Portfolio optimization using the wavelet-based Bayesian MGARCH approach

اقتصاد پولی مالی

Article 6, Volume 27, Issue 19, June 2021, Pages 133-164   

Document Type: پژوهشی

DOI: 10.22067/mfe.2020.16409.0

Authors

Seyed Ali Hoseini Ebrahimaba¹; khalil jahangiri^{* 2}; mahdi Ghaemi Asl³; hasan heidari^{* 4}

¹Ph.D. Student in Economics, Urmia University

²Assistant Professor of Financial Economics, Urmia University

³Assistant Professor of Islamic Economics and Banking, Kharazmi University

⁴Professor of Financial Economics, Urmia University

Abstract

Introduction:
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.
Methodology:
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.
Results & Discussion:
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.

Keywords

Portfolio optimization; Bayesian approach; Wavelet; Time scale; Efficient border

References

 

[1]   Abdullah, A. M., Saiti, B., & Masih, M. (2016). The impact of crude oil price on Islamic stock indices of South East Asian countries: Evidence from MGARCH-DCC and wavelet approaches. Borsa Istanbul Review, 16(4), 219-232.‏

[2] Aguilar, O., & West, M. (2000). Bayesian dynamic factor models and portfolio allocation. Journal of Business & Economic Statistics, 18(3), 338-357.‏

[3] Armstrong, J. (2018). The Markowitz Category. SIAM Journal on Financial Mathematics, 9(3), 994-1016.‏

[4]  Avramov, D., & Zhou, G. (2010). Bayesian portfolio analysis. Annu. Rev. Financ. Econ., 2(1), 25-47.‏

[5]  Billio, M., Caporin, M., & Gobbo, M. (2006). Flexible dynamic conditional correlation multivariate garch models for asset allocation. Applied Financial Economics Letters, 2(02), 123-130.‏

[6] Black, F., & Litterman, R. (1992). Global portfolio optimization. Financial analysts journal, 48(5), 28-43.‏

[7]  Black, F. (1972). Capital market equilibrium with restricted borrowing. The Journal of business, 45(3), 444-455.‏

[8]    Bauder, D., Bodnar, T., Parolya, N., & Schmid, W. (2018). Bayesian mean-variance analysis: Optimal portfolio selection under parameter uncertainty. arXiv preprint arXiv:1803.03573.‏

[9]  Bakar, N. A., & Rosbi, S. (2018). Efficient frontier analysis for portfolio investment in Malaysia stock market. A A, 2(2), 2-2.‏

[10] Bahlous, M., & Mohd. Yusof, R. (2014). International diversification among  Islamicinvestments: is there any benefit. Managerial Finance, 40(6), 613-633.‏

[11] Bala, D. A., & Takimoto, T. (2017). Stock markets volatility spillovers during financial crises: A DCC-MGARCH with skewed-t density approach. Borsa Istanbul Review, 17(1), 25-48.‏

[12] Best, M. J., & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results. The review of financial studies, 4(2), 315-342.‏

[13] Bodnar, T., Mazur, S., & Okhrin, Y. (2017). Bayesian estimation of the global minimum variance portfolio. European Journal of Operational Research, 256(1), 292-307.‏

[14] Bauwens, L., & Laurent, S. (2005). A new class of multivariate skew densities, with application to generalized autoregressive conditional heteroscedasticity models. Journal of Business & Economic Statistics, 23(3), 346-354.‏

[15] Dai, Z., & Wen, F. (2018). Some improved sparse and stable portfolio optimization problems. Finance Research Letters, 27, 46-52.‏

[16]  Dajcman, S. (2015). An empirical investigation of the nexus between sovereign bond yields and stock market returns–a multiscale approach. Engineering Economics, 26(2), 108-117.‏

[17]  De Franco, C., Nicolle, J., & Pham, H. (2018). Bayesian learning for the Markowitz portfolio selection problem. arXiv preprint arXiv:1811.06893.‏

[18] Ekstrom, E., & Vaicenavicius, J. (2016). Optimal liquidation of an asset under drift uncertainty. SIAM Journal on Financial Mathematics, 7(1), 357-381.‏

[19]  Elton, E. J., & Gruber, M. J. (1973). Estimating the dependence structure of share prices- implications for portfolio selection. The Journal of Finance, 28(5), 1203-1232.‏

[20]  Fioruci, J. A., Ehlers, R. S., & Andrade Filho, M. G. (2014). Bayesian multivariate GARCH models with dynamic correlations and asymmetric error distributions. Journal of Applied Statistics, 41(2), 320-331.‏

[21]  Fabozzi, F. J., Kolm, P. N., Pachamanova, D. A., & Focardi, S. M. (2007). Robust portfolio optimization and management. John Wiley & Sons.‏

[22]  Frost, P. A., & Savarino, J. E. (1986). An empirical Bayes approach to efficient portfolio selection. Journal of Financial and Quantitative Analysis, 21(3), 293-305.‏

[23]  Gençay, R., Selçuk, F., & Whitcher, B. (2001). Differentiating intraday seasonalities through wavelet multi-scaling. Physica A: Statistical Mechanics and its Applications, 289(3-4), 543-556.‏

[24] Greyserman, A., Jones, D. H., & Strawderman, W. E. (2006). Portfolio selection using hierarchical Bayesian analysis and MCMC methods. Journal of Banking & Finance, 30(2), 669-678.‏

[25] Huang, C. F., & Litzenberger, R. H. (1988). Foundations for Financial Economics, 1988.‏

[26] Hoseini, A., jahangiri, K., Heydari, H., Ghaemi asl, M. (2019). Study of Shock and Volatility Spillovers among Selected Indices of the Tehran Stock Exchange Using Asymmetric BEKK-GARCH Model. Journal of Applied Economics Studies in Iran, 8(29), 123-155. (in Persian)

[27] Hoseini, A., Jahangiri, K., ghaemi asl, M., Heidari, H. (2020). Investigation of the volatility spillover effect and dynamic conditional correlations in Tehran Stock Exchange using wavelet based Bayesian conditional variance heteroscedasticity. Quarterly Journal of Applied Theories of Economics, 7(1), 149-184.

[28] In, F., & Kim, S. (2013). An introduction to wavelet theory in finance: a wavelet multiscale approach. World scientific.‏

[29] Ismail, A., & Pham, H. (2019). Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix. Mathematical Finance, 29(1), 174-207.‏

[30] Jackson, M., & Staunton, M. D. (1999). Quadratic programming applications in finance using Excel. Journal of the Operational Research Society, 50(12), 1256-1266.‏

[31] Kroner, K. F., & Ng, V. K. (1998). Modeling asymmetric comovements of asset returns. The review of financial studies, 11(4), 817-844.‏

[32] Karami, S., Rastegar, M. (2018). Estimation of Return and Volatilities Spillover between Different Industries of Tehran Stocks’ Exchange. Financial Engineering and Protfolio Management, 9(35), 323-342. (in Persian)

[33] Ledoit, O., & Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of multivariate analysis, 88(2), 365-411.‏

[34] Laloux, L., Cizeau, P., Potters, M., & Bouchaud, J. P. (2000). Random matrix theory and financial correlations. International Journal of Theoretical and Applied Finance, 3(03), 391-397.‏

[35] Liu, X., An, H., Huang, S., & Wen, S. (2017). The evolution of spillover effects between oil and stock markets across multi-scales using a wavelet-based GARCH–BEKK model. Physica A: Statistical Mechanics and its Applications, 465, 374-383.‏

[36] Merton, R. C. (1980). On estimating the expected return on the market: An exploratory investigation. Journal of financial economics, 8(4), 323-361.‏

[37] Nazlioglu, S., Soytas, U., & Gupta, R. (2015). Oil prices and financial stress: A volatility spillover analysis. Energy Policy, 82, 278-288.‏

[38] Poor Ahamadi, Z., & Najafi, A.B. (2015). Dynamic optimization of the investment portfolio according to the cost of transactions. Financial Engineering and Protfolio Management, 6(22), 127-146. (in Persian)

[39] Pindyck, R. S., & Rotemberg, J. J. (1993). The comovement of stock prices. The quarterly journal of economics, 108(4), 1073-1104.‏

[40] Percival, D. B., & Walden, A. T. (2000). Wavelet methods for time series analysis (Vol. 4). Cambridge university press.‏

[41] Rahim, A. M., & Masih, M. (2016). Portfolio diversification benefits of Islamic investors with their major trading partners: Evidence from Malaysia based on MGARCH-DCC and wavelet approaches. Economic Modelling, 54, 425-438.‏

[42] Rambaud, S. C., Pérez, J. G., Granero, M. Á. S., & Segovia, J. E. T. (2009). Markowitz’s model with Euclidean vector spaces. European Journal of Operational Research, 196(3), 1245-1248.‏

[43] Sharifi, S., Crane, M., Shamaie, A., & Ruskin, H. (2004). Random matrix theory for portfolio optimization: a stability approach. Physica A: Statistical Mechanics and its Applications, 335(3-4), 629-643.‏

[44] Saiti, B., & Noordin, N. H. (2018). Does Islamic equity investment provide diversification benefits to conventional investors? Evidence from the multivariate GARCH analysis. International Journal of Emerging Markets, 13(1), 267-289.‏

[45] Salimi, M., Taqhavi Fard, M., Fallahshams, M., Khajezadeh Dezfuli, H. (2018). Evolutionary 4-Objective Optimization Portfolio Algorithms for fuzzy and non-fuzzy selection. Financial Engineering and Protfolio Management, 9(36), 1-16. (in Persian)

[46] Trichilli, Y., Abbes, M. B., & Masmoudi, A. (2020). Islamic and conventional portfolios optimization under investor sentiment states: Bayesian vs Markowitz portfolio analysis. Research in International Business and Finance, 51, 101071.‏

[47] Tu, J., & Zhou, G. (2010). Incorporating economic objectives into Bayesian priors: Portfolio choice under parameter uncertainty. Journal of Financial and Quantitative Analysis, 45(4), 959-986.‏

[48] Way, R., Lafond, F., Lillo, F., Panchenko, V., & Farmer, J. D. (2019). Wright meets Markowitz: How standard portfolio theory changes when assets are technologies following experience curves. Journal of Economic Dynamics and Control, 101, 211-238.‏

[49] Xiao, Y., & Valdez, E. A. (2015). A Black–Litterman asset allocation model under Elliptical distributions. Quantitative Finance, 15(3), 509-519.‏

[50] Yang, L., Couillet, R., & McKay, M. R. (2014, November). Minimum variance portfolio optimization with robust shrinkage covariance estimation. In 2014 48th Asilomar Conference on Signals, Systems and Computers (pp. 1326-1330). IEEE.‏

[51] Yin, K., Liu, Z., & Jin, X. (2020). Interindustry volatility spillover effects in China’s stock market. Physica A: Statistical Mechanics and its Applications, 539, 122936.‏

[52] Zhang, Y., Li, X., & Guo, S. (2018). Portfolio selection problems with Markowitz’s mean–variance framework: a review of literature. Fuzzy Optimization and Decision Making, 17(2), 125-158.‏

[53] Zhao, Y. F., Chaoliang, Z., & Zongrun, W. (2019). Portfolio Selection Based on Bayesian Theory. Mathematical Problems in Engineering, 2019, 1-11.