=Paper=
{{Paper
|id=Vol-3650/paper9
|storemode=property
|title=Quantifying Credit Portfolio Sensitivity to Asset Correlations With Interpretable Generative Neural Networks
|pdfUrl=https://ceur-ws.org/Vol-3650/paper9.pdf
|volume=Vol-3650
|authors=Sergio Caprioli,Emanuele Cagliero,Riccardo Crupi
|dblpUrl=https://dblp.org/rec/conf/aiia/CaprioliCC23
}}
==Quantifying Credit Portfolio Sensitivity to Asset Correlations With Interpretable Generative Neural Networks==
https://ceur-ws.org/Vol-3650/paper9.pdf
Quantifying Credit Portfolio sensitivity to asset
correlations with interpretable generative neural
networks
Sergio Caprioli1,∗,† , Emanuele Cagliero2,† and Riccardo Crupi2,†
1
Intesa Sanpaolo S.P.A., Milano MI - 20121, Italy
2
Intesa Sanpaolo S.P.A., Torino TO - 10138, Italy
Abstract
In this research, we propose a novel approach for the quantification of credit portfolio Value-at-Risk
(VaR) sensitivity to asset correlations with the use of synthetic financial correlation matrices generated
with deep learning models. In previous work Generative Adversarial Networks (GANs) were employed
to demonstrate the generation of plausible correlation matrices, that capture the essential characteristics
observed in empirical correlation matrices estimated on asset returns. Instead of GANs, we employ
Variational Autoencoders (VAE) to achieve a more interpretable latent space representation. Through our
analysis, we reveal that the VAE latent space can be a useful tool to capture the crucial factors impacting
portfolio diversification, particularly in relation to credit portfolio sensitivity to asset correlations changes.
Keywords
Variational Autoencoder, VAE, Credit Portfolio Model, Concentration risk, Interpretable neural networks,
Generative neural networks
1. Introduction
1.1. Credit Portfolio concentration risk
One of the most adopted models to measure the credit risk of a loan portfolio was proposed
in [1] and it is currently a market standard used by regulators for capital requirements [2].
This model provides a closed-form expression to measure the risk in the case of asymptotic
single risk factor (ASRF) portfolios. The ASRF model is portfolio-invariant, i.e., the capital
required for any given loan only depends on the risk of that loan, regardless of the portfolio
it is added to. Hence the model ignores the concentration of exposures in bank portfolios, as
the idiosyncratic risk is assumed to be fully diversified. Under the Basel framework, Pillar I
capital requirements for credit risk do not cover concentration risk hence banks are expected to
autonomously estimate such risk and set aside an appropriate capital buffer within the Pillar II
process [3].
AIABI’23: 3rd Italian Workshop on Artificial Intelligence and Applications for Business and Industries, November 9, 2023,
Rome, Italy
∗
Corresponding author.
†
These authors contributed equally.
Envelope-Open sergio.caprioli@intesasanpaolo.com (S. Caprioli); emanuele.cagliero@intesasanpaolo.com (E. Cagliero);
riccardo.crupi@intesasanpaolo.com (R. Crupi)
Orcid 0009–0005–6714–5161 (R. Crupi)
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
A commonly adopted methodology of measuring concentration risk, in the more general
case of a portfolio exposed to multiple systematic factors and highly concentrated on a limited
number of loans, is to use a Monte Carlo simulation of the portfolio loss distribution under the
assumption reported in [4]. The latter states that the standardized value of the 𝑖-th counterparty,
𝑉𝑖 , is driven by a factor belonging to a set of macroeconomic Gaussian factors {𝑌𝑗 } and by an
idiosyncratic independent Gaussian process 𝜀𝑖 :
𝑉𝑖 = 𝜌𝑖 𝑌𝑗 + √1 − 𝜌𝑖2 𝜀𝑖 = ∑ 𝜌𝑖 𝛼𝑗,𝑓 𝑍𝑓 + √1 − 𝜌𝑖2 𝜀𝑖 (1)
𝑓
through a coefficient 𝜌𝑖 . The systematic factors {𝑌𝑗 } are generally assumed to be correlated,
with correlation matrix Σ. The third term of Eq. 1 makes use of the spectral decomposition
Σ = 𝛼𝛼𝑇 to express 𝑉𝑖 as a linear combination of a set of uncorrelated factors {𝑍𝑓 }, allowing for a
straightforward Monte Carlo simulation.
The bank’s portfolio is usually clustered into sub-portfolios that are homogeneous in terms
of risk characteristics (i.e. industrial sector, geographical area, rating class or counterparty
size). A distribution of losses is simulated for each sub-portfolio and the Value at Risk (VaR) is
calculated on the aggregated loss.
The asset correlation matrix Σ is a critical parameter for the estimation of the sub-portfolio loss
distribution, that is the core component for the estimation of the concentration risk. Therefore
it is worth assessing the credit portfolio VaR sensitivity to that parameter.
1.2. Sampling Realistic Financial Correlation Matrices
As reported in [5],
“markets in crisis mode are an example of how assets correlate or diversify in times
of stress. It is essential to see how markets, asset classes, and factors change their
correlation and diversification properties in different market regimes. […] It is
desirable not only to consider real manifestations of market scenarios from history
but to simulate new, realistic scenarios systematically. To model the real world,
quants turn to synthetic data, building artificially generated data based on so-called
market generators.”
Marti [6] proposed Generative Adversarial Networks (GANs) to generate plausible financial
correlation matrices. The author shows that the synthetic matrices generated with GANs present
most of the properties observed on the empirical financial correlation matrices estimated on
asset returns. In line with [6] we generated synthetic asset correlation matrices verifying some
“stylized facts” of financial correlations.
We used a different type of neural network, Variational Autoencoders (VAE), to map historical
correlation matrices onto a bidimensional “latent space”, also referred to as the bottleneck of
the VAE. After training a VAE on a set of historical asset correlation matrices, we show that
it is possible to explain the location of points in the latent space. Furthermore, analyzing the
relationship between the VAE bidimensional bottleneck and the VaR values computed by the
Credit Portfolio Model using different historical asset correlation matrices, we show that the
distribution of the latent variables encodes the main aspects impacting portfolio’s diversification
as presented in [7].
2. Sensitivity to the Asset Correlation matrix
2.1. Data
The dataset contains 𝑛 = 206 correlation matrices of the monthly log-returns of 𝑀 = 44 equity
indices, calculated on their monthly time series from February 1997 to June 2022, using overlap-
ping rolling windows of size 100 months. Historical time series considered are Total Market
(Italy, Europe, US and Emerging Markets) and their related sector indices (Consumer Discre-
tionary, Energy, Basic Materials, Industrials, Consumer Staples, Telecom, Utilities, Technology,
Financials, Health Care), the source is Datastream.
2.2. Variational Autoencoder design
An autoencoder is a neural network composed of a sequence of layers (“encoder” E) that perform
a compression of the input into a low-dimensional “latent” vector, followed by another sequence
of layers (“decoder” D) that approximately reconstruct the input from the latent vector. The
encoder and decoder are trained together to minimize the difference between original input
and its reconstructed version.
Variational Autoencoders [8] consider a probabilistic latent space defined as a latent random
variable 𝑧 that generated the observed samples 𝑥. Hence the “probabilistic decoder” is given by
𝑝(𝑥|𝑧) while the “probabilistic encoder” is 𝑞(𝑧|𝑥). The underlying assumption is that the data
are generated from a random process involving an unobserved continuous random variable 𝑧
and it consists of two steps: (1) a value 𝑧𝑖 is generated from some prior distribution 𝑝𝜃∗ (𝑧), (2) a
value 𝑥𝑖̂ is generated from some conditional distribution 𝑝𝜃∗ (𝑥|𝑧). Assuming that the prior 𝑝𝜃∗ (𝑧)
and the likelihood 𝑝𝜃∗ (𝑥|𝑧) come from parametric families of distributions 𝑝𝜃 (𝑧) and 𝑝𝜃 (𝑥|𝑧),
and that their PDFs are differentiable almost everywhere w.r.t. both 𝜃 and z, the algorithm
proposed by [8] for the estimation of the posterior 𝑝𝜃 (𝑧|𝑥) introduces an approximation 𝑞𝜙 (𝑧|𝑥)
and minimizes the Kullback-Leibler (KL) divergence of the approximate 𝑞𝜙 (𝑧|𝑥) from the true
posterior 𝑝𝜃 (𝑧|𝑥). Using a multivariate normal as the prior distribution, the loss function is
composed of a deterministic component (i.e. the mean squared error MSE) and a probabilistic
component (i.e. the Kullback-Leibler divergence from the true posterior):
𝑛 2
1 2 2 2
𝐾 𝐿 = − ∑ ∑ (1 + 𝑙𝑜𝑔(𝜎𝑘𝑖 ) − 𝜇𝑘𝑖 − 𝜎𝑘𝑖 )
2𝑛 𝑖=1 𝑘=1
𝑛 𝑛 (2)
1 1
MSE = ∑ ∥ x − x𝑖 ∥22 = ∑ ∥ x𝑖 − 𝐷(𝐸(x𝑖 )) ∥22
𝑛 𝑖=1 𝑖 𝑛 𝑖=1
Loss = MSE + 𝛽 ⋅ KL
where 𝐸 and 𝐷 are the encoding and decoding map respectively, 𝐸 ∶ x ∈ R𝑀×𝑀 ⟶ 𝜃 =
{𝜇1 , 𝜇2 , 𝜎1 , 𝜎2 } ∈ R4 , 𝐷 ∶ z ∈ R2 ⟶ x ∈ R𝑀×𝑀 , z = 𝜇 + 𝜎 ⊙ 𝜀, 𝜇 = {𝜇1 , 𝜇2 }, 𝜎 = {𝜎1 , 𝜎2 }, 𝜀 is a
bivariate standard Gaussian variable, and 𝑛 < 𝑛 is the number of samples in the training set.
In this equation, 𝜇𝑘𝑖 and 𝜎𝑘𝑖 represent the mean and standard deviation of the 𝑘-th dimension
of the latent space for the sample x𝑖 . The loss function balances the MSE, reflecting the
Figure 1: VAE Framework: The input layer comprises 1936 nodes, corresponding to the 44 × 44 matrix
input. Subsequently, there are layers with 512, 250, and a central hidden layer with 4 nodes. These
values represent the mean and variance of a bivariate gaussian distribution. The decoder receives as
input two values sampled from the latent space and it is asked to reconstruct the input. Hence, the
architecture is symmetrically mirrored until the output layer, which also has 1936 nodes.
reconstruction quality, with 𝛽 times the KL divergence, enforcing a distribution matching in the
2-dimensional latent space. The KL divergence can be viewed as a regularizer of the model and
𝛽 as the strength of the regularization.
We trained the VAE for 80 epochs using a learning rate of 0.0001 with an Adam optimizer.
The structure of the VAE is shown in Fig. 1. We randomly split the dataset described in section
2.1 in a training sample, used to train the network, and a validation set used to evaluate the
performance. We used 30% of the dataset as validation set.
Variational Autoencoders were employed in previous works for financial applications. In
particular Brugier and Turinici [9] proposed a VAE to compute an estimator of the Value at
Risk for a financial asset. Bergeron et al. [10] used VAE to estimate missing points on partially
observed volatility surfaces. Sokol [11] applied VAEs for interest rate curves simulation.
2.3. Comparison with linear models
We compared the performances of the Variational Autoencoders with the standard Autoencoder
(AE) and with the linear autoencoder (i.e. the autoencoder without activation functions).
The linear autoencoder is equivalent to apply PCA to the input data in the sense that its
output is a projection of the data onto the low dimensional principal subspace [12]. As shown
in Fig. 2b the autoencoder performs better than VAE (Fig. 2a), while linear models have lower
(a) Variational Autoencoder (VAE) (b) Autoencoder (AE)
Figure 2: Histogram of Mean squared error (MSE) of the Autoencoder and Variational Autoencoder on
the historical correlation matrices, split into train and validation set.
(a) 2-dimensional linear autoencoder (b) 3-dimensional linear autoencoder
(PCA 2d) (PCA 3d)
Figure 3: Histogram of Mean squared error (MSE) of the linear autoencoders on the historical correlation
matrices, split into train and validation set.
performances (Fig. 3a) even increasing the dimensions of the latent space (Fig. 3b). Hence,
neural networks actually bring an improvement in minimizing the reconstruction error. The
generative probabilistic component of VAE decreases the performance when compared to a
deterministic autoencoder. On the other hand, it allows to generate new but realistic correlation
matrices in the sense of stylized facts.
2.4. Latent space interpretability
According to Miller [13] and Lipton [14]:
Interpretable is a model such that an observer can understand the cause of a decision.
Explanation is one mode in which an observer may obtain understanding in the latent space,
for instance, building a simple surrogate model that mimics the original model to gain a better
understanding of the original model’s underlying mechanics.
For the sake of our analysis, we refer to the “interpretability” of the VAE as the possibility
to understand the reason underlying the responses produced by the algorithm in the latent
space. The Variational Autoencoder projected the 206 historical correlation matrices on a two
(a) VAE, AE and 2d-PCA latent space (b) 3-d PCA latent space
Figure 4: Comparison of the latent space generated with different models. The latent space generated
by the VAE and AE are similar while the cluster of points in the middle is recovered only by the 3-
dimensional linear autoencoder.
dimensional probabilistic latent space represented by a bivariate normal distribution. As shown
in Fig. 4a, the latent space generated by the VAE and AE are similar while the cluster of points
in the middle is recovered only by the 3-dimensional linear autoencoder (Fig. 4b).
In order to understand the rationales underlying such representation, we analysed the relation-
ship of the encoded values of the original correlation matrices with respect to their eigenvectors
{𝜈𝑖 ∣ 𝑖 = 1 ∶ 𝑀} and eigenvalues {𝜆𝑖 ∣ 𝑖 = 1 ∶ 𝑀}. It turned out that the first component of the
latent space (𝜇1 ) is strongly negatively correlated to the first eigenvalue (Fig. 5).
As pointed out in [15]
“the largest eigenvalue of the correlation matrix is a measure of the intensity of the
correlation present in the matrix, and in matrices inferred from financial returns
tends to be significantly larger than the second largest. Generally, this largest
eigenvalue is larger during times of stress and smaller during times of calm.”
Hence, the first dimension of the latent space seems to capture the information related to the
rank of the matrix i.e. to the “diversification opportunities” on the market. The interpretation
of the second dimension (𝜇2 ) of the latent space turned out to be related to the eigenvectors
of the correlation matrix. In order to understand the other dimension we consider the cosine
similarity 𝛼𝑡𝑖 between the 𝑖-th eigenvector at time 𝑡 and its average over time. Formally:
1 𝑛
𝑛
(∑𝑡=1 𝜈𝑖,𝑡 )𝑇 ⋅ 𝜈𝑖,𝑡
𝛼𝑖,𝑡 = 1 𝑛 (3)
∥ 𝑛 (∑𝑡=1 𝜈𝑖,𝑡 )𝑇 ∥∥ 𝜈𝑖,𝑡 ∥
where 𝑖 is the number of the eigenvector and 𝑡 the index of the matrix in the dataset.
Figure 5: Scatterplot of the first eigenvalue 𝜆1 versus the first component of the latent space 𝜇1 , showing
a clear negative correlation.
Let us define 𝛼1 = {𝛼1,𝑡 }𝑡=1,…𝑛 , 𝛼2 = {𝛼2,𝑡 }𝑡=2,…𝑛 . The data point subgroups observed in the
space (𝛼1 , 𝛼2 , 𝜆1 ) can be traced to corresponding subgroups in the latent space (𝜇1 , 𝜇2 ), as shown
in Fig. 6.
As pointed out in [7], each eigenvector can be viewed as a portfolio weights of stocks that
defines a new index which is uncorrelated with the other eigenvectors. It follows that a change
in eigenvectors can impact portfolio diversification. We can conclude that the VAE latent space
effectively captures, in two dimensions, the main factors impacting the financial correlations,
which is determinant for portfolio diversification.
2.5. Generating synthetic correlation matrices
As explained in Section 2.2, the probabilistic decoder of the VAE allows to generate a “plausible”
correlation matrix starting from any point of the latent space. Hence, we defined a grid of 132
points of the latent space that cover approximately homogeneously an area centered around
the origin and including the historical points. For each point on the grid, we used the decoder
(described in Section 2.2) to compute the corresponding correlation matrix. Along the lines of
[6], we checked whether the following stylized facts of financial correlation matrices hold for
both the historical and the synthetic matrices.
• The distribution of pairwise correlations is significantly shifted towards positive values.
• Eigenvalues follow the Marchenko–Pastur distribution, except for a very large first
eigenvalue and a couple of other large eigenvalues.
• The Perron-Frobenius property holds true (first eigenvector has positive entries).
• Correlations have a hierarchical structure.
• The Minimum Spanning Tree (MST) obtained from a correlation matrix satisfies the
scale-free property.
We verified that the distributions of pairwise correlations are shifted to the positive and
that the distributions of the eigenvalues (each averaged respectively over the historical and
synthetic matrices) are very similar to each other and can be approximated by a Marchenko-
Pastur distribution, but for a first large eigenvalue and a couple of other eigenvalues. Regarding
the Perron-Frobenius property, we verified that the eigenvector corresponding to the largest
eigenvalue has strictly positive components. Inspecting the dendrogram of the correlation
matrices we confirmed the presence of a hierarchical structure. Finally, the distribution of the
degrees of the Minimum Spanning Tree (calculated on the mean of the matrices) is compatible
with the scale-free property, i.e. very few nodes have high degrees while most nodes have degree
1. It is worth noting that the correlation matrices analyzed for our purposes were calculated
starting from 44 equity indices (as explained in Section 2.1) instead of single stocks as shown in
[6], hence a higher degree of concentration was expected.
2.6. Quantifying the sensitivity to asset correlations
For each matrix generated with the VAE probabilistic decoder, we estimated the corresponding
VaR according to the multi-factor Vasicek model described in section 1.1. We used the VaR
metric to show a proof of concept of the methodology and to be aligned to the Economic Capital
requirements, but the same rationale can be followed adopting a different risk metric. As
mentioned in section 1.1, Vasicek multi-factor model cannot be solved in closed form solution,
hence it is necessary to run a Monte Carlo simulation for each generated matrix. We used a
stratified sampling simulation with 1 million runs. In each estimation, the parameters of the
model and portfolio exposures are held constant. Running the simulation for every sampled
point of the latent space, we derived the VaR surface of Fig. 7.
To obtain an estimate of the sensitivity of the VaR to future possible evolutions of the
correlation matrix, we “bootstrapped” (see Fig. 9) the historical time series of the points in the
2-dimensional latent space. We used a simple bootstrapping [16] and a block-bootstrapping
technique [17] on the differences’ time series of the two components of the VAE latent space,
𝜇1 and 𝜇2 (depicted in Fig. 8).
Interpolating the estimated VaR over the randomly sample grid (Fig. 7) we can derive the
Value at Risk corresponding to any point of the latent space. Hence, for each point belonging to
the distribution of correlations changes over a 1-year time horizon estimated via bootstrap, we
can calculate the corresponding VaR without recurring to the Monte Carlo simulation.
In this way, we obtained the VaR distribution related to the possible variations of correlation
matrices on a defined time horizon.
3. Conclusions
In this work we applied a Variational Autoencoder for generating realistic financial matrices
that we used as input for the estimation of credit portfolio concentration risk estimated with a
multi-factor Vasicek model. We deviated from the methodology proposed by G. Marti 2020 [6]
who adopted a Generative Adversarial Network, in order to obtain an interpretable model by
leveraging the dimensionality reduction provided by VAE. Using as a proof of concept a VAE
trained on a dataset composed of 206 correlation matrices calculated on the time series of 44
equity indices using a rolling window of 100 months, we showed how it is possible, even using
a small data sample, to derive an interpretation of the latent space that seems aligned to the
main aspects driving portfolio diversification [7].
We exploited the generative capabilities of the VAE to extend the scope of the model beyond
the necessarily limited size of the historical sample, generating a larger set of correlation
matrices that retain all the realistic features observed in the market. Therefore, the VAE has
primarily been utilised for data augmentation, assessing its efficacy in terms of the quality of the
artificially generated matrices, determined by suitably testing the stylized facts known about
the financial correlation matrices.
We computed the augmented sample of synthetic correlation matrices on a grid in the 2-
dimensional VAE latent space and for each synthetic matrix the corresponding credit portfolio
loss distribution (and its VaR at a certain percentile) was obtained via Monte Carlo simulation
under a multi-factor Vasicek model. This way we estimated a VaR surface over the VAE latent
space.
Analyzing the time series of the encoded version of the correlation matrices (i.e. the two
components of the probabilistic latent space) we easily estimated (via bootstrapping) the possible
variation of the correlation matrices over a 1-year time horizon. Finally, using the interpolated
VaR surface, we were able to estimate the corresponding VaR distribution obtaining a quantifi-
cation of the impact of the correlations movements on the credit portfolio concentration risk.
This approach gives rapid estimation of risk without depending on the extensive computations
of Monte Carlo simulation, and it does so in a compressed, easy-to-visualize space that captures
several aspects of market dynamics.
Our analysis provides clear indications that the capabilities of realistic data-augmentation
provided by Variational Autoencoders combined with the ability to obtain model interpretability
can prove useful for risk management purposes, when addressing the sensitivity of models on a
structured multidimensional market data as the correlation matrix.
Disclaimer
The views and opinions expressed within this paper are those of the authors and do not
necessarily reflect the official policy or position of Intesa Sanpaolo. Assumptions made in the
analysis, assessment, methodology, model and results are not reflective of the position of any
entity other than the authors.
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(b) The data points of Fig. (a) represented in
the parameter space defined by 𝛼1 , 𝛼2 , and
𝜆1 (also, the size of each dot corresponds to
the value of 𝜆2 ). The proximity of these data
(a) The points on the latent space (𝜇1 , 𝜇2 ) repre- points consistently mirrors the subgroups
senting the historical correlation matrices. illustrated in Fig. (a), with matching color
The latent space was conventionally parti- schemes. There is a noticeable separation
tioned in nine subgroups of data points iden- between different subgroups, and this sepa-
tified by different colors. ration is well-defined in most cases.
(c) Sampling in the latent space (𝜇1 , 𝜇2 ): each (d) The sampled points of Fig. (c), plotted with
point can be decoded into a synthetic corre- the same color schemes in the space formed
lation matrix. The latent space was conven- by 𝛼1 , 𝛼2 and 𝜆1 (also, the size of each dot
tionally partitioned in nine regions identi- corresponds to the value of 𝜆2 ). Similar ob-
fied by different colors, with the same con- servations and considerations can be drawn
vention as Fig (a). here as those made for Fig. (a) and Fig. (b).
Figure 6: The distribution of the distance of the first two eigenvectors from their respective historical
average and the distribution of the first eigenvalue characterize the regions of the latent space.
Figure 7: The surface generated from Value at Risk with respect to the points of the 2-dimensional
latent space.
Figure 8: The time series of the variations of 𝜇1 and 𝜇2 , projections of the 206 historical correlation
matrices in the 2-dimensional latent space.
Figure 9: Using simple bootstrap (on the left) or block-bootstrap (with 11 monthly steps) on the
“compressed” representation of the correlation matrices, we estimated the distribution of the possible
variation of the current matrix on 1-year time horizon.