A Note on Universal Bilinear Portfolios

TL;DR

This paper extends Cover's universal portfolio theory to bilinear trading strategies, creating a universal portfolio that asymptotically matches the best bilinear strategy's growth rate in hindsight.

Contribution

It generalizes the universal portfolio concept to bilinear strategies, providing a method to asymptotically compete with the best bilinear trading strategy in hindsight.

Findings

Constructed a universal bilinear portfolio with guaranteed asymptotic performance.

Proved the universal portfolio asymptotically dominates the original universal portfolio.

Showed the potential for an endless hierarchy of more complex universal portfolios.

Abstract

This note provides a neat and enjoyable expansion and application of the magnificent Ordentlich-Cover theory of "universal portfolios." I generalize Cover's benchmark of the best constant-rebalanced portfolio (or 1-linear trading strategy) in hindsight by considering the best bilinear trading strategy determined in hindsight for the realized sequence of asset prices. A bilinear trading strategy is a mini two-period active strategy whose final capital growth factor is linear separately in each period's gross return vector for the asset market. I apply Cover's ingenious (1991) performance-weighted averaging technique to construct a universal bilinear portfolio that is guaranteed (uniformly for all possible market behavior) to compound its money at the same asymptotic rate as the best bilinear trading strategy in hindsight. Thus, the universal bilinear portfolio asymptotically dominates the original (1-linear) universal portfolio in the same technical sense that Cover's universal portfolios asymptotically dominate all constant-rebalanced portfolios and all buy-and-hold strategies. In fact, like so many Russian dolls, one can get carried away and use these ideas to construct an endless hierarchy of ever more dominant $H$-linear universal portfolios.