10. Time Value of Money

Using Time Value of Money Tables

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## Time Value of Money Tables:Equations

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Alright. So just like I promised we're gonna learn how to find the present value of annuities and the present value of lump sums using tables. Okay let's check it out. Okay. So instead of having those formulas that we've been working with so far and like I said, those formulas for annuities are beyond the scope of this course. Well we're gonna use tables that make it a lot easier to find the present value based on the interest rate and the number of periods. Remember those were two of our variables that we used in finding present value. So those are gonna be important when we use the tables. So in this in this course we'll we're generally going to be focused on present value. Okay we're gonna be taking some future sums of money like those interest payments and those principal payments from bonds and we're gonna find what those are worth today. Okay. And that's the that's the key is we're finding those present values of those future sums of money specifically. Like I said, we're finding what those future interest payments are. And like we saw in our previous example the future interest payments, they form an annuity. Remember that the annuity that's getting equal amounts of money so the same amount of money each period. So in equal amount of time. So annual interest each each year you would be getting the same interest payment. Okay. That's what makes an annuity and then there's also gonna be principal payment at the very end and that's going to be a lump sum right, there's just gonna be the one principal payment at the end when they're when they have to pay back the bond they have to pay back the liability to the to the person who bought the bond. Well that's going to be at the end as a lump sum. They're gonna pay off the value of the bond. Okay so we're gonna be dealing with both annuities and lump sums. So let's go ahead and see how we're going to use the tables to do our present values of lump sums and present value of annuity. So remember when we used our formula for lump sums where we had some future amount of money and we wanted to know what is it worth today? That was like the example where we were saving up for a european vacation and we wanted to have $12,000 in the future. Well how much did we need to invest in the bank today? Well we can use a table to save us time rather than have that clunky formula with exponents and stuff. So remember we had the future value. So our our formula was present value equals future value divided by one plus R. To the end. Notice how I highlighted the one plus R. To the end because instead of doing the dividing by one plus R. To the end we're gonna have our future value and then we're gonna multiply by the present value factor and these present value factors, they come from the table. Okay So you'll see in the last page of this lesson, I've included the present value tables for lump sums and for annuities. Okay so we'll go over those before we're done here and before we get to the example. So notice it's very simple. Instead of doing all of those calculations, what we have is the future value times the present value factor. And we're just gonna get some number out of the table and multiply it by the future value and that's it. We found our present value. Same thing with the annuities except in this case we're going to use the annuity payment instead of the future value. So let's say you were going to get $10,000 a year for five years. Well that would be the annuity payment is $10,000. We wouldn't add them all together and say $10,000 for five years. That's 50,000 total dollars. No no no. We're gonna take the amount of each payment which is $10,000 and that's going to be the amount in this formula annuity payment. And we're gonna find the number in the table that we're going to multiply it by. Okay So let's pause here and then we're gonna go to the tables and we're gonna do a quick explanation of how to read the tables and we'll come back to the example. So flip over to the other page where you've got the tables tables listed. Alright let's do that.

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## Time Value of Money Table:Present Value of Lump-Sum

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Alright, so here we go. This is the first of our two tables and this is for finding the present value of $1. When they say the present value of $1 this is the table for lump sums. Okay. The table for lump sum payments. So remember we're looking for the present value factor and all these numbers in here. These are the present value factors all listed throughout. I think I missed when I was circling all these different numbers. These are the present value factors. So it's all about being able to pick out which is the correct number. And notice as we go down the table down this way we've got different ends. Remember we had those variables N for number of periods and are for interest rates. Well this this way going this way left to right. Those are our interest rates are right. So we've got the number of periods going down and the interest rate going to left to right. So what we're gonna do is we're gonna analyze our problem and we're gonna find out what our interest rate is, what our number of periods is. And then we'll go to the table say that we find out that there's 10 periods and the interest rate is 7%. So we would go to 10 periods and 7% let me do it in a different color 10 periods and 7%. And we would go in our table and we would find what that present value factor is. So then we would use our formula which is take the future value. Say we wanted to know what $1000.10 years from now is worth at an interest rate of 7%. Well we would take that $1000 and multiply it by this present value factor in the table. $1000 times 10000.508 will tell us what $1000.10 years from now is worth at a rate of 7%. Okay, so that's how we're gonna read this table. The important factors that we need. Our our number of periods and our interest rate. Okay. Those are the key to being able to read this table. Alright, let's pause and let's move on to the next table.

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## Time Value of Money Table:Present Value of Annuity

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So here we have a very similar looking table except notice the title. It says present value of ordinary annuity of $1. Okay, so since this is ordinary annuity of $1, this is your table for annuities. Okay. So it's very easy to tell which tables for annuities because it's gonna say annuity in the title. Alright. But why does it say of $1? Well that's because this is what $1. If the annuity was just $1 this is what it would be worth. And that's why we multiply it by the annuity payment because it's usually not just $1 it's going to be some bigger amount of money. Okay. So in the same way we read we read this table the same way we're gonna go into our problem and we're gonna analyze what is our end notice? We've got ends going down again the number of periods and we've got interest rates are going across. Okay. So we're gonna have to analyze our problem to find the interest rate and the number of periods and then we would dive into this table to find what our present value factor is. That we're gonna multiply by. Okay. So now that you've seen the tables, one of the biggest tricks is knowing which table to use right? When you're dealing with an annuity, you wanna make sure you're using the annuity table when you're dealing with a lump sum which is just one amount of money, right? That's when you use the other table, the lump sum table above present value of $1. So it's the lump sum is just $1. Right here, we're talking about an annuity getting $1 for several periods and equal intervals. Alright, so I think the best way to learn this is with an example. So let's go back to our first page of the lesson and let's do an example related to this.

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example

## Using Time Value of Money Tables

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Alright let's go ahead and get some practice using our present value tables. Right? Your grandmother is giving you a graduation gift but it's giving you two options. She's either going to give you $5000.01 year from now. So she doesn't want to cough up any money right now. She'll give you $5000.01 year from now or she'll give you $1000 per year for the next six years. And the first payment is occurring one year from now, right? And that's how we typically expect our annuities. Is that they start one year from now? Okay. So notice what we've got in the first one, she's gonna give you $5,000 1 payment one time. This is a lump sum, right, lump sum. And we're dealing with What what are the what are the factors in this first one? So let's deal with these one at a time. And I guess we'll draw a timeline just to see what's going on here. So this is right now zero and this is 11 year in the future where she's gonna give us $5,000 and we need to bring that $5,000 back in time to today to see what it's worth today. Right. We're trying to see which is the better option. Are we gonna take option one or option two? It's basically gonna be the one with the higher present value. So this $5000.01 year from now? It tells us the interest rate here is 10%. So 10% is our our our so when we go to our table we're gonna have our equal to 10%. And we're gonna have n equal to one, right? Because it's one year in the future at a rate of 10%. Okay. So remember when we use our formula we want to take that future value of 5000 and multiply it by the present value factor that we get out of the table. So our present value is going to be equal to 5000 which is our future value times the present value factor. Okay? So now let's go down to the table and find the present value factor for a rate of 10% and an end of one. Okay I'm gonna scroll down to the table and I'm gonna ask you which table are we gonna use? Are we gonna use the top table which is the present value of $1? Or the bottom table? The present value of an ordinary annuity of $1. We're gonna use the top table. Right? Because this is a lump sum. She's paying us one amount of money at one point in time And that's it. There's not gonna be a stream of payments. Remember an annuity involves a stream of payments. So here what we need to do is use our variables. It told us that it was one period at at an interest rate of 10%. So we need to find that number and that's pretty easy. It's right here, 0.909. That is our present value factor in this case. 0.909. So let's go back up to our problem and let's fill in our 0.909. Into our present value factor. Okay? So here we are, 5000 Times 0.909. This is going to tell us what the present value of $5,000 at an interest rate of 10% for one period is 5000 times .909. That comes out to $4,545. Right? For $4,545. That is what $5,000 is worth one year today if the $5,000 was received one year from now. Okay. So that's the that's the present value of option one. Now let's find the present value of option two and see what's the better option. So let's go on to two. And let's draw our timeline, draw it a little longer because there's a few more payments going on 12345 and then six. Right. And here we are today. So what's she gonna do? She's gonna give us 1000 per year for six years. So what do we have each year? Starting next year. 1000 1000 each year. Right. So it's gonna be six payments of $1000 rather than one payment of $5000. That already sounds pretty enticing, right? We're gonna get it sounds like we're gonna get $6000 rather than $5000. Sounds pretty nice. But let's go ahead and see what the present value of this annuity is. Right. In this case we have an annuity right? We have a stream of payments rather than just one payment And they're happening at equal intervals each year. A $1,000 each year in this case for six years. So we want to bring that and find the present value of this annuity. Okay So we're gonna use our formula that we had above. It's very simple. Right? We're gonna do the present value of the annuity is gonna be equal to the annuity payment. Okay. Times our present value factor, right? And our annuity payment in this case. What is the annuity payment? It's $1000 right. We don't add them all together. It's just $1000 that we're gonna get each period. So it's gonna be 1000 times the present value factor. So let's go ahead and go to our table and find out what that's gonna be. So notice what is R. R. And R. N. In this case. Before we go to the table we need to know what our R. And R N. Is R. is the same right? The interest rate didn't change from each option so the R. is still 10 and n what's n in this case it's not just one right? There's not just one year going on it's six right there's six years of payments and is going to be equal to six. So when we go to our table we're gonna have to look for our of 10% and an end of six. Let's go ahead and do that. Now scroll back down to our tables. And which one are we going to use the lump sum table or the annuity table? In this case we use the annuity table. Right? Because we're talking about an annuity we're getting $1000 per year for six years. So let's go ahead and see what the present value of that stream of payments is. Just like we said we had an interest rate of 10% and the periods were six in this case six periods and an interest rate of 10%. Yeah. Well that gets us a present value factor of 4.355. Okay. So we need to multiply our annuity payment of 1000 times 4.355. And that will tell us what the present value of the annuity is. So let's go back to our example and let's figure that out. So here we were scroll up a little bit. So we found in our table 4.355, 1000 times the 4.355. Let me get my calculator out. Pretty easy math though. It comes out to be 4355. Okay so option two is worth $4355 today. So the present value of the annuity is $4355. Which one sounds better. What's the better option we had? Option one where she gives us a $5000.01 year from now. Well that's only that's worth 4545. And option to the annuity of $1000 for six years. Which might have sounded more enticing. Had you not been a master of time value of money like you are now? Well that one's only worth 4355. So in this case option one is the better option, right? You would pick option one because you you know how to do your time value of money and you would know you get more value out of option one. Cool. Thanks grandma for the graduation gift. Let's go ahead and try the practice problem below. Alright, let's see if you guys can handle it

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Problem

ProblemYou have won the lottery! You are given two options for your payout:

1. You can receive $540,000 today

2. You can receive $50,000 annually each year for the next twenty years

Assuming the interest rate is 6%, which is the better option?

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## Time Value of Money Tables:Bonds Payable

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Alright. So now let's relate everything we've been talking about with these tables back to bonds payable. Right? Like I said, this is going to be the main way that we use time value of money in this class is with bonds payable. So remember with a bond, there's two streams of cash flow. The first one is the annual interest payments. Right? Sometimes we're going to deal with semiannual interest payments, which just means two per year every six months. Right? It's the same thing as saying every six months we're gonna pay interest is semiannual interest. So notice these are gonna be happening every six months or every year consistently and they're gonna be the same amount of money. So these are going to be an annuity But then we're also going to deal with the principal payment at the end of the bonds life, they're gonna have to pay back the principal of the bond and that's going to be a lump sum payment. They're gonna pay back the entire amount of the bonds principal as a lump sum once the bond matures, which is usually gonna be five years from now, 10 years from now, something like that. Okay. Now a very important thing when we're dealing with bonds is that we're gonna have to interest rates that we're dealing with when we're finding the interest payments of the bond, but we're also gonna have the market interest rate. Okay, So the first one is what we call the stated interest rate And the stated interest rate is used to calculate the actual cash payments of interest, okay, the bond has some interest rate attached to it. You're gonna sell a bond that says, Hey, we're gonna pay 6% interest? Well, that is going to be the stated interest rate of the bond. Now that stated interest rate does not have to be the same as the market interest rate. You can sell bonds at whatever interest rate you want. Right? You can say, hey, I'm paying 5% interest right now. Do you want to buy my bond? Well, the value that people place on your interest rate. Well, it's gotta compared to the market interest rate, right, What if you're offering 5% on bonds when everyone else offering similar bonds is offering 8% interest and yours is only 5% interest. Well, people are more likely going to buy the 8% bonds, right? Because they pay more interest. So there's gonna have to be some calculation here to to discount the value of that bond because it's not paying as much interest. Okay. So we're gonna see how this works in an example. One more note that I want to mention is that if the bond pays interest semiannually, Well, what we're gonna have to do before we go to the table because we're still going to be using our tables, right notice we're gonna be dealing with annuities where we have our annuity table and lump sum because we have our lump sum table as well. Right. So up to this point, we've been dealing with things that happen annually. But what if they pay semiannually? Well the interest rates that you're given are always annual interest rates. You're always given annual interest rates. But if interest is paid semiannually, what we're gonna have to do is we're gonna have to divide the interest rate by two. So this is our our remember our is gonna be divided by two to find the semiannual rate. And we're gonna have to multiply the number of years by two. This is R. N. Right? R. N. Is going to be multiplied by two before we go to the tables. And that's usually the biggest trick before you get to the table is having to divide by two for the r and multiplied by two for the end. Right. That makes sense because if there's 10 years, well that means there's 26 month periods. Right? And if it's 10% interest per year, Well then it means there's 5% interest per half year for six months. Okay. Let's go ahead and pause here. And then we're gonna do an example related to this

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example

## TVM Tables and Bonds Payable

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All right, let's check out this example. Your company inc issues $100,000 of 10% bonds in 10 do in 10 years. The bonds pay interest semiannually. And the current market rate of interest is 8%. What is the present value current selling price of the bonds? Alright. So whenever we talk about the present value of bonds, that's what they're worth today. What people are willing to pay for them today based on these future cash flows. Okay. So it all comes down to these time value of money calculations. So what do we have here? We're issuing $100,000 worth of bonds and they pay 10% interest. So notice this 10%. This is the stated interest rate, Right? And they're due in 10 years. So this is the number of years. But notice what happened in this problem is we're dealing with semiannual. So we're gonna have to divide these things by two because of what we talked about above, right? Because the periods are half not full years. But notice we also have one more interest rate here. 8% is the market rate. Okay, so this is where things start getting a little complicated, You have to remember which rate does what the market rate is. The one that we use to go to our present value table. However, the stated rate, that's the rate that the bond actually pays the bond says, I'm gonna pay 10% interest. So that's how we calculate the cash payments. There's gonna be cash interest payments and that's based on this 10% interest. So let's find out what that cash interest is gonna be The cash interest each semiannual period, well, it's going to be 100,000 times the 10% right? 0.10. But remember its semiannual, right? So like we said, we have to divide by two when we get when we're dealing with semiannual periods. So we're gonna multiply this by half because it's not 10%. We're dealing with half the amount of time, half periods. Okay. Half year periods. So let's go ahead and find out what the cash interest is every six months. 100,000 times 60.1 Uh times half, right, divided by two. So that's gonna tell us that we're gonna pay 5000 in interest interests. Let me get that right interest every Six months, right? Every semiannual period we're gonna pay 5000 in interest, which is $10,000 per year, the 10% per year. Okay, so let's go ahead and see this on a timeline. So I'm gonna cut out some of these periods here because they're about to be, they're gonna be the same and then we'll get to the end there. So notice when we make our timeline here, we're not gonna do it in years, we're going to do it in semi annual periods. So in half years. So this is really right now is zero, this is one, this is two and this is not one year, this is one semiannual period, right semiannual, this is to semiannual periods from now. So that's technically one year from now, is the two in this case. Right. Are you following me? We have to stay in semiannual periods because of what we said above, since there's 10, since we're talking about 10 years, we're talking about 20 semiannual periods. So our timeline is going to go all the way to 20 semiannual periods here. And the reason we do this is because we're paying the cash out every semiannual period And in this case it's 5000 being paid out each semiannual period. So 5000 in interest every six months For those 20 semiannual periods over the next 10 years. Okay, so that's gonna be the 5000 is going to be that annuity that we talked about those interest payments. Right? So this here is our annuity But we also have the principal payment right at the end of the of the 10 years, which is the 20 semi annual periods at the end of that period. We're gonna have to pay $100,000, right? We're gonna have to repay them The $100,000 that they lent to us, they lent to us $100,000. Uh we sold these bonds that said in 10 years we're gonna pay you $100,000 plus the interest. So we gotta find what those were today. This is the principle and that's going to be a lump sum. So every time we deal with payable it comes down to this we're gonna find the present value of the annuity. Which means we need to find the cash payment of interest that's going to happen each period and we have to find the present value of the principle which is a lump sum at the end of the the life of the bond. Okay. So we're pretty much done with all the tough math here now that we've got it all visualized on our timeline, we're almost ready to go to our table. Okay. So there's one more thing we have to do before we go to our table is we need to find out what R. N. And R. Are are gonna be. Okay so here is the annuity and we're gonna find the present value of the annuity and then we need to find the present value of the the principal payment as well. So what's going to be our n. And what's going to be our our in these cases when we go to the table? Well N. Is going to be equal to 20. Right? We've got 10 years times the two semi annual periods per year comes out to 20 for R. N. And how about our our our interest payments are interest rate. So let me get out of the way here our interest rate. What are we gonna use for our interest rate. They gave us to interest rates in this problem. Remember when we first introduced interest rates, we always said that the R. is gonna be the market interest rate, right? This is the market interest rate that we use when we calculate our when we go to our present value table, always remember that we use the market interest rate when we go to the table and we use the stated interest rate to calculate the cash interest. Okay, so notice how we use the stated rate already up here now it's time to use the market interest rate of 8%. So since it's 8% we're gonna use 8%. But since it's semi annual periods, well it's not 8% per it's 8% per year. So it's 4% per six months. So there we go. We've got our N and R. R. We've got end of 20 are of four. We're ready to go to our table. Okay. And we're gonna go to our table for 22 equations. So let's write those equations in real quick. Our first one is for the annuity and the annuity. We're gonna find the present value of the annuity Is going to be equal to 5000. The amount of the annuity payment, right? The interest payment times the present value factor From that, from the table for 20 and 4%. So let's go ahead and do the annuity 1 1st and then we'll come back and we'll do the lump sum payment. So let's go down to our table And let's find what the present value factor for an annuity is for 20 periods at 4% interest. So remember we're using the annuity one for the interest payments. So let me erase this previous problems data And what are we doing here? Well we said it's 4% interest per semiannual period for 20 semiannual periods. And that gets us right here 13.590 13.590. So that's what we're gonna use in our problem here. So let's go ahead and write that in for our present value factor. Let me erase that and put it in here to save space 13.590. So what's the present value of the annuity? Well that's gonna be 5000 times 13.590. It comes out to 67,950. That is the present value of the annuity today. So that's the present value of just the interest payments. But remember that's not the only oops not 590 67,950. That's the present value of the interest payments. But we also need the present value of the principal. So we're gonna use the other table to find that the present value of the principal Is equal to the 100,000 that we're trying to find what that lump sum is worth today times the present value factor. Right? So we're gonna need to go to the table again. But we've already done the hard work. We know what our end is. We know what our our is we're ready to go to the table. So remember this time we use the lump sum right because this is one payment of principal. The 100,000 is just one payment that's happening uh 10 years from now in 20 semiannual periods. Okay so let's erase this from the last problem. And what do we have? Our end was 20 because there's 20 semiannual periods. Our our our interest rate is 4%. So we go down here and we find that we're gonna use 0.456 is our present value factor. So let's go ahead and bring that up here and we're going to put equals and for a present value factor whoop sees Oh I'm writing on top of the other question. So our present value factor is 0.456. So let's see what that comes out to 100,000 times .456. It comes out to 45,600. So that is the present value of our Of our principal payment. So that principal payment of of $100,000 that's happening 10 years from now. Well that's worth $45,600 today wow this has been a lot of work so far but we're finally onto our final step. And all we gotta do is add the present value of our interest payments and the present value of our principal payment. And that will tell us the present value of the bond 9 67,050 plus 45,600. That tells us that the bond today is worth $113,550. This is the present value of the bond. Okay. So that means that if we went to the market and we said hey right now we're gonna sell $100,000 worth of bonds that pay 10% interest for the next 10 years semiannually well the market would be willing to pay us $113,000 and $550 for those bonds. Why aren't they willing to just pay us $100,000? Well that's because we're paying more interest than the market, right? The market is only paying 8% interest but we're saying hey check us out we're gonna pay 10% interest were even better than the market. So people are gonna be more willing to pay us uh because we pay out more interest per period than other similar bonds on the market. So that's what makes our value be more than 100,000. Okay so these bonds are gonna sell for 113,550 and we're going to deal more with the accounting side of this once we get to those calculations, okay so there we go this is pretty tricky. And like I said this is about as tricky as these calculations are gonna get an accounting okay? And you're in this first accounting course. So I would even suggest before going on to the next practice problem. Double check that you underst and everything that went on in this video. It came down to finding the present value of the annuity and finding the present value of the principal and adding those together. Okay? So let's go ahead and once you're ready, move on to the next one and you guys can try a practice problem yourself. Alright, let's do that.

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Problem

ProblemABC Company issues 1,000 bonds with a face value of $1,000 maturing in eight years. The bonds pay 8% interest semi-annually and the current market rate of interest is 12%. What is the total amount of cash received from the bond issuance?

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