Everyone seems to agree that there are significant interactions between corporate financing and investment decisions. The most important argument to the contrary — embodied in Modigliani and Miller's (MM's) famous Proposition I — specifically assumes the absence of corporate income taxes; but their argument implies an interaction when such taxes are recognized. Interactions may also stem from transaction costs or other market imperfections. The purpose of this paper is to present a general approach for analysis of the interactions of corporate financing and investment de-cisions, and to derive some of the approach's implications. Perhaps the most interesting implication is that the weighted average cost of capital formulas proposed by MM and other authors are not always correct. Except in certain special cases, a more general "Adjusted Present Value " rule should, in principle, be used to evaluate investment opportunities. The paper is organized as follows. Section II presents the

Presented at the American Finance Association Meeting, New York, December 1973.

Proof for a Case where Discounting Advances the Doomsday Get access Tjalling C. Koopmans Tjalling C. Koopmans International Institute for Applied Systems Analysis Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 41, Issue 5, December 1974, Pages 117–120, https://doi.org/10.2307/2296375 Published: 01 December 1974

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.