In this module we're going to discuss and introduce the Gaussian Copula Model. This model came in for 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 would be the portfolio underlying CDOs and CDO tranches 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 amount 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 the loss of Ai times 1- 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 premium. So this is t, this is the fair spread and the credit default swap, and you might see some function like this for different maturities. And so you can back off from this [INAUDIBLE] qi(t)s are. So we're going to assume that these qi(t)s are known to us. So now, let's discuss 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 + Ei, so this is the debt plus equity of the ith company. As I said, the company could be Ford, or General Motors, or Volkswagen, or any company you like. We're going to assume that Xi = Ai times M plus the square root of 1- Ai squared times Zi, where M and the Zis are standard IID normal random variables. 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 Corr(Xi, Xj) is equal to ai times aij, and this follows because the Corr(Xi, Xj) is in this case equal to just the expected value of Xi times Xj. And that is true, because the Xis are standard normal random variables, which means they 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 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 of 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. Should also be clear that the Xis are multivariate normally distributed, and 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, we'll say. That's a function of Ti, but generally we'll just refer to Xi bar. By our earlier assumption, it must therefore be the case that Xi bar equals phi inverse of qi(ti), where phi's the standard normal CDF. Why is this? Well, if you recall we said that qi(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(ti). But the probability of this occurring, we know, is equal to phi(Xi bar(ti)), because the Xis have a standard normal distribution. So this is equal to phi(Xi bar(ti)), but we also know it's equal to qi(ti). And so therefore, we see that Xi bar just apply phi inverse to both sides of this equation, we'll see that Xi bar(ti) is equal to phi inverse of qi(ti). We're going to let F(t1 up to tN) denote the joint distribution of the default times of the N credits in the portfolio. Then, we know that F(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(t1), all the way up to Xn being less than or equal to Xn bar(tn). Because this is the event that X1 up to Xn defaults before times t1 up to tn, respectively. However, X1 up to Xn has a multivariate normal distribution, so this is equal to phi P. The phi is in bold, so that it 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(t1) up to to Xn bar(tn). So that gives us this line, and now we just substitute in 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 a lots of notation and so on, but it's actually very simple to derive this expression given our assumptions. What we have is the infamous one-factor Gaussian Copula Model. So this here gives us the one-factor Gaussian Copula Model. It's one-factor because we just have one random variable M driving the dependence between the Xis. If we go back to the previous slide, we see M appearing here. M is a random variable that is in common to all of the Xis, so all of the correlation between the Xis comes from this random variable M. If we introduce the second random variable, say N, that was also common to all of the Xis, then we would end up with a two-factor Gaussian Copula Model.
