Okay, so let's get started.

Hi, good morning, good afternoon, good evening, wherever you are.

And welcome of course to this Felix Life refresher session on bonds.

And I should of course mention that this is actually part two, although it does say that on the on the slide title.

To be fair, part one was delivered two weeks ago, and the reason for this split is that we felt it was just a little bit too much content to squeeze it into one single, one hour session.

Hence the split.

Anyway, allow me to quickly introduce myself.

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

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

So, quickly run through the agenda of today.

As I said, it's another refresher on bonds, more specifically this time on interest rate, risk and sensitivities.

And in general, refresher session for me means it's gonna be relatively fast paced as we want to refresh your knowledge, on a couple of, um, items.

And today, for example, we're gonna start here with a look at the Macaulay duration and, or sorry, with an intuitive look at the drivers of interest rate sensitivity. Then we're gonna have a look at macaulay's duration and how it's actually calculated.

We will also cover the modified duration and the dollar value of a basis point, and then how they are applied in a single bond context, but also, in a portfolio context.

And then we also, towards the end, we'll have a high level look at interest rate sensitivity for floating rate notes.

But before we start, just a couple of general reminders.

You probably have received in the chat already the link to download the materials. If not, you can also access them through the resource link in Zoom, you find the materials to download on the Felix live website.

So please make sure you get your hands on those.

Also, you know, as a reminder, you can ask questions during the session, right? The one thing to remember though is that you have to use the q and a function, not the chat.

I don't monitor into the chat.

So please route all your questions through the q and a function.

And then last but not least, after the session has finished, you will be directed to a feedback form. And I would appreciate it very much if you could take a minute or two to fill those in because it is of course very important to know what you're thinking of these sessions.

Also, which other topics you might want to see covered in this format.

And then last but not least, this is also a great way to ask any potential follow up questions that you might have.

Anyway, without further ado, I suggest we get started.

And, um, first I'm gonna start with a general definition of interest rate risk, right? And this sort of connects directly to part, part one, which we did as I said two weeks ago, where we developed this intuitive understanding Of the inverse price yield relationship of fixed coupon bonds.

And we did look at the math at that time, but in general, let's quick really recap that.

For fixed coupon bonds, what we said was that because the coupon of the bond is fixed at the time that the bond is issued and then does not change over the life of the bond, the only main variable that is subject to change really is the bond price, right? And we're ignoring the reinvestment here, to reduce complexity. And also because we identified it's probably not the most significant driver in normal cases, right? So then when, for example, the demand for a specific bond increases and the clean price consequently increases because people are buying that bond, this means that investors that haven't bought that bond will now have to pay a higher price or more at spot to receive the same cash flows.

And this intuitively means that their return is gonna be lower than the return of an investor that bought the same bond with the same cashflow at a lower price.

Now, for investors that were already holding the bond before the increase this increase in bonds clean price does not only mean that they're getting a higher return than those that are buying the bond now, but there's actually a very tangible and quantifiable advantage.

So think about the following. If an investor bought the bond at a clean price of let's say a hundred percent so at par, and the same bond now trades at a clean price of 102, the investor can sell the bond for an immediate profit of 2%.

So a decline in yields and generally is advantageous for the holder of a plain vanilla fixed coupon bond, at least in a very, uh, general, uh, or generalized way right? Now, on the other hand, what this means is that an increase in yield, which means then a decline in bonds or prices will mean mark to market losses for those existing investors.

Okay? That's nothing new.

But now let's think about the risk and the definitions behind.

Now, the vast majority of bond investors approach the bond market, I would suggest from the long side. That means they're either long bond where they have no position at all. Most investors actually don't entertain outright short positions, and therefore it's very common to define interest rate risk from the perspective of those long investors.

So very often interest rate risk is understood by bond investors as the risk. And it says here on the slide that bond prices fall as yields rise.

And we're focusing on government bonds here so that again, there are no credit spreads involved.

And for that, or in that case, what it means is if yields rise, that increase in yield purely comes from an increase in the credit risk free rate.

Now of course, it's really useful to have this intuitive understanding about the relationship and it's definitely a great starting point, but I think it's also fair to say that investors re really need a lot more information in order to make educated investment decisions.

And I hope it makes intuitively sense that not all bonds will respond in the exact same way to the same change in interest rates.

So some bonds might be more risky than others, even if we ignore credit risk here for simplicity.

So that means that even in the same absolute change in yield, so let's say we have two bonds, both bond yields go up by the exact same amount.

So let's say one percentage point, that doesn't necessarily mean that the bond prices will react in the exact same way.

And what that then means in turn is that investors need tools to make these interest rate risks or the yes, of different bonds directly comparable.

And this is what leads us to the concept of interest rate sensitivity.

Now, the general concept of interest rate sensitivity, also known as duration, or more precisely as we're focusing on interest rates here, we should say rates duration.

There's nothing else in the description of the magnitude of bond price move in case of change in risk rates. So if a bond has a higher sensitivity, what that means is its price will change more dramatically, for the same absolute change in yields than a different or, otherwise well or a different or a bond with a different interest rate sensitivity.

Okay, so before we now start looking at those different ratios that you might come across in practice, I suggest we take the time to develop an intuitive understanding of the key drivers of interest rate sensitivity.

Now, on the chart in front of you, what you can see is the price yield relationship of two bonds.

And in fact, they are two zero coupon bonds.

One has a time to maturity of two years, that is a lighter blue line, and the other one has a time to maturity of 10 years.

And the reason why I chose zero bonds, even though they're not necessarily the most actively traded type of instrument in the market, is that I wanted to really isolate the time to maturity.

And also, as we discussed in part one, a zero coupon bond really is basically just a special type of fixed coupon bond. And this is where the coupon is set at zero.

Now, let's have a look at actually these, these different, uh, curves here at a yield level of 0%.

So basically this is where we start our observation, um, which of course sounds like a strange level from today's perspective, but don't remember, uh, don't forget that about two years ago in many parts of the world, that actually was where yields were.

Um, so at this level, now, both of these bonds would trade at a price of 100% or par, right? Which makes perfect sense because if the coupon of a bond equals the yield level, then the bond should trade at par.

That's the general rule, right? So if the coupon is 0% and the yield is 0%, these bonds should in fact trade at par.

Now, let's look what happens when we increase yields. Now, if yields increase, both bond prices fall in price, that makes perfect sense given the inverse price yield relationship that we just recapped on the previous slide. However, what we can see is that the bond prices here, or the prices of both bonds actually fall at very different speeds.

And if you see here, the dark blue line falls faster as a steeper decline, and that means that the 10 year bond falls much faster in price than the two year bond for the same change in yield.

That's what the graphic says.

Now, let's think about as to why this is the case.

And you can approach this obviously from two different ways.

First, you can look at the math, right? If we're thinking about this, these are zero coupon bonds, which makes it actually relatively simple to look at the mass, right? Because a zero coupon bond represents just one future cashflow.

We have the two year zero bond, which basically is representing a cashflow of a hundred percent in two years.

And we have the 10 year zero coupon bond, which represents a cashflow of a hundred percent to be received in 10 years.

Now, what we said about bond prices is that they are basically nothing else in the present value of all the bonds cash flows, right? So basically, how do we calculate the price of our two year zero bond? That's 100 divided by one plus whatever the yield to maturity level is to the power of two, because this cash flow is two years away, and we do the same thing for the 10 year, but this time it's to the power of 10 because the cashflow is 10 years away.

And now you can see that the same change in yield.

So if we're assuming yield goes up by, let's say one percentage point, this will affect the price of a two year zero coupon bond to the power of two and the price of a 10 year bond to the power of 10, which of course is a lot larger impact than to the power of two two.

So hopefully the mass is clear.

Then let's think about this.

How can we explain this intuitively? And what I like to use in this context is this concept of relative happiness, right? So let's assume that the whole group that is on this call here today believes that yields are looking fairly ish right now.

So let's say yields are the 4% level, we're assuming a perfectly flat yield curve.

So all yields are at 4%.

We look at this and say, no, it's time for yields to go down because I think this level is not sustainable.

So what we're all gonna do is now we're going out and we're investing into long or into fixed coupon bonds.

Alright? However, half of the group goes out and buys a two year fixed coupon bond.

The other half goes out and buys a 10 year fixed coupon bond.

Now, let's say that by the time we close up for the day tonight, yields have fallen across the curve and now yield levels are significantly lower.

So let's say it's 3.8%.

So every one of us has made the right decision in a way, right? So we're all feeling reasonably well because we managed to invest our money this morning at 4%.

If we would've waited a couple of hours, all we would've been able to get from a return point of view would've been 3.8. So the timing was good, everybody's happy.

I would suggest though that at least at that point tonight, some of us are happier than the other, right? So the imagine the half of the group that bought the two year bond, maybe slightly less happy than the group or the part of the group that bought a 10 year bond, simply because yes, they managed to work, we managed to invest our money for 4%, but unfortunately we chose a reasonably short time to maturity.

And those who bought the 10 year bond will enjoy this higher yield level for much longer.

IE for 10 years than we was our two years. In two years, we get our money back and the deals haven't gone up.

At that point, we will have to reinvest at a lower rate.

That, of course, is the intuition behind this, and I hope this makes sense. I'll leave it up to you, which, um, kind of way you prefer to remember this with, okay, so bottom line of all this kind of stuff we've been talking about is that in general the following rules applies, and that is the longer the time to maturity of a bond, the higher its sensitivity i.e. the faster it falls in price for the same increase in yield and the faster it drops in price for the same decrease in, for the same decrease in yield.

Okay? So, now let's have a look at the next main driver here.

And that is the coupon. Because bonds do not only differ in terms of time to maturity, they will also have very different coupon levels.

So what we're gonna do on this slide here is we're gonna isolate the coupon effect and see what the impact of the coupon is on interest rate sensitivity.

So again, we're showing you two different bonds here and their price yield relationship this time they have identical maturity.

So both are 10 year bonds, but they have different coupons.

And we're comparing a 10 year zero coupon, the one that we have seen before, and a 10 year 4% coupon bond here, right? So now the coupons are different, which means that the bond prices at the same yield levels differ quite significantly.

For example, at a 0% yield level, the zero coupon bond will trade at par as previously discussed, whereas the 4% coupon bond will trade at 140%.

And as price levels are so different, it's actually quite difficult to compare the price changes directly using this sort of charts on the left hand side. And for that reason, what we're doing on the right hand side is we're calculating the percentage change in bond price for a 0.25% change in yield for both bonds.

And then we put this in the chart as a bar, on the right.

So for example, when yields increase, and we're again looking at the first item here from 0% to 0.25%, the zero coupon bond drops by around two and a half percent of its current price, whereas the price of the 4% coupon bond will in the vert commas only drop by a bit more than 2% of its price.

And as you just kind of look further down the right, so when yields are increasing, one thing that you can observe is that what we just pointed out.

And that is that the zero coupon bond loses more for the same yield than the coupon bond is something that not only happens in the first change of 0.25, but that's basically consistent throughout all of the observed yield changes.

So the obvious conclusion from there is then that the higher the coupon of a fixed coupon bond, the lower the interest rate sensitivity of this bond is, or the lower the coupon, the higher the sensitivity.

Okay? That's the graphical observation or conclusion here from the charts.

But how can we explain that? And this time, let's start with the intuition, right? And remember we have mentioned already that there is, when you buy coupon bonds, and the bond is not a zero coupon, but the bond, uh, the fixed coupon is something positive that introduces reinvestment risk into your investment, right? Because as a holder of a fixed coupon bond you'll receive parts of your investment back via coupons in regular intervals, like for example, semi-annually or annually, right? And the assumption is that a rational investor will of course reinvest these coupons until the end of the original targeted investment period.

So if you buy a 10 year bond because you wanna lock your money away for 10 years, and then after one year you get a coupon, then the rational person with no other interest or or anything would take this money and reinvest it for nine years, then all the money is basically received in 10 years.

And I think this is where the intuition comes in.

And this explains why intuitively a bond with a higher coupon should fall less in price as yields go up than an otherwise identical bond with a lower coupon. Because in both cases, right, of course, the investor misses out on higher yields for the already invested amount.

But in case you have bought a high coupon paying bond, what this means is that you get a decent proportion of your invested money off the face value back prior to maturity.

So imagine you have a bond that pays a 10% coupon, that means 10% of your notional is repaid to you at the end of every year, right? And that amount of money you can now reinvest at higher yields, and that can be seen as almost softening the blow a little bit, right? Yes, yields overall have gone up.

You locked your money away too early at a too low yield level if you wish, but at least the good news is now the coupon that you're receiving, it can be reinvested for the remaining investment period at higher yield.

And that explains why in a rising yield environment, a bond that has a higher coupon might potentially have advantages over a bond with lower coupons. And that is reflected in the fact that the higher the coupon, the lower the interest rate sensitivity is because the same applies for a fallen yield, right? And there's just a different impact now that when yields go down, then having a, um, you know, having a large amount of money coming back to you is actually turning into a disadvantage anyway.

So, that's the intuition. And I think it's also a great example of why, you know, one shouldn't think of reinvestment risk necessarily as something bad, right? What risk generally just means is uncertainty.

And that means we don't know the exact outcome.

But as we've just discussed, you know, this risk can actually have certain advantages under certain circumstances.

So that's the right way of looking at the reinvestment risk.

And if you wish, call it reinvestment uncertainty.

Now, how about then a more numerical approach because not everybody is liking the intuitive approach, right? So what can we explain the link or how can we explain the link between coupon and sensitivity using that technique? Well, I would suggest that another way to interpret it, 10 year 4% coupon bond is just to think of it as a basket of zero coupon bonds.

Right? Now, in case of an annual coupon payment, this would mean that a 10 year, 4% coupon bond would actually consist of a one year zero bond with a notional amount of 4%, a two year zero bond with a notion of 4% three year, 4, 5, 6, 7, 8, 9, 10, and a 10 year. The notional is 104 simply because it's a redemption payment plus the last, and of course, the interest rate sensitivity of such a basket of bonds should really reflect the interest rate sensitivity of all its constituents, right? And this brings us back to our early observation.

The longer the time to maturity of the bond, the higher is its sensitivity.

Now, instead of looking at one bond, we're looking at a basket of 10 zero coupon bonds.

And some of them will mature after 1 year, 2, 3, 4.

So indirect comparison when we're looking at this 10 year zero bond versus a portfolio of zero coupon bonds with times to maturity between 1 and 10 year, this portfolio of the latter i.e., the, you know, zero coupon bonds with different times to maturity should have a lower sensitivity because as we said, some bonds or 1 bond matures after 1 another, 1 after 2, and another 1 after 3 and so on.

And to take this one step further, we can also conclude that the higher the coupon of the bond, right, the higher the proportion of the bond that is paid earlier, if you think about them as a zero coupon bond basket, the higher the weighting of those zero coupon bonds that have a maturity that is shorter than 10 years, and that again means the higher the coupon, the lower the sensitivity of the bond together.

Okay? Now let's have a look at the last main factor of interest rate sensitivity. And that is the level of yields itself.

So what we've done here on this slide is we charted the price yield relationship of a 10 year, 4% fixed coupon bond, and then we'll put a linear tangent through it, right? That's a light blue line.

And when you compare both lines you will see that the bond price yield relationship of our 10 year focus percent coupon bond is in fact not linear.

Instead, it shows the slightly convex profile.

And that's exactly why this phenomenon is referred to as convexity.

But what if exactly does convexity mean in context of fixed coupon bonds? Well, let's look at the chart, right? And let's focus on the point where the yield is 4%.

So here, and that means a bond trades at par.

Once again, coupon equals yield bond trades at par, however.

And so, if we now kind of look at what happens when yields increase, then as one would expect, we see that the bond price generally declines inverse price, yield relationship, nothing new.

But when you really look carefully, what you see is that the dark blue line, which represents a real bond price yield relationship flattens out, which basically means that as yields go up, the interest rate sensitivity of a bond decreases, right? That means each additional increase in yields leads to a smaller increase or decrease in price than the previous one.

So we can say that from the perspective of a long position of, or a, from the holder of a bond, that when the market moves in the wrong direction, i.e yields go up.

The risk position or the risk of the position we're holding automatically becomes smaller, which is clearly a good thing from the perspective of the holder of a bot.

So what happens then, if we go the other way, what happens if yields decline? Well, the bond price rises, but this time the speed increases, the further yields fall.

So each additional decline in yields now leads to a bigger increase in price than the previous one translated into the risk of a position for a long bond holder.

We can say that when the market moves in the right direction, the position automatically becomes larger.

Again, that's clearly an advantage from the perspective of a bond holder.

So what we can conclude then after this observation is that as yields increase the interest rate sensitivity of fixed coupon bonds declines and vice versa.

And this convexity, as said, is very valuable for the holder of a bond. And of course, it's a disadvantage for those who are holding a short position in a book.

Well, I'd say this is something we have just derived from, again, looking at charts, right? And it's hopefully obvious that this is the case, but it's kind of tricky to come up with the intuitive with an intuitive explanation that sort of allows us to conceptualize this.

But one way that I think might at least help in conceptualizing this idea of convexity is the following, imagine yields are at 0.01%.

Again, that was not that was the case not too long ago, right? And now let's imagine that yields increase from the current level of 0.01% to 0.02%.

Now that is an absolute change of yields of one basis point or 0.01%, but relatively speaking, yields have just doubled, right? They have increased by a hundred percent.

And now imagine that we're at a different level of yields of 1% and the yield increased from 1% to 1.01%, same absolute change than in the scenario before.

But a different relative change this time yields have only increased by 1%.

And so the higher yields are the smaller is a relative change in yields if the absolute change in yields is kept constant.

And that is maybe the first indicator as to why convexity exists.

We're looking at absolute rate oil changes in yields, which means a relative or different relative changes in yields.

As I said, it's not perfect, but it might help with just conceptualizing the concept of, uh, convexity.

Okay? But enough about general concepts and, and, and ideas.

Let's look at some concrete measures because while it's good to know the main drivers, because it gives you this intuition and the expectation of where roughly you should expect a certain ratio to, to be in practice, it won't quite be enough because what if you wanna compare two bonds with regards to the interest rate sensitivity? And one of them has a slightly longer time to maturity, but it also has a slightly higher coupon or a higher coupon.

Now the time to maturity argument indicates that this bond should have a higher sensitivity, but the higher coupon indicates it should have a lower sensitivity. So what's the overall impact? Which two bo which of the two bonds now is actually more sensitive? So investors are looking for quantifications, really, that allow them to accurately compare to bonds with regards to the interest rate sensitivity.

And those are what we generally refer to as the sensitivity ratios.

And there are three of them, which are very commonly used, or at least also shown by those different bond analytics tools like you find on Bloomberg, et cetera.

There's a Macaulay duration, the modified duration, and the DV01.

And what they all allow us to do is to compare the interest rate sensitivity of two or more bonds directly. So we can look at, for example, the Macaulay duration of two bonds, and immediately we can say, which of the two bonds is more sensitive, i.e., which bond changes more in price for the same absolute change in yields modified duration and DV01 go even one step further because they allow us to estimate the actual P&L impact of a certain interest rate scenario.

So we can, for example, look at the DV01 of two bonds and then calculate the amount of dollars or yen or euros, whichever your reporting currency is that we would expect to lose on those bond positions if yields were to increase by, for example, one basis point or five 10 or something like that.

So of course it feels like modified duration and DV01 then have a little bit more use in reality.

The good news is, as we will see, the calculation of all three is actually very closely linked.

So you frequently find all three in all the systems and additional measures even shown.

And that means investors can of course, you know, choose which measure they actually prefer for themselves.

Now we will look at the actual calculations, of all three numbers here in a minute.

But before we do that, I think it's really important to mention that none of these three ratios is actually static.

So that means we cannot just calculate the macauley duration or modify or DV01 when the bond is issued, and they never do that again because it's gonna go through constant change.

Now why is that? Well, first of all, we just talked about convexity, which meant the sensitivity of a bond changes when yields change.

And of course that means also the sensitivity ratios should change as yield change.

So basically every time the yield of a fixed income instrument changes, it's sensitivity changes and therefore we should recalculate, those ratios.

In addition to that, we have also learned that the sensitivity of a fixed income instrument is very closely leading to its time to maturity.

And we know one thing about time to maturity, and that is it shortens over time.

So you buy a 10 year bond today, that's not a 10 year bond.

Tomorrow, that's 10 year nine, sorry, 9 years, 11 months, 3 weeks, and 6 day bond tomorrow, right? So that means a time to maturity of a fixed income instrument shortens, and that means the sensitivity, all other things remaining equal or, unchanged will change as well.

So that means we will have to frequently recalculate all those ratios in reality.

But now let's have a look at how these calculations are actually done.

And we're starting with McCauley duration as I think this was the first actual duration measure, and it was developed in 1938 by Frederick McCauley.

And I think that he probably based this approach on the same observations that we have already discussed.

We said the longer the time to maturity of a bond, the higher is the sensitivity and the higher the coupon of the bond, the lower its sensitivity.

Now, if you combine these two observations what you get is the following conclusion, right? Basically it's all about when on average are all payments of the bond received, the earlier the payments of a particular bond are received, on average, the lower the sensitivity of the bond should be.

So following that logic, all we have to do to make bonds comparable with regards to their interest rate sensitivity is to calculate the weighted average time to maturity of all the bonds cash flows.

And that is exactly what Macauley duration does.

You can see the formula here shown on the slide. And what we do in calculating the macauley duration is we take the present value of all the bonds cash flows.

So that is this part here, right? Cash flows divided by one plus yield to maturity to the power of time. That's the present value of the cash flow, and we multiply that present value with the amount of years until the payment is received.

So times just a quick side comment here. By the way, all formulas shown in this presentation assume annual coupon payments, right? In case of Samuel or quarterly coupon payments, the formula changes slightly, but the concepts remain the same.

So I decided to reduce complexity.

I.e. you assume annual payments.

So that's what it's done, right? It's present value of cashflow times number of years until the cashflow is received.

Sum up all these values and then divide by the sum of the unweighted present values, which by the way, and this is the formula here, is nothing else then the dirty price of the bond, right? So that's basically, what you have to do to calculate macauley's duration.

And the result is the Macauley duration, which is typically measured in years.

Now because we're calculating averages here what can happen easily is that the Macauley duration might be something like 9.22 years, if there's a 10 year bond with annual payments.

And that means there is actually not a single payment happening at 9.2 years.

But that's not the point Macaulay is about.

It's not about the actual payment dates, but it's about the average of when you will get your money back.

Okay? And a couple of useful rules of thumb that I think I was remembering right. First one, if you have a zero coupon bond, the Macaulay duration will always be equal to its remaining time to maturity because there's only one cashflow, and that happens to be at the maturity date.

And for fixed coupon bonds that have a coupon of higher than 0%, the macauley duration will be shorter than the time to maturity as part of the bond is repaid before maturity.

And the higher the coupon, the bigger the part is that repaid earlier.

So the shorter the Macauley duration. So as you see, the maturity time to maturity matters and also the coupon level matters in macauley uh, duration.

Okay, good.

Now let's have a look at a quick concrete example here, right? So just to run through the numbers one time.

So here we're looking at a five year fixed coupon bond pays an annual coupon of 2.4%.

The yield is 2.4, it's 3%.

So the bond is actually trading slightly below par.

And that's basically what we're doing here in the PV column.

We take the cash flows of the bond, so 2.4% after year one calculate the present value.

And that is done by dividing two point, uh, four by one plus yield to maturity, which is 2.43 in that case to the power of one that gives us a PV for one year. We do the same for 2, 3, 4, and 5.

Now we're going into calculating the macauley duration. And the thing we said we need to do is wait the present values of the cash flows with the number of years until they occur.

So for the first one, this doesn't actually change anything because it's one year away we waited, was multiplying by one, so it remains unchanged. But the two year, present value is now multiplied with 2.

So it goes to 4.5750 and we do the same for three, four, and five.

We then sum up the time weighted present values, which gives us 476.4204 and divided by the sum of the unweighted pvs.

That was the bond price, 99.8603.

And then what we get is a macauley duration of 4.7709 years, which basically means it takes about four point, or on average 4.7709 years to get your cash flows, from the spot.

Now, that itself in isolation doesn't tell us very much, but what we can do here is we can use this number to quickly compare to bonds.

As I said, if we had a second bond and we did the same calculation and we get a duration for that bond of let's say four and a half years, what we can say now is that this bond that we're looking at here right now has a higher interest rate sensitivity.

Then the other bond was a 4.5 year duration.

And therefore it should change in price more for the same absolute change in yields.

Okay, let's move to modified duration.

because I always found that macauley duration is so wonderfully intuitive, but it has that shortfall and that is, it doesn't allow us to sort of calculate P&L impacts for interest rate scenarios.

And that's quite a useful thing.

In the context of risk management, you wanna quantify numbers, right? So, let's take a look at these ratios that we already have mentioned modified and, and DV01.

Now the modified duration is actually based on mathematical derivatives, right? In mathematics, the first derivative of a function measures the rate at which a function changes its input as its input changes, right? So it essentially measures the slope of the function or the steepness of a function at any given point.

Now, in the context of bonds, we can consider the, bond price a function of the bonds yield, and as yields change, the bond price changes too.

So the first derivative of the bond price with respect to its yield then gives us the bonds interest rate sensitivity.

That's the underlying theory, right? So now to the question, what exactly does modified duration tell us? Now, modified duration is actually defined as a relative change of the bond price that is caused by a change of the yield to maturity. And we're usually looking at a one percentage point change here, and that's a very useful measure because we can now, use modified durations to compare two bonds, and we can see immediately which bond will lose a higher percentage of its current price for the same given change in yield to maturity i.e. which one is more sensitive? Now the good news is well or bad news depending on how much you like to derive things, but generally speaking, we don't have to go through this whole process of deriving the bond price formula with respect to its yield, because that's a shortcut, right? If you divide the macauley duration by one plus yield to maturity, at least this is a case for annual coupon bonds, as I said, then you get the modified duration of the bond and you can, if you have time and, and you feel like it, of course do the whole derivations thing, but you get to the, the exact same result by following the simplified shortcut here on this slide.

And if we go and have a look at our previous example, then it becomes clear what you know, how to interpret these numbers.

So we calculated the macauley duration of 4.7709 and rounding. Now, and that was the Macauley duration. If we wanna calculate the modified duration out of it, as I said, all we need to do is we have to take the macauley duration and divide it by one plus yield to maturity. Yield to maturity was 2.43%, so divided by 1.0243.

And that gives us a number of 4.6577.

That is the modified duration of this bond.

Now, what this number means in essence is that we would expect the bond price to decrease to fall by 4.6577% of its current level in case that the bond yield was to increase by exactly one percentage point.

So from 2.43% where we currently are to 3.43%, remember it's a relative price change.

It falls two or 4.6577% of its current price, which also happens to be given in percent.

So that's why sometimes there's a little bit off confusion. But what if we wanted to calculate the absolute price change? What if I wanna know, okay, where do I expect my bond price to be when yields are actually at 3.43? Then I wanna kind of subtract the absolute change in price from my current bond price to give me this indication.

And to do that, what we have to do is we have to multiply the relative price change i.e. there's 4.6577% with the current bond price, right? That gives us, in this case 4.6512.

Now these values are not too different in this particular case because the bond trade's relatively close to par.

But if the bond was trading way above par or way below par, then the difference would actually be larger.

But what does this number now mean a 4.6512%? Well, what it means is that we now expect that the bond price will fall 4.6512% in absolute terms when yields increase by one percentage point.

So for a yield level of 3.4 is 3%, modified duration actually predicts a bond price of 95.2091%,

right? Because that's nothing else in the current bond price minus the absolute price change.

So let's see how accurate this is. And I'm gonna go to Excel where I have prepared a very simple bond pricing tool.

So our five year bond, 2.4% annual coupon.

Here's the PVs, here's the time weighted here, you see the calculations of duration, et cetera.

And here you see the current yield is 2.43% and the bond price is 99.8603.

Now let's change the yield simply to 3.43.

And what we see is that at that yield level, the bond price is gonna be 95.3403.

We're going back to our slides.

We see that the actual bond price is gonna be slightly higher than the one we predicted.

Question is then of course, what did we do wrong? And the answer is we didn't do anything wrong.

But what becomes obvious here or what, this is a reminder that modified duration estimates bond prices for specific yield levels using a assumed linear relationship.

But as we said before, the bond price yield relationship is actually convex.

So this difference that we're seeing here is actually just a prediction arrow.

And this slide here goes into this a little in a little bit more detail, right? So the reason for the forecast error on the previous slide, as I said was that the modified duration implicitly assumes a linear relationship when the relationship is actually convex.

Now what this means is that for small changes in yield, the forecasted value will actually be relatively close to real values, but for larger changes, and one percentage point is actually quite a significant change in yields, there will be an increasing prediction factor.

Now, of course, there are mathematical techniques to improve the forecast value. You could use a second derivative 10 expansion and so on.

But you've just observed how simple it is actually to recalculate or calculate the price of a bond for a changed yield level.

So I think instead of using second and third and fourth and fifth derivatives, you should just reprice the bond at different yield levels and that gives you the accurate price level at a specific point.

But, linear approximations, as I said, are fine for relatively small expected changes in the yield.

And with that let's look at the final ratio, which is a DV01, and that stands for dollar value of a basis point.

And uh, then obviously as an m applies looks at impact of yield changes not of a percentage point, but at one basis point. So that's a very small but very reasonable yield change simply because when you look at a busy day and as treasuries yields change by a couple of basis points a day, right? So over a reasonably short timeframe, looking at one basis point changes is actually quite a realistic measure.

And so DV01 is a very widely used measure of interest rate risk in fixed income, not just in bonds, but also for derivatives. So if you look at interest rates for portfolios, for example, the interest rate risk is usually measured in DV01s, et cetera.

So let's have a quick look at how this works.

And the essence of the DV01 is that it gives us the actual p and l impact.

So a number in dollar terms of a one basis point change in yields. And if we compare this directly to modified ation, I said some of this before, this comes with a couple of advantages, right? First of all is we can see straight away how much money we will lose on a specific position in case a certain change in yield was to occur as the DV01 does not only consider the sensitivity of a position, but also the size of the position.

So do we own 1 million in face value? A hundred million or a billion, right? So that's a very tangible number. It's very practical. It's like yield go up one basis point, I'm losing X dollars.

That's a very, very powerful and simple way of looking at risk.

Now, then the second advantage I'd say is that we're looking at a one basis point change in yields. And that's the most currencies is a more realistic assumption than a one percentage point move, right? At least as I said, over a reasonable short timeframe.

So obviously the DV01 is a very useful ratio.

Now the good news is that for a single position or a single bond position, we can again calculate it by just really mildly modify, modifying calculations that we have already done.

'cause remember that modified duration gives us the relative price change of the bond for a 1% increase in yield to maturity.

So to get to the DV01, all we have to do is really make a couple of adjustment. And the first one is we have to scale from the 1% yield change to the 0.01% yield change.

So basically what we're doing here is we're starting with the modified duration, which we have calculated out of macauley duration, if you remember.

And then we're multiplying that with not 1%, but with 0.01%.

That means we're now looking at the relative change in bond price for a one base point change in yield.

The second step is we need to now multiply with dirty price that brings us from the relative price change to the absolute price change.

And now because we want to sort of turn this into dollars and not percentages, we multiply that with the face value of the bond.

So what's the notional that we own in this position and that turns this absolute price change from a percent perspective and price to an actual dollar number.

And if we go to the next slide where we just see the calculation here, we're starting with the modified duration 4.6577, we then multiply that with the one basis point times the price of the bond times a hundred million because that's the assumed face value or notional or principle amount that we're holding.

And the outcome is minus 46,511.

Now, I put a minus there because I look at the impact of one basis point increase here, but of course it doesn't have to be. You can look at a one basis point decrease in yields. You can look at a one basis point increase and then one basis point decrease and take the average of both.

So there are different techniques, but you know, that's sort of my approach here in this slide.

Hence, what this means is that we would assume that if I own a hundred million dollars in this bond notional terms that is, and the price or the yield raises by one basis point, then I should expect to lose on a P&L or mark to market basis 46511.8.

And if I wanted to calculate my approximate P&L impact of a larger increase in yield, so for example, 10 basis points, then what I could do is I could multiply this DV01 by 10, but that of course means we're once again assuming a linear relationship.

So there might be some prediction errors, sneaking in, especially when we're looking at larger moves.

Okay? So hopefully that concept is clear.

And now what we need to do is we need to look at this on a portfolio basis, because everything we've done so far really is single bond in reality.

Most investors will hold many bonds in their portfolio and whenever this is a case rather than looking at a whole bunch of different Macaulay or modified durations or whatever they want to, or investors would have an interest in seeing the sensitivity of their portfolio rather than the individual bonds.

Now in practice, this is very often done by calculating portfolio sensitivities, and these are simply weighted averages of the individual bond position.

So for example, we can calculate the weighted macauley duration, uh, of a portfolio simply by modify multiplying each individual bond duration.

So here macaulay duration of bond A with the portfolio weight of bond A.

So if 20% of my portfolio was BA bond A, then 20% times the Macaulay duration plus x percent of you know, the, the, the weight of bond B was the duration of bond B and so on.

And then basically I get a portfolio macauley duration, and then I can use, for example, to compare the sensitivity of my portfolio with the interest rate sensitivity of, for example, the benchmark that I'm using for my portfolio manage, right? Now the same approach can be taken for modified portfolio duration, which will then allow us to approximate portfolio value changes for an overall change in yield level.

So if yields go up by one percentage point, how much is my portfolio losing in percent of the prices? And we can use the same thing for DV01 as well, which would then allow us to approximate the value change in the whole portfolio per base point increase, in yields.

Now, these approaches arguably are reasonably simple, but the bad news is they come with a, in my mind, relative significant limitation.

And that is that the weighted average duration really is only looking at impact of parallel shifts of the yield curve, right? If the bond portfolio now consists of one year bonds and 30 year bonds, then I'm not sure the weighted Macaulay duration is necessarily that useful, but it will be more useful to take a bit more of a granularity because it's probably not wise to assume that one year and 30 year yields will always move in the exact same way.

So to get a better view on where exactly in the, on the yield curve our portfolio risk lies.

And therefore to get sort of like a bit of an idea about the curve risk that we take in our portfolio, we then have to use a different and arguably more complex approach.

And that's one of the approaches we can use is called interest rate delta letters.

And I want to quickly talk you through at least the main idea.

So what we're doing in this Delta letter approach or DV01 vector or you know, whatever you wanna call this is we're breaking down the portfolio into specific segments. And these segments are usually based on time to maturity of the bond.

And then each step on the ladder, each segment basically represents specific time to maturity.

And these times to maturity are then what we call often the key rates.

And then what we do is that in a step-by-step approach, we then calculate the DV01 values for each step or key rate. That sounds a little bit abstract. So let's run through a quick example.

Let's say we want to look at our portfolio and we have identified that we that the following key rates, we have one year, two year, three year, four year, five year, and now we want to calculate the DV one values for each of these key rates.

Now, what we're gonna do is we're looking at the current yield curve, right? So let's say 1, 2, 3, 4, 5, and that's a yield curve.

So here we have 1 year yield, 2, 3, 4, and 5.

And with that current yield curve, we're now valuing our current portfolio and that gives us the current portfolio value.

Now what happens next is we're just artificially shifting the one year point of the curve, let's say make it really simple just up by one basis point.

And with this now slightly altered yield curve, we now recalculate the bond portfolio value again.

And then the difference between the original portfolio value and the new hypothetical portfolio value was a changed one year key rate.

That's then the DV01 for one year.

So let's say the difference in portfolio value is $25,000 and the after the one year shift up one basis by one basis point, then the portfolio value has declined by 205K.

Okay? So that means now that in this portfolio, if the one year yield was to go up, we're losing $25,000, we then basically bring the one year rate down to where it was.

Now we shift two year rate up by one basis point, and we do the whole thing again.

And let's say in two, we have nothing.

In three, we have nothing.

And then we see, okay, a hundred thousand here in year four and 50,000 in year three.

Now what does it tell us? It gives us a lot of information. First of all, it says that on that portfolio, right now we have a total DV01 of 170 5K.

That means if all rates that we're looking at all our five key rates increased by one basis point, we're expecting to lose 175,000 bucks.

But what it also tells us that the main risk that we're having is actually in the four and five year parts of the curve. We have no risk at all, two and three, and we have a tiny little bit of risk in year one. So that gives you an already much more granular view on where exactly does the risk lie. Because remember, when we're investing in fixed income, there's a yield curve, and the yield curve does not always move in parallel.

So you need to consider where exactly am I positioned as well.

Alright, and um, now to wrap this up we wanna just have a quick look at interest rate sensitivities of floating rate notes because we've talked, about interest rate sensitivities for fixed coupon and zero coupon bonds up to this point.

But how exactly does the interest rate sensitivity of a floating rate node compare? Now, have a look at the question on the screen where we ask you, which of the bonds shown here would you expect to have the lowest interest rate sensitivity? The lowest, right? And the choices are five years, zero bond, a 10 year, four and a half percent coupon bond, and a 30 year floating rate note linked to sulfur.

Now, if I've only shown you the first two options, I would suggest that most of you would've said B as it's a bond with the longest time to maturity.

And yes the first coupon bond is a zero coupon bond.

But I think gut feeling will tell you that the fact that bond B'S times maturity is twice than the one of A, this will outweigh the fact that bond B pays a coupon, especially as a coupon is not extraordinarily high.

However, now we're throwing a 30 year bond in the mix, and based on the maturity argument alone, this should clearly have the highest duration.

But on the other hand, it's a floating rate note.

So that brings us to the question, what actually drives the duration of a floating rate note? Now, we talked about them last time.

But quickly a reminder, one of remind you of the key mechanisms, or characteristics, right? FRNs are bonds where the coupon is fixed at issuance is is where the coupon isn't fixed at issuance.

But instead we fix a coupon formula, right? So the coupons that investors will receive are linked to some reference rate.

And so depend on where these reference rates are going to be in the future, and how often the coupon is reset is usually a function of the, term of the underlying reference rate.

So if the reference rate is, for example, three months, IBOR, it will be reset every three months.

If the reference rate is an overnight rate like sulfur, the coupon will be reset on a daily basis.

Now, to reduce complexity, what we're gonna do here is we're leaving credit spreads and discount and coded margins and all this kind of stuff aside.

So we're assuming a a for example here, a two year floating rate.

Now that pays a coupon of uor simply because it's a little bit easier to talk about, to through that.

And there's no spread added or subtracted.

So in that case, if we're thinking about a two year floating rate now that's based on three or that pays coupons every three months because that's when we reset the underlying three months arrival rate, then basically we can conceptually understand that floating rate node as a series of eight short term fixed coupon bonds.

And they have a duration of three months each.

Okay, so let's visualize this to make it a little bit less abstract.

So, um, let's assume this bond is issued this two year floating rate note.

And when the bond is issued, the first three months, your ROR is fixed at, let's say 3%.

Now this is then basically the coupon for the first three months fixed coupon bot, right? So you buy the floating rate note, your IR is set at 3%.

What this means is in three months you get synthetic, at least hypothetically your money back. You get paid at repaid. That part, that's how coupon bonds generally work.

And um, you also get the coupon, right, which is not 3% because you only get 3% for a quarter of a year, more or less.

So that's about 0.75%.

So you now have power back, you have your money back, but it's a two year bond. So basically what you will do at that point in three months time is you now will effectively buy another bond that again, has a three months time to maturity.

This time, however, the coupon is not 3% because meanwhile, over the last three months, ribo has increased to 3.5%.

So this coupon or the fixed coupon of the second hypothetical bond is 3.5%, where yields are three point a half percent. So again, you buy that bond at par, hold it for three months, get repaid at par, receive the coupon, and then you buy the third wherever your Euribor is.

So it's effectively, that's a good way to conceptually think about a floating rate node without margins.

It's basically where you buy or you commit to buy short term fixed coupon bonds, but the yield that you will get or the coupons that you will get will be reset in regular intervals and they will always reflect the current yield level and the maximum duration the investor then can face on this two year bond linked to three months, your Euribor is indeed three months because that's today, you buy the bond, it's a three months bond.

Tomorrow it's two months, three weeks, six days.

The day after it's even less.

And so duration of this bond goes from zero three months to one day, and then you buy effectively the new bond and then you get a three months duration again.

And if you expand this to the 30 year floating rate node linked to SOFAR, what you realize is that although the time to maturity is 30 years, the interest rate sensitivity of this bond is actually very low.

Because coupons, in the case of sulfur linked floating rate notes will only ever be fixed for one day.

Granted, they're not paid every day, but at least the coupon is only fixed for one business day only.

So in theory, what this means from an interest rate point of view, at least regular floating rate notes should always trade relatively closely to par as they have in fact very little interest rate sensitivity in case of SOFAR linked, it's a day right? But of course if you throw credit risk in the mix and the issue is credit quality improves or, or worsens significantly over time, that could of course lead to floating rate.

No prices deviating significantly from par, but that's credit.

So it's for another day.

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