 In this presentation, we will take a look at present value formulas related to bonds. Support accounting instruction by clicking the link below giving you a free month membership to all of the content on our website broken out by category further broken out by course. Each course then organized in a logical reasonable fashion making it much more easy to find what you need than can be done on a YouTube page. We also include added resources such as Excel practice problems, PDF files and more like QuickBooks backup files when applicable. So once again, click the link below for a free month membership to our website and all the content on it. One of the reasons bonds is so important to accounting and finance is because they're a good example of the term of present value of money. We're trying to look for an equal measure of money when we think of bonds and bonds is going to have this relationship between market rates and the stated rate, which helps us to kind of look through and figure out these types of concepts. So even if we don't work with bonds in other words, if we're not planning on issuing bonds or buying bonds or knowing anything about bonds not being important to us, the time value of money is a very important concept and bonds is going to be a major tool to help us with that. Why is bonds so useful for learning time value of money? Because there's two types of cash flows with bonds, meaning at the end of the time period we typically are going to get the face amount of the bond, the 100,000, similar to a note. And then we've got the interest payments that are going to happen on a periodic basis. And therefore we have these two different types of cash flows that we can use two different formulas for to think about how to equalize. So note when we think about time value of money, what we're doing is we're just basically saying that we want a unit of measure. That's what money is, a unit of measure. We're trying to measure value. But unlike any other unit of measure, most others, like a ruler, the time value of money or the dollar, the value of the dollar changes. So our ruler kind of changes in size as we're trying to measure value with it. And that would be similar to our ruler changing in size when we're trying to measure how big a table is. So it changes with time. So therefore, for us to really measure something, we have to know what time period we're in, what time frame are we in. So when we consider something, we have to say, well, here's the value of money at this certain time period. So in terms of a bond then, we have two types of cash flows. One, we have the 100,000, and we're going to say that it's going to be due in this case at the end of four time periods. Why? Because it's a two-year bond and we're going to say it's semi-annual payments. So that means it's going to be four periods out, not years, but four periods. And then we're going to have the amount of interest that we're going to pay on a semi-annual basis. And that'll happen periodically. So we've got the 4,000 interest happening each of those four time periods. So we have two types of cash flows happening here. We're going to pay out the 100,000 at the end, and we're going to pay out 4,000 each paper, that 4,000 being calculated as the 100,000 times the stated rate, the amount on the bond, 0.08, that would be for a year divided by two, because it's semi-annual, 4,000 each period or each six months. Those are going to be our two flows that we have. Now our question here is what really is the value of the bond, and we typically want to know the value of the bond now, today, not at the end of the bond. So we don't want to know what the cash flow is. That would be easy for us to calculate. It's going to be the 4,000 plus the 4,000 plus the 4,000 plus the 4,000 plus the 100,000. That's how much cash is going to go out. But the units of cash differ. The value of cash differs depending on what time frame we are in. In other words, this 4,000 a year from now is worth less than 4,000 today. This 4,000, this would be six months from now. 4,000 a year from now would be worth less than 4,000 today. 4,000 a year and a half would be worth less. And this two years out that we're going to get the 100,000 and the 4,000 is going to be worth less than 104,000. In other words, we would rather have the 104 today. Why? There's two factors. One is that we know that there's interest, so the value of purchasing power will actually go down. And two, there's opportunity cost. We could have put, if we had that money now, we could put it somewhere else. And therefore, the fact that we could put money to work means that it's worth more today than in the future. So for those reasons, we got to present value this information. We have to take these cash flows and somehow put it into the present value. Now there's two types of ways we can do that. One, we could try to take each number and bring it back to the present. And we will do that with this 100,000 because there's only one payment, which is two years out or four payments out. So we're just going to take that 100,000 and present value it. But here we'd have to take the 4,000, which is one period out or six months in present value it. And then present value this 4,000, which would be a little bit different. This is going to be worth a little bit less in terms of present value dollars because it's a year out or two time periods. And then this is a year and a half out. It's going to be worth a little bit less than this 4,000. So because they're all the same, we call that an annuity, the payment's going to be the same. We can simplify the formula and use an annuity type of formula to do that. There's a couple of ways we can do this in practice. So we can do this with a formula. We can do this with Excel. We can do this with a financial calculator or we can do it with tables. Just depends on the course on how they display that and what tools you have to work with it. To me, Excel is probably the easiest thing to use or a financial calculator. The formula is what we want to start with, however, in order to see how these are kind of derived. What's the work doing? So we won't try to derive the formula. We'll just show you. Here's the formula that we will be working with. So you could, if you're in a situation where you have to just work with a formula, you could memorize the formula. If you're obviously working in practice, then you're going to be using most likely Excel or a calculator to do this. We'll look at it with a formula first and then look at the other tools. Note, of course, what we're doing, what you want to know is conceptually what is happening. Why are we doing this? What does it mean? What can it tell you? So here's going to be our present value formula. We're going to take the future value divided by one plus the rate or interest to n, which is going to be the number of time. And then the present value for an annuity is going to be a bit more complicated. We're going to take p or the value of the bonds, of the payments. And then we're going to take one minus one plus r to the negative n over the rate. So this would be the math way to do it. Again, we could kind of shortcut this with a calculator or Excel or tables. So let's look at the concepts of this. So we're going to take this formula. We're going to plug our information into this formula. The value, often recorded as PV of one, is kind of a formula that we are doing here. We're going to apply this more simplified formula just to the face amount that we're going to pay out if we're issuing the bond and we're going to pay it at the end. We're going to pay it out at the end of the time period, the 100,000. In this case being four time periods because it's two years that we're going to pay this out two years in the future and it's paid semi-annually. So if we plug that into our formula then we're just going to say that the future value is 100,000. What does that mean? Future value. It means that this 100,000 is how much we're going to pay out after four time periods at the end of two years. But that's future value. That's not what it's worth right now. That's how much we're actually going to pay in dollars, but it's not the value now today. So that's the future value. If we divide that by, we're going to plug the rest in one plus 0.05, where does that come from? Well, we have a market rate, which we're going to use for this calculation, not the stated rate. We're going to use the market rate because that's the market rate. And we're going to divide it by two because the rate is per year and we want to make it per six months. So we're going to take that market rate, if we move the decimal over 10% divided by 2.05. And then we're going to take that out to the number of periods, four, two years, there's two payments per year. So this is our math that we'll have to work out. Now I recommend not doing this in Excel if you're going to do it like a formula here because it's easier to write it in paper. Even if you have Excel, it's often easier to write something like this in paper and just do the algebra meaning do one step at a time. So we're going to add the one plus 0.05 is 100,000 divided by 1.05 to the fourth. And then we're going to take the 100,000, if we do the math here, we got 1.216 rounded and that will finally get to the 82,270. That means that this 100,000 that we're going to pay out two years from now or four time periods is worth present value in today's dollars, only 82,270. So if this was the only thing that was happening, meaning we're going to pay out 100,000 in the future and there was no interest involved, then we would expect 82,272 day in order for that to be a fair transaction. But that's not the only thing that's happening. We're also paying out interest. So let's do the same thing for the interest calculation. Now we could do it one by one. We could take the interest for one six month period, the second six month period. But it's easier to use an annuity, maybe not with this formula. But if we have a Excel or a calculator or tables, the annuity table will be faster than calculating each payment, especially if it was a long annuity. So this is going to, because they're all the same, we can use this formula to calculate the annuity. So we're going to take the 4,000 then and we're just going to plug this information into our formula of times, that's P times 1 minus 1 plus 0.05, same 0.05 market rate divided by 2 to the negative 4. Why negative? We're just going to flip the sign basically, we're putting a negative to flip the sign of this final result. And then we're going to divide it by 0.05, the rate once again, 10% divided by 2. That's our formula. If we just do the math algebraically, write it down to do the math. I wouldn't recommend doing an Excel, you want to write it down and do one step at a time. So we're going to add up these inner columns one minus 0.05 to the N to the negative 4. And then we'll do that the math here and get the 0.18 rounded divided by 0.05 times the 4,000 force in both of these. And then if we finish up the math, we're going to say 4,000 times the 3.55 dividing this out. And that of course will be the 14184. So note what this means that we had four payments of 4,000, meaning we're going to pay 100,000 times the 0.08, that's somewhat divided by 2. That's how much we're actually going to pay four times, times four. So we're actually going to pay 16,000 at the end of this time period. We're going to pay 4,000 every six months, four times. But it's not really worth 16,000, it's worth 14,184. So if the only thing we were doing here, if we weren't paying the 100,000 and we were just going to pay back 4,000 each year, each six months for two years or four time periods, we would expect 14,164 today in order to make those payments, in order to have a fair transaction, a negotiable, marketable transaction. Note when we calculated the interest of 4,000, we used what we're actually going to pay. When we use this calculation, we're going to use the market rate because that's what the market is doing at this point in time. So if we add those up then, we're going to pay back 100,000 at the end of the time period. That's worth 82,270 today. And then we're going to pay 4,004 times every six months for two years. That has a current present value of 14,184 for a total of 96,454. Therefore, if we were to issue this bond for 100,000, we're going to say we want money today, we're going to give you 100,000 at the end of two years and we're going to give you 4,000 every six months for that two year time period. How much would we want for it today? We want 96,454 according to the market rate of 10% at this time. Therefore the journal entry would just be that cash is going to be that 96,454 that we just calculated. The bond is going on the books for 100,000 and then we'd have the discount of 3,546. So cash would be going up. That's why we issued the bond. The bond's going on the books for 100,000 that we owe and then we discounted it by the 3,546 in order to equate it to the market value. So this credit what we owe minus the discount is the carrying value kind of the book value of the bond.