Mean-Variance Efficiency of Optimal Power and Logarithmic Utility Portfolios

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, an application to the stock market is presented and the behavior 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.

An approximate solution for the power utility optimization under predictable returns

“…Proof of Lemma 1. According to Bodnar et al (2018), assuming that µ > 0 and σ/µ → 0, the difference between the distribution functions of N (µ, σ 2 ) and ln N ln µ, σ 2 µ 2 approaches zero.…”

“…One can find it as the typical solution for the investment strategies for different types of utility functions (see Bodnar et al, 2015a,b). However, most of the time it is hard to succeed and in order to derive an analytical solution it is required to have some sort of specific assumptions because of the difficulty to perform the calculations (see Bodnar et al, 2015bBodnar et al, , 2018Campbell and Viceira, 2002). On the other hand, one can always use numerical or approximative results if no information on returns is considered (see Brandt et al, 2005;Broadie and Shen, 2017).…”

“…The main result of the paper provides approximate single-period optimal portfolio weights for the power utility function given by the multivariate normal distribution on returns. The approximation is based on a similar behaviour of the normal N (µ, σ 2 ) and log-normal ln N (ln µ, σ 2 /µ 2 ) densities if σ/µ approaches zero (see Bodnar et al, 2018). For instance, let µ = 1 and σ = 0.04 then both densities look pretty similar (Figure 1) and in practice, simulated samples from the corresponding log-normal distribution would rarely fail to reject the normality hypothesis testing.…”