  Although maximizing median and quantiles is intuitively appealing and has an
axiomatic foundation, it is difficult to study the optimal portfolio strategy
due to the discontinuity and time inconsistency in the objective function. We
use the intra-personal equilibrium approach to study the problem.
Interestingly, we find that the only viable outcome is from the median
maximization, because for other quantiles either the equilibrium does not exist
or there is no investment in the risky assets. The median maximization strategy
gives a simple explanation to why wealthier people invest more percentage of
their wealth in risky assets.
