Before revision:  Because of the time inconsistency caused by quantiles, we consider intra-personal equilibrium strategies. We find that among the class of time-varying, affine portfolio strategies, the intra-personal equilibrium does not exist when \alpha>1/2, leads to zero investment in the risky assets when \alpha<1/2, and is a portfolio insurance strategy when \alpha=1/2.
After revision:  Because of the time inconsistency caused by quantiles, we consider 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 when \alpha>1/2, leads to zero investment in the risky assets when \alpha<1/2, and is a portfolio insurance strategy when \alpha=1/2.