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 .
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 .
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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
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
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strategies. We find that among the class of time-varying, affine portfolio strategies, the intra-personal
approach to study the problem. Interestingly, we find that the only viable outcome is from the median maximization, because for other quantiles either the
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when \alpha>1/2, leads to zero
or there is no
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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
. The median maximization strategy gives a simple explanation to why wealthier
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precentage of
percentage of their
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. Finally, we extend our model to account for multiple terminal time