The value and cost of more stages in stochastic programing: a statistical analysis on a set of portfolio choice problems

Sequential decision problems under uncertainty are commonly studied with stochastic programing. An important modeling choice is the number of stages. More stages allow additional information to be captured, but is associated with a coarser representation of uncertainty may worsen solution quality. In this paper, we study this trade-off, with the objective to advance the understanding of how the number of stages affect solution quality in stochastic programing. We show: (i) how the optimistic bounds from stochastic programing gradually suggest improved performance with more stages, while the real solution quality simultaneously deteriorates; and (ii) that real performance can be improved by adding stages, but only up to some point, after which more stages are detrimental. Further, we highlight the importance of understanding what creates the value of more stages in the problem of interest, and particularly if this can be captured in models with few stages. The numerical experiments are based on the classic portfolio choice problem of an investor with constant relative risk aversion preferences, maximizing the expected utility of terminal wealth. We study instances with proportional transaction costs and predictability in returns, which takes this problem into an inherently multi-stage nature.