Very briefly, I want to spend a little bit of time talking about Copulas. We introduced copula in an earlier module when we discussed the Gaussian copula model. But we haven't had time really to spend on Copulas more generally. I'm just going to say a little bit about Copulas here and the Gaussian copula model. So certainly, the Gaussian copula models is the most famous model for pricing structured credit securities. There has been enormous criticism aimed at this model, and most if not all of it is justified. That said, some people like to say that they didn't understand the weaknesses of the Gaussian copula model until after the financial crisis broke. Well, I simply don't think that's true. There's nothing we've learned about the Gaussian copula model that we didn't know before the financial crisis. So to say or to plead ignorance that you didn't understand the model, that the market didn't understand the weaknesses of the model until the crisis blew up or crisis took place, simply is not a well-founded statement. Many people fully understand the weaknesses of the Gaussian copula model. Just to mention a couple of them, it's a static model. By static, I mean there are no dynamics in the model. We just compute the expected tranche loss at a fixed period of time. There's no stochastic process here, we assume credit spreads are constant, we assume correlation is constant, we assume correlation is constant across the various names. There are problems with this model in terms of time consistency and so on. So I know this is a very brief aside. We don't have time to go into this in any more detail. But certainly, the weaknesses of the Gaussian copula have been well understood. There has been a lot of academic work in building better and more sophisticated models, but none of them are really satisfactory. It's an aside, but I want also make this point, a common fallacy is that the marginal distributions and correlation matrix are sufficient for describing the joint distribution of a multivariate distribution. In other words, what I'm saying here is many people think that if you've got a multivariate distribution, the only thing that you need to know about distribution are the marginals and the correlation between the random variables. Well, that's not sure. Correlation only measures linear dependence. On the next slide, we'll provide a counter example to this. So in this slide, we've got two distributions, two bivariate distributions. The first one is a bivariate normal distribution and the second is what's called a Meta-Gumbel distribution. What I want to emphasize here is that in each case, the marginals are standard normal. So the marginals in each of these are standard normal. Now the plots aren't really drawn to an appropriate scale. Maybe we should stretch the x-axis out here because really the width, the length of the horizontal piece here should be the same length as the vertical piece here. So they're both the marginals in all cases where N 0,1, the correlation in each case is 0.7. So what we have here are two bivariate distributions which have the same marginals and the same correlation, but they're very different. So we can see that the large joint moves are much more common to occur in the Mate-Gumbel distribution than they are in the multivariate normal distribution. The way to see that as the electron moves where the x and y variable are both greater than or equal to three, these are up here and up here. So over in the multivariate normal, we see there's only one move where both the x and y variable are both greater than or equal to three, whereas over here, we see there are five such situations. What we've done here by the way is we've simulated 5,000 points in each of these distributions. So five of those 5,000 points. So 0.1 percent of those 5,000 points resulted in extreme joint move, whereas only one-fifth of that resulted in an extreme joint move in the case of the bivariate normal. Now, 0.1 percent might seem like a very small number and indeed it is. But when it comes to figuring out losses on equity tranches and so on, that 0.1 percent can be substantial. So the point I just want to make here is that copulas, by the way the usage of copulas and financial engineering actually originates in insurance mathematics. So this is an example where technology and mathematical modeling technology copulas was actually borrowed from the insurance world, from the insurance mathematics and taken into the financial world, and used to price structured credit products, including CDO tranches, and entity defaults swaps which we haven't discussed at all. But the point I'm trying to make is that, these copulas encompass all the information about joint distributions and correlation isn't enough to do that. One criticism of the Gaussian copula model is that it results in joint moves that occur too infrequently. We don't see frequent enough joint moves, and that might have an impact as well for super-senior tranches which are only hit when lots of defaults occur together, and the losses in the portfolio get above the lower attachment point for the super senior joints. I can't get into much detail as I would like to, but this here just gives you a brief flavor of copulas and how correlation does not capture the dependence of multivariate distributions. Correlation only captures the linear dependence of multivariate distributions. The copula, the idea of a copula captures all of the dependence in a multivariate distribution.