Abstract
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.
Highlights
The theory of optimal portfolio choice started with the pioneering contribution of [29]
It is remarkable that the optimal portfolio in the sense of maximizing the expected power utility function is located for any relative risk aversion coefficient γ on the set of feasible portfolios, i.e., on the parabola (10)
The results of Corollary 1 are used to derive conditions on the relative risk aversion coefficient γ which ensure that the obtained portfolio (17) lies in the upper part of the parabola (10) and, the optimal portfolio that maximizes the expected power utility function is mean-variance efficient
Summary
The theory of optimal portfolio choice started with the pioneering contribution of [29]. We consider the problem of finding the optimal portfolios for power and logarithmic utilities under the assumption that the portfolio gross returns are approximately log-normally distributed. If the ratio of mean and standard deviation is large (analogue of Sharpe ratio), the log-normal distribution is close to the normal one (analytically shown in the paper) This is, for instance, the case when the mean of the portfolio is bounded and its variance is small. 4.2, the analytical results for the optimal portfolio in the sense of maximizing the expected logarithmic utility are present: the closed-form expression of its weights is provided in Theorem 3, while its mean-variance efficiency is proved in Theorem 4. We compare the portfolio with optimal weights with other portfolio strategies and show the dominance of the new approach
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