Chapter 6. Stability of Credit Migrations

This chapter is dedicated to the examination of the stability of rating migration with the focus on credit transition matrices. After a first glance at rating behavior through the business cycle, we will provide tests for two major assumptions that are often made about transition matrices: time homogeneity and Markov behavior. Generally, both assumptions should be treated with care. Several studies have shown that migration matrices are not homogeneous through time and that also the assumption of first-order Markov behavior is rather questionable; see, e.g., Bangia et al. (2002), Jafry and Schuermann (2004), Krüger et al (2005), Nickell et al (2000), Weber et al. (1998). As a major reason for this, many authors name the influence of macroeconomic variables and their effects on migration behavior (Nickell et al., 2000; Trueck, 2008; Wei, 2003). However, such business cycle effects might have a substantial influence on homogeneity of migration behavior, but they do not implicitly contradict the idea of Markov behavior. For some theoretical explanations of non-Markov behavior in credit migrations, see, e.g., Löffler (2004, 2005). For a framework on stochastic migration matrices and a study on serially correlated rating transitions in the French market, we refer to Gagliardini and Gourieroux (2005a, b). Overall, while the assumption of first-order Markov behavior for credit migrations might be a simplification of the real world, departing from this assumption makes the modeling, estimation, or simulation of rating transitions much more complicated. In this chapter we will review methods that can be used to investigate time homogeneity and Markov behavior of credit migration matrices. They might be helpful to examine deviations from these properties but can also be used to compare different rating systems with respect to their migration behavior. Several of the issues raised will also be illustrated using empirical examples of an internal rating system as well as a history of Moody’s yearly credit migration matrices.

6.1. Credit Migrations and the Business Cycle

This section tries to give a first glance at the link between the current state of the economy and default risk or migration behavior of a company. Intuition gives the following view: when the economy worsens, both downgrades as well as defaults will increase. The contrary should be true when the economy becomes stronger. Figures 6.1 and 6.2 show Moody’s historical default frequencies for noninvestment grade bonds of rating class CCC and B for the years 1984 to 2001. Clearly there is a high deviation from the average. For CCC-rated bonds the default frequencies range from 5% in 1996 in high market times to more than 45% in 2001, when there was a deep recession in the American economy. We conclude that taking average default probabilites of a longer time horizon as estimators for future default probabilities might not give correct risk estimates for a portfolio.

FIGURE 6.1. Moody’s historical default rates for rating class B and time horizon 1984–2001.

FIGURE 6.2. Moody’s historical defaults rates for rating class Caa and time horizon 1984–2001.

Note that also the second major determinant of credit risk, the recovery rate, shows large variations through time. Figure 6.3 illustrates the issuer weighted recovery rates for corporate loans from 1982–2003 according to Moody’s KMV investor services. For further results on the investigation of the relation between default and recovery rates, see Altman and Kishore (1996), Altman et al. (2005), Schuermann (2004).

FIGURE 6.3. Issuer weighted recovery rates for corporate loans (1982–2003). Source: Moody’s KMV.

Since our focus is mainly on credit migration matrices, in Tables 6.1 and 6.2 we also provided information on Moody’s average one-year transition probabilities for unsecured long-term corporate and sovereign bond ratings for the case of business cycle trough and peak from 1970–1997. The data are taken from a study by Nickell et al. (2000) and show the clear tendency of higher downgrade probability for investment and speculative grade issues during recessions compared to expansions of the economy. Similar results were obtained by Bangia et al. (2002), who considered Standard & Poor’s (S&P) historical transition matrices from 1981 to 1998.

TABLE 6.1. Average One-Year Transition Probabilities for Unsecured Moody’s Long-Term Corporate and Sovereign Bond Ratings, Business Cycle Recession Source: Nickell et al (2000)

	Aaa	Aa	A	Baa	Ba	B	Caa	C	D
Aaa	89.60	10.00	0.40	0.00	0.00	0.00	0.00	0.00	0.00
Aa	0.90	88.30	10.70	0.10	0.00	0.00	0.00	0.00	0.00
A	0.10	2.70	91.10	5.60	0.40	0.20	0.00	0.10	0.00
Baa	0.00	0.30	6.60	86.80	5.60	0.40	0.20	0.00	0.10
Ba	0.00	0.10	0.50	5.90	83.10	8.40	0.30	0.00	1.70
B	0.00	0.10	0.20	0.80	6.60	79.60	2.20	1.00	9.40
Caa	0.00	0.00	0.00	0.90	1.90	9.30	63.00	1.90	23.10
C	0.00	0.00	0.00	0.00	0.00	5.90	5.90	64.70	23.50

TABLE 6.2. Average One-Year Transition Probabilities for Unsecured Moody’s Long-Term Corporate and Sovereign Bond Ratings, Business Cycle Peak Source: Nickell et al. (2000)

	Aaa	Aa	A	Baa	Ba	B	Caa	C	D
Aaa	92.20	7.40	0.30	0.00	0.10	0.00	0.00	0.00	0.00
Aa	1.50	87.50	10.10	0.70	0.20	0.00	0.00	0.00	0.00
A	0.10	1.80	91.70	5.40	0.80	0.20	0.00	0.10	0.00
Baa	0.10	0.20	5.20	88.10	4.90	1.20	0.00	0.00	0.20
Ba	0.10	0.00	0.30	5.40	85.70	6.70	0.20	0.00	1.50
B	0.10	0.10	0.40	0.80	6.60	83.60	1.60	0.30	6.60
Caa	0.00	0.00	0.00	0.00	2.80	9.30	59.80	8.40	19.60
C	0.00	0.00	0.00	0.00	0.00	8.30	8.30	70.80	12.50

The question arises when two migration matrices or rows of these matrices differ from each other significantly and when they differ from the unconditional average transition matrix. To examine this issue more thoroughly, we will provide tests to detect significant differences between transition matrices in the next section.

Considering assigned new ratings by S&P for the period from 1984–2001 in Table 6.3, we find that in years of economical contraction, as in 1990 or 1991, the lowest percentages of new issuers in speculative grade could be observed. This could be due to the fact that in these times fewer risky businesses emerge and issue debt. Further, in these years as well as during the economic downturn in the year 2001, a very high downgrade to upgrade ratio considering all issuers could be observed. So obviously, ratings themselves but also changes in these ratings, i.e., credit migrations, seem to exhibit different behavior through time and are dependent on the overall macroeconomic situation. Considering the coefficients of variation, a study by Bangia et al. (2002) finds that most of the coefficients of variation during an economic recession are much lower than for all years considered. According to the authors for the expansion matrix, the coefficients of variation are on average reduced by only 2% compared to the unconditional matrix, while the contraction matrix exhibits about 14% less volatility. Furthermore it is striking that many of the largest reductions in variation coefficients for the contraction matrix actually stem from elements on or close to the diagonal supporting the reliability of the results. Overall, these results could be a sign for the fact that migration probabilities are more stable in economic contractions than they are on average, supporting the existence of two distinct economic regimes. The difference in the matrices observed indicates that the historical defaults develop differently during expansions and contractions.

TABLE 6.3. S&P’s Assigned Ratings in Investment and Speculative Grade to New Issuers and Downgrade to Upgrade Ratio for all Issuers Source: Standard & Poor’s

Year	% Investment Grade	% Speculative Grade	Downgrade/Upgrade Ratio
1984	54.3	45.7	1.23
1985	60.6	39.4	2.13
1986	40.3	59.7	2.21
1987	43.5	56.5	1.57
1988	59.6	40.4	1.56
1989	55.5	44.5	1.40
1990	77.6	22.4	2.56
1991	86.0	14.0	2.25
1992	62.0	38.0	1.41
1993	50.9	49.1	1.25
1994	65.0	35.0	1.63
1995	54.2	45.8	1.06
1996	53.0	47.0	0.67
1997	43.2	56.8	1.00
1998	37.0	62.5	1.81
1999	39.6	60.4	1.91
2000	49.4	50.6	2.24
2001	56.5	43.5	2.90

TABLE 6.4. Correlations Between Macroeconomic Variables and Moody’s Historical Default Frequencies for Rating Classes Baa–C 1984–2000

Rating Class	Baa	Ba	B	C
Δ GDP	−0.1581	−0.4516	−0.6414	−0.3066
Δ CPI	0.4046	0.5691	0.2271	0.2010
Annual Saving	0.4018	0.5663	0.2258	0.1983
Δ Consumption	−0.0902	−0.0132	−0.3099	−0.1686

Let us finally examine the correlations between yearly default rates in the speculative grade area and macroeconomic variables. Assuming that the variables will have an effect on default rates, we considered the GDP growth, changes in the consumer price index, annual savings, and changes in the personal consumption expenditures for the United States. The correlations were calculated based on an assumed effect of the exogenous variables on rating migrations for the same year (see Table 6.4).

We find that some of the considered variables clearly have a significant impact on the considered migrations. The sign of the correlation in Table 6.4 is in the direction we would expect for all variables. In Chapter 9 we will take a closer look at the relationship between business cycle indicators and credit migration matrices. In this chapter we will mainly focus on the assumptions of Markov behavior and time homogeneity of migration matrices and provide a study on the significant effects changes in migration behavior may have on VaR and estimated probabilities of default for credit portfolios.

6.2. The Markov Assumptions and Rating Drifts

In most rating based models for credit risk management, rating transitions are modeled via a discrete or continuous-time Markov chain. However, in several applications, [see Lando and Skødeberg (2002); Nagpal and Bahar (2000); Frydman and Schuermann (2008)] non-Markov migration behavior also is investigated, and the authors provide methods for detecting higher order Markov behavior or rating drifts in migration models. In this section we summarize tests for Markov behavior or so-called rating drift in credit migrations. The tests will be illustrated using some empirical results for an internal rating system.

The term “Markov behavior” generally denotes Markov behavior of a first order. However, later we will also investigate Markov behavior of higher order. Therefore, we denote that unless it is not explicitly indicated, the term “Markov behavior” refers to first order Markov behavior. We define

Definition 6.1 A random variable X^t exhibits (first-order) Markov behavior, if its conditional distribution of X^t on past states is a function of X^t−1alone, and does not depend on previous states {X^t−2, X^t−3,…}:

As mentioned above, the term “Markov behavior” usually describes a first-order Markov chain. If the probability of moving to a certain state in the next period is not only dependent on the current state X^t−1 but also on past states X^t−2,… X^t−n (but not on state X^t−(n+1),…), one uses the term “Markov behavior of nth order”:

6.2.1. Likelihood Ratio Tests

Following Goodman (1958), we suggest testing the Markov property using a likelihood ratio test. The test is based on the comparing two likelihood functions L^B and L^A while one of the models is assumed to show Markov behavior of a higher order. The test statistic

(6.1)

is approximately X² distributed; the number of degrees of freedom is the difference between the estimated parameters in model B minus the estimated parameters in model A.

Testing for first-order Markov behavior against independence then is equivalent to testing the hypothesis of an independent identical distribution (iid) against the hypothesis of a first­order Markov chain (MK1). We obtain the likelihood functions

and

Clearly, in order to test on the same data basis, we can include only those records in L^A having at least one period of history. With L^A and L^B the likelihood ratio becomes

(6.2)

The likelihood ratio is supposed to be X²-distributed with Δm degrees of freedom, where Δm is the difference in the number of estimated parameters in both models.

For higher order Markov behavior, a second test can be deducted on the null-hypothesis of first-order Markov property (MK1) against the hypothesis of second-order Markov property (MK2). The matter of interest is whether the actual distribution depends only on the last state X^t−1 or also incorporating the second last state X^t−2 provides higher likelihoods. The likelihood functions read

Hence, the likelihood ratio is calculated according to the expression

(6.3)

Following Goodman (1958) the expression LR also is X²-distributed with Δm degrees of freedom. Again Δm is equal to the difference of the number of estimated parameters in both models. Testing for second-order Markov behavior (MK2) against the hypothesis of third-order Markov behavior (MK3) is straightforward. We therefore exclude the test statistics and simply provide some empirical results on these tests of an empirical study on the internal rating system described above.

6.2.2. Rating Drift

In a number of publications (Altman and Kao, 1992a, b; Bangia et al., 2002; Lando and Skødeberg, 2002), first-order Markov property has been rejected by testing for rating drift or so-called path dependence. Rating changes from the previous period have continued in the actual period in most cases. In a first-order Markov chain, the rating distribution of the next period is dependent only on the present state and not on any developments in the past. If there is a so-called rating drift or path dependence, then it is assumed that loans that have been downgraded before are less frequently upgraded in the next period, while loans that have experienced prior upgrading are prone to further upgrading. Therefore, two-period changes like “Down-Down”^[1] or “Up-Up” are generally considered to be more probable than alternating rating changes like “Down-Up” or “Up-Down”—the former is the so-called rating drift.

¹ For example, a series of subsequent downgrades like AAA → AA → A.

In order to investigate if such a rating drift exists in our data, we rely on the matrix M, which includes the total number of transitions from one rating grade to another; i.e., {M(t)}^ij gives the number of transitions from rating grade i at time t to rating grade j at time t+1. The matrix M is split into the sum of three matrices, called Up-Momentum-Matrix, Maintain-Momentum-Matrix, and Down-Momentum-Matrix.^[2] These three matrices are defined element-by-element in the following way:

² This procedure is the same as in Bangia et al. (2002).

By construction we have

As mentioned above, the issues of Markov bahvior and rating drifts have been studied by various authors (Altman and Kao, 1992a; Bangia et al., 2002; Lando and Skødeberg, 2002). In most of the cases, higher order Markov behavior and rating drift can be detected. However, most of the studies conducted deal with data provided by the major rating agencies. In the following we will illustrate the use of the methods described above to an internal rating system of a financial institution.

6.2.3. An Empirical Study

In the following some empirical results for an internal rating system based on balance sheet data rating process of a German bank are provided.^[3] The default probabilities and corresponding ratings were determined based on a logit model (Engelmann et al., 2003) and were used to investigate the time series behavior of the migration matrices for the period from 1988 to 2003. We will start with overview on the observed transitions and average one-year migration matrix for the considered time period; see Krüger et al. (2005).

³ The results were originally published in Krüger et al. (2005).

Hereby, the transition probabilities were calculated by Maximum Likelihood estimation as where N^ij denotes the number of transitions from rating i to j and Nⁱ denotes the total number of transitions from rating i. The average rounded one-year number of transitions and transition probabilities for the considered rating system are reported in Tables 6.5 and 6.6. In a next step the estimated yearly migration matrices were investigated by conducting the likelihood ratio tests on Markov behavior and examining rating drifts.

TABLE 6.5. Average Rounded One-Year Transitions for the Considered Rating System

	AAA	AA	A	BBB	BB	B	CCC	D	∑	Portion
AAA	3249	679	479	263	61	13	0	2	4744	14.73%
AA	686	721	744	400	71	12	0	1	2635	8.18%
A	431	744	1805	1648	259	32	0	4	4923	15.29%
BBB	218	368	1552	6609	2288	259	0	31	11325	35.17%
BB	45	56	192	2034	3672	864	1	82	6946	21.57%
B	8	6	22	180	748	762	3	71	1800	5.59%
CCC	0	0	0	0	1	3	0	0	4	0.02%

TABLE 6.6. Average One-Year Transition Probabilities for the Considered Rating System

	AAA	AA	A	BBB	BB	B	CCC	D
AAA	68.48%	14.30%	10.10%	5.54%	1.29%	0.27%	0.00%	0.03%
AA	26.01%	27.37%	28.24%	15.19%	2.69%	0.46%	0.00%	0.04%
A	8.76%	15.12%	36.66%	33.46%	5.26%	0.65%	0.00%	0.08%
BBB	1.93%	3.25%	13.70%	58.36%	20.20%	2.29%	0.00%	0.27%
BB	0.65%	0.80%	2.76%	29.29%	52.86%	12.43%	0.02%	1.18%
B	0.45%	0.35%	1.24%	10.00%	41.54%	42.33%	0.15%	3.94%
CCC	0.00%	2.91%	0.97%	5.83%	19.90%	66.02%	4.37%	0.00%
D	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	100.00%

TABLE 6.7. Results from the LR-Tests for the Considered Internal Rating System

H0	H1	# Companies	Likelihood Ratio	f	X2(f)
iid	MK1	32,380	26426	39	54.57
MK1	MK2	27,912	1852	192	225.33
MK2	MK3	24,575	800	783	849.21

The results for tests on first or higher order Markov behavior are shown in Table 6.7; significant test results are highlighted in bold letters. The first test was on the hypothesis of independent identical distribution (iid) against the hypothesis of a MK1. The results were significant rejecting the iid hypothesis. In a second test the hypothesis of MK1 is tested against second-order Markov property. Based on the results, we also reject MK1 in favor of second-order Markov property. We conclude that for the considered rating system at least a rating history of two periods should be used to estimate the actual state distribution as precisely as possible or make forecasts on future rating distributions. In a third test we examine whether the hypothesis of third-order Markov property (MK3) leads to even better results. Again the number of companies included decreases, because only records with a three-period history could be taken into account. In that case the test is not significant, so we conclude that the hypothesis of MK2 cannot be rejected. We conclude that for the considered internal rating system, the rating state distribution seems to depend on two periods of history.

The average transition probabilities we obtained based on, and M^Maintain, M^Down for the the years 1990 until 2003 can be found in Table 6.8, Table 6.9, and Table 6.10. Note that due to the very small number of observations in the CCC rating category, we excluded the category from the analysis.

TABLE 6.8. Average Transition Probabilities Obtained from the Up-Momentum-Matrix

	AAA	AA	A	BBB	BB	B	CCC	D
AAA	49.37%	21.30%	16.67%	9.80%	2.35%	0.47%	0.00%	0.05%
AA	20.40%	24.83%	32.05%	18.64%	3.43%	0.59%	0.00%	0.06%
A	6.15%	11.15%	33.57%	40.93%	7.13%	0.93%	0.01%	0.12%
BBB	1.27%	1.86%	7.42%	51.81%	32.88%	4.31%	0.00%	0.46%
BB	0.35%	0.74%	1.90%	17.68%	52.76%	24.28%	0.06%	2.24%
B	0.00%	1.96%	0.00%	7.84%	0.00%	78.43%	11.8%	0.00%

TABLE 6.9. Average Transition Probabilities Obtained from the Maintain-Momentum-Matrix

	AAA	AA	A	BBB	BB	B	CCC	D
AAA	75.87%	12.07%	7.42%	3.66%	0.80%	0.15%	0.00%	0.03%
AA	24.42%	32.48%	28.71%	12.39%	1.80%	0.19%	0.00%	0.01%
A	6.55%	14.98%	41.60%	32.55%	3.95%	0.32%	0.00%	0.06%
BBB	1.12%	2.16%	11.90%	63.58%	19.25%	1.75%	0.00%	0.24%
BB	0.29%	0.47%	1.58%	23.66%	59.54%	13.17%	0.01%	1.27%
B	0.39%	0.26%	0.71%	5.64%	34.22%	54.06%	0.17%	4.54%

TABLE 6.10. Average Transition Probabilities Obtained from the Down-Momentum-Matrix

	AAA	AA	A	BBB	BB	B	CCC	D
AAA	–	–	–	–	–	–	–	–
AA	34.50%	26.89%	23.37%	12.70%	2.20%	0.28%	0.00%	0.04%
A	13.88%	18.95%	35.23%	27.04%	4.26%	0.55%	0.00%	0.10%
BBB	4.01%	6.44%	22.34%	51.97%	13.35%	1.66%	0.00%	0.21%
BB	0.93%	1.11%	4.10%	38.60%	45.77%	8.57%	0.02%	0.88%
B	0.38%	0.40%	1.45%	12.21%	45.89%	35.69%	0.08%	3.90%

We found an interesting result for our rating system: companies in a rating category that were upgraded in the previous period are more likely to be downgraded than companies in the same rating category that were downgraded in the previous period. Considering transition probabilities obtained from the Up-Momentum-Matrix, we find that upgrades (elements at the left side of the diagonal) have lower probabilities than downgrades (elements at the right side of the diagonal). In the Down-Momentum-Matrix we see that upgrades have higher probabilities than downgrades. To illustrate this behavior, in Tables 6.8, 6.9, and 6.10 some transition probabilities for upgrades and downgrades of AA rated records have been highlighted in bold. More formally, our observations concerning conditional upgrade and downgrade probabilities of a rating process X can be written as

To investigate whether the differences are significant for single states (rows) and for the entire matrices, we used Pearson’s X² test. We considered the values of the Maintain-Momentum-Matrix as expected events and transitions of the Up-Momentum-Matrix (Down-Momentum-Matrix) as observed events. The result both for row-wise comparison and matrix-wise comparison confirms that the matrices are significantly different.

Summarizing the results of this section, we find that for the considered rating system rating transitions tend to compensate previous-period rating changes. These results are quite different to the rating drift observed in previous studies—for example, Altman and Kao (1992b), Bangia et al. (2002) or Lando and Skødeberg (2002). The authors found a tendency that companies in a certain rating category which were downgraded in the previous period are more likely to be downgraded in the next period than other companies in the same rating category which were upgraded in the previous period. An analogous statement was found for upgrades. Our results may be a consequence of the fact that, in contrast to other studies, we investigate rating transitions which are based on changes in credit scores only. Personal judgements or so-called soft factors included in the rating procedure by the major rating agencies that might induce effects like a rating drift were not considered.

6.3. Time Homogeneity of Migration Matrices

In the following we will review some techniques that can be used to investigate time homogeneity of migration matrices. While it is well known that generally transition matrices do not exhibit homogeneous migration behavior through time (Bangia et al., 2002; Nickell et al., 2000; Trueck and Rachev, 2005; Weber et al., 1998), for the sake of simplification average historical migration matrices are used as a starting point for evaluating credit risk or also to derive risk-neutral migration matrices as in Jarrow et al. (1997). Nevertheless, this property includes some element of idealization, since different states of the economy will generally result in a different migration behavior of companies in terms of rating upgrades, downgrades, or defaults. Recall, however, that in Section 2.1 it was explained that so-called through-the-cycle ratings should take into account possible changes in the macroeconomic conditions and not be affected when the change of the creditworthiness is caused only by a change of macroeconomic variables.

We will now investigate some methods for detecting time inhomogeneity of transition matrices. The most prominent tests for comparing transition matrices were developed by Anderson and Goodman (1957), Goodman (1958), and Billingsley (1961). They use chi-square and likelihood-ratio tests comparing transition probabilities estimated simultaneously from the entire sample to those estimated from subsamples obtained by dividing the entire sample into at least two mutually independent groups of observations. In our empirical study we will focus on the chi-square test; the LR test statistic is asymptotically equivalent.

For the definition of time homogeneity, let us consider the two years s and u and the corresponding state vectors X^s, X^u. Let us denote the transition matrix which transforms X^s into X^u by P^t(s), where t : u − s denotes the time horizon in years.

Definition 6.2 A Markov chain is time homogeneous if the property

holds for the state vectors X^sand X^uat two different dates s and u, where s and u are arbitrary.

As a consequence of this definition, for t = v − u = s − r we have

(6.4)

where P^t does not depend on the initial date r or u but only on the difference t between the initial date r and s or u and v, respectively.

In the nonhomogeneous case the transition probability matrix would depend on the initial date r or u as well as on the distance t between the dates; i.e., we have X^v = P^t(u)X^u instead of (6.4), whereas in the homogeneous case the transition probability matrix is a function of the distance between dates and not the dates themselves.

For simplicity, let us denote the transition probability matrix for two subsequent years by P. The property time homogeneity offers the nice feature that the state vector x^v at any future date v can be calculated in terms of the initial state vector x^u by x^v = P^tx^u, where P^t denotes the tth power of the matrix P.

6.3.1. Tests Using the Chi-Square Distance

In tests using the chi-square distance, time stationarity is simply checked by dividing the entire sample into T periods. Then it is tested whether the transition matrices estimated from each of the T subsamples differ significantly from the matrix estimated from the entire sample.

The test statistic used is

(6.5)

Clearly, P^ij denotes the average probability of transition from the ith to the jth class estimated from the entire sample, the corresponding transition probability estimated from subsample in t. Further we should note that only those transition probabilities are taken into account which are positive in the entire sample; thus, we set and exclude transitions for which no observations are available in the entire sample. Q has an asymptotic chi-square distribution with degrees of freedom equal to the number of summands in T minus the number of estimated transition probabilities corrected for the number of restrictions. Thus, we get for the degrees of freedom.

6.3.2. Eigenvalues and Eigenvectors

Another possibility to investigate time homogeneity is obtained by considering the eigenvalues and eigenvectors of transition matrices P for different time horizons. For example, Bangia et al. (2002) use this method in an empirical study on the stability of migration matrices of Standard & Poors. Note that a transition matrix P can be decomposed into a diagonal matrix of eigenvalues diag(Θ¹) of P and the basis-transformation matrix Φ = {ϕ¹,…,ϕⁿ}

(6.6)

where the diagonal elements are given by θ¹, θ²,…, θⁿ. Without loss of generality, it can be assumed that the column indices are ordered such that

Then the ordered column indices for the average one-period transition matrix P¹ are

Note that in this notation the first index is used for the time horizon in years in the time-homogeneous case, whereas the second index is the number of the eigenvalue. Let further P^t denote the tth power of the estimated transition matrix for one period P¹. Using the same decomposition as above, we can express the tth power of P¹ as

with

Obviously, the eigenvalues of P^t are given by (θⁱ)^t. Note that due to the fact that the row sum of a transition matrix must equal one per definition, the largest of the eigenvalues θ¹ must be identically equal to 1. When the matrix is raised to the tth power, the eigenvalue of 1 persists while all the other (nonunity) eigenvalues have magnitudes less than 1. Therefore, when raised to the tth power, they eventually decay away; see Jafry and Schuermann (2004) for further details. Because of the sequence θ¹ⁱ, θ²ⁱ, θ³ⁱ,…, θ^ri of the ith eigenvalues of the matrices P¹, P², P³,…,P^r is a log-linear function of t. For example, we get for the relationship between the second eigenvalue of the matrix P¹ and.

With this in mind, it is straightforward to investigate the property of time homogeneity based on the eigenvalues of estimated k-period migration matrices: consider, for example, the average transition matrices,,, and for the time horizons of 1, 2, 3, and 4 years from an empirical data set and plot the logarithms,,, and of the eigenvalues smaller than 1 of these 4 matrices. Under the assumption of time homogeneity, one could expect that each sequence of eigenvalues can be fit by a straight regression line.

A plot of the relation between the logarithm of the eigenvalues,,, and for the considered internal rating system is provided in Figure 6.4. Obviously, the fourth and fifth eigenvalues do not show log-linear behavior, as the plotted lines are not really straight. Further the log-eigenvalues for periods of two or more years are not t-multiples of the log-eigenvalue for one year.

FIGURE 6.4. Relationship between second up to fifth log-eigenvalues of the estimated average transition matrices,,, and.

Based on the results in Figure 6.4, we rather assume that the property of time homogeneity should be rejected for the considered rating system. Note, however, that we have not provided a formal test procedure for time homogeneity here, but rather a qualitative way to investigate the issue.

Another way of approaching time homogeneity is analyzing the eigenvectors of a migration matrix P¹. Following Bangia et al. (2002), in the case of time homogeneity the matrices P¹ and any arbitrary power have the same set of eigenvectors. Thus, plotting the ith eigenvectors for different time horizons t should always yield approximately the same result, independently of t. However, when we compute the second eigenvector for t = 1,…,4 years and assign the components to the corresponding rating grades, Figure 6.5 shows that the eigenvectors are far from being equal. The curve is getting less steep as the time horizon increases. Based on Figure 6.5, the hypothesis that the process of rating distributions is a homogeneous one should be rejected. Note, however, that with the eigenvector analysis, a rather qualitative procedure to investigate time homogeneity of transition matrices was considered.

FIGURE 6.5. Second eigenvector for the average migration matrices with time horizon one year, two years, three years, and four years.

Overall, our empirical examination suggests that the considered internal rating system does not exhibit first-order Markov behavior nor time homogeneity. We provided a number of methods investigating these issues and pointed out a particular behavior of the considered internal rating system according to the detected second-order Markov behavior. Opposed to former studies for the considered rating system, downgrade probabilities were higher for companies that were upgraded in the previous period compared to those being downgraded in the period before and vice versa. We conclude that issues like time homogeneity, Markov behavior, and rating drift cannot be assumed to be the same for different rating systems and have to be examined as the case arises. In the following sections we will investigate the consequences of observed business cycle effects and time inhomogeneity in rating migrations on risk capital of credit portfolios and PD estimates through time.

6.4. Migration Behavior and Effects on Credit VaR

This section aims to illustrate how changes in credit migration matrices may have a substantial impact on the associated risk of a credit portfolio.^[4] In the previous sections we reviewed methods for investigating the stability of migration matrices and a rating system based on financial ratios and found that transition matrices showed significant changes over the years. We will now consider the effects of such changes in migration behavior on capital requirements in terms of expected losses and VaR figures for an exemplary credit portfolio. It is well known that the loss distribution for a credit portfolio as well as capital requirements vary between recession and expansion times of the business cycle (Bangia et al., 2002). We will find that for a considered exemplary portfolio these numbers vary substantially and that the effect of different migration behavior through the cycle should not be ignored in credit risk management.

⁴ The results of this section were originally provided in Trueck and Rachev (2005).

To illustrate the effects, let us consider an exemplary loan portfolio of an international operating major bank consisting of 1120 companies. The average exposure is dependent on its rating class. In the considered portfolio higher exposures could be observed in higher rating classes, while for companies with a non-investment grade rating Baa, B, or Caa, the average exposures were between 5 and 10 million Euro. The distribution of ratings and average exposures in the considered rating classes of the loan portfolio are displayed are Table 6.11.

TABLE 6.11. Ratings and Exposures for the Considered Credit Portfolio

Rating	Aaa	Aa	A	Baa	Ba	B	Caa
No.	11	106	260	299	241	95	148
Average Exposure (million Euro)	20	15	15	10	10	5	5

We further make the following assumptions for the loans. For each of the simulated years, we use the same portfolio and rating distribution to keep the figures comparable. We also assume an average yearly recovery rate of R = 0.45 for all companies. This is clearly a simplification of real recovery rates, but since we are mainly interested in the effects of different migration behavior on credit VaR and PDs, it is not a drawback for our investigation. Further, not having enough information on the seniority of the considered loans, it may be considered as an adequate assumption for empirical recovery rates.

For the investigation we use Moody’s credit transition histories of a 20-year period from 1982–2001 and a continuous-time simulation approach for determining VaR figures based on N = 5000 simulations. Note that the actual reported one-year migration matrices by Moody were in discrete form. However, using the log-expansion 5.4 for Λ and the approximation methods suggested by Israel et al. (2000), we could calculate the corresponding approximate generator matrix. To illustrate the differences in migration behavior, Figures 6.6 and 6.7 show the cumulative default probabilities or credit curves based on years belonging to two different phases of the business cycle. The year 2001 was a year of economic turmoil with high default rates and many downgrades, while in 1997 the macroeconomic situation was stable and the economy was growing. Lower default probabilities and more upgrades than downgrades were the consequences.

FIGURE 6.6. Credit curves for speculative grade issuers according to Moody’s migration matrix 1997 (solid lines) and 2001 (dashed lines).

FIGURE 6.7. Credit curves for investment grade issuers according to Moody’s migration matrix 1997 (solid lines) and 2001 (dashed lines).

Both for investment grade issuers and speculative grade issuers, we find completely different credit curves with cumulative default probabilities being substantially higher when the 2001 transition matrix is used as describing the underlying migration behavior. The graphs were plotted for a 10-year time horizon. Clearly, it is rather unrealistic that the macroeconomic situation stays in a recession or expansion state for such a long time. However, in the following we will illustrate that even for a considered shorter time horizon, differences can be quite substantial.

In a second step we investigated the effect of different migration behavior on risk figures for time horizons of six months, one year, and three years. Two typical loss distributions for the years 1998 and 2001 and an assumed one-year time horizon are displayed in Figure 6.8. The distributions have an expected loss of μ = 148.55 million Euro with a standard deviation of σ = 16.67 million for 1998 and μ = 223.15 million with σ = 21.15 million for 2001. Both distributions were slightly skewed to the right with γ = 0.1217 for 1998 and γ = 0.1084 for 2001. The kurtosis for the loss distributions with k = 2.99 for 1998 and k = 2.97 for 2001 is very close to the kurtosis of the normal distribution.

FIGURE 6.8. Typical shape of simulated loss distributions for the years 1998 and 2001.

Comparing loss distributions for different years, we find that in many cases the distributions do not even coincide. We plotted a comparison of the simulated loss distributions for the years 2000 and 2001 in Figure 6.9 and for the years with minimal (1996) and maximal (2001) portfolio risk in the considered period in Figure 6.10. While for the subsequent years 2000 and 2001 the distributions at least coincide at very low (respectively high) quantiles, we find no intersection at all for the years 1996 and 2001. This points out the substantial effect of migration behavior on risk figures for a credit portfolio.

FIGURE 6.9. Simulated loss distributions for the years 2000 (left side) and 2001 (right side).

FIGURE 6.10. Simulated loss distributions for the years 1996 (left side) and 2001 (right side).

A closer picture of the significant changes in the Value-at-Risk for the considered period is provided in Figure 6.11 and in Table 6.12. We find that both expected loss and simulated VaR figures show great variation through the business cycle for all three considered time horizons of six months, one year, and three years. While in the years 1983 and 1996 the average expected loss for the portfolio would be only 31.29 million or 28.84 million Euro in a one-year period, during the recession years 1991 and 2001, the simulated average loss for the portfolio would be 227.25 million or 258.75 million Euro, respectively. The maximum of the simulated average losses for the portfolio is about eight times higher than the minimum amount in the considered period. Similar numbers were obtained considering Value-at-Risk or expected shortfall for the portfolio. The one-year 95%-VaR varies between 45 million and 258.75 million Euro, while the one-year 99%-VaR lies between a minimum of 56.25 million in 1996 and 273.37 million in the year 2001. This illustrates the enormous effect the business cycle might have on migration behavior and, thus, on the risk of a credit portfolio. Ignoring these effects may lead to completely wrong estimates of credit VaR and capital requirements for a loan or bond portfolio. We conclude the necessity to use credit models that include variables measuring the state of the business cycle or the use of conditional migration matrices. In the next section we will further investigate the effect of different migration behavior on confidence sets for PDs.

FIGURE 6.11. Simulated one-year VaR alpha = 0.95 for the years 1982–2001 (solid line), average one-year VaR for the whole period (dashed line).

TABLE 6.12. Simulated Average Loss, 95%-, and 99%-VaR for the Exemplary Portfolio for 1982–2001. Considered was a Six-Month, One-Year and Three-Year Time Horizon

Year	1982	1983	1984	1985	1986	1987	1988	1989	1990	1991
Mean, 6 months	83.48	14.46	44.05	34.66	52.50	27.07	57.42	87.58	97.30	102.21
Mean, 1 year	157.83	31.29	84.67	69.71	105.96	53.03	108.13	158.47	183.59	191.93
Mean, 3 years	343.50	88.95	198.83	178.30	275.23	127.73	233.95	314.05	413.67	411.20
Year	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001
Mean, 6 months	60.05	35.13	42.50	61.57	14.11	30.66	85.91	79.89	79.56	121.17
Mean, 1 year	111.60	61.30	81.54	113.48	28.84	58.05	148.55	146.30	148.84	223.15
Mean, 3 years	240.16	113.84	176.37	235.14	68.33	129.68	280.51	305.39	312.74	477.26
Year	1982	1983	1984	1985	1986	1987	1988	1989	1990	1991
VaR0.95, 6 months	108.00	27.00	63.00	51.75	72.00	40.50	76.50	110.25	123.75	128.25
VaR0.95, 1 year	191.25	49.50	108.00	92.25	132.75	72.00	135.00	185.62	217.12	227.25
VaR0.95, 3 years	393.75	119.25	234.00	216.00	321.75	157.50	272.25	353.25	463.50	461.25
Year	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001
VaR0.95, 6 months	78.75	50.62	58.50	81.00	24.75	45.00	108.00	101.25	101.25	148.50
VaR0.95, 1 year	135.00	81.00	101.25	139.50	45.00	76.50	175.50	174.37	177.75	258.75
VaR0.95, 3 years	274.50	139.50	207.00	270.00	90.00	157.50	317.25	343.12	351.00	531.00
Year	1982	1983	1984	1985	1986	1987	1988	1989	1990	1991
VaR0.99, 6 months	120.37	33.75	74.25	58.50	83.25	47.25	84.37	119.25	132.75	139.50
VaR0.99, 1 year	209.25	58.50	120.37	105.75	148.50	79.87	148.50	200.25	231.75	243.00
VaR0.99, 3 years	421.87	132.75	253.12	236.25	338.62	171.00	285.75	370.12	487.12	482.62
Year	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001
VaR0.99, 6 months	85.50	56.25	65.25	90.00	31.50	51.75	115.87	112.50	110.25	164.25
VaR0.99, 1 year	147.37	87.75	112.50	150.75	56.25	87.75	186.75	187.87	190.12	273.37
VaR0.99, 3 years	288.00	153.00	218.25	283.50	99.00	172.12	333.00	357.75	371.25	553.50

6.5. Stability of Probability of Default Estimates

Another main issue of credit risk modeling is the modeling of probability of defaults. In the internal rating based approach of the new Basel Capital Accord, PDs are the main input variables for determining the risk and the necessary regulatory capital for a portfolio. Of course, regulators are not the only constituency interested in the properties of PD estimates. PDs are inputs to the pricing of credit assets, from bonds and loans to more sophisticated instruments such as credit derivatives. However, especially for companies with an investment grade rating, default is a rare event. Often high credit quality firms make up the majority of the large corporate segment in a bank’s portfolio. But with only little information on actual defaulted companies in an internal credit portfolio, observed PDs for the investment grade categories are likely to be very noisy. The question arises how reliable confidence interval estimates for PDs may be obtained. This is of particular importance, since similar to the VaR or expected shortfall of a credit portfolio, PDs and also PD confidence sets may vary systematically with the business cycle. Thus, investment grade rating PDs are also rather unlikely to be stable over time. Therefore, in this section we tackle the question of obtaining reliable estimates for default probabilities also in the investment grade sector and compare these PDs for the considered time period from 1982–2001.

Christensen et al. (2004) estimate PD confidence intervals for default probabilities for different rating classes by using a continuous-time approach similar to the one suggested in the previous section. They find that a continuous-time bootstrap method can be more appropriate for determining PD distributions than using the estimates based on actual default observations. This is especially true for investment grade ratings where defaults are very rare events.

To illustrate the advantages of the bootstrapping idea, let us first consider a binomial random variable X ~ B(pⁱ, nⁱ) where pⁱ denotes the probability of default for rating class i and nⁱ the number of companies in the rating class. Now assume that there is an investment grade rating class in the internal rating system where no actual defaults were observed in the considered time period. Clearly, the corresponding estimator for the PD in this rating class is pⁱ = 0. However, for VaR calculations a bank is also interested in confidence intervals for PDs of the investment grade rating classes. Based on the binomial distribution, one could compute the largest default probability not being rejected for a given confidence level α by solving the following equation:

Therefore, the corresponding upper value p^max of a confidence interval for a rating class with no observed defaults is

The disadvantage of this estimation technique becomes obvious in Table 6.13. Let Xⁱ be the number of observed defaults in the two rating classes. Generally, the confidence intervals are dependent on Xⁱ, the number of defaults observed, and nⁱ, the number of firms in the considered rating class. However, if no defaults are observed for a rating class, the lower nⁱ, the wider becomes the confidence interval. This is illustrated in Table 6.13 for an exemplary portfolio with 50 companies in the rating class Aaa and 500 companies in rating class Aa. We find that using the binomial distribution, the intervals for rating class Aaa are about ten times wider than those for rating Aa. From an economic point of view, this is rather questionable and simply a consequence of the fact that more companies were assigned with the lower rating.

Of course, the binomial distribution can also be used for calculating two-sided confidence intervals for lower rating classes where transition to defaults also were observed. What is needed is the total number of firms with certain rating i at the beginning of the period and the number of firms among them that defaulted until the end of the considered period. Then, for a given confidence level α, the standard Wald confidence interval is

(6.7)

where nⁱ is the total number of firms in rating class i and q^α is the α-quantile of the standard normal distribution. Unfortunately, as pointed out by Schuermann and Hanson (2004), the estimates for confidence intervals obtained by the Wald estimator are not very tight. Christensen et al. (2004) state that the only advantage in the binomial case is that using this method, one is able to derive genuine confidence sets, i.e., to analyze the set of parameters which an associated test would not reject based on the given observations. The authors point out that obtained confidence sets by a continuous-time bootstrap method are much tighter than those using the standard Wald estimator.

TABLE 6.13. Example for PD Confidence Interval Estimated Based on the Binomial Distribution

	ni	Xi	KIα = 0.05	KIα = 0.01
Aaa	50	0	[0,0.0582]	[0,0.0880]
Aa	500	0	[0,0.0060]	[0,0.0092]

To compare confidence intervals through the business cycle, we therefore used the method described in Christensen et al. (2004). An introduction to bootstrapping can be found in Efron and Tibshirani (1993), so we will only briefly describe the idea of the bootstrap and our simulation algorithm. For our continuous-time simulation, the same procedure as in Section 5.4 is used to obtain histories for each of the considered companies. Note that unlike Schuermann and Hanson (2004) who apply a nonparametric bootstrapping approach, we based our simulations on the parametric assumption of a continuous-time Markov chain with a given migration matrix. For each year we simulate N = 5000 times using a fake data set with a number of 1000 issuers in each rating category. Then the issuer’s history background Markov process is simulated using the observed historical transition matrix for each year. The simulated rating changes are translated into a history of observed rating transitions. For each replication the generator matrix of the hidden Markov chain model is re-estimated, using the companies’ rating history and the maximum-likelihood estimator described above:

From the estimated transition structure, we calculate the one-year default probability for each true state. Exponentiating this matrix gives an estimator of the one-year migration matrix, and the last column of the transition matrix provides the vector of estimated default probabilities for each replication. Thus, for each year or following Christensen et al. (2004)—each true state of the background process—we have N = 5000 one-year default probabilities for each rating class.

Another possibility to find confidence sets would have been to develop asymptotic expressions for the distribution of test statistics in the continuous-time formulation and use those for building approximate confidence sets. However, in practice the bootstrap method seems both easier to understand and to implement. The maximum-likelihood estimator does not have a simple closed form expression for its variance-covariance matrix. This makes it difficult to provide information about the confidence sets for estimated parameters. In fact, we would need to use asymptotics twice—first to find the variance of the estimated generator and additionally to find an expression for the variance of exp(). The second step again only seems feasible using an asymptotic argument. Unfortunately the asymptotic variance of is hardly a good estimator, since many types of transitions occur only rarely in the data set. Thus, the bootstrap method provides tighter intervals and is also more understandable.

As mentioned earlier, confidence intervals for PDs can alternatively be obtained by a nonparametric bootstrap as illustrated in Schuermann and Hanson (2004). Here, the resampling is directly based on the observed rating histories and not on the estimated generator matrix. It basically follows the steps described in Section 5.4 for the nonparametric simulation approach. The method can be considered as a recommendable alternative if additional information on rating transitions is available.

Based on the bootstrapped generator matrices, for each year the relevant quantiles and distribution of the PDs can be obtained. The results for investment grade rating classes Aa–Baa can be found in Table 6.14 as well as in Figure 6.12 and 6.13 where boxplots of the PDs for the whole considered period are provided. It becomes obvious that confidence intervals vary substantially through time. This includes not only the level of the mean of the bootstrapped PDs but also the width of the confidence interval. Comparing, for example, the 95% interval for rating class A, we find that the interval in 2001 KI^A,2001 = [0.0005, 0.0054] compared to the interval in 1993 KI^A,1993 = [0.000, 0.0001] is about 50 times wider. The variation of the lower and upper boundary of the intervals is illustrated for rating classes Ba and A in Figure 6.14 and 6.15. We also find that with the level the width of an estimated confidence set for the PD increases substantially. Histograms of bootstrapped PD distributions for rating class Ba and different periods—1991 and 1996—can be found in Figure 6.16. Obviously the plotted histograms for the two periods do not coincide. Note that similar results in terms of variation are observed by Schuermann and Hanson (2004) using Standard & Poor’s credit rating history.

TABLE 6.14. Descriptive Statistics of the Width of Confidence Intervals for Different Rating Classes

Rating	Aaa	Aa	A	Baa	Ba	B	Caa
mean	0.0001	0.0006	0.0019	0.0049	0.0126	0.0262	0.0473
σ	0.0002	0.0009	0.0017	0.0030	0.0053	0.0065	0.0140
min	0.0000	0.0000	0.0000	0.0002	0.0049	0.0172	0.0021
max	0.0007	0.0040	0.0057	0.0095	0.0246	0.0424	0.0605
	1.5994	1.6162	0.9361	0.6125	0.4210	0.2493	0.2955

FIGURE 6.12. Boxplot for bootstrapped confidence intervals for rating class Aa from 1982–2001.

FIGURE 6.13. Boxplot for bootstrapped confidence intervals for rating class A from 1982–2001.

For noninvestment grade ratings the variations in the level of average PDs is also extreme. However, as one can see in Table 6.15, the width of the intervals does not show the extreme variations as for the investment grade ratings. This is best illustrated by the coefficient of variation , comparing the standard deviation of the width of the confidence intervals through time to its mean. We find a decreasing coefficient of variation for deteriorating credit quality. Thus, we conclude that the fraction PD volatility divided by average PD decreases with increasing PD.

FIGURE 6.14. Mean and 95% PD confidence levels for rating class A from 1982–2001.

FIGURE 6.15. Mean and 95% PD confidence levels for rating class Ba from 1982–2001.

FIGURE 6.16. Histogram of bootstrap PDs for rating class Ba in 1991 (left) and 1996 (right).

Overall, the results point out the substantial effects of variations in the economy on the expected loss and VaR for a credit portfolio if a rating based credit risk system is used. The estimated one-year VaR for the considered portfolio was more than twice the average for the period of economic turmoil in 2001 and about eight times higher than VaR in the year 1996. The effect of changes in migration behavior on confidence sets for default probabilities is even more dramatic. Variations in the width and level of confidence intervals for investment grade rating classes were significant. In several cases the intervals did not even coincide in periods of economic expansion or recession. We further found a decreasing coefficient of variation for PD confidence sets with increasing riskiness of the loan. This may also be a useful result for credit derivative modeling, where PD volatilities are also of particular interest. Using average historical transition matrices as input for portfolio risk calculations should be handled with care. The effect of the business cycle on changes of migration behavior and therefore also on Value-at-Risk and PDs might be too imminent to be neglected. To overcome these problems, one could adjust or forecast migration behavior and PDs in credit risk models with respect to the macroeconomic situation. We will further investigate this issue in Chapter 9, where several approaches for conditioning transition matrices on business cycle effects are reviewed and compared.

TABLE 6.15. Bootstrap-95%-Confidence Intervals for Investment Grade Ratings. Figures Based on Moody’s Historical Transition Matrices 1982–2001

Year	Aaa	Aa	A	Baa
1982	[0.0000, 0.0005]	[0.0000, 0.0010]	[0.0000, 0.0057]	[0.0019, 0.0062]
1983	[0.0000, 0.0000]	[0.0000, 0.0001]	[0.0000, 0.0007]	[0.0004, 0.0073]
1984	[0.0000, 0.0001]	[0.0000, 0.0003]	[0.0000, 0.0010]	[0.0026, 0.0121]
1985	[0.0000, 0.0001]	[0.0000, 0.0014]	[0.0000, 0.0010]	[0.0008, 0.0034]
1986	[0.0000, 0.0001]	[0.0000, 0.0010]	[0.0001, 0.0050]	[0.0014, 0.0069]
1987	[0.0000, 0.0000]	[0.0000, 0.0000]	[0.0000, 0.0011]	[0.0001, 0.0015]
1988	[0.0000, 0.0003]	[0.0000, 0.0004]	[0.0000, 0.0007]	[0.0008, 0.0032]
1989	[0.0000, 0.0000]	[0.0000, 0.0001]	[0.0000, 0.0014]	[0.0026, 0.0106]
1990	[0.0000, 0.0000]	[0.0000, 0.0010]	[0.0000, 0.0018]	[0.0023, 0.0101]
1991	[0.0000, 0.0000]	[0.0000, 0.0002]	[0.0000, 0.0015]	[0.0042, 0.0128]
1992	[0.0000, 0.0000]	[0.0000, 0.0000]	[0.0000, 0.0004]	[0.0002, 0.0015]
1993	[0.0000, 0.0000]	[0.0000, 0.0000]	[0.0000, 0.0001]	[0.0001, 0.0010]
1994	[0.0000, 0.0002]	[0.0000, 0.0009]	[0.0000, 0.0039]	[0.0000, 0.0006]
1995	[0.0000, 0.0000]	[0.0000, 0.0001]	[0.0000, 0.0006]	[0.0002, 0.0074]
1996	[0.0000, 0.0000]	[0.0000, 0.0000]	[0.0000, 0.0000]	[0.0000, 0.0002]
1997	[0.0000, 0.0000]	[0.0000, 0.0004]	[0.0000, 0.0007]	[0.0002, 0.0058]
1998	[0.0000, 0.0002]	[0.0000, 0.0001]	[0.0000, 0.0008]	[0.0012, 0.0090]
1999	[0.0000, 0.0007]	[0.0000, 0.0040]	[0.0000, 0.0030]	[0.0002, 0.0052]
2000	[0.0000, 0.0000]	[0.0000, 0.0002]	[0.0000, 0.0029]	[0.0008, 0.0072]
2001	[0.0000, 0.0000]	[0.0000, 0.0003]	[0.0005, 0.0054]	[0.0024, 0.0075]