In this module we're going to discuss and introduce the Gaussian Copula model. This model came under a lot of criticism during the financial crisis. So it's very much worthwhile introducing it here, and seeing how the Gaussian Copula model actually works. We're going to use it to construct the probability distribution of the number of losses in a reference portfolio of bonds. This reference portfolio will be the portfolio underlying CDOs and CDO chances that we will discuss in later modules. We assume there are N bonds, or credits, in the reference portfolio. Now, I use the word, credit, here because sometimes that is how the underlying bonds in the pool are referred to. So a credit can refer to a bond or it can refer to a company like General Motors, or Ford, or so on. So I'm going to use bonds and credits interchangeably here. Each credit has a notional of Ai. This is in dollars. However, sometimes it can be expressed as a percentage of the overall portfolio notional. If the ith credit defaults, then the portfolio incurs a loss of Ai times 1 minus Ri. Where Ri is the recovery rate that is the percentage of the notional amount that is recovered upon default. Now, we've seen this already in the context of credit-default-swaps. From a modeling perspective Ri is often assumed to be fixed and known. In practice however, it is random and not known until after a default event has taken place. We're also going to assume that the risk-neutral distribution of the default time of the ith credit is known. And in fact this can be estimated from either credit-default-swap spreads or the prices of corporate bonds. Therefore we can compute qi(t), the risk-neutral probability that the ith credit defaults before time t for any t. So we're going to assume that we know these probabilities for all of the names, all of the bonds in the portfolio, and for any time t. Remember, for credit default swap, you can actually compute the term structure of spreads or premier, so this is t. This is the fair spread in the credit-default-swap, and you might see some function like this for different maturities. And so, you can back off from this what these qi of t's are. So, we're going to assume that these qi of t's are known to us. So now, let's discuss the the Gaussian Copula model. We're going to let Xi denote the normalized asset value of the ith credit. So, Xi you can think of if you like as referring to Di plus Ei. So, this the debt plus equity of the ith company. As I've said, the company could be Ford, or General Motors, or Volkswagen, or any company you like. We're going to assume that Xi is equal to ai times M plus the square root of 1 minus ai squared times Zi, where M and the Zi's are standard ith ID normal random variables. Note, note also that each Xi is also a standard normal random variable. Each of the factor loadings, ai, is assumed to lie in the interval 0, 1. It should also be clear that the correlation of Xi, Xj is equal to ai times aj. And this follows because the correlation of Xi with Xj is in this case, equal to just the expect value of Xi times Xj. And that is true because the Xi's are standard normal random variables, which means the have mean 0 and variance 1. So normally I compute the correlation as being the covariance divided by the square root of the product of the variances. Well the variances, in this case, are 1. So now I don't have to divide by anything. The, the covariance is the expected value of Xi Xj minus the expected value of Xi times the expected value of Xj. But the expected value of Xi and Xj is 0. So therefore the correlation is just equal to this term here, and clearly then this is equal to ai, aj, times the expected value M squared. And M is a standard normal random variable, so the expected value of M squared is equal to 1. And so that's how I get this expression here. It should also be clear that the Xi's, our multivariate normally distributed, an we'll use that as well, in a moment. We're going to assume that the ith credit has defaulted by time ti, if Xi falls below some threshold value xi bar. An that's a function of ti, but generally will just refer to xi bar. By our earlier assumption, it must therefore be the case, that Xi bar equals phi inverse of qi of ti, where phi is the standard normal cdf. Why is this? Well if you recall, we said that qi, we said qi of t is the risk-neutral probability of default by time t. We also said that the ith credit defaults by time ti if Xi is less than or equal to xi bar of ti. What the probability of this occurring, we know is equal to phi of xi bar of ti. Because the Xi's have a standard normal distribution. So this is equal to phi of xi bar of ti, but we also know it's equal to qi of ti. And so therefore we see that xi bar just apply phi inverse to both sides of this equation, we see that xi bar of ti is equal to phi inverse of qi of ti. We're going to let capital F of t1 up to t capital N denote the joint distribution of the default times of the n credits in the portfolio. Then we know that F of t1 up to tn must be equal to the following. It's equal to the probability that X1 is less than or equal to x1 bar of t1, all the way up to Xn being less than or equal to xn bar of tn. Because this is the event that X1 up to Xn defaults before times t1 up to tn, respectively. However, X1 up to Xn is a multivariant normal distribution. So this is equal to phi P, the phi is in bold so that represents a multivariate normal distribution. P refers to the covariance or in this case the correlation matrix. And it's got a mean 0 as well, and it's evaluated, the arguments of this multivariate normal CDF are x1 bar of t1 up to xn bar of tn. So that gives us this line. And now we just substitute n for these x1 bars using this expression here, and that's how we get this equation down here. So, believe it or not, even though this is very simple, there's lots of notations, so on, but it's actually very simple to, to derive this expression, given our assumptions. What we have is the infamous one factor Gaussian Copula model. So this here give us the one factor aspect Gaussian Copula model. It's one factor because we just have one random variable m driving the dependent between the Xi's. If we go back to the previous slide, we'll see m appearing here. M is a random variable that is uncommon to all of the XI's. So all of the correlation between the XI's comes from this random variable M. If we introduced a second random variable, say n that was also common to all of the XI's then we would end up with a two factor Gaussian Copula Model. The next task is to actually compute the so-called portfolio loss distribution. In order to price credit derivatives and CDOs in particular, we need to be able to compute this portfolio loss distribution. So that's our first goal here. The first thing we'll do is we'll note the following. That if we condition on the random variable M, then the n default events are independent. And in particular the default probabilities conditional on the random variable M are given to us by these expressions here. Now where does this come from? Well, this comes from the following, we know that qi t of M or t given M is equal to the probability that Xi is less than or equal to xi bar of t given M. Well, we know what Xi is equal to. So we can substitute n for Xi. This is equal to the probability Xi is equal to ai times M, plus the square root of 1 minus ai squared times Zi, is less than or equal to xi bar of t, given n. So this is then equal to the probability that Zi is less than or equal to xi bar, t minus ai, times M divided by the square root of 1 minus ai squared. And that's all conditional on M. Now if you think about it for a second conditional on M, if we condition on M as we have here then everything on the right hand side here is a constant. Zi is a standard normal running variable. So this just becomes the probability that a standard normal running variable is less than or equal to. This expression here, and so that's equal to phi of what we have inside here, so that's what the conditional risk neutral default probabilities are. There are qi of t given n's. Now let P superscript n, l of t denote the risk neuter probability that there are a total of l portfolio or defaults before time t. We may then write pl of t as being the integral for minus infinity to infinity of p superscript n l of t given M times phi of MdM where phi is the standard normal PDF. Now if you're wondering where this expression comes from, well, this is just a standard basic undergraduate expression for probability. In discrete form, you can imagine the following. Suppose we want to compute the probability that X is equal to little x. Well, a standard way of doing this is to consider another random variable y, and to sum over all possible values of y. So it's equal to the probability of X equals little x, and Y equals little y. And this is also equal to the sum over little Y of the probability that X equals little X. Given Y equals little Y by the probability that Y equals little Y. So this is a standard expression you probably all seen before in your undergraduate probability class. We're just using this here and. In density form rather than discrete probability mass function form. So M here takes the place of y. So we have our m here, taking the place of y over here, and so instead of a summation, we have an integral. And we're integrating with respect to m instead of summing over the y values here. So this is standard, so we can now write the probability of l default, and just to be consistent, I should have put a superscript n there. So the probability of l default out of the n names by time t, is given to us by this integral here. So the next task is going to be how do we compute this quantity? We know phi of M, it's just the standard normal density. So, we need to compute this quantity here. We can compute this quantity, then we can evaluate this integral numerically, and therefore compute this risk-neutral probability function here. So let's focus on how to compute this expression here. So, in fact, we can easily do it using an iterative procedure, and the iterative procedure will work as follows. So the first thing we're going to do let's initialize. We're going to have to run a couple of for loops here. So let's initialize our quantities first of all, we're going to set p 0. Given M to be equal to 1 minus q1 of M. We're going to set p1, given M to be equal to q1 of M. And we're going to set p2 given M to be equal to P3 given M all the way up to Pn given M. We're going to set all of that equal to 0. Now, what I'm doing here is, I'm dropping the dependence on n and t here. So, I don't want the slide to get too cluttered. If I did I'd have an n in all of these values here. And I'd have a t in here, and so on. But that's just going to get really cluttered. So, I won't do that. So, I'm dropping the dependence of these quantities on n, and t. So, now we're going to do the following. We're going to run the following for loop, we're going to say. For i equals 2 up as far as n. And, for j equals 1 to i. I am going to update these quantities here that i have already initialized. In particular I am going to say. P, j given M so this is the probability of j defaults, conditional on M. So this is going to represent the probably of j defaults in the first i names. So there, there are two ways we can get j defaults in the first i names. We could have j minus 1 defaults in those first i names, and having the ith name defaulting. Which is represented here or we could have had j defaults in those first i names. And then having the ith name not defaulting. So that's the end of this for loop. We must also handle the, the case of 0 defaults. So we must also update that. We get the probability of 0. Given M is equal to whatever the previous value was or the value in the previous iteration. Times 1 minus qi of M. In other words, what I'm really doing here. So this is the end of the, of the algorithm. What I'm really doing here is I'm lining up the names. I'm ordering them from say, 1 all the way up to to nth names. I know the risk neutral probability given n for all names. And I'm going through these names these credits in order starting with 1, all the way up to value capital N. I'm doing that via these two for loops here, and I'm updating the probabilities at each step. So, as I said here, so if I'm at this point, for some value of i and some value of j, what I'm doing is the following. I'm now going to update the probabillity of the j defaults. In the portfolio based on the first i names. So, the way j defaults can occur among the first i names is that j minus 1 defaults occur in the first i minus 1 names. Remember this value here is the value left over from the previous iteration. Which considers the names from 1 up to i minus 1, so I get this equals the probability of j minus 1 defaults in the first i minus 1 names. Times the probability that the ith name defaults, plus the probability of j defaults in the first i minus 1 names, by the probability that the ith name does not default. And so I can just run through these two four loops to compute pN l of t, given M. So in other words, at the very end once I get up to , N, i equals capital N and j equals i at the end of this procedure I have the probabilities that I want. And now at this point, we can perform a numerical integration on the right hand side of 2 to compute Pl of t. So this is Pl of t here. I now have computed all of this using that iterative procedure, and I can do numeric integration to calculate this quantity here. And there should be a super script in there as well. If we assume that the notional ai. And the recovery rate ri are constant across all credits, then the loss on any given credit will be either 0 or a times 1 minus r. Therefore knowing the distribution of the number of defaults is equivalent to knowing the distribution of the total loss and the reference portfolio. And that's because l losses on that case is equivalent to a loss of l times a 1 minus r and the portfolio of bonds and so knowing one implies the other. So this, the assumption of constat Ai's and Ri's actually simplifies the calculations, and we're going to be assuming that through out here. But I will mention that this assumption is easily relaxed. We could actually get by without it.