FALLACY OF THE LOG-NORMAL APPROXIMATION TO OPTIhlAL PORTFOLIO DECISION-3lAKING OVER MAhi PERIODS*

The fallacy that a many-period expected-utility maximizer should maximize (a) the expected logarithm of portfolio outcomes or (b) the expected average compound return of his portfolio is now understood to rest upon a fallacious use of the LawofLargc Nurtrbcrs. This paper exposes a more subtle fallacy based upon a fallacious use of the Cenfral-Limit Throrcnr. While the properly normalized product of indcpcndent random variables does asymptotically approach a log-normal distribution under proper assumptions, it involves a fallacious manipulation of double limits to infer from this that a maximizer of cxpcctcd utility after many periods will get a useful approximation to his optimal policy by calculating an cffrcicncy frontier based upon (a) the cxpcctcd log of wealth outcomes and its variance or(b) the cxpcctcd average compound rclurn and its variance. Lxpeclcd utilities cnlcuktted from the surrogate log-normal function dilTcr systcmaticnlly from the corrccl cxpcctcd utilities calculated from the true probability distribufion. A new concept of ‘initial wealth equivalent’ provides a transitive ordering of portfolios that illuminates commonly held confusions. A non-fallacious application of the log-normal limit and its associated mc:m-variance cllicicncy frontier is established for a limit whcrc nny/i.rcd/ horizon period is subdivided into cvcr more indcpcndcnt sub-intervals. Strong mutual-fund Separation Theorems are then shown to be asymptotically valid.