This list of lessons isn't definitive, but it highlights many of the reasons why and how financial engineers and management went astray in CDO valuation.

Lesson 1: Senior Management Knew or Should Have Known The Copula Approach Was Flawed, but They Did Nothing

Robert Rubin famously commented that he didn't have "enough experience" on Wall Street to understand the liquidity puts of CDO tranches that were at the heart of Citigroup's CDO related losses.  But I am sure Mr. Rubin reads the Wall Street Journal, which on August 12, 2005 ran an excellent story on page 1 by Mark Whitehouse, "Slices of Risk: How a Formula Ignited Market That Burned Some Big Investors."  The story described David Li's role in bringing the use of copulas into finance, and it then went on at length on how the technique had failed in the May 2005 period, causing losses of hundreds of millions of dollars.  Did the heads of the audit committees of Citi, Merrill and UBS call in the CEO and chief risk officers and grill them on the use of the copula method in their rapidly growing CDO businesses?  Did Charles Prince or Stanley O'Neal read that article and put on the brakes?  No, they did nothing.  People cause losses, not formulas, even when the world's most important business newspaper warned us all in a page 1 story.

Lesson 2: The Copula Approach Assumed Away All But One Risk Factor, but Analysts Used It Anyway

The August 12, 2005 story in the Wall Street Journal was very specific about why the copula approach caused such losses.  It implicitly assumes that there is only one common risk factor driving the returns on company assets of the reference names underlying a CDO tranche. Other than this common risk factor, all other risk is assumed to be idiosyncratic and uncorrelated with the risks of other companies.  We know this is a gross oversimplification; in fact, a March 23, 2009 press release on www.kamakuraco.com recounts how 40 macro risk factors are used for modeling correlated default on the Kamakura Risk Information Services web-based credit portfolio management tool, KRIS-cdo.  For CDO market participants, however, the assumption of one common risk factor was hugely attractive.  It made the CDO analysis "Excel friendly" and it made the math tractable to the smartest 2% of Wall Street.  The copula approach allows the kind of mathematical exposition that is attractive to academic journals and that one can find in the very interesting compilation of articles in the book The Definitive Guide to CDOs (editted by Gunter Meissner, Risk Publications, 2008).  The cost to this is that the results from this highly simplified abstraction to reality are just not accurate.  The Wall Street Journal put it plainly.  The implication of one risk factor is that you should be able to go long one tranche of a CDO and hedge it with the proper short position in other tranche of the same CDO.  This trade caused enormous losses because the implication of the copula model was plain wrong.

There was another anecdote in the story that seriously called into question the Merton model of risky debt, which is closely linked with the copula approach.  The 1974 classic by Robert Merton assumes that only one risk factor, the value of the assets of a company, drive the prices of the firm's debt and equity.  If this is true, one should be able to buy the debt of a company and hedge it with a short position in the firm's common stock.  Unfortunately, as the Journal recounted, this implication is simply not true and traders in GM and Ford lost tons of money in May 2005 using the Merton-esque trading strategy.  Multiple factors drive bond and common stock prices, but people didn't want to hear it.  In fact, Bob Jarrow and I were very proud to receive hate mail from a prominent junk bond analyst when we made exactly this point in a 1998 publication ("Integrating Interest Rate Risk and Credit Risk in Asset and Liability Management," Asset and Liability Management: A Synthesis of New Methodologies, Risk Publications, 1998).  For a detailed study of the lack of accuracy of the highly stylized Merton model of risk debt, see "In Search of Distress Risk" in the December 2008 Journal of Finance by John Y. Campbell (Harvard University), Jens Hilscher (Brandeis University and Kamakura Corporation), and Jan Szilagyi (Duquesne Capital).

Lesson 3: Lots of Correlations Matter in CDO Valuation, but Copula Users Assumed Only One Correlation Matters

If there are N reference names in a CDO collateral pool, there are N(N-1)/2 pairs of companies in the pool. The correlations between the default probabilities of each pair of companies are of course different.  The same is true for the correlation in the returns on the assets of each pair of companies.  The two concepts of correlation are mathematically linked (see Jarrow and van Deventer, RISK Magazine, 2005 for the formula).  According to the KRIS default probability service, the correlation between the 1 year default probabilities of Citigroup and Bank of America has been 95% over the last five years, but the correlation between IBM and Ford default probabilities over the same period was only 35%.  Copula analysts assume that the pairwise correlations in asset returns are the same for every one of the N(N-1)/2 pairs of companies, but this is simply not true.  The fact that it is not true was well known to traders--the proof was simple. The correlation implied by observable market prices was not the same across tranches of a CDO on the same reference collateral.  The assumption was simply wrong, like the assumption in the Black-Scholes model that implied volatility is constant and should be the same for stock options with any maturity and strike price.

Why did traders, analysts and risk managers use such a simple assumption that they knew to be wrong?  For many, the reason was because it was the easy thing to do.  For the smartest people on Wall Street, who knew how flawed the copula approach was, they pushed the technique as an easy way to induce investors to buy CDO tranches at the wrong price like the story related by my colleague Tatsuo Kishi in our FAS 157 blog post.  The smart ones used a much more realistic valuation technique and they have the profitability to prove it!

Lesson 4: Default Probabilities Vary Randomly, But Copula Users Assumed Away This Randomness

There are two closely related assumptions about the default probabilities used in a copula analysis.  The most common assumption is that they are constant.  A more rare but only slightly better assumption is to assume that they drift over time (non-randomly) in a way that matches an observable term structure of default probabilities, like those on KRIS or in the CDS market.  Unfortunately, these assumptions are also dramatically untrue.  As the current crisis shows, default probabilities rise and fall randomly over the business cycle, driven by common macro factors like home prices and interest rates.  The upshot of assuming away this randomness is dramatic errors in valuation, particularly from underestimating the "fat tails" of losses when the economy turns bad.  This was especially devastating for "super senior" tranches of CDOs.

Lesson 5:  Default Can Happen at Any Time, but Copula Users Assumed That Away

A surprisingly large number of copula users employed a single period simulation, including many third party vendors as explained by Mich Araten of JP Morgan Chase in a presentation at the ICBI Risk Conference in December in Geneva a few years ago.  In a single period simulation, defaults can happen only at time zero or at the end of the single period, not in between.  For a CDO of bank trust preferred securities, maturities can be more than 30 years, so this assumption about timing of defaults is extreme.  The right approach is obvious--to simulate default/no default on a multi-period basis, typically monthly. CDO market participants, however, generally didn't do this.  This explains the widely held but completely false belief that an increase in the single correlation in a copula model will cause the price of the equity tranche to rise (see "Synthetic CDO Equity: Short or Long Correlation" by Robert Jarrow and myself, Journal of Fixed Income, Spring 2008).  That conclusion about equity tranches is due to the assumption of a single modeling period. A multiperiod simulation was essential for accuracy, but most analysts and some vendors clung to a single period model.

Lesson 6: CDO Analysts Can Be Slaves to Fashion, but At High Costs

Gunter Meissner's fascinating compilation of articles in The Definitive Guide to CDOs (Risk Publications, 2008) is "definitive" proof of the "trendiness" of the copula approach.  Even though most of the chapters in the book were penned as the current credit crisis began to unfold, the overwhelming majority of the chapters endorse or extend the copula method.  A reviewer on amazon.com cited only two chapters in the book (Chapter 4 on hedging, and Chapter 16 by Bob Jarrow and myself entitled "CDO Valuation: Fact and Fiction") that were critical of the copula approach.  Even as events were revealing (again) the flaws of the copula approach that the Wall Street Journal highlighted in 2005, many were still singing its praises.  This is certainly not the fault of David Li, who first used the copula approach in finance.  It's the fault of all of us who came after David who failed to maintain our scientific objectivity about what works and what doesn't, and why.

A multiperiod simulation of CDO values with default probabilities driven up and down by macro economic factors is an exercise in computer science that's hard to do without third party software like Kamakura Risk Manager. It's not fashionable, because there's no closed form solution, so you won't see many academic publications on its implications.  Nonetheless, it's the only method that works.  That's perhaps the most important thing we have learned from the attempt to use the copula approach for credit portfolio management.

Donald R. van Deventer

Kamakura Corporation

Honolulu, April 9, 2009