A Note on Universal Bilinear Portfolios

Alex Garivaltis

doi:10.3390/ijfs9010011

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 Thomas Cover’s ingenious 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.

Highlights

We start by defining the concept of a bilinear trading strategy, which is a simple 2-period active strategy that generalizes the notion of a constant-rebalanced portfolio (CRP)
The original (1-linear) universal portfolios guarantee to achieve a high percentage of the final wealth that would have accrued to the best constant-rebalanced portfolio in hindsight for the actual sequence of asset prices
Inspired by the fact that a constant-rebalanced portfolio is a trading strategy whose capital growth factor in any given period is a linear function of the market’s gross return vector, we decided to consider the wider class of bilinear trading strategies, which are mini

Summary

Bilinear Trading Strategies

We start by defining the concept of a bilinear trading strategy (or bilinear portfolio), which is a simple 2-period active strategy that generalizes the notion of a constant-rebalanced portfolio (CRP). Every constant-rebalanced portfolio (cf with Thomas Cover (1991)) c := (c1 , ..., cm )0 amounts to a bilinear trading strategy that is represented by the outer product B := cc0 , e.g., bij := ci c j for all i, j ∈ {1, ..., m}. D := (d1 , ..., dm)0 ∈ ∆m in period (c1 , ..., cm )0 ∈ ∆m in period 1 and always uses the portfolio This scheme is a bilinear trading strategy that corresponds to the outer product B := cd0 , e.g., bij := ci d j for all i, j ∈ {1, ..., m}.

Universal Bilinear Portfolios

Resolution of the Motivating Example

Summary and Conclusions

Findings

Disclosures