Portfolio Design using Mean-Variance Optimization and Hierarchical Risk Parity Approach

This is the introduction to Chapter 4 in the volume titled Analysis and Forecasting of Financial Time Series: Selected Cases, authored by Jaydip Sen to be published in November 2022, by Cambridge Scholars Publishing Limited, Welbeck Road, Newcastle Upon Tyne, NE6 2PA, United Kingdom. Copyright Protected Materials. PORTFOLIO DESIGN USING MEAN- VARIANCE OPTIMIZATION AND HIERARCHICAL RISK PARITY APPROACH JAYDIP SEN Introduction The design of optimized portfolios has remained a research topic of broad and intense interest among researchers of quantitative and statistical finance for a long time. An optimum portfolio allocates the weights to a set of capital assets in a way that optimizes the return and risk of those assets. Markowitz in his seminal work proposed the mean-variance optimization approach which is based on the mean and covariance matrix of returns (Markowitz, 1952). The mean- variance portfolio (MVP) design works on an algorithm, known as the critical line algorithm (CLA). The CLA algorithm, despite the elegance of its theoretical framework, has some major limitations. One of the major problems is the adverse effects of the estimation errors in its expected returns and covariance matrix on the performance of the portfolio. Since it is extremely challenging to accurately estimate the expected returns of an asset from its historical prices, it is a popular practice to use either a minimum variance portfolio or an optimized risk portfolio with the maximum Sharpe ratio as better proxies for the expected returns. However, due to the inherent complexity, several factors have been used to explain the expected returns. The hierarchical risk parity (HRP) algorithm attempts to address three major shortcomings of quadratic optimization methods which are particularly relevant to the CLA used in the MVP approach to portfolio design (de Prado, 2016). These problems are instability, concentration, and underperformance. Unlike the CLA, the HRP Copyright protected material This is the introduction to Chapter 4 in the volume titled Analysis and Forecasting of Financial Time Series: Selected Cases, authored by Jaydip Sen to be published in November 2022, by Cambridge Scholars Publishing Limited, Welbeck Road, Newcastle Upon Tyne, NE6 2PA, United Kingdom. Copyright Protected Materials. algorithm does not require the covariance matrix of return values to be invertible. Hence, the HRP portfolios can deliver good results even if the covariance matrices are ill-degenerated or singular, which is an impossibility for a quadratic optimizer like CLA. Interestingly, even though MVP’s objective is to minimize the variance, HRP is proven to have a lower probability of yielding lower out-of-sample variance than MVP. Given the fact that future returns cannot be forecasted with sufficient accuracy, many researchers have proposed risk-based asset allocation using the covariance matrix of the returns (Jurezenko, 2015). However, this approach brings in another problem of instability that arises because the quadratic programming methods require the inversion of a covariance matrix whose all eigenvalues must be positive. HRP is a new portfolio design method that addresses the pitfalls of the quadratic optimization-based MVP approach using techniques of graph theory and machine learning (de Prado, 2016). This method exploits the features of the covariance matrix without the requirement of its invertibility. This chapter presents an algorithmic approach for building efficient portfolios by selecting stocks from fourteen sectors listed on the National Stock Exchange (NSE) of India. Based on the report of the NSE published on December 31, 2020, the top ten stocks with the highest free-float market capitalization from thirteen sectors and the 50 stocks included in the NIFTY 50 are first identified (NSE Website). Portfolios are built using the MVP and the HRP algorithms using the historical prices of the stocks from January 1, 2016, to December 31, 2020. The portfolios are backtested on the training data of the stock prices from January 1, 2016, to December 31, 2020, and on the test data from January 1, 2021, to December 31, 2021. The portfolios are evaluated based on their cumulative returns and Sharpe ratios over the training and test periods. The main contribution of the current work is threefold. First, it presents two different methods of designing robust portfolios, the MVP and the HRP methods. These portfolio design approaches are applied to fourteen sectors of stocks of the NSE including the NIFTY 50 stocks. The results may serve as a guide to investors in the stock market. Second, a backtesting method is proposed for evaluating the performances of the algorithms based on the cumulative daily returns yielded by the portfolios. Since the backtesting is done both on the Copyright protected material This is the introduction to Chapter 4 in the volume titled Analysis and Forecasting of Financial Time Series: Selected Cases, authored by Jaydip Sen to be published in November 2022, by Cambridge Scholars Publishing Limited, Welbeck Road, Newcastle Upon Tyne, NE6 2PA, United Kingdom. Copyright Protected Materials. training and the test data of the stock prices, it can identify the better performing method on both training and test samples. Hence, a robust framework for evaluating different portfolios is demonstrated. Third, the returns of the portfolios on the sectors on the test data highlight the current profitability of investment and the volatilities of the sectors studied in this work. This information can be useful for investors. The chapter is organized as follows. The section titled Related Work presents some of the existing portfolio design approaches proposed in the literature. 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(2018a) “Stock price prediction using machine learning and deep learning frameworks”, Proceedings of the 6th International Conference on Business Analytics and Intelligence (ICBAI’18), December 20-22, Bangalore, India. Sen, J. (2018b) “Stock composition of mutual funds and fund style: A time series decomposition approach towards testing for consistency”, International Journal of Business Forecasting and Marketing Intelligence, Vol 4, No 3, pp. 235-292. DOI: 10.1504/IJBFMI.2018.092781. Copyright protected material This is the introduction to Chapter 4 in the volume titled Analysis and Forecasting of Financial Time Series: Selected Cases, authored by Jaydip Sen to be published in November 2022, by Cambridge Scholars Publishing Limited, Welbeck Road, Newcastle Upon Tyne, NE6 2PA, United Kingdom. Copyright Protected Materials. Sen, J. (2017) “A time series analysis-based forecasting approach for the Indian realty sector”, International Journal of Applied Economic Studies, Vol 5, No 4, pp. 8-27. DOI: 10.36227/techrxiv.16640212.v1. Sen, J. and Datta Chaudhuri, T. (2018) “Understanding the sectors of Indian economy for portfolio choice”, International Journal of Business Forecasting and Marketing Intelligence, Vol 4, No 2, pp. 178-222. DOI: 10.1504/IJBFMI.2018.090914. Sen, J. and Datta Chaudhuri, T. (2017a) “A robust predictive model for stock price forecasting”, Proceedings of the 5th International Conference on Business Analytics and Intelligence (BAICONF’17), December 11-13, 2017, Bangalore, India. DOI: 10.36227/techrxiv.16778611.v1. Sen, J. and Datta Chaudhuri, T. (2017b) “A predictive analysis of the Indian FMCG sector using time series decomposition-based approach”, Journal of Economics Library, Vol 4, No 2, pp. 206- 226. DOI: 10.1453/jel.v4i2.1282. Sen, J. and Datta Chaudhuri, T. (2016a) “Decomposition of time series data to check consistency between fund style and actual fund composition of mutual funds”, Proceedings of the 4th International Conference on Business Analytics and Intelligence (ICBAI’16), December 19-21, 2016. DOI: 10.13140/RG.2.2.33048.19206. Sen, J. and Datta Chaudhuri, T. (2016b) “An investigation of the structural characteristics of the Indian IT sector and the capital goods sector – An application of the R programming language in time series decomposition and forecasting”, Journal of Insurance and Financial Management, Vol 1, No 4, pp 68-132. DOI: 10.36227/techrxiv.16640227.v1. Sen, J. and Datta Chaudhuri, T. (2016c) “An alternative framework for time series decomposition and forecasting and its relevance for portfolio choice – A comparative study of the Indian consumer durable and small cap sectors”, Journal of Economics Library, Vol 3, No 2, pp. 303-326. DOI: 10.1453/jel.v3i2.787. Sen, J. and Datta Chaudhuri, T. (2016d) “Decomposition of time series data of stock markets and its implications for prediction – Copyright protected material This is the introduction to Chapter 4 in the volume titled Analysis and Forecasting of Financial Time Series: Selected Cases, authored by Jaydip Sen to be published in November 2022, by Cambridge Scholars Publishing Limited, Welbeck Road, Newcastle Upon Tyne, NE6 2PA, United Kingdom. Copyright Protected Materials. An application for the Indian auto sector”, Proceedings of the 2nd National Conference on Advances in Business Research and Management Practices (ABRMP’16), pp. 15-28, January 8-9, 2016. DOI: 10.13140/RG.2.1.3232.0241. Sen, J. and Datta Chaudhuri, T. (2015) “A framework for predictive analysis of stock market indices – A study of the Indian auto sector”, Journal of Management Practices, Vol 2, No 2, pp. 1- 20. DOI: 10.13140/RG.2.1.2178.3448. Sen, J. and Dutta, A. (2022a) “A comparative study of hierarchical risk parity portfolio and eigen portfolio on the NIFTY 50 stocks”, Proceedings of the 2nd International Conference on Computational Intelligence and Data Analytics (ICCIDA’22), January 8-9, 2022, Hyderabad, India. (In press) Sen, J. and Dutta, A. (2022b) “Design and Analysis of Optimized Portfolios for Selected Sectors of the Indian Stock Market”, Proceedings of the 2022 International Conference on Decision Aid Sciences and Applications (DASA), pp. 567-573, March 23- 25, 2022, Chiangrai, Thailand. DOI: 10.1109/DASA54658.2022.9765289. Sen, J. and Dutta, A. (2022c), “Portfolio Optimization for the Indian Stock Market”, in: Wang, J. (ed.) Encyclopedia of Data Science and Machine Learning, IGI Global, USA, August 2022. (In Press). DOI: 10.4018/978-1-7998-9220-5. Sen, J. and Dutta, A. (2021) “Risk-based portfolio optimization on some selected sectors of the Indian stock market”, Proceedings of the 2nd International Conference on Big Data, Machine Learning and Applications (BigDML’21), December 19-20, 2021, Silchar, India. (In press) Sen, J. and Mehtab, S. (2022a) A comparative study of optimum risk portfolio and eigen portfolio on the Indian stock market”, International Journal of Business Forecasting and Marketing Intelligence, Vol 7, No 2, pp 143-195. DOI: 10.1504/IJBFMI.2021.120155. Sen, J. and Mehtab, S. (2022b) “Long-and-Short-Term Memory (LSTM) Price Prediction-Architectures and Applications in Stock Price Prediction”, in: Singh, U., Murugesan, S., and Seth, Copyright protected material This is the introduction to Chapter 4 in the volume titled Analysis and Forecasting of Financial Time Series: Selected Cases, authored by Jaydip Sen to be published in November 2022, by Cambridge Scholars Publishing Limited, Welbeck Road, Newcastle Upon Tyne, NE6 2PA, United Kingdom. Copyright Protected Materials. A. (eds) Emerging Computing Paradigms - Principles, Advances, and Applications, Wiley, USA, 2022. (In Press) Sen, J. and Mehtab, M. (2021a) “Design and analysis of robust deep learning models for stock price prediction”, in: Sen, J. (ed) Machine Learning – Algorithms, Models and Applications, pp. 15-46, IntechOpen, London, UK. DOI: 10.5772/intechopen.99982. Sen J. and Mehtab, S. (2021b) “Accurate stock price forecasting using robust and optimized deep learning models”, Proceedings of the IEEE International Conference on Intelligent Technologies (CONIT), pp. 1-9, June 25-27, 2021, Hubballi, India. DOI: 10.1109/CONIT51480.2021.9498565. Sen, J., Dutta, A. and Mehtab, S. (2021a) “Stock portfolio optimization using a deep learning LSTM model”, Proceedings of the IEEE Mysore Sub Section International Conference (MysuruCon’21), pp. 263-271, October 24-25, 2021, Hassan, Karnataka, India. DOI: 10.1109/MysuruCon52639.2021.9641662. Sen, J., Dutta, A. and Mehtab, S. (2021b) “Profitability analysis in stock investment using an LSTM-based deep learning model”, Proceedings of the IEEE 2nd International Conference for Emerging Technology (INCET’21), pp. 1-9, May 21-23, Belagavi, India. DOI: 10.1109/INCET51464.2021.9456385. Sen, J., Dutta, A., Mondal, S. and Mehtab, S. (2021c) “A comparative study of portfolio optimization using optimum risk and hierarchical risk parity approaches”, Proceedings of the 8th International Conference on Business Analytics and Intelligence (ICBAI’21), December 20-22, Bangalore, India. Sen, J., Mehtab, S. and Dutta, A. (2021d) “Volatility modeling of stocks from selected sectors of the Indian economy using GARCH”, Proceedings of the IEEE Asian Conference on Innovation in Technology (ASIANCON’21), pp. 1-9, August 28- 29, 2021, Pune, India. DOI: 10.1109/ASIANCON51346.2021.9544977. Sen, J., Mehtab, S., Dutta, A. and Mondal, S. (2021e) “Precise stock price prediction for optimized portfolio design using an LSTM Copyright protected material This is the introduction to Chapter 4 in the volume titled Analysis and Forecasting of Financial Time Series: Selected Cases, authored by Jaydip Sen to be published in November 2022, by Cambridge Scholars Publishing Limited, Welbeck Road, Newcastle Upon Tyne, NE6 2PA, United Kingdom. Copyright Protected Materials. model”, Proceedings of the IEEE 19th International Conference on Information Technology (OCIT’12), pp. 210-215, December 16-18, 2021, Bhubaneswar, India. DOI: 10.1109/OCIT53463.2021.00050. Sen, J., Mehtab, S., Dutta, A. and Mondal, S. (2021f) “Hierarchical risk parity and minimum variance portfolio design on NIFTY 50 stocks”, Proceedings of the IEEE International Conference on Decision Aid Sciences and Applications (DASA’21), December 7-8, 2021, Sakheer, Bahrain. DOI: 10.1109/DASA53625.2021.9681925. Sen, J., Mondal, S. and Nath, G. (2021g) “Robust portfolio design and stock price prediction using an optimized LSTM model”, Proceedings of the IEEE 18th India Council International Conference (INDICON’21), pp. 1-6, December 19-21, 2021, Guwahati, India. DOI: 10.1109/INDICON52576.2021.9691583. Sen, J., Mondal, S. and Mehtab, S. 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