How efficient are sustainability-oriented equity markets under geopolitical and climate risk? Evidence from entropy and multifractal analysis

This paper investigates the informational efficiency and cross-dependence structure of sustainabilityoriented equity indices under major geopolitical and climate-related shock events from a complex-systems perspective. We analyze four sustainability-linked S&P indices-Environmental, Clean Energy, Smart Transportation, and Agribusiness-alongside the S&P Global BMI as a broad market benchmark, using daily data from 2016 to 2025. The empirical framework combines entropy-information quantifiers based on ordinal patterns with multifractal detrended cross-correlation analysis (MFDCCA), allowing us to capture nonlinear, scale-dependent, and regime-sensitive dynamics beyond standard correlation measures. Informational efficiency is assessed through permutation entropy and Fisher information jointly embedded in the Shannon-Fisher causality plane, while cross-market dependence with geopolitical risk and climate physical risk is examined through multifractal scaling exponents and spectra across distinct crisis periods, including COVID-19, the Russia-Ukraine war, the Israel-Hamas conflict, and a recent trade-policy shock. The results reveal substantial heterogeneity across sustainability sectors. Transportation-related indices appear closer to the random-efficiency benchmark, whereas Environmental and Agribusiness indices exhibit stronger local temporal structure. Shock episodes generate pronounced, but sector-specific, departures from baseline efficiency. Multifractal analysis uncovers persistent and highly heterogeneous cross-correlations between sustainability indices and external risk factors, which intensify during periods of elevated uncertainty, particularly for Clean Energy and Transportation under geopolitical stress and for Environmental and Agribusiness sectors under climate physical risk. Overall, the findings demonstrate that sustainabilityoriented equity markets are not informationally or structurally homogeneous and that external geopolitical and climate risks transmit into these markets in a nonlinear, multifractal, and scale-dependent manner. These results have implications for portfolio diversification, risk management, and the modeling of sustainabilitylinked financial systems under extreme events.