Universal Bilinear Portfolios for Multi-Period Trading

Summary

The note extends universal portfolio theory beyond constant-rebalanced strategies. It considers bilinear strategies, whose final growth depends linearly on each of two successive market return vectors, and compares them with the best such strategy selected in hindsight. A performance-weighted averaging method, adapted from Cover’s universal portfolio approach, constructs a portfolio without knowing in advance which bilinear strategy will perform best.

The stated guarantee is asymptotic: for every possible market sequence, the constructed portfolio compounds at the same asymptotic rate as the hindsight-optimal bilinear strategy. The note also positions this as a stronger benchmark than the original universal portfolio and sketches an extension to strategies spanning more periods. The excerpt supplies a theoretical claim, not empirical tests, transaction cost analysis, or implementation details. The result concerns long-run growth comparisons, so it does not by itself establish practical profitability over finite samples or under real-world trading frictions.

Key ideas

• The note extends universal portfolios from constant-rebalanced allocations to bilinear strategies.
• A bilinear strategy’s growth is linear in the return vectors from each of two periods separately.
• Performance-weighted averaging adapts the portfolio to the realized sequence of returns.
• The stated guarantee is asymptotic dominance relative to the best bilinear strategy in hindsight.
• The framework can be extended to strategies spanning additional periods.

Tags

# A Note on Universal Bilinear Portfolios


# A Note on Universal Bilinear Portfolios









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.