# A Summary of Privacy-Preserving Data Publishing in the Local Setting

Wenjun Lin, Jiahao Qian, Wenwen Liu, Lang Wu
Department of Computer Science
National Taiwan University

###### Abstract

The exponential growth of collected, processed, and shared microdata has given
rise to concerns about individuals’ privacy. Consequently, various laws and
regulations have been established to oversee how organizations handle and
safeguard microdata. One such method is Statistical Disclosure Control, which
aims to minimize the risk of exposing confidential information by de-
identifying it. This de-identification is achieved through specific privacy-
preserving techniques. However, a trade-off exists: de-identified data can
often lead to a loss of information, which might impact the accuracy of data
analysis and the predictive capability of models. The overarching goal remains
to safeguard individual privacy while preserving the data’s interpretability,
meaning its overall usefulness. Despite advances in Statistical Disclosure
Control, the field continues to evolve, with no definitive solution that
strikes an optimal balance between privacy and utility. This survey delves
into the intricate processes of de-identification. We outline the current
privacy-preserving techniques employed in microdata de-identification, delve
into privacy measures tailored for various disclosure scenarios, and assess
metrics for information loss and predictive performance. Herein, we tackle the
primary challenges posed by privacy constraints, overview predominant
strategies to mitigate these challenges, categorize privacy-preserving
techniques, offer a theoretical assessment of current comparative research,
and highlight numerous unresolved issues in the domain.

_K_ eywords Differential Privacy $\cdot$ Information Privacy $\cdot$ Privacy
Notions

##  1 Introduction

The right to privacy has been a significant discussion topic in computing
since the late 60s. This period saw the development of legal safeguards
tailored for the digital age, reflecting the longstanding concern over
individual privacy. With the current data-intensive environment, the sheer
volume of collected and disseminated data provides detailed records on
individuals. Such microdata is indispensable for analytical tasks in data-
driven projects, assisting in research and strategy development. However, the
continuous analysis and reuse of this data on a vast scale challenges the
privacy of individuals, sparking international legal debates and heightening
concerns about how institutions handle personal data.

In response to these concerns, a range of data privacy regulations have
emerged. The GDPR[1] was introduced to streamline data privacy laws across
Europe, followed by the CCPA for Californians and the LGPD for Brazilians.
These legislations compel organizations to adopt appropriate measures for
processing personal data. While the data subject, the person to whom the data
refers, has rights such as access and erasure, data controllers, and
processors, the entities managing the data, bear obligations like ensuring
confidentiality and breach notifications.

Balancing data confidentiality with its utility is a complex endeavor. High
confidentiality might inhibit the extraction of valuable insights, whereas
minimal privacy could lead to breaches and re-identification. This balancing
act presents a dichotomy in research between data encryption and data
transformation. While encryption using cryptographic methods offers robust
security, it poses challenges in data manipulation and analysis. Data
transformation, on the other hand, modifies data for public release while
maintaining its interpretability, a crucial factor when sharing information.

This survey aims to comprehensively review the Statistical Disclosure Control
issue and privacy-preserving techniques for data transformation. We focus on
methods that prioritize interpretability and serve as foundational tools for
future computational analysis. Among our core contributions are a detailed
definition of microdata de-identification, a taxonomy of privacy-preserving
techniques, metrics for information loss and predictive performance, and
reviews of existing studies.

Previous work in this domain has either been segmented or outdated. This
survey integrates and updates prior studies, providing a nuanced analysis of
the efficacy of privacy-preservation techniques in the contemporary setting.

##  2  PRIVACY RISKS

In the realm of microdata, where specific attributes define datasets, the
choice of potent privacy-preserving techniques becomes vital for data
controllers in achieving de-identified datasets. These transformations
commonly revolve around three primary privacy threats:

Singling Out: The risk that an intruder might isolate specific records to
uniquely identify a data subject. Linkability: The danger of connecting or
correlating two or more records of an individual or a set of individuals.
Inference: The potential to derive the value of one attribute based on the
values of other attributes. Moreover, the rise in attempts to breach
confidential information has spotlighted other privacy threats, albeit less
renowned, which we’ll touch upon.

To determine the effectiveness of a chosen privacy-preserving method, data
controllers use distinct privacy measures suited for each threat mentioned
above. However, measuring disclosure risk poses challenges. Data breaches
often result from intruders leveraging external information, information that
the data controller might not have anticipated. To combat this, data
controllers must hypothesize about potential external knowledge sources to
estimate risk. This often involves exploring various scenarios, such as
different Quality Indicators (QI) that intruders might be privy to, or
considering whether intruders are aware of study participants. To err on the
side of caution, the most robust stance is assuming a maximum-knowledge
intruder, implying they’re aware of all initial attribute values. This allows
for the most accurate risk estimate.

Delving deeper, we identify four prominent disclosure risks from literature,
each with its measures: 1\. Identity Disclosure[2]: This occurs when an
intruder matches QI values to discern a specific record in the revealed
dataset is linked to a particular individual. 2\. Attribute Disclosure[3]:
Here, an intruder determines additional attributes about a data subject from
the disclosed data. 3\. Inferential Disclosure: This risk involves an intruder
deducing specific private data about a subject, backed by the statistical
characteristics of the released data. Importantly, inferential disclosure can
even jeopardize those not included in the dataset. However, since inference
techniques focus more on collective rather than individual behavior or
attributes, it’s usually not a focal point in Statistical Disclosure Control
(SDC) for microdata. As such, we won’t be elaborating on measures for
inferential disclosure in this discussion. 4\. Membership Disclosure: This is
when an intruder can discern whether data pertaining to a specific individual
is included in the dataset or not. Presently, there are only a handful of
metrics, like $\delta$-presence, which is designed to limit the probability of
deducing any potential data subject’s record within a specific $\delta$ range.
Given the limited focus on this type of risk in the current context, we’ll
omit its detailed measures moving forward.

##  3 PRIVACY-PRESERVING TECHNIQUES

The crux of de-identification lies in revealing data that aids organizations,
administrations, and businesses in informed decision-making while safeguarding
sensitive particulars of specific data subjects. The quest, therefore,
revolves around unearthing methodologies that trim down disclosure risks while
retaining the potency for statistical evaluations and data mining. This
delicate balance between safeguarding data and maintaining its utility has
fueled the quest for pioneering privacy-preserving approaches, and in some
instances, revamping of existing ones.

Pioneers in the domain, Willenborg and Waal, laid down foundational principles
for safeguarding microdata, introducing a taxonomy of privacy-preserving
methods based on microdata traits. This taxonomy segregates techniques into:

Non-perturbative Techniques: These are defined by reducing intricacy or even
omitting information altogether. The key aspect here is the absence of
inconsistencies, ensuring that intruders cannot pinpoint records that have
undergone transformation.

Perturbative Techniques: As the name suggests, these revolve around tweaking
or altering the information. The side effect of this technique is that it can
create inconsistencies in data. Such inconsistencies might pique the interest
of an intruder, drawing them to speculate on records that might have been
altered and thereby attempt to restore them to their original state.

While these classifications have dominated academic discussions, there’s a
fresh category believed to fall under Statistical Disclosure Control (SDC):
De-associative Techniques. These are designed to sever the link between
Quality Indicators (QI) and sensitive traits. Instead of releasing a unified
table, the strategy is to release two distinct tables, one with QI and the
other with sensitive attributes.

###  3.1 Differential Privacy (DP)

Differential Privacy (DP)[4, 5] aims to protect individual data privacy in
data analysis. It adds noise to the data or responses to queries to obscure
individual entries. Key Concepts: \- Epsilon ($\epsilon$): A parameter that
measures the privacy guarantee. Smaller $\epsilon$ means better privacy. \-
Delta ($\delta$): A small probability of privacy failure, usually close to
zero. \- Random Noise: Added to responses based on the sensitivity of the
query and $\epsilon$.

DP’s Basic Principle: For two datasets D and D’ differing by one individual,
and for any subset of outcomes S:

$Pr[M(D)\in S]\leq e^{\epsilon}Pr[M(D^{\prime})\in S]+\delta$

This ensures that the presence/absence of any individual in the dataset does
not significantly affect the outcome, thus preserving privacy.

###  3.2 Local Differential Privacy (LDP)

Local Differential Privacy (LDP) is a variant of DP where privacy is
guaranteed before any data leaves an individual’s device. [6, 7, 8, 9] In LDP,
each user perturbs their data locally before sending it to the aggregator.
This ensures privacy protection at the individuall evel.

Key Concepts: \- Local Randomization: Users apply a randomized algorithm to
their data locally. \- No Trust in Data Collector: Unlike traditional DP, LDP
does not require trusting the data collector with original data.

LDP’s Basic Principle: For any individual’s data x and x’, and for any subset
of outcomes S:

$Pr[LM(x)\in S]\leq e^{\epsilon}Pr[LM(x^{\prime})\in S]$

This ensures that each individual’s data remains private, even from the data
collector.

##  4 State of the Art Deployment

We delve into three tangible implementations of LDP algorithms utilized for
gathering popularity metrics. These encapsulate the evolution of ideas within
computer science research, as well as the advancements made thereafter.

1\. RAPPOR by Google[10]: Merging the concepts of Randomized Response and
Bloom Filters, RAPPOR efficiently compresses extensive sets. It’s primarily
designed to pinpoint frequently visited web destinations (URLs) while
safeguarding individual browsing patterns. The same group furthered their
research, illustrating an approach to discern popular websites without pre-
existing knowledge of their URLs.

2\. Apple’s DP Mechanism[11]: Unveiled in 2016, this implementation is
detailed in a patent application and a later white paper. The method
integrates several innovations: the deployment of the Fourier transform to
disseminate signal details and the use of sketching methods to trim down the
massive domain’s dimensionality. Concurrently, a burgeoning literature trend
has approached the task of pinpointing Differentially Private Heavy Hitters,
continually honing and enhancing these methods.

3\. Microsoft’s Telemetry Collection: This strategy utilizes histograms
coupled with set random numbers, facilitating data collection over extended
periods.

##  5 LDP variants and mechanisms

In this section, we delve into the diverse variants and mechanisms of LDP
designed to improve the trade-off between utility and privacy for various
applications.

###  5.1 Variants and Mechanisms of LDP

Geo-indistinguishability Building on the principles of differential privacy,
Geo-indistinguishability was developed primarily for location privacy. It
leverages the geographical distance inherent in data [12].

BLENDER Model Taking a step towards combining global DP and LDP, we have the
BLENDER model. This model, designed to balance data utility while maintaining
privacy, classifies users based on trust factors and optimally utilizes data
from both categories.

Utility-optimized LDP (ULDP)[13] While traditional LDP tends to add noise
uniformly, the ULDP model recognizes that not all data elements share the same
sensitivity level. This model proposes an optimization that caters to the
varying sensitivity of data points.

Input-Discriminative LDP (ID-LDP) For a more granular understanding of data
sensitivity, ID-LDP is introduced. Unlike ULDP , which classifies data into
broad categories of sensitive and non-sensitive, ID-LDP acknowledges the
varying levels of sensitivity within data. By assigning different privacy
budgets based on this sensitivity, it allows for a more nuanced privacy
mechanism.

Local Information Privacy Progressing further, the Local Information Privacy
(LIP) comes into the picture[14, 15]. Initially introduced as a version of LDP
aware of prior data, it was later adapted by Jiang et al. to assume only a
partial prior-aware state [16].

Sequential Information Privacy Building upon the concepts of LIP, the
Sequential Information Privacy (SIP)[17] focuses on the unique challenges
posed by sequential or time series data. It seeks to measure the privacy
leakage for such data, drawing similarities with the principles of LDP.

CLDP[18] Addressing one of the limitations of LDP, Gursoy et al. recognized
the decreased utility of LDP in scenarios with fewer users. This led to the
introduction of the metric-based model, condensed local differential privacy
(CLDP) [18].

Parameter Blending Privacy (PBP) Lastly, the Parameter Blending Privacy (PBP)
model offers an advanced and flexible variant of LDP. It allows for the
blending of privacy parameters to achieve an optimal privacy setting [18].

Membership Privacy [19] In Differential Privacy or Local Differential
Privacy’s definition, the likelihood ratio must be upper and lower bounded
simultaneously. However, this constraints is relaxed in the definition of
membership privacy, which only considers one side of the boundaries, and thus
providing either positive or negative memebership privacy protection.

In conclusion, the variety of LDP mechanisms and variants showcase the ongoing
efforts to balance utility and privacy in different application scenarios.
Each variant brings its own set of advantages, catering to unique challenges
in the realm of privacy preservation.

##  6 Conclusion

In conclusion, the rapid expansion of microdata usage has necessitated the
development of robust privacy protection measures. Statistical Disclosure
Control (SDC) has emerged as a critical approach to de-identify sensitive
information, balancing the privacy of individuals with the need for
maintaining data utility. Our comprehensive survey has provided an in-depth
look at the current state of SDC techniques, evaluating their effectiveness in
different scenarios, and the trade-offs they present between confidentiality
and the fidelity of data analysis. Although significant progress has been made
in the field, the quest for an ideal equilibrium between data privacy and
utility remains ongoing. As the landscape of data privacy evolves, so must the
strategies and methods for protecting individual privacy without compromising
the quality of data-driven insights. Future research directions are plentiful,
with many unresolved issues offering opportunities for innovation. The
continuous refinement of SDC techniques and the exploration of new
methodologies will be crucial in advancing the field and achieving the dual
goals of robust privacy protection and retention of data interpretability.

## References

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# Enhancing Asset Allocation in the Fixed Income Universe: A Synthetic Dataset
Approach for Robust Portfolio Construction

Szymon Kubiak  City, University of London ,  Tillman Weyde  City, University
of London ,  Oleksandr Galkin  City, University of London ,  Dan Philps
University of Warwick and  Ram Gopal  University of Warwick

(2023)

# Generating Realistic Synthetic Datasets for Improved Asset Allocation in the
Fixed Income Universe

Szymon Kubiak  City, University of London ,  Tillman Weyde  City, University
of London ,  Oleksandr Galkin  City, University of London ,  Dan Philps
University of Warwick and  Ram Gopal  University of Warwick

(2023)

# Improved Data Generation for Enhanced Asset Allocation: A Synthetic Dataset
Approach for the Fixed Income Universe

Szymon Kubiak  City, University of London ,  Tillman Weyde  City, University
of London ,  Oleksandr Galkin  City, University of London ,  Dan Philps
University of Warwick and  Ram Gopal  University of Warwick

(2023)

###### Abstract.

We present a novel process for generating synthetic datasets tailored to
assess asset allocation methods and construct portfolios within the fixed
income universe. Our approach begins by enhancing the CorrGAN model (Marti,
2020) to generate synthetic correlation matrices. Subsequently, we propose an
Encoder-Decoder model that samples additional data conditioned on a given
correlation matrix. The resulting synthetic dataset facilitates in-depth
analyses of asset allocation methods across diverse asset universes.
Additionally, we provide a case study that exemplifies the use of the
synthetic dataset to improve portfolios constructed within a simulation-based
asset allocation process.

synthetic datasets, generative neural networks, fixed income, asset allocation

††copyright: acmcopyright††journalyear: 2023††conference: 4th ACM
International Conference on AI in Finance; November 27–29, 2023; New York, NY

##  1\. Introduction

When applying asset allocation methods in practice, a crucial challenge arises
due to limited and biased empirical datasets. Conventional approaches to
constructing portfolios solely on historical data often suffer from
overfitting and neglect the potential market scenarios that could have
occurred but did not (Lopez de Prado, 2019). To address this problem and
enhance the performance of asset allocation methods during out-of-sample
periods, the utilization of synthetic financial datasets has emerged as a
promising solution.

Recent literature provides examples where synthetic correlation matrices were
used to assess asset allocation methods in simulation-based frameworks (Lopez
de Prado, 2016a; Papenbrock et al., 2021; Marti et al., 2021). However,
limitations arise when applying these methods in more complex scenarios,
especially when dealing with financial assets that exhibit diverse
characteristics or when practitioners have objective functions and constraints
that are more intricate than those of simple minimum variance portfolios. In
our paper, we analyze the fixed income universe, comprising assets with
varying volatility levels, where a frequent optimization goal is maximizing
yield within specified risk constraints. To perform a comprehensive
simulation-based assessment of asset allocation methods within this universe,
it is necessary to not only have realistic correlation matrices but also other
attributes, particularly volatilities and yield data that is coherent with the
matrices. This necessitates the use of more advanced models for data sampling.

We propose a two-step process for generating a synthetic dataset tailored
specifically to assess and fine-tune asset allocation within the fixed income
universe. First, we employ a generative adversarial network to sample
realistic correlation matrices, capturing the interdependencies among assets.
Then, we generate additional characteristics of the assets conditional on the
matrices, including volatilities, expected returns, and forward returns data,
using a proprietary Encoder-Decoder model. We hypothesize that the synthetic
dataset, with its reduced reliance on historical data, can address the
limitations of empirical datasets and improve the performance of asset
allocation methods during out-of-sample periods. Our analysis indicates that
the generative models provide a more diverse dataset that extrapolates
distributions into plausible areas and mitigates outliers, decreasing
dependency on historical events with low future probability.

We also present a case study to showcase the practical application and
potential benefits of the synthetic dataset. Our findings indicate that the
synthetic dataset enables investors to construct more robust portfolios.

In summary, our research contributes to addressing the challenges posed by
limited and biased empirical datasets in fixed income asset allocation. By
introducing a novel process for generating a tailored synthetic dataset and
demonstrating its benefits through a case study, we provide valuable insights
for practitioners and researchers in the field.

##  2\. Related Work

Asset allocation methods aim to distribute funds across different investments
to optimize return and manage risk. Lopez de Prado introduced two asset
allocation methods, Hierarchical Risk Parity (Lopez de Prado, 2016a) and
Nested Clustered Optimization (Lopez de Prado, 2016b), and to indicate their
superior performance over more traditional frameworks used a Monte Carlo
approach based on synthetic correlation and associated covariance matrices.
The matrices were formed using multivariate normal distributions with
additional random shocks in the former paper and imposed correlated blocks in
the latter article. While the performed tests provided convincing arguments
supporting the introduced techniques, the synthetic data was drawn from
relatively simple processes and may not be suited for more advanced
simulations and tests.

A more sophisticated method of sampling synthetic correlation matrices was
proposed by Papenbrock et al. (Papenbrock et al., 2021) who employed a multi-
objective evolutionary algorithm. The generated dataset was used to compare
the performance of asset allocation methods in a simulation-based framework.
The authors also applied SHAP framework (Lundberg and Lee, 2017) to find which
characteristics of correlation matrices could provide intuitive explanations
of what asset allocation method to use. The approach provided realistic
synthetic correlation matrices that exhibited many characteristics of
empirically observed ones but the algorithm was designed to optimize for the
characteristics, and as a result, it might have omitted features that were not
specifically included in the objective functions.

Generative Adversarial Networks (GANs) (Goodfellow et al., 2020) have gained
popularity in the financial domain for generating synthetic datasets. GANs
consist of a generator and a discriminator network, where the generator
generates new data samples resembling the training data, and the discriminator
distinguishes between real and fake samples. GANs have been applied in various
finance-related studies, such as generating time-series data for trading
strategy calibration (Koshiyama et al., 2021), addressing imbalanced classes
in actuarial datasets (Ngwenduna and Mbuvha, 2021), and hedging (Kim, 2021).

Marti (Marti, 2020) introduced a GAN model based on the DCGAN architecture
(Radford et al., 2015) to generate synthetic correlation matrices. (Marti et
al., 2021) extended the model to condition sampled data on market regimes and
conducted experiments to assess distributions of outcomes for selected asset
allocation strategies. The framework learns a number of characteristics of the
empirical correlation matrices in an unsupervised manner and provides high-
quality data samples. The generated matrices are visually undiscernible to
empirical matrices which was supported by results of an online poll in which
users could guess whether depicted correlation matrices are real or fake.

(Mariani et al., 2019) and (Lu and Yi, 2022) proposed alternative approaches
to generating synthetic data for portfolio construction based on GAN models
that learn a joint probability distribution of assets’ future price trends
based on historical time series. The authors indicate that portfolios
constructed using simulations derived from the GANs outperform traditional
asset allocation methods. While the approach of optimizing portfolios across
time-series simulations is promising and may potentially provide a more
comprehensive environment to assess asset allocation methods than using
correlation matrices, the papers provided examples with a low number of
assets. Also, given that the testing of correlations across the simulations is
limited, it is not clear whether the independency structure of the synthetic
data exhibits all of the characteristics found in empirical datasets and
listed in (Marti et al., 2021).

Synthetic data from generative models can also be valuable in assessing
portfolio risk, simulating extreme events, and testing the robustness of
models to different scenarios. Flaig and Junike (Flaig and Junike, 2022)
employed GAN architecture to create an economic scenario generator for
calculating market risk in insurance companies. In a separate publication
(Flaig and Junike, 2023), the authors presented methods for further validating
their model, assessing the alignment of dependencies between risk factors, and
measuring model overfitting to empirical data. (Cont et al., 2022) introduced
the Tail-GAN model, focusing on generating tail risk scenarios for user-
specified trading strategies. The model utilized a bespoke loss function to
accurately capture the tail risk of benchmark portfolios based on common risk
measures such as Value-at-Risk and Expected Shortfall.

In the later sections of our paper, we present a case study that utilizes the
concept of tracking error volatility (TEV) which measures the standard
deviation of a portfolio’s returns relative to its benchmark (Roll, 1992).
Given the diverse characteristics of fixed income assets, investors can
construct portfolios that are significantly different from their benchmarks.
TEV serves as a simple measure of relative risk, particularly relevant to
portfolio managers who have a mandate to outperform a benchmark within strict
risk limits.

##  3\. Generating synthetic data

We propose a process to generate synthetic data that consists of two parts:

  1. (1)

A generative adversarial network (GAN) model that samples synthetic
correlation matrices.

  2. (2)

An Encoder-Decoder model that provides additional data vectors (asset
volatilities, expected returns, and forward realized returns) conditionally on
a given correlation matrix.

The GAN model in 1) is supposed to provide information on market structure and
current market conditions. The Encoder-Decoder network in 2) uses correlation
matrices as input data and generates any additional information that may be
required.

| mean correl | eigen gini |  | coph corr
---
single
| coph corr
---
ward
| perron frob
---
sum neg
| power eigen
---
values
| Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std
Empirical |  |  |  |  |  |  |  |  |  |  |  |
Training Period |  0.231 |  0.063 |  0.855 |  0.011 |  0.716 |  0.118 |  0.751 |  0.071 |  2.335 |  0.389 |  1.952 |  0.093
Testing Period |  0.284 |  0.079 |  0.862 |  0.019 |  0.717 |  0.102 |  0.697 |  0.079 |  2.227 |  0.396 |  1.965 |  0.112
Generated |  |  |  |  |  |  |  |  |  |  |  |
DCGAN |  0.222 |  0.038 |  1.044 |  0.031 |  0.670 |  0.098 |  0.728 |  0.077 |  3.306 |  0.288 |  2.154 |  0.068
WGAN |  0.230 |  0.070 |  0.976 |  0.064 |  0.697 |  0.115 |  0.742 |  0.073 |  3.225 |  0.303 |  2.032 |  0.081

Table 1. Summary statistics of empirical and generated correlation matrices.
The highlighted numbers correspond to values that are closer to the
characteristics of the empirical dataset in the training period. The WGAN
model provided correlation matrices that more closely resemble the empirical
dataset compared to the DCGAN model across the analyzed metrics.

Our analysis is based on a time-series dataset spanning from April 2007 to
October 2022. This dataset includes 68 financial assets, representing 28
currencies and 40 investable fixed income indices. The fixed income indices
cover various types of treasury, corporate, government-related, and
securitized bonds in developed countries across North America, Europe, Asia,
as well as emerging market countries. The selection of the financial
instruments aims to reflect a broad spectrum of building blocks, suitable for
constructing a well-diversified global fixed income investment product.
Although all selected fixed income indices are FX-hedged to USD, our case
study allows to open FX positions by taking long or short positions directly
on the currencies. For each asset, we collected two main time-series from
Bloomberg terminal – a total return index and a series of expected returns
based on assets’ yields, adjusted to account for the yield that an investor
could have obtained by hedging FX risk to USD with forward contracts. The
generative models were trained on a transformed form of the dataset that
consists of the following four components for each day in the analyzed period:
correlation matrix based on 52 weekly returns, vector of volatilities based on
52 weekly returns, vector of 1-year expected returns and vector of 1-month
forward realized returns.

The dataset is divided into a training period from April 2007 to March 2017
and a testing period from May 2017 to October 2022. The testing period covered
a significant drawdown that the fixed income universe experienced in 2022 when
both, interest rates increased and credit spreads widened. The rare
combination of positively correlated interest rates and spreads accompanied by
high volatility provides an interesting testing environment for any asset
allocation strategy seeking to minimize risk.

###  3.1. Generative Adversarial Network

The GAN model developed for the analysis takes a random vector and outputs a
correlation matrix that exhibits characteristics found in empirically observed
correlation matrices. The architecture of the model follows the methodology
described in (Marti, 2020) but introduces several enhancements:

  * •

The model is converted to a Wasserstein GAN model following an approach
described in (Gulrajani et al., 2017). Without the change, the network in some
configurations of hyperparameters failed to converge or exhibited mode
collapse (Goodfellow, 2016). The most common reason for the issues was that
the discriminator was learning too quickly which did not allow the generator
to improve. The change has also improved the characteristics of generated
correlation matrices across the tests that were conducted.

  * •

In many hyperparameter configurations, the generated correlation matrices
exhibited a “checkerboard artifact” described in (Odena et al., 2016) and
(Aitken et al., 2017). The issue may occur when there is an overlap between
filters of transposed convolutional layers and consequently, some pixels (or
correlation coefficients in this case) receive less focus in the learning
process. To address this issue, in the generator network, we reduced the use
of transposed convolutional layers and instead favored convolutional and
upsampling layers with nearest-neighbor interpolation.

  * •

The empirical correlation matrices used for training were not sorted using a
hierarchical clustering algorithm; instead, they maintained a consistent asset
order. While (Marti, 2020) suggested sorting the matrices to emphasize the
hierarchical structure of financial markets, in our experiments we noticed
that the GAN model can learn characteristics of the empirical dataset without
reordering the empirical dataset.

We evaluate the GAN model by comparing the characteristics of the generated
correlation matrices with empirical ones. Two variants of the GAN model are
assessed: ”DCGAN,” the original model based on (Marti, 2020), and ”WGAN,” the
enhanced model proposed in this section. We sampled 64,000 correlation
matrices for both models and used metrics suggested in (Papenbrock et al.,
2021) for assessment:

  * •

mean correl – average correlation coefficient across matrices

  * •

eigen gini – Gini coefficient of matrices’ eigenvalues

  * •

coph corr – cophenetic correlation between the original correlation distance
matrix and the cophenetic matrix of the hierarchical clustering algorithm
(using single or ward linkage). This is a proxy that measures how hierarchical
correlation matrices are. The correlation distance is measured as
$D=\sqrt{2(1-Corr)}$

  * •

perron frob sum neg – measures the sum of negative entries of the first
eigenvector.

  * •

power eigen values – power exponent of correlation matrices’ eigenvalue
distribution

As shown in Table 1, most WGAN metrics closely resemble those of empirical
matrices. Notable distinctions emerge mainly in the ”perron frob sum neg”
values, which signify a higher prevalence of negative elements within the
eigenvector corresponding to the largest eigenvalue. Additionally, Gini
coefficients and power exponents of eigenvalues exhibit higher averages
compared to the empirical dataset. All WGAN variant’s mean metrics are closer
to those of the empirical training dataset compared to the DCGAN model. While
some of the differences between models are not large, it is worth noting that
training the DCGAN model necessitated hyperparameter calibration and multiple
training trials to achieve successful convergence and avoid the mode collapse
issue.

Figure 1. Sample empirical and ”WGAN”-generated correlation matrices. The
generated matrices’ are visually indistinguishable from empirically observed
correlation matrices.

The higher performance of the WGAN model can be also illustrated by comparing
the diagonals of the matrices - the average diagonal element of WGAN’s
matrices is 0.9992 whereas for DCGAN the average is 0.9719. While the original
CorrGAN model (Marti, 2020) was trained on a different dataset, the author
noted an average diagonal element of 0.998.

Similarly to (Marti, 2020), our generated correlation matrices are difficult
to be visually discerned from empirical correlation matrices as illustrated in
Figure 1.

###  3.2. Encoder-Decoder network for additional asset characteristics

The Encoder-Decoder model generates additional asset attributes from a
correlation matrix, including volatilities, expected returns, and 1-month
forward returns. The additional attributes should reflect the market
conditions that a correlation matrix corresponds to.

The encoder takes a correlation matrix as input and deconstructs it using a
block of 2D convolutional layers with LeakyReLU activation functions and
dropout layers. The output is then compressed using linear layers. The decoder
consists of three parallel blocks, each comprising linear, 1D convolutional,
and upsampling layers with LeakyReLU activation functions. The model outputs
three vectors representing volatilities, expected returns, and 1-month forward
returns of the underlying assets. Notably, the values in these generated
vectors maintain the same order of assets as the input correlation matrix.

Figure 2. Scheme of the Encoder-Decoder network. The model takes a correlation matrix as input and returns three vectors that are conditional on the correlation matrix: expected returns, volatilities, and forward returns. | Mean | Std Dev | Skew | Kurtosis
---|---|---|---|---
Volatilities |  |  |  |
Empirical Train | 7.098% | 5.475% | 1.447 | 2.807
Generated | 7.391% | 5.482% | 1.246 | 2.048
Empirical Test | 6.309% | 5.278% | 2.873 | 19.890
1-Year Expected Returns |  |  |  |
Empirical Train | 3.053% | 3.297% | 4.613 | 96.076
Generated | 2.937% | 3.000% | 3.073 | 35.917
Empirical Test | 2.397% | 3.828% | 6.857 | 107.077
4-Week Forward Returns |  |  |  |
Empirical Train | 0.251% | 2.537% | -0.758 | 10.601
Generated | 0.223% | 2.372% | 0.053 | 6.026
Empirical Test | -0.029% | 2.497% | 0.693 | 59.745
Table 2. Summary statistics of the dataset generated by the Encoder-Decoder
network. The metrics aggregate results for all assets. The characteristics of
the generated dataset are similar to the empirical dataset in training period.

During training, the model employs a loss function that consists of three Mean
Squared Error (MSE) components to compare the generated three vectors with the
original empirical dataset. Training is conducted using only empirical
correlation matrices and the final model is evaluated by comparing
distributions of the synthetic and empirical additional vectors.

The model acts as a deterministic ”translator” that converts a correlation
matrix into additional data vectors, without introducing additional
randomness. The diversity in the output stems from the variety of generated
correlation matrices.

For the further parts of the analysis, we created a “synthetic” dataset by
sampling 64,000 correlation matrices from the GAN model and corresponding
vectors of volatilities, expected returns and forward returns from the
Encoder-Decoder model.

Table 2 compares the main statistics of the vectors in the synthetic and
empirical datasets. The Encoder-Decoder model effectively captures the
distributions of the additional data vectors, with means similar to the
empirical vectors used in training. The other three moments are moderately
less pronounced which is caused by the model mitigating outliers that are
present in the empirical dataset. The kurtosis of forward returns is lower in
the generated dataset, but still reflects fat tails in the underlying assets’
returns. The generated vectors are relatively different from the empirical
dataset in the testing period. However, considering that the Encoder-Decoder
model did not directly target these metrics in the training process, such
discrepancies were expected.

In Figure 3, we compare the distributions of the empirical and synthetic
vectors for a selected asset, US High Yield, which is one of the most volatile
fixed income indexes in the dataset. Some discrepancies are observed between
the two datasets - notably, the Encoder-Decoder model smooths out rough
sections in the empirical dataset’s distributions, leading to more monotonic
shapes.

Figure 3. Distributions of additional generated vectors for US High Yield in
empirical and generated datasets. The model occasionally generates ”smoother”
distributions than those observed in the empirical training dataset. Figure 4.
Scatterplots illustrating sample relationships between dataset’s components
for US High Yield. The Encoder-Decoder model often mitigates outliers and
extrapolates into probable areas that were sparsely represented in the
empirical dataset. Figure 5. Scatterplots showing sample relationships between
components of US High Yield and long-term US Treasuries. The components
generated by the Encoder-Decoder model are conditional on a given correlation
matrix and their values are consistent across different assets.

Figures 4 and 5 demonstrate that the Encoder-Decoder network learns the
relationships across components for a single asset and dependencies between
different assets’ components. The model shows the ability to extrapolate
relationships, as evident in the second scatterplot of Figure 4, where
expected returns and volatilities extend into a less represented region.
Although this area was not extensively covered in the empirical data, it is
plausible that the asset may exhibit volatilities and expected returns within
that region in the future. Overall, a visual assessment of the components
across the entire dataset reveals consistent conditioning of the additional
vectors by the correlation matrices for all assets.

##  4\. Case Study

Let’s consider an investor who aims to build a portfolio that outperforms a
chosen benchmark over the long-term, while maintaining low tracking error
volatility (TEV; measured as the standard deviation of the portfolio’s returns
relative to the benchmark). Once the initial investment is made, the portfolio
allocations are not adjusted. To construct the portfolio, the study employs an
approach that searches for optimal asset weights across simulations using
either empirical or synthetic datasets. We hypothesize that portfolios
constructed using the synthetic dataset would outperform portfolios based only
on the empirical dataset in an out-of-sample period.

|  #Obs |  | TEV
---
(bps)
| Excess ER
---
(bps)
| Average
---
Excess ER /
TEV Ratio
t-stat |  p-value |  | Min
---
Excess ER /
TEV Ratio
| Median
---
Excess ER /
TEV Ratio
| Max
---
Excess ER /
TEV Ratio
| Share of
---
Excess ER /
TEV >
Empirical
|  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |  10
FR Sims |  |  |  |  |  |  |  |  |  |
Empirical |  335 |  74.6 |  46.6 |  0.684 |  |  |  0.106 |  0.713 |  0.943 |
Synthetic |  335 |  69.3 |  47.7 |  0.751 |  -13.77 |  <0.01% |  0.214 |  0.779 |  1.028 |  81.2%
Combined |  335 |  68.9 |  47.7 |  0.758 |  -14.85 |  <0.01% |  0.202 |  0.784 |  1.050 |  84.5%
MV Sims |  |  |  |  |  |  |  |  |  |
Empirical |  335 |  76.2 |  44.5 |  0.640 |  |  |  0.096 |  0.683 |  0.990 |
Synthetic |  335 |  74.9 |  46.4 |  0.661 |  -9.52 |  <0.01% |  0.203 |  0.680 |  1.036 |  73.1%
Combined |  335 |  75.0 |  46.3 |  0.661 |  -9.54 |  <0.01% |  0.190 |  0.680 |  1.035 |  73.1%

Table 3. Summary statistics of the case study. The table consolidates the
results from 335 experiments, comparing portfolio construction using the
empirically observed (”Empirical”), synthetic (”Synthetic”), or combined
(”Combined”) datasets. The table is divided into two sections: FR Sims and MV
Sims, representing different sources of forward returns used in simulations.
The metrics examined include (2) TEV (tracking error volatility), (3) Excess
ER (average expected returns in excess of a benchmark), t-statistics (5), and
p-values (6), which determine whether the Excess ER to TEV ratio (4) is higher
for the ”Synthetic” or ”Combined” variants compared to the ”Empirical”
variant. Additionally, the table presents the minimum, median, and maximum
values of the Excess ER to TEV ratio across experiments (7, 8, 9), along with
the share of observations in which the ”Synthetic” and ”Combined” variants
outperformed the ”Empirical” variants in terms of the Excess ER to TEV ratio
(10).

The simulation-based asset allocation method can be described with the
following process. We have an $nxm$ matrix $\mathbf{R}_{\text{assets}}$ that
represents n simulation returns for each of the m assets.
$\mathbf{r}_{\text{bench}}$ corresponds to a returns vector of the benchmark,
which is one of the columns in the matrix, and $\mathbf{r}_{\text{port}}$ is
the returns vector of the portfolio, which is obtained by multiplying the
matrix of assets’ simulation returns ($\mathbf{R}_{\text{assets}}$) by
$\mathbf{w}$, the vector of portfolio weights:
$\mathbf{r}_{\text{port}}=\mathbf{R}_{\text{assets}}\mathbf{w}$. The
optimization problem can be defined as:

(1)
$\displaystyle\underset{\mathbf{w}}{\text{minimize}}\left\|\mathbf{r}_{\text{port}}-\mathbf{r}_{\text{bench}}\right\|_{2}$

subject to: $\mu_{\text{port}}\geq\mu_{\text{bench}}+\mathbf{t}$,
where $\mu_{\text{port}}$ and $\mu_{\text{bench}}$ represent the average
expected returns of the portfolio and benchmark across simulations and $t$ is
an additional excess return target. As a result, the optimizer aims to
minimize the 1-month deviations of the portfolio from the benchmark, reducing
the tracking error volatility, while ensuring that the expected return is on
average higher than or equal to a specified target above benchmark’s expected
return. The portfolio must be also fully invested, and there are limitations
on FX positions within the range of -5% to 5%.

Simulated returns ($\mathbf{R}_{\text{assets}}$) and expected returns are
derived from three different datasets: the empirical dataset (”Empirical”
variant), the synthetic dataset (”Synthetic” variant), or a combination of
both (”Combined” variant). The simulated returns were also drawn from two
different methods:

  * •

”FR Sims” – the 1-month forward returns vectors were directly sourced from the
empirical dataset or generated using the Encoder-Decoder model discussed
earlier. This yielded 2,610 forward return sets from the empirical dataset and
64,000 from the synthetic dataset

  * •

“MV Sims” – returns were generated by drawing from a multivariate normal
distribution, with covariance matrices based on correlation matrices and
volatility vectors from either the empirical or generated datasets. In this
approach, multiple simulations were performed per covariance matrix, resulting
in 26,100 simulations using the empirical dataset and 640,000 simulations
using the synthetic dataset

The ”FR Sims” option was expected to more effectively incorporate the skewed
distributions of assets’ returns, resulting in portfolios with on average
lower TEVs.

To determine if portfolios constructed using simulations based on synthetic
datasets outperform, we conducted a series of experiments. These involved
constructing portfolios with the described method and evaluating their
performance through backtests conducted over the testing period from May 2017
to October 2022. The backtests were performed for each of the 40 fixed income
indices analyzed, with varying excess expected return targets ranging from 20
to 100 bps in 10 bps increments. Instances, where the optimizer failed to
converge, were excluded from the analysis. This typically occurred when the
portfolio’s benchmark was a high-yielding asset and there were insufficient
other assets in the dataset with expected returns meeting the excess return
target. Consequently, in total, 335 examples were used in the analysis in both
the ”FR Sims” and ”MV Sims” configurations.

The primary metric used to assess the results was the ratio of expected
returns in excess of the benchmark’s returns to tracking error volatility.
This metric directly aligns with the objectives targeted by the optimizer and
serves as an indicator of the expected information ratio in the long term.

Table 4 presents the results of the case study. The variants utilizing the
synthetic dataset (”Synthetic” and ”Combined”) demonstrated lower TEVs and
higher average excess expected returns compared to the ”Empirical” variant.
This difference was particularly notable in the ”FR Sims” options, where the
TEVs of the ”Synthetic” and ”Combined” variants were 5 bps lower than that of
the ”Empirical” variant. The combination of lower TEVs and slightly higher
excess expected returns led to a significant increase in the excess expected
returns to TEV ratios. In the ”MV Sims” options, the differences were more
subtle, with the TEV being only about 1 bps lower and excess expected returns
approximately 1 bps higher for the ”Synthetic” and ”Combined” variants
compared to the ”Empirical” variant. The improvements in the excess expected
returns to TEV ratios were confirmed by one-sided t-tests, which yielded
p-values lower than 1%. Notably, the ”Combined” variants exhibited higher
t-stats than the ”Synthetic” variants in both the ”FR Sims” and ”MV Sims”
options, suggesting that augmenting the empirical dataset may offer more
benefits to practitioners than solely replacing it with a synthetic dataset.

##  5\. Conclusion

In this work, we proposed a process to generate synthetic data for analyses of
asset allocation methods in the fixed income universe. The data generation
process was divided into two stages: firstly, we generated correlation
matrices with an enhanced CorrGAN (Marti, 2020) model and then we sampled
additional assets attributes (volatilities, expected returns and forward
returns) with an Encoder-Decoder model introduced in this paper. The generated
dataset exhibited characteristics of the empirical dataset and preserved
relationships between the generated matrices and the additional attributes.
Importantly, the generative models also demonstrated their capability in
mitigating outliers, reducing reliance on historical events with low future
probability, and extrapolating data into probable areas that were sparsely
represented in the empirical dataset. In the case study we also illustrated
that utilizing the synthetic dataset led to the construction of better-
performing portfolios.

Future research endeavors could explore additional modifications to the GAN
and Encoder-Decoder model, such as conditioning them on external factors to
improve the performance of asset allocation methods in specific market
regimes. Additionally, further investigations can focus on expanding the
application of synthetic datasets to other asset classes and considering
different practitioner requirements. For example, in an equities universe, the
model could incorporate valuation metrics alongside volatility metrics to
evaluate asset allocation methods with a value-bias.

By introducing this approach to synthetic data generation and demonstrating
its practical benefits, we aim to encourage practitioners and researchers to
embrace synthetic datasets as a valuable tool for improving asset allocation
in the fixed income universe, as well as exploring their potential
applications addressing diverse requirements of asset allocators.

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