So that's the first two observations. Another observation to note is the following, mezzanine tranches are often relatively insensitive to Rho. We can see that, for example, perhaps most easily in this plot when q equals one percent which in many cases in practice might be the most relevant example. Because in practice, depending, of course, on the nature of the names in the portfolio, you will often see a q of approximately one percent. So you see here that actually, certainly, maybe in this range here that I'm circling, you see that the expected tranche losses don't vary much with Rho at all. Actually, this is a commonly observed phenomenon. Also, what actually is implications when it comes to model calibration. Now, we're not going to get into model calibration for this Gaussian copula model. I might say few words about it in a later module, but generally we won't have time to go into it in any detail. So that's the third observation here. So I have two more observations I want to make but I'm going to make those observations when we change the problem slightly. In the remainder of this module, we're going to assume that the senior tranche is now on the hook for all losses between seven and 125. So it's not between seven and nine. It's all losses between seven and 125. What we will see then, and we'll see it on the next slide, is that the expected loss in this senior tranche, sometimes called a super senior tranche, we'll see that it's increasing in Rho, and sure enough we can see that. So this is the green here. We see it is always increasing in Rho regardless of the value of q. Actually, the reason for this is the same reason why the equity tranche expected losses are always decreasing in Rho. So the same intuition I gave will actually also apply in the opposite direction to the expected losses in the senior tranche. The final thing to note, and this might not be so obvious from the figure because I'm not giving you the individual numbers, but the total expected losses are on the three tranches that is the expected losses on the index. Remember all of the index, now, the index consists of all 125 names, and the three tranches are equity, zero to three, the mezzanine tranche, three to six, and now the senior or super senior tranche is seven to 125. So now these three tranches now all add up to the entire index, all 125 names. The point I'm making, and you can see it on the next slide, is that the total expected losses on the three tranches is independent of Rho. This is not an accident. So what am I getting at here. Here's what I'm getting at. Let's pick this figure here q equals three. If I fix a value of Rho, so Rho equals 0.1 and sum up the losses for the three tranches at Rho equals 0.1, I'll be summing up maybe this number plus this number, plus this number, and that will equal to the losses at any other value of Rho. So if I take Rho equals 0.7, then this, this, and this number here, the sum of these three will be equal to the sum of these three numbers over here. Why is this occurring? Well, we can analyze it. We can show that this is no accident as follows. The expected losses, the expected losses using the risk-neutral probabilities. So the expected total loss is equal to the expected value of the sum of the losses from i equals one to 125 of the indicator function that the ith bond or named defaults. I'll just use the cap letter D and e f to denote default. Now, I would normally multiply this by A_i times one minus R_i. But by assumption, we have said that the A_i's are equal to one, and the R_i's are zero. So in fact, this term here is just one. So in fact, I can ignore this piece here and I get that. Well, this is just equal to the sum from i equals one to 125 of the expected value of the indicator function of the ith name defaulting. This is equal to the sum from i equals one to 125 of the probability of the ith bond defaulting. The important thing to notice here is that this, the probability of the ith bond defaulting, well, this is equal to our q, but this certainly does not depend on Rho. So we see that the expected total loss in the portfolio, which is equal to the total loss in the sum of the three tranches in this case, is equal to this quantity over here and that does not depend on Rho. That's why we see this behavior I was describing a moment ago. However, it is worth pointing out that the allocation of these losses to the three tranches, the three separate tranches, does indeed depend on Rho. We can see, as before, that the expected tranche losses in the equity piece are decreasing in Rho. This will always be the case for an equity tranche with lower attachment point equal to zero. Similarly, we can see that the expected tranche losses in the senior tranche are always increasing in Rho. This is true for any senior tranche with an upper attachment point of 100 percent or in this case, 125 units.
