Publication date: 12-Nov-2001 Reprinted from RatingsDirect

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1 Publication date: 12-Nov-2001 Reprinted from RatingsDirect Commentary CDO Evaluator Applies Correlation and Monte Carlo Simulation to the Art of Determining Portfolio Quality Analyst: Sten Bergman, New York (1) Standard & Poor's new CDO Evaluator refines CDO credit criteria and analysis. This new system uses Monte Carlo statistical methodology to evaluate the credit quality of a portfolio of CDO assets. It takes into consideration the credit rating, size, and maturity of each asset, along with the correlation between each pair of assets. This article will describe in some detail the use of the CDO evaluator to evaluate the credit risk of a CDO asset portfolio. However, it is beyond the scope of this article to present CDO criteria in full. In particular, the article will not discuss any of the cash flow, structural or legal analysis or the Manager Focus product. It should be noted, however that the credit risk of the pool affects the cash flow analysis; thus some observations on the cash flow analysis are included in the discussion. CDO Evaluator The CDO Evaluator system is used to determine the credit risk of a portfolio of assets both for cash flow and for synthetic CDOs. The direct result is a probability distribution of potential default rates for the portfolio assets in aggregate. These potential default rates range from 0% (no asset in the portfolio default by maturity) to 100% (all assets in the portfolio default by maturity). The more likely outcome is that some, but not all, assets default. The portfolio default rate is computed as the total dollar amount of assets defaulted by maturity, divided by the total principal amount of the portfolio. The probability distribution describes the likelihood of the occurrence of any particular default rate of the portfolio. Chart 1 below presents an example histogram of a probability distribution for a highly diverse pool of 50 corporate bonds rated 'BB', each with a 10-year maturity and the same principal balance. It shows that the likelihood that 24% of the assets in the portfolio would default is approximately 7%, which means that the odds that exactly 12 bonds of the 50 bonds default by maturity is seven out of 100. Similarly, it shows that the probability of defaults in the portfolio exceeding 28% is less than 3% (calculated as the sum of the probabilities of default rates greater than 28%, which are represented by the green bars on the chart).

2 After calculating the probability distribution associated with a given portfolio, we can derive a set of Scenario Default Rates (SDRs). This set of SDRs is used in determining, for each credit rating, the default rate that a CDO tranche with that rating should be able to withstand under the various cash flow scenarios encompassed by Standard & Poor's rating criteria. The determination of these SDRs is a twostep process. First, for a given tranche credit rating, determine the portfolio default rate such that the probability of defaults in the portfolio exceeding this portfolio default rate is be no greater than the probability of default of a corporate bond with that rating. Second, multiply this portfolio default rate by an adjustment factor designed for the specific tranche rating. This adjustment factor, which may be either greater than or less than 1.0, depending upon the specific tranche rating, partly reflects the fact that the assumed probabilities of default for each asset are only estimates of the likelihood of default not the eventual default experience of that particular asset class prior to the maturity of the portfolio. For example, based on historical default rates, the probability of default for a 10-year 'A' rated corporate bond is estimated to be 3.0%. We therefore want to determine the portfolio default rate for which there is no greater than a 3% chance that it will be exceeded by the observed default rate by maturity. For the highly diverse pool underlying chart 1, this portfolio default rate is 28%. That is to say, the probability of exceeding a 28% default rate is no greater than 3.0%. But, since we are working with estimated probabilities of default for the assets in the portfolio, we multiply the 28% by an adjustment factor, which is 1.02 at the 'A' rating category. This yields an 'A' SDR of 28.56% for the portfolio. A consequence of this methodology for this rating category is that if a tranche can survive defaults less than or equal to the 'A' SDR, then its probability of default would be no greater than 3.0%, as would be appropriate for an 'A' rating. The CDO Evaluator replaces the Risk Tabulator Model and the CDO Structuring Model, respectively used for CDOs backed by ABS (asset-backed securities) and corporate bonds/loans. Unlike these two models, the CDO Evaluator does not use notching penalties for high industry concentration. Instead, it relies on the effects of correlation upon the SDRs to inhibit industry concentration. In addition, it can work with hybrid portfolios of both ABS and corporates. Moving beyond the traditional task of determining SDRs, the CDO Evaluator computes new CDO benchmarks, which may prove useful in describing the credit quality of a portfolio. These include industry friendly measures of default, variability, and correlation. (These measures are explained in Standard & Poor's Structured Finance Special Report entitled "New Benchmarks Overcome Shortcomings of Traditional CDO Evaluations").

3 Conceptual Framework Although the Evaluator methodology is the same for all types of collateral, the conceptual framework is best understood in the context of a specific example. For ease of reference we chose an ABS CDO transaction. Chart 2 is a schematic of an ABS CDO supported by a number of ABS securities. The ABS securities are securizations of asset-pools, consisting of credit card receivables, auto loans, mortgages, or other pools of financial instruments. For the purpose of the example, we will assume that an ABS security will default because the underlying pool of assets is experiencing too many defaults. In general, the probability of the ABS security defaulting is assumed to be the one implied by its Standard & Poor's credit rating. For example, based on historical studies Standard & Poor's uses a default probability of 8% for a 'BB' ABS security (see chart 2, and the section on Asset Default Probabilities below for a more detailed discussion). Given that ABS securities derive their performance largely from the asset pools that collateralize them, it follows that the default correlation that exists between such securities is primarily a consequence of the performance correlation between the asset-pools that support them. In general, asset-pool correlation reflects the increased likelihood that one pool will perform poorly (or well) given that another has performed poorly (or well). This may be based on the impact of general economic conditions, as well as on issuer or industry specific conditions or events. For example, the performance of auto loans and credit card receivables may be adversely affected by higher unemployment. If this is the case, then the auto ABS security and the credit card ABS security collateralized by these kinds of asset pools will tend to default together and thus are also correlated. The framework described above in the context of ABS securities is equally applicable to corporate securities. In this case, the economic performance of an obligor (typically a corporation) would generally be correlated with that of other obligors belonging to similar industry sectors or to industries that may be affected by general economic events in the same manner. The CDO Evaluator addresses correlation primarily at the underlying obligor/asset-pool level and assumes that it can be expressed in terms of a pairwise sector correlation table. The advantage of studying correlation at the obligor/asset-pool level, rather than the portfolio level, is that it allows issuers and investors to focus on the general correlation assumptions governing the performance of industries, broad asset-pool classes and the economy as a whole, rather than on the considerably less transparent relationship between securities or tranches with different positions within the capital structure of their respective issuing entities.

4 The emphasis placed on modeling correlation in the CDO Evaluator is due to the profound effect that correlation can have on the level of SDR for various credit ratings. Chart 4 vividly shows the effects of correlation on the entire probability distribution of default rates for an ABS CDO consisting of 50 assets, from five different sectors, assuming all securities are rated 'B'. As can be seen in the chart, the mean remains unchanged, but extreme values become more likely. Most affected are the SDR for the higher credit rating categories. For example, with no correlation the 'AA' SDR is 31%. Assuming our current ABS sector correlations, the 'AA' SDR increases to 49% (see chart 3). Monte Carlo Simulation To properly model the effect of correlation on the CDO asset pool, Standard & Poor's has adopted a Monte Carlo approach to estimating the probability distribution of default rates. Within this approach, a number of independent trials are simulated. Each trial generates a vector of random numbers equal in length to the number of assets and having the desired correlation structure. For each trial, each asset represented in this vector is then determined to have either defaulted or not, based on the value of its associated random number, in a manner calibrated to be consistent with the probability of default associated with that particular asset's credit rating. The total principal balance of defaulted assets is then tallied up and expressed as a percentage of the total portfolio principal balance. This result represents the default rate for the trial. Collecting all such observed default rates generates a probability distribution for default rates. (See the appendix for a more detailed description). The transparent and proven Monte Carlo methodology is no longer difficult to use, given today's fast PCs with superior computing power. The methodology is robust due to its ability to deal with complex relationships between variables. It can fully handle the effects created by portfolios containing assets that are unequal in principal balance, credit rating, and maturity. The Monte Carlo approach enables one to simulate the behavior of a system as it is modeled and then simply to observe the results, thereby avoiding the need to determine these results analytically. For example, capturing the effect of correlation, which is difficult, if not impossible to do analytically, is relatively easy by using the simulation methodology. The methodology makes it possible to include the effects of other important variables, such as concentration effects due to servicers, portfolio managers, year of origination, and shared names. These latter variables are not modeled in the current version of the CDO Evaluator, but may be included in the future, as the methodology is refined.

5 Counter-intuitively, the Monte Carlo methodology can achieve virtually the same degree of precision as a purely analytical methodology, if one were available. This can be illustrated by the problem of computing the probability of winning by betting on red in roulette. When the wheel has 38 slots of which 18 are red, one can easily determine the probability of winning analytically to be 18/38 or 47.37%. Chart 5 depicts the Monte Carlo estimate after a number of different trials. Initially, there is considerable flux, but by 10,000 trials the estimate has settled down to 47.37%. The theme in this chart is one of short-term fluctuations and long-term certainty (see chart 4). Clearly, the key to successfully using Monte Carlo simulation techniques is one of performing enough trials to capture long-term certainty. Today's PCs are fast enough to perform enough trials in a reasonable period of time. For example, it typically takes 30 seconds for 15,000 trials on a portfolio of 100 assets. It takes 2.5 minutes for 100,000 trials on the same portfolio. We recommend the smaller number of trials for initial structuring and request the larger for the final structuring run. Portfolio Inputs The CDO Evaluator allows the user to input the wide variety of corporate and ABS assets that are currently used in CDO portfolios. The basic information required of each asset is (1) the issuer ID, (2) the par amount, (3) the maturity date, (4) the industry group, and (5) the Standard & Poor's corporate issuer rating or ABS tranche rating. Presently, there are 40 industry categories, including CDOs, and approximately 20 ABS categories, with the latter consisting of four different basic ABS types in five different geographic areas. System Parameters While the portfolio inputs are the only variables directly accessible by the user, there are three other sets of system parameters that affect the results given by the CDO Evaluator. These parameter sets are the sector correlation coefficients (which measure the pairwise correlated performance of obligors and underlying pools of receivables and similar obligations within and between sectors), the table of default probabilities for assets, and the table of default probabilities for CDO tranches. Correlation Coefficients The CDO Evaluator uses a correlation coefficient of 0.3 within an ABS sector and 0.1 between ABS

6 sectors. For corporate sectors, it uses 0.3 within a given industry and 0.0 between industry sectors. Standard & Poor's believes that correlation will receive considerable attention from market participants in the coming years. As data becomes available, the correlation coefficients will be modified based on documented studies. It should be noted that the Standard & Poor's methodology of estimating correlation coefficients by sectors, rather than assets, leads to asset default correlations that decrease as the asset credit ratings become stronger. This is consistent with the historically observed correlation behavior of corporate obligors. For example, within an industry sector the default correlation between an 'AA' corporate and a 'BBB' corporate is computed to be 4.45%, while between a 'BB' corporate and a 'B' corporate it is 12.72%. Asset Default Probabilities Default probabilities for individual assets are assumed to be implied by that particular asset's type (corporate obligor, ABS, municipal security), credit rating and maturity. For example, historical ABS defaults rates are lower than corporate rates and are not as sensitive to final maturity. This may be due to the fact that many ABS securities experience a seasoning effect, as is the case with residential mortgages. All ABS securities are assumed to have a seven-year weighted-average life, with default rates that reflect the results of our ABS default studies. The default rates for corporate assets continue to be differentiated by rating and maturity as used in the previous Standard & Poor's CDO models. They reflect the results of the Standard & Poor's default study of corporate obligors. A portion of the asset default table is displayed in table 1. Table 1 Implied Asset Default Rates (%) Security Maturity AAA AA A BBB BB B ABS All Corporate Year Corporate Year Corporate Year Tranche Default Probabilities All default probabilities used for sizing of the CDO tranches to be issued by a transaction are designed to be consistent with corporate default probabilities, as given in table 1. CDOs are more like finance companies than asset pools and have the inherent risks of highly levered, actively managed products. The fact that the CDO may only manage ABS assets, in and of itself does not liken these vehicles to a structured ABS portfolio. For a given tranche rating, one should use the corporate portion of table 1, selecting the probability of default assigned to the corporate bond with the desired rating and with a maturity equal to the weighted average portfolio maturity. If the weighted average portfolio maturity is not a whole number, then interpolation is used. The subsequent steps of determining the appropriate SDR for the tranche are discussed in some detail in the previous section entitled "CDO Evaluator." Comparison of Results Because the CDO Evaluator uses specific ABS default rates for ABS portfolio assets, but continues to size tranches based on corporate default rates, the SDRs produced by the CDO Evaluator for ABS portfolios may be significantly lower than those obtained under the Risk Tabulator model. The difference is most pronounced for lower-rated tranches. However, exceptions may occur for portfolios that have heavy concentrations in a few sectors. A comparison of the SDRs generated by the CDO Evaluator and the Risk Tabulator for one typical ABS transaction is given in chart 5.

7 SDRs for corporate assets are comparable to those obtained under the previous CDO Structuring Model. Because the effects of correlation are more pronounced for higher-rated tranches, these tranches will often have SDRs that may be slightly greater than before. In contrast, lower-rated tranches often have lower SDRs. A comparison for one typical transaction is given in chart 6.

8 Cash Flow Verification As seen in the foregoing discussion, the CDO Evaluator creates for each portfolio a probability distribution of defaults and a set of SDRs. A different step in the rating process for cash flow CDOs is the cash flow analysis. The purpose of this step, which is not part of the CDO Evaluator, is to verify that each CDO tranche can continue to pay principal and interest in accordance with its terms notwithstanding defaults up to the SDR on the underlying portfolio. This is accomplished through the detailed modeling of the proposed transaction's cash flow (the waterfall), taking into consideration all structural elements of the transaction. These structural elements may include any reserve accounts, various accelerated amortization triggers, as well as overcollateralization and interest coverage tests. The cash flow analysis also incorporates the effect of various hedging instruments and contracts, such as interest rate swaps and caps. An important element of the cash flow analysis is the appropriate treatment of any recoveries on the defaulted portfolio. This means modeling both the timing of recoveries and the recovery rates. It is beyond the scope of this article to discuss cash flow modeling, other that to mention that the SDR associated with the particular rating for a given CDO tranche is an element of the cash flow analysis. Defaults on the assets, together with recoveries on such defaults, affect the cash flow available to pay off the CDO tranches. Appendix This appendix gives a more detailed mathematical exposition of how correlation is modeled and how the Monte Carlo simulation is performed. Modeling Correlation Each asset is assumed to reflect the performance of either an underlying pool of collateral (e.g. auto loans) or the obligor. Assume that there are N assets and let X(i) denote the performance the pool/obligor supporting the i-th asset, with poor performance corresponding to large values of X(i). Hence, the event that the i-th asset defaults is equivalent to the event that X(i) exceeds some quantity z (i). The quantity z(i) is chosen so that the probability of X(i) exceeding z(i) is equal to the default probability determined for the asset, given its rating and tenor, from the asset default table (see table 1 and chart 7). It is convenient to assume that the probability distribution of X(i) is the normal distribution. Without loss of generality it may be assumed that the mean is 0 and the standard deviation is 1. Otherwise, the variable X(i) may be transformed to have such a mean and variance, and the same transformation may be applied to z(i), which leaves the probability of the transformed random variable exceeding the transformed z(i) unchanged.

9 The above assumption implies that the joint distribution for the random vector X = X(1),X(2),..., X(N), which is the collective performance of the pools/obligors, is multivariate normal with a mean vector of 0's and a covariance matrix equal to its correlation matrix. The correlation matrix may be chosen to reflect the correlation structure that is assumed to exist among the industry and ABS sectors. That is to say, a value of 0.3 is chosen for the matrix if two pools or obligors come from the same sector, a value of 0.1 for two ABS pools not from the same sector, and 0.0 for all other off-diagonal cells. Chart 8 illustrates the joint bi-variate distribution of two underlying asset pools, together with their marginal distributions. Also marked are the regions of the bi-variate distribution where either or both of the two securities collateralized by their respective pools will default (see chart 8). Monte Carlo Simulation The simulation process requires that a large number T of trials be drawn. Each such trial t is an independent realization of the random vector X. For that realization, each component X(i) of X is compared to z(i) and if it is greater, then asset i is deemed to have defaulted. The principal balances of all defaulted assets are added together and the resulting sum, dividing by the total initial portfolio balance, is the observed default rate for that trial. All trials are tabulated and used to create an estimated probability density function for default rates. The process of generating random drawings from a multivariate normal distribution with a known correlation matrix is relatively easy. For example, one may begin by generating a sequence of N independent random variables drawn from a uniform distribution. Then one may convert these into a sequence of independent random variables drawn from a normal distribution with mean 0 and variance 1 by applying the inverse normal function. These N variables may then be transformed into a multivariate normal distribution by pre-multiplying by an N by N matrix M. To obtain the desired correlation structure, the matrix M is chosen to be the Cholesky decomposition of the targeted correlation matrix.

10 This report was reproduced from Standard & Poor's RatingsDirect, the premier source of real-time, Web-based credit ratings and research from an organization that has been a leader in objective credit analysis for more than 140 years. To preview this dynamic on-line product, visit our RatingsDirect Web site at Standard & Poor's. Setting The Standard. Published by Standard & Poor's, a Division of The McGraw-Hill Companies, Inc. Executive offices: 1221 Avenue of the Americas, New York, NY Editorial offices: 55 Water Street, New York, NY Subscriber services: (1) Copyright 2001 by The McGraw-Hill Companies, Inc. Reproduction in whole or in part prohibited except by permission. All rights reserved. Information has been obtained by Standard & Poor's from sources believed to be reliable. However, because of the possibility of human or mechanical error by our sources, Standard & Poor's or others, Standard & Poor's does not guarantee the accuracy, adequacy, or completeness of any information and is not responsible for any errors or omissions or the result obtained from the use of such information. Ratings are statements of opinion, not statements of fact or recommendations to buy, hold, or sell any securities.