Diversified Risk Parity Strategies for Equity Portfolio Selection

Diversified Risk Parity Strategies for Equity Portfolio Selection Harald Lohre Deka Investment GmbH Ulrich Neugebauer Deka Investment GmbH Carsten Zimmer Deka Investment GmbH May 9, 22 We are grateful to Stanimir Denev, Antti Ilmanen, and Attilio Meucci. 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; ulrich.neugebauer@deka.de Correspondence Information: Deka Investment GmbH, Mainzer Landstr. 6, Quantitative Products, 6325 Frankfurt/Main, Germany; carsten.zimmer@deka.de

Diversified Risk Parity Strategies for Equity Portfolio Selection ABSTRACT We investigate a new way of equity portfolio selection that provides maximum diversification along the uncorrelated risk sources inherent in the S&P 5 constituents. This diversified risk parity strategy is distinct from prevailing risk-based portfolio construction paradigms. Especially, the strategy is characterized by a concentrated allocation that actively adjusts to changes in the underlying risk structure. In addition, x-raying the risk and diversification characteristics of traditional risk-based strategies like /N, minimum-variance, risk parity, or the most-diversified portfolio we find the diversified risk parity strategy to be superior. While most of these alternatives crucially pick up risk-based pricing anomalies like the low-volatility anomaly we observe the diversified risk parity strategy to more effectively exploit systematic factor tilts. Keywords: Risk-Based Portfolio Construction, Risk Parity, Diversification, Entropy JEL Classification: G; D8

In the absence of estimation risk the mean-variance approach of Markowitz (952) is the method of choice to optimally trade off assets risk and return and thus generate efficient portfolios. In reality, estimation risk most often outweights the diversification benefits of mean-variance optimization rendering ex ante efficient portfolios rather inefficient ex post. Even more so, there is a large literature starting with Haugen and Baker (99) that demonstrates minimum-variance strategies to be far more efficient than capitalization-weighted benchmarks. Besides minimumvariance investing further risk-based allocation techniques have become popular given an increased desire for risk control emanating from the most recent financial crisis. For instance, Qian (26, 2) and Maillard, Roncalli, and Teiletche (2) advocate the risk parity approach that allocates capital such that all assets contribute equally to portfolio risk. Taking a different stance, Choueifaty and Coignard (28) introduce the most-diversified portfolio that maximizes their diversification ratio which is defined as the ratio of the weighted average of its underlying assets volatilities to total portfolio volatility. When it comes to diversification, minimum-variance strategies typically prove to be rather concentrated in low-volatility assets. This observation resonates with the finding of Scherer (2) that minimum-variance strategies implicitly capture risk-based pricing anomalies inherent in the cross-section of stock returns, especially the low-volatility and low-beta anomalies as evidenced by Ang, Hodrick, Xing, and Zhang (26, 29), Frazzini and Pedersen (2), or Blitz and Vliet (27). Moreover, Leote de Carvalho, Lu, and Moulin (22) extend the finding of Scherer (2) to risk parity and the most-diversified portfolio of Choueifaty and Coignard (28). The present paper is especially concerned about generating truly diversified equity portfolios. To measure portfolio diversification early studies of Evans and Archer (968) or Fisher and Lorie (97) resort to the number of portfolio assets. Woerheide and Persson (993) examine the characteristics of the portfolio weight distribution by means of entropy or concentration metrics, especially, Bera and Park (28) provide a recent account of portfolio diversification using maximum entropy. However, all of these metrics disregard the assets dependence structure. To this end, Meucci (29) provides a more comprehensive framework to measuring and managing diversification. Pursuing principal component analysis of the portfolio assets he extracts the For a comprehensive overview and evaluation of diversification metrics see the recent paper of Frahm and Wiechers (2).

main drivers of the assets variability. Especially, 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 the 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. Recently, Lohre, Opfer, and Ország (22) adopt the framework of Meucci (29) to determine maximum diversification portfolios in a multi-asset allocation study. This investment strategy coincides with a risk parity strategy that is budgeting risk by principal portfolios rather than the underlying assets. The authors demonstrate the diversified risk parity strategy to provide convincing risk-adjusted performance in the multi-asset context together with superior diversification properties when benchmarked against other risk-based investment strategies. Within this paper we translate the idea of diversified risk parity to the equity domain as represented by the constituents of the S&P 5. First of all, we find the diversified risk parity strategy to provide superior risk-adjusted performance when compared to the index and the prevailing risk-based allocation schemes like /N, minimum-variance, risk parity, and the mostdiversified portfolio. Especially, the diversified risk parity strategy is characterized by a relatively concentrated allocation that is altered quite actively whenever a significant change in risk structure calls for adjusting its risk exposure. Controlling for common risk factors we show the strategy s outperformance to be fully captured by systematic factor tilts. However, in comparison to the remaining strategies the diversified risk parity strategy seems to be more successful in dynamically managing these tilts. In addition, x-raying the risk structure of competing alternatives we find the traditional risk parity strategy to be similar to the /N -strategy or the market index in picking on concentrated risk. While the most-diversified portfolio is hardly doing better we find minimum-variance to come closest to the diversified risk parity strategy in terms of uncorrelated bets. 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 motivates the concept 2

of principal portfolios. Section III is devoted to presenting the diversified risk parity strategy and contrasting it to alternative risk-based equity strategies. Section IV concludes. I. Diversifying Risk Parity In their construction of diversified risk parity strategies for multi-asset allocation Lohre, Opfer, and Ország (22) build on the approach of Meucci (29) to measuring and managing a given portfolio s diversification. Under this paradigm it turns out that the maximum diversification portfolio is equivalent to a risk parity strategy across the uncorrelated risk sources embedded in the underlying investment universe. For further analyzing the diversified risk parity strategy in the equity domain it is instructive to briefly present the underlying framework. Consider a portfolio consisting of N stocks with weight and return vectors w and R that give rise to a portfolio return of R w = w R. Diversification especially pays when combining low-correlated assets. Hence, it is natural to construct uncorrelated risk sources by applying a principal component analysis (PCA) to the variance-covariance matrix Σ of the portfolio assets. According to the spectral decomposition theorem Σ can be expressed as a product Σ = EΛE () where Λ =diag(λ,..., λ N ) is a diagonal matrix consisting of Σ s eigenvalues that are assembled in descending order, λ... λ N.ThecolumnsofmatrixE represent the eigenvectors of Σ. These eigenvectors define a set of N uncorrelated principal portfolios 2 with variance λ i for i =,..., N and returns R = E R. As a result, a given portfolio can be either expressed in terms of its 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: Var(R w )= N i= w 2 i λ i (2) 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. 3

Normalizing the principal portfolios contributions by the portfolio variance then yields the diversification distribution: p i = w2 i λ i, i =,..., N (3) Var(R w ) Note that the diversification distribution is always positive and that all p i sum to one. 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 prompting Lohre, Opfer, and Ország (22) to dub this approach diversified risk parity (DRP). Conversely, portfolios mainly loading on a single principal portfolio display a peaked diversification distribution. Aggregating the diversification distribution Meucci (29) chooses the exponential of its entropy 3 for evaluating a portfolio s degree of diversification: ( ) N N Ent =exp p i ln p i i= (4) Intuitively, N Ent can be interpreted as the number of uncorrelated bets. For instance, a completely concentrated portfolio is characterized by p i =foronei and p j =fori j resulting in an entropy of which implies N Ent =. Conversely,N Ent = N obtains 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) andn Ent = N. In the spirit of Markowitz (952), this framework readily allows for determining a meandiversification frontier that trades off expected return against a certain degree of diversification. Taking this approach to the extreme, one 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) 3 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. 4

where the weights w may possibly be restricted according to a set of constraints C. Thus,thesolution of optimization (2) ultimately results in a diversified risk parity strategy that is potentially subject to some investment constraints. II. Understanding Principal Portfolios A. Data and Descriptive Statistics We investigate the diversified risk parity strategy for the S&P 5 constituents from October 989 to September 2. In any given month, portfolio construction is restricted to the then active 5 constituents thus mimicking a realistic investment setting that is not hampered by survivorship or forward-looking biases. As a consequence, we deal with a total of 37 companies that have been in the index over the sample period. The first column of Table II conveys the performance statistics of the S&P 5 using total return figures. 4 Its annualized return amounts to 7.5% at a volatility of 3.8% which implies a Sharpe Ratio of.28 when measured against the 3M treasury rate. 5 This moderate risk-adjusted equity performance basically bears testimony of the two severe setbacks caused by the burst of the TMT bubble in 2 and the more recent financial crisis. Especially, the latter event triggered a maximum drawdown of 47.5% over the subsequent.5 years. B. How many risk sources are embedded in the S&P 5? In theory, one can construct as many principal portfolios as assets that enter the PCA decomposition. For instance, our set of 5 index constituents gives rise to 5 principal portfolios at any given date. However, it is well-known that already a few number of principal portfolios are sufficient for explaining most of the assets variance. In computing these principal portfolios we monthly perform a PCA using a rolling window of 6 months. To assess the relevance of the principal portfolios over time we plot the first principal portfolios variances over time in the upper panel of Figure. In the figure s lower panel we boxplot their distribution with regards to the explained variance. We observe principal portfolio (PP) to typically account for some 4 Note that the S&P 5 is usually being reported as a price index as opposed to a total return index. 5 The annualized return of the 3M Euribor amounts to 3.6%. 5

3% of the total variability. Principal portfolio 2 (PP2) captures less than % on average thus leaving only single-digit fractions for the subsequent principal portfolios PP3 to PP5. All in all, the first PPs account for at least half of the data variability at any given date. [Figure about here.] Moreover, with their relevance quickly dying off it seems hardly reasonable to allocate any risk budget to higher principal portfolios. Hence, it is crucial to determine an adequate threshold for cutting off rather irrelevant principal portfolios. To this end, we rely on the PC p and PC p2 criteria of Bai and Ng (22) for determining a reasonable number of principal portfolios. Of course, this number is not constant over time given that the set of companies varies as does the underlying risk structure. Depending on the information criterion the average number of principal portfolios ranges between 2 and 8 but is typically around 5, see Figure 2. At a given date, a consistent implementation obviously calls for sticking to the then prevailing number of relevant principal portfolios. [Figure 2 about here.] C. Dismantling Principal Portfolios To foster intuition about the uncorrelated risk sources inherent in the underlying assets we investigate the 8 (static) principal portfolios arising from a PCA over the most recent 6 months period from October 26 to September 2. In particular, we disentangle the eigenvectors representing the principal portfolios weights in the underlying assets. Instead of tabulating these 8 5 = 4 weights we resort to inspecting bi-plots of the principal portfolios weights, see Figure 3. By construction these weights are standardized to lie within the [-,]-interval. To speed interpretation each pair of weights is connected to the origin by a colored line where the line color varies according to the respective stock s GICS classification. Note that the ordering of sectors is such that the sectors with the highest weights are plotted first. Therefore, evidence for sectors with smaller weights will not be obstructed. [Figure 3 about here.] 6

Notably, PP has positive weights that are relatively homogenous across all 5 companies with Information Technology, Financials, and Industrials receiving the highest weights. Obviously, PP qualifies for a common market factor. Conversely, PP2 is characterized by positive and negative portfolio weights. PP2 is essentially short Information Technology Stocks and long most of the remaining sectors. 6 PP3 is mostly long in Energy and short in Financials, Consumer Discretionary, and Consumer Staples. PP4 is long Utilities, Health Care, and Telecoms and short Materials and Industrials. Stepping on to subsequent principal portfolios it is generally less straightforward to pinpoint certain sector tilts. In this vein, PP5 may at best be long Financials and short Health Care. The distinction is less clear-cut for subsequent principal portfolios. Even more so, PP7 and PP8 both have a significant loading to two specific stocks, U.S. Bancorp and Ecolab Inc. While the investigation of portfolio weights helps shaping our understanding we further seek to characterize the principal portfolios by means of time-series regressions against a set of wellknown factor portfolios. To this end, we extend the standard approach of Fama and French (993) by additional factors and estimate a regression model of the form R PPi,t = α + β R M,t + β 2 R Size,t + β 3 R Value,t + β 4 R Mom,t + β 5 R Vola,t + β 6 R Liqui,t + ε t (6) where R PPi,t is the return of one of the principal portfolios PPi, fori =,..., 8. The excess return of the S&P 5 relative to the risk-free rate serves as the market return R M,t. The remaining factor controls are long-short portfolios that we source from the Barra Global Equity Model (GEM2). In the following, we briefly sketch the firm characteristics driving the various factor portfolios and we refer the reader to Menchero, Morozov, and Shepard (28) for a more detailed description of the factor portfolio construction. The size factor, R Size,t, represents differences in the pricing of large and small cap stocks where size is being measured by two descriptives, total market capitalization and total assets. Likewise, the value factor, R Value,t,isdrivenbytwo characteristics, i.e. price-book and price-sales ratio. The momentum factor, R Mom,t, builds on three indicators, namely 6M- and 2M-price momentum together with the stock s alpha arising from a CAPM-regression using 2 years of weekly data. Also, the volatility factor, R Vola,t, builds 6 Note that the sign of weights is of second-order importance when judging principal portfolios. One can either buy or sell a given PP and thus gain exposure to a risk factor that is orthogonal to the other PPs by design. In that regard, one may rather think of playing Information Technology against the remaining sectors when replicating PP2 rather than going long or short the respective sectors. 7

on three stock dispersion metrics: The most important one is the beta from the just mentioned CAPM-regression. The other two are a cumulative range indicator and a short-term daily asset volatility measure. Finally, the liquidity factor, R Liqui,t, subsumes the information contained in three liquidity metrics: Annual share turnover, quarterly share turnover, and monthly turnover. Equipped with the factor structure in (6) we can thus identify the common factor exposures of the principal portfolios. Table I documents that PP is indeed loading heavily on market risk together with a significant negative value tilt and a negative exposure to size, momentum, and volatility. All in all, the chosen factor structure accounts for 94.% of its time series variation. In unreported results we find the market factor to already account for 9.3%. The adjusted R 2 of the remaining principal portfolios are considerably smaller in size because one is seeking to explain the variation of long-short portfolios. Thus, it is natural to find the market factor to only have a minor influence (if any). PP2 is significantly loading on every factor with the volatility factor exposure being most pronounced as indicated by a t-statistics of -9.8 which ultimately gives rise to an adjusted R 2 of 57.6%. PP3 seems to be more balanced across factors loading negatively on size and value and positively on momentum and volatility. However, the explanatory power for PP3 is small given an adjusted R 2 of 3.%. The explanatory power for PP4 and PP5 amounts to 22.2% and 34.6%, respectively. PP4 especially loads on value and momentum. PP5 is volatility versus liquidity. PP6 only has one significant exposure, negative with respect to liquidity. PP7 is long value and size which also applies to PP8. The latter two observations hint at the suggestion that the time series regressions may be rather limited for higher order principal portfolios. This reading is further reinforced by the fact that the adjusted R 2 s do not exceed % for PP6 to PP8. Thus, most of these principal portfolios time series variation cannot be accounted for by the common factors. Obviously, the explanation may be twofold. Either the higher order principal portfolios are not meaningful (at least at some times) or the factor structure in (6) is incomplete and may be missing important factors. 8

III. Risk-Based Equity Strategies A. Risk-Based Portfolio Construction For benchmarking the diversified risk parity strategy we consider four alternative risk-based equity strategies: /N, minimum-variance, risk parity, and the most-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 (7) Second, we compute the minimum-variance (MV) portfolio building on a rolling 6 months window for covariance-matrix estimation. The corresponding weights w MV derive from w MV =argmin w w Σw (8) subject to the full investment constraint w =. We further restrict stock weights to be positive and bounded by 5%, i.e. w.5. Third, we construct the original risk parity (RP) strategy by allocating capital such that the assets risk budgets contribute equally to overall portfolio risk. Because closed-form solutions are not available we follow Maillard, Roncalli, and Teiletche (2) to computing w RP numerically by minimizing the variance of the risk contributions w RP =argmin w N i= j= Again, the above full investment and weights constraints apply. N (w i (Σw) i w j (Σw) j ) 2 (9) Fourth, we consider 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 assets volatilities. In this setting, the most-diversified portfolio (MDP) simply maximizes the distance between two distinct definitions of portfolio volatility, i.e. the distance between a weighted average of portfolio assets volatility and the total portfolio volatility. We obtain MDP s weights vector w MDP by numerically computing w MDP =argmaxd(w) () w Fifth, for constructing the diversified risk parity (DRP) strategy we first determine the principal portfolios using rolling window estimation as described in Section I. We then optimize portfolios to have maximum diversification following optimization w DRP =argmax w C N Ent(w) (2) In line with the other risk-based strategies we also enforce the full investment and weights constraints. Rebalancing of all strategies occurs at a monthly frequency. Given that the first PCA estimation consumes 6 months of data the strategy s performance can be assessed from October 989 to September 2. In implementing the DRP strategy we stick to the relevant principal portfolios according to the PC p2 criteria of Bai and Ng (22), see Figure 2. For the ease of exposition we will nonetheless opt for a constant number of 8 principal portfolios when evaluating a given risk-based strategy in terms of its risk contributions. B. Performance of Risk-Based Equity Strategies Table II gives performance and risk statistics of the risk-based equity strategies. Unsurprisingly, the lowest annualized volatility (.8%) is achieved by the minimum-variance strategy together with an annualized return of 8.%. The strategy exhibits the second smallest drawdown among all alternatives (38.2%). Conversely, the /N strategy has a higher return (9.9%) which comes at the cost of the highest volatility (7.2%) and the most severe maximum drawdown (55.9%). Reiterating Maillard, Roncalli, and Teiletche (2) we find the risk parity strategy to be a middle-ground portfolio between /N and minimum-variance. Its return is 9.2% at a 4.% volatility thus giving rise to a Sharpe Ratio of.39 which compares to.36 for /N and

.38 for minimum-variance. However, its maximum drawdown statistics are hardly reduced when compared to the /N -strategy. Notably, the MDP is fairly close to the index by yielding 7.9% at 3.% volatility, hence, its Sharpe Ratio of.33 is slightly higher than the one of the S&P 5 (.28) which yields 7.5% at 3.8% volatility. Having recovered the risk and return characteristics of the classic risk-based strategies we now inspect the diversified risk parity strategy. Its annualized return amounts to.% and is thus outperforming the remaining strategies by at least %. This extra return does not come at an overly excessive volatility (5.%) implying a high Sharpe Ratio of.49 for the DRP strategy. Even more so, its maximum drawdown is the smallest among all alternatives (35.8%). Note that the DRP strategy entails the largest turnover among the risk-based equity strategies (25.3%) suggesting that transaction costs may reduce its relative return potential though. To gauge the strategies evolution over time we plot their cumulative returns in Figure 4. First of all, we note that all of the risk-based equity strategies move pretty much in sync until the end of the Nineties. Within this time period one has to acknowledge the index to be the best performing investment, however, during the build-up of the TMT-bubble the strategies performance starts to diverge. On the lower end we find MV and MDP mingling together until the end of the sample period. Likewise, we observe RP and /N -strategy to be tightly coupled. However, while their performance is generally higher their drawdown during the financial crisis is as well. The diversified risk parity strategy especially sets itself apart within the three years surrounding the burst of the TMT bubble. [Figure 4 about here.] While the performance table and chart already provide insight into the strategies nature we additionally compare mutual tracking errors and mutual correlation coefficients in Table III. Across the board, all strategies typically show high mutual correlation above.8. The odd one out is the DRP strategy with a correlation of.75 to the S&P 5 and.79 to all of the risk-based alternatives except risk parity (.82). In terms of strategy similarity we again recover the two pairs: /N and risk parity with a tracking error of 4.6% and minimum-variance and MDP with a tracking error of 4.4%. Conversely, the DRP strategy is least related to the other strategies: It has the smallest tracking error to risk parity (8.8%) and the highest one to /N (.8%).

C. Risk and Diversification Characteristics Judging risk-based strategies by their Sharpe Ratios alone is not meaningful given that returns are not entering the respective objective functions, see Lee (2). According to standard portfolio theory investors striving for maximum Sharpe Ratios should simply stick to the (ex ante) tangential portfolio together with a fraction of cash to accommodate their level of risk tolerance. One may rationalize that investors opt for risk-based strategies hoping that these strategies prove to be more efficient ex post than the (ex ante) maximum Sharpe Ratio portfolio. Still, risk-based strategies are designed to entail certain risk characteristics, especially, the diversified risk parity strategy is designed to take (uncorrelated) risks. In a vein similar to Lee (2), we thus resort to primarily evaluating the strategies along their risk and diversification characteristics. Especially, we first decompose risk by the underlying stocks and second by the according principal portfolios. This approach provides us with a concise picture of the underlying risk structure and number of uncorrelated bets implemented in a given portfolio. To set the stage we start by analyzing the S&P 5. While we refrain from plotting stock weights we nevertheless provide some aggregate figures summarizing the characteristics of the weight decomposition on stock-level, see Panel B of Table II. Especially, we report Gini coefficients for the stock weight decomposition (Gini Weights ), the risk decomposition by stocks (Gini Risk ), and the risk decomposition by principal portfolios (Gini PPRisk ). As a reminder, the Gini coefficient is a statistic for assessing concentration which turns in case of no concentration (equal weights throughout time) and in case of full concentration (one stock or principal portfolio attracts all of the weight all of the time). For the S&P 5 the Gini Weights (.63) and the Gini Risk (.65) show the index to be rather concentrated. In Figure 5 we plot sector weights and risk contributions using the GICS classification of MSCI Barra. The left chart of Figure 5 gives the market s sector weights over time. We learn that Information Technology, Financials, and Consumer Staples are the dominant sectors whereas Materials, Telecoms, and Utilities have the smallest weights. Regarding structural shifts over time one observes the sector decomposition to be rather stable except for the time of the build-up and burst of the TMT bubble. At its height Information Technology accounts for 3% of the index. Moreover, according to the risk decomposition by sector (middle chart) Information Technology absorbs half of the risk budget at the turn of the 2

century. Except for Energy the risk decomposition of the remaining sectors basically resembles their weights decomposition. Despite having a rather constant index weight the risk contribution of Energy varies over time. Finally, the right chart depicts the risk decomposition with respect to the uncorrelated risk sources. A portfolio that reflects 8 uncorrelated bets should thus exhibit a risk parity profile along the principal portfolios, i.e. the decomposition should follow a constant /8 risk budget allocation over time. For the S&P 5 this decomposition is almost exclusively exposed to the single risk factor PP which typically accounts for more than 8% of the total risk throughout time. [Figure 5 about here.] Turning to the risk-based equity strategies we start with /N in the first row of Figure 6. While its weights and risk decomposition with respect to stocks is fairly evenly distributed (given Gini Weights of. and Gini Risk of.26, respectively), its risk decomposition with respect to the principal portfolios is not. The fraction explained by PP is even higher than the one for the index, even more so, its risk decomposition almost collapses into a blue square. [Figure 6 about here.] The weights decomposition of minimum-variance is concentrated in a few assets and is often characterized by holding the maximum position weight of 5%. On average, the minimum-variance portfolio consists of 36.2 stocks. Intuitively, minimum-variance is collecting the lowest volatility assets up to the maximum feasible weight. The traditional risk decomposition by stocks is likewise concentrated but is not overly biased towards specific stocks giving rise to a Gini Risk of.95. In terms of sector composition, the minimum-variance strategy is overweighting more defensive sectors like Utilities, Consumer Staples, and Health Care. Its risk decomposition by principal portfolios is more diverse than the one for /N or the index. Still, PP explains at least 5% of the total risk on average, however, this figure rose to more than 8% at the end of the sample. Interestingly, we find PP2 to a large part of the remaining risk budget at the turn of the century which resonates with the smaller exposure to the IT sector. Conversely, the risk parity portfolio is less concentrated in terms of weights because it has to load on all stocks for achieving its target of equally weighted risk contributions. As a consequence, risk parity is similarly dominated by 3

market risk as documented by a risk profile that is slightly more diversified than the one for /N or the index. The MDP is characterized by a rather concentrated portfolio allocation. On average, 38. stocks give rise to a Gini Weights of.95. Given its conceptual likeness to the minimumvariance approach it is not surprising to find the MDP to have a very similar defensive sector allocation which ultimately gives rise to a similar yet more concentrated profile with respect to the uncorrelated bets. Documenting all of the classical risk-based strategies to heavily load on market risk we are especially interested in testing whether the DRP strategy is providing a more diversified risk profile. Similar to minimum-variance, the DRP strategy typically builds on a rather concentrated portfolio in terms of weights (Gini Weights of.96). When compared to the other strategies the DRP strategy seems to be more active in reallocating across sectors. More importantly, the market risk factor as reflected by PP is significantly less dominant and attracts 2% to 5% of the risk budget throughout time. This is a very meaningful enhancement when compared to the alternatives. Even more so, the DRP strategy is successfully tracking the number of relevant bets as suggested by the Bai and Ng (22) criterion, see Figure 2. Especially, the DRP strategy s combination of concentrated positioning together with its active re-positioning over time seems to be key for maintaining a fairly balanced risk decomposition across the uncorrelated risk sources. 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 7. Reiterating our above interpretation of the associated risk contributions we find the /N -strategy to be dominated by the other strategies and more reflective of a -bet strategy than of an N-bet strategy. The same verdict applies to the S&P 5 which fares hardly better. Only in the 4-year period following the burst of the TMT bubble does the index resemble a 2-bet strategy. Surprisingly, the traditional risk parity strategy does hardly better with a mean number of.73 bets over the whole sample period, see Table II. Conversely, the minimum-variance strategy exhibits 2.57 bets on average which renders it more diversified throughout time when compared to the MDP which represents 2.3 bets over time. Coercing the DRP strategy s risk 4

profile into the number of bets ultimately provides the highest degree of diversification. Averaged over time, the DRP represents 5. bets. 7 [Figure 7 about here.] D. Dismantling Risk-Based Equity Strategies To further characterize the risk-based equity strategies we investigate their aggregate firm characteristics and relate the strategies returns to common risk factors. In Panel A of Table IV we aggregate the firm characteristics of the strategies constituents over the whole sample period. We find all of the risk-based strategies to invest in smaller-sized companies relative to the S&P 5. This index reflects companies with a market capitalization of 83 billion USD on average; the minimum size obtains for the MDP (7.3 billion USD). Unsurprisingly, the other risk-based strategies exhibit small average firm sizes around 2 billion USD. Interestingly, the /N -strategy has relatively small value characteristics (i.e. price to earnings, price to book value, or price to sales) and the smallest profitability statistics (return on equity, return on assets, and margin figures). Whereas the /N -strategy thus serves as the lower bound we find the minimum-variance strategy to serve as the upper bound in terms of profitability characteristics. However, note that the range of characteristics realizations spanned by these two strategies is relatively narrow. Still, the risk-based strategies significantly differ from the index that has the highest valuation ratios, growth characteristics, and profitability figures. Regarding the DRP strategy we observe its underlying stocks to have high dividend yields and P/Es and relatively high sales. Therefore, we next turn to examining the strategies exposure to well-known risk factors. To this end, we rely on the same factor structure we have used for characterizing the principal portfolios in Section II. The model thus reads: R RBS,t = α + β R M,t + β 2 R Size,t + β 3 R Value,t + β 4 R Mom,t + β 5 R Vola,t + β 6 R Liqui,t + ε t (3) 7 Still, the presented diversified risk parity strategy does not accomplish the target of equal risk contributions across principal portfolios and time. Intuitively, for attaining an equally weighted risk profile along the uncorrelated risk sources one has to mimic the prevailing principal portfolios. As we have documented these principal portfolios typically stipulate investing in long and short positions which is not feasible in the presence of a long-only constraint. The degree of diversification can thus be enhanced by relaxing the latter constraint to also allow for short sales. We refrain from reporting these results for being comparable to the other risk-based strategies. 5

where R RBS,t is the excess return of one of the risk-based strategies relative to the S&P 5, R M,t. 8 To set the stage we estimate a reduced version of factor model (3) to assess the factor exposures of the market itself: R M,t = α + β 2 R Size,t + β 3 R Value,t + β 4 R Mom,t + β 5 R Vola,t + β 6 R Liqui,t + ε t (4) Panel B of Table IV reports the according regression diagnostics. For the S&P 5 we observe the five factor portfolios to do a decent job in explaining the market s time series variation: 7.7% can be attributed to these risk factors with momentum being the least significant one. Turning to the risk-based strategies we naturally expect to obtain significantly smaller adjusted R 2 s because we are dealing with excess instead of total returns. Still, we find the risk factors to be driving a considerable amount of the time series variation in some of the strategies returns. Unsurprisingly, /N and RP turn out to positively load on the market factor. A high adjusted R 2 attains for /N (47.2%) which has additional exposure to the size, value and momentum factor. Its negative exposure to momentum is highly significant with a t-statistic of -4.92 which blends in well with the contrarian-like allocation pattern of the /N -strategy. Given the similarity of /N and RP it comes as no surprise to find RP likewise loading negatively towards momentum. In addition, RP loads on the low-volatility and low-beta anomalies as reflected by its pronounced exposure to the volatility factor. For the minimum-variance strategy we detect a significant exposure to size and volatility. Almost half of its excess returns time series variation can be attributed to common factors (43.7%). In particular, the strategy is heavily exposed to the volatility factor with a t-statistic of -5.47, hence, by loading on low-volatility and low-beta stocks the minimum-variance strategy is implicitly picking up the associated pricing anomaly thus confirming the findings of Scherer (2). Leote de Carvalho, Lu, and Moulin (22) find this rationale to also apply to the RP and MDP strategies within a global stock universe from 997 2, indeed, both strategies load on low-volatility assets. However, while the RP strategy is presenting a significant exposure to most of the risk factors, the MDP is basically mirroring the pattern of the minimum-variance strategy, albeit having a less accentuated exposure to the volatility factor. Finally, we observe 8 For the description of the factor portfolios see Section II and Menchero, Morozov, and Shepard (28). 6

the DRP strategy to give rise to the smallest adjusted R 2 across all strategies (7.%). Note that the strategy does not significantly load on the volatility factor. Instead, it seems to be exploiting the value anomaly versus a negative market exposure. Given that the DRP strategy is fairly active in reallocating its position weights we are especially interested in the evolution of its factor exposures over time. Figure 8 gives the risk-balanced strategies exposure to equity factors according to a rolling estimation of the factor models in (3) and (4). The estimation window consumes 6 months of data, thus, the factor exposures can be computed from January 22 to September 2. The S&P 5 is unsurprisingly rather steady in constantly loading on large, rather volatile and potentially value-related stocks. Conversely, the DRP strategy is exposed to smaller-sized companies and it is significantly loading on value stocks at the beginning of the century, however, this exposure has diminished towards the end of the sample period. By and large, this pattern with respect to the value factor carries over to the remaining strategies, as does the small cap exposure. For MV, RP, and the MDP we additionally demonstrate a constant and significant exposure to the volatility factor which is not present for the DRP. Conversely, the DRP strategy has a time-and-sign-varying exposure to the momentum factor. [Figure 8 about here.] IV. Conclusion Within this paper we operationalize the approach of Meucci (29) to maximizing diversification of equity portfolios. Creating uncorrelated risk sources by means of a principal component analysis we obtain maximum diversification portfolios when equally budgeting risk to each of the uncorrelated risk sources. Especially, we investigate the economic nature of this diversified risk parity strategy. When benchmarked against classical risk-based allocation schemes the diversified risk parity strategy provides the most convincing risk-adjusted performance, more importantly, the strategy is by far the most diversified portfolio among the investigated alternatives. Contrasting the other allocation schemes we find the diversified risk parity strategy to follow a rather concentrated allocation which is actively rebalanced at some dates. This behavior allows the 7

diversified risk parity strategy to constantly adapt to changes in risk structure and to maintain a balanced exposure to the then prevailing uncorrelated risk sources. 8

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Leote de Carvalho, R., X. Lu, and P. Moulin, 22, Demystifying equity risk-based strategies: A simple alpha plus beta description, Journal of Portfolio Management forthcoming. Lohre, H., H. Opfer, and G. Ország, 22, Diversifying risk parity, Working paper, Deka Investment GmbH. 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. Menchero, J., A. Morozov, and P. Shepard, 28, The Barra Global Equity Model (GEM2), Research notes, MSCI Barra. 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,. Qian, E., 26, On the financial interpretation of risk contribution: Risk budgets do add up, Journal of Investment Management 4,. Qian, E., 2, Risk parity and diversification, Journal of Investing 2, 9 27. Scherer, B., 2, A note on the returns from minimum variance investing, Journal of Empirical Finance 8, 652 66. Woerheide, W., and D. Persson, 993, An index of portfolio diversification, Financial Services Review 2, 73 85. 2

Table I Time Series Regressions of Principal Portfolios The table gives time series regression results according to factor model (6) for the principal portfolios using the period from January 997 to September 2. Coefficients are in bold face when significant on a 5%-level and in italics when significant on a %-level. PP PP2 PP3 PP4 PP5 PP6 PP7 PP8 Coefficients Alpha -.7 -.9 -.8.6 -.3.4.3.5 Market 25.95 9.59-2.2 2.5 2.23.33.42 -.3 Size -2.75 -.4-4.2-2.3-2.58 6.5 4.86 7. Value 6.6 35.79 2.62-6.23 5.36 3.65-8.87 -. Momentum -4.5-6.6 9.5 7.8 -.35-2.7 -.49 -.7 Volatility 5.6-3.63 8.36-3.3 5.66.9 -.95 2.48 Liquidity 4.2-23.96 2.7 4.89-7.35-9.4 4.43 7.6 t-statistics Alpha -2.32-4.36 -.65.85 -.25..37.68 Market 24.89 6.62 -.45 2.6 2.59.3.68 -.6 Size -5.8 -.87-2.38 -.58 -.78.53 2.4 2.5 Value 3.73 5.78 3.5-3.64.45.77-3.35-2.66 Momentum -2. -2.46 3.23 4.5 -.22 -.32 -.3 -.5 Volatility 2.3-9.8 2.53 -.47 3.4.38 -.7.3 Liquidity.74-3.3.25.86-3.69-3.5.3.57 Adjusted R 2 94.% 57.6% 3.% 22.2% 34.6% 9.6% 8.3% 8.% Durbin-Watson 2.6 2.2.9.73.67.85.93 2.3 2

Table II Performance and Risk Statistics of Risk-Based Equity Strategies The table gives performance and risk statistics of the risk-based equity strategies from October 989 to September 2. Annualized return and volatility figures are reported together with the according Sharpe Ratio. Maximum Drawdown 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 PPRisk ). The # bets is the exponential of the risk decomposition s entropy when measured against the uncorrelated risk sources. Statistic Index Risk-Based Allocations S&P 5 /N MV RP MDP DRP Risk and Return Figures Return p.a. 7.5% 9.9% 8.% 9.2% 7.9%.% Volatility p.a. 3.8% 7.2%.8% 4.% 3.% 5.% Sharpe Ratio.28.36.38.39.33.49 Maximum Drawdown M -6.8% -23.5% -4.5% -9.5% -4.4% -4.4% Maximum Drawdown -47.5% -55.9% -38.2% -47.6% -39.6% -35.8% Weights and Risk Decomposition Characteristics # Assets 5. 5. 36.2 5. 38. 43.4 Turnover.4% 2.2% 4.7% 3.7% 6.2% 25.3% Gini Weights.63..96.33.95.96 Gini Risk.65.26.95.3.95.96 Gini PPRisk.79.87.66.78.73.8 # bets.53.2 2.57.73 2.3 5. 22

Table III Comparison of Risk-based Equity Strategies The table compares the risk-based equity strategies by reporting mutual tracking errors above the diagonal and mutual correlation figures below the diagonal with all figures referring to the sample period October 989 to September 2. Tracking Error-Correlation-Matrix /N MV RP MDP DRP Market /N..5% 4.6% 9.%.8% 6.2% MV.8. 6.9% 4.4% 9.2% 8.2% RP.98.87. 6.2% 8.8% 5.% MDP.86.94.9. 9.3% 7.6% DRP.79.79.82.79..2% Market.95.8.93.84.75. 23

Table IV Characteristics of Risk-Based Equity Strategies The table gives aggregate firm characteristics together with time series regression results of the risk-based equity strategies covering the sample period from October 989 to September 2 and January 997 to September 2, respectively. The aggregate firm characteristics of Panel A are based on fundamental data provided by Factset with the ultimate data source being Worldscope. The time series regression results in Panel B arise from estimating the factor models given in (3) and (4). The coefficients are in bold face when significant on a 5%-level and in italics when significant on a %-level. Index Risk-Based Allocations S&P 5 /N MV RP MDP DRP Panel A: Aggregate Firm Characteristics Market Capitalization 82,979 9,4 2,86 2,226 7,29 2,4 Dividend Yield in % 7.9 7.74 7.46 2.59 8.25 8.5 Price/Earnings 5.49 4.49 5.4 4.44 5.53 5.4 Price/Cash Flow.9 8.22 9.44 8.56 9.8 8. Price/Book 3.2 2.28 2.76 2.42 2.67 2.4 Price/Sales.36.85.5.9.98.3 Hist. 3Y Sales Growth 2.6 9.88 9.8 9.43.35 2.6 Hist. 3Y Earnings Growth 6.22 3.3 2.8.9 5.33 2.4 Return on Assets 8.38 6.42 8.54 7.35 7.98 7.23 Return on Equity 9.54 4.34 8.3 6.27 6.76 4.4 Operating Margin 7.83 4.39 6.68 5.64 5.72 6.43 Net Margin.43 7.43 9. 8.37 7.96 7.57 Panel B: Time Series Regressions Coefficients Alpha.9% -.2% -.8% -.26% -.27% -.22% Market -.5 -.5.8 -.8 -.9 Size.55 -.65 -.53 -.59 -.67 -.5 Value.8.25.28.93.45.8 Momentum.25 -.43 -.3 -.3 -..9 Volatility.72 -.3 -.72 -.5 -.39.8 Liquidity.2 -.9.2 -.6.54 -.56 t-statistics Alpha.39 -.45 -.97 -.99 -.36 -.76 Market - 3.8 -.83.96 -.3-2. Size 5.75-3.55-2.26-3.63-2.7 -.39 Value 3.42 6.5.8 5.7.63 4.5 Momentum.78-4.92 -.2-3.92 -.87.3 Volatility 7.24 -.33-5.47-5.56-2.82.42 Liquidity 2.49 -.36.62 -.7.52 -. Adjusted R 2 7.7% 47.2% 43.7% 42.% 25.8% 7.% Durbin-Watson.99 2.6 2.2.93.95.83 24

Figure. Variance Decomposition of Principal Portfolios Variances The upper chart gives the variance of the principal portfolios and its relative decomposition over time. Each month, a PCA is performed to extract the first principal portfolios embedded in the underlying 5 stocks and the corresponding principal portfolio variances are stacked in one bar. The lower charts give the boxplots pertaining to a given principal portfolio s explained fraction of total variance over time. The results are covering the time period from October 989 to September 2..9.8.7.6 PP PP2 PP3 PP4 PP5 PP6 PP7 PP8 PP9 PP Variance of Principal Portfolios in Percent.5.4.3.2. 89 9 9 92 93 94 95 96 97 98 99 2 3 4 5 6 7 8 9.5 Boxplots of Explained Variance by Principal Portfolios.45.4.35.3.25.2.5..5 2 3 4 5 6 7 8 9 25