This note is a continuation of a previous paper on normal approximations of standard forms. The standard form treated here is the ratio of two normally distributed variables. It is found that a suitable transformation of this ratio is approximately normally distributed provided that the coefficient of variation of the denominator is less than 0.39.

Conditions of equivalence are established among the following portfolio selection criteria: (1) Mean-Variance (E-V), (2) Second-Degree Stochastic Dominance (SSD), and (3) Mean-Entropy (E-H), for portfolios whose returns are characterized by-Uniform, Normal, and Lognormal probability distributions. We also assume that all portfolios derive from the same family of distributions and consider only the cases where the cumulative density functions intersect. In comparing the three selection criteria, under the three posited probability distributions, we utilize a combination of mathematical and graphical analyses. It is concluded that the three efficiency criteria are equivalent for uniformly and normally distributed portfolio returns. For lognormally distributed portfolio returns, the SSD criterion is optimal. Its efficiency, however, is sufficient but not necessary to establish efficiency for E-V and E-H. On the other hand, given the empirical similarities between E-V and SSD derived portfolios, and the close correspondence between the E-V and E-H criteria, the potential use of E-H becomes more appealing because of its distribution free nature.

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