# Optimization and Diversification of Cryptocurrency Portfolios: A Composite Copula-Based Approach
###### Abstract
[PERSON] et al. [7], [PERSON] et al. [8] as well as [PERSON] and [PERSON] [9] construct optimal portfolios using alternative risk measures. All these methods have shown their performance compared to the results given through classical analysis. This confirms the interest in dealing with issue of portfolio selection outside the Gaussian framework. One of the approaches aimed at relaxing the Gaussian framework imposed by the mean-variance approach concerns the modelling of the asymmetric nature of the dependency structure of the portfolio [10]. One of the solutions is that of copula functions, vine copula specification based on sequential method (also based on maximum spanning tree algorithm and maximum likelihood estimation of the pair-copula parameters) used to solve the joint probability modelling problem. The use of copula functions in the context of the asset allocation problem is still relevant today. [PERSON] et al. [11], use GARCH-differential evolution t-copula method in order to optimize and analyse cryptocurrency portfolio risk and return within the framework of a multi-period setting type approach. More recently, [PERSON] et al. [12] integrated copula functions into a GARCH modelling to analyse the structural interdependencies among seventeen cryptocurrency prices and to optimise the portfolio Value-at-Risk (VaR). In the same wake [PERSON] and [PERSON] [13] used a two state Markov-switching technique combined with R-vine copula and GARCH (MSCOAGARCH) to model heavy tail dependencies and structural breaks within the states of Markov switching with the aim of achieving a maximum return with a minimum conditional value-at-risk of a portfolio of top ten virtual currencies (in term of market capitalisation). The results obtained in the aforementioned works on the selection, allocation and optimization 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, highly volatile and permanently subjected to extreme events. * In addition, these portfolios are only made up of the best performing crypto assets, which are strongly and positively correlated between themselves and in particular with Bitcoin and, therefore, cannot constitute a diversified portfolio. The fundamental objective of this paper is to improve the work of [PERSON] [11; 13] and [PERSON] et al. [12] on the optimization and selection of cryptocurrency portfolios in order to address the aforementioned shortcomings. The issue of non diversification is addressed by applying a machine learning technique known as K-means algorithm, reinforce with hierarchical clustering which groups similar assets into a cluster exhibiting a certain level of dissimilarity with other clusters. In fact, the technique will be applied to the top hundred cryptocurrencies consisting of different class of cryptoassets such as: coins, token, stablecoins, decentralised finance token (DeFi) and non fungible token (NFT). To overcome the issue of underestimation of the risk and overestimation of the returns, we preprocess the input data for the optimisation of CVaR and expected returns as follows: inverse transformation of the copula output using the quantile of the skew student-t distribution which constitutes the marginal in the copula fitting. The novelty of our integrated approach is the combination of the machine learning technique (clustering and hierarchical algorithm), econometric model (GJR-GARCH), differential evolution algorithm and vine copula model in the selection and optimisation of a cryptoasset portfolio. Our results highlight the fact that, the top ten or twenty cryptocurrencies 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 correlated to the other cryptoassets in the portfolio and could therefore be safe haven for crypto-investors when market experiences extreme events.