Multi-period Mean-Buffered Probability of Exceedance in Defined Contribution Portfolio Optimization

Abstract

We investigate multi-period mean-risk portfolio optimization for long-horizon Defined Contribution plans, focusing on buffered Probability of Exceedance (bPoE), a more intuitive, dollar-based alternative to Conditional Value-at-Risk (CVaR). We formulate both pre-commitment and time-consistent Mean-bPoE and Mean-CVaR portfolio optimization problems under realistic investment constraints (e.g., no leverage, no short selling) and jump-diffusion dynamics. These formulations are naturally framed as bilevel optimization problems, with an outer search over the shortfall threshold and an inner optimization over rebalancing decisions. We establish an equivalence between the pre-commitment formulations through a one-to-one correspondence of their scalarization optimal sets, while showing that no such equivalence holds in the time-consistent setting. We develop provably convergent numerical schemes for the value functions associated with both pre-commitment and time-consistent formulations of these mean-risk control problems. Using nearly a century of market data, we find that time-consistent Mean-bPoE strategies closely resemble their pre-commitment counterparts. In particular, they maintain alignment with investors' preferences for a minimum acceptable terminal wealth level-unlike time-consistent Mean-CVaR, which often leads to counterintuitive control behavior. We further show that bPoE, as a strictly tail-oriented measure, prioritizes guarding against catastrophic shortfalls while allowing meaningful upside exposure, making it especially appealing for long-horizon wealth security. These findings highlight bPoE's practical advantages for Defined Contribution investment planning.