The formula that Wall Street never believed in

In ‘The Formula That Killed Wall Street’? The Gaussian Copula and the Material Cultures of Modelling, Donald MacKenzie and Taylor Spears present a history of the development of the one-factor Gaussian copula model, which is used to price various structured products, including Collateralised Debt Obligations (CDOs). As the title of the paper suggests, the model has many critics and has had a lot of blame placed at its feet.

What this paper reveals that really stands out is that the quant community also didn’t, and doesn’t, rate the Gaussian copula model highly at all. In fact, we’re putting that very mildly if the statements from quants interviewed by the researchers are anything to go by.

Furthermore, this was a view held by many before the financial crisis hit. But even in the face of this rejection, the model has stayed in use through multiple crises and is still in use.

The researchers sought to investigate exactly why that might be. Surely such a discredited, ‘killer’ model should be shown straight to the door? Or maybe, just maybe, the Gaussian copula isn’t that killer at all. Naturally there must be purposes that the model served, and is still serving… but before we go there.

The dataset

Rather than rely on numbers, the paper draws its main inputs from people (who spend a lot of time with numbers):

Fortunately, we are able to control the ‘hindsight bias’ that arises from the involvement of these models in the crisis by the fact that of the 95 interviews we are drawing on, 29 took place before the onset of the crisis in July 2007. These 94 interviews form the primary empirical basis of this paper; also drawn on is the extensive technical literature on Gaussian copulas. We draw here primarily on the subset of interviews (numbering 24) that were with ‘quants’,

Talk about a unique dataset! Let’s get straight to it, shall we?

All the hedges are here

First of all, it needs to be stated that one of the main constituent groups in this story were using the model to hedge:

…despite the widespread impression of reckless risk-taking that the credit crisis created, the standard practice of derivatives departments was and is carefully to hedge their portfolios.

The hedge ratios dictated by the models were most critically needed for managing the risk created by the structuring of bespoke CDOs that were popular at the height of the credit boom (for the nerds: single-tranche CDOs, primarily mezz tranches). And from the hedge follows the price:

‘price is determined by hedging cost’ (McGinty, Beinstein, Ahluwalia and Watts, 2004: 20).

From the price follows the P&L:

…the price quoted to an external customer will be greater than that cost, the difference generating the bank’s profit and the trader’s hoped-for day-one P&L.)

A crucial point of this paper is that the Gaussian copula model, by virtue of directing hedging, allowed this “day-one P&L” whereby the profit from a trade, no matter its maturity, could be booked upfront.

The pricing, and hedging that the model directed, were given weight by the fact that it was widely used by the market as the standard model and employed market observed inputs (another one for the nerds: implied correlations). The model was the standard even though it had well-known flaws and a history of examples when it had completely failed to generate prices at all, such as in the “correlation crisis” of 2005.

Bonus culture

There was therefore an advantage in sticking with the model — it served everyone’s bonus well, as this was determined in large part by P&L. Dumping the model would have left a void that could not be filled, or would take considerable effort to fill in terms of finding a standard for all to agree on and getting auditors to give it their blessing.

The model also served as a common language when communicating with other banks and trading partners, e.g. in price negotiations.

The paper notes that the banks were so concerned to ensure that they were speaking in the same language, i.e. that their models met the market standard given that there are different possible implementations, that they used a service provided by data company Markit, called Totem.

Every month, Totem would send out examples of CDOs for the banks to price. The banks would duly price them, send their homework back, and in return get a report card of how they were pricing relative to their peers.

Everyone’s a hater

But again, this was despite the belief that the Gaussian copula model was flawed in very fundamental ways, or as one quant told the researchers, it didn’t even qualify as a model:

The quant who told us that the Gaussian copula was ‘not a model’ went on to explain what he meant: ‘it doesn’t satisfy the law of one price [in other words, the absence of arbitrage opportunities]. It … can give you inconsistencies and arbitrages very easily. You’re not computing values … as expectations under some well-defined measure.’

It is important to note that the quant who said this had made important technical contributions to the development of the Gaussian copula family of models. Others were even more critical. ‘Copulas are generally an early doodling activity in an area’, said another quant in a January 2009 interview. ‘They are a simple trick to allow yourself to preserve the marginals [default probabilities for individual corporations] and to induce some sort of coupling.’ Copulas ‘are perceived as a hack’, he said, despite having ‘superficially attractive properties like being able to perfectly reproduce markets’.

The interviews in the above took place in November 2008 and January 2009, but it was a sentiment widely expressed before the crisis too, not least because of the failure of the models in the spring of 2005, as mentioned above. For example, from an interview in January 2007:

“So, the thing that is very interesting on credit [derivatives] is … almost for the first time in finance we have models which are not so robust and are almost there as a kind of consensus, and that’s all very well until something changes and something can change quite dramatically.

Where the losses are buried

Now let’s reflect on the fact that the groups that relied the most heavily on Gaussian copula models, namely the correlation trading desks, did not experience catastrophic losses during the crisis. Billions, for sure, but not often tens of billions.

To find truly armageddon-sized losses, one must look instead to the desks involved in the manufacture of CDOs of Asset-Backed Securities (ABS). Those were more typically the products of mortgage desks, not credit, and not correlation. In addition to that, those desks were rarely employing the Gaussian copula as they didn’t have a need for it.

The assembly line

The ABS CDO desks were, rather, distribution lines and as such, their models were those of the rating agencies. All those desks needed to do was structure deals in order to achieve the desired rating/return profile, e.g. highest spread for biggest AAA-tranche. As the authors state, “market participants had ‘outsourced’ the analysis of ABS CDOs to the rating agencies”. Put event more bluntly:

The first author vividly remembers a February 2009 interview in which he asked a senior figure at a firm that managed ABS CDOs what correlation model the firm had employed, only to be met with a blank stare: no model of its own had been used.

In the major investment banks, some analysis of ABS CDOs (beyond simply checking desired ratings) was conducted, but in most cases very little by the standards of the culture of no-arbitrage modelling.

Other derivative desks, like correlation, had the culture of “no-arbitrage modelling” and the more sophisticated models that went with that. However, they weren’t involved in the toxic products that derived their value (or lack thereof) from subprime mortgages and the like:

ABS CDOs often fell outside the remit of the derivatives departments of those banks. They were frequently constructed and analyzed by other groups, such as those specializing in mortgage-backed securities:

Enter, stage left

As for what the rating agencies were up to, they weren’t even using the standard one factor Gaussian copulas — they were, in fact, several steps behind in the evolution of those models.

By the time of the crisis, the ratings agencies had moved only partially from the early one-period models to fully fledged copula models of the kind introduced by Li, and the move was not of great consequence to the processes generating the crisis.

The above quote alludes to the fact that even if the agencies had become more sophisticated, with the way they went about rating ABS CDOs, it probably wouldn’t have helped:

It was far less consequential than the way in which the evaluation of ABS CDOs was mapped onto the organizational structure of the agencies (the separate analyses of first the component ABSs and then the CDO’s structure), the estimation of the probabilities of default on ABSs using data from a period of benign economic conditions, and the fact that the CDO groups in the agencies analyzed an ABS CDO in almost the same way as a CDO based on corporate debt.

What were they even trying to do?

What looks like a colossal failure on the part of the agencies, needs to be viewed from the perspective of what they were trying to accomplish:

…Gaussian copula models as employed by the rating agencies were different in their effects from Gaussian models employed in the derivatives departments of investment banks. Not only were the goals and the ontology different (the rating agencies sought to estimate actual probabilities of default; the banks sought to extract risk-neutral probabilities and hedging ratios), but the surrounding processes also differed. Governance (risk control and the booking of profit) was certainly one aspect of the use of the Gaussian copula in investment banks, but ratings were almost entirely about governance. With many investment managers constrained either by regulation or by organizational mandate to buy only investment-grade securities, the ratings of such securities dictated the nature of the market for them.

The rating agencies didn’t need to know what the market thought, as communicated through prices and implied correlations. The agencies also didn’t need to know when a given ABS would default, but merely if it would before maturity. And, well, that was ‘determined’ by a limited, overly rosy, historical dataset.

Having done that, a rating could be assigned to each ABS or tranche thereof and this could be passed onto the colleagues would would rate the overall structure as a CDO of effectively generic bonds, i.e. where ratings were the distinguishing feature alone.

The assignment of ratings to the tranches of the CDO served the purpose of allowing those with strict investment mandates, and/or those subject to regulation, to hold these securities.

Who, particularly on the side of the banks, was an ignorant party to what was going on? Who wasn’t in on the fact that it was the models, and business practices, of the rating agencies that were being gamed here?

The crisis was caused not by ‘model dopes’, but by creative, resourceful, well-informed and reflexive actors quite consciously exploiting the role of models in governance. ‘[T]he whole market is rating-agency-driven at some level’, one of our earliest interviewees told us, a year before the crisis: ‘the game is …to create …tranches which are single-, double- or triple-A rated, and yield significantly more than a correspondingly rated [bond]’. That interviewee did not himself directly participate in that ‘game’ (his hedge fund was profiting only indirectly from the fact that, as he put it, ‘there are investors who are constrained by ratings’), but other interviewees did. Two told us how they had employed optimization programs to find the highest-yielding pools of securities that would still make possible CDOs with sufficiently large AAA tranches…

Where does all of that leave the Gaussian copula then?

Looking a bit less like the killer it was made out to be, we hope.

Related links:
Copula culture – FT Alphaville
‘The Formula That Killed Wall Street’? The Gaussian Copula and the Material Cultures of Modelling - Donald MacKenzie, Taylor Spears
Recipe for Disaster: The Formula That Killed Wall Street – Wired
The Whale and the Quants – Dealbreaker

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