Time Value of Money Equations - Video Tutorials & Practice Problems

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Time Value of Money and Using Timelines

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All right. So we're going to discuss one of the more difficult topics for this course. It's the time value of money. Okay. So 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 gonna use time value of money when we're pricing these bonds. Alright, let's go ahead and dive right into the ideas of 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 $1000 today or $1000.05 years from now, which one would you take? The $1000 today? Right. You want that money today and you can start spending it now or you could invest it and it will be worth more money in the future. So that's the main idea here. This is the big takeaway of 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 time value money, we're gonna talk about two main concepts and they're opposites of each other. The first one you might have heard of before, 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 your compounding interest into the future right? You're gonna have some current amount of money. So you're gonna take some current amount of money, say the $1000 that I offered you today and you're gonna earn interest, right? You're gonna take that money and you're gonna earn interest as time passes into the future. So you're compounding into the future, right? So you can imagine that there's gonna be an opposite to compounding and that's gonna be discounting. So we're gonna use this term discounting when we're taking some future sum of money. So now that $1000 that I offered you five years from now? Well if we were to take out the interest that would have occurred over the five 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 about back to that offer I made you write, I offered you either $1000 today or $1000.05 years from now. So that wasn't so enticing right? You want the $1000 today? But what if I offered you $1000 today or maybe $1500.05 years from now maybe? Now you'd weigh your options. What could I, how much could I earn on that $1000 or what is it worth to me to have that $1000 right now compared to the interest that I would earn or what would it be worth five 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 when we talk about time value of money, it becomes very useful to use timelines. I'm sure you've used timelines before in a math class. Um 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 three years. So um what we're doing is we're gonna take that that $100 and we're investing it for three 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 gonna draw something like this and then we're gonna draw the different years here. So we're dealing with three years. So we always start today. Today is gonna be zero 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 zero. Okay, when we deal with this concept, we deal with it as zero. And before I go on here, I want to make a point that time value of money in this class, we usually keep it pretty basic. You're gonna go into a lot of detail with time value of money. When you take a finance class, finance you're gonna buy a financial calculator and you're gonna do all sorts of complicated time value of money uh equations and transactions. But at 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 zero here, which is right now, year zero, then year 11 year from now, two years from now and three years from now. Pretty simple. Right? And then underneath the timeline you write your cash flows. So in this one is 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 gonna 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 gonna do the calculations here and you would get in one year. Well, we could do this first one, right? You have $100 and you earn 10%. So it would be worth, say 100 and $10 a year from now, right? Because if you took it and in one year would be worth 100 and 10. 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 second 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 gonna keep growing over time like that. All right. So that's what we're gonna do. We're gonna use timelines generally to visualize our cash flows. This one obviously was pretty simple. We just have one cash flow of $100 that we're investing for three years. Right? So let's pause real quick and then we're gonna discuss this time value of money equation. It's a very important equation. Let's do that in the next video

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Time Value of Money Equation:Future Value

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Alright so let's go into the time value of money equation. This is pretty much the core of time value of money and the core of your finance class once you get there. But we're gonna use it in pretty simple fashion throughout this course. Okay so the time value of money equation we're gonna have F. V. 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 some point in the future. Okay at some point in the future 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 gonna give you 100 give you $1000 right now, guess what the present value of that $1000 I'm about to hand to you is it's $1000 right? The present value is what it's worth today. But future value is what that $1000 might be worth at some future point in time when we consider interest. So talking about interest is our next variable that are right here. So notice we have future value equals the present value what it's worth today times one 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 gonna go into more detail about market interest rate versus the stated interest rate or coupon interest rate. And that's when we deal with the bonds payable. So you just want to note whenever you're using your time value of money equation, you always use your market interest rate when you plug into this equation. OK. And you want to make a note that the market interest rate, you're gonna express it as a decimal, right? So if I told you the market interest rate is 10%. Well, You want to put that in a 0.10, right? And it's always gonna be this one plus the rate. So it would be 1.10 that goes into that parentheses there. Right. So market interest rate. Well, that's the interest rate on the market, right? The common interest rate. That could be found generally the competitive rate on the market. Okay, So that's our market interest rate. And finally we have N notice N here is an exponent. So we're having exponent here. And generally in this class, you're not gonna 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 three years, four years, maybe five years if they really want to push it. But you'll 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 were talking about investing at the clutch bank for for three years. Well three 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 one plus the interest rate of 10%. So one plus 10.10. And we would raise it to the third power, right? That would be our end for three years. And that would tell us the future value. So what is it going to be worth in three years? What would be 100 times 1.10 to the third power? 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. So you do it three times right? 1.1 times 1.1 times 1.1. And then we multiply that by the 100 in present value. And we're gonna get a value of $133.10. So the $100 today? Uh compounded for three 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 100 and $33.10 three years from now. Cool. So this is a very important equation here. Future value equals present value times one plus R. To the N. Power. Okay. This is your time value of money equation. Cool. Let's go ahead and do some practice problems before we continue on with this topic.

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Problem

Problem

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

A

Compounding

B

Discounting

C

Rebounding

D

Converting

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Problem

Problem

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?

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Problem

Problem

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

A

Compounding

B

Discounting

C

Rebounding

D

Converting

6

Problem

Problem

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?

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Time Value of Money

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Alright 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 need to deposit $9917 at a 6% interest rate. What do you guys think? Do you think it's gonna be a higher number than 9009 17 Or a lower number than 9009 17? Let's think about it for a second. It's a lower interest rate. Right? So you're not gonna earn as much interest. Let's go ahead and dive in and let's see what happens. So in this case everything 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 012 years from now. So today is zero and then we've got one and two years from now and instead we're earning 6% interest. Now I'm gonna 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 gonna 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 a present value equals future value divided by one plus R. To the end. And notice not much is changing here. We've got 12,000 in the numerator. But now instead of 1.10 it's going to be one point oh six right? One plus 6% which is point oh six. So we're gonna have 1.06 instead of 1.1 in the denominator and it's still gonna be two years. So we're gonna square that denominator 1.06 to the second power. And that's easy enough. Right? We should do that part first. So let's go ahead and do 1.06. Where did it go? One point oh six times one point oh six. And that gives us a denominator. I'm not gonna round till the very end. Our denominator is gonna be 1.1236. That's one point oh six squared right, 1.6 times one point oh six gives us 1.1236. Our numerator is gonna be 12,000. So let's go ahead and do that 12,000 Divided by 1.1236. And we get 10,600 and we'll run it to 10,680. So that means today we need to deposit $10,680. So that in two years we'll have $12,000. That should make sense. Right? We're earning less interest so we're gonna need more money now to get to the $12,000 in two 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 let's see what happens. Let's increase our balance with the compounding equation. So what we're gonna do is we're gonna take the 10,006 80 and we're gonna multiply it by one point oh six. Right? This is this will tell us what it's worth after one year we're earning 6% interest so it's gonna be worth 6% more. One plus 6%. 1 point oh six. That was our compounding equation, right? We multiplied by the one plus art of the end. So 6 10,080 times one point oh six. So I'm going around here to $11,321 after one year. And then if we were to multiply it by 1.06 again for the second year. Well there we go. We're up to our 12,000 again. Remember we got a little bit of rounding errors because we just we were rounding numbers but that's exactly what happened To have $12,000 in two years at a 6% interest rate. Well, we're gonna have to deposit $10,680, right? And this is what can be confusing to students, right? Because the only thing that changed here is the interest rate. Alright. But we're still looking for that same future value. We still have that same future value and we're looking for a present value. Okay? Uh So that's about it for this problem. Let's go ahead and move on to one more topic about time value of money.

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concept

Annuity

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Alright. So the formulas we've been using so far, that future value equals present value times one plus r to the end 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 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 $1000 every time there's a payment and it has to be a 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 gonna be a $500 payment every year or every month there's gonna 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 the equal intervals for it to be an annuity. Okay, most likely you're gonna see annuities that are gonna be interest payments and interest payments that you usually make annually. So you're gonna be making an annual interest payment or semiannual interest payments where you're making interest payments every six 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 gonna be doing is the present value of an annuity. We're gonna have tables, we're gonna have what's called a present value table or a future value table that's gonna 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 gonna talk about the tables in this video, we'll talk about it in a future video. But what I wanna do is just get familiar with the topic of annuities and we're gonna do timelines here uh 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 five years, let's visualize this information on a timeline. Okay, so you've earned a five year pension that's gonna give you $10,000 each year for five years. So what's gonna happen is let's draw our timeline and we're gonna be right here at year zero right now, then there's year one year, two, year three, your four and your five. Right? So for the next five 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 one year from now. Okay. And that's called an ordinary annuity. There's other types of, But we're not going to get into those in this course. This course deals with ordinary annuities where the payments start one year from now. Okay, so that's exactly what we have here. We're going to get $10,000 annually for the next five years. So that means we're gonna get a cash flow of $10,000 here, a cash flow of $10,000 here, $10,000 here. So every year, $10,000 for five 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 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 our 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 gonna take all of the cash flows and we're gonna bring them all back using our Our table will learn how to do that and we're gonna find the present value of the annuity. So it's gonna be some amount that's worth the same amount as if you were to take $10,000 each year for five years. Okay so we're gonna find the present value of an annuity like that and it's gonna use the same kind of principles. We're gonna have some we're gonna have some n for the number of periods. So in this case would be five periods. We would have some are 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 gonna instead have the payment, whoops. B. A. P. M. T. The payment that's the annuity payment. Okay so those are the variables we'll 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 five p 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 two years and then $8,000 in three years. That wouldn't be an annuity. Okay. The annuity has to be the same amount of money each period for equal equal space in between each payment, which is usually a year like this. All right. So why don't you guys go ahead and try and build the timeline in the next problem?

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Problem

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.