Diversifying Risk Parity

Diversifying Risk Parity Harald Lohre Deka Investment GmbH Heiko Opfer Deka Investment GmbH Gábor Ország Deka Investment GmbH July 22, 23 We are grateful to Andrew Adams, Jean-Robert Avettand-Fenoel, Dan dibartolomeo, Eric Girardin, Antti Ilmanen, Jason Laws, Attilio Meucci, Edward Qian, Yazid Sharaiha, Carsten Zimmer, two anonymous referees, and seminar participants at the 22 EFM Symposium on Asset Management in Hamburg, the 22 FFM Conference in Marseille, the 22 FactSet Investment Process Symposium in Monaco, Northfield s 25th Annual Research Conference in San Diego, and the 22 European Quantitative Forum of State Street in London. Note that this paper expresses the authors views that do not have to coincide with those of Deka Investment GmbH. Correspondence Information (Contact Author): Deka Investment GmbH, Quantitative Products, Mainzer Landstr. 6, 6325 Frankfurt/Main, Germany; harald.lohre@deka.de Correspondence Information: Deka Investment GmbH, Mainzer Landstr. 6, Quantitative Products, 6325 Frankfurt/Main, Germany; heiko.opfer@deka.de Correspondence Information: Deka Investment GmbH, Mainzer Landstr. 6, Quantitative Products, 6325 Frankfurt/Main, Germany; gabor.orszag@deka.de

Diversifying Risk Parity ABSTRACT Striving for maximum diversification we follow Meucci (29) in measuring and managing amulti-assetclassportfolio. Underthisparadigmthemaximumdiversificationportfoliois equivalent to a risk parity strategy with respect to the uncorrelated risk sources embedded in the underlying portfolio assets. Our paper characterizes the mechanics and properties of this diversified risk parity strategy. Moreover, we explore the risk and diversification characteristics of traditional risk-based asset allocation techniques like /N, minimum-variance, or risk parity and demonstrate the diversified risk parity strategy to be quite meaningful when benchmarked against these alternatives. Keywords: Risk-basedAssetAllocation,RiskParity,Diversification,Entropy JEL Classification: G;D8

Diversification pays. This insight is at the heart of most portfolio construction paradigms like the seminal one of Markowitz (952). Under his mean-variance optimization diversifying portfolio weights is the key to obtaining efficient portfolios with an optimal risk and return trade-off. Unfortunately, mean-variance optimization is typically confounded by estimation risk, especially the one embedded in estimates of expected returns, see Chopra and Ziemba (993). One way to circumvent this problem is to simply refrain from estimating returns and to resort to riskbased allocation techniques. Within the framework of Markowitz (952) this approach leads to the well-known minimum-variance portfolio. However, minimum-variance portfolios are designed to load on low-volatility assets which renders them rather concentrated in a few assets. Thus, minimum-variance portfolios are hardly diversified in terms of a homogenous weights distribution. In striving for well-diversified portfolios Meucci (29) builds on principal component analysis of the portfolio assets to extract the main drivers of the assets variability. These principal components can be interpreted as principal portfolios representing the uncorrelated risk sources inherent in the portfolio assets. For a portfolio to be well-diversified its overall risk should therefore be evenly distributed across these principal portfolios. Condensing this risk decomposition into a single diversification metric, Meucci (29) opts for the exponential of this risk decomposition s entropy because of its intuitive interpretation as the number of uncorrelated bets. The contribution of this paper is to apply the framework of Meucci (29) in an empirical multi-asset allocation study. Under this paradigm the maximum diversification portfolio emerges from a risk parity strategy that is budgeting risk with respect to the extracted principal portfolios rather than the underlying portfolio assets. Therefore, we think of this approach as a diversified risk parity strategy which turns out to be a reasonable alternative when it comes to risk-based asset allocation. Moreover, the framework allows for a litmus test of competing techniques like /N,minimum-variance,orriskparity.Whileminimum-varianceisfairlywell-knownforpicking up rather concentrated risks, we find the traditional risk parity strategy to be more balanced. However, benchmarking risk parity against diversified risk parity one observes a degeneration in its diversification characteristics over time, rendering the traditional risk parity strategy a rather concentrated bet in the current environment.

The paper is organized as follows. Section I reviews the approach of Meucci (29) for managing and measuring diversification. Section II presents the data and further rationalizes the concept of principal portfolios. Section III is devoted to contrasting the diversified risk parity strategy to alternative risk-based asset allocation strategies. Section IV concludes. I. Managing Diversification According to standard portfolio theory diversification is geared at eliminating unsystematic risk. In addition, investors and portfolio managers common notion of diversification is the desire to avoid exposure to single shocks or risk factors. In either case, diversification especially pays when combining low-correlated assets. Taking this idea to extremes, Meucci (29) constructs uncorrelated risk sources by applying a principal component analysis (PCA) to the variancecovariance matrix of the portfolio assets. In particular, he considers a portfolio consisting of N assets with return vector R. Given weights w the resulting portfolio return is R w = w R. According to the spectral decomposition theorem the covariance matrix Σ can be expressed as a product Σ = EΛE () where Λ =diag(λ,...,λ N )isadiagonalmatrixconsistingofσ s eigenvalues that are assembled in descending order, λ... λ N.ThecolumnsofmatrixErepresent the eigenvectors of Σ. These eigenvectors define a set of N uncorrelated principal portfolios with variance λ i for i =,...,N and returns R = E R.Asaconsequence,agivenportfoliocanbeeitherexpressedintermsofits weights w in the original assets or in terms of its weights w = E w in the principal portfolios. Since the principal portfolios are uncorrelated by design the total portfolio variance emerges from simply computing a weighted average over the principal portfolios variances λ i using weights w i 2: V ar(r w )= N i= w 2 i λ i (2) Note that Partovi and Caputo (24) coined the term principal portfolios in their recasting of the efficient frontier in terms of these principal portfolios. 2

Normalizing the principal portfolios contributions by the portfolio variance then yields the diversification distribution: p i = w2 i λ i, i =,...,N (3) V ar(r w ) Note that the diversification distribution is always positive and that the p i ssumtoone. Building on this concept Meucci (29) conceives a portfolio to be well-diversified when the p i are approximately equal and the diversification distribution is close to uniform. This definition of a well-diversified portfolio coincides with allocating equal risk budgets to the principal portfolios. Therefore, we dub this approach diversified risk parity. Conversely, portfolios loading on a specific principal portfolio display a peaked diversification distribution. It is thus straightforward to apply a dispersion metric to the diversification distribution to obtain a single diversification metric. Meucci (29) chooses the exponential of its entropy 2 : ( ) N N Ent =exp p i ln p i i= (4) The reason for choosing N Ent relates to its intuitive meaning as the number of uncorrelated bets. To rationalize this interpretation consider two extreme cases. For a completely concentrated portfolio we have p i =foronei and p j =fori j resulting in an entropy of which implies N Ent =. Conversely,N Ent = N holds for a portfolio that is completely homogenous in terms of uncorrelated risk sources. In this case, p i = p j =/N holds for all i, j implying an entropy equal to ln(n). Taking the above approach to the extreme, we can especially obtain the maximum diversification portfolio or the diversified risk parity weights w DRP by solving w DRP =argmax w C N Ent(w) (5) where the weights w may possibly be restricted according to a set of constraints C. Note that optimization (5) does not allow for a unique solution in absence of further constraints. Intuitively, any inverse volatility portfolio along the principal portfolios is a solution 2 The entropy has been used before in portfolio construction, see e.g. Woerheide and Persson (993) or more recently Bera and Park (28). However, these studies consider the entropy of portfolio weights thus disregarding the dependence structure of portfolio assets. 3

maximizing objective function (5). Because all of the principal portfolios are uncorrelated buying and selling the same amount of a given principal portfolio gives rise to the same risk exposure. Hence, there are 2 N optimal inverse volatility portfolios that reflect all possible variations of the long and short variants of the underlying principal portfolios. The issue of multiple solutions applies to traditional risk parity strategies as well. In that regard, Maillard, Roncalli, and Teiletche (2) demonstrate that imposing positive asset weights renders the traditional risk parity strategy unique. However, imposing positive asset weights is not sufficient for ensuring a unique DRP strategy. Investigating general risk budgeting strategies Bruder and Roncalli (22) and Roncalli and Weisang (22) show that unique risk budgeting strategies obtain when imposing positivity constrains with respect to the underlying risk factors. Therefore, imposing sign constraints with respect to the principal portfolios instead of the underlying assets is key for obtaining a unique DRP strategy. These optimal portfolio weights can be computed analytically given the eigenvector decomposition of the covariance matrix Σ. However, these weights might not be feasible for a given set of investment constraints. For instance, we will enforce positive asset weights and full investment constraint later on. In this case, we numerically maximize objective function (5) using a sequential quadratic (SQP) algorithm. 3 For anchoring the numerical solution we feed the optimizer with the optimal analytic solution as starting value. II. Rationalizing Principal Portfolios A. Data and Descriptive Statistics In building risk-based asset allocation strategies we focus on five broad asset classes as represented by the following indices. We use the JPM Global Bond Index for government bonds, the MSCI World Total Return Index for developed equities, the MSCI Emerging Markets Total Return Index for emerging equities, the DJ UBS Commodity Index for commodities, and the Barclays U.S. Aggregates Index. All indices are measured in monthly local currency re- 3 Especially, we build on a variant of Meucci (29) s implementation which is available on his webpage, see http://www.symmys.com/node/99. 4

turns and we report total return figures from the perspective of an U.S. investor by employing the 3-month U.S. Treasury Rate. Table I conveys the descriptive statistics of the above asset classes. Over the whole sample period from December 987 to September 2 we observe an annualized bond return of 6.9% at avolatilityof3.8%whichhappenstobethelowestfigureacrossassetclasses. Duringthisperiod developed equities have fared slightly worse in terms of return (5.9%), however, their volatility is considerably higher (4.5%). For emerging equities, return and volatility figures are higher when compared to developed equities. Conversely, commodities are quite similar to developed equities in terms of volatility and return. Most interestingly, credit exhibits the same return as government bonds. This observation is unexpected given that credit is significantly more volatile than government bonds. However, it is important to note that the bulk of credit volatility is related to the credit crunch in 28 and the subsequent financial crisis. In terms of risk-adjusted returns the high-risk asset classes rather disappoint given the Sharpe Ratios around.2. By this metric, credit ranks second with a figure of.69 but is still underperforming bonds that exhibit a quite impressive Sharpe Ratio of.96. Further inspecting the assets dependence structure in Table I, we observe bonds to be hardly correlated to equities. Its correlation to commodities is slightly negative while the one to credit amounts to.53. All of the remaining correlation coefficients are positive and range from.9 (credit vs. commodities) to.74 (developed vs. emerging equities). Unsurprisingly, credit is more correlated to both equity indices. B. Extracting and Interpreting Principal Portfolios To foster intuition about the uncorrelated risk sources inherent in our multi-asset time series, we investigate the PCA over the whole sample period from December 987 to September 2. The economic nature of the principal portfolios is best assessed in terms of the eigenvectors that represent the principal portfolios weights with respect to the original asset classes. By construction these weights are standardized to lie within the [-,]-interval. Given that the correlation of the original assets is generally relatively low, the interpretation of the principal portfolios as collected in Panel A of Table II is straightforward. The first principal portfolio (PP) is purely driven by emerging and developed equities with emerging equity having a fairly high weight of 5

.87. Therefore, PP represents genuine equity risk which is accounting for 69.8% of the overall variance. Conversely, principal portfolio 2 (PP2) reflects the diversification potential of commodities relative to equities as indicated by a commodities weight of.96 and has a much smaller variance contributing 9.9% to the overall variance. Principal portfolio 3 (PP3) represents the difference between emerging and developed equities, thus capturing the emerging market spread. Principal portfolio 4 (PP4) mostly loads on credit and government bonds and we interpret it as an interest rate risk factor. Note that PP3 and PP4 explain most of the remaining variance, leaving aminusculefractionof.7%forprincipalportfolio5(pp5). JudgingbytheweightsofPP5we conceive it to be a mimicking factor of the credit spread. In addition, Panel B of Table II gives the principal portfolio weights pertaining to a PCA over the last 6 months of the sample period, that is from October 26 to September 2. Despite covering the most turbulent of times these weights prove to be fairly similar to those obtaining for the whole sample period. However, the variance of PP is elevated by a factor of 2 when compared to the results for the whole sample period. For estimating the principal portfolios over time, one has to make a choice with regard to the estimation window. The two most common approaches either rely on an expanding window or a rolling window for estimation. The proponents of expanding window estimation appreciate that building on all available data typically gives rise to a quite robust set of components. On the other hand, rolling window estimation is believed to be more responsive to potential structural breaks. Therefore, our main analysis focuses on the discussion of results arising from rolling window estimation using a 6-months window. In particular, we perform a PCA every month to extract the principal portfolios embedded in the multi-asset classes. Stacking the corresponding principal portfolio variances, Figure depicts the variation of the principal portfolios variances over time. [Figure about here.] We observe PP to be fairly dominant by accounting for at least 6% of the underlying time series variation at any given point in time. Given that PP2 and PP3 represent some 2% and % of the variation, the remaining principal portfolios PP4 and PP5 do only account for a minor 6

fraction. At the end of the sample period we find PP accounting for 8% of the overall variability which bears testimony of the contagion effects emanating from the financial crisis in 28. AcommonconcernassociatedwiththePCAapproachisthestabilityofthecovariancematrix eigenvectors, especially those pertaining to the smallest eigenvalues. To investigate this issue we collect the principal portfolio weights throughout time in Figure 2. In particular, we consider three different sets of principal portfolio weights depending on the length of the estimation window. For instance, the left column of Figure 2 gives principal portfolio weights when using expanding window estimation. Unsurprisingly, the corresponding weights prove to be very stable throughout the sample period mirroring the general interpretation that we have inferred from the static weights over the whole sample period (see Table II). Obviously, stepping from expanding to rolling window estimation renders the time series of principal portfolio weights more volatile. In that regard, the middle and right columns of Figure 2 give results when using rolling estimation windows of 6 months and 36 months, respectively. As for the former, we note that even though weights are more volatile there has been no switch in the economic meaning of PP (Equity Risk), PP 4 (Interest Rate Risk), and PP 5 ( Spread). Only once do we observe a switch between PP 2 (Commodity Risk) and PP 3 (EM-Spread) which takes place in the middle of the Nineties and lasts for some 3 years. By and large, this interpretation continues to hold when stepping to a shorter estimation window of 36 months. [Figure 2 about here.] III. Risk-Based Asset Allocation A. Risk-Based Asset Allocation Schemes For constructing the diversified risk parity strategy we first determine the principal portfolios using either expanding or rolling window estimation. From Section II we know that the optimal DRP strategy is an inverse volatility strategy along the principal portfolios which can be computed analytically given specific sign constraints with respect to the principal portfolios. We choose these constraints such that the sign of each principal portfolio equals the sign of its historical risk premium. We measure the principal portfolios historical risk premia using an expanding window 7

starting at the beginning of the sample period in 987. Intuitively, the strategy thus aims at capturing long-term risk premia associated with the principal portfolios. More importantly, the strategy s positioning will not establish bets that have not been rewarded historically. 4 To allow for comparison with alternative risk-based allocation techniques we also determine a constrained DRP strategy using optimization (5) where we enforce full investment and positivity constraints. Rebalancing of all strategies occurs at a monthly frequency. Given that the first PCA estimation consumes 6 months of data the strategy performance can be assessed from January 993 to September 2. For benchmarking the diversified risk parity strategy we consider four alternative risk-based asset allocation strategies: /N,minimum-variance,riskparity,andthemost-diversified portfolio of Choueifaty and Coignard (28). First, we implement the /N -strategy that rebalances monthly to an equally weighted allocation scheme, hence, the portfolio weights w /N are w /N = N (6) Second, we compute the minimum-variance (MV) portfolio either building on an expanding or rolling 6-months window for covariance-matrix estimation. The corresponding weights w MV derive from w MV =argmin w w Σw (7) subject to the full investment and positivity constraints, w =andw. Third, we construct the original risk parity (RP) strategy by allocating capital such that the asset classes risk budgets contribute equally to overall portfolio risk. 5 Note that these risk budgets also depend on either expanding or rolling window estimation. Since there are no closed-form solutions available, we follow Maillard, Roncalli, and Teiletche (2) to obtain w RP numerically via w RP =argmin w N i= j= N (w i (Σw) i w j (Σw) j ) 2 (8) 4 Note that we standardize the portfolio weights of the optimal DRP strategy to sum to %. 5 Risk parity has been put to the fore by Qian (26, 2) and Maillard, Roncalli, and Teiletche (2). 8

which essentially minimizes the variance of the risk contributions. 6 Again, the above full investment and positivity constraints apply. Fourth, we describe the approach of Choueifaty and Coignard (28) to building maximum diversification portfolios. To this end the authors define a portfolio diversification ratio D(w): D(w) = w σ w Σw (9) where σ is the vector of portfolio asset return volatilities. Thus, the most-diversified portfolio (MDP) simply maximizes the ratio between two distinct definitions of portfolio volatility, i.e. the ratio between the average portfolio assets volatility and the total portfolio volatility. We obtain MDP s weights vector w MDP by numerically computing w MDP =argmaxd(w) () w As before we enforce the full investment and positivity constraints. B. Performance of Risk-Based Asset Allocation Schemes Table III gives performance and risk statistics of the two DRP strategies as well as the alternative risk-based asset allocation strategies. In particular, the table has three panels depending on the length of the estimation window underlying the strategies. Unless stated otherwise, the reported results refer to an estimation window of 6 monhts. The optimal DRP strategy earns 7.8% at 4.4% volatility which is equivalent with a Sharpe ratio of.3. This high risk-adjusted return is mostly robust with respect to imposing positive asset weights. The constrained DRP strategy gives 7.3% at 4.% volatility. Among the competing strategies the highest annualized return materializes for the /N -strategy, but the 7.4% comes at the costofthehighestvolatility (9.4%). Moreover, the strategy exhibits the highest drawdown among all alternatives (33.5%). Conversely, the minimum-variance strategy provides a lower return of 6.3%. Given that minimumvariance indeed exhibits the lowest volatility (3.5%) its Sharpe Ratio of.86 is highly favorable. Also, its drawdown statistics are the least severe amounting to a maximum loss of 5.% during 6 Because the DRP portfolio is equivalent to a risk parity portfolio in principal portfolio space the approach of Maillard, Roncalli, and Teiletche (2) might also be used to determine the optimal DRP portfolio. 9

the whole sample period. Note that the constrained DRP strategy s maximum drawdown is only slightly higher (6.8%). Paraphrasing Maillard, Roncalli, and Teiletche (2) we then find the risk parity strategy to be a middle-ground portfolio between /N and minimum-variance. Its return is 6.8% at a 4.7% volatility thus giving rise to a Sharpe Ratio of.76. Also, the maximum drawdown statistics are significantly reduced when compared to the /N -strategy. The maximum drawdown of the MDP is even smaller (.7%), however, its return is the lowest among all alternatives (5.8%) which still allows for a fairly adequate Sharpe Ratio of.6. To gauge the strategies evolution over time, we plot their cumulative returns in Figure 3. Whereas the /N -strategy is pursuing a rather rocky path, the remaining strategies exhibit a quite steady evolution of performance. In addition, it seems as if the strategies resilience with respect to the financial crisis in 28 is the main driver in explaining the strategies overall volatility. [Figure 3 about here.] While the performance table and figure already give a good grasp of the different strategies, we additionally provide mutual tracking errors and mutual correlation coefficients in Table IV. In terms of strategy similarity we find the constrained diversified risk parity strategy to be very close to the optimal DRP strategy given a correlation of.96 and a tracking error of.27%. Concerning the alternative risk-based strategies we find the constrained DRP strategy to be highly correlated to risk parity, or minimum-variance and the MDP. Judging by a tracking error of.68% it is closest to the MDP. Unsurprisingly, tracking errors are highest for the /N -strategy with figures ranging from 6.% (vs. risk parity) to 8.74% (vs. minimum-variance). Note that the high risk-adjusted performance of the diversified risk parity strategy is to be taken with a grain of salt, since its monthly turnover is slightly higher than one of the other riskbased asset allocation strategies. It amounts to 5.6% on average which compares to 4.% for the MDP, 2.8% for minimum-variance, and.6% for risk parity. 7 By construction, the turnover of the /N -strategy is % disregarding potential rebalancing because of price movement. Also, one has 7 Note that we measure turnover based on weight changes of successive models which could understate the actual turnover of the strategies.

to acknowledge that all of the strategies risk and return statistics come with a significant degree of uncertainty. In particular, we give standard errors for the performance statistics arising from 5 bootstrap simulations using the stationary block bootstrap of Politis and Romano (994) with average block length of 2 months. Regardless of the strategy, we detect standard errors of.3 for the Sharpe Ratio making it hard to pinpoint significant differences in the risk-adjusted performance of the strategies. By and large, the above findings continue to hold when using expanding window estimation or rolling window estimation with 36 months (see Panels B and C of Table III). Across the board we then find the strategies to yield rather similar annualized returns. Interestingly, the diversified risk parity strategy is most affected when switching to expanding window estimation which renders it less responsive to changes in risk structure and thus less resilient to the 28 financial crisis. Still, the constrained DRP strategy yields 7.% at 4.2% volatility giving a Sharpe Ratio of.9 in the expanding window case. For rolling window estimation with an estimation window of 36 months, the figures are 7.3% at 4.% volatility implying a Sharpe ratio of.2. C. How Diversified are the Risk-based Asset Allocation Schemes? As argued by Lee (2), evaluating risk-based portfolio strategies by means of Sharpe Ratios is hard to reconcile with the fact that returns are not entering their respective objective function in the first place. In a vein similar to Lee (2) we rather resort to contrasting the risk characteristics of these portfolios. Thus, we turn to an in-depth discussion of the risk-based strategies weights and risk allocation in Figures 4 and 5. Risk is being decomposed not only by asset class but also by principal portfolios. Hence, the former analysis provides the well-known percentage risk contributions while the latter analysis performs theverysamedecompositionwithregardsto uncorrelated risk sources. First, we examine both diversified risk parity strategies in Figure 4. Only at the beginning of the sample period is the optimal DRP strategy characterized by positive asset weights across all asset classes. In general, the optimal DRP strategy has a large fraction of government bonds and is rather long in emerging markets and commodities. The strategy is often short in credit and sometimes short in developed equities as well. The need to go short is exacerbated in times of

crises when correlations spike and the corresponding PPs are less straightforward to implement in along-onlymanner. Thisobservationisimportantwhenitcomestorationalizingtheallocation pattern of the constrained DRP strategy. It allocates two thirds to government bonds and some 2% to commodities. At the beginning of the sample period the strategy has a 5% credit position that is driven out of the portfolio by an increase in government bonds at the end of the Nineties. Besides, there is a quite constant exposure to equities with emerging and developed equities being of equal importance. The increase of the equity exposure at the end of 28 comes at the cost of the commodities position which is completely closed. By construction, the strategy reacts timely to changes in risk structure and thus maintains a quite homogenous risk decomposition by principal portfolios throughout time. Even though facing a long-only and full-investment constraint the objective of risk parity across principal portfolios isfairlywellachieved. Notably,thisobjective turns out to be harder to realize at the end of the sample period. [Figure 4 about here.] Next, we turn to the benchmark strategies. First investigating the /N -strategy, we find more than 8% of its overall risk to be driven by equities with the highly volatile emerging equities attracting the highest share of the risk budget. Given that commodities consume most of the remaining risk budget, the other asset classes, namely bonds and credit, are close to being irrelevant. Decomposing the strategy s risk by principal portfolios instead reveals the /N - strategy to be budgeting risk mostly to PP, i.e. equity risk. Even more so, as time progresses the /N -strategy more or less emerges as a single-bet strategy as opposed to an N-bet strategy. [Figure 5 about here.] Second, we recover the archetypical weights distribution of minimum-variance that is heavily concentrated in the two low-risk asset classes bonds and credit. While equities hardly enter the minimum-variance portfolio, there is always a diversifying commodities position of some 5-% in place. This weights decomposition almost serves as a blueprint for the minimum-variance strategy s traditional risk decomposition. Conversely, the decomposition of risk with respect to the principal portfolios demonstrates minimum-variance to be heavily exposed to a single risk source, PP4, representing interest rate risk. Compared to /N,theminimum-variancestrategy 2

appears to be less concentrated because it is also exhibiting a quite marked exposure to PP3 and PP5. Third, we examine the risk parity strategy. Its weights decomposition reflects a reasonably smooth allocation over time with global bonds accounting for the highest portfolio fraction; on average, one third is being allocated to this asset class. While the bond share is increasing over time, we realize that this increase is mainly fueled by a decrease of the credit position. This observation relates to the fact that the rising credit volatility induces the strategy to limit its credit exposure for maintaining risk parity. The remaining asset classes, equities and commodities, are characterized by rather constant allocation weights over time that are approximately inversely proportional to their respective time series volatilities. By construction, the traditional risk decomposition exhibits equal weights across asset classes. 8 Interestingly, the decomposition of the risk parity strategy with respect to the principal portfolios is significantly less evenly distributed. At the beginning of the sample period PP and PP4, each attract some quarter of the risk budget while PP2 and PP3 almost absorb its remainder. However, PP2 and PP3 are constantly losing share giving rise to a 5% risk contribution of PP and some 35% of PP4. Hence, the risk parity strategy is rendered highly concentrated in terms of uncorrelated risk sources at the end of the sample period. Fourth, we examine the results for the MDP. Overall, its weights decomposition over time is in between the one of risk parity and minimum-variance. Nevertheless, the MDP s reaction to the 28 crises is more pronounced with respect to reducing the credit position in favor of global bonds. Given that the MDP s share in commodities is also slightly higher than the one for the risk parity strategy its traditional risk decomposition is slightly dominated by commodities risk. More interestingly, the risk decomposition with respect to the uncorrelated risk sources is quite evenly distributed. While we also observe a minor degeneration of the profile over time it is less severe when compared to /N,minimum-variance,orriskparity. Inarecentpaper, Choueifaty, Froidure, and Reynier (23) show that the MDP is optimal under certain conditions amongst which the homogeneity of the investment universe. This condition is hardly met in an asset allocation context where asset classes are typically characterized by very different risk-return 8 At some times risk parity does only hold approximately given that the numerical optimization may be tricky, see Maillard, Roncalli, and Teiletche (2). 3

tradeoffs. Thus, our results obtained in the MDP case should be taken with a grain of salt. Still, it is interesting to compare diversification ratios across strategies, see Table III. Relative to the MDP (.88) the traditional risk parity strategy (.79) and the constrained DRP strategy (.7) are not too far off while /N (.46) and minimum-variance (.6) exhibit considerably smaller ratios. For directly comparing the degree to which the risk-based asset allocation strategies accomplish the goal of diversifying across uncorrelated risk sources, we plot the number of uncorrelated bets over time in Figure 6. Reiterating our above interpretation of the associated risk contributions over time, we find the /N -strategy to be mostly dominated by the otherstrategies. Unsurprisingly, the constrained DRP strategy is maintaining the highest number of bets throughout time which often reaches the maximum of 5 bets. However, there is a short episode in 2/ when longonly constraints render the constrained DRP strategy with less bets than most of the risk-based alternatives. Intuitively, the latter strategies implement bets that are not deemed attractive by the principal portfolio constraints effective for the constrained DRP strategy. In between /N and diversified risk parity we find minimum-variance and risk parity to represent some 3 bets over time. MDP is close to 4 bets on average but these three strategies essentially degenerate in their degree of diversification at the end of the sample period. [Figure 6 about here.] IV. Conclusion Within this paper we embrace the approach of Meucci (29) to maximize a portfolio s diversification. His paradigm stipulates rearranging the portfolio assets into uncorrelated risk sources by means of a simple PCA. Maximum diversification obtains when equally budgeting risk to each of the uncorrelated risk sources prompting us to label the strategy diversified risk parity. Whereas risk-based asset allocation techniques generally yield superior risk-adjusted performance, judging these strategies by their returns is at odds with the fact that returns are not part of the strategies underlying objective function. Following Lee (2) we rather turn to evaluating their ex-ante risk characteristics, especially with respect to the uncorrelated risk sources. While the diversified risk parity strategy is designed to balance these risk sources, it is reassuring that it is meeting this 4

objective well even when facing long-only constraints. Unfortunately, the competing alternatives tend to be rather concentrated in a few bets. While the traditional risk parity strategy appeared to be least affected at the outset, we document a decrease in its degree of diversification over time. Also, the traditional risk parity strategy s nature is critically dependent on the choice of assets for contributing equally to portfolio risk. Conversely, diversified risk parity has a built-in mechanism for tracking the prevailing risk structure thus providing a more robust way to achieve maximum diversification throughout time. 5

References Bera, A.K., and S.Y. Park, 28, Optimal portfolio diversification using maximum entropy, Econometric Reviews 27, 484 52. Bruder, B., and T. Roncalli, 22, Managing risk exposures using the risk budgeting approach, Working paper, Lyxor Asset Management. Chopra, V.K., and W.T. Ziemba, 993, The effect of errors in means, variances and covariances on optimal portfolio choice, Journal of Portfolio Management 9, 6. Choueifaty, Y., and Y. Coignard, 28, Toward maximum diversification, Journal of Portfolio Management 34, 4 5. Choueifaty, Y., T. Froidure, and J. Reynier, 23, Properties of the most diversified portfolio, Journal of Investment Management 2, 49 7. Lee, W., 2, Risk-based asset allocation: A new answer to an old question, Journal of Portfolio Management 37, 28. Maillard, S., T. Roncalli, and J. Teiletche, 2, The properties of equally weighted risk contribution portfolios, Journal of Portfolio Management 36, 6 7. Markowitz, H.M., 952, Portfolio selection, Journal of Finance 7, 77 9. Meucci, A., 29, Managing diversification, Risk 22, 74 79. Partovi, M.H., and M. Caputo, 24, Principal portfolios: Recasting the efficient frontier, Economics Bulletin 7,. Politis, D.N., and J.P. Romano, 994, The stationary bootstrap, Journal of the American Statistical Association 89, 33 33. Qian, E., 26, On the financial interpretation ofriskcontribution: Riskbudgetsdoaddup, Journal of Investment Management 4,. Qian, E., 2, Risk parity and diversification, Journal of Investing 2, 9 27. Roncalli, T., and G. Weisang, 22, Risk parity portfolios with risk factors, Working paper, Lyxor Asset Management. Woerheide, W., and D. Persson, 993, An index of portfolio diversification, Financial Services Review 2, 73 85. 6

Table I Descriptive Statistics The table contains descriptive statistics of the multi-asset classes according to the sample period from December 987 to September 2 based on monthly local currency returns. On the left-hand side, annualized return and volatility figures are reported and the right-hand side gives the corresponding correlation matrix. Return Vola Sharpe Correlation Matrix p.a. p.a. Ratio Equities Dev. Emg. 6.9% 3.8%.96. Developed Equities 5.9% 4.5%.8.. Emerging Equities 8.3% 24.2%.2 -.5.74. 5.7% 5.5%.6 -.8.8.32. 6.9% 5.3%.69.53.25.2.9. 7

Table II Principal Portfolio Weights The table gives the eigenvectors representing the principal portfolio weights with respect to the underlying asset classes. These eigenvectors either arise from a PCA of the multi-asset class covariance matrix over the whole sample period from December 987 to September 2 (Panel A) or from a PCA over the last 6 months of the sample period (Panel B). Weights in excess of.4 are in bold face, weights in excess of.2 are in italics. The principal portfolios variance is given in absolute terms and relative to the overall data variation. Cumulative represents the fraction of variance being explained by a given number of principal portfolios (with the highest variance contributions). Asset Class PP PP2 PP3 PP4 PP5 Equity EM-Spread Interest Rate Spread Panel A: December 987 to September 2 JPM Global Bond -. -.5 -.4.5 -.86 MSCI World.43 -.23 -.86 -.3 -.3 MSCI Emerging Markets.87 -.6.47.2 -. DJ UBS.24.96 -.4. -.5 Barclays U.S. Aggr..4 -. -.2.85.5 Variance 7.7% 2.2%.8%.3%.% Percent Explained 69.8% 9.9% 6.9% 2.8%.7% Cumulative 69.8% 89.7% 96.5% 99.3%.% Panel B: October 26 to September 2 JPM Global Bond -.3 -.3 -.2.43 -.88 MSCI World.42 -.39.78.22 -.8 MSCI Emerging Markets.76 -.33 -.5 -.25 -.2 DJ UBS.49.86.4.6 -.5 Barclays U.S. Aggr..9 -.4 -.26.84.47 Variance 4.4% 2.%.5%.4%.% Percent Explained 82.4%.8% 3.% 2.3%.5% Cumulative 82.4% 94.2% 97.3% 99.5%.% 8

Table III Performance and Risk Statistics of Asset Allocation Strategies The table gives performance and risk statistics of the risk-based asset allocation strategies from January 993 to September 2. Annualized return and volatility figuresarereportedtogetherwiththeaccording Sharpe Ratio. Maximum Drawdown (MDD) is computed over month and over the whole sample period. Turnover is the portfolios mean monthly turnover over the whole sample period. Gini coefficients are reported using portfolios weights ( Gini Weights ) and risk decomposition with respect to the underlying asset classes ( Gini Risk ) or with respect to the principal portfolios ( Gini PP Risk ). The # bets is the exponential of the risk decomposition s entropy when measured against the uncorrelated risk sources. D is the diversification ratio. We give standard errors in brackets based on the stationary bootstrap of Politis and Romano (994) with average block length of 2 months and 5 simulations. Panel A gives the results for rolling window estimation using 6 months, and Panels B and C give the results for expanding and rolling window estimation using 36 months, respectively. Statistic Diversified Risk Parity Risk-Based Allocations Optimal Constrained /N MV RP MDP Panel B: Rolling Window 6 months Return p.a. 7.8% [4.%] 7.3% [.6%] 7.4% [2.4%] 6.3% [.%] 6.8% [.2%] 5.8% [.3%] Volatility p.a. 4.4% [.2%] 4.% [.4%] 9.4% [.%] 3.5% [.2%] 4.7% [.4%] 4.3% [.3%] Sharpe Ratio.3 [.37].98 [.3].45 [.3].86 [.3].76 [.3].6 [.29] MDD M -2.7% [26.4%] -2.7% [3.5%] -5.% [3.%] -2.4% [.5%] -5.9% [.7%] -4.3% [.3%] MDD -9.% [62.5%] -6.8% [7.%] -33.5% [.5%] -5.% [.4%] -3.% [4.5%] -.7% [3.5%] Turnover.3% [4%] 5.6% [5.3%].% [.%] 2.8% [.6%].6% [.%] 4.% [.7%] Gini Weights.68 [.8].68 [.9]. [.].69 [.7].33 [.2].5 [.8] Gini Risk.6 [.8].6 [.].49 [.3].69 [.7]. [.].4 [.7] Gini PP Risk. [.].7 [.4].82 [.3].39 [.5].5 [.3].4 [.4] #bets 5. [.] 4.57 [.73].6 [.7] 3.23 [.25] 3.29 [.2] 3.65 [.9] D.6 [.37].7 [.5].46 [.7].6 [.7].79 [.8].88 [.8] Panel A: Expanding Window Return p.a. 7.5% [3.9%] 7.% [.5%] 7.4% [2.4%] 6.7% [.%] 7.% [.2%] 6.7% [.2%] Volatility p.a. 4.4% [6.%] 4.2% [.3%] 9.4% [.%] 3.6% [.2%] 5.% [.4%] 4.9% [.4%] Sharpe Ratio.97 [.3].9 [.3].45 [.3].97 [.3].76 [.3].7 [.3] MDD M -2.9% [7.6%] -3.6% [3.%] -5.% [3.%] -2.3% [.3%] -7.4% [.9%] -6.6% [.5%] MDD -6.4% [4.7%] -8.% [6.4%] -33.5% [.5%] -5.7% [.3%] -5.4% [4.8%] -5.2% [4.%] Turnover 6.4% [23.7%] 3.4% [2.4%].% [.%].9% [.3%].6% [.%].4% [.3%] Gini Weights.6 [.9].63 [.2]. [.].54 [.].3 [.3].33 [.4] Gini Risk.55 [.].57 [.3].5 [.4].54 [.]. [.].24 [.2] Gini PP Risk. [.].2 [.5].75 [.4].47 [.6].47 [.4].37 [.7] #bets 5. [.] 4.55 [.83].85 [.2] 2.88 [.29] 3.55 [.32] 4.8 [.3] D.69 [.39].77 [.7].58 [.9].58 [.8].87 [.9].9 [.9] Panel C: Rolling Window 36 months Return p.a. 6.8% [4.9%] 7.3% [.7%] 7.4% [2.4%] 6.5% [.%] 7.% [.2%] 6.% [.3%] Volatility p.a. 5.2% [5.2%] 4.% [.4%] 9.4% [.%] 3.5% [.2%] 4.5% [.4%] 4.2% [.3%] Sharpe Ratio.69 [.4].2 [.3].45 [.3].93 [.32].86 [.3].64 [.29] MDD M -4.8% [2.3%] -5.% [3.6%] -5.% [3.%] -3.% [.6%] -5.% [.6%] -4.3% [.3%] MDD -2.% [5.9%] -.% [6.9%] -33.5% [.5%] -6.% [.5%] -.7% [4.2%] -.4% [3.%] Turnover 23.2% [7%].3% [6.6%].% [.%] 4.8% [.9%] 2.6% [.2%] 6.% [.%] Gini Weights.67 [.7].69 [.8]. [.].7 [.6].34 [.2].55 [.7] Gini Risk.59 [.7].6 [.9].48 [.3].7 [.6]. [.].43 [.6] Gini PP Risk. [.].2 [.3].84 [.2].39 [.4].52 [.3].4 [.3] #bets 5. [.] 4.52 [.7].5 [.5] 3.3 [.23] 3.5 [.7] 3.59 [.6] D.57 [.37].67 [.4].42 [.7].63 [.8].76 [.8].88 [.8] 9

Table IV Comparison of Risk-based Asset Allocation Strategies The table compares the risk-based asset allocation strategies by reporting mutual tracking errors above the diagonal and mutual correlation figures below the diagonal. All figures refer to the 6-months rolling window estimation over the sample period January 993 to September 2. Tracking Error-Correlation-Matrix DRP Optimal Constrained /N MV RP MDP DRP Optimal..27% 4.34% 8.22% 2.38% 2.37% DRP Constrained.96. 4.% 7.66%.96%.68% /N.5.62. 8.74% 6.% 6.79% MV.84.88.39. 2.94% 2.38% RP.78.88.86.78..27% MDP.85.92.77.83.96. 2

Figure. Variances of the Principal Portfolios The figure gives the variance of the principal portfolios int the first row and its relative decomposition over time in the second row. Each month, a PCA is performed using a 6-months window to extract the principal portfolios embedded in the multi-asset classes and the corresponding principal portfolio variances are stacked in one bar. The left column gives results when using an expanding window estimation. The middle and right columns give results when using rolling estimation windows of 6 months and 36 months, respectively. The results are ranging from January 993 to September 2. Variance of Principal Portfolios Variance of Principal Portfolios Variance of Principal Portfolios.2.8.25 PP PP PP PP2 PP2 PP2. PP3 PP4 PP5.6.4 PP3 PP4 PP5.2 PP3 PP4 PP5.8.2.5..6.8..4.6.4.5.2.2 Variance of Principal Portfolios in Percent Variance of Principal Portfolios in Percent Variance of Principal Portfolios in Percent PP PP PP.9 PP2 PP3.9 PP2 PP3.9 PP2 PP3 PP4 PP4 PP4.8 PP5.8 PP5.8 PP5.7.7.7.6.6.6.5.5.5.4.4.4.3.3.3.2.2.2... 2

Figure 2. Principal Portfolio Weights The figure gives the principal portfolios weights over time. The left column gives results when using an expanding window estimation. The middle and right columns give results when using rolling estimation windows of 6 months and 36 months, respectively. The estimation period is from January 988 to September 2. PP Weights PP Weights PP Weights.8.8.8.6.6.6.4.4.4.2.2.2 PP.2.2.2.4.4.4.6.8.6.8.6.8 PP2 Weights PP2 Weights PP2 Weights.8.8.8.6.6.6.4.4.4.2.2.2 PP2.2.2.2.4.4.4.6.8.6.8.6.8 PP3 Weights PP3 Weights PP3 Weights.8.6.4.8.6.4.8.6.4.2.2.2 PP3.2.2.2.4.4.4.6.8.6.8.6.8 PP4 Weights PP4 Weights PP4 Weights.8.8.8.6.6.6.4.4.4.2.2.2 PP4.2.2.2.4.4.4.6.8.6.8.6.8 PP5 Weights PP5 Weights PP5 Weights.8.6.4.8.6.4.8.6.4.2.2.2 PP5.2.2.2.4.4.4.6.8.6.8.6.8 22

Figure 3. Performance of Risk-Based Asset Allocation The figure gives the cumulative total return of the risk-based asset allocation strategies when using rolling window estimation of 6 months over the sample period starting January 993 to September 2. 4.5 4 3.5 /N MV RP MDP DRP Constrained DRP Optimal Rolling Window 3 2.5 2.5.5 23

Figure 4. Weights and Risk Decompositions The figure gives the decomposition of the diversified risk parity strategies in terms of weights and risk. Risk is being decomposed by asset classes and by principal portfolios, respectively. The sample period is from January 993 to September 2. The upper row gives results for the optimal DRP strategy and the lower row gives results for the constrained DRP strategy when using rolling window estimation of 6 months. Weights Volatility Contributions by Assets in % Volatility Contributions by Principal Portfolios in % 2.5 PP.9 PP2 PP3.5.8 PP4 PP5.7.6 Opt.5.5.5.4.3.2.5..5 Weights Volatility Contributions by Assets in % Volatility Contributions by Principal Portfolios in % PP.9.9.9 PP2 PP3 PP4.8.8.8 PP5.7.7.7.6.6.6 Con.5.5.5.4.4.4.3.3.3.2.2.2... 24

Figure 5. Weights and Risk Decompositions: Risk-Based Strategies The figure gives the decomposition of the risk-based allocation strategies in terms of weights and risk. Risk is being decomposed by asset classes and by principal portfolios, respectively. The results build on rolling window estimation over 6 months. The sample period is from January 993 to September 2. Weights Volatility Contributions by Assets in % Volatility Contributions by Principal Portfolios in % PP.9.9.9 PP2 PP3 PP4.8.8.8 PP5.7.7.7.6.6.6 /N.5.5.5.4.4.4.3.3.3.2.2.2... Weights Volatility Contributions by Assets in % Volatility Contributions by Principal Portfolios in % PP.9.9.9 PP2 PP3 PP4.8.8.8 PP5.7.7.7.6.6.6 MV.5.5.5.4.4.4.3.3.3.2.2.2... Weights Volatility Contributions by Assets in % Volatility Contributions by Principal Portfolios in % PP.9.9.9 PP2 PP3 PP4.8.8.8 PP5.7.7.7.6.6.6 RP.5.5.5.4.4.4.3.3.3.2.2.2... Weights Volatility Contributions by Assets in % Volatility Contributions by Principal Portfolios in % PP.9.9.9 PP2 PP3 PP4.8.8.8 PP5.7.7.7.6.6.6 MDP.5.5.5.4.4.4.3.3.3.2.2.2... 25

Figure 6. Number of Uncorrelated Bets We plot the number of uncorrelated bets for the risk-based asset allocation strategies when using rolling window estimation of 6 months for the sample period January 993 to September 2. 5.5 Rolling Window 5 4.5 4 3.5 3 2.5 2.5 /N MV RP MDP.5 DRP Constrained DRP Optimal 26