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First Significant Digits and the Credit Derivative Market During the Financial Crisis

The Credit Default Swap (CDS) market has both been lauded for its ability to stabilize the financial system through credit risk transfers and been the source of regulatory concern due to its size and lack of transparency. As a decentralized over-the-counter market, detailed information about pricing mechanisms is rather scarce. To investigate reported CDS prices (spreads) more closely, we make use of empirical First Significant Digit (FSD) distributions and analyze daily CDS prices for European and US entities during the financial crisis starting in 2007. We find that on a timeaggregated level, the European and US markets obey empirical FSD distributions similar to the theoretical ones. Surprising differences are observed in the development of the FSD distributions between the US and European markets. Whereas the FSD distribution of the US derivative market behaves nearly constantly during the last financial crisis, we find huge fluctuations in the FSD distribution of the European market. One reason for these differences might be the possibility of strategic default for US companies due to Chapter 11 and avoided contagion effects.


Introduction
The financial markets and the world economy as a whole are currently beset by huge uncertainty. What was sparked by a decrease of housing prices in the US eventually led to a near collapse of the global credit markets. In this article, we use empirical first signifi-price setters and traded volumes. We find that for Europe and the US, the first digits follow an FSD distribution pattern similar to the proposed Benford-like FSD distribution; that is, the appearance of the first digits follow a weakly monotonic decreasing pattern as provided by (Benford, 1938;Grendar et al., 2007).
Secondly, provided with daily data, we are interested in the development of the FSD distributions during the financial crisis of 2007. Here we find huge differences between the FSD distribution of US companies and that of European ones. Quite surprisingly, for US companies, the FSD distribution of the CDS spreads remained nearly stable during the financial crisis, which poses the question of why the CDS market of this region -which was the origin of the financial crisis -is more stable than the European market in terms of its FSD distribution.
This article is organized as follows: Section II briefly reviews the literature on FSD distributions and the CDS market. Section III describes our data. Section IV presents the main results of this article. Finally, our conclusions and discussions about further research follow.

Benford's Law, Benford-like Distributions and the CDS Market
Benford's Law (Benford, 1938) is an unexpected mathematical relationship that states that the FSDs of numerous examples of data follow a specific distribution and are not uniformly distributed, as one would expect. It postulates that the probability that the first digit is i = {1 . . . 9} is given by p(i) = log10 (1 + 1/i).
For example, 1 appears approximately 30.1% of the time as the first digit, and 9 appears only 4.6% of the time. Nearly 60 years later, (Hill, 1995) provides a rigorous proof of this law as well as the conditions under which it holds. Today, a wide range of data sets have been tested according to Benford's Law, e.g., (Clippe & Ausloos, 2012;Depken, 2008;Giles, 2007;Günnel & Todter, 2009) or (Ley, 1996. Ley (1996) (DeCeuster, Dhaene, & Schatteman, 1998;Giles, 2007). However, the literature also shares the common result that we rarely find a perfect match of the observed FSDs to Benford's Law.
As noted by Scott and Fasli (2001), only approximately one half of Benford's original data sets provide reasonably close fits to the Benford distribution. Therefore, Grendar et al. (2007) propose an information theoretic approach based on the first moments of the empirical FSDs to derive modifications of the Benford distribution -Benford-like distributions. The idea of this approach is to estimate a probability distribution R that minimizes the Kullback-Leibler distance to the Benford distribution and has a first moment equal to the empirical FSD mean. Distributions for different first moments of FSDs are tabulated in Grendar et al. (2007). The resulting Benford-like distribution R provides a null distribution for testing empirical FSD distributions.
Building on the work of Ley (1996), who studied the stock market using Benford's Law, and Realdon (2008), who linked the CDS market to the stock market, we study the CDS market using FSD and Benford-like distributions.

Data
As a basis to illustrate the distribution of the FSDs and to check the appropriateness of FSD distributions for studying the OTC market, we use daily Markit CDS data for European and US companies. Markit is one of the leading data providers specializing on pricing credit derivatives. According to Markit, the CDS spreads do not represent the actual traded spreads. Instead, each contributor to Markit provides data from its books of records and/or automated trading systems. The offered Markit CDS spreads are composites of these different sources. In this work, we use daily CDS spreads in basis points (bpts.) for European and US companies that run from 2006-08 to 2010-02. We are provided with spreads for 11 different maturities ranging from 6 months to 30 years.
In addition to basic CDS contract terms, such as maturity, sector and issuer, CDS contracts of-First Significant Digits and the Credit Derivative Market During the Financial Crisis ten come in four different flavors according to their restructuring mechanism (no restructuring (XR), full restructuring (CR), modified restructuring (MR), or modified-modified restructuring (MM)).
The Markit data set contains CDS spreads in all four different restructuring versions. For studying FSD distributions, we do not exclude any of these restructuring mechanisms.
Our time series runs 868 days, which results in a total of 1.44 × 108 CDS spread observations.
For the different regions, we observe, on a daily average, 78, 289 observations for European companies and 88, 346 for US ones. Table 1 summarizes some descriptive statistics of the CDS spreads for the European and the US market. It contains the observed means, medians and standard deviations for the considered time series. We split the data into two categories, investment CDS entities and subinvestment CDS entities. The first group contains entities with a credit rating -the av-erage of the Moody's and S&P ratings, provided by Markit -of at least A, and the second group contains all CDS entities rated worse than A.

Reasonability of the data:
To verify the quality of our data, we use the approach proposed by Grendar et al. (2007). Fig. 1 presents rootograms (Tukey, 1972)    Kuiper test would reject the null-hypothesis because of the huge power that any of these tests have given the large sample size. If one takes models as approximations to reality, instead of perfect data reproducers, this can be seen as a weakness of Neyman Pearson statistics (Ley, 1996). For sample sizes N less than 5000 observations, we would not reject the null at a significance level of 1%. Table 2  A well known difference between the US and European CDS market is their composition in terms of credit quality. The US CDS market is broader in the sense of credit quality; that is, we observe more bad credit quality CDS entities. To take this fact into account, we split our data into "investments" and "subinvestments". The first group only contains entities with an average credit rating quality of at least A, whereas the second group contains all non-default entities below A, i.e., from BBB to CCC/C. The results for these two groups are illustrated in Fig. 3. As we can see in both groups -investment and subinvestment companies -the FSD distribution for US entities is more stable. The higher FSD fluctuation of the investment-grade CDS market is caused by the fact that spreads are more concentrated at the lower end of the spread scale. That is, we mainly observe spreads below 100 bpts. Therefore, a change of the credit spreads is more likely to come along with a change in the FSD than for spreads above, e.g., 100bpts. However, as we can see from Figure 3, splitting the data into investment and subinvestment grades does not sufficiently explain the regional differences in the FSD distributions.       Empirical FSD for different restructuring mechanism of the CDS contracts: In search for potential explanations for the regional differences mentioned above, we investigate the FSD distributions of the CDS spreads according to their restructuring clauses, which are a key feature of every CDS contract, defining the credit events that trigger default. Table 3 presents summary statistics for the different document clauses for both European and US CDS spreads. We can infer from Table 3 that the proportion of traded CDS contracts with either full restructuring (CR) or modifiedmodified restructuring (MM) is much higher for European CDS entities than for US ones. More than two thirds of all traded European CDS entities belong to either CR or MM. For the US market, we find a majority -nearly two thirds -of traded CDS entities belonging to the other document clauses, modified restructuring (MR) and no restructuring (XR). Table 3. Proportions of restructuring mechanisms in the CDS data.

Conclusion and Discussion
In this article, we study Benford-  defaults, the protection seller is more flexible to recover claims from the reference entity. Our results nourish the notion that especially in stormy economic periods, the chapter 11 clause may affect the pricing mechanism. The absence of chapter 11 for European companies might imply that CDS spreads for European companies are more vulnerable to pricing malfunctions like herding.
Following (Trichet, 2007) and (Park & Sabourian, 2011), in times of economic uncertainty, herding is an immanent problem in financial markets. Building on the work of Jorion and Zhang (2007), who discuss contagion effects and the US bankruptcy clauses, this article sets the scene for further research, e.g., an investigation of herding in financial markets using FSD distributions.