Abstract
An insightfulproblem of passive managementis considered, where an aggregate portfolio is rebalanced annually to restore the percent weights of its strategic asset allocation. Asits annual total returns are assumed to be time uncorrelatedand lognormally distributed, multi-period optimization boils down to single period optimization. Expanding on previous theoretical results, it is shown how a minimum-variance set based on linear returns turns into a minimum-variance set based on logarithmic returns. More precisely, inefficient portfolios based on linear returns cannot turn into efficient portfolios based on logarithmic returns, whereas efficient portfolios based on linear returns can also turn into inefficient portfolios based on logarithmic returns. In the latter instance, there can be two different qualitative patterns, both of which are portrayed by using historical data. Moreover, a dynamic shortfall constraint is introduced. Each threshold return can be turned into a threshold accumulation that has the same shortfall probability; coeteris paribus, the more distant the time horizon, the smaller the shortfall probability. As our procedure is analytically tractable, it might be operationally useful, especially to financial advisors and institutional investors.