1. Introduction
The introduction of Bitcoin in 2009 by an anonymous person or group of people known as “Satoshi Nakamoto” ignited a technological revolution called blockchain, which spawned similar tokens, now called “cryptocurrencies”. By design, a blockchain is an open, distributed ledger able to record transactions efficiently between two parties and in a permanent and verifiable way. Starting from just one cryptocurrency (Bitcoin) with tokens valued at less than a cent, at the time of this writing we have around 19,500 cryptocurrencies with market capitalization reaching USD 1.2 trillion. Recent years have seen an increase popularity of cryptocurrencies used mostly as financial assets known as cryptoassets. These new type of assets are known to be highly volatile and these extreme dynamics can results in dependence shifts and portfolio losses (see, e.g.,
Bekiros et al. 2015;
Brunnermeier 2009;
Florackis et al. 2014;
Moshirian 2011). Developing technical tools that can deal with such underlying properties is contemporary among practitioners and academic community.
On the one hand, we have models developed from technical trading rules and on the other hand those built from econometrics models. A review has been provided in (
Corbet et al. 2009b) indicating the presence of a gap in the trading dynamics of these new assets, especially in the performance of trading rules. Using high-frequency trading data, moving average and the trading range break are tested in (
Corbet et al. 2009b) to check if they hold for the Bitcoin market. The result reveals that the variable-length moving average has predictive power in cryptocurrency markets and is the most beneficial trading strategy when trading Bitcoin. While testing trend-following indicators in (
Gerritsen et al. 2009), the results reveals that trading range breakout displays significant forecasting power for Bitcoin prices. This is also supported in (
Gradojevic et al. 2021) where a combination of technical analysis with non-linear forecasting models appears to be statistically significantly dominant relative to the random walk model on a daily horizon. Furthermore, a classification tree-based model for return prediction is constructed in (
Huang et al. 2019) and displays strong out-of-sample predictive power for narrow ranges of daily returns on bitcoin which is in line with the findings in (
Nair 2021) illustrating the use of technical trading indicators as control variables in the extreme value regressions significantly improves the predictive power of models for cryptocurrencies. However, quite opposite to the previous findings, no predictability for Bitcoin was found in the out-of-sample period while employing 15,000 technical trading rules from the main five classes of technical trading rules, see (
Hudson and Urquhart 2021).
In this study, we adopt a different approach to trading. Instead of focusing on trading signals through technical trading rules which looks at each asset individually, we target the behaviour of of assets collectively in a portfolio. To this end, we look at how capital should be allocated to minimize risk while maximizing the portfolio return.
Among other models are the class of multivariate GARCH models which rely on parametric multivariate distribution and are likely to be erroneously specified when the distribution of all the variables are not the same, as in the case of cryptoassets. This can be addressed through copula models famous for their ability to describe the dependence structure (inter-correlation) between random variables. Alongside the multivariate elliptical copula family (Gaussian copula and t-copula), Vine copula have been introduced with the ability to model the dependence structure between random variables using a cascade of bivariate copulas, including asymmetric Archimedean copulas, see (
Aas et al. 2009;
Allen et al. 2013;
Bedford and Cooke 2001,
2002). The introduction of Vine copula models is very interesting in the sense that suitable bivariate copulas matching the stylized facts of the concerned random variables can be used to model their dependence structure. Another important aspect of these Vine models is their ability to auto-select these bivariate copulas when modelling the dependence structure. This way, the model learn from the data to adequately assign bivariate copula to each pair of random variable, so that appropriate tail dependence coefficients can be computed. We distinguish the regular vine (R-vine) copula and its two subclasses: the canonical vine (C-vine) and the drawable vine (D-vine). The statistical inference techniques for these two subclasses can be found in (
Aas et al. 2009). In this paper, we will use C-vine copula specification for its flexibility and its ability to model the dependence around a selected central node (see, e.g.,
Embrechts et al. 2001;
Fang and Fang 2002).
The problem of portfolio optimization consists of finding the portfolio weights meeting the investor’s objectives and constraints. In 1952, (
Markowitz 1952) developed a parametric optimization model providing a fundamental basis for portfolio selection termed as standard portfolio optimization. The constraints taken into account here are full investment and no-short selling. In practice, portfolio optimization has realistic constraints such as diversification constraint, transaction cost, portfolio size or any other additional constraints to suit the needs of the investors. Adding these constraints makes the portfolio optimization problem too complex to be solved by standard optimization methods, and therefore require alternative techniques such as heuristic algorithms.
Many meta-heuristic techniques (
Chang et al. 2000) such as genetic algorithms, simulated annealing and tabu search have been used in portfolio optimization to find the cardinality constrained efficient frontier. Other recent more efficient ones are differential evolution (DE) and particle swarm optimization (PSO). Although PSO belongs to swarm intelligence, both DE and PSO can be classified into stochastic optimization algorithms. DE is a population-based search strategy developed by Storn and Price (
Price et al. 2006;
Storn and Price 1997) and found to be more efficient than genetic algorithms and simulated annealing. It has shown remarkable performance on continuous numerical problems and optimizing portfolios under non-convex settings, see (
Ardia et al. 2011;
Krink and Paterlini 2009a,
2009b;
Maringer and Oyewumi 2007;
Yollin 2009). Similar to DE, PSO is a population based stochastic optimization technique developed by Kennedy and Eberhart in 1995 (
Kennedy and Eberhart 1995). It has become a popular optimization method as it often succeeds in finding the best optimum by global search in contrast with most common optimization algorithms.
Iwan et al. (
2012) realised that generally, in terms of repeatability (robustness) and the quality of the obtained solutions, DE outperforms PSO. As the cryptocurrency market is still revealing its intrinsic properties in terms of price discovery, this study aims at first contrasting the performance of PSO against DE in combination with a copula model.
Given the ability of t-copula to successfully model the tails/extreme events (
Mba et al. 2018) introduced the Copula differential evolution (CDE)-based approac,h which outperforms the standard DE and exhibits risk control ability. However, t-copula is symmetric, meaning that it models the lower tail and the upper exactly the same way. Although the tail of financial returns data are heavy, the heaviness varies from one dataset to another. So, imposing one type of copula to model the dependence structure between all assets pairs as in (
Mba et al. 2018) may result in inaccurate risk measures. It is on this ground that this study instead employs the Vine copula, which has the ability to use a cascade of bivariate copulas to model the dependence structure between pairs of random variables in a multivariate settings. The Vine copula model is flexible enough to allow to user to determine list of copulas to be used. This has a great advantage over multivariate copula such as t copula or Gaussian copula. Inspired by this result, we aim in this paper to introduce copula particle swarm optimization (CPSO) and contrast its performance to that of copula differential evolution (CDE) in a portfolio consisting of six top cryptocurrencies by market capitalization, namely Bitcoin (BTC), Ethereum (ETH), USDT (Tether), USDC (USD coin), Binance coin (BNB) and Ripple (XRP). The rest of the paper is organised as follows:
Section 2 presents the mathematical formalism of optimization methods and portfolio strategies used in this study.
Section 3 discusses the empirical findings and the last
Section 4, concludes the work.
2. Mathematical Formalism for Optimization Methods and Portfolio Strategies
The basic Mean-Variance optimization by (
Markowitz 1952) can be formulated as in the Equation (
1) below.
where
represents the portfolio weights,
is the return of asset
i and
its expectation,
is the covariance between
and
,
is the portfolio expected return.
Let
be the portfolio vector representing the proportion of wealth invested in each of the
n financial assets. Let
be the vector of returns. Let
and
be, respectively, a confidence level and a loss function for the portfolio
and the return vector
. Then, the value-at-risk (VaR) function,
is the smallest number satisfying
, where
is the probability that the loss
does not exceed the threshold value
. Given that VaR does not satisfy the sub-additivity axiom that should be expected from any sensible risk measure, the Conditional VaR (CVaR), was introduced by (
Rock and Uryasev 2000) as an alternative measure of risk. It is more appropriate to the loss function of the tail distribution and suitable for timeSeries with fat tails than most of the cryptocurrencies. Equation (
2) gives the expression of the CVaR.
To avoid complications resulting from the implicitly defined function
, (
Rock and Uryasev 2000) provided an alternative function given by Equation (
3).
for which they show that the minimum of CVaR can be found by minimizing
with respect to
.
Given returns data
for
, the function
defined in Equation (
3) can be approximated by Equation (
4)
where
.
In this study, we intend to solve the CVaR-optimization problem defined in Equation (
5)
We intend to find the portfolio that minimizes CVaR under 95% confidence level subject to the following weight constraints: weights must sum to 1 and no short-selling is allowed .
2.1. GARCH Specifications
Let
be a d-dimensional vector of random variables. The standard GARCH is defined by Equation (
6):
where
is the conditional variance of the returns series
and
the standardized innovations/residuals which can be assumed to follow a normal, student
t, skewed-student
t or any other distribution. Giving the lack of standard GARCH model to capture the leverage effect, we opted for an asymmetric GARCH, namely GJR-GARCH introduced in (
Glosten et al. 1993) to simulate the dynamics of the conditional variance given by Equation (
7):
where
. The distribution in this GARCH model is chosen as
skewed-Student t, which is a generalization of the student-t distribution with an additional parameter to control the skewness. Its density is given by:
where
,
,
and
is the gamma function. What is interesting about this density function is that it encompasses a wide range of conventional densities. For example, when
, it reduces to student-t distribution; if
and
, it reduces to the normal distribution. This justifies our choice for skewed Student-t distribution as margins in the copula model used in this study.
2.2. Copula Particle Swarm Optimization (CPSO)
2.2.1. Particle Swarm Optimization
The behavior of one organism in a swarm is often insignificant but their collective and social behavior is of great importance. In the PSO method, a swarm of particles flies through an N-dimensional search space where the position of each particle represents a potential solution to the optimization problem. Each particle p in the swarm is characterised by its jth dimensional following components: position , velocity , personal best (pbest) position and the global best (gbest) position of the swarm .
Let
f be the objective function to be minimized in a
d dimensional space. Then, the personal best position of particle
p can be obtained iteratively using Equation (
8):
The global best
gbest is then given by Equation (
9)
where
.
For each iteration, updating of positions are performed for each particle
and along each dimensional component
as described by Equation (
10):
where
is the inertia weight,
and
are the acceleration constants, usually set to
or 2.
and
are random variables with a uniform distribution. In the first equation of Equation (
10), the first term in the summation is the memory term representing the contribution of previous velocity, the second term is the cognitive component, which represents the particle’s own experience, and the third term is the social component, guiding the particle by the gbest of the particle towards the global best of the swarm so far obtained.
2.2.2. Vine-Copula
This subsection describes the characteristics necessary for a function to be a copula, as well as some of their properties.
Let
be a vector of
n random variables with the following marginal distributions
, such as
(the probability that the measure of
be less than
). Then, the joint distribution function is given by Equation (
11)
Definition 1. An n dimensional () copula is a function satisfying the following properties:
- 1.
C is non-decreasing that is for all
- 2.
C possess one dimensional uniform margins on , that is:
for all . is an invariant non-decreasing transformation of the marginal.
Copula was introduced in (
Sklar 1959) through the following theorem.
Theorem 1. Assume is an n dimensional joint distribution function with marginal distribution function . Then there exists a copula C such that for all If are continue,then C is unique. Otherwise C is non-unique on
In addition, if are distribution function on I and if C is a copula, then the function is a joint distribution function on .
The canonical representation of the copula density function is given by Equation (
12)
To obtain the density of the
n-dimension distribution
F, Equation (
13) is used
where
is the density of the marginal distribution
.
Copula functions constitute an advantageous statistical tool for constructing and simulating multivariate distributions and therefore modelling the dependence structure. Multivariate copulas are restricted to Elliptical copula family (Gaussian and t copulas). Giving the limitations of this copula family in accounting for stylized of financial returns, Vine copula models which use bivariate conditional copulas as building blocks have been introduced. Vine models are flexible enough to capture the underlying dependence and tail dependence structure. In dimension
N, a multivariate density is constructed by
bivariate (conditional) copulas (see,
Bedford and Cooke 2001) as building blocks, thus the name Pair-Copula Construction (PCC) given to this construction process. For example, let
,
and
be three random variables with distribution functions
,
and
respectively. The joint density can be decomposed as
where
with
for every
of the vector
with
in the general case.
As this construction is not unique, all the possible constructions can be illustrated by a set of nested trees
where
are the nodes and
the edges. This set of trees is called a
vine. The advantage of Vine copula is its ability and flexibility to model the dependence structure between time series using a cascade of bivariate copulas, see (
Aas et al. 2009;
Allen et al. 2013;
Bedford and Cooke 2001,
2002). We distinguish the Regular vine (R-vine) copula and its two subclasses: the Canonical vine (C-vine) and the Drawable vine (D-vine). The statistical inference techniques for these two subclasses can be found in (
Aas et al. 2009). In this paper, we will use C-vine copula specification.
2.2.3. Implementation
The Copula Particle Swarm Optimisation (CPSO) method is implemented as follows:
- (1)
Obtain the standardized residuals from GJR-GARCH.
- (2)
Simulate from C-vine copula a sample data from the standardized residuals with skew-student’s t (sst) marginals.
- (3)
Apply the inverse transformation of sst to to obtain new data .
- (4)
Solve the optimization problem (
14):
- (i)
Minimum CVaR portfolio using PSO and DE.
where
and
.
- (ii)
Global minimum variance portfolio (PGMV).
Let be the weights vector. Let be the return vector and the positive semi-definite variance–covariance matrix of the portfolio’s assets. The portfolio return is given by and the portfolio variance . For the global minimum variance portfolio for a given portfolio return, , the optimization problem can be stated as:
where is a column vector of ones.
- (iii)
Most diversified portfolio (MDP).
The diversification ratio (DR) is given by Equation (
15)
where
and CR denote the volatility-weighted average correlation and the volatility-weighted concentration ratio, respectively. The diversification ratio measure the degree of diversification of a portfolio. The higher the DR, the more the portfolio is diversified. Portfolio solutions that are characterized by either a highly concentrated allocation or highly correlated asset returns would qualify as being poorly diversified. A most diversified portfolio (MDP) can be obtained through maximization of DR:
- (iv)
Minimum tail-dependent (MTD) portfolio.
An optimal MDP portfolio is the one which yields the greatest diversification, by definition. However, this approach is based on the symmetric correlations between the assets. Akin to the optimization of a global minimum-variance portfolio, the minimum tail dependent portfolio is determined by replacing the variance–covariance matrix with the matrix of the lower tail dependence coefficients as returned by TDC. If the limit exists, then the lower tail dependence coefficient (TDC) is given by Equation (
17)
where
is the copula of the marginal distributions.
2.3. Copula Differential Evolution (CDE)
DE uses biology-inspired operations of initialization, mutation, recombination, and selection on a population to minimize an objective function through successive generations (see
Holland 1975). Similar to other evolutionary algorithms, to solve optimization problems, DE uses alteration and selection operators to evolve a population of candidate solutions.
Let N denote the population size. To create the initial generation, the optimal guess for N is made, either by using values input by the user or random values selected between lower and upper bounds (choosing by the user).
Consider the optimization problem (
5) and let
where
.
Given the population
where
g is the generation and
. The process is achieved through the following stages:
- (1)
Initial population:
The initial population is randomly generated as
where
and
represents the lower and upper bounds of
, respectively, and
.
- (2)
Mutation:
The differential mutation is accomplished as follows: A random selection of three members of the population
and
to create an initial mutant vector parameter
, called donor vector, which is generated as
where
F is the scale vector and
.
- (3)
Recombination or Cross-Over:
Let
denotes the target vector. From the target vector and the donor vector, a trial vector
is selected as follows
where
is a random integer in
and
the recombination probability.
- (4)
Selection:
At this stage, the target vector is compared with the trial vector and the one with the smallest function value is the candidate for the next generation
where
.
4. Conclusions
The portfolios considered in this study consisted of six cryptocurrencies: four traditional cryptocurrencies (BTC, ETH, BNB and XRP) and two stablecoins (USDT and USDC). We have examined the copula particle swarm optimization (CPSO) portfolio strategy against three other portfolio strategies, namely, the Global minimum variance (GMV), the most diversified portfolio (MDP), and the minimum tail-dependent (MTD). Initially we compare the performance CPSO against its analog population-based algorithm “copula differential evolution” (CDE), which shows out-performance ability based on their Sharpe ratios. While examining the risk contribution of each asset in each portfolio, CPSO appears to be a promising strategy during extreme market conditions, while GMV seem favorable during normal market conditions. Most importantly, the hedge and safe-haven ability of the two stablecoins is clearly exhibited with CPSO, while their diversification property is inhibited. So, we may ask the following question: can the benefits of the stablecoins properties as diversifiers, hedges and safe havens be exploited in one portfolio strategy? In order words, can we have a portfolio strategy in which stablecoins risk contribution is the lowest, while the diversification remains the highest? This direction of research will be taken in our next study.