[BLANK_AUDIO] . In this short module, we're going to discuss the mechanics of a synthetic CDO tranche. While we haven't actually described for you what a synthetic CDO tranche is yet, we will actually do that in a later module. In particular, we will distinguish between a cash CDO and a synthetic CDO. But for now, we're just going to go into the details of the mechanics, of how a synthetic CDO tranche actually works. So now let's discuss the mechanics of the synthetic CDO tranche. I'm going to describe or explain for you, the distinction between a synthetic and a cash CDO in a later module. For now we're just going to discuss the details of a synthetic tranche. As I said, I will distinguished between the synthetic and a cache CDO, in the later model. So recall there are N credits in the reference portfolio. Each credit has the same notional amount, A. If the ith credit defaults, then the portolio incurs a loss of A times 1 minus r. Where R is the recovery rate which is assumed fixed, known, and constant across all credits. A tranche is defined by the lower and upper attachment points, L and U respectively. So, we've already seen examples of L and U, L and U, 0 to 3 and so on. 3 to 6, 6 to 9. Usually L and U are given as percentages of the total portfolio notional amount. In our simple example on the previous two modules, L and U were given as the number of losses. 0, 1, 2, or 3, 4, 5, or 6, 7, 8 or 9, or so on. But, typically in practice, they're given as percentages. The tranche loss function, TL for tranche loss, superscript l and u, to denote the lower and upper attachment points are parameters of this function. So its a function of the number of losses in the portfolio L, is a function given as follows. First of all we take the minimum of LA1 minus R and U. So this, here, is actually the total portfolio loss. So if the total portfolio loss exceeds U, then the tranche loss is given by U. After all, the tranche cannot lose more than U. U is the upper attachment point, it cannot lose more than U. So if the total portfolio loss exceeds U, then this minimum is given to us by U. Otherwise the minimum is given by the total portfolio loss. Now the lower attachment point is L, so we then have to subtract L from this minimum here. And finally we take the maximum of that last quantity and 0, and that gives us our tranche loss. It tells us for a given number of defaults, what loss is suffered by the tranche. So for example, suppose L is 3% and U is 7%, so we've got some sort of CDO as follows, maybe there's an equity piece here, which is the 0 to 3%, we've got a mezzanine tranche, which is maybe 3 to 7, and so on. Well, suppose the losses in the portfolio add up to 5%. Well, in that case, this piece represents the losses in the portfolio. The first 3% of losses are incurred by the equity tranche. But the next 2% of losses fall in here, and they are incurred by this tranche here, with lower attachment point l equals 3%, and upper attachment point U equals 7%. So, therefore, the tranche loss is 2%, and actually, that's 50% of the tranche notional. The tranche notional is 7% minus 3%, which is 4%. So, we've incurred 2% losses out of a total maximum loss of 4%. So, therefore, in this example we have lost 50% of the tranche notional. When an investor sells protection on the tranche, she's guaranteeing to reimburse any realized losses on the tranche to the protection buyer. In return, the protection seller receives a premium, at regular intervals from the protection buyer. These payments ticky, typically take place every three months. So, when you see the word protection here, what you might want to do is think of it as being insurance. So protection, think of this as insurance. And what's going on here, is that one person is selling insurance, and the other person is buying insurance. So the person who's buying insurance, is insuring against losses, in the underlying portfolio that impact their given tranche. In return for providing insurance, the insurance seller receives an insurance premium. And they receive that premium at regular intervals, from the protection or insurance buyer. In some cases, the protection bar may also make what is called an upfront payment, we're not going to be concerned about this. But sometimes they might make enough front payment, in addition to, or instead of the regular payment which might take place every three months. This is often the case, for example, in the case of equity tranches which, as I said earlier have a lower attachment point of zero. The fair value of the CDO tranche, is that value of the premium plus upfront payment, if applicable, for which the expected value of the premium leg equals the expected value of the default leg. So, just like a swap, the initial value of a CDO tranche position is 0. Now, if this doesn't seem very clear to you yet, that's fine, we're going to see a diagram of the next page, which will make it clearer still. And then in the next module, we're going to go through the premium leg and the default leg in more detail, so that hopefully the two legs of a CDO tranche position, should become clear to you, and you understand exactly what is going on, with a CDO. Another point I want to make is the following. We have already seen the tranche loss function, we're going to need to compute the expected tranche loss function, nor did it compute the value of the CDO. We actually already computed this, in our earlier example, our one period example. We computed the expected tranche loss in an equity tranche, a mezzanine tranche, and the senior tranche. Well, the expression we used for that was just this ex, this expression here. So, it's equal to the sum from L equals zero to capital N, T L T of L times the probability of L losses in the portfolio. And remember, we can compute this probability by conditioning on the random variable M, which makes the default events of each name independent. So, we're, we were then able to compute this quantity, either using the iterative. In that case for a simple example, we saw that this was just a binomial probability. But either way, we can compute this. This is the standard normal PDF, so we can compute this quantity using a numerical integration. So this gives us our expected tranche loss function. So now let's see what happens visually with a CDO. We've got a number of periods, so we've got a period here, here, here and so on. This is the premium leg. So these are the payments made by the person, or investor, who buys insurance, or buys protection. S is the annual premium, or spread per unit of notional. These here, are default events. We will see in the next module, that the tranche notional decreases after each default. So, we, these vertical arrows represent the size of the payment. Because S is an annual spread, we have to multiply be delta T, which is the length of an interval. Typically approximately one quarter, representing payments every three months. So the, the, the premium per quarter will be delta TS. It's paid on the notional of the tranche, and in fact it's paid on the outstanding notional on the beginning of each period. So at this point, there haven't been any defaults yet, any default's that have impacted to tranche yet. At this point we have a loss, and this loss impacts the tranche, so this actually decreases the notional of the tranche, by an amount of 1 minus RA. and so we get decreased payments, after each default event. So a second default event occurs here, this default event also impacts the tranche, and that lowers the outstanding notional in the tranche. And so the insurance payments, or the premiums, are payed only on the outstanding notional in the tranche. As I've said in the previous slide, this is just a schematic, describing basically what goes on in the CDO. We have two legs. We have the premium leg, and we have the default leg. If you like, you can think of this like being a swap, where we've go the fixed payments and we have the floating payments. The fixed payments correspond to the premium leg, they're not always fixed. The rate is fixed, the delta TS, but they're paid on the outstanding notional. The default lagger like the floating payments, you're never sure what you're going to get in each period. Most of the time you'll get nothing, if there hasn't been any default event in the tranche. But if there has been a default event that impacts the tranche, you will receive a payment. You will receive an insurance payment, for that event of 1 minus R times A. And we will see in the next module, that the way the CDO is priced, or in other words, the way the S value is calculated. It is calculated by equating the value of the premium leg with the default leg. And so the initial value of an investment in a CDO tranche would be zero, and we're going to find what value of S makes that value equal to zero.
