  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.
