Portfolio optimization using the wavelet-based Bayesian MGARCH approach

Decision making in conditions of uncertainty is one of the important features of risk asset allocation optimization models. Interconnection in stock price fluctuations or other assets is introduced as a factor in transferring price fluctuations from one or more sectors to other sectors. Since the main drawbacks of the Markowitz model are the need for a normal distribution of the return series and the impossibility of short-selling, the Bayesian DCC-GARCH model and the Huang & Litzenberger approach solve the problems of the Markowitz model, respectively. At the same time, the use of wavelet analysis makes it possible to present a suitable portfolio based on different frequency and scale domains during different sub-periods. Theoretical framework: According to Zhang, et al. (2018), the Markowitz mean-variance method is the most popular method for solving the optimal portfolio selection problem. But Trichilli, et al. (2020) point out that due to the high sensitivity of the Markowitz mean-variance process to small changes in inputs as well as the dependence of the process on past historical prices, it leads to a lack of application of the investor’s knowledge and experience in the capital market. Unfortunately, the Markowitz portfolio optimization model leads to the selection of a small number of superior assets. He suggests using the Bayesian approach to address the shortcomings of the Markowitz model. Another critique of previous models of modern portfolio theory is the assumption of a normal distribution for variance of portfolio. Hence, fat-tail asymmetric distributions such as the dynamic conditional variance heterogeneity (DCC-GARCH) approach are used in generalized Markowitz models that are closer to real-world data. But dynamic conditional heterogeneity models have limitations in asymmetric time series analysis. This led to the use of multivariate skew variance heterogeneity models such as Bayesian DCC-GARCH, which are more capable than MGARCH models in adopting the characteristics of financial time series in the process of estimating covariance and correlation matrices, used by Bala and Takimoto (2017), Fiorchi et al. (2014). Another problem with the Markowitz approach is that it assumes sales restrictions. This means that short-term sales are not possible. Therefore, Huang and Litzenberger (1988) introduced this generalized Markowitz model to remove this constraint in the model. In and Kim (2013) also consider the use of wavelet transform methods in Markowitz model to lead to more realistic results.

Rambaud, et al. (2009) argue that if an economy consists of a set of risky assets combined with a risk-free asset, then portfolios along the capital market line (CML) are superior than the efficient frontiers portfolios that contain only high-risk assets. Black (1972) imposed the possibility of short-selling (negative weight) to the basic Markowitz model by introducing mathematical relations. The period of this research is from 14/12/2008 to 16/06/2019 and according to the periods, before JCPOA, after JCPOA and the leave of the United States from JCPOA. The covariance matrix uses two different methods (unconditional and conditional derived from the Bayesian DCC-GARCH model) in the Huang & Litzenberger portfolio optimization model, in four different time scales based on the maximal overlap discrete wavelet transform (MODWT) approach. the results are compared at the end to select the best portfolio from the two covariance matrices.

By comparing the performance of the portfolios obtained from the unconditional and conditional covariance-variance matrices of the Bayesian DCC model, it is observed that in all subsections and wavelets, the efficiency of the portfolio of the Bayesian DCC model is higher than the unconditional model and the degree of efficiency varies in different subsectors. In fact, when all time-series have an abnormal distribution, the efficiency of asset portfolio derived from the variance-covariance matrix of the Bayesian conditional model is much higher than the unconditional model. The difference between the performance of asset portfolios derived from Bayesian unconditional and conditional models is less when there is a combination of normal and abnormal time series, and this necessitates the application of Bayesian models in financial markets, especially when all series are abnormal. Conclusions & Suggestions: The important result of the present study is to realize the multi-resolution nature of Huang and Litzenberger portfolio optimization theory in the Iranian capital market. The Estimation results indicate that the performance of portfolios in the medium-term and long-term scales (wavelets D3 and D4) is higher than the performance of these portfolios in the short-term scales (D1 and D2). Also, the present study clearly showed that in all subsectors, asset portfolios obtained by Bayesian distribution and by means of variance-covariance matrix extracted by Monte Carlo Markov chain (MCMC) method have higher efficiency than other portfolios which are obtained from other statistical distributions. Also, since all asset portfolios obtained under the second part are more efficient than other sub-sectors, one of the important achievements of the present study is the positive effect of lifting economic sanctions on the Iranian capital market.