Ratings Assignment Methodologies (III)

This is the third part of the article on Rating Assignment Methodologies. The previous parts, as well as other articles of this series can be accessed using the following these links:

• The Credit Decision
• Classifications and Key Concepts of Credit Risk (I)
• Classifications and Key Concepts of Credit Risk (II)
• Rating Assignment Methodologies (I)
• Rating Assignment Methodologies (II)

In the previous parts of this article, we went through the Merton model and the Linear discriminant analysis (LDA).

Another popular set of models are the Logistic regression (LOGIT), part of the set of models called Generalized Linear Models (GLM). These types of models are commonly used to predict default based on understanding the relationships between dependent and independent variables. GLMs typically have three common elements:

• A systematic component, specifying the variables used in a linear prediction function.
• A random component, identifying both the target variable and its probability function.
• A link function, which is a function tying the target variable mean with the the systematic component.

Sources of information for the independent variables in these models can include internal or external behavioural information (legal disputes, credit reports, etc.), financial reports of the counterparty company, behavioural data, and assessments covering factors such as management quality, competitiveness of the firm. Because of the heterogeneity of these data in terms of frequency and data type, extra care should be applied handling this data. Additionally, specific models are often built to manage these data heterogeneity (called modules) and integrated into a final rating model. A good source for more real-world flavour can be found here.

Assume that Pi  represents the probability that a default event takes place. The LOGIT (ie, logarithm of odds) equation is given by:

The LOGIT function associates the expected value for the dependent variable to the linear combination of independent variables, whereas the relationship between the probability of default (pi) and the independent variables is nonlinear.

If for example there is only one explanatory variable, the LOGIT function can be represented by the following:

The output of these models is a sample-based estimate of the probability of default. 

The probability output must be scaled, and the model must be calibrated. The rescaled default probability is given by:

A scaled default probability can be created for every value yielded from a logistic regression. The calibration is complete once each of these default probabilities is assigned to grades/ratings in a rating scale.

Both LDA and LOGIT methodologies are considered supervised given both have a defined dependent variable (the default event). When the dependent variable is not explicitly defined, the statistical technique is considered unsupervised.

Cluster analysis looks to identify identify groups of similar cases in a data sets, effectively aggregating and segmenting borrowers based on the profiles of their variables. A default rate can be calculated for each segment as a proxy for their default probability. 

Two approaches can be used to implement cluster analysis: 

• Hierarchical/aggregative clustering and 
• Divisive/partitioned clustering.

Various other analyses use techniques to optimally transform the variables set into a smaller but statistically significant set:

• Principal component analysis takes original tabular data and transforms it into a new derived data set which is used to determine the primary drivers of a firm’s profile and potential default. 
• Factor analysis is like principal component analysis but is used to describe observed variables in terms of fewer unobserved variables called ‘factors’ and can be seen as more efficient.
• The canonical correlation method is a technique used to address the correspondence between a set of independent variables and a set of dependent variables, determining the independent variables that have maximum correlations with the factors from the dependent variable set.

Cash flow analysis is a useful methodology for assigning ratings to companies that don’t have meaningful historical data, nevertheless it faces practical challenges due to the model risk of oversimplifying the reality, costs for building and maintaining the model, and accurately defining default.

Heuristic methods are also referred as expert systems and are designed to predict default by mirroring human decision-making processes and procedures. These are a traditional application of artificial intelligence and will typically involve the creation of a knowledge base and use knowledge engineering to gather and codify knowledge into a framework.

Qualitative information is a cornerstone for modelling default risk. Capturing historical data for qualitative information is particularly challenging and an area to which banks dedicate significant resources. 

Source: Source: Giacomo De Laurentis, Renato Maino, and Luca Molteni, Developing Validating and using Internal Ratings (West Sussex,UK: John Wiley & Sons, 2010). Chapter 3