False illusion of safety: is a handful of stress scenarios enough?

False illusion of safety: is a handful of
stress scenarios enough?
Abstract
The use of stress tests to assess bank solvency has developed rapidly over
the past few years. In particular the stress tests run by the USA authorities and by companies under the Dodd-Frank Act suggest that most large
Bank Holding Company are resilient to shocks similar to the last crisis. In
general each crisis has its particularities and banks must be prepared to
new stress scenarios. This paper aims to confront the severely adverse scenario with those based on the framework recently proposed by De Genaro
(2015) for generating maximum entropy Monte Carlo simulations and so
seek to shed some light on the false illusion of safety embedded in regulatory stress tests.
Keywords: Stress Testing; Stress Scenarios; Maximum entropy Monte Carlo
Simulation
JEL: G01; G28; G21;C15
1
1
Introduction
In the midst of the 2008 financial panic caused by the collapse of the subprime
housing market, the U.S. government responded with unprecedented measures,
including liquidity provision through various funding programs, debt and deposit guarantees and large-scale asset purchases. In February 2009, the U.S.
banking supervisors conducted the first-ever system-wide stress test on 19 of the
largest U.S. bank holding companies (BHCs), known as the Supervisory Capital Assessment Program (SCAP). Based on this success, the Federal Reserve
institutionalized the use of supervisory stress tests for establishing minimum
capital standards in 2010 through its now annual Comprehensive Capital Assessment and Review (CCAR) for the same large banking organizations. Shortly
thereafter, the Dodd-Frank Act mandated stress testing for all banking organizations with more than USD 50 billion in total assets, as well as “systemically
important non-bank financial institutions” designated by a newly established
Financial Stability Oversight Council.
The results of the quantitative assessment figure importantly in Federal Reserve policy decisions regarding whether to object to firms plans for future
capital distribution and retention for example, planned dividend payouts. More
generally, stress tests provide valuable information about the safety and soundness of individual firms, and, significantly, allow comparisons and aggregation
across a range of firms. For example, the stress tests conducted by the Federal
Reserve under the terms of the Dodd-Frank Act Stress Tests (DFAST) provide
a very useful perspective on both individual firms and the banking system.
Whereas stress testing is an important risk management tool it is inherently
challenging for a number of reasons. First, it requires specifying one or more
scenarios that are stressful but not implausibly disastrous. Scenarios must have
certain elements of realism, but are certainly ahistorical, and may represent
structural breaks in the processes that are being stressed. Second, stress testing requires forecasting earnings and capital conditional on the nature of the
scenario; not only is this more difficult than unconditional forecasting, but the
objects of interest are the tails of the distribution, which, almost by definition,
are infrequently observed in historical data.
The Federal Reserve Boards rules implementing DFAST require the Board
to provide at least three different sets of scenarios, including baseline, adverse,
and severely adverse scenarios, for both supervisory and company-run stress
tests. The adverse and severely adverse scenarios are not forecasts, but rather
are hypothetical scenarios designed to assess the strength of banking organizations and their resilience to adverse economic environments. In addition, the
baseline scenario follows a contour very similar to the average projections from
surveys of economic forecasters and does not represent the forecast of the Federal Reserve.
2
As stated above stress testing became notorious after the regulatory initiatives headed by the FED. However, the one prior U.S. experience of the usage
of stress testing for determining capital requirements was a spectacular failure:
the Office of Federal Housing Enterprise Oversight’s (OFHEO) risk-based capital stress test for Fannie Mae and Freddie Mac. Frame, Gerardi, and Willen
(2015) study the sources of failure with OFHEO risk-based capital stress test
for Fannie Mae and Freddie Mac. Their analysis uncovers two key problems
with the implementation of the OFHEO stress test. The first pertains to model
estimation frequency and specification, OFHEO had left the model specification and associated parameters static for the entire time the rule was in force.
Second, the house price stress scenario was insufficiently dire.
As the recent failures of stress test have shown, stress scenarios are vital for
stress testing and a badly chosen set of scenarios might compromise its quality
and credibility in two ways. First, the really dangerous scenarios might not have
been considered. This results in a false illusion of safety. Consequently it may
happen that banks go bankrupt although they have recently passed stress tests.
According to Breuer and Csiszár (2013) a notable example are the supposedly
successful stress tests of Irish Banks in 2010, which had to be bailed out a few
months later. Second, the scenarios considered might be too implausible. This
results in a false alarm and subsequent limited effect on risk-reducing actions
by banks and regulators.
In addition, regulators must bear in mind that what is a worst-case scenario
for one portfolio might be a harmless scenario for another portfolio. This limitation becomes more evident in standard stress testing when only a handful of
scenarios is taken into consideration. Stress testing provides relevant information when the quantity and quality of scenarios are rich enough to ensure that
dangerous scenarios have been considered regardless the portfolio’s constituents.
Christensen et al. (2015) observe that financial stress tests which consider
a small number of hand-picked scenarios have, as a benefit, the fact that the
model risk involved in the choice of a risk factor distribution is minimized. However, this advantage comes at a price. Not assuming any risk factor distribution
stress testers cannot judge whether the stress scenarios are really dangerous or
sufficiently plausible.
Therefore this paper seeks to shed some lights on the false illusion of safety
which arises when the stress test is performed with a handful of scenarios by
implementing a methodology capable of constructing a richer set of stress scenarios. To achieve this goal, this paper resort to the framework proposed by
De Genaro (2015) for generating multi-period trajectories via maximum entropy Monte Carlo simulations and its values are used as inputs for determining
the stress test of the four largest US banks. Finally the stress test figures obtained with simulated scenarios are confronted with those arising when the 2015
severely adverse scenarios are used. As a policy implication, regulatory stress
3
scenarios should be enlarged in order to avoid the false illusion of safety.
This paper contains, besides this introduction, 5 more sections. Section 2
presents the background to the regulatory stress test performed in the USA.
Next, in section 3 the main elements of the multi-period stress testing problem
are presented. Section 4 briefly reviews the two-step methodology proposed by
De Genaro (2015) for generating stress scenarios using the concept of maximum
entropy Monte Carlo simulations. Section 5 presents the outcome of a comparative study between the simulated scenarios and the 2015 severely adverse
scenario when applied to determine the stress test for the four largest US banks.
Section 6 presents the final remarks.
2
Background to Comprehensive Capital Analysis and Review (“CCAR”) and Dodd-Frank
Stress Tests (“DFAST”)
The CCAR and DFAST stress tests are separate exercises that rely on similar processes, data, supervisory exercises, and requirements, although DFAST
is far less intensive as it is designed for firms that were not included in the
original CCAR regime (the CCAR followed the 2009 Supervisory Capital Assessment Program (SCAP), a standardized stress test that was conducted for
the 19 largest U.S. BHCs at that time, and was originally conducted for those
same 19 large BHCs). Both exercises are run by the Federal Reserve and in
both cases the aim of the authorities is to ensure that financial institutions have
robust capital planning processes and adequate capital.
The CCAR, which is conducted annually, is closer in scope to micro stress
tests: when the Federal Reserve deems an institution’s capital adequacy or internal capital adequacy assessment processes unfavourable under the CCAR,
it can request it to revise its plans to make capital distributions, such as dividend payments or stock repurchases. Closer in scope to macro stress tests, DFA
stress tests in turn are forward-looking exercises conducted by the Federal Reserve and financial companies regulated by the Federal Reserve. The DFA stress
tests aim to ensure that institutions have sufficient capital to absorb losses and
support operations during adverse economic conditions. The Federal Reserve
coordinates these processes to reduce duplicative requirements and to minimize
burden.
Accordingly, the Federal Reserve publishes annual stress test scenarios that
are to be used in both the CCAR and DFAST exercises. In general, the baseline
scenario will reflect the most recently available consensus views of the macroeconomic outlook expressed by professional forecasters, government agencies, and
other public-sector organizations as of the beginning of the annual stress-test
cycle (October 1 of each year). The severely adverse scenario will consist of a
4
set of economic and financial conditions that reflect the conditions of post-war
U.S. recessions. The adverse scenario will consist of a set of economic and financial conditions that are more adverse than those associated with the baseline
scenario but less severe than those associated with the severely adverse scenario.
The Federal Reserve generally expects to use the same scenarios for all companies subject to the final rules though it may require a subset of companiesdepending on a company’s financial condition, size, complexity, risk profile,
scope of operations, or activities, or risks to the U.S. economy - to include additional scenario components or additional scenarios that are designed to capture
different effects of adverse events on revenue, losses, and capital. An example of
such an additional component is the instantaneous global market shock that applies to companies with significant trading activity. An institution may choose
to project additional economic and financial variables, beyond the mandatory
supervisory scenarios provided, to estimate losses or revenues for some or all of
its portfolios. In the last step, banks are required to use the variables in their
forecasting models to assess their effects on the firm’s revenues, losses, balance
sheet (including risk-weighted assets), liquidity, and capital position for each of
the scenarios over a nine quarter horizon.
3
The multi-period stress testing problem
Typically, a portfolio is defined as a collection of N financial instruments, represented by the vector θ (0) = [θi (0), . . . , θN (0)], where each θi (0) corresponds
to the total notional value of an instrument i at an initial time t = 0. The
uncertainty about futures states of the world is represented by a probability
space (Ω, F, P) and a set of financial risks M defined on this space, where these
risks are interpreted as portfolio or position losses over some fixed time horizon.
Additionally, a risk measure is defined as a mapping ψ : M → R with the interpretation that ψ(L) gives the amount of cash that is needed to back a position
with loss L.
In its turn, asset values are defined as a function of time and a I-dimensional
random vector of t−measurable risk factors Zt,i by f (t, Zt,1 , Zt,2 , . . . Zt,I ) where
f : R+ × RI → R is a measurable function. For any portfolio comprised of N
instruments and notional value θ (0) = [θi (0), . . . , θN (0)] the realized portfolio
variation over the period [t − 1, t] is given by:
L[t,t−1] =
N
X
h
i
θi (0) fi (t, Zt ) − fi (t − 1, Zt−1 )
(1)
i
Here fi (t, Zt ) describes the exact risk mapping function which is not only
limited to linear functions.
5
In a typical stress tests exercise the examiner is interested in assessing what
happens to the portfolio losses given an adverse shock on the risk factors. In
other words, it pictures the same portfolio at two different moments (today and
future) and then calculates the maximum loss over all possible future scenarios
Zt+m :
sup L[t+m,t] (Zt ) =
Zt+m ∈S
N
X
h
i
θi (0) fi (t + m, Zt+m ) − fi (t, Zt )
(2)
i
Therefore a more realistic formulation arises when the planning horizon is
explicitly incorporated into the loss function. In this case it is recognized that
the worst loss could occur any time during the planning horizon and so capital
requirements would be sufficient to support losses occurring any time not only
those arriving at the terminal horizon. Formally the loss function capturing the
worst-loss over planning horizon for a given stress scenario Zt can be describe
as:
#
" t+T N
XX
?
L[t,t+T ] (Zt ) = min
(3)
θi (0) fi (t + 1, Zt+1 ) − fi (t, Zt )
t
i
Therefore the multi-period stress test in line with the CCAR and DFAS can
be formally given by:
sup L?[t+m,t] (Zt )
(4)
Zt+m ∈S
Notwithstanding the quality of this kind of problem depends crucially on the
definition of the uncertainty sets, S. In stress testing the bias toward historical
experience can lead to the risk of ignoring plausible but harmful scenarios which
have not yet happened in history.
As a way of avoiding the quality of stress testing depending too much on individual skills and more on models/methods, the Basel Committee on Banking
Supervision (2005) issued a recommendation to construct stress scenarios observing two main dimensions: plausibility and severity. Severity tends to be the
easiest component to take into account, because risk managers can ultimately
look at historical data and define scenarios based on the largest movement observed for each risk factor. On the other hand, plausibility requires that after
setting individual stress scenarios Si and Sj for risk factors i and j the resulting
joint scenario (Si , Sj ) makes economic sense.
The current literature in stress test has provided different approaches for
generating plausible stress scenarios. A first attempt in this direction was made
by Studer (1999) and Breuer and Krenn (1999), who developed what is called
“traditional systematic stress tests”. In particular, Studer considered elliptical
multivariate risk factor distributions and proposed to quantify the plausibility
of a realization by its Mahalanobis distance:
6
sup
L(Z)
(5)
Z:M aha(Z)≤k
This approach was extended by Breuer et al. (2012) to a dynamic context
where multi-period plausible stress scenarios can be generated. While this approach introduced the systematic treatment of plausibility for generating stress
scenarios it has problems of its own. First, the maximum loss over a Mahalanobis ellipsoid depends on the choice of coordinates. Second, the Mahalanobis
distance as a plausibility measure reflects only the first two moments of the risk
factor distribution. It is important to notice that the argument for working
with elliptical approximations to the risk factor is the relative tractability of
the elliptical case. However, this ability to handle the problem has a cost of
ignoring the fact that an given extreme scenario should be more plausible if the
risk factor distribution has fatter tails.
Recently, Breuer and Csiszár (2013) proposed a new method to overcomes
the shortcomings of Studer‘s method. The authors introduce a systematic stress
testing for general distributions where the plausibility of a mix scenario is determined by its relative entropy D(Qkν) with respect to some reference distribution
ν, which could be interpreted as a prior distribution. A second innovation of
their approach was the usage of mixed scenarios instead pure scenarios. Putting
all elements together Breuer and Csiszár (2013) propose the following problem:
sup
EQ (L) := M axLoss(L, k)
(6)
Q:D(Qkν)≤k
This method represents an important contribution to the literature as it
moves the analysis of plausible scenarios beyond elliptical distributions. The
authors show that under some regularity conditions the solution to (6) is obtained by means of the Maximum Loss Theorem and some concrete results are
only available for linear and quadratic approximations to the loss function.
It can be observed that while elliptical distributions have been an usual
assumption for describing uncertainty on stress testing problems empirical research in financial time-series indicates the presence of a number of facts which
are not properly captured by this class of distributions. Concretely, empirical literature indicates that asset returns display a number of so-called stylized
facts: fat tails, volatility clustering and tail dependence.
4
Maximum entropy Monte Carlo Simulation
Recently De Genaro (2015) proposed a hybrid two-step approach for generating
stress scenario which has shown be flexible enough to capture all the three
stylized facts present in financial asset returns: fat tails, volatility clustering,
7
and tail dependence. In his approach the author describes the uncertainty on
S via a discrete set:
S = {Z1,t (ω), Z2,t (ω), . . . , ZN,t (ω)} ∀ t ∈ [1, Tmax ]
(7)
Where Zi represents a realization (ω) for the source of uncertainty under
consideration and Tmax is the maximum holding period.
According to his methodology the first step consists in determining envei
lope scenarios, Senv
, for each risk factor in M by using Extreme Value Theory
(EVT). One advantage of recognizing that worst-case scenarios are indeed associated with extraordinary events it is very straightforward to employ elements
from EVT literature to pave the ground for constructing S.
Basically, envelope scenarios would represent the worst-case scenario (WCS)
for each risk factor and have the highest level of severity embedded:
i
Senv
= [S(t)il , S(t)iu ] , ∀ t ∈ [1, Tmax ] and i = 1, . . . , I
(8)
Where Tmax is the maximum holding period and I is the number of risk
factors.
Once the parameters of the extreme value distribution have been estimated
the upper and lower bounds of (8) can be obtained using the percentile of (??):
S(t)il = inf{x ∈ < : F(ξ,σ) (x) ≥ αl } , ∀ t ∈ [1, Tmax ] and i = 1, . . . , I
S(t)iu
= inf{x ∈ < : F(ξ,σ) (x) ≥ αu } , ∀ t ∈ [1, Tmax ] and i = 1, . . . , I
(9)
(10)
A compelling argument for this approach is the capacity of defining the size
of the uncertainty set based on probabilistic terms instead of risk aversion parameters which can be either hard to estimate or that are attached to a specific
family of utility functions. Thus, as it is standard in the risk management literature, for daily data, and assuming a confidence level of 99.96%, it is expected
that actual variation would exceed S(t)l one out of 10 years.
While these envelopes can be viewed as a worst-case scenario so it will protect against uncertainty by setting all returns to their lowest (highest) possible
values the end points of the intervals, this approach can produce implausible
scenarios. In practice, there is usually some form of dependence among future
returns, and it rarely happens that all uncertain returns take their worst-case
values simultaneously. It may therefore be desirable to incorporate some kind
of variability and correlation of asset returns. A second limitation with this
uncertainty set comes from the fact that for some no-linear instruments, such
as options, the maximum loss may not occurs when the underlying asset hits its
envelope scenario. For instance, a stress scenario for a portfolio with an ATM
long straddle position is a scenario where the underlying remains unchanged.
8
However for an outright position this scenario generates almost no losses. The
literature calls this the dimensional dependence of maximum loss and suggests
that effective stress scenarios are not only made up of those exhibiting extreme
variation.
As a way to overcome the limitations above De Genaro (2015) suggested
the adoption of a second step. So, assuming a given dynamic data generating
α, Φ), which depends on a parameter vector α and an arbitrary distriprocess, Γ(α
bution Φ the second step consists in generating trajectories for each risk factor
along the holding period which are expected to fill as uniformly as possible the
state space comprised by the upper, S(t)u , and lower bounds, S(t)l :
S = {S(t) : S(t)il ≤ S(t) ≤ S(t)iu , ∀ t ∈ [1, Tmax ] and i = 1, . . . , I}
(11)
Where Tmax is the maximum holding period and I is the number of risk
factors.
De Genaro (2015) pointed out that given a particular realization ω there
is no sufficient condition imposed to assure that S(t, ω) ∈ S. Therefore an
additional structure is required to meet this condition. One way of assuring
that S(t, ω) ∈ S is by recurring to the concept of exit time:
τ (ω) := inf{t ≥ 0|S(t, ω) 6∈ S}
(12)
Therefore, it is possible for every trajectory generated by Monte Carlo to
construct the simulated stopped process as:
S(ω, t), if τ > t
S̄(ω, t) =
(13)
S(ω, τ ), if τ ≤ t
In this situation the use of stopped process is an artifice that assures for any
ω the simulated stopped process is by construction contained in S. Avellaneda
et al. (2000) also adopted a similar artifice in the context of option pricing
for correcting price-misspecification and finite sample effects arising during the
Monte Carlo Simulation.
An important aspect with regard to quantitative stress tests is their computational tractability. A typical Monte Carlo experiment for risk measurement
usually involves thousands of samples, which, combined with a 9 quarters horizon can yield millions of scenarios for just one portfolio. As one can easily
observe, the computational power required to solve the problem when actual
bank portfolios are used increases as the number of scenarios grow, and therefore the number of scenarios should be carefully chosen to maintain the stress
test tractable.
De Genaro (2015) proposed the use of Shannon n-gram (block) entropy for
choosing a second plausible set S ? , S ? ⊂ S, formed by those trajectories with
9
highest entropy. The n-gram entropy (entropy per block of length n) is defined
as:
X En = −
p Aλ1 , Aλ2 , . . . , Aλn logλ p Aλ1 , Aλ2 , . . . , Aλn
(14)
χ
Where Aλ1 , Aλ2 , . . . , Aλn is a sequence of states generated from real-valued
observations xi ∈ R which were discretised by mapping them onto λ nonoverlapping intervals Aλ (xi ). In the above equation the summation is done over
all possible state sequences χ ∈ Aλ1 , Aλ2 , . . . , Aλn . The probabilities p(Aλ1 , Aλ2 , . . . , Aλn )
are calculated based on all subtrajectories x1 , x2 , . . . , xn contained within a
given subvector of length L. In general, processes or variables with entropy
equal to 0 are deterministic, in our context, trajectories with block-entropy
close to zero should present low price variation.
After computing the n-gram entropy for every simulated trajectory these
values are sorted in ascending order: E1 (Z1 ) ≤ . . . ≤ Em (Zj ) where m is used
to denote an ordering based on each entropy trajectory and Ei (Zj ) denotes
the n-gram entropy for the j − th trajectory Zj calculated as defined by (14).
Therefore after raking every simulated trajectory based on its block entropy our
strategy consists in choosing the k − th largest values for every risk factor1 to
form the final stress scenarios, S ? .
To illustrate this concept, the pictures above present two sets of simulated
trajectories grouped according to their block-entropy over a 10-quarters horizon:
1 For those risk factors which are jointly modeled we repeat this calculation for every factor
in the group choosing those realization ωi for each factor l. The final set will be formed by
the union of all realizations of every factor. So, if we have two risk factors and we choose only
the two largest values for each factor, we will end up with 4 scenarios for each risk factors, in
other words, if realization ω1 is the largest for factor one and ω3 is the largest for factor two
we keep both realization in our final set of stress scenarios {ω1 , ω3 } for each factor. In this
way it is possible to preserve the dependence structure presented in the data.
10
0.25
0.2
0.2
0.15
0.15
0.1
0.1
Price variation − %
Price variation − %
0.25
0.05
0
−0.05
0.05
0
−0.05
−0.1
−0.1
−0.15
−0.15
−0.2
1
2
3
4
5
6
Holding Period
7
8
9
−0.2
1
10
Figure 1: Low entropy trajectories
2
3
4
5
6
Holding Period
7
8
9
10
Figure 2: High entropy trajectories
As one can see in Figure 1, 1,000 trajectories were simulated and later classified as low entropy according to the block-entropy estimator. It can be observed
that the vast majority of the trajectories are contained in the [−0.05, 0.05] interval, with just a few exceptions hitting the envelope scenarios. This lack of
coverage means that a number of price paths will not be taken into consideration
for determining the portfolio losses, which might lead to risk underestimation.
On the other hand, as can be seen in Figure 2, 1,000 simulated trajectories
were drawn and ranked as high entropy. In this second set one can observe, as
expected, that trajectories with highest entropy indeed presented wider variations. Additionally, these trajectories performed better in making the coverage
of state space formed by the envelopes scenarios more uniform.
5
Application: generating scenarios for S ?
Once the framework proposed by De Genaro (2015) has been presented in the
previous section, here this framework is applied for generating S ? . The first
step involves defining the envelope scenarios:
S(t)il = inf{x ∈ < : F(ξ,σ,l) (x) ≥ αl } , ∀ t ∈ [1, Tmax ] , i = 1, . . . , I
(15)
S(t)iu
(16)
= inf{x ∈ < : F(ξ,σ,u) (x) ≥ αu } , ∀ t ∈ [1, Tmax ] , i = 1, . . . , I
Here F(ξ,σ,·) (x) is the CDF of a Generalized Pareto distribution parameterized as:
ξx
F(ξ,σ,·) (x) = 1 − 1 +
σ
− ξ1
11
σ ≥ 0 , ξ ≥ −0.5
(17)
Where the last function argument denotes that this function is independently
defined to each tail.
Once the parameters of the GPD have been estimated, the upper and lower
bounds of (8) are obtained using the percentiles of (17) for the level of severity
of αl = 0.0004 and αl = 0.9996.
Next we proceed by specifying the DGP, Γ(α, Φ) that will be used for generating paths of the simulated stopped process, S̄(ω, t):
S = {S̄(ω, t) : S(t)il ≤ S̄(ω, t) ≤ S(t)iu , ∀ t ∈ [1, Tmax ] , i = 1, . . . , I}
(18)
In order to model dynamic volatility and fat tails together, the method implemented for generating asset returns follows a generalization of the two-step
procedure of McNeil and Frey (2000) where assets volatility are modeled by
GARCH-type models and tails distributions of GARCH innovations are modeled by EVT.
Therefore, following the same spirit of the model employed by McNeil and
Frey (2000), assume that the Data Generating Process describing asset returns
is given by:
rt = ω + ρrt−1 + σt t
(19)
where {t } is a sequence of independent Student’s t distribution with ν degrees of freedom to incorporate fat tails often presented in asset returns.
Furthermore, assume that:
σt2 = α0 +
p
X
j=1
2
βj σt−j
++
q
X
αj 2t−j +
j=1
q
X
ζj 1I{t−j <0} 2t−j
(20)
j=1
The indication function 1I{t−j <0} equal to 1 if t−j < 0 , and 0 otherwise.
The specification above was first developed by Glosten, Jagannathan, &
Runkle (1993). The GJR is a GARH model variant that includes leverage
terms for modeling asymetric volatility clustering. An advantage of the GJR
specification includes the asymmetric nature of the impact of innovations: with
ξ 6= 0, a positive shock will have a different effect on volatility than will a negative shock, mirroring findings in equity market research about the impact of
“bad news” and “good news” on market volatility.
In this paper we shall concentrate on (20) assuming p = q = 1. This is preferred for two reasons. First, the GJR(1,1) model is by far the most frequently
applied GARCH model variant specification. Second, we want to keep a more
12
parsimonious specification once we are handling a large scale problem.
The second step according to McNeil and Frey (2000) requires modeling
the marginal distributions for standardized innovations zt := rt /σt of each risk
factor. To accomplish that, a non-parametric smooth kernel density estimator is implemented for describing the center of the data while the Generalized
Pareto Distribution for the upper and lower tails above the threshold is adopted.
To incorporate the dependence structure among different risk factors we
adopted the “t-copula” which has received much attention in the context of
modeling multivariate financial return data. A number of papers such as Mashal
Zeevi (2002) and Breymann et al. (2003) have shown that the empirical fit of
the t-copula is generally superior to that of the so-called Gaussian copula. One
reason for this is the ability of the t-copula to better capture the phenomenon
of dependent extreme values. This is a desirable feature in risk management
because under extreme market conditions the co-movements of asset returns do
not typically preserve the linear relationship observed under ordinary conditions.
5.1
Results
In this subsection it is summarized some aspects of statistical inference, as well
as, the outcomes for the models that we have proposed and estimated. Our
dataset contains end of month closing prices for USD/EUR, USD/GBP and
Dow Jones Total Stock Market Index (Dow Jones for short) over the period
from January 1999 to October 2015. These variables were chosen among those
considered by the FED to conduct stress test in 2015. Although the dimension
of the problem is small, the example illustrates the qualitative properties of the
proposed methods.
The first step requires estimating the parameters for the AR-GARCH processes as defined by equations (19) and (20). In general non-Gaussian GARCH
models parameters are estimated by quasi-maximum likelihood (QMLE). Bollerslev and Wooldridge (1992) showed that the QMLE still delivers consistent and
asymptotically normal parameter estimates even if the true distribution is nonnormal. In our approach the efficiency of the filtering process, i.e., the construction of zt , is of paramount importance. This is so because the filtered residuals
serve as an input to both the EVT tail estimation and the copula estimation.
This suggests that we should search for an estimator which is efficient under
conditions of non-normality. Therefore, as pioneered by Bollerslev (1987) and
adopted here, model’s parameters can be estimated by maximizing the exact
conditional t-distributed density with ν degrees of freedom rather than an approximate density.
Having completed the calibration of GARCH parameters and hence obtained
sequences of filtered residuals we now consider estimation of the tail behavior
13
by using a Pareto distribution for the tails and the Gaussian kernel for the interior of the distribution. A crucial issue for applying EVT is the estimation
of the beginning of the tail. Unfortunately, the theory does not say where the
tail should begin. We know that we must be sufficiently far out in the tail for
the limiting argument to hold, but we also need enough observations to reliably
estimate the parameters of the GPD. There is no correct choice of the threshold
level. While McNeil and Frey (2000) use the “mean-excess-plot” as a tool for
choosing the optimal threshold level, some authors, such as Mendes (2005), use
an arbitrary threshold level of 90% confidence level (i.e. the largest 10% of the
positive and negative returns are considered as the extreme observations). In
this paper we define upper and lower thresholds such that 10% of the residuals
are reserved for each tail.
Estimating the parameters of a copula or the spectral measure is an important part of this framework. In this paper it is implemented the technique
proposed by Romano (2002) which is based on a semi-parametric approach
for estimating parameters of the t-copula. This method which makes mild assumptions on the margins starts by using the marginal distributions obtained
by transforming the standardized residuals to uniform variates by the semiparametric empirical distribution and afterward plug it into the likelihood function to yield a canonical maximum likelihood (CML) for estimating the t-copula
parameters.
Maximum likelihood estimates of the parameters for each estimated model
are presented in table 1 along with asymptotic standard error in parentheses:
14
ω
ρ
α0
α1
β1
ζ
t(ν)
ξ
σ
ξ
σ
USD/EUR
USD/GBP
Dow Jones
DoF
USD/EUR
USD/GBP
AR-GARCH
Dow Jones
0.7065
(0.2914)
-0.07574
(0.0867)
0.1754
(0.834)
0.212
(0.08)
0.7365
(0.1505)
0.184
(0.01247)
8.2
(1.3)
0.1376
(0.1736)
-0.0694
(0.0773)
0.4692
(0.5303)
0.0632
(0.0271)
0.865
(0.1049)
0.1159
(0.0511)
5.95
(2.66)
Lower Tail
0.70658
(0.29149)
-0.07574
(0.0873)
7.6417E-04
(2.75E-04)
0.228
(0.1296)
0.735
(0.167)
0.6861
(0.32)
9.32
(4.34)
0.507
(1.55E-2)
0.815
(0.31E-2 )
0.5413
(5.47E-3)
0.736
(3.17E-2)
Upper Tail
0.5295
(8.36E-3)
0.852
(4.26E-2)
0.5641
(1.55E-2)
0.8162
(0.21E-2 )
0.5996
(3.47E-2)
0.9107
(2.17E-2)
t-copula
0.5622
(3.43E-2)
0.3495
(3.26E-2)
USD/EUR
1
0.6258
0.2361
USD/GBP
0.6258
1
0.1464
24.96
Dow Jones
0.2361
0.1464
1
Table 1: Estimates of GARCH(1,1), tail distribution and t-copula
For each of our three variables, table 1 reports the results of the maximum
likelihood estimation of each model’s parameters. Even though the interpretation and analysis presented in Table 1 is standard in the financial econometrics
literature, some results can be pointed out. First, the autoregressive parameters
are not statistically significant, indicating a low momentum for these variables.
Second, for all variables, GARCH coefficients α1 and β1 are significant at the 1%
and their sum is less than one implying that the GARCH model is stationary,
15
thought the volatility is fairly persistent since (α1 +β1 ) is close to one. Third, the
estimated number of degrees of freedom of the conditional t-distribution for the
USD/EUR, USD/GBP and Dow Jones are smaller than 10 which suggests that
the returns on the selected variables are conditionally non-normally distributed.
In the middle of the table 1 the ML estimation of the GPD parameters fitted
to the observations in excess over the thresholds are presented. For all variable,
the estimated ξ is found to be positive for both lower tail and upper tail of
the filtered residuals distribution. This indicate that the tails on both sides of
the distribution are heavy and all moments up to the forth are finite. Finally,
the parameters describing the dependence are presented at bottom of the table.
The number of degrees of freedom estimated for the t-copula is equal to 24.96
which is a relatively high value.
Once the parameters in each model have been estimated the next step consists of generating maximum entropy trajectories for up to 9–quarters by Monte
Carlo simulation. Thus, based on parameters of table 1, 50,000 samples for the
simulated stopped process were drawn for each variable over a 9–quarter horizon
forming the set S. Finally, the uncertainty set S ? is formed by choosing from
S those 1,000 trajectories with highest block-entropy. The results can be seen
in figures 3, 5, 7 below:
16
EURUSD
0.3
0.2
0.2
0.1
0.1
Accumulated variation (%)
Accumulated variation (%)
EURUSD
0.3
0
-0.1
0
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
1
Figure 3:
EUR/USD
2
3
4
5
Quarters ahead
6
7
8
-0.4
9
Simulated and regulatory scenarios for
1
Figure 4:
EUR/USD
2
3
4
0.2
0.1
0.1
Accumulated variation (%)
Accumulated variation (%)
17
0.2
0
-0.1
-0.3
-0.3
Figure 5:
GBP/USD
4
5
Quarters ahead
9
-0.1
-0.2
3
8
0
-0.2
2
7
GBPUSD
0.3
1
6
Historical and regulatory scenarios for
GBPUSD
0.3
-0.4
5
Quarters ahead
6
7
8
9
Simulated and regulatory scenarios for
-0.4
1
Figure 6:
GBP/USD
2
3
4
5
Quarters ahead
6
7
8
9
Historical and regulatory scenarios for
Dow Jones
0.6
0.4
Accumulated variation (%)
0.2
0
-0.2
-0.4
-0.6
-0.8
1
2
3
4
5
Quarters ahead
6
7
8
9
Figure 7: Simulated and regulatory scenarios for Dow Jones
Dow Jones
0.6
0.4
Accumulated variation (%)
0.2
0
-0.2
-0.4
-0.6
-0.8
1
2
3
4
5
Quarters ahead
6
7
8
9
Figure 8: Historical and regulatory scenarios for Dow Jones
The red dotted lines represent the envelope scenario which were estimated
using Extreme Value Theory and a confidence level of αl = 0.04% and αu =
99.96%. The blue dotted lines represent the regulatory scenario prescribed by
the Federal Reserve to conduct the 2015 annual stress tests of Bank Holding
Companies. Figures 3, 5 and 7 present the scenarios simulated based on maximum entropy Monte Carlo simulation. We observe that paths in this set cover
almost all possible variation along the holding period which is desirable from a
risk viewpoint because in this case CCAR & DFAST will be determined considering a broad range of future outcomes for each variable and no potentially
harmful price variation is left aside.
For comparative purposes, figures 4, 6 and 8 present the historical scenario
for each market variable. We observe that all historical scenarios are within the
envelope scenario. In particular for the EUR/USD and Dow Jones the regula18
tory scenarios are in line with the historical price variation along the planning
horizon, on the other hand for GBP/USD the regulatory scenarios are less severe than the historical behavior for this variable. Finally, we notice that from
the 50,000 trajectories that were originally generated only 75 hit the envelopes
scenarios (boundaries of our problem) and therefore were stopped. This value
represents a very high acceptance rate for our sampling algorithm.
5.2
Discussion of results and implications
Even though the graphical results suggest that simulated scenarios outperform
the regulatory scenarios a concrete measure is required to confirm this affirmation. One way to provide irrefutable elements would be to rerun the stress test
using the simulated scenarios, however to accomplish that it will be necessary to
obtain the balance sheet informations of each bank and its proprietary positions
in each asset class, of course this is a confidential information and these data
are not available. Another alternative for assessing the impact of the simulated
scenarios on real data is via the reports released by the FED when stress test
cycle is completed. In general the FED publishes during the first quarter the
report describing the final results for the stress test conducted in the previous year. Among the results for every BHC it is available some measures such
as projected stressed capital ratios, risk-weighted assets, losses, for each bank,
however with limited disclosure of assets held and derivatives positions.
Due to the lack of real data we resort to some proxy variables. In the United
States all regulated financial institutions are required to file periodic financial
and other information with their respective regulators and other parties. For
banks in the U.S., one of the key reports required to be filed is the quarterly
Consolidated Report of Condition and Income, generally referred to as the call
report or RC report. Specifically, every National Bank, State Member Bank
and insured Nonmember Bank is required by the Federal Financial Institutions
Examination Council (FFIEC) to file a call report as of the close of business on
the last day of each calendar quarter, i.e. the report date. The specific reporting
requirements depend upon the size of the bank and whether or not it has any
foreign offices.
Among the informations provided, the call report have for each institution
the gross (positive and negative) fair value and notional amount for OTC derivatives with unaffiliated financial institutions for interest Rate, FX, Equity, commodities and credit. The nominal amount is the dollar value used to calculate
payments made on swaps and other risk management products. This amount
generally does not change hands and is thus referred to as notional. Notional
amount are not an accurate reflection of credit exposure as they do not reflect
the market value of the underlying contracts and the benefits of close-out netting and collateral.
19
Measuring credit exposure in derivative contracts involves identifying those
contracts where a bank would lose value if the counterparty to a contract defaulted today. The total of all contracts with positive value (i.e., derivatives
receivables) to the bank is the gross positive fair value (GPFV) and represents
an initial measurement of credit exposure. The total of all contracts with negative value (i.e., derivatives payables) to the bank is the gross negative fair value
(GNFV) and represents a measurement of the exposure the bank poses to its
counterparties. Banks usually hedge the market risk of their derivatives portfolios, the change in GPFV is matched by a similar increase in GNFVs, but
this hedge is not perfect and some amount is still subject to price fluctuation.
Therefore it is defined the net current exposure as the gross positive fair value
minus the gross negative fair value which represents the residual exposure when
all netting and hedging mechanisms are taken in consideration.
According to the report published by the Office of the Controller of the Currency using data from call report for all required institutions during the third
quarter 2015, the derivatives activity in the U.S. banking system is dominated
by a small group of large financial institutions. Four large commercial banks
(JPMorgan Chase, Citibank, Bank of America and Goldman Sachs) represent
91.3% of the total banking industry notional amounts and 80.3% of industry.
In table 2 it is presented the exposures for the four large banks in the US
according to the Consolidated Financial Statements for Holding Companies (FR
Y-9C) computed by the Federal Reserve and published on its National Information Center website2 .
2 Link:
http://www.ffiec.gov/nicpubweb/nicweb/HCSGreaterThan10B.aspx
20
Gross Positive and Negative Fair Values
JP Morgan & Co
Bank of America Corp
Citigroup Inc.
Goldman Sachs
GPFV (1)
GNFV (2)
Net ((1)-(2))
GPFV (1)
GNFV (2)
Net ((1)-(2))
GPFV (1)
GNFV (2)
Net ((1)-(2))
GPFV (1)
GNFV (2)
Net ((1)-(2))
FX
$185.015.000
$201.298.000
$(16.283.000)
$103.825.000
$104.986.000
$(1.161.000)
$146.831.000
$150.143.000
$(3.312.000)
$102.604.000
$107.669.000
$(5.065.000)
Equities
$46.727.000
$47.549.000
$(822.000)
$38.184.000
$35.102.000
$3.082.000
$27.208.000
$31.769.000
$(4.561.000)
$56.221.000
$54.276.000
$1.945.000
Table 2: Reported Gross Positive Fair Value (GPFV) and Gross Negative Fair
Value obtained from FR Y-9C report. Values in thousands of dollars. As of
September 30, 2015.
From table 2 it is possible to identify that after taking into consideration all
netting mechanism banks still have exposure on these risk factors and therefore
they are subjected to market risk. Whereas GPFV and GNFV bring valuable
information regarding derivatives exposures it is not possible to identify whether
the bank is long or short, so it is assumed that all banks have net long exposure
on these risk factors.
These figures provide a good perspective on how selected banks are exposed
to FX and equities price risks, however there is no breakdown of figures available,
such as which currency or stock indexes were traded. For simplicity lets assume
that 90% of all equity net exposure comes from Dow Jones and 90% of all FX
net exposure is equally split into USD/EUR and USD/GBP.
Exposure to risk factors
JP Morgan & Co
Bank of America Corp
Citigroup Inc.
Goldman Sachs
USD/EUR
$7.327.350
$522.450
$1.490.400
$2.279.250
USD/GBP
$7.327.350
$522.450
$1.490.400
$2.279.250
Dow Jones
$739.800
$2.773.800
$4.104.900
$1.750.500
Table 3: Exposure to risk factors. Values in thousands of dollars. As of September 30, 2015
21
Once defined how much each bank is exposed to FX and equities our investigation is carried out by means of comparative analysis between the regulatory
and the simulated scenarios. The exercise consists in simulating what would be
the losses for each banks with regulatory scenarios and comparing these losses
with those arising when the simulated scenarios is adopted. Table 4 exhibits the
losses calculated for each bank using the Severely Adverse Scenario informed by
the FED and 1,000 simulated scenarios using the methodology proposed in this
paper.
Stress test outcomes produced by different Stress Scenarios
JP Morgan & Co
Bank of America Corp
Citigroup Inc.
Goldman Sachs
Simulated
Scenarios
$(5.118.961)
$(2.131.354)
$(3.476.757)
$(2.429.482)
Severely Adverse
Scenario
$(1.756.187)
$(1.701.619)
$(2.648.148)
$(1.427.089)
Ratio
291%
125%
131%
170%
Table 4: Stress test results obtained using the Severely Adverse Scenario by the
FED and the Maximum Entropy Monte Carlo Scenario. Ratio represents the
the loss from the simulated scenario divided by the loss from Severely Adverse
Scenario. Values in thousands of dollars
From table 4 it is possible to confirm that the simulated scenarios are more
severe than those provided by the FED and as a result all bank would have
higher capital requirements using these scenarios. The most impacted bank in
our analysis was JP Morgan & Co whose simulated loss was 291% higher with
the simulated scenarios.
5.3
Robustness check
Even though t-copula presents desirable properties for handling high dimension distributions of financial variables, the dependence structure among pairs
of variables might vary substantially, ranging from independence to complex
forms of non-linear dependence, and, in the case of the t-copula, all dependence
is captured by only two parameters, the correlation coefficients and the number
of degrees of freedom. Due to this potential limitation, we follow De Genaro
(2015) and a robustness check is performed for assessing the t-copula results
comparing its outcomes with a more flexible structure given by a fast-growing
technique known as pair-copula originally proposed by Joe (1996).
In general, as stated in Kurowicka and Cooke (2004) a multivariate density can, under appropriate regularity conditions, be expressed as a product of
pair-copulae, acting on several different conditional probability distributions.
22
It is also clear that the construction is iterative in its nature, and that given
a specific factorization, there are still many viable re-parameterizations. Here,
we concentrate on D-vine, which is a special case of regular vines, where the
specification is given in form of a nested set of trees.
This section presents the results of a comparative study performed between
pair copulae and t-copula using the same dataset used in section 5.1. Whereas
pair-copulae is a very flexible approach for handling complex dependence among
variable it requires the imposition of significant number of pairs combinations.
To overcome this issue, we implemented an automated way to select pairs using Akaike criteria. For every pair of copula we estimate its parameters using
maximum likelihood and choose the one with smaller AIC. We tested a total of
18 different pair-copulae: independent; Gaussian; t-copula; Clayton; Gumbel;
Frank; Joe; BB1; BB6; BB7; BB8; rotated Clayton; rotated Gumbel; rotated
Joe; rotated BB1; rotated BB6; rotated BB7 and rotated BB8. We also estimated the standard t-copula using the same dataset. We depicts the results on
table below:
Log-Likelihood
AIC
Pair Copula with AIC
68.20
-128.40
t-copula
65.27
-122.54
Table 5: Fitting results
The configuration which presented the smaller AIC was given by:
Pair Copula
USD/EUR- Dow Jones
USD/GBP - Dow Jones
USD/EUR-USD/GBP given Dow Jones
Family
t-copula
survival Gumbel
Gaussian
Parameters
(0.71;4.70)
1.15
0.23
Table 6: Best pair-copula configuration
In fact, from table 5 we can observe the superior performance of pair-copulae
when considered the statistical criteria. As a reality check of our models we
simulated 10,000 samples using each configuration and computed percentiles
(0.01 − 0.99) for each risk factor and compared them with the empirical distribution.
23
USD/EUR
USD/GBP
Dow Jones
0.01
-3.37%
-2.66%
-5.89%
0.99
3.31%
2.10%
4.44%
Table 7: Empirical percentiles for each risk factor
USD/EUR
USD/GBP
Dow Jones
0.01
-3.40%
-2.82%
-5.92%
0.99
3.48%
2.41%
4.65%
Table 8: Losses percentiles for each risk factor using Pair-copulae.
USD/EUR
USD/GBP
Dow Jones
0.01
-3.38%
-2.80%
-5.87%
0.99
3.45%
2.38%
4.55%
Table 9: Losses percentiles for each risk factor using t-copula
We conclude from tables 8 and 9 that both methodologies are capable to
produce results in line with those observed in practice (table 7).
We performed a comparative study among t-copula and 18 possible pair
copulae and we found that even though a pair copula presents a better fit in
our example its results are not substantially different from a t-copula when
applied to measure tail risk in a portfolio. Additionally, despite its appealing
flexibility for describing dependence among risk factors, pair-copula requires a
additional step where one should specify the dependence structure itself which
in a real world application might be a high dimension problem there are no
enough elements to replace the parsimonious t-copula.
6
Final remarks
Stress testing has become an increasingly popular tool for assessing the resilience of financial institutions to adverse macro-financial developments. The
2008 financial turmoil and the euro area sovereign debt crisis, which exposed
the financial sector to unprecedented adverse shocks, reinforced this trend.
Regulators in different jurisdictions have implemented stricter rules and reforms so as to increase banks resilience. Stress tests held by Federal Reserve and
24
European Banking Authority allowed banks to meet strict capital requirements
ensuring that they are allegedly well-cushioned during future crises.
Despite the substantial analytical advances in stress testing techniques made
in recent years by the regulators and academics, several challenges remain.
Firstly, it is related to the fact that while the scenarios should be, by definition, stressful it should not be implausible. In other words, when designing
the scenarios, due consideration needs to be given to ensuring a level of severity
that is appropriate but it should reflect a material risk that enforce institutions to pursue risk-reduction actions. Another important point is that capital
requirements based on a handful of stress scenarios can create a false illusion
of safety. In general, bank holding companies (BHCs) have complex portfolios
with a countless number of non-linear instruments so it is important to perform
stress testing covering a wide spectrum of price variations as a way to unveil
the potential losses resulting from non-linear payoffs.
Whereas some stress testing frameworks, including the one presented here,
have already made steps towards overcoming some of these analytical challenges,
further efforts are clearly needed in coming years to improve the overall reliability and accuracy of stress test exercises.
References
[1] Avellaneda, M., Buff, R., Friedman, C. A., Grandechamp, N., Kruk, L. and
Newman, J. (2001) Weighted Monte Carlo: A New Technique for Calibrating Asset-Pricing Models (2001). International Journal of Theoretical and
Applied Finance, 4,1–29 .
[2] Bertsimas, D. and Pachamanova, D. (2008). Robust multiperiod portfolio
management in the presence of transaction costs, Computers & Operations
Research, 35, 3-17.
[3] Bollerslev, T. (1987), A Conditionally Heteroskedastic Time Series Model
for Speculative Prices and Rates of Return, Review of Economics and
Statistics, 69, 542–547
[4] Basel Committee on Banking Supervision.(2005). International Convergence of Capital Measurement and Capital Standards: A Revised Framework. Technical Report, Bank for International Settlements.
[5] Bedford, T. and R. M. Cooke (2001). Probability density decomposition
for conditionally dependent random variables modeled by vines. Annals of
Mathematics and Artificial Intelligence 32, 245-268.
[6] Bedford, T. and R. M. Cooke (2002). Vines - a new graphical model for
dependent random variables. Annals of Statistics 30(4), 1031-1068.
25
[7] Bollerslev, T. and J.M. Wooldridge (1992), Quasi-Maximum Likelihood Estimation and Inference in Dynamic Models with Time Varying Covariances,
Econometric Reviews, 11, 143–172.
[8] Breuer, T. and Csiszár, I. (2013). Systematic stress tests with entropic
plausibility constraints. Journal of Banking & Finance, 37, 1552-1559
[9] Breuer, T. and Krenn, G. (1999). Stress Testing. Guidelines on Market
Risk 5, Oesterreichische Nationalbank, Vienna.
[10] Breuer, T., Jandacka, M., Mencı́a, J., Summer, M.(2012). A systematic
approach to multi-period stress testing of portfolio credit risk. Journal of
Banking and Finance, 36(2), 332-340.
[11] Breymann, W., Dias, A. and Embrechts, P. (2003). Dependence structures
for multivariate high-frequency data in finance. Quantitative Finance 3,
1-14.
[12] Christensen, J. H.E. , Lopez, J. A. and Rudebusch, G. D. (2015) A
probability-based stress test of Federal Reserve assets and income. Journal
of Monetary Economics, 73, 26-43
[13] Committee on Payment and Settlement Systems and Technical Committee
of the International Organization of Securities Commissions (2012), Principles for financial markets infrastructures.
[14] De Genaro (2015) Systematic multi-period stress scenarios with an application to CCP risk management, Journal of Banking and Finance, forthcoming.
[15] Embrechts, P., McNeil, A. and Straumann, D. (2001). Correlation and
dependency in risk management: properties and pitfalls. In Risk Management: Value at Risk and Beyond, M. Dempster H. Moffatt, eds. Cambridge
University Press, 176-223.
[16] Embrechts, P., Resnick, S. and Samorodnitsky, G. (1999). Extreme Value
Theory as a Risk Management Tool. North American Actuarial Journal,
3(2), 30–41.
[17] Glosten, L.R., R. Jagannathan and D. Runkle (1993), On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return
on Stocks, Journal of Finance, 48, 1779–1801.
[18] Hansen, B.E. (1994), Autoregressive conditional density estimation, International Economic Review, 35(3), 705-730.
[19] Joe, H. (1996). Families of m-variate distributions with given margins and
m(m-1)/2 bi- variate dependence parameters. In L. Ruschendorf and B.
Schweizer and M. D. Taylor (Ed.), Distributions with Fixed Marginals and
Related Topics.
26
[20] Kurowicka, D. and R. M. Cooke (2004). Distribution - free continuous
Bayesian belief nets. In Fourth International Conference on Mathematical
Methods in Reliability Methodology and Practice, Santa Fe, New Mexico.
[21] Mashal, R. and Zeevi, A. (2002). Beyond correlation: extreme comovements
between financial assets. Unpublished, Columbia University.
[22] McNeil, A. (1999) Extreme Value Theory for Risk Managers. Unpublished,
ETH Zurich.
[23] McNeil, A.J., Frey R., Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach, Journal of
Empirical Finance, 7, 271-300.
[24] Mendes, B.V., 2005. Asymmetric extreme interdependence in emerging equity markets. Applied Stochastic Models in Business and Industry, 21, 483–
498.
[25] Nelsen, R. B. (1999). An Introduction to Copulas. Springer, New York.
[26] Nelson, D.B. and Cao, C. (1992). Inequality Constraints in the Univariate
GARCH Model, Journal of Business and Economic Statistics, 10, 229–235.
[27] Ou, J. (2005). Theory of portfolio and risk based on incremental entropy.
Journal of Risk Finance, 6, 31-39.
[28] Philippatos, G. and Wilson, C. (1972). Entropy, market risk, and the selection of efficient portfolios. Applied Economics, 4, 209-220.
[29] Romano, C. (2002) Calibrating and simulating copula functions: An application to the Italian stock market, Working paper n. 12, CIDEM.
[30] Studer, G. (1999). Market risk computation for nonlinear portfolios. Journal of Risk, 1(4), 33-53.
[31] Zhou, R., Ru Cai, R. and Tong, G. (2013). Applications of Entropy in
Finance: A Review. Entropy, 15, 4909–4931;
27