# Navigating Market Turbulence: Insights from Causal Network Contagion Value
at Risk

Katerina Rigana  Swiss Finance Institute (SFI), USI Switzerland, Lugano,
Switzerland  Ernst C. Wit  USI Switzerland, Lugano, Switzerland  Samantha Cook
FNA UK, London, UK

###### Abstract

Accurately defining, measuring and mitigating risk is a cornerstone of
financial risk management, especially in the presence of financial contagion.
Traditional correlation-based risk assessment methods often struggle under
volatile market conditions, particularly in the face of external shocks,
highlighting the need for a more robust and invariant predictive approach.

This paper introduces the Causal Network Contagion Value at Risk (Causal-NECO
VaR), a novel methodology that significantly advances causal inference in
financial risk analysis. Embracing a causal network framework, this method
adeptly captures and analyses volatility and spillover effects, effectively
setting it apart from conventional contagion-based VaR models. Causal-NECO
VaR’s key innovation lies in its ability to derive directional influences
among assets from observational data, thereby offering robust risk predictions
that remain invariant to market shocks and systemic changes.

A comprehensive simulation study and the application to the Forex market show
the robustness of the method. Causal-NECO VaR not only demonstrates predictive
accuracy, but also maintains its reliability in unstable financial
environments, offering clearer risk assessments even amidst unforeseen market
disturbances. This research makes a significant contribution to the field of
risk management and financial stability, presenting a causal approach to the
computation of VaR. It emphasises the model’s superior resilience and
invariant predictive power, essential for navigating the complexities of
today’s ever-evolving financial markets.

Keywords: Financial Risk Management, Robust Predictions, Causal Networks,
Value at Risk, Contagion Effects, Market Volatility, Exogenous Shocks, Forex

##  1 Introduction

Risk is a crucial aspect of financial research, encompassing its definition,
measurement, management, and pricing (Rubinstein, 2002; McNeil et al., 2015).
Whatever its precise definition, there seems to be a consensus that contagion
plays a role in the evaluation of financial risks (Forbes and Rigobon, 2001;
Okimoto, 2008; Adams et al., 2014; Caccioli et al., 2014; Glasserman and
Young, 2015; Dungey and Renault, 2018; Löffler and Raupach, 2018; Londono,
2019; Hansen, 2021).

Causal inference refers to strategies that allow one to draw causal
conclusions based on data (Pearl, 2009). This is particularly challenging when
experiments are not feasible and the researcher must rely solely on
observational data (Hill, 1965; Warren et al., 2014). Separating correlation
from causality has always been an important issue in any field of empirical
research. The increased use of machine learning and AI methods has led not
only to a major overhaul of the approach to data handling, but also to a
careful reevaluation of causal algorithms for big data (Richens et al., 2020;
Raita et al., 2021; Tchetgen and VanderWeele, 2012). In medicine, this has
already led to a renewed evaluation of causal inference methods (Gaskell and
Sleigh, 2020; Shapiro and Msaouel, 2021; Greenland, 2017; Kroenke et al.,
2016; Williamson et al., 2014). As machine learning and AI methods evolve,
more and more research fields are adopting causal inference principles for
actionable results, explainability, and safety in applications (Amodei et al.,
2016; Pearl, 2019; Siebert, 2023; Schölkopf, 2022).

The finance industry has been more reluctant to embrace new developments in
causal inference, in part as a result of the age-old conundrum of correlation
versus causality (Embrechts et al., 2002; Hoover, 2006). However, there have
recently been some examples of using causal inference to improve stress
testing (Gao et al., 2018), empirical research in the accounting field (Gow et
al., 2016) and assessing causal factors determining the success of start-ups
(Garkavenko et al., 2022). In finance, the predominant modelling approaches
include Granger causality (Granger, 1969; Eichler, 2007) and Instrumental
Variables (Angrist et al., 1996). Granger causality, which is part of transfer
entropy-based methods (Schreiber, 2000) and applicable to non-linear systems
(Hlavácková-Schindler, 2011; Montalto et al., 2014), predicates causality in
temporal precedence, assessing if past values of a variable (X) can predict
future values of another (Y). In contrast, Instrumental Variables, a facet of
quasi-experimental methods (Shadish et al., 2002), strive to extract actual
causal links from observational data, yet their application hinges on the
specific selection of exogenous instruments to address endogeneity. While both
models, particularly Granger Causality, hinge on several, often unrealistic
assumptions (Maziarz, 2015; Shojaie and Fox, 2022), the usage of Instrumental
Variables is seen to be more effective post the establishment of a causal
graph through causal inference (Sharma and Kiciman, 2020).

Most modelling approaches for the effect of contagion on VaR do not have an
explicit causal outset (Filho et al., 2020). Most contagion-based VaR
approaches, such as _CoVaR_ (Tobias and Brunnermeier, 2016), _SDSVaR_ (Adams
et al., 2014) and various other alternative attempts to integrate contagion
within risk management and the VaR measure (Pesaran and Pick, 2007; Metiu,
2012; Bae et al., 2003), focus on correlations in systemic risk rather than
causality. Furthermore, while insightful, _CoVaR_ and _SDSVaR_ are complex and
often challenging to apply universally. These methods require detailed and
extensive datasets and are highly specific, limiting their broad applicability
in diverse financial contexts.

In principle, the use of causal models offers the prospect of robust
prediction (Peters et al., 2016; Pfister et al., 2019; Heinze-Deml et al.,
2018). If cause and effect are correctly identified, then the prediction will
offer clear risk guarantees in unexpected circumstances. However, these models
are not without limitations. Model uncertainty is an inherent challenge, and
under stable financial conditions, simpler predictive methods could outperform
causal analysis. However, due to frequent continuous external shocks,
financial systems rarely remain in an ergodic equilibrium state, thereby
enhancing the benefits of causal models. External influences, such as global
events and regulatory changes, underscore the importance of robust and dynamic
modelling methods.

In this paper, we apply a causal network approach to risk management. In
Section 2 we introduce a risk measure that relies on causal network contagion
to capture volatility and spillover effects. When assessing the risk position,
we consider price changes in related assets that have been identified from the
data. Section 3 analyses the performance of this new approach in a simulation
study. Section 4 applies the method to the Forex market and Section 5
concludes.

##  2 Value at Risk and Network Contagion

|
---|---
Figure 1: The Value at Risk, $\mbox{VaR}_{\alpha}$, is the $\alpha$ quantile
of the distribution of a financial instrument. In the example above, the VaR
for $\alpha=5\%$ is depicted for a normally distributed instrument both cross-
sectionally (left) and longitudinally (right).

Value at Risk (VaR) is an established risk measurement used in finance to
evaluate and compare the risk of holding a specific financial instrument or
portfolio of different instruments. The VaR represents the $\alpha\%$ quantile
loss under normal market conditions for a specific holding time period
(Jorion, 2007, 1996; Linsmeier and Pearson, 1996; Dowd, 2007). The future
value of an asset can be considered a random variable. Identification of the
distribution of the value of this instrument $X$ gives access to the value at
risk at any level $\alpha$,
$\mbox{VaR}_{\alpha}(X)=\inf\\{x\in\mathbb{R}:F_{X}(x)>\alpha\\}$, where
$F_{X}$ is the cumulative distribution function of X (Artzner et al., 1999).
This corresponds to the $\alpha$-quantile $q_{\alpha}(X)$ of the distribution,
shown in Figure 1. In practical terms, the portfolio value or return will be
smaller than the corresponding VaR over the holding period with a probability
of at most $\alpha$ per cent.

Usually in finance, the Value at Risk is computed on the basis of log-returns
on investment instead of the price for their statistical properties and
computational ease. The returns-based VaR is then easily recomposed into the
dollar loss VaR using the current price of such a portfolio. Value at Risk is
a measure developed by investment banks to measure market risk associated with
their market positions, a task that is becoming increasingly difficult due to
higher volatility periods, increased market interconnectedness, and an
increasing use of derivatives.

Value at Risk (VaR) has evolved from its conceptual beginnings in Edgeworth’s
1888 work (Edgeworth, 1888) to becoming a key financial risk measure. Its
development accelerated in the 1980s, culminating in its adoption in the 1990s
as the official risk measure for financial institutions (Garbade, 1986; Basel
Committee on Banking Supervision, 1996). VaR’s ascent was aided by methods
such as JP Morgan’s _RiskMetrics_ TM (Longerstaey et al., 1996).

There are two types of issues in practice with the VaR. One is the lack of
reliable computational methods and the other is a limitation of what it is
capable of measuring (Artzner et al., 1999). VaR is a measure that is not
additive and does not inherently represent the absolute worst-case loss, but
is widely used nonetheless. Due to its standardised and easily interpretable
nature, it makes an ideal baseline for examining causal network contagion
effects in finance. As this paper focusses on VaR, the methodology can be
extended to other measures such as expected shortfall (ES), a coherent risk
metric that, despite its higher data requirements, can be linked to VaR
through Taylor (2022).

###  2.1 Most Common Ways to Estimate Value at Risk

There are various methods for estimating the VaR. A parametric approach is the
variance-covariance method, also called the delta-normal method, popularised
by JP Morgan as the RiskMetrics method (Longerstaey et al., 1996). The
parametric variance-covariance approach assumes a distribution for the returns
on each instrument, whose parameters are then estimated to find the VaR. The
most common method is based on the multivariate normal distribution, which
leads to an easy and quick estimate of
$\mbox{VaR}_{\alpha}^{\mbox{VC}}(X)=\mu_{x}-z_{\alpha}\cdot\sigma_{x}$, where
$\mu_{x}$ is the mean and $\sigma_{x}$ the standard deviation of past
realisations of the log-return $X$, and $z_{\alpha}$ is the Z-score of a
standard normal distribution at $\alpha\%$. The biggest drawbacks of this
method are the, not always realistic, underlying assumptions of normally
distributed log-returns, a constant volatility through time, and no
correlations among the financial instruments. There is no way to include the
effects of contagion from assets not within the considered portfolio in this
method, regardless of the distribution assumptions.

To counteract the constant volatility assumption, volatility can also be
modelled using ARCH and GARCH models. When dealing with non-linearly priced
instruments, such as options, the VaR is usually estimated using a Monte Carlo
or stochastic simulation (Linsmeier and Pearson, 1996, 2000). The Monte Carlo
VaR is computed as the quantile of the simulated returns $X^{sim}$, simulated
from a chosen stochastic model for the behaviour of X – usually a GARCH
process, $\mbox{VaR}_{\alpha}^{\mbox{GARCH}}(X)=\hat{q}_{\alpha}(X^{sim})$

In non-parametric VaR models, the joint risk factor distribution is
constructed using historical data rather than assuming a specific functional
form (Cabedo and Moya, 2003). The easiest method would be to resample the past
returns within the estimation window and pick the $\alpha$-quantile for the
VaR directly. Historical simulation (HS) further develops this idea by
bootstrapping past returns,
$\mbox{VaR}_{\alpha}^{\mbox{HS}}=\hat{q}_{\alpha}(X^{\mbox{\scriptsize
boot}})$, where $X^{\mbox{\scriptsize boot}}$ represents the increase in
sample through bootstrapping. HS is an easy and fast method. Although it is
non-parametric, it still relies on an assumption of stationarity in
distribution and specifically volatility, often violated in practice due, for
example, to volatility clusterings (Gurrola-Perez and Murphy, 2015).

Filtered historical simulation (FHS) combines the best of parametric and non-
parametric approaches (Barone-Adesi et al., 1998, 1999). FHS runs an HS method
on volatility-rescaled past returns, thus maintaining the non-parametric
nature of HS while allowing for varying volatility. Rescaling is done in two
steps within a conditional volatility model (e.g., GARCH or AGARCH). Returns
are first standardised by the estimated volatility of the day of the return
and then rescaled by the forecast volatility for the VaR holding period,
$\mbox{VaR}_{\alpha}^{\mbox{FHS}}(X)=\hat{q}_{\alpha}(X_{re-
scaled}^{\mbox{\scriptsize boot}})$. This transformation reflects current
market conditions and, therefore, requires shorter estimation windows to
simulate extreme events. In contrast to other Monte Carlo based approaches,
the correlation matrix does not have to be estimated, as all rescaled returns
at a specific event time are sampled together. Extreme observations or
correlations that vary over time may not be adequately considered by the FHS
VaR (Pritsker, 2006). Extensions of the FHS have been proposed by Hull and
White (1998) as the Volatility-weighted HS and by McNeil and Frey (2000), who
combine FHS with extreme value theory. For comparisons, see Dowd (2007),
Martins-Filho et al. (2018), and Pritsker (2006).

###  2.2 Definition of Causal NECO Value at Risk

In this section, we propose a novel VaR procedure based on causal network
contagion (causal NECO). What separates this approach from standard dependency
analysis like Jondeau and Rockinger (2006), is the ability to identify the
direction of the contagion: which assets export contagion risk and which
others import it? The method does not assume any a priori causal model and
considers only what can be gathered through observed data — through the
external manifestation through the path of returns for each asset and their
co-dependences. Our goal is to demonstrate how causal contagion from assets
outside the portfolio of interest can be identified from observational data.

#### Gaussian Causal NECO VaR.

In this paragraph, we first derive the form for the value at risk in the event
that the causal contagion system can be described by Gaussian noise. We assume
that the log-returns for each instrument i at time t ($X_{i,t}$) are described
by a structural equation model (SEM), consisting of an autoregressive part and
a contagion part,

$X_{i,t}\leftarrow\alpha_{0,i}+\sum_{\ell=1}^{L}\alpha^{\ell}_{i}X_{i,t-l}+\sum_{j\in
pa(i)}\beta_{ji}X_{j,t}+\varepsilon_{i,t}\quad\quad\varepsilon_{i,t}\sim
N(0,\sigma_{i}^{2})$ (1)

where $L$ is the number of lags considered for the autoregressive part,
$\alpha_{0}$ is the intercept, $\alpha^{\ell}_{i}$ are the autoregressive
coefficients of lag $\ell$ for asset $i$, $pa(i)$ are the causal parents of
instrument $i$, $\beta_{ji}$ are the causal effects of instrument $j$ on
instrument $i$, and $\varepsilon_{i,t}$ are the error terms, assumed to be
independent and normally distributed with variance $\sigma^{2}$. Here,
contagion is defined as the instantaneous causal structure of the financial
system according to Rigana et al. (2023), which also provides more information
on the choice of assets. The SEM can be written more concisely in matrix form
for all financial instruments in the causal contagion network:

$X_{t}=(\mathbb{1}-B)^{-1}\left(\alpha_{0}+A\cdot
X_{t-1:t-\ell}+\varepsilon_{t}\right)$ (2)

where the $\varepsilon_{t}\sim N(0,\Sigma)$ is the vector of noise components,
$A$ is the matrix of the autoregressive coefficients and $B$ the matrix of
contagion effects,

$A=\begin{bmatrix}\alpha_{11}&\ldots&\alpha_{1\ell}\\\
\alpha_{21}&\ldots&\alpha_{2\ell}\\\ &\cdots&\\\
\alpha_{p1}&\ldots&\alpha_{p\ell}\end{bmatrix},\quad
B=\begin{bmatrix}0&\beta_{12}&\cdots&\beta_{1p}\\\
\beta_{21}&0&\cdots&\beta_{2p}\\\ \vdots&\vdots&\ddots&\vdots\\\
\beta_{p1}&\beta_{p2}&\cdots&0\end{bmatrix},\quad\Sigma=\begin{bmatrix}\sigma_{1}^{2}&0&\ldots&0\\\
0&\sigma_{2}^{2}&\ldots&0\\\ 0&0&\ddots&0\\\
0&\ldots&&\sigma_{p}^{2}\end{bmatrix}$ (3)

Though resembling a traditional vector autoregressive model, this SEM model is
strictly causal, where $\beta_{i,j}$ shows the direct causal effect of $i$ on
$j$. The identifiability of this causal model is explained in Section 2.3.

Given Equation 2, the distribution of our $X_{t}$, conditional on the past
log-returns, is given as:

$X_{t}|X_{t-1:t-\ell}\sim N\left((\mathbbm{1}-B)^{-1}\left(\alpha_{0}+A\cdot
X_{t-1:t-\ell}\right),(\mathbbm{1}-B)^{-1}\Sigma(\mathbbm{1}-B)^{-\top}\right)$
(4)

The contagion matrix $B$ has a direct impact on both mean and volatility. This
is not just a theoretical point; it shows that contagion really does have a
practical impact on these critical financial metrics.

Using the distribution in Equation 4, we can incorporate causal network
contagion (NECO) into the VaR definition. The Gaussian Causal VaR for each
instrument $i$ corresponding to a risk level of $\alpha$

$\mbox{VaR}_{\alpha}^{\mbox{NECO}}(X_{(t,i)})=\left[\begin{array}[]{c}(\mathbbm{1}-B)^{-1}\left(\alpha_{0}+AX_{t-1:t-\ell}\right)\end{array}\right]_{i}-z_{\alpha}\sqrt{\left[(\mathbbm{1}-B)^{-1}\Sigma(\mathbbm{1}-B)^{-\top}\right]_{ii}}$
(5)

#### General Causal NECO VaR.

The definition of the Causal VaR above is based on the assumption of a
multivariate normal distribution for the instruments considered for the causal
networks. This assumption represents log-returns relatively well, but it is
still often violated in reality where financial instruments are concerned. In
this paragraph, we will relax the normality assumption by using the Gaussian
copula transformation to capture fat tails and other non-Gaussian behaviour
typical of the returns on financial instruments (Dobra and Lenkoski, 2009;
Abegaz and Wit, 2015). Copulas provide a flexible tool for understanding the
dependence between random variables, particularly for non-Gaussian
multivariate data (Mohammadi et al., 2017).

Given that logarithmic returns of the instrument $i$ follow a marginal
distribution $F_{i}$, we assume that there is a latent temporal process $Z$,
which can be described by the mean causal SEM in Equation 2. This SEM
describes the idealised Gaussian version of the financial process, centred
around zero with variance one. The connection between the idealised process
$Z$ and the observable $X$ is given by the copula transformation,

$X_{t,i}=F_{i}^{-1}\left(\Phi\left(Z_{t,i}\right)\right),$ (6)

where $\Phi$ is the CDF of a standard normal distribution. This gives us a
direct way to define a general causal VaR,

$\mbox{VaR}_{\alpha}^{\mbox{NECO}}(X_{(t,i)})=F_{i}^{-1}\left(\Phi\left(\left[(\mathbbm{1}-B)^{-1}AZ_{t-1:t-\ell}\right]_{i}-z_{\alpha}\sqrt{\left[(\mathbbm{1}-B)^{-1}(\mathbbm{1}-B)^{-\top}\right]_{ii}}\right)\right),$
(7)

where $Z_{t,i}=\Phi^{-1}(F_{i}(X_{t,i}))$. The copula transformation allows us
to perform our analysis in the latent space without the need to assume
lognormal distribution for the returns. The other advantage of using a
Gaussian copula is that, unlike most other copulas, it can handle high
dimensionality, i.e., many interrelated financial instruments. Gaussian
copulas gained a negative reputation due to their indiscriminate use during
the 2008 financial crisis, particularly in modelling complex financial
instruments whose risks were poorly understood. However, it is essential to
distinguish between the misuse of these tools and their inherent capabilities.
In our context, we do not employ the Gaussian copula to model risk, we use
them to enhance the process of estimating causal networks as shown in the
following section.

###  2.3 Estimation of Causal NECO Value at Risk

Given a multivariate time-series, $D_{X}=\left\\{x_{t,i}\right\\}_{it}$, of
log-returns of $p$ financial instruments across $N$ periods, the estimation of
the Causal NECO VaR is done in four steps. First, we estimate the underlying
marginal distributions $F_{i}$ for the instruments. Second, we estimate the
causal structure on the transformed scale to see what financial instruments
impact the risk of the instrument of interest. This pivotal step is
instrumental for transitioning from mere correlation analysis to a causal
model. Thirdly, given the causal structure, we then estimate the contagion
coefficients $A$ and $B$ from Equation 3. Finally, the estimated marginal
distributions $\widehat{F}_{i}$ and the coefficients $\widehat{A}$ and
$\widehat{B}$ are used to calculate the VaR as in Equation 7.

#### Step 1: Estimating the marginal distribution of financial instruments

For each instrument $i$, we estimate its marginal distribution $F_{i}$ non-
parametrically as the adjusted empirical distribution function,

$\widehat{F}_{i}(x)=\frac{0.5+\sum_{i=1}^{N}1_{\left\\{x_{i}\leq
x\right\\}}}{N+1}.$

We then use this empirical distribution, to define a transformed dataset,
$D_{Z}$, of normally distributed variables,
$D_{Z}=\left\\{z_{t}\in\mathbbm{R}^{p}~{}|~{}z_{t,i}=\Phi^{-1}(\widehat{F}_{i}(x_{t,i})),t=1,\ldots,N\right\\}.$
We use the adjusted empirical distributions, in order to avoid degenerate
values for $z$.

#### Step 2: Discovery of Causal Structure

For the discovery of the causal structure, we follow Rigana et al. (2023). We
estimate the causal structure based on transformed data $D_{Z}$ in the form of
a causal network. This method is based on the PC-stable algorithm of Colombo
and Maathuis (2014). The PC-stable algorithm is a more robust version of the
original PC algorithm (Spirtes and Glymour, 1991). We implement the PC-stable
algorithm with the R package pcalg (Kalisch et al., 2012; Hauser and Bühlmann,
2012).

The fundamental insight of the PC algorithm is that causal connections in the
structural equation model can be estimated from purely observational data. The
algorithm is based on two main insights. _Conditional independence_ : if two
variables are independent or conditionally independent given a third variable,
then they are not directly causally connected. _Collider_ : if two variables
are (conditionally) independent, but become dependent when conditioning on a
third variable, then the original two variables are the causal parents of the
third. Using these insights, we can reconstruct the complex causal dynamics
within a system in terms of a causal network and identify direct and indirect
relationships.

#### Step 3: Estimation of Causal Effects

Once the causal structure of the SEM in Equation 1 is established, the
coefficients of matrices A and B can be estimated using standard least
squares. If there are no links present in the causal network between the
variable $i$ and $j$, then the corresponding coefficients $\beta_{ij}$ and
$\beta_{ji}$ are set to zero. If the algorithm from the previous step is
unable to deliver a fully directed network — where for some couples of
instruments both directions of the contagion are just as likely given the
structure of the causal network — we obtain a multiset of possible
coefficients and combine these into a range estimator (Maathuis et al., 2009).
Given that the financial system is very big and interconnected, the chances of
finding a fully directed network are very high.

#### Step 4: Causal NECO Value at Risk

With the estimated $\widehat{F}_{i}$, $\widehat{A}$ and $\widehat{B}$ we can
then compute the VaR using Equation 7. We assume that any interim payments on
the considered assets are either zero or reinvested continuously in the asset
itself, as is done in mutual funds. Furthermore, we take the time period for
the VaR evaluation to be equal to the frequency of the recorded log-returns —
so if we consider daily log-returns, we will compute the 1-day
$\mbox{VaR}_{\alpha}$.

##  3 Performance of Causal NECO Value at Risk

In this section, we conduct a detailed analysis of the Causal NECO Value at
Risk (VaR) in its ability to accurately predict risk levels in a variety of
simulation studies. The efficacy of Causal NECO VaR must be juxtaposed with
four other established methods, as delineated in Section 2.1. We introduce the
backtest techniques used in Section 3.1. We pick backtesting measures that are
model-free and best suited for comparative analysis.

###  3.1 Backtesting Measures

Using simulations or historical data, backtesting can be used to assess how
well VaR performed. Campbell (2007) provides an overview of the most common
backtesting methods. Part of the backtesting will be performed using the R
package MSGARCH (Ardia et al., 2019). The following is a description of the
tests we will consider.

#### Average Exceedance Rate.

The computationally easiest and most immediate test for VaR performance is
looking at the number of times that the actual $X_{t}$ log-returns fell below
the VaR, called violations or exceedances. Given the VaR definition, we expect
$X_{t}$ to be smaller than $\mbox{VaR}_{\alpha}$ approximately $\alpha\%$
times. The average exceedance rate $\hat{\alpha}$ should therefore be close to
the target level $\alpha$ of $\mbox{VaR}_{\alpha}$.

#### Actual over Expected Ratio.

Actual over Expected Ratio (AE) measures whether the VaR computation method
tends to have more or fewer violations than expected given the target
$\alpha$. A good method has the number of exceedances $n_{1}$ being $\alpha
N$. Defining $AE=\frac{n_{1}}{\alpha N}$, $AE>1$ means that the method is not
restrictive enough and underestimates the risk of the underlying investment.
An $AE<1$ shows the opposite, a method that is overly conservative and
overestimates the risk. Both directions of this error can lead to costly
mistakes.

#### LR Test of Unconditional Coverage.

The Likelihood Ratio test (LR) assesses the coverage rate for the target rate
$\alpha$. This unconditional coverage test (UC), as proposed by Kupiec (1995),
evaluates whether the observed proportion of violations deviates significantly
from the expected level. The LR test statistic, defined as
$LR_{UC}=-2ln\left[\frac{(1-\alpha)^{n_{0}}\cdot\alpha^{n_{1}}}{(1-\hat{\alpha})^{n_{0}}\cdot\hat{\alpha}^{n_{1}}}\right]$,
follows an asymptotic $\chi^{2}$ distribution under the null hypothesis
$H_{0}:E\left[\hat{\alpha}\right]=\alpha$.

#### LR Test of Conditional Coverage.

The Likelihood Ratio test (LR) test for the Conditional Coverage (CC) from
Christoffersen (1998) expands the UC test for the detection of violations
clusterings in time. No violation occurrence should be informative about the
performance of the next-step VaR. CC checks that exceedance realisations
$\\{\mathbbm{1}_{\\{x_{1,i}\leq\mbox{\scriptsize
VaR}_{\alpha}(1,i)\\}},\cdots,\mathbbm{1}_{\\{x_{T,i}\leq\mbox{\scriptsize
VaR}_{\alpha}(T,i)\\}}\\}$, often called hit series, are distributed
independently and identically. It uses a likelihood ratio test with 2 degrees
of freedom.

#### Dynamic Quantile Test.

The Dynamic Quantile (DQ) test described in Engle and Manganelli (2004) and
Dumitrescu et al. (2012) tests whether the violations, i.e. exceedance
realisations, are not only uncorrelated among themselves, but also with other
lagged variables. The $LR_{DQ}$ follows asymptotically a $\chi^{2}_{p}$
distribution with $p$ degrees of freedom.

#### Absolute Deviation.

The mean and maximum absolute deviation (AD) show the actual loss that would
occur if an investor or bank had relied on the VaR prediction. As pointed out
by McAleer and Da Veiga (2008), the AD measure is of great importance as large
violations can lead to bank failures when the capital requirements implied by
the VaR threshold forecasts are not sufficient to protect against losses that
are actually realised.

#### Average Quantile Loss.

The Average Quantile Loss (González-Rivera et al., 2004) is a weighted loss
measure, defined as
$QL_{i}(\alpha)=\frac{\sum_{t=1}^{N}\left(\alpha-\mathbb{1}_{\\{x_{t,i}\leq\mbox{\scriptsize
VaR}_{\alpha}(t,i)\\}}\right)\left(x_{i,t}-VaR_{\alpha}(t,i)\right)}{N}.$ If
for two methods we have $QL^{1}<QL^{2}$, then method 1 is preferable over
method 2.

###  3.2 Comparison of NECO VaR with Other Methods

We start by comparing the performance of our causal VaR method with more
traditional methods. We simulate data for the contagion network of 5 financial
instruments, as shown in Figure 2. We choose a relatively difficult market
situation with a relatively high density and market contagion to simulate a
market in crisis. The contagion coefficients were randomly chosen to reach a
market contagion of 47%. Using the Network Contagion Factor (NECOF) measure,
we can target a particular market contagion. NECOF expresses the impact of
contagion in %, where 0% means that no contagion effects are detected and 100%
means that all changes can be explained by contagion alone (Rigana et al.,
2023). We examine the ability of various methods to deal with non-normality
using exponentially distributed returns with an added shock every 100 days.
Out-of-sample performance is tested for 100 days after training on 250 time
points. The process is repeated twenty times.

$X_{5}$$X_{3}$$X_{4}$$X_{2}$$X_{1}$ |  | Simulated Network
---
nr. links | 7
density | 0.7
market NECOF | $47\%$
Figure 2: Simulated financial network with 5 instruments related to section
3.2

Figure 3 shows both the temporal performance and the overall performance of
the various VaR methods targeting $\alpha=0.05$. It shows that the causal NECO
VaR outperforms the other methods, both in its ability to deal with the non-
normality and the external shocks to the system. Table 1 shows that the causal
NECO method has good coverage for the three tests, $LR_{UC}$, $LR_{CC}$ and
$LR_{DQ}$, achieving high acceptance rates of 94%, 96% and 88%,
respectively.From Figure 3 it is immediately clear that the two methods that
struggle the most are VarCovar and GARCH; these two methods rely heavily on
the assumption of normality. FHS-GARCH significantly improves the GARCH
method, but since it is based on merging HIST and GARCH, it is influenced by
shocks as much as HIST in underestimating the risk.

|
---|---
(a) | (b)
Figure 3: Fraction of VaR exceedances within the out-of-sample window of 100 days for each Value-at-Risk model for all of the considered financial instruments. Figure (a) illustrates how closely it follows the target of $5\%$ in different simulations, and (b) shows the overall precision with a boxplot. | Causal-NECO | VarCovar | HIST | GARCH | FHS
---|---|---|---|---|---
mean$(\hat{\alpha})$ | 0.0458 | 0.0050 | 0.0632 | 0.0168 | 0.0637
st.dev$(\hat{\alpha})$ | 0.0203 | 0 | 0.0196 | 0.0236 | 0.0257
LRuc.accept | 0.9400 | 0 | 0.9400 | 0.2100 | 0.8600
LRcc.accept | 0.9600 | 0 | 0.9300 | 0.2900 | 0.8200
DQ.accept | 0.8800 | 1.0000 | 0.7400 | 0.8800 | 0.6200
AE.mean | 0.9160 | 0.0000 | 1.1760 | 0.2400 | 1.1860
AE.sd | 0.4052 | 0.0000 | 0.3962 | 0.4774 | 0.5182
AD.mean | 0.0303 | NA | 0.0375 | 0.1249 | 0.0634
AD.max | 0.1200 | -Inf | 0.1300 | 1.9500 | 2.2600
CompareQL | 1.0000 | 2.5811 | 0.9997 | 1.9885 | 1.0329
Table 1: Simulation Backtesting Results, for $\alpha=5\%$. The best results for each measure are indicated in bold. The results reflect all 20 rounds of 100 out-of-sample predicted VaR. |
---|---
(a) | (b)
|
(c) | (d)
Figure 4: Performance of the Causal NECO VaR for different estimation windows.
(a) Comparison on the logarithmic scale between different target $\alpha$
levels and different lengths of the estimation window in terms of the
considered number of observations (N). The black line in the centre is the
target to be achieved. (b) Standard deviation of the individual
$\hat{\alpha}$. (c) Comparison at target $\alpha=1\%$ for different N. (d)
Comparison at target $\alpha=5\%$ for different N.

###  3.3 Effect of Training Window

The performance of any VaR method depends on the accuracy of the estimated
model. In this simulation study, we vary the training window from $N=50$ to
$N=500$ observations to fit the causal model. Once the causal structure is
estimated, we apply the causal NECO VaR to an out-of-sample time series of 100
time points. Figure 4 shows the results. Although the standard deviation of
the achieved VaR level does not depend much on the size of the training window
$N$ as seen in Figure 4 (b), the VaR tends to have a lower bias with
increasing estimation window as seen in Figure 4 (a), (c), (d). This impact is
especially significant when targeting an $\alpha$ below 5%, but the impact is
relatively negligible for any $N$ greater than 50.

###  3.4 Effect of Number of Variables

We then investigate the effect of the size of the contagion network, that is,
the number of financial instruments and variables that we wish to analyse to
estimate the causal structure. We simulate log-return financial networks with
$p=5,10,20,50$ instruments. In the training simulation, N is set to 250, and
the rest of the simulation is carried out as in the other simulations. From
Figure 5 we see that the impact of the size of the network is negligent,
especially for the more common levels of $\alpha$ below 10%. This finding is
significant, suggesting that our causal NECO model maintains its efficacy in
various network sizes, a crucial aspect for its application in various
financial settings.

|
---|---
(a) | (b)
Figure 5: Causal NECO VaR performance for different network sizes. (a) Log-
scale comparison between $\hat{\alpha}$ achieved and $\alpha$ target for
different numbers of instruments (p). (b) Standard deviation of the achieved
$\hat{\alpha}$ at different values of the target $\alpha$ VaR level for
different number of instruments (p).

###  3.5 Effect of Market Contagion

We compare the impact of market contagion, expressed in market NECOF, on the
achieved $\hat{\alpha}$ VaR level. The study is carried out on a network of
$p=10$ instruments and an estimation window of $N=250$. Market contagion is a
function of a number of causal links in the financial network and the size of
the causal contagion effects. We simulate different network structures with
varying effect sizes and express the market contagion with the NECOF. Figure 6
(a) and Table 2 show that there is no discernible trend in the efficiency of
VaR estimation as the levels of contagion in the market increase. For small
$\alpha$ below $5\%$, the method performs slightly better for lower contagion
levels, suggesting that the choice of the length $N$ of the training window is
especially crucial when targeting lower $\alpha$ levels in the case of extreme
contagion levels. However, extreme levels of contagion would suggest a dense
financial contagion network, which is not a typical situation for the
financial market. From the left side of Table 2 we see lower contagion levels.

|
---|---
(a) | (b)
Figure 6: Causal NECO VaR perfomance comparison at a log-scale between target
$\alpha$ and achieved $\hat{\alpha}$ VaR levels for (a) different levels of
market contagion (NECOF) and (b) different levels of volatility ($\sigma$).

| Contagion (NECOF) | Volatility (sd)
---|---|---
| 0% | 19% | 47% | 73% | 83% | 0.005 | 0.05 | 0.5 | 1 | 2
mean($\hat{\alpha}$) | 0.039 | 0.041 | 0.046 | 0.051 | 0.042 | 0.054 | 0.049 | 0.052 | 0.049 | 0.050
sd($\hat{\alpha}$) | 0.015 | 0.017 | 0.022 | 0.021 | 0.020 | 0.027 | 0.025 | 0.026 | 0.026 | 0.032
LRuc.accept | 0.960 | 0.940 | 0.920 | 0.940 | 0.900 | 0.910 | 0.880 | 0.910 | 0.860 | 0.860
LRcc.accept | 0.990 | 0.980 | 0.960 | 0.950 | 0.960 | 0.870 | 0.910 | 0.950 | 0.890 | 0.900
DQ.accept | 0.880 | 0.860 | 0.820 | 0.810 | 0.880 | 0.850 | 0.890 | 0.830 | 0.860 | 0.840
AE.mean | 0.786 | 0.819 | 0.922 | 1.017 | 0.849 | 1.074 | 0.988 | 1.036 | 0.972 | 1.098
AE.sd | 0.306 | 0.329 | 0.436 | 0.427 | 0.399 | 0.538 | 0.495 | 0.531 | 0.525 | 0.643
AD.mean | 0.024 | 0.026 | 0.029 | 0.035 | 0.035 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
AD.max | 0.150 | 0.120 | 0.130 | 0.160 | 0.240 | 0.010 | 0.000 | 0.010 | 0.020 | 0.000
CompareQL | 0.963 | 0.972 | 1.000 | 0.959 | 1.032 | 0.983 | 1.034 | 1.000 | 1.028 | 0.986

Table 2: Backtesting results for target $\alpha=5\%$ at different levels of
market contagion (left) and different levels of volatility (right).

###  3.6 Effect of Volatility

Volatility, captured by the term $\Sigma$ in Equation 4, is the stochasticity
in the system. As contagion also affects overall volatility, we keep the
contagion levels fixed at 47%. We simulate a system with $p=5$ instruments and
keep the training window at $N=250$. We keep the shock to the system every 100
days but adapt the size to be proportional to the standard deviation.The
results for five different volatility levels $\sigma$ are shown in Figure 6
(b) and Table 1. The volatility change appears to have a negligible effect on
the performance of causal NECO VaR. Regardless of volatility, the NECO VaR
tends to be somewhat liberal at low VaR target values $\alpha$. This is mainly
due to the external shocks included in the simulation.

###  3.7 Computational Time

The causal inference approach is well adapted to sparse networks that have
relatively few causal links (Le et al., 2016). This seems to be the case for
financial networks, as there are usually clear pathways through which
contagion flows (Bardoscia et al., 2021). Figure 7 shows that for such sparse
networks, the computational time is low, even for a high number of financial
instruments. It is pertinent to note that in scenarios involving extremely
dense networks or an exceptionally high number of nodes, the PC-stable
algorithm, which underpins our methodology, may encounter convergence
challenges (Kalisch and Bühlmann, 2007).

Figure 7: Average computation time in milliseconds across 100 simulations of
the causal NECO network identification and estimation as a function of the
number of financial instruments (nodes).

##  4 Measuring Risk on Forex

Forex is a very liquid and important financial market, trading $ 6.6 trillion
per day (Wooldridge, 2019). 90% of these transactions involve the US dollar
(USD). In this study, we analyse the Forex market of 20 exchange rates over
the USD, selected to provide a representative overview of the most commonly
traded currencies, as well as some less common ones to showcase different
challenges in risk estimation. Given that we consider liquid assets, we will
consider a VaR with a 1-day holding period.

Table 3 shows the summary statistics of the daily log-retuns on 20 considered
exchange rates over the USD for the years 2000-2021, as published by the
Federal Reserve of New York. Most of the currencies have considerable fat
tails in their log-returns. Most currencies show significant deviations from
normality, as evidenced by the Jarque-Bera test statistics, which are notably
high across the board. Most of the currencies have considerable fat tails and
asymmetry in their log-returns. NECOF values, which indicate the degree of the
contagion effect that impacts the specific currency, vary considerably,
suggesting various levels of interconnection and risk exposure.

| $\widehat{\mbox{NECOF}}$ | Minimum | Median | Mean | Maximum | StDev | Skewness | Kurtosis | Jarque Bera
---|---|---|---|---|---|---|---|---|---
AUD | 0% | -0.0771 | -0.0003 | -0.0001 | 0.0822 | 0.0080 | 0.6197 | 12.0299 | 31085
EUR | 0% | -0.0463 | 0 | -0.0001 | 0.0300 | 0.0059 | -0.0557 | 2.5411 | 1375
NZD | 63.9% | -0.0593 | -0.0003 | -0.0001 | 0.0618 | 0.0082 | 0.3770 | 4.8517 | 5124
GBP | 20.2% | -0.0443 | -0.0001 | 0 | 0.0817 | 0.0060 | 0.7047 | 10.8604 | 25491
BRL | 5.8% | -0.0967 | 0 | 0.0002 | 0.0867 | 0.0105 | -0.0040 | 8.1012 | 13949
CAD | 37.6% | -0.0507 | -0.0001 | -0 | 0.0381 | 0.0057 | -0.0669 | 5.4645 | 6350
DKK | 98.5% | -0.0580 | -0.0001 | -0.0001 | 0.0494 | 0.0060 | -0.1284 | 4.7956 | 4902
HKD | 0.3% | -0.0045 | 0 | -0 | 0.0033 | 0.0003 | -1.2257 | 25.3347 | 137697
INR | 1.3% | -0.0376 | 0 | 0.0001 | 0.0394 | 0.0045 | 0.1980 | 9.9731 | 21174
JPY | 6.2% | -0.0522 | 0.0001 | -0 | 0.0334 | 0.0062 | -0.3144 | 4.4245 | 4245
KRW | 13.3% | -0.1322 | -0.0001 | -0 | 0.1014 | 0.0068 | -0.5511 | 49.7260 | 525804
MXN | 37.8% | -0.0596 | -0.0001 | 0.0001 | 0.0811 | 0.0072 | 0.7485 | 11.0121 | 26250
NOK | 40.6% | -0.0644 | -0.0002 | -0 | 0.0612 | 0.0078 | 0.2380 | 4.8194 | 4985
SEK | 63.3% | -0.0530 | -0 | -0 | 0.0547 | 0.0074 | -0.0482 | 3.9502 | 3319
ZAR | 42.4% | -0.0916 | -0.0002 | 0.0001 | 0.0843 | 0.0109 | 0.2626 | 4.2637 | 3922
SGD | 59.5% | -0.0238 | -0.0001 | -0.0001 | 0.0269 | 0.0033 | 0.0313 | 4.9083 | 5121
LKR | 0.4% | -0.0339 | 0 | 0.0002 | 0.0641 | 0.0029 | 2.5329 | 76.6587 | 1254466
CHF | 65.6% | -0.1302 | 0 | -0.0001 | 0.0889 | 0.0067 | -1.1545 | 36.8482 | 289720
TWD | 39.5% | -0.0342 | 0 | -0 | 0.0248 | 0.0030 | -0.3858 | 9.8420 | 20714
THB | 37.9% | -0.0353 | 0 | -0.0001 | 0.0447 | 0.0037 | 0.1609 | 12.2860 | 32104

Table 3: Overview of the summary statistics for the dataset of log-returns on
individual 20 exchange rates over the USD, for the period January 2000 to
April 2021. The higher the Jarque Bera test statistic the less likely the data
are normally distributed — all of the statistics have a p-value of 0. Each
sample is 5101 observations long with no values missing.

###  4.1 Fitting Causal Network Contagion on Forex

We consider causal networks with different values of the lag $L$ in Equation
1. In order to choose the best number of lags, we make use of the Akaike
information criterion, given as $\mbox{AIC}(\ell)=2k(\ell)-2l_{\ell}(D_{X})$,
where $l$ is the log-likelihood, and $k(\ell)$ is the number of all non-zero
autoregressive parameters $A$ and contagion coefficients $B$. Figure 8 shows
the value of the AIC for different lags. The minimum is obtained at lag
$\ell=1$, which is used in the estimated NECO model.

Figure 8: Overall AIC for each number of lags on the whole of the Forex
dataset considered.

Given the presence of fat tails in the log-return, we use a copula
transformation, as described in section 2.2. We fit a marginal distribution
$F_{i}$ for the log-returns for each of the 20 currencies. We fit the causal
network to the transformed log-returns as explained in Section 2.3. Finally,
we obtain the 1-day ahead causal NECO VaR values via Equation 7.

###  4.2 Results and Backtesting

We will consider the Value at Risk for each individual currency rate at the
$\alpha$ level of 5%. We consider a training window of $N=250$ trading days.
We will compare the performance of the Causal NECO approach to the more
established methods HIST, VarCovar, GARCH, and FHS-GARCH with the backtesting
measures from Section 3.1 applied to 100 out-of-sample VaR predictions, for
each of the 20 currencies during 20 non-overlapping periods between 2000 and
2021.

Table 4 shows that the causal NECO VaR beats all other methods in most
categories with a much higher acceptance rate for all tests. We also see a
much lower standard deviation of the estimated $\hat{\alpha}$, in line with a
robust invariant prediction. Given that, at least during calm periods, the
overall volatility of the Forex is quite low, the losses are not extreme for
any of the methods. FHS seems to be the best competition for causal NECO VaR
at this level.

| Causal-NECO | VarCovar | HIST | GARCH | FHS
---|---|---|---|---|---
mean$(\hat{\alpha})$ | 0.0511 | 0.0597 | 0.0594 | 0.0549 | 0.0574
st.dev$(\hat{\alpha})$ | 0.0202 | 0.0603 | 0.0392 | 0.0338 | 0.0318
LRuc.accept | 0.9500 | 0.8100 | 0.6700 | 0.7600 | 0.7800
LRcc.accept | 0.9600 | 0.9000 | 0.8000 | 0.8700 | 0.8800
DQ.accept | 0.8500 | 0.8000 | 0.7800 | 0.8200 | 0.8500
AE.mean | 1.0220 | 1.1060 | 1.0995 | 1.0085 | 1.0595
AE.sd | 0.4046 | 1.2171 | 0.7928 | 0.6818 | 0.6416
AD.mean | 0.0032 | 0.0031 | 0.0032 | 0.0030 | 0.0030
AD.max | 0.1200 | 0.1200 | 0.1200 | 0.1200 | 0.1200
CompareQL | 1.0000 | 0.9804 | 1.0247 | 0.9934 | 0.9801
Table 4: Backtesting of the VaR out-of-sample predictions in the Forex market
(2000-2021) for the different methods at target $\alpha=5\%$. The best results
are presented in bold.

Figure 9 (a) shows the distribution of actual exceedances for all currencies
and periods. Causal-NECO is the most centered around the target value of 5%.
Figure 9 (b) shows how these exceedances change over time. The Causal-NECO
method seems to be able to adapt properly to the changing conditions of the
underlying network. Unlike the other methods, it is barely affected by the
various financial crises in this period and achieves the nominal 5% level
throughout the evaluation period. Even as other methods seem to underestimate
the risk during peaks such as the 2008 global financial crisis, and move all
together, Causal-NECO keeps tight on target.

|
---|---
(a) | (b)
Figure 9: (a) Overall fraction of exceedances $\hat{\alpha}$ within the out-
of-sample window of 100 days for each Value-at-Risk model across the Forex
market (2000-2021). (b) Fraction of the exceedances over time.

In addition, we consider each of the 20 currencies individually. Figure 10
shows the individual boxplots of the exceedance fractions for each of the
given methods. The causal NECO performance is consistently less variable and
more centred around the 5% level.

Figure 10: Number of actual exceedances within the out-of-sample window of 100
days for each Value-at-Risk model, expressed in terms of the $\hat{\alpha}$
computed for each period and shown for each currency separately. The red line
shows the target of $\alpha=5\%$.

##  5 Conclusion

This paper introduces an innovative way to eliminate spurious correlations in
risk management decisions. It lays the groundwork for using causal inference
in finance showing how practical advantages can be obtained by inferring the
underlying causal system from available financial data. The causal NECO VaR is
easy to compute and offers a robust and competitive addition to standard
approaches. Volatility is not modelled directly but is addressed through the
contagion effect and, in part, through the copula transformation, offering an
alternative view on modelling volatility. As a way to deal with the non-
normality of financial data, we use a Gaussian copula transformation improving
upon its historical applications and addressing some of the limitations
highlighted in past financial crises. In our case, it is also important to
note that we do not treat the Gaussian copula as a way to model risk but
rather to improve the estimation of our causal contagion network.

The Causal NECO VaR model presents numerous opportunities for expansion.
Future research could attempt to account for a wider class of distributions
using a Student-t copula instead of a Gaussian copula and time-varying errors
with an added GARCH component (Jondeau and Rockinger, 2006). However, both
extensions are beyond the scope of this paper and could theoretically be
incorporated without affecting the underlying causal structure of our model.
Future research could explore the effectiveness of the causal approach across
various financial instruments. Our model allows for the inclusion of any
measurable confounder, which could be used to assess the impact of factors
such as interest rates, inflation, or stock market returns on Forex or other
contagion networks. In cases involving unmeasured confounders and latent
variables, the Fast Causal Inference Algorithm (FCI) (Spirtes et al., 1995;
Spirtes et al., 2000) presents a suitable alternative. In addition,
incorporating nonparametric methods into our model, as discussed in the recent
literature (Li and Fan, 2020), would further enhance its adaptability.

In conclusion, the Causal NECO VaR model marks a significant advancement in
financial risk analysis. Its ability to incorporate causal relationships into
risk assessment not only broadens our understanding of market dynamics but
also paves the way for more resilient financial management strategies in an
ever-evolving economic landscape.

## Acknowledgments

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