# Portfolio Risk Measurement Using a Mixture Simulation Approach

Seyed Mohammad Sina Seyfi
Khatam University  Azin Sharifi
Sharif University
of Technology  Hamidreza Arian
Sharif University
of Technology

( September 2020)

###### Abstract

Monte Carlo Approaches for calculating Value-at-Risk (VaR) are powerful tools
widely used by financial risk managers across the globe. However, they are
time consuming and sometimes inaccurate. In this paper, a fast and accurate
Monte Carlo algorithm for calculating VaR and ES based on Gaussian Mixture
Models is introduced. Gaussian Mixture Models are able to cluster input data
with respect to market’s conditions and therefore no correlation matrices are
needed for risk computation. Sampling from each cluster with respect to their
weights and then calculating the volatility-adjusted stock returns leads to
possible scenarios for prices of assets. Our results on a sample of US stocks
show that the Gmm-based VaR model is computationally efficient and accurate.
From a managerial perspective, our model can efficiently mimic the turbulent
behavior of the market. As a result, our VaR measures before, during and after
crisis periods realistically reflect the highly non-normal behavior and non-
linear correlation structure of the market.

Keywords: Gaussian Mixture Model, Value-at-Risk, Expected Shortfall , Risk
Management, Monte Carlo Simulation

JEL classification: C61, G11.

##  1 Introduction

The stock market behaves like a black box, so that the exact causal
relationship between causes and values is ambiguous. The only accurate
insights we can make on the black box are market data such as stock price and
trading volume (Sui, 2003). We can consider two main types of risk in
portfolio management: risk that can be eliminated by investment
diversification, and risk that cannot be avoided by diversification, or often
referred to market risk. Risk management provides mathematical machinery to
measure the latter type of risk (Klaassen and van Eeghen, 2009).

JP Morgan Chase Bank, first proposed the VaR approach and is widely used for
risk estimation. As the VaR approach is becoming increasingly popular in
financial risk management, numerous methods for estimating the VaR have been
introduced (Peng et al., 2019). VaR is a quantile estimation technique for
measuring portfolio downside risk over a defined horizon. For this
calculation, the expected loss and the expected excess loss beyond some
threshold are integral. The main challenge in the estimation of these types of
risk measures is the measurement of profit and loss distribution itself,
particularly the left tail associated with large losses of this distribution
(Glasserman et al., 2002). Quantitative approaches for estimating the
distribution of changes in portfolio value depend on two forms of modeling
considerations: estimates of changes in the underlying risk factors to which a
portfolio is exposed, and a method for interpreting these changes in risk
factors into changes in portfolio value (Glasserman et al., 2002).

The variance-covariance method made popular by Morgan et al. (1996) is the
simplest and perhaps the most commonly used approach to modeling changes in
portfolio value. This approach is based on two main assumptions: the
distribution of portfolio profit and loss is conditionally normal, and the
standard deviation can be measured from the covariance matrix of the
underlying risk factors. Another important line of research is focused on
relaxing the assumption that portfolio value changes linearly with changes in
risk factors, which results in methods commonly called delta-gamma developed
in (Britten-Jones and Schaefer, 1999; Duffie and Pan, 2001; Wilson, 1999).
Offering a similar computational cost as the delta-gamma approach, Date and
Bustreo (2016) propose a novel heuristic methods by mapping non-normal
marginal distributions to normal distributions using a probability conserving
transformation. Their approach is specially designed for portfolios exposed to
non-linear functions of non-normal risk factors. Another approach for
measuring VaR is the Extreme Value Theory (EVT) approach which models the
extreme portion of a general distribution. After the financial subprime crisis
of 2007-08, EVT has gained considerable attention for modelling fat tail
properties of return distributions (Stoyanov et al., 2011; Furió and Climent,
2013; Rossignolo et al., 2012; Danielsson et al., 2016). Other than the above
developments on extreme value modeling, quantile regression frameworks such as
conditional auto-regressive Value-at-Risk (CAViAR) model support both return
asymmetry and time-varying uncertainty (Engle and Manganelli, 2004). The main
idea behind CAViaR is that the VaR, like stock market volatility, can be
predicted by auto-regressive models which are able to capture the volatility
clustering phenomenon. Several authors have proposed extensions and
modifications for CAViaR to adapt it to real market conditions such as
intraday price ranges, trading-loss limits, overnight returns and close-to-
close ranges (Chen et al., 2012; Fuertes and Olmo, 2013; Meng and Taylor,
2018).

Risk factors are believed to be normally distributed in the traditional models
of risk. Because of this assumption of normality, the factor dependency
structure is fully explained by the correlation between factors using the
statistics of second order (Haugen and Haugen, 2001). Some researchers propose
to construct more realistic models of portfolio factor to ease the normality
assumption. The stock returns have been shown to be non-normal with high peak
values and heavy tails (Fama, 1965). Chin et al. (1999), for example, suggests
an efficient solution to the VaR portfolio problem calculation. Their approach
imposes independence between factors by using the independent component
analysis (ICA) and captures the non-normality of the return distribution by
Gaussian Mixture Model (Gmm).

In the risk management community, VaR calculation methods are also divided
into full and local valuation categories. Local valuation methods produce one-
time portfolio value, and then calculate VaR based on the underlying risk
factors model. The delta-normal model, based on a linear approximation of
changes in asset values due to the underlying factors, is an example of the
local valuation method. On the other hand, by re-profiling the portfolio over
a range of scenarios, the full evaluation methods measure risk. Monte Carlo
simulation methods are among full valuation techniques (Jorion, 2000).

Monte Carlo simulation (MCS) methods are statistical techniques to solve
complex mathematical and statistical problems for capturing the uncertain
conditions by considering many different states that each of which is probable
to take place. As Reddy and Clinton (2016) mentions, simulating an assets
price is to generate some paths which the asset can follow in the future. As
for Sengupta (2004), if the price of an asset was predictable, simulation was
unnecessary, for there was only one possible future price. Needless to say
that simulating would become worthy only when the generated paths are produced
based on some rules about historical data and other dynamic information about
the prices and market factors. Furthermore, Monte Carlo simulation generates
sequences of risk factors which are then used to estimate future behavior of
portfolios value (Rollett and Manohar, 2004). There are three steps in
implementation of any Monte Carlo simulation engine: first it randomly
generates sample paths, second evaluates the payoff along each path, and
finally calculates an average along all possible trajectories to attain an
estimation (Chen and Hong, 2007).

One of the traditional models for simulating stock price, Geometric Brownian
Motion (GBM), models stochastic price movements of financial assets by log-
normal dynamics (Abidin and Jaffar, 2014). GBM makes two strong assumptions
about the price process. First, the price returns, follow a normal
distribution, and second, the past information about financial instruments
fully represents the future behavior of the portfolio. Despite popularity of
the GBM process, it has many weaknesses, including that it dose not capture
market extreme behaviors such as non-normality.

The emergence of data-driven analyses have led to the increased use of
generative modelling to learn the probabilistic dependencies between random
variables. Although their apparent use has largely been limited to image
recognition and classification in recent years, generative machine learning
algorithms can be a powerful tool for predicting behavior of financial markets
(Wong and Farooq, 2019).

A generative algorithm used for predicting stock market and generating new
data is Gaussian Mixture Model (Gmm). Gmms have several distinct advantages
that make them suitable for modeling stock returns. With enough number of
components, they can model future outcomes to any probable scenario when they
are carefully fit to train data using the Expectation-Maximization algorithm.
Furthermore, they by no means make any normality assumptions about return
distribution when the number of components is equal or greater than two
(Bishop, 2006). Kon (1984); Tan and Tokinaga (2007) studies statistical
properties of the Gaussian mixture distribution and found that it could
provide a reliable approximation of the probability distribution function for
data, capturing heavy tails and excess kurtosis specially in multi-modal
situations.

As stated in Tan (2005), statistical characteristics, such as skewed
distributional shape or heavy-tailed behavior, are easily modeled using a
class of Gaussian mixture distributions. A mixture distribution has the
ability to approximate different distribution shapes by changing the component
weights and other distribution parameters such as mean and variance.
Glasserman (2013) proposes an effective method to estimate the tail
distribution related to the rare events through a simulation approach by using
Importance Sampling (IS). Their application of importance sampling not only
reduces the size of simulation samples, but also improves the accuracy of the
estimated probabilities. However, in their framework, it is difficult to
identify the optimal parameter used for importance sampling. Nonetheless,
under the Gmm framework, importance sampling is implicitly used in generation
of new samples as the model assumes a number of components weighted to produce
extreme data. Using importance sampling by Gaussian mixture models can
substantially improve the effectiveness of simulation approaches compared to
the standard Monte Carlo simulation.

In this paper, a new price simulation model which maintains the second
assumption of GBM (future is made up by the past) and relaxes the first one
(Normality assumption) is proposed. Even though we apply our model to stock
market, it can easily be used for simulating other asset classes which exhibit
fat tail and non-normal properties. Our model provides the advantage that it
does not assume the stock market returns conform to a Normal distribution. We
then calculate an efficient and fast method for estimating VaR and ES that is
brought from the generated scenarios from a number of Gaussian interdependent
joint distributions. Our model proposes an efficient Monte Carlo approach for
simulating stock prices in a non-normal framework where return distributions
follow mixture of Gaussian.

In order to improve robustness of our VaR engine, we have re-scaled simulated
returns based on the ratio of short term versus longer term volatility levels.
We calculate VaR of an investment portfolio, compared the performance of the
Gmm-based approach with and without re-scaling, and observed that re-scaling
of the generated returns yields much better results. Our method has a number
of advantages. First, it is highly accurate and superior to some other well-
known VaR methods based on back-testing routines. Second, compared to other
Monte Carlo applications for measuring risk, our model is quite time
efficient. Last but not least, our model is capable of capturing conditional
correlation phenomenon and therefore, its performance during financial crisis
periods is quite impressive.

##  2 Gmm as a Conditional Generative Model

A generative model, as its name suggests, has this capacity to learn the
structure of the data, say, their distribution or statistical properties and
thus create new unseen samples. The new samples are generated in a way that
they have the same features as those by which the model is trained. Generative
models also are able to deal with intractable distributions. No matter how
much the data is complicated, the algorithm will learn to mimic those
structures. One of the most used methods for generating new data is Gaussian
Mixture Models (Gmm) which divides the data into segments and simulates
members of each segment independently from others. By the way Gmm segments the
train data, it can capture the non-normality of stock market, and hence it
outperforms models with normality assumptions like GBM during crisis periods.

###  2.1 Model Setup

Gaussian Mixture Models (Gmms) are probabilistic mixture models with density
functions combined of a number of Gaussian densities. More precisely, Gmm’s
density function is the weighted sum of a finite number of Normal probability
density functions:

$P\left(x\right)=~{}\underset{i=1}{\overset{{{N}_{c}}}{\mathop{\sum}}}\,{{\omega}_{i}}\text{
}\\!\\!\Phi\\!\\!\text{ }(x|{\boldsymbol{{\mu}_{i}}},~{}{{\text{
}\\!\\!\Sigma\\!\\!\text{ }}_{i}}),$ (1)

where $N_{c}$ is the number of components, $\omega_{i}$s are the weights
adding up to one,
$\underset{i=1}{\overset{{{N}_{c}}}{\mathop{\sum}}}\,{{\omega}_{i}}=1$,
$\boldsymbol{\mu_{i}}$s are mean vectors and $\boldsymbol{\Sigma_{i}}$s are
the covariance matrices for $i=1,2,...,N_{c}$. Moreover, the Normal density
function $\Phi\\!\\!\text{ }(x|{\boldsymbol{{\mu}_{i}}},~{}{{\text{
}\\!\\!\Sigma\\!\\!\text{ }}_{i}})$ is given by

$\text{ }\\!\\!\Phi\\!\\!\text{ }\left(x\text{ }\\!\\!|\\!\\!\text{
}{\boldsymbol{{\mu}_{i}}},~{}{{\boldsymbol{\Sigma}}_{i}}\right)={{\left(2\pi\right)}^{-\frac{k}{2}}}\det{{\left({{\boldsymbol{\Sigma}}_{i}}\right)}^{-\frac{1}{2}}}\text{
}\\!\\!~{}\\!\\!\text{
exp}\left\\{-\frac{1}{2}{{\left(x-~{}{\boldsymbol{{\mu}_{i}}}\right)}^{{}^{\prime}}}\text{
}\\!\\!\boldsymbol{\Sigma_{i}^{-1}}\left(x-{\boldsymbol{{\mu}_{i}}}\right)\right\\},$
(2)

with $\boldsymbol{x}\in\mathbb{R}^{k}$ being the input data.

###  2.2 Parameter Estimation

For estimating the parameters of the Gaussian Mixture Model we need to find
the parameters
$\boldsymbol{\theta}=\left\\{\boldsymbol{\omega},~{}\boldsymbol{\mu},~{}\boldsymbol{\Sigma}\right\\}$.
Assume $p(z_{j}=1|x_{i})$ denotes the probability that $x_{i}$ belongs to the
$j$th Gaussian distribution, and the sequence of
$z=\\{z_{1}\,\dots,z_{N_{c}}\\}$ are set of latent variables following a
binary distribution, $z_{i}\in\\{0,1\\}$, with ${\omega}_{j}=p(z_{j}=1)$. Gmm
in addition assumes each data sample can belong to only one cluster, so only
one $z_{i}$ can take the value one, at a time. We also have the probability
measure on the vector of latent variables

$p(\boldsymbol{z})=\prod_{j=1}^{N_{c}}p\left(z_{j}=1\right)^{z_{j}}=\prod_{j=1}^{N_{c}}\omega_{j}^{z_{j}},$
(3)

and the conditional probabilities of samples conditioned on latent variables

$p\left(x_{i}|\boldsymbol{z}\right)=\prod_{j=1}^{N_{c}}p\left(x_{i}|z_{j}=1\right)^{z_{j}}=\prod_{j=1}^{N_{c}}\Phi\left(x|\mu_{j},\Sigma_{j}\right)^{z_{j}}.$
(4)

From $p(x_{i},\mathbf{z})=p(x_{i}|\mathbf{z})p(\mathbf{z})$, the above yields

$p(x_{i})=\sum_{j=1}^{N_{c}}p(x_{i}|\mathbf{z})p(\mathbf{z})=\sum_{j=1}^{N_{c}}\omega_{i}\Phi(x_{i}|\mu_{i},\Sigma_{i}).$
(5)

According to the Bayes rule, we obtain the responsibility function as

$p\left(z_{j}=1|x_{i}\right)=\frac{\omega_{j}\Phi\left(x_{i}|\mu_{j},\Sigma_{j}\right)}{\sum_{i=1}^{N_{c}}\omega_{i}\Phi\left(x_{i}|\mu_{i},\Sigma_{i}\right)}=r_{{ij}}.$
(6)

In order to calibrate the parameters of Gmm, the Expectation-Maximization
algorithm is used as follows

  1. 1.

Initializing step: parameters
$\boldsymbol{\theta}=\left\\{\boldsymbol{\omega},~{}\boldsymbol{\mu},~{}\boldsymbol{\Sigma}\right\\}$
are randomly initialized.

  2. 2.

Expectation step: by taking an expectation from $z_{j}$ we calculate the
responsibility functions

$p(z_{j}|x_{i},\boldsymbol{\theta})=r_{ij}.$

  3. 3.

Maximization step: update parameters as by

$\begin{array}[]{c}{\omega_{j}=\frac{\sum_{i=1}^{N}r_{ij}}{N}},\\\
{\mu_{j}^{\text{new}}=\frac{\sum_{i=1}^{N}r_{ij}x_{i}}{\sum_{i=1}^{N}r_{ij}}},\\\
{\Sigma_{j}^{\text{new}}=\frac{\sum_{i=1}^{N}r_{ij}\left(x_{i}-\mu_{j}\right)\left(x_{i}-\mu_{j}\right)^{\prime}}{\sum_{i=1}^{N}r_{ij}}}\end{array},$

where $N$ is the number of all data used as input of the model.

  4. 4.

Repeat step 2 and 3 until convergence.

###  2.3 Generating new samples

A generative model is linked to a conditional probability function $p(X|Y)$,
where $X$ is sample and $Y$ is the target. These models are able to produce
new instances with respect to $Y$. Gmm is used for fitting a PDF over the
data, and using that PDF will create new samples. A generative model should at
first learn the exact structure of data in order to get to generate new ones.
As sampling from a joint distribution is a challenging, the Gmm can be used
for scenario generation (see Wang et al. (2018)). For generating new data, Gmm
simply samples separately from every component by
$\Phi(x|\mu_{i},\Sigma_{i})$. The proportion of the numbers of samples from
each distribution $\Phi(x|\mu_{i},\Sigma_{i})$ to all samples is equal to
$\omega_{i}$. In other words, Gmm samples from each class with respect to its
weight and finally integrates all samples. In this way, the generated data has
the joint distribution of
$P(x)=\sum_{i=1}^{N_{c}}\omega_{i}\Phi(x|\mu_{i},\Sigma_{i})$ which guarantees
that the new data resembles the historical one.

For any given probability density function $p(x)$, assume $N_{T}$ represents
the total number of generated samples, and $N_{0}\leq N_{T}$ indicates the
number of samples less than an arbitrary value $x_{0}$, by infinitely
increasing the number $N_{T}$

$\lim_{N_{T}\rightarrow\infty}\frac{N_{0}}{N_{T}}=\int_{-\infty}^{x_{0}}p(x)dx.$
(7)

In the case of a Gmm, $N_{0}^{*}$ the number of samples from Gmm that are less
than $x_{0}$, will be equal to the summation of all $N_{0}^{i}$,
$i=1,2,...,N_{c}$, number of samples less than $x_{0}$ in each of normal
distribution. Then we have

$\displaystyle\lim_{N_{T}\rightarrow\infty}\frac{N_{0}^{*}}{N_{T}}$
$\displaystyle=\lim_{N_{T}\rightarrow\infty}\frac{\sum_{i=1}^{N_{c}}N_{0}^{i}}{N_{T}}=\sum_{i=1}^{N_{c}}\lim_{N_{T}\rightarrow\infty}\frac{N_{0}^{i}}{N_{T}}=\sum_{i=1}^{N_{c}}\omega_{i}\int_{-\infty}^{x_{0}}\Phi\left(x|\mu_{i},\sigma_{i}\right)dx$
(8)
$\displaystyle=\sum_{i=1}^{N_{c}}\int_{-\infty}^{x_{0}}\omega_{i}\Phi\left(x|\mu_{i},\sigma_{i}\right)dx=\int_{-\infty}^{x_{0}}\sum_{i=1}^{N_{c}}\omega_{i}\Phi\left(x|\mu_{i},\sigma_{i}\right)dx.$

The equation above mathematically shows that new samples conform to a Gaussian
Mixture distribution. Therefore, the simulation algorithm for a mixture of
Gaussians can be planned in three steps. First for $i=1,...,N_{c}$, the number
of samples for each component $\Phi(x|\mu_{i},\sigma_{i})$, is chosen as
$\omega_{i}N_{T}$. Second, we generate $\omega_{i}N_{T}$ samples for $i$th
component, and finally we integrate all the generated samples from all the
components (Wang et al., 2018).

##  3 Stock Price Simulation

###  3.1 Limitations of Geometric Brownian Motion

One of the most commonly used processes in finance is Geometric Brownian
motion (Black and Scholes, 1973). However, GBM suffers from the assumption
that stock prices are log-normally distributed, while it has been observed as
an stylized fact that in many cases, stock returns have fat tail and are
skewed. Symbolically, GBM offers the following model for stock prices:
$r_{t}=\ln(\frac{S_{t}}{S_{t-1}})\sim\Phi(\mu,\sigma)$, where $r_{t}$ and
$S_{t}$ refer to return and price processes, respectively. $\Phi$ is also a
Normal distribution with parameters $\mu$ and $\sigma$ calibrated from
historical data. This assumption leads to

$\ln\left(\frac{S_{t}}{S_{t-1}}\right)=\mu\delta t+\sigma\epsilon\sqrt{\delta
t},$ (9)

where $\delta t$ is the period of time and $\epsilon\sim\Phi(0,1)$. The first
term of the price dynamics introduces the drift, and the second term presents
the diffusion effect. Rearranging the equation above gives

$\displaystyle\frac{S_{t}}{S_{t-1}}=\exp(\mu\delta
t+\sigma\epsilon\sqrt{\delta t}),$ (10) $\displaystyle
S_{t}=S_{t-1}\exp(\mu\delta t+\sigma\epsilon\sqrt{\delta t}).$

Therefore, the equation (10) enables us to simulate stock paths as many times
as needed, by sampling $\epsilon$ from a standard normal distribution.

However, as Morone (2008) mentions, there are some stylized facts in the
financial markets, which GBM is not able to produce them, such as fat-tail
distributions in the stocks’ returns. Volatility clustering, also, is another
phenomenon that GBM cannot provide any explanation for that. Takahashi et al.
(2019) mentions other stylized facts, and gain and loss asymmetric for
financial time series, which GBM is unable to explain, but some of the
generative models can capture the financial market’s pattern. Considering the
stylized facts directly affects the performance of simulated data, and
accordingly, the VaR.

###  3.2 Gmm as an Alternative to GBM

In this section, we propose a simulation approach for modeling the non-
normality features of stock return processes. Gaussian Mixture models have the
advantage of capturing fat tail effects in returns of stocks and investment
portfolios. Gmm’s probability density function (PDF) follows

$p\left(x\right)=~{}\underset{i=1}{\overset{{{N}_{c}}}{\mathop{\sum}}}\,{{\omega}_{i}}\text{
}\\!\\!\phi\\!\\!\text{ }(x|{{\mu}_{i}},~{}{{\text{ }\\!\\!\Sigma\\!\\!\text{
}}_{i}}),$ (11)

where $\phi\\!\\!\text{ }(x|{{\mu}_{i}},~{}{{\text{ }\\!\\!\Sigma\\!\\!\text{
}}_{i}})$ denotes a Normal PDF with parameters $\mu_{i}$ and $\Sigma_{i}$ as
mean and covariance matrix. Therefore it is evident that every single point
belongs to one of the $\Phi\\!\\!\text{ }(x|{{\mu}_{i}},~{}{{\text{
}\\!\\!\Sigma\\!\\!\text{ }}_{i}})$ for $i=1,...,N_{c}$. These are the number
of classes that stock market experiences during a long period. Every class is
assumed to be Normal.

Generating a new sample, with any arbitrary number of data, say T, is carried
out by sampling from components $\Phi\\!\\!\text{ }(x|{{\mu}_{i}},~{}{{\text{
}\\!\\!\Sigma\\!\\!\text{ }}_{i}})$ with corresponding number of samples
proportional to $\omega_{i}$. It assures that if in a long time window, there
exists a particular number of returns in a component, the same portion from
those classes will appear in $T$ days. By using this approach for sampling a
new path, the simulated returns resemble historical data in such a way that
unlikely events are accounted for with the exact same probability of
occurrence. In the GBM, however, the rare events are represented by a single
term $\sigma$, and therefore using GBM to simulate stock prices no longer
generates the rare historical events. Note that there is also no need for
calculating statistical features manually as it is done in the GBM framework.

In mathematical terms, to generate one path with $T$ new samples, the Gmm
allocates $\omega_{i}T$ of samples to the class $i$. The new samples,
$\\{r_{1}^{*},r_{2}^{*},...,r_{T}^{*}\\}$ are considered as a set of possible
returns for the stock. The stock prices
$\\{S_{1}^{*},S_{2}^{*},...,S_{T}^{*}\\}$ are created using generated returns
by $S_{t}^{*}=S_{t-1}^{*}\exp{r_{t}^{*}}$ for $t=1,2,...,T$. Repeating this
algorithm $m$ times, the model produces $m$ paths for stock price based on the
same feature of historical data structure. Calibrating the Gmm is the most
computationally time consuming part of our algorithm moving from one day to
the next. We run the Expectation-Maximization (EM) algorithm daily, however,
in each calibration step the parameter set $\boldsymbol{\theta}$ from previous
step is used to initialize the EM calibration routine. We will show that this
technique reduces the computational cost of our Gmm model calibration
significantly. We also mention that, in the very first day of executing the EM
algorithm, we use k-means initialization to start the calibration routine.

##  4 Calculating Value at Risk (VaR) and Expected Shortfall (ES)

VaR is about measuring the downside risk during a specific time frame and with
respect to a significance level. Symbolically, $Pr(S\geq
VaR_{\alpha,t})=1-\alpha$. There are many methods for calculating VaR among
which, Historical Simulation (HS), Parametric Method, and Monte Carlo
simulation (MCS) are three of the most popular approaches used by academics
and practitioners (Jorion, 2000). Historical Simulation approximates risk by
the $\alpha$-percentile of historical returns and uses this quantile as a
proxy for $\text{VaR}_{(\alpha,t)}$. This method presumes the historical
returns will re-occur in the future with the same pattern from past. MCS
applies the same technique for calculating VaR except that instead of
historical returns, it re-generates many returns according to a particular
dynamics like GBM. After option pricing and estimating derivatives
sensitivities to their underlyings, estimating market risk is the most common
usage of MCS (Dowd, 2007). It is not necessary to assume a closed-form
representation for stock prices when applying Monte Carlo risk assessment.
Monte Carlo simulations in general is based on the Law of Large Number (LLN),
which suggests that if $X_{1},X_{2},...,X_{N}$ are i.i.d. random variables
with mean $\mu$ and variance $\sigma^{2}$ then

$\operatorname{Pr}\left(\left|\frac{\sum_{i=1}^{N}X_{i}}{N}-\mu\right|\geq\epsilon\right)\leq\frac{\sigma^{2}}{N\epsilon^{2}}.$
(12)

As an application of LLN, $\frac{\sigma^{2}}{N\epsilon^{2}}\rightarrow 0$ when
$N\rightarrow\infty$. Accordingly, if $X_{i}$ is the last generated price for
$i$th path, then it should have the same statistical properties as the
original data.

As VaR is to measure the probability of loss in a particular time frame, MCS
focuses only on the last returns simulated. Considering all last returns as
$P^{*i}_{T}$, $i=1,...,m$, the VaR becomes
$\text{VaR}_{\alpha,T}=\text{Percentile}(P_{T}^{i},1-\alpha\%)$, or the
$\alpha$-percentile of all generated returns $P^{*i}_{T}$. Using Gmm, the
matrix of returns for $m$ paths, and $T$ evaluation days returns is

$\left[\begin{array}[]{ccc}{r_{1}^{*1}}&{\cdots}&{r^{*1}_{T}}\\\
{\vdots}&{\ddots}&{\vdots}\\\
{r_{1}^{*m}}&{\dots}&{r^{*m}_{T}}\end{array}\right].$ (13)

From this matrix, stock price paths are generated according to
$(S_{t}^{*j}=S_{t-1}^{*j}\exp{r_{t}^{*}})$

$\left[\begin{array}[]{ccc}{S_{1}^{*1}}&{\cdots}&{S^{*1}_{T}}\\\
{\vdots}&{\ddots}&{\vdots}\\\
{S_{1}^{*m}}&{\dots}&{S^{*m}_{T}}\end{array}\right].$ (14)

We then calculate the overall return for each path by
$R_{\text{Gmm}}^{i}=\ln(\frac{S_{T}^{*}}{S_{0}})$. Finally, the
$\alpha$-percentile of $R_{\text{Gmm}}^{i}$ is $VaR_{1-\alpha}$, and the
corresponding expected shortfall is calculated according to

$ES_{1-\alpha}=\frac{1}{n}\sum_{i=1}^{m}R^{i}_{\text{Gmm}}\times
I_{\\{R_{\text{Gmm}}^{i}\leq VaR_{1-\alpha}\\}}.$

where $I$ is an indicator function

$I_{\text{A}}=\left\\{\begin{array}[]{l}{1,\quad\quad\quad\text{A,}}\\\
{0,\quad\text{otherwise, }}\end{array}\right.$ (15)

and $n$ is the number of $1$s in $I_{\\{R_{\text{Gmm}}^{i}\leq
VaR_{1-\alpha}\\}}$. Gmm guarantees that the new returns matrix has the same
distribution as the historical returns up to the assumption of a mixture
distribution. One of the main advantages of our approach in estimating Value-
at-Risk using Gaussian Mixture distributions is that during crisis periods,
our simulation algorithm learns quickly that clusters of correlated losses
occur. Therefore, and by the fact that most models assume a linear constant
correlation, our technique results in superior performance when compared to
traditional VaR techniques. This is specially important since it is expected
that a robust risk measurement tool is capable of being adapted to new market
environments when going into and out of turbulent financial markets. It is
also worthwhile to note that during crisis periods both correlation and
volatilities increase significantly. In our algorithm, our model learns to
allocate appropriate weights to various return clusters representing these
periods. This way, when a crisis period starts higher weights are allocated to
extreme market scenarios which results in a systematic increase in the risk
measure result. As the market prepares to move out of the crisis period, the
corresponding weight to such scenarios start to decline resulting in reducing
the VaR result.

###  4.1 Adjusting the VaR and ES with Respect to Their Volatility and Ages

Gmms are powerful tools for capturing the entire distribution of returns,
including their tails. But VaR depends not only on the returns levels, but
also on the volatilities. The most well-known models for capturing volatility
is GARCH (Engle and Sheppard, 2001). An efficient model for estimating VaR and
ES should be able to capture volatility clustering phenomenon. Traditional
Monte Carlo risk measures assume input data is not ordered, while it is clear
that the recent returns have more effects on the futures risk profile of
investment portfolios, when compared to older returns. Here we propose a
simple and fast method for adjusting the VAR and ES with respect to their
fluctuations and their ages.

Let $\sigma_{Long}$ be the standard deviation of returns series in a given
period. For taking both age and volatility into account, we define
$\sigma_{short}$ as the standard deviation calculated in a short and recent
period, say two weeks to one month. Then, we update the calculated VaR and ES
as follows

$\begin{split}\text{VaR}_{1-\alpha}^{adj}&=\text{VaR}_{1-\alpha}\times\frac{\sigma_{short}}{\sigma_{Long}},\\\
\text{ES}_{1-\alpha}^{adj}&=\text{ES}_{1-\alpha}\times\frac{\sigma_{short}}{\sigma_{Long}}.\end{split}$
(16)

By using the volatility term, $\sigma_{short}$ versus $\sigma_{long}$, older
return patterns are updated to the current state of the market. The length of
the short time period can be optimized by back-testing obtained results using
different periods. Generally, based on our experience, the short period can be
half or less than half of the long period. Equivalently, the above technique
can be applied directly on the simulated matrix of returns

$\left[\begin{array}[]{ccc}{r_{1}^{*1(adj)}}&{\cdots}&{r^{*1(adj)}_{T}}\\\
{\vdots}&{\ddots}&{\vdots}\\\
{r_{1}^{*m(adj)}}&{\dots}&{r^{*m(adj)}_{T}}\end{array}\right]=\frac{\sigma_{short}}{\sigma_{Long}}\times\left[\begin{array}[]{ccc}{r_{1}^{*1}}&{\cdots}&{r^{*1}_{T}}\\\
{\vdots}&{\ddots}&{\vdots}\\\
{r_{1}^{*m}}&{\dots}&{r^{*m}_{T}}\end{array}\right].$ (17)

and calculating VaR and ES according to adjusted matrix.

##  5 Portfolio Value Simulation

In this section, we explain our algorithm for using Gaussian Mixture
distributions to estimate the risk level of an investment portfolio. To
highlight the advantages Gmm makes in capturing the profile of the return
distribution, we first provide the classical approach based on Geometric
Brownian motion and then we discuss the improvements based on our Gmm-based
approach.

###  5.1 Simulating Returns based on Geometric Brownian Motion

Monte Carlo simulation on the Geometric Brownian motion requires correlated
random variables to generate future stock prices in a portfolio setting.
Supposing that there are $N_{s}$ stocks in a portfolio. Then we can write the
equation (10) in the form

$\left[\begin{array}[]{c}{S_{1_{t}}}\\\ {\vdots}\\\
{S_{N_{S_{t}}}}\end{array}\right]=\left[\begin{array}[]{c}{S_{1_{t-1}}\left(1+\mu_{1}\delta
t\right)}\\\ {\vdots}\\\ {S_{N_{S_{t-1}}}\left(1+\mu_{N_{S}}\delta
t\right)}\end{array}\right]+\left[\begin{array}[]{c}{S_{1_{t-1}}\sigma_{1}\epsilon_{1}\sqrt{\delta
t}}\\\ {\vdots}\\\ {S_{N_{St-1}}\sigma_{N_{S}}\epsilon_{N_{S}}\sqrt{\delta
t}}\end{array}\right].$ (18)

Here $\xi=\\{\epsilon_{i}\\}_{1}^{N_{s}}$ is a matrix of random variables from
a Normal standard distribution with $N_{s}$ columns and $T$ rows. As each row
should be correlated based on the correlation structure of stocks returns, one
may use Cholesky decomposition for creating $\xi$. This way, we can write
$\rho$, correlation matrix as $\rho=AA^{\prime}$, and put $\xi=A\epsilon$,
with i.i.d. random variables $\epsilon$. However, as we are using a
correlation matrix $\rho$ we are implicitly making an assumption that there
exists a linear dependency structure between stocks. Repeating this process
$m$ times generates $m$ different paths for each stock, and consequently $m$
different values for value of the portfolio. Portfolio’s value with weights of
$[w_{1},...,w_{N_{s}}]$ in the $i$th simulation path becomes

$S_{p_{t}}=[w_{1},...,w_{N_{s}}]\times\left[\begin{array}[]{c}{S_{1_{t}}}\\\
{\vdots}\\\ {S_{N_{S_{t}}}}\end{array}\right].$ (19)

Accordingly, the return from $i$th simulation is
$r^{i*}_{p}=\ln(\frac{S^{i*}_{p,T}}{S_{P,0}})$. The $\alpha$-percentile of
these generated returns is the risk measure $VaR_{T,1-\alpha}$ calculated
based upon GBM assumptions. Considering a portfolio of varying assets,
correlation between these assets is an important element from investment point
of view. It is not rational to simulate all of the stocks independently due to
the fact that returns of stocks are correlated with each other. The main point
is that the correlations between stocks do not remain constant all along.
Therefore, making assumptions on a single number as correlation between two
stocks or equivalently a correlation/covariance matrix between $n$ stocks does
not give realistic decision making results in most market environments. It has
been observed that during crisis periods, stocks are highly correlated while
in other conditions they are not (Aït-Sahalia and Xiu, 2016; Frank et al.,
2009)). This joint modelling of financial assets behavior during varying
market environments has always posed great challenges to practitioners working
in industry. A very well-known model for providing a structural framework for
correlation between stocks and the market portfolio is the Capital Asset
Pricing Model (CAPM) which considers one risk factor as the market risk and
measures risk premium of all stocks according to a coefficient named $\beta$
(Merton et al., 1973).

###  5.2 Simulating returns based on Gaussian Mixture Model

One of the main advantages of using Gaussian Mixture models to simulate stock
prices is that it does not require a correlation matrix driving the dynamics
of stock returns. Instead, Gmm directly models joint distributions of stock
returns by a finite number of Gaussian components. Allocating proper clusters
to data with different parameters leads to different correlation structures
according to the co-variation $\Sigma$ in each cluster. Therefore, there is no
need for calculating correlation matrix since there are multiple covariance
matrices with their associated weights in Gmm. By using a sampling method
discussed in section 3.2, every single sample $R^{*}$ will contain $N_{s}$
elements which denote each stock return. Sampling stock returns $m$ times will
produce the $m\times N_{s}$ matrix

$\left[\begin{array}[]{ccc}{r_{s_{1}}^{1*}}&{\cdots}&r_{s_{N_{S}}}^{1*}\\\
{\vdots}&{\ddots}&{\vdots}\\\
{r_{s_{1}}^{m*}}&{\cdots}&{r_{s_{N_{S}}}^{m*}}\end{array}\right].$ (20)

As Gmm looks at all historical data through one lens, we suggest to adjust all
simulated returns, with respect to their current volatility. We use the
coefficient $\frac{\sigma_{short}}{\sigma_{long}}$ and adjust the returns by
multiplying it with this coefficient. This results in a new matrix with
elements

$r_{s_{j}}^{i*}:=r_{s_{j}}^{i*}\times(\frac{\sigma_{short}}{\sigma_{long}})_{s_{j}}.$
(21)

Let $T$ be the time horizon at which we plan to simulate stock prices. We
repeat our simulation algorithm for $T$ times, resulting in $T$ matrices as
20. The $i$th simulated prices $(i=1,\cdots,m)$ for stock $s_{j}$,
$(j=1,\cdots,N_{s})$, at $T$ is

$S^{i*}_{s_{j},T}=S_{s_{j},0}\times\exp{(\sum_{t=1}^{T}r^{i*}_{s_{j},t})}.$
(22)

The $T$-holding period return for the stock $s_{j}$ becomes
$\sum_{t=1}^{T}r^{i*}_{s_{j},t}$. The portfolio return considering the fixed
weights $\left[w_{1},\dots,w_{N_{s}}\right]^{\prime}$ is given by

$\left[\begin{array}[]{c}{r^{1*}_{p}}\\\ {\vdots}\\\
{r_{p}^{m*}}\end{array}\right]=\left[\begin{array}[]{ccc}{\sum_{t=1}^{T}r^{1*}_{s_{1},t}}&{\cdots}&{\sum_{t=1}^{T}r^{1*}_{s_{N_{s}},t}}\\\
{\vdots}&{\ddots}&{\vdots}\\\
\sum_{t=1}^{T}r^{m*}_{s_{1},t}&{\cdots}&\sum_{t=1}^{T}r^{m*}_{s_{N_{s}},t}\end{array}\right]\times\left[w_{1},\dots,w_{N_{s}}\right]^{\prime},$
(23)

and the $i$th simulated value for the portfolio in at $T$ is

$S_{p,T}^{i^{*}}=S_{p,0}\times\exp{{r_{p}^{i*}}},\quad\text{for all}\quad
i=1,2,\dots,m.$ (24)

Having the generated price series, the $\alpha$-percentile of portfolio
returns is $\text{VaR}_{T,1-\alpha}$. Similarly, other risk factors can be
calculated accordingly. For example, the expected shortfall,
$ES_{T,1-\alpha}$, can be calculated as
$\frac{1}{n}\sum_{i=1}^{m}R^{i}_{Gmm}\times I_{\\{R_{Gmm}^{i}\leq
VaR_{1-\alpha}\\}}$, where $I$ is the indicator function.

##  6 Empirical Analysis

In this section, we provide the numerical results based on our Python
implementation for calculating the VaR for individual stocks and a portfolio
including those stocks. We use Christoffersen back-testing for investigating
validity of calculated VaR. We also use a quadratic loss function for making a
comparison between our model and other models such as historical simulation,
parametric (variance-covariance), Monte Carlo simulation and the GARCH
methods. In addition, a Gaussian Copula Monte Carlo approach is employed for
our selected portfolio and compared with our Gmm-based VaR model. We also
compare our model with different types of VaR named CaViaR introduced by Engle
and Manganelli (2004).

###  6.1 Data

The data for this study is from a number of S&P500 stocks, namely 3M Company
(MMM), American Express Co (AXP), Advanced Micro Devices Inc (AMD), American
International Group (AIG), AFLAC Inc (AFL), Abbott Laboratories (ABT),
Microsoft Corp (MSFT), Apple Inc. (AAPL), Amazon.com Inc. (AMZN), Bank of
America Corp (BAC), JPMorgan Chase & Co (JPM), Johnson & Johnson (JNJ), Exxon
Mobil Corp (XOM), Mastercard Inc (MA) and the S&P500 index itself from Yahoo!
Finance. The dates include the sub-prime crisis of 2007-2008. The start date
for the stocks returns time series is $7/24/2006$. The time horizon for our
VaR model, $T$, is 1, which means we have calculated daily VaR. We calculate
the daily VaR for 1000 days. The logarithmic returns are used from $7/24/2006$
to $7/23/2007$. In the next step, the window is shifted one day till VaR is
calculated up to $7/11/2011$ (For 1000 days). The descriptive statistic of
returns series is illustrated in table 1. The jarque bera statistic has been
rejected and shown with (R), when the associated p-value is under 0.01. As
shown, no one of the stocks during a crisis follow a normal distribution. For
calculating the VaR, Two confidence levels of $1\%$ and $5\%$ are considered
for daily VaRs. Furthermore, we calculate $1\%$ and $5\%$ VaR of an equally
weighted portfolio from these stocks. We used $N_{c}=3,4,5$, and $6$ for our
Gmm’s number of components. For the parameters $\sigma_{long}$ and
$\sigma_{short}$, we choose 252 and 70 days, respectively. The results for
different $\sigma_{short}$ is also shown in 12, 13, 14 and 15 in appendix. The
number of simulations in our VaR model is 3000. However, considering 1000
number of simulations also gave us the same results. For calculating an
accurate VaR result, the model should be able to capture the future
distribution of the data. A comparison between Gmm and two other
distributions, namely Normal and Student’s t-distributions, on each stock is
made.

Table 1: Descriptive statistics of the return series

| S&P500 | MMM | AXP | AMD | AIG | AFL | ABT | MSFTF
---|---|---|---|---|---|---|---|---
skewness | -0.247 | -0.115 | 0.093 | 0.024 | -2.150 | -1.507 | -0.174 | 0.350
kurtosis | 8.417 | 5.193 | 6.605 | 3.702 | 41.435 | 28.815 | 6.095 | 9.001
Jarque-Bera
statistic | 3671.8 (R) | 1394.3 (R) | 2254.2 (R) | 706.9 (R) | 89721.2 (R) | 43391.9 (R) | 1924.3 (R) | 4210.0 (R)
max | 0.110 | 0.094 | 0.188 | 0.202 | 0.507 | 0.265 | 0.092 | 0.171
min | -0.095 | -0.089 | -0.194 | -0.185 | -0.936 | -0.460 | -0.096 | -0.125
| AAPL | AMZN | BAC | JPM | JNJ | XOM | MA |  Portfolio
returns
skewness | -0.452 | 0.694 | -0.179 | 0.350 | 0.660 | 0.105 | 0.189 | -0.404
kurtosis | 5.793 | 12.413 | 13.613 | 9.838 | 15.655 | 13.216 | 13.529 | 8.474
jarque bera
statistic | 1775.1 (R) | 8061.8 (R) | 9582.4 (R) | 5025.4 (R) | 12755.1 (R) | 9026.7 (R) | 9465.2 (R) | 3743.2 (R)
max | 0.130 | 0.239 | 0.302 | 0.224 | 0.115 | 0.159 | 0.189 | 0.115
min | -0.197 | -0.246 | -0.342 | -0.232 | -0.080 | -0.150 | -0.133 | -0.129

###  6.2 Investigating Goodness of Fit

We use two goodness of fit approaches to verify how much the model is capable
of being fitted on the data. Firstly we employ a root-mean-square error (RMSE)

$RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(PDF_{i}^{\text{model
}}-PDF_{i}^{\text{real}}\right)^{2}},$ (25)

between real PDF of data and various models, where $n$ is the number of points
of PDF. The table 2 shows the RMSE for different stocks during the period from
$7/24/2006$ to $7/11/2011$. Since different number of components $N_{c}=3,4,5$
and $6$ have the same results we only show the Gmm for $N_{c}=3$ components.
As results show, it is clear that the RMSEs for Gmm is significantly lower
than Normal and Student’s t-distributions, which means that the Gmms with
$N_{c}=3,4,5$ and $6$ are better models than traditional distributions.

Secondly, The log-likelihood function per sample is also shown in table 3
which illustrates the greater amount for Gmms with different $N_{c}$ rather
than Normal and Student’s t-distribution. The results are showing the average
of log-likelihoods for samples from $7/24/2006$ till $7/11/2011$. It can be
seen that all Gmm log-likelihoods are higher than those for Normal and t
student. On the other hand, table 3 suggests that increasing the number of
components from $3$ to higher numbers may lead to either a higher log-
likelihood or to a same.

Thirdly, we use a Kolmogorov-Smirnov (KS) test between real data and various
models. Table 4 shows the p-values of testing the null hypothesis in the KS
test, where the $H_{0}$ hypothesis tests whether the data conforms to the
assumed distribution or not. The statistic for this test is
$D_{n}=\sup|CDF_{i}^{model}-CDF_{i}^{real}|$ where CDF is the cumulative
distribution function. In this test an instance randomly is sampled from the
determined distribution and its CDF is presumed as $CDF_{i}^{model}$. If at
the significance level of $1\%$, the p-value is higher than $1\%$, we fail to
reject $H_{0}$, as shown in table 4.

Figure 1 including some of our stocks gives an illustration for a comparison
between Normal and Student’s t-distribution with Gmm’s number of components
$N_{c}=3$. The Gmm fits a separate Normal distribution on the tails of the
data in order to capture their fat-tail profile. The weights of these
distributions are relatively low, due to the fact that they are less likely to
occur and less data is available to fit their behavior. The graph for all
stocks are shown in appendix in figures 8 and 9. As table 4 shows there is no
significant difference between efficiency of Gmm distribution with different
number of components $N_{c}$, leading to the conclusion that the market can be
explained well enough with $3$ different Normal distributions.

Table 2: RMSE comparison between different distributions

| S&P500 | MMM | AXP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | XOM | MA
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
GMM ($N_{C}=3$) | 0.27 | 0.21 | 0.23 | 0.22 | 0.31 | 0.24 | 0.17 | 0.23 | 0.19 | 0.19 | 0.30 | 0.25 | 0.23 | 0.20 | 0.16
Normal | 0.56 | 0.39 | 0.41 | 0.29 | 0.60 | 0.53 | 0.30 | 0.40 | 0.27 | 0.30 | 0.52 | 0.45 | 0.51 | 0.37 | 0.32
t-student (d.o.f = 10) | 0.57 | 0.40 | 0.41 | 0.29 | 0.60 | 0.53 | 0.31 | 0.41 | 0.28 | 0.30 | 0.53 | 0.45 | 0.52 | 0.37 | 0.33

Table 3: LogLikelihood per sample for different distributions |  GMM
$N_{C}=3$ |  GMM
$N_{C}=4$ |  GMM
$N_{C}=5$ |  GMM
$N_{C}=6$ | Normal
|  t-student
d.o.f = 10
---|---|---|---|---|---|---
S&P 500 | 2.91 | 2.90 | 2.93 | 2.93 | 2.73 | 2.79
MMM | 2.81 | 2.81 | 2.82 | 2.82 | 2.70 | 2.75
AXP | 2.18 | 2.19 | 2.20 | 2.20 | 2.01 | 2.07
AMD | 1.92 | 1.92 | 1.93 | 1.93 | 1.84 | 1.87
AIG | 1.82 | 1.83 | 1.83 | 1.82 | 1.26 | 1.41
AFL | 2.27 | 2.29 | 2.29 | 2.29 | 1.90 | 2.02
ABT | 2.95 | 2.95 | 2.95 | 2.96 | 2.87 | 2.91
MSFTF | 2.63 | 2.63 | 2.64 | 2.64 | 2.50 | 2.55
AAPL | 2.36 | 2.36 | 2.36 | 2.36 | 2.30 | 2.33
AMZN | 2.21 | 2.21 | 2.21 | 2.21 | 2.06 | 2.13
BAC | 2.02 | 2.03 | 2.04 | 2.05 | 1.65 | 1.75
JPM | 2.17 | 2.18 | 2.18 | 2.18 | 1.93 | 2.01
JNJ | 3.24 | 3.24 | 3.24 | 3.25 | 3.06 | 3.14
XOM | 2.71 | 2.71 | 2.71 | 2.71 | 2.54 | 2.62
MA | 2.64 | 2.64 | 2.64 | 2.65 | 2.48 | 2.55
Table 4: KS test’s statistics and p-values

|  GMM
$N_{C}=3$ |  GMM
$N_{C}=4$ |  GMM
$N_{C}=5$ |  GMM
$N_{C}=6$ | Normal |  t-student
d.o.f = 10
---|---|---|---|---|---|---
| p-value | statistic | p-value | statistic | p-value | statistic | p-value | statistic | p-value | statistic | p-value | statistic
S&P | 0.149 | 0.046 | 0.513 | 0.033 | 0.679 | 0.029 | 0.865 | 0.024 | 0.00 | 0.119 | 0.00 | 0.122
MMM | 0.341 | 0.038 | 0.149 | 0.046 | 0.865 | 0.024 | 0.451 | 0.034 | 0.00 | 0.086 | 0.00 | 0.090
AXP | 0.394 | 0.036 | 0.513 | 0.033 | 0.838 | 0.025 | 0.451 | 0.034 | 0.00 | 0.111 | 0.00 | 0.115
AMD | 0.136 | 0.046 | 0.293 | 0.039 | 0.890 | 0.023 | 0.679 | 0.029 | 0.00 | 0.074 | 0.00 | 0.079
AIG | 0.084 | 0.050 | 0.341 | 0.038 | 0.149 | 0.046 | 0.102 | 0.049 | 0.00 | 0.186 | 0.00 | 0.191
AFL | 0.316 | 0.038 | 0.513 | 0.033 | 0.271 | 0.040 | 0.913 | 0.022 | 0.00 | 0.160 | 0.00 | 0.164
ABT | 0.865 | 0.024 | 0.367 | 0.037 | 0.612 | 0.030 | 0.777 | 0.026 | 0.00 | 0.062 | 0.00 | 0.069
MSFTF | 0.316 | 0.038 | 0.230 | 0.042 | 0.976 | 0.019 | 0.712 | 0.028 | 0.00 | 0.090 | 0.00 | 0.097
AAPL | 0.777 | 0.026 | 0.367 | 0.037 | 0.933 | 0.022 | 0.913 | 0.022 | 0.00 | 0.066 | 0.00 | 0.072
AMZN | 0.394 | 0.036 | 0.865 | 0.024 | 0.482 | 0.034 | 0.950 | 0.021 | 0.00 | 0.085 | 0.00 | 0.091
BAC | 0.032 | 0.058 | 0.093 | 0.050 | 0.149 | 0.046 | 0.394 | 0.036 | 0.00 | 0.158 | 0.00 | 0.163
JPM | 0.124 | 0.047 | 0.341 | 0.038 | 0.422 | 0.035 | 0.578 | 0.031 | 0.00 | 0.124 | 0.00 | 0.128
JNJ | 0.195 | 0.043 | 0.913 | 0.022 | 0.745 | 0.027 | 0.808 | 0.026 | 0.00 | 0.096 | 0.00 | 0.103
XOM | 0.645 | 0.030 | 0.808 | 0.026 | 0.178 | 0.044 | 0.545 | 0.032 | 0.00 | 0.093 | 0.00 | 0.100
MA | 0.712 | 0.028 | 0.513 | 0.033 | 0.712 | 0.028 | 0.777 | 0.026 | 0.00 | 0.082 | 0.00 | 0.092
results | ✓ | ✓ | ✓ | ✓ | X | X

Figure 1: Mixture of Normal distribution for individual stocks during a crisis
period

Figure 2 shows how the Gmm with $N_{c}=3$ can model joint distribution of a
stock against the market index. The horizontal axes are index of S&P500, while
the latter axes illustrate the individual stock returns. Every single cluster
has a unique correlation structure.

Figure 2: Clustering stock returns against S&P500 with Gmm $N_{c}=3$

Since in higher dimensions correlation structures of stock returns become more
complicated, Gmm can be trained under changing market conditions to simulate
future returns. The graph for all stocks are shown in appendix in figures 10
and 11. Figure 3 also shows clustering of time series returns with $N_{c}=3$
components. All of stocks are shown in appendix in figures 12 and 13.

Figure 3: Clustering stock returns during the time with $N_{c}=3$

###  6.3 Christofferson test and Quadratic Loss Function

For checking the validity and accuracy of VaRs, we use the proposed method by
Christoffersen (1998). The VaR result should satisfy both unconditional
coverage and independence properties of exceeding returns to be accepted. Two
statistics are

$LR_{uc}=-2\ln{[(1-p)^{n-x}p^{x}]}+2\ln{[(1-\frac{x}{n})^{n-x}(\frac{x}{n})^{x}]},$
(26)

and

$\begin{array}[]{rl}{LR_{ind}=-2}&{\ln\left[\left(1-\hat{\pi}_{2}\right)^{n_{00}+n_{11}}\hat{\pi}_{2}^{n_{01}+n_{11}}\right]+2\ln\left[\left(1-\hat{\pi}_{01}\right)^{n_{00}}\hat{\pi}_{01}\left(1-\hat{\pi}_{11}\right)^{n_{10}}\pi_{11}^{n_{11}}\right]}\end{array},$
(27)

where

${\hat{\pi}_{01}=}{\frac{n_{01}}{n_{00}+n_{01}},\quad\hat{\pi}_{11}=\frac{n_{11}}{n_{10}+n_{11}},\quad\hat{\pi}_{2}=\frac{n_{01}+n_{11}}{n_{00}+n_{10}+n_{01}+n_{11}}}.$

Both statistics follow a $\chi^{2}_{df=2}$ distribution. Here, $n_{ij}$,
($i,j\in\\{0,1\\}$) are the number of days with $i$ exceedance and $j$
exceedance observed in the previous day.

For a loss function $L$, we use a quadratic loss, proposed by Lopez et al.
(1999) and Martens et al. (2009), such that
$L=\frac{\sum_{t=1}^{T}QL_{t}}{T}$, where

$QL_{t}=I_{r_{t}\leq VaR_{t}}(1+(r_{t}-VaR_{t})^{2})$ (28)

and $I$ is an indicator function.

A VaR result is valid if the p-value for both unconditional statistics (UC)
and independent statistics (ID) are greater than $0.01$.

###  6.4 Calculating VaR and back-testing

####  6.4.1 Individual Stocks’ VaR

The VaR for 1000 days, starting from $7/24/2007$ for both individual stocks
and the equally weighted portfolio, are calculated. The results of VaR for
historical, parametric (variance-covariance or Normal), GBM Monte Carlo
Simulation (MS), CaViaR adaptive (C-AD), CaViaR asymmetric (C-AS), CaViaR
symmetric-VaR (C-SAV), CaViaR GARCH (C-GARCH) proposed by Engle and Manganelli
(2004), and GARCH(1,1) are compared with Gmm ($N_{c}=3,4,5,$ and $6$) in
figure 4 and 5 for $1\%$ and $5\%$ levels of significance. See figures 14, 15,
16 and 17 in appendix for all stocks. The calculated VaR are back-tested based
on both unconditional and independence statistics of Christofferson back-
testing, as well as a quadratic loss function. The decision for rejecting a
VaR output depends on the corresponding p-value compared to the significance
level of $1\%$. The results are shown in tables 7 and 8 (1% and 5%) with their
associated p-values. Not rejected VaRs are shown in green, while the rejected
ones are in red. Where $\alpha=1\%$ all of VaRs for different stocks and
$N_{c}$s are not rejected. As table 7 shows the number of failed VaRs for
Historical Simulation (HS), Parametric Variance-Covariance VaR (Normal), Monte
Carlo (MC), GARCH(1,1), CaViaR symmetric-VaR (C-SAV), CaViaR asymmetric
(C-AS), CaViaR GARCH (C-G), CaViaR adaptive (C-AD), are $6,12,12,10,0,2,1$ and
$5$, respectively. Table 9 shows the quadratic loss $L$ for different VaRs.
The Gmms-VaR are lower than other models for almost all stocks in quadratic
loss. In table 8, the number of failed VaR cases for HS, Para, MC, GARCH(1,1),
C-SAV, C-AS, C-GARCH and C-AD are $8,4,4,4,0,1,1\text{ and }5$, while for the
Gmms with different numbers of components, $N_{c}$, are $3,1,3$ and $2$.
Figures 4 and 5 plot some of the individual stocks with HS VaR, Para VaR, MC
VaR, GARCH(1,1) VaR C-SAV, C-AS, C-G, C-AD and Gmm VaR with $N_{c}=3,4,5$ and
$6$ for levels of 1% and 5%. Results of quadratic loss function $L$ for
$VaR_{5\%}$ are also shown in table 10, suggesting a lower amount of losses
for Gmms VaR, in many cases.

We also apply christofferson’s test on our model when $\sigma_{short}$ is
changing in range $\\{10,20,...,90\\}$. The rejected VaRs are shown by R and
marked by red in tables 12, 13, 14 and 15. Otherwise they are illustrated by
NR (not rejected) and shown by green. We conclude that for small
$\sigma_{short}\leq 30$ the VaRs are mostly rejected, while for greater
amounts they have better results. Moreover, there is no significant
differences between $\sigma_{short}=\\{60,70,80\\}$. We do not use
$\sigma_{short}\geq 100$, because we want to apply the effect of recent days
(closer than 100) on the model’s output.

Figure 4: Value at Risk 1 % for individual stocks during a crisis period

Figure 5: Value at Risk 5 % for individual stocks during a crisis period

####  6.4.2 Portfolio of Stocks’ VaR

We consider a portfolio consisting of all the 14 stocks in our analysis and
the S&P500 index itself. Clearly, for a portfolio of stocks, the correlation
structure between constituent assets is not negligible. As Gaussian Copulas
(GC) are strong tools for modeling joint distribution of random variables, we
compare our proposed model with the Gaussian Copula. Portfolio returns are
simulated for 1000 times, and figure 6 shows a comparison between GC and Gmm
($N_{c}=3$) against real data. The vertical axes represent real returns while
the horizontal axes show the simulated ones. Clearly, during a crisis a GC is
unable to capture the fat-tail distributions, in contrary to a Gmm. KS test is
also employed for checking goodness of fit, and the corresponding p-values are
shown in table 5, showing that simulated returns in Gmm conform to a real
distribution, unlike those came from a GC.

Table 5: KS test results between real data and simulated one

|  GC |  GMM
$N_{c}=3$ |  GMM
$N_{c}=4$ |  GMM
$N_{c}=5$ |  GMM
$N_{c}=6$
---|---|---|---|---|---
statistic | 0.1214 | 0.0639 | 0.0575 | 0.0535 | 0.0615
p-value | 1.86E-08 | 0.0120 | 0.0318 | 0.0554 | 0.0175
| X | ✓ | ✓ | ✓ | ✓

Note: This table shows that if the simulated returns from different models are
following the same distribution as real returns. GC denotes Gaussian Copula.
The threshold for not rejecting the null hypothesis based on the p-value is
considered 0.01.

Figure 6: Comparing Simulated portfolio returns between Gmm $N_{c}=3$ (left)
and Gaussian Copula (right)

The VaRs are calculated with the same parameters we used in calculation of
individual stocks, and shown in figure 7. The p-values for testing the VaR in
unconditional coverage (UC), Independence (ID) and Conditional coverage (CC)
tests are demonstrated in table 11. When all these values are higher than the
threshold $0.01$ the VaR is not rejected and shown by ”NR”, and otherwise, it
is rejected and shown by ”R”. In the $5\%$ confidence level, all Gmms VaR and
the GARCH(1,1) VaR are not rejected, whereas other VaRs are being rejected. In
the $1\%$ level, HS VaR, Gmm $(N_{c}=4)$ VaR and Gmm $(N_{c}=6)$ VaR are not
rejected and other VaRs are rejected. We rank the 9 VaR methods for each
level, firstly according to whether they are rejected or not, and secondly
based on their quadratic losses. As it can be seen, Gmm results are superior
according to their UC, ID, and quadratic losses. The execution time for
calculating VaRs in 1000 days are shown in table 6. Although our Gmm model is
based on Monte Carlo approach, and simulates 3000 returns for each day, its
execution time is significantly lower than other models such as GARCH(1,1),
GC, Parametric VaR and Monte Carlo simulation based on a GBM, as well as
different CaViaR types. It does not necessitates of any correlation matrix,
therefore the computation is fast, and Gmm can model the joint distributions
using EM algorithm.

Table 6: Execution time for different VaR models

Model |  GARCH
(1,1) |  GMM
$N_{c}=3$ |  GMM
$N_{c}=4$ |  GMM
$N_{c}=5$ |  GMM
$N_{c}=6$ |  GC
---|---|---|---|---|---|---
time (seconds) | 260.60 | 26.94 | 29.54 | 31.75 | 33.93 | 604.60
Model |  Para |  MC |  C-sy |  C-as |  C-G |  C-ad
time (seconds) | 247.00 | 2693.43 | 520.56 | 726.74 | 789.97 | 726.08

Note: This table compares the computational time between different models. GC
denotes the Gaussian Copula, Para is parametric (variance-covariance), MC is
Monte Carlo Simulation, C-sy is CaViaR Symmetric, C-as is CaViaR Asymmetric,
C-G is CaViaR GARCH, and C-ad is CaViaR Adaptive model.

On the other hand, for one single time parameter training
$\boldsymbol{\theta}$ in a Gmm, by initializing with respect to k-means and
previous parameters, the execution time for convergence is 0.0572 and 0.0305
seconds and the number of iterations is 20 and 2, respectively. This technique
for parameters initialization lessens the computation cost as well as
execution time. Overall, according to the algorithm speed, Gmm outperforms
other models.

Table 7: VaR 1 % for individual stocks

| test | S&P500 | MMM | AXP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | XOM | MA
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
GMM
$N_{c}=3$ | UC | 0.75 | 0.36 | 0.14 | 0.23 | 0.08 | 0.08 | 0.08 | 0.36 | 0.36 | 0.01 | 0.02 | 0.08 | 0.75 | 0.08 | 0.02
ID | 0.62 | 0.56 | 0.5 | 0.19 | 0.23 | 0.25 | 0.23 | 0.56 | 0.54 | 0.37 | 0.33 | 0.47 | 0.62 | 0.47 | 0.3
GMM
$N_{c}=4$ | UC | 0.23 | 0.02 | 0.08 | 0.36 | 0.04 | 0.04 | 0.36 | 0.36 | 0.08 | 0.01 | 0.01 | 0.04 | 0.54 | 0.08 | 0.01
ID | 0.53 | 0.04 | 0.23 | 0.56 | 0.29 | 0.29 | 0.14 | 0.56 | 0.23 | 0.37 | 0.39 | 0.44 | 0.59 | 0.25 | 0.34
GMM
$N_{c}=5$ | UC | 0.01 | 0.14 | 0.14 | 0.75 | 0.02 | 0.08 | 0.02 | 0.02 | 0.36 | 0.02 | 0.01 | 0.08 | 1 | 0.02 | 0.23
ID | 0.4 | 0.02 | 0.2 | 0.62 | 0.03 | 0.25 | 0.3 | 0.33 | 0.54 | 0.42 | 0.34 | 0.47 | 0.65 | 0.33 | 0.19
GMM
$N_{c}=6$ | UC | 0.14 | 1 | 0.14 | 0.23 | 0.01 | 0.02 | 0.08 | 0.14 | 0.54 | 0.02 | 0.02 | 0.08 | 0.75 | 0.14 | 0.14
ID | 0.5 | 0.65 | 0.2 | 0.53 | 0.04 | 0.33 | 0.23 | 0.5 | 0.57 | 0.42 | 0.3 | 0.47 | 0.62 | 0.22 | 0.22
HS | UC | 0 | 0.02 | 0 | 0.04 | 0 | 0.01 | 0.01 | 0 | 0.14 | 0.01 | 0 | 0.04 | 0.01 | 0.08 | 0.14
ID | 0.31 | 0.04 | 0.33 | 0.29 | 0.03 | 0.39 | 0.39 | 0.34 | 0.5 | 0.37 | 0.75 | 0.45 | 0 | 0.25 | 0.22
Normal | UC | 0 | 0 | 0 | 0.01 | 0 | 0 | 0.04 | 0 | 0 | 0.14 | 0 | 0 | 0 | 0 | 0
ID | 0.29 | 0.18 | 0.22 | 0.41 | 0 | 0.71 | 0.27 | 0.6 | 0 | 0.02 | 0.15 | 0.58 | 0.01 | 0.12 | 0.75
MC | UC | 0 | 0 | 0 | 0.04 | 0 | 0 | 0.02 | 0 | 0 | 0.14 | 0 | 0 | 0 | 0 | 0
ID | 0.33 | 0.18 | 0.18 | 0.44 | 0 | 0.71 | 0.3 | 0.65 | 0 | 0.02 | 0.17 | 0.58 | 0 | 0.16 | 0.75
GARCH(1,1) | UC | 0 | 0 | 0.01 | 0.02 | 0 | 0.01 | 0 | 0 | 0.02 | 0.75 | 0 | 0 | 0 | 0 | 0
ID | 0.64 | 0.04 | 0.39 | 0.33 | 0.81 | 0.4 | 0.46 | 0.51 | 0.42 | 0.62 | 0.32 | 0.09 | 0.28 | 0.89 | 0.14
C-SAV | UC | 0.23 | 0.04 | 0.75 | 0.36 | 0.04 | 0.23 | 0.01 | 0.54 | 0.36 | 1 | 0.08 | 0.08 | 0.51 | 0.14 | 0.04
ID | 0.17 | 0.44 | 0.59 | 0.56 | 0.03 | 0.53 | 0.37 | 0.59 | 0.54 | 0.65 | 0.47 | 0.47 | 0.72 | 0.5 | 0.29
C-AS | UC | 0 | 0.01 | 0.54 | 0.23 | 0.23 | 0 | 0.01 | 0.02 | 0.36 | 0.75 | 0.54 | 0.02 | 0.75 | 0.08 | 0.02
ID | 0.33 | 0.41 | 0.57 | 0.53 | 0.19 | 0.07 | 0.37 | 0.42 | 0.54 | 0.62 | 0.59 | 0.03 | 0.62 | 0.23 | 0.42
C-GARCH | UC | 0.02 | 0.02 | 1 | 0.75 | 0.02 | 0.23 | 0.36 | 0.04 | 1 | 0.36 | 0.23 | 0.01 | 0.54 | 0.14 | 0
ID | 0.42 | 0.33 | 0.61 | 0.62 | 0.04 | 0.53 | 0.56 | 0.44 | 0.61 | 0.56 | 0.53 | 0.34 | 0.59 | 0.5 | 0.3
C-AD | UC | 0 | 0.36 | 0 | 0.08 | 0 | 0.08 | 0.14 | 0.04 | 0.23 | 0.08 | 0 | 0 | 0.14 | 0.04 | 0.02
ID | 0.33 | 0.56 | 0.35 | 0.47 | 0.06 | 0.47 | 0.2 | 0.44 | 0.17 | 0.25 | 0.51 | 0.16 | 0.02 | 0.29 | 0.3

Note: This table compares the performance of VaR 1% between different models
based on christofferson unconditional coverage (UC), Independence (ID) test.
The p-values more than 0.01, are considered as ”good” VaRs and shown with
green. VaRs with at least one p-value less than 0.01 are shown by red, and are
not satisfying christofferson’s criteria. HS shows historical simulation VaR,
GC denotes the Gaussian Copula VaR, Normal is variance-covariance VaR, MC is
Monte Carlo Simulation VaR, C-SAV is CaViaR Symmetric VaR, C-AS is CaViaR
Asymmetric VaR, C-GARCH is CaViaR GARCH VaR, and C-AD is CaViaR Adaptive model
VaR.

Table 8: VaR 5 % for individual stocks

| test | S&P500 | MMM | AXP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | XOM | MA
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
GMM
$N_{c}=3$ | UC | 0.32 | 0.07 | 0.09 | 0.16 | 0.02 | 0.66 | 0.25 | 0.88 | 0.47 | 0.39 | 0.01 | 0.56 | 0.16 | 0.25 | 0.32
ID | 0.67 | 0.32 | 0.77 | 0 | 0.47 | 0.45 | 0.39 | 0.16 | 0 | 0.3 | 0.05 | 0.52 | 0 | 0.39 | 0.05
GMM
$N_{c}=4$ | UC | 0.32 | 0.05 | 0.05 | 0.16 | 0.04 | 0.32 | 0.2 | 0.47 | 0.66 | 0.56 | 0.03 | 0.66 | 0.32 | 0.09 | 0.39
ID | 0.43 | 0.36 | 0.64 | 0.01 | 0.7 | 0.32 | 0.43 | 0.04 | 0.01 | 0.23 | 0.03 | 0.48 | 0 | 1 | 0.04
GMM
$N_{c}=5$ | UC | 0.09 | 0.12 | 0.05 | 0.05 | 0.01 | 0.32 | 0.32 | 0.39 | 0.99 | 0.39 | 0.03 | 0.39 | 0.2 | 0.12 | 0.09
ID | 0.93 | 1 | 0.64 | 0.01 | 0.3 | 0.32 | 0.34 | 0 | 0 | 0.29 | 0.09 | 0.62 | 0 | 1 | 0.12
GMM
$N_{c}=6$ | UC | 0.07 | 0.16 | 0.12 | 0.16 | 0.02 | 0.32 | 0.39 | 0.66 | 0.47 | 0.47 | 0.07 | 0.39 | 0.25 | 0.25 | 0.47
ID | 0.99 | 1 | 0.85 | 0.01 | 0.04 | 0.32 | 0.31 | 0.02 | 0 | 0.26 | 0.02 | 0.62 | 0 | 0.39 | 0.03
HS | UC | 0.07 | 0 | 0 | 0.04 | 0 | 0.01 | 0.32 | 0.03 | 0.88 | 0.2 | 0 | 0 | 0.07 | 0.2 | 0.09
ID | 0.14 | 0.22 | 1 | 0.01 | 0.02 | 0.06 | 0.14 | 0 | 0 | 0.41 | 0 | 0.87 | 0 | 0.19 | 0.12
Normal | UC | 0.01 | 0.25 | 0 | 0.09 | 0.01 | 0.07 | 0.78 | 0.2 | 0.89 | 0.01 | 0 | 0.2 | 0.56 | 0.25 | 0.09
ID | 0.96 | 0.72 | 1 | 0.04 | 0.02 | 0.29 | 0.1 | 0 | 0.03 | 0.42 | 0.03 | 0.19 | 0 | 0.06 | 0.12
MC | UC | 0.01 | 0.2 | 0 | 0.09 | 0.02 | 0.12 | 0.89 | 0.12 | 0.99 | 0.01 | 0 | 0.16 | 0.77 | 0.25 | 0.04
ID | 0.96 | 0.77 | 1 | 0.04 | 0.01 | 0.46 | 0.12 | 0 | 0.04 | 0.42 | 0.01 | 0.21 | 0 | 0.06 | 0.07
GARCH(1,1) | UC | 0 | 0.09 | 0.07 | 0.25 | 0.32 | 0.56 | 0.99 | 0.47 | 0.56 | 0.01 | 0.32 | 0.47 | 0.89 | 0 | 0
ID | 0.03 | 0.27 | 0.65 | 0.16 | 0.67 | 0.72 | 0.36 | 0.11 | 0 | 0.88 | 0.34 | 0.57 | 0.32 | 0.61 | 0.39
C-SAV | UC | 0.39 | 0.05 | 0.66 | 0.01 | 0.39 | 0.25 | 0.03 | 0.47 | 0.39 | 0.56 | 0.66 | 0.66 | 0.32 | 0.07 | 0.56
ID | 0.16 | 0.96 | 0.02 | 0.12 | 0.62 | 0.62 | 1 | 0.16 | 0.12 | 0.09 | 0.07 | 0.59 | 0.14 | 0.7 | 0.55
C-AS | UC | 0.66 | 0 | 0.77 | 0.09 | 0.47 | 0.32 | 0.02 | 0.88 | 0.66 | 0.05 | 0.12 | 0.32 | 0.04 | 0.09 | 0.32
ID | 0.19 | 0.25 | 0.02 | 0.27 | 0.99 | 0.32 | 1 | 0.24 | 0.07 | 0.16 | 1 | 0.88 | 0.38 | 0.77 | 0.74
C-GARCH | UC | 0.56 | 0.03 | 0.66 | 0 | 0.56 | 0.05 | 0.18 | 0.39 | 0.38 | 0.16 | 0.2 | 0.56 | 0.12 | 0.16 | 0.47
ID | 0.04 | 0.85 | 0.22 | 0.31 | 0.52 | 0.69 | 0.1 | 0.3 | 0.46 | 0.21 | 0.77 | 0.96 | 0.04 | 1 | 1
C-AD | UC | 0.01 | 0.03 | 0.01 | 0.05 | 0 | 0 | 0.39 | 0.01 | 0.89 | 0.56 | 0 | 0 | 0.05 | 0.03 | 0.12
ID | 0.75 | 0.45 | 1 | 0.02 | 0.53 | 0.08 | 0.31 | 0.01 | 0 | 0.09 | 0.2 | 0.24 | 0.01 | 0.45 | 0.25

Note: This table compares the performance of VaR 5% between different models
based on christofferson unconditional coverage (UC), Independence (ID) test.
The p-values more than 0.01, are considered as ”good” VaRs and shown with
green. VaRs with at least one p-value less than 0.01 are shown by red, and are
not satisfying christofferson’s criteria. HS shows historical simulation VaR,
GC denotes the Gaussian Copula VaR, Normal is variance-covariance VaR, MC is
Monte Carlo Simulation VaR, C-SAV is CaViaR Symmetric VaR, C-AS is CaViaR
Asymmetric VaR, C-GARCH is CaViaR GARCH VaR, and C-AD is CaViaR Adaptive model
VaR.

Table 9: Losses for VaR 1%

|  GMM
$N_{c}=3$ |  GMM
$N_{c}=4$ |  GMM
$N_{c}=5$ |  GMM
$N_{c}=6$ | HS | Para
---|---|---|---|---|---|---
S&P500 | 0.01100 | 0.01400 | 0.01900 | 0.01500 | 0.02301 | 0.04001
MMM | 0.01301 | 0.01801 | 0.01501 | 0.01001 | 0.01801 | 0.02601
AXP | 0.01501 | 0.01601 | 0.01501 | 0.01501 | 0.02202 | 0.02803
AMD | 0.01401 | 0.01301 | 0.01101 | 0.01401 | 0.01702 | 0.02002
AIG | 0.01645 | 0.01847 | 0.01849 | 0.01944 | 0.02684 | 0.04001
AFL | 0.01608 | 0.01709 | 0.01608 | 0.01808 | 0.01913 | 0.02620
ABT | 0.01600 | 0.01300 | 0.01800 | 0.01600 | 0.02001 | 0.01701
MSFTF | 0.01300 | 0.01300 | 0.01800 | 0.01500 | 0.02101 | 0.02401
AAPL | 0.01302 | 0.01602 | 0.01302 | 0.01202 | 0.01502 | 0.02403
AMZN | 0.02001 | 0.02001 | 0.01801 | 0.01801 | 0.01901 | 0.01501
BAC | 0.01804 | 0.02004 | 0.01903 | 0.01803 | 0.02711 | 0.03514
JPM | 0.01701 | 0.01801 | 0.01701 | 0.01701 | 0.01703 | 0.02404
JNJ | 0.01100 | 0.01200 | 0.01000 | 0.01100 | 0.01900 | 0.02301
XOM | 0.01601 | 0.01601 | 0.01801 | 0.01501 | 0.01602 | 0.02402
MA | 0.01801 | 0.01901 | 0.01401 | 0.01501 | 0.01502 | 0.02702
| MC |  GARCH
(1,1) | C-SAV | C-AS | C-G | C-AD
S&P500 | 0.04101 | 0.04000 | 0.01400 | 0.02200 | 0.01800 | 0.02201
MMM | 0.02601 | 0.02701 | 0.01701 | 0.02001 | 0.01801 | 0.01301
AXP | 0.03003 | 0.02002 | 0.01101 | 0.01202 | 0.01001 | 0.02102
AMD | 0.01702 | 0.01802 | 0.01302 | 0.01402 | 0.15000 | 0.01601
AIG | 0.03901 | 0.02850 | 0.01837 | 0.01422 | 0.01907 | 0.03074
AFL | 0.02620 | 0.02111 | 0.01408 | 0.02110 | 0.01409 | 0.01612
ABT | 0.01801 | 0.02100 | 0.02000 | 0.02000 | 0.02000 | 0.01501
MSFTF | 0.02501 | 0.02201 | 0.01200 | 0.01800 | 0.01701 | 0.01700
AAPL | 0.02402 | 0.01802 | 0.01302 | 0.01302 | 0.01002 | 0.01402
AMZN | 0.01501 | 0.01101 | 0.01001 | 0.01102 | 0.01301 | 0.01601
BAC | 0.03613 | 0.02204 | 0.01603 | 0.01203 | 0.01403 | 0.02210
JPM | 0.02404 | 0.02302 | 0.01601 | 0.01802 | 0.01902 | 0.02602
JNJ | 0.02500 | 0.02400 | 0.00800 | 0.01100 | 0.01200 | 0.01500
XOM | 0.02602 | 0.02901 | 0.01501 | 0.01601 | 0.01500 | 0.01701
MA | 0.02702 | 0.02501 | 0.01701 | 0.01801 | 0.02301 | 0.01802

Note: This table compares the performance of VaR 1% between different models
based on Quadratic loss function. The color red shows that if the hypothesis
that VaR is satisfying the christofferson’s criteria is rejected. HS shows
historical simulation VaR, GC denotes the Gaussian Copula VaR, Para is
parametric or variance-covariance VaR, MC is Monte Carlo Simulation VaR, C-SAV
is CaViaR Symmetric VaR, C-AS is CaViaR Asymmetric VaR, C-GARCH is CaViaR
GARCH VaR, and C-AD is CaViaR Adaptive model VaR.

Table 10: Losses for VaR 5%

|  GMM
$N_{c}=3$ |  GMM
$N_{c}=4$ |  GMM
$N_{c}=5$ |  GMM
$N_{c}=6$ | HS | Para
---|---|---|---|---|---|---
S&P500 | 0.05801 | 0.05801 | 0.06301 | 0.06401 | 0.06402 | 0.07102
MMM | 0.06302 | 0.06402 | 0.06102 | 0.06002 | 0.07202 | 0.05802
AXP | 0.06205 | 0.06405 | 0.06405 | 0.06105 | 0.07608 | 0.07407
AMD | 0.06105 | 0.06106 | 0.06506 | 0.06106 | 0.06507 | 0.06207
AIG | 0.06922 | 0.06723 | 0.07128 | 0.06925 | 0.08447 | 0.06922
AFL | 0.05419 | 0.05821 | 0.05820 | 0.05820 | 0.07031 | 0.06427
ABT | 0.05801 | 0.05901 | 0.05701 | 0.05601 | 0.05702 | 0.04801
MSFTF | 0.05102 | 0.05502 | 0.05602 | 0.05302 | 0.06603 | 0.05902
AAPL | 0.05504 | 0.05304 | 0.05004 | 0.05504 | 0.05105 | 0.04905
AMZN | 0.05704 | 0.05504 | 0.05704 | 0.05605 | 0.06006 | 0.03304
BAC | 0.06917 | 0.06717 | 0.06717 | 0.06416 | 0.08831 | 0.07625
JPM | 0.05507 | 0.05407 | 0.05706 | 0.05707 | 0.07612 | 0.06009
JNJ | 0.06001 | 0.05701 | 0.05901 | 0.05801 | 0.06301 | 0.05401
XOM | 0.05803 | 0.06203 | 0.06103 | 0.05803 | 0.05904 | 0.05804
MA | 0.05703 | 0.05603 | 0.06203 | 0.05503 | 0.06205 | 0.06204
| MC |  GARCH
(1,1) | C-SAV | C-AS | C-G | C-AD
S&P500 | 0.07102 | 0.08201 | 0.05601 | 0.05401 | 0.06810 | 0.06902
MMM | 0.05902 | 0.06202 | 0.06401 | 0.07402 | 0.06602 | 0.06602
AXP | 0.07407 | 0.06305 | 0.05304 | 0.05204 | 0.05304 | 0.06907
AMD | 0.06207 | 0.05806 | 0.06805 | 0.06205 | 0.07506 | 0.06407
AIG | 0.06821 | 0.05874 | 0.05770 | 0.05671 | 0.05565 | 0.09841
AFL | 0.06227 | 0.05517 | 0.05921 | 0.05818 | 0.06519 | 0.07730
ABT | 0.04901 | 0.05001 | 0.06601 | 0.06701 | 0.06010 | 0.05601
MSFTF | 0.06102 | 0.05502 | 0.05502 | 0.05102 | 0.05602 | 0.06802
AAPL | 0.05005 | 0.05404 | 0.05604 | 0.05303 | 0.50100 | 0.04905
AMZN | 0.03304 | 0.03402 | 0.05503 | 0.06503 | 0.06104 | 0.05505
BAC | 0.07725 | 0.05813 | 0.05309 | 0.06114 | 0.06011 | 0.09329
JPM | 0.06109 | 0.05605 | 0.05406 | 0.05807 | 0.05507 | 0.07411
JNJ | 0.05201 | 0.04901 | 0.05701 | 0.06501 | 0.06101 | 0.06401
XOM | 0.05804 | 0.06902 | 0.06302 | 0.06204 | 0.06002 | 0.06604
MA | 0.06504 | 0.08202 | 0.05401 | 0.05702 | 0.05502 | 0.06104

Note: This table compares the performance of VaR 5% between different models
based on Quadratic loss function. The color red shows that if the hypothesis
that VaR is satisfying the christoffeson’s criteria is rejected. HS shows
historical simulation VaR, GC denotes the Gaussian Copula VaR, Normal is
variance-covariance VaR, MC is Monte Carlo Simulation VaR, C-SAV is CaViaR
Symmetric VaR, C-AS is CaViaR Asymmetric VaR, C-GARCH is CaViaR GARCH VaR, and
C-AD is CaViaR Adaptive model VaR.

Table 11: The back-testing results for VaRs of a portfolio

rank | Model (VaR 1%) | UC | ID | CC | result | quadratic loss
---|---|---|---|---|---|---
1 | HS | 0.17 | 0.79 | 0.17 | NR | 0.0060
2 | GMM $N_{c}=4$ | 0.01 | 0.4 | 0.01 | NR | 0.0190
3 | GMM $N_{c}=6$ | 0.01 | 0.37 | 0.01 | NR | 0.0210
4 | C-GARCH | 0 | 0.42 | 0 | R | 0.0180
5 | C-ad | 0 | 0.47 | 0 | R | 0.0160
6 | GMM $N_{c}=3$ | 0 | 0.34 | 0 | R | 0.0220
7 | GMM $N_{c}=5$ | 0 | 0.3 | 0 | R | 0.0240
8 | C-sav | 0 | 0.29 | 0 | R | 0.0240
9 | C-as | 0 | 0.25 | 0 | R | 0.0260
10 | Garch | 0 | 0.18 | 0 | R | 0.0300
11 | GC | 0 | 0 | 0 | R | 0.0360
12 | Para | 0 | 0.79 | 0 | R | 0.0400
13 | MC | 0 | 0.39 | 0 | R | 0.1471
rank | Model (VaR 5%) | UC | ID | CC | result | quadratic loss
1 | C-sav | 0.56 | 0.63 | 0.56 | NR | 0.0540
2 | C-as | 0.39 | 0.14 | 0.39 | NR | 0.0570
3 | C-GARCH | 0.56 | 0.55 | 0.56 | NR | 0.0550
4 | GMM $N_{c}=4$ | 0.12 | 0.3 | 0.12 | NR | 0.0620
5 | GMM $N_{c}=6$ | 0.05 | 0.54 | 0.05 | NR | 0.0650
6 | GMM $N_{c}=3$ | 0.03 | 0.46 | 0.03 | NR | 0.0670
7 | Garch | 0.01 | 0.15 | 0.01 | NR | 0.0690
8 | GMM $N_{c}=5$ | 0.01 | 0.15 | 0.01 | NR | 0.0690
9 | HS | 0 | 0.79 | 0 | R | 0.0060
10 | GC | 0 | 0.68 | 0 | R | 0.0720
11 | C-ad | 0 | 0.82 | 0 | R | 0.0750
12 | Para | 0 | 0.33 | 0 | R | 0.0790
13 | MC | 0 | 0.39 | 0 | R | 0.1951

Note: This table compares the performance of portfolio VaRs different models
based on both Christoffersen’s test and Quadratic loss function. If
Christoffersen’s test rejects goodness of a VaR it is shown by R, while
otherwise it is shown by NR (Not Rejected). P-values for unconditional test
(UC), Independence test (ID), and Condition Coverage (CC) are shown. HS shows
historical simulation VaR, GC denotes the Gaussian Copula VaR, Normal is
variance-covariance VaR, MC is Monte Carlo Simulation VaR, C-SAV is CaViaR
Symmetric VaR, C-AS is CaViaR Asymmetric VaR, C-GARCH is CaViaR GARCH VaR, and
C-AD is CaViaR Adaptive model VaR.

Figure 7: Value at Risk 1% (up) and 5% (down) for a portfolio of stocks during
a crisis period

##  7 Conclusion and Future Works

A Gmm-based Monte Carlo approach for simulating stock prices with applications
in value-at-risk has been introduced. Our model can be an alternative to GBM
and Copula approaches. Our results were mostly approved by Christoffersen’s
test and using a quadratic loss function in a crisis period. While the
traditional methods are time-consuming and fail the validity tests, our
portfolio’s VaR algorithm is by far less time consuming and its results were
mostly approved by Christoffersen’s test. The power of Gmm’s simulation
algorithm can be seen when estimating the risk of a portfolio of stocks. Gmm
can produce the portfolio structure, with the same joint distribution and
preserves non-linear correlation between individual stocks quite well.

Since our model’s predictive ability is approved by back-testing routines,
using other forecasting approaches like GARCH models is not necessary.
However, a combination with these models might also provide accurate and
robust results. Another direction to do further research is to apply our
technique for valuation of derivatives. Predicting the trend of market is also
another application of our model which might be used in algorithmic trading
machines. Without a shadow of a doubt, the algorithm speed is vital for
professional traders working in financial institutions. While being highly
accurate, our model’s computational cost is quite low. This is particularly
interesting that as our results show, three number of Gmm components is
sufficient for capturing complex market conditions, even during a crisis.

This method can be applied for simulation of not only stocks, but many other
financial instruments such as plain bonds or bonds with embedded options.
Luckily, Gmm is capable of capturing complex data structure, specifically the
tail of return distributions. Another advantage of using Gmm for calculating
VaR is that there is no assumption for a linear correlation between different
assets.

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## Appendix

##  Appendix A Appendix

Figure 8: Mixture of Normal distribution for individual stocks during a crisis
period

Figure 9: Mixture of Normal distribution for individual stocks during a crisis
period

Figure 10: Clustering stock returns against S&P500 with Gmm $N_{c}=3$

Figure 11: Clustering stock returns against S&P500 with Gmm $N_{c}=3$

Figure 12: Clustering stock returns during the time with $N_{c}=3$

Figure 13: Clustering stock returns during the time with $N_{c}=3$

Figure 14: Value at Risk 1 % for individual stocks during a crisis period

Figure 15: Value at Risk 1 % for individual stocks during a crisis period

Figure 16: Value at Risk 5 % for individual stocks during a crisis period

Figure 17: Value at Risk 5 % for individual stocks during a crisis period
Table 12: Christofferson results for different $\sigma_{\text{short}}$ with
$N_{c}=3$

| $N_{c}=3$
---|---
| VaR 1 %
| S&P500 | MMM | ARP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | ROM | MA
10 | R | R | R | R | R | R | R | R | R | R | R | R | NR | NR | R
20 | R | R | NR | NR | R | NR | R | R | NR | NR | R | NR | NR | NR | NR
30 | R | NR | R | NR | R | NR | NR | NR | NR | NR | R | R | NR | NR | NR
40 | NR | NR | NR | NR | NR | NR | NR | NR | R | R | R | NR | NR | NR | NR
50 | NR | NR | NR | NR | R | NR | NR | NR | NR | R | R | NR | NR | NR | NR
60 | NR | NR | NR | NR | R | NR | NR | NR | NR | NR | R | NR | NR | NR | NR
70 | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR
80 | NR | NR | NR | R | R | NR | NR | NR | NR | NR | NR | NR | R | NR | NR
90 | NR | NR | NR | NR | R | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR
| VaR 5 %
| S&P500 | MMM | ARP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | ROM | MA
10 | NR | R | R | R | R | R | R | R | R | R | R | R | R | R | R
20 | NR | NR | R | R | R | NR | R | NR | R | NR | R | NR | R | R | R
30 | NR | NR | NR | R | R | NR | NR | NR | R | NR | R | NR | R | R | NR
40 | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR | NR | NR | NR | NR
50 | NR | NR | R | R | R | NR | NR | NR | R | NR | NR | NR | NR | NR | NR
60 | NR | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
70 | NR | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
80 | NR | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
90 | NR | NR | NR | R | NR | NR | NR | R | R | NR | NR | NR | R | NR | NR

Table 13: Christofferson results for different $\sigma_{\text{short}}$ with
$N_{c}=4$

| $N_{c}=4$
---|---
| VaR 1 %
| S&P500 | MMM | ARP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | ROM | MA
10 | R | R | NR | R | R | R | R | R | R | R | R | R | NR | R | NR
20 | R | R | NR | NR | NR | NR | R | R | R | R | R | NR | NR | NR | NR
30 | R | R | NR | NR | NR | NR | NR | NR | NR | NR | R | R | NR | NR | NR
40 | R | R | NR | NR | NR | NR | NR | NR | NR | NR | R | R | NR | NR | NR
50 | NR | R | NR | R | NR | NR | NR | NR | NR | R | R | R | NR | NR | NR
60 | NR | R | NR | NR | NR | NR | NR | NR | R | NR | R | NR | R | NR | NR
70 | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR
80 | R | NR | NR | R | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | NR
90 | R | R | NR | R | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | NR
| VaR 5 %
| S&P500 | MMM | ARP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | ROM | MA
10 | R | R | R | R | R | R | R | R | R | R | R | R | NR | R | R
20 | R | NR | NR | R | R | NR | NR | NR | R | R | R | NR | NR | R | R
30 | R | NR | NR | R | R | R | NR | R | R | NR | R | NR | R | NR | NR
40 | R | NR | NR | R | R | NR | NR | NR | R | NR | NR | NR | NR | NR | NR
50 | NR | NR | NR | R | NR | NR | NR | R | R | NR | NR | NR | R | NR | NR
60 | R | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
70 | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR |  | NR | R | NR | NR
80 | NR | NR | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
90 | NR | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR

Table 14: Christofferson results for different $\sigma_{\text{short}}$ with
$N_{c}=5$

| $N_c= 5$
---|---
| VaR 1 \%
| S&P500 | MMM | ARP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | ROM | MA
10 | R | R | NR | NR | R | R | R | R | R | R | R | R | R | R | NR
20 | NR | R | NR | NR | NR | NR | NR | R | NR | R | R | R | NR | NR | NR
30 | NR | NR | NR | NR | NR | NR | NR | R | NR | NR | R | R | NR | NR | NR
40 | NR | NR | NR | NR | NR | NR | NR | R | NR | R | NR | R | NR | NR | NR
50 | NR | NR | NR | NR | NR | NR | NR | NR | NR | R | NR | R | NR | NR | NR
60 | R | NR | NR | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | NR | NR
70 | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR
80 | R | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR | NR
90 | R | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR | NR
| VaR 5 %
| S&P500 | MMM | ARP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | ROM | MA
10 | R | R | NR | R | R | R | R | R | NR | R | R | R | NR | NR | R
20 | NR | NR | NR | NR | R | NR | NR | NR | R | NR | R | NR | NR | R | R
30 | NR | R | NR | NR | R | NR | NR | R | R | NR | R | NR | R | NR | NR
40 | NR | NR | NR | NR | R | NR | NR | NR | R | NR | R | NR | NR | NR | NR
50 | NR | NR | NR | R | NR | NR | NR | R | R | NR | NR | NR | R | NR | NR
60 | NR | NR | NR | R | NR | NR | NR | R | R | NR | NR | NR | R | NR | NR
70 | NR | NR | NR | NR | NR | NR | NR | R | R | NR | NR | NR | R | NR | NR
80 | NR | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
90 | NR | NR | NR | R | NR | NR | NR | R | R | NR | NR | NR | R | NR | NR

Table 15: Christofferson results for different $\sigma_{\text{short}}$ with
$N_{c}=6$

| VaR 1 \%
---|---
| S&P500 | MMM | ARP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | ROM | MA
10 | R | R | NR | R | R | R | R | R | R | R | R | R | NR | R | NR
20 | NR | R | NR | NR | NR | NR | R | R | NR | R | R | NR | NR | NR | NR
30 | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | R | R | NR | NR | NR
40 | NR | NR | NR | NR | NR | NR | NR | NR | NR | R | R | R | NR | NR | NR
50 | NR | NR | NR | NR | NR | NR | NR | NR | NR | R | R | NR | NR | NR | NR
60 | NR | NR | NR | NR | NR | NR | NR | NR | NR | R | R | NR | NR | NR | NR
70 | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR
80 | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR | NR
90 | NR | NR | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | NR | NR
| VaR 5 %
| S&P500 | MMM | ARP | AMD | AIG | AFL | ABT | MSFTF | AAPL | AMZN | BAC | JPM | JNJ | ROM | MA
10 | R | R | NR | NR | R | R | R | R | R | R | R | R | NR | R | R
20 | NR | NR | NR | NR | R | NR | NR | NR | R | NR | R | NR | NR | R | R
30 | NR | NR | NR | NR | R | NR | NR | NR | R | NR | R | NR | R | NR | NR
40 | NR | NR | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | NR | NR | NR
50 | NR | NR | NR | R | NR | NR | NR | R | R | NR | NR | NR | R | NR | NR
60 | NR | NR | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
70 | NR | NR | NR | NR | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
80 | NR | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR
90 | NR | NR | NR | R | NR | NR | NR | NR | R | NR | NR | NR | R | NR | NR