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Time series patterns in credit ratings
Finance Research Letters 4 (2007) 217–226 www.elsevier.com/locate/frl
Time series patterns in credit ratings Dror Parnes ∗ Finance Department, College of Business, BSN 3127, University of South Florida, 4202 E. Fowler Ave., Tampa, FL 33620-5500, USA Received 29 May 2007; accepted 19 September 2007 Available online 22 September 2007
Abstract This article offers a substitute setting to simulate credit rating migrations. The internal correlations model tracks time-series movements within credit rating entries, rather than cross-ratings correlations. The proposed nonhomogeneous process is authenticated through the likelihood ratio Dickey–Fuller test, and is found to be statistically and economically significant, by better fitting observed cumulative default rates. Several nonlinear regression models assist to better identify these time-related patterns. The economic structure underlying the time dependency often corresponds to changes in GDP, business cycles, and market risk. Furthermore, significant positive autocorrelation is detected mostly among downgrade probabilities. © 2007 Elsevier Inc. All rights reserved. JEL classification: G21; G24; G32; G33 Keywords: Credit ratings; Time-series patterns; Dickey–Fuller test; Homogeneous Markov chain; Serial correlation
1. Introduction Developments in credit ratings commonly reveal an improvement or deterioration in the credit-worthiness of a firm. Although rating agencies primarily meant to utilize credit ratings to signify the current credit quality, several theories use the rating transitions phenomenon to forecast default events, or to price risky bonds. Academicians and practitioners often use the homogeneous Markov chain, first proposed by Jarrow et al. (1997), to describe the dynamics of credit ratings. Yet, in reality, this process does not correspond to actual yield curves, as well as violates several observed dependencies. * Fax: +1 813 974 3084.
E-mail address: [email protected]. 1544-6123/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.frl.2007.09.001
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D. Parnes / Finance Research Letters 4 (2007) 217–226
So far, few attempts have been made to step away from the homogeneity assumption of the Markov chain transition matrix. The CreditMetricsTM methodology offered of Gupton et al. (1997) examines cross-ratings correlations. The momentum scheme proposed by Bahar and Nagpal (2001) considers lagged rating transitions. The generator matrix investigated by Israel et al. (2001) offers procedures to simulate further transition matrices. Bangia et al. (2002) distinguish between transition matrices depending on the business cycle, and the density-dependent model of Parnes (2007) associates firm’s survivability and transitivity to the market-density. These nonhomogeneous models consider the credit ratings migration not to be a memory-less process any more. This article offers a different approach to track rating transitions while accounting for timeseries movements within credit entries, rather than cross-rating dependencies. We substantiate this nonhomogeneous dynamic through a Dickey–Fuller statistical test, associate it with market variables, and demonstrate its economic importance. The article proceeds as follows. Section 2 presents the alternative model. Section 3 discusses the data and methodologies. Section 4 explains the empirical findings and their economic meaning, and Section 5 concludes. 2. Time series patterns in credit ratings An evidence for time series correlation in credit ratings migration can be found in Altman and Kao (1992a, 1992b). The authors divide the examined time frame into two sub-periods, from 1970 to 1979, and from 1980 to 1985, and reveal a tendency for a downgrade in rating to be followed by a second downgrade, indicating on a serial correlation when the initial rating changes was a downgrade. However, weak autocorrelation is found when the first change is an upgrade. The following model makes an attempt to track trends in transition probabilities over time, and to generate future transition probabilities that follow the same affine trend. This scheme explicitly assumes that each transition probability is univariate autoregressive correlated with previous transition probabilities within the same entry in the credit ratings matrix.1 By that, the model implicitly assumes that transition probabilities are cross-correlated with other transition probabilities within the same column. The cross correlation evolves from the fact that the sum of entries within each column always remains one. Thus, when a specific entry in the transition matrix tends to increase (decrease), other entries in the same column tend to decrease (increase). The model estimates AR(1) time-series regressions for each entry in the transition matrix as sij (t + 1) = αij + βij sij (t) + εij (t)
∀i, j,
(1)
where entry sij (t) denotes the transition probability from state j to state i at time t in the survival sub-matrix S. The β coefficient of the regression equation determines whether a transition probability is stationary. Transition probabilities must remain in their domain, thus, unless α is close to 1 and β is positive, or α is close to 0 and β is negative, the model requires no constrains.2 1 Describing the life cycle of a company requires the distinguishing between different survival states and an absorbing
state of default. Transition matrix T is a nonnegative square matrix of dimensions 22 × 22, where its entries represent probabilities of all credit ratings migration. Matrix T contains a survival sub-matrix S of dimensions 21×21, representing transitions among livelihood states, an absorbing row vector D of dimensions 1 × 21, representing transitions from livelihood states into the default state, and an arbitrary column vector of zeros and one. 2 The empirical investigation has found no evidence for these extreme situations.
D. Parnes / Finance Research Letters 4 (2007) 217–226
219
To better identify the nature of time-related patterns we further explore several nonlinear techniques. To examine whether an entry in the survival matrix display time-varying volatility clustering, i.e. periods of quiet and periods of fluctuating residuals in Eq. (1), we test for AutoRegressive Conditional Heteroskedasticity (ARCH(1)) effect. To further inspect whether lagged volatility is as important as lagged residuals, we test for Generalized ARCH (GARCH(1, 1)) effect. To investigate whether a time series pattern exhibit asymmetric reaction to positive and negative shocks, we test the Quadratic GARCH model (QGARCH(1, 1)). 3. Data and methodologies The Compustat database reports on 130,559 quarterly S&P long-term credit ratings for 4510 industrial companies from 1985 to 2004. Each credit rating is tagged by a number between 2 for ‘AAA’ and 27 representing a bankruptcy filing. Within this period, there are 123,849 valid credit rating transitions in consecutive quarters. These transitions are collected and assigned to a frequency matrix.3 Since there are 21 states in the life cycle of a company and an additional absorbing state of default, the frequency matrix is of dimensions 22 × 21, where each entry represents the number of credit ratings migrations observed from one state to another. Several observations are reported as ‘not rated’ (NR). These observations are classified as 1, 3, 22, 25, 28, 29 and 90 in the database. This study considers transitions to and from NR status as missing information, by eliminating them from the data sample.4 We first construct homogeneous transition matrices for each year over the examined period, from 1985 to 2004. These twenty different tables are used to explore the time series dependencies within each of the 22 ×21 = 462 entries. Next we conduct the autoregressive time-series analyses for each entry in the transition matrix as illustrated in Eq. (1). To confirm whether the transition matrix entries follow some trends over time, and by that to validate the time-series internal correlations model, the Dickey–Fuller test is undertaken. Dickey and Fuller (1981) present a likelihood ratio statistical test for autoregressive time-series regression, to examine whether the dynamic follows a random walk with zero drift, or some trends occur. The likelihood ratio null hypothesis admits the coefficients: H0 : αij = 0; βij = 1, against the alternative hypothesis H1 : not H0 . The likelihood ratio statistic is n−1 2 Φ1 , 1+ (2) 2(n − 3) where n is the number of intervals in the time-series analysis, and the statistics 2 −1 2 Φ1 = 2Seμ (n − 1)σ02 − (n − 3)Seμ , 2 Seμ = (n − 3)−1
σ02 = (n − 1)−1
n
t=2 n
2 sij (t) − αˆ ij − βˆij sij (t − 1) ,
2 sij (t) − sij (t − 1) .
(3) (4)
(5)
t=2
3 We observe merely few transitions out of the default state back to the survival sub-matrix. Therefore, the assertion of
an absorption state of default is rational. 4 Bangia et al. (2002) discuss other alternatives to treat NR observations, but ultimately choose to exclude them as well.
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D. Parnes / Finance Research Letters 4 (2007) 217–226
2 and σ 2 are consistent with Dickey and Fuller (1981), and s (t) denotes The notations Φ1 , Seμ ij 0 the survival transition probabilities at time t . The likelihood ratio test rejects the null hypothesis for larger values of Φ1 , relative to critical values from Dickey–Fuller tables.5 To better identify the economic structure of time dependency, we examine the relations between time-related changes in transition probabilities within the survival matrix and changes in three relevant macroeconomic variables.6 The first two factors signify general business conditions. These are the scaled Gross Domestic Product (GDP) in billions of chained 2000 dollars, taken from the Bureau of Economic Analysis (BEA) web site, and the National Bureau of Economic Research (NBER) own definition for contractionary and expansionary stages.7 More formally, we regress
sij (t) = υij + μij GDP(t) + γij NBER(t) + ωij (t) 0 if contractionary cycle, where NBER(t) = 1 if expansionary cycle.
∀i, j, (6)
The third macroeconomic variable is the volatility index (VIX) of the Chicago Board Options Exchange (CBOE). The VIX is meant to depict the market’s expectation of thirty-day volatility.8 It is constructed through a simple average of implied volatilities of call and put options on a large sample of shares included in the S&P 100 index until 2003, and in the S&P 500 index after that. Since its first appearance, the VIX is widely used as a measure of market risk.9 We aim to explore the link between changes in market risk and changes in credit-rating transition probabilities over time, thus we examine the following relation10 : sij (t) = δij + λij VIX(t) + ηij (t)
∀i, j after 1993.
(7)
The economic importance of the proposed internal correlations model is examined through a comparative analysis with the commonly used Markov chain dynamic and the actual cumulative default rate (CDR). We perform a back-testing where the two techniques are calibrated over the first ten-year period, from 1985 to 1994. We then run a Monte Carlo simulation for both models over the next ten-year period, from 1995 to 2004, and compare these expected CDR with observed ones in the later time frame. We further examine serial correlations among upgrade and downgrade probabilities according to Altman and Kao (1992a) sign-test methodology. We also generate the complete homogeneous transition matrix over the entire period, from the beginning of 1985 to the end of 2004, by dividing each entry in the frequency matrix by the sum of its column, as reported in Table 1. 5 The Dickey–Fuller tables are described in Dickey and Fuller (1981, p. 1063). 6 Since this analysis is time dependent, we are required to maintain a stationary dataset, and to eliminate possible
tendencies. We therefore compute changes in transition probabilities. 7 The NBER expansionary and contractionary quarters are constructed by examining a larger number of business indicators in addition to the GDP. 8 Since the VIX is originally formed in 1993, in this specific test we ought to truncate the sample of transition probabilities from 1993 to 2004. 9 For example, see Connolly et al. (2005), Ang et al. (2006), and Corrado and Miller (2006). 10 Collin-Dufresne et al. (2001) and Huang Kong (2003) investigate the association between changes in the VIX as a proxy for market risk and changes in bankruptcy risk quantities.
AA
AA−
A+
A
A−
BBB+ BBB
BBB− BB+
BB
BB−
B+
B
B−
CCC+ CCC
CCC− CC
C
0.16 0.29 95.27 3.00 0.65 0.35 0.08 0.12 0.06 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.02 0.01 0.00 0.04 0.01 0.00 0.02 0.03 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.01 0.02 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.79 0.06 0.04 0.04 0.00 0.02 0.01 0.00 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 94.49 0.82 0.09 0.06 0.01 0.03 0.06 0.03 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.34 94.98 1.19 0.15 0.15 0.05 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.07 0.00 0.00 0.00 0.00 0.81 2.77 95.04 1.81 0.33 0.12 0.07 0.07 0.00 0.02 0.02 0.00 0.00 0.00 0.12 0.00 0.00 0.00 0.25 0.85 2.32 93.71 1.72 0.43 0.17 0.10 0.01 0.04 0.02 0.08 0.00 0.00 0.12 0.00 0.00 0.00 0.13 0.24 0.78 2.80 93.45 1.83 0.47 0.12 0.08 0.04 0.07 0.02 0.04 0.00 0.12 0.00 0.00 0.00 0.04 0.08 0.27 1.12 3.06 94.04 2.52 0.89 0.32 0.12 0.02 0.07 0.00 0.07 0.00 0.00 0.00 0.00 0.04 0.01 0.05 0.12 0.77 2.32 92.90 3.40 0.86 0.16 0.10 0.02 0.07 0.15 0.00 0.00 0.00 0.00 0.00 0.02 0.07 0.01 0.14 0.57 2.03 90.86 2.39 0.66 0.13 0.07 0.07 0.00 0.23 0.24 0.00 0.00 0.00 0.02 0.04 0.03 0.22 0.22 1.20 2.44 91.43 2.20 0.54 0.15 0.14 0.22 0.12 0.24 0.00 0.00 0.00 0.01 0.04 0.01 0.03 0.10 0.33 1.26 2.99 92.21 1.94 0.42 0.29 0.29 0.23 0.72 0.93 0.00 0.02 0.04 0.02 0.04 0.04 0.16 0.09 0.37 1.17 2.98 92.85 2.69 0.98 1.11 0.58 0.96 1.86 0.00 0.05 0.04 0.00 0.01 0.03 0.03 0.07 0.16 0.32 0.76 2.57 89.82 2.17 1.47 1.05 0.72 0.62 10.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.07 0.16 0.29 0.86 2.54 86.64 2.36 1.40 0.96 2.79 0.00 0.00 0.00 0.00 0.00 1.01 0.02 0.01 0.00 0.05 0.16 0.31 1.64 3.03 82.96 1.63 1.20 1.24 10.00 0.00 0.01 0.00 0.00 0.00 0.01 0.00 0.03 0.07 0.10 0.22 0.66 2.46 2.51 82.09 2.16 2.17 10.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.03 0.04 0.07 0.05 0.40 1.05 2.58 1.63 81.29 1.55 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.03 0.05 0.04 0.08 0.43 1.12 1.99 2.09 3.36 73.68 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.00 0.07 0.47 0.24 0.00 50.00
D. Parnes / Finance Research Letters 4 (2007) 217–226
Table 1 The transition matrix Transition AAA AA+ matrix (%) AAA 98.04 0.83 1.05 94.36 AA+ AA 0.32 3.71 AA− 0.32 0.76 0.11 0.14 A+ A 0.00 0.14 0.04 0.00 A− BBB+ 0.00 0.00 0.07 0.07 BBB BBB− 0.00 0.00 0.00 0.00 BB+ BB 0.04 0.00 0.04 0.00 BB− B+ 0.00 0.00 0.00 0.00 B B− 0.00 0.00 0.00 0.00 CCC+ CCC 0.00 0.00 0.00 0.00 CCC− CC 0.00 0.00 0.00 0.00 C
0.00 0.00 0.00 0.00 0.02 0.01 0.01 0.01 0.05 0.01 0.09 0.03 0.13 0.16 0.96 1.95 4.13 8.14 7.91 15.17 20.00 D Notes. Nonarbitrary part of the transition table as calculated directly from the frequency matrix. The upper sub-matrix is the survival part S, and the lower row vector is the absorbing default state D. Entries inside the matrix stated as percentages. Each column sums to 100%. Like the frequency matrix, this matrix is of dimensions 22 × 21.
221
222
D. Parnes / Finance Research Letters 4 (2007) 217–226
4. Results This section reports the empirical findings and summarizes their economic meaning. Table 2 presents the values of Φ1 , derived from the likelihood ratio Dickey–Fuller tests, for all the entries in the transition matrix along with the statistical significance levels of each entry, after comparing the entry values to the critical values from Dickey–Fuller tables. Out of the 462 possible entries in the transition matrix, 168 are classified as not meaningful (N/M), since no transitions were observed throughout the whole time frame, from 1985 to 2004. Among the remaining 294 meaningful entries, 194 entries, or 66.0%, reject the null hypothesis with a significance level of 0.99, 29 entries, or 9.9%, reject the null hypothesis with a significance level of 0.975, 18 entries, or 6.1%, reject the null hypothesis with a significance level of 0.95, and 21 entries, or 7.1%, reject the null hypothesis with a significance level of 0.90. Only 32 entries, or 10.9%, cannot reject the null hypothesis of a random walk with zero drift. These findings assess the claim that most transition probabilities follow trends over time, and thus, validate the internal correlations model. Table 3 identifies the linear and the nonlinear time-series patterns for each entry in the survival matrix S. In addition, this table designates the economic structure of these dependencies; whether time variability in each entry is associated to changes in the GDP, to the business cycle as defined by the NBER, or to the VIX as a proxy for market risk. Out of the 262 entries that follow time-series patterns of type AR(1), only 11 exhibit ARCH(1) effect, indicating on almost no time volatility clustering. Yet, 87 entries demonstrate GARCH(1, 1) relation, showing on some volatility autocorrelation. None of the entries prompt QGARCH(1, 1), suggesting on similar response to positive and negative shocks. In addition, economically and statistically significant correlations are found between 39 of the entries and changes in the GDP, between 39 of the (same or other) entries and business cycles as defined by the NBER, and between 26 of the (same or other) entries and developments in market risk as captured by the VIX. The lower panel of Table 3 presents the serial correlation test results. We identify 849 rating transitions in consecutive quarters from 1985 to 2004, and classify them as ‘up–up,’ ‘up–down,’ ‘down–up,’ and ‘down–down’ movements originated at different credit ratings. We recognize robust positive serial correlation between subsequent downgrades, but sporadic positive autocorrelation among succeeding rating upgrades. Figure 1 demonstrates a comparative analysis of the expected CDR, as derived from the homogeneous Markov chain and from the internal correlations model, as well as the actual CDR observed from 1995 to 2004. The results reveal that the existing Markov chain scheme overestimates the credit ratings migration probabilities while projecting much higher CDR than is perceived in practice. Furthermore, the proposed time-series approach improves the predictive strength by better fitting observed patterns. 5. Summary and conclusions This article offers an alternative approach to credit ratings migration analysis. The internal correlations model tracks time-series movements within credit rating entries. The proposed nonhomogeneous dynamic is validated through the likelihood ratio Dickey–Fuller test, and is found to be statistically and economically significant. It improves commonly used methodologies by better fitting observed cumulative default rates.
Table 2 Dickey–Fuller likelihood ratio test AA+
AAA
5.32**
8.91***
7.25**
AA+
8.68***
36.74***
21.13***
4.15* 21.24*** 11.92*** 358.19*** 11.13*** 9.73*** 10.76*** 10.76***
AA
BBB−
8.32*** 2.15 92.08*** 10.01*** 12.28*** 8.15*** 4.47* 10.57*** 5.43** N/M 3.34 6.43** 9.5*** N/M 10.29*** N/M N/M 9.89*** 10.46*** 9.50*** 11.87*** N/M N/M 9.50***
BB+
AA AA− A+
AA−
A+
A
A−
BBB+
BBB
BBB−
9.50***
9.50***
N/M
11.00***
9.50***
N/M
10.70***
3.30
9.50***
9.50***
9.50***
9.50***
N/M
N/M
6.49** 8.13*** 17.81*** 6.47** 5.07* 7.79** 7.06** 9.50***
4.45* 11.68*** 15.99*** 13.86*** 4.25* 4.28*
10.56*** 15.10*** 12.54*** 5.81*** 18.86*** 8.96*** 3.65 5.88** 3.89 9.54*** 13.48*** 9.50*** 5.52** 9.50*** 13.67*** 9.50*** 10.76*** 11.37*** N/M 9.50***
N/M 10.53*** 9.50*** 11.91*** 11.17*** 6.63** 15.45*** 8.61*** 6.55** 4.67* 9.02*** 4.08 5.64** 16.70*** ** 6.67 10.33*** 12.85*** 15.06*** 9.25*** 6.91** 11.81*** 16.57*** 4.16* 6.98** 12.41*** 3.90
BB+
BB
BB−
B+
B
B−
CCC+
CCC
CCC−
CC
C
10.77***
N/M
N/M
9.50***
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
10.61*** 9.50*** 12.39*** 10.74*** 11.12*** 12.2*** 4.19* 12.37*** 5.81** 5.34** 8.99***
N/M
9.50*** N/M N/M 14.27*** 10.47*** 9.50*** N/M
N/M
N/M
N/M
CCC−
N/M
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
CC
N/M
N/M
N/M
N/M
8.50***
−8.50
N/M
N/M
7.4** 17.07*** 10.34*** 8.18*** 3.89 12.00*** 11.12*** 18.91*** 58.42*** 3.24 8.58*** 9.91*** 16.90*** 14.32*** 10.43*** 15.17*** N/M 12.94*** 13.79*** 10.41*** 9.50*** N/M 9.50*** N/M 10.77*** N/M N/M 10.36*** N/M N/M 10.77***
C
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
7.40*** 5.36** 9.50***
D
N/M
N/M
N/M
N/M
9.50***
9.50***
9.50***
9.50***
7.19**
9.50***
4.63*
3.30*
6.63**
6.39**
A A− BBB+ BBB
N/M
N/M
N/M
N/M
1.99
BB
9.50***
N/M
N/M
N/M
9.50***
BB−
9.50***
B+
N/M
N/M
N/M
B
N/M
N/M
N/M
N/M
N/M
N/M 9.50*** 10.48***
9.50*** 7.90*** 4.30*
B−
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
CCC+
N/M
N/M
N/M
N/M
N/M
N/M
8.50***
−8.50
CCC
N/M
N/M
8.50***
−8.50
N/M
N/M
N/M
N/M
10.56*** 4.31* 16.92*** 13.54***
N/M 10.52*** 9.50*** 11.96*** 16.57*** 13.72*** 195.21*** 9.75*** 4.05 6.51** 9.84*** 3.43 9.16*** 5.68** ** 5.40 12.28*** 6.50** 3.74 5.06* 12.59*** *** 15.09 5.06* 9.57*** 5.16* 13.12*** 15.02*** 10.73*** 5.63** 5.57** 3.68
2.35 8.60*** 4.18* 2.70
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M 13.59*** 9.50*** 12.26*** 9.50*** 13.97*** 5.34** 9.68*** 6.08** 16.74***
N/M
N/M
N/M
N/M
9.50***
N/M
N/M
9.50*** 10.55*** 10.83*** 5.12* 8.31*** 10.62*** 1.35 12.98*** 18.74*** 7.54** 11.28*** 7.69** 13.62*** 9.38*** 8.69*** 3.87 9.50*** N/M 5.37**
327.22***
9.50*** 9.50*** 9.50***
9.50*** N/M 10.71*** N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M 10.74*** 12.04*** 9.50*** 9.77*** 10.60*** 2.25 8.51*** 6.94** 7.07** 2.42 10.42*** 17.55*** 10.33*** 17.04*** 21.85*** 2.83 5.19** 7.68** 6.33** 9.50*** 7.95***
9.50*** 9.50*** 10.58*** 7.25*** 9.73*** 4.14* 7.65** 10.83*** 13.60*** 7.21** 9.50***
N/M
N/M
N/M
N/M
3.57
7.94***
2.24
5.99** 8.38*** 4.44* 9.52*** 6.61** 4.46* 6.64** 6.60**
N/M N/M 0.00 N/M 0.00 0.00 N/M N/M
N/M
0.00
4.88*
8.50***
Notes. The values of the statistics Φ1 of the Dickey–Fuller likelihood ratio test, for a random walk with zero drift. Entries classified as not meaningful (N/M) represent entries with zero observations in the transition matrix, throughout the whole time frame, from January 1985 to December 2004. The likelihood ratio test rejects the null hypothesis for larger values of Φ1 , relative to critical values from Dickey and Fuller (1981) tables. This table also presents the significance levels for each Φ1 value, after comparing the entry values to the critical values from Dickey and Fuller (1981) tables. The statistical levels are classified as:
D. Parnes / Finance Research Letters 4 (2007) 217–226
AAA
Dickey– Fuller test
* 0.90. ** 0.975 or 0.95. *** 0.99.
223
224
Table 3 Linear and nonlinear time-series patterns and serial correlation test results AAA AA+ AA
AA−
A+
AAA AA+ AA AA− A+ A A− BBB+ BBB BBB− BB+ BB BB− B+ B B− CCC+ CCC CCC− CC C
AG A AG AG A
A ABV AV AG AV A AG A AG
A A A A AGB AB A AD A A
A A
AG AG AD A
AG A A A
AG A
AG
AGB A A AC ADV A AG AG AG AG
AC AC
A
A A AGB AG AG AGB A
AD AD A A
A−
BBB+
AG AG A AV AG ADV ADV ABD AV ABD A A A A A
A A A AGV AD AG A AGB AGBD AV AG A AG A
A
BBB
A A AV A A A AGBD ABD AG A A
A A
A
BBB−
BB+
AG
A
A AG
AG
A A AD AGV AB AG AB A AG AG A
A
D
BB−
B+
B
B−
CCC+
CCC CCC− CC
C
ACG A AG
A A A AD AG ACG ABD A AGB A A A A
A
BB
A
AG
A
A
AV
A
A
AG
A
A A AG
A AG AG A A
AB ABDV ABDV AG AD AB AD A AG A
ABV ADV AB AB AD A ABD
AG A A AG AD AG A AV ABD AB AD A A
AC A A A AG A AGB AG ABDV AB
A AG A
A ACG AG A AD A AD AG AB ADV AG
AB ADV A
AB AD AB AV A
AV
AB
AD
AGB
ACG A A AC AG ABD AG ACG AGV AG AG A AG AGV AGBDV AG A A
AG A
AC AG A AD AG AG AG AG AG AG ACG
AG AGD AG AGD A AV AGB AG
A
ABD A
#UU #UD Ratio: UU/UD Sign-test
0 0 N/M N/M
0 0 N/M N/M
0 0 N/M N/M
0 0 N/M N/M
0 0 N/M N/M
0 1 0.00 −2.00
0 2 0.00 −2.00
2 0 N/M 2.83
2 1 2.00 1.15
5 3 1.67 1.41
3 0 N/M 3.46
6 1 6.00 3.78***
6 1 6.00 3.78***
10 6 1.67 2.00*
6 2 3.00 2.83**
5 4 1.25 0.67
0 1 0.00 −2.00
4 1 4.00 2.68*
0 0 N/M N/M
3 2 1.50 0.89
0 0 N/M N/M
#DD #DU Ratio: DD/DU Sign-test
0 0 N/M N/M
3 0 N/M 3.46
15 0 N/M 7.75***
7 0 N/M 5.29***
17 4 4.25 5.67***
35 7 5.00 8.64***
17 0 N/M 8.25***
48 9 5.33 10.33***
47 0 N/M 13.71***
45 2 22.50 12.54***
33 1 33.00 10.98***
52 7 7.43 11.72***
61 7 8.71 13.10***
113 17 6.65 16.84*
68 19 3.58 10.51**
62 11 5.64 11.94***
24 10 2.40 4.80***
10 8 1.25 0.94
2 7 0.29 −3.33
0 4 0.00 −4.00
0 0 N/M N/M
Notes. The table identifies for each of the entries in the survival matrix the existence of a time series pattern of AR(1) type (marked as ‘A’), ARCH(1) effect (marked as ‘C’), GARCH(1, 1) effect (marked as ‘G’), QGARCH(1, 1) relation (marked as ‘Q’), statistically significant relation to changes in the GDP (marked as ‘D’), robust association to the business cycle as defined by the NBER (marked as ‘B’), or substantial correlation to changes in the VIX index (marked as ‘V’). Each entry is examined through Eq. (1) along with the complementary nonlinear models, Eq. (6), and Eq. (7). The lower panel counts the number of two consecutive upgrades (‘UU’), downgrades (‘DD’), an upgrade followed by a downgrade (‘UD’), or a downgrade followed by an upgrade (‘DU’) rating transition initiated at different credit categories. The ratios and the sign-tests are computed similar to Altman and Kao (1992a) to authenticate the existence of positive or negative serial correlations. * Significance at the 0.10 level or better. ** Significance at the 0.05 level or better. *** Significance at the 0.01 level or better.
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Regression analyses
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Fig. 1. Comparative analysis of the cumulative default rates (CDR) derived from the homogeneous Markov chain, the proposed internal correlations model, and the actual CDR. We perform a back-testing where the two techniques are calibrated over the first ten-year period, from 1985 to 1994. Then, we run a Monte Carlo simulation for both models over the next ten-year period, from 1995 to 2004, and compare the expected CDR with the actual observed CDR within the later time frame.
Nonlinear regression techniques assist to identify these time-related patterns, where in most cases time-varying volatility does not cluster, but volatility is occasionally auto-correlated through time. The economic structure underlying the time dependency often reveals relations to changes in GDP, business cycles, and market risk. In addition, we discover significant positive autocorrelation primarily among rating downgrades, hence a credit downgrade is likely to be followed by a subsequent downgrade. Acknowledgments I thank an unknown referee of the Financial Research Letters journal for helpful comments and suggestions throughout this study. References Altman, E.I., Kao, D.L., 1992a. Rating drift in high yield bonds. The Journal of Fixed Income, 15–20 (March). Altman, E.I., Kao, D.L., 1992b. The implications of corporate bond ratings drift. Financial Analysts Journal 48, 64–75. Ang, A., Hodrick, R.J., Xing, Y., Zhang, X., 2006. The cross-section of volatility and expected returns. The Journal of Finance 61, 259–299. Bahar, R., Nagpal, K., 2001. Dynamics of rating transition. Algo Research Quarterly 4, 71–92. Bangia, A., Diebold, F.X., Kronimus, A., Schagen, C., Schuermann, T., 2002. Ratings migration and the business cycle, with applications to credit portfolio stress testing. Journal of Banking and Finance 26, 445–474. Collin-Dufresne, P., Goldstein, R.S., Martin, J.S., 2001. The determinants of credit spread changes. The Journal of Finance 56, 2177–2207. Connolly, R., Stivers, C., Sun, L., 2005. Stock market uncertainty and the stock–bond return relation. Journal of Financial and Quantitative Analysis 40, 161–194. Corrado, C.J., Miller, T.W., 2006. Estimating expected excess returns using historical and option-implied volatility. The Journal of Financial Research 29, 95–112. Dickey, D.A., Fuller, W.A., 1981. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072. Gupton, G.M., Finger, C.C., Bhatia, M., 1997. CreditMetricsTM —Technical Document. J.P. Morgan, New York. Huang, J.Z., Kong, W., 2003. Explaining credit spread changes: Some new evidence from option-adjusted spreads of bond indexes. Journal of Derivatives 11, 30–44.
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Israel, R.B., Rosenthal, J.S., Wei, J.Z., 2001. Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings. Mathematical Finance 11, 245–265. Jarrow, R.A., Lando, D., Turnbull, S.M., 1997. A Markov model for the term structure of credit risk spreads. The Review of Financial Studies 10, 481–523. Parnes, D., 2007. A density dependent model for credit ratings migration dynamics. The Journal of Fixed Income 17, 26–37.
Time series patterns in credit ratings Dror Parnes ∗ Finance Department, College of Business, BSN 3127, University of South Florida, 4202 E. Fowler Ave., Tampa, FL 33620-5500, USA Received 29 May 2007; accepted 19 September 2007 Available online 22 September 2007
Abstract This article offers a substitute setting to simulate credit rating migrations. The internal correlations model tracks time-series movements within credit rating entries, rather than cross-ratings correlations. The proposed nonhomogeneous process is authenticated through the likelihood ratio Dickey–Fuller test, and is found to be statistically and economically significant, by better fitting observed cumulative default rates. Several nonlinear regression models assist to better identify these time-related patterns. The economic structure underlying the time dependency often corresponds to changes in GDP, business cycles, and market risk. Furthermore, significant positive autocorrelation is detected mostly among downgrade probabilities. © 2007 Elsevier Inc. All rights reserved. JEL classification: G21; G24; G32; G33 Keywords: Credit ratings; Time-series patterns; Dickey–Fuller test; Homogeneous Markov chain; Serial correlation
1. Introduction Developments in credit ratings commonly reveal an improvement or deterioration in the credit-worthiness of a firm. Although rating agencies primarily meant to utilize credit ratings to signify the current credit quality, several theories use the rating transitions phenomenon to forecast default events, or to price risky bonds. Academicians and practitioners often use the homogeneous Markov chain, first proposed by Jarrow et al. (1997), to describe the dynamics of credit ratings. Yet, in reality, this process does not correspond to actual yield curves, as well as violates several observed dependencies. * Fax: +1 813 974 3084.
E-mail address: [email protected]. 1544-6123/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.frl.2007.09.001
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So far, few attempts have been made to step away from the homogeneity assumption of the Markov chain transition matrix. The CreditMetricsTM methodology offered of Gupton et al. (1997) examines cross-ratings correlations. The momentum scheme proposed by Bahar and Nagpal (2001) considers lagged rating transitions. The generator matrix investigated by Israel et al. (2001) offers procedures to simulate further transition matrices. Bangia et al. (2002) distinguish between transition matrices depending on the business cycle, and the density-dependent model of Parnes (2007) associates firm’s survivability and transitivity to the market-density. These nonhomogeneous models consider the credit ratings migration not to be a memory-less process any more. This article offers a different approach to track rating transitions while accounting for timeseries movements within credit entries, rather than cross-rating dependencies. We substantiate this nonhomogeneous dynamic through a Dickey–Fuller statistical test, associate it with market variables, and demonstrate its economic importance. The article proceeds as follows. Section 2 presents the alternative model. Section 3 discusses the data and methodologies. Section 4 explains the empirical findings and their economic meaning, and Section 5 concludes. 2. Time series patterns in credit ratings An evidence for time series correlation in credit ratings migration can be found in Altman and Kao (1992a, 1992b). The authors divide the examined time frame into two sub-periods, from 1970 to 1979, and from 1980 to 1985, and reveal a tendency for a downgrade in rating to be followed by a second downgrade, indicating on a serial correlation when the initial rating changes was a downgrade. However, weak autocorrelation is found when the first change is an upgrade. The following model makes an attempt to track trends in transition probabilities over time, and to generate future transition probabilities that follow the same affine trend. This scheme explicitly assumes that each transition probability is univariate autoregressive correlated with previous transition probabilities within the same entry in the credit ratings matrix.1 By that, the model implicitly assumes that transition probabilities are cross-correlated with other transition probabilities within the same column. The cross correlation evolves from the fact that the sum of entries within each column always remains one. Thus, when a specific entry in the transition matrix tends to increase (decrease), other entries in the same column tend to decrease (increase). The model estimates AR(1) time-series regressions for each entry in the transition matrix as sij (t + 1) = αij + βij sij (t) + εij (t)
∀i, j,
(1)
where entry sij (t) denotes the transition probability from state j to state i at time t in the survival sub-matrix S. The β coefficient of the regression equation determines whether a transition probability is stationary. Transition probabilities must remain in their domain, thus, unless α is close to 1 and β is positive, or α is close to 0 and β is negative, the model requires no constrains.2 1 Describing the life cycle of a company requires the distinguishing between different survival states and an absorbing
state of default. Transition matrix T is a nonnegative square matrix of dimensions 22 × 22, where its entries represent probabilities of all credit ratings migration. Matrix T contains a survival sub-matrix S of dimensions 21×21, representing transitions among livelihood states, an absorbing row vector D of dimensions 1 × 21, representing transitions from livelihood states into the default state, and an arbitrary column vector of zeros and one. 2 The empirical investigation has found no evidence for these extreme situations.
D. Parnes / Finance Research Letters 4 (2007) 217–226
219
To better identify the nature of time-related patterns we further explore several nonlinear techniques. To examine whether an entry in the survival matrix display time-varying volatility clustering, i.e. periods of quiet and periods of fluctuating residuals in Eq. (1), we test for AutoRegressive Conditional Heteroskedasticity (ARCH(1)) effect. To further inspect whether lagged volatility is as important as lagged residuals, we test for Generalized ARCH (GARCH(1, 1)) effect. To investigate whether a time series pattern exhibit asymmetric reaction to positive and negative shocks, we test the Quadratic GARCH model (QGARCH(1, 1)). 3. Data and methodologies The Compustat database reports on 130,559 quarterly S&P long-term credit ratings for 4510 industrial companies from 1985 to 2004. Each credit rating is tagged by a number between 2 for ‘AAA’ and 27 representing a bankruptcy filing. Within this period, there are 123,849 valid credit rating transitions in consecutive quarters. These transitions are collected and assigned to a frequency matrix.3 Since there are 21 states in the life cycle of a company and an additional absorbing state of default, the frequency matrix is of dimensions 22 × 21, where each entry represents the number of credit ratings migrations observed from one state to another. Several observations are reported as ‘not rated’ (NR). These observations are classified as 1, 3, 22, 25, 28, 29 and 90 in the database. This study considers transitions to and from NR status as missing information, by eliminating them from the data sample.4 We first construct homogeneous transition matrices for each year over the examined period, from 1985 to 2004. These twenty different tables are used to explore the time series dependencies within each of the 22 ×21 = 462 entries. Next we conduct the autoregressive time-series analyses for each entry in the transition matrix as illustrated in Eq. (1). To confirm whether the transition matrix entries follow some trends over time, and by that to validate the time-series internal correlations model, the Dickey–Fuller test is undertaken. Dickey and Fuller (1981) present a likelihood ratio statistical test for autoregressive time-series regression, to examine whether the dynamic follows a random walk with zero drift, or some trends occur. The likelihood ratio null hypothesis admits the coefficients: H0 : αij = 0; βij = 1, against the alternative hypothesis H1 : not H0 . The likelihood ratio statistic is n−1 2 Φ1 , 1+ (2) 2(n − 3) where n is the number of intervals in the time-series analysis, and the statistics 2 −1 2 Φ1 = 2Seμ (n − 1)σ02 − (n − 3)Seμ , 2 Seμ = (n − 3)−1
σ02 = (n − 1)−1
n
t=2 n
2 sij (t) − αˆ ij − βˆij sij (t − 1) ,
2 sij (t) − sij (t − 1) .
(3) (4)
(5)
t=2
3 We observe merely few transitions out of the default state back to the survival sub-matrix. Therefore, the assertion of
an absorption state of default is rational. 4 Bangia et al. (2002) discuss other alternatives to treat NR observations, but ultimately choose to exclude them as well.
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D. Parnes / Finance Research Letters 4 (2007) 217–226
2 and σ 2 are consistent with Dickey and Fuller (1981), and s (t) denotes The notations Φ1 , Seμ ij 0 the survival transition probabilities at time t . The likelihood ratio test rejects the null hypothesis for larger values of Φ1 , relative to critical values from Dickey–Fuller tables.5 To better identify the economic structure of time dependency, we examine the relations between time-related changes in transition probabilities within the survival matrix and changes in three relevant macroeconomic variables.6 The first two factors signify general business conditions. These are the scaled Gross Domestic Product (GDP) in billions of chained 2000 dollars, taken from the Bureau of Economic Analysis (BEA) web site, and the National Bureau of Economic Research (NBER) own definition for contractionary and expansionary stages.7 More formally, we regress
sij (t) = υij + μij GDP(t) + γij NBER(t) + ωij (t) 0 if contractionary cycle, where NBER(t) = 1 if expansionary cycle.
∀i, j, (6)
The third macroeconomic variable is the volatility index (VIX) of the Chicago Board Options Exchange (CBOE). The VIX is meant to depict the market’s expectation of thirty-day volatility.8 It is constructed through a simple average of implied volatilities of call and put options on a large sample of shares included in the S&P 100 index until 2003, and in the S&P 500 index after that. Since its first appearance, the VIX is widely used as a measure of market risk.9 We aim to explore the link between changes in market risk and changes in credit-rating transition probabilities over time, thus we examine the following relation10 : sij (t) = δij + λij VIX(t) + ηij (t)
∀i, j after 1993.
(7)
The economic importance of the proposed internal correlations model is examined through a comparative analysis with the commonly used Markov chain dynamic and the actual cumulative default rate (CDR). We perform a back-testing where the two techniques are calibrated over the first ten-year period, from 1985 to 1994. We then run a Monte Carlo simulation for both models over the next ten-year period, from 1995 to 2004, and compare these expected CDR with observed ones in the later time frame. We further examine serial correlations among upgrade and downgrade probabilities according to Altman and Kao (1992a) sign-test methodology. We also generate the complete homogeneous transition matrix over the entire period, from the beginning of 1985 to the end of 2004, by dividing each entry in the frequency matrix by the sum of its column, as reported in Table 1. 5 The Dickey–Fuller tables are described in Dickey and Fuller (1981, p. 1063). 6 Since this analysis is time dependent, we are required to maintain a stationary dataset, and to eliminate possible
tendencies. We therefore compute changes in transition probabilities. 7 The NBER expansionary and contractionary quarters are constructed by examining a larger number of business indicators in addition to the GDP. 8 Since the VIX is originally formed in 1993, in this specific test we ought to truncate the sample of transition probabilities from 1993 to 2004. 9 For example, see Connolly et al. (2005), Ang et al. (2006), and Corrado and Miller (2006). 10 Collin-Dufresne et al. (2001) and Huang Kong (2003) investigate the association between changes in the VIX as a proxy for market risk and changes in bankruptcy risk quantities.
AA
AA−
A+
A
A−
BBB+ BBB
BBB− BB+
BB
BB−
B+
B
B−
CCC+ CCC
CCC− CC
C
0.16 0.29 95.27 3.00 0.65 0.35 0.08 0.12 0.06 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.02 0.01 0.00 0.04 0.01 0.00 0.02 0.03 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.01 0.02 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.79 0.06 0.04 0.04 0.00 0.02 0.01 0.00 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 94.49 0.82 0.09 0.06 0.01 0.03 0.06 0.03 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.34 94.98 1.19 0.15 0.15 0.05 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.07 0.00 0.00 0.00 0.00 0.81 2.77 95.04 1.81 0.33 0.12 0.07 0.07 0.00 0.02 0.02 0.00 0.00 0.00 0.12 0.00 0.00 0.00 0.25 0.85 2.32 93.71 1.72 0.43 0.17 0.10 0.01 0.04 0.02 0.08 0.00 0.00 0.12 0.00 0.00 0.00 0.13 0.24 0.78 2.80 93.45 1.83 0.47 0.12 0.08 0.04 0.07 0.02 0.04 0.00 0.12 0.00 0.00 0.00 0.04 0.08 0.27 1.12 3.06 94.04 2.52 0.89 0.32 0.12 0.02 0.07 0.00 0.07 0.00 0.00 0.00 0.00 0.04 0.01 0.05 0.12 0.77 2.32 92.90 3.40 0.86 0.16 0.10 0.02 0.07 0.15 0.00 0.00 0.00 0.00 0.00 0.02 0.07 0.01 0.14 0.57 2.03 90.86 2.39 0.66 0.13 0.07 0.07 0.00 0.23 0.24 0.00 0.00 0.00 0.02 0.04 0.03 0.22 0.22 1.20 2.44 91.43 2.20 0.54 0.15 0.14 0.22 0.12 0.24 0.00 0.00 0.00 0.01 0.04 0.01 0.03 0.10 0.33 1.26 2.99 92.21 1.94 0.42 0.29 0.29 0.23 0.72 0.93 0.00 0.02 0.04 0.02 0.04 0.04 0.16 0.09 0.37 1.17 2.98 92.85 2.69 0.98 1.11 0.58 0.96 1.86 0.00 0.05 0.04 0.00 0.01 0.03 0.03 0.07 0.16 0.32 0.76 2.57 89.82 2.17 1.47 1.05 0.72 0.62 10.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.07 0.16 0.29 0.86 2.54 86.64 2.36 1.40 0.96 2.79 0.00 0.00 0.00 0.00 0.00 1.01 0.02 0.01 0.00 0.05 0.16 0.31 1.64 3.03 82.96 1.63 1.20 1.24 10.00 0.00 0.01 0.00 0.00 0.00 0.01 0.00 0.03 0.07 0.10 0.22 0.66 2.46 2.51 82.09 2.16 2.17 10.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.03 0.04 0.07 0.05 0.40 1.05 2.58 1.63 81.29 1.55 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.03 0.05 0.04 0.08 0.43 1.12 1.99 2.09 3.36 73.68 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.00 0.07 0.47 0.24 0.00 50.00
D. Parnes / Finance Research Letters 4 (2007) 217–226
Table 1 The transition matrix Transition AAA AA+ matrix (%) AAA 98.04 0.83 1.05 94.36 AA+ AA 0.32 3.71 AA− 0.32 0.76 0.11 0.14 A+ A 0.00 0.14 0.04 0.00 A− BBB+ 0.00 0.00 0.07 0.07 BBB BBB− 0.00 0.00 0.00 0.00 BB+ BB 0.04 0.00 0.04 0.00 BB− B+ 0.00 0.00 0.00 0.00 B B− 0.00 0.00 0.00 0.00 CCC+ CCC 0.00 0.00 0.00 0.00 CCC− CC 0.00 0.00 0.00 0.00 C
0.00 0.00 0.00 0.00 0.02 0.01 0.01 0.01 0.05 0.01 0.09 0.03 0.13 0.16 0.96 1.95 4.13 8.14 7.91 15.17 20.00 D Notes. Nonarbitrary part of the transition table as calculated directly from the frequency matrix. The upper sub-matrix is the survival part S, and the lower row vector is the absorbing default state D. Entries inside the matrix stated as percentages. Each column sums to 100%. Like the frequency matrix, this matrix is of dimensions 22 × 21.
221
222
D. Parnes / Finance Research Letters 4 (2007) 217–226
4. Results This section reports the empirical findings and summarizes their economic meaning. Table 2 presents the values of Φ1 , derived from the likelihood ratio Dickey–Fuller tests, for all the entries in the transition matrix along with the statistical significance levels of each entry, after comparing the entry values to the critical values from Dickey–Fuller tables. Out of the 462 possible entries in the transition matrix, 168 are classified as not meaningful (N/M), since no transitions were observed throughout the whole time frame, from 1985 to 2004. Among the remaining 294 meaningful entries, 194 entries, or 66.0%, reject the null hypothesis with a significance level of 0.99, 29 entries, or 9.9%, reject the null hypothesis with a significance level of 0.975, 18 entries, or 6.1%, reject the null hypothesis with a significance level of 0.95, and 21 entries, or 7.1%, reject the null hypothesis with a significance level of 0.90. Only 32 entries, or 10.9%, cannot reject the null hypothesis of a random walk with zero drift. These findings assess the claim that most transition probabilities follow trends over time, and thus, validate the internal correlations model. Table 3 identifies the linear and the nonlinear time-series patterns for each entry in the survival matrix S. In addition, this table designates the economic structure of these dependencies; whether time variability in each entry is associated to changes in the GDP, to the business cycle as defined by the NBER, or to the VIX as a proxy for market risk. Out of the 262 entries that follow time-series patterns of type AR(1), only 11 exhibit ARCH(1) effect, indicating on almost no time volatility clustering. Yet, 87 entries demonstrate GARCH(1, 1) relation, showing on some volatility autocorrelation. None of the entries prompt QGARCH(1, 1), suggesting on similar response to positive and negative shocks. In addition, economically and statistically significant correlations are found between 39 of the entries and changes in the GDP, between 39 of the (same or other) entries and business cycles as defined by the NBER, and between 26 of the (same or other) entries and developments in market risk as captured by the VIX. The lower panel of Table 3 presents the serial correlation test results. We identify 849 rating transitions in consecutive quarters from 1985 to 2004, and classify them as ‘up–up,’ ‘up–down,’ ‘down–up,’ and ‘down–down’ movements originated at different credit ratings. We recognize robust positive serial correlation between subsequent downgrades, but sporadic positive autocorrelation among succeeding rating upgrades. Figure 1 demonstrates a comparative analysis of the expected CDR, as derived from the homogeneous Markov chain and from the internal correlations model, as well as the actual CDR observed from 1995 to 2004. The results reveal that the existing Markov chain scheme overestimates the credit ratings migration probabilities while projecting much higher CDR than is perceived in practice. Furthermore, the proposed time-series approach improves the predictive strength by better fitting observed patterns. 5. Summary and conclusions This article offers an alternative approach to credit ratings migration analysis. The internal correlations model tracks time-series movements within credit rating entries. The proposed nonhomogeneous dynamic is validated through the likelihood ratio Dickey–Fuller test, and is found to be statistically and economically significant. It improves commonly used methodologies by better fitting observed cumulative default rates.
Table 2 Dickey–Fuller likelihood ratio test AA+
AAA
5.32**
8.91***
7.25**
AA+
8.68***
36.74***
21.13***
4.15* 21.24*** 11.92*** 358.19*** 11.13*** 9.73*** 10.76*** 10.76***
AA
BBB−
8.32*** 2.15 92.08*** 10.01*** 12.28*** 8.15*** 4.47* 10.57*** 5.43** N/M 3.34 6.43** 9.5*** N/M 10.29*** N/M N/M 9.89*** 10.46*** 9.50*** 11.87*** N/M N/M 9.50***
BB+
AA AA− A+
AA−
A+
A
A−
BBB+
BBB
BBB−
9.50***
9.50***
N/M
11.00***
9.50***
N/M
10.70***
3.30
9.50***
9.50***
9.50***
9.50***
N/M
N/M
6.49** 8.13*** 17.81*** 6.47** 5.07* 7.79** 7.06** 9.50***
4.45* 11.68*** 15.99*** 13.86*** 4.25* 4.28*
10.56*** 15.10*** 12.54*** 5.81*** 18.86*** 8.96*** 3.65 5.88** 3.89 9.54*** 13.48*** 9.50*** 5.52** 9.50*** 13.67*** 9.50*** 10.76*** 11.37*** N/M 9.50***
N/M 10.53*** 9.50*** 11.91*** 11.17*** 6.63** 15.45*** 8.61*** 6.55** 4.67* 9.02*** 4.08 5.64** 16.70*** ** 6.67 10.33*** 12.85*** 15.06*** 9.25*** 6.91** 11.81*** 16.57*** 4.16* 6.98** 12.41*** 3.90
BB+
BB
BB−
B+
B
B−
CCC+
CCC
CCC−
CC
C
10.77***
N/M
N/M
9.50***
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
10.61*** 9.50*** 12.39*** 10.74*** 11.12*** 12.2*** 4.19* 12.37*** 5.81** 5.34** 8.99***
N/M
9.50*** N/M N/M 14.27*** 10.47*** 9.50*** N/M
N/M
N/M
N/M
CCC−
N/M
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
CC
N/M
N/M
N/M
N/M
8.50***
−8.50
N/M
N/M
7.4** 17.07*** 10.34*** 8.18*** 3.89 12.00*** 11.12*** 18.91*** 58.42*** 3.24 8.58*** 9.91*** 16.90*** 14.32*** 10.43*** 15.17*** N/M 12.94*** 13.79*** 10.41*** 9.50*** N/M 9.50*** N/M 10.77*** N/M N/M 10.36*** N/M N/M 10.77***
C
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
7.40*** 5.36** 9.50***
D
N/M
N/M
N/M
N/M
9.50***
9.50***
9.50***
9.50***
7.19**
9.50***
4.63*
3.30*
6.63**
6.39**
A A− BBB+ BBB
N/M
N/M
N/M
N/M
1.99
BB
9.50***
N/M
N/M
N/M
9.50***
BB−
9.50***
B+
N/M
N/M
N/M
B
N/M
N/M
N/M
N/M
N/M
N/M 9.50*** 10.48***
9.50*** 7.90*** 4.30*
B−
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
CCC+
N/M
N/M
N/M
N/M
N/M
N/M
8.50***
−8.50
CCC
N/M
N/M
8.50***
−8.50
N/M
N/M
N/M
N/M
10.56*** 4.31* 16.92*** 13.54***
N/M 10.52*** 9.50*** 11.96*** 16.57*** 13.72*** 195.21*** 9.75*** 4.05 6.51** 9.84*** 3.43 9.16*** 5.68** ** 5.40 12.28*** 6.50** 3.74 5.06* 12.59*** *** 15.09 5.06* 9.57*** 5.16* 13.12*** 15.02*** 10.73*** 5.63** 5.57** 3.68
2.35 8.60*** 4.18* 2.70
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
9.50***
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M 13.59*** 9.50*** 12.26*** 9.50*** 13.97*** 5.34** 9.68*** 6.08** 16.74***
N/M
N/M
N/M
N/M
9.50***
N/M
N/M
9.50*** 10.55*** 10.83*** 5.12* 8.31*** 10.62*** 1.35 12.98*** 18.74*** 7.54** 11.28*** 7.69** 13.62*** 9.38*** 8.69*** 3.87 9.50*** N/M 5.37**
327.22***
9.50*** 9.50*** 9.50***
9.50*** N/M 10.71*** N/M
N/M
N/M
N/M
N/M
N/M
N/M
N/M 10.74*** 12.04*** 9.50*** 9.77*** 10.60*** 2.25 8.51*** 6.94** 7.07** 2.42 10.42*** 17.55*** 10.33*** 17.04*** 21.85*** 2.83 5.19** 7.68** 6.33** 9.50*** 7.95***
9.50*** 9.50*** 10.58*** 7.25*** 9.73*** 4.14* 7.65** 10.83*** 13.60*** 7.21** 9.50***
N/M
N/M
N/M
N/M
3.57
7.94***
2.24
5.99** 8.38*** 4.44* 9.52*** 6.61** 4.46* 6.64** 6.60**
N/M N/M 0.00 N/M 0.00 0.00 N/M N/M
N/M
0.00
4.88*
8.50***
Notes. The values of the statistics Φ1 of the Dickey–Fuller likelihood ratio test, for a random walk with zero drift. Entries classified as not meaningful (N/M) represent entries with zero observations in the transition matrix, throughout the whole time frame, from January 1985 to December 2004. The likelihood ratio test rejects the null hypothesis for larger values of Φ1 , relative to critical values from Dickey and Fuller (1981) tables. This table also presents the significance levels for each Φ1 value, after comparing the entry values to the critical values from Dickey and Fuller (1981) tables. The statistical levels are classified as:
D. Parnes / Finance Research Letters 4 (2007) 217–226
AAA
Dickey– Fuller test
* 0.90. ** 0.975 or 0.95. *** 0.99.
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224
Table 3 Linear and nonlinear time-series patterns and serial correlation test results AAA AA+ AA
AA−
A+
AAA AA+ AA AA− A+ A A− BBB+ BBB BBB− BB+ BB BB− B+ B B− CCC+ CCC CCC− CC C
AG A AG AG A
A ABV AV AG AV A AG A AG
A A A A AGB AB A AD A A
A A
AG AG AD A
AG A A A
AG A
AG
AGB A A AC ADV A AG AG AG AG
AC AC
A
A A AGB AG AG AGB A
AD AD A A
A−
BBB+
AG AG A AV AG ADV ADV ABD AV ABD A A A A A
A A A AGV AD AG A AGB AGBD AV AG A AG A
A
BBB
A A AV A A A AGBD ABD AG A A
A A
A
BBB−
BB+
AG
A
A AG
AG
A A AD AGV AB AG AB A AG AG A
A
D
BB−
B+
B
B−
CCC+
CCC CCC− CC
C
ACG A AG
A A A AD AG ACG ABD A AGB A A A A
A
BB
A
AG
A
A
AV
A
A
AG
A
A A AG
A AG AG A A
AB ABDV ABDV AG AD AB AD A AG A
ABV ADV AB AB AD A ABD
AG A A AG AD AG A AV ABD AB AD A A
AC A A A AG A AGB AG ABDV AB
A AG A
A ACG AG A AD A AD AG AB ADV AG
AB ADV A
AB AD AB AV A
AV
AB
AD
AGB
ACG A A AC AG ABD AG ACG AGV AG AG A AG AGV AGBDV AG A A
AG A
AC AG A AD AG AG AG AG AG AG ACG
AG AGD AG AGD A AV AGB AG
A
ABD A
#UU #UD Ratio: UU/UD Sign-test
0 0 N/M N/M
0 0 N/M N/M
0 0 N/M N/M
0 0 N/M N/M
0 0 N/M N/M
0 1 0.00 −2.00
0 2 0.00 −2.00
2 0 N/M 2.83
2 1 2.00 1.15
5 3 1.67 1.41
3 0 N/M 3.46
6 1 6.00 3.78***
6 1 6.00 3.78***
10 6 1.67 2.00*
6 2 3.00 2.83**
5 4 1.25 0.67
0 1 0.00 −2.00
4 1 4.00 2.68*
0 0 N/M N/M
3 2 1.50 0.89
0 0 N/M N/M
#DD #DU Ratio: DD/DU Sign-test
0 0 N/M N/M
3 0 N/M 3.46
15 0 N/M 7.75***
7 0 N/M 5.29***
17 4 4.25 5.67***
35 7 5.00 8.64***
17 0 N/M 8.25***
48 9 5.33 10.33***
47 0 N/M 13.71***
45 2 22.50 12.54***
33 1 33.00 10.98***
52 7 7.43 11.72***
61 7 8.71 13.10***
113 17 6.65 16.84*
68 19 3.58 10.51**
62 11 5.64 11.94***
24 10 2.40 4.80***
10 8 1.25 0.94
2 7 0.29 −3.33
0 4 0.00 −4.00
0 0 N/M N/M
Notes. The table identifies for each of the entries in the survival matrix the existence of a time series pattern of AR(1) type (marked as ‘A’), ARCH(1) effect (marked as ‘C’), GARCH(1, 1) effect (marked as ‘G’), QGARCH(1, 1) relation (marked as ‘Q’), statistically significant relation to changes in the GDP (marked as ‘D’), robust association to the business cycle as defined by the NBER (marked as ‘B’), or substantial correlation to changes in the VIX index (marked as ‘V’). Each entry is examined through Eq. (1) along with the complementary nonlinear models, Eq. (6), and Eq. (7). The lower panel counts the number of two consecutive upgrades (‘UU’), downgrades (‘DD’), an upgrade followed by a downgrade (‘UD’), or a downgrade followed by an upgrade (‘DU’) rating transition initiated at different credit categories. The ratios and the sign-tests are computed similar to Altman and Kao (1992a) to authenticate the existence of positive or negative serial correlations. * Significance at the 0.10 level or better. ** Significance at the 0.05 level or better. *** Significance at the 0.01 level or better.
D. Parnes / Finance Research Letters 4 (2007) 217–226
Regression analyses
D. Parnes / Finance Research Letters 4 (2007) 217–226
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Fig. 1. Comparative analysis of the cumulative default rates (CDR) derived from the homogeneous Markov chain, the proposed internal correlations model, and the actual CDR. We perform a back-testing where the two techniques are calibrated over the first ten-year period, from 1985 to 1994. Then, we run a Monte Carlo simulation for both models over the next ten-year period, from 1995 to 2004, and compare the expected CDR with the actual observed CDR within the later time frame.
Nonlinear regression techniques assist to identify these time-related patterns, where in most cases time-varying volatility does not cluster, but volatility is occasionally auto-correlated through time. The economic structure underlying the time dependency often reveals relations to changes in GDP, business cycles, and market risk. In addition, we discover significant positive autocorrelation primarily among rating downgrades, hence a credit downgrade is likely to be followed by a subsequent downgrade. Acknowledgments I thank an unknown referee of the Financial Research Letters journal for helpful comments and suggestions throughout this study. References Altman, E.I., Kao, D.L., 1992a. Rating drift in high yield bonds. The Journal of Fixed Income, 15–20 (March). Altman, E.I., Kao, D.L., 1992b. The implications of corporate bond ratings drift. Financial Analysts Journal 48, 64–75. Ang, A., Hodrick, R.J., Xing, Y., Zhang, X., 2006. The cross-section of volatility and expected returns. The Journal of Finance 61, 259–299. Bahar, R., Nagpal, K., 2001. Dynamics of rating transition. Algo Research Quarterly 4, 71–92. Bangia, A., Diebold, F.X., Kronimus, A., Schagen, C., Schuermann, T., 2002. Ratings migration and the business cycle, with applications to credit portfolio stress testing. Journal of Banking and Finance 26, 445–474. Collin-Dufresne, P., Goldstein, R.S., Martin, J.S., 2001. The determinants of credit spread changes. The Journal of Finance 56, 2177–2207. Connolly, R., Stivers, C., Sun, L., 2005. Stock market uncertainty and the stock–bond return relation. Journal of Financial and Quantitative Analysis 40, 161–194. Corrado, C.J., Miller, T.W., 2006. Estimating expected excess returns using historical and option-implied volatility. The Journal of Financial Research 29, 95–112. Dickey, D.A., Fuller, W.A., 1981. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072. Gupton, G.M., Finger, C.C., Bhatia, M., 1997. CreditMetricsTM —Technical Document. J.P. Morgan, New York. Huang, J.Z., Kong, W., 2003. Explaining credit spread changes: Some new evidence from option-adjusted spreads of bond indexes. Journal of Derivatives 11, 30–44.
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Israel, R.B., Rosenthal, J.S., Wei, J.Z., 2001. Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings. Mathematical Finance 11, 245–265. Jarrow, R.A., Lando, D., Turnbull, S.M., 1997. A Markov model for the term structure of credit risk spreads. The Review of Financial Studies 10, 481–523. Parnes, D., 2007. A density dependent model for credit ratings migration dynamics. The Journal of Fixed Income 17, 26–37.