Hi. Good morning, good afternoon, good evening, everyone, and, uh, very well welcome to this, uh, first Felix live market session of the year.

Uh, and today's session is going to be about bond market fundamentals.

My name is, uh, Thomas Carls, and I have the honor to guide you through, uh, this introductory session today. And it's always great to start with something fixed income related, because that means I'm, uh, back at my home turf, um, as this is, um, you know, especially rates trading where I spend, um, most of my practitioner's life.

Um, so hopefully some of the passion that I bring to the subject will, uh, be, uh, coming through, uh, today.

But what exactly, uh, are we going to talk about? Well, we're gonna start with an overall introduction to bonds.

And while we're talking about those instruments, we're gonna clarify some of the key terminology.

Uh, then we're gonna have a look at the different coupon types, while not all of them, obviously, but at least some of the, uh, key types that everybody should be familiar with.

Uh, and then we're gonna take an intuitive look at, uh, bond pricing, which will then also will lead us nicely into the discussion, uh, around the yield to maturity and this, uh, famous inverse price yield, uh, relationship that you find in a fixed coupon box.

Without further ado, let's go into, uh, the, uh, topic of today. And as I said, we're gonna start, we with an introduction to bonds.

We're gonna start with a bonds overview.

And here I'm probably not telling you anything new when I say that bonds typically are debt instruments.

Yes, there are a few hybrid, uh, instruments out there, but the vast majority of, uh, the bonds are this classic debt instruments, meaning we're talking about borrowed money here.

That is money that has to be repaid at some, uh, point, which is typically in bond world, a fixed maturity, and between obviously the point in time where the issue of the bond receives the money.

And once they repay it to investors, they usually have to be regular interest rate payments.

That does in itself sound very much like, uh, any debt instrument.

So what's the different then between, for example, bonds and loans? And I think the key difference here is that bonds have been designed with tradability in mind, right? If you think about a loan that's usually a bilateral contract between the borrower and, uh, the lender.

And as such, it's not impossible to sell a loan from one lender to another one, um, as a secondary market transaction, but obviously because transparency isn't necessarily there.

And, um, and there's this bilateral contract, um, originally been signed by two counterparties. Here it is somewhat a little bit more complex.

And so if you think about the simplicity of, uh, selling, um, a debt instrument before it reaches its maturity date, I would say in most cases, bonds, um, because they have been designed as securities that are transferable, um, or relatively easily transferable, um, the liquidity in secondary market transactions should be higher, um, for bonds than it is in loans.

That doesn't mean loans are impossible to trade, we think about securitization, for example, but also loan, uh, trading.

Uh, and it also doesn't mean that all bonds are liquid, right? There's differences. There are some fairly illiquid bond issuance out there.

So that means, yes, theoretically you can sell them on the secondary market, but once you try to sell a larger size over a short period of time, you might actually find some limitations in there.

But overall, generally speaking, uh, bonds should be easier.

Tradable have higher liquidity than most loans out there.

Okay. Then, uh, we talked about, uh, some of those terms here already.

Just let's go through them one by one.

Um, and we probably should use an example for that, which I'm basically making up here right now.

So, um, don't necessarily look for this bond on Bloomberg or, or similar, but let's say somebody, um, has offered us, uh, to buy a hundred million, um, of the, uh, 2036 3.25% US Treasury, um, at a price of 98.5%.

So what does that mean? Well, first of all, one thing that's relatively clear here that stands out is who the issuer is.

That is the US Treasury.

That's basically the legal entity, if you wish, that is borrowing the money, right? That is ultimately very important for any, uh, bond investor, because you always wanna know who you exactly you're lending your money to, because that determines, uh, to a great deal the credit risk that you are taking.

Now, in this case, we're talking about, uh, government bonds, where we're thinking about the sort of, uh, at least, uh, default risk free in domestic currency, uh, debt instruments here, credit, we leave this, uh, for a different session.

But of course, if this was a corporate bond, uh, then you need to obviously be extremely careful because in the modern world and global, um, you know, operating, uh, corporate, we'll have a very complex legal entity structure.

So even if you recognize the name on the bond, you need to check very, very carefully, which legal entity specifically am I lending my money to, and what's the relationship between this legal entity and the, uh, holding and the operating company and all those sort of things will become very, very important.

But as I said, that's for another day maturity date.

That's simply the day on which the bond will be redeemed.

Now, we haven't been given the explicit date here.

Uh, all we know is that this bond will mature in 2036.

And although there's obviously a lot of government bonds outstanding, there will probably just be one or two, uh, benchmark bonds that mature in 2036.

So we would, as, you know, informed market practitioners probably know which bonds specifically, um, is being talked about.

Also, the coupon helps us to identify, but you know, where it comes to worst, we just check which is the exact date, and that would probably be something like the 15th of August or something, uh, like that.

So that's the maturity, uh, information we have received.

Coupon is 3.25 in this case. And what is the coupon? The coupon is just simply the interest rate that is paid in, um, regular intervals to the, uh, bond holder that's given as a per annum rate.

So if you buy this bond, you get 3.25% of the notional amount, and that's gonna be explained next, uh, per year.

But because US treasuries pay their interest rate semi-annually, that means you get roughly half, no, in this case, you get half of it e every, um, six months.

Why is it called a coupon that's historically driven? Because when bonds were still existing in physical format, it was one piece of paper basically stating that the issuer promises to repay a hundred million to the is, uh, to the, um, hold of this piece of paper on the maturity date.

And then there were little pieces of paper attached to this that you could add coupon debts, cut off, go to the bank, give it to the bank, and get your interest rate payment. That was just basically in exchange of a piece of paper called the coupon, um, for the money.

Nowadays, uh, this no longer is really relevant.

You obviously, this is all, uh, taken care of electronically nowadays, but, um, you know, the term still, um, is, is used, um, notional amount, face value, par value, all effectively, meaning more or less the same here.

And what we're talking about when we're using that, uh, one of those, um, terms is that's the amount that actually will be repaid to the bond holder at maturity.

This will be our notional amount of face value, uh, in this particular example.

So meaning in 2036, whenever this exact maturity date happens, if we're still holding that bond, the US treasury will repay us a hundred million dollars.

Um, and then, uh, you know, the debt has been repaid.

And now all important is obviously the price that we pay for any financial instrument.

And in bond world, the price is usually expressed as a percentage of face value.

On the trading screens, you don't necessarily see the percentage sign, but what it basically says, if you see 98.5 as a bond price, that means per hundred notional, you would pay $98 and 50 cents, and that's practically the definition of percent, right, per hundred.

So, um, what the bond price of 98.5 now effectively means is that if we were to buy a hundred million notional in that bond today, we wouldn't have to pay a hundred million, we would have to pay 98.5% of that, meaning 98 and a half million we need to put on the table today.

In exchange for that, we get, uh, a bond that will re or that will pay us 3.25% interest on a hundred million, um, on a per annum basis, semi-annually.

And then in a bit more than a decade from now, IE at that maturity date in 2036, we will receive a hundred million of, um, uh, well a hundred million from the us uh, treasury.

So this bond price, right? And we're gonna talk obviously, uh, about, um, the concept of this price and where it comes from, et cetera.

But, uh, you know, generally not gonna tell you anything. New prices are driven by supply and demand, right? So this bond price is just the, um, point at which at that point in time, the demand and supply for this bond are balanced.

That seems to be the fair market price, uh, at the time.

Then, if people decide to buy this bond, this price will go up.

And if people decide to sell the bond, this price will obviously go down.

So the bond price, per se, uh, most often is the, uh, result of supply and demand.

Okay, so, um, before we go further into this, let's, uh, talk a little bit more about some of the bond characteristics.

Uh, and the first thing we want to, uh, briefly touch upon is, uh, how are bonds generally redeemed? And I already said that the vast majority of bonds has a fixed maturity, right? That means we know the day on which the money is gonna be repaid, and we're just putting aside all sorts of credit risk, and we are just ruling out any sort of, um, restructuring or anything like that.

So we're just talking about the, uh, credit risk free world here.

So, um, meaning if we bought that treasury bond from the previous slide, uh, we know when we're gonna get repaid.

And, um, how exactly is the bond then repaid? Usually, and again, this applies to the vast majority of bonds, uh, they follow a bullet repayment structure, meaning they are repaid with one single payment at the maturity date.

So no interim amortization, uh, or anything down payment or something that if it's a 10 year bond, 10% is repay per year or anything like that.

The vast majority of bonds pay everything back in one goal at maturity.

However, there are of course exceptions to all of these points that are made, right? There's, for example, callable bonds where the issuer has a right to, uh, decide to repay the bond earlier than the original stated maturity date. Then there's some sort of flexibility or optionality around the maturity date.

There are amortizing bonds that have a particular down payment structure, uh, and so on and so forth.

If you think about securitization, for example, mbss, right? They, there will be some sort of prepayment, uh, that's not even known when you invest in this bond.

So, you know, different, uh, structures do exist, but if we are thinking about the classic bond, the classic fixed coupon bond that is, uh, so well known in fixed income, uh, then these things, uh, generally halt.

Um, next thing then, um, is to think about in the classic debt, um, or bond, uh, world, what are the, uh, coupon structures that exist? And of course, there's a whole range we can think about.

You know, obviously the, the more, um, vanilla types, then there's obviously a lot of structured products out there where the coupon is dependent on performance on all sorts of assets.

But if we're just thinking again on the classic debt instruments, then there's usually three main coupon types that one should be familiar with in the world of government bonds. There's also a forced one that's inflation linked, um, sort of, uh, bond the, the, the tips if you wish, uh, in the US for example, but we'll leave that out here. That's for another session as well, because they are, you know, technically quite interesting, uh, to look at, but probably goes a little bit beyond what we're trying to do here, uh, today as an introduction.

So therefore, we're limiting this to a, uh, three types, the fixed coupon bonds, the zero coupon bond, and the floating rate note starting with fixed coupon bonds, simply because once again, that's the vast majority of bonds, uh, that are out there.

And it's relatively, um, intuitive to understand is, uh, the, it's, and it's sort of in the name, the, uh, coupon of the bond is fixed.

IE it doesn't change over the life of, uh, the bond.

And it's determined when the bond is issued and then basically is paid in regular intervals until the bond is being, um, repaid.

How do we determine it? Usually it reflects the general level of interest rates when the bond is issued and it's adjusted by some sort of like credit spread dependent. Then on the issuer example of such a structure, um, here, a 10 year bond is paying a 3% fixed coupon that's paid annually.

And then it's very simple to map out the cash flow structure.

That means, as an investor in such an instrument, you will get 3% on your notional every year.

So after one year, you get 3%, after two, you get three, and so on and so forth.

And after 10 years, you get 103 simply because you get the last coupon of 3% and you get your notional amount back.

Bonds are repaid at pie, as we said.

Um, therefore 103%.

So this is a relatively straightforward instrument.

Um, you, if we push credit risk aside, uh, can sort of argue that this is a predictable stream of cash flows, you know how much you're gonna get paid at which point in time. Therefore, it's relatively straightforward to kind of get an idea about the return of such an instrument.

Um, there is one caveat though, and that is that, um, while you have a very good sense about what your return most likely would be, um, you don't know with absolute certainty what your finite return is going to be until, uh, you know, you have received the final coupon.

That sometimes is a little bit of a surprise for people because they say, well, it says fixed income, it's a fixed coupon instrument.

How can the return not be known, um, when we're investing in such an instrument? And, uh, I think this is maybe a little bit also, um, too much of a technical point, but what I would like to, uh, point out is that, yes, of course, you know which coupons you're gonna receive, but there is an additional component of your return that you do not necessarily know is absolute certainty at this point in time.

So what am I talking about? I'm talking about the so-called reinvestment of coupons and sort of, you got to think about, um, this in, in, in context of a practical example.

And we're not using a large portfolio manager here, but let's use, um, a single investor here that is buying a 10 year bond just because maybe they have a 10 year retirement plan, right? So they think in 10 years, I want to retire, I have saved up some money, um, and I do not want to, uh, put it in stocks right now because, you know, I don't wanna lose it or see a, a, a draw down If there was a, you know, ill-timed correction in stock markets, so I'm gonna play it safe and I'm gonna put it into, um, fixed coupon bonds, then I know exactly what I'm gonna get.

And the reason why they chose a 10 year instrument is because they want to retire in 10 years.

That's the story, right? Simple.

Uh, and, uh, now, um, this, um, retiree has invested in this 10 years, 3%, um, fixed coupon bond.

And let's say, you know, we're just, for the sake of round numbers, they have invested a hundred million, um, which might be unrealistic for most of us.

But anyway, um, you know, a hundred million in that bond.

And, uh, now after one year, they get the first 3% coupon, which means 3 million.

Of course, this is a nice, uh, problem to have in a way.

But, um, you know, the, the potential retiree here is now in a situation where they have 3 million, um, of their money already, but what they wanted is they wanted all this money to come back to them at the 10 year point.

So in a way, they don't want the money at that point where they receive it.

So at this point, you know, what they probably are looking to do is take this money that they have received and then put it back to work.

IE reinvest this first coupon for now a nine year period, simply because now the retirement is nine years away.

So what this means is, in a year's time, they are looking to re in reinvest 3 million or 3% of the notional at the then prevailing nine year rate in two years, at eight year rates in three years, at seven year rates, et cetera, et cetera.

And there's simply no way of knowing with absolute certainty where one year rates in, uh, sorry, nine year rates in one year are going to be where eight year rates are going to be in two years, and so on and so forth.

So there is a question mark about the achievable reinvestment rates.

Of course, when rates are going up, then the return is gonna be influenced in the right way, right? Because our reinvestment rates are higher and we get more money than we thought.

If rates will go down, then, uh, the opposite applies, right? But what I'm pointing out here is just there's no hundred percent certainty about the return of such an instrument, which sometimes is a bit of a surprise because they pay a fixed coupon.

And now then the question of course is, is there no way to prevent this reinvestment uncertainty to be there? And of course, if you wanted to have absolute certainty, if you wanted to know precisely to the cent how much money you will have in a decade from now without any sort of, um, you know, uh, uh, without making any sort of assumptions about rates, et cetera, um, is that possible at all? And, uh, the answer to that question is yes, it is.

You will just have to move away from the standard fixed coupon bond to something we call a zero coupon bond and a zero coupon bond in a way, is nothing else, then a specific type or special type of fixed coupon bond in the sense that the coupon is fixed, but it's fixed at zero.

And that of course, means that if you, um, were receiving a coupon of 0% and nothing else happens, you would basically accept, um, you know, a return of 0%, which of course is not really the case.

Um, unless of course the interest rate level is at zero, which has been the case not too long ago.

But, uh, usually, uh, the way that these zero coupon bonds generate a return is just by the fact that, that they are issued at what we call it discounted price.

So as this example here on the screen shows, you would buy the 10 year zero coupon bond issued at a price of 74.51%, which then is repaid at a price of a hundred percent or add par add maturity 10 years later.

There's no interim coupon payments, no money comes back to you before the end of your investment horizon.

There's no reinvestment.

You have to do IE no uncertainty, but also no opportunity, right? That's something to be, um, aware of.

You know, you put $74.51 million on the table today and a decade later you get a hundred million dollars back, and then you have the present value, you have the future value, and then you can simply work out what's the implied annual rate of return.

And that's a guaranteed return right there, assuming no default of the issuer, no thing, um, in between.

Uh, the one thing, um, to note there then, and you can see that this is potentially an instrument, um, that is attractive for, for some, uh, investors that are really looking for, um, no uncertainty.

Um, the issue in reality is that despite the fact that these type of coupon bonds might be, or zero coupon bonds might be particularly attractive in certain environments, um, they don't necessarily exist, um, straight away because a lot of, uh, typical bond issuers don't directly issue zero coupon bonds.

But you could easily see that there was demand for zero coupon bonds, for example, issued by the US Treasury.

And whenever this is a case, when there is a potential demand for a financial instrument that in such a form doesn't exist, uh, then obviously there is a mismatch.

And that's where the financial services industry comes in and creates solutions, um, to meet this demand.

And, um, that had led to, um, you know, the, the, the, the principle of, of bond stripping or the strips is sec, uh, separately tradable interest and principle securities.

And basically what happens here is that the US treasury issues, regular fixed coupon bonds, and then these, those can be stripped.

IE split up into the different coupons and the redemption payment and all those single cash flows can then be traded separately, which basically means we have turned a fixed coupon bond into a basket of zero coupon bonds.

So, um, that's how we have solved that, um, problem.

Now anyway, we have coupon bonds, we have zero coupon bonds, and arguably, um, both bonds, um, have advantages and disadvantages, right in, in the sense that one has reinvestment risk, but the other, um, uh, doesn't then of course, but obviously, you know, the risk is not always, um, gonna cost us, but it might actually bear an opportunity as well.

But what I hopefully intuitive is that both these, um, type of bonds, the normal fixed coupon bond and the zero coupon bond would be investments that, you know, investors would probably find interesting, uh, when they, they have a particular expectation with regards to, uh, changes of interest rates going forward.

Because you can see that if you think that over the next 10 years, interest rates in general will go up, um, that you didn't necessarily want to invest your money in a 10 year fixed coupon bond today, because why would you lock in the low level of interest rates as you think it is, um, today for the next decade? Um, on the other hand, if you were believing that interest rates over the next decade would only go down, then investing in a fixed coupon bond where zero coupon one, um, but locking in the currently high level of interest rates for the next decade, that might be a particularly attractive, uh, proposition, right? So that's basically the point.

Um, you would, um, I have to think about your expectation, and then depending on what your expectation on rates is, uh, fixed coupon bonds might actually be a good choice, or you might look for something else.

So coming back to the example where we said, our expectation is that interest rates are about to go up in quite meaningful ways over the next, uh, couple of years, then, you know, as as said, buying a 10 year fixed coupon bond might not be the best choice, right? Because we think rates will go up.

We want to, um, not lock our money in at a 10 year fixed rate right now.

And then there's a couple of strategies, um, to get around this.

If we want to invest in bonds, for example, you could buy just relatively short maturity, fixed coupon bonds, right? And then if rates go up, yes, um, you might miss out, um, opportunity wise over the first couple of months, but after, let's say half a year you get your money back, you just buy another bond other than higher level of rates. That's one way.

Another way is to not buy a fixed coupon bond, but invest in something called a floating rate note.

And what a floating rate note is, is basically a bond where the coupon of the bond will be reset in the future to reflect current level of interest rates.

And that is achieved by, uh, when the bond is issued.

Not setting a fixed coupon, but instead determining a coupon formula.

Here's an example, actually two examples.

One in the, um, IBOR framework, right into bank offered rates that, uh, were pretty much the standard all around the world until a couple of years ago when we had the, uh, IBO transition, but it's still, um, being used, right? U io, for example, is still pretty heavily used in the Eurozone, uh, whereas other currencies have predominantly moved on to those risk free rates, like SR et cetera.

So I have included examples of both worlds, but you know, they are just having small differences in, in how they work mechanically. So I'm just gonna talk you through the six months, um, u IOR example here.

So how do we, how does that work with the coupon formula, et cetera? So basically here, as you can see, we have a 10 year floating rate node, and the coupon is determined to be six months UIO plus 0.5% or plus 50 basis points as this will be called.

And the interest rate will be paid semi-annually, that means two times a year, so six months, your IOR is now of course, a rate that is reset every day.

So how exactly do we get from this coupon formula then to the actual interest rate payment? Well, that is a relatively straightforward process.

So when the bond is issued, uh, then obviously we will take the, uh, six months UIO fixing of that particular, uh, day.

And assuming this was 3.96%, we will just have, have to add those, uh, 50 basis points on top. That gives us a rate of 4.46%, and that's gonna be then the coupon rate for the first six months period.

And to get from that rate to the actual payment, we multiply with approximately 0.5, simply because the six months period is roughly half a year. We simplify it here a little bit, but, uh, you get the idea.

Then, um, six months later, this is the first, uh, coupon that will be paid.

Um, and now we're gonna take the six months UIO at, um, the six months point, and assuming this is now, uh, 5%, that means the second coupon will not be 4.46, but instead 5.5% because your arrival plus 0.5%.

So now for the second six months period, it will be 5.5% times 0.5, and you can see your rib or has gone up over the, uh, six months period.

And as a result, the coupon that the investor will receive for the second six months period will increase as well.

So if your assumption was that interest rates in general, more particularly short term interest rates will go up over the next, uh, couple of years, then maybe investing in a floating rate node will be, uh, a good decision.

And just very quickly on the, um, sulfur, uh, format, which obviously is now being used more and more, not just sulfur, but also in pounds.

Uh, we have Sonia, we have toner, and you know, we have As and euros, et cetera.

So this is sort of, um, you know, getting ever more, uh, the norm.

The point to be aware of is that all those rates, sulfur, toner, esta, and so on, are overnight rates, meaning they don't apply for six months term per uh, period.

But for a one day period, IE from today to the next business day, that means these rates will only ever apply for a day.

And so technically speaking, the interest rate payment on such an instrument should be happening on a daily basis, which of course is gonna be way too, uh, cumbersome service, way too many, too small payments that we have to make and investors have to check if they receive the money or not, and so on and so forth.

So to be more efficient, this, uh, general market convention is to increase, um, or sorry, to decrease the frequency of those payments. And what we very often see, for example, are quarterly, uh, payments. So while the interest rate is reset every day, the coupon is then paid, um, slightly delayed every three months, and then are some adjustments, uh, that are happening to the payment.

But that, again, is for another day when we're looking at, uh, those mechanics in a little bit more detail.

Okay? Um, what I wanna spend the second half of, uh, this session on is really, um, the pricing, um, of a bond. And here we're going to focus on fixed, uh, coupon bonds.

And, um, the interesting thing about bond pricing is that, you know, very often, um, you, you have sessions like this where we talk about this is how you price a bond and, and, and, and, and so on and so forth.

But the interesting thing about this is that a bond doesn't really need to be priced all that often, right? Because as we said earlier, once a bond has been issued and then people are starting to buy and sell it, then their activity is driving the price.

It's not really when you, for example, work as a market maker in, in bonds and you know, you have a, you have a, uh, a customer or a client calling you asking for a particular price, you don't necessarily start pricing, uh, that bond because you know where the market price is.

What you will do is you will just kind of think about your access and how you position and how aggressive you wanna be, and how tight your bid office should be, and all this sort of, uh, questions. So there's obviously a lot of work that goes into this, but it's not gonna be pricing the bond, IE discounting future cash flows, right? However, conceptually it's very important to understand this.

And also there is one point in the life of every bond where it needs to be priced, where we need to come up with, uh, a price.

And that is when the bond is issued, because then there is per definition, no market price for this particular instrument.

So we have to come up with the first price, the issuance price.

So how can we then, um, achieve this conceptually? Well, um, bonds are securities, right? And that means, um, the fair price of a bond, um, can be seen as the sum of the present values of all those future cash flows that a bond generates.

And in case of a fixed coupon bond, the good thing is that those cash flows are very well known.

Again, credit risk aside, right? Because we know when the coupon is gonna be paid, we know how big the coupon payments are going to be, so we really know the future cash flows because we also know the redemption.

We know when the redemption should take place.

So we can relatively straightforward the, um, map out the future cash flows of this bond.

Then all we have to do is discount all those future payments back to today and sum up the present values, and we have the bond price.

And the tricky part in all this is, of course, to find the appropriate discount rate.

Now, if you look at this, uh, slide here, um, you see, see that's for the coupon fv is the face value, uh, and is the number of maturity.

So all of that relatively straightforward and r yeah, okay, it has to do something with rate.

And then you read that this has required rate of return that sounds very theoretical and very, um, you know, academic almost, right? And, and, and so obviously now, um, that that was not meant to be, uh, disrespectful, but it sort of sounds like, okay, there's not really a very practical approach.

And that's something I thought for years, right? Um, but actually the more I've been sort of teaching those sessions, the more I realized it's actually exactly the right way of looking at it.

If you think about when we need to price the bond, because as I said, that is, uh, usually when the bond is issued.

So why don't we have a look at a, uh, bond issued example that then showcases why calling this a required rate of return is actually exactly right.

I'm skipping the next slide. There's just some, uh, little bit of, uh, formula here. We might come back to this, but that's a slide that I actually want to, uh, talk about.

So here, what we're showing is the general, um, idea and process of a government issuing a bond in a domestic currency.

And we're assuming here a, uh, developed, um, government.

So this, uh, usually then obviously a com a country with a fairly frequent bondages of, of, of regular sizes.

Um, and, um, so in, in those cases, bonds often are issued via public auction.

So we're not using underwriters, um, you know, per se.

And so what we have is, uh, our own, well, those countries have their own debt management offices or whatever you wanna call them, have their own, um, auction platforms.

And so what then happens is that we, or that they inform, um, potential investors that they are planning to raise a particular amount of money at a particular point of time for a particular maturity.

So here the announcement was, uh, that a particular government is looking to raise 10 billion of their currency, um, for a five year period.

And then, uh, there are details as to when the auction will start and stop and when the allocation will happen, et cetera, et cetera.

Uh, and then when the time comes and the auction, uh, starts, uh, then obviously the different investors can, uh, go onto these platforms and, and, and place their bits, right? Um, and what they bid is an interest rate.

It's basically a return.

And so what the investors are saying here is, for me to find this an attractive investment that makes me want to invest in it, I require a rate of return of X, right? And so, um, that's basically as to why the required return, um, you know, thing earlier makes absolute sense.

So here's how it goes, then investors will then tell us how much they're willing to buy at, you know, uh, the specific, um, level of yield.

And so then we can just download all the data and, and, uh, you know, short those bits and, and aggregate and, and, and sort of, um, you know, more things like that.

And we get with, uh, we get to, you know, a table, um, like maybe the one on the slide here where you say, okay, add a yield or take this as return, um, rate is, uh, of 4.522, uh, we have 3.4 billion demand for the bonds.

So there's investors saying, yeah, for that return, I'm buying 3.4 billion.

And, um, that's of course not gonna be enough simply because the, uh, government is looking for 10 billion, but we now go to the next higher yield level.

And there we see that at a yield of 4.523.

So for a return of five point, uh, sorry, 4.523, there is an additional demand of 4.9 billion, uh, to buy this bond. And now we can do the following. Someone that's willing to buy a bond at 4.522 is of course gonna be happy to buy the same bond at a higher return as well.

So what we can say is, if we have 3.4 billion demand at the, for, uh, 52 2, and we have 4.9 uh, demand on the 52 3, then we have 8.3 IE, the sum of both in total at that yield level, still not gonna be enough because they wanna raise 10 billion.

So we have to go up to the next level, and that's a 4.524%.

There's an extra two and a half billion of demand that brings us to, in total aggregated demand of 10.8 billion.

And so that is enough to clear the whole 10 billion that we're planning to do, and this is then the highest accepted yield.

And the way these options usually work is, regardless of the yield you bid, I mean, if you have bid the highest accepted yield or lower, you will get the highest accepted, um, yield.

And so now we know that this bond is being issued at a yield of 4.524%.

And now, of course, coming back, uh, to, to this, because you now say, okay, so why do we have to price anything? Um, if we know the required rate of return, well, I mean, of course, you know, the investor could get this, uh, required rate of return in, in very different ways, right? Remember, we've already said there's a price of the bond, there's a coupon, et cetera.

So of course, we could just put this 5.42, no, 4.524% as a coupon, and then the bond gets issued at par.

Uh, the investors will receive 4.524% every, you know, six months or a year or whatever.

Uh, and then the bond gets repaid at par.

That would give investors a return.

Uh, they've asked for, but a coupon of 4.524, that's not a very memorable number.

So what happens in reality, we take coupons that are slightly, um, more, uh, round.

So this case would, for example, be a coupon of, um, 4.5%.

But that of course, causes a, a challenge Now, because investors said, look, I want a return of 4.5 to 4%. If we're now just gonna pay them 4.5% coupon, that's obviously not enough. That's not what they ask for. They want a 4.524% return.

What is it that we can do? Well, of course, we can give them, um, some extra return, just not through coupon, but for example, through price, meaning we have to sell the bond at a price slightly below par, because if it comes out slightly below par, investors buy it at 99.8, let's say, then they get a hundred back at maturity, that's a return component right there. Money has grown from 99.8 to a hundred, and then in addition to the four point a half percent coupon, that might give a 4.524% return. And that's exactly where the pricing comes in, because now what we have here, sorry, going into the wrong direction, see, it's Friday, uh, late afternoon here.

Um, so here, now what we know, we know the coupon is 4.5, uh, and we know the required rate of return.

So we have this, just kind of put this here, 4.5%. We have this as 4.5 to 4%, and now we can do the mass here because we know all the values and simply discount future cash flows back to today. And that gives us the price at which the bond needs to be issued.

So that with a coupon of 4.5%, uh, investors receive, um, 4.5 to 4% return, and that's gonna be then the issuance price.

So calling it required rate of return now, um, makes actually perfect sentence, very, very practical, right? So, um, that's, as I said, the the way to price bond here is, uh, just as a reminder because, um, you know, the math is slightly different dependent on the coupon payment frequency.

So the top example here is when the coupon is paid annually.

Uh, the bottom example is when the coupons are paid, uh, semi-annually, and if it was quarterly, there would be another adjustment, et cetera.

It's not really mission critical for today's session.

I just wanted to put it out there for com completeness.

But, uh, be aware the math is slightly different depending on the coupon frequency.

Again, if you have any particular interest in discussing this in more detail, reach out, uh, and let's have a chat about this.

Um, relatively easy to show all of this.

Um, anyway, so what I then really, uh, want to, um, do to wrap this up is bring this concept, um, now that we've said, okay, bond is issued, uh, and, and use it to kind of, or extend it a little bit to, you know, how we actually communicate about bonds.

And that's often you hear bond yields being, um, you know, talked about not so much bond prices, right? So, you know, when you read, um, you know, the, the macro market commentary hardly ever do they say, oh, the 10 year treasury dropped in price by X.

But we're saying, you know, yields have gone up by couple of basis points because that's really what investors, um, think, uh, about, right? It's a return. Um, you know, of course, you know, ultimately from a p and l point of view, the price is very relevant, but you know, if you are just observing the market, you care more about the returns or the, the, the, the yield of an instrument rather than the actual price.

And that's where the yield to maturity comes in. Because whenever you hear the word bond yields, um, you can relatively safely assume that people are talking about this yield to maturity or YTM.

Um, now what's that? Um, it's basically, um, you know, a, a quick and dirty, what you call it, um, decision making tool.

The problem is, um, you know, for bond investors that there's many, many different bonds, often by, uh, the same issuer and relatively similar times to maturity, um, and relatively similar coupons, but similar, not identical.

So it's actually, um, quite difficult to compare two bonds, um, directly with each other, because one might have a slightly higher coupon, but it also comes at a higher price.

So it's not quite clear, uh, which bond is now the better one to invest in. Because as I said, one thing that ultimately matters for investors is not so much the price.

It's not so much the coupon, but it's the return they're gonna generate, right? So if I invest my money in this instrument, how fast will my wealths grow? And if I invested in this other instrument, how fast will my wealths grow? And of course, we're, um, ignoring any sort of like tax differences here for simplicity.

But generally, you know, if none of it existed as an investor, you would just purely be interested in the return.

And the return is something that the bond doesn't really show on.

Its, uh, you know, in its perspectives or something, simply because it's driven by several components.

Some of them will change over time, like for example, the bond price.

Uh, and so, um, the yield to maturity is then really the attempt to estimate the return that an bond investor will, uh, make, uh, under a following, uh, under the following assumptions.

A buy the bond at its current market price, which of course is absolutely unproblematic, that's where you would buy the bond, right, at its current market price.

The second assumption is we're calculating this return, uh, based on this bond being held to maturity.

And while it's not guaranteed that investors will hold the bond to maturity, it's a fair enough assumption because, um, some of them do, and that's obviously just easy to caveat this, um, yield to maturity by saying, look, if you don't hold it to maturity, you don't know what the return's gonna be because you don't know at which price you're gonna sell it at, right? Um, and then last but not least, there's another assumption.

This brings us all the way back to reinvestment rate, uh, reinvestment risk that we talked about earlier, and we already made the point there to calculate the return of a bond investment to know exactly what the return is going to be a decade from now, not only do you have to know the coupon and the price, but you also need to know what the reinvestment rates of all the interim coupon payments are gonna be.

Now, we don't know all these rates, but what we can do, we can make assumptions.

And the inherent assumption in the yield to maturity concept simply is that all coupons that the bond investor will receive between buying the bond and maturity of the bond will be reinvested at the same level of interest rate.

And that is identical to the yield to maturity.

So it's a bit of an internal rate of return, um, approach here.

And, uh, that is of course, an assumption that in reality, often may not hold.

So, and this is a critical point then to realize about the yield to maturity.

This is a number that we generate based on a certain set of assumptions, but one of these assumptions, um, will most likely not hold.

That means the yield to maturity is not guaranteed return.

That under all circumstances, fixed income investors will generate.

But it's a approximation.

It's under the assumption that reinvestment rate is x, this is gonna be your return.

If the reinvestment rates are different, your return is gonna be different, it might be higher, it might be lower.

Now, the good news is that this reinvestment risk only in inver affects the coupon that is paid, right? So, and in today's environment where interest rates still, at least in comparison to historical standards are reasonably low, coupons are reasonably low.

So this reinvestment race is only affecting relatively small parts.

But don't be mistaken, this yield to maturity is an approximation.

It's a gauge of the return, despite the fact that this is called fixed income or fixed coupon or whichever, uh, way it's not a hundred percent known.

Um, of course there's ways to eliminate those risks, but, um, you know, if you just buy the bond and do nothing else, you have a pretty good idea about the return, but you don't know with absolute certainty to the second decimal point what your return is gonna look like.

And then, of course, how do we get to the yield to maturity in the first place? Well, we have remember, uh, seen the formula that, um, allows us to calculate the bond price based on the cash flows of the bond and a discount rate.

Uh, now when we are trading a bond in the secondary markets, what we actually have is the bond price, right? We see the bond price on screen.

We can make this say, you know, a statement about it, which price are you going to buy the bond? So we know this component here now, and we still know all the cash flows, the bond generates, and what we can do then obviously solve this equation, um, for r or, you know, the interest rate that is used for discounting. And that is then basically what the yield to maturity approach does.

There's no close form solution. You have to iterate, um, or to take an iterative approach.

Um, but that's obviously relatively, uh, simply done.

And that's basically where you get the yield to maturity then from, and that's, uh, when you see government bond yield curves, et cetera, uh, what you see there, it's not the coupons of those bonds, it's the yield to maturity.

Um, next thing then, and this is, uh, gonna be, um, the last two slides that, uh, I'm gonna share here with you today, is we want to understand, uh, the components of the yield.

And when I say yield, I mean yield to maturity, by the way, right? Just, uh, an abbreviation.

Um, and then we want to, um, understand that inverse price, yield relationship for fixed coupon, um, bonds.

So a lot of this we have already discussed just in slightly different context, but we should have learned by now that the return of a bond, the yield of a bond has three, uh, components.

One, and the most obvious one is the coupon, right? IE the regular, uh, amount of money that you receive as an investor, the regular, uh, amount of interest that is paid to you in form of those coupon payments.

Then there's a second, uh, source of return, and these are those reinvestment returns.

IE you get the first coupon after a year, you reinvest that money for nine years, that creates extra return, and that, of course, will contribute to your overall return, right? But as we said, that only affects a relatively small part of your, uh, total investment.

So what we are gonna do here for simplicity is we're just gonna push that component aside, right? So we're just gonna pretend it's not there. That's of course a simplification, but it's okay at this point.

Um, so we then have coupons being the most obvious part, and we have already said that there's another one, um, beyond the reinvestment returns, and that is the price of the bond that could lead to capital gains or capital losses.

Going back to our initial example where we bought this, uh, bond at 98.5.

So let's say a bond pays a 4% coupon, um, it was I think 3.25.

Uh, so let's make this consistent.

So it pays a 3.25% coupon, and you are able to buy it at 98.5%, and at, um, maturity, you will get a hundred back.

Then hopefully it's clear, uh, to see that your actual return in this bond investment will be higher than, uh, 3.25 IE yield term maturity should be larger than 3.25%.

Why? Well, because 3.525% is the coupons, and then you get a capital gain of 1.5, uh, percent here on your notional. So that's an extra one and a half million on a one, uh, hundred million investment that you get through the, you know, appreciation of, of price, right? You're paying 98.5 million, now you get a hundred million back, that's an extra one and a half million.

That is something that you, uh, should consider when calculating your return.

And, um, as much as you know, capital gain will contribute positively, uh, to your, uh, yield.

A capital loss will of course do the opposite.

And sometimes bonds trade above par, right? So this bond might trade at, uh, 101.5, right? Rather than, um, at, uh, 98.5 or at par.

And then this capital gain will turn into a capital loss because now, um, to get the a hundred million bond, you have to pay 101.5 million, and you will only in inver commas get a hundred million back at maturity.

So you have lost one and a half million that will take away some, uh, return from your, uh, coupon income.

So as a result, the yield term maturity in this case, uh, will be smaller than the coupon because you're losing some of the coupon income through the capital losses that you're experiencing by buying the bond above, um, par.

And now this then, uh, can be taken, um, to, you know, really, uh, come up with this, uh, price yield relationship that is, um, an inverse one.

And, uh, going back one more time to this, if the coupon is set when the bond is issued and then never, ever changes, then really the only component if we leave reinvestment returns aside, that influences the yield, is the price of the bond, because that determines the capital gains or capital losses that a new investor, um, would face, um, over time, right? So, um, when you, with that in mind, let's conceptually go through what we expect would happen to the yield of a bond when the price of a bond drops.

And that is relatively straightforward, right? If the bond, uh, price drops, that means that you will see, uh, or new investors will experience higher capital gains because they buy the bond at a lower price.

That means they get the same coupon, that means their return will be higher than the one of an investor who bought at a higher price.

So basically, prices down, yields up.

That's a relationship for fixed coupon bonds.

And just to showcase that this is not some, um, you know, nice, um, you know, thought out, but practically irrelevant concept, what you can see here is two lines.

One showing the, uh, price of a 10 year treasury bond, and the other one showing the yield of a 10 year, uh, treasury bond over a certain time period. And what we can observe here in the first, um, period of, uh, that, that we're looking at here, the price of the bond declined and, uh, in the sa at the same time the yield went up.

So in fact, those two values are in fact, extremely negative, or not extremely, but, um, very, uh, negatively correlated. And it makes perfect sense simply because as we said, if the price of the bond goes down, the next investor that buys a bond now at the lower price will have paid a lower price for the exact same cash flows that we have, uh, bought at a higher price. And so their return will naturally be higher.

Now, the important thing to remember in all of this is of course, if you buy the bond at par and then you hold it to maturity, um, you know, your yield won't change just because the bond prices is, is is falling or rising, right? Um, if you're still holding it to maturity, you will lock in this, uh, 4%.

Again, we're ignoring the reinvestment risk here.

Um, but, um, obviously you will probably experience some p and l pressure because, you know, if the bond price drops because you know, you bought a bond at a hundred, now the market value is 98, and your p and l, even if unrealized will, will be, uh, negative $2 million, right? So depending on what kind of investor you are, what your investment horizon is, et cetera, uh, these things might obviously be more or less important.

But, um, generally speaking, that's the important thing to take away.

This change in yield does not affect existing investors in the sense that they now get a different return.

Um, it just, uh, means that if you're buying this bond at its current price, you now get higher or lower returns than previously.

And then of course, uh, as the relationship is inverse, uh, what this also means when bond prices go up, the yields of these bonds, um, will, uh, come down.

And then obviously the next steps will be to think about, uh, all bonds changing, um, the yield for the same change in price, in the same, you know, at the same speed, or are there any differences? And that brings us to the concept of interest rate sensitivities, durations, Et cetera. But that ladies and gentlemen is also, uh, for another day. And with that, we're at the end of today's session.

I hope you found it beneficial.

Thank you so much for taking the time to attend.

Remember to tell us what we can do to improve, remember to tell us which other topics you would like to have, uh, covered.

Very interested to hearing from you.

If there's anything you wanna follow up, be in touch, look forward to hearing from you.

Um, but for now, I wish you a great rest of your Friday.

Have a wonderful weekend, and, uh, take good care of yourself, and I hope to have you back, uh, on our next session in a couple of weeks.

Until then, take good care of yourselves and bye-bye.