In this workout, we're asked to identify the profit made by a steepening yield curve twist with an upward shift. Otherwise referred to as a bear steepener and a downward shift. Otherwise referred to as a bull steepener and also a flattening twist with an upward shift a bare flattener or a downward shift a bull flattener given the available bonds below and a desire to have within the portfolio a target modified duration of 4.45. So we've got our target modified duration here the 4.45 and we're given three available bonds. So the three available bonds that we have all going to be purchased all available for purchase and assessment date of the 1st of January of 2015. The first one is a two year bond that will mature two years later and I've got a five-year bond and also an eight year bond available for us as well. Each of these bonds have a coupon of 4% So the same in that regard, but they have a differing yield because we have an upward sloping yield curve. So to your bonds got a 3% yield maturity 4% yield for the five year bond and it's a 5% yield for the eight year bond. Now if we look at the duration metrics for these bonds, the important one is the modified duration because our target duration is expressed on a modified duration basis and the five-year bond has a modified duration of 4.45 so we could construct a portfolio. Just holding the five-year bond, which would match our target modify duration. This is effectively going to be our bullet portfolio. Equally, we could construct a barbell portfolio by investing some of our portfolio into the two-year bond with its modified duration of only 1.9 and some of our portfolio into the eight-year bond that's got a modified duration of 6.63. Now if we just jumped down over the changes in the yield curve that we might be modeling. We are given down here in Rose 30 and 31 some portfolio weights. Now in terms of the portfolio weights these have been given to us for very particular reason if we try to calculate our overall portfolio duration by using the sum product function and by multiplying together. The weights that we have in the two and five and ten year bonds with the modified durations of those same particular bonds and if we lock onto the modified duration numbers, we can see that for the bullet portfolio where we've got 100% invested into the five-year bond. We get a portfolio duration of 4.45. That is the modified duration of the five-year bond. But if I just copy that formula down you'll see that we'll get the same portfolio duration where we have 46.1% of the portfolio in the two-year bond and 53.9% of the portfolio invested in the eight-year bond.

To give us the overall portfolio duration of 4.45. So the aim here is to demonstrate that so long as we have the same duration between these two portfolios. Then it doesn't matter whether we're getting that upward shift a bear movement or a downward shift a full movement. It is only the changing the slope of the yield curve that we need to think about in terms of preference for a bullet portfolio or a barbell portfolio. So let's get some of the numbers in we've got the changes the yield curve that we've got. Don't forget that our yield curve at the moment is upward sloping.

So let's first of all look at the bare sleeper. The better means that bond prices are falling in other words interest rates are going up, but it's steepling. So that the two-year bond has only gone up in its yield from three to 3.5%. Whereas the eight year bond has seen its yield go up from 5% to 6.5. So we're getting a sleeping effect there. Let's just go down and see what impact this has on our portfolio. So if we're looking at the impact of our bear steepener, what we need to do is to identify the impact that the changes in yields has on our bond prices in our portfolio and we can see that through the modified duration metric What we're going to need to do is to take the weight that we have in our portfolio. So here the portfolio waits for the two-year bond for the bullet portfolio is going to be 0%. We then need to multiply that by the modified duration. of that particular security and then multiply this by the change in yield that we're investigating so our yield has Gone up to 3.5 where it started up from.

3% I'm going to lock onto the Row 18 only here I want this to move across as I look at my other bonds in my portfolio.

one other adjustment just before we finish off on this we need to make sure that we make this all negative to pick up the inverse relationship between interest rates and use I want to be able to copy this formula down to the barbell portfolio. So I want to move the wings, but I always want to look at. row 20 for the duration metrics I also want to look at row 23. I also want to look at row 23. For the new yield that we're looking at. So if we then copy this formula to the right.

We should be able to see that not a huge surprise. The yield has gone up by 1% on the five-year bond, so from 4 to 5% to the yield 1% change in yield gives us a 4.45% change in the bonds price in the opposite direction as you'd expect told to us by a modified duration metric So they just add up all these changes. We should therefore get the overall impact on our portfolio. If I then copy this formula where we're looking at row 20 and row 23 and row 18 only for the new yield and the modify duration and the existing yield then we can just copy that formula down change the portfolio weights. So we now pick up the barbell portfolio weights and we can see that if we copy this to the right but overall. We will get a worse outcome here. As a result of this steepening change in the slope of a yield curve. When interest rates are all going up bond prices are coming down, but it's worse for the barbell portfolio because of the increased weight that we have in the eight year bond that has that higher duration and is more sensitive to our changes in interest rates. The key thing here is that we have a steepening yield curve. So the change at the longer end of the curve the eight year bond as we have it where duration numbers are higher as a bigger magnitude of effect and the smaller change in yields at the shorter end of the yield curve where the duration metric is lower itself. So overall the barbell portfolio does worse here in the steepening yield curve scenario where it's a bear steepener in comparison to the bullet portfolio. If we're now going to go on to consider another form of a steepening yield curve, but now a bull steepener that is where interest rates are falling. We can see that information is that they're on row 24. So we're getting a steeping of the yield curve but where all interest rates are falling. We're getting a bigger magnitude of fall in the two-year bond and a smaller magnitude of fall in the eight year bond. If we turn that into the actual valuations in terms of the value of the portfolio changes and copy this formula. I still want to be looking at row 30 for the portfolio weights. And in terms of the new situation for the yield curve, we don't want row 23, but row 24 instead. But then copy this to the right we can see that again. We've had a 1% decrease in interest rates for the five year bond. So not a huge surprise that we therefore get this 4.45% increase in the portfolio value since that is the modified duration of the five-year bond. but if we then copy this down for the barbell portfolio we can see that at the long end for our eight year bond. There's only been a small decrease in interest rates. And even though duration is higher at those longer dated bonds. Overall we get a smaller positive impact for the barbell portfolio, then we had for the bullet to portfolio. So in this scenario, we're getting more benefits from holding a bullet portfolio. We can see here that it doesn't matter whether we have a bear steepener or a bull steepener. the bullet portfolio performs better in a steepening yield curve environment and the barbell portfolio If we then go on to think about the impact of a flattening yield curve, we could be looking at a scenario where we have a bear flattening that is an increase in the yield curve at the same time that it is flattening. So we're getting more of an increase at the short end of the curve and the long end. Or we could have a bull flat and up which is interest rates falling generally, but again a bigger falling interest rates at the long end of the curve and the short end. And if we go through and calculate the impact on our portfolios where we have a 46.1% in the two year bond and a 53.9% in the eight year bond to give us the same portfolio duration as a bullet portfolio of our five year bond that has the 4.45 portfolio duration itself. We can then see what the impact is going to be of the portfolio as a whole. So the formula we're going to need here we're going to need an inverse relationship with then going to need to pick up a portfolio weight. So the two year bond so the two-year bond bullet portfolio weights and then going to multiply this by the duration of this bond the modified duration of this bond and I'm going to hit F4 to lock onto that row. I then want to multiply this by the change in the portfolio that we're modeling or we're looking for the bare flattener. So a flattener when you get interest rates rising you get a bigger movement at the short end of the curve. I always want to be looking at row 25 for this and I need to subtract from this the original yield curve values. They are there on row 18 and again always want to look at row 18 for those.

So no impact in this flattening curve. Shift where interest rates are going up for the two-year bond if I copy this to the right. We can see that there is an impact on the five-year bond where we have 100% of our weights. And then if we add up all of those weights, we can see that the portfolio as a whole will have fallen in value by this 5.56% if we carry this over to the barbell portfolio. We can see that there will be an impact on the two-year bond and also on the eight year bond because we have some portfolio holdings there and overall. There is a benefit here to holding the barbell portfolio a flattening yield curve movement where interest rates are rising. It is better to have the barbell portfolio because you have less exposure at the long end of the marketplace where interest rates have moved the lower amounts and the fact that we have higher exposure. Where there's been a big interest rate movement doesn't really matter so much because we have a low duration on those short-stated bonds. Similarly if we copy this formula down to the full flattener yield curve shift. We need to go back to row 30 for the barbell portfolio weights. We need to go to row 26 for our new yield curve shape. we can then see on row 26 at the yield curve has flattened but at the same time interest rates have fallen, but we're getting a big fall from our 5% original eight year bond yield down to only 3% Whereas the two-year bond is only fallen in terms of its yield from 3% down to two and a half. So a bigger movement at the long end of the curve. And what that means then is that interest rates are falling that's our full word, but the flattening impact means that there is a bigger impact at the long end of the curve. Where duration is highest? as a result the barbell portfolio performance better when we have a bull flattening movement because there is a bigger fall in interest rates at the long end of the curve where there's more duration. So as a result the barbell portfolio performs better in a flattening yield curve scenario, whether we have a bear flattener interest rates generally increasing but at the same time the yield curve flattening or if we have a bull flattener that is interest rates generally falling but again flattening of a yield curve. The barbell portfolio will perform better no matter where interest rates move in terms of their direction. So long as there is a flattening of the yield curve.