My name is Thomas Krause. I'm head of financial products here at Financial Edge, and I have an hour to take you through this session here today. So what exactly are we going to, um, cover then today? Because bond mass certainly is a relative extensive subject. So what we're gonna spend our time with is we're going to basically talk about interest rate, risk, and sensitivity. That means we're going to start with a quick look at the concept of interest rate risk, and what sensitivity measures in general are all about. We will then discuss, uh, duration, starting with intuition behind, um, macole duration, and then moving on to modified duration from there. And towards the end, we'll give a high level introduction to the concept of convexity, uh, and also of the dollar value of a basis point or DV oh one. Um, before we start though, couple of important reminders. As always, you are first of all, able to access the course materials via, um, or for download. But without further ado, let's get, uh, started with today's content. And as I said, we're going to look at the general concept of interest rate risk, um, to start with. And, um, we've known, or we've learned in previous sessions or, you know, uh, at another day that, um, bond prices and yields are generally inversely related, at least if we're looking at, uh, the classic plain vanilla fixed coupons, um, bonds, right? And then, because most investors, I would say approach the bond market from the long side, so they are either long bonds or they have no positions in bonds. And yes, of course there are investor types that, um, also approach the bond markets through outright short positions, but I'd say the vast majority of investors is probably on the long side. Um, it does make sense then that, um, the most often read definition of interest rate risk that you come across with regards to, uh, bond investments is that interest rate risk is generally seen as the risk that bond prices will fall IE leading to a reduction in value of existing positions. And that is, will be the result, obviously, of a rise in yields because of that inverse price yield relationship. So what we're gonna do here as well, we're gonna adapt, adopt this, um, you know, this definition and we're gonna have a look at the risk that bond prices will drop when yields, um, rise. Now this of course, is, uh, very important for bond investors to be aware of this mechanism, but I think, um, that is relatively obvious. I would also suggest, though, that it doesn't suffice, um, in terms of information bon deon's price behavior because what investors in addition, um, certainly want to know is if yields move up, um, then how much will my bond price actually drop? IE what is the magnitude that I should expect of, um, the price move for a certain move in yield? And that's then where the interest rate sensitivity or generally refer to as duration. And if you wanna be precise here, you should call it rates duration because there's also a credit duration that then looks at the bond price change as a change in credit spread. But what we wanna do here today is we're ignoring credit risk as such. So we're assuming basically that we're looking at government bonds only here. And then the only market risk that you take there, uh, from that perspective is the risk of a change in the credit risk free rate. IE we're talking about rates duration only credit duration will come in a different, um, session. Okay? So the idea of all those sensitivity, uh, ratios of rates duration is to give investors information on how much a bond price is going to respond to a certain move in yields. Um, before we then go and have a look at the different ratios that, um, are more or less regularly used here in practice. Um, I'd like to approach this a little bit from an intuitive point of view. So before we look at the mass, let's have a look at how bond prices, um, behave. And so what we're gonna do over the next three slides is we're gonna have a look at different bonds, and we're gonna see how their prices behave when the underlying yield to maturity, uh, changes. And the first thing we're going to do is we're going to have a look at two bonds that have different types or times to maturity. We're comparing a 10 year bond with a two year bond. And so that we're isolating the impact of time to maturity here. Um, we are also using zero coupon bonds, so fixed coupon bonds that pay a coupon of exactly zero. And that means obviously that the return is generated by investors by usually buying these bonds at a discount. Then at maturity you get the, uh, p amount back, and the difference is then the return that you have achieved over the holding period, you can turn that into an annual number, uh, and so on so forth. But sort of we are just, as I said, looking at those zero coupon bonds here just to make sure we're only focusing on the impact of a different time to maturity. So what we're then doing on this, uh, graph here is we're calculating the, um, price of those two bonds for different scenarios of yield to maturity in, uh, 25 basis points, steps from zero to 7%. So let's start on the left hand side here. We're looking at the prices of these zero, uh, Coupon bonds at a yield level of zero. And then it does make sense that both bonds trade at par, because remember, when the yield equals the coupon, then the bond price should be at par. And here of course, also another way to look at this is just the case that there's only one cash flow that these bonds will have, and that's a repayment of the notional amount at maturity. And when yields are zero, then there's no discounting effect. So you can obviously discount the payments with a yield of zero and there will be not any change. So the future value equals the present value, and as the future value is a hundred percent, then the present value, uh, either price of the bond, in this case we'll be a hundred percent as well. So both bonds at a hundred at that point, then we start shifting yields upwards and no surprises here. In general, both bonds are falling in price because of that inverse price yield relationship we have referred to earlier. But what is also very obvious is that not both bonds are falling at the same rate or speed, right? So we can see that the decline of this line here is much steeper than the decline, um, of the green line. And so if we look at the bottom in the legend here, we see that the blue line, the one that declines faster is the 10 year zero coupon bond, and the green line is a two year zero coupon bond. And what we can conclude from the data that we have in front of us at that point is, at least when we're just looking at these two, um, that the longer the time to maturity is the higher is the interest rate sensitivity of a fixed coupon bond, meaning for the same change in yield. So let's say from zero to 1%, the blue line, um, drops a significantly faster. IE the price of the 10 years zero coupon bond drops significantly faster than, uh, the green line, which is a two year, uh, zero coupon bond. And that's what we said, uh, is what interest rate sensitivity is actually all about. So how can we, uh, you know, obviously we can, we can conclude this from the data, but what's the rationale, uh, behind it? And I think in this particular case of zero coupon bonds, it really makes a lot of sense to look at it from a mathematical point of view because it's relatively simple, uh, to explain what's going on. We said already that a zero coupon bond has one cashflow only after we paid for it, obviously in initially. So that means, um, the future value of this bond, IE that repayment at redemption is gonna be par at least, um, in the majority of all cases, that should be the case. So we can then, um, say that any bond, you know, um, zero coupon or, or, or a non-zero coupon bond, um, you know, what we can say about the price of any bond is that it's basically nothing else than the sum of the present values of all the cash flows that the bond generates. So we can say, let's discount all future cash flows back to today, sum up those present values, and that should be the bond price. Now, because a zero coupon one has only one cash flow, this operation gets fairly simple because in general we can say a future value divided by one plus yield to maturity to the power of time and t is obviously gonna be two for the two year bond and 10 for the 10 year bond. And that's where we can see as to why the response is different because the 10 year bond price will impacted by the change in yield to the power of 10, whereas the two year bond will only be impacted to the power of two. So relatively obvious, just looking at the, uh, simple bond price formula here, um, that the 10 year bond will have a larger, uh, sensitivity. So that is something we should remember. The predominant factor, uh, of interest rate sensitivity is for fixed coupon bonds, I should say, uh, is the time to maturity. Now let's have a look at two other bonds. This time we're using two bonds that have the same time to maturity, so we're keeping the time to maturity the same, but we're chain or looking at two bonds with different coupon levels. And this way we can now take an isolated look at the impact of the size of the regular cash flows, IE the size of the coupon. And what we're doing here specifically is we're looking at a 10 year zero and a 10 year bond that pays an annual coupon of 4%. Now we're doing the same thing again, we're calculating the bond price, um, for different levels of yield to maturity, and we're plotting, uh, those prices here in the chart. And the first thing that we can see is that obviously both lines, um, never reach, right? So they, they, well, well, at least not across the range that we're, that we're looking at. And that's very, um, or not really surprising simply because we're comparing two bonds with fairly different payment patterns here, right? We have bond one, the two 10 year zero bond here that just pays a hundred, um, at maturity. And then we have bond two that pays not just a hundred at maturity, but it also pays 4% after year one, year two, year three, and so on and so forth. So that is a very different set of cash flows. Arguably the 4% coupon paying bond gives investors much higher cash flows than the other one. And therefore, all else being equal, the price for this stream of cash flows should be higher. And that's what we see. The green line is above the dark blue line, meaning that the 10 year 4% coupon bond is actually more expensive than the zero coupon, uh, bond. Again, not surprising but worse. Um, just, you know, um, considering this. And then we can look at where will both bond prices be at a yield level of 0%. Um, the zero coupon Bond we have already talked about this, it's gonna be at par. The, um, coupon paying bond is gonna trade at 140%. And that's because not only is the redemption payment not having any discount impact, but also all interim coupon payments will, um, basically not see any impact of discounting. So we're simply summing up all the, um, quick cash flows and of the bond, and that gives us in total a hundred and um, 40. So that's, uh, the starting point. And now when you just look at the lines here, what's relatively tempting, uh, to say is that the green line, uh, falls steeper or falls, um, you know, or declines in a steeper fashion than the blue line. And that's why this is an indication that the, um, that the, uh, bond price of the coupon bond actually falls faster than, um, the zero coupon bond. And while this is easy to sort of, um, get to that conclusion by looking at the chart here, it's maybe a little bit misleading because what you have to consider is that both bonds will trade at fairly different prices to start with. So yes, we're seeing a, a, a larger decline in terms of, uh, prices here, but we're starting at 140, um, in the case of a coupon paying bond rather than at a hundred percent. So we're having a different basis to start with. So what I think is a better approach to a look at this is then, rather than just looking at the chart on the left hand side, we're now gonna, uh, look at the, uh, right hand side where we created another, um, graph or set of graphs for you. And so basically what we're doing here is we're just looking at the percentage change of the bond price for those different steps in, um, yield to maturity. So basically what we're doing here for this one is we're changing yields from zero to 0.25%. We then calculate the new bond price for both bonds, and then we calculate how much has the price declined in percentage terms from a yield of zero to a yield of 0.25. And when you look at this, um, graph here, what you see is that the blue bond, the 10 year zero coupon bond for also scenarios that we're looking at here, has actually seen a larger percentage decline in price than, uh, the green bond. And arguably what you care most about, um, as an investor is the amount of money that your, the value of your portfolio has gone down and you don't care that much about some sort of like change in, in price that might have absolutely been a little bit bigger here in the, uh, in, in, in one bond over the other. So that then allows us to conclude that the higher the coupon, because obviously we're now looking at the 4%, uh, coupon bond here, the lower the interest rate sensitivity, or if you wanna, uh, spin this the other way around, we can say the lower the coupon, the higher the interest rate sensitivity of a fixed coupon bond, um, is going to be. And that's an important, uh, thing to remember, um, as we will sort of come back, um, to that as well. Before we, uh, then go to the next thing, I just want to maybe, um, give you some, um, intuitive explanation as to why the coupon size, um, leads to a reduction of, uh, interest rate sensitivity. And I think the best way to think about it is by just really, um, going back to what you've been, uh, explain in the context of zero coupon bonds versus fixed coupon bonds. And that is the reinvestment risk, right? So the main difference between a zero coupon and a fixed coupon bond that doesn't pay a coupon that's not zero, is that, well, one difference at least, um, is that this fixed coupon bond has reinvestment risk, right? So after, I don't know, six months or 12 months, whatever the payment frequency of the coupon is, um, you will receive as an investor in that bond, a specific amount of your investment back. That's the coupon payment. Nothing wrong with that, but what you have to think about is if you had originally a 10 year investment horizon, and now after six months, you will get the first, um, you know, couple of percentage points back of your investment as a coupon, you might wanna, you know, as an in, at least as a rational investor that wants to, uh, have a 10 year or uh, investment horizon for all of their investment full time is, um, basically you have to reinvest that, um, first interim payment at the nine and a half year yield that is then observable in six months time. And as we don't know from today's perspective where exactly that's gonna be, that is, uh, leading to reinvestment risk and strictly speaking, the final return of a fixed coupon bond position and will only be known once a last coupon payment has, no, sorry, the second to last coupon payment has been reinvested, that's when we can finally or calculate, uh, the, the overall return of our bond investment, um, you know, with absolute certainty. And that is obviously, um, the reinvestment risk that, that people are talking about. Now, um, important though to remember that reinvestment risk, like any risk doesn't necessarily mean something is, is purely, you know, bad is it's not purely negative because it's just uncertainty, right? Because what could happen, and that's a place where, where, where, where we're going back to the slide now is that yields could go up, right? And then of course, um, while this will have a negative impact on your overall present value of cash flows, What at least will, um, mean for a coupon bond is that the interim payment that you have received, you will now be able to reinvest at higher interest rates. And if you had a zero coupon bond, then you don't get any money between, um, making the investment and the redemption of that bond. Um, we're assuming we're holding the bonds to maturity here in everything we say, uh, then, um, you know, you will not benefit from that increase in yield. All you will see is a decline in, uh, the present value of your cash flows. And so basically you could argue that the higher the coupon is, the more than the amount of money that you will be able to reinvest at higher rates when yields go up. So that softens the blow of an increase in yields to a certain degree, and I hope that's, um, intuitive. Okay. Um, then the last thing that we need to look at is, um, this chart here, and that is where we're looking at how the, um, sensitivity of a bond changes when the level of yields change. So what I've done here is I've taken the, uh, price yield, um, function, um, for the 10 year, 4% coupon bond, calculated the price of these bonds at the different levels of yield. And then I just put a green dotted linear line through this because all we try to do here is to demonstrate that actually the bond price yield relationship is not giving us a linear line. So if we calculate the yield or the prices of bonds at different yields, that doesn't give us a linear, um, you know, line. It is actually a, uh, curve and that has a convex shape. It's not extreme, but it certainly shows elements of, um, convex pieces here. And, um, what do we have to, um, do then in terms of, you know, interpreting this chart? And I'm just starting here at a yield level of, um, 4%. And, um, what I'm going to do is, um, now saying, okay, if I have a 4% annual coupon paying bonds, then um, there's, um, you know, at a yield level of 10%, uh, this bond will trade at par because the coupon equals the yield. If we're now kind of looking at, um, what happens when yields go up all the way to 7% as we see, and we would expect that the bond price declines, but not in a linear fashion. So the curve, sorry, the, the, the, the curve that we're looking at here, um, flattens out a little bit. And what that means is that it's relatively good news for the bond investor because in the sort of bad case or in the, in the, in the undesired case, that yields will go up after you bought a fixed coupon bond. The one good news that there is for you is that the sensitivity of your position will go down automatically, meaning that any additional increase in yields will lead to a smaller loss, um, relatively to the previous one, right? So the position automatically downsizes not dramatically, but by certain, uh, to a certain extent as yields go up. On the other hand, when yields decline, what we see is that the increase in price, um, gets faster and faster and faster. So every additional decline in yield leads to a higher increase in a price of the bond than the previous one. And that is, um, then again, um, beneficial because when the market starts moving in the right direction, then the size of your position, the risk of your position automatically, uh, increased. Um, just want to interrupt this for, uh, answering a a, a quick question here. There was a question of how does the coupon rate factor into the, uh, present value formula? And I think, you know, obviously we've looked at the zero coupon, um, formula. If we had now a coupon, uh, then we will obviously have more than one cashflow. So if I just take, you know, for simplicity here, now a, a two year, um, bond, and let's say we have an annual coupon of, I don't know, uh, 3% here, then we would have the following, um, calculation to do, we would have the 3% coupon divide that by one plus a yield to maturity to the power of one. And that is because, um, you know, this coupon is paid in one year, so that's the present value of the first coupon. And then we have 103, which is basically the, you know, redemption payment of par plus the 3% coupon of the second year. And that has to be divided by one plus the yield to maturity to the power of two, because that payment is two years away. Sum up the pvs and then basically that will be, uh, the bond price. And of course if you had semi-annual coupons, the formula would change a little bit and quarterly as well. And, and if there was more than these two, there would just be more cash flows to discount. But that's basically how you would, uh, consider coupons in that, um, present value formula or in the bond price formula, I should say. Okay, so we've been through this and because, um, this is a convex shaped curve, we call the whole concept convexity. And that is an important one because, um, what we can take away from this is that when yields go up, the interest rate sensitivity of a bond declines and vice versa. And because yields change, you know, almost constantly, even though not maybe in, in, in major amounts, but there's daily change in yields almost, um, then, uh, what we can say is that obviously we have learned from this that we cannot, no matter which of the sensitivity ratios that we're gonna discuss next, we are going to use, we cannot just calculate them once and then, uh, never ever do this again. Because strictly speaking, every time the yield changes, those sensitivity ratios or the interest rate sensitivity of a bond in general is going to change as well. Not in an extreme way, but there will be an impact. And if we want to be a hundred percent accurate, then we have to recalculate at any time the yield changes. And there's also another reason, uh, why we need to recalculate, and we're gonna see this on the next, um, slide where we now start to talk about the actual, um, sensitivity ratios. Because again, um, like I said earlier, you know, having that understanding of what drives interest rate sensitivity, like what, uh, drives prices and stuff is obviously absolutely important for bond investors. But again, it's probably not, um, a very useful tool to just, uh, look at these three charts that we've seen and, um, you know, remember how bond prices or how, how sensitivity generally depends on certain factors, because sometimes you want to compare bonds that are sort of having conflicting, um, sort of indicators here. So for example, there's uh, two bonds we want to compare. One is a 5%, uh, sorry, a five year bond. That is a zero coupon bond. Then the other one is a six year bond, but pays a coupon of 2.5% every year. So then of course, you know, you attempted to say, well, um, the six year bond is longer, so that indicate has a longer time to maturity. So that indicates it has a higher sensitivity, but it also pays a higher coupon, which sort of might indicate it has a lower sensitivity. So then obviously just knowing these general concepts don't get you very far. You want this one number or one indicator to look at. That then allows you to make a qualified statement here, um, with regards to which of these two bonds has a higher interest rate sensitivity, IE which price will move more for the same change in yields. And that's exactly where those interest rate sensitivity numbers are coming in. They make bonds with different maturities and different payment patterns. IE different coupons, um, directly comparable. And there's many of them and probably the three most often mentioned and used ones are here on the slide, the Macau duration, the modified duration and DV O one. Now it's not an exclusive list, there are others, but let's, uh, focus on these three here. And, um, Macaulay duration maybe is not something that is very heavily actively used anymore, but it's a nice concept to understand the intuition behind, uh, those ratios. And there's also a good starting point for them extending out to others. Hence we're covering this here. So what all those three numbers allow you to do is to compare directly the interest rate sensitivity of different bonds. So if you have those two bonds that I just described, five, six with zero and 2% coupon, you can calculate the Macaulay duration or the modified duration or the DVO one and the, uh, you will, with just looking at the numbers for these two different bonds, be able to say, this bond has a higher interest rate sensitivity. That's what they all allow us to do. However, two of them go one step further, and that is the modified duration and the DV oh one. And they also allow us within certain restrictions to estimate the p and l impact or the monetary impact of a specific interest rate scenario. So say if yields will go up one percentage point, how much will this five year bond lose? And how much will the six year bond lose? Right? That's sort of like the extra piece of information we can quantify, uh, the outcome of the price, uh, or sorry, the outcome of the yield change here with modified and DV oh one. And then I said already on the previous slide, um, before we start looking into the calculation, it's important, really important to remember that none of these sensitivity ratios is gonna be static. And the first thing we've already mentioned there is convexity. So the sensitivity partially depends on the level of yields, and when yields change, then the sensitivity changes and therefore the, um, sensitivity ratios change. But another factor is that over time, as time goes by, and not only do yields change, but also the time to maturity, right? Because every day that passes the bond gets one day closer to its, um, maturity date, all cash flows are moving one day closer, uh, to us. And therefore we have to recalculate the sensitivity at least daily, even if yields haven't changed simply because then, uh, time has shortened, and with that, the sensitivity has shortened and any fixed coupon bond will, um, you know, converge to an interest rate sensitivity of uh, zero. Uh, and that's gonna be achieved on the day of, um, maturity. Okay, enough about, uh, the general stuff. Let's have a look at, um, the first number here, and that is macauley's duration. So Macauley was probably the first duration measure. It goes back to the 19 late thirties, I believe. And um, and it's based on these observations that we have made earlier by just looking at those different bonds and, and how their prices behaved. Um, because remember what we concluded, we said first of course, the longer the time to maturity of a bond, the higher is its sensitivity. And then we also said the lower the coupon of a bond, the higher its sensitivity. That's just the flip side of the conclusion that was on the slide earlier. And so, um, what both of these conclusions have in common is that the more money of the bond's cash flow, though the, the bigger the part of the bond's cash flow that's paid later, um, the higher the interest rate sensitivity actually is. And so that was for one zero coupon bond was a large or a longer time to maturity has higher. And then if a bond is, uh, having a lower coupon, that means more money is paid relatively late because relatively small amounts are repaid early and both end up with, uh, having a higher interest rate sensitivity. And that led McColey, um, to the following approach saying, what I have to do is basically I have to, to, to make bonds with different payment patterns comparable, I have to calculate the weighted average time to maturity of the bond's cash flows, IE when on average, uh, all the payments of the bond made. Okay? And so that number is measured in years because we are actually measuring the distance in time between today and when on average we receive all those cash flows. And then of course, following what we said earlier, what we concluded by looking at the charts earlier, we can say that the longer the macauley duration IE the longer it takes to receive on average all the cash flows of the bond, the more sensitive this bond, um, should be to a change in interest rates. So, um, how does it look from a mathematical point of view here? As we said, what we need to do, we need to calculate the weighted average time to maturity of each of the bonds cash flow. So what we're seeing here is first, um, you know, in this, in this small red dotted, uh, box here, the cash flows of the bond, that will be the coupon payments, each coupon payment as seen on the, um, earlier slide, and then plus the redemption payment of a hundred or par at maturity. We are then taking all those cash flows and discount them by one plus a year to maturity, which is here expressed as multiplying with one plus yield to maturity to the power of negative time. And that is nothing else but calculating the present values of each cashflow. And then each of these cash flows will be weighted with the time in years until the cashflow is received. Now, one point I should, uh, I want disclaimer here, this formula only applies for annual coupon paying bonds. If they were semi-annual or quarterly payments, the mass is gonna look slightly different, but we're not here to show you the complete set of formula. We just want to talk about the general ideas behind. So let's stick with this annual example. So take the cash flows, discount them back, and then multiply or wait them with the number of years until they will be received, sum this whole thing up and then divide by the present value, the sum of the present value of the unweighted, uh, cash flows of the bond. And that ladies and gentlemen, is nothing else but the dirty price of the bond. Hence we can simplify this expression, uh, to, um, the dirty price of the bond. So that's the idea. And let's have a look at a concrete example simply because that always makes things somewhat less abstract. So we're looking at a five year bond here, annual coupon payments. So we're reducing, or we have five cash flows here that are happening, or cash flows happening at five different points in time. Annual coupon is 2.4% yield to maturity, 2.43. Now, before we do anything, let's remind ourselves where would we expect the bond price to be relative to par inverse price yield relationship, right? So we have a coupon of the bond here of 2.4%. The yield is slightly higher, that means the bond must trade at a, you know, small discount. So the price should be below, um, par accrued interest. Doesn't matter here because we have, um, an exact time to maturity of five years in the annual coupon bond, so there's no accrued interest. But what do we get then when we, um, build the cashflow table here, obviously 2.4 after year, 1, 2, 3, 4, 5. And then in uh, five we get also the a hundred back. So 102.4. And now what we're gonna do with each of these cashflow here, so example for the first one, 2.4 divided by one plus 2.43% to the power of one, and then the second cash flow is 2.4% divided by one plus 2.43 to the power of two and so on and so on and so on. And so basically that gives us then these PVS here, um, which then we can sum up. And that gives us 99.8603. That will be the bond price, that's the price of the, um, you know, as a sum of the unweighted cash flows. And that's in line with what we were expecting. It's below par because the yield is above the coupon, right? Then we do the whole weighting bit that we have set, because that's now missing. We have to multiply this present value with one for the first cash flow with two for the second, with three for the third. And so on, sum that up. And now we have the two numbers that we need to, uh, divide by each other and we take, uh, the big number first, obviously that was the, uh, time weighted, um, present value of cash flows divided by the bond price. And that gives us a result of 4.7709. And I said Macaulay duration is expressed in years. And so this bond will have a macaulay duration of 4.7709 years. And now the way to use this number in practice, if we had another bond, so bond two here that has a duration of let's say 5.1, then what we can say is that this bond here was a 5.1 year macauley duration has a higher interest rate sensitivity than the bond we're looking at here on this chart, uh, on the slide. And that means it will change in price more dramatically for the same change in yields. Okay? Um, that's how we can then interpret the Macaulay duration. So we're going back to the slide because there's two pieces of information that I wanted to also share with you. Um, and then we can obviously think what should be the macaulay duration of our zero coupon bonds that we've seen earlier. So a 10 year zero coupon bond, what's the Macaulay duration? Yes, it's right, it will be 10 years precisely, because remember, there's only one cashflow and that is happening 10 years from now. So all money is received in exactly 10 years. Macaulay duration will be 10 years. However, as soon as the coupon is not zero, but it's a fixed coupon of above zero, then some money will be repaid before 10 years are over. And that leads then to the shortening of Macau duration. And of course, the higher the coupon is, the more money will be received relatively earlier by the investor, and that leads to a shortening of duration. Okay? So that's, um, the Macaulay duration. Now it's obviously a fairly intuitive solution, but it has that one downside, and that is that other than comparing bonds with regards to the interest sensitivity, we cannot really do much more with it. And, um, it's, you know, important for investors to have as much information as possible, um, not just with what's gonna happen in the markets, but also, uh, with regards to how does my risk, um, situation currently looks like. And so investors might be keen on getting additional information, and that's why then we probably in practice tend to use modified and DV L one, um, a little bit more often because as I said, what they allow us to do is sort of to get a sense about the p and l impact of a certain yield scenario. Okay? So let's talk about modified duration now. And in general, what it does is it approximates the percentage change of a bond price that results from a change in yield to maturity. And it's called modified duration. At least this is my theory, and I think, um, that because we can calculate it by just taking macauley's duration that was just calculated on the previous slide, and then divided by one plus the yield to maturity. Now, so long and hard, whether or not there's some sort of like intuitive link on how to explain how we get from the coli, um, to modified duration just by dividing by one plus heel to maturity, I haven't really succeeded yet. If you have any sort of ideas, let me know. But generally speaking, um, I think, you know, what you can do if you really want to prove this to yourself is you can, um, think about this from a mathematical point of view and see what modified duration does. It basically attempts to forecast, uh, really the value of a function or the output of a function and the function being here, the, um, bond, uh, price formula, um, for a variable or varying, uh, input of the yield to maturity. And whenever you hear something like that, you think probably the term about the term derivatives, not the financial kind, but this time the mathematical kind. So what modified duration, general underlying concept is, is the first derivative of the bond price yield formula, uh, with regards to the yield to maturity. And then it's practically basically just drawing a linear tangent line through that, um, price yield of work function that we have, uh, seen earlier here. And so, um, that's sort of, um, how you could approach this. And then when you do this mathematically correctly, you end up with something that looks like Macaulay duration divided by one plus yield, um, two maturity, or you just accept it and move on. Um, now the important thing, um, that one needs to remember when we're looking at modified duration is that it calculates the percentage change of a bond price, right? And this is a little bit confusing because bond prices generally are quoted in percentage terms. And so relative change of something that's quoted percentage, that's a source of confusion. So again, let's look at a practical example, and here we go. Our same bond here, five years, 2.4, 2.43 yield McColey duration, we have calculated 4.77, uh, and now we're calculating the modified duration out of it by just dividing by one plus a yield to maturity. And that gives us a modified duration of 4.6577. And what this means is if yields were to change by one percentage point, so yields go from 2.43 at the moment, two, 3.43, then the bond is going to change by a 4.6577% of its carrying price. Why negative? Well, I've added a negative here in front of the a modified duration, and that's because remember first page or second page, I said we're gonna look at the impact of an increase in yields, and that will be a decline in, um, bond price, right? So take the modified duration, put a negative sign in front of it, multiply it with the expected yield change, uh, that was one or we see fo uh, simulated yield change, if you wish, that was one percentage point, and that gives you the relative change in price. So the bond price will drop 4.6577% off its current price, which is, um, not shown here, but we have it on the previous slide. That was 99.8603. So if you want to calculate the absolute change, IE how much money will we effectively lose express the notional terms is then we take the modified duration times the yield scenario, IE, um, one percentage point times the dirty price of the bond, and then you get 4.6512. So basically what you now can do with this number is calculate the predicted bond price for a yield to maturity level of 3.43. And that is, this is a price where we're currently at. This is then the absolute change in price, um, still in percentage because bond prices are coded in percentage terms. And so we will see that, um, or the predicted bond price will be 95.209 and a bit. And so if you now say if I have a hundred million notional, then I pay 99.86 million for this, now it will then be worth 95.209 million. That means we are from a mark to market point of view down 4.6512 million, um, dollars, right? Okay. So that's, um, how you could or how you, um, generally apply the modified duration. Um, but there's one thing I just wanted to highlight, and for that we just, or, you know, remind you of. And for that, I just go to our, um, government bond, um, tool here. So here what you can see is I've basically rebuilt this bond. So that's, we're trading this today. Um, one day settlement brings us to Monday, uh, it's a five year bond, so it matures then on the 16th of June, 2030 2.4% annual actual, actual yield to maturity being 2.43%. Um, and now what I'm doing here is I calculate the clean price. Um, there's no accrued interest as I said, um, before that gives us then the dirty price of the bond here is 99.8603. Again, as we have on, uh, our slides, uh, settlement amount is this, and you see here the duration and you see the modified duration and DV O one we're gonna talk about next. But what I wanted to show you now is obviously what happens to the bond price, um, when we're typing the, or changing the yield manually here to 3.43%, so have an eye on, on the clean and dirty price here. What happens? They both drop as expected yields are going up, but what we can see is that the dirty price at a yield level of 3.43 is not the 95.2091, as we have predicted by using modified duration, but it's actually higher than that. It's 95.34 and a little bit, uh, of percentage points. And of course, the answer as to why this is the case is that modified duration, and I now just go back to to the slides here, um, is attempting or is basically, um, estimating, um, the bond price or approximates of bond price for give certain yield level, um, using a linear relationship because that's basically what we're doing with the ative, right? We're putting a linear tangent through the function. And, uh, in reality, as we've seen the, um, relationship between bond price and yield is actually convex now. And as it says here on the slide, that doesn't matter all that much for relative small changes in yield, but of course for relative large changes and one percentage point is a significant change in yield, there will be some, um, prediction error and that will ever grow, um, or only grow with, um, the larger of a yield change we are looking at. So that's just a reminder of convexity, and this is important obviously to remember. So what convexity then really means in practice when we're using modified duration, um, we're from a long bonds, uh, perspective, we're overestimating losses that, um, are expected to occur when yields go up, but we're underestimating, um, profits. So from a risk management point of view, probably a fairly, um, careful, um, approach overestimating losses underestimating, um, profits. Alright, cool. So, um, that's modified duration. Let's go then to the last one that we want to discuss. And that's a DVO one or dollar value of a basis point. Now, how does that differ from modified duration? Well, first of all, it's basically only looking at relative small changes in yield more precisely for one basis point change in yield. And in addition, not only do we want to, or we don't really, uh, want any sort of absolute or relative percentages here, um, we want to look at the p and l impact in an actually currency amount. Uh, and so one basis point is chosen because that's a relative, um, you know, useful, um, metrics given that yields at least in developed markets don't tend to move. Um, you know, um, that much on a, on a busy day. Now, of course, recently we had a couple of events where this was, um, more exciting than usual, but we had, um, you know, on average probably like, you know, five to 10 basis points. That's a busy day, right? Anyway, so, uh, one basis point is then a reasonably sized, uh, risk metrics to look at. So we're looking at one base point changes here, and how do we get this, um, calculated and to reflect a specific, uh, currency amount? Well, we're already halfway there because what modified duration allows us to do, as we've seen, is the absolute price change of a bond price, um, you know, with, uh, for, uh, you know, certain yield scenario. So we're kind of using modified duration here as a starting point, um, and then multiply it with one basis point this time, not one percentage point because we're looking at a one base point yield change, then we're multiplying with the, uh, sorry, with the dirty price of the bond that gets us the absolute price change. And now we are multiplying this with the face value of the bond, and that will then give us the actual monetary impact in currency, um, for a change of a yield by one basis point. So that means the DB one considers the sensitivity of the bond and also the position size, right? And if we just do this for our example here, and you will recognize all these factors. Uh, this one here is new though. It's a hundred million face value, that's the size of the position we're looking at. So we're doing the calculation that we've seen on the previous slide, 4.6577, that's the modified duration, uh, times 0.01, that's a one basis point change in yield times the dirty price of the bond times the a hundred million, and then the result is minus 46511.8. And what that means is if we bought a hundred million of that bond at the price of 99.8603 IE, our yield to maturity is 2.43, and then subsequently yield to maturity goes from 2 43 to 2 44, then we're going to lose $46,511 and 80 cents. And in my view, this is probably the most useful of those measures that we have discussed because it doesn't require us any further calculation. I can immediately see if on this particular position that I'm holding, um, the yield would go up by one basis point, then that's gonna be the monetary impact on my, um, portfolio or the monetary impact on my, um, p and l. So that's why DVO one is a fairly, uh, popular concept in reality. And we have to, um, add just one more thing, and that is because so far, um, we have only looked at, um, you know, the interest sensitivity for a single bond position, which of course, you know, um, is helpful. But in reality, most investors will have portfolios that consists of more than one bond, right? Sometimes several hundreds. And of course, the calculation of all those ratios that we've gone through for single bonds is definitely possible with today's analytic capabilities that we have. Um, but I am not a hundred percent sure that just looking at a hundred different macauley durations or modified durations or even DV oh one, um, for each bond in, uh, you know, or for, for all the different bonds that we're holding, is giving us a lot of useful information. So what we might wanna do is to calculate, Um, some sort of portfolio duration. We wanna know exactly more about how is, uh, what is maybe the average, um, macauley duration on our portfolio, what is the average modified duration of our portfolio? And that brings us then obviously to a weighted average because, you know, we don't want equal weights of all different bond positions if the position size, uh, may be, uh, quite different. So how can we do this? We can do, again, a weighted average approach. So what we can do here, for example, is, um, for macauley duration, um, for a portfolio, we take the portfolio weight of bond A, so let's say that's 2% of our portfolios in bond A and the macauley duration of bond A is 4.77 years, and then we have a 5% weight, uh, of bond B, and that has the duration of 3.8 years just making this stuff up. Uh, and then we continue that, and that will obviously then all the way up to a hundred percent give us then our weighted average, um, macauley duration of our portfolio, and we can use the exact same approach then obviously for the, um, for the, um, modified, um, duration, uh, of the portfolio. And then for DV O one, because we have already the position size, uh, included in the calculation, we can just sum up the DV o one of bond A with and the DV O one of bond B and C and whatever else is in there. And then we get the, um, portfolio DV O one. And while this is certainly useful information, I'm also, um, keen to say that we can improve, uh, this, um, or the amount of information that we have can be extended. Because, you know, what I want to, um, just well refer to here is now the fact that obviously, you know, we're talking at bonds, uh, we're looking at term structure of interest rates, yield curves, um, and, and, and the likes. And so when you are looking at this, then obviously not, uh, only yields different for different times to maturity, but also yields across different times to maturity will not change necessarily in the same way. And so basically, let's say we have a portfolio and that for simplicity purposes consists of two bonds, right? A two year bond and a 10 year bond. And we've calculated the DVO one for the single, uh, bonds here. And let's say this is 20,000 negative. So we're long that bond and 50,000 negative for the 10 year bond. And so if we do this approach, then that would give us a DV one of 70,000 in total. And while this is helpful, um, all that it does tell us is if yields across the yield curve, IE all different maturities increase by one basis point in parallel, right, yields will move up, then our portfolio loses 70,000 uh, dollars. What it doesn't tell us which yields precisely are we exposed to, because, you know, if we expect, for example, the yield curve to steepen significantly, then we are expecting, you know, maybe the longer term rates to increase more. And then it's important to know that a large chunk of our risk is actually at the 10 year point, which can be considered a relative, uh, long term rate, right? So, um, I'd say the more information we have about where our risk is, not just how much the risk is on a parallel move, the better it is. And so for that, we need to use a different technique because we cannot just simply sum up the DV O one values. Now again, if we have two bonds, this is probably not requiring anything else because we can remember which bonds we have and where our risk is. But for complex portfolios, uh, we need, uh, something else. And that is bringing us to the last slide here. And that is, um, how we can calculate at the interest rate sensitivity, uh, or can calculate the interest rate sensitivity for bond portfolios in a more meaningful way. And that's kind of the interest delta letter that's also, you know, something you see, for example, in interest rate swaps and, and and things like that. So calculated in different ways, and we're using a very simplified generic approach here. But that, um, general idea behind how to do this is, um, uh, laid out here on, on the slide. And we're gonna quickly talk, uh, through this. So, um, first thing is we need to define what we call key rates. IE what are the interest rates that we want to analyze here with regards to, uh, our portfolio sensitivity? And this could depend a little bit on, you know, what's the maturity in your portfolio. So at the short end, you have maybe more frequent, um, key rates, IE the gap between the different key rates, it's gonna be smaller. Um, we're just gonna use an annual approach here. So let's say we have key rates 1, 2, 3, and then all the way up to 10 years. So we wanna see if the one year yield changes, how much does that impact our portfolio value, two years, three years, and so on. And let's say, right, we have those two bonds, um, that we, um, have uh, discussed on the previous slide. So two year and a 10 year bond position. And let's say this is the current yield curve that I've drawn here. So what are we going to do now? We're gonna take the current yield curve, so the as is yield curve and calculate the portfolio value. And, uh, let's, um, assume this is around nice $100 million. So that's the value of our portfolio right now as yield curve looks right now. So now we wanna calculate the impact of a, um, you know, change in the one year yield. So what we're gonna do is we're gonna take the yield curve and bump the one year yield up by one basis point, and then this new yield curve we're now going to use to revalue our portfolio value. And let's say this is then a hundred million as well, which then shows there's zero DV o one in our one year, um, uh, or in the one year bucket and kind of makes sense, at least you wouldn't expect to see meaningful, um, impact here because the two year bond is driven by the two year yield, not by the one year yield, as I said, in in in interest rates. So that will be a little bit different. And if you bootstrap and, and, and do the zero, uh, uh, zero rate valuation technique, then yes, but nothing meaningful will will pop up here. And then, um, what we're then going to do is we're gonna bring the one year rate down to where it was, and now we're, uh, bringing the two year yield up by, um, a percentage point, uh, sorry, uh, one basis point. And then, um, because we're having our two year bond here and that has a 20,000 dvo one, as we said earlier, then the portfolio value should go down to 99,980,000. And then we know, okay, that this will be a 20,000, uh, negative DV O one at the two year point and so on. We're now bringing the two year yield down and then shifted three year yield and repeat this until we reach 10 year point at which we are gonna see, uh, the 50,000 that we've just talked about on the previous slide. Now, what we can do is of course, we can sum up all those DV O one, uh, here, and we get to the same negative 70,000 that we have set as the portfolio DV O one, and that's against the parallel shift. So if all rates we're looking at here, uh, will go up in parallel by one basis point, the indeed the combined loss of the portfolio is gonna be $70,000. However, we can see as well that 20,000 of the DV one is at the two year point of the curve, whereas 50,000 of the risk is at the 10 year point of the curve. So we have much more granular view on where exactly our risk is on the yield curve. And that's can be obviously, um, very useful in making, um, decisions with regards to hedging or repositioning or anything like that. And that, ladies and gentlemen, it brings us to the end of today's session. I hope you found it beneficial. Thank you very much, um, for your attendance. I hope, um, you know that as I said, you found it beneficial. And if you have any further questions, feel free to ask them now. I still have a couple of minutes, um, for you. Um, if I don't hear from you, have a great rest of your Friday, uh, and obviously, uh, a great weekend ahead. Take good care of yourselves and hope to see you again, um, very, very soon on one of the future sessions. Goodbye for now.