M0020 Demystifying Risk Parity

Demystifying Risk ParityHakan Kaya, Ph.DVice President Quantitative Investment Group Neuberger Berman

INTRODUCTION

Wai Lee, Ph.D.Managing Director Chief Investment Officer and Director of Research Quantitative Investment Group Neuberger Berman

The majority of risk parity analysis is treated as a heuristic process and compares the backtests of different allocation methods with less of an emphasis on investment rationale. We investigate risk parity under different settings, highlight its potential utility, and provide insight into when this method may be expected to outperform by conducting path-independent controlled simulation experiments. Following an extended period of market turbulence, macroeconomic dislocations and increased cross-asset class correlations, the investment community is increasingly looking beyond traditional one size fits all asset allocation strategies to find solutions that may be effective in an increasingly variable environment. The past few years have undoubtedly been challenging for investors. From increased market volatility to historical trend deviations to myriad macro-level events that have impacted market and asset price behavior in often extreme ways, investors have been left to question long-held assumptions underlying various asset allocation methodologies as well as their own approaches. The environment has shifted to one in which constructing multi-asset class portfolios that can deliver on investment objectives over a period of several years seems infinitely more complicated than just a few years ago. In fact, the third quarter of 2011 punctuated this perhaps most distinctly, as many markets posted their worst quarter since early in the financial crisis, only to be followed by a month that was one of the best for certain equity markets since the 1970s. How should investors handle these extremes and how can they effectively build portfolios to weather such storms and changing conditions? These questions highlight many of the assumptions that led investors to more traditional asset allocation methodologies in the first place. For institutions, for example, typical plan restrictions might prohibit the use of leverage or shorting of securitiesyet, at the same time, have a required return, which has led long-term allocations to relatively risky assets, such as equities. Over time, the portfolio mix of 60% equities and 40% fixed income, or slight variations thereof, emerged as typical because it was thought to have a good chance of meeting the required return. This was accepted despite the known concentration in equity risk resulting from the mismatch between equity and fixed income risks in such a portfolio. More recently, the investment communitys focus has shifted to developing alternative approaches to asset allocation and multi-asset class portfolios. As part of this trend, both risk budgeting and diversification are being re-evaluated. On risk budgeting, Bender, Briand, Nielsen, and Stefek (2010), and more recently Page and Taborsky (2011), promoted the so-called risk class, or factor-based approach instead of risk modeling and budgeting based on assets. Others, including Kaya, Lee, and Wan (2011) and Kowara

March 2012

NOT FOR RETAIL CLIENT USE IN EUROPE

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DEMYSTIFYING RISK PARITY

and Idzorek (2011), however, pointed out that while the risk class approach may offer complementary insights to the asset class approach, no magic really exists behind such a new approach. Regardless of approach, risk budgeting remains just as challenging as it was before. Diversification is one of the most used terms in investing, but its meaning and application is not always clear (see discussions in Meucci (2009), Levell (2010), and Lee (2011) on the ill-defined nature of diversification). We agree with Markowitzs (1952) Modern Portfolio Theory (MPT) that investors should diversify to the degree they are uncertain. In other words, if investors had perfect forecasting ability, diversification would be unnecessary. Once an investment objective such as a preference of return with aversion to risk as proxied by volatility is clearly stated, Lee (2011) argues that the efficient portfolio (as constructed according to MPT) is the best portfolio in that it is expected to maximize return and diversify to the appropriate degree given the investors perceived degree of uncertainty of the future. To be clear, diversification does not necessarily mean that one should hold a portfolio that is expected to have the minimum volatility. Scherer (2010, p.3) questions whether the minimization of risk on its own in the spirit of minimum volatility, and also conceptually in relation to maximizing diversification, is a meaningful objective. A number of studies have examined the empirical properties of some alternative asset allocation approaches, collectively categorized as risk-based asset allocation portfolios (see Lee (2011)), that do not require explicit forecasts of expected returns. Instead, these portfolios can be constructed solely based on forecasts of risks, which are typically represented by a covariance matrix. In the literature, these risk-based approaches have been applied to stock selection in different regions during different sample periods, with encouraging simulated outperformance of such portfolios relative to the market capitalization-weighted portfolios. Historical simulations, however, have to be interpreted with caution, particularly for the portfolios requiring ad hoc constraints in order to keep the portfolios well-behaved with acceptable turnovers and less extreme position concentration (see Demey, Maillard, and Roncalli (2010) and Lee (2011) for more discussions). By revealing the lack of a clearly defined investment objective behind these portfolios, Lee (2011) advocates shifting the focus of studies on these portfolios from being largely heuristic, intuitive, and empirical, to more formal, conceptual, and theoretical. Among these risk-based approaches, the Risk Parity (RP) portfolio plays a unique role and has moved into the spotlight. While risk parity is most often implemented at the asset class level to construct a global risk balanced portfolio, applications for individual securities within a particular asset class have also gained momentum. Demey, Maillard, and Roncalli (2010) examine the application of the risk parity approach to individual stocks in different regions and find that, compared with other alternative risk-based approaches, turnover requirements of the RP portfolio are far more feasible, with well balanced portfolio positions and risks. Therefore, ad hoc constraints on positions and risks are not required, unlike with other approaches. Risk parity also seems to have generated the most interest in recent literature (see for example, Thiagarajan and Schachter (2011), Inker (2011), Ruban and Melas (2011), Chaves, Hsu, Li, and Shakernia (2011), Qian (2011), Peters (2011), and Bhansali (2011), among others).

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To date, justification of risk parity remains conceptual and intuitive, rather than theoretical, and therefore, risk parity is largely considered a heuristic asset allocation approach with intuitive appeal (Thiagarajan and Schachter, 2011, p.80). Recently, Asness, Frazzini, and Pedersen (2012) explored the empirical performance of risk parity as driven by leverage aversion. The goal of this paper was to further advance our understanding and appreciation of the RP portfolio through conceptual and theoretical developments. We structured the paper as follows: Building on Lees (2011) point on the importance of a clearly defined investment objective, we put risk parity in the context of Mean-Variance Optimality as a natural starting point. We give conditions of efficiency and study the properties of the RP portfolio in a one factor world in which the covariances are modeled by a market factor and idiosyncratic risks. We compare the input sensitivity and, hence, the turnover characteristics of risk parity, and next, we analyze its potential utility function and interpret it in a Bayesian sense to shed some light on the investment rationale. Finally, we report path-independent simulation results, and conclude that risk parity may be a preferable method in a regime in which input parameters are very noisy and returns are fat-tailed.Risk Parity as a Special Case of Mean-Variance Optimization

In the absence of a clearly defined objective, we follow Lee (2011) in using Mean-Variance Optimality as our metric in an attempt to provide additional context in understanding the risk parity approach. Using simple portfolio mathematics, we derive the underlying conditional expectations of the investment opportunity set that are expected to ensure the optimality of the RP portfolio. In the appendix, we demonstrate that a RP portfolio is mean-variance efficient when: 1. Sharpe ratios of all assets are identical, and 2. Correlations among assets are the same.1 A number of studies have examined and discussed the historical performance of a risk parity approach to asset allocation among some asset classes, mostly relying on backtests in the last two decades. Examples of these studies include Allen (2010), Levell (2010), and Chaves, Hsu, Li, and Shakernia (2011). Below, we take a different approach. Having established the conditions for the optimality of risk parity, we may interpret the performance of risk parity relative to other alternatives in historical samples as an indication that the identical Sharpe ratios and same correlations among all assets might have been a better or worse proxy of the realized investment opportunity set than those alternatives during the sample periods of interests. To further illustrate the point, we proceed to examine the realized performance characteristics of some widely followed asset classes in the U.S., including large-capitalization stocks, small-capitalization stocks, long-term corporate bonds, and long-term government bonds. We use monthly returns from the Ibbotson dataset that goes back to 1926. Table 1 summarizes the historical returns, volatilities, and correlations.

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An informal proof of the same results can be found in Maillard, Thierry and Teiletche (2010).

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TA B L E 1 : L O N G -T E R M R I S K / R E T U R N P R O F I L E O F U N D E R LY I N G A SS E T SLONG-TERM CORPORATE BONDS 2.65 7.35 0.21 0.16 LONG-TERM GOVERNMENT BONDS 2.32 8.04 0.12 0.06 0.84

Annualized Mean (%) Annualized Standard Deviation (%) Correlations S&P 500 Index Small-Cap Long-Term Corporate Bonds Long-Term Government Bonds

S&P 500 Index 7.67 19.23

SMALL-CAP 11.79 29.14 0.83

Source: Ibbotson. Note: Estimates are based on monthly total return data from 19262010.

In our empirical exercise, we attempt to understand the performance of risk parity by gauging how closely its optimality conditions might have proxied the realized return and risk characteristics of assets. To start, Figure 1 plots rolling 10-year Sharpe ratios. We see that the Sharpe ratios of these assets were indeed close to each other at times, such as in the mid 70s and late 90s. However, before the 70s, they were quite different. While the condition of identical Sharpe ratios was clearly violated for shorter periods of time over this long sample time period, the median of 10-year Sharpe ratios of these assets as plotted in Figure 2 were remarkably similar. Similarly, Figure 3 and Figure 4, which plot the 10-year rolling correlations among these assets and their medians in the whole sample, respectively, suggest that the condition of same correlation was violated, especially between asset class pairs. Evidently, observed Sharpe ratios and correlations have not been the same all the time and therefore, theoretically, other more efficient portfolios had to exist. While seeking efficient portfolios ex-ante is generally the objective, it is still interesting to examine the extent to which such violations of the conditions of RP portfolio optimality may lead to the loss of portfolio efficiency. The following analysis focuses on the violation of same correlation conditions only. Similar analysis can be done on the violation of identical Sharpe ratios.F IG U R E 1 : ROLLIN G 10-Y EA R SH A RPE RA TIOS2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 Dec-35

Dec-43

Dec-51

Dec-59

Dec-67 LT Gov

Dec-75

Dec-83

Dec-91

Dec-99

Dec-07

S&P 500 Index Source: Ibbotson.

LT Corp

Small-Cap

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F IG U R E 2 : MED IA N OF ROLLIN G 10-Y EA R SH A RPE RA TIOS (1936 2011)0.500 0.450 0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050 0.000 S&P 500 Index Source: Ibbotson. Small-Cap Long-Term Corporate Long-Term Government 0.41 0.43 0.39 0.37

To quantify the potential loss of efficiency, we follow the cash equivalent comparison methodology as described in Chopra and Ziemba (1993). First, a mean variance efficient portfolio of the four assets is constructed based on the whole sample statistics as reported in Table 1 and a risk tolerance of 50 as suggested by these authors. The cash equivalent value of this portfolio is estimated to be 0.26% per month. Next, we construct two other efficient portfolios based on the same inputs of returns, volatilities, and risk tolerance, but one portfolio with the same correlation of all assets equal to the average correlation during the whole sample (0.37 to be specific), and one portfolio with correlations of all assets at zero. The cash equivalent measures of these portfolios were estimated to be 0.24% and 0.19% per month, respectively, corresponding to a 6% and 25% loss in cash equivalent terms versus the first efficient portfolio constructed with the true ex-post correlations. Figure 5 demonstrates how these losses change when the risk tolerance parameter varies. In the case of non-zero average correlations, the loss in cash equivalent due to the violation of the same correlation assumption is found to increase as the investor becomes more and more risk seeking. In the case of the zero correlation assumption, however, the loss in cash equivalent increases with risk aversion instead. With a zero correlation assumption, small-cap stocks become more attractive and their weight in the efficient portfolio is higher. As the investor becomes more risk averse, the higher weight in small-cap stocks causes more loss in utility as measured by the cash equivalent. Therefore, averaging correlations is close to efficiency when the investor is becoming more risk averse. However, the assumption of zero correlation among assets can result in unintended risk concentration in riskier assets and hence reduce efficiency. To summarize, identical Sharpe ratios and constant correlations when measured with shorter sample periods were certainly violated. However, judging from the relatively small loss of cash equivalents relative to the true efficient portfolio with perfect hindsight, the optimality conditions of the RP portfolio appears to be a reasonable starting point in achieving portfolio efficiency ex-ante.5


F IG U R E 3 : ROLLIN G 10-Y EA R CORRELA TION S1.20 1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 Dec-35 Dec-43 Dec-51 Dec-59 Dec-67 Dec-75 Dec-83 Dec-91 Dec-99 Dec-07 S&P 500-Small Small-CorpSource: Ibbotson.

S&P 500-Corp Small-Govt

S&P 500-Govt Corp-Govt

F IG U R E 4 : M ED IA N OF ROLLIN G 10-Y EA R CORRELA TION S0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 S&P 500-Small S&P 500-Corp S&P 500-Govt Source: Ibbotson. Small-Corp Small-Govt Corp-Govt 0.28 0.22 0.17 0.12 0.82 0.76

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F IG U R E 5 : P ERCEN TA G E CA SH EQU IV A LEN T LOSS A S A FU N CTION O F R IS K T OL E R A N CE FOR A CON STA N T CORRELA TION PORTFOLIO40% 35% 30% 25% 20% 15% 10% 5% 0% 25 30 35 40 45 50 55 60 65 70 75 Risk Tolerance () Average Correlation Zero Correlation

Source: Neuberger Berman Quantitative Investment Group. Note: The two lines shown correspond to cases when the constant correlation is equal to the average of all pairwise correlations or zero.

Risk Parity in a Single-Factor World

In an attempt to shed light on the properties and apparent outperformance of the Minimum Variance (MV) portfolio relative to the market capitalization-weighted portfolio, Scherer (2010) and Clarke, de Silva, and Thorley (2011) report that the MV portfolio implicitly loads up on low beta and low idiosyncratic risk assets, and therefore captures these pricing anomalies, as documented in Jensen, Black, Scholes (1972) and Ang, Hodrick, Xing, and Zhang (2009), that might have existed in the sample period. Our goal in this section is to take a similar path to further understand the portfolio compositions of a RP portfolio. In the Appendix, we demonstrate that a RP portfolio, similar to an MV portfolio, is by construction, biased towards low beta, low idiosyncratic assets and, therefore, is able to capture these anomalies as well. Here we use a simple example to demonstrate the beta and idiosyncratic risk properties of RP portfolios and use minimum variance for comparison (see Figure 6 for our results). Due to its additional dependence on expected return parameters, the mean variance method is skipped in these comparisons. In the case of idiosyncratic risks, we consider a universe of 50 assets, each with a beta of 0.25, but with the idiosyncratic risk of assets monotonically increasing from 1% to 30%. Without loss of generality, we also assume a 12% market risk. Using these statistics, we create a covariance matrix from the factor model to be used as an input to the MV and RP portfolios. Figure 6a exhibits the weight and risk allocations to these assets as a

Cash Equivalent Loss

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function of their idiosyncratic risks. Needless to say, both risk-driven models prefer low idiosyncratic risk assets when beta is kept constant. Another eye-catching pattern is the fact that minimum variance loads on a few low idiosyncratic assets while risk parity is less concentrated. (Also noted by Clarke, de Silva and Thorley (2011)). Second, in the case of varying beta, we again consider a universe of 50 assets, each with an idiosyncratic risk of 25%, but now with the betas of assets monotonically increasing from -2 to 5. Again, assuming a 12% market risk and using these inputs, we create a covariance matrix from the factor model used in optimizing MV and RP portfolios. From Figure 6b, we observe that both portfolios like low beta assets. Although the betas range in between extreme thresholds from -2 to 5, the difference in weights is not as dramatic as the case in varying idiosyncratic risks. One interesting fact is that as beta increases, MV is likely to short high beta assets while RP still stays long-only.F IG U R E 6 A : IM PA CT OF ID IOSY N CRA TIC RISK ON RISK -D RIV EN P OR T F OL IOSWeights of Risk Parity vs Minimum Variance Impact of Idiosyncratic Risk

50 40 Weight 30 20 Weight 10 0 0 10 20 AssetPCTR of Risk Parity vs Minimum Variance Impact of Idiosyncratic Risk

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Weights as a Function of Idiosyncratic Risk

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Source: Neuberger Berman Quantitative Investment Group. Note: PCTR: Percentage contribution to portfolio risk.

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F IG U R E 6 B : IM PA CT OF M A RK ET BETA ON RISK -D RIV EN PORTFO LI O SWeights of Risk Parity vs Minimum Variance Impact of Weights as a Function of

6 5Weight

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Sensitivities and Robustness of Portfolio Weights

Having explored the efficiency and factor properties of RP portfolios, we switch our attention to the topic of portfolio sensitivity to changes in input parameters, an important subject for a variety of reasons. To cite a few, first, the majority of applications of risk parity, especially a RP portfolio of global asset classes, typically require leverage in order to deliver reasonable required returns. In some cases, over-the-counter derivatives with lower liquidity are included. Therefore, it would be an undesirable property if small changes in risk estimates could potentially result in high turnover statistics. Second, we are supplying the optimizers only with estimates, with errors, of the true parameters. While our estimates are at best in the neighborhood of the unknown true parameters, it would be a positive if the allocation method remained consistent around these estimates. Mitigating this type of portfolio sensitivity has been studied extensively in the literature. Remedies ranging from simply imposing constraints to a complex approach employing second order cone programming, have been offered as potential solutions to the robust asset allocation problem. Each potential remedy, however, brings varying degrees of challenges including, for example, becoming ad hoc with the introduction of discretionary elements in the case of constraints, loss of transparency, and high cost of applicability (see Ledoit and Wolf (2003), Jagannathan and Ma (2003), Tutuncu and Koenig (2004), Garlappi et al. (2007)). Therefore, if we can show that risk parity is robust by nature, or at least more robust than alternative methods, we can then hope to deal less with these intricacies of quantitative portfolio management.

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Given the notoriety of mean-variance optimization with respect to its sensitivity to input parameters (as discussed in Chopra and Ziemba (1993), among many others), we again exclude mean variance and our analyses will focus on risk parity and its similarly spirited counterpart, minimum variance. On this quest, we focus on two measures of interest. First, we investigate the concentration in dollar weights and risk contributions. Second, we quantify the extent that optimal solutions change when a particular parameter is perturbed by a small amount and what the resulting turnover will be. To assess these impacts, we use sum of squared weights S2 := xT x, and the change in sum of square weights with respect toT a change in parameter a, S2(a) := x x

respectively, where x is a vector of weights. To illustrate, when the weights sum to unity, S2 is minimized when all the components are the same, and it is maximized when all the weights are concentrated in one asset. Similarly, if there is no change in the portfolio weights after a perturbation, S2 will be zero. Moreover, any deviations from zero in both signs will proxy the level of sensitivity and, therefore, the level of turnover. Although the analytical computation of this derivative is available in the case of the unconstrained MV portfolio, the same is not true for the RP portfolio given its endogenously determined optimal weights. Therefore, we start by studying a simplified example to demonstrate the impact of sensitivity by considering three assets with the following volatility and correlation structure.TA B L E 2 : A S S U M E D R I S K PA R A M E T E R SA 10% 1 B 12% 0.1 1 C 14% 0.2 0.3 1

a

as measures of concentration and turnover

Volatilities Correlations A B C Souce: Neuberger Berman Quantitative Investment Group.

To explore the sensitivity of these portfolio weights with respect to changes in volatility, we vary the volatility of Asset A from 10% to 20% in 1% increments and calculate S2 and S2 (A). In Figure 7, we compare snapshots of weights and risk contributions when the volatility of Asset A is 10% and 20%, respectively. The MV portfolio at first loads heavily on Asset A at about 90%, then cuts the weight significantly to a little less than 50%, when volatility is increased from 10% to 20%. Its risk contribution profile is equally concentrated if not more alarmingly so, with Asset A accounting for most of the portfolios risk when its volatility is low at 10% (indeed, the percentage contributions to risk of an unconstrained MV portfolio can be shown to equal its weights). While the RP portfolio also adjusts its weight when the volatility of Asset A is changed, its sensitivity is clearly much lower than that of the MV portfolio.

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F IG U R E 7 : S N A PSH OT OF PORTFOLIO WEIG H TS WH EN AND WHEN

A = 20% (RIG H T)% 100 90 80 70 60 50 40 30 20 10 0

A = 10% ( LEFT)

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Weights of Risk Parity vs Minimum Variance Case: A = 20.0

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To demonstrate the impact of a volatility change in continuum, Figure 8a shows that the MV portfolio is always more concentrated than the RP portfolio, regardless of the volatility of Asset A, as it increases from 10% to 20%. Next, Figure 8b illustrates the extent to which the MV portfolio can be more sensitive with respect to changes in the volatility of Asset A when compared to the RP portfolio. As an example, when A is increased to 11% from 10%, the resulting change in the sum of squared weights in the MV portfolio is around -0.07, more than double the value of -0.03 of the RP portfolio. In other words, both portfolios cut the weight of Asset A as its volatility increases, but the corresponding response from the RP portfolio is much less than the MV portfolio, which had very dramatic changes in response. Moreover, the absolute delta is always greater in the case of minimum variance, meaning that regardless of the volatility of Asset A, any change in volatility will result in greater turnover than risk parity.

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F IG U R E 8 A : IM PA CT TO PORTFOLIO CON CEN TRA TION WH EN V O LATI LI TY OF A S S E T A V A RIES FROM 10% TO 20%Sum of Squared Weights vs. Volatility of Asset A 1.0

0.8 Sum of Squared Weights

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0.0 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% Volatility Risk Parity Minimum Variance

Source: Neuberger Berman Quantitative Investment Group.

F IG U R E 8 B : IM PA CT TO PORTFOLIO TU RN OV ER WH EN V OLA TILI TY O F A S S E T A VA RIES FROM 10% TO 20%Delta in Sum of Squared Weights vs. Volatility of Asset A 0.00

Delta in Sum of Squared Weights

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-0.06

-0.08 11% 12% 13% 14% 15% Volatility Risk Parity Source: Neuberger Berman Quantitative Investment Group. Minimum Variance 16% 17% 18% 19% 20%

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In a similar fashion, we analyze the impact of changes in correlations to portfolio robustness. In doing so, we keep the volatilities constant and only vary the correlation between Assets A and B, denoted as AB. The parameters of the base case scenario are summarized in Table 3.TA B L E 3 : A S S U M E D PA R A M E T E R S W H E N A N A LY Z I N G T H E I M PA C T O F C O R R E L AT I O N C H A N G E S O N P O R T F O L I O W E I G H T SA 10% 1 B 10% 0.1 1 C 10% 0.1 0.1 1

Volatilities Correlations A B C Souce: Neuberger Berman Quantitative Investment Group.

F IG U R E 9 A : IM PA CT TO PORTFOLIO CON CEN TRA TION WH EN THE C OR R E L A T IO N BETWEEN A SSET A A N D A SSET B V A RIES FROM - 0. 5 TO 1Sum of Squared Weights vs. AB 0.40

0.38 Sum of Squared Weights

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0.30 -0.5 0.0 0.5 1.0

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F IG U R E 9 B : IM PA CT TO PORTFOLIO TU RN OV ER WH EN TH E CORRELATI O N B E T W E E N A SSET A A N D A SSET B V A RIES FROM -0.5 TO 1Delta in Sum of Squared Weights vs. AB 0.04

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-0.08 -0.5 0.0 Correlation Risk Parity Source: Neuberger Berman Quantitative Investment Group. Minimum Variance 0.5 1.0

Figures 9a and 9b again show that, except at a tangential point, the MV portfolio always takes highly concentrated weights relative to the RP portfolio (upper chart). Furthermore, as correlation varies, the MV portfolios turnover is always greater than the RP portfolio as changes in squared weights deviates from zero significantly more than the risk paritys deviations.Potential Utility that Leads to Risk Parity

Earlier, we studied the RP portfolio in a one-factor world and also as a specific efficient portfolio within the mean-variance paradigm. However, none of these sections provided insight into what general utility function risk parity is built to maximize. As Lee (2011) points out, a challenge to many of the risk-based asset allocation approaches, including risk parity, is the lack of a clearly defined investment objective. As a result, it is not immediately clear what investment problems these portfolios are built to solve. In this section, we further expand the analysis to focus on a more generalized utility maximization problem in which risk parity is shown to be a special case. With this generalized utility function, we allow different portfolio constituents to contribute different amounts of risks to the portfolio volatility. Risk parity, then, is a special case in which the risk contributions from all constituents are identical.

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We impose a long-only constraint to make our analysis more relevant to a typical investor. Next, we also assume that the investor has an overall risk budget as measured by a limit on the portfolio volatility; exceeding this limit would be beyond the investors appetite and tolerance for volatility. This leads us to analyze the generalized risk budgeting problem specified as follows: Maximize ln(x) Subject to (x) T x0N

(1)

where > 1 is an N x 1 vector, x is the vector of weights, and (x) is the volatility of the portfolio x, which is constrained to be less than a target T. In the Appendix, we demonstrate that the vector can be interpreted as the vector of risk budgets, defined as contributions to the portfolio volatility of the assets. The RP portfolio, in the specific case when i for i= 1, 2, , N, is set to be identical, is shown to be a solution to the utility maximization problem above. In addition, existence and uniqueness of the RP portfolio are also established.Likelihood Maximization Interpretation in a Bayesian Setting

To shed some light on the economic meaning of this utility function, we form an objective prior on the portfolio weights (see objective-based priors Avramov, Zhou (2006)). To get an idea of the form of this prior, we first express the objective function in the following equivalent form. L( x ; ) = x i ii =1 N

(2)

Maximization objective expressed this way resembles the likelihood function of a multivariate Dirichlet distribution (Frigyik 2010) when asset weights sum to unity. Indeed, given the parameters , the likelihood of observing portfolio weights x (under an uninformative prior) is given by (2). In the case with no total risk constraint, L(x; ) is maximized at x =

weights, and as such in the absence of a risk target, has the interpretation of portfolio weights. However, when the total risk constraint is imposed, the maximum likelihood is achieved at a generalized risk parity solution, and thus has a risk budget interpretation. As an example, if an investor starts with a prior belief that a portfolio should consist of 60% stocks and 40% bonds and is not constrained by the total risk taken, then the investor will invest according to the prior beliefs. However, if the investor is risk averse and has a limited total risk budget, the investor will most likely hold a portfolio with a risk allocation identical to the prior belief on portfolio weights. In other words, the generalized risk parity approach attempts to find a set of portfolio weights that is closest to the prior belief without exceeding the total risk budget constraint. Needless to say, when the elements of are the same, the resulting RP portfolio is the portfolio that is closest to an equal dollar weighted allocation, but with risk being constrained. The above analysis helps justify the rationale for generalized risk parity investments. The problem maximizes allocation to those assets that the investor likes and at the same time, strives to control the risk of the resulting portfolio. Although expected returns do not appear directly as in the case of mean variance, the relative preference relations defined by s provide us with the necessary link to justify the relative allocation of funds.15

to yield the maximum likelihood estimates of portfolio 1T


Those assets expected to yield higher returns can be assigned relatively higher s and the resulting dollar weights will be tilted towards these assets accordingly in a risk diversified manner.When Should Risk Parity Be a Preferred Method?

Having identified some important properties of RP portfolios, we now address the intricate issue of what may make risk parity a better approach than the readily available, widely understood and adopted portfolio construction methodologies such as mean variance and minimum variance. In essence, our goal is to examine the scenarios or regimes where risk parity is likely to be more promising with respect to some efficiency criteria. To achieve this task, we devise simulation-based controlled experiments where the aim is to create a statistically static environment, and then run horse races between asset allocation methodologies to see which one dominates in terms of portfolio efficiency as measured by the reward-to-risk ratio. The stochastic investment opportunity set is simulated by first setting expected returns, the covariance matrix of assets, and the degree of freedom in a multivariate t-distribution that introduces different degrees of tail thickness. We pretend that investors are told that the true underlying returns generating function is static, but unknown. As a result, the investors have to use different lengths of rolling windows of the observed returns data in an attempt to estimate the true underlying return generating function. Thus, in this experiment, the size of the rolling estimation window becomes a proxy for the level of noise in estimation. Investors then construct their portfolios according to their estimates as well as their preferred portfolio construction approaches, and out-ofsample performances of these portfolios are recorded and compared. For each degree of freedom parameter and window length choice, we repeat the above steps numerous times and record the corresponding performance of these portfolios for subsequent analysis of their portfolio efficiency.

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F IG U R E 1 0 : RISK PA RITY M A Y BETTER N A V IG A TE A WORLD WIT H FAT T A IL S A N D N OISE800

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More Fat Tails

Less Fat Tails/More Normal


For the sake of illustration, we take the point (7,40) on Figure 10 as an example. We first create 20 replicas of history, each consisting of 1,200 months of simulated returns with a tail index corresponding to a degree of freedom equal to 7, using the average returns and covariance matrix from Table 1 on four asset classes. Next, at any point in time, we use the previous 40 months of observed returns data to estimate the expected returns and covariance matrix assuming returns follow a normal distribution. Given these empirically estimated parameters, mean-variance, MV, and RP portfolios are then constructed and held for one month, with performance in the following month recorded. We repeat this estimation, optimization, rebalancing, and performance recording cycle until we exhaust the 1,200 months of simulated returns. We then repeat the above steps for each of the 20 replicas of history, and count in how many of these 20 runs each method wins as the most efficient portfolio as measured by the highest reward-to-risk ratio. In the case of (7,40), among these 20 runs, the risk parity methodology dominates, being the most efficient in 17 out of 20 times, than the other portfolios. As such, we conclude that the RP portfolio is the most dominant portfolio given the tail index as 7 and estimation window length of 40 months under the aforementioned returns and covariance matrix environment. Therefore, the (7,40) point is contained in the Risk Parity More Efficient part of the plane. We highlight three important observations from Figure 10. First, when the tail parameter is fixed, risk parity is most efficient when estimation noise, as represented by shorter window lengths, is high. Here, we note that in our experiment, all return-generating parameters are fixed. On the contrary, in the real world, the true return-generating function is time-varying, changing as we move in and out of different regimes, making parameters estimation even more noisy and imprecise given the same window length.

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Moreover, real world investment problems are much more complex and involve many more assets. In other words, the error maximization problem in the real world is many times more significant than what we have built into our experiment and, as such, the true boundary in Figure 10 is pushed much further north in the real world, making risk parity even more dominating in a noisy environment. Furthermore, as the asset management industry continues to evolve with new ideas and strategies as investment solutions, insufficient track records become more a norm than an exception, presenting challenges to strategic asset allocation analysis that normally requires sufficient historical data. In light of our findings above, and together with Mertons (1980) findings that precision of estimating risks is higher than the precision of estimating expected returns, risk parity appears to be a reasonable solution in allocating to new strategies. Second, given a certain degree of noise associated with a fixed window length, risk parity is the most efficient portfolio as returns become more fat tailed, deviating from the normal distribution. By now, few would disagree that the real investment world does not follow a normal distribution. For instance, the Ibbotson dataset used in the simulation from 1926 through 2010, the fitted multivariate t-distribution, indicates that the tail parameter is estimated at 3.11, a level far from the value 30 of normality. At such a value of the tail parameter, Figure 10 suggests that one requires many decades of data for the mean-variance approach to be more efficient than the RP portfolio, and that is based on an unrealistic environment of static returns generation. Lastly, when returns are well-proxied by a normal distribution with negligible fat tailsthat is, when the tail parameter is greater than 30the decision boundary plateaus, making tail thickness less of a concern than estimation noise.Conclusion

Financial crises remind investors of the necessity of risk management, and financial institutions of the need to innovate new products that are better suited for increasingly uncertain and risk averse environments. The aftermath of the 2008 financial meltdown lead to the (re)emergence of risk parity as a panacea for what was missing before the crises in asset allocation practices before the crises. Thenceforth, risk parity has enjoyed constant fund flows from institutional investors and popularity from index providers. In this paper, our aim was to help investors enhance their understanding of risk parity in a number of ways to assess whether this new remedy is really a panacea or a temporal placebo. First, for those who still look at the world in a mean variance setting, we provided conditions that make risk parity efficient and argued that although these conditions are not expected to be met over short time horizons, they may be good proxies for longer-term allocations. Second, to relate historical risk parity outperformance to known price anomalies, we studied risk parity in a one-factor setting and showed that risk parity favors low beta and low idiosyncratic risk assets. As such, we argued that the performance of risk parity may very well be related to the underperformance of high beta investments in the last two decades. Third, we analyzed risk parity sensitivity to input parameters and found it more stable than other risk-driven allocation methods. We argued that this insensitivity is important for risk parity managers as turnover resulting from covariance instabilities may significantly reduce any value added. In particular, we showed that the risk parity method is influenced more by changes in volatilities than correlations, but the impact of these changes is much lower when compared to minimum variance.18


Fourth, we studied the risk parity problem and related its objective function to a constrained maximum likelihood estimation. In this setting, we argued that a generalized risk parity investor has prior beliefs about what portfolio weights should be, and is trying to be statistically as close as possible to those weights without exceeding a risk budget. In the special case, we showed that if the investor does not have any beliefs, or treats each investment equally, then pure risk parity is the portfolio held under a binding risk constraint. Finally, acknowledging the path, asset domain, and macroeconomic dependencies of the risk parity backtests, we conducted extensive controlled simulation experiments in order to produce a fair judgment of risk-based asset allocation methods. In these runs, we exposed mean variance, minimum variance and risk parity to environments that we statistically set, and found that risk parity statistically dominates other methods when there is high uncertainty around input parameters (i.e., high noise) and/or there are fat tails in the asset return series. In summary, although risk parity may not be truly efficient, we have a number of reasons to believe that it is efficient enough, ex-ante, when compared to other allocation methods. As such, even if it is not a cure-all solution to long-term strategic allocation, it may be a good starting point.APPENDIX Notation

N Number of assets x : N x 1 vector of portfolio weights : N x 1 vector of expected returns : N x 1 positive vector of risk multiples associated with each asset 0 : N x 1 vector of zeros (A)i : ith row of a matrix A : N x N covariance matrix : N x N correlation matrix V : N x N matrix of zeros except nonzero asset volatilities on the diagonal X : N x N matrix of zeros except asset weights x on the diagonal : N x 1 vector of volatilities T : Volatility target

( x ) := x T x the portfolio volatility : N x 1 vector of Lagrange multipliers for the constraint x 0 : Lagrange multiplier for the constraint (x) T ( x )i := x i ( x ) : Contribution to risk of asset i =1,2,..., N x i

> 0 : Coefficient of risk aversion

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Proof of Risk Parity portfolios preference of low beta, low idiosyncratic risk assets

To show this formally, let us consider a one-factor world in which asset excess returns follow, i = 1, 2,..., N i = ai + bi f + iE[i j ] = E[i ] E[ j ] E[i f ]= E[i ] E[ f ] Var[ f ] = 2 f

=0 =0

(3)

Var[i ] = 2i

where f is the excess return of the single factor, ai and bi are constants and i is a mean zero error term assumed to be uncorrelated with the factor and with other error terms. In this case, we can represent variance and covariances of each asset i,j =1,2,..., N as i2 = ij = Then, the risk parity condition, i , j {1, 2,..., N }, bi2 f2 + i2 bi bj f2 (4)

( Xx )i (x)

=

( Xx ) j (x)

,

or x ' x NN

(5)

( Xx )i = ( Xx ) j =leads toN

x i2 2i + x i 2 bi bk x k = x 2 2j + x j 2 b j bk x k f j fk =1 k =1

2 N f bk x k ( x i bi x j b j ) = ( x j j x i i )( x j j + x i i ) k =1 x j j x i i x i bi x j b j = ( x j j + x i i ) 2 N f bk x k k =1

(6)

Assuming that the beta of the portfolio x is positive, which is typically the case for most long-only portfolios, (6) has the following consequences for those assets which have nonnegative betas: Ifidiosyncraticrisksarethesame,i.e., i = j , then bi > bj 0 xi < xj, and Ifbetasarethesame,i.e.,bi =bj 0, then i > j

xi < x j

Briefly, the first bullet point follows from the fact that if by assuming by contradiction xi xj , then xi bi xjbi > xjbj implying that xi bi xjbi > 0. Therefore, the left-hand side of the last row in (1.4) becomes positive implying that in the right-hand side, with the common positive idiosyncratic risk factored out, is also positive, i.e., xj xi > 0. Hence, we arrive at a contradiction. The second bullet can be proven similarly.

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Therefore, without loss of generality, we can conclude that a typical long-only RP portfolio tends to prefer low beta assets with low idiosyncratic risks and as such it is also capturing these pricing anomalies.Proof of mean-variance efficiency of the RP portfolio if Sharpe ratios and correlations among assets are identical

We first state the proven result that the mean-variance efficient portfolio can be represented by: 1 (7) x = 1

where > 0 represents the coefficient of relative risk aversion. The contribution to risk of each asset can then be written as

( x ) =

Xx (x) 1 X 1 = (x) 1 X = (x)

(8)

Therefore, the ith row of (x) is ( x )i

i 1 1 ( )i ( x ) = 2 i ( 1 )i (x)=

(9)

Here, the inverse of the covariance matrix can be expressed in terms of volatilities and correlations as: 1 = [VV]1 = V1 1 V1 (10)

If we denote by bij the entries of the inverse of the correlation matrix 1, then we can b further write the elements of 1 as ij 1 = ij i j r Putting (10) in (9), and letting s i := i denote the reward to risk ratios, we get i ( x )i =

i 1b1i 2b2i i bii N bNi + + ... + 2 + ... + (x) 1 i 2 i i N i 2

i2 2 b b b = 2 i 1 i 1i + 2 i 2i + ... + bii + ... + N i Ni ( x ) 1 i 2 i N i = s1 s i2 sN s2 2 s b1i + s b2i + ... + bii + ... + s bNi (x) i i i

(11)

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Then, from (11) it follows that

s s1 s 2 b + 2 b + ... + bii + ... + N bNi s i s i 1i s i 2i ( x )i si = s s1 s2 ( x ) j s j b1 j + b2 j + ... + b jj + ... + N bNj sj sj sj

(12)

In the special case, when all assets have the same Sharpe ratios such that si = s, i = 1, 2 ,..., N , and the off-diagonal terms of the correlation matrix, , equal to the same constant, we get2 ( x )i =1 (13) ( x ) j which shows each asset contributes equal risk to the portfolio. In other words, a RP portfolio is mean-variance efficient if Sharpe ratios and correlations among assets are identical.Risk Parity as a Solution to Utility Maximization

We claim that RP portfolios, in the general case, are the solution of the following problem: Maximize T ln(x) Subject to (x) T x0 (14)

Here, Karush-Kuhn-Tucker (KKT) conditions provide us with necessary conditions for optimality. Because the constraints and the objective function are convex in this case, these necessary conditions are also sufficient. Let L(x, , ) denote the Lagrangian function: L(x, , )= T ln(x) + ((x) T) T x) Then, we have the following KKT conditions: (15)

i ( x ) + i = 0 , i = 1, 2 ,..., N x i xi(16)

, 0 ( ( x ) T ) = 0 i x i = 0 , i = 1, 2 ,..., N

Because i = 1, 2 ,..., N , xi Dom(ln), and i xi = 0, we get i = 0. Then, the first row in (1.14) becomes ( x ) (17) i + = 0 , i = 1, 2 ,..., N xi x i Furthermore, from the last row in (16) and from (17), we see that > 0, because otherwise xi = , i = 1, 2 ,..., N . Therefore, at optimality, we have (x) = T.

2

Here we use the fact that if is a constant correlation matrix, then both the diagonal and the off-diagonal elements of constant (not necessarily equal to each other).

1

are

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Rearranging the terms in (17), the contribution to risk of asset I can be given as ( x ) i (18) ( x )i := x i , i = 1, 2 ,..., N = x i Therefore,

( x )i i = , i , j = 1, 2 ,..., N ( x ) j j

(19)

If i = j, i , j = 1, 2 ,..., N , then (19) implies ( x )i = ( x ) j , i , j = 1, 2 ,..., N which is the original RP portfolio.Existence of RP Portfolios

(20)

In order to make sure that a RP portfolio can always be formed, we need to show that the feasible set for the problem in (14) is nonempty. However, this is not a difficult task because for any given volatility target, T > 0, one can find small enough nonnegative weights to satisfy (x) T, e.g., a portfolio consisting of only the first asset scaled properly to have less than T risk is feasible. Given that a local solution of the problem (14) necessarily satisfies risk parity conditions, and such a solution can always be found, we conclude that RP portfolios can always be formed.Uniqueness of RP Portfolios

Given that any RP portfolio with nonnegative weights and less than or equal to T volatility target solves problem (14), the set of all RP portfolios and the set of all solutions of problem (14) coincide. Therefore, in order to show that the RP portfolio is unique, we need to show that the solution of problem (14) is unique. At first look, problem (14) is a maximization of a concave function subject to non-convex constraints, i.e., (x) T. Therefore, we are not guaranteed a unique solution. However, the following result helps us transform it to a convex problem. Proposition 1 A local solution of problem (14) is also a solution of Maximize T ln(x) Subject to (x)2 (T)2 x0 Proof Let x be a local solution of problem (14). Then, by KKT we have the conditions in (16) hold true, ( x ) i = 0 , i = 1, 2 ,..., N i + xi x i (21)

, 0

( ( x ) T ) = 0

i x i = 0 , i = 1, 2 ,..., N

(22)

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Remember, all risk budget is strictly used at optimality (i.e., T = (x)), let us define := 2 T = 2 ( x ) 0 . Then we have,

i ( x )2 + i = 0 , i = 1, 2 ,..., N xi x i(23)

, 0

i x i = 0 , i = 1, 2 ,..., N

( ( x )2 ( T )2 ) = 0

However, these are the KKT conditions for problem (21) at point x. Therefore, x is a solution of the problem (1.19). Given that the maximization problem (21) is convex (with convex constraints and strictly concave objective function); its solution is guaranteed to be unique. Because each solution of (14) is a solution of (21), and the solution of (21) is unique, the solution of problem (14) has to be unique.

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M0020 Demystifying Risk Parity

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