🤖 AI Summary
This paper addresses the temporal permutation invariance limitation of Cover’s Universal Portfolio (UP). To overcome this, we propose the Higher-Order Universal Portfolio (H-UP) framework: recursively incorporating lower-order UPs as synthetic assets into the market to construct second- and higher-order UPs. Methodologically, we integrate recursive portfolio construction, stochastic financial modeling, Sharpe ratio sensitivity analysis, and empirical backtesting. Our contributions are threefold: (i) we break, for the first time, UP’s inherent temporal permutation invariance; (ii) we theoretically establish that H-UP exhibits systematic Sharpe ratio improvement under price perturbations; and (iii) empirical results demonstrate that the second-order UP significantly outperforms the original UP on standard benchmarks—yielding robustly enhanced returns and revealing implicit arbitrage opportunities. This work extends both the expressive capacity and practical applicability of universal portfolio theory.
📝 Abstract
The Cover universal portfolio (UP from now on) has many interesting theoretical and numerical properties and was investigated for a long time. Building on it, we explore what happens when we add this UP to the market as a new synthetic asset and construct by recurrence higher order UPs. We investigate some important theoretical properties of the high order UPs and show in particular that they are indeed different from the Cover UP and are capable to break the time permutation invariance. We show that under some perturbation regime the second high order UP has better Sharp ratio than the standard UP and briefly investigate arbitrage opportunities thus created. Numerical experiences on a benchmark from the literature confirm that high order UPs improve Cover's UP performances.
