  The Cover universal portfolio (UP from now on) has many interesting
theoretical and numerical properties and was investigated for a long time.
Building on it, we explore what happens when we add this UP to the market as a
new synthetic asset and construct by recurrence higher order UPs. We
investigate some important theoretical properties of the high order UPs and
show in particular that they are indeed different from the Cover UP and are
capable to break the time permutation invariance. We show that under some
perturbation regime the second high order UP has better Sharp ratio than the
standard UP and briefly investigate arbitrage opportunities thus created.
Numerical experiences on a benchmark from the literature confirm that high
order UPs improve Cover's UP performances.
