The CDO business is reaching out to banks, investors and high net worth individuals all over the World. Investment banks and CDO structurers from London, Paris, New York, Singapore or Sydney would put together a CDO of roughly 100 obligors, achieve some kind of a model-driven diversification, achieve a convenient first loss piece of x% that is sufficiently rewarded by the inherent arbitrage earnings of the CDO, and lo, the CDO is in the market. If it is a single tranche transaction, as most CDOs today are, you don't need to wait until all the tranches are sold off, as just one tranche is enough to successfully sell the CDO in the market. Inherently, all the CDOs are cast in a model - unlike the portfolio of a usual balance sheet transaction, CDO portfolios are completely synthetic. "Synthetic" is close to "unreal", that is, the portfolio is completely virtual. It is constructed not by actually originating credits, but simply by synthetically selling protection on the target names. Therefore, the idea of a synthetic CDO is that of calendar beauty - it is perfect in every respect. It is an idealized portfolio where everything is only as much as you would love to have. This idealized perfection is attained to fit into rating agency models that compute the expected losses of the CDOs, and therefore, in a not very discrete way, it is the rating agency models that have been instrumental in the spurt of CDOs in the market. Briefly, the extent of credit enhancement at any tranche level of a CDO is such as to reduce the probability of the defaults exceeding the level of subordination to an equivalent of the probability of losses at that tranche level. For instance, if Moody's idealized probability of default for a Baa2 piece is 1.58%, there must be such credit enhancement (meaning subordination) at the BBB piece level that the probability of wiping out the same is reduced to 1.58%. Each rating agency has its own model to work out this probability distribution. Arguably, the most transparent of the rating methodologies has been the Moody's binomial expansion technique (BET). The binomial expansion method comes from probability distributions where there is a definite number of outcomes of an event. For example, if we are tossing a fair coin, there is 50% chance of getting a head, and 50% of getting a tail. If we toss it 50 times, what is the probability of getting n heads, say, 7 heads? This is given by the binomial distribution. One may mathematically compute the probability using a formula, or find it on Excel with function binomdist. When we have n number of harmomised obligors in a pool, there is a probability, for every one of these obligors, that the obligor may be in a state of default or state of performance. Therefore, there are two possible outcomes per obligor, and the probability of default of each of the obligors is given by the estimated probability. That probability of default per obligor may itself be drawn from several sources - such as historical probabilities implied by the rating transition histories, or prevailing cash market spreads, or structural study of each obligor based on financial data, such as in Merton model. If n number of obligors default, and there is a loss (1-recovery rate) per obligor of x amount, then the total amount of loss of the CDO is nx. As long as nx is not more than the subordination at the tranche level, there is no default as for the tranche. So, the magic of the model lies in computing the probability of nx exceeding the level of subordination. The critical inputs that go into estimating the probability of the losses exceeding the level of subordination for the tranche are: * Probability of default of each obligor * The notional value each obligor - in synthetic transactions, the notional value per obligor is harmonized * The recovery rate, which is reciprocal of the loss per obligor * The inter-obligor correlation, that is, the degree to which losses of one obligor will be associated with losses in other obligors too. Moody's single binomial method The simplest of approaches is the Moody's single binomial expansion. This is by far the most simple approach. It reconstructs the actual CDO portfolio into an idealized portfolio that completely zeroes out the correlation in the pool. This is done by computing the diversity score of the portfolio. The diversity score having been computed, it is as if there are as many obligors in the pool as indicated by the diversity score. For example, if in a portfolio of 100 obligors of $ 10 million each (notional of $ 1 billion), the diversity score is 45, we assume as if there are 45 obligors in the pool with a notional of $ 1000/45 million. All the obligors have no correlation, and have the same probability of default. Now, we know from the binomial distribution the probability of n number of defaults out of 45, from which we may compute the probability that losses will exceed a particular level. The cumulative probability for say, 5 defaults out of 45 will indicate the probability that the losses will be limited to the loss of 5 obligors, and the tail risk is that the probability that the loss will exceed 5 obligors. Moody's multiple binomial method Later, Moody's came out with its multiple binomial method. Here, it is still the binomial method, but with the pool broken into several subsets with different probabilities of defaults for each sub set. This method marked an improvisation, as, instead of assuming the probabilities of default of each obligor in the harmonized pool to be the same, the multiple binomial method allows for different probabilities per subset. The multiple binomial method reflected the tail risk inherent in the CDO as the higher probabilities of default inherent in lower-rated obligors were not being adequately considered in the single binomial method. The tail risk of the sub-sets was more than that of the whole. Moody's correlated binomial method Recently, the rating agency came out with a correlated binomial methodology. A special report of August 10, 2004 (Moody's Correlated Binomial Default Distribution) explains the method. Unlike the earlier assumption where, the diversity score having been computed, the correlation in the pool was taken to be zero, this model allows for a correlation to be present even after computation of the diversity score. The new method is obviously triggered by one of the most dreaded problems in CDOs - fat tails. "Fat tail" implies a more than nominal probability of losses at the far end of the distribution - that is, high degree of probability of several defaults in the pool. The diversity score itself has been adjusted after taking into account the correlation, that is to say, the correlated diversity is higher than the independent diversity score. S&P's CDO Evaluator approach Standard and Poor's CDO Evaluator is based on a Monte Carlo simulation approach. As for the user, most of the underlying computations take place inside the "gray box". The required inputs the issuer's ID, par amount, industry classification, and S&P rating for the issuer. Internally, S& P assumes correlations - 30% intra industry. Fitch VECTOR model: Fitch, on the other hand, uses a different sector-by-sector correlation on its Vector model. Fitch model is also based internally on Monte Carlo simulation. The default rates come from a new CDO Default Matrix (giving asset default rates by rating and maturity), which is based on historical bond default rates and can be modified to take account of "softer" default definitions when used for rating synthetic CDOs. Pairwise asset correlations, similar to what's done by Moody's, are based on estimates of cross- and intra-industry, and geographical correlations of equity returns. As a result, Fitch will assign an internal and external correlation for each of the 25 industry sectors used. In the past, Fitch did not explicitly model correlations, but applied penalties for high obligor, industry and country concentrations in CDO collateral pools. http://www.vinodkothari.com/cdoratingmodels.htm

The model risk Intuitively, it would not be difficult to understand that correlation is the key million-dollar input in estimating the expected loss probability at any tranche level. As we move up the correlation assumption, the loss distribution curve shifts its peak to the left, and the tail at the right hand becomes fatter and fatter. In case of CDOs, it is the tail that moves the dog - therefore, the real risk is the risk of fat tails. Inherently, there are several risks still not being captured by the rating agency models. First, the probability of defaults of obligors are being mapped along the historical probabilities of given ratings. There is a huge difference between the historical probabilities of default, and those implied by the cash market spreads, and this difference becomes more acute for lower rated obligors. The intuitive argument for this widening difference is that the market tends to exaggerate the risks of default of lower rated obligors. While computing probabilities of default, the rating agencies are still influenced by the historical ratings. Besides this, the credit spreads in the market for obligors of the same rating may be widely different. Motivated by arbitrage considerations, a CDO structurer may choose obligors on the upper fringes of credit spreads though with a given rating. The CDO business is booming - structurers are adding inputs like interest rate swaps, equity default swaps, etc., in a bid to provide higher spreads to investors. Investors have looked at CDOs as not a part of a hard core investment pool but like a bit of venturesome portfolio allocated to provide a yield-kicker. In this environment of spread-peddling, it is likely that some one would like to play smarter than the investment bank next door, and this would lead to a race of outsmarting. The casualty may be that the rating agency models may be overexploited, which might eventually lead to a loss of credibility of the ratings information.