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Citation: Tenkam, H.M.; Mba, J.C.;
Mwambi, S.M. Optimization and
Diversification of Cryptocurrency
Portfolios: A Composite
Copula-Based Approach. Appl. Sci.
2022, 12, 6408. https://doi.org/
10.3390/app12136408
Academic Editor: Arcangelo
Castiglione
Received: 21 March 2022
Accepted: 7 June 2022
Published: 23 June 2022
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applied sciences
Article
Optimization and Diversification of Cryptocurrency Portfolios:A Composite Copula-Based Approach
Herve M. Tenkam 1,* , Jules C. Mba 2 and Sutene M. Mwambi 2
1 Department of Mathematics and Applied Mathematics, North West University, P.O. Box 209,Potchefstroom 2520, South Africa
2 School of Economics, University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa;[email protected] (J.C.M.); [email protected] (S.M.M.)
* Correspondence: [email protected]; Tel.: +27-182852494; Fax: +27-182992570
Abstract: This paper focuses on the selection and optimisation of a cryptoasset portfolio, using theK-means clustering algorithm and GARCH C-Vine copula model combined with the differentialevolution algorithm. This integrated approach allows the construction of a diversified portfolio ofeight cryptocurrencies and determines an optimal allocation strategy making it possible to minimizethe conditional value-at-risk of the portfolio and maximise the return. Our results show that stable-coins such as True-USD are negatively correlated to the other cryptoassets in the portfolio and couldtherefore be a safe haven for crypto-investors during market turmoil. Our findings are in line withprevious studies exhibiting stablecoins as potential diversifiers.
Keywords: multivariate t-copula; CVaR; differential evolution algorithm; K-means clustering; vinecopula; cryptocurrency
1. Introduction
In the modern theory of portfolio selection, investor preferences are defined in termsof profit and risk. The return of a portfolio is a combination of the returns of the weightedassets that compose it. The risk of a portfolio is a function of the correlation between theassets that compose it. It is therefore important to diversify your portfolio so as not tosuffer from large fluctuations in the asset prices due to the reoccurrence of shocks in thefinancial sphere and the increase in geopolitical and macroeconomic uncertainties. Thesefluctuations in assets prices are also linked to the general health of the sector in whichone invests. It is well known that, for example, the technology, telecommunications andcryptocurrency markets are marked by strong fluctuations. Diversification therefore obeysthe famous adage in portfolio management which says: “you must not put all your eggs inone basket”. Thus, according to the modern portfolio theory developed by Markowitz [1],the ultimate goal of portfolio managers is to combine a set of assets with maximum profitfor a given level of risk or alternatively, with minimum risk for a given level of profit. Thisis the efficient portfolio.
Indeed, the frantic search for diversification and the creation in 2009 of new digitalfinancial assets based on the highly secure “distributed ledger technology” (DLT) andcryptography, have led portfolio managers and many financial institutions to successfullyintegrate this new class of crypto-assets into the financial world. Cryptocurrencies orvirtual currencies (VC) are the main components of crypto-assets defined as a type ofunregulated or decentralised digital currency, created and generally controlled by itsdevelopers, used and accepted by members of a virtual community. Therefore, it is easyto become a cryptocurrency market player as long as one has some basic knowledgeabout their functionality. VCs have many unusual characteristics compared to otherfinancial instruments such as lack of centralised control, (pseudo-) anonymity, difficulties ofestimating their value, their hybrid characteristics combining aspects of traditional financialinstruments with those of intangible assets, and the rapid evolution of technology which
Appl. Sci. 2022, 12, 6408. https://doi.org/10.3390/app12136408 https://www.mdpi.com/journal/applsci
Appl. Sci. 2022, 12, 6408 2 of 18
underpins them. Those erratic characteristics have contributed to their popularity andthe rapid growth of their total market capitalisation which was estimated to 2.37 trillionUS dollars in May 2021 (https://coinmarketcap.com/all/views/all/, accessed on 20 May2021) with Bitcoin, the oldest and most traded and valuable cryptocurrency representing44% and other altcoins (alternative coin to Bitcoin) share the rest. The abovementioned factsand the potential shown by cryptocurrency to become a cheap alternative to conventionalcurrencies can justify the interests of investors, portfolio managers, financial institutionsand researchers on the opportunities offered by the cryptocurrencies markets.
A good portfolio is one that gives a maximum return for a given level of risk orone that gives the minimum risk for a given level of return. Thus, a good portfoliomust combine different assets satisfying a set of given restriction in order to achieve thatgoal. Hence, this situation requires mathematical modelling for portfolio selection andoptimization. In practice, the portfolio selection (or allocation) and optimisation problemsmust take into account real characteristics of the assets that compose it, such as correlationand dependencies that may exist between assets, market risks, quantity constraint whichimposes a limit on the number of assets in the portfolio, weigh allocation constraintslimiting the proportion of each asset in the portfolio and transaction cost. This leads to acomplex optimization problem.
Markowitz’s model [2] was the first to formalise an analytical response to the assetallocation and selection problem. Indeed, Markowitz considers the particular case ofinvestors with risk preferences adjusting to quadratic utility functions. The analyticalsolution to this problem gives all the portfolios that form the “so-called efficient frontier”,which represents the optimal returns to be achieved for each level of risk. In order toovercome the shortcomings of the Markowitz approach, alternative approaches have beendeveloped. Bares et al. [3] discuss portfolio optimization within the framework of theexpected utility approach using iso-elastic utility functions. Javed et al. [4], Khan et al. [5]as well as Jurczenko et al. [6] proposed the analysis based on moments of higher orders.Hunjra et al. [7], Krokhmal et al. [8] as well as Agrawal and Naik [9] construct optimalportfolios using alternative risk measures. All these methods have shown their performancecompared to the results given through classical analysis. This confirms the interest indealing with issue of portfolio selection outside the Gaussian framework.
One of the approaches aimed at relaxing the Gaussian framework imposed by themean-variance approach concerns the modelling of the asymmetric nature of the depen-dency structure of the portfolio [10]. One of the solutions is that of copula functions, vinecopula specification based on sequential method (also based on maximum spanning treealgorithm and maximum likelihood estimation of the pair-copula parameters) used to solvethe joint probability modelling problem. The use of copula functions in the context of theasset allocation problem is still relevant today. Mba et al. [11], use GARCH-differentialevolution t-copula method in order to optimize and analyse cryptocurrency portfoliorisk and return within the framework of a multi-period setting type approach. More re-cently, Boako et al. [12] integrated copula functions into a GARCH modelling to analysethe structural interdependencies among seventeen cryptocurrency prices and to optimisethe portfolio Value-at-Risk (VaR). In the same wake Mba and Mwambi [13] used a two stateMarkov-switching technique combined with R-vine copula and GARCH (MSCOGARCH)to model heavy tail dependencies and structural breaks within the states of Markov switch-ing with the aim of achieving a maximum return with a minimum conditional value-at-riskof a portfolio of top ten virtual currencies (in term of market capitalisation).
The results obtained in the aforementioned works on the selection, allocation andoptimization of crypto-asset portfolios present some shortcomings, namely:
• These portfolios have a very low VaR or Conditional VaR and high return (above 50%)which is not in line with the crypto-market dynamics which is unregulated, highlyvolatile and permanently subjected to extreme events.
Appl. Sci. 2022, 12, 6408 3 of 18
• In addition, these portfolios are only made up of the best performing crypto assets,which are strongly and positively correlated between themselves and in particularwith Bitcoin and, therefore, cannot constitute a diversified portfolio.
The fundamental objective of this paper is to improve the work of Mba [11,13] andBoako et al. [12] on the optimization and selection of cryptocurrency portfolios in order toaddress the aforementioned shortcomings. The issue of non diversification is addressedby applying a machine learning technique known as K-means algorithm, reinforce withhierarchical clustering which groups similar assets into a cluster exhibiting a certain levelof dissimilarity with other clusters. In fact, the technique will be applied to the top hun-dred cryptocurrencies consisting of different class of cryptoassets such as: coins, token,stablecoins, decentralised finance token (DeFi) and non fungible token (NFT). To overcomethe issue of underestimation of the risk and overestimation of the returns, we preprocessthe input data for the optimisation of CVaR and expected returns as follows: inverse trans-formation of the copula output using the quantile of the skew student-t distribution whichconstitutes the marginal in the copula fitting.
The novelty of our integrated approach is the combination of the machine learningtechnique (clustering and hierarchical algorithm), econometric model (GJR-GARCH), dif-ferential evolution algorithm and vine copula model in the selection and optimisation of acryptoasset portfolio. Our results highlight the fact that, the top ten or twenty cryptocurren-cies cannot constitute a diversify portfolio since they are highly and positively correlated.Another take away of our findings is that stablecoin such as True-USD is negatively cor-related to the other cryptoassets in the portfolio and could therefore be safe haven forcrypto-investors when market experiences extreme events.
The roadmap of the contributions of this work is organised as follows. First, inSection 2, a non-exhaustive literature review on the application of copula in the portfoliooptimisatio is presented; a survey of different risk measures that can be useful in cryp-tocurrency portfolio optimization problems is also discussed. Section 3 is devoted tothe methodology that was used to achieve the objective of this paper. In Section 4, weimplement the methodology and present the empirical results. In the last section, wesummarise the main results obtained in the previous section and we will also point outtheir weaknesses and strengths.
2. Methodology
This section presents the theoretical framework of the paper. We introduce the objectiveand the methods used to present our results.
2.1. Machine Learning: K-means Clustering
Definition 1. K-means is an unsupervised algorithm for non-hierarchical clustering. It allows theobservations of the data set to be grouped into distinct clusters. Thus, similar data will be found inthe same cluster. In addition, an observation can only be found in one cluster at a time (membershipexclusivity). The same observation cannot therefore belong to two different clusters.
K-means is an iterative algorithm that minimizes the sum of the distances between eachobservation and the centroid. The initial choice of number K of the centroids determines thefinal result. Admitting a cloud of a set of points (data or observations), K-means changesthe points of each cluster until the sum can no longer decrease. The result is a set of compactand clearly separated clusters, provided the researcher chooses the right K value for thenumber of clusters.
The convergence of the K-means Algorithm 1 can be one of the following conditions:
• A pre-set number of iterations, in this case K-means will perform the iterations andstop regardless of the shape of compound clusters.
• Stabilization of cluster centers (centroids no longer move or change during iterations).
Appl. Sci. 2022, 12, 6408 4 of 18
Algorithm 1: K-means Algorithm.Input:• K the number of clusters to be formed• X the Training Set (m× n data matrix)
Output:
1. Randomly choose K < m points (K rows of the data matrix).These points are the centers of the clusters (called centroid).
2. Assign a cluster to each point (or observation), randomly.3. Calculate the centroid of each cluster (i.e. the vector of the means of the
different variables)4. For each point calculate its Euclidean distance with the centroids of each of
the clusters5. Assign the closest cluster to the object6. Calculate the sum of the intra-cluster variability7. Repeat steps 3 to 5, until an equilibrium is reached, that is convergence:
no more change in clusters, or stabilization of the sum of theintra-cluster variability.
2.2. Vine-Copula
This subsection describes the characteristics necessary for a function to be a copula, aswell as some of their properties. Before mentioning them, some preliminary definitionsand results are useful. In fact, here is the idea behind the copula approach.
Let X = (X1, . . . , Xn) be a vector of n random variables for which we want to constructthe joint distribution function. Let assume that, X1, . . . , Xn are random variables withthe following marginal distributions F1, . . . , Fn, such as Fi(xi) = P(Xi ≤ xi) = ui (theprobability that the measure of Xi be less than xi). Then the joint distribution function isgiven by
F(x1, . . . , xn) = P(X1 ≤ x1, . . . , Xn ≤ xn). (1)
Definition 2. An n dimensional (n ≥ 2) copula is a function C : In −→ I satisfying thefollowing properties:
1. C is non-decreasing that is C(0, . . . , xi, . . . , 0) = 0, for all xi ∈ I = [0, 1]2. C possess one dimensional uniform margins on Ci, that is:
Ci(xi) = C(1, . . . , 1, xi, 1, . . . , 1) = xi for all xi ∈ I. Ci is an invariant non-decreasingtransformation of the marginal.
We can now show the link between copulae and random variables. Copulas are ofinterest in statistics thanks to the theorem proposed by Sklar [14].
Theorem 1. Assume F = (F1, . . . , Fn) is an n dimensional joint distribution function withmarginal distribution function Fi (i = 1, . . . , n). Then, there exists a copula C such that for allxxx = (x1, . . . , xn) ∈ In
F(xxx) = C(F1(x1), . . . , Fn(xn))
If F1, . . . , Fn are continue, then C is unique. Otherwise C is non-unique on In.In addition, if F1, . . . , Fn are distribution function on I and if C is a copula, then the function
F(xxx) = C(F1(x1), . . . , Fn(xn)) is a joint distribution function on In.
The canonical representation of the copula density function is given as
c(u1, . . . , un) =∂nC(u1, . . . , un)
∂u1, . . . , ∂un(2)
Appl. Sci. 2022, 12, 6408 5 of 18
To obtain the density of the n-dimension distribution F, the following relation is used
f (x1, . . . , xn) = c(F1(x1), . . . , Fn(xn))n
∏i=1
fi(xi) (3)
where fi is the density of the marginal distribution Fi.Copula functions constitute an advantageous statistical tool for constructing and
simulating multivariate distributions. The literature devoted to the copula approachprovides us with different types of density functions which can be summarized in twocategories of families: the family of elliptical copulas and that of Archimedean copulas.Tables 1 and 2 show different properties of popular copulas functions.
Table 1. Copula component.
No. Elliptical Parameter Range Kendall’s τ Tail DependenceDistribution
1 Gaussian ρ ∈ (−1, 1) 2π arctan(ρ) 0
2 Student-t ρ ∈ (−1, 1), ν > 2 2π arctan(ρ) 2tν+1(−
√ν + 1
√1−ρ1+ρ )
No. Name Generator Parameter Kendall’s τ Tail DependenceFunction Range (Lower, Upper)
3 Clayton 1θ (t−θ − 1) θ > 0 θ
θ+2 (2−1/θ , 0)4 Gumbel (− log t)θ θ ≥ 1 1− 1
θ (0, 2− 21/θ)
5 Frank − log[
e−θt−1e−θ−1
]θ ∈ R/0 1− 4
θ + 4 D1(θ)θ (0, 0)
6 Joe − log[1− (1− t)θ ] θ > 1 1 + 4θ2
∫ 10 log(t)(1− t)
θ(1−θ)θ dt (0, 2− 21/θ)
7 BB1 (t−θ − 1)θ θ > 0, δ ≤ 1 1− 22(θ+2) (2−1/θδ, 2− 21/θ)
8 BB6 (− log[1− (1− t)θ ])θ θ > 0, δ ≤ 1 1 + 4∫ 1
0 (− log(−1(1− t)θ + 1)) (0, 2− 21/θδ)
× (1−t−(1−t)−θ+t(1−t)−θ)θδ dt
9 BB7 [1− (1− t)θ ]−δ − 1 θ ≥ 1, δ > 0 1− 2θ(2−θ)
+ 4θ2δ B( 2−θ
θ , δ + 2) (2−1/θ , 2− 21/θ)
10 BB8 − log[ 1−(1−θt)θ
1−(1−θ)θ ] θ ≥ 1, δ ≤ 1 1 + 4∫ 1
0 (− log( (1−tδ)θ−1(1−δ)θ−1 ) (0, 0)
× 1−1δ−(1−tδ)−θ+tδ(1−tδ)−θ
θt dt
Table 2. Popular Archimedean copulas functions.
Name Bivariate Copula Cθ(x, y) Parameter Range
Clayton [max{x−θ + y−θ − 1; 0}]−1/θ θ ∈ [−1, ∞ )− {0}Gumbel exp[−
((− log x)θ + (− log y)θ
)1/θ] θ ∈ [1, ∞ )
Frank − 1θ log
[1 +
(exp(−θx)− 1)(exp(−θy)− 1)exp(−θ)− 1
]θ ∈ R− {0}
Joe 1−[(1− x)θ + (1− y)θ − (1− x)θ(1− y)θ
]1/θθ ∈ [1, ∞ )
Independent xy
In practical applications, modelling using copula with a large set of high-dimensionalvariables, such as in this paper (a set of eight cryptocurrencies), has some limitations.These limitations can be parameters restriction and the selection of the appropriate copulafunction. Joe [15] proposed an alternative which is the use of vine-copulas method forhigh-dimensional data. Thus vine copulas, described by Bedford and Cooke [16,17] areflexible graphical models to describe multivariate copulas constructed using a cascade ofbivariate copulas or pair-copulas. In this paper we will use C (canonical) and R (regular)vine copula specification.
Appl. Sci. 2022, 12, 6408 6 of 18
2.3. GARCH-Copula Differential Evolution (DE): (GJR-GARCH-DE-C-Vine-Copula)
In many works, the copula approach is directly applied to returns of financial assets.Nevertheless, we recognize the stylised facts related to the fat tail distributions and thephenomenon of volatility clustering observed in financial time series data. Given this fact,it is useful to introduce GARCH models in order to filter these distributions. Then, thecopula model can be used to model the dependency structure between the standardisedinnovations considered here as iid observations.
Here, we opt for GJR-GARCH(1,1) because it accounts for the leverage effect thatcreates asymmetry in the dynamics of the variance of the financial returns. More details onthis model can be found in the works of Glosten et al. [18]. We recall that if a n-dimensionalvector of time series variables (returns) xt = (x1,t, , x2,t, . . . , xn,t) is assumed to follow aGARCH-Copula modelling type, by construction the joint distribution function is given inthe following form
F(xxxt|µµµt, σσσt) = C(F1(x1,t|µ1,t, σ1,t), . . . , Fn(xn,t|µn,t, σn,t))
where C is the n-dimensional copula, Fi are the conditional marginal distribution functionsrelative to xi,t, whose dynamics are described by the intercept µi,t which corresponds tothe conditional mean and by an error term εi,t =
√σi,tνi,t, which can be constant or time
dependent as in a GJR-GARCH(1,1) specification
xit = µi,t + εi,t
εi,t =√
σi,tνi,t (4)
σ2t = α0 + β1σ2
t−1 + (α1 + γ1Iεt−1<0)ε2t−1
where σi,t is the conditional variance of the series xi,t, given the prior information to timet, and νi,t =
εi,t√σi,t
are iid random variables characterising the standardised innovation,considered as white noise with zero mean and variance equals 1 (νt ∼ N(0, 1)). Iεt−1<0 = 1if εt−1 < 0, otherwise Iεt−1<0 = 0 and γ1Iεt−1<0ε2
t−1 is the term related to the leverage effect.The differential evolution initiated by Storn and Price [19] is a nonlinear optimization
algorithm that has been extremely successful since its conception and was originallycreated to solve continuous problems. The algorithm is inspired from evolutionary biologyoperations which follow the following steps: initialisation, mutation, recombination andselection on a given population to minimise an objective function through successivegenerations. The algorithm uses alteration and selection operator to transform progressivelya population of candidate solution.
Consider a population ω = (ω1, . . . , ωn) of size n and the objective functionh(ω1, ω2, . . . , ωn) to be optimised. The optimisation process is as follows [20,21]:
1. Initial populationThe initial generation is created randomly, either by the computer or the user.Given the population ω
gki = (ω
gk1, ω
gk2, . . . , ω
gkn), where the number g represents the
generation order and k = 1, 2, . . . , N. The initial population or first generation iscreated as
ωki = ωLki + rand()(ωU
ki + ωLki)
where ωLki and ωU
ki are lower and upper bounds of ω respectively, rand() a randomlygenerated number and i = 1, 2, . . . , n.
2. MutationThe evolutionary process of a generation follows a simple cycle, making it possible tosequentially improve each of the N individuals. Thus, each of the N individuals iscalled upon to be the target vector in turn. An initial mutant vector uuug+1
k is createdusing a mutation process which simply involves adding the weighted difference of
Appl. Sci. 2022, 12, 6408 7 of 18
two other individuals randomly selected in the population to a third party vector asdemonstrated by Equation (5).
uuug+1k = ω
gjk + c1(ω
glk −ω
gsk), j 6= l 6= s. (5)
where k = 1, 2, . . . , N. In Equation (5), the coefficient c1 is the mutation coefficientwhich can be adjusted to control the amplitude of the mutations.
3. RecombinationLet us assume that each individual of the population will become a target vector.Assuming ω
gki is the target vector a discrete recombination process then creates a
new or trial vector vvvg+1ki by crossing the newly created mutant vector uuug+1
k to thetarget vector.
vvvg+1ki =
{uuug+1
ki , if rand() ≤ Cp or i = Irand i = 1, 2, . . . , n;ω
gki, dif rand() > Cp or i 6= Irand k = 1, 2, . . . , N
where Irand is an integer, randomly selected in [1, n] and the recombination probabilityCp makes it possible to manage the level of involvement of both the target and mutantvector in the creation of the new vector.
4. Selection
ωg+1ki =
{vvvg+1
ki , if h(vvvg+1ki ) < h(ωg
ki)
ωgki, otherwise.
(6)
The condition in Equation (6), helps to avoid creating clones by ensuring that the newvector or solution has at least a dimension resulting from the mutant vector. Then, aselection will allow one to choose the best of the two solutions between vvvg+1
ki and ωgki.
By observing the stages of mutation and recombination well it is obvious that to befunctional, the number of individuals in the population must be at least 4.
2.4. Efficient Portfolio and Optimisation
An efficient portfolio is a portfolio whose expected return µp is maximum for a givenlevel of risk, or whose risk is minimal for a given return. Efficient portfolios are on the“efficient frontier” of the set of portfolios in the plane (σ2
p , µp). The first question an investorasks himself is obviously to know: Which efficient portfolio offers the lowest level of risk?Our aim, therefore, is to determine this efficient frontier or at least to find a function whichallows to determine the optimal portfolio for a target level of return µp. This problem canbe formulated as below:
min(ωωωTσσσωωω
)subject toωωωTE(Rp) = µp
ωωωT111 = 1
(7)
where 111 is a n vector column of 1.In this paper, we chose the conditional value at risk (CVaR) as risk measure and follow
Rockafellar and Uryasev [22] optimisation approach for problem (7).Let l(ωωω, R) : Rn×Rn → R denote a loss function characterise par the decision (weight)
vector ωωω and the return (random) vector R. The probability that the loss function l(ωωω, R)never exceeds the threshold α ∈ (0, 1) is given by
F[l(ωωω, R), α] = P{l(ωωω, R) ≤ α}
:=∫
l(ωωω,R)≤αP(R)dR
Appl. Sci. 2022, 12, 6408 8 of 18
and the value at risk for a certain level of confidence β ∈ (0, 1), is given by:
VaRβ[l(ωωω, R)] := inf{R ∈ Rn : F[l(ωωω, R), α] ≥ β},
and the formula of the conditional value at risk by
CVaRβ[l(ωωω, R)] = E[l(ωωω, R)|l(ωωω, R) ≥ VaRβ[l(ωωω, R)]
]:=
11− β
∫l(ωωω,R)≥VaRβ [l(ωωω,R)]
l(ωωω, R)P(R)dR
Since it depends by construction on the function VaRβ[l(ωωω, R)] which itself dependson ωωω, the optimization of CVaR can sometimes be difficult to approach. Without havingrecourse to an analytical representation of VaR, Rockafellar and Uryasev [22] formulate thefollowing auxiliary function
F[l(ωωω, R), α] := α +1
1− βE[max{l(ωωω, R)− α, 0}] (8)
and demonstrate thatCVaRβ[l(ωωω, R)] = inf
α∈RF[l(ωωω, R), α] (9)
The advantage of using the auxiliary function F[l(ωωω, R), α] is twofold: Firstly it isjointly convex with respect to α and ωωω, provided that the loss function l(ωωω, R) is alsoconvex with respect to ωωω. Secondly we do not have to choose a value for α beforehand,which can be difficult in practice. This is naturally derived during the optimization processbased on the chosen confidence level.
Rockafellar and Uryasev [22] finally show that minimizing CVaRβ[l(ωωω, R)] with re-spect to ωωω is equivalent to minimizing F[l(ωωω, R), α] with respect to (ωωω, α) ∈ Rn × (0, 1),that is
minωωω∈R
CVaRβ[l(ωωω, R)] = min(ωωω,α)∈Rn×(0,1)
F[l(ωωω, R), α] (10)
Since Rn is convex by definition, (10) is therefore a convex optimization problem.The optimal value of the conditional value at risk optimization problem (10) of a
crypto-asset portfolio, can be found by solving the following convex optimization problem:
min(ωωω,α)∈Rn×(0,1)
F[l(ωωω, R), α] (11)
subject to ωωωTE(Rp) = µp
ωωωT111 = 1ωi ≥ 0, 1, . . . , n.
(12)
2.5. Snapshot of the Methodology
The nine steps below are followed for the analysis on the CVaR:
step 1 Compute log-returns of the top 100 cryptoassets.step 2 portfolio selection
Machine learning: K-means and hierarchical clustering deployment in orderto group assets that appear to be reasonably similar versus those that sharelarge dissimilarities
step 3 Extract Standardized Residuals from AR-GJR-GARCH(1,1) with Student-t inno-vations to convert the log returns into an IID series.
step 4 Use the residuals from Step 3 and standardise them with the deviations obtainedin Step 3.
Appl. Sci. 2022, 12, 6408 9 of 18
step 5 Convert these residuals to student-t marginals for the estimation of copula. Thesesteps are repeated for all the cryptocurrencies to obtain a multivariate matrix ofuniform marginals.
step 6 Fitting C-vine. to multivariate data obtained in step 5 and Benchmark GaussR-vines using sequential estimation with restricted pair copula family set of first14 copulas.
step 7 Step 6 is repeated with R-vine.step 8 Inverse transform of the C-vine copula output using skew student-t distribution
(marginal in the copula fitting).step 9 Use outputs from step 8 to generate a series of simulated monthly portfolio
returns to predict 5% CVaR.
3. Data Analysis and Empirical Results
In this section, we propose to illustrate empirically the main objective of this pa-per, namely, to select a diversified portfolio of cryptocurrencies using a machine learn-ing algorithm: K-means supported by an hierarchical clustering method, investigate theperformance of GJR-GARCH Differential evolution t-copula approach in modelling theco-dependence and CVaR of the selected portfolio and finally minimizing the latter riskmeasures in order to propose under which method a portfolio of cryptocurrencies is morerisky or profitable than the other.
3.1. Data Description
We chose to apply our methodology to cryptoassets portfolios. The latter are sourcedfrom the top 100 cryptoassets by market capitalisation (which ensures that the analysiswill not be affected by liquidity risk issues) from yahoo finance (https://finance.yahoo.com/, accessed on 3 May 2022) powered by coinmarketcap (https://coinmarketcap.com/,accessed on 3 May 2022) form 1 May 2017 to 30 April 2022. We then eliminate those withless than 700 observation and this operation has produced 58 cryptoassets with 1830 dailyprices. The machine learning algorithm K-means (supported by the hierarchical clusteringmethod) is used to select a diversified portfolio of eight cryptocurrencies and cryptotokens,representing about 50% of the whole crypto-market share (see Table 3). It is evident fromTable 3 that Bitcoin reigns supreme over the cryptoassets market. As a result it controls theprice formation of all altcoin.
Figures 1 and 2 show respectively the evolution of the price and return dynamicsfor the eight cryptoassets under investigation. Visual inspection shows many signs ofdiscrepancies than commonality signs in the returns and prices of all the eight virtual assets,testifying thus a certain dissimilarity in their behaviours over the considered period. Volatilityclustering patterns are observed in the return dynamics of all eight cryptocurrencies.
The descriptive statistics of the selected cryptocurrencies examined in this paper areshown in Table 4. All currencies display positive standard deviation and mean close tozero. With the exception of LEO, FIL, WAVES and ONE coin, the other cryptocurrencies areskewed left. In addition, the Jarque–Bera (JB) test and kurtosis for each series are positiveand far from zero; that is, they possess heavy tailed and non normal distribution, which isconsistent with the behaviour of most financial assets.
Appl. Sci. 2022, 12, 6408 10 of 18
Table 3. Description of selected cryptoassets.
Rank Symbols Category Consensus Value Maximum % of TotalMechanism Proposition Supply (106) Marketcap
1 BTC crytocurrency PoW digital gold 21 43.33%20 LEO utility token PoS fund reimbursement 985 0.36%25 LINK application token N/A oracle network 1000 0.65%38 FIL storage network PoRep secure decentralised 2000 1.83%
token storage network48 THETA application token PoS video streaming 1000 1.93%
network49 TUSD stablecoin N/A digital fiat N/A 0.06%74 WAVES application token LPoS Web 3.0 application & 108,325 0.13%
decentralised solution80 ONE crypto-token FBFT deep sharding technology 12,600 1.77%
PoRep: proof-of-Replication. LPoS: Leasing Proof-of-Stake is an enhanced version of Proof-of-Stake (PoS).FBFT: Fast Byzantine Fault Tolerance.
Figure 1. Cryptoasset prices evolution from May 2017 to April 2022.
Figure 2. Cryptoasset returns in the sample spanning from May 2017 to April 2022.
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Table 4. Descriptive statistic.
Dependent Variable
BTC LEO LINK FIL THETA TUSD WAVES ONE
Mean 0.169 0.185 0.215 0.178 0.368 −0.0005 0.307 −0.107min −46.500 −20.000 −61.500 −60.500 −60.400 −5.100 −48.700 −52.100max 17.200 36.300 27.600 76.900 34.700 4.570 44.800 84.000sd 4.000 3.210 6.680 10.600 7.410 0.363 7.040 5.800asd 63.200 50.800 106.000 168.000 117.000 5.750 111.000 91.700Kurtosis 21.700 27.300 10.900 8.980 8.410 74.800 8.280 60.100Skewness −1.670 1.990 −1.090 0.547 −0.770 −0.621 0.114 2.970JB 18,528.000 29,309.000 4,746.000 3,161.000 2820.000 215,881.000 2651.000 140,674.000Q10 21.200 41.900 23.900 43.100 288.000 7.190 24.800 22.200Q102 16.100 26.700 48.500 97.000 258.000 53.300 58.300 43.100ACF 0.040 0.035 0.086 0.210 0.077 0.473 0.116 0.080
3.2. Empirical Findings
The empirical results are analyse in three main axes: The first step of the analysis is todeploy a machine learning algorithm to form a diversified portfolio. The second step ismodelling the interdependence structure of the selected cryptoassets using the C-vine andR-vine copulas and estimating the parameters of the associated pair-copula. The Last stepis modelling and estimating the CVaR of the weighted portfolio of the selected cryptoassetsusing C-vine and R-vine copula coupled with the differential evolution algorithm.
Machine Learning: K-means and Hierarchical Clustering
K-means in particular and clustering algorithms in general all have one common goal:to group similar items into clusters. These elements can be any type of data, as long as theyare encoded in a matrix form.
Choosing a number of clusters K is not necessarily intuitive, especially when thedataset is large and the researcher does not have a priori or assumptions about the data.The elbow method is the most common for choosing the number of clusters. This methodfirst calculates the variance which is the sum of the distances between each centroid (centre)of a cluster and the various observations included in the same cluster. Then, it seeks to finda number of clusters K so that the clusters retained minimize the distance between theircenters (centroids) and observations in the same cluster, that is, minimizing the intra-classdistance or total intra-cluster variance also known as total within-cluster sum of square(wss). Generally, by putting in a graph wss as a function of K like in Figure 3, one findsa graph similar to an arm where the highest point represents the shoulder and the pointwhere K is 14, represents the other end: the hand. The optimal number of clusters is thepoint representing the elbow (knee). Here, the bend can be represented by K being 4 or 5.This is the optimal number of clusters. Generally, the knee point is that of the number ofclusters from which the variance no longer decreases significantly.
The K-means algorithm result is recorded in Table 5. The algorithm has allowed data of58 cryptoassets to be grouped into 4 clusters, based on their similarities and dissimilarities.Cluster 1 contains 4 assets which are stablecoins having the same characteristics andbehaviours. Cluster 2 contains 23 assets, mainly in the category of application token withPoS consensus mechanism or close to it, except ZIL, ICX, BAT and XVG. Cluster 3 contains2 assets, LEO and LUNA which belong to the category of utility token PoS(Proof-of-Stake) and DPoS (Delegated-Proof-of-Stake) as consensus mechanism respectively. Cluster4 contains 29 assets, mainly in the category of digital token with Pow (Proof-of-Work)consensus mechanism or close to it, except BNB, ETH, XRP, TRX and EOS. The dynamic ofthe members of this group is highly influenced by that of Bitcoin compared to the membersof other groups.
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Figure 3. Elbow method result for determining the optimal number of clusters.
Table 5. K-means result.
Clusters 1 2 3 4
USDP USDC FIL THETHA RUNE LINK LEO ONE BSV XMR NEXO BTCUSDT TUSD QNT TFUEL WAVES XTZ LUNA TRX NEO DASH DOGE WBTC
Crypto HOT CHZ FTM MATIC BTT ETC ZEC CRO BNBCEL MANA LRC VET ADA EOS HT KCS ETHXDC ENJ ATOM ZIL XRP LTC OKB GNO FTTXEM MIOTA BAT XLM BCH DCR MKR
wss 0.65 0.8 0.85 0.75
The hierarchical clustering is used to confirm the results of the K-means method, andthe dendrogram obtained on the basis of the intra-cluster and inter-cluster variance isdepicted in Figure 4 and the level of similarity in each cluster is presented in Figure 4.As one moves up the dendrogram, assets that are similar to each other are merged intobranches, which subsequently fuse at higher height. The lower the height of the fusion, themore similar the assets are and clusters with high height have greater variability withintheir assets.
Finally, the selection of the cryptoassets is done as follows: one stablecoin: True-USD (TUSD) is chosen in cluster 1. Since the level volatility associated to stablecoins isconsistently close to zero, they are therefore “simply” a tool for dematerializing our fiatcurrencies, the parity of which remains stable with them and allows repeated transactionswith cryptocurrencies without going through a traditional bank account. In addition, ithas a negative correlation with other cryptoassets in the portfolio (see Table 6). Four assets
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(FIL, THETA, LINK, WAVES) are selected from cluster 2 based on their higher heightof dissimilarity (low correlation) with Bitcoin. In cluster 4, Harmony (ONE) is the mostdissimilar coin to Bitcoin (see Figures 2 and 4), so both are chosen.
Table 6. Correlation coefficients.
BTC LEO LINK FIL THETA TUSD WAVES ONE
BTC 1 0.124 0.609 0.374 0.505 −0.087 0.465 0.564LOE 0.124 1 0.138 0.035 0.077 0.055 0.100 0.115LINK 0.609 0.138 1 0.371 0.487 −0.062 0.474 0.446FIL 0.374 0.035 0.371 1 0.287 −0.051 0.263 0.265THETA 0.505 0.077 0.487 0.287 1 −0.054 0.410 0.339TUSD −0.087 0.055 −0.062 −0.051 −0.054 1 −0.050 −0.014WAVES 0.465 0.100 0.474 0.263 0.41 −0.050 1 0.370ONE 0.564 0.115 0.446 0.265 0.339 −0.014 0.370 1
Figure 4. Dendrogram of the cryptoassets.
3.3. Modelling the Residual Dependencies Using Vine Copula
Econometric models coupled with vine copula are commonly used in the field ofmultivariate modelling of financial returns. The GARCH marginal time series model is firstfitted to each cryptoasset returns and standardised residuals are formed. These residualsare subsequently transformed to marginally uniform data using parametric probabilityintegral transformation and used as inputs for the selection of the appropriate bivariate
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copula. The Alkaike information criterion (AIC) is used to select the copula that best fitsthe data.
At the first step the univariate AR(1, 1)-GJR-GARCH(1, 1) model with Student-tinnovations is fitted to the log return series of each cryptoasset to extract standardisedresiduals and their independence are verified using the Ljund-Box test. Subsequently,the empirical probability integral (since the data size is large) is used to transform thestandardised residuals to obtain marginally uniform data (iid residuals) for the estimationof copula. These steps are repeated for the eight cryptoassets to obtain a multivariate matrixof uniform marginal. The selected bivariate copula and their estimated Kendall’s tau forthe eight cryptoassets are recorded in Table 7 for C-vine and Table 8 showing differentstrengths and signs of pairwise dependencies.
Table 7. Parameter estimate of C-vine copula.
Tree Edges Copula par1 par2 Kendall’s τ utd ltd
1
BTC-TUSD G90 0.02 – −0.06 – –BTC-FIL SG 0.03 – 0.27 – 0.35BTC-LEO t 0.03 2.00 0.09 0.03 0.03BTC-THETA SBB7 0.07 0.07 0.38 0.04 0.57BTC-WAVES SJ 0.06 - 0.33 - 0.56BTC-LINK SBB6 0.19 - 0.45 - 0.6ONE-BTC SG 0.05 – 0.41 – 0.49
Table 8. Parameter estimate of R-vine copula.
Tree Edges Copula par1 par2 Kendall’s τ utd ltd
1
LINK-LEO t 0.03 2.37 −0.09 0.02 0.02BTC-TUSD G90 0.02 – −0.06 – –LINK-FIL SG 0.03 – 0.27 – 0.34BTC-THETA SBB7 0.07 0.07 0.38 0.04 0.57LINK-WAVES SBB8 0.06 0.00 0.33 – –BTC-LINK SBB6 0.19 0.15 0.45 – 0.60ONE-BTC SG 0.05 – 0.41 – 0.49
3.3.1. Canonical Vine Copula
Figure 5 represents the structure of the C-vine. The node that maximises the sum ofpairwise dependencies is chosen as the root of the tree. In tree 1 all altcoins are connectedto Bitcoin. This confirms the leading role Bitcoin plays in the price formations of othercryptoassets, as it accounts for about 44% of the total crypto-market shares.
A range of 14 bivariate copula functions was available for selection to model thedependences. The following four copula functions were selected based on the AIC criterion(key characteristics are given in Table 9): Survival Joe-Gumbel and Clayton (SBB6, SBB7),survival Gumbel or rotated Gumbel by 180 degree (SG), survival Joe (SJ), student-t and ro-tated Gumbel by 90 degree (G90): The predominance of survival and mixed copula indicatethe presence of lower-tail dependences (ltd) which characterise cryptoasset markets duringextremes. The structure in Figure 5 reflects the expected relationship among cryptoassets.The resulting pairs of cryptoassets captured by the appropriate copula are recorded inTable 7 together with their corresponding estimated parameters and Kendall’s tau valuesindicating different strengths of dependencies. We observe from the Kendall’s tau columnthat Bitcoin is positively and moderately correlated with altcoins except with True USDwith whom it has a strong negative correlation (with the negative correlation with BTC,whom is positively correlated with other altcoin. True USD can be used for hedging duringmarket turmoil).
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Table 9. Characteristic of copula functions.
Name Dependence Structure Strength Weakness
S. Gumbel (SG) only negative lower tail dependence upper tail dependenceS. Joe (SJ) only negative lower tail dependence upper tail dependenceRotated 270 Gumbel (G270) only negative tail-asymmetry no tail dependenceS. Joe-Gumbel (SBB6) independent or positive tail symmetricS. Joe-Clayton (SBB7) independence or negative lower tail dependence upper tail dependenceS. Joe-Frank (SBB8) positive or negative symmetrical & tail dependence
Figure 5. C-vine tree 1.
3.3.2. Regular Vine Copula
Since R-vine copula by construction is more flexible than C-vine, we propose in thissection to deploy R-vine to the eight selected cryptoassets and compare the result to thatof C-vine. Figure 6 shows tree 1 for R-vine and the flexibility can be seen at first glance.Two categories of cryptoassets emerge from tree 1, LINK, ONE, THETA and TUSD are themost connected to Bitcoin and LEO, FIL and WAVES are most connected to LINK but haveno direct dependency with Bitcoin. This subdivision clearly underscores the flexibility ofR-vine structure over that of C-vine and can be partly explained by the fact all the altcoindirectly connected to LINK are application token and with almost the same consensusmechanism. We observe from Table 8 that tree 1 of R-vine, three mixed copula families(SBB6, SBB7, SBB8) compared to two mixed copula (SBB6, SBB7) for C-vine. This shows thepower of R-vine to capture complex dependencies.
Examining Tables 7 and 8 we observe that edges that are similar to C-vine and R-vinehave been captured by the same rotated or survival copula functions and have equaldependency strength (Kendall’s tau). In addition, the results in the last two columns of bothtables show the evidence of capturing fat-tailed distribution and left tail risk in economicand financial downturns as SG, SBB6,7,8 copula, which capture lower, asymmetric andsymmetric behaviour are dominant.
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Figure 6. R-vine tree 1.
Comparison results between C-vine and R-vine copula is summarised in Table 10.The log-likelihood obtained after optimisation of the chosen copula type are in column 1.The Akaike’s Information Criteria (AIC) and Bayesian Information Criteria are recorded incolumns 2 and 3, respectively. We observe that the values of those three comparison factorsfor C-vine are greater than the ones of R-vine. Overall, this result shows the usefulnessof C-vine copulas with individually selected type of copulas for each term of the pair-copula. In addition, the selection procedure of C-vine gives a result that is consistent withcrypto-market dynamics.
Table 10. Model comparison.
loglik AIC BIC Number of Copula Types on Tree1
C-Vine 1240 −2435 −2324 6R-vine 1238 −2427 −2311 6
3.4. Portfolio Optimisation
Table 11 shows a fairly weighted (between 0 and 0.33) portfolio, with LEO and TUSDhaving the largest weight throughout the various period. This is consistent with the factthat, the return dynamic of LEO is opposite to the one of Bitcoin (see Figure 2) and cantherefore be used together with a stablecoin such as TUSD to mitigate or hedge against therisk. FIL and Bitcoin are the third and fourth weighted assets across the considered periodfor optimal return. The less weighted assets are LINK, THETA and WAVES which belongto the same category of application token and are strongly correlated in the lower tail (seeTable 7) with Bitcoin, making them more riskier than the remaining assets. This may justifytheir lower weight allocation.
The optimisation result in Table 12 shows that the CVaR is high across all the periodwhich is in line with the well-known fact that cryptomarket is highly volatile. This animprovement as compared to previous studies exhibiting a CVaR over 100%. We can alsoobserve that the CVaR is almost constant throughout the period and the same movement
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can be noticed for the mean return with about 4% per period. It seems less than expected,but remains constant. An investor looking for a portfolio which can generate a fixed cashflow rather than the returns that fluctuate between gains and losses across period, shouldconsider such portfolio.
Table 11. Cvine: weighs.
Rebal Periods BTC LEO LINK FIL THETA TUSD WAVES ONE
month1 0.09 0.22 0.02 0.16 0.14 0.22 0.09 0.06month2 0.04 0.31 0.10 0.13 0.06 0.24 0.02 0.10month3 0.09 0.33 0.02 0.10 0.08 0.20 0.07 0.11month4 0.11 0.22 0.06 0.17 0.03 0.21 0.08 0.12month5 0.02 0.29 0.02 0.10 0.05 0.25 0.08 0.19month6 0.12 0.33 0.02 0.12 0.08 0.20 0.06 0.07month7 0.10 0.29 0.02 0.22 0.02 0.02 0.05 0.10month8 0.03 0.20 0.06 0.18 0.10 0.27 0.02 0.14
Table 12. CVine: CVaR and mean.
Rebal Periods CVaR Mean
month1 0.720 0.0421month2 0.732 0.0405month3 0.718 0.0415month4 0.731 0.0414month5 0.740 0.0415month6 0.740 0.0382month7 0.747 0.0406month8 0.750 0.0434
4. Conclusions
The goal sought through this paper was to develop a method that could allow a crypto-investor or manager to select a diversified portfolio of cryptocurrencies and to estimate therisk and profitability of the portfolio. Our approach to diversification combined similarityand tail dependence structure through K-means algorithm and Vine copula respectively,thus, achieving wealth/weights allocation which is not concentrated only on few assets.
During this study, we first tried to illustrate a method that relies on the K-means algo-rithm and which made it possible to select a diversified portfolio of eight cryptocurrencies.Subsequently, we opted for a GARCH-C-Vine copula approach combined with the differ-ential evolution algorithm for co-dependence analysis in order to estimate the return andCVaR of the selected portfolio. The method also makes it possible to determine an accurateand reliable result in a very short computing time, which facilitates its implementation inpractice. Our results show:
• consistency with the risky characteristics of the cryptocurrency market-unregulated,anonymity of the transaction and highly volatile.
• that stablecoin such as True-USD is negatively correlated to the other cryptoas-sets in the portfolio and could therefore be safe haven for crypto-investors duringmarket turmoil.
On the other hand our findings are in line with previous studies exhibiting stablecoinsas potential diversifiers.
Contrary to the previous studies which focused mainly on the top twenty, we buildour portfolio from a pool of hundred cryptocurrencies to take advantage of possibledissimilarity that may exists among them. The top twenty cryptocurrencies consideredin previous studies appear to be be highly correlated, so that a diversified cryptocurrencyportfolio cannot be formed from these top twenty.
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Several extensions of this work can be considered later. For example, it would beinteresting to consider the use of diversification measures such as Diversification ratio orEntropy. The higher these measures, the well diversified a portfolio will be. We could alsoconsider a diversification approach through risk contribution with risk measures beingeither CVaR or any other coherent risk measures such as spectral risk measure or distortionrisk measure. One could also consider a possibility of a multi-period horizon for the portfo-lio optimisation with rebalancing with additional constraints such as transaction costs.
Author Contributions: Conceptualization, H.M.T. and J.C.M.; methodology, H.M.T.; software,S.M.M.; validation, H.M.T. and J.C.M.; formal analysis, S.M.M.; investigation, H.M.T.; resources,J.C.M.; data curation, S.M.M.; writing—original draft preparation, H.M.T.; writing—review andediting, H.M.T.; visualization, H.M.T.; supervision, J.C.M.; project administration, J.C.M.; fundingacquisition, J.C.M. and S.M.M. All authors have read and agreed to the published version of themanuscript.
Funding: This research received no external funding.
Data Availability Statement: https://finance.yahoo.com/ powered by coinmarketcap (https://coinmarketcap.com/(accessed on 3 May 2022)).
Conflicts of Interest: The authors declare no conflict of interest.
References1. Markowitz, H. Portfolio selection. J. Financ. 1952, 7, 77–91.2. Markowitz, H. Portfolio Selection: Efficient Diversification of Investments; John Wiley and Sons: Hoboken, NJ, USA, 1959.3. Bares, P.; Gibson, R.; Gyger, S. Hedge Fund Allocation with Survival Uncertainty and Investment Constraints. Soc. Sci. Res. Netw.
2002, 1–50. [CrossRef]4. Javed, F.; Mazur, S.; Ngailo, E. Higher order moments of the estimated tangency portfolio weights. J. Appl. Stat. 2021, 48, 517–535.
[CrossRef] [PubMed]5. Khan, K.I.; Naqvi, S.M.W.A.; Ghafoor, M.M.; Akash, R.S.I. Sustainable Portfolio Optimization with Higher-Order Moments of
Risk. Sustainability 2020, 12, 2006. [CrossRef]6. Jurczenko, E.; Maillet, B.; Merlin, P. Hedge Funds Portfolio Selection with Higher-Order Moments: A Non-Parametric Mean-Variance-
Skewness-Kurtosis Efficient Frontier; Social Science Research Network (SSRN): Rochester, NY, USA, 2005.7. Hunjra, A.I.; Alawi, S.M.; Colombage, S.; Sahito, U.; Hanif, M. Portfolio Construction by Using Different Risk Models: A
Comparison among Diverse Economic Scenarios. Risks 2020, 8, 126. [CrossRef]8. Krokhmal, P.; Palmquist, J.; Uryasev, S. Portfolio optimization with conditional value-at-risk objective and constraints. J. Risk
2002, 4, 43–68. [CrossRef]9. Agarwal, V.; Naik, N.Y. Risks and portfolio decisions involving hedge funds. Rev. Financ. Stud. 2004, 17, 63–98. [CrossRef]10. Babaei, S.; Sepehri, M.M.; Babaei, E. Multi-objective portfolio optimization considering the dependence structure of asset returns.
Eur. J. Oper. Res. 2015, 244, 525–539. [CrossRef]11. Mba, J.C.; Pindza, E.; Koumba, U. A differential evolution copula-based approach for a multi-period cryptocurrency portfolio
optimization. Financ. Mark. Portf. Manag. 2018, 34, 399–418. [CrossRef]12. Boako, G.; Tiwari, A.K.; Roubaud, D. Vine copula-based dependence and portfolio value-at risk analysis of the cryptocurrency
market. Int. Econ. 2019, 158, 77–90. [CrossRef]13. Mba, J.C.; Mwambi, S. A Marvov-Switching COGARCH approach to cryptocurrency portfolio selection and optimization. Financ.
Mark. Portf. Manag. 2020, 34, 199–214. [CrossRef]14. Sklar, A. Functions de Répartition a n dimensions et leurs Marges. Publ. Inst. Stat. L’Université Paris 1959, 8, 229–321.15. Joe, H. Multivariate Models and Dependence Concepts; Chapman & Hall: London, UK, 1997.16. Bedford, T.; Cooke, R.M. Probability density decomposition for conditionally dependent random variables modeled by vines.
Ann. Math. Artif. Intell. 2001, 32, 245–268. [CrossRef]17. Bedford, T.; Cooke, R. Vines—A new graphical model for dependent random variables. Ann. Stat. 2002, 30, 1031–1068. [CrossRef]18. Glosten, L.R.; Jagannathan, R.; Runkle, D.E. On the relation between the expected value and the volatility of the nominal excess
return on stocks. J. Financ. 1993, 48, 1779–1801. [CrossRef]19. Storn, R.; Price, K. Differential Evolution—A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces;
TR-95-012; Intemational Computer Science Institute (ICSI): Berkeley, CA, USA, 1995.20. Ababio, K.A.; Mba, J.C.; Koumba, U. Optimisation of mixed assets portfolio using copula differential evolution: A Behavioural
approach. Cogent Econ. Financ. 2020, 8, 1780838. [CrossRef]21. Das, M.; Roy, A.; Maity, S.; Kar, S.; Sengupta, S. Solving fuzzy dynamic ship routing and scheduling problem through new genetic
algorithm. Decis. Making Appl. Manag. Eng. 2021. [CrossRef]22. Rockafellar, R.T.; Uryasev, S. Optimization of conditional value-at-risk. J. Risk 2000, 2, 21–42. [CrossRef]
