**1. Introduction**

One of the topics at the heart of complex systems analysis is the study of financial markets. Financial markets have a diverse range of participants ranging from extremely sophisticated investors leveraging a technological and information advantage to retail investors who may purchase securities based on other fundamental intuitions. One asset class that has seen a significant degree of variance in the sophistication of the investor base is the cryptocurrency market. Over the last few years, the cryptocurrency market has gathered meaningful interest from institutional and retail investors alike. Despite exhibiting tumultuous changes in aggregate assets under management, the overall market has produced substantial growth in total assets since its inception. Given the relative immaturity of the cryptocurrency market, it is important to study the underlying dynamics of the market and contrast optimal trading and portfolio management strategies with that of more traditional asset classes such as the equity market. The main motivation of this paper is to investigate the next stage of the cryptocurrency market's evolution. Although the cryptocurrency market is young, we feel that it may be coming of age and exhibiting

**Citation:** James, N.; Menzies, M. Collective Dynamics, Diversification and Optimal Portfolio Construction for Cryptocurrencies. *Entropy* **2023**, *25*, 931. https://doi.org/10.3390/ e25060931

Academic Editors: Stanisław Drozd˙ z,˙ Jarosław Kwapie ´n, José F. F. Mendes and Marcin W ˛atorek

Received: 18 April 2023 Revised: 7 June 2023 Accepted: 12 June 2023 Published: 13 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

signs of maturity, becoming more like the equity market. To assess this, we tactically assess whether certain phenomena such as collective movement, uniformity and diversification benefits are similar to that of the equity market.

It is worth commenting more broadly on financial market dynamics and the wealth of work that has been conducted on that topic before we focus on the cryptocurrency market most specifically. There are a variety of academic communities that have studied financial market dynamics and evolutionary changes in structural dynamics such as those in applied mathematics, complex systems and econometrics [1–3]. A wide range of data scientific methodologies has been used to study evolutionary dynamics in financial assets such as linear algebraic-inspired techniques [2,4–6], spectral methods such as random matrix theory [1,7–10], a variety of unsupervised learning methodologies [11,12], change point detection [13,14] and a litany of statistical modeling techniques [15].

Another topic of substantial interest to the financial markets community is that of nonstationarity, regime switching and the time-varying nature of model parameters for phenomena such as volatility. Such research dates back to autoregressive conditionally heteroskedastic (ARCH) models [16], generalized ARCH (GARCH) [17] and stochastic adaptations such as stochastic volatility models [18–20]. Recently, many researchers have explored adaptions to these fundamental models explicitly capturing dynamics exhibited by various time series. Some of these models include exponential general autoregressive conditional heteroskedastic models [21], Glosten–Jagannathan–Runkle GARCH [22], Threshold GARCH [23] and T-SV [24], Markov switching GARCH [25–27] and MS-SV [28]. Many financial mathematicians have also adopted Bayesian estimation methodologies [29–32], generally citing the need for uncertainty quantification in estimating model parameters. These modeling techniques have been widely applied to the study of several asset classes including equities, cryptocurrencies and fixed income [33–38]. Finally, we would be remiss not to mention the wide range of techniques in time series analysis that have been used to study financial problems [39–48], including cryptocurrencies [49–58] and diverse fields in socio- and econophysics [59–77].

Another topic of great interest across asset classes is the topic of portfolio optimization, and more generally, the essence of portfolio construction. The quantitative finance and econometrics communities have studied core issues related to portfolio diversification, where portfolios are optimized with respect to different objective functions [78–85]. More broadly, financial market dynamics are universally difficult to model. The seminal work of Markowitz in 1952 [78] proposed the concept of diversification as a superior framework for investing in multiple securities at a time. The principle underpinning diversification is built upon disassociating the risk of an individual and particular financial asset into a market (systematic) risk component and an asset-specific risk, called unsystematic risk. Diversification essentially equates to smoothing (or averaging over) unsystematic risk by investing in an appropriately large number of individual assets, which leads to candidate investment portfolios' only exposure being inherently due to market risk.

In recent work, the authors of this work and collaborators [86] perform a thorough inspection of diversification properties from the perspective of a pure equity portfolio. Precisely, they explore the changing diversification benefit of various portfolios spread across a range of industry sectors. While in more recent years investor composition has broadened to include the likes of quantitative and high-frequency investors, active investment management has historically been dominated by fundamental investors who make investment decisions based on the future potential of companies relative to market valuations (most commonly, the earnings the company produces relative to its share price). The authors hypothesized that there is more substantial diversification benefit investing across sectors, rather than within them. Indeed, different sectors exhibit varying performance during distinct market periods: some sectors may outperform in buoyant equity markets (such as information technology and often energy), while other sectors outperform in declining equity markets (such as healthcare, consumer staples and utilities).

The authors confirmed this hypothesis, producing four primary findings. First, they use time-varying PCA to highlight that the collective behavior of equities spikes during market crises, rendering diversification far less effective. Second, they demonstrate that various community detection algorithms such as modularity are unable to distinguish between heterogeneous equity sector dynamics during times of crisis. By contrast, during more buoyant equity market periods, equity sector behaviors are more easily distinguished. Third, they introduce a new metric to quantify the uniformity of market impact across equity sectors. There, they show substantial variance across the uniformity of market impact across independent equity sectors. Finally, they demonstrate that a best value equity portfolio exists with respect to evolutionary diversification benefits. They show that a portfolio of size 36, where 4 equities are sampled randomly from 9 different equity sectors, provides comparable diversification benefit to a portfolio of size 81, where 9 equities are randomly sampled from 9 equity sectors. Our critical focus is exploring diversification benefits for cost-conscious retail investors. These are investors who are intelligent, and may be financially interested but lack the resources to trade frequently in an efficient manner.

With respect to the signature of maturity, the cryptocurrency market is very much in its infancy when compared to the equity market. Cryptocurrency sectors are not well defined, and it is often hard to differentiate behaviors between cryptocurrency sectors [14]. If one explores candidate cryptocurrency sector themes online, categories such as wallet, web3, yield farming, play to earn, energy, decentralized finance, distributed computing and cybersecurity can be found. However, these categories frequently overlap or differ from source to source, and it is not necessarily clear how the behaviors of these cryptocurrency sectors relate to the underlying economy. In fact, it is unclear just how often cryptocurrencies are purchased with the underlying coin sector or thematic within the digital ecosystem in mind. We suspect that this phenomenon is especially pronounced among less sophisticated retail investors—where coins may be bought and sold based on factors such as their recent price and volume movements, and overall macroeconomic trends. Accordingly, in this work, we turn to the cryptocurrency market and adapt our experiments to test for alternative diversification strategies among retail cryptocurrency investors. Rather than testing diversification effectiveness among equity sectors, we use tranches of cryptocurrency market capitalizations to proxy sectors. We suspect that many cryptocurrency investors buy securities from platforms where they simply scan a list of assets that are ordered by market capitalization, and are unaware of many coins' association with a deeper role in the digital economy. We feel that this is an original and suitable measure of different "classes" of cryptocurrencies. Here, we apply the same fundamental methodologies to the cryptocurrency market as a means of testing the levels of maturity and sophistication in the cryptocurrency market.

This paper is structured as follows. In Section 2, we outline the data that we use in this paper. In Section 3, we study the evolution of the collective dynamics of the cryptocurrency market. We compare these findings to what has been observed over 20 years in the equity market and draw conclusions regarding the cryptocurrency market's signatures of maturity. In Section 4, we turn to the theme of optimal portfolio construction among cryptocurrency portfolios. There we study marginal diversification benefits as additional cryptocurrency sector deciles, and cryptocurrencies within deciles are sequentially added to a portfolio. In Section 6, we conclude and summarize our findings regarding recent signatures of maturity in the cryptocurrency market.