Comparing Optimal Portfolio Performance Based on Skew-Normal Distribution and Skew-Laplace-Normal Distribution: A Mean-Absolute Deviation-Entropy Approach

Investors typically seek to strike the optimal balance between potential returns and associated risks in their trades. Various models have been presented to choose the optimal portfolio using different approaches. one of these methods is based on the statistical distribution of asset return. In these methods, the type of distribution of returns is first identified, and a suitable portfolio selection method is then applied based on this identified distribution type. This study compares the effectiveness of the mean-absolute deviation-entropy model utilizing both Skew-Normal Distribution and Skew-Laplace-Normal Distribution for constructing an optimal portfolio in the Tehran Stock Exchange over 36 months from April 2018 to March 2020.

The data used in this study comprises the monthly returns of 181 companies listed on the Tehran Stock Exchange. These returns were gathered from a statistical population of 338 members utilizing Morgan's table and Cochran’s formula. After fitting density functions for Skew-Normal and Skew-Laplace-Normal distributions to the returns, maximum likelihood estimates were obtained using the Stats package and the optim Function in R software. The reliability of these estimates was then checked using bootstrap sampling with 1,000 repetitions. Subsequently, relationships corresponding to the mathematical expectation of return distribution and the objective function representing the risk of absolute deviation were estimated using numerical methods. Therefore, this paper aimed to propose a multi-objective optimization model, namely a mean-absolute deviation-entropy model for portfolio optimization by using a goal-programming approach based on Skew-Normal Distribution and Skew-Laplace-Normal Distribution. The objective functions of the model were to maximize the mean return, minimize the absolute deviation, and maximize the entropy of the portfolio.

It can be inferred from the observed values ​​of the descriptive statistics of the monthly stock returns corresponding to the stock exchange symbols that some stocks have different skewness and kurtosis values ​​compared to the normal distribution. For example, The symbol "Shepna" exhibits negative skewness, indicating a left-skewed distribution. Similarly, the distribution of the "Basama" symbol exceeds the normal distribution. These instances suggest that the normal distribution is inadequate for describing monthly return distributions. Instead, distributions with parameters should be employed to account for skewness and kurtosis. According to the obtained results, the model utilizing the Skew- Laplace- Normal distribution has a higher performance ratio than the model based on the Skew-Normal distribution.

The reason for this superiority, where the model utilizing the Skew-Laplace-Normal distribution outperforms the model based on the Skew-Normal distribution, is the incorporation of both skewness and kurtosis criteria within the former. Additionally, upon analyzing the descriptive statistics of the symbols, it's evident that the kurtosis of most stock symbols is substantial. Therefore, integrating a combination of higher-order moments (skewness and kurtosis) along with entropy leads to enhanced performance.