Equal Risk Contribution Research Articles

Small Portfolio Construction with Cryptocurrencies

In this paper, we describe and apply different models of portfolio construction in the selection between a small number of big-cap cryptocurrencies. Our purpose is to select the minimum riskiness between cryptocurrencies, comparing different risk measures and maximum diversification. We build our models without the constraints of the expected returns. Without relying on expected returns, we have the same condition on the comparison between them. Cryptocurrencies are not common stock or other assets indexed in the market but it is interesting to study how diversification can significantly improve investment performance. We first give the methodology to use high-frequency observation data, in the numeral approximation especially in the novel application of the Risk parity models, used with different risk measures we can achieve a very good result, from the position of gaining and variation. Since Risk parity models divide the weights of the asset in equal risk contribution proportion, it is suggested to use a small number of cryptocurrencies, otherwise their performance will be close to the uniform portfolio. To the traditional Mean Variance model, and the alternative, Expected shortfall/Conditional Value at Risk, we use three versions of Risk Parity with two different risk measures and a naive risk parity. The uniform portfolio is used as a benchmark for selection comparison with the other portfolio models. We give the conditions for the Risk Parity with the Expected shortfall/Conditional Value at Risk (CVaR) to guarantee convergence with the numerical approximation. In the end, we study the tradeoff between each model and which is more suitable for a small cryptocurrency portfolio.

Centrality-Based Equal Risk Contribution Portfolio

This article combines the traditional definition of portfolio risk with minimum-spanning-tree-based “interconnectedness risk” to improve equal risk contribution portfolio performance. We use betweenness centrality to measure an asset’s importance in a market graph (network). After filtering the complete correlation network to a minimum spanning tree, we calculate the centrality score and convert it to a centrality heuristic. We develop an adjusted variance–covariance matrix using the centrality heuristic to bias the model to assign peripheral assets in the minimum spanning tree higher weights. We test this methodology using the constituents of the S&P 100 index. The results show that the centrality equal risk portfolio can improve upon the base equal risk portfolio returns, with a similar level of risk. We observe that during bear markets, the centrality-based portfolio can surpass the base equal risk portfolio risk.

A Novel way to Diversify Portfolio Weights

There is broad agreement among academics and practitioners that, in general, a diversified portfolio delivers more robust out-of-sample performance than a concentrated one. This paper develops a new method to diversify portfolio weights, which does not rely on expected returns or volatilities and is therefore robust to measurement error in these variables. It uses pair-wise regressions to estimate how much variation each asset explains in terms of another asset. The optimization function finds the portfolio weights that maximize the unexplained variation of the portfolio. We benchmark our method against established methods from the literature, such as the minimum volatility and the equal risk contribution portfolios. We demonstrate that our method adheres to established diversification properties and it performs well in empirical tests.

Enhancing portfolio resilience during crisis periods: Lessons from BRICS indices and multi asset strategies

This paper uses Markowitz’s mean-variance model to construct an investment portfolio incorporating multiple assets – BRICS equity indices, Gold, crude oil, bonds, and cryptocurrencies. The optimally created risky portfolios outperform alternative portfolio optimization methods – the naive portfolio and the equal risk contribution portfolio; and established indices – the S&P 500 and the MSCI Emerging Equity Index in terms of metrics – adjusted Sharpe ratio, modified Sharpe ratio, and the modified Value at Risk. The findings are validated across different periods, including the COVID-19 period and the Russian invasion of Ukraine, including various in and out of sample periods. The findings highlight the benefits of portfolio diversity, mainly using BRICS indices, Gold, and Brent Crude oil, and challenge the notion of limited diversification benefits in BRICS indices found in previous studies. This paper further suggests the potential of emerging market bonds ETF as a diversification option during turbulent economic periods and highlights the limitations of cryptocurrencies in optimizing multi asset portfolios. By adopting the recommended multi asset portfolios, investors can enhance their risk-return trade-offs and achieve superior performance compared to the S&P500 and MSCI emerging indices. Lastly, the paper recommends future research opportunities in measuring portfolio performance and hedging strategies considering risk-adjusted return measurements, transaction expenses, and dynamic rebalancing techniques.

Does commodity exposure benefit traditional portfolios? Evidence from India

Commodities and commodity futures are expected to benefit stock and bond portfolio diversification because traditional asset types like equities and bonds have low correlations with commodities. During periods when stocks and bonds may underperform, commodities may provide a hedge against inflation and other economic uncertainties. This study investigates the diversification benefits of adding commodities to a traditional portfolio of stock and bonds from the perspective of an Indian investor. It employs several commonly used asset allocation strategies such as mean-variance, equal risk contribution, most diversified portfolio, and equal weight portfolio on different commodity derivative groups. The performance of various portfolios indicates that not all commodity groups provide substantial diversification benefits to a traditional portfolio. Agricultural commodities enhance performance (with an Omega ratio of 1.654), whereas metal and energy-related commodities do not diversify the traditional portfolio significantly (Omega ratio of 1.087 and 0.945, respectively). Gold and different equity sectors also provide some diversification benefits. This study also supports the hypothesis that the behavior of different commodity groups is quite different. AcknowledgmentThe infrastructural support provided by the FORE School of Management, New Delhi, in completing this paper is gratefully acknowledged.

Risk Budgeting portfolios: Existence and computation

AbstractModern portfolio theory has provided for decades the main framework for optimizing portfolios. Because of its sensitivity to small changes in input parameters, especially expected returns, the mean–variance framework proposed by Markowitz in 1952 has, however, been challenged by new construction methods that are purely based on risk. Among risk‐based methods, the most popular ones are Minimum Variance, Maximum Diversification, and Risk Budgeting (especially Equal Risk Contribution) portfolios. Despite some drawbacks, Risk Budgeting is particularly attracting because of its versatility: based on Euler's homogeneous function theorem, it can indeed be used with a wide range of risk measures. This paper presents mathematical results regarding the existence and the uniqueness of Risk Budgeting portfolios for a very wide spectrum of risk measures and shows that, for many of them, computing the weights of Risk Budgeting portfolios only requires a standard stochastic algorithm.

Prediction-based mean–variance portfolios with risk budgeting based on neural networks

The integration of return prediction and portfolio models has been widely investigated and proven effective. The derived prediction-based portfolios can be classified into five main categories. This paper comprehensively compares the out-of-sample performance of these categories based on the traditional mean–variance (MV) model for the first time, where different neural networks are used for return prediction. The MV model and equal-weighted portfolio are used as comparisons. Due to the high volatilities of these prediction-based portfolios, the risk budgeting method is applied to advance these portfolios, where stocks with higher predicted volatilities own higher proportions of portfolio risk contribution. The equal risk contribution is used as the benchmark. Further, the out-of-performance of these advanced portfolios is investigated after deducting transaction fees since their high turnovers can seriously affect the obtained profits in practical investments. Empirical tests are implemented with component stocks of the Shanghai Stock Exchange (SSE) 50 index, where daily data range from 2011 to 2021. Experimental results indicate that the prediction-based MV portfolio built with forecasted errors (MV-FE) is significantly superior to the others under different risk aversion levels, followed closely by the prediction-based MV portfolio built with forecasted returns (MV-FR). In particular, medium risk aversion is more suitable for these portfolios. Moreover, the risk budgeting method considerably reduces the volatilities of these portfolios and improves their risk-adjusted returns. The risk budgeting method is more efficient than the equal risk contribution method. After subtracting their transaction fees of turnovers, the MV-FR portfolio with risk budgeting, i.e., the MV-FR-RB portfolio, outperforms the others and still obtains satisfying performance. Also, there is little difference between the MV-FR and MV-FE portfolios with risk budgeting.

Using Deep Reinforcement Learning with Hierarchical Risk Parity for Portfolio Optimization

We devise a hierarchical decision-making architecture for portfolio optimization on multiple markets. At the highest level a Deep Reinforcement Learning (DRL) agent selects among a number of discrete actions, representing low-level agents. For the low-level agents, we use a set of Hierarchical Risk Parity (HRP) and Hierarchical Equal Risk Contribution (HERC) models with different hyperparameters, which all run in parallel, off-market (in a simulation). The information on which the DRL agent decides which of the low-level agents should act next is constituted by the stacking of the recent performances of all agents. Thus, the modelling resembles a statefull, non-stationary, multi-arm bandit, where the performance of the individual arms changes with time and is assumed to be dependent on the recent history. We perform experiments on the cryptocurrency market (117 assets), on the stock market (46 assets) and on the foreign exchange market (28 pairs) showing the excellent robustness and performance of the overall system. Moreover, we eliminate the need for retraining and are able to deal with large testing sets successfully.

Risk parity: An alternative formulation for risk-averse stochastic optimization in presence of heavy-tailed distribution of losses

The concept of Risk Parity (or Equal Risk Contribution), which has been widely used in financial portfolio management, aims at explicitly enforcing diversification in a portfolio by ensuring equal contribution from each asset to the total volatility. While the Risk Parity condition has a straightforward use case in finance, several other application areas can be found in engineering and operations research. In these settings, the Risk Parity condition can be interpreted as enforcing the fairness of a decision or as a way to balance between a number of candidate solutions. In this paper, we consider Risk Parity in conjunction with modern risk-averse stochastic optimization (namely coherent measures of risk), study a generalized Risk Parity model, and propose a combined two-stage diversification-risk framework. We also introduce a bi-level formulation in a case when hierarchical decision-making is enforced. An approach to reformulate the Risk Parity problem as second-order cone programming is also proposed. We assess the performance of the proposed models based on a case study in hazardous materials transportation. The results show their effectiveness in terms of fairness and risk equity for decision-making under uncertainty with heavy-tailed distribution of losses.

The best-fitting model(s) of equal risk contribution: evidence from environmental-friendly portfolio

PurposeThis research aims to select the best-fitting model(s) of equal risk contribution portfolios (ERC). ERC is a robust estimation in the absence of reasonable expectations about future returns.Design/methodology/approachThe portfolio consists of five environmental-friendly exchange-traded funds (ETFs). It applies equal risk optimization, beneficial when the assets are firmly linked, such as the ETFs. This paper operationalizes 20 covariance models in portfolio construction, and a portfolio with classic covariance is the benchmark to beat. To select the best-fitting model(s), the paper applies statistical inferences of the model confidence set. This research also constructs the newly-developed minimum connectedness optimization method and utilizes maximum drawdown as the primary evaluation tool.FindingsThe outbreak of COVID-19 hugely impacts the portfolio drawdown. The results also show that the classic covariance is hard to beat, partly explained by estimation error and model misspecification. This paper suggests that equal risk contribution can benefit from copula-based covariance. It consistently and significantly outperforms the other models in various robustness tests.Practical implicationsIn the absence of substantial predictions about future returns and the existence of strongly linked assets, selecting appropriate portfolio components by risk contribution is a sound choice.Originality/valueThis is the first paper to select the best-fitting model(s) of ERC portfolio during the COVID-19.

Improved iterative methods for solving risk parity portfolio

Risk parity, also known as equal risk contribution, has recently gained increasing attention as a portfolio allocation method. However, solving portfolio weights must resort to numerical methods as the analytic solution is not available. This study improves two existing iterative methods: the cyclical coordinate descent (CCD) and Newton methods. The authors enhance the CCD method by simplifying the formulation using a correlation matrix and imposing an additional rescaling step. The authors also suggest an improved initial guess inspired by the CCD method for the Newton method. Numerical experiments show that the improved CCD method performs the best and is approximately three times faster than the original CCD method, saving more than 40% of the iterations.

Machine learning portfolios with equal risk contributions: Evidence from the Brazilian market

We investigate the use of machine learning (ML) to forecast stock returns in the Brazilian market using a rich proprietary dataset. While ML portfolios can easily outperform the local market, the performance of long-short strategies using ML is hampered by the high volatility of the short portfolios. We show that an Equal Risk Contribution (ERC) approach significantly improves risk-adjusted returns. We further develop an ERC approach that combines multiple long-short strategies obtained with ML models, equalizing risk contributions across ML models, which outperforms, on a risk-adjusted basis, all individual ML long-short strategies, as well as alternative combinations of ML strategies.

Evaluating Hierarchical Equal Risk Contribution Portfolios in the Chinese Stock Market

This paper investigates the usefulness of the Hierarchical Equal Risk Contribution algorithm to exploit correlation structure in China’s equity market over 2001-2020. By running a horse race of different combinations of metrics and linkages, we demonstrate that the winner strategy always beats traditional portfolio construction techniques. Better-performing risk-based hierarchy strategies vary with stock-sorting methods by size, mean return, volatility, and Sharpe ratio. However, our treatment results in extremely imbalanced asset allocation, implying that we capture information other than the standard Chinese industrial classification.

Interpretable Machine Learning for Diversified Portfolio Construction

In this article, the authors construct a pipeline to benchmark hierarchical risk parity (HRP) relative to equal risk contribution (ERC) as examples of diversification strategies allocating to liquid multi-asset futures markets with dynamic leverage (volatility target). The authors use interpretable machine learning concepts (explainable AI) to compare the robustness of the strategies and to back out implicit rules for decision-making. The empirical dataset consists of 17 equity index, government bond, and commodity futures markets across 20 years. The two strategies are back tested for the empirical dataset and for about 100,000 bootstrapped datasets. XGBoost is used to regress the Calmar ratio spread between the two strategies against features of the bootstrapped datasets. Compared to ERC, HRP shows higher Calmar ratios and better matches the volatility target. Using Shapley values, the Calmar ratio spread can be attributed especially to univariate drawdown measures of the asset classes. <b>TOPICS:</b>Quantitative methods, statistical methods, big data/machine learning, portfolio construction, performance measurement <b>Key Findings</b> ▪ The authors introduce a procedure to benchmark rule-based investment strategies and to explain the differences in path-dependent risk-adjusted performance measures using interpretable machine learning. ▪ They apply the procedure to the Calmar ratio spread between hierarchical risk parity (HRP) and equal risk contribution (ERC) allocations of a multi-asset futures portfolio and find HRP to have superior risk-adjusted performance. ▪ The authors regress the Calmar ratio spread against statistical features of bootstrapped futures return datasets using XGBoost and apply the SHAP framework by Lundberg and Lee (2017) to discuss the local and global feature importance.

Risk Parity for Multi-Asset Futures Allocation – A Practical Analysis of the Equal Risk Contribution Portfolio

Since the early beginning of investing as it was commonly seen as a form of gambling for the rich and wealthy, the idea of Harry Markowitz was revolutionising the way of thinking and how portfolios should be constructed. However, today the traditional mean-variance portfolios are still not fully adopted by practitioners. After the financial crisis of 2008, a type of portfolio called risk parity arise and attracted the attention of numerous investors. In this paper, a risk parity portfolio named equal risk contribution portfolio is constructed based on a rolling window of 300 days. The portfolio is built on 21 future contracts downloaded from Quandl. It includes assets from four asset classes with a data range from June 2005 to March 2020. A performance and risk analysis is made for each asset, asset class and year. Many findings from the literature are reflected in our results, such as strong diversification and a higher Sharpe Ratio than the equally weighted Benchmark. The impact of the financial crisis of 2008 and the good performance of risk parity during this period can also be seen. To a certain extent, the COVID-19 crisis, in which our risk parity portfolio performs well, can also be observed.

Long-term dependency between sovereign bonds and sectoral indices of India: evidence using Hurst exponent and wavelet analysis

PurposeThe purpose of this study is to provide a new way to optimize a portfolio and to show that combining the Hurst exponent and wavelet analysis may help to increase portfolio returns.Design/methodology/approachThe authors use the Hurst exponent and wavelet analysis to study the long-term dependencies between sovereign bonds and sectoral indices of India. The authors further construct and evaluate the performance of three portfolios constructed on the basis of Hurst standard deviation (SD) – global minimum variance (GMV), most diversified portfolio (MDP) and equal risk contribution (ERC).FindingsThe authors find that an ERC portfolio generates positive superior return as compared other two. Since our sample includes periods of two crisis – post-2007 financial crisis and the ongoing pandemic, this study reveals that combining government bond with equities and gold provides a higher returns when the portfolios are constructed using the risk exposures of each asset in the overall portfolio risk.Practical implicationsThe findings provide guidance to portfolio managers by helping them to select assets using the Hurst approach and wavelet analysis thereby increasing the portfolio returns.Originality/valueIn this study, the authors use a combination of Hurst exponent and wavelet analysis to understand the long-term dependencies among various assets and provide a new methodology to optimize a portfolio. As far as the authors’ knowledge, no study in the past has attempted to provide a joint framework for portfolio optimization and therefore this study is the first to apply this methodology.

Dynamic risk-based optimization on cryptocurrencies

PurposeIt is crucial to find a better portfolio optimization strategy, considering the cryptocurrencies' asymmetric volatilities. Hence, this research aimed to present dynamic optimization on minimum variance (MVP), equal risk contribution (ERC) and most diversified portfolio (MDP).Design/methodology/approachThis study applied dynamic covariances from multivariate GARCH(1,1) with Student’s-t-distribution. This research also constructed static optimization from the conventional MVP, ERC and MDP as comparison. Moreover, the optimization involved transaction cost and out-of-sample analysis from the rolling windows method. The sample consisted of ten significant cryptocurrencies.FindingsDynamic optimization enhanced risk-adjusted return. Moreover, dynamic MDP and ERC could win the naïve strategy (1/N) under various estimation windows, and forecast lengths when the transaction cost ranging from 10 bps to 50 bps. The researcher also used another researcher's sample as a robustness test. Findings showed that dynamic optimization (MDP and ERC) outperformed the benchmark.Practical implicationsSophisticated investors may use the dynamic ERC and MDP to optimize cryptocurrencies portfolio.Originality/valueTo the best of the author’s knowledge, this is the first paper that studies the dynamic optimization on MVP, ERC and MDP using DCC and ADCC-GARCH with multivariate-t-distribution and rolling windows method.

Matrix Evolutions: Synthetic Correlations and Explainable Machine Learning for Constructing Robust Investment Portfolios

In this article, the authors present a novel and highly flexible concept to simulate correlation matrixes of financial markets. It produces realistic outcomes regarding stylized facts of empirical correlation matrixes and requires no asset return input data. The matrix generation is based on a multiobjective evolutionary algorithm, so the authors call the approach <i>matrix evolutions</i>. It is suitable for parallel implementation and can be accelerated by graphics processing units and quantum-inspired algorithms. The approach is useful for backtesting, pricing, and hedging correlation-dependent investment strategies and financial products. Its potential is demonstrated in a machine learning case study for robust portfolio construction in a multi-asset universe: An explainable machine learning program links the synthetic matrixes to the portfolio volatility spread of hierarchical risk parity versus equal risk contribution. <b>TOPICS:</b>Statistical methods, big data/machine learning, portfolio construction, performance measurement <b>Key Findings</b> ▪ The authors introduce the matrix evolutions concept based on an evolutionary algorithm to simulate correlation matrixes useful for financial market applications. ▪ They apply the resulting synthetic correlation matrixes to benchmark hierarchical risk parity (HRP) and equal risk contribution allocations of a multi-asset futures portfolio and find HRP to show lower portfolio risk. ▪ The authors evaluate three competing machine learning methods to regress the portfolio risk spread between both allocation methods against statistical features of the synthetic correlation matrixes and then discuss the local and global feature importance using the SHAP framework by Lundberg and Lee (2017).

Matrix Evolutions: Synthetic Correlations and Explainable Machine Learning for Constructing Robust Investment Portfolios

In this article, the authors present a novel and highly flexible concept to simulate correlation matrixes of financial markets. It produces realistic outcomes regarding stylized facts of empirical correlation matrixes and requires no asset return input data. The matrix generation is based on a multiobjective evolutionary algorithm, so the authors call the approach matrix evolutions. It is suitable for parallel implementation and can be accelerated by graphics processing units and quantum-inspired algorithms. The approach is useful for backtesting, pricing, and hedging correlation-dependent investment strategies and financial products. Its potential is demonstrated in a machine learning case study for robust portfolio construction in a multi-asset universe: An explainable machine learning program links the synthetic matrixes to the portfolio volatility spread of hierarchical risk parity versus equal risk contribution.

Adaptive Optimal Risk Budgeting

In this article, the authors suggest Bayesian-style adaptive enhancement to a popular equal risk contribution (ERC) portfolio construction technique they call adaptive optimal risk budgeting (AORB). The enhancement has the potential to bring portfolios closer to mean–variance efficiency when Sharpe ratios and correlations of assets vary while retaining some of the ERC’s robustness to estimation errors. The authors test AORB’s viability by putting it in competition with ERC itself and with a version of the Bayesian shrinkage mean–variance technique in a carefully simulated setting. They find that the new method appears to deliver measurable advantages over its competition in a broad range of realistic settings. Multiple possible applications to portfolios of risk premia strategies and a multi-asset universe more generally are discussed by the authors. TOPICS:Portfolio construction, analysis of individual factors/risk premia Key Findings • We suggested a computationally simple yet powerful portfolio construction approach that is formulated in terms of risk contributions but is designed to prescribe an approximately mean-variance efficient solution even when Sharpe ratios and correlations between assets vary in magnitude and over time. We called the new method Adaptive Optimal Risk Budgeting (AORB). • When tested against popular Equal Risk Contributions and “classical” Bayesian shrinkage methods, AORB was competitive in all cases and emerged as a clear winner when Sharpe ratios and correlations between assets were substantially differentiated. • AORB can also be trivially expanded to incorporate tail risk considerations.