 Personal finance practice problem using OneNote. Graphing bond price. Prepare to get financially fit by practicing personal finance. You're not required to, but if you have access to OneNote, would like to follow along. We're in the icon left-hand side, practice problems tab in the 11220 graphing bond price tab. Also take a look at the immersive reader tool, practice problems typically in the text area too. Same name, same number, but with transcripts. It can be translated into multiple languages, either listened to or read in them. We want to get a better understanding of what happens to the price of a bond as we get closer and closer to maturity. Remembering that a bond can be thought of as, in essence, a slowning money to the issuer of the bond, typically being a government entity or a corporation. And in return for that, we're gonna be receiving interest payments, usually semi-annual or annual, and we're gonna get a lump sum or face amount at maturity at the end of the term of the bond, which might be equal to what we paid for the bond unless we bought the bond at a premium or a discount. When we price the bond, we usually think about it as those future cash flows. We're gonna receive an annuity stream for the interest payments, a lump sum for the amount we receive at maturity, discounting them back using the current market rate to help figure out what the price will be. Now, when we wanna think then, well, what's gonna happen to the price if the market rate kind of stays the same as we get closer to maturity? That can be a little bit confusing to visualize because we have these two different kinds of streams of cash flow, one being an annuity, a series of payments. So as we get closer to maturity, we're gonna get less of those, less series of payments because we're gonna be closer to maturity. And the other being that lump sum, which is a bigger dollar amount, but at the beginning, it's gonna be way out into the future. Therefore, discounting being worth less, but when we get closer to it, then it's gonna be worth more and more. So we'll kind of graph this out and we'll do it in a couple of different ways. First, we'll think about just like a zero coupon bond where we just have that lump sum that we're gonna receive at the end and see what the graph for that looks like. And then we'll consider an annuity situation and see what a graph for that looks like. And then we'll graph them both basically together in a typical bond kind of scenario. Now we do this in Excel too. So if you do it in Excel, you can then change your data and see, and then the graphs will change and whatnot and possibly get a better understanding of the relationships between time passing and basically the market price. Okay, so to get an idea of this, let's say we got the bond and we're gonna start off with a face value bond. It's a 30 year bond. It's gonna have annual payments as opposed to semi-annuals, but there's no rate because it's a zero coupon bond. So we're not gonna be receiving interest payments on it. Instead, we're just gonna get the $1,000 in the future. Therefore, we would purchase it at a discount. So we would still have interest that we would be receiving, but the interest is kind of imputed in a way. It's not gonna be explicit interest payments. We're gonna pay a discount and then we're gonna get a bigger amount back at the face at the end. What this helps us do is look at a bond where we only have the one component. We don't have the present value of interest payments, in other words, because we're not gonna get the interest payments, but rather only have the present value of the face amount. So in other words, if I took this $1,000 and I discounted it back, present value in it at the discount rate we're gonna use is the 9.5% for 30 years, then we get to the 65.70. That would be the price of the bond, meaning we're gonna be paying the 65.70 to buy the bond. 30 years later, we're gonna get $1,000. The imputed interest then would be 1,000 minus the 65.70, right? But it's not gonna be in the form of annual interest payments. So if we were to then think, okay, well, what would happen if I keep the market rate the same and we get closer and closer to maturity? So if I calculated the price at year zero, we get the 65.70. If I did the same kind of calculation up top, but now I'm gonna do it for 29 years, assuming one year has passed, discount rate is the same. So we're gonna take that $1,000 discounted 29 years at the 9.5, we get to the 71.94. A difference, 71.94 minus the 65.70 of the 6.25. If we do it two years past, now we're at 28 years, we discount the 1,028 years at the 9.5, we get to the 78. And obviously as we get closer to that 1,000 endpoint, you would expect that we would pay more. Like if we were only 28 years or 27 years away from that 1,000 and the discount rate was the same, we'd be willing to pay more money for the bond because we're closer to getting that $1,000 in time. The time value of money isn't having as big an impact. So you can see, of course, then with this kind of structure, where you're just getting that lump sum at the end, you would expect the price basically to go up as we get closer to that endpoint, to the point where it's basically at 1,000 if you're gonna be buying the bond, like right at the point where it's gonna mature, then you'd pay $1,000 for it, right? Because it would be worth $1,000 because tomorrow they're gonna give you $1,000 for the bond because it's gonna mature. So you can graph that and notice it's not a straight line. You've got this upward trend, which you would expect basically to be happening given the fact then that once again, you are getting closer to that maturity date where you're getting that lump sum. So that's, and you can see that change if you do this in Excel and you adjust your discount rate here, for example, then you can graph this as this is good practice, the practice graphing in Excel. So I highly recommend going through those practice problems. But now let's consider a situation which is the reverse where we're gonna say, let's look at an annuity. Now we've got it structured as a bond, but you typically will not have a bond that doesn't pay you a face amount at the end. You could have a zero coupon bond like we saw, which would be similar to this scenario, but usually a bond isn't an annuity. But let's pretend we just have an annuity, meaning we're not gonna get the face amount of the bond, but instead we're gonna get just the interest payment. So we're basically investing in an annuity. So it's gonna be 30 years, annual payment, the rate is 10%, meaning we're gonna get 10% times 100 or a coupon payment of $100. So how much would we pay for, if we were to price this for a series of $100 payments, an annuity? So if we were to value that, we could say the price, if we were to start off on the price, so in other words, if I present valued $100 annuity payment that we're gonna be receiving every year for 30 years, discounting it back at that 9.5%, it would be worth, we're gonna say $983. Now, if I was to take it for 29 years, now we're discounting $2,900 payments for the next 30 years, discounting at the 9.5, it's at the 977. Notice that this one is going down because as we get closer to maturity, as we get closer to 30 years, although those future payments that are way out, 30 years out there are being discounted, they're worth less because of time value of money, we have more payments, right? So we're getting a payment each year. So the fact that we have more payments means that it's more valuable here. So we would expect as we get closer to maturity, we have less of these annuity payments, these series of $100 payments, and therefore you've got the price, you've got the price as we get closer to maturity going down. So you can see the price going down and then of course, if you're not getting any interest payments, it's gonna go down to zero. So we've also kind of grafted this way as well and this is really good practice to do in Excel to kind of work on these tables. So we basically said that we could graph it out this way and see the series of payments that we're gonna be receiving in the future. So if I was to present value the $100 payment one year out, it would be worth 91, present value in the $100 payment two year out, it would be worth 83, present value with that $100 payments to three years out, that'd be 76. And then if I was to purchase the bond in year two, then this would be like year one, right? We'd say, okay, now if I purchased it then, it would be worth 91 and then the next year, 83, that would be the series of payments. And you can kind of graph this out and get this table, which gives you kind of a lot of information about it, but I won't go into that too much more detail here. We'll just look at the graph looking like this, right? So now you've got the graph as you get closer to maturity, then you can see the graph is going down, of course, because every time you get closer to maturity, even though these future payments are worth less, you're getting less annuity payments as we get closer. So these two graphs are kind of doing the opposite kind of thing here. And if we put those two things together in a typical kind of bond situation, which has that lump sum at maturity, as well as the interest payments, then you can try to get an idea of what is actually happening with the bond. So now we've got the full bond, 1,030 year, 10% rate, the discount rate or market rate 9.5%, so the coupon is still $100. So now if we price the bond, we can take our full bond calculation, the present value of the interest payments, that would be us taking the present value of the rate 9.5 number of periods, 30, the payment is gonna be the $100 at the 983, same thing we got up top there, and then for present value in the face amount, the $1,000 we're gonna get at the end here for the rate would be 9.5, number of periods would be 30, and then not payment, but future value, the $1,000 discounting it back 30 years gives us around the 66, adding that together we've got a bond price of the 1049. Now if we did that each year, we'd get 1049 and then year one similar, year two 1048, meaning now we're adding those two things together in one formula here and saying, okay this number represents the present value of interest payments for 28 time periods that we're gonna be receiving $100 discounted back at the 9.5 and the $1,000 lump sum that we're gonna receive at the end year 30 discounting it back 28 time periods. And you've got those two things that are offsetting each other to some degree, right? So you've got those two things happening with our difference here as we get closer to maturity. And then again, you can kind of do it this way as well. So you could try to build this table and try to give you the series of discount payments that you're gonna get into the future. And you can see the lump sum 30 years out that includes that $1,000. That's why it went up over here. And so you can do it that way and you can see the series of payments if you started in year two for like 28 or 29 periods, right? And the lump sum at the end is a little larger. So and you can graph it that way and then if I graph this out, now you've got our graph that's including those both things together to try to get a grasp on what's happening with the price as we get closer to maturity when we have these two kind of canceling things happening because these two different streams of cash flows, one with an annuity, the other with the present value of one. So if you were to compare them, this was the first one where we focused in on the coupon, zero coupon bond. So we just looked at the face amount we're gonna receive at the end after we get closer to maturity, then the price goes up because we're gonna get, we're just looking at that lump sum payment that we're gonna get. This was the annuity where we said we're just looking at that series of payments that we're gonna be receiving. And in that case, the price is gonna go down because we're getting less dollar amount in terms of full dollar amount as we get closer to maturity for annuity, which is a series of payments. And then we kind of put them together and looked at the bond here and got a graph looking like this. Now again, we work this in Excel so that you can then change your data on the right hand side and see, well, what would happen if I adjust my market rates to something that's higher than the rate on the bond, for example, and so on. And then use that to kind of get a feel of what is happening with these items. So great practice problems to practice your graphing, getting an understanding of the bonds, work on your present value, future value tables and then work on something that you can then adjust to see differences as you work through your Excel worksheets.