  In this paper, we consider a frequency-based portfolio optimization problem
with $m \geq 2$ assets when the expected logarithmic growth (ELG) rate of
wealth is used as the performance metric. With the aid of the notion called
dominant asset, it is known that the optimal ELG level is achieved by investing
all available funds on that asset. However, such an "all-in" strategy is
arguably too risky to implement in practice. Motivated by this issue, we study
the case where the portfolio weights are chosen in a rather ad-hoc manner and a
buy-and-hold strategy is subsequently used. Then we show that, if the
underlying portfolio contains a dominant asset, buy and hold on that specific
asset is asymptotically log-optimal with a sublinear rate of convergence. This
result also extends to the scenario where a trader either does not have a
probabilistic model for the returns or does not trust a model obtained from
historical data. To be more specific, we show that if a market contains a
dominant asset, buy and hold a market portfolio involving nonzero weights for
each asset is asymptotically log-optimal. Additionally, this paper also
includes a conjecture regarding the property called high-frequency maximality.
That is, in the absence of transaction costs, high-frequency rebalancing is
unbeatable in the ELG sense. Support for the conjecture, involving a lemma for
a weak version of the conjecture, is provided. This conjecture, if true,
enables us to improve the log-optimality result obtained previously. Finally, a
result that indicates a way regarding an issue about when should one to
rebalance their portfolio if needed, is also provided. Examples, some involving
simulations with historical data, are also provided along the way to illustrate
the~theory.
