  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 returns 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.
