The battle of informational efficiency: Cryptocurrencies vs. classical assets

This research applies the Martins, Fernandes, and Nascimento (MFN) method for estimating statistical confidence intervals in information theory, focusing on two key quantifiers: Permutation entropy $\left(H_s\right)$ and Fisher information measure $\left(F_s\right)$. Our study focuses on the daily closing price time series of five major cryptocurrencies — Bitcoin (BTC), Ethereum (ETH), BNB, Solana (SOL), and XRP — alongside two stock market indexes (S&P 500 and NYSE Composite), one commodity (Gold), and one exchange rate (EUR/USD). Based on the values of $H_s$ and $F_s$, we construct the Shannon–Fisher Causality Plane (SFCP), which allows us to quantify disorder and evaluate randomness in the daily closing prices of various financial assets. Also, we provide novel insights related to the SFCP with density contours. Our findings reveal that XRP, BNB, and BTC are positioned close to the random ideal position $\left(H_s=1,F_s=0\right)$ on the SFCP, which suggests they exhibit higher disorder, lower predictability, greater informational efficiency, and reduced informational asymmetry and speculative activity. In contrast, the S&P 500, NYA, and Gold are positioned further from this ideal point, indicating increased market inefficiencies and speculation. Also, cryptocurrencies demonstrate less dense density contours with high $\left(H_s\right)$ and low $\left(F_s\right)$, while traditional financial assets show denser contours with low $\left(H_s\right)$ and high $\left(F_s\right)$. XRP, BNB, and BTC have less dense contours than other assets. The densest contours are observed for Gold, NYA, and S&P 500. Principal Component Analysis (PCA) supports these findings by confirming that cryptocurrencies, S&P 500 and Gold, function as safe-haven assets. Overall, the study highlights the potential of cryptocurrencies to provide more reliable investment signals, thereby mitigating risks associated with information asymmetry and speculative trading.