Well, good morning, good afternoon, potentially good evening, everyone, and a very, very warm welcome to this Felix live session on bond market essentials. My name is Thomas Carls, and I have the honor to guide you through today's session.

And of course, one of the most important things at the beginning of such a session is always to learn what exactly are we going to talk about.

So let's have a quick look at the agenda.

We're going to start with a quick look at the concept of interest rate risk and sensitivity.

So just really the generic foundation of those numbers that we're going to look at. And then we will discuss duration more precisely.

We will discuss the rates duration, simply as we are going to focus on changes in interest rates that result in price changes, not looking at credit risk here.

We will talk about Macaulay duration.

We will move on to the modified duration, and then we're going to end up with a high-level introduction to convexity and also to the dollar value of a basis point or DV01.

And that should give us a pretty complex overview about the different risk metrics that are being used by various kind of market practitioners. But without further ado, let's get started with today's content. And as I said, we're going to start with a generic look at interest rate risk and sensitivity.

Now, we all probably will know already that when it comes to fixed coupon bonds, at least, there's this inverse relationship between bond prices and yields.

And that's reasonably intuitive because, in a way, the bond price is just what an investor pays to receive a certain set of predetermined cash flows in the future.

And as I said before, we're ignoring credit risk here.

So if now a bond has been issued, and at the time the bond was issued, let's say the appropriate yield level for such a bond and that maturity would have been around 4%, and the bond would have been issued with a 4% fixed coupon, then the bond would have been issued pretty much at par. So if now subsequently, for example, because of increased concerns about inflation or anything like that, the yield level that investors demand to invest in such a bond now should rise to, let's say, 4.2%, because the coupon of the bond doesn't change.

But investors demand the yield to total return of this investment to be 4.2%, which is above the bond's coupon.

The bond price must drop below par because that's the only way in which a bond with a coupon of 4% can actually pay a return of 4.2%. So in addition to the regular interest rate payments, you do get some sort of capital appreciation as well. So that means yields go up, bond prices go down.

Nothing new here. With the caveat, of course, that this is only really strictly speaking true for fixed coupon bonds. And of course, zero coupon bonds, which if you attended the bond mechanic session, you know that I see zero coupon bonds pretty much as a special type of fixed coupon bond, simply because it's a bond where the coupon is fixed, but at 0%. Anyway, so that's a quick reminder of the inverse price yield relationship.

Now, then I think it's true that the vast majority of bond investors or investors in general approach the bond market from the long side, right? There's investors that are either long the bond, or they might have a neutral bond position. But having outright short positions in bonds like in the equity world as well, that's only really observable for a relatively small group of market participants. And I think this is also why it makes sense that when we look at interest rate risk, it's most often generically defined as the risk that, yes, the bond price will drop, so after we bought the bond, the bond price will drop when yields are on the rise for whatever reason this might be the case. And now this, of course, I think is very important knowledge.

You know the inverse price yield relationship.

You know that when yields go up, bond prices go down, and if you're having a long position, this will lead to negative P&L impact.

And this is very important qualitative knowledge.

But really, when you are an investor, qualitative understanding necessarily isn't enough, right? You also want to quantify the risk that you're taking. And this is exactly where then those interest rate sensitivity numbers come in, that when we talk about interest rates here, we just generally call rates duration or just in short, duration. And what these numbers do, or what this concept of sensitivity really does, it describes the magnitude of the price move. In other words, if yields were to go from a certain level to another certain level, how much is the bond price going to drop? And it's intuitive that not all bonds will respond to the same change in yields in the exact same way, right? And so what we're now going to do over the next three slides is we're going to have a look at what are the most important influencing factors on this magnitude, and then we're going to look at how we can actually quantify these things using the ratios like Macaulay duration, et cetera.

Okay, so the first thing that I did here is I compared two bonds with regards to their interest rate sensitivity andThese bonds differed by one characteristic only, and that was the time to maturity. So as you can see here, I have taken two zero coupon bonds, one 10-year and one two-year maturity, and then I have calculated the price for both bonds, assuming different scenarios for the yield to maturity. And as you would not be surprised to see when the yield to maturity is 0%, the zero coupon bond, both of them actually trade at par, and that sort of is perfectly aligned with what I said earlier.

When the yield equals the coupon, then the price is par. And if a zero coupon bond is basically a fixed coupon bond with a coupon of zero and the yield is zero, then it must be a par price that's observable here.

Now, of course, some might challenge the 0% yields. How realistic is that? Well, not too long ago, we were facing those yield levels, just as a reminder.

Anyway, so then what I'm doing here is I'm just going to shift the yield to maturity up and recalculate the bond price.

And as we would expect, both bond prices decline as yields go up. That's the inverse price-yield relationship.

But what gets pretty obvious here is that they don't decline at the same speed. Right? In other words, the blue line here is much steeper in decline than the green line.

Now let's have a look at the legend.

That is the 10-year zero coupon bond.

So the conclusion we can draw from just looking at those two lines here is that obviously the longer the time to maturity, at least in this specific case here, the higher the interest rate sensitivity of a fixed coupon bond is, i.e. for the same change in yield, we're just changing yield from here to there.

The price drops relatively little for the two-year zero, but quite a bit more for the 10-year zero.

So same change in yields, bigger change in price for the bond which has the higher time to maturity. And that's why I use zero coupon bonds here. So that the coupon, the cash flow structure of the bond is identical. And now of course, we could either draw in more bonds here to just see if this actually is the case. As we go and draw a 30-year bond, for example, then it should look even steeper.

Or you can try to understand why this should be the case.

And I think here also, having used zero coupon bonds make the whole thing relatively easy.

So if you think about the price of a two-year zero coupon bond, that's basically the cash flow of the bond, and there's only one cash flow in a zero coupon bond, and that's a redemption at par at maturity, divided by one plus yield to the power of two, because that cash flow is two years away. The price of a 10-year zero coupon bond is 100% divided by one plus yield to maturity to the power of 10, assuming annual coupon payments here for simplicity.

And now we see as to why the same change in yield really impacts the price in different ways.

Because for the two-year bond, it impacts the present value to the power of two. For the 10-year bond, it impacts the present value to the power of 10, which of course is a much more significant impact after all. And that explains mathematically why a longer term bond has a higher sensitivity to a change in interest rates.

Okay. So let me just reshare the materials download in the chat because some request came in. Okay, cool. So now that's the first factor or conclusion there.

As I said, longer time to maturity, the higher the interest rate sensitivity of a bond. And that for fixed coupon bonds, just sharing no secret here, is usually the dominating factor.

However, there's more than that. And the second part or the second contributing factor is the cash flow structure, i.e., when are the cash flows being paid, how is the payment pattern of the bond looking like? And what we're doing here now is we're this time looking at two bonds.

But now they don't differ in times to maturity.

They're both 10-year bonds, but instead they are differing in the size of coupon that's going to be paid.

Right? And we're now comparing a 10-year bond with a zero coupon with a 10-year bond that pays a 4% coupon annually.

And the starting point is exactly the same.

I calculate the bond price for both bonds at different yield scenarios.

And this time, because we're looking at different coupons, they won't have the same price when yields are at zero, because when yields are zero, the zero coupon bond trades at par. But when yields are zero, then the 10-year 4% coupon doesn't trade at par, but instead by the sum of its coupon payments, because when we have yields of zero, discounting doesn't matter.

We just can sum up the cash flows, and that means we get a redemption payment at par and we get 10 coupons of 4%.

That equals 40%. So the price is at 140%.

Now we're doing again a shift up in yields, and we recalculate the bond prices, and as we can see, once again, bond prices go down, inverse price-yield relationship.

And it looks a little bit like the green line here has a steeper declineAnd that is certainly the case. However, what I would also like to point out is that it starts at a much higher value. So it starts at a higher basis, if you wish.

So the green line starts at 140, and the blue line starts at 100. And I would argue, rather than the absolute change in price, what is maybe of more relevance here to most investors is the relative change in price.

Okay, so what I've done on the right-hand side here is that I have calculated the percentage change in price for both bonds for certain yield scenarios.

So basically took the price at a yield of zero and then recalculated the price for a yield of 0.25%, bond price is lower, and I've just calculated then what percentage amount of the original price the bond declined.

And as we can see here, is that the blue bar, in absolute terms, has a higher value than the green bar, and the blue bar is actually representing the percentage change of the zero coupon bond, whereas the green bar shows the 4% coupon bond. So as we can observe, so at all yield scenarios, on a relative basis, when yields go up, the 10-year zero coupon bond loses more percentage-wise of its value than the 10-year 4% bond. And that leads to the conclusion that the higher the coupon of a bond is, the lower the interest rate sensitivity becomes.

Because here we have the green bars having a higher coupon and also changing by smaller amounts, i.e., that describes the interest rate sensitivity.

So conclusion so far, the longer the time to maturity, the higher the interest rate sensitivity, and the higher the coupon, the lower the interest rate sensitivity, and of course, vice versa. Now, why is this intuitive? So we looked at the math on the previous slide.

On this slide, I want to focus on the intuition.

Why does it make sense that a higher coupon bond actually falls in price less significantly than a lower coupon bond as yields go up? And that has something to do with reinvestment.

Think about it as follows. When you have bought a 10-year zero coupon bond, that's just one cashflow.

You will get all your money back at the end, at maturity of the bond. You have no interim cash flows.

That, of course, has a couple of advantages.

For example, that you don't have reinvestment risk, but it also has a disadvantage in the case that interest rates are generally rising over the lifetime of your bond investment, simply because you don't get any money back before maturity that you now can reinvest in order to benefit from higher rates.

In contrast to that, the 4% annual coupon bond, you get 4% of your money back after year one. And if now interest rates subsequently over the last year have been risen or have risen significantly, then this gives you an opportunity. Yes, of course, rising interest rates still mean the timing of your bond purchase was not ideal, so you will see a decline in bond price.

But at least the fact that you do get the coupon, the fact that you can reinvest at least part of your money at higher yields now, that softens the blow somewhat, and that's expressed automatically in the lower interest rate sensitivity of a higher coupon bond.

Think about it as it's sort of like softening the pain when you have at least some coupons that are coming towards you that you now can reinvest at higher yield levels. That also means, though, that not only will you benefit relatively to zero coupon bond as yields go up, it also means you're losing out relatively to the zero coupon bond when yields go down.

Because now, in a zero coupon bond, there is no money coming out that you have to reinvest at lower yields. In the coupon bond, that's exactly what happens. You get a coupon back.

If you want to reinvest, you now will have to reinvest it at lower rates. So not only does the price of a high coupon bond fall slower when yields go up, it also rises slower than the zero coupon bond when yields go down.

Again, that's the concept of interest rate sensitivity works for normal bonds in the same way, in both ways.

Okay. And now, third piece, third factor.

And what you see here, it says the yield level.

And so what we've done here is we're not really comparing two bonds here, but instead, we're just calculating the price of our 10-year 4% annual coupon-paying bond and compare it with a linear line. And that's the green dotted line here.

And then you see towards the edges that obviously the blue line moves away quite substantially from the green line. And that basically tells us that the bond price yield relationship is not a linear line, but it's actually a function that is curved.

And more specifically, this is a convex shaped curve, and therefore we call this phenomenon convexity.

And what convexity generally means in this context is that the interest rate sensitivity of a bond changesWhen the yield level changes. So that means for the ratios that we're going to calculate later on, that we cannot just calculate them once when the bond is issued and then walk away.

We actually have to recalculate them, strictly speaking, every time there has been a change in yields, because it is not just dependent on time to maturity, it's not dependent on the cash flow structure alone, but it also dependent on the level of yields. Now, what can we observe here? How's the relationship looking like? Well, see, we have a 4% bond, so let's say we bought this when it was issued at a price of par and yields were at 4%. Now, subsequently, yields are rising, and what we can observe is that obviously, we're losing money simply because the price of the bond is going down, but the blue line, relatively to the green linear line, flattens out a little bit, right? So that means the more yields increase, the less sensitive the bond becomes. And that, in turn, means that every additional basis point increase in yields will cause less financial damage than the previous one.

Right? So here what you can see, we're flattening out and we're losing less for each additional increase in yields.

How does it look like when yields go down? Then we observe the opposite, and that is that the blue line steepens. That means, again, it's not linear, but what happens when yields go down is that the interest rate sensitivity of our bond actually increases. That means every additional basis point decline in yields now gives us a higher gain than the previous one. Right? And that is the phenomenon of convexity, and so basically the way you can remember it for long bond positions, as yields rise, the interest rate sensitivity of a bond declines and vice versa. So that's the idea.

And that's just by looking at charts and lines, and we haven't really done a lot of bond math as of now. So let's change this, and let's actually start looking at the numbers that are shown to us in our risk management systems, et cetera.

And before we go into the different ones, here's just a quick overview. Right? And there's obviously more than the ones that I'm sharing here on the screen, but I think these are the most important ones for everybody to know.

And what the purpose of all those numbers really is, is to make bonds directly comparable with regards to their interest rate sensitivity.

Because, again, the qualitative understanding that we've gained on the previous slides is important, but it doesn't really allow us to always make a quick comparison.

What, for example, if you had two bonds, and you wanted to see which one has the higher sensitivity, and the first bond has a longer time to maturity, which means it should be, if it was only about maturity, the higher sensitivity bond. But it also has a significantly higher coupon, which then means from a coupon argument's point of view, it should be the bond with the lower sensitivity. So how can you now sort of untangle those things and make these bonds directly comparable? And that's exactly where those interest rate sensitivity ratios are coming in. And the ones that we're going to have a look at is Macaulay duration, the modified duration, and the DV01. Now, they all work slightly differently. They have different types of information they deliver, and I think the most significant difference between Macaulay duration on the one hand, and then modified and DV01 on the other hand, is that Macaulay duration really only allows us to compare two bonds. Right? We can look at the Macaulay duration of two bonds and immediately decide whether Bond One or Bond Two has the higher interest rate sensitivity.

It allows us to do that, and that's the main purpose, if you remember, of those numbers. However, again, sometimes we do not just want to see which bond is more sensitive, but we also want to be able to perform some sort of simple scenario analysis, right? Saying if yields go from the current level to a different level, then what is going to be the P&L impact on our portfolios? And that's something that we can only do with modified duration and DV01. Also not in the same way, and we're going to see the differences as we go along, but at least they allow us to do this. The Macaulay duration doesn't really, in by itself, allows us to do that. And then just one more time, the reminder that it's really important to remember that those sensitivity ratios here, neither of them is actually static. That means we need to constantly recalculate that. Why? Well, first of all, remember the convexity slide.

When the yield to maturity level changes, the sensitivity changes. That's number one.

Number two, however, is that time simply goes by, right? And that means when we bought a 10-year bond today, in a year's time, this is not a 10-year bond anymore.

This is a nine-year bond. And as we've seen, the time to maturity is one of the most significant drivers of the sensitivity.

So as time goes by and the cash flows are moving closer to us, or the redemption of the bond comes closer to us, the sensitivity ratios will change as well. So it's something that we constantly need to calculate. Then the question is, how do we actually do this? And we're going to start with the Macaulay duration because conceptually, I think it's the most intuitive one. Right? SoWhat have we observed? We have observed that the sensitivity of a bond, interest rate sensitivity that is, is higher the longer the time to maturity. And remember, we kind of thought, yeah, that makes sense because we have the cash flow divided by one plus yield to maturity to the power of t. And the bigger t is, the more impactful this whole thing becomes. Okay.

Second point is that we also have said that the lower the coupon is, the higher the interest rate sensitivity becomes.

Or if we want to switch this around, the higher the coupon is, the lower the interest rate sensitivity becomes.

And why is that? Or how can we now combine these two? We know that when we get the redemption later, that's generally positive for interest rate sensitivity, right? It increases then.

And the more money we get paid earlier, because that's what a higher coupon means, or the higher proportion we get paid earlier, the lower the interest rate sensitivity is.

And I think it does make sense when you think about this because the redemption is clearly, this is the factor that drives this. But when you think about coupons, one way to think about it is that basically a bond that pays an 8% coupon is not really a coupon bank bond, but you can actually think about it as a basket of zero coupon bonds, right? You can say, well, I have a one zero coupon bond that matures in year one. I have another zero coupon bond that matures in year two, year three, year four, year five, and so on and so forth.

And the higher the coupon is, the more weight, the more weighting is assigned to those shorter dated zero coupon bonds. And if you then think about the interest rate sensitivity of a coupon bond, which we said is basically just a basket of zero coupon bonds, then you can say, well, the interest rate sensitivity of a coupon bond should be the same than the interest rate sensitivity of this underlying basket of zero coupon bonds. And the more or the higher the coupon is, the more weight is put on shorter times to maturity, and therefore those ones, the t is smaller and therefore that should work. So what Frederick Macaulay then have done based on these or similar thoughts, was to just develop a way to make bonds comparable by just making those sort of different cash flow patterns comparable.

And the way he's done that is to calculate the weighted average time to maturity of all the bonds' cash flows, i.e., when on average is every single dollar that bond investors will receive in the future going to be paid? And this is why Macaulay duration is actually measured in years, because it is thinking about time to payment, right? And that is expressed in years or months if it's a lot shorter, or days even, but it's certainly a time component in here. So then the question is, how do we do this, right? How can we calculate the weighted average time until investors get paid? Well, the first thing is we're going to start with when are payments happening and what is the current value of these payments, i.e., we are just rolling out the cash flows of the bond and discount them back to today so that we have the present values of the cash flows. Then each present value of each cash flow is then weighted with time, i.e., in this case, the number of years until the cash flow is received.

And I should say, although it says this at the bottom here, that the formula you see here, strictly speaking, only works when there's annual coupon payments. The math is slightly different when we have semi-annual quarterly, but that's very well documented, and I don't think it's really important for understanding the concepts here.

So we then weight the present value of the cash flow with the amount of time, or with the amount of time it takes to receive this.

And then we sum up all those time-weighted present values, and that is then divided by the simple sum of the unweighted cash flows, or unweighted present values of the cash flows. And now, of course, what is the unweighted present value sum of cash flows? This expression here, take the cash flow, discount them by the yield to maturity to the power of time, sum it all up, and that of course is the price of the bond. So taking these things together, we take the cash flows, discount them, multiply them with the number of years until we receive them, and divide by the bond price, and that then gives us, in fact, the weighted average time to payment. And that is what Macaulay's duration calculates.

And the higher this value is-The more time passes until, on average, the investor gets paid, the more sensitive this bond is to a change in interest rates.

And then there's a couple of useful things to remember.

When we have fixed coupon bonds, and the coupon is not zero, but higher than that, what is a necessity then is that the Macaulay duration will be smaller than the time to maturity because some money will be paid earlier, and that means the weighted average needs to be smaller than the actual time to maturity.

This is very different for a zero coupon bond, where the general rule is that the Macaulay duration should be exactly the same than the time to maturity, simply because there's only this one cash flow, there's no averaging or anything required.

So that's the theory. Now let's have a look at how this works in practice. So we're now taking a five-year bond, paying annual coupon of 2.4%. The yield level was 2.43% when we took that example, and again, annual coupon payments. So we have here the cash flows, one, two, three, four, five years, the cash flows of 2.4% in the column next to it. Then the present value, so this is basically 2.4% divided by one plus 2.43% to the power of time, time being one in this specific case. And then we end up with the PV that's shown here, and then we can recalculate that for all the other cash flows.

So these are the PVs. We calculate the sum of the PVs.

That's the bond price, 99.86. Quick sense check.

Is this approximately right in our view? And I would say we agree with that simply because the yield to maturity is 2.43.

The coupon is 2.4, and as we said before, for the investor to receive a return that exceeds the coupon, the bond must trade below par, and that's clearly the case in this example. So that's the normal bond valuation piece here. And now to the new bit, and that is calculating the Macaulay duration, and that was, we said we're going to take the present value here, multiply it with one, this with two, this with three, and so on.

Again, sum up all those values and then divide the time-weighted present values by the unweighted present value sum. That's this part here, and we get a result of 4.7709 years. What does it mean? It means that on average, every single dollar that the investor is going to receive out of this bond is paid in 4.7709 years. As you can clearly see, not a single dollar is actually paid at that point, but that's the idea of an average calculation.

Okay, and now what we can do is we can calculate these numbers for different bonds, and then we can immediately compare them.

So if we had a second bond here, and that has a Macaulay duration of 4.5 years, what we can say is that this bond that you see on this slide has a higher interest rate sensitivity than the one that has a Macaulay duration of five.

And that's it.

Okay, next number, the modified duration. And as we said, here we're now moving into the area where we can actually do some sort of quantification. We can calculate the impact of a certain yield scenario. And more precisely, what the modified duration does is it approximates the percentage change of the bond price due to a change in yield to maturity.

So usually, what we look at is, in this case, is a one percentage point increase.

And so then we can say, well, if the modified duration is five, then what this means is that when yields go from currently 2.43 to 3.43%, the bond is going to lose 5% of its current value.

Okay, and so we're going to see that in an example in a minute.

The question is, of course, how do we get to this number? Where does this number come from? How do we calculate the modified duration? Well, the good news is it's actually relatively simple to do that because all we need to do is we need to take the Macaulay duration that we have already calculated and then divide it by one plus yield to maturity. Again, this only holds when there's annual coupon-paying bonds.

We might have to make small adjustments if it's semi-annual, quarterly, but leave that aside. So then we get this number.

Now, of course, sometimes people look at this and say, "I don't get it.

How do we get from here to there?" Well, the one thing I think that one could do if one is really not happy with just taking this formula for granted, is really thinking about what does modified duration actually conceptually do.

We know that there's obviously a function that describes the relationship between the bond price and the yield of the bond, and this is something we can express in the yield to maturity formula.

We can then obviously calculate for different yield scenarios and then basically draw this function up as we've done here on this slide.

And so then you think about mathematics many, many years ago in school.

You realize that financial-- Oh, no, sorry, not financial derivatives, but actually the mathematical derivatives now allow you to calculate, for example, the steepness of this function in one particular point.

Let's say we're taking this point here.

So at a yield of 2.43%, we just calculate the steepness of the function at that point.

And then we assume a linear relationship, and then we approximate the bond price for a different yield scenario, so something like this here, and then using this linear relationship that we have forecasted. And if you do take the bond price yieldFormula and then derive it to the yield to maturity and then do a little bit of further rearrangement, then you end up with something that looks exactly like the Macaulay duration divided by one plus yield to maturity. Now, couple of things then are important, and I think the most important one is that it gives us relative change in price.

If we want to know the absolute change in price, we need to take modified duration and multiply it with the current bond price. Now, what difference does it make, right? Relative and absolute change, what's all that about? Well, here we're going back to our example, right? We have our five-year bond, 2.4 coupon, 2.43 yield.

We have the Macaulay duration. I round it here now, but anyway.

So next step is to calculate the modified duration, which is done by dividing the Macaulay duration by 1.0243, given that we have a yield to maturity of 2.43%. That then gives us a modified duration of 4.6577. Now, how to read this number.

This number now basically tells us that if yields go from 2.43 to 3.43%, i.e. 1 percentage point higher, then our bond price is going to lose 4.6577% of its current value.

That's how to interpret this.

That, however, means we cannot really directly conclude how much money we're losing, because for that, we just have to know where's the bond price to start with.

So remember, when we started our observation, the bond price was 99.8603.

So now what we have learned is that when yields go from 243 to 343, then the bond price should drop 4.6577% off 99.8603%, which then means it should drop by 4.6512%. And now it's utterly confusing because this is percent, this is percent. What's going on? Well, the problem is, the reason why this is a little bit confusing when you look at it for the first time is bond prices usually are quoted in percent, right? So if we would just take the percentage off here and just say, well, let's lose absolute numbers, then we can take that percentage off there as well. And then you see, okay, that means if I invested $100 in this bond, then I'm going to lose $4.6512, and there we are. Okay.

What this also means is now that we can calculate the predicted bond price for a yield to maturity of 3.43%, and that is, we take the 99.8603, which is where we are right now, and then we subtract the 4.6512.

That's the absolute change in bond price.

And then we say, okay, the prediction using modified duration is 95.2091.

Is that accurate, though? And of course, you know the answer. It is not.

Because what we are ignoring here is convexity.

As I said when we discussed modified duration, it approximates the value of, or the price of a bond at different yield levels using a linear or assuming a linear function. And we have already discussed that the function is not linear, it's actually convex, meaning this value here that we're predicting is going to be okay for relatively small changes in yields. And it's also going to be relatively okay for bonds that are not really that convex.

And convexity, by the way, is driven by the same factors then duration, i.e. the longer the time to maturity, the higher convexity is, the smaller or the lower the coupon, the higher convexity, et cetera. So that's then the point. These results, these predictions should be taken with a pinch of salt, especially when you kind of want to forecast bond prices at a much higher or much lower level of yields than where they currently are, then the prediction error might actually become quite significant.

So how would one of course do this in reality? Well, we would just build a bond valuation application and just type in the bond's cash flows and discount them with whatever yield we want to assume, and that would give us the yield level for-- Oh, sorry, the bond price for that exact yield level. That's I think the easiest way.

We could of course use Taylor expansion and similar things here as well, but in reality, I think just repricing the bond for a different level of yield seem to be the much easier way of doing this in this particular case.

Okay, so that's convexity.

And now to the, I'd say, way in which practitioners are normally focused on interest rate sensitivity.

As we've seen before, modified duration allows us to get a view on how significant a certain yield move is going to be in P&L terms.

It should only be used, though, for relatively small changes in yields. And then the DV01 takes that small change argument here and looks at the impact on the value here of our investment by-- Oh, that's driven by a change of yields by one basis point only.

So, a relatively smallChange in yields here.

So this linear approximation thing that I said earlier is not really an issue in this specific case.

So that's fine.

The other thing that DV01 does, and that is different from the modified duration, is that it doesn't show us the percentage change in price or anything like that, but it actually shows us the actual P&L impact in monetary units. So it tells us how many dollars are we going to lose, or how many dollars is the negative P&L, or is the P&L impact should yields go up or down by one basis point, right? And so that means not only does it take the absolute change into consideration, but in addition to modified duration, it also includes the size of the position, right? Because obviously, to calculate a relative change in price into an actual monetary number, not only do we need the current bond price, but we also need to understand how much notional are we holding.

And that's then why you can calculate the DV01 of a bond by taking the modified duration and just basically modifying it a little bit further.

We use the modified duration, as I said, as a starting value, which shows us the relative change in bond price. We're then multiplying with the bond price, which gives us the absolute change of a bond price. We're then weighting that with 0.01% to basically say we want to understand the impact of a one basis point change in yield. And then we have the absolute change in price, and now we want to weight this with the face value, with the notional amount, to get to an actual monetary value. And that means that DV01 really takes both factors into consideration, the sensitivity of the bond and the size of the position, and therefore is, of all those ratios that we've seen here, probably the most complete number.

Okay, so DV01. There's a question here if DV01 assumes notional is USD, what if the bond is in another currency? Well, it's a bit misleading the term DV01, i.e.

dollar value of a basis point, because even in a euro portfolio or pound sterling or Japanese yen portfolio, you would calculate the DV01 then strictly speaking not the dollar value, but the yen value or the euro value or the pound value of a basis point. The generic reference, however, is dollar. It doesn't mean we're always calculating this in US dollars. You could, of course, do that if you, for example, are a US-based bond investor that is also investing in euros, then you could calculate the DV01 in euros first and then translate that using the current FX spot into a dollar number. But that is probably not the best way of looking at it. So yes, DV01, while strictly speaking stands for dollars, it doesn't mean we're always calculating in US dollars. Usually, we use a local currency to express the risk. But good question. Okay, cool.

DV01, just to wrap this up, let's have a look at how we're calculating this. And as we've seen, we use the modified duration. We then multiply it with one basis point, the price of the bond, and the notional value.

We're assuming here a face value that we've bought of 100 million.

And then when you do the multiplication there, you get to a value of -$46,511.80, negative sign, meaning if we're holding 100 million face value of that bond currently trading at 99.8603% and yields go from 243 to 244, then the bond price is going to drop in a way that the value of our bond position will have decreased by exactly $46,511.80.

Now, that is obviously the most practical, the most workable risk number there is. One thing that I have to add here, though, is none of the ratios we've discussed actually assigns any probability to a specific yield change.

It's just basically a scenario analysis.

If yields change in a certain way, then impact on price is X. It doesn't say how likely yields are to move a whole percentage point, what's a certain probability, so 95% or 99%, the worst case shift in yields. It doesn't do anything like that.

So it's not something that allows us to forecast the value at risk directly, but of course, it's closely related.

Because all we now have to think about to get from DV01 to the value at risk is what is the sort of worst case change was if 95% or 99% probability over a certain time frame, and we can express it a number of basis points and then linearly scale this upward down.

So that is possible. However, as I said, linear scaling assumes linear lines, so it ignores convexity, but over relatively small changes in yield, it doesn't matter all that much.

Okay, so now next step we have to do is think about how do we do this on a portfolio level? Because so far we have just looked atWell, single bond positions, right? And which is fine.

Before you invest money in an instrument, you want to know the sensitivities of it. But normally, most investors do not just have one bond on their portfolios, but actually quite a significant number.

And so then managing this or being aware of the risk on a single bond position or on a single bond basis is probably not going to be the most efficient way. So how can we express this on a portfolio level? First of all, what comes to mind is we calculate averages, right? So we know the Macaulay duration of every bond we have.

We've calculated that, and we know the modified duration of every bond we have.

And so now what we can do is just calculate what's the average Macaulay duration of the portfolio. So we take the portfolio weight of bond A and multiply that with the Macaulay duration of bond A, and then we add the portfolio weight of bond B times Macaulay duration of bond B, and that gives us then the average Macaulay duration of our portfolio.

We can do the same thing for modified duration, and that means we're taking the portfolio weight of bond A, the modified duration of bond A, and so on and so forth. And then we have the average modified duration of bonds of our portfolio, and if that is seven, then we know that, on average, when the yield goes up one percentage point, then our portfolio will lose around 7% of its current value.

That's fine, but it sort of ignores the fact that or is not really telling us exactly where our risk sits. Because when you talk about fixed income, then immediately you think about the yield curve, right? So you could now construct a portfolio that has a modified duration on average of seven, by just investing in bonds that have a modified duration of seven that are all sort of having around seven-year maturity and low coupons, right? That's one way.

You could also get to a seven-year modified duration by, for example, buying a few bonds that have two years time to maturity and then a couple of bonds that have 30 years time to maturity. The average gives you seven maybe, but clearly the risk you're exposed to is very different, right? So the problem with all these averages here, as you can see, is that they will just sort of give you a sense of the portfolio dynamics assuming a parallel shift in the yield curve, i.e. the same change in yields all across the maturity spectrum, which of course isn't really a very practical assumption. So it's not the most ideal thing, so you might want to look for a little bit more detail in the risk report. And so in reality, what we often see is that those risk reports give us not just an average number here or total number, but they give us something that is called interest rate delta letter or DV01 vector or something like that. So it basically shows us the DV01 values for specific yields with a specific time to maturity.

And the way these reports are then typically created in reality is that, first of all, we design the report, and we think carefully about which interest rates do we actually want to look at. And then, of course, it depends a little bit on how the granularity with which you're going to look at these rates will depend a little bit on which part of the curve you're focused on.

So when you're thinking about the short end of the curve, you will naturally be a little bit more granular than at the much lower end, longer end of the curve. But let's say here we have basically decided we want to see our DV01 values for one year, for two years, for three years, four years and five years.

And then what we're going to do is we're going to pull the current yield curve. So one, two, three, four, five, let's say, right? And then with that yield curve, we're now valuing our whole portfolio using these yields, and calculate the starting value of the portfolio, and let's assume that's $100 million. And now what we're going to do is we're going to manually shift the one-year point of the curve up by one basis point, right? That generates a new yield curve, which I'm drawing in here now, right? And then with that new yield curve, we're now recalculating the value of our portfolio.

And let's say in the one year, we now see 99.98 million. Then what we can say is obviously when the one-year yield goes up by one basis point, we're losing $20,000. So the one-year DV01 is -20,000. Now we bring this yield back down, and now we do the same with the two year, so bump it up by one basis point.

And then again we have another yield curve that we then use to revalue, and if the two-year point has now 99.99, then we can say, okay, here is obviously a 10,000 DV01 at the two-year point, and we can do this for all further maturities and then look at the risk in this specific way, then what we can say is that not only do I know the risk to a parallel shift in the yield curve, but I can also really see where to which parts of the yield curve are we really exposed, right? Because here, if this was indeed zero, then we have 30,000 DV01 on our portfolio in total, but we can also see that this risk is concentrated entirely at the short end of the curve, so in the one and two year sector.

And that's then really how in reality practitioners would look at the risk or would get their risk reports generated.

Now, of course, we need to think about some sort of interpolation techniques being used because of course we will have bonds that are somewhere between one and two years and so on and so forth. It's not quite as simple, but at least generically speaking, that is the idea. And that, ladies and gentlemen, is really all I wanted to share with you here today. So thank you so much for your participation. Hope you found this session beneficial