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. Please, like my lecture, share them, subscribe to the channel, put them in playlists. If they benefit you, it means they might benefit other people, especially these days with the coronavirus out there, connect with me on Instagram. On my website, farhadlectures.com, you will find additional resources such as PowerPoint slides, practice questions through false multiple choice that will help you supplement your education and or your CPA exam. Check out my web. The prerequisite for this session is the present value of a single amount. The link is in the description. It's very helpful that you understand how the present value of a single amount work because it will help you understand the present value of an annuity. In this lesson, we're gonna be working at the present value of an annuity in specifically ordinary annuity because there's an annuity due. We don't cover annuity due in a financial accounting course. Just wanna let you know, if you are looking for an annuity due, look up my intermediate account. So what are we looking for? We are looking to find the present value of an annuity. No, the first thing is what is an annuity? An annuity is a series of equal payment occurring at equal intervals. So simply put, you're gonna have to pay or receive $100, $100, $100, $100 and $100. You're gonna receive $1, 2, 3, 4, 5, $100. The question is, how much will you pay for the series of payments? So how much will you pay with this present value of an annuity? Annuity is a payment or a receipt of the same amount of money over a period of time. For example, here the series is three annual payment, $100, $100 and $100. An ordinary annuity is defined as equal as equal end of period payment at an equal interval. What does that mean? It means you don't receive the first payment until a period from now, we assume the period is here, but at period one, then period two, then period three, the same payment at regular interval. So what we need to find out now is how much will we pay for these payments? Now we should know how to do so. If we know the present value, we know how to do so. How do we compute the present value of this annuity? Well, we have to assume a certain interest rate. So let's assume 15%. What we will do is we will discount this $100 at 15% and equal to one, and this is how we do so. So we'll take the $100 divided by one plus the interest rate raised to the end power, which is the present value of a single amount. How do we discount this second $100? Well, we'll take $100 divided by one plus I raised to the second power. We discount the second $100. Then we have a third $100. How do we discount the third $100? $100 divided by one plus I raised to the third power. What does that mean? It means if you put away $228.32 today, if you invest this money today in an investment that's gonna pay you 15%, you can withdraw $100, $100, $100, year one, year two, and year three. So the best way to show you is to prove it. I like to show the math proof. So let me show you that if you put $228.32 today, you can take out the $100. So that computation is correct. So let's put 228.32, and here's what's gonna happen. It's gonna grow after a year. After a year, this money's gonna grow at 15%. The balance will be 262.56. Then you're gonna take out, you're gonna withdraw. Remember, we're gonna take out $100. Why do we take out $100? Because that's the annuity. The annuity says you take out $100. Then you're gonna have left 162.568. This is year two. This money's gonna grow also at 15% times 1.15. It's gonna become 186. Then you're gonna withdraw $100 from this money. And you're gonna, this is year two. Let me just say this is year two. Then you're gonna withdraw $100 of that money. Then at the beginning of year three, you're gonna have $86.95. And this money's gonna grow at 15% as well, 1.15. And you will withdraw $100 and you will have zero. And the balance is technically zero. So this is beginning of year three. Then it's end of year three. You will have this money. Then you will take out the money and you will have zero balance. So notice, indeed, if we invest 228, if we invest 228 and at 15% and take $100 every year, we will be able to take $300. So this is how we found the present value of this annuity. Now, let's assume we need to find the present value of an annuity with 20 payments. Well, this is gonna become cumbersome. This is gonna become cumbersome. So what we do, there is a present value table just like there is a present value table for an annuity, like there's a present value table for a single payment, the present value for a future payment. So there's a present value table to compute the present value of an annuity. This computation is identical to computing the present value of each payment in table B1, if you looked at the present value of a single annuity. However, they did all the computation for us. They did all the computation for us. And here's how it works. So if we want to find the present value of an investment that the interest rate is 15% and the period is three years, here's how we find out. We'll take from table B1, the present value of the single payment is 0.8696, which is this factor here is 0.8696. Then we'll take the present value of the second payment, 0.7561, 0.7561. This is what we did. Then on the third 1.6575, 0.6575. If we add up those factors, if we add up those factors, they will add up to 2.2832. So if we want to find the present value of any amount invested at 15% after three years, all that we have to do is take the payment, multiple payment, which is the payment, multiply it by the present value annuity factor, which is the payment is $100. For our example, the present value annuity factor is 2.2832, which is 228.32. Now, rather than looking at table B1 and adding all those payment, we have a table that's called the present value of an annuity. Notice here, we have to be careful which table we are using. In my textbook that I'm using, it's table B3. But you have to look at the title, the present value of an annuity. It's an ordinary annuity. And this is how we find the factor, but you don't have to worry about this. So if we're looking at an investment that's three years, three periods, three periods and interest rate is 15%, the factor is point, I'm sorry, 2.2823. So you take the payment, multiply it by the factor, multiply it by the factor, and it's gonna give you the present value of the annuity. So we find the present value of the annuity. So this is how we use the present value of an annuity. Now, how is it used? How the present value of the annuity used? A common way to do it, let's assume you want to, you want to invest money so you can take out $20,000, $20,000, $20,000 and $20,000 for your kids' education for the next four years. But you want to know how much to put today, okay? So we know N equal to four. Every year you need to take out 20,000 so you can pay your son or daughter's tuition, N equal to four. And the interest rate is what interest rate are you going to use? Well, what are you gonna invest your money in? So let's assume the interest rate is 5%. How much money you will need to invest? Obviously you will need to invest less than 80,000 because you are taking out 80,000. To find out exactly, well, N equal to four, I equal to five, N equal to four, N equal to four right here, I equal to five right here. So the factor is 3.546 zero. So if we take 20,000 times 3.546 zero, today you will need to put away, if we take 20,000 times 3.546 zero, you will need to put away $70,920. If you put this money away and you let it grow at 5%, every year you can take out $20,000 for the next four years. There's a many usage for the present value computation. I just showed you one example. Also, if you want to value an investment, this is how you value an investment, what is the present value of the future cash payment? This is a very important concept in accounting, extremely important because you will see that when you need to find the price of the bond, you will need the present value. So how is this concept used in accounting? Well, if you want to find the price of the bond, if you want to find the pension obligation, if you want to find the present value of the loan, I mean, it's endless bonds, pension, loans, anything that's long-term, anytime, simply put, anytime you are going to be paying, paying or receiving a future amount of money, every time you are paying or receiving, you need to find the present value of that payment. You need to find the present value. So you would record everything at the present value. Also, what's gonna happen sometimes, sometimes you might be asked to compute the interest, sometimes you might be asked to compute the period. Sometimes you'll be given the payment, you'll be given the present value, you'll be given the interest rate, you find the period, or you're gonna be giving the present value, the payment and the interest you need to find the period. And we can do so just like what we did in the, using the tables. Let's start to illustrate, work some examples to illustrate what else can we do with using the present value table. Jones expect an immediate investment of $57,466 to return $10,000 for eight years. So here's what this problem would look like. Jones was offered an investment. If you paid today, $57,466, you paid off the day, you paid that money today. And one, two, three, four, five, six, seven, eight. One, two, three, four, five, six, seven, eight. You will be able to receive $10,000, $10,000. Each one is $10,000 for the next eight years. So we know N, N equal to eight. We know the payment equal to, N equal to eight. The payment is $10,000. And we know the present value of the annuity is $57,466. So the question is, what interest rate at what interest rate do we need to invest this $50,466? So we can take out $10,000 for the next eight years. So what's missing is I. Well, here's what we do. We'll remember, just remember this formula that if you take the payment, remember we talked about this, if we take the payment times the factor, we'll give us the present value. We have the payment here is $10,000. We don't know the factor. We don't know the factor. We know the present value of the payment of all the payment is $57,466. Now we can find the factor, which is the factor is $57,466 divided by $10,000. All what I did is I rearranged this formula. So the present value annuity factor, if we look at the factor, that's 5.7466. 5.7, sorry, 5.7466. So that's the factor. Now if I have the factor, 5.7466, and I have n equal to eight, well, let me go to the table of the present value annuity factor. If I go to the table, again, you wanna make sure you're working in the right table, present value annuity factor, n equal to eight, I know the n equal to eight. I don't know the interest rate, but if I go across and find the closest one to 5.7466, and it's right here. So you need to invest this money at 8%. So this investment, so the answer for this problem, the I equal to 8%. So if you invest your money, if you invest 57,466 today, keep this money for eight years taken out, 10,000 every year, earning 8%. Your investment will earn 8% for eight years. This is what we're saying here. Let's take a look at the second example. Keith Riggins expect an investment of $82,014 to return 10,000 for several years. If Riggins earn 10%, how many annual payment? Now we need to know the payment that he will receive. We need to know n. What are we looking at here? Here's what we're looking at. If Keegan pays today $82,014, Keegan's will be able to receive, we don't know the payment. We don't know how many payments, we know Keegan can earn 10%. We'll do the same concept and the payment is 10,000. But we don't know how many times we're gonna be receiving this payment. So if we take $82,014 divided by 10,000 to find the factor, and the factor is 8.2014. Now we know the interest rate is 10%. We'll go to the present value annuity table. And here we are looking at 10%. And we go down until we find the factor 8.2 or the closest thing to 8. Actually, it's right here, 8.024. We go across, well, this investment will take 18 years. So if you put away today, so n equal to 18. What we're saying is this, Keith, if Keith put away today $82,014, invest this money at 10% for 18 years, Keegan can take away every year $10,000. And this is what you usually do when you get closer to your retirement. What you do is you sell all your investments, all your stocks, all your bonds, and you buy an annuity. And this is basically what an annuity. So you pay $82,014 for that annuity and you'll be able to take out $10,000 every year for the next 18 years. So you would receive in total $180,000 but over 18 year period, which is, you paid for it $82,014. So this is how we use it, okay? Let's take a look at more examples. This exercise here, David Gregg finances a new automobile by paying 6,500 cash and agreeing to make 40 monthly payment of $500 each, the first payment to be made one month after the purchase, the loan bears an annual interest of 12%, what's the cost of the automobile, okay? Well, let's find the cost of this automobile. So they're buying, David is buying an automobile and David will have to pay upfront 6,500. So 6,500 times one, which is the factor is one, equal to 6,500. So today, Dave will have to pay 6,500. In addition to the 6,500, Dave will have to make a payment of $500. Listen to me carefully, this payment is monthly. It's a monthly payment of $500. And he's gonna have to make this payment for 40 periods. So N equal to 40, so we are giving in and the interest rate is 12%. Well, if the payment is monthly, it means we have to take the interest rate and divide it by 12. Remember, if the payment is not annually, we have to adjust the interest rate. The payment is monthly, we divide by 12. If the payment is semi-annually, we divide by two. If the payment is quarterly, we divide by four, okay? So here the payment is monthly, so we divide by 12. So the interest rate, I equal to 1%. Now we know N, we know I. Now we need to find out the present value of those $500, 40 payments. So I equal to 1% and we're gonna go down to 40 and the factor is 32.838347. So the payment of the car is 6,500 plus the present value of the $500 annuity. So we're gonna take 500 times 32.8347, which is equal to 16,417.35. So he's gonna pay 6,500 plus 16,417.35. So this is the price of the car. Let me see if I can find another pin. So this is the price of the car. And let me highlight. So it's 6,500 plus the present value of the $500 payment, okay? Let's take a look at exercise 10, okay? C&H Ski Club recently borrowed money and agreed to pay it back with a series of six annual payment of $5,000. So this looks like an annuity, okay? C&H subsequently borrowed more money and agrees to pay it back with a series of annual payment of 7,500. The annual interest rate for both loans is 6%, okay? So here it says use table, to find the present value of these two separate annuities, use table B1 and use table B3. So what they want us to do, they want us to use table B1, which is to find the present value individually, okay? Then use table B3 to find the present value all in one shot. So let me, let's use both tables. This way, hopefully it will help you understand. So for the first, if we're using table B1, table B1 is the present value of a single payment. So they borrowed money and agreed to pay it back with a series of six annual payment. 5,000, 5,000, 5,500. This is what they have to pay back. Now, the interest rate on this loan, 6%, on both loans, 6%. So I equal to 6%. Now, they have to pay this payment one year, one year from now, two year from now, three year from now, four year from now, five year from now, six year from now. Now I'm gonna go to table the present value of B1, and find the factor for n equal to one, I equal to six. So notice, I have to go to table B1, which is the present value of a single amount, 6%. This is the factor, and this is, I would use this factor, this factor, this factor. So notice n equal to one, n equal to two, n equal to three, four, five, six. So those are the factors, that I'm gonna be using, so you can write them down, if you'd like to. And I'm gonna take each payment separately, each 5,000 multiplied by the factor. So times 0.9, 4, 3, 4, times 0.8, 9, 0, 0, times 0.8, 3, 9, 6, times 0.7, 9 to 1, times 0.7, 4, 7, 3, times 0.7050. And you have to find the individual answers, and if you find the individual answers, all the answers, they should add up to 24,588, okay? Now, I'm gonna move on to table B3 and do the same computation. Well, table B3, I have an annuity of 5,000, n equal to six, yes, and i equal to six as well. Let's go to table B1 and see what we find. n equal to six, n equal to six, and i equal to six, the table is 4.9173. So if I take 5,000 times 4.9173, that's gonna equal to 24,588. Notice, it's the same answer. Well, you need to know if you add up all these factors and you can add them up, they will add up to this present value annuity factor. So this is basically showing you that an annuity can be found using table B1 the long way, discount each payment separately, but since it's an annuity, we can go to the annuity table and find the factor all at once. So, let's go to table B1, and find the factor all at once in one shot. Now, the same thing for number two, sorry, the same thing for the 7,500. Well, for the 7,500, the answer is 24,587. Whether you used table B1 or table B3, you can do the computation yourself. But this is a good exercise to show you that if you have a series of payment, you can use it, you can find the answer using the present value of a single payment, present value of a single payment. Let's take a look at this exercise. Auro borrows money on April 30th by promising to make a four annual payment of 13,000 each on November 1st, 2019, May 1st, 2020, November 1st and May 1st, 2021. So, notice here that we're gonna be making four payments and the payment is 13,000. What you need to notice here is, well, let's look at the question first. How much money is Auro able to borrow if the interest rate is 8% compounded semi-annually? So, Auro will have to make 13,000, 13,000, 13,000 and 13,000. This looks to me like an annuity. We are told the annual I is 8%, but the interest is compounded semi-annually. It means you have to divide I by two, so the interest rate is 4%. N equal to four, we have four payments. One, two, four periods, one, two, three, four. Now, all I have to do is go to the table, 13,000 times the present value factor, I equal to four, N equal to four. So, let's go to the table. 4% and 4% they meet at 3.6 299. So, I'm gonna come back here, 3.6 299 and Auro can borrow 47,189. Auro will borrow this money and Auro will pay 13,000, 13,000, 13,000 and this loan will be paid off. Now, how much money is Auro able to borrow if the interest rate is 12% compounded semi-annually? So, now what happened is the interest rate went up. Can he borrow less or would he be able to borrow more? Well, the interest rate is higher, you're gonna be able to borrow less because the present value it's gonna be lower. So, let me show you. So, again, we're dealing with the same payment, 13,000, 13,000, 13,000. Now, the interest rate is 12% semi-annually, we have to divide by two. So, 6% and equal to four. Now, we have to find the factor, we have to find the factor, which is the payment is 13%, I equal to six, I equal to six and equal to four and the factor is 3.4651. If we multiply that by 13,000, Auro can borrow 45,046 dollars. Now, let's take a look at the third scenario, how much money can Auro be able to borrow if the interest rate is 16% compounded semi-annually? Now, the only thing, the difference is since 16% semi-annually, I equal to 16% semi-annually divide by two. So, we use the 8%. So, now we're using the 8%. Fourth period and the factor is 3.12. So, if we take 13,000 times 3.3121, the amount is going to be less $43,057. So, this is how we found the present value of those payment, the present value of those loans, okay? Let's take a look at exercise 12, exercise 12. Spiller plans to issue, this is a bond, is it a bond? To issue a 10%, 15 year, oh yeah, it's a bond. This is what we need to know how to compute the price of the bond. So, Spiller corporation plan to issue 10%, 15 year, half a million power value bond payable that pays interest semi-annually on June 30th and December 31st. So, the annual rate is 10, the semi-annual rate is five because we're paying the interest semi-annually. The bond are dated December 31st, 2019, and are issued on that date. So, it's a 15 year, it means 30 payments because this bond makes payments twice a year. So, we have N equal to 30, the payments, the semi-annual rate is so, so the payment is 5%, the payment on the bond. The bond are dated, we talked about this, if the market interest rate for the bond is 8%, what is the total cash proceeds from the bond? Interesting, now we have to go back to the bonds and we looked at bonds, you don't know how the bonds work if you don't know what is a bond, I'm gonna review real quick how a bond work but you need to know how a bond work. When you buy a bond, you will get two things. When you buy the bond, you're gonna get two things. When you buy the bond, you're gonna get your par value plus payments. So, when you buy the bond, you're gonna get two things, the par value plus the payment. So, today, so let's take a look at today, the question is how much will you pay for this bond? This bond's gonna pay you half a million, the par value 15 years from now, 15 years from now. Also, the bond is gonna be making 30 payments. So, 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, the bond is gonna making 30 payments. Now, what is the payment? Well, the payment, the bond has a par value of half a million. You multiply the payment by 10% times one half. Why one half? Because the bond pays interest summa annually. So, if we take half a million times 10%, it's gonna be 50,000 times one half is 25,000. So, each payment, each of these axes, which are 30 axes, each of these axes equal to 25,000. So, when you pay for this bond, you're gonna pay for two things. You're gonna pay for the future value of the, so you're gonna pay for this half a million, and you're gonna pay for the 25,030 payments. So, to find how much you will pay for this bond today, you will discount everything. So, listen to me carefully. You will discount everything using the market, using the market. So, when you go to the table to find the present value of the bond, you would always use the market rate because the investor wants to earn the market rate. The market rate is 8%. That's annually, because we're doing everything semi-annually. So, I equal to four. So, for this exercise, we're gonna be using both tables, B1 and B3. So, let's first finish table B1. So, for table B1, the interest rate is 4%. The period is 30 period, and the factor is 0.3083. 0.3083. So, what does that mean? It means this half a million, you're gonna discount the half a million at 0.3083, which this half a million by itself is worth 100, 54,150. Now, before I find the final answer, I'm gonna ask you a question here. Will my final answer for the bond, will my final answer be more than half a million or less than half a million and why? Well, let me tell you. You can find how much the answer should be. The answer should be more than half a million. Why? Well, if you looked at my bond lecture, it will tell you that if you're offering 10% more than the market of eight, your bond will sell at a premium. Therefore, the bond price will be more than half a million. So, the half a million by itself is worth 154,150. Now, we have to discount the $25,000. This is an annuity. Therefore, I have to go. So, I use table B1 for this one. I have to use table B3 for the annuity. Now, I have to find the annuity factor. Again, I equal to 4%, N equal to 30. Let's find, go to table B3. I equal to 4, N equal to 30. And the factor is 17 point. The factor is 17.2920, 17.2920. 17.2920. And if we'll take 25,000 times this amount, it's gonna give us 432,300. If I add the present value of the part value plus the present value of the payment, I will pay for the bond 586,450. So, this is how much you will pay for this bond today. And as I told you, it's gonna be a premium bond. You're gonna pay more than half a million because what the company is offering, 10% on the bond is greater than the market value. The market only requiring 8%. So, this is an important computation, finding the price of a bond. Finding the price of a bond, okay? Let's take a look at this exercise. Compute the amount that can be borrowed under each of the following circumstances. So, how much can you borrow? A promise to repay $90,000, seven years from now at an interest rate of 6%. Well, it's good to like just see this on a graph, see this on a graph. So, you promise to repay 90,000. So, you borrowed money from someone, tell them, look. I will give you back $90,000, seven years from now. And I'm gonna pay you 6%. How much will you loan me today? Obviously, they're gonna loan you less than 90,000. Specifically, how much less than 90,000? This individual wants to earn 6% for seven years. Well, we have to find the present value of this single payment. You're only gonna pay them the $90,000 only once. So, we go to table B1 because you're only paying them once. And you're gonna pay this money seven years later and the promise is 6%. Therefore, the factor is 0.6651. So, if we take 90,000 times 0.6651, this is a 5-1, you would lend them today, $59,859. And what's gonna happen, you wait and this money will grow at 6%. Well, let me show you, but that's the case. So, if you gave someone 59,859, 59,859. If you gave someone 59,859, and this money's gonna grow for seven years. So, year one, year two, it's gonna grow for seven years and it's gonna grow at 6%. So, we'll take the prior amount times 1.06 and if we let that money grow for seven years and notice you will get exactly 90,000. You'll get exactly 90,000. Okay, let's go back to the second exercise. I'm in the second scenario. An agreement made on February 1st to make three separate payment of 20,000. On February 1st, a year from the borrowing, 20,000, another year and another year at an interest rate of 10%. So, notice here what we have is, we have an annuity. We have an annuity, why? Because you're gonna lend them the money today and they're gonna pay you 20,000, 20,000, 20,000, 20,000 and 20,000, okay? So, N equal to three, the I equal to 10%. Now, the question is how much will you lend them today? So, you will make 10% on that investment when they pay you back 20, 20, 20. Well, what do I have to do? Find the present value of those payments. N equal to three, N equal to three, I equal to 10. The present value of an annuity. N equal to three, I equal to 10 and the factor is 2.4869. So, I'm gonna take 20,000 times 2.4869 and you will give them today 49,738. And what's gonna happen? A year from now, they'll pay you 20,000 that money grows at 10%, they'll pay you 20,000 and 20,000. Well, why don't we also show you how this work? Because maybe if you see that, it will help you understand this concept. So, today you're gonna give them 49,738. It's gonna grow at 10%. So, this money a year from now, you're gonna multiply it by 1.1 and it's gonna become 54,711. Then, they're gonna pay you back 20,000. What's left after a year is 34,712. Then this money, so this is after year one, okay? So, this is balancing when you start year two. So, this is after one year, after year one, this is how much money they will take the 20,000, this is the remaining balance. Then this balance will grow at 10% times 1.1. Then you're gonna take, they're gonna pay you 20,000. What's left is 18,182. This money that remained at the beginning of year three will grow at 10%. Then you will take away, then they will pay you 20,000 and notice it worked perfectly. Then the balance is zero. So, you can kind of in a sense confirm your computation and increase your understanding of this concept. As always, I would like to remind you to like the recording. If you have any questions, please let me know. In the next session, we would look at the future value of an annuity, like the recording, share them. During the coronavirus, most people are relying on online lectures. Please make sure to share the wealth if they benefit you. It means they benefit others and check out my website for additional resources. Good luck and study hard. Most importantly, stay safe.