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August 2004 | Feature

Credit guide: pricing

The price of default correlation

The development of basket credit products such as single-tranche CDOs and first-to-default baskets has produced a market in default correlation. However there is little consensus on exactly how default correlation should be measured or priced

The idea of trading how closely the prices of different shares are correlated is well established. But in recent years there has been a push by banks and some hedge funds to create a similar market for credit based on the correlation between defaults.

The development has been made possible by the introduction of various structured credit products. One such product is the single-tranche collateralised debt obligation (CDO), where only one tranche of a CDO is created in the credit default swap market. Other products include first-to-default baskets, where the buyer loses their investment if any one name in a portfolio defaults, and most recently the tranching of tradable credit default swap indices such as iTraxx, the product of a merger between iBoxx and Trac-x.

There are a number of ingredients involved in pricing individual tranches: default probability, default severity (or recovery) and default correlation. The first two are self-explanatory. Correlation, however, deals with the distribution of defaults throughout a portfolio and the likelihood of a single default causing a succession of defaults.

The extent to which defaults between two different companies are correlated can be seen as a measure of the general health of the credit market. When there are market risks such as accounting scandals and concerns over corporate governance, investors and banks are reluctant to lend and so the extent to which corporate defaults are correlated increases. When market sentiment is positive, investors are able to analyse companies on a case-by-case basis and default correlation is lower. So in an untranched portfolio, the correlation between names defaulting has an effect on the value of the basket, but only insofar as the whole market is performing poorly when correlation is high. It is only when portfolios are tranched that the relative value of default correlation becomes meaningful.

Until recently the issue was not a problem as the entire capital structure of a CDO would be placed for the full amount and thus could be hedged by selling the respective cash bonds for each credit. But in late 2001 and early 2002 a number of banks introduced single-tranche collateralised debt obligations such as Deutsche Bank’s Credit Select and JPMorgan’s Merit.

Calculating correlation

The development of a bespoke tranche market in recent years has made the pricing and risk management of correlation imperative. As a result each bank has its own model of how this default correlation should be calculated. Essentially, the value of the lowest tranches of a CDO, say an equity tranche, increases as the correlation between defaults falls, and decreases as default correlation rises. The more the defaults within a basket become correlated, the more the portfolio behaves like a single credit. So the probability of the equity tranche being wiped out becomes more similar to the probability of the most senior tranches being wiped out.

The relationship is the other way round for the most senior tranches: as correlation decreases, value decreases because the probability of a large number of defaults decreases.

No credits defaulting is obviously the best case, but for the holder of the equity tranche many credits defaulting has similar loss implications to a few credits defaulting. As it is a first-loss tranche, the holder of an equity tranche doesn’t care whether, say, seven or 70 credits are affected. However, if a tranche is assessed as having a low correlation between defaults, this means there is little chance of a high number of defaults. The assets referenced by the tranche can be seen as largely independent of one another. So, for subordinate tranches, the risk and spreads decrease as correlation between defaults increases, while for senior tranches the risk and spreads increase as default correlation increases.

When prices of CDO tranches are quoted in the market, they incorporate a correlation calculation. For example, a junior mezzanine tranche (6%–9% of loss) might be quoted at a bid offer of 75/95bp, with a 16.5% correlation. An equity tranche, however, might be seen as 1,400/1,600bp with a 25% correlation on the bid and a 20% correlation on the offer. In practice, equity tranches often trade with an upfront premium, as do distressed credits in the CDS market.

Until last year this correlation pricing of first-to-default baskets and single-tranche CDOs was dependent on each bank or hedge fund’s assessment of correlation. However, in 2003 the banks behind iBoxx and Trac-x started trading tranched versions of the indices. This standardisation in tranches has created a market where bank desks and hedge funds are assessing value and placing prices on the same products rather than on portfolios’ bespoke single-tranche CDOs and first-to-default baskets. Rather than the price of correlation being based on a model, it is now being set by the market.

The nine lives of a senior tranche



JPMorgan explains default correlation trading as analogous to a cat walking blindfolded through a room filled with mousetraps. If the cat has only one life, he would prefer the traps to be located in clusters. The cat will lose his life whether he hits a single trap or a cluster. At least with the traps in clusters, there will be paths between them. The same is true of a lower-rated tranche of a CDO: the holder of such a tranche would prefer high correlation, or clustered traps.

If the cat is a more traditional cartoon cat, one with nine lives, he is happy for the traps to be scattered evenly round the room. He can afford to hit a few traps, but does not want to hit a large cluster which would wipe out all his nine lives. Likewise investors with senior tranches prefer low correlation.

Source: JPMorgan

Accurate pricing

But not everyone is comfortable that just because the market is now pricing default correlation, default correlation is being priced accurately. Most models used for this purpose incorporate a copula, or formula devised to calculate the probability distribution. There are several different types of copula, but most in the market at the moment appear happy to use a normal, or one-factor, copula.

But as the market is very young, it is too early to say whether these models will be proved accurate. Historical data provides part of the input to the copula, but reliance on historical data is a haphazard game. For example, factors such as recovery rates are crucial to the calculation of correlation but are not readily agreed in the market. Each one is dependent on idiosyncratic factors. Neither does looking at the stock market afford many clues to how credits behave on default. Stock correlation is completely unrelated to default correlation, argue bankers.

In an August 2003 research document, Merrill Lynch notes that correlation is the “fundamental driver of pricing” of tranched products but adds: “While in practice there is a lack of historical data to reliably extract default correlations, empirical evidence shows that default correlation is linked to credit rating and varies with time.”

Neither is the often-made comparison with implied volatility entirely convincing. For a start, there is no relatively simple and well-understood single model of default correlation like the Black-Scholes model, which is used in the interest rate and equity option markets.

Bankers admit that in the world of credit derivatives, the market is only just beginning to settle on a standardised model of correlation, while Black-Scholes has been around for two decades. These models of correlation are not only inherently flawed because the historical data is insubstantial and often untrustworthy, but they are also likely to make assumptions about the market that do not capture anything like its full dynamic and complexity.

This, of course, is the nature of models. Any model of a concept as complex as correlation that is structured to take full account of market complexity is likely to be too cumbersome to be used effectively in a front-office environment. Decisions about pricing a deal have to be made in seconds, or at least minutes, but very rarely can they be allowed to take hours. Even using the standard one-factor copula, pricing a bespoke CDO tranche can be a very time-consuming business.

But there are other difficulties associated with correlation pricing as well. The over-the-counter credit default swap market is relatively illiquid compared with other major asset classes, so dynamic delta hedging becomes that much more difficult. In the option market, a dealer can buy a stock option and short the individual stock, leaving him simply exposed to the implied volatility of the option. This is less easy in the CDO market.

Although the market is growing substantially, it is by no means as large or as commoditised as, say, the foreign exchange option market. This relative illiquidity is reflected in wide bid-offer spreads. And as with any market that is prey to comparative illiquidity, it is also prey to periodic bouts of pricing eccentricity.

Supply and demand can of course also alter prices irrespective of other factors, although this is also true of the option markets. In fact, all things considered, some bankers believe that shops have begun trading correlation without a firm grasp of its differences and particular difficulties it imposes. And some fear that the blind use of equity market approaches to the credit market is a particular problem.

Please email Matthew Attwood, Editor, to comment on this article.