Time Value of Money Equations - Online Tutor, Practice Problems & Exam Prep

The time value of money illustrates that a dollar today is worth more than a dollar in the future due to potential interest earnings. Key concepts include compounding, where current money grows over time, and discounting, which determines the present value of future cash flows. The fundamental equation is FVn=PV0(1+r)n, where FV is future value, PV is present value, r is the interest rate, and n is the number of periods. Understanding annuities, which are equal payments over time, is also essential for financial calculations.

Time Value of Money and Using Timelines

Alright, so we're going to discuss one of the more difficult topics for this course; it's the time value of money. Okay? So, the time value of money, this is how some amount of money today is going to be worth a different amount in the future, okay? And we use this mostly when we deal with bonds payable. So the liability, bonds payable, well, we're going to use the time value of money when we're pricing these bonds. Alright? Let's go ahead and dive right into the ideas of the time value of money. So I got a quick pretest for you to set your mind in the right direction. So it's my money and I want it blank. Now or some other time? It's my money and I want it now, right? It's my money and I want it now. So think about it, if I offered you $1,000 today or a $1,000 5 years from now, which one would you take? The $1,000 today, right? You want that money today, and you can start spending it now or you could invest it, and it'll be worth more money in the future. So that's the main idea here. This is the big takeaway of the time value of money, is that a dollar today is worth more than a dollar tomorrow, right? That same dollar could have earned some interest and have been worth more in the future.

So when we talk about the time value of money, we're going to talk about 2 main concepts and they're opposites of each other. The first one, you might have heard of before, is compounding. So you might have heard of compounding interest, right? You might have talked about compounding interest at some point in a math class. And you're compounding interest into the future. Right? You're going to have some current amount of money. So you're going to take some current amount of money, say the $1,000 that I offered you today, and you're going to earn interest. Right? You're going to take that money and you're going to earn interest as time passes into the future. So you're compounding into the future. Right? So you can imagine that there's going to be an opposite to compounding and that's going to be discounting. So we're going to use this term discounting when we're taking some future sum of money. So now, that $1,000 that I offered you 5 years from now. Well, if we were to take out the interest that would have occurred over the 5 years, it would have been worth some lower amount of money today. Right? So what we're taking is some future sum of money and removing the interest that occurs over time to find its value today.

So think back to that offer I made you, right? I offered you either $1,000 today or $1,000 5 years from now. So that wasn't so enticing, right? You want the $1,000 today, but what if I offered you $1,000 today or maybe $1,500 5 years from now. Maybe now you'd weigh your options. What could how much could I earn on that $1,000 or what is it worth to me to have that $1,000 right now compared to the interest that I would earn or what it would be worth 5 years from now, right? So now you have a little more options to weigh, and that's because of the time value of money. And it all comes down to that interest. So we talk about the time value of money, it becomes very useful to use timelines. I'm sure you've used timelines before in a math class, but it's very useful to see visualize these cash flows on a timeline so you see them all in one place. Okay? So let's go ahead and do an example and build a timeline here.

Today you invest $100 at Clutch Bank at a 10% interest rate for 3 years. So, what we're doing is we're going to take that $100 and we're investing it for 3 years, and it'll be worth some amount of money in the future. We're compounding the $100 today into some future sum of money. So let's go ahead and draw a timeline, so we can visualize what's happening here. So this is how we usually do a timeline. We're going to draw something like this and then we're going to draw the different years here. So we're dealing with 3 years, so we always start today. Today is going to be 0 and I like to put my years above the graph. So these are the years right here. And we always talk about right now as 0, okay? When we deal with this concept, we deal with it as 0.

And before I go on here, I want to make a point that the time value of money in this class, we usually keep it pretty basic. You're going to go into a lot of detail with the time value of money when you take a finance class. In finance, you're going to buy a financial calculator and you're going to do all sorts of complicated time value of money, equations and transactions. For this point, we keep it generally pretty simple and we mostly use it for valuing bonds payable, that liability bonds payable. Alright. So let's go ahead and finish up our timeline. So we had year 0 here, which is right now year 0. Then year 1, 1 year from now, 2 years from now, and 3 years from now. Pretty simple, right? And then, underneath the timeline, you write your cash flows. So in this one, it's pretty simple; we only had one cash flow of $100 and the thing is, once you start doing bonds payable and once you get into higher-level courses and doing more difficult transactions, you're going to have multiple cash flows at different points in time. So the timeline helps you a lot to visualize and see where all these different cash flows are happening. Okay? So that $100 is what you invest today and then we like to put our interest rate right here. So the interest rate was 10% in this case. So you can imagine, you would earn 10% and we're not going to do the calculations here, and you would get in 1 year. Well, we could do this first one, right? You have a $100 and you earn 10%. So it would be worth, say, $110 a year from now, right? Because if you took it and in 1 year, it'd be worth $110 and then you keep compounding, right? There's another 10% and keep going in the future, 10% And you keep earning interest, right? And that's the whole thing. When we're compounding, we're earning interest on the interest. Because in the 2nd year, you're earning interest on the $110, not just the $100. So you're getting a little extra interest, so you can imagine it's going to keep growing over time like that.

Time Value of Money Equation:Future Value

Alright, so let's go into the time value of money equation. This is pretty much the core of the time value of money and the core of your finance class once you get there, but we're going to use it in a pretty simple fashion throughout this course, okay? So the time value of money equation, we're going to have fv, this is our first variable and FV is future value. So future value, this is some future amount of money right? So we're talking about our example above, the $100 is what the present value is, what it's worth today and the $110 a year later, well that was the future value of that $100 today. So future value is the value of money at some point in the future, okay? Compare that to present value, right? PV is present value and that's the value of a sum of money right now. So if I told you I'm going to give you $100 or give you $1,000 right now, guess what the present value of that $1,000 I'm about to hand to you is? It's $1,000, right? The present value is what it's worth today. But future value is what that $1,000 might be worth at some future point in time when we consider interest.

So talking about interest is our next variable, the r right here. So notice we have future value equals the present value, what it's worth today times 1 plus R which is the interest rate. And when we talk about the interest rate here, we're talking about the market interest rate. We're going to go into more detail about the market interest rate versus the stated interest rate or coupon interest rate, when we deal with bonds payable. So just take note, whenever you're using your time value of money equation, you always use your market interest rate when you plug it into this equation, okay? And you want to make a note that the market interest rate, you're going to express it as a decimal, right? So if I told you the market interest rate is 10%, well, you want to put that in as 0.10, right? And it's always going to be this one plus the rate. So it would be 1.10 that goes into that parenthesis there.

And finally, we have n. Notice n here is an exponent. So we have an exponent here and generally in this class, you're not going to be compounding for 20 years or something because most of the time you can only use a very simple calculator that doesn't do exponents. So for the most part, they'll maybe do 3 years, 4 years, maybe 5 years, if they really want to push it. But you'll be able to use this exponent for n and n is our number of periods. Okay? The number of periods which is usually years. So above in our example that we're talking about investing at the Clutch Bank for 3 years, well, 3 would be our exponent here for n, right? So if we're trying to find the future value up here, the future value of our investment of $100. So notice that $100 that would have been our present value times 1 plus the interest rate of 10%. So 1+0.10 and we would raise it to the 3rd power, right? That would be our n for 3 years and that would tell us the future value. So, what is it going to be worth in 3 years? Well, it would be 100 times 1.103. Let's go ahead and do that real quick just to solidify that example here. So 1.1 and let's say we don't have an exponent, right? We don't have an exponent button. Well, we would do 1.1 times 1.1 times 1.1 times 1.1 and then we multiply that by the 100 in present value and we're going to get a value of $133.10. So the $100 today, compounded for 3 years at 10% interest will be worth $133.10 3 years from now. Cool? And that's because it earned interest of 10% each year. So the future value of the $100 today is $133.10 3 years from now. Cool? This is a very important equation here. FV=PV⋅1+rn This is your time value of money equation. Cool? Let's go ahead and do some practice problems before we continue with this topic.

The formula FV = PV * (1 + r)ⁿ is best used for:

• A. Compounding
• B. Discounting
• C. Rebounding
• D. Converting

You invest $4,545 in Clutch Bank today earning a juicy 10% annual interest. What is the value of your investment in one year? What is the value of the investment after two years?

Alright. Let's try this. You invest $4,545 in Clutch Bank today, earning a juicy 10% annual interest. What is the value of your investment in 1 year? What is the value of the investment after 2 years? Alright? So we're going to deal with these separately, but what I'm going to do first is I'm going to draw this on a timeline, just because it helps us visualize what's going on. Okay? So this is right now, year 0, 1 year from now, and then 2 years from now. So right now, you're investing $4,545 and it's going to earn 10% and what's it going to be worth in 1 year? And then we also want to know what it's going to be worth in 2 years, right? So that's kind of what's going on here. We're taking this money and we're compounding it into the future, right? So let's go ahead and use our time value of money equation and let's fill in our timeline here. So we've got the future value which is what we're going to solve for. Let me first, before we get into that, let me say, 1 year. So this is 1 year when we're compounding for 1 year and then we will do it for 2 years. Okay? So the future value is going to equal the present value of 4,545 times 1 plus r, right? Present value times 1 plus r and our r in this case, was 10%, right? This, we invested this is our present value, right? Because we invested it right now. You invested it today so its present value is 4545 and that is our interest rate is our 10% which we need to put in as a decimal. 1 plus 0.10 and what's going to be our n in this case? Our exponent of n. Well, since it's 1 year, our exponent is 1, right? So there's not really an exponent in this case because it's just 1 year. So future value it's just going to be 4545 times 1.10. Okay. Let's see what that gives us. 4545 times 1.1, well, it's approximately $5,000 we're going to say $5,000 here we're going to round it off and that's what it's going to be worth in 1 year, okay? So that's what we'll put in right here. After 1 year, it's worth $5,000. Let's do the same thing for 2 years. So same equation, future value equals we have the same present value of 4545 times the the same interest rate 0.10, but now we're compounding it for 2 years, right? It's not just 1 so our exponent is 2 in this case. So what I would do is I would do the exponent part first, right? We want to do the parenthesis 1.1 when we do our order of operations. 1.1 times 1.1, right? That's our exponent. So 1.1 squared is 1.1 times 1.1 and then we're going to multiply that by 4,545 and it comes out to approximately $5500 here right if we were to round it off. Okay? So after 2 years it's worth 5500 and that should make sense. Also, if you think about at 1 year, if we look at our 1 year amount of 5,000, right? We had 5,000 at one point in time and we're going to earn interest for 1 year, right? From 5,000 in year 1 to year 2, well, it's earning 10% interest, right? So the 1 plus the 10%. Whoops, 1 plus that 10%. Well, 5,000 times the 1.1, guess what? That's also 5500. So that makes sense, right? Because we compounded it for 1 year and then we had 5,000. We compounded for 1 more year and we get to $5500. So we could we have that shortcut where we can just compound for 2 years when we use the exponent. Alright? But that's just how how interest works and time passes and we just earn interest on the interest. So that's about it for this one, right? We started with $4545, after 1 year, it's worth $5,000 and after 2 years, it's worth $5,500. Cool? So one more, kind of logical approach to this. If you know that this interest rate is 10%, right now I make you the offer. If I were to say to you, based on this information right here, I'm going to offer you $4,545 today or I'll offer you $5500 2 years from now. Which one do you want? Do you want $4545 today? Or $5500 2 years from now? Well, the answer here is actually that you should be indifferent. To you, they should pretty much mean the same thing to you. Maybe you'll take the $4545 today, but the idea here is that they're worth the same amount essentially. Because that $5500 2 years from now, if you were to discount it 2 years, well it would be worth $4545 when you take out the interest. If you take the $4545 and you invest it for 2 years, guess what? It's worth the same thing as if I paid you $5500 2 years from now. So the idea is, that these are basically the same amount of money. At this interest rate, me offering you $4545 today or me offering you $5500 2 years from now, they're basically the same thing. Alright? That's kind of the tricky concept here that a lot of students struggle with. Is that this is actually the same amount of money when we deal with the interest being put into it, alright? So let's go ahead and move on to the next video.

The formula PV = FV (1 + r)ⁿ is best used for:

• A. Compounding
• B. Discounting
• C. Rebounding
• D. Converting

You are saving up $12,000 for a luxurious European vacation two years from now. How much money would you need to invest today at Clutch Bank, earning their juicy 10% annual interest, to have enough for your vacation? How much would you need to invest today, if instead you could only earn 6% interest?

Alright, so let's see how you guys did with this one. You are saving up $12,000 for a luxurious European vacation 2 years from now. How much money would you need to invest today at Clutch Bank earning their juicy 10% annual interest to have enough for your vacation? Alright, so notice what they're asking us for in this problem. They're asking us for an amount today, right? They're saying, how much would you need to invest today? So what we're looking for today, we're looking for a present value, right? The question is asking us to find a present value. But what information did they give us? They told us $12,000. What is that $12,000? That's an amount of money that we want to have at a future point in time, right? 2 years from now, we want to have $12,000 so we can go on this vacation. So this $12,000, well, that's the future value, right? We want to have at some future point in time, $12,000. So how much do we need today? So we want to have a future value of $12,000 and what other variables did they give us? They gave us 2 years, right? 2 years from now, well, that's going to be our n, right? N, the number of periods that need to pass between the present value and the future value. And they also gave us the interest rate. They said, there's a 10% annual interest rate. 10%, that's going to be our R, our interest rate R. Okay? So notice what they did. They gave us 3 out of the 4 variables, right? They're always going to do that. They gave us the future value, they gave us the interest rate, and they gave us the number of periods, so we can solve for present value, alright? So remember, in this equation in this problem, what we're looking for is the present value today. So let's go ahead and do real quick a timeline just to visualize this. So we've got right now, year 0, then 1 year from now, and 2 years from now. Alright? So 2 years from now, we want to have $12,000 and we're earning 10% interest, right? So there's 10% interest every year. So what is that worth today? If we can earn 10% each year, we can bring this back in time 2 years using our formula and it'll tell us what this $12,000 is worth today and that's the amount that we're going to have to put in the bank to earn that interest for 2 years, alright? So let's go ahead and use our formula and that is the present value. Actually, I'll do it down here. PV = FV 1 + r n So let's fill that in. We know the future value was $12,000 and our denominator, we've got 1+r, so that's one plus 0.10, right? 10% as a decimal is 0.10. So 1 +0.10 and it's 2 years. So our n is 2. So it's going to be $12,000 divided by 1.1 squared. So the first thing we want to do is find out what that denominator is. What is 1.1 times 1.1? It's 1.21. So $12,000 divided by 1.21 that's going to give us our answer here, and that comes out to I'm going to round it here to $9,917. Right? If we were to invest $9,917 today, well, it would earn interest for 2 years and it would be worth $12,000 after 2 years and we can confirm that, right? Let's go ahead and see what happens with the $9,917 after 1 year. So if we take 9917 times 1.1 and remember, we're rounding here, right? So we're going to earn interest for 1 year, so 9917 and we're going to add the 10% interest, right? So 9917 times 1.1 and that tells us what it's worth after 1 year. $10,908. And that's just basically using our compounding equation, right? Where now, instead of dividing by 1 plus r to the n, we're multiplying by 1 plus r to the n like we did previously when we were compounding. So now that if we take that $10,908 and we multiply that times 1.1, Well, there we go. We're back up to $12,000. Obviously, we're off by a few pennies here and there because we're rounding. But that's essentially what's happening here, right? We take that $12,000 in the future and we bring it back 2 years and it's worth $9,917 today. And that's exactly what we would have to do. If we deposit $9,917 in Clutch Bank today, after 2 years, we'll have the $12,000 that we need for our vacation. So the answer here is $9,917 is what we need to invest today to have $12,000 in 2 years. Cool? Let's pause and let's see if you guys got the second one. Now that you've seen it go through one time, the next one we're just changing the interest rate. So let's go ahead and see what happens there.

Time Value of Money

All right. So let's see how you guys did. How much would you need to invest today if, instead, you could only earn 6% interest? Now let's think logically before we dive into all the numbers. Let's think. At a 10% interest rate, we needed to deposit 9,917 dollars. At a 6% interest rate, what do you guys think? Do you think it's going to be a higher number than 9,917 or a lower number than 9,917? Let's think about it for a second. It's a lower interest rate, right? So you're not going to earn as much interest. Let's go ahead and dive in and see what happens. So, in this case, everything is staying the same. We still have the same future value of 12,000, we have the same number of periods, and we're still looking for a present value. However, we've got a new interest rate, right? The interest rate \( r \) is now 6% instead of 10%. So, let's go ahead and draw our timeline real quick right here. And it's 0-12 years from now. So today is 0 and then we've got 12 years from now. And instead, we're earning 6% interest now. I'm going to put it in a different color, so it stands out. 6% interest rather than 10% interest. Right? But we're still looking for a future value. This future value is still \( \$12,000 \). Right? And we want to know what it's worth today. So we're going to bring it back in time and find out what it's worth today, okay? So let's go ahead and use our formula just like we did before.

We had present value = future value ∕ 1 + rN. I noticed not much is changing here. We've got 12,000 in the numerator, but now instead of \( 1.10 \), well, it's going to be \( 1.06 \), right? \( 1 + 6\% \) which is \( 0.06 \). So we're going to have \( 1.06 \) instead of \( 1.1 \) in the denominator. And it's still going to be 2 years. So, we're going to square that denominator, \( 1.06^2 \), and that's easy enough, right? We should do that part first. So let's go ahead and do \( 1 \times 0.06 \). Oops, where did it go? \( 1.06 \times 1.06 \) and that gives us a denominator. I'm not going to round till the very end. Our denominator is going to be \( 1.1236 \). That's \( 1.06^2 \), right? \( 1.06 \times 1.06 \) gives us \( 1.1236 \). Our numerator is going to be \( 12,000 \). So let's go ahead and do that. \( 12,000 \) divided by \( 1.1236 \) and we get \( 10,600 \), we'll round it to \( 10,680 \). So that means today, we need to deposit \( \$10,680 \) so that in 2 years we'll have \( \$12,000 \). That should make sense, right? We're earning less interest. So we're going to need more money now to get to the \( \$12,000 \) in 2 years, right? Before we were earning more interest, so more of that 12,000 was made up of interest. But now, we need to deposit more, so that we can accumulate up to 12,000 just the same. So let's do just like we did above and see what happens. Let's increase our balance with the compounding equation. So what we're going to do is we're going to take the \( \$10,680 \) and we're going to multiply it by \( 1.06 \), right? This will tell us what it's worth after 1 year. We're earning 6% interest, so it's going to be worth 6% more. \( 1 + 6\%, 1.06 \). That was our compounding equation, right? We multiply it by the \( 1 + r \) to the \( n \). So \( \$10,680 \times 1.06 \). So I'm going to round here to \( \$11,321 \) after 1 year. And then if we were to multiply it by \( 1.06 \) again for the 2nd year, well, there we go. We're up to our \( \$12,000 \) again. Remember, we got a little bit of rounding errors because we were rounding numbers, but that's exactly what happened. To have \( \$12,000 \) in 2 years at a 6% interest rate, well, we're going to have to deposit \( \$10,680 \), right? And this is what can be confusing for students, right? Because the only thing that changed here is the interest rate, alright? But we're still looking for that same future value or we still have that same future value and we're looking for a present value, okay? So that's about it for this problem. Let's go ahead and move on to one more topic about time value of money.

Annuity

Alright. So the formulas we've been using so far indicate that the future value equals the present value times 1+r raised to the power n, or the other one where we rearrange that same formula, those were for lump sums of money. That was for when we had a lump sum of money, right? We were talking about, okay, I'm trying to save $12,000 at a future point in time or I'm depositing $100 in the bank, right? There's just one lump sum of money and it's not a stream of cash flows, okay? When we deal with the time value of money, we also talk about a specific stream of cash flows called an annuity, okay? We talk about annuities. So an annuity, well, these are payments of the same amount of money. So when I say the same amount of money, it has to be $500 every time or $1,000 every time there's a payment. And it has to be at regular, and I want to say equal intervals. So regular intervals and it's the same amount of money. So we're going to say something like, okay, every year there's going to be a $500 payment every year. Or every month, there's going to be a $500 payment. As long as it's equal intervals, it could be every day a $500 payment. Whatever it is, it has to be the same amount of money at equal intervals for it to be an annuity, okay? Most likely, you're going to see annuities that are going to be interest payments. And interest payments that you usually make annually, so you're going to be making an annual interest payment or semi-annual interest payments, where you're making interest payments every 6 months rather than every 12 months, okay? So finding the present value of an annuity. So when I say the present value of an annuity, how much is this stream of payments? So it's not just how much is this one lump sum of money worth today? No, how much is this stream of payments worth today? Or the future value? How much is this stream of payments worth at some future date? Well, those formulas, they're beyond the scope of this class. Luckily, you're not going to have to use a formula, when we're calculating the present value of an annuity, which is usually what we're going to be doing. It's a present value of an annuity. We're going to have tables. We're going to have what's called a present value table or a future value table. That's going to give us some ratio that we use rather than have to do a whole formula. And it makes the calculation a lot easier, okay? We're not going to talk about the tables in this video; we'll talk about it in a future video. But what I want to do is just get familiar with the topic of annuities, and we're going to do timelines here to get familiar with annuities. So let's start with this example here. The example is you have reached retirement and have earned a pension that will pay you $10,000 annually for the next 5 years. Let's visualize this information on a timeline. Okay? So you've earned a 5-year pension that's going to give you $10,000 each year for 5 years. So what's going to happen is, let's draw our timeline, and we're going to be right here at year 0, right now. And there's year 1, year 2, year 3, year 4, and year 5, right? So for the next 5 years, you're going to get $10,000 payments. And generally, when we talk about annuities, there's no payment right away. All these payments start 1 year from now. Okay? And that's called an ordinary annuity. There are other types of annuities, but we're not going to get into those in this course. This course deals with ordinary annuities where the payments start 1 year from now. Okay? So that's exactly what we have here, we're going to get $10,000 annually for the next 5 years, so that means we're going to get a cash flow of $10,000 here, a cash flow of $10,000 here, $10,000 here. So every year, $10,000 for 5 years, right? So notice, before when we were drawing our timelines, there was just one cash flow. We just drew, okay, we need this $12,000 2 years from now. Or whatever the cash flow was. There was always just one cash flow that we wrote in, and then we found out what that was worth at a different point in time. So when we find the present value of an annuity, we don't find the present value of each of these cash flows separately. Yes, it's possible to discount each of these cash flows to today's date separately, but that would take a long time, right? We would have to use the present value formula over and over again for each of these cash flows at different points in time. So what we do with the present value of an annuity, we're going to take all the cash flows and we're going to bring them all back using our table. We'll learn how to do that, and we're going to find the present value of the annuity. So it's going to be some amount that's worth the same amount as if you were to take $10,000 each year for 5 years, okay? So we're going to find the present value of an annuity like that, and it's going to use the same kind of principles. We're going to have some n for the number of periods. So in this case, it would be 5 periods. We would have some r for our interest rate. But now, instead of a future value or a present value, well, we're searching for a present value. Right? What is that annuity worth today? We're going to instead have the payment. Oops. Not a b. A PMT. The payment. And that's the annuity payment. Okay? So those are the variables we'll be working with once we get to the tables and stuff, but at this point, I just want you to get familiar with what an annuity looks like. Notice how this follows the rules of an annuity. That we're getting $10,000 each year for five years. So it's the same amount of money for 5 periods, for equal periods, right? For each year, so annually. It's not, okay, you're going to get $10,000 in a year and then $5,000 in 2 years and then $8,000 in 3 years. That wouldn't be an annuity, okay? The annuity has to be the same amount of money each period for equal spacing between each payment, which is usually a year like this. Alright? So why don't you guys go ahead and try to build the timeline in the next problem.

Today, you purchased a $1,000 bond that matures in 5 years. The bond pays annual interest of 10%. Visualize these cash flows on a timeline.

All right. Let's see what we have here. Today you purchased a $1,000 bond that matures in 5 years. The bond pays annual interest of 10%, visualize these cash flows on a timeline. All right? So it's going to be paying annual interest of 10%, and this is the annuity that we're generally going to be working with in this class, is an annuity of interest payments. Alright? So let's go ahead and draw our timeline and then we're going to fill it in. We've got 0, which is right now when we bought the bond. I want to make a note that even though it's a $1,000 bond, that's the principal amount of the bond. It's not necessarily what we paid for it today. That could sound a little confusing, and we're going to dive into bonds a lot more once we get to the liabilities, okay? But at this point, I just want you to know that even though it's a $1,000 bond, that's just the principal amount that's going to be paid out in 5 years. The value of the bond today could be some different number, all right? So what's going to happen is that we need to find the value based on our variables, right? We need to find what all the cash flows are worth today, and that's going to be the price of the bond, the present value of the bond. So what are the cash flows in this case? Well, there's the cash flows for the interest payments, right? There's going to be annual interest payments of 10% of the $1,000, and then there's also going to be the $1,000 principal payment at the end of the 5 years. So let's go ahead and find out what that interest is going to be. The interest, right? There's going to be cash interest that's being paid, and it's going to be a $1,000 times the 10%, right? Each year, this bond pays 10% annual interest based on the $1,000 principal amount. So $1,000 times 10%, that tells us that the interest is $100 per year, right? This is an annual interest that gets paid once per year. And it's going to be paid every year for 5 years while it's maturing. So notice what we're going to have here. We're going to have $100 here in 1 year, $100 here in year 2, in year 3, $100, year 4, $100, and year 5, $100 as well, right? Because time is passing and we're earning that interest. So we're going to get $100 each year. And what does that look like to you already? We've got the same amount of money at equal intervals. We've got an annuity, right? These interest payments are an annuity. But those aren't the only cash flows involved in this bond. There are the interest payments, but remember there's the principal payment at the end, right? There's the $1,000 that this bond is worth. Well, after 5 years, we're going to get that $1,000 back, right? We purchased the bond for some amount, and in 5 years, there's going to be this $1,000 payment. So this $1,000 is another cash flow. There's going to be this $1,000 cash flow in year 5. So it's not just the annuity of the interest payments, but there's also the principal which is a lump sum, okay? So there are two types of cash flows that we've dealt with. We've dealt with lump sums, and that's when we were using our formulas earlier. The present value equals the future value divided by 1 plus r to the n. Right? Those formulas that we learned earlier in this lesson, well, those were for lump sums, such as this $1,000 principal payment in 5 years. But then we also learned about annuities, right? And annuities is going to be these interest payments. So once we get to our tables in a following lesson, you're going to see how we can use those tables to find what the value of this bond is today. Okay? So what that would take is it would take finding, one, the present value of the annuity, which is the present value of interest payments, but we also need to know the present value of the principal payment, okay? And those together, when we add the present value of the interest payment and the present value of the principal, that is going to be the present value of the bond, okay? And that's going to be dependent on the same factors, right? We're going to have the number of periods, which in this case was 5 years. We're going to have the market interest rate. Remember when I told you r is the market interest rate? Well, in this problem they didn't tell us what the market interest rate was because we're not going to do all the calculations in this case. But there's going to be the market interest rate and there's going to be the future value of the principal payments and the annuity here, okay? So it gets a little tricky, right? There's a lot of moving pieces at this point. But I promise once we start getting to the calculations of finding what the present value of the bond is, it's not so tough once we're using the tables, alright? So let's go ahead and move on to that lesson now.

Here’s what students ask on this topic:

What is the time value of money and why is it important?

The time value of money (TVM) is a financial concept that states a dollar today is worth more than a dollar in the future due to its potential earning capacity. This principle is crucial because it underpins many financial decisions, such as investment appraisals, loan amortizations, and retirement planning. By understanding TVM, you can evaluate the worth of future cash flows in today's terms, making it easier to compare investment opportunities and make informed financial decisions. The core idea is that money can earn interest, so the sooner you have it, the more you can potentially earn.

How do you calculate the future value of an investment using the time value of money equation?

To calculate the future value (FV) of an investment using the time value of money equation, you use the formula:

${\mathrm{FV}}_n={\mathrm{PV}}_0(1+r)n^{}$

Here, ${\mathrm{FV}}_n$ is the future value, ${\mathrm{PV}}_0$ is the present value, $r$ is the interest rate, and $n$ is the number of periods. For example, if you invest $100 at an annual interest rate of 10% for 3 years, the future value would be calculated as:

${\mathrm{FV}}_3=100(1+0.10)3^{}$

This results in a future value of $133.10.

What is the difference between compounding and discounting in the context of time value of money?

Compounding and discounting are two fundamental concepts in the time value of money. Compounding refers to the process of calculating the future value of a present amount of money by applying interest over multiple periods. Essentially, it shows how an investment grows over time. The formula used is:

${\mathrm{FV}}_n={\mathrm{PV}}_0(1+r)n^{}$

Discounting, on the other hand, is the process of determining the present value of a future sum of money by removing the interest that would have been earned over time. It helps in understanding what a future amount is worth today. The formula used is:

${\mathrm{PV}}_0={\mathrm{FV}}_n/(1+r)n^{}$

Both concepts are crucial for making informed financial decisions.

How do you calculate the present value of an annuity?

To calculate the present value (PV) of an annuity, which is a series of equal payments made at regular intervals, you typically use a present value annuity table or the formula:

${\mathrm{PV}}_0=\mathrm{PMT}\times [1-(1+r\left){-}^n\right]/r$

Here, $\mathrm{PMT}$ is the annuity payment, $r$ is the interest rate per period, and $n$ is the number of periods. For example, if you receive $10,000 annually for 5 years at an interest rate of 5%, the present value would be calculated as:

${\mathrm{PV}}_0=10000\times [1-(1+0.05\left){-}^5\right]/0.05$

This results in a present value of approximately $43,295.50.

What are the key variables in the time value of money equation?

The key variables in the time value of money equation are:

• Future Value (FV): The amount of money at a future date.
• Present Value (PV): The current value of a sum of money.
• Interest Rate (r): The rate at which money grows per period, expressed as a decimal.
• Number of Periods (n): The number of time periods the money is invested or borrowed for.

The fundamental equation is:

${\mathrm{FV}}_n={\mathrm{PV}}_0(1+r)n^{}$

Understanding these variables helps in calculating how much an investment will grow over time or what a future sum is worth today.