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 Law of Large Numbers. This paper exposes a more subtle fallacy based upon a fallacious use of the Central-Limit Theorem. While the properly normalized product of independent 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 expected utility after many periods will get a useful approximation to his optimal policy by calculating an efficiency frontier based upon (a) the expected log of wealth outcomes and its variance or (b) the expected average compound return and its variance. Expected utilities calculated from the surrogate log-normal function differ systematically from the correct expected utilities calculated from the true probability distribution. 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 mean-variance efficiency frontier is established for a limit where any fixed horizon period is subdivided into ever more independent sub-intervals. Strong mutual-fund Separation Theorems are then shown to be asymptotically valid.

This paper develops general overtaking techniques for studying the asymptotic properties of portfolio policies optimal with respect to a terminal utility valuation. For a restricted class of utility functions the sequence of optimal constant (non-revised) portfolio policies formed as the horizon recedes into the future is shown to converge. Furthermore, for utility functions unbounded above and below, this turnpike policy need not be the policy associated with the minimal constant relative risk aversion function that bounds the valuation function from above. Finally, an analogy between the portfolio turnpike problem and the turnpike problem of growth theory is studied.