In this module, we're going to continue on with the example we introduced in the last example. That was the simple model. It was a one-period model with identical risk-neutral, default probabilities for each bond in the reference portfolio and so on. So what we're going to do here is, we're going to look at some of the tranche losses, or the expected tranche losses, and we're going to look at the characteristics of these expected tranche losses as a function of the correlation parameter rho and so on. We're also going to see how the total expected losses in the portfolio does not depend on the correlation parameter rho. So recall the details for our simple example, we want to find the expected losses in a CDO or in CDO tranches with the following characteristics. The maturity is one year, and it's just a one-period CDO. There are 125 bonds in the reference portfolio. Each bond pays a coupon of one unit after one year if it has not defaulted, and the recovery rate on each bond is zero. So in particular, if a bond has defaulted before the one year is up then that bond will pay a coupon of zero. There are three tranches of interest the equity, mezzanine, and senior tranches. The equity tranche is exposed to defaults zero to three. The mezzanine tranche is exposed to defaults 46, and the senior tranche is exposed to defaults 79. We saw in the last module that we could actually compute the expected tranche losses. We saw our expressions for the expected loss in the equity tranche, the expected loss in the mezzanine tranche, and the expected loss in the senior tranche. So on this slide, we're actually going to see what these expected losses are as a function of rho which is the common pairwise correlation between the various names in the portfolio, and the individual risk-neutral default probabilities. So this is the q equals one percent. So this corresponds to the case where the probability of each individual named defaulting is one percent. So there's a 125 bonds in the portfolio. We assume each of them defaults with probability q equals one percent here. Over here, each of them defaults with probability q equals two percent, down here four percent, and over here three percent, and on the y-axis we have the expected losses. Note, we have a 125 names in the portfolio. Each name will pay one unit if it hasn't defaulted after one year. So therefore, the total portfolio notional is 125. The equity tranche is on the hook for the first three losses. So the equity tranche can lose a maximum of three. The mezzanine charge is on the hook for losses 4, 5, and 6. So the maximum the mezzanine tranche can loose is also three. Likewise the maximum the senior tranche can lose is also three. It's on the hook for losses 789. So the maximum loss in any of these tranches is three, and that's why you don't see in any of these cases, anything greater than three on the y axis. Across the X axis, we're actually plotting these expected losses as a function of the correlation parameter rho. So this is that value rho we saw in the last module which defines the normalized asset value Xi. So there are some important observations we should make here. The first observation is the following. Regardless of the individual default probability q, and correlation parameter rho, we see that the expected equity tranche loss is greater than or equal to the expected mezzanine tranche loss is greater than or equal to the expected senior tranche loss. Now this only holds when each tranche has the same notional exposure in this case three units. So we see it here. We see the blue line which is the expected tranche losses in the equity tranche, is always greater than or equal to the red line which corresponds to the mezzanine tranche which in turn is always greater than or equal to the senior tranche expected losses, and that makes sense. After all, we can see that the equity charges the riskiest. The mezzanine tranche can only loose, if the equity tranche has already lost everything. In other words, if you have four defaults, so that the mezzanine tranche looses one unit. Then that means we've had more than three losses, and the equity tranche is to lost everything. Similarly, if the senior tranche is to loose anything, then that can only be the case if we have seven or more losses which means the equity and mezzanine tranche have already lost everything as well, and so we should be absolutely certain and understand why the expected losses in the equity tranche must be greater than or equal to the expected losses in the mezzanine tranche, must be greater than or equal to the expected losses in the senior tranche. Another important observation is that the expected losses in the equity tranche are always decreasing in this correlation parameter rho. So we see it here. So let's pick q equals two percent. For example, if we look at q equals two percent, we can see that the equity tranche losses is a decreasing function of rho, and in fact it's true in all four graphs. It's always a decreasing function of rho. Why might this be the case? Well, one easy way to see this perhaps is the following. Imagine two possibly different values of rho. So imagine rho being equal to zero, or rho being equal to one. Well, if rho is equal to one, well then what happens either all of the names, all of the credits default together, or none of them default together. So in that case, the equity tranche will actually be as risky as the senior tranche because either all of the names default or none of them default. The probability of all of them defaulting will therefore just be q equals one percent, and the one percent case or two, and the two percent case and so on. So in other words, let's consider this example here. So the q equals one percent case. For numerical reasons, we didn't take a value of rho all the way equal to one. But we can look at the value rho equals 0.99 to see what's going on. In this case, either all of the names default together, or none of them default. In that case, the three tranches are all equally risky, and so the probability here of any one defaulting is one percent. So the expected losses in the tranche is going to be one percent times three which is 0.03. So this value here is roughly 0.03. On the other hand, if rho equals zero. Well, because there's no correlation among default events we will always expect there to be maybe just one default, or two defaults, or zero defaults. But what it means is that most of the time, we're actually going to see a default, maybe one, maybe zero but sometimes two or three, and so with such a low correlation we'll always expect to see some credit event happening, and because it's the equity tranche that is on the hook for that first credit event. We expect the equity tranche to incur losses most of the time, and that's why we see this number being much higher, for a low value of rho, and in fact we will see this behavior for all values of q. Down here for example, we see q equals four percent, and we see the expected tranche losses is now almost three, and that's because we expect with q equals four percent. We expect there to be four percent of 125 which is five. So we expect to see on average five losses in the portfolio. Because correlation is very low down here. We're always going to expect to see almost five losses in the portfolio which means that most of the time the equity tranche will be wiped out. We're going to see more than three losses. Most of the time we're going to see four or five losses maybe. So that's why it will be wiped out. Down here for example, when rho equals 0.99. Sure, we do on average expect to see four percent losses which corresponds to five port names defaulting. But they're all going to default together or not at all which means 96 percent of the time. In fact the equity tranche won't be hit at all, and only four percent of the time will it see a loss of three. Four percent of three is 0.12. So this number here is roughly at the 0.12 level.
