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1、A On e-Parameter Represe ntati on of Credit Risk and Tran siti on MatricesLawrence R. Forest, Jr., KPMG Peat Marwick LLPBarry Belk in and Stepha n J. Suchower, Daniel H. Wag ner Associates1. OverviewThis paper prese nts a on e-parameterreprese ntati onof credit risk and tran siti on matrices. Westar

2、t with the CreditMetrics view that ratings transition matrices result from the“ binning of asta ndard no rmal ran dom variable X that measures cha nges in creditworth in ess. We further assume here that X splits into two parts: (1) an idios yn cratic comp onentY, uni que to a borrower,and (2) a syst

3、ematic comp onentZ, shared by all borrowers. Broadly speak ing,Z measures the“ credit cycle ” , meaning the values of default rates and of end -of-period risk ratings not predicted (us ing historical average tran siti on rates) by the in itial mix of credit grades. In good yearsZ willbe positive, im

4、pl ying for each in itial credit rat ing, a lower tha n average default rate and a higher than average ratio of upgrades to downgrades. In bad years, the reverse will be true. We describe a way of estimatingZ from the separate transition matrices tabulated each year byStandard & Poor (S&P) and Moody

5、 s. Conversely, we describe a method of calculating transition matrices con diti onal on an assumed value for Z.The historical patter n of Z depicts past credit con diti ons. For example, Z rema ins n egative for most of 1981-89. This mirrors the general decline in credit ratings over that period. I

6、n 1990-91,Zdrops well below zero as the US suffers through one of its worst credit crises since the Great Depression.The relatively high proportionof lower grade credits inheritedfrom the 1980 stogether with the 1990-91 slump (Z0) accou nts for a high number of defaults. Over 1992-97,Zhas stayed pos

7、itive and credit con diti ons have rema ined benign. The moveme nts ofZ over thepast 10 years correlate closely with loan pricing.Our focus here is on how Z affects credit rati ng migratio n probabilities. However, one can also model the effect of Z on the probability distributi on of loss in the ev

8、e nt of default (LIED), credit par spreads, and ultimately the value of a commercial loa n, bond, or other in strume nt subject to credit risk. By parametrically vary ing 乙 one can perform stress testi ng to assess the sen sitivity of the value of an in dividual credit in strume nt or an en tire cre

9、dit portfolio to cha nging credit con diti ons.One can also qua ntify how volatility inZ tra nslates into tran sacti on and portfolio value volatility.2. Defi ning Z RiskFollow ing the CreditMetrics approach described by Gupt on, Fin ger, and Bhatia (1997), we assume that rati ngs tra nsiti ons refl

10、ect an un derl ying, con ti nuous credit-cha nge in dicatorX. Onefurther assumes that X has a standard normal distribution. Then, conditional on an initial creditGG1rati ng G at the beg inning of a year, one partiti onsX values into a set of disj oint bins (xg , Xg 訂.To simplify refere nces, the in

11、dices G and g represe nt seque nces of in tegers rather tha n letters orother symbols. One defines the bins so that the probability that X falls within a given interval equals the corresponding historical average transition rate (see Exhibit 1).We observe an inconsistency among the bins for differen

12、t initial ratings. For an initial borrower rating of Go, consider successive yearly values for X of X1 and x2, in which X1 implies a rating s chang eiittd G a change to G2. We won t find, in general,that an X value of X1 +x2 in the first year implies a rating change from G 0 to G2.A One Parameter Re

13、prese ntati on of Credit Risk and Tran siti on Matrices10/02/1812:45 PM DRAFT13Exhibit 1: Relati on ship Betwee n Con ti nuous Credit In dex X and Rati ng Tran siti onsWe write the conditions defining the bins as follows:(1)P(G,g)。；1)- ：心冷in which P(G,g) denotes the historical average G-to-g transit

14、ion probability and 4、()represents the sta ndard no rmal cumulative distributi on fun cti on. The default bin has a lower threshold of -The AAA bin has an upper threshold of +；. The remaining thresholds are fit to the observed tran siti on probabilities.Suppose there are N ratings categories includi

15、ng default. Then there are N-1 initial grades, which represent all the ratings excluding default. For each of those initial grades, we observe N-1 historical average transition rates.(The other (Nth) value derives from the condition that theprobabilities sum to 1.) We must determine N-1 threshold va

16、lues defining the bins. Thus, we can solve for all of the bin bou ndaries.We illustrate this process below. The starting point is the smoothedversion of the 1981-97historical average tran siti on matrix tabulated by S&P for 8 grades, in clud ing default (see Exhibit22). The corresp onding bins are c

17、omputed using the above formula.Con sider tran siti ons from BBB. We observe a 15 bps default rate. Using the in verse probabilityfunction for a standard normal distribution, we compute a value of about -2.97 for the upper threshold for the default bin. Next con sider the CCC bin. We get a value of

18、about 25 bps for the sum of tra nsiti on rates to CCC or to default. Again appl ying the in verse probability fun cti on, weget an upper threshold value for CCC of about281. Now con sider B. We compute a probabilityof about 1.3 per cent for tra nsiti ons to B or to lower grades. Once aga in appl yin

19、g the in verse2 The smoothing applied to the matrix enforces default rate monotonicity, row and column monotonicity and several of the other regularity conditions listed in the CreditMetrics documentation. Default rate monotonicity means that default rates rise as credit ratings go down. Row and col

20、umn monotonicity means that transition rates fall as one moves away from the main diagonal along either a row or a column. We note one exception to this rule. Default is a trapping state. Thus, the default rate may rise above the probability of transition to neighboring non-default babilit

21、y function,we get an upper threshold value of -2.23. Continuingin this way for eachterminal and each in itial grade, we derive all of the bin values.As in Belk in, Suchower, and Forest (1998), we decomposeX into two parts:(1) a (scaled)idiosyncratic component Y, unique to a borrower, shared by all b

22、orrowers. Thus, we writeand (2) a (scaled) systematic component Z,Exhibit 2: Smoothed Historical Average Tran siti on Rates and Associated Bi nsinitialRatingEnd of Year Credit RatingAAAAAABBBBB 1CCCSmoothedHistorical Average Transition RatesAAA91.13%8.00%0.70%0.10%0.05%0.01%0.01%0.01%AA0.70%91.03%7.

23、47%0.60%0.10%0.07%0.02%0.01%A0.10%2.34%91.54%5.08%0.61%0.26%0.01%0.05%BBB0.02%0.30%5.65%87.98%4.75%1.05%0.10%0.15%BB0.01%0.11%0.55%7.77%81.77%7.95%0.85%1.00%B0.00%0.05%0.25%0.45%7.00%83.50%3.75%5.00%CCC0.00%0.01%0.10%0.30%2.59%12.00%65.00%20.00%Bins Corresponding to Smoothed Historical Average Trans

24、ition RatesAAA(如.35)-1.35,-2.38)-2.38,-2.93)-2.93,-3.19)-3.19,-3.54)-3.54,-3.72-3.72,-3.89)-3.89,-oq)AA(*2.46)2.46,-1.39)-1.39,-2.41)-2.41,-2.88)-2.88,-3.09)-3.09,-3.43)-3.43,-3.72)-3.72,-oq)A33.10)3.10,1.97)1.97,-1.55)-1.55,-2.35)-2.35,-2.73)-2.73,-3.24)-3.24,-3.29)-3.29,-00)BBB(比3.50)3.50,2.73)2.7

25、3,1.56)1.56,-1.55)-1.55,-2.23)-2.23,-2.81)-2.81,-2.97)-2.97,-00)BB(尖3.89)3.89,3.05)3.05,2.48)2.48,1.38)1.38,-1.29)-1.29,-2.09)-2.09,-2.33)-2.33,-00)B34.11)4.11,3.29)3.29,2.75)2.75,2.43)2.43,1.42)1.42,-1.36)-1.36,-1.64)-1.64,-oq)CCC34.27)4.27,3.72)3.72,3.06)3.06,2.64)2.64,1.88)1.88,1.04)1.04,-0.84)-0

26、.84,-oq)3and mutually independent.TheZ and X. Thus Z explains aWe assume that Y and Z are unit normal random variables parameter (assumed positive) represe nts the correlati on betwee n fracti on of the varia nee ofX.In any year, the observed tran siti on rates will deviate from the n orm (Z =0). We

27、 can the n find avalue of Z so that the probabilities associated with the bins defined above best approximate the given year s observed transition rates (see Exhibit 3). We label that value ofZ for year t, Zt. Wedetermine Zt so as to minimize the weighted, mean-squareddiscrepanciesbetween the modelt

28、ran siti on probabilities and the observed tra nsiti on probabilities.For this we defi neThis is the model value for the G-to-g transition rate in year t. Then for a fixed least-squares problem takes the form!-: and a fixed t, themi上吐翼（x：1产?* ,Z G g厶此1用,乙）1此.1用乙）where P t(G,g) represents the G-to-g

29、transition rate observed inyear t and nt,G is the number oftran siti ons from in itial grade G observed in that year. In this formula, we weight observati ons by the inverses of the approximate sample variances of Pt(G,g ).Exhibit 3: Illustration of theZ Value for a Particular Initial Rating in a Gi

30、ven YearThis Year s ZF(X)0.4Distribution BestDistribution BestApproximatingTransitions in a NormalApproximating This Year s Transitions(Z=0) Credit YearD CCC BBBBAA AAAWe don t know the value of a priori. We estimateas follows. We apply the mini mizati on in (4)for 1981-97 using an assumed

31、value of二 We the n obta in a time series forZt, con diti onal onWe compute the mean and variance of this series. We repeat this process for many values ofIn the formula (3), we normalize each squared deviation by the factor(x；卄x；, 4)(1 A(x；卄痔,zO)nt,GThis weighting factor represents the sample varian

32、ce for the G -to-g transition rate under a binomial sampling approximation such that“ success ” is the occurreGctoeg transition and “failure ” is any other transition. Afull multinomial treatment would account for the constraint that the sample transition rates across a row must sum to one.:value fo

33、r which theZt time seriesand use a numerical search procedure to find the particular has varia nee of one.We illustrate this process of solving for Zt at a single time t. We start with the S&P transition matrix observed for 1982 (see Exhibit 4). We hold fixed the bi ns determ ined from the historica

34、l average matrix and fixp at the value (.0163) determ ining by the search process. The in dicatedvalue for Zt of -0.89 provides the best fit to the observed 1982 transition rates.Exhibit 4: S&P Tran sition Matrix for 1982 and Calculatio ns Lead ing toZ EstimateInitialRating#Obs.End of Year Credit Ra

35、tingAAAAAABBBBBBCCCDObserved Transition MatrixAAA8592.94%4.71%2.35%0.00%0.00%0.00%0.00%0.00%AA2200.46%92.52%6.08%0.47%0.47%0.00%0.00%0.00%A4800.00%4.45%84.95%9.54%0.64%0.00%0.00%0.42%BBB2980.37%0.37%3.26%85.52%9.78%0.37%0.00%0.34%BB1680.00%0.68%0.00%2.68%82.42%10.05%0.00%4.17%B1610.00%0.00%0.72%0.72

36、%2.89%87.50%5.06%3.11%CCC160.00%0.00%0.00%0.00%0.00%7.39%73.86%18.75%Fitted Transition MatrixAAA8589.34%9.54%0.89%0.13%0.07%0.01%0.01%0.01%AA2200.48%89.56%8.93%0.77%0.13%0.09%0.03%0.01%A4800.06%1.72%90.88%6.14%0.78%0.34%0.01%0.07%BBB2980.01%0.20%4.39%88.03%5.72%1.33%0.13%0.20%BB1680.00%0.07%0.38%6.1

37、9%81.63%9.39%1.06%1.29%B1610.00%0.03%0.17%0.32%5.56%83.41%4.38%6.14%CCC160.00%0.01%0.06%0.20%1.94%10.09%64.53%23.16%Z Value-0.89Broadly speaking,Zt measures the “ credit cycle, ” meaning the values of default rates and of end -of-period risk ratings not predicted (using historical average transition

38、 rates) by the initial mix ofcredit grades. In good years Zt will be positive, impl ying for each in itial credit rat ing a lower tha n average default rate and a higher tha n average ratio of upgrades to dow ngrades. In bad years, the reverse will be true.3. Zt s Historical Patter nsn the initial m

39、ix of ratingscredit cycle ” suggests. In particular,Zt s historical moveme nts describe past credit con diti ons not evide nt i (see Exhibit 5).Zt s history is erratic, more so tha n the termthe fluctuations don t show a stable sinusoidal pattern.Exhibit 5: Zt as Estimated from S&P Ann ual Tran siti

40、 on Matrices2.001.501.000.500.00-0.50-1.00-1.50-2.001981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997Zt is mostly n egative over1981-89. Credit rat in gs gen erally decli ned over that period as manycorporati ons in creased leverage. In 1990-91,Zt drops below zero,

41、 as the US suffers through oneof its worst credit crises since the Great Depression. The relatively high proportion of lower grade credits inherited from the 1980 s together with the 1990 credit slump ( Zt0) accounts for a highn umber of defaults. Over the period 1992-97,Zt has stayed positive and c

42、redit con diti ons haverema ined benign.Loa n prices over the past 10 years correlate quite closely with the credit in dicatorZt (see Exhibit6). One observes that loa n spreads have gen erally lagged abrupt cha nges in credit con diti ons.Exhibit 6:Zt In dex and BB SpreadsBB Par Spreads, left scalei

43、 Zt, right scale, in vertedZt -乙亠=0.54Zz 1.04 t.(5)140130120110100908070601987 1998 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998Loan spreads in North America and Europe over the past 2-3 years have remained near record lows. This suggests that the past 6 years of favorable credit con diti ons h

44、ave made many len ders optimistic about the future. One might ask whether the past patter ns exhibited byZt justify thisoptimism or any other forecast of credit con diti ons.Applying theweighted least-squares scheme for estimating theZt, we get a ：? value of 0.0163.Thus, systematic credit migration

45、risk accounts for only about 1.6% of total credit migration risk over the period 1981-97. This con trasts with equity price data, which suggest that systematic risk accounts for about 25 percent of the total varianee in an average company s stock price. Weshall see below, however, that small variati

46、ons in Zt can translate into substantial swings in default and dow ngrade rates.The discrete time cou nterpart to the O-U model is a first-order autoregressive process. We fitted such a model forZt to the data for the 1982-97 period and obta ined the follow ing:Here * is a standardized white noise s

47、equenee. From (5), the sample estimates for the O-U model parameters are P=.54yr-1 and c = 1.04 yr-1/2 . Thus, the Zt process has an estimated mean relaxation time of about 2 years and an estimated annual volatility of about 100%.We performed several statistical tests on (5) for model goodness of fi

48、t. The sample estimate of the mean of the Zt process mean is 0.16, with a standard error of 0.25. As a result, there is no statistical basis to reject the hypothesis that theZt process has zero mean. Based on a t-statisticvalue of 2.18, the hypothesis that there is no mean reversi on (i.e., that: =0

49、) can be rejected atthe .025 significance level. The Kolmogorov-Smirnov test statistic for the model residuals has a value of d =17, indicating that the residuals are statistically indistinguishable from a white noise sequence ( ： =.71).2The calculated R for the model in (5) is .24, indicating that

50、a first-order autoregressive model for the Zt has modest predictive power. 2The R for a second-order autoregressive model is only .33, so going to a higher order model adds little in the way of predictive power. The simply reality is that the Zt process, at least over the 17-year historical period a

51、nalyzed, is quite volatile. However, the utility of the model is not in predicting future values of Zt. Rather, it is to quantify how the variability inZt that is predictable and the variabilityin Zt that is not predictable each in flue nce credit risk and the pric ing of that risk.4. Determ ining t

52、ran siti on matrices as fun cti ons ofZtWe ve already desc ribed a way of imputing the Zt variable from observed transition matrices. By in vert ing this process, one may determ ine tran siti on matrices from values ofZt.GWe again use the bin values xg for each initial grade G and end-of-year grade

53、g. Now, con diti onal on Zt, we compute the probability of a G to g tran siti on as:职Zt 口丿，口丿Pt(G,g)八We show matrices below for a good year (Zt =1), an average year ( Zt =0), and a bad year ( Zt =-1)(see Exhibit 7). Note that an absolute value of 1 represe nts a 1-sta ndard deviati on variati on fro

54、m“normal ” credit conditions.Exhibit 7: Tran siti on Matrices Computed UsingZ Parameterizati onInitialRatingEnd of Year Credit RatingAAAAAABBBBBBCCCDCalculated Transition Matrix tor Good Year (Z=1)AAA93.17%6.25%0.47%0.06%0.03%0.01%0.00%0.00%AA0.95%92.72%5.81%0.41%0.06%0.04%0.01%0.01%A0.14%3.02%92.33

55、%3.88%0.42%0.17%0.01%0.03%BBB0.03%0.41%7.03%88.00%3.65%0.73%0.06%0.09%BB0.01%0.15%0.73%9.50%82.00%6.32%0.61%0.67%B0.00%0.07%0.34%0.59%8.58%83.68%3.03%3.70%CCC0.00%0.01%0.14%0.40%3.30%14.12%65.60%16.42%Calculated Transition Matrix for Average Year (Z=0)AAA91.31%7.87%0.67%0.09%0.05%0.01%0.00%0.00%AA0.66%91