STANDARD & POOR’S MARKET DERIVED SIGNALSHow Standard & Poor’s Arrives At Market Derived SignalsMay 2009Sten Bergman, ., Director, S&P QARG, 212-438-2478, sten_bergman@ Hampel, ., Director, S&P QARG, 212-438-8025, martin_hampel@ Rome, Associate, S&P QARG, 212-438-7519, jayson_rome@ Shi, ., Associate, S&P QARG, 212-438-6050, iris_shi@ Taralli, ., Associate, S&P QARG, 212-438-6421, lauren_taralli@ Yang, ., Associate, S&P QARG, 212-438-6843, sean_yang@
S&P Market Derived SignalsBy Sten Bergman, ., Martin Hampel, ., Jayson Rome, Iris Shi, ., Lauren Taralli, ., and Sean Yang, . IntroductionStandard & Poor’s is introducing S&P Market Derived Signals (MDS). The signals are derived from a statistical model that evaluates credit default swaps (CDS) to create a measure that facilitates the interpretation of market are credit derivative contracts between two parties that are meant to insure a buyer against a default or credit deterioration. The buyer pays a premium to the seller in return for credit protection. The premium, or spread, is often expressed as a fraction of a notional amount in basis and sellers of the CDS contracts include numerous types of firms, including investment funds, hedge funds, insurance companies, and banks, to name a few. CDS contracts are quoted and sold through various broker-dealers, but as of now are not traded on an organized exchange. While rating agencies do not rate CDS, they do rate most of the underlying debt obligations that are the focus of the CDS CDS contracts provide protection against default, we believe these spreads can provide inferences about the market’s view of credit risk. To better understand the signals and to facilitate making such inferences, Standard & Poor’s has developed the Market Derived Signals the Market Derived Signals Model, the key parameters observed for a firm are its current five-year credit default swap spread, Standard & Poor’s long-term issuer credit rating and CreditWatch/Outlook status, ®Global Industry Classification Standard (GICS) sector, CDS document type, and currency denomination. We compile this information on a large number of firms on a daily basis, and then at the end of each day, we estimate a linear model that regresses the observed log of the CDS spread on each of the other variables. For purposes of the model, the credit rating on the underlying obligation is assigned a numerical score that corresponds to the credit ratings scale, . ‘AAA’ = 1, ‘AA+’ = 2, etc. The resulting statistical relationship between CDS spreads and the other variables may be used in a number of different ways, including the following: 1. We can use it to establish generic benchmark spreads for a given rating, GICS, and currency. 2. We can compute from a statistical perspective the expected spread for each firm, given its rating, GICS, and other features. This may be subtracted from the actual spread to obtain the difference between the firm’s actual spread from the expected spread. 3. The statistical relationship can be used to solve for the numerical score that gives exactly the spread observed for the CDS, given the other features of the firm and the CDS contract. This will determine the MDS of the explore the methodology Standard & Poor’s uses to create MDS in detail below. First, we present it heuristically, and then we give a more mathematical treatment in Appendix I. 2
Regression RelationshipOn a daily basis, Standard & Poor’s accesses a database of CDS spreads provided by CMA, a company that provides credit market pricing data, to observe the relationship between CDS spreads, Standard & Poor’s ratings, and other important variates. The number of firms for which quoted CDS spreads are available varies from day to day. For example, on Feb. 2, 2009, spreads were available on the five-year dollar-dominated CDS contract for 590 Standard & Poor’s rated . domiciled firms. Table 1 gives a breakdown of the number of contracts by rating and GICS sector (see Appendix II for the corresponding sector codes). Table 2 gives the average spread for each corresponding rating and GICS cell. As can be seen, not all cells have observations, averages differ substantially by both rating and GICS, and the relationship is neither simple nor exact; rather, it exhibits considerable 1 Number of available USD CDS Contracts on 2/2/2009 GICS GICS GICS GICS GICS GICS GICS GICS GICS GICS 10 15 20 25 30 35 40 45 50 55 SubtotalAAA 1 2 5 8AA+ 2 2AA 1 1 1 2 7 12AA- 2 2 2 1 7A+ 3 3 4 2 2 4 12 2 2 34A 4 4 8 4 9 1 17 4 6 3 60A- 11 3 6 6 3 5 20 3 1 4 62BBB+ 9 6 13 9 8 6 12 1 2 9 75BBB 14 16 9 13 6 4 13 2 18 95BBB- 11 6 3 15 3 2 22 3 4 4 73BB+ 4 6 2 4 1 1 5 2 1 26BB 4 5 2 7 1 4 1 6 30BB- 3 5 4 12 3 3 2 2 34B+ 1 2 8 4 6 3 24B 1 2 9 1 2 1 1 1 18B- 3 7 2 1 1 2 16CCC+ 1 2 4 1 8CCC 3 1 1 1 6Subtotal 66 59 66 101 41 35 118 31 25 48 590HOW STANDARD & POOR’S ARRIVES AT MARKET DERIVED SIGNALS 3
Table 2 Average USD CDS Spreads (in basis points) on 2/2/2009 GICS GICS GICS GICS GICS GICS GICS GICS GICS GICS 10 15 20 25 30 35 40 45 50 55 AverageAAA 77 90 814 541AA+ 114 114AA 99 114 106 83 617 400AA- 113 103 108 325 139A+ 190 133 307 152 74 63 310 94 134 208A 99 180 141 103 111 87 406 131 182 274 218A- 301 240 178 310 141 200 513 177 334 99 321BBB+ 366 455 369 186 121 121 523 410 239 224 312BBB 445 330 253 358 103 198 939 299 321 404BBB- 412 641 365 424 192 378 1031 284 292 383 594BB+ 453 577 881 496 200 986 596 303 850 563BB 594 885 1394 752 389 1762 460 883 935BB- 1615 858 1334 831 370 3314 1388 626 1163B+ 837 2259 2051 924 1953 676 1634B 4244 1896 2828 1072 946 3600 768 1018 2324B- 2188 5217 2965 4782 1774 1656 3680CCC+ 7407 3736 6360 1274 5199CCC 10569 4037 19473 5213 10072Average 430 1173 734 1542 306 285 804 949 471 403 Key Variables To derive the S&P Market Derived Signals Model, Standard & Poor’s first identified key firm-specific ratings and CDS contract-related variables that are statistically significant for predicting CDS spreads. Other than ratings and GICS, these variables include CreditWatch/Outlook status, CDS currency denomination, and document type, which describes the events that constitute a default in the CDS contracts (see Appendix III). Figure 1 shows that the relationship between spreads and ratings on Feb. 2, 2009, is primarily exponential, and it is useful to transform CDS spreads by taking their logarithm. In Figure 2, a piecewise linear function, which consists of a sequence of linear segments with connecting points, or knots, at ‘AAA’, ‘AA’,’ A’, ‘BBB’, ‘BB’, ‘B’, and ‘CCC’, is found to fit the log spread data well. The values at the connecting points determine the benchmark log spread for each rating 4
Spread (bp)Five-year CDS120001000080006000400020000 AAA AA+ AA AA− A+ A A−BBB+ BBBBBB− BB+ BB BB− B+ B B−CCC+ CCCRating Figure 1: USD CDS Spreads Versus S&P Ratings on 2/2/2009HOW STANDARD & POOR’S ARRIVES AT MARKET DERIVED SIGNALS 5
DataPiecewise fitLog spreadFive-year CDS 10987654 3 AAA AA+ AA AA− A+ A A−BBB+ BBBBBB− BB+ BB BB− B+ B B−CCC+ CCCRatingFigure 2: Piecewise Linear Fit of Log Spreads Versus S&P Ratings on 2/2/2009Basic Model After examining three years of data, Standard & Poor’s determined that for the logarithm of . dollar-denominated CDS spreads of . domiciled corporate firms (excluding financials), a piecewise linear model with an adjustment for each two-digit GICS sector, CreditWatch/Outlook status, and document type provide a good fit. In particular, the basic model stipulates that: log(CDSspread)= Standard & Poor's rating benchmark log spread+adjustment for CreditWatch/Outlook status+adjustment for GICS sector+adjustment for document type+ the end of each day, values for the parameters of the model are determined to reflect the CDS spreads observed that day. These parameters include spreads at each rating category and the adjustments listed above. 6
For example, on Feb. 2, 2009, FedEx Corp. had a CDS spread of 205 basis points (bps), or equivalently a log 1spread of . Noting that FedEx Corp. had a Standard & Poor’s rating of ‘BBB’, a CreditWatch/Outlook status of Stable, a GICS sector of 20, and that the contract had a Modified Restructuring clause, the model was fitted as:By taking exponents of both sides of the equation, this relationship becomes:205 bp= (benchmark spread for a 'BBB'-rated firm)* (0% adjustment CreditWatch/Outlook status = Stable)* (% adjustment for GICS = 20)* (0% adjustment for document type = Modified Restructuring)*(-13% adjustment for residual).Extension of Basic Model The basic model may be extended to encompass nonindustrial firms, sovereigns, and currencies other than the . dollar. For example, through historical analysis of CMA data, we noted that financial firms (GICS 40), unlike other firms, could not be modeled by a simple shift in overall log spread level. Rather, we determined by a statistical test that extending the model to include financial firms required a combined shift and tilt adjustment of the piecewise linear function of the ratings in the basic sovereign credits presented a different challenge. Some countries had 10-year CDS spreads, others had five-year CDS spreads, while a third group had both. As a consequence, we added an adjustment for a 10-year CDS spread in contrast with a five-year spread. If a country had both spreads, then we recorded two observations for that country for the purpose of computing the regression coefficients. Many CDS spreads are denominated in euros or yen. Typically, the CDS spreads associated with these currencies behave differently from those associated with . dollar spreads. (See Figure 3 for an example of log spreads on five-year CDS contracts denominated in euros, yen, and . dollars.) We found we could model the spreads of euro and yen contracts without loss of statistical significance only by adding a shift and tilt to the basic . dollar piecewise linear function of ™CDS spread data provided by CMA STANDARD & POOR’S ARRIVES AT MARKET DERIVED SIGNALS 7
USDEURUSD benchmarkEUR benchmarkJPY benchmarkJPYLog spreadFive-year CDS 10987654 3 AAA AA+ AA AA− A+ A A−BBB+ BBBBBB− BB+ BB BB− B+ B B−CCC+ CCCRatingFigure 3: Log Spreads, after adjustments for GICS and CreditWatch/Outlook, of USD, Euro, and Yen Denominated CDS Contracts (excluding Financial CDS and Sovereign CDS) on 2/2/2009Linear Constraints There are some natural constraints upon the model parameters that we concluded were appropriate to include explicitly. For example, due to day-to-day variation of CDS spreads and the natural overlapping nature of spreads associated with adjacent ratings, a benchmark ‘BBB’ spread could be estimated on some days to be greater than a benchmark ‘BB’ spread. But such a result would violate the implicit expectation that higher credit spreads and lower ratings reflect increasing probabilities of default. Hence, our analysis requires that benchmark spreads increase as ratings deteriorate. Similarly, the model requires that the adjustment for CreditWatch Negative or a Negative Outlook increases the log spread and, conversely, adjustment for CreditWatch Positive and a Positive Outlook decreases the log spread. With some exceptions, the imposition of such linear constraints does not change the regression parameters of the model because the desired monotonic relationships are typically reflected in the 8
Robustness We have given considerable attention to the robustness of the estimation process to ensure that outliers and a paucity or lack of observations for some combinations of ratings, GICS, CreditWatch/Outlook status, document types, or currencies do not create misleading parameter begin with, a two-step process carefully controls outliers. First, the regression is run with all observations, and the parameters are estimated. Those observations found to have excessively large residuals are then removed and not used in the final estimation of the parameters. However, these outliers are subsequently analyzed and become part of the final set of , we improve the accuracy and stability of the daily estimators for the parameters of the model by taking advantage of the information contained in the prior day’s estimators. In particular, we utilize a Bayesian estimation procedure, with the prior mean equal to the estimated parameter values for the prior day. The accompanying prior covariance matrix is based on historically observed correlations between the parameters and their historically observed variances, with the latter adjusted for each parameter each day to reflect the number of days that have passed since the last relevant observation. For example, if no firm of a specific rating or currency is observed for a number of days due to holidays or a technical interruption, then the prior variances for the related parameters are increased to reflect this passage of time. The strength of the Bayesian procedure is that it strikes an appropriate balance between today’s and yesterday’s observations, giving greater weight to today’s data if they are plentiful and less when they are not. In addition, it guarantees that there will always be a unique solution to the regression equation, regardless of the number and completeness of the current data. Standard Error and Confidence Intervals After noting that historically the residuals have an approximately normal distribution, the standard error of the regression can measure the regression equation’s goodness of fit. In addition, we can compute the approximate confidence intervals for each parameter, as well as the correlation between them, by noting that the constraints often are not binding. Therefore, the usual regression analysis concerning confidence intervals the parameters are determined in the linear regression, the resulting relationship between CDS spreads and explanatory variables allows one to determine generic (benchmark) spreads for different ratings and the adjustments to those spreads, given the presence of certain features, such as GICS sector or document type. The historical sequence of benchmark spreads for different ratings is given in Figure 4, where we can see the effects on spreads of three significant market events. Figures 5 and 6 are plots of historical adjustments for various GICS and CreditWatch/Outlook status. The size of the adjustment depends upon the slope of the piecewise linear function between the log spreads and numerical scores. When translated back to an adjustment of the benchmark numerical score on the credit ratings scale, on Feb. 2, 2009, the value of the average adjustment for a GICS sector and a Positive or Negative CreditWatch/Outlook was approximately one and two notches, STANDARD & POOR’S ARRIVES AT MARKET DERIVED SIGNALS 9
??????Log spreadSpread ?p? ?????07/2007: Bear Stearns hedge funds collapse???? ??? ?? ?09/2008: Lehman Brothersfiles for bankruptcy? 03/2008: JP Morgan?buys Bear Stearns ?????????????????DateFigure 4: Historical Benchmark Spreads ?CS ? ?nergy??CS ? ?aterias??CS ? ?ndstrias??CS ? Cons?er discretionary??CS ? Cons?er stapes??CS ? ?eat care??CS ? ?nforation tecnoogy??CS ? ?eeco?nication??CS ? ?tiities?Log spread ad?stent d?stent ??? ??? ? ???????? ????????????????DateFigure 5: Historical GICS 10
Outlook−NegativeOutlook−PositiveCreditWatch−NegativeCreditWatch−PositiveLog spread adjustment% Adjustment 1150 100 00 ()(50) (1)01/01/0601/01/0701/01/0801/01/09DateFigure 6: Historical CreditWatch/Outlook Adjustments S&P Market Derived Signals The estimated linear regression also allows one to determine the MDS for each firm, including those originally excluded as outliers. The procedure is to solve for the numerical score in the estimated regression that gives the observed spread exactly, given adjustments for the firm’s attributes. When rounded to the nearest integer, the corresponding credit signal on the credit ratings scale associated with the numerical score is the MDS. For example, in the previous FedEx Corp. example, one may solve for the firm’s adjusted log spread that is equal to the observed log spread of , once taking into consideration adjustments for CreditWatch/Outlook status, GICS sector, and document type, and assuming a zero residual term. This is the value , as seen in the equation (205)= (firm's adjusted log spread)+ (adjustment CreditWatch/Outlook status = Stable)− (adjustment for GICS = 20)+ (adjustment for document type = Modified Restructuring)+ (remaining residual).Using the estimated piecewise linear rating function, which is redisplayed in Figure 7, on the y-axis corresponds to a numerical score of on the x-axis, which rounds to 8, or an MDS of ‘bbb+’. Note that the MDS is expressed in lowercase to distinguish it from a Standard & Poor’s credit rating, which analysts determine through a completely different process. HOW STANDARD & POOR’S ARRIVES AT MARKET DERIVED SIGNALS 11
DataPiecewise fitLog spreadFive-Year CDS 10987FedEx Corp Log Spread = Corp Derived Rating = 3 AAA AA+ AA AA− A+ A A−BBB+ BBBBBB− BB+ BB BB− B+ B B−CCC+ CCCRatingFigure 7: Illustration of S&P Market Derived Signal Calculation Appendix I: A Matrix Representation Of The Regression Equation And Its MinimizationLet denote the parameter vector whose values will be determined by the solution to the regression. For this implementation M=40 and contains seven segments. The first seven positions describe the log spread for a generic firm for respective ratings ‘AAA’, ‘AA’, ‘A’, ‘BBB’, ‘BB’, ‘B’, and ‘CCC’, without adjustment for GICS sector, but with the assumption that the CreditWatch/Outlook status is Stable, the document type is Modified Restructuring, and the currency is in . dollars. The second six positions of the parameter vector are log spread adjustments to the model for a CreditWatch/Outlook value other than Stable. The third eight positions are log spread adjustments for GICS sectors. The next position of is the adjustment for a No Restructuring type. The next four positions represent shift and tilt log spread adjustments for European and Japanese industrial firms, respectively. Positions 26 through 32 are shift and tilt adjustments to financial firms with currencies of . dollars, euro, and yen, respectively. The final eight positions of the parameter vector are shift and tilt adjustments for ratings for X be a matrix with NC rows and M columns, where Xi,j is the j attribute characterizing the []ththi CDS contract and where NC is the number of contracts observed. For example, if the i CDS contract is denominated in USD and has a No Restructuring clause and is for a firm with a BBB rating, a GICS sector 12
j=4,13,17of 30 and a CreditWatch Positive, then Xij=1 for and Xi,j=0 otherwise. More [][]complicated assignments for X are required for some CDS contracts, such as those denominated in euro and yen, with GICS=40, or the Standard & Poor’s ratings modified by a “+” or a “-”.Let y denote the vector y[i],i=1,...,NC, where y[i] is the observed natural log mid-spread for CDS {}contract i. With the above notation, the regression equation is , where is a residual vector term that has a prior multivariate normal distribution with mean and covariance matrix C. Note that the prior mean is set equal to the estimate of on the previous constraints in the MDS can be expressed as:where L is a K×M matrix and c is a vector of length K, where K is the number of the prior and the linear constraints, the objective is to minimize the Bayesian residual sum of squares:thwith respect to , subject to . Here, U is a diagonal matrix with the i diagonal element equal to 2thVS1 or 0 depending upon whether the i observation is an outlier that should be excluded or not. is a scalar weight that determines the amount of weight that based on history should be given to the prior in the estimation process. Note that the implicit weight given to the current data depends upon the number of data points contained in the first term in the above equation for . T−1Since C is a positive definite covariance matrix, there exists a full rank matrix A such that AA=C. By setting:where Q and z both have NC+M rows, the quantity to be minimized can be written A has full rank, Q also has full rank and there exists a unique minimum to the above equation subject to the constraints. In particular, if there are no observations, will be equal to . This function can be easily minimized using a quadratic minimization routine with an interior point subject to satisfying the constraints given STANDARD & POOR’S ARRIVES AT MARKET DERIVED SIGNALS 13
Appendix II:Global Industry Classification Standard (GICS) Codes10 Energy35 Health Care15 Materials40 Financials20 Industrials45 Information Technology25 Consumer Discretionary50 Telecommunication Services30 Consumer Staples55 UtilitiesAppendix III:Document Types CR Full RestructuringMR Modified RestructuringMMR Modified Modified RestructuringXR No RestructuringNo comment or representation is intended or should be inferred regarding Standard & Poor’s ratings descriptive “ratings” used in this document are intended only to describe relative credit quality and sensitivity to parameters on a familiar credit scale with respect to this specific modeling framework. Unless explicitly stated, they do not refer to actual Standard & Poor’s ratings or credit opinions on any company, actual deal, or classes of 14
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