When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to its volatility and liquidity. In a model with one safe and one risky asset, with constant investment opportunities and proportional costs, we find strategies that maximize long term return given average volatility. As leverage increases, rising rebalancing costs imply declining Sharpe ratios. Beyond a critical level, even returns decline. Holding the Sharpe ratio constant, higher  volatility leads to superior returns through lower costs . For funds replicating benchmark multiples, such as leveraged ETFs, we identify the strategies that optimally trade off alpha against tracking error, and find that they depend on the target multiple and the benchmark's liquidity, but not its volatility .
When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to its volatility and liquidity. In a model with one safe and one risky asset, with constant investment opportunities and proportional costs, we find strategies that maximize long term return given average volatility. As leverage increases, rising rebalancing costs imply declining Sharpe ratios. Beyond a critical level, even returns decline. Holding the Sharpe ratio constant, higher asset volatility leads to superior returns through lower costs .
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. For funds replicating benchmark multiples, such as leveraged ETFs, we identify the strategies that optimally trade off alpha against tracking error, and find that they depend on the target multiple and the benchmark's liquidity, but not its volatility