Using Principal Components Analysis (PCA) for Stochastic Credit Spread Modelling in Risk Management
Using Principal Components Analysis for Stochastic Credit Spread Modelling
As part of their internal models for Solvency II or Credit Spread Risk in the Banking Book (CSRBB), insurers and banks model credit spread risk stochastically. This is a challenging task, especially since dependence between sub-categories (combinations of credit categories, ratings, and maturity buckets) is non-trivial. Standard copula choices such as the Gaussian copula or the t copula are not sufficiently flexible to capture this dependence. Principal Component Analysis (PCA) is a relatively simple approach that is able to capture the joint dependence of credit spreads for a large number of sub-categories.
What Is Principal Component Analysis (PCA)?
PCA is a mathematical procedure that transforms a number of correlated variables into a smaller number of uncorrelated variables called Principal Components (PCs). The first PC accounts for as much of the variance in the data as possible, and each succeeding component accounts for as much of the remaining variance as possible. PCA is a well-known technique and generally used for modelling yield curves, but its suitability for modelling credit spreads is often overlooked.
Applying PCA to Credit Spread Data
PCA starts with determining either the covariance matrix or the correlation matrix of the spread changes. Generally, PCA is applied to the covariance matrix if the underlying variables are homogeneous. Since credit spreads of various credit categories (for example, sovereign bonds, corporate financial bonds, corporate non-financial bonds) and ratings have significantly different characteristics and volatility, this data cannot be regarded as homogeneous. Therefore, the PCA should be applied to the correlation matrix.
Mathematical Foundation
PCA can be done by performing an eigenvalue decomposition on a correlation matrix C:
C = QΛQᵀ
where Q is an m × m matrix (with m the number of sub-categories on which the PCA is applied) whose columns are the eigenvectors, and Λ is a diagonal matrix whose diagonal elements are the corresponding eigenvalues. The eigenvalue reflects the (normalized) variance in all the risk factors that is explained by the corresponding eigenvector. The eigenvectors, which are the PCs, should then be sorted descending based on the value of the eigenvalue, and a number of PCs should be selected that accounts for a sufficient part of the variance.
Generating Stochastic Scenarios
The PCA procedure above allows determining spread changes for each sub-category as a linear combination of the relevant PCs and their scores. To produce stochastic scenarios for the future in this way, the historical time series of PC scores have to be determined. To be consistent with the use of a correlation matrix as a basis for the PCA, the spread changes have to be normalized.
If the normalized spread changes are collected in an n × m matrix S, where n is the number of historical observations, the historical PC scores Y can be determined as follows:
Y = SQ*
where Q* is the dimension-reduced version of Q. Simulations of the credit spreads are determined by:
Fitting a suitable distribution to the historical PC scores Y;
Sampling from this distribution to obtain simulated Y’s;
Backing out the simulated normalized spread changes from Y and Q*;
De-normalizing these simulated normalized spread changes to spread changes.
This procedure provides simulated spreads for all sub-categories reflecting the historical individual distributions and joint dependence. Note that the fit to the historical PC scores has to be very good to translate to adequate distribution fits of the individual risk factors.
In case it is desired to have the flexibility to fit the distributions of sub-categories individually, the simulated spreads coming from the procedure above could be translated into uniform variables. The resulting so-called “PCA copula” (see also Gubbels et al (2024)) is then used to capture the dependence between sub-categories, while the individual distributions are fitted with the distribution class of choice.
PCA vs Gaussian Copula
The figure below shows a realistic example of the flexibility of PCA (right) versus a Gaussian copula (left) in modelling the dependence between credit spreads of two sub-categories. The black dots represent the historical observed dependence, while the yellow dots are the simulations from either the Gaussian copula or PCA.
PCA captures asymmetric and tail dependence more accurately than a Gaussian copula.
The figure shows that PCA captures the asymmetric shape of the historical dependence adequately, while it also captures the tail dependence between the two sub-categories. The Gaussian copula, however, fails to capture the historical dependence due to its fixed structure.
Conclusion
Given the above, it can be concluded that PCA is a relatively simple and practical approach that is flexible enough to capture the complex dependence between credit risk variables.
Written by Richard Plat, partner of Risk at Work. To learn more about modelling credit risk (dependence), feel free to contact at richard@riskatwork.nl
