Abstract: An enterprise's default behavior may not only cause its bankruptcy, but also lead to the financial distress of other enterprises, and even induce systemic risks. Due to the complex web of links between financial enterprises, such as an extensive use of mutual credit guarantee, large capital borrowing and complex cross-shareholding relations among themselves, financial stresses to one part of the group can spread to others, leading to a system-wide threat to financial stability. The global financial crisis that began in 2008 was triggered by the Lehman Brothers bankruptcy, which led to a comprehensive collapse of US financial system. In fact, non-financial enterprises may also withstand a financial default contagion. A specific instance is the knock-on effects of Evergrande Real Estate default behavior in 2021 that caused the deterioration of financial conditions of many real estate companies. The accurate estimation of default correlation between interested companies is quite important for the subsequent default risk measurement. However, it has always been challenging due to two main reasons. Firstly, the joint credit migration is less likely to be directly observed; and thus, the link needs to be inferred to observable correlations, such as equity returns, which may lead to inaccurate assessment of correlation. Secondly, the degree of mutual influence on default behavior from firm A to firm B and from firm B to A is different and lack of covariance-like symmetry, owing to the big difference among companies in terms of industry dominance and competitiveness.
Under the existing framework of the multivariate Markov chain model, this paper utilizes a group decision-making method to adjust the key coefficients associated with the existing approach, proposing an improved multivariate Markov chain model for default prediction. The primary distinction between this model and existing multivariate Markov chain models lies in the estimation of the credit transition influence coefficients. The current study constructs a linear optimization model with the credit transition influence coefficients as decision variables and an objective function that minimizes the distance between the probability distributions of credit statuses in two consecutive periods, reaching or approaching a stable state. Once these optimal coefficients are determined, the model can predict the default status of different firms in the next period by utilizing the current credit status of the firms and the one-step transition probability matrix. This paper introduces a novel approach within the existing framework for solving credit transition influence coefficients by incorporating experts' opinions to rank the influence of different enterprises on one company's credit change. It employs an improved G1 method to assign weights to the opinions of various experts, forming a set of inequality constraints of credit transition influence coefficients that reflect the authoritative expert wisdom in the process of the coefficients forming. This approach reveals the role of different companies in the joint credit changes, altering the optimal values of key parameters in the existing model.
The innovations and contributions of this paper are twofold: First, it integrates expert judgments on the default correlations with the determination of key parameters in the multivariate Markov chain model, addressing the issue of correlations, unobservable and difficult to measure, in the credit status changes of multiple firms. Second, by using the credit transition influence coefficients to reflect the impact of different companies on the formation of a firm's credit status in the next period, it addresses the limitations of current research that only considers the mathematical significance of these parameters in terms of normalization and non-negativity. This approach enhances the key parameter's interpretability, thereby improving the model's explanatory power.
In the empirical analysis, this paper utilizes a dataset of credit state transitions from publicly listed financial companies to construct the existing multivariate Markov chain model and the proposed improved model and compare their default prediction capabilities. The results indicate that the improved model proposed in this paper outperforms the existing model in terms of both predictive accuracy and interpretability. The optimal credit transition influence coefficient matrix derived from the existing approach shows that each row contains only one element of 1, with all other elements being 0, and the element of 1 occurs randomly and follows no discernible pattern. This suggests that a firm's future credit status is only related to the current credit status of other firms, and surprisingly, not to its own historical credit status, which contradicts common understanding. In contrast, the distribution of the optimal influence coefficients in the proposed model reveals two key characteristics: first, the diagonal elements are the largest, and second, there are more instances where non-diagonal elements are non-zero. This indicates that while a company's credit change is related to the credit changes of other companies, it is most influenced by its own credit status in the previous period. All these findings demonstrate again that human expertise and judgment introduced by this paper as a new dimension of information can correct the key parameters of the traditional multivariate Markov chain model and further enhance its predictive performance.

Key words: multivariate Markov chain model, correlation in credit risk changes, group decision-making, default prediction, listed companies

摘要： 本研究通过运用群组决策方法为信用状态变化过程中不同企业的影响力排序,进而建立信用转移概率影响系数的不等式约束,提出了一个改进的多元马尔可夫链违约预测模型。本文的创新和特色:一是将专家对企业信用变化相关程度的判断与多元马尔可夫链模型重要参数确定结合起来,解决了多家企业信用状态变化过程中相关性不可观测且难准确度量的问题。二是利用信用转移影响系数反映不同企业对下一时期同一家企业信用状态形成的影响比重,改变了现有研究仅考虑了该参数满足归一性和非负性权重的数理含义、忽略其能差异化地反映不同企业信用影响程度的弊端,提高了模型的可解释性。本文运用改进后的马尔可夫链模型对中国上市金融业企业实施了违约预测。结果显示,改进模型预测表现均高于现有模型,且一家企业未来信用状态不仅与自身当前的信用分布有关,也受到专家认为的关系密切的其他企业信用变化的影响,反映了同行业企业信用状态变化过程中的不对称影响特征。

关键词: 多元马尔可夫链模型, 信用变化相关性, 群组决策, 违约预测, 上市企业