 Hello and welcome to this session in which you would look at the future and present value of a single amount. The prerequisite for this recording is the prior recording which is the concept of time value of money. Specifically you want to make sure you are familiar with simple interests as well as compound interests in order to take advantage of this lecture. This topic future value and present value of a single amount is covered on the CPA exam, FAR section, the EC section definitely covered in your intermediate accounting as well as your finance courses. Whether you are an accounting student or a CPA candidate, I strongly suggest you take a look at my website farhatlectures.com. I don't replace your CPA review course. I'm a useful addition. I'm a supplemental material to your CPA review course. I help you understand the material better. I explain the concept behind the theory. I make things easier for you so you can take advantage of your CPA review course and you'll do well on the exam. My subscription is for one month. It's a nominal subscription. You can try it. It helps you. You keep it otherwise you can cancel. If not for anything, take a look at my website to find out how well or not well your university doing on the CPA exam. My supplemental CPA review courses are aligned with your Becker course, with your Roger course, with your Wiley course, with your Gleam, so you can move back and forth between my material and your CPA review course. I also have resources for other accounting courses such as governmental, advanced, intermediate, managerial, so on and so forth. Also, you will find all the AI CPA previously released questions, which is approximately 1500 broken down by topics with detailed solution. Also, if you have not connected with me on LinkedIn, please do so. Take a look at my LinkedIn recommendation, like this recording, share it with others connected with me on Instagram, Facebook, Twitter, and Reddit. So the time value of money is important. Specifically, we're going to be starting with the future value FV. And the reason I start with the future value because it's easier for the student to start with the future value concept. What is the future value of a single amount? Basically, we're looking how much our investment is worth after a period of time invested at a particular date. So simply put, we are standing here today at day zero and maybe after 10 years. How much would our investment be 10 years from now? This is what we're looking for. We're looking for the future value. So when we're looking for the future value, usually we are giving the present value, we are giving the period, how long it's going to take, the period, and we have to be giving the interest rate at what rate our interest is being compounded. So let's take a look at a quick example to see how this all work. What is the future value of 50,000 invested today? So today we're going to invest 50,000. This is the present value amount. We're going to invest this money for five years. So this is a five-year period and the interest rate is 6%. So how much is that money worth compounded annually? We need to know three out of the four factors. What are the three out of the four factors? We have to know the present value, which is 50,000. We know the interest rate is 6%. We know the number of periods is five. So all we're looking for is the future value. How much is the future value of this investment? Well, the formula is this. You will take your present value, multiply it by 1 plus the interest rate raised to the period, which is, if we take a look at it, it means we're going to take 50,000, the present value multiplied by 1 plus 0.06 raised to the fifth power. It means we're going to take 50,000 multiplied by 1.33823. This is called the future value factor. This is called the future value factor, which is 1.33823. And the answer is our money will be valued at 66,912,000. Now, the best way to prove this is to show you that it is indeed, this is the amount. Let's take a look at this. If we started with 50,000, multiply the first year at 6%, it's going to give us 53,000. So when we start next year, we're going to start with 53,000. I'm sorry, this is not 6%, 1.06. Or if you're going to multiply it by 6%, which is equal to 3,000 plus the 50,000. So simply put, after one year, we will have 53,000. This is compounded. It means we're going to keep the interest. When we start here, too, we're going to start at 53,000. Again, we're going to multiply it by 1.06. And it's going to give us 56,180. 56,180, it's going to grow at 1.06. It's going to come up to 59,551. 59,551 multiplied by 1.06 to 63,124. And 53,124 multiplied by 1.06 equal to 66,900 rounded 912. So I just showed you that we can prove that indeed this formula works and showing you that the factor is this much. Now, in your classroom, on your exam, most likely you don't have to know the formula. What's going to happen is you're going to be provided with what's called time value table. In the time value table, this is the present value of a single amount of time value. So simply put again. Daddy, can you do a different color? What color? A green. Great. Daddy, daddy, but how does the color get on the computer? It's recorded, daddy. Okay. Thank you. Thank you. Okay. So now what's going to happen? We're going to take 50,000, multiply it by the present value factor at i equal to 6% and n equal to 5. So 6% is right here. This is the 6%. And 5 period is here. And we're going to go across and we find out that it's 1.3382. 1.3382. i equal to 5 at the top and i equal to 6 and equal to 5. And notice what happened is it's the same factor. So it does not matter. We don't have to do the formula as long as you are using the proper table. And this is the present value of a single amount. Now we can compute the future value. This is not the present value table. This is the future value table of a single amount. And how much is 23,450 worth seven years from now invested at 9%? Well, simply put, all we have to do is take 23,450, multiplied by the future value factor, n equal to 7, i equal to 9. And all we have to do is look in the table, 7 is 7 periods. And if we go across, 9 is 1.880. So if we take 23,450, multiplied by this amount, and let me get my calculator and do this computation, we will find the future value of that amount, which is, let's see, 23,450 times 1.880. And that's 42,866, 42,866. And obviously, you can prove it to yourself if you'd like to just do like what I did, take this amount, multiply it by 1.09, then keep adding to this amount 1.09. And you will see after seven years, it should come up to 42,866. So this is how we compute the future value of a single amount. Let's take a look at this example. Adam invested, which is Adam just asked me to change the color to green $20,000 at an annual interest rate of 8% compounded for eight years. So we need to find out the future value of that amount. First thing we're going to do is we're going to assume that Adam invests this money to earn simple interest. Remember, we have to understand the difference between simple interest and compound interest. At single interest, simple interest, it's going to be 20,000. Every year, it's going to earn 0.08. Every year, it's going to earn 1,600. And if it's simple interest, it's going to earn exactly 1,600 every year because we're going to take the money out. Therefore, we can multiply this by eight. And we know that the amount will be 12,800. So the interest will be 12,800 plus the original amount 20,000 after eight years. So after eight years, 12,800 plus 20,000. After eight years, Adam will have 32,800. Let's assume this money is invested earning interest compounding interest and it's compounded annually. Now we have to use the future value table. The interest is 8% on the top and the period is eight right here. So they're going to meet at 1.8509. So this is our factor. So we're going to take 20,000 multiplied by 1.8509. And I'm going to do this multiplication on my calculator here times 1.8509, 20,000 times 1.8509. This is my calculator is skipping 20,000 times 1.8509. And that's going to give us 37,000 $18. Notice a huge difference whether the interest is a simple interest or a compound interest. The difference is approximately five more than $5,000. So it makes a difference. Now we're going to compute the future value assuming that the compounding is semi annually. Remember compounding semi annually, it means every six months, we're going to compute the interest at the interest of the principle and add more interest to the principle and the interest. Then after six months, we'll do the same thing. We'll add the interest of the principle and the previous interest, so on and so forth. However, when we go to the table, remember we have to adjust the I. The I now is 4%, not eight, but the period rather than eight, it's going to be 16. So we're going to come across and the factor is 1.8730. Now we're going to take 20,000 times 1.8730. And let's see how much is that. 20,000 times 1.8730. And that's 37,460, which is more than compounded annually. And it's much more than simple interest. So in this exercise, I showed you how to compute the future value of an amount, including compounding. The next topic we're going to look at is the present value computation, the present value specifically for a single amount. What is the present value? Is the amount needed to invest today to obtain a desired future value given an interest rate in a period of time? On a timeline, it looks something like this. You know how much you need in the future. You need 100,000. You have N equal five years, I equal 6%. So how much money you'll have to invest today? So this money will grow for five periods at 6% and it will become 100,000. So 100,000 is the future value. We have the future value. We have N, we have I. We're looking now for the present value. And this is basically what you are responsible for. On the exam and on the CPA exam in your classes, you will be more likely involved with computation of present value. So how much money you will need to invest today? So you have 30,000 to start your own business five years from now, giving a 7% interest rate compounded annually. So simply put, you want to start the business five years from now, but you need to start with $30,000 of capital. Well, you have some money now. You need 30,000. You need to ask yourself, how much will I have to invest today? So I will have that 30,000 five years from now. Well, there's a formula for that. And the formula is you'll take the future value divided by 1 plus I raised to the N. The future value is 30,000 divided by 1.07 raised to the N. It's 30,000 divided by 1.4025. You need today 21,390. So if you let that money grow for five periods, what we're going to assume the period is five years, five years, earning 7%, it will get you there. It will get you to 30,000. Or you can use the present value table, which I will show you in a moment. But before we use the present value table, let me prove to you that if you invest today 21,390, let it grow at 7%. It's going to give you at the end of the first year, 22,877. You'll let this money grow the 22,887 at 7% times 1.07. It's going to grow to be 24,489. You're going to keep this money growing at 7% for year four and year five. By year five, you will have 30,000 and a dollar. And this is the number that we need. So notice I prove to you that if you do invest your money at 7% at 21,390, you will get your 30,000 that you need when you are ready to open your business. As I told you, also you can use the present value table rather than going through the computation. And let me show you how you could use the present value. Again, you would look at something called the present value of $1 of a single amount. So your textbook might say a single amount. And the interest rate we are looking for is 7%. And i equal to 5. And i equal to 5, they meet at 0.731. So if you take 30,000 times 0.713, not 731, times 0.713 will give us 21,390. And that's the number that we got using this formula. But rather than going through the formula, rather than going through the formula, the formula is already computed for you right here. But remember when you are looking up, you have to be careful about two things when you're looking things up in the table. One is you're looking at the right table. Here we are looking for the present value. Therefore, we are looking at the present value table. That's the first thing you need to know. Two is you are using the proper factors. What do I mean by the proper factors? You are using the proper n and the proper i. Make sure if it's not compounded annually, you have to make adjustments. You have to, for example, if it's semi-annually, you have to take your i divided by 2. And you have to take your n multiplied by 2. So be careful when you are dealing. And we talked about this in the prior session, because this has to do with simple versus compound interest. The best way to illustrate this concept is to actually work an example. Adam's lifelong dream is to take a trip in outer space. The cost of the trip is a million. How much will Adam need to invest today to take the trip in five years, giving an interest rate of 8% compounded semi-annually? You have to be careful when the compound is not annually. You have to make adjustment to when you go to the tables. Now we are dealing with a situation where the compounding is semi-annually. So for discussing five years, it means semi-annually. It means 10 periods, because every year we'll have two periods, 10 periods. And the interest rate, rather than 8, it's going to be 4. So the factor is 0.676. Therefore, Adam will need to invest approximately 0.767, 676,000. Now let me prove it to you. If Adam invests away, 676,000, let it grow every year at 4% and keep that money invested at 4%. After 10 years, he would have approximately a million dollars. Because this is rounding, we said 1,645, but rounded, it's going to be approximately a million dollars. Again, this is a proof that if you invest today in an investment and you pay 676,000, let it grow, you will earn a million dollars. Sometime what we have to do is to find the interest rate or find the period in a single sum problem. So rather than asking you to find the present value of an investment or the future value of an investment, sometimes you'll be asked to find the interest rate or the number of periods. So let's assume Sam wants to take a vacation around the world. He wants to accumulate 50,000 for this trip. Right now, Sam has $33,330. And Sam can earn 7% on his investment. How long would it take to accumulate the 50,000? So we know the present value. We know we have 33,330. We know we need 50,000. We know we can invest this money at 7%. Okay, so we know all of this. The question is how long is this investment going to take? There are two ways to complete this problem. There's the present value method and there's the future value method. First, I'm going to show you the present value. What do you do with the present value? Under the present value, you will take the present value amount, divide in the present value amount divided by the future value. And that's going to give us 33,330 divided by 50,000. And that's going to give us 0.6666. Now, here we go. We go to the present value table since the present value amount and the numerator. And we know it's 7%. So we're going to look at this column and we're going to go and come to the closest number to 0.6666, this factor. So the factor we find 0.666, we go across and guess what? It's going to take us approximately six years to accumulate 50,000. Now, let me show it to you. If we take 33,330 today, let it grow at 7%, it's going to grow to 35,663 a year one. Starting year two, 35,663, let it grow at 7%. It's going to grow to 38,160. And if we keep doing this for six years, we will have 50,019 dollars to spare. I just showed you the proof. So this is called the present value way because we use the present value in the numerator. Well, so it's going to take us six years. There's the present value method and there's the future value method. The future value method is the exact opposite of the present value method. What we do in the future value method, we're going to take the future value and divide it by the present value to find the factor. So we're going to take 50,000, divide 50,000 by 33,330 and that's going to give us 1.5. Again, we're going to go to 7% column because we know we're dealing with 7%. We're going to go down and the closest thing to 1.5 is right here, 1.507. Again, it's going to take us six years. I just showed you the proof that it's going to take us six years. So there are two ways to find this, to find this answer, either the present value way or the future value method or way. Maggie's pharmaceutical needs 10 million to research at your fortinitis four years from now. So that's what they need four years from now. The company has right now 6,830,134 dollars set aside for future research. How much Maggie's pharmaceutical will need to earn to accumulate the 10 million? So again, we know we need 10 million. We already have 6.83 million. We know that. We know we have four years, but we don't know at what interest rate we need to accumulate this money. Guess what? It's the exact same concept. What we're going to do, we're going to go to the present. I'm going to do the present value, but you know you can do the present value method or the present value method or the future value method. And basically what I'm going to do, I'm going to take the present value 6,830,134 divided by 10 million and that's going to give me the factor that's closest to 0.683. I know I need four years. So you'll go to the period four and you go across and the closest thing to 0.683 is seems we need to earn 10 percent. It means we need to earn 10 percent. Now, the best way is to prove it to yourself that indeed it's 10 percent and let's prove it. If you take 6 million, 830,134 dollars, let it grow at 10 percent for year one, you're going to get 7,513,147. Let this money grow at 10 percent and if you let this money grow at 10 percent for the following four years, you're going to be short a dollar. You're going to earn at four years 9,999,999. The reason I show you the proof is I want to make sure you don't only understand the formula, you see that it makes sense. I want you to be convinced that it works and you should try it yourself. Use Excel sheet. I built those tables using Excel sheet to show it to you. Once you are convinced, everything will be good. At the end of this recording, I'm going to remind you whether you are an accounting student or a CPA candidate, especially if you're a CPA candidate. Keep your course. I don't replace your course. I'm a useful addition. I explain the material differently, specifically a little bit more in depth. I'll give you additional explanation, additional exercises, additional resources that's going to help you understand with your CPA review course and from there, you can pass your exam. Good luck, study hard and of course stay safe. The CPA is worth it.