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Bitcoin versus S&P 500 Index: Return and Risk Analysis

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  • A. H. Nzokem

Abstract

The S&P 500 index is considered the most popular trading instrument in financial markets. With the rise of cryptocurrencies over the past years, Bitcoin has also grown in popularity and adoption. The paper aims to analyze the daily return distribution of the Bitcoin and S&P 500 index and assess their tail probabilities through two financial risk measures. As a methodology, We use Bitcoin and S&P 500 Index daily return data to fit The seven-parameter General Tempered Stable (GTS) distribution using the advanced Fast Fractional Fourier transform (FRFT) scheme developed by combining the Fast Fractional Fourier (FRFT) algorithm and the 12-point rule Composite Newton-Cotes Quadrature. The findings show that peakedness is the main characteristic of the S&P 500 return distribution, whereas heavy-tailedness is the main characteristic of the Bitcoin return distribution. The GTS distribution shows that $80.05\%$ of S&P 500 returns are within $-1.06\%$ and $1.23\%$ against only $40.32\%$ of Bitcoin returns. At a risk level ($\alpha$), the severity of the loss ($AVaR_{\alpha}(X)$) on the left side of the distribution is larger than the severity of the profit ($AVaR_{1-\alpha}(X)$) on the right side of the distribution. Compared to the S&P 500 index, Bitcoin has $39.73\%$ more prevalence to produce high daily returns (more than $1.23\%$ or less than $-1.06\%$). The severity analysis shows that at a risk level ($\alpha$) the average value-at-risk ($AVaR(X)$) of the bitcoin returns at one significant figure is four times larger than that of the S&P 500 index returns at the same risk.

Suggested Citation

  • A. H. Nzokem, 2023. "Bitcoin versus S&P 500 Index: Return and Risk Analysis," Papers 2310.02436, arXiv.org.
    • Handle: RePEc:arx:papers:2310.02436
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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. A. H. Nzokem, 2022. "Pricing European Options under Stochastic Volatility Models: Case of five-Parameter Variance-Gamma Process," Papers 2201.03378, arXiv.org, revised Jan 2023.
    3. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    4. A.H. Nzokem, 2021. "SIS Epidemic Model Birth-and-Death Markov Chain Approach," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(4), pages 1-10, July.
    5. Young Kim & Svetlozar Rachev & Michele Bianchi & Frank Fabozzi, 2009. "Computing VAR and AVaR in Infinitely Divisible Distributions," Yale School of Management Working Papers amz2569, Yale School of Management.
    6. Peter H. Westfall, 2014. "Kurtosis as Peakedness, 1905-2014. R.I.P," The American Statistician, Taylor & Francis Journals, vol. 68(3), pages 191-195, April.
    7. Aubain Hilaire Nzokem, 2023. "Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process," JRFM, MDPI, vol. 16(1), pages 1-28, January.
    8. Küchler, Uwe & Tappe, Stefan, 2013. "Tempered stable distributions and processes," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4256-4293.
    9. Dirk G. Baur & Thomas Dimpfl, 2021. "The volatility of Bitcoin and its role as a medium of exchange and a store of value," Empirical Economics, Springer, vol. 61(5), pages 2663-2683, November.
    10. A. H. Nzokem & V. T. Montshiwa, 2022. "Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach," Papers 2205.00586, arXiv.org, revised Jun 2022.
    11. A. H. Nzokem, 2023. "European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis," Papers 2304.06060, arXiv.org, revised Feb 2025.
    12. Conghui Chen & Lanlan Liu & Ningru Zhao, 2020. "Fear Sentiment, Uncertainty, and Bitcoin Price Dynamics: The Case of COVID-19," Emerging Markets Finance and Trade, Taylor & Francis Journals, vol. 56(10), pages 2298-2309, August.
    13. Baur, Dirk G. & Hong, KiHoon & Lee, Adrian D., 2018. "Bitcoin: Medium of exchange or speculative assets?," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 54(C), pages 177-189.
    14. repec:dau:papers:123456789/1392 is not listed on IDEAS
    15. Conghui Chen & Lanlan Liu & Ningru Zhao, 2020. "Fear Sentiment, Uncertainty, and Bitcoin Price Dynamics: The Case of COVID-19," Emerging Markets Finance and Trade, Taylor & Francis Journals, vol. 56(10), pages 2298-2309, August.
    16. Rodrigo Hakim das Neves, 2020. "Bitcoin pricing: impact of attractiveness variables," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 6(1), pages 1-18, December.
    17. Young Kim & Svetlozar Rachev & Michele Bianchi & Frank Fabozzi, 2009. "Computing VAR and AVaR in Infinitely Divisible Distributions," Yale School of Management Working Papers amz2569, Yale School of Management.
    18. Dimitrios Koutmos, 2023. "Investor sentiment and bitcoin prices," Review of Quantitative Finance and Accounting, Springer, vol. 60(1), pages 1-29, January.
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