# CDO evaluator and portfolio benchmarks.

Standard & Poor's new CDO Evaluator refines CDO default analysis. This new model uses Monte Carlo statistical methodology to evaluate the credit quality of a portfolio of CDO assets and to provide scenario-default rates for the portfolio at each rating level. The CDO Evaluator system is used to determine the credit risk of a portfolio of assets both for cash flow and for synthetic CDOs.

I. STANDARD & POOR'S CDO EVALUATOR

Standard & Poor's new CDO Evaluator refines CDO default analysis. This new model uses Monte Carlo statistical methodology to evaluate the credit quality of a portfolio of CDO assets and to provide scenario-default rates for the portfolio at each rating level.

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 assets 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 bars on the Exhibit 1).

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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 two-step 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").

A. Conceptual Framework

Although the CDO 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. Exhibit 2 is a schematic of an ABS CDO supported by a number of ABS securities. The ABS securities are securitizations 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 Exhibit 2).

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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 pair-wise 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.

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. Exhibit 3 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 exhibit, 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%.

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II. 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.

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.

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%. Exhibit 4 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 exhibit is one of short-term fluctuations and long-term certainty.

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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.

III. 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 the issuer ID, the par amount, the maturity date, the industry group, and 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.

A. 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.

B. Correlation Coefficients

The CDO Evaluator uses a correlation coefficient of 0.3 within an ABS sector and 0.1 between ABS sectors. For corporate sectors, it uses 0.3 within a given industry and 0.0 between industry sectors. Exhibit 5 lists the corporate industry and ABS 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%.

C. ASSET DEFAULT PROBABILITIES

Default probabilities for individual assets art 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 Exhibit 6.

D. Tranche Default Probabilities

All default probabilities used for sizing of the CDO tranches to be issued by a transaction arc designed to be consistent with corporate default probabilities. 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 select 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.

E. 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 Exhibit 7.

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SDRs for corporate assets are comparable to those obtained under the previous models. 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 Exhibit 8.

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F. 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 than 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 oft the CDO tranches.

IV. MODELING CORRELATION

The following discussion gives a more detailed mathematical exposition of how correlation is modeled and how the Monte Carlo simulation is performed. 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 (sec Exhibit 6 and Exhibit 9).

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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.

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 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. Exhibit 10 illustrates the joint bivariate distribution of two underlying asset pools, together with their marginal distributions. Also marked are the regions of the bivariate distribution where either or both of the two securities collateralized by their respective pools will default.

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A. 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.

V. CDO MONITOR

Standard and Poor's CDO Monitor is a software program designed to monitor the total dollar loss that the CDO transaction has already or may incur in the future, relative to the maximum amount of losses the transaction can support at each original rating level. The program is based on the CDO Evaluator, but goes further and is specific to each deal and each rated tranche. Once the transaction has become effective and the effective portfolio is verified to make sure that it meets the modeled parameters done prior to closing, Standard & Poor's will provide each transaction structured with its own customized CDO Monitor.

Hard coded in this program is the size of the asset pool and the original default rate that each tranche can sustain without violating its assigned rating. Going forward in the transaction, each time the program is run, the program looks at the current portfolio balance and the current portfolio default rate at each rating level. If the current potential default amount on the current portfolio exceed the total dollar amount of losses calculated through cash flows at closing for that particular rating, minus the losses incurred to date on the portfolio, then the program indicates a failure of the test. For example, if the cash flows at closing showed that for that rating level the transaction could sustain up to X amount of dollar losses, the Monitor looks at the current portfolio default rated times the current portfolio balance and compares this with X minus adjusted par losses incurred to date. If the potential losses on the current portfolio were greater than X minus par adjusted par losses to date, then the Monitor would indicate a failure. Obviously, if there were gains in portfolio par since closing, the potential loss on the current portfolio could increase without failing the test.

This comparison is known as the CDO Monitor Test for each rated tranche. If the manager chooses to use this test in the transaction, the test is required to be run on every measurement and determination date. Furthermore, the collateral manager for sensitivity analysis may use the Monitor on a more frequent basis.

The primary input for the CDO Monitor is the current collateral pool of the particular transaction. The Input page is identical to that of the CDO Evaluator. For a description of each required field, please refer to the CDO Evaluator section.

The CDO Monitor also requires an "As of Date," from which the tenor of the Assets is computed. The "As of Date" will be the determination or measurement date for all indenture required CDO Monitor Tests and any specified date for a collateral manager running hypothetical portfolios. The "As of Date" allows the program to calculate the maturity exposure period for each assets based on the asset's specific maturity date. Thus the portfolio manager can run different future dates and see what will happen with the Monitor Test until the end of the reinvestment period, assuming no further changes in par due to gains or losses.

The output of the CDO Monitor Test consists of three components for each tranche: the Break-Even Default Rate, the Scenario Default Rate (SDR), and whether the test is a pass or a fail.

The break-even default rate is the maximum percentage of defaults a collateral pool can sustain and still pay ultimate principal and all due interest (timely or capitalized) to the tranche. Each tranche will have a different break-even default rate. This rate is initially set during the rating process by Standard & Poor's, based on the variety of cash flow runs performed.

Each time the CDO Monitor test is performed the break-even default rate is adjusted to reflect any par gains or losses in order to reflect that there are sufficient funds to meet timely or capitalized payment of principal or interest. Thus, as losses occur the break-even default rate for that rating changes.

The scenario default rate is the level of defaults the collateral pool is expected to experience during a given period of economic stress. For example, in an 'AAA' scenario, this collateral pool is expected to have X% default. This X% is known as the 'AAA' SDR.

The CDO Monitor Test Result for a particular tranche is a pass if the break-even default rate is higher than the SDR, otherwise, the test is a fail. Using the CDO monitor allows the collateral manager a great deal of flexibility in selecting securities because the test always factors in the underling default characteristics and lumpiness of the assets. For example, a large number of CDOs have a 5% 'CCC' collateral bucket limit, which is not imposed by Standard & Poor's. If the manager exceeds this, they may be required to take certain actions. But this limitation does not factor in the maturity and the associated risk for such collateral.

A 5% 'CCC' concentration with assets maturing one year from now will likely add less default risk to the transaction than a 4% 'CCC' concentration maturing 10 years from now. The CDO evaluator captures and reflects such risks. By using the CDO monitor, the collateral manager can fine-tune the portfolio. If he has gone long on 'CCC' exposure, then he will need to improve the ratings or shorten the exposure on other assets to maintain the same default probability. For these reasons, most collateral managers find the CDO Monitor to be an effective and flexible tool for monitoring portfolio risk and doing what-if reinvestment analyses.

VI. PORTFOLIO BENCHMARKS

A. Introduction

To assist issuers, managers, and investors in evaluating CDOs, Standard & Poor's introduced a set of new CDO benchmarks. Among these are statistics that describe three key portfolio characteristics: the Expected Portfolio Default Rate (EPDR), the Standard Deviation of the Portfolio Default Rate (SD), and the Weighted Average Correlation (WACorr) of the assets in the portfolio. For the purpose of assessing the credit quality of a tranche issued by a CDO, Standard & Poor's has developed a new type of statistic called the Rated overcollateralization, or ROC. It calculates the effective overcollateralization of a tranche, given its credit rating. It overcomes many of the shortcomings of the traditional overcollateralization (OC) method of measuring overcollateralization. Going forward, Standard & Poor's will seek to make available these benchmarks upon pricing and monthly thereafter on all CDO transactions rated by Standard & Poor's.

B. Expected Portfolio Default Rate

The statistic Expected Portfolio Default Rate, or EPDR, is the weighted average portfolio default rate. As such, it has a clear, easily understood meaning: it is the expected default rate of the portfolio, based on the asset default rates specified in the Standard & Poor's CDO default table, which in turn is based on historical default rates as complied by Standard & Poor's. Unlike other measures of average default currently in use, it encompasses all assets in the portfolio, including defaulted securities and cash. Moreover, it reflects the actual remaining life of the assets, rather than some fixed hypothetical remaining life. The use of the latter would increasingly distort the default probabilities of the portfolio as the actual average remaining life shortens over time. In introducing the EPDR, Standard & Poor's has sought to provide the investment community with a simple, intuitive, and unbiased estimation of portfolio credit quality.

Standard & Poor's will also provide an annualized version of EPDR that will measure the expected annual default rate. Since the expected annual default rate is a parameter frequently used in projecting the performance of a CDO, we believe that it will be constructive for the CDO markets to have an independent, Standard & Poor's derived estimate of this parameter for each CDO.

The EPDR may formally be translated back to give a Weighted-Average Rating (WAR) for the portfolio. To do this, it is necessary to compute the Weighted-Average Maturity (WAM) of the portfolio, weighting by par amounts, and to compute for that maturity the interpolated default probabilities for each rating category. The portfolio WAR is then defined to be the rating that has the least probability of default that is equal to or greater than the EPDR.

It is important to note that a WAR for a CDO portfolio is significantly different from the rating of a corporate bond. While both may have the same EPDR, their respective risk profiles are apt to be substantially different. Similarly, two portfolios with the same EPDR and therefore WAR may also be quite different. Standard & Poor's quantifies these differences thorough the statistic discussed next.

C. STANDARD DEVIATION OF PORTFOLIO DEFAULT RATE

The statistic Standard Deviation of Portfolio Default Rate, or SD, measures the variability of the default rate about its expected value. Many professionals in finance are familiar with standard deviation and its use in measuring the likelihood of various default rates. It is also a key input parameter in value at risk systems. As applied by Standard & Poor's, this statistic captures the effects created by asset variability in size, credit rating, and maturity. Moreover, it explicitly takes into account the correlation between assets in the portfolio. In doing so, it reflects the effective diversity of the portfolio, but does so in a manner that creates a clear direct connection between the value of this statistic and the degree of variability in the portfolio's default rate.

The formula utilized by Standard & Poor's is a general statistical formula that is independent of any distributional assumptions. As seen in the "Mathematical Descriptions" section following, SD is a function of each asset's size, probability of default, and pair-wise correlation. The probability of default is a function of rating and maturity.

The estimation of the pair-wise correlation coefficients requires some care, as they depend not only upon the asset types, but also upon the ratings of the instruments. Assuming that macro economic activity may be modeled as a multivariate normal process, Standard & Poor's has derived analytical formulas that permit the correlation between two securities to be derived flora their respective credit ratings and an estimate of the correlation of the economic activity of their respective industry sectors. This methodology allows the user to focus on the general correlation assumptions governing industries 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.

Standard & Poor's is conducting research on historical correlation rates within and between industry and ABS sectors. Based on preliminary results, Standard & Poor's is using a correlation coefficient of 0.3 within both the ABS and corporate sectors, 0.1 between ABS sectors, and 0.0 otherwise. The 0.3 within coefficient has been incorporated into models used by some major investors in these sectors. The 0.3 corporate coefficient is also consistent with the implicit correlation produced by the notching methodology of the previously used CDO Default Model.

It should be noted that the Standard & Poor's methodology for estimating correlation coefficients leads to default correlation coefficients that decrease as the probability of defaults decreases. This is consistent with the historically observed correlation behavior of corporate bonds. For example, within an industry sector the default correlation between an 'AA' corporate and a 'BBB' corporate is computed to be 0.0448, while between a 'BB' corporate and a 'B' corporate it is 0.1272. This is in marked contrast to some other diversity measures for corporates that implicitly hardwire correlation to be the same across all credit ratings.

D. WEIGHTED AVERAGE CORRELATION

As expected, correlation has a substantial effect upon the Standard Deviation of Portfolio Default Rates and places limits on the ability of the portfolio manager to diversifty away variability in the default rates. Consequently, Standard & Poor's has developed a statistic to quantify the amount of correlation in a portfolio. It is defined as the one correlation coefficient applied to all pairs of assets that gives the same value for the standard deviation as obtained when using the actual correlation coefficients between assets. For a typical high-yield CDO, this Weighted-Average Correlation may be in the range of 0.007. To measure the impact of correlation upon the standard deviation, Standard & Poor's utilizes the ratio of the standard deviation (computed with correlation) to the standard deviation computed without correlation. For a typical high-yield CDO it may be in the range of 1.3. That is to say, the standard deviation is 30% larger due to correlation. For a highly rated CDO tranche, the required Stressed Default Rate that a tranche must be able to sustain without defaulting is often several standard deviations greater than the EPDR. Hence, a 30% larger standard deviation may increase the Stressed Default Rate by a multiple of this increase.

E. RATED OC OR ROC

While monitoring portfolio statistics gives direct insight into the initial or current credit risk profile of a portfolio and how it has changed over time, it does not provide direct insight into the level of support that a tranche truly enjoys. For example, a tranche with an EPDR of 10% and a SD of 5% may or may not be better collateralized than a tranche with all EPDR of 20% and an SD of 8%.

To address this issue Standard & Poor's developed the Rated OC or ROC statistic, which estimates the effective over-collateralization of a tranche. Unlike the traditional OC, the Rated OC explicitly takes into consideration three of the most important components of support for a tranche: the credit quality of the portfolio, the recovery rate, and the excess coupon available to support additional principal. A value equal to 1.0 or greater is an indication that there is sufficient support in the CDO to maintain the tranche's rating.

The Rated OC or ROC for a tranche is determined by dividing the amount of tranche debt that can be supported by the collateral at the tranche credit rating by the actual amount of debt that must be supported to avoid a default for the tranche. The result may be interpreted in several different ways. For example, an 'AA' Rated OC may be viewed as the ratio of the 'AA' equivalent par value of the portfolio to the principal amount of the tranches that are rated 'AA' or better. It may also be interpreted as specifying the proportion of the portfolio that may be discardcd without affecting the 'AA' rating of the tranche. This proportion, called the Excess Percentage, is approximately the amount that the statistic exceeds 1.0. More details on the computation can be found in the section "Mathematical Descriptions".

The Rated OC effectively addresses many of the shortcomings of using a portfolio's principal balance as a measure to assess the degree of overcollateralization of a tranche. It is much less vulnerable to several portfolio management strategies that game the OC: selling strong credits and purchasing weaker credits at a discount to improve OC, trading down in coupon to improve OC, and trading into assets with lower recoveries to improve OC. It is less vulnerable because it explicitly takes into account credit quality, excess interest coupon, and recoveries. Consequently, the Rated OC contributes to solving the problem of how to capture the true overcollateralization of a tranche without making a cash flow CDO into a market value CDO.

While the Rated OC has significant strength and is conceptually simple, it is not intended to be a substitute for the more in-depth rating process required when rating or surveilling bonds. Such activities must explicitly consider the cash flow waterfall and the many different scenarios that affect the ability of the collateral to support a tranche, including various default timing patterns and interest rate paths. Consequently, a Rated OC of a value greater than 1 is no guarantee that the tranche will not be downgraded. Nevertheless, this statistic provides valuable insight into the effective degree of support enjoyed by a tranche.

VII. MATHEMATICAL DESCRIPTIONS

The previous discussion can be given greater clarity by using mathematical notation. For computing the Expected Portfolio Default Rate, let P(i) be Standard & Poor's default rate for the i-th asset, let B(i) be the principal balance of the i-th asset, let N be the total number of assets, and let TB be the total principal balance of the portfolio. Then

EPDR = {P(1)*B(1) + P(2)*B(2) + ... + P(N)*B(N)} / TB

For determining the Standard Deviation of the Portfolio Default Rate, also let R(i) be the ratio of the principal balance of asset i to the total portfolio principal TB, let C(i,j) be Standard & Poor's correlation coefficient between the i-th and j-th assets, and let S(i) be the square root of P(i)*(1-P(i)). Then the standard deviation SD for the portfolio default rate is

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The portfolio's Weighted Average Correlation, WA Corr, may be obtained by the formula

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The effect of correlation upon the portfolio standard deviation, SD, may be measured by the Correlation Ratio, CR, which gives the proportional increase in standard deviation due to correlation. It may be computed by the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Rated OC or ROC for a tranche is determined by dividing the principal amount of debt that can be supported by the collateral at the tranche rating to the actual amount of debt that must be supported to avoid a default for the tranche. The amount that can be supported is estimated as the sum of the amount of collateral that will not default at the rating, the amount of principal recovered after default, and the additional principal that can be supported by the available excess coupon interest. Each component involves its own calculation. The amount not defaulted is the principal amount remaining after subjecting the collateral to the Stressed Default Rate (SDR) for the tranche rating, as computed by Standard & Poor's CDO Evaluator. The SDR represents the maximum default rate that the tranche must be able to sustain without defaulting, given its rating and the credit quality of the portfolio. The amount recovered is the present value at default (or at the As-of-Date, if already defaulted) of future recoveries at the applicable Standard & Poor's recovery rates. The value of the excess coupon spread depends upon the coupons of the tranche and all senior and pari passu tranches, and how quickly the Stressed Default Rate is realized for the portfolio. For simplicity, it is assumed that the SDR level of defaults will be realized at a constant rate over the first five years of the remaining maturity of the collateral.

The proportion of the collateral that may be discarded without adversely affecting the rating is estimated to be (1 -(1/ROC)). For example, a ROC of 1.05 suggests that 4.8% of the portfolio is not needed for the rating.

Sten Bergman

Standard & Poor's

STEN BERGMAN is a director in Structured Finance Ratings. He has previously held senior positions at Salomon Brothers, Wasserstein Perella, and Koch Asset Management. Prior to coming to Wall Street, Sten taught at Cornell University and the University of Oxford. Sten holds a B.A. in mathematics and economics flora the University of California at Berkeley and a Ph.D. in statistics from Yale University.
```Exhibit 5. Standard & Poor's Industry and ABS Sectors

Table 1: Standard & Poor's Industry and ABS Sectors

Corporate Industry Sectors ABS Sectors

Aerospace and defense CDO
Air transport ABS consumer
Automotive ABS commercial
Beverage and tobacco CMBS Diversified (conduit and CTL)
Radio and television CMBS (large loan, single borrower,
Brokers, dealers, and investment and single property)
houses REITs and REOCs
Building and development RMBS A
Business equipment and services RMBS BandC, HELs, HELOCs, and
Cable and satellite television tax lien
Chemicals and plastics Manufactured housing
Clothing/textiles U.S. agency
Conglomerates (explicitly guaranteed)
Containers and glass products Monoline/FER guaranteed
Cosmetics/toiletries Non-FER Company Guaranteed
Drugs FFELP student loans
Ecological services and equipment (Over 70% FFELP)
Electronics/electrical Project finance
Equipment leasing
Farming/agriculture
Financial intermediaries
Food/drug retailers
Food products
Food service
Forest products
Health care
Home furnishings
Lodging and casinos
Industrial equipment
Insurance
Leisure goods/activities/movies
Nonferrous metals/minerals
Oil and gas
Publishing
Rail industries
Retailers (except food and drug)
Steel
Surface transport
Telecommunications
Utilities

Exhibit 6. Implied Asset Default Rates (%)

Security Maturity AAA AA A BBB BB B

ABS All 0.25 0.50 1.00 2.00 8.00 16.00
Corporate Year 4 0.19 0.57 0.81 1.81 9.49 21.45
Corporate Year 7 0.52 1.20 1.81 3.94 14.20 26.15
Corporate Year 10 0.99 1.99 3.04 6.08 17.47 28.45
```