Hi, good morning, good afternoon, good evening, and, very welcome of course to this Felix live refresher session on bonds.

And actually, I should say part one of the session because part two will be delivered in, I think two weeks from now, as it was just a little bit too much content to be squeezed into a single one hour session.

But allow me to quickly introduce myself.

My name is Thomas Crower and I'm head of financial products here at Financial Edge.

I started my career in fixed income, um, mostly trading rates and FX in cash and derivatives, but then I also had the opportunity to work with the cross asset mandate, which gave me some good insights into credit and equity markets as well.

So, what's on the agenda today? As I said, it's gonna be a refresher session on bonds, and this in general means that it's gonna be relatively fast paced as we want to refresh your knowledge on a couple of, uh, bond specific items.

And that means we're gonna start with some of the main principles of bonds and recap some of the terminology.

We will also cover the main coupon types and the general mechanics.

We will of course talk about the yield to maturity and this inverse price yield relationship in, uh, fixed coupon bonds.

And we will also have a very high level, uh, look at the, pricing of a bond issuance and how the yield can be derived from a bond secondary market price. That's pretty much the agenda.

But before we start, a couple of reminders.

First of all, you can access the course materials via the link that you will find in your zoom window, so under resources but you can also later access them via the Felix Live website.

And please don't forget, you can ask questions during the session.

The one thing to remember though, is that you have to use the q and a function for this.

So please don't use the chat.

I'm not gonna monitor it at all.

But if you want to ask a question, just put it in the q and a function, I will see it and then pick up on it.

And then after the session, you shall be directed to a feedback form.

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So please take, you know, a couple of seconds really, or maybe a minute or two to fill it in.

And of course, if you want to ask some follow-up questions, a feedback form is a great way to do that.

And also, if you would like to see other topics covered in this format from a market's point of view then of course the feedback form is a great way to let us know as well.

But then without further ado, let's get started.

And as I said, we're gonna start with a quick walkthrough.

The main principles of bonds that apply regardless of whether you talk about government bonds, corporate bonds, or any kind of bond really and bonds are in that regard, a form of debt instrument.

Very much like in the case of loans, the bond investor lends money to the bond issuer and the issuer promises to repay the amount at a specific date in the future.

And that's what we call the maturity date.

Now, between the issue date and the maturity date, bond investors then typically receive regular interest rate payment.

So what's then the difference between a loan and a bond? Well, there are actually many differences.

I believe, though, that the most important one in this context that we're talking about here today is that bonds usually are a little bit easier to trade in and out than loans.

So for many bonds, especially when you're looking at government bonds here, for example, there is a relatively liquid secondary market, which means that investors who lent money to the issuer initially for let's say seven years, when they bought a seven year bond on issuance, they can decide to sell the bond to another investor on any business day and therefore get their money back even if the maturity date has not been reached away, might actually be still quite far away from that.

Now, of course, yes, there is a possibility to trade loans as well, at least certain kinds of loans are traded.

But overall, I would say that selling a loan is a little bit more complex because this is a bilateral relationship between, you know, the bank and the client. There's a lot more, um, documentation, et cetera, et cetera, to go through.

And so this will generally require more time to completion and is therefore likely to be coster from a resource perspective than selling a bot.

So in general, what that means is loans trade in comparison to bonds rarely in the secondary markets, while bonds often change hands multiple times actually before they mature, and therefore can be considered as being, um, more liquid.

Now terminology that we use in bond markets might also differ somewhat from the loan markets.

So let's go through some of the key terms that you find here at the bottom of the slide and clarify what they mean.

Issue, I don't think there's gonna be any really difficulty around that. That's simply the organization or that refers to the organization here that is borrowing the money.

Then the maturity date, we already talked about that, that's the due date for the borrowed money.

Then there is a coupon, there's nothing else but the interest that's paid to the borrower in regular intervals.

Quick reminder here.

If the coupon is given to you as a percentage number then this means this number or that, you know, this is always given to you as a per annual number.

So if you have a bond that pays semi-annually, 5%, that means in total you get 5% over the year, but you get it paid twice a year and you need to make adjustments. Then when you're thinking about rolling out the cash flows, we look into that later on, then there is the par value, which also often referred to is as the face value or even the notional amount.

And that's basically the amount that will be paid back to the bond holder at maturity.

So when you read that the US Treasury has issued 5 billion, you know, 10 year bonds, what this means is that in 10 years, they will have to repay $5 billion to the investors in that particular bond.

And then there's a bond price because if bonds are traded on the market there, it will be a price attached to this. And the bond price is basically then the price that an investor currently would have to pay for buying that particular bond.

It's most often quoted at a percentage of the par value, and that means it can be below par i.e.

below a hundred percent.

This, as I said, refers is referred to as a bond trading below par, it can trade at a hundred percent. That's the bond trading at par or above a hundred percent.

That is then referred to as a bond trading above par.

So let's have a quick look at the actual bond price quotation because there has been some changes over, you know, many the last couple of years.

So bond markets, as I said, have evolved over time, right? In terms on how they quote prices, traditionally there were quite significant differences based on, for example, the geographical location and the types of bonds that you were trading.

For example, US treasury bonds were always quote or have always been quoted in fractions specifically in 30 seconds of a dollar.

So for example, if you see a bond quoted at 107, that basically means that it's a 100 -7 over 32.

And that means in decimal terms, 100.2188 Now, as I said, that's the way treasury used to be quoted. And on Bloomberg for example, you still find this, but there's this ongoing transition to decimal pricing and that has been initiated somewhere in the early two thousands.

And this is an attempt to modernize and simplify really bond trading on a global level.

And it's sort of like generally reflecting the, I'd say broader use of the metric system, et cetera, et cetera, right? So however, still in some ways or in some cases you do find that quote, just remember it's really no big deal, just divide the seven by 32 and you can quickly transform it into the decimal point in European and most other international markets we have used decimal pricing throughout.

So here, if you think about a Japanese government bond, for example, that would be quoted in a hundred, you know, in decimal terms.

So a 100.9290 is the example here.

And this is, as I said, in line with the system with the metric system systems broader usage, and this global, I'd say trend towards digitalization in financial markets.

Okay, that more as a side note, but let's have a quick look at the bond redemption now.

So how are bonds typically repaid? Now, the vast majority of bonds have a fixed maturity date.

This means that the date on which investors will get their money back is actually said when the bond is issued and then doesn't change over the life of the bond.

That's number one. Number two is that the vast majority of bonds are what we call bullet bonds.

And bullet bonds are bonds that are repaid in one single payment at the maturity date.

And there's no amortization or partial repayment between issuance date and maturity date, anything like that.

And that's, you know, really as I said, the vast, vast majority of all bonds that work like that.

But of course, there are exceptions to these rules.

There are, for example, amortizing bonds, right? And these are bonds where the face value declines over the life of the bond as part of the face value is actually repaid before maturity.

But that usually happens on a preset schedule that is agreed, you know, basically at the issuance of the bond.

And then another variation is, for example, the callable bond.

Callable bonds have a preset maturity date, like any other bond.

But in addition to that, the issuer has the right to repay investors prior to that date and then stop paying coupons accordingly because they have repaid the money.

Now, in most cases, the bond can then only be called at one or at least a limited range of specified dates of the life of the bond. And you can easily imagine as to when the issue of the bond would take advantage of that call rate.

That's usually when funding costs have moved in favor of the bond issuer.

So for example, if they had a 10 year bond that's Caldwell after five years, and now after five years they realize that they can refinance for another five years at a lower level than what they're currently paying on their callable bond, the bond will be called and a new bond will be right, which is an advantage for the issuer, but a disadvantage for the investor because they get their money back and they have to reinvest.

Now at lower interest rates, extendable bonds then work in the opposite way.

There is a preset maturity date as well, but the issuer has the right to extend the maturity of the bonds on one or numerous future dates.

Now, in case of an extension, what usually happens is that the coupon rate is generally increased by, again, a preset amount.

So in both cases, callable and extendable, the investor provides the issuer with an option to change the maturity of the bond.

And you can easily see as we went through the callable example, that the issuer will always use that option if it means an advantage for them and potentially a disadvantage for the investor in that bond.

And why would the investor take this position? This kind of on the face of it looks slightly disadvantageous.

Well, of course, in exchange for basically selling an option to the investor there, they will receive a higher return on that bond.

Okay? So that's the bond redemption in a nutshell.

And next what we're gonna do is we're gonna have a look at those general coupon types.

Now, as I said, between issuance and maturity, the issuer generally makes regular interest rate payments to the investor.

And we are now gonna have a look at the most important ones and the main characteristics and the vast majority of bonds, again, are what we call fixed coupon bonds, okay? And this means that the coupon is set when the bond is issued, for example, here at 3%, and then doesn't change over the life of the bond.

So if you look at the cashflow here that you can see on the slide, it's a 10 year bond, paying 3% every year.

It's really as simple as that.

Every year the investor receives 3% interest on their investment on the notional, we should say, and then at maturity.

So in 10 years from now, they get their money back.

IEA hundred percent, that's the face value, that's a notional amount, excuse me, being repaid plus a final coupon payment of 3%.

That's really as plain plain vanilla as it gets in bond world.

Now let's have a look at a, which I always sort of like to think about a special case of fixed coupon bonds and that are zero coupon bonds or zero bonds, right? And the reason why I call them a special case of fixed coupon bonds is that in case of zeros, there is no interim interest rate payment between the issuance of the bond and the maturity.

So you could really think about this as a fixed coupon bond that pays a coupon of 0%, hence zero coupon bond.

However, while you know, this might have been different back in, you know, 2021, et cetera, when interest rates were so super low generally just because you are buying a bond that pays zero coupon, that of course doesn't mean that you won't get a positive return as a bond investor, right? And the return is just generated in a slightly different way because zero coupon bonds then are usually sold at a discount price and then repaid at a hundred or at par at maturity.

And then this difference between the purchase either discounted purchase price and the redemption price that you get back at maturity, that is the return that an investor will realize over the life of the bond if they hold it to maturity, of course.

Now, here is an example.

We have a 10 year zero coupon bond that is issued at a price of 74.51.

Then there's not a single interest rate payment over the 10 year term, and the bond will be repaid at 100.

So the difference between those two prices, that's the total return over this 10 year period.

And of course, we could somehow work this into an annual, return using the appropriate mathematics.

Now, one point should be made though, as I said, this is a special case of fixed coupon bond, but there's one very, very significant difference then between the first one we looked at 3% per annum and this one 0% per annum. And the key difference is that in case of a zero coupon bond ignoring credit risk right now, right? We're not looking at credit here, we're just assuming that this is a credit risk free bond.

So let's say we're talking about government debt here but assuming there's no credit risk, then in case of a zero coupon bond, we know exactly the total return that this investment is gonna realize over the next decade because we know we pay 74.51% today and we know we're getting a hundred back at maturity and there's nothing else.

In case of the previous example though, in case of our super percent coupon bond, what happens after one year, and let's go back to that slide, is we're actually getting 3% of our investment back.

Now as rational investors that have a 10 year investment horizon, we probably wanna reinvest it, right? Because we said, okay, I wanted to invest the money for 10 years.

Now after one year I get 2% of it back. What am I gonna do with it? I'm gonna reinvest it for how long? Nine years, right? But at which rate we don't know because for that, you know, we need to know the nine year rate in one year's time.

And of course, that's not something we can really forecast was a hundred percent accuracy.

So there is something kicking in that we call the reinvestment risk.

So in case of coupon payments, you don't know exactly what the return is gonna be, you know, the major components, right? The coupon, the price, et cetera. We're gonna talk a little bit more about that later, but, it's not a hundred percent known at time of the investment what your return in total is gonna be because of that uncertainty around reinvestment rates, okay? That is then obviously different in zero coupon bonds, but zero coupon bonds, especially when you're looking at corporate bonds come with their own challenges.

So that's for another day. Credit ss. Cool.

So now let's have a look at not fixed coupon bonds, but actually floating, coupon bonds.

And those are referred to as floating rate nodes, RNs, or floaters.

And they are bonds where the interest rate is variable. And what this means is that when we issue floating rate nodes, it's not the coupon itself that is fixed i.e. 3% for the next decade, but we fix a coupon formula and that then doesn't change over the life of the bond.

And the coupon formula in most cases refers to a money market benchmark rate, like for example, ior. Yeah, Euribor, for example, is still around, or then nowadays more often of course new risk for rates i.e. SOFR, SONAR, et cetera.

Thank you for pointing that out.

Floating right notes slide coming up, sorry.

So you now have, this situation where, you know, a formula, it's some RFR plus minus a spread.

And what this means is that, you know, the actual interest rate payments that you're gonna get on this bond are not known in advance because they will depend on future developments of the reference rate, right? If SOFR now goes up, your coupon will increase. If sofa declines, it will decrease, et cetera, et cetera.

Now let's have a look at these high level mechanics of how this floating rate concept generally works.

And here we have to really distinguish between IBOR and, the IFR linked FRNs because they work actually quite differently, okay? And the reason for that is that IBOR rates like your IBOR for example, are what we call forward looking term rates, right? Which means we know them at the beginning of the period and they will be paid at the end, whereas IRS, like sulfur, for example, are overnight rates that are transaction based and therefore consequently backwards looking.

So basically what that means is SOFR that applies for today.

i.e. for borrowed money from today to Monday.

That will be known on Monday because it's based on the actual transactions that we have, made during the day.

So let's have a look at the Euribor, first because that's a traditional way, as I said, it's fading out, but it's still around.

So here we go. An investor bought a 10 year floating rate note that pays six months Euribor plus 0.5% payments are made semi-annually.

And let's say when the flowing rate node was fixed, IBOR was or was issued, IBOR was actually fixed at 3.96.

Okay? The first interest that is then used to calculate this six months payment is then 4.46%.

And the reason for that is that this is just LIBOR or Euribor plus, the spread of 0.5. However, coupon is paid for six months period.

So half for half the year basically.

Now interest rates, as I said, always quoted per annum.

So the actual payment will be around half of that.

We're ignoring count conventions for simplicity here. Now, okay, if now over the next six months your arrival, for example, will go up then in six months.

So at this point in time, we're gonna look at where your arrival is there.

Suppose it's, let's assume it's gone up to 5%, then 5% is gonna be the, sorry it's gonna be the new or the new fixing level for your rib.

We are adding the spread of five and a half on top.

So that gives us an interest rate of 5.5 for the next period, divide by two, and so on and so on.

In any case, however, the coupon that you, that an investor will receive at the end of the interest rate period will always be known at the beginning of the coupon period, right? So at this point, we know the coupon we're gonna receive here, and at that point, we know the coupon we're gonna receive here because that's when the underlying arrival rate has been fixed.

Okay? Let's have a look at the newer way of dealing with this.

So let's see how floating rate notes generally work when the underlying is, or the reference rate is not IBOR, but now one of those overnight rates, right? And that works quite differently because whereas in IBOR linked FRNs, we usually have sort of like the same reset and payment frequency of the coupon. So in our example here, the coupon is reset every six months.

So there's a new IBOR fixing being used every six months, and there's also a payment every six months.

That's the same frequency, right? In RFR linked floating rate nodes.

We usually have a difference here between the reset frequency and the payment frequency.

So the coupon is reset daily because the sulfur and toner and all those things are overnight rates.

Now, you can easily imagine that it won't be operationally very efficient if the issuer of a floating rate note now makes daily interest rate payments to investors, right? That's not good for anyone because investors have to transfer small amounts of money every day.

Sorry, the issue is the investors have to check if the payment has arrived, et cetera.

So to make it operationally a little bit more efficient, we have in many cases the agreement that yes, it's a daily reset of the interest rate, but payments of made, for example, quarterly.

And let's kind of go through the mechanics of this.

So if an investor now bought this 10 year floating rate, now that pays sulfur plus 0.5% with agreed upon interest rate payments every quarter, that means not all relevant sulfur fixings for the first coupon will be known at this point, right? In fact, this point was what I'm referring to, in fact, most likely not even the first sofa fixing will be known because as I said, fixing or the SOFR fixing for the bonds issuance state will be known on the next business day. So our case here on Monday.

So in case of an FRN, what that or, or of an RFR linked FRN, what this means is that the exact coupon will only be known once the last RFR fixing for the coupon period has been published. And that may the last business day of the coupon period.

So here it might actually be a day later that depends a little bit on the exact structuring of, uh, the floating rate. Now, and that's why we call it backwards looking because we know at the end of the coupon period all relevant rates that we need to know to calculate the actual coupon payment.

So how's it then actually done? How do we get from all the individual sofa fixings that we have seen over this three months period to the actual coupon payment? And again, there are different possible methods, but most often we use something called daily compounding here, right? And, this method basically takes into consideration that the borrower does not pay interest daily, but also the interest on a daily basis, right? But as we said, for operational efficiency, everybody's kind of better off if we make a larger payment at the end of the coupon period. But technically speaking, today's sofa fixing, if I borrow money today and I'm supposed to pay SOFR on it, I should make the sofa payment today, not in three months time.

That would put the issuer into an advant advantageous position because they basically get a free delay of interest rates and the investor, of course, will not enjoy that that much.

So what we're doing instead is that the accumulated interest that is owed, but that has not been paid.

I everything that sort of accrues between those coupon payments that are ever happening every three months here, they are tracked and then the daily sulfur fixing is applied to both the principal amount and then the accumulated, but unpaid interest. So basically today's SOFR, if the bond was issued today, would be paid on a hundred percent if that was, you know, there was no accrued interest and then Monday sulfur would be paid on a hundred percent plus the accrued interest that should have really been paid on Monday, but then wasn't.

So that means yes, the issuer pays the interest with a delay, but the investor receives interest on interest. That's what we mean by compounding method and that's why no one really is better or worse off, at least in theory.

Okay? So that's floating rate notes in a nutshell.

Now let's have a look at inflation linked bonds because what all the coupon types that we looked at so far, uh, had in common is that the amount that is repaid by the bond issuer at maturity is the originally issued paramount.

Right? Now, in case there is high inflation over the time to maturity of the bond for investors, this could lead to a significant erosion in purchasing power of the investment, right? So you imagine you invest a hundred million today in a bond, that bond pays 3% interest, then inflation is gonna be 10% over the next, you know, couple of years, and then you get a hundred percent back.

That's now significant or worse, significantly less in terms of purchasing power then, you know, then it was when you entered into this bond, right? So inflation link bonds are than bonds that have been designed to really combat this risk of inflation eroding this purchasing power, right? So example here, treasury inflation protected securities or tips in the they are what we call principle linked bonds.

This means that when tips are issued, a fixed coupon is set like in case of regular fixed coupon bonds, that the actual inflation protection is then achieved by changing the f or adjusting the face value regularly to reflect changes in, you know, US price levels, you know, looking for example at the US consumer price index CPI.

And so the face value of the bond would then generally increase over time if there is positive inflation and decreases over time if there was deflation.

So we're adjusting the principal amount and as this adjusted face value or principle or notional amount is then basically the calculation basis for regular coupon payments.

What we can say is that yes, they pay a fixed coupon percentage wise, but they pay on variable notions if you wish.

So that actual payments that you received in terms of coupons will differ over time as the face value increases or decreases driven by inflation.

And now what happens at maturity of the bond? Well, investors will either receive the inflation adjusted face value or the original face value in case of tips, whichever is greater.

So what that means is tips do not only offer protection against inflation.

So if there was inflation, you get more back than the face value was originally.

But if there's a net decrease in price levels over the life of the bond i.e. deflation over the whole period of the bond's life, the face value will not be adjusted downwards to reflect deflation. And we call this a deflation floor because basically you are protected against deflation as the bond holder.

Word of caution, though not all inflation protected securities that are issued around the world contain this.

So-called deflation floor.

So it's definitely something to read the small print for.

Okay, so that's sounding a little bit abstract. Let's have a look at a very simplified example of how cash flows of an inflation linked bond would look like roughly, because in reality the mechanisms are much more complex.

So for example, tips pay coupon semi-annually, not annually, and they're saying a time lag between the official inflation measurement and the adjustment of payments, et cetera, et cetera. But we wanna focus on the general high level mechanics here. And for this, this example actually works really well.

So let's say we have a five year principle inflation linked bond that pays a coupon of 2%, right? And as described, the coupon is paid on the inflation adjusted principle.

And now during the first year of the bonds life, the inflation is positive.

So CPI increased from a hundred to 104, that is equal to a 4% annual inflation, which means that at the end of year one, the principle here in the simplified case is adjusted to 104%, and then the coupon of 2% is paid on 104%, which then gives you 2.08% as a coupon that you receive in day or at the end of year one.

Then in year two, we see another positive inflation, so 2.88%.

Now that's the CPI had gone to 107.

We adjust the principle to 107, we have a higher annual coupon payment, and then we actually have two years of negative inflation or deflation.

So the CPI comes down, and so the adjusted principal amount comes down, stays above 100 though.

But the annual coupon payment sort of like decline in those two years and then at maturity after, and because in year five we had another bout of positive inflation and we're ending, the CPI index ends at 105 when the bond actually matures.

That's then the final principle adjusted amount, and that is what's gonna be repaid to investors.

So basically investors will have received five coupons, plus they get another 5% increase in capital back through the inflation adjustment on the principle, and that's the main idea.

So in times of high inflation, such an investment would at least protect you in some ways against the erosion of purchasing power.

Okay? Now, let's switch and talk about bond pricing, right? Now as mentioned earlier, bonds are often talked about and quoted in terms of price.

So I said earlier they're quoted in decimals or even in fractions, but you know, it's percent of the notion that you would have to pay to get the bond at the moment.

So prices are obviously very important for participants, but the question is, what does a bond price actually come from? Now, the theoretical foundation for this, lies in time value of money, right? And this is this fundamental concept that suggests that the present value of a cash flow is different from a suture value because, you know, if you have money readily available that has earning potential.

And so it's worth more than the same amount of money being received in a decade from now.

Right? Now, from a pure technical point of view a bond is really nothing else but a series of cash flows, right? And the good thing is if we're looking at fixed coupon bonds, those cash flows are actually known with certainty over the life of the bond.

Again, if we're ignoring credit risk here for simplicity, and let's do that generally throughout this session.

Now, what we know is now we have a set of payments that this bond basically represents, but these payments will obviously occur in the future.

And because of time value of money, we cannot simply add the coupons and redemption payments to get to the bond price.

We have to discount all outstanding payments, um, first and then sum of those pvs and the sum of the pvs.

That should then be the bond price.

That is the theory, right? And discounting then usually is done by dividing whatever cashflow we're looking at by one plus the desired rate of return, or generally speaking, the appropriate rate of return per period to the power of the relevant number of the period.

So how would that look in a concrete case here we're kind of seeing two different bonds and they differ in the payment frequency, right? So first we have a 10 year bond that pays an annual coupon of 3%, let's say the desired rate of return or the appropriate rate of return here for this case, I'm gonna dig a little bit deeper into that.

Now, in a, in a minute is 4%.

So what that means is we have to discount each of these remaining coupon payments and the redemption payment with one plus 4%, which is a discount rate to the power of the period.

4% is the period rate because interest is paid annually.

And then the period number, because we have annual payments, is identical to the number of years this payment is away.

So that's basically here then the general, um, approach on how one would calculate the price of this bond.

Now, if we have a bond that doesn't pay the coupon annually, but pay semi annually, then two things change.

First thing is of course the cash flows are no longer 3% because now the interest is paid twice a year.

So we said already, that's then roughly gonna be one or half of the actual annual coupon ignoring the account conventions.

So you can say, well, that's twice a year, and that should equal 3%.

So that's two times 1.5%.

Now, assuming that coupons are also paid exactly every half year for simplicity, the desired or, you know, appropriate discount rate per period, not per year, but per period, which is now six months is 2%, so half the 4% per annum rate.

And so we're seeing a discount rate here at 2%.

And then another difference is now because we're having semi-annual coupon payments, we don't have 10 periods, but we actually have 20 half year periods in a decade, right? So, um, that's the general idea.

Now, let's have a look at where this actually, becomes relevant, right? So because very often, you know, and, and, and I include myself there, when you look at this for the first time, you think, okay, what, that's cool, the masses, you know, I'm sure, okay, but what's this desired rate of return? Where does that come from, right? How do we know? Well, in case of bond pricing, um, this actually is quite easy to explain and hopefully quite intuitive to grasp.

So because, or, or generally speaking, taking a step back, there's one point in the life of every bond where it has to be priced, and that is the issuance date.

Right? Now, depending on if we're talking about government bonds or corporate bonds, the process leading to this pricing might differ somewhat. For example, governments often use via public auctions, corporates might use underwriters, et cetera.

But in general, what we can say is that investors tend to put orders in, based on a return rather than based on the price they're willing to pay for a bond.

And this makes, a lot of sense simply because, very often critical details of the bond, like for example, the actual coupon rate that will be paid, they will not be known until before, you know, very, very shortly before the actual issuance.

So that makes it difficult to bid for a price if you don't know what the coupon is gonna be, right? So you say, okay, I want, I'm bidding every term.

So let's go through this example that's here on the screen.

Let's say the US government has announced to issue $10 billion, five year bond, right? This is the one where apologies, this is the one we're talking about.

And, uh, then we have collected the bids in this issuance or in this auction process, right? The following bids have been received, and you can see them here on the table.

That is first thing, the lowest yield that have been bid is 4.5222.

And for that yield level or return level, there was a demand of 3.4 billion.

So that itself is of course not enough to get all 10 billion sold.

Now for the slightly higher yield level of 4.523, there was demand for in total of about $4.9 billion.

Now, we're making an assumption here, and that is that there are no multiple orders from market participants at different yield levels.

So that we can say that investors that are willing to buy the bond at a yield of 4.522 will also be happy to buy the same bond at the higher yield of 4.523.

And if we make that assumption, then we can basically say that at a yield level of 4.523, there is an aggregated demand of 8.3 billion, which is nothing else than the sum of these two numbers.

Okay? Now, now we go to the next higher yield level because, you know, uh, the aggregated demand isn't enough to get the whole 10 billion issued.

So now we go to a yield of 4.524, and that's where the aggregated demand exceeds the plant issuance.

So let's say that then this is the yield at which the bond is issued, and remember what this yield is that's the yield investors bid for.

So it's basically the yield that an investor requires to receive in order to be interested in buying the bond.

And so that's why you can sort of call this actually the desired rate of return or the appropriate discount rate or whichever, way you wanna call it.

And that's very critical because now we know that the bonds return for investors will actually be 4.524, but we don't know the issuance price yet, right? So that's what we have to figure out now, and the, the piece that we have to do is, of course, we have to determine the bond's coupon, right? Because without knowing the coupon, it will be extremely difficult to actually calculate the price of the bond.

Now, let's say that in this case, the coupon of the bond was determined to be 4.9, uh, 4.5%, right? So that means that actually the cash flows of this bond are gonna look, as you can see here in this column, it's a five year bond paying 4.5% semi-annually, that means 2.25% every six months.

Now, we know that the desired rate of return was 4.526, which means the period yield is exactly half of that, which is 2.262%.

And now basically what we have to do is, in this case, let's do that here very briefly, 2.25% divided by one plus 2.262% to the power, uh, of one that will give us the PV of 2.2002.

We do that for all the remaining cash flows, and we sum up the present values, and that gives us the issuance price of 99.893672.

And that's how you calculate the issuance price.

Now, the bond has been priced, so investors now get the bond, they pay the price for it, and then secondary market trading starts, right? And then things somewhat change because, as we've seen that obviously here at bond issuance, it's important to be able to price a bond.

But in the secondary market, the price is often given in practice, right? And it's driven by supply and demand dynamics.

So pricing in a narrower sense actually isn't necessary, right? Um, but we still use this formula because investors often just want to know the return or total return that in bond investment would realize for them if they buy a bond at a particular price.

And the formula we've just seen, or the one that's also again given here on this slide, is actually the one that we can use to determine this return, which is also called yield, or, much better known as the yield to maturity.

Now, unfortunately, we have to say there's no closed form solution, to calculate the yield of a bond that has more than one cashflow directly.

So in practice, what we have to do is we use iterative techniques.

So in other words, we change the discount rate that we use in this formula here until the sum of the present values of the bonds cash flows equal the current bond price.

And then that is what we display as the yield to maturity.

And that is one of the most commonly used types in fixed income markets of yield.

So usually when you hear talking about yields, they are actually meaning yield to maturity.

And the reason why this is such a popular measure is it's a great comparison tool across different bonds, right? Because, you know, sometimes you, you can clearly see that one bond is more attractive than the other in terms of return potential, right? So, but how about the following? You have two bonds, one has a slightly higher price than the other, but also pays is slightly higher coupon.

So you pay a little bit more, but then you get a little bit more.

Now, which of the two bonds is actually the better investment from a pure return point of view? And that's where the yield to maturity comes in because it gives you this comparison tool, it calculates the return of both bonds, and then you can use this measure as a direct comparison, great tool.

It just needs to be taken with a pinch of salt because it's based on a capital of assumptions.

And some of them simply might not hold in reality.

And then, you know, you need to be aware that this yield to maturity is maybe not 100% accurate.

So what are the assumptions? And you find them listed on the bottom of the slide here.

The first one, which is really unproblematic, is the bond is purchased at its current market price, right? Where else would you purchase it? The second assumption is that the bond is held to maturity.

And that might not be necessarily the case for every investor, but I just think, I don't think it's an, it's a, it's a problematic assumption either, right? So it's just something that you could say, okay, if I hold this bond to maturity, then that's the return I will generate.

The third assumption is that all coupons that you will receive in the interim will be reinvested until the bond maturity.

Now that itself, again, not really a problematic assumption, that's what rational investors should do, but what the formula does before, but sorry, but what the formula does it is assuming a constant reinvestment rate for all the coupons, and that is gonna be the same reinvestment rate, then the current yield to maturity.

So basically what the assumption is that interest rates for all times to maturity are identical and they will not change over the life of the bond. And that of course, is not very realistic.

There's a yield curve.

So rates for different maturities are different.

Yields change all the time. Great question here. If we solve for r how do we get the market price of the bond? Well, that needs to be given to you, right? So you either need the yield to calculate the price or you need the price to calculate the yield.

And as I said, usually if you want to kind of really take a simplistic view on this, then when you issue the bond, you start with the yield and you calculate the issuance price, and then people start buying and selling.

And you look at your screen and you see where the market is, you see the price of the bond trading in on whichever platform. And you say, okay, if I bought that bond at 99.8%, then I can solve for r Hopefully that answers that question.

Now, that is limitation that I just wanted to make sure everybody is aware of, but of course, investors are generally aware of those limitations, and that's why the yield to maturity remains one of those main anchor points in the fixed income world.

Okay, so here on the next slide we see, basically the general idea of this iteration process here we have a five year bond annual coupon of 2.4%, uh, trades at a price of 99.8603.

And then the question is, what is the yield to maturity? Now I'm gonna quickly show you how you can use Excel, for example, to iterate this yield.

We're going to use goal secure.

So you can see now and I'm just gonna increase this a little bit, the general setup, we have a five year bond paying cashflow of 2.4% every year.

So this is one year, 2, 3, 4, 5 redemption, add five years.

And then, we're also having the PV column, which is basically the discounted rate, right? And so basically what we're doing here is take the cashflow, divide it by one plus the yield to maturity, which currently is empty to the power of one in this case because it's a one year cashflow. Right? Now, if the yield box is empty, that basically implies a yield of 0%.

That means the present value equals the future value because there has no interest.

That of course is not realistic.

And what we see then, if we apply a yield of 0%, the bond price indeed is 112%.

So that's nothing else but the sum of all the pvs, right? However, what we wanna do now is obviously find the yield that needs to be put in this box so that this price here, the one we calculate is equal to this one.

And then what I do, and you know, them are I'm sure much more elegant solutions to that, but I just basically put in a sale the difference between the bond price that's observable in the market, and then the bond price that I calculate in my simplistic model here.

And then I multiply that with a hundred thousand or a million or whatever just to get the you know, the accuracy a little bit bigger.

And now what I can do is I can simply go to what if analysis goal, see? And then basically what I wanna do is I want the difference in price.

IE delta price here, I call it to a value of zero.

And that will be achieved by changing the yield press. Okay? And then I get a yield to maturity of 2.43%.

And what that means is if we now discount all these cash flows, and I'm going back to my slide now, if we discount all those cash flows here with a yield of 2.43%, then the bond price will indeed 99.8603. So it's exactly what you've done.

You started with the bond price that was given, and then we just iterate for the for the actual discount rate that we need to use or that is implied, let's put it this way, in this bond price.

Okay? So, as I said, yields are talked about literally all day long in fixed income.

And so I think it's useful to have just a slightly closer look at this.

More specifically the ultra maturity as it is kind of like the key measure, right? So, we haven't really listed them explicitly, but I think, you know, it has become clear that the yield to maturity has three main components.

The most obvious one is the coupon, right? You get regular payments and then we talked about the reinvestment returns, right? i.e. you get 3%, for example, after year one, you're gonna reinvest this, that gives you return, you just don't know how much, right? And that's where the model has its challenges.

And then of course, there's also the bond price that matters for the yield.

Why? Well, because if an investor buys a bond at a price below par, and then the bond is redeemed at par at maturity as its market standard, this will simply result in capital gains for the investors.

And on the other hand, if you buy above par and the bond is redeemed at par, then you have capital losses which will reduce, your overall return.

So that hopefully is quite intuitive.

And now we can extend that observation and sort of just, you know, come back to this inverse price yield relationship that I'm sure you have heard about.

And the idea here is that there is a close link between bond prices and yields as we've just sort of established.

And if we just exclude this reinvestment return for a moment to reduce the complexity then we can derive that relationship very, very easily. Because in case of a fixed coupon bond, I said one more time, the coupon is set at bond issuance and then usually doesn't change over the life of the bond, but the bond trades in the secondary market and its price changes driven by supply and demand dynamics, right? And as the yield of the bond now as we've heard, does not only reflect the coupon payments, but also these capital gains and losses, a change in bond price will change the yield to maturity.

Now, of course, not for investors that have already bought that bond, but for those who are going to buy the bond at the new price, and if we just, you know, based on this, we can derive this very simple rule for the relationship between bond prices and yields, and that is when bond prices increase, the yield of bonds fall and vice versa.

Why is that the case? Well, simply, if the price of a bond rises, that means for any investor that now buys it at a higher price, they are paying a higher price for the same cash flow that they are gonna receive in the future, which means their return will be lower than for someone that had paid and lower price for that.

And it's hopefully quite intuitive that the further away from par the price moves, the further away from the coupon, the yield will actually move.

So if a bond trade significantly below par, then the yield yield to maturity will be, be significantly higher than the coupon.

And if it trades significantly above par, then the opposite applies.

Great. One more thing.

That's really just adding one more detail to our understanding of bond prices, okay? So far, um, we've only spoken about the bond price in reality there are two bond prices and we have to understand the difference between them.

And there's a dirty price and there's also something called the clean price.

Now, the dirty price is the price we have looked at so far, it's nothing else but the sum of the present values of all the bonds cash flows when we're using the yield to maturity as discount rate, right? And that represents the current value of the bond and is basically the price that has to be paid by investors to buy the bond, right? But then there's also the clean price and the clean price.

And you can see this here on the, on the slide is something that we can derive from the dirty price by subtracting something called accrued interest.

And that raises a couple of questions.

First one, of course, what exactly is accrued interest? Now this is basically interest that has accumulated on the bond since the last interest payment date, all the way up to the purchase date of the bond.

And it represents, in other words the amount of interest that the previous bond holder is owed for the period they held the bond.

But that has not been paid as coupons are typically paid, you know, in regular intervals, like for example, every six months, every year.

And whenever the bond is sold between those coupon payment dates, the seller of the bond is of course entitled to this interest that accrue accrued during their ownership period, and that's accrued interest.

Next question then. Which price is quoted in the market? Is it the dirty price or is it the clean price in general? What's quoted in the markets is the clean price.

So the price, excluding the crude interest, which means if you pick up the phone and you call a bond market maker and you ask for the bond price, you get quoted the clean price, you negotiate the clean price, you agree on the clean price, but that's not necessarily the price you're gonna pay because if there is accrued interest, then what you have to do is you have to pay the clean price plus ac crude interest, i.e. you have to pay the dirty price, which then raises the question of why do we quote not directly the dirty price? Why do we do this? Detour, if you wish, negotiate on a clean price basis, then add the accrued interest and then pay. In general this practice, I believe is rooted in the clarity and, and also simplicity for market practitioners if you wish, because when you quote clean prices, it's actually I think, a little bit clearer and easy to understand what exactly bond investors are paying for. Because the clean price really purely reflects the market perception of the bond's value.

And it considers things like credit risk, interest rate, risk, time to maturity, et cetera.

Accrued interest on the other hand, is simply a function of time and the bond's coupon rate.

And we know that tomorrow's accrued interest, if the coupon is not zero, will be higher than today's, right? There's nothing risky about it, that's what's going to happen.

So that means that the dirty price of a bond changes every day as interest accrues.

And this then leads to daily fluctuations in the dirty bond price that do not necessarily reflect changes in the market's valuation of the bond, right? And when we use clean prices, then we're focusing on the bonds value without this noise of daily variations that are caused simply by accrued interest.

So here, quick example.

Let's say we have a five year bond pays an annual coupon of 2.4%.

You remember that bond of course, and trades at a yield of 2.43%.

We know that dirty price and clean price in this case is the same.

Why? Because there's exactly five years to maturity.

So either this bond has been issued today, or it's, uh, just had a coupon payment, so there is no accrued interest on that bond, okay? Price, dirty price equals clean price.

Now, um, one day goes by and we're repricing this bond using the same yield to maturity of 2.43%.

The difference is all cash flows are now one day closer, um, to, you know, to the present.

That means we have the same cash flows we're discounting with the same yield, but to the power not of one, but now 0.9973.

Unsurprisingly, the sum of the present values now has increased slightly because simply, you know, if you just kind of like to look at the formula, we're now dividing by a small number.

Now that means a dirty price of the bond has gone up. And we know why this is the case, because one day interest has accrued, right? Yesterday, there was no accrued interest.

But now one day later, there's one day of accrued interest in the dirty price, and we can calculate that, right? Because we know the coupon of the bond is 2.4.

We can basically say, okay, let's assume here actual actual, so we're basically saying 2.4% times one over, I don't know, 365 in normal cases, yeah, I know this is a leap year, but generally not.

And so this gives us then the 0.0066% that's accrued interest here for one day.

And if we want to calculate the clean price, now all we have to do is take the dirty price, subtract the accrued interest, and the clean price is virtually unchanged to where it was yesterday.

And so this is here basically where you see the dirty price has changed because time has gone by the clean price because the yield has not changed, has stayed virtually the same and so I'm assuming a constant yield to maturity of the bond's life.

Then this slide basically shows the development of clean and dirty prices, uh, how they would compare over time.

And the clean price will practically remain unchanged here.

Not quiet, but you know, we're keeping things simple here.

But the, um, dirty price will follow a very different pattern.

And that's a sous pattern, right? So we're starting here today with zero crude interest, then time progresses and the dirty price starts to increase as interest accrues.

And that basically is yeah, the accumulation of interest and that continues linearly, uh, until the next coupon date. And then the coupon is paid and the dirty price drops back to the clean price level because there's no accrued interest.

And then the whole thing repeats.

And this is repeated, as I said, until the bond's maturity date when the bond has completely been repaid.

That's it. Thank you very much for your participation, ladies and gentlemen. Hope you found it beneficial.

Um, please remember to fill out this feedback forms and hopefully I'll see you in well, or, um, be with you again in two weeks from now when we're talking about durations, interest rate sensitivities and things of that nature. Thank you very much. Have a great rest of your day.

Bye for now.