Financial Engineering is a multidisciplinary field involving finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part II will be on the use of simple stochastic models to (i) solve portfolio optimization problems (ii) price derivative securities in various asset classes including equities and credit and (iii) consider some advanced applications of financial engineering including algorithmic trading and the pricing of real options. We will also consider the role that financial engineering played during the financial crisis. We hope that students who complete the course and the prerequisite course (FE & RM Part I) will have a good understanding of the "rocket science" behind financial engineering. But perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism.
Pricing and Risk Management of CDO Portfolios
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Из курса от партнера Columbia University
Financial Engineering and Risk Management Part II
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Credit Derivatives and Structured Products
Mechanics and pricing of CDOs; exotic structured credit securities including CDO-squared’s and CDO-cubed’s. Risk management of these products and their role in the financial crisis.
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Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
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.
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