Dynamic Mean-LPM and Mean-CVaR Portfolio Optimization in Continuous-time

摘要

Instead of controlling "symmetric" risks measured by central moments ofinvestment return or terminal wealth, more and more portfolio models haveshifted their focus to manage "asymmetric" downside risks that the investmentreturn is below certain threshold. Among the existing downside risk measures,the lower-partial moments (LPM) and conditional value-at-risk (CVaR) areprobably most promising. In this paper we investigate the dynamic mean-LPM andmean-CVaR portfolio optimization problems in continuous-time, while the currentliterature has only witnessed their static versions. Our contributions aretwo-fold, in both building up tractable formulations and deriving correspondinganalytical solutions. By imposing a limit funding level on the terminal wealth,we conquer the ill-posedness exhibited in the class of mean-downside riskportfolio models. The limit funding level not only enables us to solve bothdynamic mean-LPM and mean-CVaR portfolio optimization problems, but also offersa flexibility to tame the aggressiveness of the portfolio policies generatedfrom such mean - downside risk models. More specifically, for a general marketsetting, we prove the existence and uniqueness of the Lagrangian multiplies,which is a key step in applying the martingale approach, and establish atheoretical foundation for developing efficient numerical solution approaches.Moreover, for situations where the opportunity set of the market setting isdeterministic, we derive analytical portfolio policies for both dynamicmean-LPM and mean-CVaR formulations.