Extending The Risk Parity Approach To Higher Moments: Is There Any Value-Added?

Eduard Baitinger

FERI Trust GmbH

André Dragosch

FERI Trust GmbH

Anastasia Topalova

FERI Trust GmbH

October 28, 2015

Abstract:

The popular risk parity approach is based on volatility as the sole risk measure and therefore lacks the consideration of tail risk. This fact makes risk parity portfolios vulnerable to tail events. We address this issue by showing how higher risk moment terms can be consistently incorporated in the risk parity optimization. In addition, we present a novel optimization approach, whereby optimal moment weightings (preferences) in the risk parity optimization are imputed from the data. In a broad-based empirical out-of-sample study and simulation analysis, we find superior performance of higher moment risk parity portfolios when the underlying data exhibits significant higher moments and co-moments. This fact renders higher moment risk parity portfolios ideal candidates for worst case regimes.

Extending The Risk Parity Approach To Higher Moments: Is There Any Value-Added? – Introduction

The well known risk parity approach to portfolio management, which is widely used in professional asset management (Qian (2005, 2013), Chaves et al. (2011)), is based on an equally-weighted portfolio, whereby the weights are measured in terms of absolute risk contributions of individual assets to portfolio volatility (Maillard et al. (2010)). Therefore, risk parity portfolios can be considered as ad hoc extensions of the popular equally-weighted strategy to the risk dimension defined in terms of variance. However, this implies that the \classical” risk parity approach solely focuses on equal risk contributions with regard to the second moment (variance) and thereby neglects higher moments of return distributions altogether. This can result in portfolio choices that are vulnerable to tail events. To put it differently, second moment risk parity portfolios can exhibit significant risk disparity with respect to higher moments. Since higher moments (e.g. skewness, kurtosis, when standardized) quantify tail risks, a higher moments disparity can lead to inferior portfolio performance in negative tail events. This concern can be addressed by incorporating higher moments in risk parity optimization and in this way avoiding risk contribution concentrations with regard to higher moments.

Especially, after the recent subprime crisis (2008/09) the \classical” risk parity approach gained considerable attention in practical asset management. Due to its defensive character in a multi asset class context, risk parity portfolios weathered the volatile period surrounding the subprime crisis rather well. The risk parity approach to investing was initially established by Qian (2005). The first analytical formalization of the risk parity approach was provided by Maillard et al. (2010). Numerous empirical studies prove the competitiveness of the risk parity approach. In this regard, Chaves et al. (2011) demonstrate that risk parity portfolios outperform on a risk-adjusted basis (Sharpe ratio) the minimum  variance portfolio and mean-variance efficient portfolios over a 30 year dataset of US stocks and bonds. Furthermore, they show that the outperformance of risk parity portfolios is stable with regard to various subsamples. The empirical study of Clarke et al. (2013) utilizes US stock data over the period 1968-2012. According to their results, the Sharpe ratio of the risk parity portfolio exceeds those of the value-weighted, equally-weighted and maximum diversification1 portfolios. On the other hand, the risk parity portfolio slightly underperforms the minimum variance benchmark in terms of Sharpe ratio. However, the empirical results of Clarke et al. (2013) reveal that the minimum variance portfolio is much more concentrated as it effectively2 selects just 35.7 out of 1000 assets on average and is hence highly unpractical. In contrast, the risk parity portfolio selects by construction all assets and therefore exhibits a much higher average effective number of selected assets of 934.1 (out of 1000). This implies that the risk parity portfolio is much more diversified in terms of effective number of selected stocks than the minimum variance portfolio, implying that it can be easier transferred to a practical implementation (e.g. comply with regulatory requirements). Lastly, Asness et al. (2012) run a horse race between the risk parity approach, the 60/40 (60% equities, 40% bonds) investment strategy and the value weighted index. Using various international and US datasets, they demonstrate superior Sharpe ratios and terminal wealth levels of the risk parity investment approach.

However, all the above mentioned papers establishing and testing the risk parity approach are limited in one important aspect. They exclusively focus on the second (co-)moment of return distributions as the ultimate measure of risk. To be more precise, Qian (2005) and Maillard et al. (2010), respectively, establish and analytically formalize the risk parity framework considering asset return variances/volatilities as the sole measures of risk. On the one hand, this procedure is reasonable as the risk parity approach has its roots in the practical asset management industry. Practical asset managers and investors regard portfolio volatility as one of the most important risk measures besides the Value-at-Risk metric. Nonetheless, exclusively focusing on the second (risk-)moment can lead to suboptimal portfolio choices with respect to higher (risk-)moments in the presence of significant higher (co-)moments. Higher moments of return distributions convey information with respect to tail behavior of asset returns and in this way their characteristics in extreme up and down (crash) scenarios. Especially, the crash scenario attributes of asset returns are of strong importance for (real-life) risk/loss averse investors and should therefore be incorporated in the portfolio formation process. Furthermore, the above-mentioned performance studies of risk parity portfolios (Chaves et al. (2011), Asness et al. (2012), Clarke et al. (2013)) solely focus on portfolio volatility as a source of risk when measuring performance and thereby neglect tail behavior of implemented investment strategies.

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