Efficient Frontier, Lognormal Returns, and Shortfall Constraint
19 Pages Posted: 27 Jun 2015
Date Written: June 26, 2015
An efficient frontier model is derived within a problem of passive management. An aggregate portfolio is rebalanced annually to restore the percent weights of its strategic asset allocation; its annual total returns are assumed to be independently and lognormally distributed. Expanding on previous theoretical results, we show how a minimum-variance set based on ordinary returns turns into a minimum-variance set based on logarithmic returns. According to our theoretical results, which have a general validity, inefficient portfolios based on ordinary returns cannot turn into efficient portfolios based on logarithmic returns, whereas efficient portfolios based on ordinary 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, the shortfall constraint approach is extended to the case of lognormal portfolio returns. 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.
Keywords: Portfolio management, passive management, efficient frontier, lognormal returns, shortfall constraint
JEL Classification: G11
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