 Hello and welcome to the session. This is Professor Farhad. In this session we will look at the present value of a single amount. This topic is covered in an introductory accounting course as well as in advanced accounting courses also covered on the CPA exam, the FAR section. As always I would like to remind you to connect with me on LinkedIn if you haven't done so. YouTube is where you would need to subscribe. I have 1600 plus accounting, auditing, finance, and tax lectures. This is a list of all the courses that I covered, including many CPA questions. If you like my lectures, please like them, share them, put them in playlists, subscribe. If they benefit you, it means they might benefit other people, so share the wealth and connect with me on Instagram. On my website you will find additional resources to supplement your accounting education such as multiple choice truffles and exercises to help you reinforce what you have learned. On a graph the present value would look something like this. We have P today. This is the present value is today and we have F into the future, the future value. What we're looking in this explanation for is this amount today. What do we mean by this? Well today you ask yourself how much money will I need to invest today? How much will I need to put away today to have a certain amount of money in the future? So today I have to invest and in the future I'll have a certain amount of money. Now oftentimes the issue is you know how much money you need in the future. For example, you need to buy a boat, a car, an airplane, a vehicle. You need to invest money for your kids. Whatever that amount is, let's assume I need $10,000 into the future. That's what I need. I know my future amount. How much will I need to invest today? That's what we're looking at today and I'm looking only for one amount. How much will I need to invest today? So I will have the $10,000 into the future. Well that depends on two things. It depends on the time, how long you're going to invest this money for and it depends on the interest rate. Well you're going to invest this money and it's going to grow from now to day into the future. How fast it's going to grow? Well depends on the interest rate. If your interest rate is higher your money will grow at a faster rate and if you started early your money will grow bigger because you started early. So the formula to compute the present value of a single amount is shown below. So to compute the present value of a single amount here's what we do. We're going to take P is the present value. So this is the unknown in this formula. We don't know how much we need to invest today. This is the unknown. We know the future value. This is going to be known. What else do we need to know? We need to know two things. We need to know the interest rate and the interest rate depends on many things, the type of the investment. For example, if you're going to put your money in the bank, well that's not going to be, that's not going to earn you a lot of interest but if you put money in your, in a risky bond that might earn you a higher interest and N is the number of period. Now for our purposes I'm going to simplify and say the number of years but it doesn't have to be years. The number of periods could be semi-annually, it could be monthly, it could be quarterly, but to simplify this we're going to call it number of periods in years for our purposes. But remember N is the number of periods, I is the interest rate. Again, how long you're going to invest this money for depending on your time frame. Are you going to, do you need this money in three years or in five years? If you need the money five years from now you will need to put less money because you will have two extra years of interest accumulated. So let's illustrate this concept. To illustrate the present value concept, assume we need $220, one period from today. So we're going to keep it simple, not simple just the future amount is 200, the period is one period, one period, assume one year, it doesn't have to be a year, we're going to assume it's one year. We want to know how much we need to invest today. So this is the question, we need to know how much we need to invest a day for one period and the interest rate is 10% to get $220. So the problem reads something like this, what amount I need to invest today, it will grow at 10% for one year or one period and will give me $220. Well, if we take the numbers and plug into the formula, so the future amount is 200 and I'm going to explain what the one is coming from, interest rate is 0.1 and the rate and n equal to one. So this is the formula. So conceptually, we must know P must be less than 220. Of course, what we need to invest today must be less than 220 because this to whatever the amount is, whatever we amount to invest today acts, we're going to add to it the interest that's being compounded over time and it's going to give us 220. So this number definitely is less than the future. Of course, you have to know this upfront. So this is clear from the answer. Would we rather have 220 today or 220 at some future date? If we have 220 today, we could invest it today and grow at something larger than 220. So of course, it has to be less than 220 because if we have $220 today, what we do is we'll take this 220 invest and later we'll have more than 220. Therefore, definitely the answer is less than 220. So to compute the answer, we'll take the future value and we'll divide the future value by one plus i raised to the n. n is the number of periods happens to be one. So if we perform this computation, the answer is $200. What does that mean? It means if you take $200 today, multiply this $200,000 by, well, let's first do it in two steps so I can explain where the one is coming from. So if I take $200 today multiplied by .1, which is 10%, it's going to give me $20. If I take my original amount plus $20, it's going to give me $220. So notice I proved to you that if you invest $200 today at 10% for one period, for one period of time, you will get 220. Now, the other way to do this computation is to take 200 times one plus .1, which is a 10%. Now, why do include one? Include the one to say I want the answer to be 200 times one equal to 200. 200 times basically from a mathematical perspective, you're taking 200 times one, which is 200 plus 200 times .1, which is 20 equal to 220. So that's why we add the one, we take a 200 times one plus one plus the interest rate, this way to take the original amount, to take the original amount into account. So that's why we say one plus I, one plus the interest rate. Therefore, the formula is one plus I, one plus I. So notice, indeed, if we invest this money, if we invest $200, we let it sit in the bank at 10% for one period, we will get $220. We interpret this result to say that giving an interest rate of 10%, we are indifferent between a $200 today or 201 period from now. So if somebody asks you, would you prefer to accept $200 today? Or if you wait one year, one period, let's assume the period is one year, one period. And I will give you $220. Well, if you can earn 10% on your money, those both amounts are the same. If you give me $200 today, I would let it grow to $220. And if I wait a year from now, I will get my $220. So the amount technically the same. So the $200 today is the same as the $220 a period from now. So let's use this formula to do more computation and see if we can come up to a general formula or general conclusion. Okay, let's assume we need $242 at two periods from now and we can invest this money at 10%. Simply put, now we are looking at two periods. In two periods, we need $242. This is today. Today is 0.0 or 0. This is one period and this is two period. Now we need this money, two periods from now, $242. We can invest this money at 10%, whatever money we have. How much do we need to invest today? What's the present value today? What's the amount that we need to invest today to get $242? Well, we'll use the formula. We know the future value that we need $242. We can divide $242 by 1 plus the interest rate raised to the second power. Why to the second power? Because we have two periods. Simply put, we need $200. Simply put, let me show you that if we do so, if I take $200 times 1 plus 0.1 raised to the second power, let me show you 200 times 1.1 raised to the second power. Let's do the computation. Let me show you what would happen. 1.1 times 1.1 is 1.21. So 200 times 1.21 equal to times 200 equal to 242. So notice, I just proved to you that indeed if you take $200 invested in two periods for two periods, let it grow, it will grow to $242. Therefore, the answer is $200. So this is just an application of what we just did, what we just did a minute ago. As I mentioned earlier, the number of periods, remember I said this does not have to be expressed in years, although I just said in years, just to give the example simple. Any period of time, such as day, month, or quarter, or a year can be used. Whether period is used, the interest rate must be compounded the same period for the same period. What does that mean? This means if a situation express n in month and i equal to 12%, in other words, if the problem says the n is expressed in month. So the number of periods n in month is, so the number of periods is a monthly basis and they give you an interest rate i equal to 12%. Now, most of the time, not most of the time, the interest rate is quoted annually. Well, if n in month is, how many months do we have in a year? We have 12 months in a year. It means the interest rate 12% will have to be divided by 12. It means the interest rate that we will use is 1%. So if the number of periods is other than a year, then the interest rate will have to be adjusted. If the period is semi-annually, semi-annually is what? It's twice, twice a year. So if they said the period is twice a year and i, the interest rate equal to 12%, then the interest rate really that we're going to have to use i equal to 6%. Why? Because 12% divided by 2 equal to 6%. Let's go a little bit further. If they told you the period is quarterly, how many quarters do we have in a year? Four quarters. It means if they told you it's one year compounded quarterly, it means we have four periods. The same thing. We're going to have to take the interest rate and divide the interest rate by 4. Therefore, the interest rate equal to 3. Therefore, i equal to 3. So it's very important to know this. So in this case, let's just read further. Interest is set to be compounded monthly for the 12% monthly here. For example, the present value of a dollar when n equal to 12 month and i is 12, this is what we'll use. So if you are being told that the period is one month, so you will take the amount, whatever the, this is the future value, whatever the amount is divided by, you don't use 12%. You would use 1%, sorry, 1% and you n equal to 12. This is n, n equal to 12. So if you want to get a dollar, if you want to have a dollar a year from now and the interest rate is 12% compounded monthly, it means you have 12 periods because that's compounded monthly and i equal to 1%. It means you need 88 cent, 88, 88, almost 0.8874 pennies in order to get a dollar a year from now. Okay. Now, knowing all this information, we could use the present value table to compute the present value of a single amount. So rather than doing the computation or I showed you the computation, you could use tables to find the present value of an amount. Okay. A present value table, which we're going to see in a moment, helps us with present value computation. It gives us the present value factor for a variety of both interest rate, i and n, because when to compute the present value, you have to have the interest rate and you have to have the number of period. Each present value and a present value table assume that the future value equal to 1. When the future value is different than 1, we could simply multiply the present value from the table by the future value giving. So the table assumes that your future value is 1, which is 1 dollar, but it doesn't have to be 1. Your future value could be anything, but since we have the factor, we can take that anything multiplied by the factor and get the present value. Now, the best way to show you this is to basically illustrate this concept. To illustrate this concept, let me show you the present value table. This is the present value table. It will be the end of your textbook, depending on which textbook you are using. It happens to be table B1 in my textbook, and this is the formula P equal to 1 dollar divided by 1 plus i raised to the end. Let's go back to the 200 dollar example. I'm sorry, not the 200, the 220. We said, how much money do we need if n equal to 1 and i equal to 10 percent? And remember, we use this formula. Now, rather than using the formula, all what I have to do is I know the number of periods right on the side, 1, 2, 3, 4, 5, 6. So n equal to 1 and i across, the rate is across and 10. So my factor is 0.9091. All what I have to do now, I know the future value is 220. If I multiply 220 by 0.9091, I will get, voila, 200 dollar. Let me show you on the calculator, this way you will see it, that 220 times 0.9091 will give me 200 dollar and 2 pennies that's rounding 200 dollars. Do you guys remember when we looked at the example that said, what if I need 242 dollars two years from now, so n equal to 2, or two periods from now, and i equal to 10 percent? All what I have to do now, is to go to my table, n equal to 2, now it's a new problem, n equal to 2 after two periods, interest rate is 0.0826. All what I have to do is take 242 multiplied by 0.8264. Let's do that and show you, show you the answer, how we come up with the answer, 242 times 0.8264 will give me 199 dollars and 98 cents rounding 200 dollars. So notice, at this point, you can give me any future value, you can give me the number of periods in the interest rate, I can find the present value, I can find the present value. Let's assume I need, just work an example, I need $40,000, n equal to 8 in eight years, I need this money and I can invest this money at 6 percent. So this is the future, this is the future value, this is how much I need and I'm going to n equal in years, I'm going to keep n equal to years, it doesn't have to be, I'm going to tell you it's n equal to years and obviously this is yearly rate. So how much will I need to put today? What's my, what's the amount that I need to invest today? What's my present value? I only need to put this amount only once. Well obviously it has to be less than $40,000, let me show you how I will find this. So the periods, let me erase this, the periods equal to 8, let me just erase this. Okay so for the sake of my illustration, I need this money eight years from now, I can find an investment that's going to give me 6 percent. So the factor is 0.6274, 0.6274, 0.6274 times $40,000 and I need $25,096. 0.6274, 0.6274 and that's going to give me $25,096. Now you know the best way to show you this, to even illustrate this further, I need to show you the actual computation. Let me confirm my number, let me make sure I'm doing this correctly. Will I get 25, if I invest $25,096, will I get $40,000 eight years from now? Let's do a computation of this and show you that it works. So to illustrate this concept, I'm going to show you this in an excel sheet because I need to do this computation. So let me just show you how it works. Okay a year from now, remember I have eight years to come up with $80,000. After year one, after year one, here's what's going to happen. This amount that I have, it's going to grow at 6 percent. So I'm going to take the previous number, multiply it by 1.06. So the, my original amount times 0.6, this amount becomes $26,000. Let me use the commas here. This amount becomes $26,601. Let me show you what's going to happen in year two. In year two, I'm going to keep this money invested and it's going to grow at 1.06. This money is going to grow to, let me just fix my formula, it's going to grow to 28,197. I'm going to keep this money invested and it's going to grow at 1.06 and it's going to grow to 29,000. I'm just going to pull the formula down. It's going to pull down and notice year eight, I'm going to have $39,999 which is approximately grounding 40,000. So notice I just showed you, if I take this amount that I found and I can confirm it to myself, if I keep investing it at 6 percent for eight years and this and it's invested, the investment is compounded. It means I'm not taking out the interest, I'm keeping the interest. So the interest is earning interest therefore I will get 40,000. So simply put, I just showed you that indeed I can find any present value if I'm giving I the interest rate and if I'm giving N the number of periods. If I have those, I can find the answer. Now what about if you want to know how long do you need to invest your money? If you know how much money you have now, the interest rate and future value. So what are we looking at here? We're looking at, remember we have P present value, we have F future value and the P it's going to grow at N and it's going to grow at I. It's going to grow for a period of time using the interest rate. Let's assume we know the future value. We know how much we need. We know how much money we have today. We know the interest rate that we need, that we can invest the money at. The only question that we have is we don't know how long we need to invest this money. Can we find out? Can we solve for N? And the answer is yes we can. Let me show you. Let's assume a $100,000 future value. Simply put, we know we need to have 100,000. That's worth $13,000 today. Today we have $13,000. We have $13,000 in the future. We want to have 100,000. We know we can invest this money at 12%. So notice we have the future value. We have the present value. We have the interest rate, 12%. But what's unknown is N. How long do we have to invest this money in order to get in order to get 100,000? How do we find the answer for such a question? Here's what we do. We can use the present value table, which is table B1 that I was just working with. I'm going to go back and look at it. And what we'll do, we'll take the $13,000, the present value, put it in the numerator and divide $13,000 by $100,000. So take the present value divided by the future value. Okay, let's do that. So if I take $13,000, $13,000 divided by $100,000, I'm going to get 0.13. The answer is 0.13. What is 0.13? 0.13 is the factor. Now, how do I find the answer? 0.13 is 0.13 of what? 0.13 is the factor. Now, what do I need to do? I know the interest rate that I can earn is 12%. So I'm going to go to my table and I know the 12%. I know my I, 12%, right here. I is 12%. I know my I, let me highlight, if I can highlight my I, I know my I is 12%. So what I would need to do now, I need to go down across 12% until I get to 0.13. And if I keep on going down across, I see 0.13 right here. Now, I look on the other side, I go across and it seems, let me highlight the 0.13 as well. I highlighted the 0.13, I move across and it seems I need 18 periods. So it's going to take me, for the $13,000, it's going to take me 18 periods, assuming in years, it's going to take me 18 years. So simply put, so simply put, simply put, it's going to take me 18 years to get the money that I will need, I will need to get. So the $13,000 that I have today, that's going to grow to $100,000, give an interest rate of 12%, it's going to take me, we found the answer, it's going to take me 18 years. Now, obviously, you can prove this to yourself, you can take this $13,000 multiplied by 1.18. It's going to be your year one balance, take your year one balance multiplied by 1.18, so on and so forth. And you will find out after 18 years, it's going to give me 100,000. So what I did is I find the I, I found the, I'm sorry, I found the N giving the I, giving the P and giving the F. Let's look at another scenario, Sol for I. Let's assume we know the future value that we need is 120,000. So we know the future value is 120,000. We know we have $60,000 today, $60,000 present value. And we know we have nine years, we have, we have nine years for this money to double and equal to nine, because we have $60,000 today, we need 120, nine years from now. The question becomes, what interest rate, what investment do we need an order, what interest rate at what rate do we need to, to, to invest this money, the $60,000, so we will get, so this money will double. Well, we use the same concept, we'll take the present value divided by the future value and that's easy, that's going to give us 0.5. 0.5 is the factor. Now we have the factor, 0.5, we know N equal to nine, we're going to do the same thing, we're going to go to the present value table, N equal to nine, and we're going to go across to find which one is the closest to 0.5, so we're back at the tables, now we know, now we know N equal to nine, so we know N equal to nine, so we know this number equal to nine, let me highlight this number, we know this number equal to nine, now we need to go across and find the closest thing to 0.5, the closest thing to 0.5, the closest thing to 0.5 is 0.5002, that's the closest thing, so approximately we need, let me highlight this number, we need to earn 8% on our investment, we need to earn 8% on our investment, the earning is 8%, so what we did is we found the interest rate, we found the interest rate, now the best way to illustrate all these concepts is to actually work a few examples to show you how the present value work, my company expect to earn 10% per year, so this is I equal to 10%, on an investment that will pay 606,773, which is the future value, F, six years from now N equal to six, can you find the present value of this investment? Simply put, if you need to have 606,773 dollars, six years from now, how much money you will need to invest today, so what's the present value today? Okay, what's the present value today? Well, all what I have to do is I have to go to my tables, I equal to 10 and equal to six and they meet at the factor 0.5645, 0.5645, so I'll take 606, 773, 773 multiplied by this factor, so 606, 773 multiplied by 0.5645 and I need obviously less than this amount, 342, 523, 342, 523, so this is the answer, this is the present value, simply put, if I, what I'm saying is this, if I invest this money today and let it sit for six years earning 10%, I will have 606,773, so I found the present value. Let's take a look at another example, a company agrees to pay 20,000 in three years, okay, so they pay, so the future value equal to 20,000, that's what they need to pay and they need to pay in three years. If the annual interest rate is 10%, how much cash will the company can borrow with this agreement, so how much cash will they need today? How much cash they will need today, so what's the present value? Well, obviously it has to be less than 20,000 and equal to three, they need this money three years from now and 10% is right here, so the factor I'm going to be using is 0.7513, so if I take 20,000 multiplied by 0.7513, let's do this, if I take 20,000 multiplied by 0.7513, I will need 15,026 dollars. Again, you can prove it to yourself, since this is only three periods, I'm going to go ahead and show you that it will be 20,000 if you invested it at 10% for three years, so it's today we need to come up with 15,000 and 26 dollars, 15,000 and 26 dollars, we have year one, year two and year three and it's going to grow at 10%, so I'm going to take the previous amount multiplied by one plus 0.01, the original amount and add to 0.1, add to a 10%, a year from now it will be 16,528, if I grab the formula in three years I will have 19,999 dollars and 61 cent, if I round it will give me 20,000, hopefully I just showed you that if I proved it to you, I showed it to you. Now, Thompson expect to invest 10,000 at 12%, so at the end of a certain period, at the end of a certain period received 96,463, how many years will it be before Thompson received the payment, so simply put Thompson today he got 10,000 dollars, he knows the present value, he knows the future value, 46,463, the future value, he knows he can invest this money at 12%, the only question he does not know, how long it's going to take him, how long it's going to take him to have this money, how long, well, again I'm going to be using the tables in order to compute this, how do I compute this, first I will need to take my 10,000, my present value divided by the future value, let's do that, if I take 10,000 divided by 96,463, that's going to give me 0.1036, 0.1036, this is the present value factor, this is called the factor, the PV factor, now I know I equal to 12, so I know I'm going to be looking into this equal to 12, so I'm going to go down until I get the closest to 0.1036, I'm going to keep going down below 0.10, I have here 0.1037, that's the closest, if I look across it's going to take Tom n equal to 20 years, n equal to 20 years, well, you know what, let me just also confirm this, I just want to show you that it works, I just want to show you one more time, so we have 10,000 and I want to make it 96,463, I equal to 12%, okay, so if I go to my excel sheet, I have $10,000 and it's going to be 20 years, year one, year two, let me grab it all the way to year 20, year 20 and this money is going to grow at 12%, so I'm going to take the previous amount, multiply it by 1 plus 0.12 and if I take this formula and I grab it and in 20 years notice I will have 96,000, let me round it, 963 and this is the answer, it will take me 20 years if I let this money grow at that amount, so notice we prove the computation, just to kind of show you that it works, just so you have more confidence. Bill, expect to invest 10,000 for 25 years, so we know we have 10,000, this is the present value, we know n equal to 25, after which he wants to receive 108, I don't know what he needs this money for, but he needs 108,347, the question here is what interest rate, what is the i, what is the interest rate that we will need to invest this money at, so we know PV, we know the present value, we know the future value, we'll do the same thing, we'll take the 10,000 divided by 108,347, so 10,000 divided by 108,347 and that's going to be 0.0922, 0.0922, so this is the factor, I know n equal to 25, do I have 25, I do have 25, so n equal to 25 and I'm going to go across until I find the closest thing to 0.0922, well I have it right here, 0.0923, it looks I need to earn i equal to 10%, simply put, if you take this 10,000 invested at 10% for 25 years, you will have 108,347, 108,347, so hopefully after you look at these exercises, it's starting to make more sense, we're looking to find out how much we need to invest today, so simply put, how do we use this in the real world, well how much do I need to lend you money today if I want to earn a certain amount of money, okay, so that's how it works, like how much will I need to give out today in order to get a certain amount of money in the future, so that's the importance of present value computation is a very important concept in accounting and you will see why, now we did the present value, we're going to also look on how to compute the future value, we're going to learn how to compute the present value of an annuity and the future value of an annuity, so we're going to have to compute few things, if you have any questions, any comments about this recording, please let me know, as always I would like to remind you to visit my website for additional resources, like my recording, share them, put them in playlists, in the next session we'll look at the future value of a single amount, stay safe during those coronavirus days, good luck.