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As a result, we use the two periods as the unit of analysis. Period 1 is made up of the trading days from to while Period 2 is from to The clustering results guarantee that the correlation structure stays the same, which is critical for the analysis in the next sections. The clustering results are illustrated by shadows, and the line represents the average spreads. In this section, we will first analyze the localization properties of the eigenvectors to identify the global components, localized components and noise components.

We calculate IPRs of the eigenvectors through the two periods, respectively. The results are shown in Fig 3. During each period, we conduct the shuffling operations mentioned in Section 2 for times. The shuffled IPRs are also shown in Fig 3 by green points, which are distributed uniformly around 0. The y-axis represents the corresponding IPR I k. The green points represent the I k s of the eigenvectors derived from the shuffled data. The blue solid lines represents the mean values of the I k s derived from the shuffled data. As a result, the first 2 components are identified as the GCs in both periods, while the 3 rd to the th R th components in Period 1 and the 3 rd to the th R th components in Period 2 are the LCs.

The 2 GCs explain It is unreasonable to ignore the LCs. Then we look into the eigenvectors of the GCs to find out the interpretations. We use the box plots to illustrate the distributions of the eigenvectors, respectively, during the two periods. According to the credit risk theory, especially the reduced form model [ 24 ], the credit risks mainly depends on the terms to maturity and bond ratings [ 25 , 26 ].

Fig 4 illustrates the distributions of the two GCs grouped by terms to maturity and ratings in Period 1, respectively. All of the coefficients of the first GC are negative except a few outliers, which means that the spreads move in the same direction. In the second GC, the box plot shows that longer terms to maturity lead to lower distributions of the coefficients, which means the long term bonds move in the opposite direction with the short term bonds. The box plots in the first row illustrate the distributions of the two GCs grouped by terms to maturity, respectively, in Period 1 from to The box plots in the second row illustrate the distributions grouped by ratings.

The second GC is relevant to terms to maturity, while neither component is relevant to ratings. Fig 5 shows the interpretations of the two GCs during Period 2, which are consistent with those of Period 1. The coefficients of the first GC are negative, which reflects co-movements. Besides, the first GC is also relevant to ratings. The bonds with lower ratings have higher dispersions in the first GC distribution. That is, the lower the bond ratings are, the more dispersed its coefficient distributions become. The second GC is relevant not only to terms to maturity, but also bond ratings.

In the second PC, the lower the ratings are, the higher the coefficient distributions are. It means the spreads of high rated bonds mainly AAA-rated bonds move in different direction with the spreads of low rated bonds mainly AA and AA- rated bonds. The reason why both of the GCs in Period 2 are relevant to ratings is that many default events broke out then due to the declining increasing rate of macro economy, and the investors managed to sell the low-rated bonds.

The box plots in the first row illustrate the distributions of the two GCs grouped by terms to maturity, respectively, in Period 2 from to While the second row are the distributions grouped by ratings. Besides the global components, LCs are also very important in describing market characteristics. The economic meanings of the separate LCs are obscure and chaotic. It is hard to extract meaningful information from separate LCs, especially the higher number of components, using the same way as PCA.

Besides, there are hundreds of LCs in the complex system, and it is impossible and unnecessary to study all the LCs separately. So We construct the LC portfolios by the following formula, and apply network analysis to identify the cluster information:. The reason why we use the weighted summation of LCs rather than the equal-weighted summation is that we would like to include the eigenvalue information into consideration.

Econophysics of Markets and Business Networks

The eigenvectors contain the structural information of the system, and the eigenvalues indicate the influence power of the components to the system. It is necessary to combine both of the eigenvalue and eigenvector information. In order to demonstrate the results clearly, we construct networks based on C LC by the threshold method.

The LC portfolios contain not only useful structure information, but also noise correlation. As a proper threshold should be able to filter the noise links in the network, it inspires us to select the thresholds based on the shuffled data C shuffle derived in Section 4. The correlations of the shuffled data C shuffle are considered as the noise correlations, and we choose the 95th percentile of the shuffled correlations as the threshold in each period.

The threshold is 0. Fig 6 shows the PDFs of the shuffled correlations, the raw-data correlations and the LC portfolio elements. The black line represents the probability density function PDF of the shuffled correlation coefficients. The red line represents the PDF of the correlation coefficients derived from the raw data. We construct networks by filtering the edges according to their weights.

Only the edges whose weights are higher than the threshold can be kept. We define the weighted degree in the same way as Barrat et. WD measures the strength of a vertex in terms of the total weights of all its connections [ 27 ]. The larger the WD is, the more important the vertex is to the system. We apply the Louvain method to the LC networks to detect clusters.

The Louvain method is a heuristic method based on modularity optimization and outperforms other methods in terms of computation time and modularity quality [ 28 ]. In order to interpret the clusters derived from the LC networks, we define the weighted degree proportion WDP of an economic property as.

These three properties are considered to have great influence on the spreads. There are three types of enterprise ownerships, i. There are four levels of bond ratings, i. The OCI bonds are defined as the bonds whose issuers belong to iron or mining industry, because Chinese government put iron and mining industries on the list of overcapacity industries OCIs in February The developments of the OCIs were to be constrained by the government, which was an important step of the supply-side reform in China and its impact on the corporate bond market was significant.

Additionally, we compute the average term to maturity of each bond cluster to help us interpret the networks. In this part, we will present the LC analysis results of Period 1. The left plot of Fig 7 demonstrates the LC network during Period 1. The sum of the edges in the LC network is , which is 7.

The Louvain method is applied to the network and 4 clusters are identified. The WDPs of ownership, rating and OCI, and the average term to maturity of every cluster are computed to help us to understand the network. All of the results are demonstrated in Table 1. The left plot shows the LC network of spreads in Period 1. The colors and the numbers on the nodes denote which clusters the nodes belong to.

The sizes of the vertices are measured by WDs. The right plot shows the ordinary correlation network of spreads in Period 1, whose nodes cluster according to ratings. In the LC network, the spreads form 4 clusters in Period 1. Among them, most of the bonds in Cluster 1 are long-term, AAA-rated bonds.

It will threaten the safety of Chinese economy and national defense if these CSOEs default and go bankrupt. The investors consider these bonds to have the implicit governmental guarantees so that their chances to default are much lower. That is why they are often called the super AAA bonds by the investors. Cluster 2 is similar to Cluster 3 in respect to the ownerships and ratings, but the average term of Cluster 3 is longer.

The right plot of Fig 7 shows the ordinary correlation network of spreads during Period 1 to demonstrate the advantage of the LC network. We tried to apply the same threshold as the LC network, and then the ordinary correlation network includes edges, which is The ordinary correlation network is too dense to discover the correlation structures. So we keep the same number of the edges as the LC network, which is 7. We find that the clusters are divided according to the bond ratings. Though rating is an important factor for spread pricing according to the reduced form approach, spreads are influenced by many factors, and they change with the economic states.

This is where the advantage of LC networks comes in, as it sheds light on other factors which are usually ignored such as ownership as well. Finally, we demonstrate the LC analysis results of Period 2. The sum of the edges in the LC network is , which is 9. The left plot of Fig 8 demonstrates the LC network during Period 2.

Cluster 1 is the collection of super AAA bonds with long term to maturity. This cluster has been consistently identified in both of the periods. Since the Chinese government was planning the supply-side reform, the OCIs were expected to reduce production by merging and acquisition, and the firms in these industries were encouraged to deleverage.

Victor Yakovenko: Econophysics

The bonds issued by firms of these industries were facing huge default risk, though the average spread of the whole market was declining. The diversification of the ratings shows that investors would like to sell the OCI bonds no matter what levels of their ratings were. However, China Shenhua Energy Company Limited, which is the biggest mining corporation and a CSOE, is not included in this cluster, due to the fact that it is of vital importance to the Chinese economic development and can also obtain implicit governmental guarantee.

Other bonds are split into two clusters, i. The two clusters are connected to each other, which indicates that most of the bond spreads fluctuate following the main part of the market.

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This fact is widely accepted by the bond market participants. The left plot is the LC network of spread in Period 2 and the right plot is the ordinary correlation network of spreads in Period 2.

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The denotations of these plots are the same as Fig 7. We compare the GCA results with that of ordinary correlation network analysis at first. The right plot of Fig 8 shows the ordinary correlation network of spreads during Period 2. If We use the same threshold as the LC network, the ordinary correlation network will include edges, which is It is too dense to discover the inner correlation structure. So we keep the same number of the edges as the LC network, which is 9.

In the ordinary correlation network, the clusters are still divided according to ratings, the same as Period 1. Throughout the two periods, the clustering results of the ordinary correlation networks reflect only the ratings of the bonds, without information related to macro-economic events or important policies. The results of the LC networks perform better than the correlation network approach in this regard as it shall reflect the dynamic evolution of market and economic state.

In addition, it shall identify at least two collections of bonds worth attention. Investors considered these bonds to have implicit governmental guarantees, and they tended to buy these bonds if the expectation of default was high. The other cluster is comprised of the OCI bonds, due to the fact that investors tended to sell the OCI bonds because of the supply-side reform.

PCA selects the dominant PCs by the proportion that the PCs contribute to the total variance, which is arbitrary in some extent. We identify the first 3 components in Period 1 which contribute to Recall that we have analyzed the first two PCs in Section 5. The first PC reflects the co-movements of the spreads.

The third components in the two periods are relevant to neither terms nor ratings, which are shown in Fig 9. The third components in the two periods are relevant to neither terms nor ratings. In the first row, the distributions are grouped by terms to maturity, and in the second row the distributions are grouped by ratings.

Citations Publications citing this paper. Liu , Songnan Xi. Four types of emergence: a typology of complexity and its implications for a science of management Benyamin M. Bergmann Lichtenstein , Bill McKelvey. References Publications referenced by this paper. Beyond Gaussian averages: Redirecting organization science toward extreme events and power laws. Andriani , B. It is written in plain language so that it can be understood by anyone with basic econometrics but zero knowledge of nonparametric methods. And it contains enough specifics that clearly spell out steps to implement those methods.

It provides an extremely thorough coverage of our knowledge in the area of nonparametric and semiparametric methods as they apply to economic models and economic data. And it makes accessible, for the first time, a body of relatively new material relating to discrete and 'mixed' data.

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