An insightful problem of passive management is considered, where an aggregate portfolio is rebalanced annually to restore the percent weights of its strategic asset allocation. As its annual total returns are assumed to be time uncorrelated and 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, it is found that there can be two different qualitative patterns, one of which is unprecedented and striking. Both patterns are tentatively portrayed by using historical data. The resulting efficient frontier is readily complemented by a dynamic shortfall constraint. 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 individual investors.
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