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Cosmos Coinage Code
Jamila
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The Cosmos Coinage Code:
The Multicurrency Suitcase Hedging Protocol
Author
Jamila Awad
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August 2013
Paper: “The Cosmos Coinage Code” (2013)
Author: Jamila Awad
Date: August, 2013
Rights Reserved: JAW Group
JAW Group, 3440 Durocher # 1109
Montreal, Quebec, H2X 2E2, Canada
Mobile: (1) 514 799-4565
E-mail: [email protected]
Cosmos Coinage Code
Jamila
Awad
Executive Summary
The international financial stability sphere reclaims a global coinage code in conjunction
with imminent economic reforms. The research paper strives to deliver a coherent and
impartial multicurrency hedging protocol. The paper is partitioned in three sections.
Section 1 articulates the environmental asymmetric monetary regimes. Section 2
integrates the mathematical, theoretical and atmospheric components of inward-looking
and global fiscal cooperation schemes into the engineered prototype. Section 3
exemplifies the implementation of the brainstorm with a combined devise investment
portfolio. In brief, the universal economic system shall be immunized from divergent
monetary policies and abrasive exchange-rate practices.
Paper: “The Cosmos Coinage Code” (2013)
Author: Jamila Awad
Date: August, 2013
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Cosmos Coinage Code
Jamila
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Introduction
The international monetary cooperation phenomena depicts a predominant foundation in
cementing a stable global coinage environment immune from liquidity crises, market
collapses, credit-based instruments and fiscal regime divergences. Nations adopt inwardlooking monetary guidelines however ponder interest in universal flexible exchange-rate
arrangements to safeguard their economic system from perilous downturns. Ergo, the
dissertation aspires to deliver a coherent and impartial multicurrency hedging protocol in
line with forthcoming financial reforms.
Financial market meltdowns condemn abrasive practices and force rejuvenated fund
infrastructures to shield the confidence of common-man economic participant as well as
to armor the health of economic frames. Therefore, engineered financial risk management
designs shall minimize the ever-present risks involved in financial markets.
The copulae regime analyzes the interdependencies of investment instruments in the
cosmos of financial markets. The copula approach portrays a modeling strategy whereby
a joint distribution is induced by specifying marginal distributions and a copula function
that binds them together. The copula parameterizes the dependence structure of random
variables thereby captures all the joint behavior. In addition, the copula function executes
multidimensional frameworks and enables the evaluation of extreme events. The
investigation of contagion and interconnectedness between currencies deriving from
divergent monetary policies is performed with D-vine as well as Archimedean copula
families. The selected copula classes unravel tail and asymmetric dependencies to
overcome limitations engendered by standardized model risk techniques. They also
strengthen the hedging prototype by encapsulating multivariate parameterization
distributions and by facilitating a docile dependency analysis.
The Value-at-Risk (VaR) quantum metric portrays a widespread financial risk
management tool that is complemented with the expected shortfall to determine the
capital required to buffer unanticipated losses. The VaR is not considered coherent: It
fails to comply with the subadditivity axiom whereas the summed risk of subportfolios
fails to generate an inferior result compared to the sum of individual subportfolio risks
and it neglects to estimate the size of losses when the VaR threshold is exceeded (Artzner
& al., 1999). Thus, a multivariate setting customized with a copula credit risk frame
provides an alternative solution to remedy VaR drawbacks.
The research paper is partitioned in three sections. Section 1 formulates the various
concepts about inward-looking and international monetary policies. Section 2
amalgamates the mathematical, theoretical and atmospheric components of inwardlooking and global fiscal cooperation schemes into the engineered prototype. Section 3
exemplifies the implementation of the hedging technique with a multicurrency
investment portfolio.
Paper: “The Cosmos Coinage Code” (2013)
Author: Jamila Awad
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Jamila
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In brief, coherent and transparent exchange-rate transactions in line with harmonized
monetary policies shall secure the commencement of an unprecedented epoch in
prudential financial risk management.
Paper: “The Cosmos Coinage Code” (2013)
Author: Jamila Awad
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1. The Universal Monetary Cooperation and the InwardLooking Economic Policies.
Section 1 describes the international monetary collaboration regime based on
considerations of structural asymmetries across countries as well as the inward-looking
fiscal policy. In precise terms, the first section articulates the environmental components
that secure the enforcement of a sound global coinage framework.
The universal monetary cooperation depicts a theory widely analyzed during periods of
economic meltdowns as well during episodes of market booms. Nations generally adopt
for domestic policies however consider multi-country devise models to optimize their
home country’s macroeconomic performance. Money holds a strong symbolic content
whereas a coinage portrays a traditional attribute of sovereignty and economic power.
The process of securitization and other financial innovations have engineered new
financing routes for the non-financial sector. In addition, the prices and risk spreads
across all types of financial products encapsulate relevant information to brainstorm as
well as administer monetary policies.
The silent yet ever-present fear of currency wars and competitive devaluations upsurge a
widespread disquietude for all nations regardless their ruling proxy power over the global
economic system.
Central banks meddle during episodes of market disruptions by executing dynamic policy
measures that engender an environment whereas the tacit conventional monetary actions
are constrained and come close to the domain of fiscal governance. Thus, the
methodology to separate untraditional monetary from fiscal activism requisites to
examine the nature of the assets acquired. A second technique demands to investigate the
character of the liabilities that finance the operations. In precise terms, exercises financed
with monetary base are considered monetary and those furnished with debt are esteemed
fiscal. Moreover, practices that involve government debt exclusively are regarded as debt
management.
Central banks generally succeed in administrating operations to steer short-term interest
rates in vision to stimulate economic activity as well as to oversee inflation. Nonetheless,
interventions during shock absorption episodes quest for drastic balance sheet expansions
and significant alterations in asset composition. For example, central banks intervened to
cushion the adverse effects of the past financial crisis by enforcing conventional and
tailored actions to manage consequences derived from distant party operations such as
investment banks as well as insurance firms. The confinement of risk in monetary policy
independence is complex due to the entanglement between central banks’ balance sheet
assets placed with national treasuries and its operations in relation to traditional as well as
unconventional methods.
Paper: “The Cosmos Coinage Code” (2013)
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In game theory, a nation’s macroeconomic fulfillment can be accomplished by
implementing a harmonized multinational coinage plan or by executing a Nash noncollaborative monetary doctrine. For example, central banks shift towards rules-based
protocols with enhanced emphasis on domestic price stability and deflate their activism
towards international monetary guidelines to ameliorate international economic
conditions following financial collapses. Precisely, the entanglement of the past global
economic plunge with the infectious liquidity crisis has exhibited the complexity of a
universal economic system whereas emerging entities as well as developing countries had
experienced a latent recovery. The forthcoming new epoch in prudential financial risk
management has commenced by foreign affairs and central banks closely examining
adverse spillover and contagion effects derived from host countries’ discretionary fiscal
stratagems in order to buffer hazardous repercussions in home soil and to shield the
health of the global financial system.
The upcoming financial reforms reclaim a revision of the policy assumptions behind the
theories and models enforced in rule-like monetary doctrines. The timeframe whereas
interest rates are frozen to a certain benchmark level depicts an example of a deviation
conducted by central banks in a rule-like fiscal regime (Ahrend, 2010). In addition, the
monetary conduit bifurcation is enhanced when countries synchronize their currency
policies with other jurisdictions.
The collaboration of nations in transparent coinage frame of references can therefore
deflate contagion effects induced from host monetary arrangements and inward-looking
currency approaches.
Central banks adopt open market operations to promote capital mobility and to cushion
arbitrage forces that align the rate of return in different currencies. The central banks
policies on interest and funds rate aim to encompass future coinage depreciation or
appreciation as well as to mitigate risk appetite. For example, entities in foreign countries
borrow host currency to finance their ventures even though the anticipated returns are
denominated in local devise. However, the debt carried remains subject to counterparty,
default, exchange-rate and credit risk. Financial institutions acting as liquidity
intermediaries therefore examine crusade projects with a Value-at-Risk (VaR) protocol
whereas the following parameters are infused in the equation: the probability of
insolvency, the interest rate, and finally, the size of the loan. Hence, a prolonged duration
in low interest rates may lead to more risk bearing endeavors in various geographical
locations. The effect might also be magnified when a dynamic interaction occurs between
the lending cycle and the exchange-rates. The conformism behavior must also be
integrated in the picture frame to diagnose the interdependencies between the actions of
reacting institutions to the maneuvers of central banks (Hofmann and Bogdanova, 2012).
The universal equilibrium of interest rates absorbs collisions such a financial meltdowns
or liquidity razes and shall therefore be immunized from divergent monetary policy rules.
Paper: “The Cosmos Coinage Code” (2013)
Author: Jamila Awad
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The magnifying contagion and adverse spillover apparatus between two distinct nations
(a primary nation and a reacting nation) is summarized with the mathematical frame:
Rr = β + (γ*σ)
Definition of variables:
Rr: The interest rate of the reacting nation denominated in their distinctive currency
(domestic currency for the primary nation and foreign devise for the reacting nation).
β: The domestic parameters established by central banks and that are incorporated to set
the interest rate (such as Real Gross Domestic Product and the inflation) for the primary
nation.
σ: The standard deviation that represents variations in monetary policies between the
primary and the reacting nation transposed in their distinctive devise.
γ: The homogeneity level of similarities in monetary policies between the primary nation
and the reacting country transposed in their distinctive currencies.
The mathematical formula demonstrates that central banks of reacting countries adjust
their interest rates following the behavior of a primary nation. The inflammation
propaganda occurs from infectious spillovers effects induced and thus shifts the
international equilibrium of interest rates following the cascade reactions triggered by
host countries in response to alterations in monetary strategies of the primary nation.
Consequently, the cosmos of coinage is impacted by the asymmetric shocks and the
global equilibrium of interest rates absorbs the described magnified mechanism.
On the other hand, central banks also complement monetary approaches with capital
controls to deflate enhanced scales of volatile risk borrowing levels and to moderate
devise appreciation. The currency intervention can however engender drawbacks such as
an accumulation of international reserves that requisite safe and sound investment
techniques in a globally amplified contagion monetary policy atmosphere (Bordo and
Lane, 2012). In brief, central banks and foreign affairs are bound to thwart unrestrained
exchange-rate fluctuations and disproportionate risk bearing schemes to bulwark
divergence in monetary doctrines.
In a floating exchange-rate economic atmosphere, coinage and non-interest bearing
deposits at central banks motion roles in a similar fashion to capital. Central banks apply
stress-tests to balance sheet items to ascertain the capital requirements. In precise terms,
they measure the total capital to be chambered in order to buffer anticipated losses by
determining the level of coinage plus the capital necessary to sustain their operational
expenditures. In light, the size of the central bank’s balance sheet predominates to weigh
extra-budgetary financial market intervention.
In essence, unconventional monetary stratagems shall facilitate economic recoveries
following traumatic episodes and enhance global growth in a harmonious bona fide
mindset. Nations considered individually or collectively portray entities that can distort
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optimal universal equilibrium levels through unilateral activism (Gagnon and Henderson,
2002). The analysis of cross-country strategies to recoup from depressed economic
periods unravels a correlation relationship between the timing of monetary policy
initiatives and the timing of economic rehabilitation. In specific terms, central banks
engaged in forward-looking governance such as preserving interest rates at target levels
as well as expanding money and credit supplies are held responsible for initiating
permanent changes in monetary regimes. A macroprudential set of regulations destined to
limit the capability of financial entities to lend foreign deposits can be counterbalanced
by the array of foreign funds into nonbank channels in the presence of efficient
international coordination. Furthermore, the frontward mode induces a positive impact on
asset prices and consequently on investments as well as export expansions. Thus, the
proactive alterations to monetary guidelines improve competitiveness and moderate
cross-border spillover effects.
The undesirable externalities from the absence of international fiscal harmonization
engender the following consequences: artificially weak devise to attain a competitive
advantage, aggressive policies to depreciate currencies in turbulent periods that trigger
competitive devaluations, and finally, spurious strong coinage that trigger a contagion
crisis and meltdown the global financial system.
The rationale for harmonized international fiscal policies aims to buffer external
counterproductive effects of independent domestic monetary stratagems and to stimulate
a healthy global growth factor after collisions to the economic system. Nations shall opt
for optimal balances between inward-looking guidelines and international coordination to
sustain a sage position in the era of globalization. Coinage stability shall promote global
economic exchanges and shall deflate adverse currency volatility. In light, exchange-rate
misalignments depict a source of hefty disputes for all nations reacting to the fiscal plans
of a primary country. For example, a depreciated devise can trigger competitive pressure
on a nation’s trading partners and can stimulate protectionist sentiments in foreign soil.
The sequel may lead to commercial discords between countries and disrupt trade accords.
In fact, disordered exchange-rates may also engender the following events: contagion
effects in neighboring countries, creation of conditions for speculative attacks, and lastly,
large real appreciations that are reversed by nominal devaluations.
Banking institutions curtail their tendency to pass on lower interest rates to other
participants in a morose economy especially when they are prone to absorb downturn
shocks. Moreover, a lower elasticity of output growth in relation to real interest rates
induces a pronounced incision in rate polices required to yield a predetermined increase
in demand. The process of deleveraging is often initiated during downspin economic
episodes however elicits significant benefits to enhance recoveries only after periods of
traumatic events rather than after normal intervals of financial plunges. In fact, debt
levels are not pushed to excessive levels in normal business cycles which boost lending
habits with more leverage to finance profitable investment ventures and consumption. In
brief, the leveraging parameter evaluates the effectiveness of monetary guidelines.
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Therefore, the optimal timing of executing monetary policies in regards to deleveraging
levels during periods of downturns depicts a primordial action to set in motion strong
recoveries.
However, the collocation of the operational central banks’ policies and direct legislative
control over traditional fiscal guidelines resorts to tailored balance sheet management in
order to achieve monetary goals in periods of economic distress. In line, central banks
play a prevalent intermediary role to secure conventional and emergency actions in order
to stabilize the economic system during episodes of market crashes.
Banking entities opt for target leverage ratios that derive from structural models to
capture tradeoffs between constraints and resources and thus secure optimal operations.
The deviations from predetermined leverage levels impact loan interest rates and provoke
consequences in the blooming of real economy such as altering borrowing conditions.
Banking institutions are constricted in maximizing profits when they face balance sheet
restrictions. In light, the profit ratio of banking firms is measured by the net interest
margin inferred from a loan whereas the deposit of interest payments and the bank’s
deviation degree from its target ratio are deducted. The cost of a loan is extrapolated from
a central bank’s monetary policy and from the reacting entity’s spread related to the
degree of leverage.
The credit supply conduit is initiated by loan provision movements that alter the
equilibrium level of loan financing and real output. Credit intermediaries administer
supply requirements by proactively participating in lending spreads and in modifying
leverage ratios. Therefore, banking entities adjust grant contributions in response to their
balance sheet conditions. Precisely, banks tighten their credit requirements in presence of
capital constraints and loosen their preconditions in absence of capitalization targets.
The credit reservoir channel induces a grant supply schedule that triggers the
measurement of a loan rate demanded by a reacting banking firm:
Ir = Ic + [(σ*δr3) / (1 + Ic)]*Lr
Definition of variables:
Ir: The loan rate for the reacting banking firm towards a monetary policy set by a central
bank.
Ic: The interest rate derived from a monetary policy dictated by a central bank.
σ: The reacting banking institution’s deviation degree from its target capital-to-asset
ratio.
δr: The reacting banking firm’s target capital-to-asset ratio.
Lr: The reacting banking entity’s leverage ratio.
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The elasticity of supply moves proportionally with the reacting bank’s target capital-toasset ratio as well as with the cost derived from the deviation degree. The capital
accumulation of a banking institution relies on its profits thus an increase in its
profitability level triggers a reduction in its loan rate. Moreover, the provision supply that
shifts with monetary policy adjustments is induced by central banks. In summary, credit
supply alterations repercuss the equilibrium level of loan market interest rates. It also
unravels the interaction between granting provision and the credit reservoir channels. The
magnifying effect is therefore accentuated by lending rates whereas procyclical loan
supply shifts impact consumption and investment projects.
In conclusion, the international currencies that play a benchmark role in the global
financial bubble shall safeguard the coinage system from hazardous economic frames.
The imminent reform in international monetary policies shall lay a foundation in securing
stable value, rules-based and manageable devise. In light, the reconstitution of cosmos
monetary arrangements strives to deflate the ever-present risks involved in financial
markets. Moreover, the cosmos currency remodeling code shall complement sage
liquidity management practices in line with credit-based transactions.
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2. The Multicurrency Suitcase Hedging Prototype
The second chapter describes the logic inferred to model the multicurrency hedging
prototype with a copula approach and delivers the cascade steps to implement the
protocol.
2.1 The Copulae Framework
The interlinkage between monetary policies and financial market transactions quest for
sage risk management quantum methodologies that underpin all structural behaviors
induced by the underlying traded instruments. Consequently, the adverse spillover and
contagion effects arising from divergent fiscal as well as economic activism are then
buffered by an impartial and transparent multicoinage suitcase hedging protocol. The
copulae frame of reference apprehends the joint behavior of marginal pairs such as the
combination of currencies in an investment portfolio. Moreover, the copula function
executes multidimensional frameworks and evaluates extreme events.
The theorem of Sklar (1959) decomposes an n-dimensional joint distribution into its nmarginal distributions and a copula that describes the dependency between the nvariables. The copula theory sides the Gaussian assumption to facilitate financial
modeling with a multifaceted approach. The copula function is generated from a random
vector with a cumulative distribution function.
The following mathematical expression describes a copula function, noted as (X1,….,XN)
Є N, presented as a random vector with a cumulative distribution function ( ) and
marginal functions ( n(xn)):
(X1,….,XN) =
n(xn)
=
( X1 ≤ x1,…..,XN ≤ xN) and
(Xn ≤ xn), 1 ≤ n ≤ N
In addition, a copula function of vector is described as a cumulative distribution
function of a probability measure bounded to [0,1]N such:
n(un)
=
(1,….,1,un,1,….,1) = un whereas
≤ n ≤ N, 0 ≤ un ≤ 1 and
(x1,….,xN) = ( 1(x1),…., n(xN)) for the interval
continuity point of n (1 ≤ n ≤ N).
(x1,….,xN) and whereas xn is a
The copula function is characterized with axioms:
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
Every distribution function holds at least one copula function that is uniquely
defined in the dimension (F1(x1),….,FN(xN)) of the points (x1,….,xN), thus all
figures included in the interval
≤ n ≤ N become a continuity point of . The
copula function of is unique when all the marginal functions are continuous.

Every copula
(
is continuous and conforms to the following inequality:
≤ n ≤ N, 0 ≤ un, vn ≤1): | (u1,….,uN) – ( v1,….,vN) |
| un – vn | .

The ensemble of all copula functions is convex and follows one set of
convergence: punctual, uniform or weak.

All copula functions of (X1,….,XN) are also copula functions of
(h1(X1),….,hN(XN)) when (h1,….,hN) that are monotone and non-decreasing
mappings of .

If { (m), m ≥1}is a sequence of probability cumulative distribution function in
the convergence of F(m) to a distribution function with continuous margins
when m→∞ is equivalent to the following states:
(a)
(b)
(m)
→
for all figures included in the interval
,
≤ n ≤ N.
→ when is a unique copula function associated to
illustrates a copula function linked to (m).
(m)
N
and when
(m)
The theory of the conditional copula states that the conditional distribution of (X,Y)
given W, be denoted by the parameter H. Also, the conditional marginal distributions of
X|W and Y|W are denoted F and G respectively. All previously stated variables are
supposed continuous. The cumulative distribution functions of a random parameter as
well as the corresponding probability density function are then schematized.
The copula approach facilitates the analysis of two random variables by unraveling the
dependency between arbitrary variables in a general form. The copula contains
information from the joint distribution that is not enclosed in the marginal distributions.
Hence, the transformation of X and Y to u and v coordinates filters out information
incorporated in the marginal distributions. In precise terms, the copula contains all of
the information on the dependency between X and Y, but disregards the univariate
individual characteristics of X and Y.
A two-dimensional conditional copula depicts a function that conforms with the
following property; [0,1]* [0,1]*Z→[0,1] whereas Z Є and k is a finite integer in line
with the maxims:
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The Sklar theorem for continuous conditional distributions is described with the
parameters: F is the conditional distribution of X|Z, G is the conditional distribution of Y
|Z, and finally H is the joint conditional distribution of (X,Y)|Z. If parameters F and G are
continuous in x and y, then a unique conditional copula is generated: H(x,y|Z) =
(( (F(x|Z),G(y|Z)|Z)).
Conversely, if the parameter F illustrates a conditional distribution of X|Z, G describes a
conditional distribution of Y|Z and portrays a conditional copula, then the function H
becomes a conditional bivariate distribution function enclosing the conditional marginal
distributions F and G.
The Archimedean class of copula encompasses many families due to its achievement in
reducing dimensionality. It contains the following copula names: Ali-Mikhail-Haq
(AMH), AP, Clayton, Frank, Gumbel and Joe. The mathematical properties of the
Archimedean class are captured by an additive generator function φ: II → [ 0,∞] which is
continuous, convex and a decreasing function (φ’(t) < 0 ). Moreover, the additive
generator function may also be indexed by the association parameter θ enabling an entire
family of copulas to be Archimedean. Therefore, any function φ that satisfies the above
stated conditions can be utilized to generate a valid bivariate cumulative distribution
function (cdf).
Table I: The general parameters of the Archimedean class of copulas
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Definition of variables:
θ: The association parameter of the copula.
u: The first coordinate density of the copula.
v: The second coordinate density of the copula.
φ: The additive generator function of the copula
Hence, the parameters of the copulas map the dependency structure of instruments
incorporated in the investment portfolio. The contour scatter plots generated by the
Archimedean class of copula trace the asymmetric and skewed distribution of products
pooled in a multicurrency suitcase.
The rotated flexibility of the Archimedean family of copulas renders an appealing
solution to investigate marginal distributions by surmounting standard dilemmas arising
from the biased assumption that returns are normally distributed. In addition, financial
engineering methodologies with Archimedean copula functions also enable to capture fattail features of the underlying and dependence structures. The D-Vine copula foundation
magnifies the malleability to decapsulate pairs of marginals in the multicurrency
investment suitcase. The combination of D-Vine and Archimedean copulas strengthens
the model without imposing symmetric dependence on variables.
Tail dependence captures the behavior of random variables during extreme events.
Precisely, it measures the probability of observing an extremely large positive or negative
realization of one variable, given that the other variable also took on an extremely large
positive or negative value. Therefore, the assembled copula design enables upper and
lower tail dependency to range anywhere from zero to one. Clayton's copula illustrates an
asymmetric copula, exhibiting greater dependence in the negative tail than in the positive
tail. The generator of the Clayton copulae is determined by ΦC(u) = (u-α – 1) and hence Φ1
(t) = ((t+1) -1/ α). The Clayton copula is suitable for describing dependencies in the left
tail and during financial market meltdowns (Longin and Solnik, 2001). The relationship
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between Kendall’s tau (ρτ) and the Clayton copulae parameter α is computed by: 2 ρτ /(1ρτ).
From the practical point of view, the Archimedean copulas depict a highly malleable
copulae family due to their capabilities to generate a number of copulas from
interpolating between certain copulas and capture various dependency structures. The
Archimedean copulas may be constructed by using a continuous, decreasing or convex
generator function Φ: I →R+ such that Φ(0) = 1. The generator function is called strict
whenever Φ(0) = + ∞.The pseudo inverse of the generator function, subtitled Φ-1, is
continuous, does not increase in boundaries [0, ∞] and strictly decreases in interval [0,
Φ(0)]. Given a generator and its inverse, an Archimedean copula entitled CArchimedean is
generated according to the Kimberling theorem such: C(u1,u2) = Φ-1 (Φ(u1) + Φ(u2)) in
interval [0,1]2 → [0,1] for quantiles of two random variables X and Y.
The generator of the Gumbel copulae is provided by ΦG(u) = ((-ln(u))α and therefore
ΦG(u)-1(t) = exp(-t1/ α). The Gumbel copulas schematize multivariate extreme value
distributions and they are often used to model extreme distributions. They depict
asymmetric Archimedean copulas, exhibiting greater dependence in the positive tail than
in the negative tail. The relationship between Kendall’s tau (ρτ) and the Gumbel copulae
parameter α is illustrated by: 1/(1- ρτ).
A Vine copulae (V(n)) portrays a copula containing n variables that is nested in a set of
trees (T1, ..., Tn−1) whereas the edges of the trees (j) represent the nodes of the trees ((j+1)
where (j = 1,…,n-2)) and each tree contains a maximum number of edges. Thus, a
conventional Vine on n variables in which two edges in a tree are joined by edges sharing
a common node (Bedford and Cooke, 2001).
The Vine copulae conforms to the following axioms:



A tree is characterized (T = (N,E)) with nodes (N) and edges (E) when the edges
represent a subset of unordered pairs of nodes with no cycle. In addition, a
sequence of elements surges for any variable a and b bounded in interval N.
A regular Vine (V) on n elements is defined as: V(n) = (T1, ..., Tn−1) whereas T1
illustrates a tree with nodes (N1 = {1,…,n}) and edges E1.
The regular Vine complies with the proximity property whereas edges (E)
consider the symmetric differences between the numbers of elements in a set.
A regular Vine copula is considered canonical if each tree Ti has a unique maximum node
of degree n-1. A conventional Vine copula is pronounced D-Vine when all nodes in T1
hold a degree attaining a maximum of 2. There are [(n*(n-1))/2] edges in a regular Vine
on n variables. Each edge in a conventional Vine is linked to a conditional copula that
unravels the conditional bivariate distribution with uniform margins.
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Conventional Vine copulas aim to pinpoint a set of conditional bivariate constraints
associated with each edge. The process is initiated with variables that are attainable from
a given edge via a bond known as the constraint set of that edge. Afterwards, when two
edges are joined by the edge of the next tree, the intersection of the respective constraint
sets is parameterized as the conditioning variables. The symmetric differences of the
constraint sets are optimized as the conditioned variables. The order of an edge or of a
node is defined by the cardinality of its conditioning set.
The use of copulas is however challenging in higher dimensions, therefore the
implementation of Vine and Archimedean copula families surmount standard limitations
arising from conventional financial risk regimes. In light, the engineered protocol designs
a complex dependency structure pattern by benefiting from the abundant array of
bivariate copulas as building blocks.
The Vine copula procedure depicts a flexible graphical technique to describe multivariate
copulas modeled from a cascade of bivariate copulas entitled pair-copulas (PCC). The
PCCs decompose a multivariate probability density into bivariate copulas whereas each
pair-copula can be selected independently from the others and thus magnify the
malleability in the dependency arrangement. In particular, the asymmetries and the tail
dependency can be integrated in the conditional independence to inaugurate robust
coherent hedging techniques. Lastly, the Vine copula frame combines the advantages of
multivariate copulas and amplifies the docility of bivariate copulas.
Fitting a Vine copula frame is executed in steps. Primo, the first command identifies an
appropriate Vine tree structure from examined sets of data or through expert knowledge.
The second operation selects copula families for the isolated Vine structure. The third
instruction estimates the parameters of the copula families selected for the given Vine
copula fit. The final procedure evaluates and compares other potential copula pair family
sets.
The copula values for a predetermined array of copula families are performed with the
maximum likelihood estimation. Parametric estimation with the maximum likelihood
portrays a sound methodology to prescribe the optimal copula choice by fitting the data
into a copula and afterwards analyzing the parameters to pursue further investigation.
The maximum likelihood technique is derived from the following logic. The density of
the joint distribution for the copula and the margins is illustrated:
f(x1,….,xn,….,xN) = c( (x1),…., (xn),….,
whereas fn is the density of the margin
c(u1,….,un,….,uN) = [∂
n
(xN))*
N
fn(xn)
and c is the density of the copula described by:
(u1,….,un,….,uN)] ÷ [∂u1,.... ,∂un,....,∂uN] .
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The logarithm likelihood (l(θ)) estimator is schematized with the equation:
l(θ) =
lnPr{(X1,….,XN) = (xt1,….,xtN)}.
Sklar's theorem decomposes a bivariate distribution, Ht, into three components: the two
marginal distributions, Ft and Gt, and finally the copula, . The maximum likelihood
analysis is generated with the differentiation of Ft and Gt as well as the double
differentiation of .: ht(x,y|z) ≡ ft(x|z)*gt(y|z)*ct(u,v|z) where u ≡ Ft(x|z) and v ≡ Gt(y|z).
Reformatting the equation on both sides with the logarithm results: LXY = LX + LY + LC.
The joint log-likelihood is equal to the sum of the marginal log-likelihoods and the
copula log-likelihood. For the purposes of multivariate density design, the copula
representation enhances greater flexibility in the specification by modeling individual
variables using whichever marginal distributions with best fitting properties. In addition,
the maximum likelihood estimator is the most efficient estimator because it attains the
minimum asymptotic variance bound.
In conclusion, the copula function stands an attractive loop computing tool in financial
risk management due to its flexibility to divulgate information relating to the structure of
dependence.
2.2 Risk Statistic Techniques
Risk statistic methodologies are generated from parametric simulations. Prudential risk
management is favored to encompass statistic test flaws whereas a framework of risk
statistic tools enhances an accurate setting to quantify the total risk. The Basel guidelines
emphasize on accurate and robust risk mapping instruments such as the Value-at-Risk
(VaR) and the Expected Shortfall (ES). Ergo, the following sub-division describes the
selected financial quantum management tools.
2.2.1 The Value-at-Risk
The VaR estimates the likelihood of recognizing a monetary loss exceeding a specific
amount for a determined time horizon and at a stated confidence interval. It depicts a
percentile of a profit & loss portfolio distribution enumerated either as a potential loss
from current portfolio value or as an expected loss in the forecast horizon. However, the
VaR does not conform to the subadditivity axiom and it is consequently complemented
by the coherent risk measure entitled the expected shortfall (Artzner & al., 1999).
In light, a coherent risk statistic instrument must satisfy four distinct properties:
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



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Monotonicity: The risk of a portfolio will sequently oscillate with the gravity of
losses in a portfolio.
Positive Homogeneity Degree 1: A size amendment of a position in a portfolio
will uniformly impact the portfolio risk.
Subadditivity: The risk summing subportfolios stands inferior or equal to the
addition conglomeration of their distinct risks.
Translation Invariance: A cash supplement to a portfolio will reduce its risk by
an identical proportion.
2.2.2 The Expected Shortfall
The VaR is therefore shepherd by the coherent expected shortfall statistic to remedy its
incompatibility with the subadditivity maxim. In consequence, the expected shortfall
illustrates the deepness of losses to overstep the VaR benchmark. It also delivers relevant
information about the tail traced from the profit & loss distribution. The expected
shortfall mathematically expresses the conditional expectation of a portfolio loss greater
than a VaR level. In brief, the expected shortfall instrument complies with the
subadditivity postulate and complements the VaR approach due to its convex function of
portfolio weights. It thus enhances optimization while minimizing uncertainty related to
specific restraints.
2.3 The Cascade Steps to Execute the Cosmos Coinage Hedging
Protocol
The engineered protocol aspires to deliver market participants an impartial and robust
multicurrency hedging framework.
The R and the Matlab software depict efficient programs for statistical computing and
graphics. The R program runs on all platforms and provides excellent interfaces with
codes. In addition, the R software offers a one-directional communication channel with
the Matlab software. The Archimedean and the Vine copulae cosmos coinage hedging
technique was designed from the above stated programs. In brief, the archetype has been
modeled with docility to enable participants to scissor the protocol rather than emulate
the prototype to genuinely comprehend the brainstorm.
The prototype is executed in eight cascade commands:
1- The first step requisites to gather the following data: The spot exchange-rate currencies
pooled in the investment portfolio, the data interval length and the weights attributed to
each devise.
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2-The second operation transforms the spot exchange-rate into logarithm returns.
3-The third procedure examines the data with statistical tests: the mean, the median, the
standard deviation, the minima, the maxima, kurtosis and the Jarque-Bera test.
4-The fourth command traces the QQ-plots to visually investigate dependency structures
and behavior in the tails of the distribution.
5-The fifth step migrates the pairs of marginals obtained from the logarithm spot
exchange-rate returns into the D-Vine copula procedure with an optimization restriction
to Archimedean copulas. The D-Vine copula methodology examines the sequences of
marginal pairs and selects the best suited Archimedean copula with the maximum
logarithm likelihood.
6-The sixth operation analyses currency pairs of marginals from the D-Vine copula
optimization then investigates the currency pairs of marginals with the selected
Archimedean copulas.
7-The seventh step incorporates the parameters for the pairs of marginals and the
currency weights into an engineered loss distribution simulation script. The simulation is
repeated 10,000 times with loop computing to retrieve the quantum risk metrics: the
Value-at-Risk and the Expected Shortfall for a 95% confidence level.
8-The final command repeats step 7 for a 99% confidence level.
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3. The Implementation of the Pioneered Hedging Protocol with
a Multicurrency Investment Suitcase.
The final chapter delivers an example of the engineered cosmos coinage hedging
protocol.
A random devise investment suitcase has been assembled: The portfolio contains four
currencies of equal weights for the period from January 1st 2000 to January 1st 2010. The
timeframe sample therefore includes a shock absorption event such as the financial crisis
of 2008. The spot exchange-rates for the coinage combination are denominated against
the U.S. dollar (USD): the Euro (EUR), the Swiss franc (CHF), the Japanese yen (JPY)
and finally the pound sterling (GBP).
Table II: The summary of statistic tests for logarithm spot exchangerate returns.
Test statistic
EUR
CHF
JPY
GBP
currency
currency
currency
currency
Number of
observations
2516
2516
2516
2516
Mean
5,94701E-05
-7,29734E-05
-1,5288E-05
-1,09623E-06
Median
3,12193E-05
0,0000000
1,79557E-05
4,84126E-05
Maximum
0,020067846
0,015647629
0,011762278
0,019260342
Minimum
-0,013042301
-0,021623582
-0,022651271
-0,021568149
Standard
deviation
0,002856582
0,003039597
0,002908287
0,002741144
Kurtosis
2,554672881
2,374487908
4,141374211
6,022398799
Jarque-Bera
p-value
2,2E-16
2,E-16
1,317E-12
2,2E-16
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The Jarque-Bera p-values confirm the rejection of the null hypothesis at 95% level of
confidence. Therefore, the logarithm spot exchange-rates are not normally distributed.
Ergo, a copula framework is enforced to unravel the behavior of multicurrency
interdependencies between each other and in global financial markets.
The QQ-plots are examined to determine the distribution sampling between logarithm
spot exchange-rate returns. The scatter plots support the inference of non-normal
distributions. The schematizations confirm the inferred asymmetric distribution and tail
behavior in the coinage cosmos. In conclusion, a copula approach is retained to
encapsulate multifaceted dependency structures for all instruments pooled in the
investment suitcase.
Figure 1: The QQ-plot for the logarithm returns for EUR/USD and
CHF/USD.
The figure 1 illustrates the deviations in the tails of the distribution for the logarithm
returns for EUR/USD and CHF/USD sample.
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Figure 2: The QQ-plot for the logarithm returns for EUR/USD and
JPY/USD.
The figure 2 exhibits the driftage in the tails of the distribution for the logarithm returns
for EUR/USD and JPY/USD sample.
Figure 3: The QQ-plot for the logarithm returns for EUR/USD and
GBP/USD.
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The figure 3 exemplifies the deflection in the tails of the distribution for the logarithm
returns for EUR/USD and GBP/USD sample.
Figure 4: The QQ-plot for the logarithm returns for JPY/USD and
CHF/USD.
The figure 4 demonstrates the deviations in the tails of the distribution for the logarithm
returns for JPY/USD and CHF/USD sample.
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Figure 5: The QQ-plot for the logarithm returns for JPY/USD and
GBP/USD.
The figure 5 illustrates the driftage in the tails of the distribution for the logarithm returns
for JPY/USD and GBP/USD sample.
Figure 6: The QQ-plot for the logarithm returns for GBP/USD and
CHF/USD.
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The figure 6 unravels the deflections in the tails of the distribution for the logarithm
returns for GBP/USD and CHF/USD sample.
In brief, the test statistics and the QQ-plots converge to an identical conclusion; the
distribution of logarithm spot exchange-rate returns of the currencies pooled in the
investment suitcase is non-normal, asymptomatic and heavily tailed. Therefore, a copula
strategy delivers a flexible arrangement to unravel the structural dependency in line with
divergent monetary policies and with the interlinkage behavior of currencies in global
financial markets.
The D-Vine copula optimization restricted to Archimedean copula enabled to underpin
the best suited Archimedean copulas for the pairs of marginals examined.
Table III: The D-Vine copula optimization results for the six pairs of
marginals examined.
Pairs of
marginals
EUR
EUR
-----
CHF
Clayton
------
JPY
Frank
Clayton
-----
GBP
Gumbel
Frank
Clayton
CHF
JPY
GBP
-----
The D-Vine optimization examined the maximum logarithm likelihood for the sequences
of marginal pairs to select the best suited Archimedean copula:
EUR+JPY → EUR,JPY|CHF
CHF+GBP → CHF,GBP|EUR
JPY+GBP → JPY,GBP|EUR,CHF
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Table IV: The risk quantum results
Risk quantum
metric
VaR 95%
Cosmos Coinage
Suitcase
0.00167758 0.002118918 0.002367995
ES 95%
VaR 99%
ES 99%
0.002783725
The results were obtained by integrating the devise parameters with their respective
selected Archimedean copula following the D-Vine optimization into a loss distribution
script with equal weights for all currencies and then loop computed 10,000 times. The
cosmos coinage hedging technique enabled to reduce the VaR and ES results
comparatively to conventional methods without compromising the adroitness in
execution.
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Conclusion
In conclusion, the hedging multicurrency suitcase protocol delivers a flexible and docile
framework to buffer omnipresent risks involved in transacting coinage in line with
divergent monetary strategies and global financial markets. Vine and Archimedean
copulas model malleable multivariate dependency arrangements. They also construct
joint distributions by specifying second order structures without imposing any algebraic
constraints. Nations are deemed to reckon international monetary coordination and
exponential caution about self-interest policies to enhance best outcomes in a
macroeconomic standpoint. In brief, countries shall harmonize the paradoxical nature of
the coinage complexion: On one side, the currency armors a nation’s economic power in
the global financial playground, and on the other side, it secrets a nation’s fear of threat
of damage.
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