Risk Parity Portfolio Research Articles

Gains from Markowitz Optimization: Evidence from Reoptimization of Mutual Fund Holdings

Prior studies have challenged the practical usefulness of Markowitz portfolio optimization in improving the return–risk trade-off in portfolio management. The authors approach this question from a unique angle by examining whether one can improve the performance of a large sample of actual mutual fund portfolios by reoptimizing the holdings using simple mean–variance optimization methods. The analyses produce compelling evidence of the benefits of Markowitz optimization. Simple portfolio optimization improves mutual fund portfolios’ risk-adjusted performance despite noisy expected return estimates inferred from mutual fund portfolio weights. Several alternative optimization strategies, including the risk-parity portfolio, minimum variance portfolio, mean–variance portfolio, and Sharpe ratio maximization portfolio, all outperform actual mutual fund portfolios in terms of the Sharpe ratio and other risk-adjusted performance measures. Moreover, the results are robust to subsamples partitioned on various dimensions. In contrast to the findings of DeMiguel et al. (2009), the authors find that the 1/N portfolio performs the worst.

Word Portfolio Optimization in the Environment of Zero Interest Rate

This paper provides a deep analysis of ten globally diversified portfolios, composed of different financial instruments: bonds, shares, ETF’s, commodities, indexes, currencies, constructed applying various optimization techniques. Statistical moments, such as mean, standard deviation, kurtosis and skewness of portfolios are compared and discussed. Moreover, performance of the portfolios within the time horizon of one year estimating Sharpe ratio, Treynor ratio, Sortino ratio is presented. Furthermore, a risk analysis of created portfolios is evaluated in terms of historical VaR and CVaR applying confidence interval 95%. The main results of this paper reveal that the portfolio, which is optimized to minimize VaR produces high expected shortfall. Secondly, the Risk Parity portfolio, despite reducing volatility, has delivered the highest kurtosis of the return, which may indicate the possible tail loss. Furthermore, the maximum Sharpe ratio portfolio has delivered extremely high kurtosis in comparison with the kurtosis of the other portfolios. Finally, it is observed that for the Naïve diversification portfolio it has been typical to have the highest downside deviation. 

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.

A Machine Learning Approach in Regime-Switching Risk Parity Portfolios

The authors present a machine learning approach to regime-based asset allocation. The framework consists of two primary components: (1) regime modeling and prediction and (2) identifying a regime-based strategy to enhance the performance of a risk parity portfolio. For the former, they apply supervised learning algorithms, including the random forest, based on a large macroeconomic database to estimate the probability of an upcoming recession or a stock market contraction. Out-of-sample tests show the reliability of these predictions, especially for recessions in the United States, over the period 1973 to 2020. The probability estimates are linked to a dynamic investment overlay strategy. The combined approach improves risk-adjusted returns by a substantial amount over nominal risk parity in two-asset and multi-asset test cases, even during rising interest rates in the late 1970s. <bold>TOPICS:</bold> <ext-link>Big data/machine learning</ext-link>, <ext-link>portfolio construction</ext-link>, <ext-link>performance measurement</ext-link> <bold>Key Findings</bold> <list><list-item> ▪ We examine a regime prediction problem with supervised learning approaches and implement regime-switching risk parity portfolios. </list-item><list-item> ▪ All recession periods after 1973 are captured by the random forest model, and stock market regime predictions lead to better portfolio performance. </list-item><list-item> ▪ Regime-switching models enhance risk parity portfolios, even during a rising interest rate period. Regime-based overlay strategies provide higher risk-adjusted returns in risk parity strategies. </list-item></list>

In Defence of Cyclical Coordinate Descent for Computing Risk Parity Portfolios

In this note, we review the conclusions of a paper comparing a cyclical coordinate descent (CCD) algorithm with other algorithms for solving risk parity portfolio optimization problems. In particular, we show that a proper numerical implementation of the CCD algorithm is fast, robust and convergent, confirming that the CCD algorithm is one of the most efficient algorithms to solve risk parity portfolio optimization problems.

Turning Tail Risks into Tailwinds

This study compares a broad range of risk models for managing multi-asset portfolios. The investment universe is extended to a range of systematic strategies with varying risk and return profiles. Focusing on risk parity portfolios, the authors show that considering tail risks can successfully reduce negative asymmetry and sharp losses. Extreme risk theory is of particular help in finding the right allocation to defensive systematic strategies in the portfolio. <b>TOPICS:</b>Portfolio construction, VAR and use of alternative risk measures of trading risk, tail risks <b>Key Findings</b> ▪ This study compares a broad range of risk models for managing multi-asset portfolios. ▪ The investment universe is extended to a range of systematic strategies with varying risk and return profiles. ▪ Focusing on risk parity portfolios, the authors show that considering tail risks can successfully reduce negative asymmetry and sharp losses.

Shrinkage Estimation in Risk Parity Portfolios

We investigate the impact of shrinkage estimation techniques for the moments of asset returns on risk-parity portfolios. In contrast to mean-variance portfolios, the risk contributions of individual assets in risk-parity portfolios are fixed a priori. This additional restriction stabilizes empirical portfolio weights in time. We show that the marginal risk budget for each portfolio asset indeed serves as a natural shrinkage target. Hence, we provide a new perspective on risk-parity portfolios. In an extensive empirical application, we compare and combine the various shrinkage strategies to popular risk-based approaches from the literature. We find that while using shrinkage estimators in risk-parity portfolios enhances out-of-sample performance based on several criteria, traditional covariance shrinkage estimators dominate all other strategies in high-dimensional settings.

Enhanced Index-Tracking Strategies Based on Systemic Financial Shocks: A Comparison of Countries Versus Sectors Investments

Since the launch of the first replicating funds in the 1970s, index-linked strategies have captured the interest of equity investors. On the one hand, active investing does not generally lead to higher risk-adjusted returns than a passive approach. On the other hand, passive vehicles cannot optimize the risk exposure to the equity markets without introducing elements of active management. The enhanced index-tracking offers all the advantages of traditional passive investing but aims to generate better returns than the reference benchmark. However, in the last decades, the financial system integration increased, reducing the diversification opportunities across markets, and meanwhile, more and more frequent extreme events affected the world. This amplified systemic instability caused unsatisfactory results even for these investment strategies. To better manage the markets turmoil and reduce losses, we propose alternative portfolio designs to improve the traditional index-tracking techniques. We include the systemic risks directly into the enhanced indexation problem and impose a minimum guaranteed extra-return on the benchmark with turnover control. The analysis builds on country and industry allocation policies in selected European markets from January 2004 to October 2021. Our findings prove that the proposed strategies generate consistent excess returns over the benchmark and outperform other indexing strategies and the equally weighted and risk parity portfolios.

An Analysis of Machine Learning Risk Factors and Risk Parity Portfolio Optimization

An Analysis of Machine Learning Risk Factors and Risk Parity Portfolio Optimization

Cardinality Constrained Risk Parity Portfolios

The risk parity optimization problem produces portfolios where each asset contributes an equal amount to the overall portfolio risk. While most work has investigated the problem using all assets, minimal work has investigated the cardinality constrained variant, which reduces the associated portfolio transaction overhead. In this work, we present the first formulations that can be solved to global optimality by off-the-shelf solvers. Specifically, we propose two new quadratically constrained quadratic integer programs, a non-convex and a convex one, that can be solved to global optimality without the need for specialized algorithms, heuristics, or approximations. We strengthen our formulations by adding tighter variable bounds and valid constraints. Computational experiments on real-world financial data indicate the effectiveness of our formulations at producing portfolios with equal risk contributions of chosen cardinality size. Specifically, the convex formulation is shown to be very efficient in terms of both speed and accuracy, while producing portfolios with great out-of-sample performance.

Investing in the year of Corona: The Modified Risk Parity Portfolios

Extended examples of long-term, annually rebalanced portfolio performance using the modified risk parity (MRP) approach are presented. The analysis considers three distinct diversified portfolio holdings, comprising index funds, sector funds and a blended portfolio of index and sector funds.The analysis considers the time period 2000-mid November 2020, which includes drawdown from the March 2020 worldwide outbreak of coronavirus 2019 disease.Comparisons of MRP return performance versus competitive portfolios of index-following or balanced funds are presented. Results indicate that for sufficiently long holding periods, substantial out-performance of MRP allocation strategies relative to passive buy-and-hold benchmarks can be realized.

Data-Driven Distributionally Robust Risk Parity Portfolio Optimization

We propose a distributionally robust formulation of the traditional risk parity portfolio optimization problem. Distributional robustness is introduced by targeting the discrete probabilities attached to each observation used during parameter estimation. Instead of assuming that all observations are equally likely, we consider an ambiguity set that provides us with the flexibility to find the most adversarial probability distribution based on the investor’s confidence level. This allows us to derive robust estimates to parametrize the distribution of asset returns without having to impose any particular structure on the data. The resulting distributionally robust optimization problem is a constrained convex-concave minimax problem. Our approach is financially meaningful and attempts to attain full risk diversification with respect to the worst-case instance of the portfolio risk measure. We propose a novel algorithmic approach to solve this minimax problem, which blends projected gradient ascent with sequential convex programming. By design, this algorithm is highly flexible and allows the user to choose among alternative statistical distance measures to define the ambiguity set. Moreover, the algorithm is highly tractable and scalable. Our numerical experiments suggest that a distributionally robust risk parity portfolio can yield a higher risk-adjusted rate of return when compared against the nominal portfolio.

Risk parity with expectiles

A recent popular approach to portfolio selection aims at diversifying risk by looking for the so called Risk Parity portfolios. These are defined by the condition that the risk contributions of all assets to the global risk of the portfolio are equal. The Risk Parity approach has been originally introduced for the volatility risk measure. In this paper we consider expectiles as risk measures, we refine results on their differentiability and additivity, and we show how to define Risk Parity portfolios when the expectiles are used. Furthermore, we propose three different classes of methods for practically finding Risk Parity portfolios with respect to expectiles, and we compare the accuracy and efficiency of these methods on real-world data. Expectiles are also used as risk measures in the classical risk-return approach to portfolio selection, where we present a new linear programming formulation.

Risk Parity with Expectiles

A recent popular approach to portfolio selection aims at diversifying risk by looking for the so called Risk Parity portfolios. These are defined by the condition that the risk contributions of all assets to the global risk of the portfolio are equal. The Risk Parity approach has been originally introduced for the volatility risk measure. In this paper we consider expectiles as risk measures, we refine results on their differentiability and additivity, and we show how to define Risk Parity portfolios when the expectiles are used as coherent risk measures. Furthermore, we propose several methods for practically finding Risk Parity portfolios with respect to expectiles and we compare their accuracy and efficiency on real-world data. Expectiles are also used as risk measures in the classical risk-return approach to portfolio selection, where we present a new linear programming formulation.

Risk Return Trade-Off in Relaxed Risk Parity Portfolio Optimization

This paper formulates a relaxed risk parity optimization model to control the balance of risk parity violation against the total portfolio performance. Risk parity has been criticized as being overly conservative and it is improved by re-introducing the asset expected returns into the model and permitting the portfolio to violate the risk parity condition. This paper proposes the incorporation of an explicit target return goal with an intuitive target return approach into a second-order-cone model of a risk parity optimization. When the target return is greater than risk parity return, a violation to risk parity allocations occurs that is controlled using a computational construct to obtain near-risk parity portfolios to retain as much risk parity-like traits as possible. This model is used to demonstrate empirically that higher returns can be achieved than risk parity without the risk contributions deviating dramatically from the risk parity allocations. Furthermore, this study reveals that the relaxed risk parity model exhibits advantageous traits of robustness to expected returns, which should not deter the use of expected returns in risk parity model.

Evaluation of Risk Parity Asset Allocation Strategy in Indonesia Mutual Fund During 2010–2019 Period

Indonesia as the largest country in Southeast Asia have the highest GDP among the other countries in the region. In 2011, McKinsey &amp; Company conducted a research to understand the annual consumer spending trend in Indonesia. According to the research, financial services will be the main consumption in Indonesia in 2030. Despite those promising situations, mutual fund industry in Indonesia was facing a big challenge in 2019. During 2019, Indonesia Financial Authority (OJK) has suspended 37 mismanaged mutual fund products from several investment managers. And actively managed mutual funds will have more exposure on this mismanagement risk.&#x0D; Aside from mismanagement and underperforming risk of actively managed mutual fund, investor also facing high risk of equity asset class. Recently, IDX Composite (IHSG) experience a tremendous -28% drop in Q1 2020. Biggest peak to bottom drawdown during that period was -37,49% from 14 January 2020 until 24 March 2020. This situation creates a challenge for mutual fund sales agent (APERD).&#x0D; But a challenging situation also offers opportunities. Mutual fund sales agent (APERD) that can offer optimum asset allocation suggestion that can give maximum risk protection can take advantage of these situation. One such unique strategy that has not been applied by other mutual fund sales agent (APERD) is risk parity that became popular after 2008 financial crisis.&#x0D; This research applied risk parity framework in passively managed mutual fund because most of actively managed mutual funds in Indonesia underperform the benchmark in the 10-year period. Because of that, passively managed mutual funds provide better investment opportunity. Four equity indexes (LQ45, Jakarta Islamic Index, Bisnis 27, and Sri Kehati) and one passively managed fixed income mutual fund (ABF Indonesia Bond Index Fund) that have existed for 10-year period was chosen to apply the risk parity method. Sharpe ratio calculation is used to evaluate the portfolio performance compared to commonly used portfolio strategy in the world according to 2018 Legg Mason Global Investment Survey.&#x0D; Result of the analysis is that risk parity portfolio offer better Sharpe ratio compared to commonly used portfolio strategy in the world. Result of 2009 - April 2020 back-testing process also show that risk parity can protect investor from equity market downturn in 77.3% of the case. With above result, risk parity can offer better risk protection for Indonesia individual investor.

A robust framework for risk parity portfolios

We propose a robust formulation of the traditional risk parity problem by introducing an uncertainty structure specifically tailored to capture the intricacies of risk parity. Typical minimum variance portfolios attempt to introduce robustness by computing the worst-case estimate of the risk measure, which is not intuitive for risk parity. Instead, our motivation is to shield the risk parity portfolio against noise in the estimated asset risk contributions. Thus, we present a novel robust risk parity model that introduces robustness around both the overall portfolio risk and the assets’ marginal risk contributions. The proposed robust model is highly tractable and is able to retain the same level of complexity as the original problem. We provide a general procedure by which to create an uncertainty structure around the asset covariance matrix. We quantify this uncertainty as a perturbation on the nominal covariance estimate, which allows us to intuitively embed robustness during optimization. We then propose a specific procedure to construct a robust risk parity portfolio through a factor model of asset returns. Computational experiments show that the robust formulation yields a higher risk-adjusted rate of return than the nominal model while maintaining a sufficiently risk-diverse portfolio.

Generalized risk parity portfolio optimization: an ADMM approach

Generalized risk parity portfolio optimization: an ADMM approach

Generalized Risk Parity Portfolio Optimization: An ADMM Approach

The risk parity solution to the asset allocation problem yields portfolios where the risk contribution from each asset is made equal. We consider a generalized approach to this problem. First, we set an objective that seeks to maximize the portfolio expected return while minimizing portfolio risk. Second, we relax the risk parity condition and instead bound the risk dispersion of the constituents within a predefined limit. This allows an investor to prescribe a desired risk dispersion range, yielding a portfolio with an optimal risk-return profile that is still well-diversified from a risk-based standpoint. We add robustness to our framework by introducing an ellipsoidal uncertainty structure around our estimated asset expected returns to mitigate estimation error. Our proposed framework does not impose any restrictions on short selling. A limitation of risk parity is that allowing of short sales leads to a non-convex problem. However, we propose an approach that relaxes our generalized risk parity model into a convex semi-definite program. We proceed to tighten this relaxation sequentially through the alternating direction method of multipliers. This procedure iterates between the convex optimization problem and the non-convex problem with a rank constraint. In addition, we can exploit this structure to solve the non-convex problem analytically and efficiently during every iteration. Numerical results suggest that this algorithm converges to a higher quality optimal solution when compared to the competing non-convex problem, and can also yield a higher ex post risk-adjusted rate of return.

Practical Applications of Business Cycle–Related Timing of Alternative Risk Premia Strategies

Practical Applications In Business Cycle–Related Timing of Alternative Risk Premia Strategies, from the Spring 2020 issue of The Journal of Alternative Investments, Bernd Scherer and Matthias Apel relate how they developed a business cycle model to explore its predictive power in a tactical allocation framework. They created an aggregate of economic indicators that identifies economic regimes in order to determine the sensitivities of a host of risk premia strategies across various business cycle phases. Many of these sensitivities, measured as conditional Sharpe ratios, are statistically different from their unconditional, overall risk-adjusted returns. During each regime, a long–short tactical overlay strategy takes a long position in the four alternative risk premia strategies with the highest Sharpe ratios and shorts the four with the lowest. The authors show that the overlay strategy improves the performance of a risk-parity portfolio and provide evidence that the observed return patterns are not a beta-driven result. While their primary interest lies within a universe of investable alternative risk premia strategies, cross-validation tests also applied the model within three additional investment universes: global asset classes, smart betas, and Fama–French risk factors.