 Personal finance practice problem using OneNote Bond Price Calculation. Prepare to get financially fit by practicing personal finance. You're not required to, but if you have access to OneNote, we'd like to follow along. We're in the icon on the left-hand side, the Practice Problems tab, down in the 11110 bond price using Excel tab. Also, take a look at the Immersive Reader tool. The practice problems typically in the text area too with the same name, same number, but with transcripts, transcripts that can be translated into multiple languages either listened to or read in them. We can basically think about the investment in a bond as though we are lending money to an organization. The issuer of the bond typically being either a government entity or a corporation and in order for us to lend them money we're going to want in return earnings on it, kind of like rent on it. That's going to be then the interest payments that we will be receiving. The structure of the loan is a little bit different, however, than when we borrow money. For example, when we borrow money, say in a mortgage situation, we typically structure it so that we pay back the mortgage in equal installments on a monthly basis, each of those installments having a portion of interest and principal related to them. On the bond side of things, when we get paid back from the bond, you could think of it more like a situation where we're renting, say, a home or an apartment. We're renting out an apartment. We expect to be receiving rent on it. We would typically be receiving the rent monthly in terms of an apartment and then we would get the return of the apartment itself, the property itself, at the end of the rental term. In the case of a bond, we have a similar kind of situation. However, we usually get paid instead of monthly, we get paid the rent, the equivalent of rent, which would be the interest payments, usually on a semi-annual basis every six months or on an annual basis. And then at the term of the bond, at the end of the bond, we get what's kind of like the principal back of the bond. Now, there's another little wrinkle that we have there, meaning we get back the face amount of the bond, but we might purchase the bond at either a discount or a premium, which we'll dive into a little bit later. But the general idea that we want to see is that we basically have two streams of income that are coming in, or two streams of cash flow that are happening when we purchase the bond. One is a stream of cash flow related to the interest, which we typically get either semi-annually or annually. And the other cash flow is the face amount that we're going to get at the maturity or end of the bond. So if we're trying to figure out what the market price of the bond is, what we can do is take those cash flows, discount them back to the current day, and that should give us basically what the purchase price or the value of the bond is or what the bond might sell for at this point in time. So let's think of our data here. We've got the bonds. We're going to say 100,000 of bonds. We've got the years are going to be five years until maturity. It's going to mature, in other words, in five years. Now, we're going to say that these are semi-annual payments, meaning they pay the rent on the bonds. They pay the interest on the bonds every six months, every half year. The stated yearly rate is 7%. That's the rate that would be actually on the bond, how much interest they're going to pay basically from the bond. And then there's a market rate which would not be connected to the bond, but would be determined by the market, possibly by looking at currently market conditions and the value of the bond or the risk related to the bond. And that could be determined, for example, by looking at rating agencies, for example. So we're going to be saying that the market rate is 9%. The stated yearly rate is 7%. So what we want to do to get the current price of the bond, the current value of the bond is take those two future streams of income and bring them back to the current day. Now, I'm first going to skip this and say it's not annual here. I mean, it's not semi-annual. Let's first think about it as though it was annual, and then we'll add the added wrinkle of it being semi-annual. So let's pretend here that we were getting interest payments on an annual basis as opposed to semi-annual. What we would do is take the present value of the interest payments. Those are an annuity type of payment because we're getting the same payment on an annuity basis if we're saying it's annual. It's going to be for every five years we'll get that payment. So we're going to take the present value. It's going to be a present value calculation. We do do this in Excel, but this is an Excel function. It's the way we're doing it here or visualizing it. The rate would be if it's an annual rate, it would be that 7% annual rate and then comma the number of periods. The number of periods would be five. If it was annual, we're just going to keep it at the five. And then comma the payment, which is an annuity payment, would be the 100,000 times the rate on the bond, not the market rate. And that would be 7,000. So that stream of 7,000 payments for five years, we're going to get that times five, 7,000 payment times five years. That's how much money we would actually get. But if we discount it back to the current period based on the 9% interest rate, we would be at 27 to 28. And then we're going to get the face amount, which is the 100,000 of the bond at maturity at the end of the five years. So in that case, we would say that cash flow would look like this present value. The rate would be 9%. Number of periods would be five. Once again, if it's not semi-annual, we're doing it for just annual first off. And then we're not going to have a payment. That's why there's two comments. The future value would be the 100,000. So that 100,000 we get five years later discounted back at 9%. Over five years would only be worth the 64993. As we discount it back, if we add those two up, we're at the value of 92 to 21. So the idea then being, if we've got a bond, there was 100,000 annual bond in this case, 5%, and it had a stated rate of interest of 7%, meaning they're going to give us $7,000 every year for the rent for purchasing the bond. But the market rate, meaning I can find other bonds of a similar nature that would pay us 9%, then I'm not going to invest. I'm going to buy this bond unless you sell it for something less. Discount it, in other words, if you discount it down to 92 to 21, then we might be able to buy that bond as opposed to some other bond that we might be able to make a 9% return on. So if we paid 92 to 21 for it, then we can hold on to the bond. We can get payments of that 7,000 per year if it was annual, and then we get the 100,000 basically at the end or term of the bond. So let's think about it in terms of semi-annual. So we're going to add a little bit of complexity here. Now, note that oftentimes when you look at the actual bonds, there might be like a coupon rate or the payments of the bond would be determined on a semi-annual basis, meaning they're going to pay every six months. Now, it gets a little bit tricky, and we'll try to demonstrate this a bit more explicitly in a second here. But when we move from a semi-annual to annual, that can get a little bit tricky. You might see like book problems or you might try to estimate, for example, if they give you the yearly rate and then say, well, the semi-yearly rate when I do my calculation would be half of that. So that's what we're going to basically assume now. We're going to assume that they gave us a yearly rate and we're going to break it down to a semi-yearly. Now, when we compare different rates, I'm trying to figure out the yield of a bond versus the yield of say stocks or something. We usually represent the rate in years and we want to be as specific as possible when we compare those rates. So we'll talk more about that in future presentations. But that's going to be our assumption. This is a yearly rate that we got to break down to semi-annual because we're getting paid every six months. So if I did the same present value calculation, the rate now would be the market rate at 9%, but that's a yearly rate. So we'd have to divide it by 2 to get to the semi-yearly rate if they gave us the yearly rate. And then the comma, the number of periods, the number of periods is going to be five years, but that's in years. That's why we're going to divide it by 2. I'm sorry, to multiply it times 2 to get to 10. So 10, 6 month or half year periods. And then comma, the payment in the annuity would be the 100,000 times the rate on the bond, 0.07%, 7,000. But that rate is usually expressed in years. So we're actually going to get that divided by 2, 3,500 every six months would be the idea. So that means 3,500 times 10 periods would be the 35,000. And we're going to get, if we discount it back, we get to the 27,695. Notice it's slightly different than what we got on the yearly payments. Down here, the face value, the 100,000, we're going to get back. Once again, same kind of thing. But now we're going to adjust the periods for six month periods. The rate, 9%. That's a yearly rate. Therefore, we divide it by 2. And then comma, the number of periods would be 10. But that's in years. So we're going to multiply it times 2 to have, I'm sorry, it would be 5 in years. But we multiply it 2 to have 10 six month periods, half year periods, comma, comma, to get to the future value. That 100,000 discounted back to the current day would be worth the 64,393. Again, notice it's a slightly different than the number up here, given the fact that we're using the semi-annual at this point. So that would be, then if I add those together, we get the bond at the 9287, which again, slightly different than the 92221. So let me show a little bit more about this, how we would break this out in terms of a series of payments. Because when we visualize this, we typically, we can do it this way, but this looks like a little magical kind of way, because Excel gives us these formulas. So what we can do is we can map it out. We can say, okay, let's break this out on a year by year basis. There's 10 periods of six months, or what we did before was five periods of one year. Let's think about the 10 periods of six months first. And we're going to say we've got interest payments, which are 3,500 for each period for 10 semi-monthly periods. That's the rent that we're going to get. That's the annuity of payments that we're going to get. And then we've got the face amount that we're going to get at the very end. So it's zero all the way through until the maturity at the end. We're going to get that original amount back at the 100,000. So if we think about the cash flows in total from a period by period perspective, cash flows for the first six months, we get the 3,500 all the way through. And then at the end, we get the 3,500 plus the 100,000 that we're going to get back. And that's the 103,500. So notice up here, we calculated this using one and annuity calculation. And then to the present value of one calculation because we have those two kinds of streams of cash flow. But we could do it just on an each year by year category. And this is a useful tool to use because if we have say projections out into the future that are not uniform in terms of an annuity or the present value of one, we might just break out those on a year by year basis and then present value each of them for each year. So for example, this first one, we have the 3,500. If I just present value it back to the current timeframe, we would be taking the rate. The rate would be the market rate, the 9% divided by two because the six month period, comma, the number of periods would be one six month period. And then comma, the payment would be no payment because now we're not doing an annuity. And so two commas, the future value would be that 3,500. Therefore, if I discount that 3,500 back one period, one six month period at the rate of 9% or half of 9% for the six month period, we're going to get to the 3,349. I could do the same thing here, discounting this 3,500 back to six month periods at a rate of 9% divided by two. I'd get to the 3,205 and so on and so on. 3,067, you can see as we go forward into the future, it's worth less in terms of the current day periods because we're discounting them back further due to the time value of money. And this last payment, we got that balloon payment here because we also received this 100,000, but it's 10 periods into the future or five years into the future. So it's more heavily discounted than the first one up top, which is the interest payment. And then if we add all that up, if we add this stream of payments together, we get to the 9,287, the 9,287. So this is a way that you can kind of visualize it a little bit more easily, I think, than when we're basically using them as a present value of an annuity. You could break out the stream of payments that you're going to get out into the future and then pull them back into the current day and the sum of that future stream of payment, that's going to be your present value. That's another way that you can see it. So then if I look at this in terms of the yearly payments, this 3,500 plus the 3,500 is what we're going to get per year, right? 3,500 plus 3,500 is 7,000. Or in other words, we're going to get the 100,000 times 0.077,000 per year. So if I thought about that on a yearly payment, and this is one of the kind of the problems when we're using these time value of money payments, we have to assume that all the payments happen at the end of the period or the beginning of the period, usually the end of the period. So I'm assuming that whole 7,000 happened at the end of the period when I'm using the yearly assumption. So we have a compounding kind of issue when we go to yearly to semi-annual, for example, right? So if I said 7,000, 7,000, 7,000 for five years, then we'd get to the, and then the 100,000, we'd get to the same 135,000, but now the assumption is when we use this method that this 7,000 was received at the end of each time period. And so if I present value each of them using a yearly method, so if I had the 7,000 here, I take the negative present value of the rate, which I'm going to use 9% now, and then comma, the number of periods is now going to be one year instead of one six month period. And then comma, comma, the future value is now the 7,000. I discount the 7,000 back one year using 9%. I get to the 6,422. If I discount this 7,000 back two years using the 9%, I get to the 5892 and so on and so forth. And you would think, again, that these two would be kind of the same, right? Because this one, this one, I took six months periods, but I took half the yearly rate and so on. You could see, of course, they're different and there's kind of, so you've got this kind of compounding issue. So one of the problems with that would be why that's kind of an issue, which we'll talk about a little bit more. You could see that we came out to the two different sums here. The 9,274 was the sum of these, which was that one. And this one comes out to, if I sum these out, 9,221, which is this one on the yearly basis. So part of the reason there's a difference there is because you can see here, if I got this 3,500 sooner, like within a six month period, instead of at the end of the period, I got the full 7,000, I could theoretically invest it and get a return on it. So you would think if you had these two streams of payments to 3,500 and so on, on, out, which adds up to a total of 135 or this stream of payments, which is still within the five year timeframe at the 135, this top one would be better for us as an investor because we're going to get our money sooner. And if we get the money sooner, we can basically reinvest it. So you can kind of approximate it this way to look at the yearly and the semi-yearly and we can kind of approximate the rates here if I was to look at like market rate on a yearly basis versus a semi-yearly bond. But that causes a little bit of a confusion as well, which we'll talk about a bit more in future presentations.