- Andreas Steiner
- To order reprints of this article, please contact Dewey Palmieri at dpalmieri{at}iijournals.com or 212-224-3675.
Abstract
After a short introduction to risk party, this article introduces robust risk parity, a simple formula that allows the construction of portfolios with equal contributions to volatility for arbitrary large asset universes. The empirical performance of robust risk parity is evaluated with two historical datasets, with equity ETFs and indices representing a multiasset class universe.The analysis of the in-sample and out-of-sample results illustrates how to properly evaluate ex post risk parity. Three practical recommendations are of importance to risk parity investors and investment managers alike.
Risk parity investment strategies are relatively new. Their current actuality is not shaped by their novelty but by recent market experiences: Disappointed with the performance of market-weighted benchmark portfolios and the merits of active portfolio management, many investors in recent years developed an interest in alternative index definitions and portfolios not driven by promised alpha skills, but transparent rules considering risk. The term “risk parity” is mainly used in the Anglo-Saxon world, whereas in Continental Europe, the strategy is known as “equal risk contribution.” We use the expressions synonymously. Originally, risk parity strategies were designed to fit the risk budgets of defensive investors. Over time, creative product managers developed leveraged versions that can be tailored to any risk appetite.
The analytics of the risk parity portfolio have been studied in detail by Maillard, Roncalli, and Teiletche [2008]. The economics of the equal risk contribution portfolio are discussed, for example, in Lee [2011] and Scherer [2011]. In our article, we give a short introduction to risk parity and introduce an algorithm to construct risk parity portfolios that is not only simpler to implement than exact solutions, but more robust to estimation error. The performances of the exact and simplified algorithms are compared in two in-sample and out-of-sample experiments with real-world data. We conclude with three practical recommendations for risk parity investors.
INTRODUCTION TO RISK PARITY
Risk is nonadditive: portfolio volatility does not equal the sum of asset volatilities because asset returns are correlated. As a consequence, the variance of a portfolio consisting of n assets is not calculated as the weighted sum of asset volatilities, but the weighted sum of all mutual asset covariances s_{i,j}:
1Here, and in the rest of this article, we assume that the asset weights, w, in the portfolio sum to one. This is equivalent to saying that the investor is fully invested and not leveraged. Implicitly, we also assume that asset returns follow a multivariate normal distribution, making volatility the only relevant risk measure. Using elementary algebra for covariances, it is possible to express portfolio variance in terms of covariances s_{i,P} and correlations ?_{i,P} of an asset with the overall portfolio:
2Dividing by portfolio volatility, we can express portfolio volatility as the weighted sum of a multiplicative term arising from asset correlations with the portfolio and asset volatility:
3Despite the fact that portfolio volatility is nonadditive in asset volatility, we were able to derive an additive decomposition. Note that the partial derivative of portfolio volatility with respect to the weight of one asset is
4Therefore, the additive decomposition in Equation (3) can also be interpreted as a weighted sum of partial derivatives:
5To summarize, the contribution of a single asset to portfolio volatility, rc_{i} , is defined as
6 7The risk parity portfolio is defined as a portfolio in which all assets have equal contribution to volatility:
8The parity in contributions is achieved by setting the asset weights accordingly, as all other inputs (correlations and volatilities) are not under the control of the investment manager. A closed-end formula to calculate the asset weights in risk parity portfolios does not exist, but solutions for long-only portfolios can be found with numerical solvers. Calculations for larger universes such as the S&P 500 can become a considerable computational burden.
The portfolio construction principle underlying the risk parity concept is not limited to portfolio volatility per se. In fact, it can be applied to any portfolio characteristic that can be expressed as the weighted sum of the asset characteristics For example, it would, theoretically, be possible to build equal contribution to kurtosis; if we accept kurtosis as a tail risk indicator, the resulting portfolio would exhibit “tail risk parity.”
ROBUST RISK PARITY
If we assume that all asset correlations are equal, then the asset weights resulting in risk party can be calculated from asset volatilities with a simple formula:
9The weight of asset i in the risk parity portfolio, assuming equal correlations, is the percentage of the asset’s inverse volatility of the sum of all inverse asset volatilities. The formula can be easily applied to very large asset universes. It is also possible to use proven estimators (e.g., GARCH) to estimate univariate asset volatilities.
We call the resulting portfolio the “robust risk parity portfolio” because it can be related to the average correlation portfolio in the literature on mean variance portfolio optimization (or an overview, see, e.g., Elton et al. [2007]). Average correlation models assume that all asset correlations are identical. This assumption is justified when sufficient information to differentiate asset correlations is not available either due to lack of data or generally high levels of model risk surrounding the estimation of correlations. Replacing a full correlation matrix with a single correlation coefficient dramatically reduces estimation risk, making the average correlation model a robust estimator. Empirically, the average correlation model is known to perform rather well and often results in a portfolio with superior out-of-sample risk-adjusted performance compared to mean variance efficient portfolios with a full correlation matrix.
Theoretically, the difference between the robust and exact risk parity portfolios can be significant. But interestingly, in all of our out-of-sample experiments with real-world data, the differences between the robust and exact risk parity portfolio were negligible. Below, we present two cases featuring a multi-asset class and a single-asset class portfolio. This observation implies that complex numerical solvers can be avoided and risk parity can be implemented for arbitrarily large investment universes. Investors and investment managers alike can construct robust risk parity portfolio for arbitrary large asset universes with simple spreadsheet formulas—risk parity for the masses—at least from the viewpoint of availability of portfolio construction techniques for a wider audience.
SOME CHARACTERISTICS AND MISCONCEPTIONS
Knowing how to calculate risk parity portfolios does not answer the question of why we should be interested in doing so in the first place. The arguments usually given are related to diversification and performance: First, risk parity portfolios are considered better diversified because the exposures are spread across assets. Second, risk parity portfolios are believed to outperform inefficient cap-weighted strategies, that is, passive investing.
The first argument is based on a common misunderstanding about diversification as used in the context of modern portfolio theory (MPT). MPT does explain why it is wise not to put all your eggs in one basket, but it does not imply the opposite, which would be to put your eggs in as many baskets as possible. Diversified portfolios are portfolios located on the efficient frontier. Such frontier portfolios offer the highest return for a given amount of risk and lowest risk for a given amount of a return. Rational investors select diversified portfolios because they are not prepared to bear any risk that can be eliminated for free by varying the asset weights, not because they have a preference for exposures that exhibit low concentration in terms of asset exposures. Whether efficient portfolios are concentrated or not depends on empirical asset returns and covariances.
The construction of risk parity portfolios does not require expected asset returns, but otherwise uses the same inputs as in the calculation of mean variance efficient portfolios. Risk parity per se does not imply low asset concentrations, only low concentration in risk contributions. Whether low risk contribution concentration implies low asset exposure concentration depends on empirical asset covariances. For example, given three assets with volatilities and correlations of
This risk parity portfolio is clearly concentrated in asset A. It is easy to construct even more extreme theoretical cases by varying the inputs. On the other hand, empirical studies show that when using real-world asset volatilities and correlations, the resulting risk parity portfolios tend to be much less concentrated than frontier portfolios.
Not surprisingly, risk parity portfolios are typically not located on the efficient frontier, as shown in Exhibit 1.
The grey area represents the opportunity space spanned by six equity indices over a certain period in time. The risk parity portfolio is an “undiversified” portfolio in an MPT sense, because it is possible to achieve a much higher return at the same level of risk by tweaking asset weights.
Note that in this particular example calculated from real-world data, the risk parity underperforms the equal-weighted as well as minimum-variance portfolio (located at the lower left end of the efficient frontier).
The fact that the risk parity portfolio is located to the left of the equal-weighted portfolio in mean volatility space is not a coincidence. It can be shown that the volatility of the equal weight portfolio is an upper bound, and the volatility of the minimum variance portfolio a lower bound (Maillard, Roncalliz, and Teiletche [2008]). This tilt of the risk parity portfolio toward the minimum variance portfolio is the main reason why risk parity is often perceived as a low-risk investment strategy. But as with concentration, risk parity per se does not guarantee low volatility because the location of the risk parity portfolio relative to the passive cap-weighted portfolio in mean volatility space is determined by the data. This means that reducing risk by adding cash to a cap-weighted passive portfolio is a more reliable risk management tool than switching from a passive cap-weighted portfolio to a risk parity portfolio: While the former strategy guarantees lower volatility risk when compared to the cap-weighted portfolio, the latter does not.
When comparing the asset allocation between the risk parity and the equal-weighted portfolios in the equity example given earlier in this article, we see that both portfolios exhibit low levels of asset concentration (Exhibit 2).
Low concentration on its own, however, has no value for investors. To make risk parity an attractive investment strategy, superior performance is necessary. Justifying the risk parity strategy with performance arguments is very interesting because return considerations are not part of the construction process of risk parity portfolios at all. Therefore, any measured or expected superior performance—often called alpha—must be an ex ante unintended side effect.
The persistence of these side effects is not guaranteed, which means that risk parity is an active investment strategy like any other. The superiority of the strategy depends on the stability of its alpha sources. As a consequence, risk parity becomes part of the zero sum game that every active investment management strategy is subject to: for every outperforming portfolio, at least one underperforming portfolio has to exist (Sharpe [1991]). The zero sum characteristic of active management is a reason why risk parity will never become an investment strategy for the masses.
On the other hand, attractive performance characteristics of low-risk strategies, such as minimum variance, have been documented in many empirical studies. Scherer [2011] explains the success of low-risk, single-equity strategies by the fact that they indirectly pick up risk-based pricing anomalies. He distinguishes two types of risk anomalies:
1. Beta anomaly: Low beta stocks tend to earn more than their beta implies.
2. Volatility anomaly: Stocks with high residual risk exhibit returns too low compared to their equilibrium risk-adjusted returns.
Scherrer [2011] also shows that the minimum-variance portfolio tends to be concentrated in low beta and low residual risk assets. If the beta of the minimum-variance portfolio is close to one, the weight of asset i in the minimum-variance portfolio can be approximated by
10The asset weights in both the minimum-variance portfolio and the robust risk parity portfolio are inverse functions of asset volatility. Therefore, both risk parity and minimum-variance investing favor low-volatility stocks and benefit from the volatility anomaly. Maillard, Roncalli, and Teiletche [2008] show that asset weights are inversely related to the asset’s beta with a market proxy. Therefore, both risk parity and minimum variance benefit from the beta anomaly.
The alpha of risk parity strategies unintentionally exploits known risk anomalies.
IN-SAMPLE AND OUT-OF-SAMPLE PERFORMANCE OF RISK PARITY PORTFOLIOS
In this section, we present some findings of various in-sample and out-of-sample experiments with asset return time series. As we do not consider several important features of real-world investing, such as transaction costs, we prefer to speak of “experiments” rather than “backtests.”
In in-sample experiments, the estimation period is equal to the investment period: volatilities and correlations are estimated over a certain period and risk parity portfolios are constructed with these inputs that are invested over the same period of time. In-sample experiments assume perfect foresight: the risk characteristics in a certain period of time are known in advance. In out-of-sample experiments, the estimation and investment periods do not overlap. For example, we estimate volatilities based on the previous 36-month return and then determine the weights of the risk parity portfolio for the next month. Out-of-sample tests exhibit estimation risk if the past is not repeated perfectly.
One question that arises immediately is how to evaluate risk parity portfolios. Asset volatilities and correlations are not constant, but change over time; so do asset weights. Both of these effects will cause realized risk contributions to deviate from exact parity. Ex post risk contributions, RC_{i} , are calculated from ex post asset return contributions, C_{i} , as follows (see Steiner [2010] for more information)
11 12The ex post volatility contribution of asset i is calculated as the product of the volatility of ex post asset contributions with the correlation of asset ex post contributions with the overall portfolio. The asset weight does not appear in the expression because it is already contained in C_{i} .
The first portfolio we analyze consists of seven equity exchange-traded funds in Europe. We look at daily price returns over the period from 1995 to November 2011. The results of an in-sample test can be summarized in a mean volatility plot (Exhibit 3).
As before, the grey area represents the investment opportunity space spanned by the seven ETF products. The minimum-variance and maximum Sharpe ratio portfolios are located on the efficient frontier. The equal-weighted, risk parity, and robust risk parity portfolios are inefficient and located very close to one another. This can be explained by the similarity of the correlations and volatilities of the funds. All assets belong to the same asset class and are therefore rather homogeneous in their risk and return characteristics. As a consequence, the volatility contributions in the equal weighted and risk parity portfolios are very similar (Exhibit 4).
In an out-of-sample test, we build daily risk parity portfolio constructed from historical correlations and volatilities of the previous 180 days. The cumulative returns of the calculated strategies are plotted in Exhibit 5.
The similarity between the equal weights and risk parity portfolios shows again. The two portfolios are very similar to the minimum-variance portfolio in terms of risk and return characteristics. The weights of the portfolio constituents in the risk parity portfolio over time are shown in Exhibit 6.
The portfolio composition is remarkably stable and asset concentration relatively low. The ex post risk contributions are shown in the Exhibit 7. As expected in out-of-sample experiments, the risk parity is not perfect, but varies between 17% and 11% (perfect risk parity would require all volatility contributions to equal 1/7 = 14.29%).
A second portfolio that we analyzed consists of seven asset classes represented by popular indices. The asset classes exhibit very diverse volatilities and correlations. Their cumulative monthly returns from January 2001 to October 2011 is shown in Exhibit 8.
The in-sample results are shown in Exhibit 9. In this example, relevant differences exist between the equal-weighted and the risk parity portfolios, reflecting the dispersion of risk characteristics of the portfolio constituents.
Note that the volatilities of both risk parity portfolios are higher than the volatility of the maximum Sharpe ratio portfolio, an important reminder that “risk parity” per se does not mean “low risk.”
The comparison of allocations shown in Exhibit 10 reveals that the two portfolios located on the efficient frontier exhibit high asset concentrations. The concentration of the risk parity portfolio is tilted towards the same dominating asset, but is clearly less extreme.
The cumulative returns of an out-of-sample test (using the last 36 monthly observations to estimate the inputs, rebalanced monthly) are shown in Exhibit 11.
The high correlation between the two risk parity strategies is striking. Overall, the risk parity strategies are positioned between the minimum-variance and the equal-weighted portfolios both in terms of risk and return characteristics.
CONCLUSIONS
We summarize our brief analysis of risk parity with three conclusions having practical implications for risk parity investors as well as investment managers.
First, we introduced the concept of robust risk parity, which allows the construction of risk parity portfolios with a simple formula. In all of our out-of-sample experiments, there is very little difference between robust risk parity and the exact solution. This makes risk parity accessible for an audience beyond specialist quants and allows for the construction of risk parity portfolio for arbitrary large asset universes.
Secondly, the promise of the risk parity strategy is risk parity. Parity of risk contributions has no implications on total risk or asset concentrations. When evaluating risk parity strategies in performance and risk appraisal processes, properly calculated ex post risk contributions should be considered.
Last, risk parity per se has no implications on expected returns. In fact, the portfolio construction process does not consider expected asset returns at all, turning any superior risk-adjusted performance of risk parity into an unintended side effect. As risk parity strategies tend to overweight low-volatility and low beta assets, risk anomalies are possible sources of alpha. If all investors underweight in the same defensive stocks, the anomalies and risk parity alphas can be expected to vanish over time. This makes risk parity investing an active investment strategy like any other.
- © 2012 Pageant Media Ltd