Small Portfolio Construction with Cryptocurrencies
DENIS VELIU1, MARIN ARANITASI2
1Department of Finance and Bank, Faculty of Economy,
University Metropolitan Tirana,
Street “Sotir Kolea”
ALBANIA
2Department Basics of Informatics,
Polytechnic University of Tirana,
ALBANIA
Abstract: - In this paper, we describe and apply different models of portfolio construction in the selection
between a small number of big-cap cryptocurrencies. Our purpose is to select the minimum riskiness between
cryptocurrencies, comparing different risk measures and maximum diversification. We build our models
without the constraints of the expected returns. Without relying on expected returns, we have the same
condition on the comparison between them. Cryptocurrencies are not common stock or other assets indexed in
the market but it is interesting to study how diversification can significantly improve investment performance.
We first give the methodology to use high-frequency observation data, in the numeral approximation especially
in the novel application of the Risk parity models, used with different risk measures we can achieve a very
good result, from the position of gaining and variation. Since Risk parity models divide the weights of the asset
in equal risk contribution proportion, it is suggested to use a small number of cryptocurrencies, otherwise their
performance will be close to the uniform portfolio. To the traditional Mean Variance model, and the alternative,
Expected shortfall/Conditional Value at Risk, we use three versions of Risk Parity with two different risk
measures and a naive risk parity. The uniform portfolio is used as a benchmark for selection comparison with
the other portfolio models. We give the conditions for the Risk Parity with the Expected shortfall/Conditional
Value at Risk (CVaR) to guarantee convergence with the numerical approximation. In the end, we study the
tradeoff between each model and which is more suitable for a small cryptocurrency portfolio.
Key-Words: - Bitcoin, Cryptocurrency, Asset allocation, Portfolio optimization, Risk diversification, Risk
Parity, Markowitz, Marginal Risk Contribution, Robust Optimization, Risk Management.
Received: April 9, 2023. Revised: January 5, 2024. Accepted: January 27, 2024. Published: February 23, 2024.
1 Introduction
Trading with Cryptocurrencies, as a non-regulated
market, has achieved a lot of focus from the point of
view of the view of speculation and research. The
most famous cryptocurrency, built using blockchain
technology, is Bitcoin with a market price of about
16.700 US dollars per coin and a market
capitalization of about 320 billion dollars
(December 2022), which has decreased from 359
billion USD from the last year.
The Cryptocurrency market capitalization has
reached in 2022 above 2 trillion U.S; until 2016 the
total market capitalization was below 18 billion U.S.
dollars, (Yahoo Finance 2020).
Trading is easily accessible in more than 100
different cryptocurrencies to be used as currency or
as financial assets. Thus, cryptocurrencies can be
seen as an alternative asset of investment, since they
obtained the increasing attention of many investors
for very high gains in an observed short time.
The price of Bitcoin ranges between 13.00 USD
and about 62.000 USD, thus, some professional
investors must not invest in cryptocurrency due to
its unpredictable price movements and high
volatility, not to mention the fact that there is no
reliable way to value a crypto asset. The price is
very sensitive to the authority's prohibition in
accepting Bitcoin as a currency, but also, the price
can go up if any public figure such as Elon Musk,
declares positive things about this method of
payment. However, several studies show that
including Bitcoin in a portfolio has significant
benefits, [1], [2]. The crucial problem with
Cryptocurrency markets is that they are not under
the market classic regulation, thus, we should use
models that rely on the minimum risk.
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.57
Denis Veliu, Marin Aranitasi
E-ISSN: 2224-2899
686
Volume 21, 2024
A few cryptocurrencies have large market
capitalization and in terms of millions or even
billions, while they usually provide lower
transaction costs to individuals demonstrating an
efficient financial market characteristic that
indicates immediate liquidity, [3].
If we consider cryptocurrencies as a new class
of assets or as traditional currency, [4], we should
first see their statistical characteristics in the
distribution of the returns such as the skewness,
kurtosis, and heteroscedasticity. Many works on the
study of cryptocurrencies found recently are
common in financial assets and long-memory, [5].
Moreover, researches on cryptos further
demonstrate potential diversification in this
emerging market for institutional and retail
investors.
There are cryptocurrencies that evolvement is
relatively isolated from the others, [6], which may
offer diversification benefits for speculators and the
variety of cryptocurrencies is still uprising, thus the
cryptocurrency market has an increasing place in
diversification and portfolio composition, thus, the
research on the portfolio diversification of
cryptocurrencies has been increased. Today, you
may find more than two thousand different
cryptocurrencies, but do we trust all of them?
Speaking of optimization models, many other
portfolio optimization models, such as , as
a maximum potential loss of a portfolio in an
interval of time, have been proposed in the literature
after the Nobel prize H. Markowitz, [7], with his
first step in modern portfolio theory. Numerous
studies on a similar risk measure, the Conditional
Value at Risk , [8], demonstrate why it
is preferred to Value at Risk because the
later does not allow diversification. The most crucial
characteristics are that is a convex and
coherent risk measure demonstrated in the model
function, [9], a model that supports diversification.
All of these models rely on the estimated
expected return of the assets as an input, which
causes them to concentrate heavily on a restricted
set of assets and perform badly outside of the
sample, [10]. Additionally, these models generate
extremely high weights and demonstrate large
fluctuations over time. So, comparable to within a
Mean Variance portfolio, a major adjustment in the
input parameters can affect the portfolio's
composition significantly.
The Risk Parity approach's ability to avoid
requiring the estimation of expected returns is one
of its main advantages. The Risk Parity
methodologies divide the entire risk of the portfolio
into the risk contributions of each asset in the same
proportion
Using the Euler breakdown for the first order
homogeneous function, we will be able to apply the
Risk Parity technique to the Expected shortfall or
more common .
By observation, we know that the idea that the
returns are a normal multivariate distribution is less
credible due to the lack of reality. Other authors,
[11], use a Mixed Tempered stable distributed for
the source of risk in the Risk Parity models. An
alternative approach, called Equal Risk Bounding
(ERB), requires the solution of a nonconvex
quadratically constrained optimization problem. The
ERB approach, while starting from different
requirements, turns out to be firmly connected to the
RP approach, [12]. In this paper, we will treat
cryptocurrencies as usual stocks or bonds, with a
purpose of studying how the novices'
digital currency, which operates without a financial
system or government authorities, will behave in
these conditions. We first describe the selected
small crypto portfolio, justifying our selection on
these, to analyse the performance in a out of sample
period with the use of a rolling window. Another
important step is the analysis of riskiness, portfolio
turnover, and diversification.
2 Cryptocurrency Datasets and
Models Used
The cryptocurrency dataset selected, includes the
period from 1/1/2018 to 31/01/2021 with 1123
trading days in total (remember that you can trade
each day of the year 24/7). We chose this time span
because it does not included, the moment in which
Bitcoin reached its highest peak in November 2021,
avoiding the unusual distortion of the data.
We collect ten cryptocurrencies with a market
capitalization larger than half a billion. To avoid
any currency fluctuation, all the prices are in dollars
as they are listed on Yahoo Finance.
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.57
Denis Veliu, Marin Aranitasi
E-ISSN: 2224-2899
687
Volume 21, 2024
Table 1. Data of the distribution of the daily returns
(in %)
Cryptocurrency
Mean
Median
Range
Skewness
Kurtosis
Cardano USD
(ADA-USD)
-0.07
-0.01
82.54
-21.40
924.72
Ethereum
Classic USD
(ETC-USD)
0.24
0.06
81.99
78.10
1436.37
Binance Coin
USD (BNB-
USD)
0.15
0.07
102.4
-13.64
2083
Bitcoin USD
(BTC-USD)
0.08
0.15
63.18
-147.85
2081.29
Dogecoin USD
(DOGE-USD)
0.13
-0.10
183.8
592.56
10552
Chainlink USD
(LINK-USD)
0.31
-0.10
109.5
2.31
1097.80
Litecoin USD
(LTC-USD)
-0.05
-0.07
73.97
-30.90
1021.65
Tether USD
(USDT-USD)
0.00
-0.01
10.60
27.23
2885.30
Stellar USD
(XLM-USD)
-0.04
-0.18
96.91
112.39
1671.9
Monero USD
(XMR-USD)
-0.09
0.07
72.11
-99.16
1153.8
Source: Authors' calculation
As we see in Table 1, except for Dogecoin, the
other daily returns are almost normally distributed,
even if we compare the skewness and the kurtosis.
We decided to also include Dogecoin to see the
difference between the cryptocurrencies.
Table 2. The correlation matrix of the returns of the
cryptocurrencies
1.00
0.73
0.61
0.73
0.45
0.59
0.76
-0.06
0.79
0.72
0.73
1.00
0.62
0.79
0.44
0.54
0.81
-0.05
0.63
0.74
0.61
0.62
1.00
0.68
0.39
0.50
0.67
-0.06
0.55
0.65
0.73
0.79
0.68
1.00
0.49
0.56
0.82
-0.03
0.64
0.79
0.45
0.44
0.39
0.49
1.00
0.34
0.48
-0.02
0.45
0.45
0.59
0.54
0.50
0.56
0.34
1.00
0.55
-0.02
0.51
0.53
0.76
0.81
0.67
0.82
0.48
0.55
1.00
-0.06
0.65
0.77
-
0.06
-
0.05
-
0.06
-
0.03
-
0.02
-
0.02
-
0.06
1.00
-
0.03
-
0.03
0.79
0.63
0.55
0.64
0.45
0.51
0.65
-0.03
1.00
0.64
0.72
0.74
0.65
0.79
0.45
0.53
0.77
-0.03
0.64
1.00
Source: Authors' calculation
In Table 2 we notice that Tether (row/column 8)
is the only crypto with returns negatively correlated
with the other nine cryptocurrencies. The mean of
returns is centered around zero and close to the
median which is a good condition. The next step is
to measure the performance of the portfolio.
The correlation matrix is important to
understand the ongoing market, and how the
behavior of the models is based on the variance of
the portfolio.
The portfolio models considered in this article are:
1/N equal weighted rule (Naive Portfolio),
minimum variance (MV),
minimum CVaR,
Risk Parity with standard deviation as risk
measure (RP-std),
Risk Parity with conditional value at risk
measure CVaR (RP-CVaR),
Naive Risk Parity CVaR (RP-CVaR naive).
The last one is a special case in which we have
the worst-case scenario (highest CVaR, useful as an
upper bound, [13].
In all these methods, we do not use expected
returns, so at the minimum variance i.e. we don’t
have the constraints of the return of the portfolio, to
have the smallest possible variance.
In all the cases we do not allow short selling, so
the weights allocated at each cryptocurrency can
assume only positive values.
We will try different rolling windows to see
how do they perform by measuring the cumulated
return of the portfolio.
The first indicator of performance is the following:
so that
is the total compounded return for
the whole period.
All portfolios will have n assets, for weight xi
assigned and as a measure of risk for the
portfolio x = (x1, x2,…,xn).
In other works, [14], the more common use of
Risk Parity is as a risk measure of the standard
deviation. Considering the weights x = (x1, x2,.....,xn)
assigned to n assets, the risk measure, in this case,
standard deviation is given by:
where is the covariance matrix. For the i asset, the
marginal risk contribution is:
and the total risk contribution:
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.57
Denis Veliu, Marin Aranitasi
E-ISSN: 2224-2899
688
Volume 21, 2024
The following optimization problem can be used to
represent the Risk Parity model:
Remind that the Mean Variance Markowitz
model equalizes the marginal Risk
Contribution.
For the numerical approximation in the proof of
the partial derivatives of , some
conditions are needed on the distribution of the
return vector R = (r1, r2,.....,rn) .
The denote with
the
portfolio return, which must be differentiable to the
weights to apply the Euler decomposition.
The return from t to time t+1 is measured as
follow:
We can compute the partial derivatives of the
from partial derivatives for the Value at
Risk. Starting from the definition of ,
[8], we have
Thus, using the needed assumptions, [9], from
the mathematical point of view, we can proceed
with the differentiation to the variable weights.
The Total Risk is given from the following
expression for the asset i:
In case of continuous returns distribution, we
can pass to the following presentation:
For the discrete variables in the numerical
finding for and Risk Parity
using times series observations we have to do the
following assumption. Assuming that the i-th asset
return rji with i=1,.....,n and j=1,...,T where n is the
quantity of elements considered at our portfolio and
T the laps of time of observation. The vector of the
created portfolio returns is in
which:
with j=1,....,T where .
Using the central limit theorem for the number
of observation large enough, the approximation of
the empirical distribution of the observed portfolio
returns:
So, approximation for the and
of portfolio returns will be as follows:
where α is the significance level and
are the
sorted portfolio returns that must satisfy
For more details, [15].
Using times series observations, the discrete
values of the partial derivatives for each
asst i becomes:
i=1,...,n
and then the total risk contribution of asset i is
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.57
Denis Veliu, Marin Aranitasi
E-ISSN: 2224-2899
689
Volume 21, 2024
where
are the related portfolio returns to the
ordered from the smallest to the largest value.
We consider three diversification measures, to
control, if the portfolios are well diversified.
Consider a portfolio x = (x1, x2,.....,xn) satisfying
the budget constraint
with no short
selling () .
The Bera and Park measure, [16], is very
important to understand the entropy of the portfolio,
for strategies no short selling strategies.
The assumes values between 0 (all in one)
and log(n) for uniform allocation: the
diversification measures only accurately reflect
diversity in terms of weights allocated and do not
consider the fact that different types of assets will
have changing effects on the high change of the
entire portfolio.
The turnover of the portfolio is another helpful
amount for determining transaction costs:
,
where is the amount assigned in i the
observation t.
For the computation, we use MATLAB
software running on a Laptop with Windows 10
Home operation system, Intel(R) Core(TM) i7-
7500U CPU @ 2.70GHz 2.90 GHz, 12 RAM and
NVIDIA GeForce 930MX graphic card. The timing
of optimization is very short compared to the other.
3 Results in Performance out of
Sample and Diversification
To succeed in the numerical approximation, we
create a rolling window using the data of the past 2
years (from 1/1/2018 to 1/1/2020 so L=365*2=730
observation) for the estimation of the weights of
each of the portfolio models and move the rolling
window in the period from 1/1/2020 to 31/1/2021 to
measure the out of sample for the next 7 days
(Holding period). This will take 56 iterations for the
calculation of the optimal portfolios. Remembering
that short selling is not allowed. The expected
returns constraint is eliminated, for the Minimum
Variance and Conditional Value at Risk model: the
minimum risk is achieved for each risk measure.
Let us see the weights only for the first optimization
(L=730 days).
Table 3. Weight allocation of the first optimization
(in %)
Portfolio Model
Crypto.
Uniform
R. P. S. D.
R. P. CVaR
R. P. CVaR N.
Min-CVaR
Min-Var
Cardano
10
2.97
4.38
2.07
0.00
0.07
Ethereum
10
2.76
4.07
1.98
0.00
0.03
Binance
10
3.48
4.94
2.70
0.00
0.06
Bitcoin
10
4.26
6.32
2.96
0.00
0.03
Dogecoin
10
3.62
4.96
2.78
0.00
0.07
Chainlink
10
2.99
3.96
2.13
0.00
0.01
Litecoin
10
3.28
5.13
2.47
0.00
0.05
Tether
10
70.3
57.1
78.5
100
100
Stellar
10
3.18
4.60
2.28
0.00
0.02
Monero
10
3.20
4.56
2.22
0.00
0.02
TOTAL
100
100
100
100
100
100
Source: Authors' calculation
As it is clear from Table 3, most of the
portfolios focus on the Tether USD, in which,
Minimum variance and CVaR are fully allocated.
The Risk parity models a range between 57% to
78% in the same cryptocurrency. This is very
interesting if we compare it with the matrix of
covariance, Tether is the only one negatively
correlated, as we know, it will have a smaller
volatility. Thus, most of the weights are higher for
this cryptocurrency.
The better way to show the compound return
rate is from the graph below.
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.57
Denis Veliu, Marin Aranitasi
E-ISSN: 2224-2899
690
Volume 21, 2024
Fig. 1: The compound return rate
Source: Authors' calculation
In Figure 1, we observe that the uniform
portfolio (naïve portfolio in blue color) and the Risk
Parity with CVaR Naïve, which do not consider the
risk in the first place, outperform compared to the
other risk measures but on the ongoing have also a
higher drawdown (April 2020). The uniform at the
end of the observation has doubled his compound
return rate.
The Mean Variance Model and the CVaR, have
almost the same performance by having the
minimum risk possible but a very small value in
compound return.
In between, we have the Risk Parity with CVaR
and Risk Parity with standard deviation as a risk
measure. As we explained, these come for the better
diversification of the risk (Figure 1).
If we measure the riskiness of each portfolio
using the standard deviation, we will notice that the
uniform and the Risk Parity with naïve CVaR will
have a very high risk.
Fig. 2: The standard deviation of portfolio
selections
Source: Authors' calculation
The Mean Variance and CVaR models have
almost the same results (Figure 2), as we notice in
the lower part of the graph. We will have similar
results if the holding period changes by one or two
days.
In a recent paper, [17], they created portfolios
comparing cases with and without Bitcoin only as
one asset in portfolios with usual assets such as gold
and other indexes. They use the mean variance
model and the risk parity only with the standard
deviation as a risk measure. They are pointing out
that the allocation to Bitcoin in most of the
unconstrained or semi-constrained frameworks was
minimal. They insist since Bitcoin observers have
substantial value variations, stakeholders must
exercise attentiveness and limit their acquaintance to
Bitcoin, a superfluous exposure to Bitcoin may not
principally lead to development in portfolio
performance qualities, [16].
To have a close estimation of what happens in
case we consider the transaction costs, we can see
the portfolio turnover (Figure 3).
Fig. 3: Portfolio weekly turnover
Source: Authors' calculation
Knowing that the number of cryptocurrencies
(ten) is a small one, Mean Variance and CVaR are
focused on a smaller number of cryptocurrencies,
thus the portfolio turnover will be higher (Figure 3).
Considering that cryptocurrencies are easily
accessible by different competitive platforms, we
still may face different costs, for instance,
subscriptions costs and small commissions.
The last point to discuss is diversification by
measuring Bera Park, remember that the higher is
the value, the better are diversified the portfolios.
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.57
Denis Veliu, Marin Aranitasi
E-ISSN: 2224-2899
691
Volume 21, 2024
Fig. 4: Diversification by Bera Park.
Source: Authors' calculation
As we notice from the Figure 4, both measures
have similar results. Focusing on a smaller number
of crypto, M-V, and CVaR are concentrated
compared to the models.
The values of the diversification are better with
the Risk parity methods, especially with the
Conditional Value at Risk as a Risk measure (Figure
4).
4 Conclusion
The novelty of this paper consists in comparing
different methods of portfolio optimization in the
cryptocurrency market. Without using the expected
returns all the models for portfolio selection are in
the same condition, The Mean Variance and CVaR
are at the minimum risk, as the other models without
the use of the expected returns constraint. We have
chosen ten cryptocurrencies with the highest
capitalization and a lap of time for the observed data
large enough to give a significant conclusion. We
described the observed data, to make sure that they
are suitable for the conditions of the Mean Variance
and other models that have conditions on the
distribution of the returns and the correlation matrix.
After we described the methodology for the
numerical approximation, we passed from the
continuous case to the discrete observation with
high-frequency data.
Considering cryptocurrencies as an asset class
we faced an issue of increased volatility, and were
very sensitive to the market information. For that,
there is a necessity to develop particular models for
asset allocation in cryptocurrencies. Traditional
approaches, like the Markowitz model, solely
concentrate on assets that carry an absolute minimal
amount of risk. Therefore, if the investor tries to
rebalance the portfolio, this high concentration will
likewise have significant transaction costs.
Additionally, relying on predicted returns during a
downturn in the economy would result in an
unrealistic and pessimistic asset allocation. Some
investors may have a large collection of
cryptocurrencies in their financial holdings for a
variety of reasons, most notably for speculation.
When associated with CVaR and Mean
Variance, the cryptocurrency portfolio built using
the Risk Parity criteria showed higher diversity and
less focus on high weights. This results in lower
costs for recalibrating the portfolio due to the low
turnover.
From the perspectives of performance and
volatility, the Risk Parity techniques in each of these
situations represent a good compromise between the
CVaR, Mean Variance, and the uniform portfolio.
The importance of the cryptocurrency has been seen
lately as Bitcoin is a novice to the world of
exchange-traded funds. Bitcoin ETFs allow
investors to get exposure to the tempting potential of
BTC without having to directly own it or safely
store it, [18]. Some investors may feel safer getting
exposure to Bitcoin in their portfolios by purchasing
a professionally managed ETF than they do owning
an actual BTC. This is to show the importance of
cryptocurrencies in the post COVID 19 market and
more studies need to be done by including these
assets in the investments.
References:
[1] Platanakis E. and Urquhart A, Portfolio
management with cryptocurrencies: The role
of estimation risk, Economics Letters, Vol.
177, 2019, pp.76-80
[2] Letho, L., Chelwa, G. and Alhassan, A.L.
Cryptocurrencies and portfolio diversification
in an emerging market, China Finance Review
International, Vol. 12 No. 1, 2022, pp. 20-50.
[3] Weiyi L., Portfolio Diversification across
Cryptocurrencies, Finance Research Letters,
Vol. 29, 2019, pp. 200-205,
[4] Baek, C., Elbeck, M., Bitcoins as an
investment or speculative vehicle? A first
look. Appl. Econ. Lett., Vol 22, 2015, pp.30-
34.
[5] Phillip A, Chan J, Peiris P, A new look at
Cryptocurrencies, Economics Letters,
Vol.163, 2018, pp. 6-9,
[6] Corbet S, Meegan A, Larkin C, Lucey B,
Yarovaya L, Exploring the dynamic
relationships between cryptocurrencies and
other financial assets, Economics Letters,
Volume 165, 2018, pp. 28-34
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.57
Denis Veliu, Marin Aranitasi
E-ISSN: 2224-2899
692
Volume 21, 2024
[7] Markowitz H.M. Portfolio selection. Journal
of Finance, Vol. 7, 1952, pp. 77-91.
[8] Rockafellar, R. T., & Uryasev, S.,
Optimization of conditional value-at-
risk. Journal of risk, Vol. 2, 2000, pp.21-42.
[9] Artzner, P., Delbaen, F., Eber, J.M. and
Heath, D., Coherent measures of
risk. Mathematical finance, Vol 9(3), 1999,
pp.203-228
[10] Merton, R.C., On estimating the expected
return on the market: An exploratory
investigation. Journal of financial
economics, Vol. 8(4), 1980, pp.323-361.
[11] Rroji, E. and Mercuri, L., 2015. Mixed
tempered stable distribution. Quantitative
Finance, Vol. 15(9), pp.1559-1569.
[12] Cesarone, F. and Tardella, F., Equal risk
bounding is better than risk parity for
portfolio selection. Journal of Global
Optimization, Vol . 68(2), 2017 pp.439-461.
[13] Colucci, S., A quick introduction to
quantitative models that discard estimation of
expected returns for portfolio construction.,
2013., Available at SSRN 2140673.
[14] Maillard, S., Roncalli, T. and Teïletche, J.,.
The properties of equally weighted risk
contribution portfolios. The Journal of
Portfolio Management, 36(4), 2010, pp.60-70.
[15] Veliu, D., Numerical approximation in the
application of Risk Parity with Conditional
Value at Risk in case of mixed
portfolios. European Journal of Economics,
Law and Social Sciences, Vol. 7(3), 2023,
pp.22-34.
[16] Bera, A.K. and Park, S.Y., Optimal portfolio
diversification using the maximum entropy
principle. Econometric Reviews, Vol. 27(4-6),
2008, pp.484-512.
[17] Rashid, A., Bakry, W. and Al-Mohamad, S.
Are cryptocurrencies a future safe haven for
investors? The case of Bitcoin. Economic
Research-Ekonomska Istraživanja, Vol. 36(2),
2023 pp.2140443.
[18] Adams M, Curry B, 6 Best Bitcoin ETFs Of
January 2024, Forbes, 2024, [Online].
https://www.forbes.com/advisor/investing/cry
ptocurrency/best-bitcoin-etfs/ (Accessed Date:
January 28, 2024).
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Denis Veliu has worked on the theoretical
background and literature review of the paper and
the overview of the model developments.
- Marin Aranitasi has conducted the quantitative
data analysis and the optimization of the
problems, by using MATLAB optimization.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This publication is made possible with financial
support of UPT. Its content is the responsibility of
the authors, the opinion expressed in the paper is not
necessarily the opinion of UPT.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.57
Denis Veliu, Marin Aranitasi
E-ISSN: 2224-2899
693
Volume 21, 2024
