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 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.

PORTFOLIO SELECTION AND ONLINE LEARNING

This paper studies a new strategy for selecting portfolios in the stock market. The strategy is inspired by two streams of previous work: (1) work on universalization of strategies for portfolio selection, which began with Thomas Cover's work on constant rebalanced portfolios, published in 1991,4 and (2) more general work on universalization of online algorithms,17,21,23,30 especially Vladimir Vovk's work on the aggregating algorithm and Markov switching strategies.32 The proposed investment strategy achieves asymptotically the same exponential rate of growth as the portfolio that turns out to be best expost in the long run and does not require any underlying statistical assumptions on the nature of the stock market.