No abstract

No abstract

A strategy is developed which minimizes the expected time (or plays) to reach a given financial goal when "time rebates" are given if the goal is more than attained in the last investment period. It is shown that the optimal strategy is the one which maximizes the expected logarithm of the investment relative (for discrete distributions this is equivalent to maximizing the geometric mean). This strategy is optimal for all goals and levels of capital. When no time rebates are given, however, the proposed strategy will generally not be optimal. In this case the theory shows the strategy is uniformly good. Further, at least in a limited sense, the true optimal will generally differ significantly only in endgame play.

The advantages of the Geometric Mean or Log Strategy have been well documented and recommended by many because a follower of this strategy will ultimately outperform in the long-run any other significantly different strategy, almost surely. This paper provides both theory and evidence to indicate that the "long-run" can be quite long in risky situations. These cases are typified by many business ventures and undiversified speculative investments in OTC securities. On the other hand, it is shown that the Log Strategy can "virtually dominate" in a moderate number of plays in cases when risk is low.geometric mean return, log utility, log strategy

Safety stock is often calculated to accommodate a given service level (SL), where service is defined in some manner by the user. In this regard, the two favorite definitions of service are the percent of good cycles and the fraction of demand satisfied by off‐the‐shelf inventory. Judging from its increased appearance in the literature, especially in textbooks, the latter definition apparently is gaining in popularity. Unfortunately, this definition requires a little more effort to use. A procedure originally worked out by Brown (Brown's Method) shows how the calculations are to proceed. The partial expectation, E(z), is first calculated as a function of the expected lead time demand (μ), the standard deviation (σ) of lead time demand, and the required service level. Once E(z) is found, a table is consulted (Brown's Table) to determine z. Finally, the safety stock is calculated from the formula, zσ. There are several problems with using Brown's Table. First, z is given as the dependent variable rather than the independent variable, so the user has to search a bit to find the nearest pair of entries that straddle the required value of E(z). Then the user must perform an awkward interpolation, something many business students find very confusing. An additional problem with the table is that it is both sparse and inaccurate for high values of z. In fact, there are no entries at all for z 4.50, and there is only one significant digit of accuracy for z 3.93. Finally, Brown does not supply a computer program that calculates z directly from E(z). While this lack of a program is not important to the casual user of the tables, it can be important when one wishes to set safety stock levels automatically and internally by the computer. This article presents a new table, in which E(z) is the dependent variable rather than the independent variable. The computer program used to evaluate the entries yields solutions that are accurate to about fourteen decimal places. With this kind of precision the authors were able to extend Brown's Table to high values of z, while at the same time providing for virtually any number of significant digits. Incidentally, the entry density in this table is nonuniform (somewhat logarithmic) to accommodate usage over various orders of magnitude of z and E(z). The article also includes a simpler version of our computer program (single precision), which yields about four decimal place accuracy. The double precision program actually used is available on request, but it is somewhat complicated and depends on the computer used (the authors used a TRS‐80).