A levered ETF anomaly explained

Abstract Counterintuitively, the S&P 500 Index rose between January 3, 2022, and December 29, 2023, while exchange-traded funds (ETFs) seeking to deliver $$2\times $$ and $$3\times $$ daily returns of the index delivered substantially negative returns. Roughly two-thirds of the difference between the returns of the index and the levered ETFs can be attributed to a well-known effect: volatility over a long horizon. The remainder is explained by a timing effect that is more obscure: the covariance between the ratio of daily price return of the ETF to the index and the index return.