Can Including Cryptocurrencies with Stocks in Portfolios Enhance Returns in Small Economies? An Analysis of Fiji’s Stock Market

Abstract

The market for digital assets, and more specifically cryptocurrencies, is growing, although their adoption in small island countries remains absent. This paper explores the potential benefits of integrating cryptocurrencies into portfolios alongside stocks, with a focus on Fiji’s stock market. This is the first study on a small market like Fiji, which emphasizes the role of cryptocurrencies in portfolio management. We analyze the outcomes (returns and risks) of combining cryptocurrencies with stocks using 12 different techniques. We use monthly stock returns data of 18 companies listed on the South Pacific Stock Exchange from Aug-2019 to Jun-2025 (71 months) and nine cryptocurrencies from Sept-2019 to Jun-2025 (70 months). Our main analysis shows that only one cryptocurrency, albeit with a small exposure, consistently appears in the stock-cryptocurrency portfolios in the 12 methods. Using the return-to-risk ratio across methods as a guide, we find that the stocks-cryptocurrencies portfolio based on EQW, MinVar, MaxSharpe, MinSemVar, MaxDiv, MaxDeCorr, MaxRMD, and MaxASR offers better outcomes than the stock-only portfolios. Using high returns as a guide, we find that six out of 12 methods (EQW, MaxSharpe, MaxSort, MaxCEQ, MaxOmega, and MaxUDVol) support the stocks-cryptocurrencies portfolios. Portfolios satisfying both conditions (high return-risk ratio and high return) are supported by the EQW and MaxSharpe portfolios. The consistency of assets in both stock and stock−cryptocurrency portfolios is further confirmed by 24-month out-of-sample forecasts and Monte Carlo simulations, although the latter supports small exposures in two out of the nine cryptocurrencies. Based on the results, we conclude that a small exposure to certain cryptocurrencies can strengthen diversification and improve potential returns.

1. Introduction

Investment analysis based on portfolio optimization methods has received significant attention in academia, especially drawing insights from and extending the pioneering work of Markowitz (1952a, 1952b, 1991, 2010) on optimization techniques. Earlier studies have mainly focused on traditional assets such as equities, bonds, and commodities markets (Yan & Garcia, 2017; You & Daigler, 2013; Salo et al., 2024), while recent studies have begun to explore the inclusion of digital assets, specifically cryptocurrencies, in portfolio optimization (Almeida & Gonçalves, 2022). The rise of blockchain technology and cryptocurrencies has generated considerable interest among academics, researchers, and analysts. Despite the highly volatile nature of cryptocurrencies, several studies have noted that including cryptocurrencies can improve portfolio outcomes (Ma et al., 2020; Symitsi & Chalvatzis, 2019; Li et al., 2021; Platanakis & Urquhart, 2020). However, studies noting promising results from extending traditional assets with cryptocurrencies mainly focus on developed markets and emerging market economies.

The role of blockchain technology and cryptocurrencies in small economies has received mixed opinions. On the one hand, studies like Jutel (2021, 2022, 2023) offer a critical view of the role of blockchain technology and cryptocurrencies, particularly linking technology to colonial imperialism and computational capitalism. A few studies document positive views on the potential of blockchain technology and cryptocurrencies in small island economies (Waller & Johnson, 2022; Kwok & Treiblmaier, 2024). Waller and Johnson (2022) argue that cryptocurrencies have become increasingly important as a solution to the financial and economic challenges of developing countries. Moreover, the technology (cryptocurrencies) can offer a pathway for small island developing states (SIDS) to overcome poverty, realize economic transformation, and engender financial inclusion. Kwok and Treiblmaier (2024) provide a comprehensive discussion on the potential of blockchain technology based on a dynamic capability framework. Their study analyzed the role of blockchain technology in overcoming systemic constraints and strengthening small economies. Within this framework, they argue that blockchain can support higher-order (processes) and lower-order (operations) and that the benefits of blockchain can extend to business and investment, financial system enhancement, and regulatory improvements, which are important for economic development, efficiency, competitive advantage, and diversification.

Despite the potential benefits of cryptocurrencies, some analysts note that many countries where participation in cryptocurrency (especially Bitcoin) trading is restricted or constrained also have huge debts with the International Monetary Fund and/or the World Bank (Gladstein, 2021; Batten, 2025). In the case of small island economies in the Pacific, the adoption of digital ledger technology and cryptocurrencies remains significantly low. Nevertheless, there are a few small countries like Vanuatu and Tuvalu that have embraced blockchain and crypto trading1 with their vision to become a ‘crypto’ paradise and realize financial transformation. However, other small island countries like Samoa and Tonga have declared cryptocurrency a risky proposition, and others like Fiji have declared it illegal to trade in cryptocurrencies using domestic funds.2

The slow adoption of blockchain technology and cryptocurrency for investment purposes is exacerbated by a lack of necessary infrastructure, education and regulations. A recent report by UNDP (2024) on Small Island Digital States: How digital can catalyze SIDS development, invoking SDG 9—Industry, innovation and infrastructure—highlights that the interest to learn about blockchain technology in small economies is more common among young populations. In addition, the report documents that the slow pace of progress and infancy of technologies like artificial intelligence (AI), blockchain, and Big Data in SIDS are due to the absence of formal and/or ethical standards that could potentially support greater innovation and digital transformation, although these are areas worth prioritizing.

Against this background, we situate the study from a financial and investment perspective on the small island economy of Fiji, which has a more developed stock market than other small island economies in the region. Nevertheless, Fiji’s stock market, which started in 1979 (about 46 years ago), remains less developed. While there have been some basic upgrades in terms of websites and information on stocks, trading continues to rely on traditional broker services, thus restricting real-time trading and making the market less efficient. Domestic stock price appreciation (relative to the inflation rate) remains slow. In addition, due to low participation in the stock market, daily trading volumes and trading frequency remain low. The market is generally small, and sophisticated trading strategies, including short-selling (trading of derivatives like options and futures), are largely impractical. Moreover, in terms of investment opportunities, especially in highly liquid assets, the scope is limited. However, integrating Fiji’s investment landscape with opportunities to trade digital assets can support financial development and complement investment in conventional assets like stocks, given that the market for digital assets like cryptocurrency is global, operational 24/7, and highly liquid, allowing for sophisticated trading strategies (including options and futures).

Subsequently, considering the current characteristics of Fiji’s stock market, analyzing and comparing stocks-only and stocks-cryptocurrencies portfolios would be best analyzed from a long-only perspective.3 Our study addresses the following research question: Does the inclusion of cryptocurrencies with traditional assets (stocks) in portfolios offer superior outcomes? To answer this question, we compare portfolios based only on stocks with those based on both stocks and cryptocurrencies. We analyze portfolios using 12 different methods: (1) naïve or equally weighted (1/N), (2) minimum variance (MinVAR), (3) mean-variance based on Sharpe (MaxSharpe), (4) semi-variance (MinSemVar), (5) target return based on Sortino (MaxSort), (6) certainty equivalent (MaxCEQ), (7) Omega (MaxOmega), (8) ratio of upside-to-downside volatility (MaxUDVol), (9) diversification (MaxDiv), (10) decorrelation (MinDeCorr), (11) (8) robust maximum diversification (MaxRMD), and 12 adjusted Sharpe (MaxASR). Additionally, we examine the general stability of the portfolios from the stocks-cryptocurrencies sample using 24-month out-of-sample forecasted asset prices and subsequent returns.

Specifically, following Markowitz’s (1952a, 1952b) theory of portfolio optimization, we examine whether there are any benefits to including cryptocurrencies with traditional assets like stocks. Although previous studies have conducted portfolio analyses based on Fiji’s stock market (Kumar & Stauvermann, 2022; Kumar et al., 2022, 2024), they are restricted to conventional assets (stocks and bonds). We extend the analysis to incorporate cryptocurrencies into the portfolio analysis. Our paper contributes to the literature on small economies and cryptocurrencies in the following ways. We present a comprehensive portfolio analysis based on multiple approaches and criteria, comparing portfolios that comprise only stocks with those that comprise both stocks and cryptocurrencies. Noting that trading of cryptocurrencies is not permitted in Fiji, this paper offers important information to policymakers, financial regulators, and educational institutions on the possibility of promoting education on cryptofinance and investment, both at the individual and institutional levels. Furthermore, we present in-sample and out-of-sample return forecasts and examine the respective portfolios to offer additional insights. In addition, the analyses presented in this study can be easily extended to more sophisticated methods and alternative assets.

Following a recent study by Kumar et al. (2024), we consider 18 stocks of firms listed on Fiji’s South Pacific Stock Exchange.4 In our digital assets, we consider nine crypto assets. We consider Bitcoin (BTC) and Ethereum (ETH), since they are the top two tokens by market capitalization and are often considered in the literature (Symitsi & Chalvatzis, 2019; Charfeddine et al., 2020; Bouri et al., 2020; Schellinger, 2020; Platanakis & Urquhart, 2020). Moreover, BTC has the features of a store of value, and it is the most decentralized digital currency (sometimes also called digital gold (Baur et al., 2024)); and ETH has received significant attention among spot Exchange-Traded Funds (Krause, 2024), and several stablecoins have been built on the Ethereum network. Other tokens like XRP (Ripple), ADA (Cardano), ALG (Algorand), XLM (Stellar Lumen), QNT (Quant Network), HBAR (Hedera Foundation), and XDC (XDC Network) were chosen because they are more aligned with the ISO20022 standards (https://www.iso20022.org/about-iso-20022, accessed on 9 July 2025) (see also Huffman, 2025; Tangem Team, 2025; Kalash, 2025), offering interoperability in the financial system.5 These tokens are noted to be less resource-heavy (in terms of energy use and carbon footprint),6 and they can be integrated into the central bank payment system, thus supporting efficient cross-border payments and supporting financial innovation and transformation of the traditional financial landscape. Notably, XRP facilitates faster and more efficient cross-border payments and money transfers, supporting financial institutions in terms of cheaper transactions (compared to traditional legacy systems); XLM focuses on small payments, supporting people in unbanked areas to carry out low-cost financial transactions; ADA supports a safe and efficient environment for building decentralized Apps (dApps) and smart contracts, improving older blockchains; HBAR provides high-speed security and fairness, supporting building and running decentralized dApps in finance, supply chain, gaming, and social networking; XDC is a blockchain platform for secure and efficient cross-border transactions, trade finance, and supply chain management; ALG network handles a high volume of transactions quickly and cheaply, making it ideal for financial services, decentralized finance (DeFi) and asset tokenization; and QNT was developed to improve the interaction between different. Although there are many cryptocurrencies, we believe that these nine are potentially critical for financial innovation based on their utility, use case, presence in the market, and market capitalization. Moreover, with some cryptocurrencies having close links with ISO20022, they are likely to influence the global financial system and receive greater regulatory support in the future.

The remainder of this paper is organized as follows. Section 2 provides a comprehensive literature review, with a particular focus on cryptocurrencies. Section 3 outlines the methodology, optimization procedures, and detailed description of the data used in the study. Section 4 is divided into three sub-sections: Section 4.1 presents an analysis of portfolios consisting solely of stocks and those incorporating both stocks and cryptocurrencies. Noting the results from Section 4.1, in Section 4.2, we examine the robustness and stability of the portfolio comprising stocks and cryptocurrencies using 24-month out-of-sample forecasted data. In Section 4.3, we further confirm the results using Monte Carlo simulations. Finally, Section 5 concludes the paper by integrating limitations and possible directions for future research.

2. Literature Review

Among the different approaches to investment analysis, portfolio optimization is a widely applied method. Portfolio optimization dates back to several works by Markowitz (1952a, 1952b, 1991, 2010), Sharpe (1966), and others (Elton & Gruber, 1997; Maillard et al., 2010; Booth & Broussard, 2016; Surtee & Alagidede, 2023), and the method has evolved significantly with more complex approaches (Bielstein et al., 2023; Ghanbari et al., 2024). Earlier studies on optimization techniques have focused mainly on traditional assets (stocks, bonds, and commodities). However, with emerging innovations in financial technology, cryptocurrencies have become of interest, especially among investors and fund managers (Aliu et al., 2022). So much so that many Exchange-Traded Funds (ETFs),7 and companies are holding specific cryptocurrencies on their balance sheet.8

A few studies have examined gains from portfolios comprising only cryptocurrencies (Platanakis et al., 2018; Schellinger, 2020; Čuljak et al., 2022). Platanakis et al. (2018) use weekly data of four cryptocurrencies (BTC, LTC (Litecoin), XRP, and DASH (Dash)) between February 2014 and January 2018, and portfolio methods like 1/N, Markowitz (1952a) mean-variance (Sharpe ratios), and omega (gain-loss) ratio (Keating & Shadwick, 2002). They find very little difference in performance between portfolios based on 1/N and Sharpe ratios, thus supporting the idea that naïve diversification is equally preferable to optimal diversification. In another study, Schellinger (2020) considers 20 cryptocurrencies, separating them into coins (medium of exchange) and tokens (security or utility tokens). The results showed that, based on the Sharpe ratio, both coins and tokens performed poorly. Čuljak et al. (2022) analyze 65 cryptocurrencies from August 2019 to February 2020, noting that five of six portfolio strategies perform better after including sector-based (financial, exchange, and services) cryptocurrencies.

Studies have shown that portfolios comprising traditional assets and cryptocurrencies tend to provide better returns, and in some instances, cryptocurrencies can act as a haven and hedging instrument for investors (Aliu et al., 2021; Petukhina et al., 2021; Letho et al., 2022; Almeida & Gonçalves, 2022). Noting the prominence of BTC, a few studies have included only BTC with traditional assets (stocks, bonds, and commodities). An instance of a BTC-only study is Symitsi and Chalvatzis (2019), who explore the effect of a single cryptocurrency, BTC, on benchmark and multi-asset portfolios comprising currencies, gold, oil, and stocks, and extend these with real estate and bonds, respectively. Additionally, the authors examine the out-of-sample performance of BTC within portfolios of various asset classes and strategies to check the stability of the outcomes. Overall, it was noted that BTC offered significant diversification benefits to the portfolios, and this was more notable for portfolios incorporating commodities. Li et al. (2021) show that BTC has an enormous tendency to improve an investor’s risk-return profile (and the Sharpe ratios) when it is combined with investable assets like stocks, fixed income, and commodities.

The inclusion of multiple cryptocurrencies with traditional assets has been noted to provide better outcomes, especially in terms of diversification. To examine this, Corbet et al. (2018) analyze returns of both cryptocurrencies (BTC, XRP, LTC) and other returns (MSC GSCI total return index, the USD broad exchange rate, the S&P500 Index, the COMEX gold price, VIX, and the Markit ITTR110 index). Their analysis shows that cryptocurrencies can offer diversification benefits to investors with short investment horizons. Considering the cryptocurrency index (CRIX), Brauneis and Mestel (2019) apply the traditional Markowitz mean-variance framework to assess risk-return benefits of cryptocurrency portfolios. They use daily data of the 500 most capitalized cryptocurrencies from January 2015 to December 2017. The study notes that combining cryptocurrencies enriches the set of ‘low’-risk cryptocurrency investment opportunities. They note that the 1/N portfolio outperforms a portfolio of single cryptocurrencies when compared with the Sharpe ratio and certainty-equivalent returns.

Charfeddine et al. (2020) consider two cryptocurrencies (BTC and ETH) and conventional assets (S&P500, gold, and oil) over the period July 2010–October 2018, for diversification and hedging. They conclude that while BTC and ETH are suitable for diversification, they are not suitable for hedging. Moreover, the relationship between the two asset classes can be affected by external factors. However, Akhtaruzzaman et al. (2020) analyze BTC data from August 2011 to November 2018 with 11 industry sectors and conclude that BTC offers efficient hedging mechanisms for a broad number of industrial sectors and bonds. Ma et al. (2020) is yet another study that mixed stocks and five cryptocurrencies (BTC, ETH, XRP, BCH (Bitcoin Cash), and LTC) from November 2015 to November 2019—separating the stocks between technology ((MSFT, AAPL, GOOGL, AMZN and FB) and traditional assets (Berkshire Hathaway Inc., Omaha, USA (BRKa), JPMorgan Chase & Co., Johnson & Johnson (JNJ), Procter & Gamble Company (PG), and Visa Inc Class A)—to investigate the potential for diversification. The authors analyzed the portfolios by optimizing the Sharpe ratio and expected returns based on mean-variance optimization (Markowitz, 1952a). The results show that diversification increased returns in most cases and reduced portfolio volatility in all portfolios, including cryptocurrencies. Moreover, their analysis showed that the addition of multiple cryptocurrencies to a portfolio provides enhanced results for diversification, and it was noted that ETH provided a better diversification opportunity than BTC.

Bouri et al. (2020) analyze five cryptocurrencies (BTC, ETH, LTC, XLM, and XRP) in the Asia Pacific and Japan equity markets. They note that BTC and LTC are suitable for hedging, diversification benefits from cryptocurrencies are permissible in the hedged compared to unhedged (stocks) portfolios, and although they find that BTC and ETH are the best choices in the Japan and Asia Pacific markets, investors interested in cryptocurrency-stock portfolios should also consider other large cryptocurrencies. Matkovskyy et al. (2021) formulate portfolios based on the top 10 cryptocurrencies (BTC, ETH, XRP, BCH, EOS (EOS network has been rebranded as Vaulta), LTC, BNB (Binance Coin), USDT (Tether), XLM, ADA) in terms of capitalization (from 7 June 2019) and indices like S&P600, S&P400, and S&P100. The study noted that only USDT enhanced stock returns in the 250-day time horizon; however, when excluding USDT from the sample, only BTC, ETC, and LTC improved stock returns in the longer time horizon. Petukhina et al. (2021) investigate the benefits of adding cryptocurrencies to well-diversified portfolios of conventional financial assets for different types of investors (risk-averse, return-maximizing, and diversification-seeking) trading at different frequencies (daily, weekly, or monthly). Their results show that out of 7/52 cryptocurrencies contributed to enhancing the risk-return profile of portfolios, although the gains are conditional on investor objectives and monthly rebalancing.

3. Methodology

This section is organized into two sub-sections. The first sub-section introduces the foundational concepts and models used in the study, and the second details the data employed for the analysis, providing a comprehensive overview of the study’s dataset.

3.1. Basic Knowledge of the Utilized Portfolio Models

In this section, we discuss various techniques used in this study to construct portfolios from the perspective of portfolio theory. We assume long-only portfolios in all cases, hence constrain the weights to be non-negative (${\omega }_i\ge 0$. This assumption is practical in the case of small markets like Fiji, given that the stock market is still developing, training volumes and frequency remain low, and access to crypto asset trading is prohibited. Moreover, we compute the monthly returns of each asset on a continuously compound basis as $r_t=\mathit{ln}\left({\displaystyle \frac{p_{t_i}}{p_{{\left(t-1\right)}_i}}}\right)$ where $p_{t_i}$ and $p_{t_{i-1}}$ are the prices of asset $i$ at month $t$ and $t-1$, respectively. For optimization, the respective asset returns and variances are annualized. We consider the following methods for the portfolio.

1. Equally Weighted (Naïve Diversification): This approach is basic, in that we assume equal weights across assets (securities), assuming the assets are uncorrelated (c.f. Demiguel et al., 2009; Platanakis et al., 2018). Accordingly, the weights are determined as follows:

$${\omega }_{EQW}={\displaystyle \frac{1}{N}}$$

(1)

where N is the total number of assets (i.e., securities).

2. Minimum variance (MinVar): In the minimum variance portfolio, we derive the weights by minimizing the variance (standard deviation) of the portfolio. This portfolio relies on the covariance matrix; hence, it can be affected by estimation errors. Moreover, the weights may be concentrated in a few dominant assets (Clarke et al., 2006; Haugen & Baker, 1991). To derive the MVP weight, we minimize the following program:

$$\min \left\{{\sigma }_p=\sqrt{\omega \Sigma \omega }|,{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(2)

where, ${\sigma }_p$ is the portfolio standard deviation and $\mathsf{\Sigma }$ is the $N\times N$ covariance matrix of the asset returns; and the constraints are: ${\mu }_p$ (the portfolio returns) computed as the sum product of each asset’s respective weights ${\omega }_i$ and its annualized return $r_i$; ${\sum }_{i=1}^N{\omega }_i=1$ (assuming full allocation of the wealth), and ${\omega }_i\ge 0$ (no short-selling).

3. Market portfolio (MaxSharpe): The weights are derived by maximizing the Sharpe (1966) ratio (c.f. Bai & Newsom, 2011), as follows:

$$\max \left\{SR={\displaystyle \frac{{\mu }_p-r_f}{{\sigma }_p}}|{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\sigma }_p=\sqrt{{\sum }_{i,j}^N{\sigma }_{i,j}{\omega }_i{\omega }_j};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(3)

where $SR$ is the Sharpe ratio, ${\mu }_p$, $r_f$ and ${\sigma }_p$ are portfolio mean, risk-free rate, and portfolio standard deviation, respectively.

4. Semi-variance (MinSemVar): Optimizing the Sharpe ratio entails both upside and downside volatility (symmetric normal distribution); however, given that investors are generally more concerned with downside risk (to preserve capital), the need for a target shortfall or semi-variance portfolio is recommended (Harlow & Rao, 1989; Estrada, 2008; Jarrow & Zhao, 2006; Markowitz, 2010; Wang & Yan, 2021; Kumar et al., 2022). Accordingly, the semi-variance portfolio is set up as

$$\min \left\{{\overline{\sigma }}_D|{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\sigma }_p=\sqrt{{\sum }_{i,j}^N{\sigma }_{i,j}{\omega }_i{\omega }_j};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(4)

where ${\overline{\sigma }}_D=\sqrt{{\displaystyle \frac{{\sum }_{t=1}^T\mathit{max}{\left({\mu }_p-{{\mu }_p}_t,0\right)}^2}{T-1}}\times 12}$ is the downside volatility; ${{\mu }_p}_t=$ monthly negative return at each time period $t$, ${\mu }_p$ is the average monthly portfolio returns over the total observation (months), $T$.

Moreover, the above measures may face difficulties in providing robust and correct rankings when portfolio returns are not normally distributed (Smetters & Zhang, 2013; Platanakis & Urquhart, 2020), and they depend on the first two moments (mean and standard deviation). Alternative measures for portfolio comparisons, such as the Sortino ratio (Sortino & van der Meer, 1991) and the Omega ratio (Keating & Shadwick, 2002), are proposed by Platanakis and Urquhart (2020). Accordingly, we note the following:

5. Sortino: This measure maximizes adjusted returns, accounting for the target return and downside volatility (Sortino & van der Meer, 1991) (MaxSort). The program is defined as follows:

$$\max \left\{SORT={\displaystyle \frac{{\mu }_p-T_g}{{\overline{\sigma }}_{D_S}}}|{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\overline{\sigma }}_{D_S}=\sqrt{12\left({\displaystyle \frac{{\sum }_{t=1}^T\max {\left(\tau -{{\mu }_p}_t,0\right)}^2}{T-1}}\right)};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(5)

where $T_g$ is the annual target rate, $\tau$ is the monthly target rate, and ${\overline{\sigma }}_{D_S}$ is the downside volatility adjusted for the target rate.

6. Certainty equivalent (CEQ): The measure captures the riskless return that an investor would consider of equal utility as the risky return under evaluation, with a given risk aversion ($\gamma$). A higher $\gamma$, say $\gamma >+1$ could imply greater tolerance for risk. As noted by Petukhina et al. (2021), $\gamma =1$ yields the closed form of Markowitz’s (1952a, 1952b) optimization problem. Accordingly, we formulate the following program for maximization:

$$\max \left\{CEQ={\mu }_i-{\displaystyle \frac{\gamma }{2}}{\sigma }_i^2|{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\sigma }_p=\sqrt{{\sum }_{i,j}^N{\sigma }_{i,j}{\omega }_i{\omega }_j};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(6)

7. Omega: A measure defined as the ratio of average gain to average loss (Keating & Shadwick, 2002; Platanakis et al., 2018), with the advantage that the measure does not rely on strict assumptions on distribution (Platanakis & Urquhart, 2020). The portfolio is calculated as follows:

$$\max \left\{\mathsf{\Omega }\left(r_i\right)={\displaystyle \frac{{\int }_{\tau }^{+\mathrm{\infty }}\left[1-F(r_i\right]d{r}_i}{{\int }_{-\mathrm{\infty }}^{\tau }F\left(r_i\right)d{r}_i}}={\displaystyle \frac{E\left[{\left(r_i-\tau \right)}^{+}\right]}{E\left[\left(\tau -r_i\right)\right]}}={\displaystyle \frac{E\left(r_i\right)-\tau }{E{\left[\tau -r_i\right]}^{+}}}+1|{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(7)

where $r_i$ = random return of asset $i$; $F\left(r_i\right)$ is the cumulative density function, $\tau \ge 0$ is a threshold, which can be arbitrarily chosen (either equal to zero, the risk-free rate, or some arbitrary non-negative value (Kapsos et al., 2014).9

8. Upside-to-downside volatility (UDVol): This measure is computed as the ratio of upside variance to downside variance and generally complements the results of semi-variance and Sortino ratios, which mainly focus on downside volatility and target ratio. To maximize UDVol $\left({\sigma }_{\nu }\right)$, we set up the program as follows:

$$\max \left\{{\sigma }_{\nu }={\displaystyle \frac{{\sigma }_U^2}{{\sigma }_D^2}}|{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\sigma }_p=\sqrt{{\sum }_{i,j}^N{\sigma }_{i,j}{\omega }_i{\omega }_j};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(8)

where ${\sigma }_{\nu }$ is the ratio of upside volatility to downside volatility, adjusted for the target return, $\tau$; ${\sigma }_U^2=\left({\displaystyle \frac{{\sum }_{t=1}^T\max {\left({{\mu }_p}_t-\tau ,0\right)}^2}{T-1}}\right)$ and ${\sigma }_D^2=\left({\displaystyle \frac{{\sum }_{t=1}^T\max {\left(\tau -{{\mu }_p}_t,0\right)}^2}{T-1}}\right)$.

9. Diversified Portfolio (Div): Following Choueifaty and Coignard (2008), we set up and maximize the DP as

$$\max \left\{Div={\displaystyle \frac{{\omega }_{DP}\sigma }{\sqrt{{\omega }_{DP}^T\mathsf{\Sigma }{\omega }_{DP}}}}|{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\sigma }_p=\sqrt{{\sum }_{i,j}^N{\sigma }_{i,j}{\omega }_i{\omega }_j};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(9)

10. Decorrelation (DeCorr): is closely related to the minimum variance portfolio; however, instead of using a variance-covariance matrix, the strategy assumes that individual asset volatilities are identical but heterogeneous correlations, hence using a correlation matrix. The maximum decorrelation portfolio maximizes diversification ratio (Sharpe ratio) when all assets have equal volatility (risk and return) (c.f. Christoffersen et al., 2012; Burggraf, 2019; Ardia et al., 2017). Accordingly, the program is set up as follows:

$$\min \left\{DeCorr={\omega }^T\mathit{C}\omega |\mathit{C};{\sigma }_p=\sqrt{{\sum }_{i,j}^N{\sigma }_{i,j}{\omega }_i{\omega }_j};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(10)

where $\mathit{C}$ is the correlation matrix, and DeCorr is the ‘synthetic correlation’, which is minimized, subject to the given constraints.

11. Robust Maximum Diversified (RMD): We invoke the method of Bielstein et al. (2023), applied in Kumar et al. (2024), where diversified portfolio returns are based on unexplained variations between asset pairs. The robust maximum diversified programming problem is set up as follows:

$$\max \left\{MD={\omega }^{T}R\omega |R;{\sigma }_p=\sqrt{{\sum }_{i,j}^N{\sigma }_{i,j}{\omega }_i{\omega }_j};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(11)

where $R=\left[\begin{array}{ccc}0& \dots & 1-{\rho }_{1,N}^2\\ ⋮& \ddots & ⋮\\ 1-{\rho }_{N,1}^2& \dots & 0\end{array}\right]$, and $\rho =$ pairwise correlations between asset returns. Hence, ${\rho }_{\mathrm{1,2}}\in \left[-1,1\right]$ and $1-{\rho }_{\mathrm{1,2}}^2\in \left[0,1\right]$.

12. Adjusted Sharpe Ratio (ASR): Noting the plausibility of a non-normal distribution of returns, we invoke Pézier and White’s (2008) measure (ASR), which extends the conventional Sharpe ratio by accounting for the third and fourth moments, i.e., skewness and kurtosis. The ASR accounts for positive skewness and negative kurtosis by including a penalty factor for negative skewness and positive excess kurtosis. Accordingly, we set up the maximization problem as follows:

$$\max \left\{ASR=S{R}_i\left[1+\left({\displaystyle \frac{S_i}{6}}\right)S{R}_i-\left({\displaystyle \frac{K_i}{24}}\right)S{R}_i^2\right]|{\mu }_p={\sum }_{i=1}^N{\omega }_i{r}_i;{\sigma }_p=\sqrt{{\sum }_{i,j}^N{\sigma }_{i,j}{\omega }_i{\omega }_j};{\sum }_{i=1}^N{\omega }_i=1;{\omega }_i\ge 0\right\}$$

(12)

where, $S{R}_i$ is the Sharpe ratio, $S_i$ is the skewness and $K_i$ is the excess kurtosis of asset $i$.

3.2. Data Description

Initially, we compiled month-end stock price data from May-2019 to Jun-2025 (74 months) for 20 firms listed on Fiji’s South Pacific Stock Exchange (SPX, 2025; Kumar et al., 2024),10 and computed the monthly returns for each stock. However, because the share price of Pacific Green Industries Ltd. (PGI) (Sigatoka, Fiji) remained constant (FJ$1.08) throughout the sample period, and Sun Insurance Company Ltd. (SUN) was listed recently (15 August 2024), and we dropped the two stocks from the analysis. Moreover, Port Denarau Marina (PDM) was listed in mid-August (on 14 August 2019), with a share price of FJD1.31 and closed at FJD1.54 on 30 August 2019. We used these two prices to compute the month-end return data for Aug-2019. Hence, the final month-end logarithmic returns of 18 listed companies (APP, ATH, BCN, CFL, FBL, FHL, FIL, FMF, FTV, KFL, PBP, PDM, RBG, RCF, TTS, VBH, and VIL) (c.f. Kumar et al., 2024), are from Aug-2019 to Jun-2025, resulting in 71 data points. For the digital assets, we obtained month-end price data in USD for nine cryptocurrencies (BTC, ETH, ALGO, ADA, HBAR, QNT, XLM, XDC, and XRP) from Yahoo Finance (https://finance.yahoo.com/) (accessed on 9 July 2025). We noted that the start date for the price of HBAR was 1-Sept-2019 (0.038021 USD). Accordingly, we set this price data as the month-end price data for 31-Aug-2029. Since our month-end price data for cryptocurrencies from Aug-2019 to Jun-2025, the combined sample’s (stocks and cryptocurrencies) return data are from Sept-2019 to Jun-2025 (71 months).

For the analysis, we imposed certain constraints and assumptions to ensure robust portfolios. For instance, we assume there is no short-selling (${\omega }_i\ge 0$), and the sum of the proportion of wealth allocation is 100% (${\sum }_{i=1}^N{\omega }_i=1$). Moreover, for the CEQ portfolio (6), following Petukhina et al. (2021), we assume that investors are highly risk-averse, hence setting $\gamma =1$. This is a reasonable assumption in a small market like Fiji, given the low participation in the stock market and high pessimism toward cryptocurrency trading, as noted by the Reserve Bank of Fiji (RBF, 2024). Moreover, noting the importance of achieving a higher positive Sharpe ratio (SR) on all optimized portfolios,11 in cases where the Sharpe ratio is below 2, (based on the initial investigations), we include the constraint Sharpe ratio to be either closer to 2 or above 2.12 For the risk-free return ($r_f$), and target return ($T_g$), we applied the 1-year government bond rate of 5%, and the dividend rate offered by the Fiji National Provident Fund (FNPF), as of Aug-2019.

We derive portfolios using 12 techniques (as mentioned above). These are: (1) EQW (equally weighted 1/N, naïve), (2) MinVar (minimum variance), (3) MaxSharpe (mean-variance, maximum Sharpe), (4) MinSemVar (semi-variance), (5) MaxSort (maximum target return, i.e., Sortino ratio), (6) MaxCEQ (certainty-equivalent), (7) MaxOmega (maximum Omega), (8) MaxUDVol (maximum upside-to-downside volatility), (9) MaxDiv (maximum diversification), (10) MinDeCorr (minimum decorrelation), (11) MaxRMD (robust maximum diversified), and (12) MaxASR (maximum adjusted Sharpe). We divide our analyses as follows to analyze various portfolios. First, we apply the 12 methods to stocks-only and stocks-cryptocurrency returns to derive portfolios. We compare the key metrics and report the outcomes. Next, we forecast 24-month forward out-of-sample price data and compute the 24-month out-of-sample returns of stocks and cryptocurrencies, resulting in 94 monthly data points (T = 94). Using the extended dataset of stocks and cryptocurrencies, we then derive different portfolios using the 12 measures. The descriptive statistics, correlation, and covariance matrices of the stocks-only and stocks-cryptocurrencies are provided in Appendix A (

Table A1. Descriptive statistics of stocks-only sample (T = 71).

Ticker	Mean	Standard Error	Median	Mode	Standard Deviation	Sample Variance	Kurtosis	Skewness	Range	Minimum	Maximum	Sum
APP	0.00881	0.00532	0.00000	0.00000	0.04480	0.00201	14.42227	2.49697	0.35398	−0.09237	0.26161	0.62527
ATH	−0.00779	0.00685	0.00000	0.00000	0.05775	0.00333	2.41866	−0.35220	0.36593	−0.19637	0.16956	−0.55339
BCN	0.00485	0.00526	0.00000	0.00000	0.04431	0.00196	4.52847	0.75084	0.29380	−0.14630	0.14750	0.34421
CFL	0.00044	0.00164	0.00000	0.00000	0.01382	0.00019	26.77761	3.84237	0.12586	−0.03489	0.09097	0.03092
FBL	0.00607	0.00695	0.00000	0.00000	0.05857	0.00343	14.23349	0.29018	0.54382	−0.27871	0.26511	0.43078
FHL	−0.00856	0.04942	−0.00976	0.00000	0.41644	0.17343	29.05617	−0.26836	4.71966	−2.40795	2.31172	−0.60799
FIL	0.03372	0.00637	0.01380	0.00000	0.05370	0.00288	1.59472	1.26046	0.28535	−0.07128	0.21407	2.39393
FMF	−0.00287	0.01079	0.00000	0.00000	0.09094	0.00827	29.59665	2.33435	0.99330	−0.41567	0.57763	−0.20375
FTV	−0.02492	0.01161	0.00000	0.00000	0.09780	0.00956	7.43202	−0.31108	0.78885	−0.39354	0.39531	−1.76929
KFL	0.00363	0.00691	−0.00858	0.00000	0.05819	0.00339	2.82057	0.57758	0.35148	−0.18443	0.16705	0.25783
KGF	0.00546	0.00254	0.00000	0.00000	0.02143	0.00046	26.73681	4.91766	0.14310	0.00000	0.14310	0.38745
PBP	0.01572	0.00717	0.00000	0.00000	0.06039	0.00365	17.89386	3.59763	0.49627	−0.12921	0.36706	1.11640
PDM	0.00665	0.00604	0.00000	0.00000	0.05094	0.00259	5.46851	1.96378	0.30316	−0.09309	0.21007	0.47191
RBG	0.00849	0.00783	0.00000	0.00000	0.06598	0.00435	36.40201	5.44725	0.55593	−0.08264	0.47329	0.60309
RCF	0.00244	0.00438	0.00000	0.00000	0.03687	0.00136	32.18154	4.56608	0.35341	−0.09963	0.25378	0.17290
TTS	0.00337	0.00446	0.00000	0.00000	0.03759	0.00141	29.19315	4.29627	0.35341	−0.09963	0.25378	0.23944
VBH	0.00697	0.00569	0.00000	0.00000	0.04798	0.00230	11.98466	2.55580	0.36108	−0.10976	0.25131	0.49508
VIL	−0.00312	0.00306	0.00000	0.00000	0.02582	0.00067	8.57772	−0.79285	0.19783	−0.11778	0.08004	−0.22143

Note: The table presents key statistics for each asset return in the stocks-only sample. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

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Table A2. Correlation matrix (stocks-only sample).

Ticker	APP	ATH	BCN	CFL	FBL	FHL	FIL	FMF	FTV	KFL	KGF	PBP	PDM	RBG	RCF	TTS	VBH	VIL
APP	1.0000
ATH	0.0654	1.0000
BCN	0.1657	0.0416	1.0000
CFL	0.5825	0.2299	−0.0397	1.0000
FBL	0.3841	0.1264	−0.0331	0.0925	1.0000
FHL	−0.0298	0.0767	0.0431	0.1009	−0.2065	1.0000
FIL	−0.0561	−0.1046	−0.0462	−0.0318	−0.0373	0.0095	1.0000
FMF	−0.1464	0.0404	0.0618	0.0038	−0.3365	0.1307	0.1547	1.0000
FTV	−0.0304	0.1051	0.0070	0.2165	0.0639	−0.0697	−0.0558	0.0269	1.0000
KFL	−0.0027	0.0061	0.1968	−0.0655	0.0552	−0.1019	−0.1079	0.1339	0.1167	1.0000
KGF	−0.0458	0.0060	0.2502	−0.0939	0.0067	−0.0902	−0.1078	−0.0241	−0.1905	0.0065	1.0000
PBP	0.1959	−0.0358	−0.0143	0.0380	0.0750	−0.0065	−0.0090	−0.0807	0.0018	0.0037	0.0539	1.0000
PDM	0.0084	−0.1353	0.0380	0.0964	0.0381	−0.0398	−0.1060	−0.0180	0.1074	0.1265	−0.0717	−0.0275	1.0000
RBG	−0.0460	0.1571	0.0935	0.0361	0.0964	−0.0741	−0.0270	−0.0213	0.2379	0.1220	−0.0433	−0.0425	0.1228	1.0000
RCF	0.0637	−0.0983	0.0103	−0.0194	−0.0424	0.0076	0.0671	0.0334	0.0562	0.0653	−0.0508	−0.0181	0.0087	0.0920	1.0000
TTS	0.0575	−0.0800	0.0309	0.0088	−0.0442	0.0080	0.0500	0.0361	0.0652	0.0543	−0.0562	−0.0244	0.0832	0.0869	0.9790	1.0000
VBH	0.0604	−0.0680	0.0072	0.0013	0.0965	−0.0043	0.0065	0.0706	0.0594	0.2184	0.0084	0.0076	0.1113	−0.0197	0.0833	0.0780	1.0000
VIL	0.0732	0.0178	−0.0354	0.0285	0.0087	−0.0553	−0.0708	−0.0268	0.0871	0.1060	−0.0110	0.0157	−0.0247	−0.0415	−0.4702	−0.4581	−0.1195	1.0000

Note: The correlation coefficient is for the stock-only sample. The correlation coefficient ranges from −1 to 1. A (high) positive value implies that the respective pair of assets is (strongly) positively correlated, that is, the respective asset pair moves in the same direction. A (high) negative value implies that the respective asset pair is (strongly) negatively correlated, that is, the respective asset pair moves in the opposite direction.

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Table A3. Covariance matrix (stocks-only sample).

Ticker	APP	ATH	BCN	CFL	FBL	FHL	FIL	FMF	FTV	KFL	KGF	PBP	PDM	RBG	RCF	TTS	VBH	VIL
APP	0.0241
ATH	0.0020	0.0400
BCN	0.0039	0.0013	0.0236
CFL	0.0043	0.0022	−0.0003	0.0023
FBL	0.0121	0.0051	−0.0010	0.0009	0.0412
FHL	−0.0067	0.0221	0.0095	0.0070	−0.0604	2.0811
FIL	−0.0016	−0.0039	−0.0013	−0.0003	−0.0014	0.0025	0.0346
FMF	−0.0072	0.0025	0.0030	0.0001	−0.0215	0.0594	0.0091	0.0993
FTV	−0.0016	0.0071	0.0004	0.0035	0.0044	−0.0341	−0.0035	0.0029	0.1148
KFL	−0.0001	0.0002	0.0061	−0.0006	0.0023	−0.0296	−0.0040	0.0085	0.0080	0.0406
KGF	−0.0005	0.0001	0.0029	−0.0003	0.0001	−0.0097	−0.0015	−0.0006	−0.0048	0.0001	0.0055
PBP	0.0064	−0.0015	−0.0005	0.0004	0.0032	−0.0019	−0.0003	−0.0053	0.0001	0.0002	0.0008	0.0438
PDM	0.0002	−0.0048	0.0010	0.0008	0.0014	−0.0101	−0.0035	−0.0010	0.0064	0.0045	−0.0009	−0.0010	0.0311
RBG	−0.0016	0.0072	0.0033	0.0004	0.0045	−0.0244	−0.0011	−0.0015	0.0184	0.0056	−0.0007	−0.0020	0.0050	0.0522
RCF	0.0013	−0.0025	0.0002	−0.0001	−0.0011	0.0014	0.0016	0.0013	0.0024	0.0017	−0.0005	−0.0005	0.0002	0.0027	0.0163
TTS	0.0012	−0.0021	0.0006	0.0001	−0.0012	0.0015	0.0012	0.0015	0.0029	0.0014	−0.0005	−0.0007	0.0019	0.0026	0.0163	0.0170
VBH	0.0016	−0.0023	0.0002	0.0000	0.0033	−0.0010	0.0002	0.0037	0.0033	0.0073	0.0001	0.0003	0.0033	−0.0007	0.0018	0.0017	0.0276
VIL	0.0010	0.0003	−0.0005	0.0001	0.0002	−0.0071	−0.0012	−0.0008	0.0026	0.0019	−0.0001	0.0003	−0.0004	−0.0008	−0.0054	−0.0053	−0.0018	0.0080

Note: The covariance matrix is for the stock-only samples. A positive value implies that the respective pair of assets moves in the same direction, whereas a negative value implies that the respective pair of assets moves in the opposite direction. Low covariance implies that the asset pair is weakly associated with each other.

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Table A4. Descriptive statistics (stocks-cryptocurrencies, Sample, T = 70).

Ticker	Mean	Standard Error	Median	Mode	Standard Deviation	Sample Variance	Kurtosis	Skewness	Range	Minimum	Maximum	Sum
APP	0.0089	0.0054	0.0000	0.0000	0.0451	0.0020	14.1811	2.4737	0.3540	−0.0924	0.2616	0.6253
ATH	−0.0083	0.0069	0.0000	0.0000	0.0580	0.0034	2.3988	−0.3283	0.3659	−0.1964	0.1696	−0.5842
BCN	0.0043	0.0053	0.0000	0.0000	0.0444	0.0020	4.6189	0.7860	0.2938	−0.1463	0.1475	0.3016
CFL	0.0002	0.0016	0.0000	0.0000	0.0138	0.0002	27.8076	3.9577	0.1259	−0.0349	0.0910	0.0145
FBL	0.0062	0.0071	0.0000	0.0000	0.0590	0.0035	14.0082	0.2838	0.5438	−0.2787	0.2651	0.4308
FHL	−0.0087	0.0501	−0.0098	0.0000	0.4194	0.1759	28.6339	−0.2656	4.7197	−2.4079	2.3117	−0.6080
FIL	0.0342	0.0064	0.0151	0.0000	0.0539	0.0029	1.5349	1.2388	0.2854	−0.0713	0.2141	2.3939
FMF	−0.0030	0.0109	0.0000	0.0000	0.0916	0.0084	29.2021	2.3252	0.9933	−0.4157	0.5776	−0.2129
FTV	−0.0256	0.0118	0.0000	0.0000	0.0983	0.0097	7.3537	−0.2910	0.7888	−0.3935	0.3953	−1.7918
KFL	0.0040	0.0070	−0.0082	0.0000	0.0585	0.0034	2.7544	0.5590	0.3515	−0.1844	0.1671	0.2776
KGF	0.0055	0.0026	0.0000	0.0000	0.0216	0.0005	26.3253	4.8804	0.1431	0.0000	0.1431	0.3874
PBP	0.0159	0.0073	0.0000	0.0000	0.0608	0.0037	17.6021	3.5674	0.4963	−0.1292	0.3671	1.1164
PDM	0.0044	0.0057	0.0000	0.0000	0.0477	0.0023	6.8212	2.0534	0.3032	−0.0931	0.2101	0.3102
RBG	0.0086	0.0079	0.0000	0.0000	0.0664	0.0044	35.8606	5.4068	0.5559	−0.0826	0.4733	0.6031
RCF	0.0025	0.0044	0.0000	0.0000	0.0371	0.0014	31.7047	4.5328	0.3534	−0.0996	0.2538	0.1729
TTS	0.0071	0.0058	0.0000	0.0000	0.0483	0.0023	11.7715	2.5328	0.3611	−0.1098	0.2513	0.4960
VBH	−0.0032	0.0031	0.0000	0.0000	0.0260	0.0007	8.4220	−0.7825	0.1978	−0.1178	0.0800	−0.2214
VIL	−0.0008	0.0082	0.0000	0.0000	0.0682	0.0047	10.6244	0.5281	0.5570	−0.2806	0.2764	−0.0590
BTC	0.0344	0.0220	0.0308	-	0.1839	0.0338	0.1151	−0.2131	0.8648	−0.4743	0.3905	2.4091
ETH	0.0381	0.0283	0.0261	-	0.2367	0.0560	0.0564	−0.0438	1.1767	−0.5988	0.5779	2.6684
ALG	−0.0114	0.0394	−0.0460	-	0.3299	0.1088	3.1597	0.9222	2.1239	−0.7702	1.3537	−0.7991
ADA	0.0363	0.0399	−0.0374	-	0.3335	0.1112	3.6756	1.6692	1.7752	−0.4421	1.3331	2.5429
HBA	0.0073	0.0450	−0.0334	-	0.3762	0.1415	3.0586	1.0880	2.1880	−0.8911	1.2968	0.5090
QNT	0.0423	0.0338	0.0000	-	0.2824	0.0798	1.2999	0.6498	1.6175	−0.6128	1.0047	2.9585
XLM	0.0190	0.0396	−0.0237	-	0.3314	0.1098	10.3836	2.5090	2.1927	−0.4576	1.7351	1.3313
XDC	0.0689	0.0509	−0.0017	-	0.4260	0.1815	4.0744	1.6462	2.2352	−0.6159	1.6193	4.8228
XRP	0.0308	0.0422	−0.0081	-	0.3534	0.1249	4.1655	0.9939	2.4457	−1.1060	1.3397	2.1561

Note: The table presents key statistics for each asset return in the stocks-cryptocurrencies sample. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

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Table A5. Correlation matrix (stocks-cryptocurrencies, Sample, T = 70).

Ticker	APP	ATH	BCN	CFL	FBL	FHL	FIL	FMF	FTV	KFL	KGF	PBP	PDM	RBG	RCF	TTS	VBH	VIL	BTC	ETH	ALG	ADA	HBA	QNT	XLM	XDC	XRP
APP	1.000
ATH	0.068	1.000
BCN	0.169	0.034	1.000	−0.055
CFL	0.592	0.222	−0.055	1.000
FBL	0.384	0.128	−0.032	0.095	1.000
FHL	−0.030	0.077	0.043	0.102	−0.206	1.000
FIL	−0.058	−0.099	−0.039	−0.022	−0.038	0.010	1.000
FMF	−0.146	0.039	0.061	0.002	−0.336	0.131	0.156	1.000
FTV	−0.029	0.101	0.001	0.211	0.065	−0.070	−0.052	0.026	1.000
KFL	−0.004	0.010	0.203	−0.059	0.055	−0.102	−0.112	0.135	0.120	1.000
KGF	−0.047	0.008	0.255	−0.091	0.006	−0.090	−0.111	−0.024	−0.189	0.005	1.000
PBP	0.195	−0.033	−0.011	0.043	0.075	−0.006	−0.011	−0.080	0.004	0.002	0.053	1.000
PDM	0.018	−0.178	0.001	0.049	0.046	−0.044	−0.084	−0.026	0.093	0.155	−0.065	−0.017	1.000
RBG	−0.046	0.159	0.096	0.039	0.096	−0.074	−0.028	−0.021	0.239	0.121	−0.044	−0.043	0.138	1.000
RCF	0.064	−0.098	0.011	−0.018	−0.043	0.008	0.067	0.034	0.057	0.065	−0.051	−0.018	0.013	0.092	1.000
TTS	0.060	−0.067	0.009	0.004	0.096	−0.004	0.005	0.071	0.060	0.218	0.008	0.007	0.127	−0.020	0.083	1.000
VBH	0.074	0.017	−0.037	0.027	0.009	−0.055	−0.070	−0.027	0.086	0.107	−0.011	0.016	−0.032	−0.041	−0.470	−0.119	1.000
VIL	0.034	0.108	0.106	0.014	0.081	0.010	−0.066	−0.023	0.112	−0.312	0.105	0.000	−0.075	0.267	−0.151	−0.341	−0.022	1.000
BTC	−0.053	−0.173	−0.027	−0.067	−0.117	−0.042	0.235	0.108	0.002	0.166	−0.013	0.063	−0.063	−0.173	0.087	0.214	0.047	−0.072	1.000
ETH	−0.037	−0.107	−0.045	−0.042	−0.159	−0.101	0.142	0.097	0.044	−0.040	−0.084	0.036	−0.011	−0.136	0.104	0.152	0.047	−0.081	0.778	1.000
ALG	0.201	−0.102	0.035	0.057	−0.069	−0.041	0.082	0.058	−0.004	−0.007	−0.084	0.087	−0.152	−0.050	0.045	0.120	0.108	0.014	0.614	0.667	1.000
ADA	0.004	−0.096	0.069	−0.067	−0.066	−0.087	0.237	0.037	0.003	0.164	−0.085	−0.035	−0.023	−0.089	0.165	0.157	0.105	−0.148	0.603	0.635	0.732	1.000
HBA	0.138	−0.131	0.133	0.035	−0.148	−0.006	0.078	0.085	−0.060	−0.037	0.119	0.042	−0.084	−0.295	−0.015	0.057	0.157	−0.175	0.556	0.596	0.752	0.547	1.000
QNT	−0.014	−0.203	0.126	0.073	−0.247	0.083	0.186	0.041	−0.129	−0.104	−0.034	0.000	−0.085	−0.108	−0.013	−0.068	0.182	−0.085	0.419	0.508	0.517	0.433	0.524	1.000
XLM	−0.106	−0.222	−0.113	−0.137	−0.081	−0.058	0.023	0.020	−0.004	0.112	−0.126	−0.034	−0.002	−0.138	0.059	0.286	0.086	−0.302	0.585	0.661	0.750	0.754	0.584	0.448	1.000
XDC	0.073	−0.185	0.008	−0.034	−0.016	−0.091	0.249	−0.075	−0.015	0.001	0.021	−0.045	0.022	−0.066	0.094	0.002	0.048	−0.193	0.356	0.405	0.440	0.594	0.262	0.330	0.480	1.000
XRP	−0.123	−0.183	−0.013	−0.184	−0.069	−0.065	−0.096	−0.003	−0.057	0.062	0.038	−0.020	0.019	−0.199	0.029	0.296	−0.024	−0.376	0.433	0.595	0.594	0.549	0.511	0.413	0.846	0.481	1.000

Note: The correlation coefficient is for the stocks-cryptocurrencies sample. The correlation coefficient ranges from −1 to 1. A (high) positive value implies that the respective pair of assets is (strongly) positively correlated, that is, the respective asset pair moves in the same direction. A (high) negative value implies that the respective asset pair is (strongly) negatively correlated, that is, the respective asset pair moves in the opposite direction.

and

Table A6. Covariance matrix (stocks-cryptocurrencies, sample, T = 70).

Ticker	APP	ATH	BCN	CFL	FBL	FHL	FIL	FMF	FTV	KFL	KGF	PBP	PDM	RBG	RCF	TTS	VBH	VIL	BTC	ETH	ALG	ADA	HBA	QNT	XLM	XDC	XRP
APP	0.024
ATH	0.002	0.040
BCN	0.004	0.001	0.024
CFL	0.004	0.002	0.000	0.002
FBL	0.012	0.005	−0.001	0.001	0.042
FHL	−0.007	0.022	0.010	0.007	−0.061	2.111
FIL	−0.002	−0.004	−0.001	0.000	−0.001	0.003	0.035
FMF	−0.007	0.003	0.003	0.000	−0.022	0.060	0.009	0.101
FTV	−0.002	0.007	0.000	0.003	0.005	−0.035	−0.003	0.003	0.116
KFL	0.000	0.000	0.006	−0.001	0.002	−0.030	−0.004	0.009	0.008	0.041
KGF	−0.001	0.000	0.003	0.000	0.000	−0.010	−0.002	−0.001	−0.005	0.000	0.006
PBP	0.006	−0.001	0.000	0.000	0.003	−0.002	0.000	−0.005	0.000	0.000	0.001	0.044
PDM	0.000	−0.006	0.000	0.000	0.002	−0.011	−0.003	−0.001	0.005	0.005	−0.001	−0.001	0.027
RBG	−0.002	0.007	0.003	0.000	0.005	−0.025	−0.001	−0.002	0.019	0.006	−0.001	−0.002	0.005	0.053
RCF	0.001	−0.003	0.000	0.000	−0.001	0.001	0.002	0.001	0.002	0.002	0.000	0.000	0.000	0.003	0.017
TTS	0.002	−0.002	0.000	0.000	0.003	−0.001	0.000	0.004	0.003	0.007	0.000	0.000	0.004	−0.001	0.002	0.028
VBH	0.001	0.000	−0.001	0.000	0.000	−0.007	−0.001	−0.001	0.003	0.002	0.000	0.000	0.000	−0.001	−0.005	−0.002	0.008
VIL	0.001	0.005	0.004	0.000	0.004	0.004	−0.003	−0.002	0.009	−0.015	0.002	0.000	−0.003	0.015	−0.005	−0.013	0.000	0.056
BTC	−0.005	−0.022	−0.003	−0.002	−0.015	−0.039	0.028	0.022	0.000	0.021	−0.001	0.009	−0.007	−0.025	0.007	0.023	0.003	−0.011	0.406
ETH	−0.005	−0.018	−0.006	−0.002	−0.027	−0.120	0.022	0.025	0.012	−0.007	−0.005	0.006	−0.002	−0.026	0.011	0.021	0.004	−0.016	0.407	0.672
ALG	0.036	−0.023	0.006	0.003	−0.016	−0.069	0.018	0.021	−0.002	−0.002	−0.007	0.021	−0.029	−0.013	0.007	0.023	0.011	0.004	0.447	0.625	1.306
ADA	0.001	−0.022	0.012	−0.004	−0.016	−0.146	0.051	0.014	0.001	0.038	−0.007	−0.008	−0.004	−0.024	0.024	0.030	0.011	−0.040	0.443	0.601	0.966	1.334
HBA	0.028	−0.034	0.027	0.002	−0.039	−0.012	0.019	0.035	−0.027	−0.010	0.012	0.011	−0.018	−0.089	−0.002	0.013	0.018	−0.054	0.461	0.637	1.120	0.823	1.698
QNT	−0.002	−0.040	0.019	0.003	−0.049	0.118	0.034	0.013	−0.043	−0.021	−0.003	0.000	−0.014	−0.024	−0.002	−0.011	0.016	−0.020	0.261	0.408	0.578	0.489	0.668	0.957
XLM	−0.019	−0.051	−0.020	−0.008	−0.019	−0.098	0.005	0.007	−0.002	0.026	−0.011	−0.008	0.000	−0.037	0.009	0.055	0.009	−0.082	0.428	0.622	0.984	1.000	0.874	0.503	1.318
XDC	0.017	−0.055	0.002	−0.002	−0.005	−0.194	0.069	−0.035	−0.008	0.000	0.002	−0.014	0.005	−0.022	0.018	0.000	0.006	−0.067	0.334	0.490	0.741	1.012	0.503	0.477	0.813	2.178
XRP	−0.024	−0.045	−0.002	−0.011	−0.017	−0.116	−0.022	−0.001	−0.024	0.015	0.003	−0.005	0.004	−0.056	0.005	0.061	−0.003	−0.109	0.337	0.597	0.831	0.776	0.816	0.494	1.188	0.869	1.499

Note: The covariance matrix is for the stocks-cryptocurrencies sample. A positive value implies that the respective pair of assets moves in the same direction, whereas a negative value implies that the respective pair of assets moves in the opposite direction. Low covariance implies that the asset pair is weakly associated with each other.

).

4. Experimental Results

This section is devoted to two sub-sections. In Section 4.1, we compare portfolios consisting solely of stocks with those that incorporate both stocks and cryptocurrencies. The second Section 4.2 focuses on the robustness of the results using out-of-sample forecasting and Monte Carlo simulations.13

4.1. The Stocks-Only Versus Stocks-Cryptocurrencies Portfolios

The results of stocks-only portfolios are presented in

Table 1. Stocks-only portfolios (18 stocks).

Portfolio	EQW	MinVar	MaxSharpe	MinSemVar	MaxSort	MaxCEQ	MaxOmega	MaxUDVol	MaxDiv	MinDeCorr	MaxRMD	MaxASR
Ave. Ret.	0.040	0.206	0.224	0.206	0.244	0.247	0.244	0.243	0.209	0.211	0.213	0.222
Std. Dev.	0.093	0.072	0.081	0.073	0.090	0.092	0.090	0.090	0.074	0.075	0.076	0.080
Skewness	−0.596	0.395	0.530	0.421	0.692	0.670	0.684	0.692	0.392	0.372	0.392	0.550
Kurtosis	12.635	−0.076	0.181	−0.062	0.462	0.423	0.436	0.459	−0.077	−0.071	−0.047	0.061
Sharpe	−0.112	2.150	2.159	2.150	2.150	2.150	2.150	2.150	2.150	2.150	2.150	2.150
Sortino	−0.394	5.355	5.762	5.430	6.238	6.175	6.198	6.237	5.359	5.292	5.333	5.855
Ret-Risk Ratio	0.426	2.840	2.779	2.838	2.704 *	2.695 *	2.704 *	2.707 *	2.827	2.819	2.810	2.776
Assets
APP	0.056	0.094	0.091	0.094	0.087	0.087	0.089	0.087	0.096	0.100	0.099	0.100
ATH	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BCN	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.005	0.000
CFL	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FBL	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FHL	0.056	0.000	0.000	0.001	0.000	0.000	0.001	0.000	0.002	0.000	0.000	0.001
FIL	0.056	0.359	0.409	0.361	0.465	0.472	0.466	0.463	0.366	0.369	0.373	0.400
FMF	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FTV	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
KFL	0.056	0.020	0.014	0.018	0.002	0.008	0.010	0.002	0.024	0.025	0.022	0.018
KGF	0.056	0.273	0.227	0.282	0.205	0.168	0.193	0.211	0.243	0.223	0.213	0.247
PBP	0.056	0.096	0.108	0.095	0.116	0.123	0.117	0.115	0.102	0.104	0.109	0.113
PDM	0.056	0.080	0.079	0.075	0.064	0.077	0.063	0.062	0.083	0.087	0.083	0.048
RBG	0.056	0.048	0.048	0.048	0.047	0.047	0.043	0.048	0.053	0.055	0.055	0.061
RCF	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
TTS	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
VBH	0.056	0.029	0.024	0.027	0.013	0.016	0.020	0.013	0.031	0.036	0.041	0.011
VIL	0.056	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
% of Asset	100%	50%	44%	50%	44%	56%	50%	50%	50%	50%	56%	50%

Note: * indicates that the returns-to-risk (RR) ratio based on the optimization method is higher for the respective stocks-only portfolio than for the stocks-cryptocurrencies portfolio. The rows highlighted in gray indicate the assets that consistently appear in portfolios based on the 12 methods. The risk-to-return ratio (RR) is computed as the average return divided by the standard deviation. For comparison, a higher RR is preferred to a lower RR. Each portfolio’s skewness and kurtosis are computed using the SKEW and KURT functions in Excel^®, respectively. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

, based on the month-end return data from Aug-2019 to Jun-2025 (71 observations). We find that, except for the equally weighted (1/N) portfolio, all other portfolios (portfolios based on different measures) perform well. The equally weighted portfolio (with 18 stocks) provides an average annual return of 4% and a standard deviation of 9.3% (with low and negative Sharpe (−0.112) and Sortino (−0.394) ratios). Notably, the return from the 1/N portfolio is much lower than the optimized portfolio, and it is lower than the risk-free rate of 1-year government bond of 5%. Interestingly, applying different optimization methods, we note that adding between 44% and 56% of all the assets (stocks) in the sample (noted in Figure 1) offers reasonably high returns, somewhere We also note comparable Sharpe ratios (see Figure 2) and Sortino ratios (see Figure 3), and a similar composition of assets (stocks), indicating consistent results across different methods. The returns (summarized in Figure 4) and standard deviations (Figure 5) are between 21% and 25%, and 7.2% and 9.2%, respectively. Moreover, within the stocks-only portfolios, we find that even with different optimization methods, there are similar return-risk (risk-reward) ratios (return-risk ratios are summarized in Figure 6). As noted from Table 1, out of the18 stocks in the sample, we note that, just eight stocks consistently appear in the portfolios based on the 12 measures (see Table 1, gray highlighted rows)—these are, APP, FIL, KFL, KGF, PBP, PDM, RBG and VBH—a similar composition of stocks was noted by Kumar et al. (2022).14

Next, we analyze the stocks-cryptocurrencies sample. To ensure the results of stocks-cryptocurrencies portfolios are comparable with the results of stocks-only portfolio, we apply the same (12) methods, i.e., the same risk, returns and diversification methods (Equations (1)–(12)), the same risk-free rates and the level of risk-averse ($\gamma =1$), assume long-only portfolios, and report the outcomes on same set of metrics (average return, standard deviation, Sharpe and Sortino ratio, risk-reward ratio, and the % of assets). The results are presented in

Table 2. Stocks-cryptocurrencies portfolio (18 stocks, nine cryptocurrencies).

Portfolio	EQW	MinVar	MaxSharpe	MinSemVar	MaxSort	MaxCEQ	MaxOmega	MaxUDVol	MaxDiv	MinDeCorr	MaxRMD	MaxASR
Ave. Ret.	0.141	0.181	0.235	0.183	0.299	0.338	0.302	0.292	0.191	0.194	0.197	0.207
Std. Dev.	0.284	0.061	0.083	0.062	0.116	0.134	0.117	0.113	0.066	0.067	0.068	0.073
Skewness	0.928	0.108	0.394	0.248	0.888	0.777	0.885	0.888	0.003	0.007	0.006	0.517
Kurtosis	2.106	−0.313	0.300	−0.284	0.902	0.949	0.943	0.903	−0.162	−0.138	−0.035	−0.197
Sharpe	0.321	2.150	2.238	2.150	2.150	2.150	2.150	2.150	2.150	2.150	2.150	2.150
Sortino	0.442	4.596	5.626	4.896	6.997	6.467	6.907	6.991	4.437	4.440	4.410	5.866
Ret-Risk Ratio	0.497 **	2.969 **	2.844 **	2.958 **	2.582	2.523	2.576	2.593	2.913 **	2.899 **	2.881 **	2.836 **
Assets
APP	0.037	0.108	0.114	0.107	0.105	0.059	0.132	0.105	0.108	0.107	0.100	0.118
ATH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BCN	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.016	0.000
CFL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FBL	0.037	0.003	0.000	0.000	0.000	0.000	0.000	0.000	0.014	0.009	0.021	0.000
FHL	0.037	0.003	0.001	0.003	0.001	0.000	0.000	0.001	0.007	0.003	0.002	0.003
FIL	0.037	0.287	0.416	0.294	0.608	0.695	0.616	0.593	0.310	0.314	0.317	0.354
FMF	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FTV	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
KFL	0.037	0.039	0.020	0.034	0.000	0.000	0.004	0.000	0.055	0.056	0.042	0.020
KGF	0.037	0.315	0.222	0.342	0.102	0.000	0.083	0.129	0.223	0.177	0.164	0.341
PBP	0.037	0.075	0.107	0.072	0.121	0.160	0.107	0.114	0.088	0.096	0.106	0.086
PDM	0.037	0.055	0.031	0.040	0.000	0.000	0.000	0.000	0.053	0.064	0.058	0.000
RBG	0.037	0.056	0.072	0.056	0.056	0.058	0.047	0.053	0.064	0.063	0.065	0.074
RCF	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.009	0.014	0.000
TTS	0.037	0.020	0.000	0.014	0.000	0.000	0.000	0.000	0.010	0.035	0.040	0.000
VBH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
VIL	0.037	0.026	0.000	0.027	0.000	0.000	0.000	0.000	0.048	0.048	0.035	0.000
BTC	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ETH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ALG	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ADA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
HBA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
QNT	0.037	0.000	0.000	0.000	0.000	0.000	0.003	0.000	0.000	0.002	0.003	0.000
XLM	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XDC	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XRP	0.037	0.013	0.018	0.012	0.007	0.027	0.009	0.006	0.021	0.018	0.016	0.003
% Assets	100%	44%	33%	41%	26%	19%	30%	26%	44%	52%	56%	30%

Note: ** indicates that the returns-to-risk ratio based on the optimization method is higher for the respective stocks-cryptocurrencies portfolio than for the stocks-only portfolio. The rows highlighted in gray indicate the assets that consistently appear in portfolios based on the 12 methods. The risk-to-return ratio (RR) is computed as the average return divided by the standard deviation. For comparison, a higher RR is preferred to a lower RR. Each portfolio’s skewness and kurtosis are computed using the SKEW and KURT functions in Excel^®, respectively. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

and graphed in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 for ease of comparison with stock-only portfolios.

Starting with the naïve (1/N) portfolios, we note that the stocks-cryptocurrencies (with 18 stocks and nine cryptocurrencies) portfolio yields a higher average return (14.1%), positive Sharpe (0.321) and Sortino (0.442) ratios, and a higher return-risk ratio than the stocks-only portfolio (see Table 1, and for comparison, refer to Figure 4, Figure 5 and Figure 6, respectively). Based on the risk-return ratio, the results imply that equally allocating funds across the 18 stocks and nine cryptocurrencies (BTC, ETH, ALG, ADA, HBA, QNT, XLM, XDC, and XRP) can improve portfolio outcomes (Figure 1) compared to the stocks-only equally weighted portfolio. This result is consistent with some earlier studies on developed and emerging markets (Symitsi & Chalvatzis, 2019; Brauneis & Mestel, 2019; Ma et al., 2020; Bouri et al., 2020; Petukhina et al., 2021). However, although the Sharpe ratio is positive (0.321), it is very low, indicating the possibility of optimization.

Moreover, in the stocks-cryptocurrencies portfolios, although several stocks and just two cryptocurrencies (XRP and QNT) appear in different proportions under different optimization methods (based on Equations (2)–(12)), we find that only four stocks (APP, FIL, PBP, and RBG), and only one cryptocurrency (XRP) consistently appear in all the portfolios (under different methods) (see rows highlighted in gray in Table 2), and that the exposure of XRP (based on the weights) is relatively small, ranging from 0.3% to 2.7%.

We compare the portfolios based on the following two criteria. The first criterion is the average return of the portfolio based on the respective measures. Criterion I: If the average return of the portfolio based on a specific measure is higher in the stocks-cryptocurrencies portfolio than in the stocks-only portfolio, then the former offers better performance than the latter. Based on specific optimization methods, we find that stocks-cryptocurrencies portfolios constructed using the minimum variance (MinVar) approach yield slightly lower returns (μ = 18.1%, σ = 6.1%) than stocks-only portfolios (μ = 20.6%, σ = 7.2%). Similarly, portfolios based on the MinSemVar, MaxDiv, MinDeCorr, MaxRMD, and MaxASR methods within stocks-cryptocurrencies tend to yield slightly lower returns than stock-only portfolios. Specifically, the returns and standard deviations for each method are as follows: MinSemVar: μ = 18.3%, σ = 6.2% vs. μ = 20.6%, σ = 7.3%; MaxDiv: μ = 19.1%, σ = 6.7% vs. μ = 20.9%, σ = 7.4%; MinDeCorr: μ = 19.7%, σ = 6.8% vs. μ = 21.3%, σ = 7.6%; MaxRMD: μ = 20.7%, σ = 7.3% vs. μ = 22.2%, σ = 8.0%. However, from the five optimization methods (MaxSharpe, MaxSort, MaxCEQ, MaxOmega, MaxUDVol), we observe that portfolios from the stocks-cryptocurrencies sample (respectively, based on the methods, these are: μ = 23.5%, σ = 8.3%; μ = 29.9%, σ = 11.6%; μ = 33.8%, σ = 13.4%; μ = 30.2%, μ = 11.7%; 29.2%; σ = 11.3%) have relatively higher returns than the portfolios from the stocks-only samples (respectively, these are: μ = 22.4%, σ = 8.1%; μ = 24.4%, σ = 9.9%; μ = 24.7%, σ = 9.2%; μ = 24.4%, σ = 9.0%; μ = 24.3%, σ = 9.0%). Nevertheless, we note that in all the portfolios from the stocks-cryptocurrencies sample, out of the nine cryptocurrencies, only XRP appears uniformly in all the optimization scenarios, although the weight remains very small (between 1 and 3%), indicating that a small exposure to this cryptocurrency can improve portfolio returns or lower risks (compared to the naïve portfolio with 27 assets).

Furthermore, to gain deeper insight into the return-risk profiles of different portfolios based on the optimization results, we compare the respective return-to-risk (risk-reward $=RR$) ratio of the stocks-only and stocks-cryptocurrencies portfolios (see Figure 6) as our second criterion. The ratio is simply computed as the average return divided by the standard deviation and appears in the 6th row of Table 1 and Table 2 (and is graphed in Figure 6). Criteria II: if $RR$ (of a portfolio based on the respective method and the type (stocks-only versus stocks-cryptocurrencies) is higher, then this implies the type of portfolio (under the respective method) offers better performance. From Table 2 and Figure 6, we can observe that $RR$ is higher for stocks-cryptocurrencies portfolios (Table 2) in eight out of the 12 methods: EQW ($RR=0.497$), MinVar ($RR=2.969$), MaxSharpe ($RR=2.844$), MinSemVar ($RR=2.958$), MaxDiv ($RR=2.913$), MinDeCorr ($RR=2.899$), MaxRMD ($RR=2.881$), and MaxASR ($RR=2.836$), implying that portfolios from stocks-cryptocurrencies sample based on these methods offer tend to have superior performance. However, for the remaining (four) methods (MaxSort, MaxCEQ, MaxOmega, and MaxUDVol), we find that the stocks-only portfolios (Table 1) have a higher risk-reward ratio (MaxSort ($RR=2.704$), MaxCEQ ($RR=2.695$), MaxOmega ($RR=2.704$) and MaxUDVol ($RR=2.707$), implying that portfolios from the stocks-only sample in these four cases show better performance. Moreover, two portfolios in the stock-cryptocurrencies sample are at the intersection of these two criteria (higher returns and risk-reward ratios). These are the portfolios based on EQW (1/N) and MaxSharpe (Table 2).

4.2. Stocks-Cryptocurrencies Portfolio: Out-of-Sample Forecast Analysis

Next, we examine the stability of the portfolios (weights) by first forecasting 24-month asset prices (stocks and cryptocurrencies) from the historical data. We use the forecasting function from Excel^®, i.e., $V_{t+1}^F=FORECAST.ETS(TD,Val,Time,0$), where, $V_{t+1}^F$ is the one period forward (t + 1) future asset price, $TD$ is the target date, $Val$ is the actual sample range, $Time$ is the respective sample’s period range, and 0 is included to exclude other conditions. Using the forecasted price data, 24-month returns were computed. The data were then merged with the actual returns data (T = 70), giving us a total of 94monthly data. Accordingly, the updated sample returns’ ($T=94)$ descriptive statistics, correlation, and covariance matrices are calculated. The respective tables are presented in Appendix A (see

Table A7. Descriptive statistics (stocks-cryptocurrencies including 24-month forecast, T = 94).

Ticker	Mean	Standard Error	Median	Mode	Standard Deviation	Sample Variance	Kurtosis	Skewness	Range	Minimum	Maximum	Sum
APP	0.0086	0.0040	0.0000	0.0000	0.0389	0.0015	19.8068	2.8738	0.3540	−0.0924	0.2616	0.8106
ATH	−0.0075	0.0052	−0.0048	0.0000	0.0500	0.0025	4.2101	−0.4319	0.3659	−0.1964	0.1696	−0.7014
BCN	0.0039	0.0039	0.0000	0.0000	0.0383	0.0015	7.1565	0.9412	0.2938	−0.1463	0.1475	0.3627
CFL	0.0004	0.0012	0.0000	0.0000	0.0119	0.0001	37.3237	4.5075	0.1259	−0.0349	0.0910	0.0370
FBL	0.0054	0.0052	0.0000	0.0000	0.0508	0.0026	19.4842	0.3743	0.5438	−0.2787	0.2651	0.5041
FHL	−0.0051	0.0373	0.0051	0.0000	0.3613	0.1306	38.7581	−0.3363	4.7197	−2.4079	2.3117	−0.4807
FIL	0.0281	0.0049	0.0104	0.0000	0.0476	0.0023	3.2869	1.6868	0.2854	−0.0713	0.2141	2.6395
FMF	−0.0040	0.0082	−0.0044	0.0000	0.0791	0.0063	39.2578	2.6940	0.9933	−0.4157	0.5776	−0.3742
FTV	−0.0531	0.0359	0.0000	0.0000	0.3477	0.1209	35.6753	−4.7873	3.6106	−2.6532	0.9574	−4.9903
KFL	0.0030	0.0052	0.0000	0.0000	0.0505	0.0025	4.6984	0.6997	0.3515	−0.1844	0.1671	0.2853
KGF	0.0048	0.0019	0.0000	0.0000	0.0186	0.0003	36.2969	5.7043	0.1431	0.0000	0.1431	0.4525
PBP	0.0142	0.0054	0.0000	0.0000	0.0525	0.0028	24.6313	4.1935	0.4963	−0.1292	0.3671	1.3305
PDM	0.0031	0.0042	−0.0006	0.0000	0.0412	0.0017	10.2599	2.4511	0.3032	−0.0931	0.2101	0.2958
RBG	0.0058	0.0059	−0.0025	0.0000	0.0574	0.0033	48.8431	6.3128	0.5559	−0.0826	0.4733	0.5428
RCF	0.0012	0.0033	0.0000	0.0000	0.0321	0.0010	43.2604	5.3091	0.3534	−0.0996	0.2538	0.1082
TTS	0.0066	0.0043	0.0000	0.0000	0.0416	0.0017	16.6756	2.9533	0.3611	−0.1098	0.2513	0.6194
VBH	−0.0031	0.0023	0.0000	0.0000	0.0224	0.0005	12.1441	−0.9126	0.1978	−0.1178	0.0800	−0.2898
VIL	−0.0010	0.0061	−0.0011	0.0000	0.0588	0.0035	15.0311	0.6163	0.5570	−0.2806	0.2764	−0.0937
BTC	0.0278	0.0164	0.0089	-	0.1588	0.0252	1.1097	−0.1179	0.8648	−0.4743	0.3905	2.6176
ETH	0.0312	0.0211	0.0124	-	0.2043	0.0417	1.0775	0.0523	1.1767	−0.5988	0.5779	2.9371
ALG	−0.0286	0.0296	−0.0579	-	0.2866	0.0822	5.2014	1.2099	2.1239	−0.7702	1.3537	−2.6861
ADA	0.0288	0.0297	0.0045	-	0.2876	0.0827	6.0494	1.9970	1.7752	−0.4421	1.3331	2.7034
HBA	0.0065	0.0334	0.0040	-	0.3240	0.1050	5.0762	1.2614	2.1880	−0.8911	1.2968	0.6067
QNT	0.0342	0.0251	0.0106	-	0.2436	0.0594	2.8176	0.8455	1.6175	−0.6128	1.0047	3.2191
XLM	0.0152	0.0294	0.0038	-	0.2855	0.0815	14.8622	2.9303	2.1927	−0.4576	1.7351	1.4244
XDC	0.0540	0.0379	0.0100	-	0.3678	0.1353	6.6393	2.0064	2.2352	−0.6159	1.6193	5.0748
XRP	0.0247	0.0314	0.0070	-	0.3046	0.0928	6.5972	1.2041	2.4457	−1.1060	1.3397	2.3251

Note: The table comprises key statistics for each asset return in the stocks-cryptocurrencies sample, including forecasted returns. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

,

Table A8. Correlation matrix (Stocks-cryptocurrencies sample including 24-month forecast).

Ticker	APP	ATH	BCN	CFL	FBL	FHL	FIL	FMF	FTV	KFL	KGF	PBP	PDM	RBG	RCF	TTS	VBH	VIL	BTC	ETH	ALG	ADA	HBA	QNT	XLM	XDC	XRP
APP	1.000
ATH	0.067	1.000
BCN	0.169	0.033	1.000
CFL	0.591	0.222	−0.055	1.000
FBL	0.384	0.127	−0.031	0.094	1.000
FHL	−0.030	0.077	0.043	0.102	−0.207	1.000
FIL	−0.054	−0.103	−0.033	−0.027	−0.031	0.006	1.000
FMF	−0.146	0.039	0.061	0.001	−0.335	0.130	0.156	1.000
FTV	−0.006	0.020	0.003	0.048	0.019	−0.019	0.016	0.008	1.000
KFL	−0.003	0.009	0.204	−0.060	0.055	−0.102	−0.102	0.135	0.033	1.000
KGF	−0.045	0.006	0.256	−0.092	0.008	−0.091	−0.093	−0.022	−0.037	0.007	1.000
PBP	0.196	−0.035	−0.010	0.041	0.076	−0.007	0.002	−0.079	0.008	0.004	0.057	1.000
PDM	0.019	−0.179	0.002	0.048	0.047	−0.045	−0.070	−0.024	0.030	0.157	−0.061	−0.014	1.000
RBG	−0.045	0.156	0.097	0.036	0.098	−0.075	−0.009	−0.019	0.070	0.124	−0.038	−0.038	0.142	1.000
RCF	0.064	−0.100	0.013	−0.020	−0.041	0.006	0.081	0.035	0.023	0.067	−0.046	−0.014	0.016	0.097	1.000
TTS	0.060	−0.067	0.009	0.003	0.097	−0.005	0.009	0.071	0.017	0.218	0.009	0.008	0.127	−0.018	0.084	1.000
VBH	0.074	0.017	−0.037	0.027	0.009	−0.055	−0.070	−0.027	0.020	0.107	−0.011	0.016	−0.033	−0.042	−0.469	−0.119	1.000
VIL	0.034	0.108	0.106	0.014	0.081	0.010	−0.063	−0.021	0.027	−0.312	0.105	0.000	−0.074	0.267	−0.150	−0.341	−0.022	1.000
BTC	−0.052	−0.174	−0.026	−0.069	−0.115	−0.043	0.244	0.109	0.010	0.168	−0.008	0.067	−0.059	−0.166	0.091	0.215	0.047	−0.072	1.000
ETH	−0.037	−0.108	−0.044	−0.044	−0.157	−0.101	0.151	0.099	0.018	−0.038	−0.080	0.039	−0.008	−0.130	0.107	0.152	0.047	−0.080	0.779	1.000
ALG	0.201	−0.104	0.037	0.054	−0.065	−0.043	0.103	0.058	−0.005	−0.003	−0.076	0.093	−0.145	−0.041	0.051	0.121	0.107	0.014	0.614	0.666	1.000
ADA	0.005	−0.097	0.069	−0.068	−0.065	−0.088	0.241	0.037	0.007	0.165	−0.081	−0.032	−0.020	−0.085	0.167	0.157	0.104	−0.148	0.604	0.635	0.730	1.000
HBA	0.139	−0.131	0.133	0.035	−0.148	−0.006	0.077	0.085	−0.014	−0.037	0.119	0.042	−0.084	−0.294	−0.014	0.058	0.157	−0.175	0.555	0.595	0.746	0.546	1.000
QNT	−0.013	−0.204	0.127	0.071	−0.245	0.082	0.194	0.042	−0.024	−0.102	−0.030	0.003	−0.082	−0.102	−0.009	−0.066	0.181	−0.084	0.421	0.510	0.518	0.434	0.524	1.000
XLM	−0.105	−0.223	−0.113	−0.138	−0.081	−0.059	0.028	0.020	0.002	0.113	−0.124	−0.033	−0.001	−0.136	0.061	0.286	0.086	−0.302	0.585	0.661	0.746	0.754	0.584	0.449	1.000
XDC	0.073	−0.187	0.010	−0.035	−0.014	−0.092	0.258	−0.074	0.006	0.003	0.025	−0.041	0.025	−0.060	0.098	0.003	0.048	−0.192	0.359	0.407	0.442	0.595	0.261	0.333	0.481	1.000
XRP	−0.122	−0.184	−0.012	−0.185	−0.068	−0.066	−0.086	−0.002	−0.009	0.063	0.040	−0.018	0.021	−0.195	0.031	0.297	−0.024	−0.376	0.434	0.595	0.592	0.549	0.511	0.414	0.846	0.482	1.000

Note: The correlation coefficient is for the stocks-cryptocurrencies sample, including forecasted returns. The correlation coefficient ranges from −1 to 1. A (high) positive value implies that the respective pair of assets is (strongly) positively correlated, that is, the respective asset pair moves in the same direction. A (high) negative value implies that the respective asset pair is (strongly) negatively correlated, i.e., the respective asset pair moves in the opposite direction.

and

Table A9. Covariance matrix (Stocks-Cryptocurrencies sample including 24-month forecast).

Ticker	APP	ATH	BCN	CFL	FBL	FHL	FIL	FMF	FTV	KFL	KGF	PBP	PDM	RBG	RCF	TTS	VBH	VIL	BTC	ETH	ALG	ADA	HBA	QNT	XLM	XDC	XRP
APP	0.018
ATH	0.002	0.030
BCN	0.003	0.001	0.018
CFL	0.003	0.002	0.000	0.002
FBL	0.009	0.004	−0.001	0.001	0.031
FHL	−0.005	0.017	0.007	0.005	−0.046	1.567
FIL	−0.001	−0.003	−0.001	0.000	−0.001	0.001	0.027
FMF	−0.005	0.002	0.002	0.000	−0.016	0.045	0.007	0.075
FTV	−0.001	0.004	0.000	0.002	0.004	−0.029	0.003	0.002	1.450
KFL	0.000	0.000	0.005	0.000	0.002	−0.022	−0.003	0.006	0.007	0.031
KGF	0.000	0.000	0.002	0.000	0.000	−0.007	−0.001	0.000	−0.003	0.000	0.004
PBP	0.005	−0.001	0.000	0.000	0.002	−0.002	0.000	−0.004	0.002	0.000	0.001	0.033
PDM	0.000	−0.004	0.000	0.000	0.001	−0.008	−0.002	−0.001	0.005	0.004	−0.001	0.000	0.020
RBG	−0.001	0.005	0.003	0.000	0.003	−0.019	0.000	−0.001	0.017	0.004	0.000	−0.001	0.004	0.040
RCF	0.001	−0.002	0.000	0.000	−0.001	0.001	0.001	0.001	0.003	0.001	0.000	0.000	0.000	0.002	0.012
TTS	0.001	−0.002	0.000	0.000	0.002	−0.001	0.000	0.003	0.003	0.006	0.000	0.000	0.003	−0.001	0.001	0.021
VBH	0.001	0.000	0.000	0.000	0.000	−0.005	−0.001	−0.001	0.002	0.001	0.000	0.000	0.000	−0.001	−0.004	−0.001	0.006
VIL	0.001	0.004	0.003	0.000	0.003	0.003	−0.002	−0.001	0.007	−0.011	0.001	0.000	−0.002	0.011	−0.003	−0.010	0.000	0.041
BTC	−0.004	−0.017	−0.002	−0.002	−0.011	−0.030	0.022	0.016	0.006	0.016	0.000	0.007	−0.005	−0.018	0.006	0.017	0.002	−0.008	0.303
ETH	−0.003	−0.013	−0.004	−0.001	−0.020	−0.090	0.018	0.019	0.015	−0.005	−0.004	0.005	−0.001	−0.018	0.008	0.016	0.003	−0.012	0.303	0.501
ALG	0.027	−0.018	0.005	0.002	−0.011	−0.053	0.017	0.016	−0.006	−0.001	−0.005	0.017	−0.020	−0.008	0.006	0.017	0.008	0.003	0.335	0.468	0.986
ADA	0.001	−0.017	0.009	−0.003	−0.011	−0.109	0.040	0.010	0.009	0.029	−0.005	−0.006	−0.003	−0.017	0.019	0.023	0.008	−0.030	0.331	0.448	0.722	0.992
HBA	0.021	−0.026	0.020	0.002	−0.029	−0.009	0.014	0.026	−0.019	−0.007	0.009	0.009	−0.013	−0.066	−0.002	0.009	0.014	−0.040	0.342	0.473	0.831	0.611	1.260
QNT	−0.002	−0.030	0.014	0.002	−0.036	0.087	0.027	0.010	−0.024	−0.015	−0.002	0.000	−0.010	−0.017	−0.001	−0.008	0.012	−0.014	0.196	0.305	0.434	0.365	0.496	0.712
XLM	−0.014	−0.038	−0.015	−0.006	−0.014	−0.073	0.005	0.006	0.003	0.020	−0.008	−0.006	0.000	−0.027	0.007	0.041	0.007	−0.061	0.318	0.462	0.732	0.743	0.648	0.374	0.978
XDC	0.013	−0.041	0.002	−0.002	−0.003	−0.146	0.054	−0.026	0.009	0.001	0.002	−0.009	0.005	−0.015	0.014	0.001	0.005	−0.050	0.251	0.367	0.559	0.755	0.374	0.358	0.606	1.624
XRP	−0.017	−0.034	−0.002	−0.008	−0.013	−0.087	−0.015	−0.001	−0.012	0.012	0.003	−0.003	0.003	−0.041	0.004	0.045	−0.002	−0.081	0.252	0.445	0.620	0.577	0.605	0.368	0.883	0.648	1.113

Note: The covariance matrix is for the stocks-cryptocurrencies sample with forecasted data. A positive value implies that the respective pair of assets moves in the same direction, whereas a negative value implies that the respective asset pair moves in the opposite direction. Low covariance implies that the asset pair is weakly associated with each other.

).

For comparison with different portfolios, we applied the same 12 portfolio methods as before (EQW, Min Var, MaxSharpe, MaxSort, MaxCEQ, MaxOMega, MaxUDVol, MaxDiv, MinDeCorr, MaxRMD, and MaxASR). Except for the EQW (naïve) portfolio, as before, we set the Sharpe ratio at the maximum feasible value while ensuring that we obtain convergence (a feasible solution). We discovered that a Sharpe ratio of at least 1.981 provided converging solutions. The results are presented in

Table 3. Stocks-cryptocurrency portfolios based on a 24-month forward forecast.

Portfolio	EQW	MinVar	MaxSharpe	MinSemVar	MaxSort	MaxCEQ	MaxOmega	MaxUDVol	MaxDiv	MinDeCorr	MaxRMD	MaxASR
Ave. Ret.	0.091	0.208	0.212	0.209	0.217	0.2190	0.218	0.217	0.210	0.211	0.209	0.211
Std. Dev.	0.250	0.080	0.082	0.080	0.084	0.0853	0.085	0.084	0.081	0.081	0.080	0.081
Skewness	1.141	0.938	0.943	0.945	1.006	0.996	1.005	1.005	0.934	0.954	0.934	0.960
Kurtosis	3.608	1.482	1.506	1.495	1.653	1.649	1.660	1.646	1.476	1.528	1.476	1.501
Sharpe	0.165	1.981	1.981	1.981	1.981	1.981	1.981	1.981	1.981	1.981	1.981	1.981
Sortino	0.157	5.282	5.291	5.298	5.456	5.426	5.453	5.455	5.280	5.319	5.271	5.346
Ret-Risk Ratio	0.365	2.606	2.593	2.605	2.574	2.567	2.572	2.576	2.601	2.596	2.603	2.596
Assets
APP	0.037	0.144	0.147	0.144	0.145	0.146	0.144	0.145	0.145	0.141	0.144	0.150
ATH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BCN	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
CFL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FBL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FHL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000
FIL	0.037	0.449	0.459	0.449	0.477	0.481	0.477	0.475	0.452	0.458	0.450	0.456
FMF	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FTV	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
KFL	0.037	0.012	0.013	0.011	0.007	0.008	0.006	0.007	0.013	0.013	0.012	0.015
KGF	0.037	0.189	0.167	0.190	0.168	0.157	0.167	0.171	0.180	0.178	0.181	0.177
PBP	0.037	0.126	0.129	0.126	0.132	0.135	0.133	0.131	0.128	0.126	0.128	0.127
PDM	0.037	0.005	0.009	0.004	0.000	0.000	0.000	0.000	0.005	0.006	0.006	0.000
RBG	0.037	0.054	0.053	0.054	0.053	0.053	0.053	0.053	0.056	0.054	0.055	0.055
RCF	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
TTS	0.037	0.002	0.004	0.002	0.000	0.000	0.000	0.000	0.000	0.004	0.004	0.000
VBH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
VIL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BTC	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ETH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ALG	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ADA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
HBA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
QNT	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XLM	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XDC	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XRP	0.037	0.019	0.019	0.019	0.019	0.020	0.019	0.019	0.020	0.019	0.019	0.019
% Assets	100%	33%	33%	33%	26%	26%	26%	26%	33%	33%	33%	26%

Note: The rows highlighted in gray indicate the assets that consistently appear in portfolios based on the 12 methods. The risk-to-return ratio (RR) is computed as the average return divided by the standard deviation. For comparison, a higher RR is preferred to a lower RR. Each portfolio’s skewness and kurtosis are computed using the SKEW and KURT functions in Excel^®, respectively. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

. Comparing the results in Table 3 with the respective portfolios in Table 1 (stocks-only) and Table 2 (stocks-cryptocurrencies), we note that in most cases, especially in the stocks-cryptocurrencies sample (Table 2), we obtain the same set of stocks, weights, and similar risk-reward ratios. These results lead to similar conclusions: a portfolio comprising stocks and a small allocation to a single cryptocurrency (XRP in this case) may yield better returns for investors. Additionally, the naïve (1/N) portfolio appears to generate higher returns in the stocks-cryptocurrency sample than in the stocks-only sample.

4.3. Stocks-Cryptocurrencies Portfolio: Monte Carlo Simulation Analysis

To run the Monte Carlo simulations, we followed the following procedure. We derived the Cholesky decomposition matrix for both samples from the correlation matrix (see Appendix A,

Table A10. Cholesky decomposition based on correlation (stocks-cryptocurrencies sample).

Ticker	APP	ATH	BCN	CFL	FBL	FHL	FIL	FMF	FTV	KFL	KGF	PBP	PDM	RBG	RCF	TTS	VBH	VIL	BTC	ETH	ALG	ADA	HBA	QNT	XLM	XDC	XRP
APP	1.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ATH	0.068	0.998	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BCN	0.169	0.022	0.985	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
CFL	0.592	0.182	−0.161	0.769	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FBL	0.384	0.102	−0.101	−0.217	0.886	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FHL	−0.030	0.079	0.047	0.146	−0.188	0.966	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FIL	−0.058	−0.095	−0.027	0.034	−0.002	0.012	0.993	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FMF	−0.146	0.049	0.085	0.121	−0.283	0.049	0.151	0.922	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FTV	−0.029	0.103	0.004	0.273	0.141	−0.096	−0.051	0.039	0.939	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
KFL	−0.004	0.010	0.207	−0.033	0.077	−0.096	−0.104	0.176	0.101	0.943	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
KGF	−0.047	0.012	0.266	−0.029	0.049	−0.095	−0.103	−0.018	−0.219	−0.053	0.924	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
PBP	0.195	−0.047	−0.044	−0.093	−0.023	0.015	−0.003	−0.045	0.049	0.016	0.092	0.967	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
PDM	0.018	−0.179	0.001	0.093	0.087	−0.027	−0.104	0.018	0.070	0.138	−0.059	−0.018	0.955	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
RBG	−0.046	0.162	0.101	0.069	0.138	−0.080	−0.014	−0.008	0.186	0.068	−0.048	−0.018	0.126	0.934	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
RCF	0.064	−0.103	0.003	−0.048	−0.075	0.011	0.063	0.024	0.102	0.067	−0.012	−0.047	−0.008	0.110	0.971	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
TTS	0.060	−0.071	0.000	−0.025	0.085	0.023	0.002	0.118	0.065	0.198	0.039	−0.014	0.080	−0.050	0.062	0.955	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
VBH	0.074	0.012	−0.051	−0.035	−0.038	−0.055	−0.065	−0.007	0.100	0.104	0.024	−0.013	−0.056	−0.054	−0.500	−0.119	0.831	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
VIL	0.034	0.106	0.099	−0.013	0.072	0.014	−0.051	−0.002	0.099	−0.374	0.079	−0.004	−0.018	0.263	−0.153	−0.265	−0.106	0.799	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BTC	−0.053	−0.170	−0.015	−0.009	−0.094	−0.047	0.217	0.059	0.040	0.192	0.036	0.060	−0.093	−0.149	0.054	0.170	0.085	0.166	0.875	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ETH	−0.037	−0.105	−0.037	−0.009	−0.158	−0.124	0.131	0.046	0.076	−0.036	−0.054	0.040	−0.008	−0.123	0.087	0.158	0.118	0.048	0.733	0.561	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ALG	0.201	−0.116	0.004	−0.052	−0.164	−0.051	0.086	0.045	0.052	−0.004	−0.056	0.041	−0.163	0.011	−0.007	0.128	0.109	0.090	0.576	0.265	0.657	0.000	0.000	0.000	0.000	0.000	0.000
ADA	0.004	−0.097	0.071	−0.053	−0.070	−0.091	0.234	−0.008	0.043	0.176	−0.072	−0.042	−0.041	−0.097	0.128	0.117	0.197	0.045	0.487	0.269	0.426	0.550	0.000	0.000	0.000	0.000	0.000
HBA	0.138	−0.141	0.114	−0.004	−0.198	−0.034	0.076	0.040	−0.015	−0.047	0.115	0.002	−0.077	−0.246	−0.027	0.056	0.156	−0.115	0.527	0.200	0.443	−0.087	0.505	0.000	0.000	0.000	0.000
QNT	−0.014	−0.203	0.134	0.181	−0.190	0.031	0.164	−0.069	−0.126	−0.068	−0.072	0.023	−0.093	−0.037	−0.028	−0.023	0.242	−0.045	0.385	0.256	0.217	−0.033	0.056	0.690	0.000	0.000	0.000
XLM	−0.106	−0.216	−0.092	−0.065	−0.047	−0.041	−0.003	0.022	0.037	0.130	−0.100	−0.029	−0.060	−0.112	0.035	0.262	0.132	−0.117	0.518	0.298	0.463	0.180	−0.033	0.030	0.417	0.000	0.000
XDC	0.073	−0.191	0.000	−0.054	−0.041	−0.076	0.239	−0.101	0.038	0.039	0.057	−0.087	0.014	−0.039	0.048	−0.021	0.075	−0.168	0.326	0.186	0.235	0.318	−0.252	0.058	0.021	0.681	0.000
XRP	−0.123	−0.175	0.012	−0.101	−0.027	−0.047	−0.116	0.012	−0.023	0.046	0.011	−0.014	−0.017	−0.180	0.039	0.286	0.016	−0.263	0.424	0.374	0.364	0.055	−0.088	0.090	0.326	0.117	0.385

Note: The table contains the Cholesky decomposition matrix for the stocks-cryptocurrencies sample, which is used to generate simulated returns to conduct Monte Carlo simulations.

). Next, we generated uncorrelated z-statistics using the Excel^® function = NORM.S.INV(RAND()). To compute the correlated z-statistics, we multiplied the uncorrelated z-statistics by the Cholesky decomposition matrix. Finally, the simulated returns of each asset $i$ for each period (month) are computed as: ${\overline{\mu }}_i+\overline{{\sigma }_i}{z}_i$, where ${\overline{\mu }}_i$, $\overline{{\sigma }_i}$ and $z_i$ are the mean and standard deviation, and the correlated z-statistic at each period (month), respectively. Accordingly, using the simulated returns of 18 stocks and nine cryptocurrencies (27 assets), we applied the 12 portfolio methods. For illustration purposes, we ran three simulations and applied each set of simulated returns using the respective methods. We present the results of the first simulation below (see

Table 4. Stocks-cryptocurrencies portfolios based on Monte Carlo simulation (Simulation 1).

Portfolio	EQW	MinVar	MaxSharpe	MinSemVar	MaxSort	MaxCEQ	MaxOmega	MaxUDVol	MaxDiv	MinDeCorr	MaxRMD	MaxASR
Ave. Ret.	0.044	0.141	0.175	0.145	0.196	0.223	0.194	0.191	0.150	0.158	0.161	0.164
Std. Dev.	0.274	0.043	0.055	0.044	0.067	0.081	0.070	0.066	0.046	0.050	0.052	0.053
Skewness	−0.268	−0.147	−0.091	−0.100	−0.023	−0.085	0.107	0.028	−0.094	−0.084	−0.052	−0.003
Kurtosis	−0.170	−0.504	−0.316	−0.432	−0.600	−0.249	0.391	−0.654	−0.027	0.019	0.191	−0.884
Sharpe	−0.023	2.120	2.256	2.150	2.179	2.150	2.050	2.150	2.150	2.150	2.150	2.154
Sortino	−0.117	3.686	4.436	3.851	4.598	4.308	4.046	4.574	3.825	3.924	3.974	4.355
Ret-Risk Ratio	0.160	3.281	3.160	3.283	2.927	2.771	2.762	2.911	3.229	3.149	3.120	3.102
Assets
APP	0.037	0.166	0.164	0.169	0.176	0.124	0.038	0.192	0.172	0.147	0.136	0.183
ATH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BCN	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.026	0.000
CFL	0.037	0.009	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.018	0.011	0.000
FBL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FHL	0.037	0.012	0.018	0.014	0.022	0.029	0.026	0.024	0.014	0.016	0.017	0.018
FIL	0.037	0.182	0.254	0.180	0.304	0.388	0.250	0.290	0.182	0.196	0.195	0.225
FMF	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FTV	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
KFL	0.037	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.000
KGF	0.037	0.235	0.154	0.235	0.017	0.000	0.056	0.022	0.170	0.123	0.120	0.223
PBP	0.037	0.054	0.084	0.059	0.072	0.132	0.150	0.058	0.071	0.086	0.099	0.022
PDM	0.037	0.073	0.096	0.073	0.150	0.159	0.162	0.147	0.088	0.097	0.114	0.124
RBG	0.037	0.111	0.136	0.116	0.164	0.144	0.125	0.167	0.120	0.138	0.121	0.128
RCF	0.037	0.106	0.077	0.107	0.078	0.000	0.169	0.085	0.104	0.100	0.105	0.064
TTS	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
VBH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
VIL	0.037	0.039	0.001	0.033	0.000	0.000	0.000	0.000	0.058	0.055	0.035	0.000
BTC	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ETH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ALG	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ADA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
HBA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
QNT	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000
XLM	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XDC	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XRP	0.037	0.013	0.016	0.013	0.018	0.024	0.021	0.015	0.020	0.023	0.021	0.012
% Assets	100%	44%	41%	41%	44%	30%	67%	33%	52%	44%	44%	67%

Note: This table contains results based on the first simulation. The rows highlighted in gray indicate the assets that consistently appear in portfolios based on the 12 methods. The return-to-risk ratio (RR) is computed as the average return divided by the standard deviation. For comparison, a higher RR is preferred to a lower RR. Each portfolio’s skewness and kurtosis are computed using the SKEW and KURT functions in Excel^®, respectively. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

), and the results based on the remaining two simulations are included in Appendix A (see Appendix A,

Table A11. Stocks-cryptocurrency portfolios based on Monte Carlo simulation (Simulation 2).

Portfolio	EQW	MinVar	MaxSharpe	MinSemVar	MaxSort	MaxCEQ	MaxOmega	MaxUDVol	MaxDiv	MinDeCorr	MaxRMD	MaxASR
Ave. Ret.	0.181	0.198	0.260	0.209	0.260	0.354	0.222	0.247	0.217	0.222	0.232	0.256
Std. Dev.	0.272	0.070	0.095	0.074	0.098	0.141	0.084	0.092	0.078	0.080	0.084	0.096
Skewness	−0.170	0.542	0.275	0.566	0.438	−0.008	0.592	0.666	0.524	0.438	0.461	0.345
Kurtosis	−0.006	2.249	1.841	2.349	1.786	1.332	1.872	2.109	2.573	2.120	2.390	1.024
Sharpe	0.482	2.120	2.199	2.150	2.150	2.150	2.050	2.150	2.150	2.150	2.150	2.150
Sortino	0.631	4.558	4.828	4.752	4.878	4.531	4.598	5.216	4.680	4.708	4.710	4.940
Ret-Risk Ratio	0.666	2.835	2.723	2.828	2.661	2.504	2.647	2.695	2.793	2.774	2.742	2.671
Assets
APP	0.037	0.128	0.101	0.128	0.177	0.039	0.193	0.132	0.123	0.123	0.116	0.103
ATH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BCN	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
CFL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FBL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FHL	0.037	0.006	0.008	0.006	0.007	0.012	0.010	0.002	0.011	0.009	0.010	0.001
FIL	0.037	0.277	0.371	0.292	0.396	0.524	0.369	0.364	0.285	0.299	0.292	0.408
FMF	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
FTV	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
KFL	0.037	0.000	0.006	0.017	0.000	0.000	0.003	0.000	0.029	0.027	0.030	0.064
KGF	0.037	0.281	0.157	0.257	0.138	0.000	0.160	0.138	0.219	0.183	0.153	0.136
PBP	0.037	0.099	0.152	0.103	0.130	0.226	0.110	0.102	0.119	0.121	0.144	0.100
PDM	0.037	0.027	0.000	0.005	0.000	0.000	0.000	0.000	0.004	0.033	0.014	0.002
RBG	0.037	0.078	0.070	0.079	0.113	0.037	0.100	0.099	0.091	0.088	0.083	0.047
RCF	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.011	0.000
TTS	0.037	0.088	0.104	0.093	0.000	0.110	0.029	0.128	0.091	0.085	0.112	0.110
VBH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.005	0.000	0.000
VIL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.001	0.001	0.006	0.000
BTC	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ETH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ALG	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ADA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
HBA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
QNT	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XLM	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XDC	0.037	0.014	0.029	0.015	0.021	0.053	0.000	0.025	0.018	0.022	0.026	0.028
XRP	0.037	0.002	0.002	0.006	0.018	0.000	0.027	0.010	0.008	0.003	0.003	0.000
% Assets	100%	41%	37%	44%	41%	30%	56%	33%	78%	59%	48%	70%

Note: The results are based on the second set of simulated returns from the Monte-Carlo simulation. The rows highlighted in gray indicate the assets that consistently appear in portfolios based on the 12 methods. The risk-to-return ratio (RR) is computed as the average return divided by the standard deviation. For comparison, a higher RR is preferred to a lower RR. Each portfolio’s skewness and kurtosis are computed using the SKEW and KURT functions in Excel^®, respectively. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

and

Table A12. Stocks-cryptocurrency portfolios based on Monte Carlo simulation (Simulation 3).

Portfolio	EQW	MinVar	MaxSharpe	MinSemVar	MaxSort	MaxCEQ	MaxOmega	MaxUDVol	MaxDiv	MinDeCorr	MaxRMD	MaxASR
Ave. Ret.	0.109	0.155	0.261	0.162	0.279	0.360	0.242	0.278	0.174	0.191	0.187	0.244
Std. Dev.	0.249	0.050	0.088	0.052	0.098	0.144	0.083	0.101	0.058	0.065	0.064	0.083
Skewness	0.181	−0.001	0.145	0.262	0.454	0.284	0.214	0.536	−0.096	0.276	0.219	0.331
Kurtosis	0.132	−0.266	−0.035	−0.212	−0.419	−0.402	−0.315	−0.334	−0.414	−0.547	−0.577	−0.882
Sharpe	0.236	2.120	2.397	2.150	2.332	2.150	2.304	2.257	2.150	2.150	2.150	2.341
Sortino	0.246	4.036	5.949	4.643	7.115	5.591	5.878	7.016	4.174	5.105	4.882	6.863
Ret-Risk Ratio	0.436	3.125	2.965	3.112	2.840	2.497	2.903	2.751	3.015	2.914	2.932	2.943
Assets
APP	0.037	0.045	0.000	0.058	0.055	0.000	0.033	0.098	0.030	0.021	0.052	0.017
ATH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BCN	0.037	0.018	0.034	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.056	0.000
CFL	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.002	0.000
FBL	0.037	0.095	0.183	0.095	0.165	0.186	0.258	0.141	0.124	0.109	0.098	0.180
FHL	0.037	0.004	0.003	0.002	0.000	0.000	0.000	0.000	0.010	0.001	0.000	0.000
FIL	0.037	0.128	0.278	0.105	0.205	0.149	0.143	0.212	0.131	0.108	0.112	0.126
FMF	0.037	0.031	0.033	0.037	0.038	0.000	0.050	0.041	0.063	0.053	0.029	0.058
FTV	0.037	0.000	0.000	0.001	0.000	0.000	0.002	0.000	0.007	0.000	0.012	0.000
KFL	0.037	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.000
KGF	0.037	0.270	0.000	0.286	0.000	0.000	0.047	0.000	0.183	0.102	0.083	0.124
PBP	0.037	0.097	0.169	0.103	0.189	0.220	0.161	0.176	0.116	0.133	0.125	0.173
PDM	0.037	0.023	0.000	0.000	0.000	0.000	0.000	0.000	0.028	0.047	0.034	0.000
RBG	0.037	0.134	0.248	0.143	0.232	0.302	0.178	0.222	0.154	0.152	0.123	0.201
RCF	0.037	0.074	0.000	0.068	0.000	0.000	0.067	0.000	0.031	0.077	0.083	0.000
TTS	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.026	0.047	0.000
VBH	0.037	0.025	0.000	0.032	0.000	0.000	0.000	0.000	0.000	0.044	0.023	0.000
VIL	0.037	0.024	0.001	0.030	0.045	0.000	0.000	0.038	0.077	0.066	0.062	0.047
BTC	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ETH	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ALG	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
ADA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
HBA	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
QNT	0.037	0.008	0.000	0.012	0.013	0.068	0.032	0.005	0.001	0.017	0.021	0.028
XLM	0.037	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
XDC	0.037	0.001	0.007	0.001	0.024	0.044	0.000	0.031	0.006	0.010	0.011	0.014
XRP	0.037	0.024	0.044	0.025	0.035	0.032	0.027	0.035	0.037	0.034	0.026	0.031
% Assets	100%	63%	41%	59%	48%	30%	81%	37%	63%	74%	70%	56%

Note: The results are based on the third set of simulated returns from the Monte-Carlo simulation The rows highlighted in gray indicate the assets that consistently appear in portfolios based on the 12 methods. The risk-to-return ratio (RR) is computed as the average return divided by the standard deviation. For comparison, a higher RR is preferred to a lower RR. Each portfolio’s skewness and kurtosis are computed using the SKEW and KURT functions in Excel^®, respectively. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

, respectively). From the simulated returns (Table 4), we observe that similar assets appear in the portfolio based on the stocks-cryptocurrencies sample (APP, FIL, KFL, KGF, PBP, and RBG (stocks), and XRP (cryptocurrency)). In addition to XRP, from the other simulated returns (see Table A11 and Table A12), we noted a small exposure to XDC and QNT. Based on the simulated results, we can further confirm the consistency of similar stocks and cryptocurrencies, as noted from the actual sample (in Table 2).

Based on the two criteria (Criterion I: return-to-risk and Criterion II: average return), we noted that the EQW and MaxSharpe methods yield consistent results when comparing the results between stocks-only and stocks-cryptocurrency portfolios. Accordingly, we ran 12 additional simulations of returns, six for each method. The results are presented in Appendix A,

Table A13. Stocks-cryptocurrency portfolios based on Monte Carlo simulations (EQW and MaxSharpe).

Portfolio	EW-1	EW-2	EW-3	EW-4	EW-5	EW-6	MS-1	MS-2	MS-3	MS-4	MS-5	MS-6
Ave. Ret.	0.191	0.401	0.139	0.257	0.271	0.189	0.289	0.310	0.250	0.201	0.217	0.217
Std. Dev.	0.291	0.336	0.306	0.275	0.285	0.307	0.096	0.106	0.078	0.039	0.071	0.050
Skewness	0.324	0.036	0.376	0.327	−0.255	−0.074	0.155	−0.331	0.137	−0.042	0.128	0.248
Kurtosis	−0.462	−0.026	−0.192	0.462	−0.662	0.053	−0.581	−0.144	−0.210	0.360	0.348	−0.284
Sharpe	0.485	1.045	0.291	0.751	0.775	0.454	2.485	2.450	2.571	3.836	2.371	3.317
Sortino	0.715	1.760	0.372	1.207	1.152	0.607	6.772	5.366	6.761	11.784	5.360	8.402
Ret-Risk Ratio	0.657	1.194	0.455	0.932	0.951	0.617	3.005	2.922	3.214	5.106	3.079	4.312
Assets
APP	0.037	0.037	0.037	0.037	0.037	0.037	0.108	0.001	0.001	0.147	0.027	0.066
ATH	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.000
BCN	0.037	0.037	0.037	0.037	0.037	0.037	0.101	0.000	0.042	0.000	0.000	0.000
CFL	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.000
FBL	0.037	0.037	0.037	0.037	0.037	0.037	0.033	0.000	0.052	0.026	0.157	0.000
FHL	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.017	0.008	0.005	0.006
FIL	0.037	0.037	0.037	0.037	0.037	0.037	0.429	0.345	0.335	0.263	0.266	0.212
FMF	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.017
FTV	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.000
KFL	0.037	0.037	0.037	0.037	0.037	0.037	0.089	0.000	0.000	0.111	0.004	0.028
KGF	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.258	0.043	0.152	0.278
PBP	0.037	0.037	0.037	0.037	0.037	0.037	0.136	0.316	0.099	0.006	0.149	0.085
PDM	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.134	0.057	0.093
RBG	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.251	0.132	0.000	0.073	0.023
RCF	0.037	0.037	0.037	0.037	0.037	0.037	0.031	0.000	0.000	0.085	0.000	0.075
TTS	0.037	0.037	0.037	0.037	0.037	0.037	0.007	0.000	0.038	0.040	0.002	0.000
VBH	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.000
VIL	0.037	0.037	0.037	0.037	0.037	0.037	0.043	0.001	0.001	0.101	0.093	0.090
BTC	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.000
ETH	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.017	0.012	0.008	0.009	0.000
ALG	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.000
ADA	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.000
HBA	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.014	0.000	0.000	0.000
QNT	0.037	0.037	0.037	0.037	0.037	0.037	0.010	0.055	0.000	0.000	0.000	0.000
XLM	0.037	0.037	0.037	0.037	0.037	0.037	0.000	0.000	0.000	0.000	0.000	0.000
XDC	0.037	0.037	0.037	0.037	0.037	0.037	0.013	0.000	0.000	0.001	0.000	0.008
XRP	0.037	0.037	0.037	0.037	0.037	0.037	0.001	0.015	0.000	0.028	0.006	0.021
% Assets	100%	100%	100%	100%	100%	100%	44%	33%	52%	52%	48%	48%

Note: The results are based on the 12 sets of simulated returns from the Monte-Carlo simulation. The rows highlighted in gray indicate the assets that consistently appear in portfolios based on the EW and MS methods. The risk-to-return ratio (RR) is computed as the average return divided by the standard deviation. For comparison, a higher RR is preferred to a lower RR. EW and MS indicate that the portfolio is based on the equally weighted method and the MaxSharpe method, respectively. A total of 12 simulations were run, six each pertaining to the method. Each portfolio’s skewness and kurtosis are computed using the SKEW and KURT functions in Excel^®, respectively. A positive (negative) skewness implies that the returns of each asset are concentrated on the left (right), indicating an asymmetric distribution. A relatively higher kurtosis implies that the respective asset returns have heavier (lighter) tails and a sharper (flatter) peak than the normal (Gaussian) distribution.

. From the results based on equally weighted portfolios (EQ-1 to EQ-6), we find that the average return and standard deviations are consistent (at least for those simulations with positive Sharpe ratios). Moreover, from the results based on MaxSharpe (MS-1 to MS-6) portfolios, we note that, similar to the results based on the actual sample (in Table 2), the stocks and cryptocurrencies in the respective portfolios are generally consistent. Moreover, we do note that in addition to XRP, in a few of the simulated portfolios, a very small exposure to ETH (MS-2 to MS-5), QNT (MS-1 and MS-2), and XDC (MS-1, MS4, and MS-6) is identified. Based on these results, we further confirm the consistency of assets in the stock-cryptocurrency portfolios.

5. Conclusions

Studies on portfolio analysis with respect to cryptocurrencies have mainly been undertaken in advanced markets (developed and emerging economies). Recently, a few studies have qualitatively explored the potential of blockchain technology and cryptocurrencies (Kwok & Treiblmaier, 2024; Jutel, 2021, 2022; Waller & Johnson, 2022; UNDP, 2024). However, for small economies, the literature on portfolio analysis has mainly focused on stocks (Kumar & Stauvermann, 2022; Kumar et al., 2022, 2024). In this study, we extend the literature by examining the potential of including cryptocurrencies with stocks in portfolios with respect to the small island economy of Fiji. We analyze 18 stocks listed on Fiji’s South Pacific Stock Exchange and nine cryptocurrencies (BTC, ETH, ALGO, ADA, HBAR, QNT, XLM, XDC, and XRP) from the Yahoo Finance database using 12 portfolio methods. Our analysis examines the returns and risks of investing in stock-only portfolios versus investments in both stocks and cryptocurrencies. Noting that the latter yields higher returns (relative to risk), we examine the robustness of the results using a 24-month out-of-sample analysis and Monte Carlo simulations. Interestingly, we find that the results are generally consistent with our earlier analysis, supporting the idea of a portfolio comprising predominantly certain equities and only a small exposure to a few selected cryptocurrencies. Our results show that while stock-only portfolios can offer reasonably good returns (relative to the risk-free rate of 5% and a target return of 6.5%), a small exposure to cryptocurrencies can offer even better returns and return-to-risk ratios. The results have important implications for investors interested in including cryptocurrencies in their investment strategies, with possible upside gains.

Our study applied a comprehensive set of portfolio methods and optimization techniques based on the average return and risk of assets (stocks and cryptocurrencies). These measures are widely used in the literature, especially when analyzing stocks with cryptocurrencies, and provide useful insights into the possibility of a small economy like Fiji adopting cryptocurrencies and other digital assets. However, this study had some limitations. There are other risk-based measures like the Treynor ratio, Jensen’s alpha, and the information ratio, which we did not explore in the analysis.15 These measures, nevertheless, require asset-specific betas relative to the market to create a benchmark. While we note a few studies in relation to Fiji’s stock market based on stocks-only portfolios that have computed the betas (see Kumar et al., 2022), they are based on the domestic stock price index (localized beta). In addition, given the smallness and remoteness of Fiji’s stock market, the performance of stocks is remotely linked to international and digital asset (especially cryptocurrencies) markets, making it unrealistic to apply advanced countries’ stock indices to compute betas. In addition, identifying a suitable index to compute beta for cryptocurrencies is challenging and can be misleading, especially when applied to a remotely located stock market. However, future studies could explore these possibilities after carefully adjusting for remoteness and the degree of development in financial markets.

Moreover, our study has certain practical limitations. Although we show the potential for adding cryptocurrencies, especially tokens like XRP, to Fiji’s market, there are several constraints and restrictions on investment in or trade of digital assets in Fiji. Chief among these is the lack of regulation and institutional support for digital assets and the prohibition of the use of domestic funds for digital asset trading, making it almost impractical to establish even a basic exchange for digital assets.

In addition, while there is robust Internet infrastructure in the central business districts (CBDs), remote areas and smaller islands often lack a reliable supply of electricity and general infrastructure, which potentially affects the inclusion and integration of financial products and services. Moreover, recurring natural disasters, such as tropical cyclones, further damage the technological and electrical infrastructure essential for efficient trading. Nevertheless, recent projects, like Starlink services (a new Internet Service Provider), have been implemented to support reliable Internet access, especially in the smaller and dispersed islands of Fiji. However, in addition to the high cost (set-up costs and monthly subscription fees), the current focus of the project is not on digital assets or on supporting financial market transformation.16 The advancement of the stock market, technology, and the general financial markets remains slow. For instance, recent developments in the investment market have generally focused on improving the uptake and scaling up of investment in stocks—by creating awareness of the stock market in remote communities and initiating discussions on the potential of cross-listing stocks on a neighboring stock exchange (the Papua New Guinea Stock Exchange).

Our findings, although not financial advice, aim to promote deeper financial inclusion and emphasize that small island economies like Fiji should review their policies and positions regarding digital assets and digital infrastructure. We find that the results show promising outcomes of adding cryptocurrencies to portfolios. However, it must be noted that in Fiji, the Reserve Bank of Fiji has declared that trading of digital assets in general and cryptocurrencies using domestic funds is illegal. Given the limited options for investment and diversification, small island economies like Fiji should review the potential of digital assets to transform the financial system and investment landscape. Moreover, digital assets (cryptocurrencies) often carry huge volatility; hence, the possibility of investing in stablecoins can be a reasonable starting point to minimize strong volatility and generate a relatively stable return.17 Moreover, while our analysis focused on some of the major cryptocurrencies, it is important to recognize that the digital asset and blockchain technology markets are rapidly evolving, and there are literally thousands of cryptocurrencies and other digital assets (including non-fungible tokens). Additionally, while we explored long-only portfolios with a view to a long-term horizon, which is more reflective of Fiji’s investment and financial landscape, we understand that frequent rebalancing of portfolios, short-selling, and sophisticated trading approaches (derivatives, futures, and margin trading) can give rise to high transaction costs for both stocks and cryptocurrencies. Analyzing the plausible effects of these factors in a portfolio setting for small island economies would generate interesting insights, albeit beyond the scope of this study. Future research could also explore the role of stablecoins and other (new) digital assets. At best, the study shows the potential of including cryptocurrencies to improve investor returns, confirming that small exposure can improve long-term gains. Since Fiji’s stock market is mainly underdeveloped and unsophisticated, we have only considered a long-only portfolio with the view that investors prefer to hold assets over the medium to long term. Moreover, even in the selected cryptocurrencies, we note that there are short-term price swings with general price appreciation over the long term, which could potentially support long-only portfolios. However, future studies could consider the potential of short-selling, which would be more applicable in the digital asset market (futures and margin trading) and for investors who predominantly focus on cryptocurrencies. Future research could also investigate the role of blockchain technology and cryptocurrencies in advancing sustainable development goals (SDGs) in small island economies. Another interesting area of research would be to delve into the behavioral finance aspects of cryptocurrency adoption in small island nations by analyzing local investors’ risk perceptions, biases, and decision-making patterns when incorporating cryptocurrencies into their portfolios.

From a policy perspective, key stakeholders, such as the central bank, large institutions (pension funds, trust funds, and banks), and government bodies (financial regulators), should review policies regarding cryptocurrencies and the entrance of digital platforms. Furthermore, financial and investment education should include the role and use (cases) of digital assets and technology and their subsequent implications for investment. Finally, a few countries in the Pacific region, especially Vanuatu, have already shown interest and a positive stand toward digital assets (technology and cryptocurrencies), although they have no (traditional) stock markets, which could potentially leapfrog in financial development. Small countries like Fiji and other more developed countries with relatively better technology and financial infrastructure should collaborate to modernize technology infrastructure and markets for digital assets.