The End of Mean-Variance? Tsallis Entropy Revolutionises Portfolio Optimisation in Cryptocurrencies

Abstract

Has the mean-variance framework become obsolete? In this paper, we replace traditional variance–covariance methods of portfolio optimisation with relative Tsallis entropy and mutual information measures. Its goal is to enhance risk management and diversification in complicated finance ecosystems. We utilize the S&P 500 and Bitwise 10 cryptocurrency indices’ daily returns (2019–2024 data) and conduct our analysis to the year 2020 under extreme shocks. Many models were trained with different configurations, like mean-variance (MV), mean-entropy (ME), and mean-mutual information (MI) traders and their corresponding variants, using Sharpe’s ratio, Jensen’s alpha, and entropy value of risk (EVAR). The findings indicate that entropic models outperform conventional models in terms of diversification and, especially, extreme risk management. Because the appropriate normalization conditions often fail to be satisfied, we can informally see that after a recalibration of the effective frontier, we obtain from EVAR an accumulated resilience aspect to these rare events while also observing the great potential of entropy-based models to replicate non-linear dependencies between assets. The results show that models combining entropy and mutual information optimise the gain–loss ratio (GLR), providing stable diversification and improved risk management, while maximising returns in complex and volatile market environments.

1. Introduction

Capital markets, as vital investment instruments, offer investors unparalleled opportunities to optimise their returns, particularly at historically low interest rates. These markets furnish real-time information on security prices, liquidity, and associated risks, enabling investors to diversify their portfolios in an increasingly complex universe. Nonetheless, the fundamental question remains: how can assets be allocated optimally in a financial environment characterised by accumulated opportunity and non-linear distributions? As Fama (1970) observes in his Market Efficiency Hypothesis, investors’ capacity to capitalise on this information depends on their ability to comprehend and respond expeditiously to evolving market dynamics, particularly in volatile environments.

It is a fallacy to presume, however, that ’everything that can be counted does count or that everything that counts can be counted’, according to (Whittaker, 1955) Albert Einstein (1879–1955): a German theoretical physicist known for developing the theory of relativity, among other scientific achievements. Today, portfolios are run in accordance with Einstein. The usual risk and reward metrics to measure the swerve of complex financial markets do not make financial sense. Our analysis combined the more non-traditional risk measures of Tsallis entropy and mutual information to directly challenge long held conventional thinking in the functional areas of risk assessment and to fully understand the movement of financial data (which are always non-linear and unpredictable) by building more general as well as precise risks and opportunities in traditional asset management and cryptocurrency portfolios. Thus, the goal of this study is to be able to show, by going deeper than the visible numbers, denominator performance and risk indicators. It will also make sure you have the material for best practice for your personalized portfolio management in the digital age.

Markowitz’s (1952) modern portfolio theory (MPT) established a pioneering mathematical framework with risk being embodied by variance and covariance, which constitute the kernel of this theory. This idea, in combination with Sharpe’s (1964), relates to the Capital Asset Pricing Model (CAPM), which made the computation of optimal portfolios possible using the Sharpe ratio, which seeks to optimise returns per level of risk accepted. Even so, CAPM’s foundational assumptions (e.g., that return distributions are gaussian and that assets are independent in a linear sense) have been criticized. This is particularly true in settings rife with asymmetries and volatile shocks; e.g., cryptocurrencies and emerging markets.

Recent work exploring entropy and mutual information (MI) offers a promising alternative in this context. In contrast to variance and covariance, measures stemming from information theory (Shannon, 1948) such as entropy and mutual information are better suited to heavy queuing distributions sustained in modern markets because they capture non-linear dependencies between assets. Entropy, as a measure of diversification, highlights the uncertainty and complexity of a portfolio. Mutual information quantifies the structural relationships between assets, acting on assumptions of normality. These tools have been the focus of extensive research by Bariviera and Merediz-Solà (2021) and Tsallis (1988), and enable more robust portfolio optimisation, a subject of particular relevance to cryptocurrencies.

Although mutual information is recognised as a powerful tool for capturing non-linear dependencies between financial assets, it still requires a more quantitative approach to risk assessment than traditional methods. Indeed, in contrast to variance and covariance, mutual information facilitates the exploration of complex interdependencies, a particular advantage in financial markets characterised by non-Gaussian behaviour and extreme events (Dionisio et al., 2004). While the theoretical advantages of mutual information are well established, there is a paucity of empirical evidence demonstrating its effectiveness in real-world portfolio management (Zhou et al., 2013).

For example, Lahmiri and Bekiros (2020) show that incorporating entropy measures, like mutual information, with portfolio diversification helps managing risks in fast-paced settings, like cryptocurrency markets. However, these studies are generally restricted to specific laboratory environments, letting rise questions about the generality of this strategy in effectively solving a multitude of market conditions.

Moreover, previous work, such as that by Granger and Lin (1994), emphasises the benefit of mutual information for uncovering complex dynamic relationships, but this has not yet been applied in practice to optimisation models. This gap highlights the need for meticulous real-world studies to validate the effectiveness of mutual information in improving risk-adjusted returns or minimising queueing risk relative to standard covariance methods (Ledoit & Wolf, 2004).

Consequently, while mutual information shows promise, it necessitates systematic empirical investigation and the formulation of practical guidelines to substantiate its efficacy across diverse market contexts, notably those characterised by non-linear dynamics and extreme behaviour, such as emerging markets and cryptocurrencies (Urquhart, 2016; Vidal-Tomás, 2021).

Entropy, as a measure of diversification, highlights the uncertainty and complexity of a portfolio (Chortane & Naoui, 2022), while mutual information quantifies the structural relationships between assets, transcending beyond normality assumptions (Cover & Thomas, 2006). These tools, which have been extensively studied by Bariviera and Merediz-Solà (2021) and Tsallis (1988), enable more robust portfolio optimisation, a matter of particular relevance for cryptocurrencies, which exhibit non-linear characteristics and potential accumulation (Urquhart, 2018; Le Tran & Leirvik, 2020).

Active portfolio management is an important task in today’s financial world, seeking balance between maximising returns and minimising risk for an investor. The modern portfolio theory is a change in investment theory introduced in 1952 by Harry Markowitz. It also introduced the idea of optimal diversification depending on asset return variance and covariance. But the rise of new asset classes, particularly cryptocurrencies, and increased complexity of financial markets mean that traditional approaches need to be re-evaluated. Our study proposes to revisit the fundamentals of portfolio theory and introduce this ambitious but not utopian objective using advanced measures such as Tsallis’s entropy and mutual information, providing a new way to look at risk management and return maximisation.

In this study, we examine whether adding Tsallis entropy mutual information and value-at-risk entropy could provide a valuable add-on to portfolio management against variance–covariance-based models, particularly in highly volatile and interconnected markets. The theory is that these alternative measures will better represent complex risks and interdependencies between assets, giving them higher adjustment returns. The present examination assesses elective risk measures such as Tsallis entropy, mutual information, and entropy value at risk. This is accomplished by overseeing mixed portfolios linked to S&P 500 and Bitwise 10 Crypto Index assets. This period is from 1 April 2019 to 31 May 2024, except 2020.

The results of our analysis proved the efficacy of these proposed risk estimators, quantitatively measured by Tsallis entropy (and mutual information), in diversified portfolio formation, which included traditional assets and cryptocurrencies. Mutual information, Tallis entropy, and value-at-risk entropy-based models of history can increase diversity while adjusting to an unanticipated market setup. In both indices, this led to the poorest performance of the MV model, with the best performance from models based on EVAR, suggesting ancillary protection from tail events using entropy-driven models.

This ‘graphical approach’ and the new perspectives in portfolio science it encompasses outperformed both broad market variance–covariance views of risk management and return based on unique risks across a modern diversified markets portfolio, according to the analyses. The bottom line is that the authors show that these models can improve Muller’s Sharpe ratio and Jensen’s alpha of test portfolios.

The specific contributions of this research are formulated across four majors’ axes. In the first contribution, we show how, in the presence of non-linear dependencies between assets, Tsallis entropy and the mutual information between the assets give us a more representative measure of risk for diversified portfolios. This is slightly better than what traditional Markowitz can offer diversification in regard to risk-adjusted performance. Second, an in-depth comparison of the models, based on different entropy and mutual information measures, to a series of traditional portfolios is carried out. Historical data from equity (S&P 500) and cryptocurrency (Bitwise 10) markets are used. Lastly, we demonstrate how more advanced means can shift the efficient frontier of portfolios, providing better Sharpe ratios and risk–return profiles than standard methods. Finally, this research provides actionable steps for integrating these techniques into the existing portfolio management framework. This is to support financial professionals under uncertain and interconnected market conditions.

The relevant literature is systematically reviewed in Section 2. The final section explains the data and methodology used in the study. The results of the analysis are shown in Section 4, while the conclusion is presented in Section 5.

2. Literature Review

Modern portfolio theory (MPT), established by Markowitz (1952), seeks to optimise expected return for a given level of risk, measured by variance. For many decades, this framework reigned over finance and offered one vehicle for structured portfolio selection and risk management. Yet its dependence on strong assumptions such as normality of returns and linear independence of assets has come under increasing scrutiny. According to Chopra and Ziemba (2013), the MV model is highly susceptible to estimation error for octant and suboptimal results, making it crucial to correct for such error in the mean vector of returns for asset allocation. These attributes have generated interest among researchers investigating more resilient alternatives selected for complexity and non-linearity in the context of financial systems.

MV models have certain limitations that have led to innovative approaches. In contrast to classical MV models, Ledoit and Wolf (2003a) proposed improved covariance estimators to reduce estimation errors, and DeMiguel et al. (2009) questioned the superiority of MV models by demonstrating that naive strategies, including 1/n portfolios, frequently outperform MV models in terms of out-of-sample performance, particularly when the number of assets greatly exceeds the number of observations. In contrast, Fama and French (2004, 2015) argued that the CAPM model, which was obtained directly from the TPM assumptions, would not be able to use successful observed returns. They recommended factor models—in particular, the five-factor model—to include firm size and financial ratios in return evaluations. Despite widening the lens through which expected returns are modelled, these models are still narrow in their reliance on historical data and particular statistical assumptions.

Stepping back, entropy, introduced by Boltzmann in physics and later applied to information theory by Shannon (1948), has recently shown promise as an alternative in this area. Entropy measures uncertainty without any assumptions about returns distribution, unlike variance. Among the earlier work employing entropy for portfolio optimisation was that of Philippatos and Wilson (1972). They showed that the ME optimal portfolios were usually close to the variance optimal portfolios, but more robust to non-normality and higher return moments. Lassance and Vrins (2019) expanded on this approach through a generalization using Rényi entropy to calibrate the model flexibility to focus on general patterns or extreme events. Their work demonstrates that entropy offers novel degrees of freedom in parameterising such models, allowing for asymmetric distributions and non-linear relationships between assets.

Another notable development is the introduction of mutual information (MI), which extends the concept of entropy to measure non-linear dependencies between two random variables. Dionisio et al. (2004) showed that MI captures complex relationships between financial time series, a capability that traditional time series correlation cannot provide. Lahmiri and Bekiros (2020) used Rényi entropy and MI to analyse the impact of the COVID-19 pandemic on financial markets, showing that these measures can serve as early warning indicators of market crashes. These results highlight the increasing usefulness of MI in modern market environments where non-linear dependencies and extreme events are common.

Information theory is addressed from several perspectives, but there is a gap in the literature as there is not a risk definition based on Tsallis entropy. Notably, Ormos and Zibriczky (2014) used the 150 daily returns of random stocks over 27 years, showing that entropy decreases with the number of stores in a portfolio, which also applies to the standard deviation, while efficient portfolios position on a hyperbola regarding expected return–entropy ratios. Consequently, in capital asset pricing models, entropy explains expected returns better than does beta. A novel fuzzy portfolio selection model is proposed, which is to maximise the return and skewness and minimise their variance and cross-entropy and a mean-entropy-skewness equity portfolio selection model with transaction costs. (Bhattacharyya et al. (2013, 2014), among others). MacLean et al. (2022), evolve a methodologically driven and quantitatively rigorous framework of portfolio selection to present an entropy-based dynamic portfolio selection model. Profits have to be greater than or equal to the deficit of a target. For the model to work, it must also have the probability of the shortfall being less than or equal to a given level. To do so, they formulate a regime-switching regression model that predicts the risk and return profiles of the individual assets using the exponential Rényi entropy for modelling portfolios.

For instance, Chortane and Naoui (2022), analyse the applicability of Information Entropy Theory (IET) in the valuation of assets and the choice of a portfolio. This suggests that optimum returns with the lowest risk are what portfolio managers are after. Compared to standard models, the entropy-based portfolio selection model has the advantage of providing a more comprehensive understanding of asset and distribution probabilities, which is particularly beneficial when return distributions are non-Gaussian. Entropy captures diversification better than pure variance. Another application of entropy can assist in forming models with non-linear dependencies to the time series of stock returns, and to obtain plausible relative/absolute value estimates for systematic and specific risk in linear equilibrium models. Investors and decision-makers will then better understand market efficiency, portfolio selection, and asset valuation resulting from this theoretical framework.

To address the limitations of the mean-variance framework, Markowitz (1952) and Mahmoud and Naoui (2017) suggest the use of the Shannon entropy as an alternative measure of financial risk. They note that the classical approach makes some strong assumptions, notably with respect to the normality of returns and linearity of relationships between assets. These assumptions do not match the empirical properties of financial markets, which are non-Gaussian and exhibit non-linear relationships. By conducting their empirical analysis, Mahmoud and Naoui demonstrate how portfolios optimised with respect to the entropy associated with Shannon are less vulnerable to market shocks and provide better risk-adjusted returns in the long term, particularly in volatile environments. Their study points to the relevance of entropy in diversification and risk management whilst at the same time debunking the over-reliance on traditional models. Notably, such a research insight lays ground for new ways to describe the state of systemic risk in modern financial markets based in part on information theory.

Entropy has been further employed as a measure of diversification for the purpose of portfolio optimisation. It has been emphasised by Zhou et al. (2013) that entropy’s prowess in harvesting higher-order moments can be superior to variance at estimating overall portfolio uncertainty. Entropy, unlike conventional measures, does not assume a linear relationship between the assets in your portfolio, allowing for increasingly efficient diversification. The maximum entropy principle has been shown to mitigate bias in portfolio construction (Bera & Park, 2008). This is especially true when the available data are sparse or uncertain. Furthermore, Haluszczynski et al. (2017) prove that MI essentially grounds the diversification in a more state-dependent fashion that accounts for dynamic relationships among all risk factors at play, in the process reducing certain unsystematic risks.

Assaf et al. (2022), find that Bitcoin and Ripple share a bidirectional information transmission. According to Rényi’s measure, there is no non-linear information transmission. To study their dynamics before COVID-19 and during the pandemic, they use the mutual information approach and approximate entropy to investigate information sharing between cryptocurrencies during the COVID-19 crisis. The results of the mutual information measure indicate increased information sharing and order in cryptocurrency markets during the pandemic. The evidence from the approximate entropy estimates suggests an increase in randomness during COVID-19. Ahn et al. (2019) reveal strong spillover effects from stock market uncertainty to economic fundamentals. Specifically, an uncertainty shock generates (i) a short-term decline in industrial production, (ii) a rapid drop and rebound in the composite leading indicator, and (iii) an increase in systemic risk. Barbi and Prataviera (2019) use mutual information minimum spanning trees to explore non-linear dependencies in the Brazilian equity network in two distinct pollical periods. Minimum spanning trees from mutual information and linear correlation between stock returns were obtained and compared. Mutual information minimum spanning trees present a higher degree of robustness and evidence of power-law tail in the weighted degree distribution, indicating more risk in volatility transmission than is expected by the analysis based on linear correlation. Moreover, these results emphasise the usefulness of mutual information network analysis for identifying financial market features due to non-linear dependencies.

However, these models remain practical. Zhang et al. (2012) explored the integration of entropy into a multi-objective framework, including transaction costs and diversification constraints. Despite these advances, further research is needed to test the effectiveness of these models in real-world contexts, particularly for emerging asset classes such as cryptocurrencies. Work by Fiedor (2014) and Haluszczynski et al. (2017) has shown that MI can also be used to detect market anomalies and measure the impact of extreme shocks on portfolios. This opens up novel perspectives for risk management and portfolio optimisation.

To summarise, entropy and mutual information offer powerful new approaches to portfolio optimisation. They provide powerful cures to the pitfalls of traditional models, capture non-linear dynamics, and adapt to contemporary market structures. However, extensive research is needed to confirm their usefulness in broader financial settings. Also, it is required that they learn how to exercise their prowess in digital asset management and diversified portfolios. These developments will set the stage for a newly emerging generation of investment strategies that are better suited to the conditions of contemporary financial markets.

3. Data and Methods

3.1. Data

For this study, we leverage a data set that includes the daily prices of the S&P 500 components and the Bitwise 10 Crypto Index. Except for the year 2020, these data were collected between 1 April 2019 and 31 May 2024. The period in question has been excluded here to mitigate distortions stemming from the global pandemic in 2020 and related volatility. This offers a more stable measure of the performance of an asset. S&P 500 Bloomberg data consist of adjusted daily pricing for index constituents, with adjustments for any dividends or stock split. The data on the Bitwise 10 Crypto Index are sourced from CoinMarketCap and represent the prices of the most famous cryptocurrencies adjusted for market capitalisation on a daily basis. These are BTC-USD, ETH-USD, SOL-USD, XRP-USD, ADA-USD, AVAX-USD, LINK-USD, DOT-USD, BCH-USD, and NEAR-USD. We began with a data set of the 1302 instances for which each company was treated, followed by outlier and missing value filtering (especially for 2020), which cleaned the number of valid observations to 433320. Nevertheless, this cleaning process did ultimately give us a more representative data set. To review the returns, we calculated logarithmic returns that normalise the variance and are comparison friendly for inter-temporal and inter-asset comparison. In addition, a seasonal adjustment was made to eliminate distortions caused by seasonality, which is a significant improvement in the analysis of the underlying trends. The cleaned data were then subject to further analysis to assess different portfolio models using Python 3.10.11 executed via Thonny software. We leveraged new tools such as Tsallis entropy, mutual information, and EVAR (entropic value-at-risk) to obtain complete dynamics of our portfolio, with a risk–return assessment in various market regimes.

3.2. Methods

3.2.1. Markowitz’s General Framework

Markowitz (1952) developed the mean-variance model and transformed investment portfolio management by introducing a quantitative approach to optimising the trade-off between risk and return. The model is predicated on a quadratic objective function, whereby the variance of returns measures risk and the average of asset returns represents the expected return.

The optimisation problem is formulated as follows:

$$Min(xT ⅀x), \mathrm{under} \mathrm{constraints} xT\mu \ge lmin, {\displaystyle {\sum }_{je=1}^n}xje=1, xje \ge 0, \forall je$$

(1)

where

• $X=[x1,x2,\dots ,xn]$ is the vector of weights allocated to assets 1, 2…, n,
• $⅀$ is the (n*n) covariance matrix of asset returns,
• $\mu =[\mu 1,\mu 2,\dots ,\mu n]$ is the vector of expected returns,
• $l min$ is the minimum return desired by the investor,
• ${\displaystyle {\sum }_{je=1}^n}xje=1,$ guarantees that all capital is invested,
• $xje \ge 0$ prevents short selling (long-only).

Efficient Frontier

At this point, let us recall the efficient frontier, which is one of the most important concepts in the Markowitz model, as it is the one that states the portfolios that offer the highest return for a given (mandatory) level of risk or, conversely, the portfolios that offer the least risk for a given expected return. From a mathematical standpoint, this frontier is found by solving the following problem for varying delmin values:

$$Min(xT ⅀x), \mathrm{under} \mathrm{constraints} xT\mu =lmin, {\displaystyle {\sum }_{je=1}^n}xje=1, xje \ge 0,$$

(2)

There is a convex curve in a risk–return space, where risk is standard deviation ($\sigma p=xT⅀ x$) and return by ($lp=xT\mu )$. These portfolios are considered optimal on this frontier, as they are efficient or dominate all other portfolios.

The model operates under the assumption that asset returns are normally distributed; however, this is not invariably the case in practice, particularly in financial markets, where distributions are frequently found to be asymmetric and to exhibit thick queues (Mandelbrot, 1963). This could result in an underestimation of extreme risks. The model’s input parameters ($\mu \mathrm{and} \mathrm{\Sigma }$) are estimated from historical data; nevertheless, these estimates may be subject to bias or imprecision. As demonstrated by Chopra and Ziemba (2013), errors in the estimation of $\mu$ have a significantly increased impact on portfolio performance in comparison to errors in $\mathrm{\Sigma }$. One proposed solution is the use of robust estimators for the covariance matrix, such as the shrinkage method (Ledoit & Wolf, 2004). However, this model does not take into account realistic constraints such as transaction costs, sectoral constraints, or diversification. These constraints render the optimisation problem non-convex and difficult to solve (Kolm et al., 2014). The model’s focus on the mean and variance neglects aspects such as skewness and kurtosis. These aspects are essential in capturing the non-Gaussian nature of returns (Harvey & Siddique, 2000).

3.2.2. Proposed Models

A novel approach is thus proposed to address the limitation of the classical model by incorporating entropy and mutual information (MI) into the risk matrix.

Why Entropy?

Entropy is a measure of uncertainty and diversity. Entropic measure is, however, a robust measurement of financial data. Usually, financial data do not follow normality; therefore, approches are defined according to the given distribution.

We employ Tsallis entropy, a generalisation of Shannon entropy, to measure uncertainty and diversity in portfolios including S&P 500 constituents and the Bitwise 10 Crypto Index for this purpose. This is especially so for the non-Gaussian distributions of financial returns. Entropy, according to Tsallis (1988), can be useful in capturing the complex dynamics of financial markets by integrating long-range dependencies between assets and the existing non-linear interactions.

The Tsallis entropy is given by the following equation:

$$Sq={\displaystyle \frac{1-{\displaystyle {\sum }_{i=1}^W}{P}_I^q}{q-1}}$$

(3)

where $Pi$ represents the probability of state $i$ and q is a natural (non-integer) parameter that quantifies the degree of non-additivity of the entropy. Thus, $q$ allows one to capture dependent behaviour between system components.

Why Mutual Information?

Mutual information (MI) is a measure of dependence between two variables, whether linear or non-linear (Cover & Thomas, 2006). Mutual information (MI) is a non-linear measure of dependence that can be used to identify dependencies between two random variables that are not captured by cyclical relationships identified by covariance. This capacity to model intricate dependencies renders MI particularly advantageous for portfolio optimisation, especially in markets exhibiting non-linear dynamics and non-normal returns (Kraskov et al., 2004).

Shannon’s (1948) concept of mutual information serves as a foundational metric for quantifying the degree of information shared between two random variables X and Y. It is defined by the following equation:

$$MI(X,Y)=H\left(X\right)+H\left(Y\right)-H(X,Y)$$

(4)

where $H\left(X\right)$ and $H\left(Y\right)$ are the marginal entropies and $H(X,Y)$ is the joint entropy. In contradistinction to traditional correlation coefficients, MI is capable of capturing both linear and non-linear dependencies, thus rendering it a powerful tool for the examination of complex interdependencies between financial assets.

Marginal entropy quantifies the uncertainty associated with a single random variable $X$.

$$H(X)=-\sum _{x\in X}P\left(X\right)logP\left(X\right)$$

(5)

Or $H\left(X\right)$ denotes the entropy of the random variable $X$,

$P\left(X\right)$ is the probability of the event X belonging to the set of possible values of $X$,

The ${\displaystyle {\sum }_{x\in X}}\dots$ crosses all possible states of the variable $X$,

The logarithm is often taken in base 2 to express entropy in bits.

Joint entropy measures the combined uncertainty of two random variables $X$ and $Y$.

$$H(X,Y)=-\sum _{x\in X}\sum _{y\in Y}P(X,Y)log P(X,Y)$$

(6)

$P (X,Y)$ represents the joint probability of simultaneous occurrence of events $X$ and $Y$,

We are $\sum x \in X$ and $\sum y \in Y$ parkouring, respectively, all possible states of $X$ and $Y$.

This entropy measures the overall uncertainty when the two variables are considered together (Cover & Thomas, 2006).

The conditional entropy measures the uncertainty of X given Y.

$$H(X/Y)=-\sum _{x\in X}\sum _{y\in Y}P(X,Y) logP(X/Y)$$

(7)

where

$P(X/Y)$ is the conditional probability of $x$ and $H(X/Y)$ quantifies how much uncertainty remains about x when y is known.

3.2.3. Construction of the Modified Risk Matrix

Let X = {X1, X2, …, Xn} be a portfolio containing n assets. We adopt an approach where the Tsallis entropy $Hq\left(X\right)$ for each asset $X$ is used as an analogue of the variance and is placed on the diagonal of the matrix. The mutual information $\left(MI\right)$ between two assets $X$ and $Y$ replaces the covariance between these assets in the upper and lower triangles of the matrix:

$$\sum Modified risk matrix=\left[\begin{array}{ccc}Hq\left(X1\right)& MI(X1;X2)\cdots & MI(X1;Xn)\\ ⋮& \ddots Hq\left(X2\right)& ⋮\\ MI(Xn;X1)& MI(Xn;X2)\cdots & Hq\left(Xn\right)\end{array}\right]$$

(8)

Normalisation of the mutual information:

The mutual information values in the triangles of the matrix are normalised according to a function f, which can take different forms depending on the entropies of the individual assets:

$$\mathrm{Sum} \mathrm{of} \mathrm{entropies} fx+y=Hq\left(X\right)+Hq\left(Y\right)$$

(9)

$$\mathrm{Minimum} \mathrm{of} \mathrm{the} \mathrm{entropies} fMIN=MIN \left\{Hq\right(X),Hq(Y\left)\right\}$$

(10)

$$\mathrm{Maximum} \mathrm{of} \mathrm{the} \mathrm{entropies} fMAX=MAX\left\{Hq\right(X),Hq(Y\left)\right\}$$

(11)

$$\mathrm{Joint} \mathrm{entropy} fXY=Hq(X,Y)$$

(12)

$$\mathrm{Product} \mathrm{of} \mathrm{the} \mathrm{square} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{entropies} fSQRT=\sqrt{Hq\left(X\right) Hq\left(Y\right)}$$

(13)

This model is used to formulate standard portfolio optimisation problems similar to the Markowitz and Sharpe models but with an adapted risk measure incorporating non-linear dependencies and a parameter estimate more robust to the data’s non-normality.

The objective function of portfolio optimisation is as follows:

$$Max wT \mu -\lambda (wT ⅀ w+\alpha {\displaystyle {\sum }_{i=1}^n}H\left(Xi\right))$$

(14)

The term $w$ portfolio weights vector includes the proportion of all assets found in the portfolio. $\mu$ is the expected (mean) vector of asset returns, or the expected return for each investment. In this case, $\lambda$ is the risk aversion coefficient, determining how much risk an investor will accept. This is in the context of using $\alpha$ to adjust the relative importance between the diversification aspect and avoid its effects in the asset allocation decision when controlling the weight of entropy in the objective function.

3.2.4. Entropic Value-at-Risk

Introduced by Ahmadi-Javid (2012), entropic value-at-risk (EVAR) is a risk measure that dominates both value-at-risk (VAR) and conditional value-at-risk (CVAR) by utilizing the information from Chernoff’s inequality. This ensures that it properly captures extreme risks while being coherent with the principles of convex optimisation. This could provide greater extensibility of financial applications benefitting complex, non-linear market environments.

$${\mathrm{EVAR}}_{\alpha }\left(X\right)=\underset{z>0}{\inf }\left\{z\log \left({\displaystyle \frac{1}{\alpha }}{M}_X\left({\displaystyle \frac{1}{z}}\right)\right)\right\}$$

(15)

where $M_X\left(t\right)=\mathrm{E}\left[e^{tX}\right]$ is the moment-generating function, $\alpha \in \left[0,1\right]$ is the confidence level, and j is a minimised optimisation parameter.

Where α = 0.1 the robust and efficient characteristics of this measure have been demonstrated, especially in scenarios involving asymmetric distributions or heavy queues (Krokhmal et al., 2002).

Discretisation of EVAR

Discretisation of EVAR is imperative to facilitate its practical application. The discretisation framework utilised is based on a series of financial returns:

The discretisation of continuous asset returns is conducted into k = 101 states within the interval $[-50\%,+50\%]$.

This choice of granularity is guided by the recommendations of Zhu and Fukushima (2009), who advocate a balance between accuracy and computational complexity.

The aforementioned discretisation guarantees a faithful representation of the non-normal distributions that are often referred to in financial markets (Mandelbrot, 1963).

EVAR Minimisation

From the function perspective, we can discretise EVAR using the exponential cone in CVXPY. Cajas (2021) proposed the disciplined convex programming (DCP) problem of EVAR minimisation, which is formulated as:

$$\begin{array}{cc}\underset{x,\hspace{0.17em}z,\hspace{0.17em}t,\hspace{0.17em}u}{\min }& t+z\ln \left({\displaystyle \frac{1}{\alpha T}}\right)\hfill \\ \mathrm{s}.\mathrm{t}.& \mu x^{\tau }\ge \bar{\mu }\hfill \\ & {\displaystyle {\sum }_{i=1}^N}{x}_i=1\hfill \\ & z\ge {\displaystyle {\sum }_{j=1}^T}{u}_j\hfill \\ & \left(-r_j{x}^{\tau }-t,z,u_j\right)\in K_{\exp }\hspace{0.33em}\forall \hspace{0.33em}j=1,\dots ,T\hfill \\ & x_i\ge 0\hspace{0.33em};\hspace{0.33em}\forall \hspace{0.33em}i=1,\dots ,N\hfill \end{array}$$

(16)

where $t$ is an auxiliary variable that represents the perspectives of the log_⅀_exp function, Z is the factor of perspective function, $u_j$ is an auxiliary variable, $x$ are the degrees of freedom assets weights, $\mu$ is the mean vector of expected returns, $\bar{\mu }$ is the minimum of expected return of the portfolio, $K_{\exp }$ is an exponential cone, and $r$ is the matrix of the observed returns.

We exploit an exponential cone reformulation of the exponential cone EVAR, which allows us to integrate EVAR into a disciplined convex programming (DCP) framework (Grant & Boyd, 2008).

$$Kexp\left\{\right(t,x,y)\in R3:and>0,and\cdot exp (x / y)\le t \}$$

(17)

This method ensures that the optimisation problem is convex, a requirement for use of interfaces like CVXPY or CVXOPT. The use of convex frameworks for financial risk management has been explored in substantial works, including (Nocedal & Wright, 1999).

The Construction of the Objective Function

The integration of EVAR in our model is achieved via a weighted objective function:

$$max\{ mT\mu -\lambda \cdot ÉVaR\alpha (wT\left)\right\}$$

where

• $m$ is the vector of portfolio weights.
• $M$ is the vector of expected returns.
• $\lambda$ is the risk aversion coefficient.

This function has been shown to maximise returns while minimising extreme losses, in line with the principles outlined by Rockafellar and Uryasev (2000) for CVAR but extended to EVAR. Empirical tests of EVAR were conducted on asset portfolios of the S&P 500 and Bitwise 10 crypto indices. The outcomes were then compared with those of traditional mean-variance models (Markowitz, 1952) and CVAR (Rockafellar & Uryasev, 2000).

The findings indicated that EVAR exhibited superior performance in the management of extreme risks, particularly for assets characterised by high availability, such as cryptocurrencies (Cont, 2001). The use of portfolio diversity in conjunction with entropy measures, such as Tsallis entropy (Tsallis, 1988), has been demonstrated as a means of capturing non-linear dynamics and complex correlations.

In this work, we assume that the optimisation function is a convex function, whereas the entropy presents concavity (Zhou et al., 2013), which preserves the essential framework of the mean-variance model. The CVXOPT tool (Boyd & Vandenberghe, 2004), based on conic programming and interior point methods, and the Python programming language were two publicly accessible optimisation tools used. MINIMISE is employed, which actualises the Sequential Least Squares Programming (SQSLP) method (Kraft, 1988). The efficient frontier was calculated using the CVXOPT method, while the SQSLP method was used to solve the PO models. This is because it is flexible enough to deal with constraints.

We have also included measures of Tsallis entropy and mutual information, which encapsulate non-linear dynamics and complex correlations among assets (Song & Chan, 2020; Granger & Lin, 1994). Similar techniques support portfolio optimisation in unstable market conditions, especially if crypto assets are part of considered assets. Estimating the efficient frontier incorporating entropic value-at-risk (EVAR) is shown to be more effective in managing extreme risks (Mercurio et al., 2020).

The results obtained by these methods have been validated by backtesting over different periods and for several indices (S&P 500, Bitwise 10), as detailed below. To ensure the proposed model’s practical relevance, we subjected it to empirical tests on stock market indices such as the S&P 500 and Bitwise 10. These tests covered five years (April 2019 to May 2024). We excluded 2020 due to COVID-19 pandemic anomalies. This exclusion allows us to focus on volatility periods more representative of average market conditions. It also avoids biases linked to extreme exogenous events. For robustness, we develop an efficient frontier for 2020 to highlight and enhance our results in extreme events.

Backtests were performed with several model configurations, including mean-variance (MV), mean-entropy (ME), and mutual information (MI) models, as proposed by Dionisio et al. (2004) and (Huang, 2008). These models have been evaluated using measures such as Sharpe’s ratio (Sharpe, 1964), Jensen’s alpha (Jensen, 1968), and EVAR (Mercurio et al., 2020). Tsallis entropy has been used to analyse portfolio diversity as a measure of dispersion and balance of asset weights (Cover & Thomas, 2006).

4. Results

4.1. Performance Analysis

As shown in

Table 1. Cryptocurrency daily returns analysis (S&P 500 daily returns).

	Return%	Alpha	Stdev	P1	P99	Sharpe	Entropy	GLR
MV	0.045	NA	0.006	0	4.99	0.070	29.133	0.183
MI	0.091	0	0.354	0	1.14	0.003	190.063	0.106
MI X+Y	0.091	0	0.137	0	1.14	0.007	190.071	0.106
MI Min	0.091	0	0.194	0	1.14	0.005	190.072	0.106
MI Max	0.091	0	0.194	0	1.14	0.005	190.064	0.106
MI XY	0.081	0	0.122	0	0.64	0.007	323.368	0.117
MI Sqrt	0.091	0	0.194	0	1.14	0.005	190.071	0.106
EVAR	0.055	0	0.016	0	7.18	0.034	17.038	0.203

Note: Table 1 aims to determine the efficacy of various portfolio optimisation models when applied to S&P 500 Index components. Tsallis’s entropy is integrated into the analysis. This generalised entropy measure captures non-linear dynamics and subtle interdependencies between assets, providing an exclusive perspective on risk management and diversification.

, the classic Markowitz model leads to a rather low return of 0.045%, highlighting the urgent need for robust risk management. Conversely, the mutual information (MI) models and their variants achieve an impressive return, maximising at 0.091%. This improvement of performance indicates a better ability to combine connections between assets and utilize these dependencies to produce better results. This finding implies that mutual information approaches can be a new candidate for portfolio management, especially in settings where non-linear relationships dominate between the assets.

This is also supported by the entropy results calculated by the Tsallis entropy method. For example, the MI model has an entropy value of 190,063, which is significantly higher than that of the MV model (29,133). Growing diversification is an essential factor in complexity markets to avoid risk. Sharpe ratios, on the other hand, are low in all models, meaning there is limited improved returns to risk. This could suggest that S&P 500 assets are conservative and that it is not easy to get fantastic returns.

EVAR, which adopts a non-linear risk–loss viewpoint, channels unique benefits as well. It has a lower entropy (17,038) and is sufficiently capable of absorbing distribution queues and extreme risks. This makes it especially useful for conservative investors trying to reduce catastrophic losses. Yet better risk management comes at the cost of a lower risk premium (0.055%), which strikes a reasonable balance between extreme risk management and expected returns.

The observations made here echo Sharpe’s (1964) criticisms about the limitations of conventional Markowitz models. However, these models underperform with non-normal return distributions (see Ledoit & Wolf, 2004, for an in-depth analysis). Ledoit and Wolf (2004), a major work, showed that research-tuned techniques with respect to mutual information and Tsallis entropy yield higher performance in market conditions with anomalies and non-linear dependencies. The output of these models is consistent with the contemporary theoretical background, which emphasizes the advantages of non-parametric risk measures that can better grasp the complexities of current markets (Zhou et al., 2013; Lassance & Vrins, 2019). Therefore, stemming mutual information and entropy to portfolio optimisation models could be a powerful lever for enhancing management.

The returns from the application of models to cryptocurrencies are much larger than those from the application to traditional assets, as seen in

Table 2. Cryptocurrency daily returns analysis (Bitwise 10 Crypto Index components).

	Return%	Alpha	Stdev	P1	P99	Sharpe	Entropy	GLR
MV	0.264	NA	0.034	0.00	83.49	0.078	1.451	0.765
MI	0.361	0	0.886	7.56	13.06	0.004	9.830	0.519
MI X+Y	0.361	0	0.352	7.56	13.06	0.010	9.830	0.519
MI Min	0.361	0	0.497	7.56	13.06	0.007	9.830	0.519
MI Max	0.361	0	0.497	7.56	13.06	0.007	9.830	0.519
MI XY	0.353	0	0.396	8.98	11.29	0.009	9.970	0.524
MI Sqrt	0.361	0	0.497	7.56	13.06	0.007	9.830	0.519
EVAR	0.280	0	0.041	0.00	76.27	0.069	1.759	0.646

Note: Table 2 aims to determine the efficacy of various portfolio optimisation models when applied to the Bitwise 10 Crypto Index components. Tsallis’s entropy is integrated into the analysis. This generalised entropy measure captures non-linear dynamics and subtle interdependencies between assets, providing an exclusive perspective on risk management and diversification. (Alpha refers to Jensen’s measure).

, with MV taking 0.264% and the MI variants 0.361%. This realization emphasizes the extreme randomness of crypto assets and the presence of outsized returns in a palpitating market climate. Urquhart’s (2016) finding supported these results—that cryptocurrencies, with their relative inefficiency, will allow for generated returns, albeit with higher risk.

The high entropy scores associated with MI models (e.g., 9.970 for MI XY) permit accumulated diversification and an enhanced capacity to absorb market shocks, which is consistent with Tsallis’s (1988) observations regarding the advantages of generalised entropy measures in complex systems. Furthermore, Bariviera and Merediz-Solà (2021) also highlighted that cryptocurrencies are secure tools to capture their non-linear dynamics, such as those provided by entropy and mutual information.

Higher Sharpe ratios for cryptocurrencies (e.g., 0.078 for MV and 0.069 for EVAR) generate more attractive risk-adjusted returns, albeit with a possible upside. These observations concur with the findings of Le Tran and Leirvik (2020), who demonstrated that cryptocurrencies, despite their high availability, can offer a significant liquidity premium, making them attractive to risk-tolerant investors.

The application of EVAR to cryptocurrencies, with a higher GLR (0.646 for EVAR versus 0.519 for MI), highlights its effectiveness in managing extreme risk while offering stable diversification. As Ledoit and Wolf (2004) have argued, the improvement of traditional models with robust measures such as entropy is crucial to better represent modern financial market dynamics.

This is to be expected with the highly speculative nature of these new digital assets, which are settled out in both markets and return more than traditional S&P 500 assets. Showing slightly higher Sharpe ratios, cryptos have better risk-adjusted returns despite having a similar volatility. To our knowledge, when all crypto reference modes are considered combined, entropy is in the 0.1–0.8 range for most coin models; the clustering of both ends of the spectrum is somewhat similar in theory, but in practice, you get bias in this volume which collects upper middle vector codes. Less impacted by market disruptions: Because they are not traditional S&P 500 assets, they could be less impacted by market disruptions.

Looking through this, we see that while the cryptocurrency returns are appealing, accessing them takes a complicated risk management regime. Strategies such as entropic diversification should be included to generate robust portfolios (Lahmiri & Bekiros, 2020; Zhou et al., 2013). They also need to be aware of the Sharpe ratios that show risk/return trade-offs in different markets.

Entropy-based portfolio management is an alternative to conventional methods of Markowitz (1952) frameworks and aims to better represent the operational nature of financial markets in the modern age. Fama and French (2004, 2015) highlighted a wide criticism of the traditional CAPM models that are often seen to perform poorly in accounting for other market factors, such as entropy, which alternative metrics can easily capture. Moreover, as discussed later, we can also provide superior estimates compared to traditional techniques for the covariance matrix, as seen by Ledoit and Wolf (2003b, 2004), which can play a crucial role in markets as turbulent as cryptocurrency markets.

According to risk measures, the difference between the S&P 500 and the Bitwise 10_ cryptocurrency index is not as powerful among the shear types in the development of risk coverage, as well as diversification. Equipped with this knowledge, and so better able to understand and navigate the modern marketplace, these observations provide investors and fund managers actionable insights to hedge their strategies as financial conditions are on the move.

In Figure 1, we make a comparison of three axes: volatility, returns, and Sharpe ratio. The mutual information (MI) and entropy (EVAR) models reflect better returns, while the Markowitz model (MV) reflects lower returns.

Bitwise crypto models show high returns with little variation, indicating strong performance in this extremely volatile landscape. The MV (S&P 500) model exhibits especially low volatility relative to the others, per the Markowitz risk-minimising philosophy and actually conservative nature of the approach. This great uncertainty (Bitwise crypto) is, of course, huge in all models, and avoiding this is undoubtedly more the wild nature of cryptos.

Even though the S&P 500 volatility is much higher, as quantified by the previously mentioned metrics, one would prefer the risk-adjusted return for EVAR over SH, which is also demonstrated by a markedly superior Sharpe ratio. In any case, Bitwise crypto believes that the highest Sharpe ratio for cryptocurrencies is with the EVAR model. It also validates that some more creative entropy-based schemes can be robust in high-variation markets.

Markowitz (1952) posited that diversification should theoretically reduce risk without compromising returns. Nevertheless, Ledoit and Wolf (2004) and Bera and Park (2008) contend that conventional methodologies, such as Markowitz’s, may underestimate risk when applied to highly diversified portfolios or in markets characterised by elevated volatility.

The MI and EVAR models indicate that using these measures can result in better risk-adjusted returns. This aligns with the principles of information theory described by Shannon (1948), which maintain that information is an effective reducer of uncertainty. Zhou et al. (2013) investigated entropy; Granger and Lin (1994) capture non-linear risks and portfolio diversification better (and are thus more relevant to cryptocurrency markets).

This, in turn, suggests that entropy-type models like the EVAR model can offer an additional buffer against tail risk. As Dionisio et al. (2004) argued, introducing abnormal distributions and asymmetric risks makes entropy-based risk measures more suitable for these new types of financial markets.

As we conclude, this visual figure shows empirical proof that the traditional Markowitz approaches are still fundamental; however, including mutual information and Tsallis entropy-based techniques can amplify portfolio management under complex and ever-challenging market environments.

Trajectory of gain-loss ratio (GLR) (Figure 2). The correlation-corrected measure of relative simple diversity of multiple kinds of portfolio optimisation models (0%, 10%, 20%, and 30% constrained in return) returns the result shown in the last figure. Some widely used models are the Markowitz (MV) model, entropy-Sharpe model, MV-Sharpe model, many variations of the mutual information (MI) model, and a combined model. This includes those based on traditional approaches and those based on information.

In the case of no return constraints—that is, a 0% return constraint denoted as blue bars—the average GLR obtained is generally lower. This implies a more cautious balance of future benefits and risks. Return constraints are represented by green, red, and yellow bars at 10%, 20%, and 30%, respectively. With the return constraint, the tendency of the GLR is to increase, especially in the ‘combined’ model at the 30% level.

The values show that mutual information-based models (MI and its derivatives) and entropy-based models have relatively stable diversity among several return constraints. The LR usually falls between 0.33 and 0.37 in most situations. Please note that this implies the possibility of further market environment-independent risk mitigation, in parallel with higher returns to follow additional capital and security of portfolios.

This is emphasised by the enormous rise in GLR (0.90) seen in the combined model when constrained to a return of 30%—resulting from a dangerous or very bullish strategy that seeks profits at the expense of downside protection. This model might be justified as long as it is part of an optimistic market environment that may facilitate enhanced gains. Nevertheless, it is less tactful in a volatile market environment because diversification is thrown out the window.

These insights are based on criticisms of standard Markowitz models as set forth by Sharpe (1964) and reaffirmed in later works such as Ledoit and Wolf (2004), which pointed out that Markowitz models, if naively implemented in complex markets, may tend to underestimate risks. An alternative way of capturing the details of risk dynamics is with entropy measures. This is well grounded in information theory (Bera & Park, 2008). Their approach emphasised the importance of adaptive investment diversification, whereby investors factor in the appropriate potential losses and gains to balance their portfolios optimally.

4.2. Analysis of the Efficient Frontier

4.2.1. Basics of Portfolio Theory

Figure 3 shows the classical Harry Markowitz modern portfolio theory portfolio optimisation based on a variance minimisation for a given return (Markowitz, 1952). These two laws demonstrate how the best portfolio appears from the perspective of both risk and returns, as evidenced by the tangency point and global minimum. Simple models cannot describe the information stored in the asset itself as the returns of a cryptocurrency are non-Gaussian (Mandelbrot, 1963).

The efficient frontiers of the S&P 500 and the Bitwise 10 show us some important aspects of portfolio management through an information-theoretic lens. Following Urquhart (2016, 2017), one can think of each portfolio as an information channel, considering that the expected return acts as a signal and standard deviation contributes to noise. In the same way as the Sharpe in finance, it is meant to identify an optimal signal-to-noise ratio. This ratio underlines the importance of efficiently transmitting financial information (Martin & Nagel, 2022).

Concerning the concepts of diversification and risk, the S&P 500 index has a tighter and more diffuse efficiency frontier compared with the Bitwise 10. This gap indicates that the reduction in risk for non-traditional assets is against a larger backdrop of diversification and a less risky profile for traditional offerings, as per Markowitz’s seminal portfolio diversification theory (Markowitz, 1952).

On the Bitwise 10 Crypto Index, this would be like driving up a cliff. This would imply an increased expected return, appealing to investors willing to take on risk. This is also consistent with Le Tran and Leirvik (2020), which show that cryptocurrencies can experience significant volatility yet have a considerable liquidity premium.

This can be due to their dynamic impact on market adaptation (Urquhart, 2018)—economic and financial problems also influence cryptocurrencies. This analysis can be carried out by adaptive trading systems utilising entropy, making them a tool to capitalise on market regime shifts. This was titled the adaptive markets hypothesis (Lo, 2004). Guided by a task-tailored mindset, Vidal-Tomás (2021) investigates the potential for entropy measures in detecting cryptocurrency market dynamics.

This highlights the fundamental differences in risk characteristics between traditional assets and cryptocurrencies. It underscores the importance of mapping information theory in constructing and managing portfolios. Investors and portfolio managers can use this information to realign their strategies based on the risk appetite and return profile and adapt investment decisions to changing market dynamics and new theories (Chokor & Alfieri, 2021).

4.2.2. Comparison of Approaches

Figure 4 highlights the differences between these two largest cryptocurrencies in their relative response to that of the S&P 500 by applying measures of entropy and mutual information conditioned on their underlying market states as controls. A more ordinary diversified index, the S&P 500, offers its upward slope in exchange for the risk taken. MI and entropy measure this. The Bitwise 10 index takes faster action to those same measures, probably because it is far riskier and far more concentrated than the asset class as a whole.

In this respect, it may also be concluded that the applicability of Tsallis entropy and mutual information in cryptocurrencies would be useful, as it would be able to correlate more non-linear dependencies and complex market dynamics, which can be observed, mostly, in cryptocurrencies. As commented by Tsallis (1988), at first, generalised entropy gives better insight for systems presenting non-equilibrium probability distributions, which is frequently true in the cryptocurrency markets.

Once we fit alternatives in this system of measures constructing efficiency frontiers, the unique risk–return relationship alternatives enable standard investment assets to become much more straightforward. This methodological improvement provides a more consistent picture of these assets’ market circumstances and associated risk profiles. This methodological improvement benefits portfolio managers and financial theorists, as it not only better predicts the risk level of an investment but is also a more accurate model than previous methods with respect to describing the reality of financial markets—particularly important in rapidly shifting cryptocurrencies.

4.2.3. Deep Border Analysis

Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 are a further deep dive into portfolio optimization using various formulations of mutual information (MI) and Tsallis entropy. The novel methodology we refer to here is different than the classic Markowitz variance–covariance methodology. The numbers show that these novel metrics do well in portraying the complex interdependence of the modern financial markets. This functionality is particularly applicable to hot assets like cryptocurrency.

Specifically, Figure 5 shows the sum of Tsallis entropies per each asset. That introduces a logic in which each individual part of the portfolio adds diversity to the overall mix independently. This idea fits in with the work of Shannon (1948) on the ability of information systems to reduce uncertainty. The extremes of mutual information effects of minimum and maximal entropy on the effective frontier are explored in Figure 6 and Figure 7. These models are justified by Mandelbrot’s (1963) theory of extreme diversification and risk. According to this theory, heavy-tailed distributions and anomalies must be studied beyond the primary dispersion measures.

The entropy of the joint (Figure 8) and geometric mean (Figure 9) show a smoother and more synergistic picture. These numbers should give you an impression of how different assets can work against one another to achieve the maximum from some point at a point of risk and return. Tsallis (1988) generalizes the classical Boltzmann–Gibbs statistics, which are only an approximation in chaotic dynamic systems, to a more general PDF—an idea with further support in the diversity literature as a method for diminishing uncertainty (See Cover & Thomas, 2006).

Each point on the curve can be interpreted as an optimal portfolio corresponding to a specific risk level. The curves are more regular and demonstrate a constant evolution of risk and return. This reflects the diversification and relatively stable correlation of the S&P 500 assets.

Through the principles of contemporary portfolio theory, as articulated by Markowitz in 1952, the efficient frontier represents a set of portfolios that optimise returns for a given level of risk or reduce risk for a given level of return. The discrepancies observed between the various curves may be attributed to using disparate mutual information measures to assess correlations between assets. This observation suggests that some measures may better capture aspects of non-linear dependence than conventional measures such as Pearson correlation.

Below (for the Bitwise 10 crypto index), we plot efficient frontier curves that could have more volatility than those with S&P 500 constituents. Perhaps this is due to the high volatility and low correlation of crypto assets. The curvatures also suggest that the addition of cryptocurrencies to optimal portfolios has the potential of higher returns at higher levels of risk. Sure enough, this is consistent with the general idea that cryptocurrencies are a risky but potentially rich type of investment.

These differences between curves might explain how various mutual information measures capture the complications and evolutions of cryptocurrency interrelationships. This analysis is essential to portfolio management in such volatile and non-linear markets.

This conclusion is supported using mutual information to enhance correlation and risk assessments. Joe’s (1989) research examines non-linear measures of association as evidence. To provide a more specific context for cryptocurrencies, research conducted by Corbet et al. (2020) on analysing dynamic correlations in cryptocurrency markets may also serve as an appropriate framework for comparison.

We propose that the benefits of each optimisation method are contingent on the asset characteristics to be optimised, as well as the target returns and risk. For investors looking to optimise returns through calculated risk, entropy optimisation can be a beautiful approach to take, especially when applied to volatile assets like cryptocurrencies. But none of them optimally train to the average of smoothing variations to maximise diversification, so for investors who want a balanced, consistent way of reducing variation vs. maximising diversification, geometric mean and joint entropy are excellent candidates. This whole bit is for diversified portfolios, especially crypto, and cannot include traditional equities like the S&P 500.

4.2.4. The Influence of 2020 on Efficient Frontiers

The year 2020 will forever be remembered for the pandemic caused by the novel coronavirus (SARS-CoV-2) and other unprecedented events that affected essentially all countries around the world, in addition to the economic and financial crises that markets will probably face. To understand these changes more deeply, we applied a Tsallis entropy and mutual information-based methodology for 2020. This gives a strong alternative to the traditional variance–covariance method of Markowitz (1952) that is used to study the risk–return profile of portfolios in a non-linear context, as, for example, in the case of cryptocurrencies. The effective frontiers of the S&P 500 and the Bitwise 10 index in 2020 highlight several fundamental differences between traditional assets and crypto assets. Building upon the seminal work of Urquhart (2018), each portfolio can be conceptualised as an information channel, wherein the expected return functions as the signal and the possible contributions to the noise. In this context, Tsallis entropy provides a robust framework for measuring the efficiency of financial information transmission (Martin & Nagel, 2022).

As depicted in Figure 10a, the efficient frontier of the S&P 500 describes a specific portfolio that is optimally maximised for risk (standard deviation) and expected return. We use a colour gradient for Tsallis entropy levels (q = 1.5), which represents portfolio diversity and stability in this context. Adding the year 2020 to the analysis shows that the market dispersion has risen, marking the effect of macro shocks and unexpected shocks on traditional assets.

One interesting segment of the data shown in Bitwise 10 can be viewed in Figure 10b and its efficient frontier. This frontier shows greater volatility and higher potential returns compared to the S&P 500. The results are similar to Le Tran and Leirvik (2020), who found that cryptocurrencies have a sizable liquidity premium yet are still susceptible to tail risks. Tsallis entropy has shown to be a potentially useful measure to describe the complexity of market dynamics and non-linear dependencies (Tsallis, 1988).

In the analysis, adding in 2020 highlights some stark differences between conventional assets and cryptocurrencies. The S&P 500 index, maintaining a stable frontier, shows effective asset diversification even during crises. On the contrary, the enhanced reactivity of the Bitwise 10 index affirms the adaptive markets hypothesis (Lo, 2004).

As these numbers demonstrate, mutual information can quantify relationships between assets where the relationship is non-linear. Unlike classical Pearson correlation, mutual information can uncover non-linear relationships, as illustrated by Bariviera and Merediz-Solà’s (2021) study within the domain of cryptocurrency markets.

5. Conclusions

We define a new framework for the portfolio optimisation problem for constructing optimal asset portfolios, substituting the standard metrics of variance and covariance with Tsallis entropy and mutual information (MI). These tools enable a more accurate risk judgment on the one hand and a more profound insight of non-linear interactions among them in the context of assets on the other hand. The analysis uses daily returns for S&P 500 components and for the Bitwise 10 cryptocurrency indices from April 2019 to May 2024. As there is an exception for the year 2020 being hugely influenced by shocks caused by the severe acute respiratory syndrome (SARS-CoV-2) pandemic, it was deliberately omitted from the study to avoid distortions caused by this exceptional crisis. In the second phase, an efficient frontier for 2020 was formed in order to see how models are working in the case of extreme shocks.

The performance analysis indicates that Tsallis entropy/mutual information-based models outperform the traditional models based on mean-variance (MV) in several aspects. Compared to others, MI models and their variants deliver far higher returns, stronger overall diversification, and risk-adjusted performance (via higher Sharpe ratios and greater Jensen alphas). Additionally, it has been shown that the application of value-at-risk entropy (EVAR) for loss is particularly useful for managing extreme risks and providing an additional level of protection against the tail of the loss distribution.

Frontier analysis works well to identify the important differences considered among the indices studied. Our MI and EVAR S&P 500 models produce a relatively stable efficient frontier showing efficiency even in moderate market conditions only in 2020. In contrast to cryptocurrencies, the efficient frontiers of Bitwise 10 are more volatile with higher realistic returns, which at the end of the day leads to the collection of the risk accumulated as animal spirits dominate the dynamics of digital assets. To create an additional efficient frontier for 2020, this year was included in the analysis. This illustrated the robustness of the MI and EVAR models against extreme shocks but also validated their application in very unstable environments.

The study does have some limitations, the authors acknowledge. Finally, the analysis was limited to two indices (S&P 500 and Bitwise 10), which can limit the scope of the conclusions around the generalisability of the models. Nonetheless, by including the year 2020 and building a risk–return efficient frontier for the most extreme period shocks, the proposed approaches have been shown to perform robustly under highly volatile environments. Investors in the disposal of the asset class behave in accordance with idiosyncratic events during crises, which is supported by contrasting dynamics among different asset classes.

A future extension of this work could expand this analysis to a broader array of assets and other crisis situations. In contrast, this study is already an important addition to the literature by introducing advanced tools such as Tsallis entropy and mutual information that mitigate the pitfalls of classical methods. Adopting these innovations is critical in tackling existing issues in portfolio management in an increasingly complex and interlinked financial environment.

To summarize, there is some room to explore the implementation of Tsallis entropy and mutual information by funds and institutional investors in their portfolio management process. Not only are these tools capable of modelling non-linear dependencies and asymmetric distributions more correctly, but they also make extreme risk more manageable and provide more opportunities for diversification and resilience in complicated and volatile market environments.