Portfolio Selection under Median and Quantile Maximization

In this paper, we study a portfolio selection problem in which an agent trades a risk-free asset and multiple risky assets with deterministic mean return rates and volatility and wants to maximize the $\alpha$-quantile of her wealth at some terminal time. 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 $\alpha=1/2$. We then compare the intra-personal equilibrium strategy in the case of $\alpha=1/2$, namely under median maximization, to some other strategies and apply it to explain why more wealthy people invest more precentage of wealth in risky assets. Finally, we extend our model to account for multiple terminal time.