In this module, we're going to discuss the pricing and risk management of CDO portfolios. We will focus mainly on the risk management of CDO portfolios. Unfortunately, the risk management of CDO portfolios is an enormous topic in and of itself, and we won't have time to do it justice in this module. But however, we will discuss some of the issues that arise. We will discuss some of the weaknesses with the Gaussian copula model. We will also mention as an aside, that linear correlation, the correlation coefficient in and of itself is not enough to describe the dependent structure for multivariate distribution. So, this fact is lost on many people and we will emphasize it at the end of this module. Here's an example of a sample synthetic CDO portfolio. Across the top, we've got various columns. So, the first column index, this contains a name, so CDX IG A, this actually refers to a reference portfolio. So, this is an index with a certain number of names in the index and these names are known. Typically, there would be a 100 names or a 125 names in the index. So, CDX IG A IG stands for investment grade, A is typically actually numbered, it could be eight, nine, ten, eleven and so on and different numbers can contain different reference portfolios. The second column is just a tranche description, it could be an equity tranche, a mezzanine tranche, senior trance, or it could be an index. This can be viewed as a tranche with a lower attachment point of 0 and an upper attachment point of 100 as we have here. And the next two columns indeed contain the attachment points, L for lower, U for upper attachment point. We have the maturity that is 10 years, 7 years, 5 years, 3 years, 4 years and so on. So, the materials can vary and we have the notional amount so this is the notional amount of protection that we are buying or selling. And these are the current prices and basis points. Now, these are given, these are the spreads. The spreads, current market spreads of these tranches. the equity tranche prices are often quoted in a different format to reflect upfront payments and so on. But we're not going to concern ourselves with that. IG as I said, will refer to investment grade, HY, for example, refers to a high yield. So, these would be riskier credits or riskier bonds that are more likely to, to default. very often, there's substantial overlap in these portfolios. So, for example, IG A and IG B, each portfolio or reference portfolio will contain 125 names or 120 names or so on. And in the case of A and B here, typically, there would be a very large overlap between the two portfolios. So, most of the names in A will also be in portfolio B, and so on. In practice, structured credit portfolios could contain many, many positions with different reference portfolios, different maturities, and counter-parties. They also can have different trading formats, so as I said earlier, sometimes these tranches trade in the, in the form of a spread, a spared that is paid quarterly, or a premium that is paid quarterly. This is the insurance rate if you'd like for buying protection or buying insurance on the, the tranche, but sometimes, they can also trade in an upfront format and R, the running spread format. The ultimate payoff of such, of such a portfolio is very path-dependent with substantial idiosyncratic risks. They are very difficult to risk manage. they can also be very expensive to unwind and that is due to why bid-offer spreads. Over here, what I'm giving here is a current price but this should really be interpreted as the midpoint of the bid-offer spread. So, the bid on the offer will be on either side of, say, 223. Maybe the, we'd have 210 and 240. And so, if you want to sell protection or if you will hid one side of the bid-offer spread, if you want to buy protection, you would hit the other side. So, if I'm, if I'm buying protection, I'm going to have to pay 240 basis point for that protection. If I'm selling protection, I'm going to receive 210, so that's the bid-offer, okay? And sometimes these bid-offer spreads can be very wide so actually unwinding such a portfolio can be very expensive, especially in times of market stress or when these portfolios, these synthetic CDO tranches aren't trading very often as would be the case today. Computing the mark-to-market values of these portfolios can also be very difficult because market prices maybe non-transparent. I don't think I've used the phrase mark-to-market yet in this course. But just to be clear, mark-to-market is referring to the current value of portfolio using current prices in the marketplace. So you're not using the historical price at which you purchased a portfolio of securities, instead, you're using the current market price for these securities. So, that refers to mark-to-market. And on this note, you might be interested in the Belly of the Whale series on the Alphaville blog of the Financial Times. You can actually get to that blog via this link here. And while the Financial Times does have a paywall, so most of the articles aren't available for viewing freely, their blogs are. And so, the articles in this series can be found here. This series refers to the so called London Whale. And in fact, the London Whale first came to attention because price levels in the CDX IG9 index. So, this an example of where we're using a number. So it's the IG9 index. It diverts too much from other related price levels. In particular, it diverts too much given the CDX prices of the credits in the IG9 portfolio. So, you'll see a lot of interesting material in the articles that have been published on this series, on this Alphaville blog. You won't be able to understand everything in this series and that's in part because there's a lot of jargon and there's a lot of references to positions that we can see. And indeed, there's references to communications that we can't see. Maybe there weren't e-mail communication but verbal communication between some of the players and so you won't always understand what's going on. But you will see a lot of discussion of value at risk, and Gaussian copula and the synthetic CDOs in risk management. By [UNKNOWN] you've said this at the beginning. But this London Whale came to attention because of ultimately massive losses that occurred, in the, synthetic credit portfolio of the Chief Investment Office of JP Morgan. So, this is a very recent situation, where they lost 7 billion dollars out of the Chief Investment Office on synthetic credit portfolios. And reading this series is certainly of interest and certainly relevant to what we've been discussing in these modules. Risk management for structured credit portfolios is also very challenging. we've seen two types of risk management to date, we haven't gone into either one in any real detail for time reasons but they're certainly both very important. The first the scenario analysis where what we did was we stressed the important risk factors for portfolio. We moved these risk factors to different level. We reevaluated the portfolio in these stress scenarios. Computed the profit and loss that would therefore arise, and figure out or evaluate overall risk of a portfolio based on the PNLs in the scenarios. So if we wanted to do a scenario analysis with the synthetic CDO portfolio, we'd have to figure out first of all what are the main risk factors. Well that's a very difficult question to answer. There are so many moving parts here. it'll be hard to figure out what are the risk factors. Of course, overall credit spreads are important because credit spreads drive the individual default probabilities. And certainly, the riskiness of these CDO tranches increases as individual default probabilities increase. So certainly, the overall level of credit spreads is important. But, what about the individual credit spreads? Some CDS spreads may increase, some CDS spreads may decrease, and depending on what happens, you will get very different outcomes for given CDO tranches. That, correlation, of course, is a hugely important factor. In fact, the trading of synthetic CDO tranches is often called correlation trading because correlation as we saw, it drives the value in particular of equity tranches, also super senior tranches. and so it's very important here to stress correlation appropriately. But in fact measuring correlation, even understanding correlation, what correlation is, what this correlation of default times actually means, that's, they are difficult questions to answer. And it's very difficult to determine what correlation risk factors are there and how you should stress them. Moreover, you need to determine, what are reasonable stress levels? How far should you stress a given factor? What's reasonable? What's not reasonable? What's like to happen, what's very unlikely to happen, what's almost impossible to happen? You have to be able to answer all these questions in order to do a scenario analysis. Finally, suppose you could figure out what appropriate risk factors are and you could figure out what reasonable stress levels for these factors are. Well then, how you going to reevaluate the portfolio, your synthetic CDO portfolio in a given scenario where you've stressed these factors. Well to do that, you need some sort of model. And as I mentioned before, it is very difficult to find a good model. In fact, I think it's fair to say that there isn't a satisfactory model for pricing CDOs out there in the marketplace. The Gaussian copula model has been the standard model but it is certainly a flawed model and has many, many weaknesses. So, scenario analysis is certainly very difficult. What about the Greeks? Well, we saw the Greeks when we discussed equity derivatives. We saw delta, gamma, vega, theta, and so on. Well, you can also come up with Greeks for synthetic CDO portfolios. You can figure out how much the value of the CDO tranche will increase if an individual credit spread or default probability increases or decreases, and so on. So, you can certainly come up with Greeks but there are many, many Greeks, you could argue you've got a separate Greek for every individual default probability. You've got Greeks to correlation and so o. but you've basically got too many of them. You've got too many moving parts here. The Greeks are model-dependent and it would be very difficult really to risk manage a portfolio based on the concept of the Greeks. In fact, there's an interesting article you can read here. It's from the Wall Street Journal back in 2005, which discusses some of the fallout of the downgrading of Ford and General Motors in May 2005. Certain investors, in some of these synthetic CDOs, found out that their hedging, using the Greeks didn't work nearly as well as they anticipated when Ford and General Motors were downgrade, downgraded. Just as in the side, Ford and General Motors were part of, were members in the reference portfolio for very commonly traded [UNKNOWN] at that time, and so there are inside, the reference portfolio for CDO tranches. and so, certainly some market people lost a lot of money when they thought they were actually hedged when Ford and General Motors were downgraded. I should mention that, in fact, the Wall, the Wall Street Journal is behind the payroll and so you may not be able to read this article but it depends. On one or two occasions I've been able to read it and I just Google it, other times I can't, so maybe you'll be able to see it. And that said, don't believe everything you read in this article. in my experience, some of these articles which talk about fairly arcane and complicated details in, in finance, don't always get all the facts right. But the overall picture is pretty accurate and it's certainly an interesting read. Liquidity risk and market endogeneity are also key risks that must be considered, and that should have been considered in the trading of CDOs and the risk management of CDOs. I've mentioned market endogeneity before. It basically refers to the idea, for example, that if everybody is holding the same position, then that's a much riskier situation to be in, than if only a few people hold a position. And that's because if everybody needs to exit that position at the same time, or in other words, if everyone wants to run for the exits at the same time, then everybody wants to sell the security at the same time. There's no demand for it and the price will collapse. That's an example of this concept of market endogeneity. we've got a trade that's too crowded, too many people holding a position. Too many people wanting to get out of it at the same time, prices collapse. This certainly happened during the financial crisis. Other problems that arose in the whole structured credit area during the financial crisis were the massive overreliance on ratings agencies to rate some of these tranches. And overreliance on models that really weren't worthy or that were, at best, a poor approximation to reality. you had issues related to just the behavior of organizations, the incentives of individuals making big decisions in these organizations. All of these obviously played a very important role in the financial crisis. Very briefly I want to spend a little bit of time talking about copulas. We introduced copula in an earlier model 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 model 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. With 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 after the crisis blew up or the crisis took place, simply is not a well-founded statement. Many people fully understand the weaknesses of the, 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, 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 on building better and more sophisticated models, but none of them are really satisfactory. It's an aside but I want to 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 others, 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 true. 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 one 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, 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 are were in 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 correlations, but they're very different. And one way to see they're different is the following. The Meta-Gumbel distribution is much more likely to see large joint moves. And the way to see that is to look for moves where the x and y variable are both greater than or equal to 3. 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 3. 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 from each of these distributions. So, five of those 5,000 points, or 0.1% of those 5,000 points resulted in extreme joint move. Whereas only 1 5th of that resulted in an extreme joint move in the case of the Bivariate Normal. Now, 0.1% might seem like a very small number. And indeed it is. But when it comes to figuring out losses, that 0.1% can be substantial.