Following on from last week's discussion of the impact of Standard & Poor's new synthetic collateralized debt obligations ratings model, S&P describes the workings of the model itself.

Introduction

Recently the synthetic collateralized debt obligation market has seen the introduction of many innovative structures, including single-tranche CDOs with short credit-default swap positions, forward starting CDOs, nth-to-default baskets, leveraged super senior structures, and constant proportion portfolio insurance--CPPI--structures. Whatever the type and source of innovation, rating analysis requires the determination of the long-term default risk associated with the CDO transaction, which means that the evolution of credit losses within the asset portfolio and the structural complexity of the CDO transaction must both be accurately represented by the model.

The CDO Evaluator Model

The main purpose of CDOe is the computation of the loss distribution of a portfolio of N assets, and the allocation of these losses to the CDO transaction. This is carried out through the simulation of the correlated default times of all assets in the portfolio, which will be described in more detail later. There are three key assumptions required by CDOe, namely a default probability and recovery for each asset, and the degree of dependency between the defaults of each pair of assets in the portfolio. We will first briefly discuss each of these in turn, focusing on the data and methods used to determine them, and then bring them all together to show how the resulting loss distribution is used to assign ratings to CDO transactions.

Transition And Default Probabilities

For rated companies, we make use of our global database of rating transitions and defaults over the period 1981-2003, which contains a ratings history of 9,740 companies from Jan. 1, 1981 to Dec. 31, 2003, including 1,386 default events. The method used by Standard & Poor's to estimate credit curves involves two stages. The first stage is the estimation of the probabilities of transitions between different ratings--the transition matrix. The second stage is the repeated application of this matrix to determine the credit curves. In both cases, rating transitions are assumed to follow a Markov process, in which transition probabilities are constant over time, and do not depend on the previous rating of the firm, e.g., whether the firm was recently upgraded or downgraded.

A straightforward method for estimating a discrete transition matrix from empirical data involves observing the transition of cohorts of firms with the same initial rating. An alternative to the cohort method--which compares the initial and final rating over a certain period--is the duration method, which takes into consideration the exact points in time at which rating transitions take place, using the instantaneous probability of transition, the transition intensity. By comparing the results of the two methods, and making certain qualitative adjustments, we have derived a single one-year transition matrix that, in our view, produces the best overall agreement with the average long-term historical default behaviour of rated firms. This one-year transition matrix is then used to determine the long-term credit curves for each rating category, simply by raising the matrix to higher powers, and extracting the default column of each N-year matrix (N = 1 to 30). These are shown in Chart 1.

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Recoveries

In order to properly model the different types of recovery mechanisms included in CDOs, CDOe treats recoveries in two ways: fixed and variable. Although recoveries usually exhibit high variability, there are two main reasons for using fixed recovery assumptions. The first is that recovery can in certain transactions be set at a fixed percentage of the amount at risk, e.g., 50%. The second is that historical data is not always sufficient to allow precise determination of the degree of variability in recoveries. For this reason, a fixed recovery that incorporates some degree of conservatism is the best compromise. As this clearly involves some level of qualitative judgment, these assumptions are normally determined through a committee process.

In some cases, sufficient historical data exists to allow the degree of variability in recoveries to be explicitly modelled. For example, Standard & Poor's runs a database which contains recovery information for more than 500 non-financial public and private U.S. companies that have defaulted since 1988. It contains information on more than 2,000 defaulted bank loans and high-yield bonds, and other debt instruments. This extensive data has allowed us to create recovery distributions for certain types of assets, which are based on the beta distribution. Specification of the mean and standard deviation of the beta distribution is sufficient for CDOe to simulate the full range of potential recoveries for each type of asset. For example, in the case of U.S. senior unsecured bonds, CDOe assumes a mean of 38% and a standard deviation of 20%.

Correlation

In addition to the individual default and recovery of each asset in the portfolio, the dependency between defaults of different assets must also be modelled. The standard dependency model in the market is the Gaussian copula model, originally proposed by Li (2000). In this approach, a term structure of survival probabilities is assumed for each asset, which we obtain from the cumulative default probabilities for each asset. Dependency is then introduced via the Gaussian copula function *C*(*u*,...,*u _{N}*)=*...(

*y*1,....,

*y*), where

_{N}*y*represents the normally distributed asset values of each asset, and *... is the multivariate standard normal distribution function with correlation matrix....

Note that ... above represents the asset value correlation between each pair of assets, not the default correlation, which is the correlation between (binary) default events. While default correlation and equity correlation are both observable in the market, asset correlation cannot be directly measured. There are ways, however, in which asset correlation can be estimated, such as using equity return correlations as proxies for asset correlation, or estimating asset correlations from empirical default observations using a model that links one to the other, such as the Gaussian copula itself. We have chosen the latter route, as this is likely to be less prone to the inherent market noise within equity return data.

We used several default statistical methods to ensure stability, ranging from maximum likelihood methods and factor models to simple methods based on empirical joint default events. Across all corporate industry sectors, these techniques reveal average asset correlations in the range 14% to 16%. In addition, we find average correlations between industry sectors of approximately 4% to 6%. The correlations currently used within CDOe for pairs of corporate assets in the same country are 15% within a sector and 5% between sectors.

The copula function therefore links the asset values together to create a joint distribution of correlated uniform random variables u. These uniform random variables are then compared to the survival probabilities of each asset to determine the default time * of each asset. If * is less than the maturity of the CDO transaction, the loss *L* is determined as *L = E* x (1-*), where *E* and * are the exposure-at-default and recovery respectively for each asset. The loss distribution of the portfolio is then built up by repeated simulation of the total loss arising from all assets up to a given point in time, which is normally the final maturity of the CDO transaction. This simulation is repeated until satisfactory convergence is achieved, i.e. the statistical error associated with the tail of the loss distribution is sufficiently low. We have found that 500,000 simulations provide satisfactory convergence for most CDO portfolios.

CDO Risk Analysis

*Scenario loss rate*

The primary risk measure used in S&P's analysis of CDO transactions is the scenario loss rate, which is a quantile of the portfolio loss distribution consistent with a given rating and maturity. For example, if the rating quantile corresponding to a certain rating and maturity is 0.5%, the required percentile of the loss distribution will be 99.5%. Note that the rating quantiles have been developed specifically for CDO tranches, and are not identical to the corporate credit curves as in previous versions of CDOe. For a synthetic CDO, the SLR is equivalent to the attachment point or credit enhancement required for a tranche with the relevant rating and maturity.

*Synthetic Rated over-collateralization*

Once a CDO transaction has been structured, it is possible to determine the extent to which available credit enhancement exceeds the required level. This is conveniently provided by the SROC, which is given simply by:

In the case of cash CDO tranches, the value of any excess spread must also be included, which requires the additional modelling of the transaction cash flows.

*This week's Learning Curve was written by* *Arnaud de Servigny**, head of quantitative analytics at Standard & Poor's in London.*