Credit model meltdown

Dealers are trading increasingly high volumes of bespoke tranches of synthetic credit risk with each other, yet there still appears to be little consensus on the application of credit models. Is there a danger the house of cards may come tumbling down?

The credit derivatives market is growing at a pace unprecedented in financial markets. From a non-existent business 10 years ago, the value of outstanding notionals stood at more than $26 trillion as of June, according to the latest survey from the International Swaps and Derivatives Association. That's more than four times the size of the over-the-counter equity derivatives market.

The development of the market has proven to be a major boon for loan portfolio managers that can synthetically sell on their over-concentrated exposures to reduce counterparty risk. And the fusion of derivatives technology and credit derivatives has offered end-investors the opportunity to purchase customised credit portfolio exposures at pre-specified risk/reward trade-offs. To provide these bespoke portfolios, dealers have delta-hedged their exposures using single-name credit default swaps (CDSs), credit indexes and index tranches.

In fact, the engine of growth in the credit derivatives market has been in the use of indexes. A Fitch Ratings study published in September says trades related to indexes and index-related products grew by 900% in 2005 to $3.7 trillion by the end of that year.

A report released by the British Bankers' Association (BBA) in September, meanwhile, says full index trades will represent 30.1%, or $6.1 trillion, of the market by the end of 2006, compared with just 9% in 2004. The report adds that tranched index trades have grown in market share from just 2% at the end of 2004 to a likely 7.6% by the end of 2006, or $1.53 trillion. That means, in aggregate, that the index and standard tranche businesses are probably already larger than the single-name CDS market - and the BBA expects the market share of CDSs to tumble from 51% in 2004 to an expected 32.9% by the end of this year.

This increase in volumes has meant bid/offer spreads in the index market have contracted markedly. It is no longer unusual to obtain bid/offer spreads of just 12.5 basis points on a reasonably large five-year equity tranche of a major credit index, such as the iTraxx in Europe or the Dow Jones CDX in the US. These tight spreads and apparent high levels of liquidity have made standardised tranche and full index prices useful benchmarks for calibrating credit models and valuing bespoke collateralised debt obligation (CDO) tranches.

So what is driving these large-scale increases in volumes and the contraction in spreads? According to the BBA, hedge funds and bank proprietary desks, rather than loan portfolio managers or long-term real-money investors, are primarily responsible for the increase in full index and tranche index products. Much of that can be attributed to the hedging of bespoke tranches sold to investors.

The hedging activities based on the model requirements of bespoke trades are significant. That means dealers active in the bespoke market need to be extremely confident in their deployment of credit models. As David Benichou, a portfolio manager at structured credit hedge funds Avendis Capital in Geneva, says: "Active dealers believe in their models. If they didn't, then they shouldn't trade in structured credit."

To provide an illustration of the extent of leverage levels witnessed in the observable standardised market, Benichou says a EUR100 million equity index tranche with a maturity of five years could require a dealer to transact EUR2.5 billion in the underlying European iTraxx index to act as a hedge due to a leverage level of 25 times.

Despite the seemingly buoyant relative-value trading business by hedge funds and dealers in the index business, there are concerns that the perceived liquidity in the index and CDS markets might not be as deep as initially thought. And that's starting to make some parties nervous.

"We worry about how much the apparent liquidity in the credit derivatives market is being driven by structured trades," says a New York-based senior risk manager at a US securities dealer. "Our sense is the active trading of structured credit is actually confined to a fairly small number of market participants. There are not hundreds and hundreds of people trading tranches. Quite a lot of that trading actually happens on banks' proprietary desks as far as we can see, and that's a little weird."

What's more, there is considerable disagreement among academics and quants about the best approaches to modelling correlation, with little in the way of consensus emerging on the most accurate methodology.

It's an area that is catching the attention of regulators, including the Financial Services Authority (FSA) in the UK. The regulator has launched what it calls 'three streams of work' that it hopes will give a better understanding of the structured credit market and ensure high standards of industry practice in this area.

The FSA is specifically targeting the use of mark-to-model practices in the bespoke tranche business. This move appears to be in response to the breakdown in market correlation assumptions in May and June last year following the downgrade of Ford and General Motors' credit ratings to junk, which caused millions of dollars in mark-to-market losses for a number of hedge funds and dealer proprietary trading desks (Risk June 2005, pages 46-48; August 2005, pages 36-38).

Its first focus area ties in with more general concerns expressed by the FSA and other leading regulators: what exactly are hedge funds trading and are they a danger to the financial system?

The second stream is looking at how investment banks value complex, illiquid instruments held on their trading books. The FSA is concerned about the sources of pricing, the independence of this source from the trader, and whether or not model inputs are plausible and assumptions made about liquidity hold true.

The third prong of the FSA's investigation has led to the UK financial watchdog assembling an unnamed panel of banks to analyse a hypothetical portfolio of complex, bespoke and, potentially, illiquid instruments. The FSA says this went a lot further than some of the commercial market consensus pricing tools, as it is looking at valuation, risk analysis, reserving practices and treatment of the portfolio within a firm's value-at-risk model.

Market replication

Bespoke CDO tranches first emerged in the early part of this decade, giving investors the ability to choose the credits in the collateral, the trade maturity, the attachment points (the amount of subordination below the tranche), the tranche width, the rating, the rating agency and the format (funded or unfunded). And the emergence of a liquid, two-way index tranche market in 2003 gave the prospect of efficient price discovery for tranches, enabling participants to infer implied correlation from market prices to determine relative value.

Dealers typically use base correlation for pricing CDO tranches. This approach emerged in 2004, championed by JP Morgan, and was widely seen as responding to shortcomings of compound correlation. Under the compound correlation approach, a model (usually a Gaussian copula) takes all single-name spreads and a single-asset correlation as inputs and produces a tranche spread. As in the options market, where implied volatility is calculated by backing market prices through the Black-Scholes model, an implied correlation can be calculated from traded spreads, using the Gaussian copula model. In essence, compound correlation can be defined as the single correlation that matches the value of a tranche to a market spread.

The problem with compound correlation is that the relationship is not monotonic for mezzanine tranches. In other words, spreads on mezzanine tranches are not an ever-increasing or ever-decreasing function of correlation. That means there is more than one implied correlation value for a given spread on mezzanine tranches.

In putting forward base correlation as a more accurate basis for pricing CDO tranches, JP Morgan argued that tranches of credit portfolios can be thought of as options on portfolio losses. So, for a 3-7% tranche, the payout profile resembles a call spread with strikes at the 3% and 7% attachment points. Just as it would not make sense to quote an average implied volatility for an equity option call spread, JP Morgan argued that it is inappropriate to calculate a single implied correlation for a tranche.

Base correlation is the value of implied correlation for an equity tranche that combines all the tranches up to a certain detachment point. For example, tranches in the CDX index would be expressed as 0-3%, 0-7%, 0-10%, and so on. While the 0-3% is the only standard tranche, the value of other tranches can be calculated using a bootstrapping process. So, the expected loss for the 0-7% tranche is equal to the sum of the expected losses for the 0-3% tranche and the 3-7% tranche. Once the expected loss has been calculated for each first-loss tranche, the base correlation can be calculated by backing out the relevant inputs through a homogeneous large pool model - essentially a simplified version of the Gaussian copula.

Unlike compound correlation, base correlation only has one solution for a particular spread level. That's because each tranche is effectively a first loss that combines all the tranches up to the detachment point, and the equity tranche spread is a monotonic function of correlation - the spread on the equity tranche falls as correlation rises. Base correlation also exhibits a well-defined skew - where implied correlations differ according to tranche. One of the benefits of this approach over compound correlation is that it is easy to interpolate the base correlation curve to value non-standard tranches.

These base correlations are used in the valuations of bespoke tranches. In simple terms, the dealer typically attempts to find an equivalent index tranche and uses that base correlation in the pricing of the bespoke CDO tranche.

Top academics and quants, however, are far from happy with the results of the base correlation approach. For a start, the large pool model uses a number of simplified assumptions for calculating base correlation - for instance, the model assumes an equally weighted portfolio of credits with the same default probability and a constant recovery rate, normally 40%. The model does not consider individual spreads for all the credits in the portfolio, and so does not properly account for blow-ups in a few names - something that occurred last May with the downgrade of Ford and General Motors. Bespoke portfolios are also likely to contain different credits from the indexes on which the base correlation measure is based - for instance, bespoke portfolios could include a mix of European and US names - which could create imprecisions in valuations.

In addition, because the base correlation is calculated as the implied correlation of an equity tranche that includes all tranches up to the detachment point, it's difficult to extract relative-value information among individual tranches. It can also sometimes cause the base correlation curve to move in counter-intuitive ways as spreads of individual tranches fluctuate.

Speaking at Risk's Credit Risk Summit Europe conference in London on October 4, Barclays Capital's London-based global head of credit derivatives research, Jon Gregory, described the base correlation approach as "quick, easy and cheap". While the implied correlation of a bespoke tranche can be represented in terms of index tranches, meaning the Greeks are easy to calculate, the approach is not arbitrage-free, and there is no ability to model structures depending on the dynamics of correlation, he said.

And while it is still used by many prop desks and hedge funds as a default model to ensure their more sophisticated in-house models are not too far out of kilter with the market, Gregory dismissed this basic form of the base correlation model, saying quants need to "tweak it around".

Indeed, some dealers initially resisted the base correlation approach. They claimed that, as a pricing concept, it is fundamentally flawed because in arbitrage pricing theory there is a link between price and the unique dynamics imposed in the model. Base correlation, and more generally 'mixture models', cannot be linked to a unique choice of dynamics and therefore the concept of price is weakened. "In derivatives pricing, a model with reasonable dynamics in an economic sense leads to a replicating (hedging) portfolio, which supports the concept of the price," Gregory tells Risk. "In a base correlation approach, we mix prices and therefore destroy the above ideas, which means the price itself is harder to justify."

The problem is that finding the correct correlation assumptions for a bespoke portfolio is tricky. And there appears to be little consensus on the correct approach to modelling customised CDO portfolios, despite the development of the index market. Indeed, one global head of exotic derivatives at a leading US financial institution says that while everyone hoped the consensus would fall on their own internal models, it could take up to 10 years before such a model emerges.

"The raw computing power needed to break down this problem in any comprehensive way is a big challenge," adds Wilson Ervin, chief risk officer at Credit Suisse in New York. "The advent of some more liquid tranche trading has helped - at least you have some benchmarks - but, for quant people, this is one of the areas where you have got the heaviest burden to bear. There may well not be a consensus for a long time, because for different types of portfolios, you might want to take different trade-offs to get a better micro view of industry risk or name risk, or tranche attachment issues."

According to an as yet unpublished paper, Bespoke CDO Pricing, co-written by Julien Turc, head of quantitative credit strategy at Societe Generale (SG) in Paris and his team members, David Benhamou, Benjamin Herzog and Marc Teyssier, there are five ways that dealers and academics have utilised to try to fix the credit conundrum in the Gaussian copula context: 'moneyness' matching, probability matching, equity spread matching, senior spread matching and expected loss ratio matching.

Moneyness matching occurs where the bespoke and index equivalent tranches have the same 'moneyness', defined as the ratio between the attachment point and the expected loss of the portfolio. So, a 0-6% bespoke tranche is equal to a 0-3% index tranche if the bespoke portfolio expected loss is twice as wide as the index expected loss. Probability matching involves the bespoke tranche and the index tranche equivalent having the same probability of getting wiped out.

The equity spread matching approach assumes the bespoke and index equivalent have the same equity spread. Senior spread matching is similar, but instead of using the equity spread it involves using the senior spread. Meanwhile, the expected loss ratio matching requires the expected losses of the two equivalent equity tranches to represent the same percentage loss of the expected loss of their respective portfolios.

These approaches yield different results when matched against criteria such as ease of implementation, accounting for dispersion, efficiency for tight and wide portfolios, and the continuous case for default (see table A). Overall, SG's research found that the probability spread matching and equity spread matching approaches are the best approaches as they "work for most portfolios and have consistent results when the dispersion of the portfolio increases".

In the event that the bespoke portfolio contains reference entities from different indexes, for example US and European names, the SG analysts put forward two propositions: a weighted average method, where the correlation assumption for the bespoke tranche is the weighted average of the correlations of the iTraxx and CDX equivalent tranches; and the beta method, where a beta is assigned to each name in the portfolio, and is the squared root of the correlation of the equivalent strikes.

Flawed models

Nonetheless, few participants are expecting a resolution to the correlation modelling issue in the short term. "Everyone does it pretty much the same way, although we are currently going through another generation of modelling at the moment. It's a standard copula model, and people have slight tweaks on it depending what underlying process they use," says the New York-based risk manager. "But it's pretty standard, and it has significant issues."

He says the implied volatility analogy - that implied correlation is backed out from the price of a tranche in the same way that implied volatility is backed out from the price of an option - is a bit of a stretch. "With the volatility of a stock and an index, you can go back and see how volatile it has been in the past year or two - although that doesn't necessarily tell you where the implied volatility should be. But you can't go back and look at where the historical time-to-default correlation has been," the risk manager adds.

There are models that attempt to link time-to-default correlation to spread volatility or equity price volatility, which would give dealers more information. The problem is that these models make a lot of assumptions about default barriers, which introduces a new layer of complexity, the risk manager says: "That's not completely divorced from something you can go and look at, but it is a lot harder to get a sense of what a sensible level of implied correlation should be."

The reason this is potentially disturbing is that in the event of a market dislocation, it would be difficult for investors to determine relative value. "I don't think they would do it based on some level of implied correlation. I think they would wait for an extreme move where they would get paid almost as much for a mezzanine tranche as for an equity tranche," says the risk manager. "But that is a huge dislocation."

Liquidity concerns

Industry concerns are exacerbated by the risk premium in many standard trades currently being close to zero. While many market participants expected the low spread environment to reverse in 2003 and 2004 - and so bought protection and have suffered losses as a result - any potential widening of spreads has yet to take place.

Indeed, some participants believe that although there will inevitably be a turn in the credit cycle, there has been a long-term secular and permanent shift in average levels of credit spreads. Effectively, people now know the real cost of counterparty credit risk is considerably lower than they previously believed.

But the continued tight spread environment has led to an increasing number of market participants selling credit protection, or going long credit risk. This has raised concern about the leverage involved in many tranche trades and whether hedging techniques are adequate given the imprecisions in correlation modelling.

"For synthetic CDO tranche trades, investors have historically been paid well for taking on pure default risk and got rid of their mark-to-market volatility by doing a delta hedge on spreads," says the New York-based risk manager. "So if spreads moved, there is no net mark-to-market movement and the investor would get paid for default risk as long as there weren't defaults."

But there have been problems with this approach, particularly last May, when correlation assumptions broke down. "It hasn't worked out that well because you got rid of the spread risk and just replaced it with this correlation risk, which nobody knows how to measure or deal with. So people still suffer a lot of mark-to-market volatility on those trades, and that is why I think a lot of the hedge funds have moved away from it."

Establishing which hedge funds have pulled out of the market is tricky - dealers are reluctant enough to name their active exotic credit clients. But a number of parties stated that Greenwich, Connecticut-based Amaranth Advisors, which is currently closing its multi-strategy fund due to losses in the natural gas market in September, was forced to pull the plug on its credit correlation trading business earlier this year.

However, hedge fund managers insist they get paid a healthy premium for taking bespoke risk at longer durations than they would typically take in the standardised tranche market.

And Avendis Capital's Benichou believes the liquidity in the single-name swaps and index tranche market are buoyant and growing despite the large size of trades dealers sometimes need to execute to hedge their positions. "It can have an impact," he says. "But the market for correlation is really a technical market. It is reacting to issuance. So you have to be aware what is the fashionable trade at the time ... you don't go against the market except if it is for hedging purposes."

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