We derive new results related to the portfolio choice problem for power and logarithmic utilities. Assuming that the portfolio returns follow an approximate log-normal distribution, the closed-form expressions of the optimal portfolio weights are obtained for both utility functions. Moreover, we prove that both optimal portfolios belong to the set of mean-variance feasible portfolios and establish necessary and sufficient conditions such that they are mean-variance efficient. Furthermore, we extend the derived theoretical finding to the general class of the log-skew-normal distributions. Finally, an application to the stock market is presented and the behaviour of the optimal portfolio is discussed for different values of the relative risk aversion coefficient. It turns out that the assumption of log-normality does not seem to be a strong restriction.

中文翻译：

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最优功率和对数效用组合的均方差效率

我们得出与电力和对数公用事业的投资组合选择问题有关的新结果。假设投资组合收益遵循近似的对数正态分布，则对于两个效用函数，都获得了最佳投资组合权重的闭式表达式。此外，我们证明两个最优投资组合都属于均值方差可行投资组合集，并建立了必要条件和充分条件，使得它们具有均值方差有效。此外，我们将导出的理论发现扩展到对数偏态正态分布的一般类别。最后，介绍了一种在股票市场上的应用，并针对相对风险规避系数的不同值，讨论了最优投资组合的行为。