 Hello and welcome to this session. This is Professor Farhad in which we would look at the time value of money and I'm going to call this session for capital budgeting purpose. What does that mean? It means I'm teaching you this concept, time value of money, specifically present value of a dollar of a single payment and the present value of an annuity. Now there is no such a thing as only for capital budgeting purpose. I am telling you it's for capital budgeting purpose because I do have time value of money covered in the tails in my other courses and my intermediate accounting course as well as my CPA courses. So for this session, it's a specifically focusing on the present value. I just want to make this clarification at the beginning in case you wonder why I did not cover the future value of a single payment, the future value of an ordinary annuity. The purpose is we are focusing on the present value computation. Let's start to focus on the explanation. I'm going to start by saying a dollar received today is not the same as a dollar received a year from now. It's not the same. So it's the same dollar today. It's not the same if we're looking into the future. Why? Well, there are many reasons. The first reasons we should all be feeling now and that's inflation. And what's inflation? You are losing purchasing power. I'm going to give you a true example which has happened with me recently and that's a Wawa gift card. One of my viewers like yourself, I help him pass the CPA exam and I always give examples about a place called Wawa and he gave me a gift card for $100. He gave it to me December 16, 2021. He gave me this card. I was going to Florida. I forgot to use it to fill out my card with gasoline. Then I just forgot about the card altogether. Six months later, this $100 card, which is worth $100 when he gave it to me December 16, it's not the same as today. Back then, I'm just going to give up some kind of use some numbers. And the reason I know this is because when I went to Florida, so just let me tell you how many gallons I could have purchased. The gasoline was approximately $3.50. So with that money, I could have bought just kind of so you will see the effect of inflation around 28.5 gallons of gasoline. Recently, I went to Michigan. If I wanted to use this $100 gift card to purchase gasoline from Wawa, the average price when I was up there specifically was $5. So now I can only buy 20 gallons of gasoline and the same concept would apply to coffee, so on and so forth. Simply put, the $100 that I received on December 16, it's not worth $100 today, which is only six months later. The point, it was eating by purchasing power. So there is a time value to the money. And this is an example. The other example is risk. If someone asks you, would you receive a dollar today or a dollar into the future? Well, if you're going to wait for that individual to give you a dollar in the future, you are taking a risk. So you want to be compensated for that risk, you would say, okay, either give me a dollar today or $2 into the future. Now you are willing to wait if that's if you think your risk is worth a dollar. So simply put, a dollar today is not worth the same. Although there is no inflation, there is no eroding of your purchasing power, nevertheless, that individual may not come back and give you that $2. So a dollar today in your hand, a better than a promise of $2 because we have risk involved. So keep that in mind as well. We have risk. That's why they are not the same. The third reason is you can invest this dollar. If someone gave you a dollar today, you can invest it and you could have more than a dollar a year from now, or you could have less, you could lose it all. The point here, we're trying to stay positive is you can invest this money. So when we are comparing money, and this is an important concept in this capital budgeting purpose, when we compare money, we compare money in the same time period. So we cannot compare money. We cannot say year three, we have $100 and year one, we have $100. Those two $100 are not the same because they live in two different time periods. We have to bring them to the same time period, which is bring year three to year one. Now we can compare year three money to year one money, or we can move year one money to year three money and compare them. Well, how do we do that? This is what we're going to be learning, the concept of time value of money. Let's start illustrating the concept by working a simple example. Assume Bank of America pays 10% on $1,000 deposit made today. I hope we're not going to get to that point. Bank of America paying 10%, but the way inflation is going, we can get there. But I hope not. How much this money will be worth one year from now? I'm pretty sure all of you already know that we're going to have an additional $100. Therefore, we're going to have $1,100. But let's clarify, let's illustrate the formula because we could have multiple years. The formula is as follows. F is for the future value, the future value of money equal to the present value times one plus R. R is the interest rate raised to the nth power. How many periods? So N is the number of periods. R is the interest. One is the original amount, which is whatever amount we have. We have to multiply the present value times one plus P is the amount invested today, which is the amount invested today is $1,000. And F is the future value. If we plug in all the numbers 1000 times one again, why do we add one? Because we want to keep the initial amount, the principal amount plus 10.1 raised to the first power. Therefore, the future value of this amount is $1,100. So as an individual, if you want, if you have $1,000 today, okay, this is what we're saying, if you have $1,000 today, and if you believe Bank of America is paying market rate and someone asks you, will you take, will you accept a $1,000 today or $1,100 a year from now? For you, it doesn't make a difference. Why? Because the $1,000 today is a $1,000 today and a year from now $1,100 is equivalent to you because you are waiting a year, you are being compensated for that wait. Now, what if we left the $1,100 in the bank for a second year? How much the original $1,000 will be worth at the end of the second year? Now what we're saying, let's keep it for two consecutive years, keep that money. Well, after year one, we're going to have $1,100 times one plus I raised to the first power, which we will have most most students would say $1,200, which is not correct. It's $1,210. Why? Simply the formula, if you really want to go with the whole formula, the future value equal to $1,000 if you want to use the formula plus 0.1 raised to the second power. So what we are doing here is we are compounding and this is an important concept in finance. We are compounding the interest. So one plus one raised to the second power, this is called the compounding factor. So we are doing it for two years. But what I did here is I told you, let's keep the money and plus the interest that we earned, which is the $100 and keep it for another year at the bank, we will get $1,210. You might be saying, why not $1,200? The reason is because the additional $100 stayed at the bank and this additional $100 for year two earned you another $10 and this is called compound interest. We have compound interest versus simple interest. So when you invest, when you put your money in the bank or when you borrow money on your credit card, you have to understand how you are being charged. Are you being charged or are you being rewarded using a compound interest? If you are being rewarded, you want it to be a compound. You want your interest to earn interest or you are compensating using simple interest. Simple interest where once you made this $100, they will send it to you and you'll keep the $1,000. So the only the original amount is making profit. Now when you have a credit card, well, that's bad. In most credit card in the US, it's a compound interest. So they will charge you interest on the interest. But again, if you want to learn more about this, go to my finance course. This is not the course that we need to look for. Now looking at this formula here, okay, the future value equal to the present value 1 plus r raised to the end. We can rearrange this formula and basically if you want to find the present value, we can see the present value equal to the future value 1 plus r raised to the end. Now in your textbook, they may call the r i which is the interest. It doesn't matter. It's the interest factor. What I'm trying to say is this, we can look at the present value and the future value as the same. So the present value and the future value, they're basically the same because remember they should equal to each other in terms of value. We can use the same formula. So let's find out how we can compute the present value. I showed you the formula, but let's apply it. If a bond will pay $1,000 two years from now, what is the present value of the $1,000 if an investor can earn a return of 12%. Here's what we are saying on a timeline. This is year 0, this is year 1, this is year 2. Someone came to you and said, look, I'm willing to give you $1,000, $1,000 two years from now. Now two years could be two periods. The period could be a quarterly, but here we are assuming years to keep it simple for capital budgeting purposes. So year 1, year 2, how much will you invest today? So in two years, you will make, you will receive $1,000 and you are happy with that. Now you already know you want to earn 12%. So what you do now is you use the formula and you will discount. You would say, okay, my future value, I know I want $1,000 two years from now. If I divide $1,000 by 1 plus r raised to the nth power, my interest rate is 12 raised to the nth power, that's going to give me a present value, a present value amount, present value of $797.20. We call the 12% the discount rate. Now how do we come up with discount rates? So I'm talking about the real world. How do companies or investors come up with the discount rate? This is what you want to earn. What does that mean? It's mean based on your risk tolerance. If you want to earn more, if you want to take more risk, you would earn more. If the investment is less risky, you would command lower rate of return because the investment does not require. So the higher the interest rate, the riskier is the investment. That's what I'm trying to say. But in the real world, you will determine this based on your risk tolerance, based on your time horizon. How long are you willing to wait? Now, this process is called discounting. So we have discounted the $1,000 to its present value. Let me just show you the math. This way, hopefully, you will be more comfortable with this idea. So if you invest the money today, $797.20 and you let it grow at 12%, simply put, you said, well, I wanted to grow at 12%. If we take this amount multiplied by 12%, not 1.12, just 12%, you're going to earn $965.66. At the end of year one, you will have $892.86. If you keep this money for year two and you let it grow at 12%, it's going to earn you $107.14. And in two years, you will have the $1,000. The amount that you were promised and the amount that you wanted to earn based on your risk, based on your market rate, based on your discount rate, which happens to be 12%. Now, this is how we can do this. But think about if you have an investment that's going to be there in five years or six years or 12 years and the interest rate is, I don't know, 6%. So we can change this. Mathematically, it becomes cumbersome to compute. Now, in the real world, you can use Excel, you can use software. But in college courses on the CPA exam, you need to know how to use something called the time value table. This is a time value table. This is the time value, the present value of a single payment or the present value of a dollar. And what happens is this, you're going to have these tables, you're going to have the periods and the rate. And what you do is this, you would say, if I want to find the present value of $1,000, the periods, two periods, now we're assuming the periods are years, the periods could be other than years. I'm emphasizing this point because to really learn about the time value of money, you have to go to my time value, time value, money lectures and my intermediate accounting, the more advanced or to my finance course. But this is good enough for now, two years from now, earning 12%, the factor is 0.797. So if you take the $1,000 times 0.797, 0.2, which is just surrounding and your tables might look a little bit different. So the present value is $797.20. So simply put, I showed you how you could compute this by looking at the tables, by being able to read the tables. And this is an example of the tables. This is the formula, the present value equal to 1, which is this is the future value, divided 1 plus r raised to the nth. Now your table might look a little bit different, but this is the period. So I can find the present value of any amount, eight, eight periods, 10%. That's the factor, 11 periods, I don't know, 17%, someone and so forth. So this is how you would use the table. Now we're going to work an example, just to kind of to practice this table. But before we work an example, I would like to remind you most likely that's, this is who you are, a student or a CPA candidate. Go to farhatlectures.com where I have additional resources, lectures, multiple choice, true false exercises, that's going to help you with your CPA review course, as well as your accounting courses. If you have not connected with me on LinkedIn, please do so. Take a look at my LinkedIn recommendation, like this recording, the mere fact that you are watching, it means it's helping you like it, it will help others connect with me on Instagram, Facebook, Twitter and Reddit. Let's take a look at this example, how much you would have to put in the bank today to have $1,000 at the end of five years, 10% interest. Okay, let's look at a timetable, at a timeline here. Today is zero, one, two, three, four, five. You're going to have $1,000 five years from now. How much will you invest today? That's the question. Assuming you want to earn a 10% interest rate. Now one way to do it is to take the 1,000 divided by 1 plus i, 1 plus r, I always use i because when I learned it initially it was i, 1 plus r raised to the second tower. R is 10%, not the second tower, the fifth tower and you will find the answer or most likely you would not have to do so unless you're in a finance course and if you are in a finance course you'll have to input everything in your financial calculator. You need to learn how to use this. For capital budgeting you're going to have to use the tables. Here what we are looking at is five periods. We are looking at 10% rate of return and the factor is 0.621. If you take $1,000 times the factor 0.621 we find out that the amount you will need to invest today is $621. If you invest this money today and you let it grow for five years it will give you this amount and let me show you. I did put it here to prove the math. So 621 you multiply this by 1.1, 10%. It's going to grow to 683. You'll take the 683. You keep it in the bank. You multiply it by 1.1. It's going to give you 751. You multiply it by 1.1. You multiply it by 1.1. You multiply it by 1.1 and in five years you will get the amount that we said you will get $1,000. Again what we did is we computed the present value of one single amount which is good. That's something you need to know but that's not the only thing you need to know. You need to find the present value of a series of cash flow which is called an annuity. What is an annuity? An annuity is when rather than receiving $1,000 once you might have to receive the $1,000 once, twice, three times, four times, five times, six times. This is very common, three, four, five, six. For example when people retire, when they're close to retirement, what they do, they sell their older stocks, their bonds, they usually liquidate any real estate investments they have and they put all the money in annuity. What is an annuity? They will pay an amount here and they would say we want to receive for the next 20 years the same amount every month. It's called an annuity. Also for capital budgeting, what's going to happen is this. You might have to either pay or usually you're going to either pay means you're going to save some money or you're going to receive additional income from undertaking a project and that income it's going to be the same every period. So you want to know how much is the present value of all these payments. Now you can take each one separately and discount them as we just learned each one separately. For example if this is a 5% discounting you would say okay period n equal to one r equal to 5% and you will do this $1,000, n equal to two r equal to 5% and you will discount this $1,000, n equal to three r equal to 5%, so on and so forth. But this is really time-consuming. Obviously in the real world you can use a calculator, financial calculator, Excel or a software but for my purpose, for the purpose of capital budgeting in a managerial accounting, cost accounting, CPA exam purposes, you need to know how to find the present value using a table and this is a present value of an annuity table. So in your textbook you're going to have the present value of a dollar which is a single amount or the present value of annuity. Now also you have to be careful. You have two types of annuity. You have ordinary and you have annuity though. For my purpose we don't cover annuity though which is basically I'll explain what it is but just be careful in your textbook depending which textbook you are using whether you are looking at the ordinary or annuity though. Let's work an example to illustrate this concept. Adam purchased an equipment on which $60,000 payment will be due each year for the next five years so this is what it looks like. One, two, three, four, five. One, two, three, four, five and what Adam says, Adam's going to make payment on this asset, on this equipment. $60,000, $60,000, $60,000, $60,000 and $60,000. Now we want to find out what's the present value of this annuity discounted at 10%. Well as I said we can take each $60,000, we can take each one of these $60,000 and find the present value separately or we can go to the present value table. We know we are dealing with a five, five payments. We know the interest rate that we are commending is 10 which is 10 right here. Five and 10 they intersect at the factor 3.791. Well that's easy. Now all I have to do is take 60,000 times 3.791 and the present value of the payment is $227,460 and as I said this is an ordinary annuity. What is an annuity do? An annuity do when Adam needs to pay the first $60,000 today at point zero. $60,000, $60,000, $60,000 and $60,000. So again we have five payments of $60,000 however the first payment is due immediately at point zero. Under those circumstances you will have to use the present value of an annuity do or manipulate the formula. I'm not going to go into the manipulation here. I go over it in my intermediate accounting because again I'm going to keep this simple because it's for capital budgeting purposes. So you would use the annuity do formula. Now also in your tax book you're going to be giving a present value annuity table. Remember you have to be very careful. This is OA ordinary annuity. Again you're going to have the periods as well as the rates. So you have to make sure you are using the proper table. It could be a present value of a single amount. It could be the present value of an annuity. Well let's work an example to see how this all fits together. If the if the interest rate is 6% how much you will need to put in a bank account today so you'll be able to withdraw 1,000 at the end of the next five years. And another way another usage of this is when parents send their kids to school. For example parents plan they would say okay we would like to have for our kid to be able to take out I was going to say 30 40,000 you know tuition is just getting out of control 40,000 and let's assume they have CPA five-year program 40,000. So one, two, three, four, five. So let's assume parents want to make an investment today. Today they want to make an investment and they want the investment to pay them simply put they want to be able to withdraw 40,000 dollars for the next five years so so they pay the tuition for their kids or they want you know they want it to you know they want to live of it. It doesn't matter. Okay so how do we find this? Well again we can use we can find each 40,000 dollars separately or we can go to the present value table six percent five periods. So let's go to five periods this is five periods I'm using different colors here five periods six percent and the factor is 4.21236 okay so we're going to take the 1,000 times that factor 4.21236 we're going to round and it's going to be 4,212 just going to ignore the 0.36 and your textbook they might go you know depending on how many decimals they round they're just going to edit not do it. So simply put if you want if you want to make an investment today let me show you this on a timeline because let's use the numbers that we did here this is five years you want to take $1,000 per year out $1,000 per year how much will you pay today if you make if you put an investment for $1,212 and that investment is paying you 5% you'll be able to take $1,000 every year and let's let me prove it this way you can see what I'm talking about you go to the bank today you will deposit $4,212 you will keep you will wait for a year this money will grow at 6% so we'll take this amount times 1.06 and it will give us at the end of the year we will have $4,464.72 we will take the first 1,000 out we will withdraw the first 1,000 so the balance at the end of the first year will be $3,464.72 we let it in the bank account it will grow again this amount will grow at 6% 6% means you will take this amount times 1.06 and it will grow to $3,672 you will take the 1,000 out now you have the new balance it will grow at 6% you will take the 1,000 out and you keep keep doing this for five years at the end of year five after you take the last five thousand last thousand your balance will be zero the point is i'm trying to prove the math for you that it will work this is the value of this investment $4,214 now this is you know i explain the time value i explain how to find the present value is this good enough well it's good enough to start but what you need to do now go to farhat lectures and work mcqs because this all this information that we learned today is needed for the capital budgeting which we're going to be work with capital budgeting in the next session so that's why i'm doing this that's why i'm explaining the time value now so when i'm covering capital budgeting i'm going to say you know how to discount and i'm going to take that for granted anyhow at the end of this recording i'm going to remind you to study hard invest in yourself invest in your career and of course stay safe