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, 2012
∗
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. 16, 60325 Frankfurt/Main, Germany; harald.lohre@deka.de
‡
Correspondence Information: Deka Investment GmbH, Mainzer Landstr. 16, Quantitative Products, 60325
Frankfurt/Main, Germany; ulrich.neugebauer@deka.de
§
Correspondence Information: Deka Investment GmbH, Mainzer Landstr. 16, Quantitative Products, 60325
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 500 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 1/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: G11; D81
In the absence of estimation risk the mean-variance approach of Markowitz (1952) 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 (1991) 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 (2006,
2011) and Maillard, Roncalli, and Teiletche (2010) advocate the risk parity approach that allocates capital such that all assets contribute equally to portfolio risk. Taking a different stance,
Choueifaty and Coignard (2008) 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 (2011)
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 (2006, 2009), Frazzini and Pedersen (2011), or Blitz and Vliet
(2007). Moreover, Leote de Carvalho, Lu, and Moulin (2012) extend the finding of Scherer (2011)
to risk parity and the most-diversified portfolio of Choueifaty and Coignard (2008).
The present paper is especially concerned about generating truly diversified equity portfolios. To measure portfolio diversification early studies of Evans and Archer (1968) or Fisher and
Lorie (1970) resort to the number of portfolio assets. Woerheide and Persson (1993) examine
the characteristics of the portfolio weight distribution by means of entropy or concentration metrics, especially, Bera and Park (2008) provide a recent account of portfolio diversification using
maximum entropy.1 However, all of these metrics disregard the assets’ dependence structure.
To this end, Meucci (2009) provides a more comprehensive framework to measuring and managing diversification. Pursuing principal component analysis of the portfolio assets he extracts the
1
For a comprehensive overview and evaluation of diversification metrics see the recent paper of Frahm and
Wiechers (2011).
1
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 (2009) 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 (2012) adopt the framework of Meucci (2009) 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 500. 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 1/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 1/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 (2009) 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 (2012) build on the approach of Meucci (2009) 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 Rw = 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
(1)
where Λ = diag(λ1 , ..., λN ) is a diagonal matrix consisting of Σ’s eigenvalues that are assembled in
descending order, λ1 ≥ ... ≥ λN . The columns of matrix E represent the eigenvectors of Σ. These
eigenvectors define a set of N uncorrelated principal portfolios2 with variance λi for i = 1, ..., N
and returns R̃ = 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 w̃ = 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 λi using weights w̃i2 :
V ar(Rw ) =
N
w̃i2 λi
(2)
i=1
2
Note that Partovi and Caputo (2004) 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:
pi =
w̃i2 λi
,
V ar(Rw )
i = 1, ..., N
(3)
Note that the diversification distribution is always positive and that all pi sum to one. Building
on this concept Meucci (2009) conceives a portfolio to be well-diversified when the pi 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 (2012) 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 (2009) chooses the exponential
of its entropy3 for evaluating a portfolio’s degree of diversification:
NEnt = exp −
N
pi ln pi
(4)
i=1
Intuitively, NEnt can be interpreted as the number of uncorrelated bets. For instance, a completely
concentrated portfolio is characterized by pi = 1 for one i and pj = 0 for i = j resulting in an
entropy of 0 which implies NEnt = 1. Conversely, NEnt = N obtains for a portfolio that is
completely homogenous in terms of uncorrelated risk sources. In this case, pi = pj = 1/N holds
for all i, j implying an entropy equal to ln(N ) and NEnt = N .
In the spirit of Markowitz (1952), 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 wDRP by solving
wDRP = argmax NEnt (w)
w∈C
3
(5)
The entropy has been used before in portfolio construction, see e.g. Woerheide and Persson (1993) or more
recently Bera and Park (2008). 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, the solution of optimization (12) 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 500 constituents from October
1989 to September 2011. In any given month, portfolio construction is restricted to the then
active 500 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 1037 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 500 using total return figures.4 Its annualized return amounts
to 7.5% at a volatility of 13.8% which implies a Sharpe Ratio of 0.28 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 2000 and the more recent
financial crisis. Especially, the latter event triggered a maximum drawdown of 47.5% over the
subsequent 1.5 years.
B. How many risk sources are embedded in the S&P 500?
In theory, one can construct as many principal portfolios as assets that enter the PCA decomposition. For instance, our set of 500 index constituents gives rise to 500 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 60 months. To assess the relevance of the
principal portfolios over time we plot the first 10 principal portfolios’ variances over time in the
upper panel of Figure 1. In the figure’s lower panel we boxplot their distribution with regards
to the explained variance. We observe principal portfolio 1 (PP1) to typically account for some
4
5
Note that the S&P 500 is usually being reported as a price index as opposed to a total return index.
The annualized return of the 3M Euribor amounts to 3.6%.
5
30% of the total variability. Principal portfolio 2 (PP2) captures less than 10% on average thus
leaving only single-digit fractions for the subsequent principal portfolios PP3 to PP500. All in
all, the first 10 PPs account for at least half of the data variability at any given date.
[Figure 1 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 P Cp1 and P Cp2
criteria of Bai and Ng (2002) 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 60 months
period from October 2006 to September 2011. In particular, we disentangle the eigenvectors
representing the principal portfolios’ weights in the underlying assets. Instead of tabulating these
8 · 500 = 4000 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 [-1,1]-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, PP1 has positive weights that are relatively homogenous across all 500 companies with
Information Technology, Financials, and Industrials receiving the highest weights. Obviously, PP1
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 (1993)
by additional factors and estimate a regression model of the form
RP P i,t = α + β1 RM,t + β2 RSize,t + β3 RV alue,t + β4 RM om,t + β5 RV ola,t + β6 RLiqui,t + εt
(6)
where RP P i,t is the return of one of the principal portfolios PPi, for i = 1, ..., 8. The excess return
of the S&P 500 relative to the risk-free rate serves as the market return RM,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 (2008) for a more detailed
description of the factor portfolio construction. The size factor, RSize,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, RV alue,t , is driven by two
characteristics, i.e. price-book and price-sales ratio. The momentum factor, RM om,t , builds on
three indicators, namely 6M- and 12M-price momentum together with the stock’s alpha arising
from a CAPM-regression using 2 years of weekly data. Also, the volatility factor, RV ola,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, RLiqui,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 PP1 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 94.0% of its time series variation. In
unreported results we find the market factor to already account for 91.3%. The adjusted R2 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 -9.80 which ultimately gives rise
to an adjusted R2 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 R2 of 13.0%. 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 R2 s do not exceed 10% 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: 1/N , minimum-variance, risk parity, and the most-diversified portfolio of
Choueifaty and Coignard (2008). First, we implement the 1/N -strategy that rebalances monthly
to an equally weighted allocation scheme, hence, the portfolio weights w1/N are
w1/N =
1
N
(7)
Second, we compute the minimum-variance (MV) portfolio building on a rolling 60 months window for covariance-matrix estimation. The corresponding weights wM V derive from
wM V = argmin w Σw
w
(8)
subject to the full investment constraint w 1 = 1. We further restrict stock weights to be positive
and bounded by 5%, i.e. 0 ≤ w ≤ 0.05 · 1.
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 (2010) to computing wRP numerically
by minimizing the variance of the risk contributions
wRP = argmin
w
N N
(wi (Σw)i − wj (Σw)j )2
(9)
i=1 j=1
Again, the above full investment and weights constraints apply.
Fourth, we consider the approach of Choueifaty and Coignard (2008) to building maximum
diversification portfolios. To this end the authors define a portfolio diversification ratio D(w):
w · σ
D(w) = √
w Σw
9
(10)
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 wM DP by numerically computing
wM DP = argmax D(w)
w
(11)
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
wDRP = argmax NEnt (w)
w∈C
(12)
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 60 months of data the strategy’s performance can be assessed from October
1989 to September 2011. In implementing the DRP strategy we stick to the relevant principal
portfolios according to the P Cp2 criteria of Bai and Ng (2002), 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 (11.8%) is achieved by the minimum-variance strategy
together with an annualized return of 8.1%. The strategy exhibits the second smallest drawdown
among all alternatives (38.2%). Conversely, the 1/N strategy has a higher return (9.9%) which
comes at the cost of the highest volatility (17.2%) and the most severe maximum drawdown
(55.9%). Reiterating Maillard, Roncalli, and Teiletche (2010) we find the risk parity strategy
to be a middle-ground portfolio between 1/N and minimum-variance. Its return is 9.2% at a
14.0% volatility thus giving rise to a Sharpe Ratio of 0.39 which compares to 0.36 for 1/N and
10
0.38 for minimum-variance. However, its maximum drawdown statistics are hardly reduced when
compared to the 1/N -strategy. Notably, the MDP is fairly close to the index by yielding 7.9% at
13.1% volatility, hence, its Sharpe Ratio of 0.33 is slightly higher than the one of the S&P 500
(0.28) which yields 7.5% at 13.8% 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 11.0% and is thus
outperforming the remaining strategies by at least 1%. This extra return does not come at an
overly excessive volatility (15.1%) implying a high Sharpe Ratio of 0.49 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 1/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 0.8. The odd one
out is the DRP strategy with a correlation of 0.75 to the S&P 500 and 0.79 to all of the risk-based
alternatives except risk parity (0.82). In terms of strategy similarity we again recover the two
pairs: 1/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 1/N (10.8%).
11
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 (2011). 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 (2011), 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 500. 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 (GiniW eights ), the risk decomposition by stocks (GiniRisk ), and
the risk decomposition by principal portfolios (GiniP P Risk ). As a reminder, the Gini coefficient
is a statistic for assessing concentration which turns 0 in case of no concentration (equal weights
throughout time) and 1 in case of full concentration (one stock or principal portfolio attracts all of
the weight all of the time). For the S&P 500 the GiniW eights (0.63) and the GiniRisk (0.65) 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 30% 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
12
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
1/8 risk budget allocation over time. For the S&P 500 this decomposition is almost exclusively
exposed to the single risk factor PP1 which typically accounts for more than 80% of the total risk
throughout time.
[Figure 5 about here.]
Turning to the risk-based equity strategies we start with 1/N in the first row of Figure 6.
While its weights and risk decomposition with respect to stocks is fairly evenly distributed (given
GiniW eights of 0.0 and GiniRisk of 0.26, respectively), its risk decomposition with respect to the
principal portfolios is not. The fraction explained by PP1 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 GiniRisk of 0.95.
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 1/N or the index. Still, PP1 explains at least 50% of
the total risk on average, however, this figure rose to more than 80% 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
13
market risk as documented by a risk profile that is slightly more diversified than the one for 1/N or
the index. The MDP is characterized by a rather concentrated portfolio allocation. On average,
38.1 stocks give rise to a GiniW eights of 0.95. 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 (GiniW eights of 0.96). 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 PP1 is significantly less dominant and attracts 20% to 50% 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 (2002) 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 1/N -strategy to be dominated by the other strategies and more reflective of a 1-bet
strategy than of an N -bet strategy. The same verdict applies to the S&P 500 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 1.73 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 2.13 bets over time. Coercing the DRP strategy’s risk
14
profile into the number of bets ultimately provides the highest degree of diversification. Averaged
over time, the DRP represents 5.10 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
500. This index reflects companies with a market capitalization of 83 billion USD on average;
the minimum size obtains for the MDP (17.3 billion USD). Unsurprisingly, the other risk-based
strategies exhibit small average firm sizes around 21 billion USD. Interestingly, the 1/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 1/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:
RRBS,t = α + β1 RM,t + β2 RSize,t + β3 RV alue,t + β4 RM om,t + β5 RV ola,t + β6 RLiqui,t + εt
7
(13)
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.
15
where RRBS,t is the excess return of one of the risk-based strategies relative to the S&P 500,
RM,t .8 To set the stage we estimate a reduced version of factor model (13) to assess the factor
exposures of the market itself:
RM,t = α + β2 RSize,t + β3 RV alue,t + β4 RM om,t + β5 RV ola,t + β6 RLiqui,t + εt
(14)
Panel B of Table IV reports the according regression diagnostics. For the S&P 500 we observe
the five factor portfolios to do a decent job in explaining the market’s time series variation: 71.7%
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 R2 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,
1/N and RP turn out to positively load on the market factor. A high adjusted R2 attains for
1/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 1/N -strategy. Given the similarity of 1/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
(2011). Leote de Carvalho, Lu, and Moulin (2012) find this rationale to also apply to the RP
and MDP strategies within a global stock universe from 1997–2010, 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 (2008).
16
the DRP strategy to give rise to the smallest adjusted R2 across all strategies (17.0%). 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 (13)
and (14). The estimation window consumes 60 months of data, thus, the factor exposures can be
computed from January 2002 to September 2011. The S&P 500 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 (2009) 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
17
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.
18
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20
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 1997 to September 2011. Coefficients are in bold face when significant on
a 5%-level and in italics when significant on a 10%-level.
PP1
PP2
PP3
PP4
PP5
PP6
PP7
PP8
-0.07
25.95
-20.75
16.61
-4.05
5.16
4.21
-0.19
9.59
-10.41
35.79
-6.60
-30.63
-23.96
-0.08
-2.21
-14.02
20.62
9.15
8.36
2.07
0.06
2.15
-2.31
-16.23
7.80
-3.31
4.89
-0.03
2.23
-2.58
5.36
-0.35
5.66
-17.35
0.04
0.33
6.50
3.65
-2.70
0.91
-19.04
0.03
0.42
4.86
-8.87
-1.49
-0.95
4.43
0.05
-1.03
7.00
-10.10
-0.07
2.48
7.61
Alpha
Market
Size
Value
Momentum
Volatility
Liquidity
-2.32
24.89
-5.18
3.73
-2.10
2.30
0.74
-4.36
6.62
-1.87
5.78
-2.46
-9.80
-3.03
-1.65
-1.45
-2.38
3.15
3.23
2.53
0.25
1.85
2.06
-0.58
-3.64
4.05
-1.47
0.86
-1.25
2.59
-0.78
1.45
-0.22
3.04
-3.69
1.10
0.30
1.53
0.77
-1.32
0.38
-3.15
1.37
0.68
2.04
-3.35
-1.30
-0.71
1.31
1.68
-1.16
2.05
-2.66
-0.05
1.30
1.57
Adjusted R2
Durbin-Watson
94.0%
2.16
57.6%
2.02
13.0%
1.90
22.2%
1.73
34.6%
1.67
9.6%
1.85
8.3%
1.93
8.0%
2.13
Coefficients
Alpha
Market
Size
Value
Momentum
Volatility
Liquidity
t-statistics
21
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 1989 to
September 2011. Annualized return and volatility figures are reported together with the according Sharpe
Ratio. Maximum Drawdown is computed over 1 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 (GiniW eights ) and risk decomposition with respect to the underlying asset classes
(GiniRisk ) or with respect to the principal portfolios (GiniP P Risk ). The # bets is the exponential of the
risk decomposition’s entropy when measured against the uncorrelated risk sources.
Statistic
Index
S&P 500
1/N
7.5%
13.8%
0.28
-16.8%
-47.5%
9.9%
17.2%
0.36
-23.5%
-55.9%
8.1%
11.8%
0.38
-14.5%
-38.2%
9.2%
14.0%
0.39
-19.5%
-47.6%
7.9%
13.1%
0.33
-14.4%
-39.6%
11.0%
15.1%
0.49
-14.4%
-35.8%
Weights and Risk Decomposition Characteristics
# Assets
500.0
Turnover
0.4%
0.63
GiniW eights
GiniRisk
0.65
0.79
GiniP P Risk
# bets
1.53
500.0
2.2%
0.00
0.26
0.87
1.21
36.2
14.7%
0.96
0.95
0.66
2.57
500.0
3.7%
0.33
0.13
0.78
1.73
38.1
16.2%
0.95
0.95
0.73
2.13
43.4
25.3%
0.96
0.96
0.08
5.10
Risk and Return Figures
Return p.a.
Volatility p.a.
Sharpe Ratio
Maximum Drawdown 1M
Maximum Drawdown
22
Risk-Based Allocations
MV
RP
MDP
DRP
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
1989 to September 2011.
Tracking Error-Correlation-Matrix
1/N
MV
RP
MDP
DRP
Market
1/N
1.00
0.81
0.98
0.86
0.79
0.95
MV
10.5%
1.00
0.87
0.94
0.79
0.81
RP
4.6%
6.9%
1.00
0.90
0.82
0.93
23
MDP
9.0%
4.4%
6.2%
1.00
0.79
0.84
DRP
10.8%
9.2%
8.8%
9.3%
1.00
0.75
Market
6.2%
8.2%
5.1%
7.6%
10.2%
1.00
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 1989 to September 2011 and January 1997 to
September 2011, 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 (13) and (14). The coefficients are in bold face
when significant on a 5%-level and in italics when significant on a 10%-level.
Index
S&P 500
1/N
Risk-Based Allocations
MV
RP
MDP
Market Capitalization
82,979
19,400
20,816
21,226
17,290
21,140
Dividend Yield in %
Price/Earnings
Price/Cash Flow
Price/Book
Price/Sales
17.90
15.49
10.19
3.12
1.36
17.74
14.49
8.22
2.28
0.85
17.46
15.41
9.44
2.76
1.15
12.59
14.44
8.56
2.42
0.91
18.25
15.53
9.08
2.67
0.98
18.51
15.41
8.01
2.41
1.03
Hist. 3Y Sales Growth
Hist. 3Y Earnings Growth
12.16
16.22
9.88
13.13
9.08
12.18
9.43
11.91
10.35
15.33
12.06
12.04
Return on Assets
Return on Equity
Operating Margin
Net Margin
8.38
19.54
17.83
10.43
6.42
14.34
14.39
7.43
8.54
18.30
16.68
9.01
7.35
16.27
15.64
8.37
7.98
16.76
15.72
7.96
7.23
14.41
16.43
7.57
Coefficients
Alpha
Market
Size
Value
Momentum
Volatility
Liquidity
0.09%
1.55
1.08
0.25
1.72
1.02
-0.21%
0.15
-0.65
1.25
-0.43
-0.03
-0.09
-0.18%
-0.05
-0.53
0.28
-0.13
-0.72
0.21
-0.26%
0.08
-0.59
0.93
-0.31
-0.51
-0.16
-0.27%
-0.08
-0.67
0.45
-0.10
-0.39
0.54
-0.22%
-0.19
-0.50
1.80
0.19
0.08
-0.56
t-statistics
Alpha
Market
Size
Value
Momentum
Volatility
Liquidity
0.39
5.75
3.42
1.78
17.24
2.49
-1.45
3.18
-3.55
6.15
-4.92
-0.33
-0.36
-0.97
-0.83
-2.26
1.08
-1.12
-5.47
0.62
-1.99
1.96
-3.63
5.17
-3.92
-5.56
-0.71
-1.36
-1.30
-2.70
1.63
-0.87
-2.82
1.52
-0.76
-2.00
-1.39
4.51
1.13
0.42
-1.10
Adjusted R2
Durbin-Watson
71.7%
1.99
47.2%
2.16
43.7%
2.02
42.1%
1.93
25.8%
1.95
17.0%
1.83
DRP
Panel A: Aggregate Firm Characteristics
Panel B: Time Series Regressions
24
Figure 1. 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 10 principal portfolios embedded in the underlying
500 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 1989 to September 2011.
Variance of Principal Portfolios in Percent
1
PP1
PP2
PP3
PP4
PP5
PP6
PP7
PP8
PP9
PP10
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
Boxplots of Explained Variance by Principal Portfolios
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1
2
3
4
5
25
6
7
8
9
10
Figure 2. Number of Principal Portfolios
The figure gives the number of principal portfolios over time as suggested by the P Cp1 and P Cp2 criteria
of Bai and Ng (2002). The results are ranging from October 1989 to September 2011.
# Factors: On average 4.77 (PCp1) or 4.62 (PCp2)
11
PCp1
10
PCp2
9
8
7
6
5
4
3
2
1
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
26
Figure 3. Portfolio Weights of Principal Portfolios
The figure gives the portfolio weights of the principal portfolios arising from a PCA over the period from
October 2006 to September 2011. Within each bi-plot the weights pairs are connected to the origin by
colored lines with the color corresponding to the respective stock’s GICS sector classification: Financials
are light green (81 stocks), Materials are red (31), Telecoms are magenta (8), Industrials are yellow (60),
Consumer Discretionary is dark blue (77), Consumer Staples is dark green (42), Information Technology
is light blue (74), Energy is greyish blue (42), Health Care is black (52), and Utilities is orange (33).
0.1
0.15
0.1
Principal Portfolio 4
Principal Portfolio 2
0
0.05
0
−0.1
−0.05
−0.1
−0.2
0
0.01
0.02
0.03
0.04
0.05
0.06
Principal Portfolio 1
0.07
0.08
0.09
0.1
−0.1
0.25
0.2
0.2
0.15
−0.05
0
0.05
Principal Portfolio 3
0.1
0.15
0.2
0.1
0.15
0.05
Principal Portfolio 8
Principal Portfolio 6
0.1
0.05
0
0
−0.05
−0.1
−0.05
−0.15
−0.1
−0.2
−0.15
−0.2
−0.25
−0.1
−0.05
0
0.05
0.1
Principal Portfolio 5
0.15
0.2
0.25
27
−0.2
−0.1
0
0.1
0.2
0.3
Principal Portfolio 7
0.4
0.5
0.6
0.7
Figure 4. Performance of Risk-Based Equity Strategies
The figure gives the cumulative total return of the risk-based equity strategies together with the S&P 500
over the sample period starting October 1989 to September 2011.
12
10
8
6
1/N
MV
RP
MDP
DRP
Market
4
2
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
28
Figure 5. Weights and Risk Decompositions: S&P 500
The figure gives the decomposition of the S&P 500 in terms of weights and risk. Risk is being decomposed
by asset classes and by principal portfolios, respectively. The sample period is from October 1989 to
September 2011.
S&P 500: Sector Weights
S&P 500: Volatility Contributions by Sector in %
1
Volatility Contributions by Principal Portfolios in %
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.9
0.8
0.7
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.9
0.8
0.7
0.9
0.8
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
94
95
96
97
98
99
00
01
02
03
04
05
06
07
08
09
10
0
PP1
PP2
PP3
PP4
PP5
PP6
PP7
PP8
0.1
97
98
99
00
01
02
03
29
04
05
06
07
08
09
10
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
Figure 6. Weights and Risk Decompositions: Risk-Based Equity Strategies
The figure gives the decomposition of the risk-based equity strategies in terms of weights and risk. Risk
is being decomposed by asset classes and by principal portfolios, respectively. The first row contains the
results for the 1/N-strategy, the second row is for minimum-variance, the third row for risk parity, the
fourth row for MDP and the last row for diversified risk parity. The sample period is from October 1989
to September 2011.
1/N: Sector Weights
1/N: Volatility Contributions by Sector in %
1
Volatility Contributions by Principal Portfolios in %
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.9
0.8
0.7
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.9
0.8
0.7
0.9
0.8
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
94
95
96
97
98
99
00
01
02
03
04
05
06
07
08
09
10
0
0.1
97
98
99
00
MV: Sector Weights
01
02
03
04
05
06
07
08
09
10
Volatility Contributions by Principal Portfolios in %
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.8
0.7
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
MV: Volatility Contributions by Sector in %
1
0.9
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.9
0.8
0.7
0.9
0.8
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
94
95
96
97
98
99
00
01
02
03
04
05
06
07
08
09
10
0
98
99
00
01
02
03
04
05
06
07
08
09
10
0.7
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
RP: Volatility Contributions by Sector in %
Volatility Contributions by Principal Portfolios in %
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.8
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.9
0.8
0.7
0.9
0.8
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
94
95
96
97
98
99
00
01
02
03
04
05
06
07
08
09
10
0
98
99
00
01
02
03
04
05
06
07
08
09
10
0.7
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
MDP: Volatility Contributions by Sector in %
Volatility Contributions by Principal Portfolios in %
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.8
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.9
0.8
0.7
0.9
0.8
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
94
95
96
97
98
99
00
01
02
03
04
05
06
07
08
09
10
0
98
99
00
01
02
03
04
05
06
07
08
09
10
0.7
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
DRP: Volatility Contributions by Sector in %
Volatility Contributions by Principal Portfolios in %
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.8
1
Cons. Discr.
Cons. Stapl.
Energy
Financials
Health Care
Industrials
IT
Materials
Telecom
Utilities
0.9
0.8
0.7
0.9
0.8
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
94
95
96
97
98
99
00
01
02
03
04
PP1
PP2
PP3
PP4
PP5
PP6
PP7
PP8
0.1
97
DRP: Sector Weights
1
0.9
PP1
PP2
PP3
PP4
PP5
PP6
PP7
PP8
0.1
97
MDP: Sector Weights
1
0.9
PP1
PP2
PP3
PP4
PP5
PP6
PP7
PP8
0.1
97
RP: Sector Weights
1
0.9
PP1
PP2
PP3
PP4
PP5
PP6
PP7
PP8
05
06
07
08
09
10
0
PP1
PP2
PP3
PP4
PP5
PP6
PP7
PP8
0.1
97
98
99
00
01
02
03
30
04
05
06
07
08
09
10
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
Figure 7. Number of Uncorrelated Bets
We plot the number of uncorrelated bets for the risk-based equity strategies for the sample period October
1989 to September 2011.
8
7
6
1/N
MV
RP
MDP
DRP
Market
5
4
3
2
1
0
89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10
31
Figure 8. Equity Factor Exposures over Time
The figure gives the risk-based strategies exposure to equity factors according to a rolling estimation of
the factor models given in (13) and (14). The estimation window encompasses 60 months and the factor
exposures can thus be computed from January 2002 to September 2011. The first row gives the results for
the S&P 500 and 1/N , the second row gives the results for minimum-variance and risk parity, the third
row gives the results for the MDP and diversified risk parity.
3
2.5
2.5
2
1.5
2
1
1.5
0.5
1
0
0.5
−0.5
0
−0.5
−1
02
Alpha
SP500
Size
Value
Mom
Vola
Liqui
03
−1
−1.5
04
05
06
07
08
09
10
−2
02
11
1.5
Alpha
SP500
Size
Value
Mom
Vola
Liqui
03
04
05
06
07
08
09
10
11
04
05
06
07
08
09
10
11
2
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1.5
−2
02
−1
Alpha
SP500
Size
Value
Mom
Vola
Liqui
03
−1.5
04
05
06
07
08
09
10
−2
02
11
4
Alpha
SP500
Size
Value
Mom
Vola
Liqui
03
4
3
3
2
2
1
1
0
0
−1
−1
−2
02
Alpha
SP500
Size
Value
Mom
Vola
Liqui
03
−2
04
05
06
07
08
09
10
11
−3
02
32
Alpha
SP500
Size
Value
Mom
Vola
Liqui
03
04
05
06
07
08
09
10
11