Abstract
Portfolio optimization has been an area of research that has attracted a lot of attention from researchers and financial analysts. Designing an optimum portfolio is a complex task since it not only involves accurate forecasting of future stock returns and risks but also needs to optimize them. This article presents a systematic approach to portfolio design using two risk optimization approaches, the hierarchical risk parity algorithm and the hierarchical equal risk contribution algorithm on seven critical sectors and the NIFTY 50 stocks listed in the National Stock Exchange of India of the Indian stock market. The portfolios are built following the two approaches on historical stock prices from Jan 1, 2016 to Dec 31, 2020. The portfolio performances are evaluated on the test data from Jan 1, 2021 to Nov 1, 2021. The backtesting results of the portfolios indicate that the performance of the HRP portfolio is superior to that of the HERC portfolio on both training and the test data for the majority of the sectors studied in this work.
TopIntroduction
The design of optimized portfolios has remained a research topic of broad and intense interest among the 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 algorithm, known as the critical line algorithm (CLA), despite the elegance in 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.
The hierarchical risk parity (HRP) and hierarchical equal risk contribution (HERC) portfolios are two well-known approaches of portfolio design that attempt to address three major shortcomings of quadratic optimization methods which are particularly relevant to the CLA (de Prado, 2016). These problems are, instability, concentration, and under-performance. Unlike the CLA, the HRP algorithm does not require the covariance matrix of return values to be invertible. The HRP is capable of delivering good results even if the covariance matrix is ill-degenerated or singular, which is an impossibility for a quadratic optimizer. On the other hand, the HERC portfolio optimization adapts the HRP approach to achieve an equal contribution to risk by the constituent stocks in a cluster after forming an optimal number of clusters among a given set of capital assets. Interestingly, even though CLA’s objective is to minimize the variance, portfolios formed based on HRP and HERC methods are proven to have a higher likelihood of yielding lower out-of-sample variance than the CLA. The major weakness of the CLA algorithm is that a small deviation in the forecasted future returns can make the CLA deliver widely divergent portfolios. Given the fact that future returns cannot be forecasted with sufficient accuracy, some researchers have proposed risk-based asset allocation using the covariance matrix of the returns. However, this approach brings in another problem of instability. The instability arises because the quadratic programming methods require the inversion of a covariance matrix whose all eigenvalues must be positive. This inversion is prone to large errors when the covariance matrix is numerically ill-conditioned, i.e., when it has a high condition number (Baily & de Prado, 2012). The HRP and HERC portfolios are two among the new portfolio approaches that address the pitfalls of the CLA using techniques of machine learning and graph theory (de Prado, 2016). While HRP exploits the features of the covariance matrix without the requirement of its invertibility or positive-definiteness and works effectively on even a singular covariance matrix of returns, the HERC portfolio leverages the formation of an optimal number of clusters among a set of capital assets in a manner that ensures equal risk contribution by the assets in the same cluster (Raffinot, 2018).
Despite being recognized as two approaches that outperform the CLA algorithm, to the best of our knowledge, no study has been carried out so far to compare the performances of the HRP and the HERC portfolios on Indian stocks. This chapter presents a comparative analysis of the performances of the HRP and the HERC portfolios on some important stocks from selected eight sectors listed in the National Stock Exchange (NSE) of India. Based on the report of the NSE on Oct 29, 2021, the most significant stocks of seven sectors and the 50 stocks included in the NIFTY 50 are first identified (NSE, 2021). Portfolios are built using the HRP and the HERC approaches for the eight sectors using the historical prices of the stocks from Jan 1, 2016, to Dec 31, 2020. The portfolios are backtested on the in-sample data of the stock prices from Jan 1, 2016, to Dec 31, 2020, and also on the out-of-sample data of stock prices from Jan 1, 2021, to Nov 1, 2021. Extensive results on the performance of the backtesting of the portfolios are analyzed to identify the better-performing algorithm for portfolio design.
Key Terms in this Chapter
Backtesting: It is the general method for seeing how well a strategy or model would have done ex-post. Backtesting assesses the viability of a trading strategy by discovering how it would play out using historical data. If backtesting works, traders and analysts may have the confidence to employ it gone forward.
Portfolio Risk: It refers to the chance that the combination of assets within a portfolio will fail to meet the financial objectives of the portfolio. Each asset within a portfolio carries its own risk, with higher potential return typically meaning higher risk.
Critical Line Algorithm: It is a computationally efficient method for tracing out the entire efficient frontier of a set of candidate portfolios of assets by finding successive critical values. It is applied to mean-variance optimization problems in portfolio design.
Portfolio Optimization: It is a process of selecting the best combination from a set of all portfolios being considered, according to some pre-defined objective. The objective typically maximizes factors such as expected return and minimizes costs like financial risk. The factors being considered may range from tangible (such as assets, liabilities, earnings, or other fundamentals) to intangible (such as selective divestment).
Minimum Variance Portfolio: It is a portfolio that minimizes the risk while maximizing the return. It involves diversifying the holdings to reduce the volatility, or finding a combination of assets that may be risky on their own but balance each other out when held in combination.
Agglomerative Clustering: It is a bottom-up type of hierarchical clustering in which each data point is defined as a cluster initially. Pairs of clusters are merged successively as the algorithm moves up in the hierarchy till the topmost point in the hierarchy is reached.
Sharpe Ratio: Originally developed by Nobel laureate William F. Sharpe, it is a metric used to help investors understand the return of an investment compared to its risk. The ratio is the average return earned over the risk-free rate per unit of volatility or total risk. Volatility is a measure of the price fluctuation of an asset or portfolio.
Portfolio Return: It refers to the gain or loss realized by an investment portfolio containing several types of investments. Portfolios aim to deliver returns based on the stated objectives of the investment strategy, as well as the risk tolerance of the type of investors targeted by the portfolio.
Hierarchical Risk Parity Portfolio: It is a risk-based portfolio optimization algorithm, which has been shown to generate diversified portfolios with robust out-of-sample performance without the requirement of a positive-definite return covariance matrix of the assets.
Hierarchical Equal Risk Contribution Portfolio: It aims at diversifying capital allocation and risk allocation in a portfolio by clustering the assets in a portfolio in such a way that the assets in a given cluster contribute equally to the overall risk of the portfolio.