Credit rating analysis based on the network of trading information

Ximei Wang, Boualem Djehiche and Xiaoming Hu

Abstract

In this paper, we investigate a credit rating problem based on the network of trading information (NoTI). First, several popular tools, such as assortativity analysis, community detection and centrality measurement, are introduced for analyzing the topology structures and properties of the NoTI. Then, the correlation between the characteristics of the network and the credit ratings is investigated to illustrate the feasibility of credit risk analysis based on the NoTI. Sovereign rating based on the world trade network is analyzed as a case study. The correlation between the centrality metrics and the sovereign ratings conducted by Standard & Poor’s clearly shows that highly ranked economies with vigorous economic trading links usually have higher credit ratings. Finally, a simulation is conducted to illustrate the degree of improvement in credit rating prediction accuracy if the NoTI is considered as an additional attribute.

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Introduction

1 Introduction

Credit rating has been extensively investigated by academics and practitioners alike as one of the key topics of financial risk management analysis for individual customers, giant commercial loans and even governments (Beaver 1966). The purpose of credit rating is to assess the ability and willingness of individuals to meet their financial obligations on time. For banks or lending institutions, it is important to evaluate whether the applicants or borrowers have a good credit rating. In addition, the ratings for debt issuers (eg, financial institutions or governments) play an important role in investors’ decisions.

Current credit rating methods can be divided into two categories (Li et al 2006; Huang et al 2004): traditional credit modeling and statistics-based modeling. The traditional credit method includes the CreditMetrics model of JP Morgan (1997), the CreditRisk$+$ model proposed by Credit Suisse Financial Products (Credit Suisse 1997) and the KMV model studied by Moody’s KMV Company (Bohn and Crosbie 2003). The CreditMetrics and CreditRisk$+$ models are based on estimating how the forward-looking distribution changes in value for loan portfolios and bond-type products during a given period. For the CreditRisk$+$ model, the default of an individual bond or a loan is assumed to follow an exogenous Poisson process. The KMV model focuses on computing default probabilities under the assumption that a firm’s asset returns follow the Black and Scholes dynamics (Crouhy et al 2000). In this model, default occurs when the total value of the firm’s assets falls below a certain value. Under the statistics-based approach, the credit rating is formulated as a binary or multivalued classification problem on which several algorithms, such as the logistic regression model, the ordered probit model, artificial neural network (ANN) algorithms and support vector machines (SVMs), are widely used in order to predict the probability of default (Bennell et al 2006; Nehrebecka 2018; Jones et al 2017; Xu et al 2009). For these classification models, around ten to thirty factors (called attributes), which generally consist of key financial measures, demographic information and government policy, are considered as independent variables. For example, Jones et al (2017) predict the probability of default for a company using accounting-based indicators such as liquidity ratios, solvency ratios, earnings and profitability measures, and cashflow measures. Mellios and Paget-Blanc (2006) study some important factors that determine sovereign ratings, which include per capita income, gross domestic product (GDP) growth, inflation rate, and other macroeconomic and microeconomic variables.

However, neither the model-based method nor the statistics-based method explicitly includes the network of trading information (NoTI) as a factor for credit rating, although all economic agents, ie, individual customers, companies, institutions and countries/areas, belong to some networks with common activities.

To address this issue, there are a few obvious questions that need to be answered.

• •

  Why should we bother using network analysis to study the credit rating problem?

• •

  Does network information have any effect on the credit rating problem?

• •

  If the answer is affirmative, how do we quantify the network effect?

A short answer is that networks are about relations (De Benedictis et al 2014) that can reflect some economic activities. Further, the ranking of economic agents in the network according to the denseness of their links to other economic agents may provide some insight into financial contagion and default spread. The economic agents in these networks are not independent. They are grouped or clustered following some specific interaction structure information. Hence, it is intuitive that the actions of each agent should be analyzed from a network perspective.

Recently, social networks have been considered as an effective way to analyze credit ratings for economic agents (Wei et al 2015). For example, data from users’ social network profiles, such as their education, employment history and number of friends, may determine the rating for each person (Rusli 2013). Like social networks, the networks formed by financial or trading activities among economic agents should also be relevant to credit ratings, since economic agents are more likely to take part in trading activities with trusted counterparties or with partners who have better credit ratings (or at least similar ones) except for in some fraud situations. Thus, trading, transaction and financial networks should be considered vital factors that are highly related to credit rating problems.

In this paper, we study a credit rating problem based on the NoTI. First, several popular tools are introduced to analyze the topology structures and properties of the network, such as assortativity analysis, community detection and centrality measurement. The former two are used to analyze the connection structure of the NoTI, while centrality measurement – which includes the PageRank algorithm, betweenness and closeness – focuses on the trading values (weights) in the network. Linking the relations and the trading values conveyed by those linkages are key factors in calculating centrality measurements. Second, an empirical investigation of sovereign rating analysis based on the world trade network (WTN) is conducted as a case study. The tools introduced are adopted to analyze the WTN. A combination of both binary and weighted network analysis provides a comprehensive understanding of the WTN. In particular, the correlation coefficient (CC) between the three centrality measurements and the sovereign ratings suggested by Standard & Poor’s (S&P) are computed for around 120 economies over the years 2006–17. The obtained results show that highly ranked economies with vigorous economic trading links usually have a higher credit rating. In particular, the average CC between the betweenness centrality and the credit ratings of the WTN is around 0.6 for seven years, which indicates that credit ratings are highly correlated to the WTN. Third, we conduct a simulation experiment to illustrate the impact on credit scoring prediction accuracy of adding the NoTI as an attribute. Using the logistic regression model, we show that the prediction accuracies for both the training and test data sets are higher when the NoTI is included as an attribute.

The rest of this paper is organized as follows. In Section 2, we give a brief overview of the tools used to analyze the network structure. Section 3 provides a case study of the sovereign rating and the WTN, including a data set, topology statistics, properties of the network, and an analysis of centrality and its relation to the sovereign rating conducted by the S&P rating agency. Section 4 presents a simulation in which we predict the credit rating using the logistic regression model considering the NoTI as an additional attribute. Finally, some possible extensions and future work for credit rating based on the NoTI are discussed in Section 5.

2 Methodology

In the trading and transaction networks, each agent (including individuals, companies/institutions and countries/areas) is regarded as a node, and the trading or transaction values are the weights between the nodes. Let $G=(N^t,{?}^t,{?}^t)$, where $N^t$ is the number of economic agents at time $t$; ${?}^t=(a_{ij}^t)$, where $a_{ij}^t=1$ if and only if there is a trade flow from node $i$ to $j$; and ${?}^t=(w_{ij}^t)$ is the weight from node $i$ to $j$, which is defined as $w_{ij}^t:=e_{ij}^t/\mathrm{NF}(i)$, where $e_{ij}^t$ is the outflow from node $i$ to $j$. Here, $\mathrm{NF}(i)$ stands for a normalization factor, which is used to avoid the influence of agent $i$’s intrinsic status. Specifically, NF can be the GDP of a country/area, the asset of a company/institution or the income of an individual.

To illustrate the feasibility of credit rating analysis based on the NoTI, the topology structures and properties of the network are studied. In addition, the correlation between the characteristics of the NoTI and the credit ratings is analyzed.

There are several important features of the network to be investigated. First, the assortativity and community structure are analyzed to find out the connectivity relation in the network. The assortativity analysis (Schiavo et al 2010; Serrano and Boguná 2003) is used to measure whether the nodes in the network are mostly connected to similar or dissimilar ones. If highly connected nodes prefer to connect to other highly connected nodes in a network, it is called an assortative graph; otherwise, it is a disassortative graph. Assortativity can give us a hint as to how the financial crisis will spread in a network. A simple way to analyze the assortativity of the network is to compute the CC between the node degree (ND) and the average nearest node degree (ANND). If the CC is positive, the nodes have a high probability of connecting with similar nodes; otherwise, the graph is disassortative.

The community detection technique helps us to understand how the network is organized. In this paper, we use the fast unfolding of communities based on modularity optimization for the weighted network (Blondel et al 2008) to detect communities (clusters). Modularity is an indication of whether the community assigned to a network is good or not. It measures the density of the links in the communities and the links between them, defined as (Newman and Girvan 2004; Newman 2004)

$$Q=\frac{1}{2}\sum _{i,j}\left[A_{ij}-\frac{k_i{k}_j}{2m}\right]\delta (c_i,c_j),$$

(2.1)

where $A_{ij}$ is the weight between $i$ and $j$, $k_i={\sum }_j{A}_{ij}$ is the sum of weights between $i$ and $j$, $c_i$ is the community to which agent $i$ is assigned, and $m=\frac{1}{2}{\sum }_{i,j}{A}_{ij}$ is used for normalization. The function $\delta (u,v)$ is 1 if $u=v$, and 0 otherwise. The higher the modularity $Q$, the better the communities detected in the network. The maximum value of $Q$ is 1; a value of around 0.3 usually implies a good community division.

Another important feature in the study of networks is centrality analysis, which aims to rank the importance of economic agents with different metrics. In this paper, three kinds of centrality are presented. The first is PageRank centrality, proposed by Brin and Page (1998), which uses the link structure of the web to calculate webpage rankings. This algorithm works on the principle that if a webpage has important incoming links, then it must also be considered important. Thus, it relies on backlinks to calculate the rankings of webpages.

In this paper, the weighted PageRank (WPR) proposed by Sharma et al (2010) is used for the weighted network. This algorithm uses the number of visits from inbound links to webpages in order to display the most valuable pages at the top of a results list. These rankings are calculated as follows:

$$\mathrm{PR}(u)=\frac{(1-d)}{N}+d\sum _{v\in B(u)}\mathrm{PR}(v)W_{\mathrm{vol}}(v,u),\sum _{u\in G}\mathrm{PR}(u)=1,$$

(2.2)

where $N$ is the total number of agents in the network, and $\mathrm{PR}(u)$ and $\mathrm{PR}(v)$ are the ranking scores of agents $u$ and $v$, respectively. $d$ is a damping factor (Google uses $d=0.85$) and $B(u)$ is the set of agents that links to $u$,

$$W_{\mathrm{vol}}(v,u)=\frac{\mathrm{weight}(v,u)}{{\sum }_{p\in N(v)}\mathrm{weight}(v,p)},$$

where $\mathrm{weight}(v,u)$ is the numbers of links from $v$ to $u$, and $N(v)$ represents the neighboring sets of node $v$. The ranking scores can be calculated using an iterative method.

There is also the betweenness centrality, which measures the extent to which a node lies on the paths between other nodes. It quantifies the number of times a node acts as a bridge along the shortest path between two other nodes (Freeman 1977). Nodes with high betweenness may have considerable influence within a network by controlling the spread of information. They are also the nodes whose removal from the network will cause the most disruption of communication between other nodes, since they lie on important paths conveying significant messages (Kanno 2015). Betweenness is computed as

where $g_{j,k}$ is the number of shortest paths between agents $j$ and $k$, and $g_{j,k}(a_i)$ is the number of shortest paths between agents $j$ and $k$ along which agent $i$ acts as a bridge. If this ratio (after a normalization) is close to one, then agent $i$ acts as a bridge along most of the shortest paths connecting agents $j$ and $k$, whereas if it is close to zero, then agent $i$ is less important to agents $j$ and $k$.

Finally, we have the closeness centrality. This is defined as the function of farness, which represents the reciprocal of the sum of the shortest path distances to all other nodes (Newman 2008). The closeness centrality for agent $i$ is as follows:

$$C(a_i)=\frac{N-1}{{\sum }_{j\ne i}^{N}d(a_i,a_j)},$$

where $d(a_i,a_j)$ is the number of edges in the shortest path between agents $i$ and $j$, and $N$ is the number of nodes in the network.

3 A case study: the world trade network

In this section, we investigate the topology structure and properties of the WTN as a case study. The methods we mentioned in the previous section are used in this analysis of the WTN. The relation between the characteristics of the WTN and sovereign rating conducted by the S&P is also analyzed. The obtained results show that the credit ratings are closely related to the NoTI.

3.1 Data and network topology

We collect data from the UN ComTrade database, which is the largest repository of international trade data available online. It provides international merchandise trade statistics, including data on imports, exports, re-imports and re-exports, for about 220 reporting economies from 1962 to 2017 in US dollars. Our sample refers to $t=2006,2008,2010,2012,2014,2016,2017$, and $N=120$ economies. (We consider economies with credit ratings provided by the S&P agency as well as GDP data from the reporter.) In this paper, we follow the receipt flow, not the flow of goods, since imports (exports) actually represent the outflow (inflow) of money. The WTN is considered a directed and weighted graph: see Figure 1 for the WTN of 2017. Nodes are economies and edges are the trading receipt between them.

Figure 1: The WTN of 2017. Each node represents a different economy. Edges are trading relations between economies. Eleven economies are marked in the network, ie, United States (USA), Netherlands (NLD), France (FRA), Germany (DEU), China (CHN), United Kingdom (GBR), Italy (ITA), Indonesia (IND), Belgium (BEL), Spain (ESP) and Sweden (SWE).

In the WTN, $a_{ij}^t=1$ if and only if there is a trade (receipt) flow from economy $i$ to $j$; $w_{ij}^t=e_{ij}^t/{\mathrm{GDP}}_i^t$, where $e_{ij}^t$ is the exports/imports from economies $i$ to $j$; and ${\mathrm{GDP}}_i^t$ is the GDP values of economy $i$ in year $t$. $e_{ij}^t$ is normalized by the GDP to eliminate the impact of different economic development on trading values. In the WTN, we have four trade flows. If economy A is the reporter, we have (1) exports from economy A to economy B and (2) imports from economy B to economy A. Correspondingly, if economy B is the reporter, we have (3) exports from economy B to economy A and (4) imports from economy B to economy A. In this paper, we calculate $e_{ij}^t$ as the trade volume between $i$ and $j$ at time $t$; specifically, $e_{ij}^t$ is the mean value of imports from $i$ to $j$ and the exports from $j$ to $i$ at time $t$. Finally, $W^t=\{w_{ij}^t\}$ is renormalized with the min–max normalization method.

Table 1 presents the statistics of the WTN for some intermittent years from 2006 to 2017.¹¹ 1 The links between economies with trading values less than one million are omitted. Note that there is a general uptrend in the trade links between economies, network density and ND. The network density is around 60% for each year (network density is calculated as the existing links over the maximum possible links). The average in-degree is the same as the out-degree, while the standard deviation (SD) of the in-degree is ten more than the out-degree, which implies that in most economies there is both import and export trading, while the distribution of the in-degree is more dispersed.

Table 1: Statistics of the WTN.

	2006	2008	2010	2012	2014	2016	2017
Number of edges	8667	9143	9197	9475	9404	9431	9074
Density	0.607	0.640	0.644	0.664	0.659	0.660	0.646
In-degree (SD)	72 (34)	76 (33)	77 (33)	79 (33)	78 (32)	78(33)	76 (33)
Out-degree (SD)	72 (24)	76 (23)	77 (23)	79 (23)	78 (23)	78(22)	76 (24)

Figure 2 shows a histogram of the ND and node strength (NS) of the WTN in 2017. The ND, defined as $\mathrm{ND}=A^t\mathrm{?}$, is the number of partners linked to each node, where $\mathrm{?}$ is a unary vector. From the histogram of the ND in Figure 2(a), we can see that most of the economies have many partners. This indicates a high level of economic globalization. However, this phenomenon is quite different when analyzing the distribution of NS, which is the summation of all the neighbors’ weights for each node and can be calculated as $\mathrm{NS}=W^t\mathrm{?}$. NS tells us how intense these links are in the network. Figure 2(b) shows us that NS almost follows a power law distribution, which means that the majority of the nodes have weak links to others, and only a few nodes have strong links to their partners. Thus, the hierarchy of the network is clearly presented.

Figure 2: (a) ND and (b) NS distribution of the WTN (2017).

3.2 Network properties

In this section, some properties of the WTN are studied in order to analyze the relation between the NoTI and sovereign ratings. The results related to the analysis of the WTN confirm findings in previous works, including Serrano and Boguná (2003), Fagiolo et al (2008), Schiavo et al (2010) and De Benedictis et al (2014).

First, the degree–degree correlation of the WTN is calculated to see whether the network is assortative or disassortative. For the weighted WTN, the CC between the NS and the average nearest NS (ANNS) is analyzed. Panel A of Table 2 shows the CC between the ND (NS) and the ANND (ANNS). The results are different for the binary and weighted network analysis. The CC is negative between the network attribute (NA) and the ANND, suggesting that nodes with a smaller number of links tend to connect to those with a larger number of nodes. However, in view of weighted network analysis, the WTN is assortative since the CC between the NS and the ANNS is positive.

Second, we detect whether a community structure exists in the WTN. The modularity calculated by (2.1) for each year is presented in Panel B of Table 2. Figure 3 shows the community structure of the WTN in 2017. The size of nodes and width of links stand for the betweenness scores and weights between nodes, respectively. Note that the whole WTN can be divided into four groups according to the number of links and weights distributed on the links. Highly connected economies are grouped together. The representative economies of the four groups are listed in Table 3. There is an evident geographical effect on the community measures. Trading behaviors happen actively inside each community. Generally, the WTN of 2017 can be divided into four clusters: Europe, America, Asia and Africa. This makes sense as the distance between economies is an important factor in worldwide trading. It is also worth noting that there are only six economies in Africa and that although Spain and Portugal are European economies, they are more connected to some African economies than to Europe.

Table 2: Correlation coefficient between ND (NS) and ANND (ANNS) and modularity.

	2006	2008	2010	2012	2014	2016	2017
Panel A
ND-ANND	$-$0.950	$-$0.947	$-$0.952	$-$0.947	$-$0.937	$-$0.952	$-$0.951
NS-ANNS	0.059	0.063	0.032	0.035	0.074	0.065	0.088
Panel B
Modularity	0.370	0.351	0.341	0.347	0.350	0.328	0.356

Table 3: Representative economies for different communities (2017).

	First	Second	Third	Fourth
	(blue)	(green)	(yellow)	(red)
1	Netherlands	US	China	Spain
2	France	Canada	India	Portugal
3	Germany	Mexico	Japan	Morocco
4	UK	Brazil	Republic of	Seychelles
			Korea
5	Italy	Argentina	Malaysia	Cabo Verde
6	Belgium	Chile	Hong Kong	Andorra
			(China)

Figure 3: Community structure of the WTN (2017). The WTN is divided into four communities (colors). The size of the node represents the activity of that economy in the world (the larger the node size, the more active it is in the WTN). The width of the links stands for the trading volume between economies.

In addition, Table 2 and Figure 3 show how financial contagion evolves world wide (Schiavo et al 2010; Kali and Reyes 2010). The WTN displays a disassortative structure, ie, economies with few links have a strong motivation to connect to those with many economic partners. These connections are highly influenced by the geographical distribution of the economies. An uptrend or downtrend in an economy that has lots of links in one region will soon be transferred to the connected economies. Thus, a typical crisis in an advanced economy will spread around the world.

Third, we analyze the centrality of the weighted WTN using the PageRank, betweenness and closeness centralities, respectively. Table 4 shows the top six economies of the WTN (2017) ranked by our three metric measurements.²² 2 The closeness for the top six economies is 1.000, which means that these economies are connected to everyone who takes part in the trading network. The rankings of economies in the WTN indicate their order of importance in the network. Different metrics have different ranking results for the WTN, while the core countries in the network, such as the United States, China, Germany and France, rank highly for all of the metrics. The United Kingdom ranks highly for both betweenness and closeness centrality, which means that the United Kingdom plays an important role in the WTN regarding the bridge and distance characteristics. Table 5 shows the S&P ratings for the economies included in Table 4. Noting that the centrality rankings for these economies are closely related to their sovereign ratings (highly ranked economies have higher credit ratings according to the S&P), it is necessary to analyze the importance of network information in credit rating problems.

Table 4: Top six economies with different centrality metrics (2017).

	WPR (score)	Betweenness (score)	Closeness (score)
1	China (0.119)	US (0.014)	France (1.000)
2	US (0.093)	Netherlands (0.013)	China (1.000)
3	Germany (0.079)	France (0.013)	Germany (1.000)
4	Japan (0.040)	Germany (0.013)	Italy (1.000)
5	Republic of Korea (0.033)	China (0.011)	Netherlands (1.000)
6	France (0.033)	UK (0.011)	UK (1.000)

Table 5: S&P rating for top six economies (2017).

Top six	S&P
economies	rating
China	A$+$
France	AA
Germany	AAA
Italy	BBB
Japan	A
Netherlands	AAA
Republic of Korea	AA$-$
UK	AA
US	AAA

Table 6: Rating standards of different agencies.

	Rating
Grade	number	S&P
Prime	21	AAA
High grade	20	AA$+$
	19	AA
	18	AA$-$
Upper medium grade	17	A$+$
	16	A
	15	A$-$
Lower medium grade	14	BBB$+$
	13	BBB
	12	BBB$-$
Noninvestment grade	11	BB$+$
speculative	10	BB
	09	BB$-$
Highly speculative	08	B$+$
	07	B
	06	B$-$
Substantial risks	05	CCC$+$
	04	CCC
	03	CCC$-$
Extremely speculative	02	CC
	01	C
In default with little	00	SD
prospect of recovery
In default	00	D

Table 7: Correlation between sovereign rating and centrality metrics.

	Rating–	Rating–	Rating–
	WPR	betweenness	closeness
2006	0.474	0.621	0.364
2008	0.474	0.631	0.340
2010	0.488	0.661	0.354
2012	0.479	0.625	0.277
2014	0.497	0.591	0.271
2016	0.458	0.618	0.298
2017	0.444	0.591	0.293
Average	0.473	0.620	0.314

3.3 Sovereign credit rating analysis

The above analysis of the properties of the network gives us a comprehensive view of the structure of the WTN. In this section, we study the relations between sovereign ratings and network information. One intuitive way is to compute the CC for these two variables.

Hill et al (2010) categorizes the S&P ratings into corresponding numbers, as shown in Table 6; thus, each rating for an economy corresponds to a specific number. For example, the S&P’s highest rating AAA is 21, while its lowest rating D is 0. The CC is calculated between these numbers and the scores of the three metrics for all of the economies.

The CC between the three metrics and the S&P ratings for all of the economies over seven years are presented in Table 7. Compared with the other two metrics, betweenness has the strongest correlation with the credit ratings, since the average CC is 0.620. Thus, it should be considered an important metric in analyzing the credit rating problem. The year 2014 seems to be a turning point for the correlation between rating and WPR. Before the year 2014, the CC increases from 0.474 to 0.497 and then drops to 0.458 and 0.444 in 2016 and 2017, respectively. The CC between rating and betweenness drops in 2014 and 2017.

4 A simulation study

Figure 4: Subnetwork of creditworthy and credit-worthless economic agents. This subnetwork is randomly picked from the group of economic agents. The width of the link between two economic agents stands for their trading amount.

Table 8: Prediction accuracies without and with NoTI.

	Without	With
Prediction accuracy	NoTI	NoTI
Accuracy of training set	0.613	0.875
Accuracy of test set	0.550	0.750

In this section, we simulate a group of economic agents with both individual attributes and the NoTI.³³ 3 Due to its confidentiality policy, real data cannot be used for the purposes of public verification. The data on individual attributes comes from the Australian credit approval data set (Xu et al 2009). We select 100 of 690 credit card applicants randomly, and only six out of fourteen attributes are used. To validate the credit rating problem based on the NoTI, the prediction accuracies with and without the attribute of the NoTI are compared using the logistic regression model. The simulation results show that the prediction accuracies for both training and test data sets can be improved if the NoTI is considered as an additional attribute.

Specifically, there are 100 economic agents in the group; each agent has six individual attributes and is labeled “1” or “0”, where “1” stands for a creditworthy economic agent (an agent with a high credit level), “0” stands for a credit-worthless economic agent (an agent with a low credit level). The network structure between economic agents obeys the following rules: a creditworthy (credit-worthless) economic agent has a probability of $p_{10}(p_{00})$, $p_{11}(p_{01})$ of trading with a credit-worthless and a creditworthy economic agent, respectively, where $p_{10}+p_{11}=1$, $p_{00}+p_{01}=1$. Thus, there are four kinds of trading relations between economic agents with different credit levels. The trading amounts, which are the weights in the network for the four relations, are assumed to follow the uniform distribution with different upper and lower bounds. In this paper, the probabilities are set as follows: $p_{10}=0.2$, $p_{11}=0.8$, $p_{00}=0.8$ and $p_{01}=0.2$. The parameters for the uniform distribution are, respectively, $U(13,23)$, $U(10,20)$, $U(1,10)$ and $U(0,8)$ for “1” $\to$ “1”, “1” $\to$ “0”, “0” $\to$ “1” and “0” $\to$ “0”, where “1” $\to$ “1” represents a transaction between two creditworthy economic agents.

In Figure 4, we illustrate a subnetwork with thirty economic agents. It shows that the connections between economic agents with similar credit levels are more intense than those between economic agents with different credit levels. In this study, the logistic regression model is implemented to distinguish creditworthy economic agents from credit-worthless ones. We apply this model to the data set by randomly partitioning the latter into training and test sets. Table 8 shows the prediction accuracy results without and with the NoTI. Note that in the case with the NoTI as an additional attribute, the prediction accuracies for both training and test data sets have improved, with 42.741% and 36.364%, respectively.

5 Discussion and possible extensions

In this paper, we investigated the properties of WTN and analyzed the correlation between the sovereign ratings of S&P and WTN. More precisely, we found that the economies in the WTN are connected to most of the economies in the world, but only a few of the links are intense. Nodes with fewer links have a tendency to connect with nodes with more links, which means that the structure of the network contains a core and a periphery; each core in the communities is linked actively with their neighbors in the same community. The correlation between credit ratings and NoTI was the basis for the conducted simulation study of credit ratings based on the NoTI, which showed a clear improvement in credit rating prediction accuracy compared with actual metrics.

Credit rating is a complex problem in which many high-dimensional variables related to credit need to be considered. For example, credit risk analysis for economies depends on a multiplicity of influences – including financial ratios; economic, political and regulatory environments; and government budgets (Bennell et al 2006) – which are self-dependent attributes (SAs). Both quantitative and qualitative methods are used to analyze this information. Our paper provides some evidence of the high relevance of network information for credit rating through, for example, trading or transaction information, which can be called the NA. So far, the NA has not been considered for credit ratings in the literature. As we have analyzed in this work, a natural way to improve the credit rating of economic agents would be to extend the existing (logistic or logit) regression analysis of the probability of default to include trading network information such as

$$\mathrm{logit}(\mathrm{PD}(y_i))=\sum _{j=1}^m{\alpha }_{ij}{\mathrm{SA}}_{ij}(y_i)+{\beta }_i\mathrm{NA}(y_i),$$

(5.1)

where $\mathrm{PD}(y_i)$ is the probability of default for agent $y_i$ (the credit rating is high if the probability of default of an agent is low), ${\mathrm{SA}}_{ij}(y_i)$ is self-attributive information and $\mathrm{NA}(y_i)$ is network-attributive information for agent $y_i$, and $m$ is the number of self-dependent attributes. In this paper, the scores of centrality act as a kind of NA.

In particular, if ${\beta }_i=0$, $\mathrm{PD}(y_i)$ satisfies the traditional classification method using the logistic regression model. If ${\alpha }_{ij}=0$ for $j=1,2,\mathrm{\dots },m$, the probability of default is only decided by the NA information.

Our findings suggest that network analysis plays an important role in the credit rating problem. More work is needed to quantify the influence of network information. The next step would be to focus on how network analysis can be used to improve the credit ratings of economic agents, such as individuals, companies/institutions and countries/areas. Further, the parameter estimation for (5.1) has not yet been solved. The least squares and maximum likelihood estimation methods are often used since the ${\mathrm{SA}}_{ij}(y_i)$ for $j=1,2,\mathrm{\dots },m$ are, in general, independent attributives. However, these methods may not be suitable when NA information is considered.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

Acknowledgements

This research was supported by the Verg Foundation.

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• Keywords
• Centrality measures
• Credit ratings
• Network theory
• Sovereign debt
• Original research