# Optimising Portfolio Risk by Involving Crypto Assets in a Volatile Macroeconomic Environment

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

## 3. Discussion

## 4. Materials and Methods

- α = confidence interval
- μ = average
- Z = standard score
- σ = standard deviation

- S = skewness
- K = Peakedness
- z = standard normal distribution
- μ = mean
- σ = standard deviation

- Monte Carlo method
- Parametric method
- Non-parametric (historical) method (Bugár and Uzsoki 2006)

Expected return = portfolio amount × expected return % × (time horizon/trading days)

Standard score = the inverse of the standard normal distribution function, where the seed value is a random number uniformly distributed between 0 and 1

## 5. Conclusions

## 6. Limitations and Further Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Almeida, José, and Tiago Cruz Gonçalves. 2022. Portfolio diversification, hedge and safe-haven properties in cryptocurrency investments and financial economics: A systematic literature review. Journal of Risk and Financial Management 16: 3. [Google Scholar] [CrossRef]
- Antonopoulos, Andreas. 2017. Mastering Bitcoin: Programming the Open Blockchain, 2nd ed. Sebastopol: O’Reilly Media, Inc. [Google Scholar]
- Arias-Calluari, Karina, Fernando Alonso-Marroquin, Morteza N. Najafi, and Michael Harré. 2021. Methods for forecasting the effect of exogenous risks on stock markets. Physica A: Statistical Mechanics and Its Applications 568: 125587. [Google Scholar] [CrossRef]
- Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1999. Coherent measures of risk. Mathematical Finance 9: 203–28. [Google Scholar] [CrossRef]
- Banihashemi, Shokoofeh, and Sarah Navidi. 2017. Portfolio performance evaluation in Mean-CVaR framework: A comparison with non-parametric methods value at risk in Mean-VaR analysis. Operations Research Perspectives 4: 21–28. [Google Scholar] [CrossRef]
- Bhuiyan, Rubaiyat Ahsan, Afzol Husain, and Changyong Zhang. 2021. A wavelet approach for causal relationship between bitcoin and conventional asset classes. Resources Policy 71: 101971. [Google Scholar] [CrossRef]
- Bouri, Elie, Syed JawadHussain Shahzad, and David Roubaud. 2020. Cryptocurrencies as hedges and safe-havens for US equity sectors. Quarterly Review of Economics and Finance 75: 294–307. [Google Scholar] [CrossRef]
- Bugár, Gyöngyi, and Máté Uzsoki. 2006. Befektetések kockázatának mérése. Statisztikai Szemle 84: 876–98. [Google Scholar]
- Chaum, David, Amos Fiat, and Moni Naor. 1990. Untraceable electronic cash. In Advances in Cryptology—CRYPTO’ 88. Lecture Notes in Computer Science. Edited by Shafi Goldwasser. New York: Springer, p. 403. [Google Scholar]
- Fang, Tang, Su Zhi, and Libo Yin. 2020. Economic fundamentals or investor perceptions? The role of uncertainty in predicting long-term cryptocurrency volatility. International Review of Financial Analysis 71: 101566. [Google Scholar] [CrossRef]
- Favre, Laurent, and José-Antonio Galeano. 2002. Mean-modified value-at-risk optimisation with hedge funds. The Journal of Alternative Investments 5: 21–25. [Google Scholar] [CrossRef]
- González, Maria de la O., Francisco Jareño, and Frank S. Skinner. 2021. Asymmetric interdependencies between large capital cryptocurrency and gold returns during the COVID-19 pandemic crisis. International Review of Financial Analysis 76: 101773. [Google Scholar] [CrossRef]
- Hsu, Shu Han, Chwen Sheu, and Jiho Yoon. 2021. Risk spillovers between cryptocurrencies and traditional currencies and gold under different global economic conditions. The North American Journal of Economics and Finance 57: 101443. [Google Scholar] [CrossRef]
- Jiang, Yonghong, Lanxi Wu, Gengyu Tian, and He Nie. 2021. Do cryptocurrencies hedge against EPU and the equity market volatility during COVID-19? New evidence from quantile coherency analysis. Journal of International Financial Markets, Institutions and Money 72: 101324. [Google Scholar] [CrossRef]
- Jorion, Philippe. 1996. Risk
^{2}: Measuring the risk in value at risk. Financial Analysts Journal 52: 47–56. [Google Scholar] [CrossRef] - J.P.Morgan/Reuters. 2018. RiskMetrics™. In Kovács Edina: A Value at Risk és az Expected Shortfall összehasonlítása és utótesztelési módszerei. Budapest: Corvinus-ELTE. [Google Scholar]
- Kumah, Seyram Pearl, and Jones Odei Mensah. 2020. Are cryptocurrencies connected to gold? A wavelet-based quantile-in-quantile approach. International Journal of Finance and Economics 27: 3640–59. [Google Scholar] [CrossRef]
- Lee, Yongjae, Woo Chang Kim, and Jang Ho Kim. 2020. Achieving Portfolio Diversification for Individuals with Low Financial Sustainability. Sustainability 12: 7073. [Google Scholar] [CrossRef]
- Lintner, John. 1965. The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47: 13–37. [Google Scholar] [CrossRef]
- Liu, Mengyao, Hiroaki Jotaki, and Hiroshi Takahashi. 2021. A Study of the Impact of Crypto Assets on Portfolio RiskReturn Characteristics Before and After COVID-19 Outbreak (2014–2020). In Proceedings of 15th KES International Conference. Singapore: Springer. [Google Scholar]
- Macrotrends.net. 2022. Available online: https://www.macrotrends.net/ (accessed on 5 January 2024).
- Maier-Paape, Stanislaus, and Qiji Jim Zhu. 2018. A general framework for portfolio theory—Part I: Theory and various models. Risks 6: 53. [Google Scholar] [CrossRef]
- Markowitz, Harry. 1952. Portfolio selection. Journal of Finance 7: 77–91. [Google Scholar]
- Markowitz, Harry. 1959. Portfolio Selection. Hoboken: Wiley. [Google Scholar]
- Mossin, Jan. 1966. Equilibrium in a capital asset market. Econometrica 34: 768–83. [Google Scholar] [CrossRef]
- Nakamoto, Satoshi. 2009. Bitcoin: A Peer-to-Peer Electronic Cash System. Available online: https://bitcoin.org/bitcoin.pdf (accessed on 28 March 2024).
- Reid, Fergal, and Martin Harrigan. 2011. An analysis of anonymity in the Bitcoin system, security and privacy in social networks. In Security and Privacy in Social Networks. Edited by Yaniv Altshuler, Yuval Elovici, Armin B. Cremers, Nadav Aharony and Alex Pentland. New York: Springer, pp. 197–223. [Google Scholar]
- Romero, Pilar Abad, Sonia Benito Muela, and Carmen López Martin. 2013. A comprehensive review of value at risk methodologies. Documentos De Trabajo 711: 1–50. [Google Scholar]
- Sharpe, William F. 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19: 425–42. [Google Scholar]
- Song, Ruiqiang, Min Shu, and Wei Zhu. 2022. The 2020 global stock market crash: Endogenous or exogenous? Physica A: Statistical Mechanics and Its Applications 585: 126425. [Google Scholar] [CrossRef]
- Sornette, D., Yannick Malevergne, and J.-F. Muzy. 2004. Volatility fingerprints of large shocks: Endogenous versus exogenous. In The Application of Econophysics: Proceedings of the Second Nikkei Econophysics Symposium. Tokyo: Springer. [Google Scholar]
- Stoyanov, Stoyan V., Svetlozar T. Rachev, and Frank J. Fabozzi. 2013. Sensitivity of portfolio VaR and CVaR to portfolio return characteristics. Annals of Operations Research 205: 169–87. [Google Scholar] [CrossRef]
- Tertina, Kat, and John Schmidt. 2022. How to Buy Cryptocurrency. Forbes. Available online: https://www.forbes.com/advisor/in/investing/cryptocurrency/how-to-buy-cryptocurrency/ (accessed on 28 March 2024).
- Tobin, James. 1958. Liquidity preference as behaviour towards risk. The Review of Economic Studies 26: 65–86. [Google Scholar] [CrossRef]
- Treynor, Jack L. 1999. Toward a theory of market value of risky assets. In Asset Pricing and Portfolio Performance: Models, Strategy and Performance Metrics. Edited by Robert A. Korajczyk. London: Risk Books, pp. 15–22. [Google Scholar]
- Umar, Zaghum, Francisco Jareño, and María de la O González. 2021. The impact of COVID-19-related media coverage on the return and volatility connectedness of cryptocurrencies and fiat currencies. Technological Forecasting and Social Change 172: 121025. [Google Scholar] [CrossRef] [PubMed]
- Vo, Au, Tomas A. Chapman, and Yen-Sheng Lee. 2021. Examining Bitcoin and economic determinants: An evolutionary perspective. Journal of Computer Information Systems 62: 572–86. [Google Scholar] [CrossRef]
- Wei, Dai. 1998. B-Money. Available online: http://www.weidai.com/bmoney.txt (accessed on 28 March 2024).
- Wu, Wanshan, Aviral Kumar Tiwari, Giray Gozgor, and Leping Huang. 2021. Does economic policy uncertainty affect cryptocurrency markets? Evidence from Twitter-based uncertainty measures. Research in International Business and Finance 58: 101478. [Google Scholar] [CrossRef]

**Figure 9.**Risked values of the Bitcoin–Apple stock portfolio at a 95% confidence interval in period 2.

**Figure 11.**Risked values of the Bitcoin–Apple stock portfolio at a 99% confidence interval in period 2.

**Figure 13.**Risked values of the Bitcoin–Apple stock portfolio at a 99.9% confidence interval in period 2.

**Figure 14.**Impact of the coronavirus outbreak and the Russian-Ukrainian war on the S&P500 index (Source: Authors’ construction based on Macrotrends.net 2022).

**Table 1.**Descriptive statistics of log returns during periods 1 and 2 (source: authors’ construction).

1st Period | ||||

Average | Standard Deviation | Skewness | Kurtosis | |

Bitcoin | 0.0055 | 0.0497 | −0.1749 | 1.4394 |

Gold | −0.0003 | 0.0103 | −1.1916 | 4.4728 |

Apple stock | 0.0005 | 0.0197 | −0.4058 | 1.7355 |

2nd Period | ||||

Average | Standard Deviation | Skewness | Kurtosis | |

Bitcoin | −0.0034 | 0.0427 | −1.0092 | 5.708 |

Gold | −0.0002 | 0.0092 | −0.2661 | 0.4231 |

Apple stock | 0.0001 | 0.0193 | −0.1554 | 0.4444 |

1st Period | 2nd Period | |
---|---|---|

Bitcoin | 0.000 | 0.000 |

Gold | 0.000 | 0.0883 |

Apple stock | 0.000 | 0.2138 |

**Table 3.**Correlation matrix of the log returns of the instruments tested in periods 1 and 2 (source: authors’ construction).

1st Period | |||

Bitcoin | Gold | Apple | |

Bitcoin | 1 | ||

Gold | 0.0192 (p = 0.7620) | 1 | |

Apple stock | 0.1608 (p < 0.05) | 0.2008 (p < 0.05) | 1 |

2nd Period | |||

Bitcoin | Gold | Apple | |

Bitcoin | 1 | ||

Gold | 0.0247 (p = 0.6963) | 1 | |

Apple stock | 0.4313 (p < 0.05) | −0.1352 (p < 0.05) | 1 |

Bitcoin–Gold | VaR | CVaR (Averaged) | CVaR (Mills) | MVaR |

95% | 0% | 0% | 0% | 0% |

99% | 0% | 0% | 0% | 0% |

99.90% | 0% | 0% | 0% | 0% |

Bitcoin–Apple Stock | VaR | CVaR (Averaged) | CVaR (Mills) | MVaR |

95% | 0% | 0% | 0% | 0% |

99% | 0% | 3% | 0% | 0% |

99.90% | 0% | 0% | 0% | 0% |

Bitcoin–Gold 95% Confidence Int. | Bitcoin–Apple Stock 95% Confidence Int. | ||||||

Model | Bitcoin Weight | Minimum Loss | Model | Bitcoin Weight | Minimum Loss | ||

VaR | 0.00% | 4.53% | 45,294 EUR | VaR | 0.00% | 17.14% | 171,426 EUR |

CVaR (averaged) | 0.00% | 6.41% | 64,127 EUR | CVaR (averaged) | 0.00% | 20.63% | 206,332 EUR |

CVaR (Mills) | 0.00% | 5.49% | 54,937 EUR | CVaR (Mills) | 0.00% | 22.95% | 229,483 EUR |

MVaR | 0.00% | 2.48% | 24,809 EUR | MVaR | 0.00% | 15.75% | 157,481 EUR |

Bitcoin–Gold 99% Confidence Int. | Bitcoin–Apple Stock 99% Confidence Int. | ||||||

Model | Bitcoin Weight | Minimum Loss | Model | Bitcoin Weight | Minimum Loss | ||

VaR | 0.00% | 7.45% | 74,512 EUR | VaR | 0.00% | 22.96% | 229,577 EUR |

CVaR (averaged) | 0.00% | 8.90% | 88,963 EUR | CVaR (averaged) | 3.00% | 25.82% | 258,163 EUR |

CVaR (Mills) | 0.00% | 8.34% | 83,403 EUR | CVaR (Mills) | 0.00% | 26.99% | 269,877 EUR |

MVaR | 0.00% | 5.58% | 55,847 EUR | MVaR | 0.00% | 23.03% | 230,269 EUR |

Bitcoin–Gold 99.9% Confidence Int. | Bitcoin–Apple Stock 99.9% Confidence Int. | ||||||

Model | Bitcoin Weight | Minimum Loss | Model | Bitcoin Weight | Minimum Loss | ||

VaR | 0.00% | 10.63% | 106,271 EUR | VaR | 0.00% | 28.17% | 281,693 EUR |

CVaR (averaged) | 0.00% | 11.59% | 115,863 EUR | CVaR (averaged) | 0.00% | 29.76% | 297,567 EUR |

CVaR (Mills) | 0.00% | 11.45% | 114,469 EUR | CVaR (Mills) | 0.00% | 31.06% | 310,638 EUR |

MVaR | 0.00% | 11.15% | 111,520 EUR | MVaR | 0.00% | 31.86% | 318,567 EUR |

Bitcoin–Gold | VaR | CVaR (Averaged) | CVaR (Mills) | MVaR |

95% | 26% | 18% | 26% | 19% |

99% | 21% | 13% | 21% | 10% |

99.90% | 16% | 13% | 16% | 4% |

Bitcoin–Apple Stock | VaR | CVaR (Averaged) | CVaR (Mills) | MVaR |

95% | 81% | 55% | 81% | 78% |

99% | 47% | 45% | 47% | 47% |

99.90% | 29% | 31% | 29% | 24% |

Bitcoin–Gold 95% Confidence Int. | Bitcoin–Apple Stock 95% Confidence Int. | ||||||

Model | Bitcoin Weight | Minimum Loss | Model | Bitcoin Weight | Minimum Loss | ||

VaR | 26.00% | 1.56% | 15,566 EUR | VaR | 81.00% | 5.75% | 57,477 EUR |

CVaR (averaged) | 18.00% | 3.73% | 37,312 EUR | CVaR (averaged) | 55.00% | 11.93% | 119,275 EUR |

CVaR (Mills) | 26.00% | 1.78% | 17,808 EUR | CVaR (Mills) | 81.00% | 6.59% | 65,949 EUR |

MVaR | 19.00% | 0.00% | 21 EUR | MVaR | 78.00% | 0.00% | 2 EUR |

Bitcoin–Gold 99% Confidence Int. | Bitcoin–Apple Stock 99% Confidence Int. | ||||||

Model | Bitcoin Weight | Minimum Loss | Model | Bitcoin Weight | Minimum Loss | ||

VaR | 21.00% | 4.83% | 48,291 EUR | VaR | 47.00% | 14.46% | 144,638 EUR |

CVaR (averaged) | 13.00% | 6.15% | 61,510 EUR | CVaR (averaged) | 45.00% | 17.58% | 175,812 EUR |

CVaR (Mills) | 21.00% | 5.29% | 52,931 EUR | CVaR (Mills) | 47.00% | 15.97% | 159,723 EUR |

MVaR | 10.00% | 4.99% | 49,921 EUR | MVaR | 47.00% | 10.49% | 104,904 EUR |

Bitcoin–Gold 99.9% Confidence Int. | Bitcoin–Apple stock 99.9% Confidence Int. | ||||||

Model | Bitcoin Weight | Minimum Loss | Model | Bitcoin Weight | Minimum Loss | ||

VaR | 16.00% | 7.55% | 75,537 EUR | VaR | 29.00% | 21.12% | 211,159 EUR |

CVaR (averaged) | 13.00% | 8.68% | 86,753 EUR | CVaR (averaged) | 31.00% | 22.92% | 229,234 EUR |

CVaR (Mills) | 16.00% | 8.05% | 80,526 EUR | CVaR (Mills) | 29.00% | 22.68% | 226,849 EUR |

MVaR | 4.00% | 8.47% | 84,651 EUR | MVaR | 24.00% | 17.08% | 170,802 EUR |

Bitcoin–Gold | VaR | CVaR (Averaged) | CVaR (Mills) | MVaR |

95% | 0.9987 (p < 0.05) | 0.9984 (p < 0.05) | 0.022 (p < 0.05) | 0.9995 (p < 0.05) |

99% | 0.9981 (p < 0.05) | 0.9978 (p < 0.05) | 0.9963 (p < 0.05) | 0.9988 (p < 0.05) |

99.90% | 0.997 (p < 0.05) | 0.9964 (p < 0.05) | 0.9969 (p < 0.05) | 0.9947 (p < 0.05) |

Bitcoin–Apple Stock | VaR | CVaR (Averaged) | CVaR (Mills) | MVaR |

95% | 0.9961 (p < 0.05) | 0.995 (p < 0.05) | −0.0577 (p < 0.05) | 0.9966 (p < 0.05) |

99% | 0.9942 (p < 0.05) | 0.9934 (p < 0.05) | 0.9959 (p < 0.05) | 0.993 (p < 0.05) |

99.90% | 0.9915 (p < 0.05) | 0.9902 (p < 0.05) | 0.9931 (p < 0.05) | 0.9835 (p < 0.05) |

Bitcoin–Gold | VaR | CVaR (Averaged) | CVaR (Mills) | MVaR |

95% | 0.7896 (p < 0.05) | 0.9381 (p < 0.05) | 0.7713 (p < 0.05) | −0.6647 (p < 0.05) |

99% | 0.9563 (p < 0.05) | 0.9681 (p < 0.05) | 0.9539 (p < 0.05) | 0.9682 (p < 0.05) |

99.90% | 0.9703 (p < 0.05) | 0.971 (p < 0.05) | 0.9695 (p < 0.05) | 0.9671 (p < 0.05) |

Bitcoin–Apple Stock | VaR | CVaR (Averaged) | CVaR (Mills) | MVaR |

95% | −0.9049 (p < 0.05) | −0.4983 (p < 0.05) | −0.8971 (p < 0.05) | −0.9717 (p < 0.05) |

99% | 0.1232 (p < 0.05) | 0.6175 (p < 0.05) | −0.0478 (p < 0.05) | 0.4998 (p < 0.05) |

99.90% | 0.7766 (p < 0.05) | 0.8182 (p < 0.05) | 0.7435 (p < 0.05) | 0.8933 (p < 0.05) |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bányai, A.; Tatay, T.; Thalmeiner, G.; Pataki, L.
Optimising Portfolio Risk by Involving Crypto Assets in a Volatile Macroeconomic Environment. *Risks* **2024**, *12*, 68.
https://doi.org/10.3390/risks12040068

**AMA Style**

Bányai A, Tatay T, Thalmeiner G, Pataki L.
Optimising Portfolio Risk by Involving Crypto Assets in a Volatile Macroeconomic Environment. *Risks*. 2024; 12(4):68.
https://doi.org/10.3390/risks12040068

**Chicago/Turabian Style**

Bányai, Attila, Tibor Tatay, Gergő Thalmeiner, and László Pataki.
2024. "Optimising Portfolio Risk by Involving Crypto Assets in a Volatile Macroeconomic Environment" *Risks* 12, no. 4: 68.
https://doi.org/10.3390/risks12040068