 Hello and welcome to the session in which we would look at the present value of an annuity. This topic is related obviously to the present value of a single amount. In other words, if you don't understand the present value of a single amount, I strongly suggest you take a look at the prerequisite because if you don't understand how to compute the single amount, you will find difficulty understanding the present value of an annuity. Also based on the prerequisite session, you need to understand the difference between simple and compounding interests. The present value of an annuity is an extremely important topic for accounting students, whether you are dealing with bonds, leases, notes, payable. You have to understand how the present value of an annuity works. So it's critical for your success also as an accounting student. Whether you are an accounting student or a CPA candidate, I strongly suggest you take a look at my website, farhandlectures.com. 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Take a look at my LinkedIn recommendation, like this recording, share it with other, connect with me on Instagram, Facebook, Twitter and Reddit. So let's talk about the present value of an annuity and what is an annuity because the assumption here is you know how to compute the present value of a single amount. So we're going to change the single to an annuity. An annuity is a periodic payment or receipt of the same amount of money at the same interval. Basically, we are looking at an interval. Let's assume this is year year one, two, three, four, you are either paying or receiving the same amount of money at the same interval. This is what an annuity is. And all we're going to try to do is to find the present value of the annuity, which is what? Which is discounting dose payment to the present value. This is what the present value of an annuity. It's a discounting process to find what's the value of those serious payment giving a certain interest rate. Now we're going to have two types of annuities. We're going to have the present value of an ordinary annuity and as I just showed you, year one, two, three, four, basically the payments or the receipts will start at the end of the first period or at the end of the first year. And we have the present value of an annuity due and we dealt with the same concept with the future value. And the present value of an annuity due, the first payment or the first receipt will start today. So we would still have one, two, three, four payments, but one of the payment, it's either received earlier or paid earlier. So we have to deal with both starting with the present value of an ordinary annuity. And the best way to illustrate this is just to work an example to show you the concept. How much will you pay for an investment that will pay you $10,000 for five years, assuming you want to earn 5% interest. So this is five years. So I define the period as five years. So the question is, here we go, I'm offering you an investment and I'm telling you, here's what's going to happen. I'm going to pay you $10,000, $10,000, $10,000, $10,000 and another $10,000, so those $5,000 in the next five years. How much will you pay for this investment? How much is this investment worth? What's the question to you? How much is it worth? Well, if you want to earn 5%, it means I want to earn 5% on my money. So I have to discount those investments. In each one of them, it has to earn 5%. Well guess what? I can do that. I can go to the present value table that we learned about, and I can discount the 5, the 10,000 separately, the second 10,000 separately, the third, the fourth and the fifth. And I can do that. I will discount them. The first 10,000 I'm going to be receiving, which is going to be a year from now. If I'm earning 5% a year from now, I'm going to discount it at 0.95238 or 0.952. It doesn't matter. This is rounding. So I will pay for that $9,523. For the second 10,000, I'm willing to pay 0.907, 0.9073, 10,000 times 0.9073 will give us 9,000, I'm willing to pay for that second 10,000, $9,070. For the third 10,000, I'm willing to pay 0.86384 or rounding 0.864, I'm willing to pay $8,634.40 and the fourth payment, I'm willing to pay 0.823 and the fifth payment, I'm willing to pay 0.784. So notice what I did. I discounted those payments individually, like if it's a single payment. But what do we know about this series of payment? It looks like an annuity. It's the same payment every time. And guess what? If I add all those single factors, they're going to give me 4.3295. And this is the present value of an annuity factor. So simply put, if I take $10,000 multiplied by this factor, the 4.3295, I will find out that the present value of this series of payment is 43,294. Now simply put, if I want to compute this 4.3925, this is the formula. The present value factor of an ordinary annuity. I just have to compute this formula, which is 1 minus 1 divided by 1 plus i raised to the number of periods, which is 5 and divided by i. Then what I would do is I will take my payment times what I found in this formula to find the present value of the annuity. This is the long way. This is if you want to do it manually. The good thing about your accounting course or your finance course, you might be using a calculator, which I don't show you the calculator here. I show you how to use the tables. Using the tables, you will go to the present value of the annuity table. This is ordinary annuity. Ordinary annuity. Make sure you are dealing with the proper table, present value of an ordinary annuity. And it's five years. So the period is 5 and the interest rate is 5%. And they cross at 4.3295. And this is the 4.3295. And we find the present value of the annuity. So rather than going through the computation, those factors will be giving to you. And if you really want to prove this, that this is what would happen, well, we're going to invest 43295 today, give it to someone, to an investor. This money is going to grow at 5%. And it's going to accumulate the 45459.54 after a year. We're going to take out the 1,000. What's left is 35459.54. I'm going to round it. So this is going to be 35,460 at the beginning of the second year. It's going to earn 5%. And the amount will be 37,232.52. I'm going to take out 10,000. I'm going to be remaining balances 27,232.52. And if you keep going, by year five, you should have zero. This is four pennies, it's surrounding, but you'll have zero. So indeed, if you invest 43295, let it grow at 5%, take in 10,000 out, you will earn exactly 5% on this investment. Let's take a look at another example. What is the present value obligation? What's the present value obligation? This is basically, we're looking to find the value of an obligation, you have to pay a five annual payment of 6,000 paid at the end of the period given 6% interest rate. So here we go. We have an obligation to pay 6,000, 6,000, rather than receiving, we're going to be paying. So we have an obligation 6,000, and it's 1, 2, 3, 4, 5, and another 6,006. And in the interest rate, we're going to be charges 6%. So this is basically, we can say what's the value of this or what's the obligation because we are discounting payment. So we have an obligation. It doesn't matter. The point is you have to discount those $6,000 back to the present value. How do we do so? Well, we just learned. We can go to the tables and we have five annual payments. So five is the end and we are earnings. We have to pay 6%, so the factor is 4.2124. So it will take 6,000 times 4.2124. We record the obligation at 25,274, and we have to pay it back in $6,000 increment. Let me show you the proof. If we start with 25,274, we're going to be charged interest 6%. It's going to accumulate to 26,790, we're going to pay 6,000, and we're going to the remaining balance will be 20,791, rounding 7,91. Well, we're going to be charged 6%. The balance will be $22,038, and $0.32, we're going to pay 6,000 again. The balance will go down to $16,038, then the $16,038 will increase at 6%. The balance will be $17,061, we're going to pay off $6,000, so on and so forth. By the end of the fifth year, we'll pay off this loan again, the 29 pennies is a rounding issue. Now let's talk about the present value of an annuity due. Well, as I told you, the difference is you're going to start either to receipt a payment earlier or you're going to pay the first payment immediately today, which is a payment earlier. So payment or receipt is starting today. So you're going to have one fewer period to discount. So the cash flow comes one period sooner if you're receiving the money, and the cash payment will come one period sooner. To find the present value of an annuity due, most tax book, they do have this table, but sometimes they don't have the table. So what you do is you'll take the present value factor of an ordinary annuity, what you learned on the prior slide, you multiply it by 1 plus i. So if it's not giving, you can find it. So it's the present value obligation of five annual payment of 6,000, giving 6% interest rates, starting today, not staring, starting today, starting today. And here's what's going to happen. We're going to find the factor. This is for an ordinary annuity. So this table is ordinary, ordinary annuity, because I'm going to assume your textbook, don't provide an annuity due. So you'll take your ordinary annuity, which is we already figured it out, 6%, 6 period, 4.214, and you multiply it by 1.06. So the present value of the annuity due, the present value factor of the annuity due is 4.651. You would record the obligation initially at 26,790.86. And annuity due, you will see it later on when you are dealing with leases, with the chapter of leases, because the first payment of the leases usually do immediately. Now, we're going to have to find sometime the interest rate in an annuity situation, okay? So it will be something like this. Assume that you own $527, $28.77 on your Visa card. The lender offer you 12 equal monthly payments of $50 to pay off the balance. What interest rate is the lender charging you? So here we're looking for I. Well, let's look at it from a present value annuity problem. The present value of the annuity equal to the payment times the present value factor of the ordinary annuity. We know, what do we know? We know the, let's look at the numbers. We know the present value of the obligation right now, $528.77. $50 is the payment. We have to multiply it by the present value of an ordinary annuity. And this is how we find $528.77. So we have everything except the factor. Can we find the factor? Sure we can. We are told, you have to be careful here. You are told that the payments are monthly payments, okay? So it's 12 monthly payments. Therefore, the period equal to 12, okay? And we are dealing with $50 payment. So if we have to find the factor, we have to go through the computation. $528 divided by 50, the factor is $10.575. Now dealing with 12 periods, we're going to go across and find the closest thing to this factor. So the closest thing to this factor is $10.573. It's right here. Therefore, we implied that the interest rate we are being charged is 2%. So you are being charged 2% on this loan. And this is a monthly rate. Remember the monthly, you can convert it into, you know, this is the nominal rate 2%. So monthly 2 is 24%. They're going to quote you 24%, then the 24%. You know, if it's compounded monthly, you have to find the effective rate because this is the nominal rate, okay? So it's 2% monthly. And this is the proof that this will happen. So if you have rounding $529, you are being charged 2%. At the end of the first month, your balance will accumulate to $539.35. You're going to pay $50. The remaining balance is $489.35. Then this amount, $489.35, it's going to accumulate at 2%. It's going to end up to be $499.13. You're going to pay $50. The balance will go down to $449.13, so on and so forth. And if you do this for 12 months, you will pay off the balance after 12 months. You really don't want to do that because that's a very expensive credit card balance. But the point is to show you that if you do pay $50 in the interest rate we computed 2%, you will be able to pay off your loan. Sometime you have to find the payment. So how do you find the payment and an annuity problem? Let's assume Marjan Homer have saved $36,000 to finance leases for your education and a local bank earning 4%. So they have $36,000. They're going to invest it at 4%. What amount can Lisa take out every six months during her four-year college year? So basically what we're looking for is what's the payment? What payment can Lisa afford every six months? So remember, the period is semi-annually every six months. So let's take a look at our general format. The present value of an ordinary annuity equal to the payment times the present value factor of an ordinary annuity. Let's see what we have. What we have here is the following. We have 36,000. The present value of an ordinary annuity. We don't know the payment. We know the factors are 7.325. How did we find the factors? Well, if it's a four-year, four-year times two is eight. So the period we are dealing with is eight. And we have to find the factor at 4%. But remember, 4%, that's annual rate. It's this. This is semi-annually. Therefore, we have to look at two. And we find the factor is 7.3255. So 36,000, which is the present value of the annuity, equal to the payment, which is we don't know the payment times the factor should equal 36. Well, if we solve for the payment, we see that the payment is approximately 4,914. So Lisa again would draw 4,914 when she attended her school. So basically something like this. So every one, two, three, four, five, six, seven, eight, Lisa again would draw this amount every period. If Marge and Homer can put that money in an account earning 36,000, earning 2% semi-annually. And this is the proof for it. So they start, they deposit the money, 36,000. They earn 2%. That money will accumulate to 36,720. Lisa would withdraw this balance, 4,900, and $35. What's left in the balance, 36,805.65. This amount, 3,805.65 will earn 2%. The balance will become at the end of the year, 32,441.76. Lisa will withdraw 4,914. The balance will become 27,527. And if we go like this for the next four years, the withdrawal every six months, will have two pennies in the account, which is technically zero, this is a rounding issue. So here you see also that you can prove it. The best way to learn the present value is to kind of just see if you can, see if you understand it. And if you understand it, you should be able to do it in an Excel sheet real easily. At the end of this recording, I cannot emphasize the importance of the present value of a single amount and the present value of an annuity. It's extremely important understanding this discount process, whether you are an accounting student, a finance student, and especially if you are a CPA candidate. If you are a candidate, I don't replace your CPA review course, you keep it. I am a useful addition. I explain the material differently so you can take advantage of your review course to prepare for fully for the exam. Studying for the exam is a one-time investment in your life. Take it seriously. The CPA exam is worth it. Good luck, study hard, and of course, stay safe.