Leverage and risk relativity: how to beat an index

In this paper we show that risk associated with leverage is fundamentally relative to an arbitrary choice of reference asset or portfolio. We characterize leverage risk as a drawdown risk measure relative to the chosen reference asset. We further prove that the growth optimal Kelly portfolio is the only portfolio for which the relative drawdown risk is not dependent on the choice of the reference asset. Additionally, we show how to translate an investor’s viewpoint from one choice of reference asset to another and establish conditions for when two investors can be said to face identical leverage risk. We also prove that, for a given reference asset, the correlation between two arbitrary portfolios with identical leverage risk equals the ratio of their Sharpe ratios if and only if the leverage risk is consistently traded. More surprisingly, we observe that leverage applied to the growth optimal Kelly strategy affects the drawdown risk in much the same way as the speed of light affects velocities in Einstein’s theory of special relativity. Finally, we provide details on how to trade in order to beat an arbitrary index for a given leverage risk target.

Leverage and risk relativity: how to beat an index 2021_1