Global Dimensions & Rating Anchors in Credit Risk

• Global dimensions are principal components extracted via PCA that capture shared variance among financial, economic, and ESG indicators.
• Rating anchors are empirical reference vectors that summarize the most influential features in a rating model for objective credit assessment.
• The methodology integrates relative attribute ranking and clustering to assign sovereign ratings with high concordance to established agency assessments.

Global dimensions and rating anchors are core constructs in quantitative credit risk modeling and sovereign rating methodologies. “Global dimensions” refer to the key latent factors or directions—often discovered through dimensionality reduction or statistical selection—that capture shared variance or explanatory power across a set of financial, economic, or extra-financial (ESG) indicators relevant for credit assessment. “Rating anchors” are those variables or derived constructs that serve as reference points or principal determinants in the assignment of ratings, summarizing which features rating agencies or quantitative systems assign the highest weight within the rating process.

1. Formation of Global Dimensions via Principal Component Analysis

The RELARM (RELative Attributes Rating Model) framework defines “global dimensions” as the principal directions obtained by principal component analysis (PCA) applied to a normalized data matrix $B \in \mathbb{R}^{M \times N}$, where $M$ is the number of rating objects (e.g., sovereign countries) and $N$ the number of input factors (e.g., financial/economic and expert variables). Each $b_i$ is linearly normalized such that all features $b_{ij} \in [0,1]$.

PCA is performed on the mean-centered $B$:

• The covariance matrix $C = (1/(M-1)) B^\mathrm{T} B$ is formed.
• The eigenvalue problem $Cv_k = \sigma_k^2 v_k$ ($k = 1,...,N$) is solved, or SVD is applied $B = U \Sigma V^\mathrm{T}$.
• The top $d$ principal directions $v_k$ ($k = 1,\dots, d$) are selected so that their cumulative explained variance $\sum_{k=1}^d \sigma_k^2$ exceeds a desired fraction (e.g., 95–96%).

To obtain interpretable axes, each $v_k$ is normalized by the $\ell_1$ norm:

$W_k = \frac{v_k}{\|v_k\|_1}, \quad k=1,\dots,d.$

The resulting $W_k$ provide the canonical “global dimensions”—linear combinations of the original features that best explain the data variance and subsequently serve as axes for rating comparison (Irmatova, 2016).

2. Construction of Relative Attribute Ranking Functions

On these “global dimensions,” RELARM constructs relative attribute ranking functions:

• For each object $b_i$, and each global dimension $p$, the attribute vector is $A_{i,p} = (b_{i1}W_{1p}, ..., b_{iN}W_{Np})$.
• The ranking is determined by the $\ell_1$-norm:

$\sum_{k=1}^N |b_{ik}W_{kp}| > \sum_{k=1}^N |b_{jk}W_{kp}| \implies \text{object $i$exceeds$j$along$p$}.$

• The ranking vector $w_p = (W_{1p}, ..., W_{Np})$ defines the linear ranking function:

$r_p(b_i) = w_p^\mathrm{T} b_i.$

Stacking all $d$ ranking functions yields a transformation $f: b_i \to a_i = b_i W \in \mathbb{R}^d$ mapping each rating object to the $d$-dimensional “relative attribute” space.

This construction enables pairwise and global ordinal comparisons of all objects along principal directions that are maximally informative for the rating problem (Irmatova, 2016).

3. Rating Anchors and the Role of the Rating Vector

A “rating anchor” or “rating vector” is defined in RELARM as the vector of PCA variances associated with the $d$ retained global dimensions:

$A = (\sigma_1^2, ..., \sigma_d^2)\in\mathbb{R}^d.$

In practice, $A$ encapsulates the information content of each principal component, serving as the canonical reference for mapping clusters (from k-means, see below) onto a rating scale. Projection of cluster centers onto $A$ assigns a scalar which orders clusters by presumed credit quality.

This “anchor” is not an empirical rating, but a meta-criterion constructed from the data’s variance structure: higher weights correspond to directions with greater dispersion and thus informational contribution. In this sense, “rating anchors” operationalize the principle that dominant axes of variation are the most salient for objective rating systematization (Irmatova, 2016).

4. Clustering and Category Assignment via k-Means and Projection

Following transformation to the $d$-dimensional global attribute space, k-means clustering is applied:

1. Cluster centers $CC_q\in\mathbb{R}^d$ ($q=1,\dots,k$) are found by minimizing within-cluster variance.
2. Each cluster center is projected onto the rating anchor $A$ to assign a “rating score”:

$PR_q = CC_q^\mathrm{T} A$

1. Clusters are sorted by $PR_q$, mapping to ordinal rating categories (e.g., AAA, AA, ..., CCC).
2. Each object inherits its cluster’s assigned category.

This process eliminates the need for expert-driven weight specification, instead using the eigenstructure of the data and the clustering configuration to derive an empirical, data-driven rating (Irmatova, 2016).

Empirical Evaluation

In a sovereign ratings application (30 countries, 10 input features, 6 principal components, 7 clusters), RELARM ratings matched S&P/Moody’s/Fitch broad categories in 86% of cases, demonstrating the efficacy of PCA-derived global dimensions and the rating anchor in replicating established ordinal scales (Irmatova, 2016).

5. Empirical Selection of Global Dimensions and Rating Anchors from Extra-Financial (ESG) Data

A distinct but complementary approach is provided in the analysis of sovereign risk through extra-financial (ESG) indicators (Semet et al., 2021):

• “Global dimensions” correspond to those ESG features which, after controlling for macroeconomic fundamentals and ratings, offer the greatest incremental explanatory power for sovereign bond spreads.
• Empirically, lasso-based multi-factor models identify key predictors as governance (exporting-across-borders cost, innovation capacity, corporate ethics, kidnapping severity), environmental (severe-storm hazard, temperature change, drought hazard), with social factors largely non-material for yields in the global sample.

Conversely, “rating anchors” are the ESG indicators that display direct, significant predictive power in statistical models of actual agency ratings—dichotomized into investment grade (A– or better) versus lower grades. Logistic regression models by pillar reveal governance (contract enforcement, tax procedures, government effectiveness), social (years of schooling, urbanization, health expenditure, food-price inflation), and to a lesser extent environment (climate vulnerability, water import security).

Notably, minimal overlap was found between global dimensions explaining market pricing and those functioning as rating anchors for rating agencies, with only one indicator (biodiversity threat score) significant in both contexts (Semet et al., 2021).

Table: Empirically Identified Global Dimensions and Rating Anchors

Application Context	Principal Global Dimensions (ESG)	Main Rating Anchors (ESG)
Sovereign Bond Yields	Export cost, storm hazard, innovation, ethics...	–
Credit Rating Prediction	–	Contract enforcement, schooling, etc.

6. Methodological Implications and Analytical Distinctions

The difference in composition between global dimensions and rating anchors has practical ramifications:

• For pricing sovereign risk (yields), the governance and environmental pillars are dominant at the global level, but environmental factors rise in importance for high-income countries, and social metrics become material for middle-income states.
• For rating predictions, governance and social metrics serve as the primary rating anchors, while environmental metrics are notably underweighted.

This divergence underscores the necessity to tailor ESG selection and weighting to the intended application—hedging/trading versus credit assessment. In particular, ESG-only models frequently replicate established ratings to a high degree (up to 96.7% accuracy for dichotomized categories), demonstrating that extra-financial analysis is intrinsic rather than peripheral to contemporary sovereign rating methodologies (Semet et al., 2021).

7. Conclusions and Cross-Context Relevance

Global dimensions and rating anchors serve as foundational constructs in statistical rating model design. RELARM formalizes the extraction of principal global dimensions for object ranking and category assignment using variance-based anchors, eliminating the need for manual weighting. In empirical macro-financial contexts, global dimensions correspond to those ESG and governance factors most material for yield prediction, while rating anchors are those features most predictive of agency rating outcomes.

There is minimal overlap between these sets, highlighting differing evaluative logics between markets and agencies. Alignment of investment, risk, and rating systems thus requires explicit consideration of which dimensions and anchors are most appropriate to each context, especially when employing extra-financial data-derived models (Irmatova, 2016, Semet et al., 2021).