In the risk class factor model described in Section 3.5, exposures of a risk class are assumed to follow identically specified asset value processes and have homogenous asset correlation to other exposures. Attributes used to specify the risk class affiliation of exposures are either its rating or economic sector, or both.
A brief look at the data set reveals that rating-sector-based risk classes subject to the BBG sector classification do not provide enough bond price observations for a robust fitting of yield curves. To concentrate the available data, sectors are grouped into sector-classes of assumed homogenous credit risk using an agglomerative cluster procedure based on a credit spread related similarity measure.4
Definition of Similarity Measure
The objective of clustering is to achieve a classification that groups bonds exhibiting the most homogenous evolution of credit risk within sector-classes and that provides most heterogenous spreads between classes. The indicator for the evolution of credit risk in a sector is set to be the time series of log-returns of the average credit yield spread in a sector, denoted as log-returns of sector spreads. Log-returns are preferred to the first differences of average credit yield spreads because they are less sensitive to outliers. Thus, they smooth out extreme obligor-specific effects and better represent the dependence of the sectors’ credit risk, as can be seen in the case of the Parmalat default in December 2003 in the CNC sector.
A hierarchical iterative clustering procedure is used. The prospective joining of two sector- classeskj andlj in iteration stepj is assessed using a similarity measure which is based on an indicator for the homogeneity of credit risk within the joint sector-class and an indicator to assess the heterogeneity of credit risk against the remaining classes. Both indicators rely on the time seriesrcyskj = (rcyskj
t )t=2,...,T of log-returnsrcys kj t = ln(cys kj t /cys kj t−1) of
the average credit yield spreads cyskj
t = ( P s∈Is kj P i∈Iscys i t)/ P s∈Kjn i s,t in a sector-class
kj including setIksj of sectors. In each iteration j, n
i
s,t denotes the number of bond yield spreads available in sector s at time t, ns
kj gives the number of sectors included in sector-
class kj and nscj = 13−j represents the number of sector-classes. Is denotes the set of bonds in sector s. The credit yield spread cysit is defined as the difference between the internal yield-to-maturity of bondi, derived from the observed bond price and the yield of an equivalent non-defaultable bond determined using the term structure of riskless rates fitted in Section 4.3. The dependence of credit spreads of two different sector-classes
kjand lj is measured by the correlation
ρsckjlj = Cov(rcys
kj, rcyslj)
p
V ar(rcyskj)pV ar(rcyslj)
(4.1)
of the time series of log-returns of average sector-class spreads. For a join of sector-classes
kj andlj, the homogeneity of credit spreads within the joint class is assessed by averaging the correlations ρscs1r1 of spread returns of included sectors s1, r1 ∈ {/ Iksj, I
s lj}, s1 6=r1 : ρsck jlj = P s1{Ikjs ,Iljs} P r1{Ikjs ,Iljs};r16=s1ρ sc s1r1 (ns kj+n s lj)(n s kj+n s lj −1) (4.2)
The heterogeneity of credit spreads between a joint sector-class of kj and lj and the remaining sector-classes is measured by the average
ρsck jlj = P hj6=kj,ljρ sc kjhj+ρ sc ljhj 2(nsc j −2) (4.3)
of all correlations ρsckjhj and ρscljhj between the joining sector-classes and all remaining classeshj ∈ {1, ..., nscj }\{kj, lj}of the current classification at cluster levelj. Note that in each iteration j, the inner-class credit risk homogeneity ρsc
kjlj is derived from correlations
ρsc
s1r1 of sector spread returns rcys
s1 and rcysr1, which do not change throughout the
clustering, while for the calculation of heterogeneity ρsck
jlj series rcys
kj and rcyslj of
average sector-class spread returns need to be aggregated from sample data according to the current classification of sectors. Ultimately, the similarity measure
SIMkjlj =
ρsc
kjlj
ρsckjlj (4.4)
is used to determine which sector-classes to join at iteration level j.
Clustering Procedure
For the classification of sectors, an agglomerative hierarchical clustering procedure is used. The financial sector provides by far the largest number of price observations and is therefore set to be a sector-class on its own in the final cluster. The other 12 BBG
sectors are partitioned into three sector-classes to provide a sufficient number of price observations in each sector-class without smoothing out diversification effects imposed by the heterogenous evolution of the credit spreads of different sector-classes. The sector clustering is accomplished in a four-step iterative procedure:
Step 1: Initialize sector-classes to represent individual sectors. Set iteration count j = 1. Step 2: CalculateSIMkjlj for each eligible pair of sector-classes (kj, lj), kj < lj.
Step 3: Join pair of sector-classes (k∗j, lj∗) with similaritySIMk∗jlj∗ = maxkj<lj≤nscj SIMkjlj
Step 4: If nsc
j >3, then set j =j+ 1 and proceed with step 2, else end.
Results of Clustering
Using the clustering procedure described above, the sector-classification received after
j = 10 iterations is given in the right column of Table 4.15. The denomination of sector- classes in Table 4.1 is set based on the cyclicity of sectors. Market participants typically consider the cyclicity of business sectors on the basis of lead-lag effects of stock returns relative to the economic cycle. A similar cyclicity is assumed to determine the evolution of credit risk markets. It is conjectured that the basic material (BMA) sector and the technology sector (TEC) lead the business-cycle, while the industrial sector (IND) and the construction sector (CON) lag behind. The non-cyclic consumer (NCC), utility (UTY) and energy (ENY) sectors do not show a distinct cyclicity. Correspondingly, sector- classes are designated as early-cyclic (ECY), late-cyclic (LCY), and non-cyclic (NCY). The
ρsc
k10l10\ρ
sc
k10l10 ECY FIN LCY NCY Sector Set
ECY 50.4 41.6 73.3 57.5 {AUT, BMA, COM, MED, TEC}
FIN 34.7 100.0 46.4 29.3 {FIN}
LCY 37.7 24.6 39.9 40.0 {CCY, CON, IND}
NCY 41.0 41.2 29.2 45.2 {CNC, ENY, TRA, UTY}
Table 4.1: Sector Classification
correlation matrix in Table 4.1 represents inter-class correlationsρsc
k10l10 above the diagonal,
average inner-class sector correlationsρsc
k10k10within sector-classes on the diagonal, and the
average inter-class sector correlations ρsck10l10 of sectors classes below the diagonal. Inner- class sector correlations ρsck10k10 with an average of 45.17%, excluding the FIN class, are higher than the 34.73% average of inter-class correlationsρsck10l10, which confirms that the clustering procedure effectively classifies sectors, so that the evolution of credit spreads is more heterogenous between sector-classes than within classes. Considering the correlation of log-returns of average credit yield spreads of rating class in Table B.1, correlations are high for neighboring ratings and decrease with enhanced rating distance, which suggests
5 An analogous clustering procedure was used to generate risk classes specified by the sector and rating
the existence of common background factors not included in the rating. In contrast, no clear-cut pattern of sector spread correlations reveals from Table 4.1, so that it is assumed that sector affiliations present an appropriate classification of obligors to ensure a maximum heterogeneity of credit risk between risk classes and to implement a maximum diversification of credit risk in a portfolio.
p-values ECY FIN LCY NCY
ECY – 0.3899 0.1469 0.0457 FIN 0.0521 – 0.9242 0.4126 LCY 0.0281 0.5903 – 0.5215 NCY 0.0165 0.7415 0.7866 –
Table 4.2: Granger Test of Sector-Class Causality
Table 4.2 presents the p-values of a 2-lag Granger causality test of the column sector- class leading the row class on the basis of monthly log-returns of average sector-class spreads. For example, the p-value of 0.3899 indicates that non-causality of sector-class FIN to sector-class ECY is not rejected at a 5% level of significance, so that a causality of sector-class FIN to sector-class ECY in the Granger-sense is denied. The sector-class causalities ECY→LCY, ECY→NCY, NCY→ECY are significant at a 5% level. The fact that monthly spread returns of the ECY sector-class lead those of the LCY and NCY sectors gives additional support to the designation of the ECY sector as early-cyclic, even though the direction of the causality between the ECY and NCY classes is not unambiguous.
Causalities on a sector level provide additional insights. In Table B.3, results of Granger tests for sector causalities of monthly spread returns are presented. Figure B.1 graphs sector causalities that are significant at a 5%-level.
Avg. no. of causalities per sector ECY FIN LCY NCY
Lag causalities 2.80 2.00 3.67 2.75
Lag causalities from different class 1.00 2.00 3.00 1.75
Lead causalities 4.80 0.00 1.67 2.25
Lead causalities to different class 3.00 0.00 1.00 1.25 Net lead-lag causality per sector 2.00 -2.00 -2.00 -0.50
Table 4.3: Causality Analysis per Sector-Class
Summary statistics on sector causalities in Table B.3 reveal that lead causalities and lag causalities exist predominantly for sectors of a different sector-class. Netting the number of the average lead and lag causalities per sector for each sector-class reveals that sectors of the ECY class show two more lead causalities than lag causalities, whereas the net
lead-lag causality is negative for the LCY class. The NCY sectors show an almost neutral net causality. Thus, net causalities confirm that the ECY sector-class is early-cyclic, the LCY class is late-cyclic and the NCY class is almost neutral with respect to the serial cross-dependencies of the sectors. In the following, sector-classes will be used instead of sectors to specify risk classes.
The increased number of price observations in sector-classes will affect the fitting of yield curves in the next section. Four effects are expected: (1) a reduced variation of sector- class yield curves in time, (2) residuals of yields against sector-curve-induced yields will show wider spreading, (3) a more synchronous co-movement of sector-class spreads, i.e. a higher dependence of the credit risk of sector-classes as compared to a sector setting, so that systematic factors show elevated correlations. The conjectured effects, however, will not be subject to an empirical assessment, since the fitting of yield curves for sectors is not possible due to lack of data.