Modeling the rebalancing slippage of Leveraged Exchange-Traded Funds

Leveraged exchange-traded funds are designed to track a multiple of the daily return of an underlying benchmark index. In order to keep a fixed exposure to the benchmark index, leveraged ETFs have to rebalance their positions everyday, generating a structural ’rebalancing slippage’ which has been documented in several empirical studies. This paper quantifies the rebalancing slippage of leveraged ETFs by developing a tractable model for the dynamics of leveraged funds, which takes into account the impact of active management by leveraged ETFs. We characterize the rebalancing strategy of the leveraged fund and its impact on the value of the leveraged ETF and we model its dynamics in discrete-time. We show that the rebalancing impact systematically diminishes the daily return of the leveraged ETF and that, over a holding period of more than one day, leveraged ETFs develop a tracking-error which can be decomposed between a compounding deviation – that has already been documented and quantified in previous studies – and a rebalancing deviation. The study of the continuous-time limit of the multi-period model allows us to obtain analytical formulas for the rebalancing slippage and the tracking-error of the leveraged ETF. Our theoretical results are consistent with empirical studies which find that tracking-error and rebalancing impact are larger in periods of high volatility and for leveraged ETFs with negative leverage ratios.