Application of Bayesian networks in analysing tanker shipping bankruptcy risks

2.1 Background of understanding of bankruptcy law

The federal bankruptcy court provides the basic information about how the bankruptcy laws are designed in assisting the troubled businesses to have access to restructure their debts to pay their creditors using possible liquidation:

Bankruptcy laws are normally designed to protect firms experiencing short term financial difficulties that can be overcome by restructuring. A written disclosure statement and a plan of reorganization must be filed with the court. The disclosure statement is a document that must contain information concerning the assets, liabilities, and business affairs of the debtor sufficient to enable a creditor to make an informed judgment about the debtor’s plan of reorganization (www.pacer.gov/psc/eresources.html; www.uscourts.gov).

For example, to keep business survived and pay creditors over time, illiquid and/or insolvent tanker shipping firms that filed bankruptcy – Chapters 11 or 15 – in the USA are able to save costs in ways non-bankrupt firms cannot. The purpose of disclosure is to provide sufficient information to creditors to make fair judgement regarding the feasibility of restructuring process and possible liquidation. Thus, other than the standard information such as debtor’s name, tax identification, location, etc., the debtor also needs to disclose the important information about financial and operating performance such as the schedules of assets and liabilities, current income and expenditures, contracts and unexpired leases and financial affairs in a voluntary petition.

2.2 Studies on methodology of predicting bankruptcy

The earliest bankruptcy prediction was in 1932 when FitzPatrick compared 13 financial ratios to distinguish failed and successful firms (FitzPatrick, 1932). Statistical methods have been applied in bankruptcy prediction since 1960s. Knowing that accounting ratios might have linear relationship, Altman (1968) built a multivariate discriminant analysis model. Then, “Z-score” measuring liquidity, profitability, earnings, solvency and management was introduced to the bankruptcy study. Since then, research on the bankruptcy prediction has dramatically increased, and logit analysis, probit analysis and neural networks have become the main stream.

Sun and Shenoy (2007) described the development of bankruptcy prediction models from the simple univariate analysis (Beaver, 1966) and multiple discriminant analysis (Altman, 1968) in the 1960s, to logit and probit models in the 1980s (Ohlson, 1980, Zmijewski, 1984), to neural network models (NN) (Tam and Kiang, 1992), rough set theory (McKee, 1998), discrete hazard models (Shumway, 2001), BN models (Sarkar and Sriram, 2001) and genetic programming (McKee and Lensberg, 2002). Among these techniques, BN models have many attractive features, including easiness to interpret, incorporation of both objective and subjective data and capability of conducting bi-directional analysis (i.e. forward risk prediction and backward risk diagnosis) (Yang et al., 2009). Use of BNs in financial risk prediction in general and bankruptcy in specific has been seen in the literature. According to Thomson Reuters Web of Science, 29 journal papers were found (Yan and Suo, 2013; Andersen et al., 2012) using the keywords BN and financial risk, and 22 results (Sun and Shenoy, 2007) using the keywords BN and bankruptcy. Such works have created an invaluable platform for further research in BN aiming to study financial risk analysis.

Among those recent studies on BN prediction and classification, most of the work is related to applications in finance and banking. Focusing bankruptcy predictions in a probabilistic way, Pena et al. (2011) compared traditional statistical methods such as discriminant analysis and logistic regressions to the machine-learning techniques such as Gaussian process and Bayesian Fisher discriminant in an application of computational finance. Obtaining knowledge of causal relations and significant conceptual patterns of attributes from the credit scoring models, Wu (2011) found that classification accuracy can be improved using a hybrid model of different machine learning techniques. While using Gaussian marginal densities from Bayesian model to select new variables, Feki et al. (2012) concluded a significant improvement of prediction performance in an application of bank risk prediction. Shao et al. (2011) applied BN classification to assess and forecast enterprise risk of Chinese listed companies. To identify causes and influencing factors contributing to financial crisis, Andersen et al. (2012) applied BN to four types of financial organisations and found that the industry widely failed to manage operational risks. Using accounting information, Salehi et al. (2016) compared data mining techniques such as artificial neural networks and naive Bayesian classifier in predicting corporate financial distress. Some earlier studies can be found in Gestel et al. (2006), Yin and Peng (2006) and Zhang et al. (1999) while discussing BN in risk prediction in comparison with the use of liner statistical models of discriminant analysis and logistic regression. They applied BN with posterior class probabilities of bankruptcy to analyse corporate clients, and concluded that the technique outperformed liner models with cross-validation applied.

Applications of bankruptcy and failure prediction using quantitative statistical tools on financial institutions can also be found (Gorton and Souleles, 2007; Brunnermeier and Pedersen, 2009; Shleifer and Vishny, 2010; Acharya and Yorulmazer, 2007; Cox and Wang, 2013). However, studies focusing on the failure prediction of shipping in general and tanker shipping in specific are rarely to be found in the literature, revealing the necessity of this work. Few statistical applications in risk prediction on the shipping and maritime segments are as follows. Given that macroeconomic factors impact heavily the performance of maritime industry, Drobetz et al. (2013) found that leverage is counter-cyclical, meaning slower adjustment back to the equilibrium during recession. Leverage ratios provided an implication of liquidity and solvency status. Similarly, Yeo (2016) identified the trade-off between liquidity and leverage of 130 shipping companies. A higher leverage ratio impacted cash holdings and the access to debt markets.

3.1 Bayesian network

A BN (also called belief network or probabilistic network) is a graphical presentation of probability combined with mathematical inference calculation. It is used to represent dependencies between random variables. To form a BN, each variable or a node is connected by directed links of arrows or arcs with CPT that is filled with values of probability assigned to the variables. The nodes in a BN are called chance nodes. Chance nodes represent uncertain events or variables. Each of them can be a continuous or discrete random variable or a set of events. A deterministic node is a special case of chance nodes, which operates deterministically on other nodes. The arrows are the directed links between nodes, and this direction represents the conditional dependent relationship of these nodes.

The graphical representation makes BNs a flexible tool for modelling causal impacts between events, in particular when the causal impact has a random nature. Also, specification of probabilities can be calculated to a very detailed set of the variables and the parent nodes. Having constructed the model, it can be used to compute effects of information as well as interventions from deterministic nodes. When the states of some variables are fixed, the posterior probability distributions for the remaining variables can be computed. Algorithms based on the Bayes’ rule and Chain rule (Jensen, 2001) are developed for probability updating, and they perform efficiently on a large variety of models. This makes BNs well suited to forecasting and diagnosing risks (Yang et al., 2009).

Bayes’ theorem has so far been proven to be a coherent method of mathematically expressing a decrease in uncertainty gained by (or proportional to) an increase in knowledge. As an imperative phase of probability analysis, this is achieved by combining probability distributions or functions of different parameters (such as specific outcomes or events) and revising their probabilities when new information/data are obtained.

The Bayes’ theory can be expressed by a parameter θ and given observed data x, as follows:

Generally Bayes’ rule can be considered for the problem of estimating values of k parameters (causes), θ = (θ₁,…, θ_k), using n observations (effects), x = (x₁,…, x_n). In the rule, then, given the observations x = (x₁,…, x_n), the posterior probability distribution of θ can be computed as:

Alternative forms of Bayes’ rule can also be applied to occurrence probabilities of events. In other words, the probability that an event A occurs given the condition that another event B occurs results in:

It then follows that given the situation where event B is represented by (B₁,…, B_m), the posterior probability of event A can be computed using Bayes’ rule as:

3.2 Logistic regression

In this research, logistic regression is used to provide the CPT for a discrete BN. There are two types of logistic regression: binary logistic regression and multinomial regression.

In a multinomial regression, the dependent variable y is multinomial and is modelled with different range of values for different status. Usually, the discrete-dependent variable is specified in the form of unobserved but continuous variable y*, where y* ∈ (–∞, +∞).

Consider an independent variable set X = (x₁, x₂, …, x_n) leading to the dependent variable y, where each independent variable has several status (j). Defining the unobserved variable y* as a function of X:

Therefore, we can get the conditional probability of y under a configuration of independent variable set X⁰ through multinomial regression (Rijmen and Vomlel, 2008):

Figure 1 illustrates a simple example as to how we calculate CPTs in a multi-level BN:

First, we calculate the CPT for node C based on logistics regression, then the CPT for node C is used to calculate the CPT for node E. In terms of nodes A, B and D, the percentage of each state is used as the conditional probability. In other words, we calculate the CPTs layer by layer, until the CPT of the dependent variable E is obtained.

The binary logistic regression is a special case of multinomial regression. Hence, the calculation process used in multinomial regression also fit in a binary case.

3.3 Shipping bankruptcy prediction using Bayesian network

The process of developing a data-based model consists of four phases: data acquisition, BN structure learning, BN monitoring and analysis and model validation. With respect to the four steps, a new conceptual shipping financial risk analysis methodology is developed, including the following six steps.

4.1 Data

Table I lists sound tanker shipping companies and the suspicious firms that failed, ceased trading in the stock exchange or were close to file bankruptcy in the study period from 2000 to 2010.

Macro-level variables such as annual gross domestic product (GDP) growth rate and industrial production are obtained from the World Bank. Oil prices and US stocks of crude oil and gasoline are collected from the US Energy Information Administration (EIA). Supply of oil tankers and world existing tanker fleets is obtained from the Review of Maritime Transport from UNCTAD. Tanker shipping indices such as the Baltic dirty/clean tanker indices and the Bloomberg tanker shipping index are from Bloomberg. Firm-level financial data and the decision of mergers and acquisition (M&A) strategy are available from the Thomson One. A panel data set containing cross-sectional and time-series data is applied. The sequence of the data is summarised by a number of shipping firms over the study period. If multiple financial filings are available in the same year for a company because of further adjustment, we adopt updated reclassified or restated financial data rather than original filings. We also drop the observations of a particular year out of the sample if a firm had missing data for that year.

4.2 Risk variables

This research studies why certain publicly traded shipping companies were underperformed and chose to file bankruptcy in the recent years. Given the highly unpredictable dynamics with continuous fluctuations in tanker operation, we identify factors that affect tanker performance and possible determinants of filing Chapters 11 or 15 as follows.

4.3 A proposed model

The prototype of a EWS model for analysing and predicting the failure rate of an oil tanker firm is developed by taking the risk variables as well as their interdependent relationships into consideration. The model is shown in Figure 4. In this process, domain experts are interviewed to rationalise the grades used to define each variable.

Four root nodes, five intermediate nodes and one leaf node are presented in the model. The states of each node are shown as follows:

• supply: high (≥150), low (<150);

• demand: high (≥80), low (<80);

• GDP growth: high (≥3), medium (0-3), low (<0);

• freight rate: high (≥150), medium (70-150), low (<70);

• sales growth: high (≥0.5), medium (0-0.5), low (<0);

• EBITDA: high (≥1), medium (0-1), low (<0);

• ROA: high (≥0), low (<0);

• debt ratio: high (≥0.6), medium (0.4-0.6), low (<0.4);

• current ratio: high (≥1), medium (0-1), low (<0); and

• failure: yes (1), no (0).

4.4 The conditional probability table and prior probabilities for each node

Once the BN model is developed, the next step is to establish the CPT for each node. When executing the model, the CPT of each node will be calculated through the logistics regression method.

Table VI presents the logistic regression estimates of the contribution, β_ij in Equation (5), of different variables.

When putting these coefficients β_ij into Equation (6), it is possible to calculate the conditional probabilities of different nodes in this network. With regard to the root nodes, the proportion of each defined state is used as their conditional probabilities. The relevant conditional probabilities of the other (i.e. leaf) nodes are listed in Appendices 1-6.

4.5 Results

With the established CPT of each node in the BN model, the marginal probability of each child node can be obtained using Bayes’ rules and the associated computing software packages (e.g. Netica). Figure 5 indicates that the estimated failure rate (bankruptcy rate) of an oil tanker shipping firm is approximately 30.2 per cent given the input data covering the period of 2001-2010. Such a result is consistent with the tanker company failure ratio of 26 per cent (seven failures out of 27 companies) to a large extent. It therefore partially verified the model.