In the last module we saw the mechanics of a synthetic CDO tranche. We saw there was a premium leg and a default leg. Well, in this module, we're going to price the premium leg and we're going to price the default leg. We're then going to set these two prices equal to each other. And as a result, we will be able to calculate the fair premium of a CDO tranche. So let's start with calculating the fair value of the premium leg. The premium leg represents the premium payments that are paid periodically by the protection buyer to the protection seller. These payments are made at the end of each time interval and they're based upon the remaining notional in the tranche. In this sense it is different to a CDS since the latter contract ends as soon as a default occurs. Here in the CDO. A CDO can survive well beyond an initial default. Multiple names can default and the CDO, the CDO tranche, may need to pay out upon each default event. Formerly the time T equals 0 value of the premium leg, which we're going to call P subscript 0, 0 denoting T equal 0 on superscript, L U which refers to the lower and upper attachment points respectively. So this quantity is equal to what's on the right inside here of expression six. So we have s which is the annualized spread or premium pay to the protection seller. So this would be a percentage, it might be 2%, it might be 3%, or it might be 10% for a chance where you expect to see many losses. But maybe it'll just be a quarter percent or some very small number as well for a chance which is relatively safe. DT is the risk free discount factor for payment day T. Delta t is the accrual factor for date t. So typically, delta t would be approximately one quarter corresponding to quarterly payments and is the total number of periods in the contract. So for example, if the CDO lasts or has a maturity of ten years and payments are made quarterly. Then this implies n will be equal to 40. So what's going on here is that the fair value of the premium leg. This is the fair value of the premium payments made over the n payments. It's equal to S times delta t. The sum from t equals 1 to n of S times delta t. Remember, S is an annual spread or premium, so we've gotta multiply it by delta t to get the payment made per period. So it's S times delta t. Times the expected notional remaining between periods T(-1) and T. So, remember the total notional of the is U minus L. So if U is 7% and L is 3%, then the total notional of the is 7 minus 3 equals 4%. The total losses in the tranche can't exceed U minus L. So this expression here is equal to the expected. Notional remaining at time period t- 1. And so the insurance, the s times delta t is made on this expected notion. Those payments, which occur at time T, they must be discounted back to time zero, and that's why we have this risk free discount factor here. Now I should mention I'm not going to worry about accrued payments and so on. And practice default events don't take place at the beginning or end of a three month period. They might take place in the middle. And maybe you might have some accrued payments as well but we're not going to get into that. The other leg of the CDO tranche is the default leg. The default leg represents the cash flows paid to the protection buyer upon loses occurring in the tranche. Formerly, the time t equals zero value of the default leg, which we're going to call DL subscript zero for t equals zero, superscript LU, satisfies this equation here. Again, DT is the discount factor. Be the sum from t equals 1 to n, representing the end periods in the CDO, and so payments occur when there is a default in the tranche. And the expected payments at time t, is given to us by this. Because if you think about it, this is the expected tranche loss at time T minus 1. This is the expected tranche loss at time T. So the difference is the expected tranche losses in the period from t minus 1 out to t. So these are the expected payments that must be paid by the seller of protection or the seller of insurance by time t minus 1 and t so these are the risk mutual expectant losses and the charge between these two periods. And we have to discount them back to time zero, using the discount factor in Dt. So while some programming is required, we can actually calculate these quantities very quickly. If I go back to the previous slide, I can see that I also have an expectation of a chance loss appearing here as well. So the key to computing the premium leg value and the default leg value, is being able to compute these expectations here. Now if you recall, just a reminder since we know, that the Tranche Larousse function is given to us as follows. We know that TL subscript t superscript L, U, it's a function of the number of defaults in the underlying pool of bonds or pool of credits. We know that that is equal to the maximum of the minimum lA(1-R) comma u minus L and 0. Now the only random variable in here is L, the number of losses in the portfolio. We also note that the expected value. Of the tranche loss, at time t, is equal to the sum, from l = 0, up to capital N TL Lu, T(l) times pl(t), pl(t) we saw in the one-factor Gaussian model. This is equal to the integral from minus infinite to infinite of p superscript n, l, t, given M times the probability density function from standard normal random variable, phi M, dM. And finally we saw that in general we could compute this. Using our iterative algorithm. That was the algorithm with the nested for loops, if you recall. We had a for, I think it was for i = 1, up as far as n. And then we had for j = 1 to i, and so on. So if you think about it, the big picture here is as follows. We want to be able to compute the fair value of a CDO tranche. In order to compute the fair value of a CDO tranche, we need to be able to compute the fair value of the premium leg, the fair value of the default leg. And what we're going to do is we're going to set those two fair values equal to each other to get the fair value of the spread s. But before we do that, we need to be able to compute the expected tranche loss function, and we can see, as we've written here, that this expected tranche loss function ultimately comes down to computing this integral here. And we can do this numerically. So, while we do need some programming, we do need to write some code to actual compute this quantity, we can actually get it to run very quickly. And, in fact, this is the principle reason for the Gaussian Copula model's popularity. It has many flaws, we may discuss a couple of them in later module if we have time. But the reason it's so popular is because it enables us to price very quickly a security that is in fact very complex. Remember, in a CDO tranche, we might have a 100 names or a 125 names in the underlying portfolio. Each of these names have different risk neutral probabilities of default at various times. They all have different pair wise correlations and so on. It's a very complicated product, a very complicated security with many moving parts and this Gaussian copula model enables us to price the premium leg and the default leg tranche very efficiently in practice. Okay let's move on to the final slide here. How do we compute the fair value of a tranche? Well, just like a regular swap, what we do is we determine the fair premium S star say, so that the premium leg is equal in value to the default leg. And so all we do is we equate six, which is this, this is the fair value of the premium leg, we're going to equate this leg, With the fair value of the default leg which is given to us by 7, and we're going to solve for S. And we're going to call that solution S*. And so we get S* equals to this expression here. Now one thing to keep in mind with this expression here. This expression here is modeled independent. We did not need a model, we did not need a model to compute S*. S* follows just by grading the premium leg with the default leg, and the quantities we need to actually compute S*, are these expectations here. I mentioned in the previous slide that we can compute these expectations numerically very easily using the Kopler model. So this is where a model comes in. So we need our model for this piece. Okay, and it's the Kopler model which became the standard in industry for computing these quantities. So as is the case with swaps and forwards, the fair value of the tranche to the protection buyer and seller at initiation, at the beginning of the contract, is therefore zero. It is easy to incorporate any possible up-front payments that the protection buyer must pay at time t equals zero. In addition to the regular premium payments. So that's also easy to handle but we won't get into it. Actually, it's also possible to incorporate recovery values and notional values that vary with each credit in the portfolio. Here we've assumed that R i is equal to a constant R for all i. And we've also assumed that notional value A i is equal to A for all i. But again I just want to emphasize that we could still use the Kopler model even if these assumptions didn't hold. In practice, the way it works is similar to what we saw when we were looking at the implied volatility surface for equity options. In practice, we actually don't compute S star. We actually see S star in the marketplace. There would be a market for these tranches in the marketplace. We would see the fair spread as star in the marketplace. And what we would try and do is back out a correlation parameter, let's call it rose say, which meant that this right inside, or that the model fair price S* or row was equal to the S* that we saw in the marketplace. And then row would be called an implied correlation parameter. So very analogous with what we saw with equity options and computing implied volatilities for a given strike and maturity. One point that's worth emphasizing is that if this Gaussian [INAUDIBLE] model was somehow correct, then you should see the same correlation value for different tranches. In other words, suppose I compute the fair correlation for a chance with attachment points, say 3% and 7% and then I can back out a correlation parameter row. Well I should see the same row if I change 3 to be 7 and I make 7 equal to 10 say. In other words, if the model was correct I should see the same value of rho regardless of what values of L and U I use. But in practice this is not the case. We see a different applied correlation for different attachment points L and U. And in fact it's possible in some circumstances to not be able to back out and apply the correlation parameter at all.