Skip to content
All library documents

Bitcoin and S&P 500 Return Tails Compared with a Tempered Stable Model

Article arXiv papers · Author: A. H. Nzokem
Read the original

Summary

The study compares daily return distributions for Bitcoin and the S&P 500, focusing on tail probabilities and the severity of losses and gains. It fits a seven-parameter General Tempered Stable distribution using a numerical procedure that combines a Fast Fractional Fourier Transform with composite Newton–Cotes quadrature. The reported distributional characteristics are greater peakedness for S&P 500 returns and heavier tails for Bitcoin returns.

The fitted model places 80.05% of S&P 500 returns within −1.06% to 1.23%, compared with 40.32% of Bitcoin returns. It also estimates more frequent extreme daily Bitcoin returns and average value-at-risk about four times that of the index at the stated risk level. For both assets, estimated downside severity exceeds upside severity. These are model-based findings from daily return data; the excerpt does not specify the sample period, parameter uncertainty, or whether the differences persist out of sample.

Key ideas

  • The study fits daily Bitcoin and S&P 500 returns with a seven-parameter General Tempered Stable distribution.
  • The reported S&P 500 distribution is more peaked, while Bitcoin returns are heavier-tailed.
  • The model estimates a higher prevalence of large daily Bitcoin moves than of comparable index moves.
  • Estimated downside average value-at-risk exceeds upside severity for both assets.
  • The excerpt does not report the sample period or uncertainty around the fitted estimates.

Tags

CryptoEquitiesStatisticsRisk managementUS markets
Full text
# Bitcoin versus S&P 500 Index: Return and Risk Analysis


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









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 ($α$), the severity of the loss ($AVaR_α(X)$) on the left side of the distribution is larger than the severity of the profit ($AVaR_{1-α}(X)$) 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 ($α$) 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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.