 Personal finance practice problem using OneNote. Yield to maturity and effective annual yield. 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 on the left-hand side. Practice problem tab in the 11130 yield to maturity and effective annual yield tab. Also, take a look at the immersive reader tool. 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. Information on the left-hand side. We have the bond, the face value at the 1000, the coupon rate which in a prior presentation, we practice to calculate at the 10%. We have semi-annual payments. That means that we're going to get the interest payments, the maturity portion of the bond on a semi-annual, six-month period, half-year period, as opposed to annual. We've got the coupon payments then are going to be $50. That calculation we can do real quickly. Let's say we have the 1000 face amount. The coupon rate is 10%. So times 0.1, that would give us 100. We're going to divide that by two because we're talking about the semi-annual. Note that this coupon rate is often presented even in a semi-annual bond on an annual basis. An annual rate that we then need to divide by two to think about what the semi-annual payments will be. That's part of the confusion that we're going to be looking into here, noting that oftentimes when we're doing comparisons of rates, we like to have them on an annual type of basis as basically a typical standard. But it gets a little bit confusing when we've got these semi-annual or other time period payments such as we have here with the semi-annual bonds. We've got the years to maturity at one. Also just realize that if you have this $50 coupon, you can then calculate the coupon rate by taking the $50 times two to get to $100 per year divided by the 1000 and that would give us our 10% the coupon rate. Now we're going to say the price of the bond is at the 1009.36. That is over or higher than the face value. That means that the bond is at a premium at this point in time. So we're going to dive a little bit further into these different kind of rate issues here. So the first one we might calculate is the yield to maturity. Note that the yield to maturity is kind of like the market rate. It is often we want to represent it or present it on an annual basis, a yearly type of basis. But when we calculate the rate because the payments are happening from the unannuity calculation present value standpoint every six months, then usually we're going to calculate the rate on a six month time period. So we could take that six month time period and multiply it times two to get the yield to maturity. So this is something we've got to kind of keep straight in our mind because when we try to figure the bond, actually the correct rate to use would be the 4.5 that matches the periods of the bonds, which would be every six months. But when we present it and compare it to other kind of investments, for example, we often present it in a yearly amount, the yield to maturity. But this number is not quite precise in some ways because we really have to present these two numbers saying well, the yield to maturity is 9%, but it's a semi-annual payment bond at the 4.5%. So then we have this third calculation, the effective annual yield, which can be used for comparison purposes. So let's see if we can get this kind of straight. Let's do the calculation here. Remember that the coupon rate is the rate that's actually on the bond. Now we're going to calculate the yield to maturity, but we're going to figure first the yield to maturity based on the current periods in place, which in this case is semi-annual, so we'll call it yield to maturity divided by 2, ytm divided by 2. You can do that with a rate function. We do this in Excel, so we're just going to go over it in theory here if you want to work it in Excel, highly recommend doing so. But the rate function, we would say rate, number of periods would be just one period out, but we're going to multiply it times two because we're talking semi-annual periods, six-month periods, therefore there will be two of them, comma, the payment. The payment that we're going to have would be the $50, which would be the 1,000 times 10% divided by 2. And so that's going to be the $50, and then comma, the present value is going to be a negative number in our calculation here, note, but that's going to be the amount that we're going to be paying. If we bought the bond, $1,936, and then comma, the future value, the amount that we're going to get at the end would be the $1,000. So that would give us the 4.5 rate that we could use, which is great, works great, but that's on a semi-annual basis now, so to get the yield to maturity on an annual basis, we can multiply it times two and possibly then use this kind of annual basis to compare to other investments. But again, it's kind of tricky because it's like, well, it's really 9%, but it's a semi-annual bond paying at the 4.5, which is a little bit different than an annual bond at the 9%. So if you wanted to get the effective annual yield, which is an attempt to get a number that better approximates the comparison to other kinds of investments, you can use this function in Excel, Effect, Brackets, the nominal rate, and then the number of periods, which is going to be one times two. So we'll pick up the rate times, and then the number of periods and the function. In other words, the first number, E2, representing the nine, and then the number of periods is going to be one times two. And that should give us a number that's a little bit closer to comparisons when I'm comparing, but we don't really use it to calculate the bond. So let's see if we can kind of understand that a little bit further. We can also get to this number by using a formula, which is here. So it's one plus the yield to maturity divided by two squared minus one. So you can do that in Excel. It would be one plus the yield to maturity, which is this divided by two, which would give us to that 4.5 squared minus one. We can calculate it that way. And we already have, when we calculate it up top, this 4.5 is the yield to maturity divided by two. That's kind of what we calculated first. So we can simplify it even further, just saying one plus that 4.5, the six month rate, squared minus two, and we get the same result. Okay, so then if you were to then think about, okay, let me figure the actual bond price and see if this makes sense. This rate right here is kind of like the market rate, but it's a market rate aligned to the period, the six month time period. So if I use that to calculate our bond price, which we've seen in prior presentations, it would look something like this. We would take the present value of the interest payments, which would be the present value of the rate, which is that six month rate, the 4.5 comma number of periods, which would be the one times two comma the payment. The payment would be the $50 here. That's the 94. Then we've got the present value of the face amount, the 1000 we would receive at the end, present value of the rate. We're using the six month rate, 4.5 percent comma number of periods two, one times two comma comma future value, the 1000 adding that up. That gets us to the 1009 36 as we would expect given. So we just went back and forth with this with this 4.5 and kind of proved getting back to basically the bond price. Now, if you did this with the yield to maturity, you might think, well, wait a sec. Shouldn't I be able to take then this rate just multiply it times two and act like we got everything at the end of the year, right? Use it as one year. So if we tried that, we could say, okay, the present value of the interest payments would be the rate then, which this time we're taking the 9% the yield to maturity and then comma the number of periods would just be one period because we're taking an annual year old instead of six month period. Comma the payment would be not 50 but 50 times two or 100 because we would have 100 for the two six month payments we would be getting. And that would be 92 slightly different than the 94 and then the face value. If we took the rate of the 9% comma number of periods, which would be just one year and then we took comma comma the future value of the 1000. That would be the 917 slightly different than the 916 adding those up. We get something slightly different than we have up top. It's a small difference, but if you're talking about bigger dollar amounts, it becomes more and more significant. So one more calculation on this annual yield. I'm just going to put this annual yield into like a format in a table. I think it's useful to be able to do this in Excel because it helps you to kind of build your Excel tables possibly being able to standardize things and they kind of calculations or budgets that you can then work on the left instead of like changing your data on the left. Instead of reworking an algebraic problem basically by hand all the time, which could be a little bit slow if you're trying to run different scenarios. So if I put this into like a table format, I'd say one, I'm putting that on the on the right side here, then I'm going to take the yield to maturity divided by two. I'm going to do an inner calculation to do that. That's going to be the nine percent, which we got up top divided by two, which is going to then give us now I'm putting this on the outside four point five nine percent divided by two. That's going to be the four point five. So now we've got the one plus that so one plus the yield to mature divided by two is the four point one oh four point five percent about we square that. That gives us the one oh nine twenty. Then we subtract out the one and that gives us the effective annual yield. I think if you can build like little tables like this, it helps you to kind of transparently see things and basically adjust the data on the left hand side and see what is happening as the math is being calculated out in a table format. All right, let's try to understand this difference a little bit more and say like why would that be? Let's imagine we've got we've got one year which is broken out into two time periods. And we're just going to imagine that we've got a thousand dollars here that we're going to be receiving a return on and we're going to receive the return at the at the four point five. So let's say that we've got the thousand dollars and we're getting we're getting this rate every six months the six month rate. So a thousand times the point oh four five. So every six months we're getting forty five forty five return. So forty five and one six month time period second six month time period at the end of the year we've got 90 that we received if we try to take the 90 divided by the thousand. That would give us our annual kind of return point oh nine nine percent. So but however note that from an annual basis you when we do this we're basically kind of assuming that these two payments happened at the end of the year. It would be kind of like we've got the 90 dollars at December thirty first at the end of the year which isn't really the case. Because we got this forty five in the middle of the year and that's typically more beneficial than us getting the full 90. At the end of the year because theoretically we can take that forty five dollars for example and if you reinvested it and got it got a return for six months forty five times the times the. What would we say it was times the rate point oh four five times point oh four five we get another two dollars or so right. So really I mean if you take that into consideration you you might get a return if you were able to reinvest of the 90 plus the two point oh six if you were able to reinvest this amount for you know the six months which would be ninety two oh three on our yearly you know return. And if I took that and divided it by the one thousand we get something like the nine point two oh two nine. So that's you know kind of kind of the thought process when you look at this when you look at this stream of payments that are that are on an on an annual basis getting 90 dollars at the end of the year is not as good as getting forty five dollars and forty five dollars every six months because you're getting this forty five dollars sooner. And in theory you can invest it and get an added return on it. Now the bonds get a little bit a little bit more confusing when you when you try to break this all down because you've really got two streams of income happening or cash flows. One is going to be that face amount that you get at the end of the period which you can think about in this case either being at the end of one year or at the end of two periods right. And then you also have the annuity kind of component which is the stream of payments which is the fifty dollars every six months that you're that you're going to be getting. But you get you get an idea that when you try to when you try to just take the four point five rate that you would actually be using to calculate the bond price and just double it to get the yield to maturity that could be useful for the for the calculations is easy to kind of do. But when you try to then say that nine percent if I'm comparing that to other investments possibly other bonds that are yearly bonds or other similar investments that are getting other returns. Then you can't quite use just like that one number you'd have to think well it's nine percent return which is which is paying out semi annual at the four point five or you can try to approximate it to be using this calculation. The nine point two oh two nine which is hopefully a more comparable number. So if we break down all these kind of percentages we've got the coupon rate which is actually on the bond which is often presented on a yearly basis. Therefore if it's a semi annual bond you got to divide it by two often times to get to the coupon payment on a semi annual basis when you calculate the actual rate on the bond using like a rate function in Excel. You're usually first calculating it on the periods that are used which in this case is semi annual half year six month time periods. You can then double that to get to the yield to maturity the nine percent the nine percent also being something that you would have to kind of use when you calculate the bond price because you would take the nine percent in essence yield to maturity divided by two which would be the four point five. But it's not the one that you can compare as readily to other investments because you'd have to basically and that's where the effective annual yield would come in the effective annual yield not typically being something that you're going to use to calculate the bond price. But hopefully it's a better approximation when you're making comparisons between different bonds as you're taking the semi annual for example here and making it into like an annual comparable number. So that you can compare yields to other investments on an annual basis.