In the last module, we saw the mechanics of a synthetic CDO Tranche. We saw there was a premium leg and the 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 are based upon the remaining notion 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 tranche may need to pay out upon each default event. Formerly, the time t equals zero value of the premium leg which we're going to call P subscript zero, zero denoting t equals 0 and superscript LU, which refers to the lower and upper attachment points respectively. So this quantity is equal to what's on the right hand side here of expression six. So we have S which is the annualized spread or premium paid to the production seller. So this will be a percentage, it might be two percent, it might be three percent or it might be ten percent for a chance where you expect to see many losses, but maybe it'll just be a quarter a percent or some very small number as well for chance which is relatively safe. Dt is the risk-free discount factor for payment date t, Delta t is the accrual factor for day t. So typically Delta t would be approximately one quarter corresponding to quarterly payments, n is the total number of periods in the contract. So for example if the CDO lasts or has a maturity of 10 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 got to 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 minus 1 and t. So remember the total notional of the tranche is U minus L. So if U is seven percent and L is three percent, then the total notional of the tranche is 7 minus 3 equals 4 percent. 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 minus 1. So the insurance, the S times Delta t is made on this expected notional. Those payments which occur at time t then 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. In practice, default events don't take place at the beginning or end of a three month period, that might take place in the middle and maybe you'll 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 cashflows paid to the protection buyer upon losses occurring in the tranche. Formerly, the time t equals 0 value of the default leg which we're going to call DL subscript zero for t equals 0 superscript LU satisfies this equation here. Again, dt is the discount factor. We have a sum from t equals 1 to n representing the n 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 to t. So these are the expected payments that must be paid by the seller of protection or the seller of insurance between times t minus 1 and t. So these are the risk-neutral expected losses in the tranche 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 will see that I also have an expectation of a tranche 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 to remind ourselves, we know that the tranche loss function is given to us as follows. We know that TL_ t_ LU is 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 minus R, U minus L and 0. Now, the only random variable in here is L, the number of losses in the portfolio. We also know that the expected value of the tranche loss at time t is equal to the sum from L equals 0 up to capital N, TL_LU t of L times PL of t. PL of t we saw in the one-factor Gaussian model, this is equal to the integral from minus infinity to infinity of P_ n L of t given M times the probability density function for standard normal random variable Phi_M dm. Finally, we saw that in general, we could compute this using our iterative algorithm, that was the algorithm with the nested four loops if you recall. We had a four, I think it was for i equals 1 up as far as n and then we had for j equals 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 to 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. 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 actually compute this quantity, we can actually get it to run very quickly. 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 a later module if we have time, but the reason it is 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 pairwise correlations and so on. So 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 default leg of a CDO tranche very efficiently in practice.
