Leverage, drawdowns and risk relativity

We develop a continuous-time framework in which the risk associated with leverage is parameterized by a dimensionless drawdown-risk index relative to an arbitrary reference asset. We further show that the growth-optimal Kelly portfolio is the unique portfolio whose relative drawdown-risk index is invariant to the reference asset. Additionally, we characterize when portfolios have equal relative drawdown-risk indices and how these indices transform under changes of reference asset. Finally, for a prescribed positive risk target, we derive the portfolio that maximizes instantaneous logarithmic outperformance relative to an arbitrary benchmark: a linear combination of the benchmark and the growth-optimal Kelly portfolio.