 Hello and welcome to this session. This is Professor Farhad and this session, we would look at the future value of a single amount. This topic is covered in a financial introductory course as well as the CPA exam far. And this topic might be covered on BEC a little bit as well. 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 1,600 plus accounting, auditing, finance and tax lectures. This is a list of all the courses that I cover, including many CPA questions. If you like my lectures, please like them, share them, put them in playlist, subscribe, share them with others. If they benefit you, it means they might benefit other people. So share the wealth, connect with me on Instagram. On my website, you will find additional resources to supplement your accounting education, such as practices, true, false, multiple choice, as well as other resources, especially if you are studying for your CPA exam, I strongly suggest you check it out. A prerequisite for this session is the present value of a single amount. So it's helpful if you understand how we compute the present value of a single amount, because in this session, we are computing the future value of a single amount. The link is in the description. So what's the idea behind the future value of a single amount? Simply put, we are looking into the future and finding out how much our money will be worth in the future. And that's very intuitive. It's much easier to understand the future value than the present value. Why? Because as you are working now, for example, right now you have some money, you want to put that money today, let's assume you have $10,000. Somebody gave you $10,000. And you have no need for it today. So what you'll be interested in, you'll be interested in knowing if you invest this money and you are 30 years away from your retirements or 50 years, you want to know after 30 years how much this money will be worth. So the period, you are looking at a period of 30. Now we have to determine what interest rate, what investments are you going to invest your money in? For example, are you gonna invest in stocks, in bonds? Are you gonna put that money in the bank? Are you gonna buy treasury bond? And that's gonna determine your interest rate, your rate of return, your interest rate. It could be 3%, it could be 5%, it could be 7%, depending on your risk tolerance. So the future value is this amount here. How much this money will be worth when you retire? So that's what we're looking for. That's what we're looking at now. Now to compute the future value, there's this formula here. To compute the future value, you will take the present value, the PV times one plus I the interest rate raised to the nth power, which is the period. The PV is the present value. This is how much money you have today. For example, if we're gonna apply this formula to the number up there, we'll take $10,000 times one plus 0.07 raised to the 30th power. I'm not gonna compute this because you're gonna see we can compute this number real easily later on. It just, this is how you put the formula in. Now a lot of students ask, why do I have to add the one? Because you need your original amount, times one, which is 10,000. Your original amount, 10,000, times 0.07, all raised to the third power. So the one is there just, so your original money is there. So let's look at a simple example to see how this formula work. Let's assume you have $200 today and you're gonna invest this money for one period. Remember, one period is, we're gonna assume it's one year. It doesn't have to be a year. We're gonna assume the period is a year here at an interest rate of 10%. So if I take $200 times one plus 0.1, raised to the first power, okay? Again, why do I put one? Because I'm gonna take 200 times one, 200 times 0.1 and add them together, okay? So otherwise, if I know that 200 times 0.1 is $10, but I have to take the $10 and add it to the 200 to get me 210. And that's why we put the one in the formula to account for the original amount. So if you take 200 times 1.1, raised to the first power, it's 200 times 1.1 equal to 220. What we are saying is $200 invested for one period at a rate of 10% will give you a future value of $200, okay? So this formula can be used to compute the future value for any number of period into the future, okay? Let's assume we're looking at $200 invested for three periods at 10%. Again, I'm gonna take the $200 times one plus 0.1, raised to the third power. Again, if I do this computation, 200 times 1.1 raised to the third power, 200 times 1.3310, which is 200 times 0.13310 equal to 26620. Again, notice I use the same formula and I find this number. Now, your N, which is the number of period, this could change a lot and the I could change a lot. So rather than doing this computation, we do have future value table that de-computed all the factors. So simply put, if you want to find the future value of any amount, N equal to three, I equal to 10%, all you have to do is to take this amount and multiply it by this factor, 1.3310 and you'll be able to find out the amount. So you could switch the 200 to 10,000 to 10 million to 50 million to any amount you want to, okay? And what happened is the tables, they already computed all those factors for you. So here's the table, the future value table and this is not a complete table. For example, let's go back to N equal to three periods. So the period is right here, the period is down here. So N equal to three and we were using 10%, 10% right here and notice the factor is 1.3310. Therefore, if we take 200 times 1.3310, it will give us 266.20. So we don't have to go through the formula. Now, if I want to go back and compute how much the future value of $10,000 invested at 7% for 30 years, very easy. I'm keeping, I'm having this money for 30 years. I'm earning 7% and I'm gonna take the 10,000 multiplied by 7.6123 and that's gonna be, let me do the computation here. It's gonna be $10,000, $10,000 times 7.6123 and that's gonna give me $76,123. So this is the future value. Notice, as I told you, I'm not gonna do this, I'm not gonna take 1 plus 07 raised to the nth power. I can do that. Now, let's talk about the tables, okay? Because remember, we looked at, this is the future value table, which is in your textbook, it's table B2 and this is the present value table. This is what we did in the prior session, which you can find in the description and here's what we need to know about the tables. Here's what we need to know about the table. There are some important relationship between the table B1 and B2, which is the present value and the future value for the rows where n equal to zero, the future value is one for the interest rate. This is because no interest is earned when the time do not pass. So notice, when the time is zero, when the time is zero, there's no interest. When the time is zero, your money is worth today. Your money is worth what it is worth today, okay? Because you have no future value. We also see that table B1 and table B2 report the same information but in a different manner. So what's in table B1 and table B2? They're kind of the same information reported in a different manner and I'll explain in a moment. In particular, one table is simply the reciprocal of the other and I'll show you what does that. To illustrate the inverse relation, let's say we invest $100 for a period of five years at 12% per year. So here's what we're looking for. We're looking at $100 and equal to five, I equal to 12%. How much do you expect after five years? Well, let's do that. So let's find out how much you would expect after five years at 12%. So let's find the future value for this amount. N equal to five, I equal to 12 and the factor is 1.623. So I'm gonna take $100 times 1.7623. So notice 1.7623. It's gonna give me 176,023 cents. So let's keep on reading this. We can answer this question by looking at table B2 by finding the future value of five period at 12%, which is 1.7623. If we start with 100, the amount that accumulate after five years is 176, which is we found right here. Now, we can alternatively use table B1. Here we find the present value of $1 discounted at 12% for five periods. Let's look at the other table. If we look at table B1, which is the present value and we find out that 12% five periods, the factor is 0.5674. Let's see what does that tell us? So the factor is 0.5674. Recall the inverse relationship relation between the present and the future value. So this value is basically, if you really think about it, it's one, if you take one divided by 0.5, 0.5, five, six, seven, four. Let's see what we find. One divided by 0.5674. And notice it's 1.7623, 1.7623. So notice they are the inverse of each other. So if we only have the present value table, if I only have the present value table, I can find the future value. What I do, I'll take the present value 0.5, always put 0.0, 0.5674. This is the factor. I'll take one divided by 0.5674 and it's gonna give me the future value factor, the future value factor. So notice they're the inverse. So there's the inverse of each other, whatever. If you have one table, you can find the value of the other. You can find the value of the other. Make sure you're aware of this, okay? Make sure it's a very important relationship, okay? Also, as we learn in the present value, sometime you might be looking for something other than the future value. So we can solve for future value when I and N are known, just like what we did earlier. How much is $100 worth N equal to five, I equal to 12%. So that's the typical future value, which is 176, 176, 23. Also, what we can do, just like what we did with the present value, we can solve for N when the future value is known and when the interest rate is known. Okay, let's look at an example. Let's assume we have $2,000 today. We know we have $2,000 today. And we want to know how many periods it will take to make it 3,000. So we know we have 2,000 today. And we want to have 3,000 in the future. So we know the present value, we know the future value. And we know we can earn 7%. So the question is, how long it's gonna take us to find out how long it's gonna take us to accumulate this money to 3,000? What's N, what's the period? Well, we use the same technique that we use in the prior session. Here, I'm gonna use the future value technique. What you do is you take your future value, which is 3,000 divided by 2,000. If you put the future value in the numerator, you would use the future value table. So let's find the future value, let's find the factor. So if you take three, it's actually three divided by two, but let's take 3,000 divided by 2,000 is 1.5. Since the future value, since the future value in the numerator, I'll go to my future value table. And I know my interest rate is seven. So I'm gonna go to the seven here, right here, seven. And I'm gonna go across until I find the closest thing to 1.5. The closest thing to 1.5 is this number here. I go across and the period is six. So it's gonna take me six years. It's gonna take me six period to make this money, to make this money. Now, I'm gonna prove it to you. I'm gonna show you that if you invest $2,000 for six period at 7%, you will get this number. Let me show you this. So I have today $2,000 and I'm gonna invest this for six years. Let me do this, invest this for six years. So after one year and it's invested at 7%, I'm gonna take this number, multiply it by 1.07, which is gonna grow at 7%. It's gonna become 2,140. I'm just gonna take this and drag it. And after six years, notice it will become $3,000, $3,001. It's due to rounding, but $3,000. So notice I just proved it to you that six years is the amount that you need. Sometime, what happens is you have to solve for I. What type of investment do I need to get a certain amount of money? Let's assume we have $2,001, it doesn't matter, $2,000. We have nine years to double this money to 4,000. So we have 2,000 now, that's the present value. And we want to double this money to make it 4,000 and we have nine years to do so. What type of investment, what interest rate do we need? What type of investment do we need to make this money equal to 4,000? Again, I use the future value techniques, 4,000 divided by 2,000, it give me a factor of two. This is the factor. I'm gonna take the factor, go to the present value factor and look at N equal to nine. N equal to nine is right here, N equal to nine. N equal to nine is right here. I'm gonna go across until the closest thing that's gonna give me the number two and that's 1.9990 and I move up and it seems I'm looking at 8%. So if I invest $2,000 at 8% for nine years it's gonna give me $4,000 and let me prove it to you. I have $2,001 and I'm gonna invest it for, see how many years, I believe nine years, nine. So this is year two and I'm gonna go all the way to year nine and I'm gonna invest this money times 1.08, so every year it's gonna grow at 1.08. And let me take this formula and drag it and in nine years I will have $4,000, $4,000. So notice it does work, you will have $4,000. Now the best way to illustrate this is to look at additional exercises to show us how to just to kind of confirm what we just learned. Let's take a look at few examples to illustrate this concept. So let's take a look at this example. Mark deposited 7,200 today that earn interest at a rate of 8% compounded quarterly. So this is important. This is not compounded annually, it's compounded quarterly. It means every quarter, the money will earn interest and that interest is added to the following quarter, okay? And the 7,200 must remain in the account for 10 years. So we're looking at 10 years. Let me show you what's happening here. So this is yearly, one, two, three, four, five, six, seven, eight, nine, and one more 10. Now here's what's gonna happen. Every year it's gonna be separated into four quarters, one, two, three, four, one, two, three, four. And this year it's gonna be separated into one, two, three, four. So notice if we add up all of the periods, they will add up to 40 ends, 40 periods. Why? Because the interest rate is compounded quarterly, not annually quarterly. So in for this exercise equal to 40, now the interest rate is quoted, always the interest rate is quoted annually. We have to do the same thing with the interest rate. If we divide it, if we multiply the period by four, because remember we have four quarters in a year, we have to divide the interest rate by four as well by the same amount. Therefore, I in this exercise equal to 2%. So to find out how much 7,200 worth at 8% compounded quarterly for 10 years, we have to go to the future value table. This is the future value table. Let's erase everything from the future value table. And N, notice it's the future value, N equal to 40 and I equal to 2%. So the rate is 2.2080. Therefore, I'm gonna take 7,200 times 2.2080. So the future value of this money is 7,200 times 2.2080. The future value is 15,897. Let's make it 898, 15,898. That's the future value of the 7,200. So remember, if the problem says it's compounded semi-annually, let's assume it says compounded semi-annually. It means every year is compounded into two, every year is compounded into two pieces. Therefore, eight years and we'll be, I believe we're dealing with how many years? 10 years, 10 years N will equal to 20, N will equal to 20. If it says compounded semi-annually. If it says N, you have to do the same thing with the interest rate. You have to take the interest rate and divide the interest rate by two and we'll use 4%. Now if it says compounded monthly, then we have to take 10 multiplied by 12 and it'll be 120 and equal 120. And I don't have 120 in the table. Then the interest rate will be 8% divided by 12. It will be less than 1%. But the point is to remember that if they change the compounded period, you have to change N and you have to change I. So if you multiply N by two, you have to divide, you have to divide I by two. If you multiply N by four, you have to divide I by four because you use the proper period. C company invests 163,170 today, earning 7% for nine years. Since they don't tell us anything, we assume it's compounded annually. Compute the future value of this investment nine years from now. So they did not tell us whether it's compounded monthly or semi-annually. So we have to assume it's annually. So we're gonna have I equal to 7% for this problem and equal to nine and we have 163,170. So I'm gonna go to the future value table and N equal to nine, I equal to seven. So N equal to nine, I equal to seven, I equal to seven, I equal to seven and it's 1.8385, 1.8385 times 1.8385. All what I have to do now is take my present value, 163,170 times 1.8385 and that's equal to 200. We see 299,988, 299, 988. So that's the future value of this amount. Now, in the next session, we would look at the present value of an annuity. And this is important, the present value of an annuity. Important, then we would look at the future value of an annuity. As always, I would like to remind you to like the recording, connect with me, visit my website for additional resources, subscribe, study hard and stay safe during those coronavirus days. Good luck.