This study investigates the influence of structural change on the diversification, hedging, and safe-haven characteristics of Bitcoin and Ethereum against various financial assets such as gold, the US Dollar Index, stock indices, oil, and commodity indices from August 7, 2015, to August 15, 2022, using the DCC–ARMA–GARCH models with the CUSUM test. Our results indicate that cryptocurrencies have the same characteristics vis-à-vis financial markets during the entire sample period and periods tied to the date of major international events (COVID-19 and the early-2022 Russia–Ukraine War). However, we find that cryptocurrencies play different roles against specific asset markets in different periods separated by structural change models. Our findings suggest that incorporating structural changes into a model accounts for higher volatility and may better describe the real-world capabilities of cryptocurrencies against financial assets.