Risk Parity Strategies For
Equity Portfolio Management
Can an asset-class strategy translate to equities?
By Frank Siu
18
May / June 2014
R
isk-based strategies have gained popularity amid
market uncertainty, and many are now being
touted as “smart beta,” providing a systematic
way to outperform traditional capitalization-weighted
benchmarks. Here we examine the notion of risk parity,
taking what has almost exclusively been discussed in
an asset allocation context and applying its concepts to
equity-only portfolio construction.
Risk parity seeks to equalize sources of risk such that the
relative marginal contribution to risk (RMCTR) from each
source is equal. Historically, research has treated asset
classes as the sources of risk, because these are typically
quite distinct and relatively uncorrelated (for example,
interest rate versus equity market risk). Given an n-vector
of loadings on sources of risk W and their covariance
matrix Q, a risk parity portfolio satisfies:
In a traditional asset allocation setting where n represents the number of asset classes, n is “small” (<10),
and this problem is easy to solve, even using odd formulations and mediocre solvers. In equity portfolios, n is
usually larger, and the sources of risk may be individual
stocks, groupings of stocks (sectors, countries, regions
or combinations thereof) or quantitative risk factors. By
modeling risk parity as a set of optimization constraints,
we are able to find risk-parity portfolios efficiently for
large n. In addition, we will demonstrate how certain
formulations of risk parity can in fact be used as an overlay on top of an existing investment process. The following discussion is divided into three parts. We begin with
the simplest case, where the elements of w represent
individual stocks, so each stock constitutes a source
of risk (dubbed “asset risk parity” henceforth). Next,
the analysis will expand to include risk parity between
groupings of stocks (such as country/sector). Finally,
we introduce the idea of risk parity between risk factors.
Rather than presenting a series of backtests with the
goal of promoting risk-parity strategies and validating
their possible superior performance, we discuss methods for constructing risk-parity portfolios and analyze
how each variant of risk parity affects the resulting portfolio composition.
Constructing Asset Risk-Parity Portfolios
Despite its straightforward definition and prevalence
in research, there is little discussion—let alone consensus—on exactly how risk-parity portfolios are built.
Methods discussed include model simplification (e.g.,
ignoring correlations, per Anderson et al. (2012)), adhoc/trial-and-error approaches using rudimentary statistical software (Kazemi, 2012) and iterative numerical
algorithms (Chaves et al. 2012). Finding a portfolio that
satisfies Equation 1 is sometimes tackled by using a nonlinear optimizer to minimize a loss function measuring
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the portfolio’s deviation from risk parity, for example,
the sum of squared differences between stocks’ RMCTR,
as in Maillard et al. (2010):
In the equity world where n is typically large, contrary
to this conventional method, it is preferable to model risk
parity as optimization constraints. In the case of a longonly portfolio where w represents individual stock weights,
Equation 1 can be expressed as a set of convex constraints
on the RMCTR of each stock:
Here Li is a diagonal matrix with Lii =1 and zero elsewhere, and n represents the total number of stocks. Equation
3 can be formulated and solved as a convex optimization
problem. Additionally, when minimizing risk-parity violations using the objective function, per Equation 2, depending on the quality of the search algorithm, one may end up
with portfolios that are in fact not risk parity but have simply
exhausted that particular optimizer’s ability to find a better
solution. Finally, minimizing departure from risk parity is
dangerous because depending on the loss function, it may
be difficult to differentiate between large violations involving a few stocks and small violations across the board.
Kaya and Lee (2012) express risk parity as the solution
to a utility maximization problem and show that in a longonly and fully invested setting, there exists a unique set of
weights w that satisfies Equation 1. This result only applies
to asset risk parity in this restricted case; such a claim cannot
be made for risk parity between groupings of stocks or risk
factors, where the mathematics of finding a solution are far
more complex and are discussed later. Moreover, if additional constraints are imposed, the problem may sometimes
not have a solution at all. Nevertheless, establishing uniqueness is of paramount importance because it suggests that
asset risk parity fully defines a portfolio strategy. As a set of
portfolio constraints, asset risk-parity is akin to a weighting
scheme: It contains the necessary and sufficient information
for uniquely specifying the final portfolio composition.
Asset Risk Parity In ‘Small’ Blue Chip Universes
If uniqueness is established, the constraints in Equation
3 can be combined with any objective function to yield the
same solution. In the following examples, we use a “minimize variance” objective. Figure 1 shows the performance of
such an implementation of asset-risk parity, applied to various benchmark universes, with generally positive results.
Asset-risk parity seems well suited to blue chip benchmark
universes containing relatively small numbers of names,
where there is typically concentration risk that can be diversified. Of particular interest is the relatively manageable
May / June 2014
19
Figure 1
Performance Of Asset Risk Parity Strategies In Select Markets,
Quarterly Backtest, 2005-2013
EUR.STX.50
Eurozone
AEX
ATX
Netherlands
Austria
BEL
Belgium
CAC
France
DAX
IBEX
Germany
MIB
Spain
SMI
Italy
Switzerland
Total Return
6.88%8.22%5.82%5.76% 6.39%
10.38%6.52%2.91%8.56%
16.29%18.23%20.65%14.06% 18.32%18.00%19.11%19.29%13.97%
Realized Risk
Sharpe Ratio
Active Return 1.77%3.30%3.15%1.83% 1.94%1.50%1.07%3.00%1.49%
Specific
0.87%4.51%2.87%2.11% 1.51%0.87%2.48%3.70%1.17%
Factor
0.91%-1.21%0.28%-0.28% 0.43% 0.62%-1.41%-0.71% 0.32%
0.420.450.280.41 0.350.580.340.150.61
Tracking Error 2.17%5.48%7.67%6.18% 3.39%4.98%5.08%4.94%5.10%
Information Ratio0.820.600.410.30 0.570.300.210.610.29
Hit Rate
62.96%63.89%59.26%61.11% 57.41%67.59%56.48%58.33%61.11%
Average Beta
0.950.920.860.80 0.970.910.910.890.95
Average Names 50252020 4030354020
Average Turnover36.43%35.85%41.00%35.40% 28.05%29.98%35.37%33.33%29.50%
Sources: Axioma, Stoxx, Thomson Reuters
Note: Using monthly gross returns in EUR (CHF for SMI). Risk and return figures are annualized.
Figures 2a and 2b
Ex-Ante
Ex-AnteTracking
TrackingError
ErrorDecomposition
DecompositionFor
ForAsset
AssetRisk
RiskParity
ParityOn
OnBlue-Chip
Blue-ChipVs.
Vs.Broad
BroadMarket
MarketUniverses,
Universes,2005
2005-2013
To 2013
9%
8%
a)
Euro Stoxx 50
b)
9%
Euro Stoxx (~300 Names)
8%
7%
7%
6%
6%
5%
5%
4%
4%
3%
3%
2%
2%
1%
1%
0%
0%
2005
2006
2007
2008
2009
2010
2011
2012
2013 2005
2006
2007
2008
2009
20010
2011
2012
2013
■ Systematic (Market) ■ Size ■ Styles ex-Size ■ Country ■ Sector ■ Currency ■ Specific
Sources: Axioma, Stoxx
turnover (figures are annualized one-way) when compared
with other risk-based strategies. In most cases, risk parity achieves diversification by down-weighting mega-cap
names with very large weights, resulting in a slight low-beta
tilt; Kaya and Lee (2012) provide an analytical proof of why
this is so. Risk reduction compared with the capitalizationweighted benchmark is present but to a lesser extent than,
say, minimum-variance strategies, where aggressive lowvolatility positioning often results in high tracking error, a
form of “risk” some investors find difficult to accept.
Asset Risk Parity In Broad-Market Universes
Moving to broader benchmark universes, the optimization problem becomes more difficult, because both the
number of constraints and the dimensionality of each constraint increases. Furthermore, the risk profile of the strategy
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May / June 2014
becomes increasingly dependent on benchmark construction and market structure, due to the implicit requirement
that every stock be held. To illustrate, consider applying assetrisk parity to the Euro Stoxx 50 and the Euro Stoxx (about 300
names) universes. Their corresponding ex-ante tracking error
decompositions are presented in Figures 2a and 2b.
There are large differences between the risk profiles of
the two portfolios. In the full Euro Stoxx universe, roughly
two-thirds of names are mid- and small-caps. Maintaining
risk parity will systematically overweight many such names,
resulting in a large risk contribution from the small size
bias (the red area in Figure 2b), whose share of risk in the
Euro Stoxx 50 case was almost zero. Similarly, significant
underweighting of the largest blue chip names creates large
negative-beta and low-volatility tilts. Figure 3 shows the
relationship between the number of names in the universe
Figure 3
Effect Of Universe Breadth On Average Factor
And Specific Contributions To Tracking Error
1.00
6.00%
Tracking Error
5.00%
0.95
4.00%
0.90
3.00%
0.85
2.00%
0.80
1.00%
0.00%
0
50
100
150
200
250
300
0.75
350
Number Of Names
Factor
Specific
Beta (RHS)
Source: Axioma
Note: Quarterly backtest using pan-European universe, 2005 to 2013
and the resulting risk breakdown as well as portfolio beta.
To summarize, by increasing universe breadth, the drivers
of risk/return shift disproportionately toward systematic
risk factors rather than stock-level differences vis-à-vis the
benchmark as a result of “reshuffling” weights to comply
with risk parity. When the universe becomes very broad,
such as in Figure 2b, the portfolio risk profile approaches
that of a low-beta, small-cap strategy. Increasing universe
size may not necessarily detract from returns, but changes
the investment thesis considerably. Returns become “polluted” by factor effects; most notably, low beta and small
size dominate the strategy’s exposure profile, dwarfing the
effects from risk diversification. Figure 4 shows the risk and
performance of asset-risk parity applied to several large-cap
benchmark universes from the previous section compared
with their broad-market equivalent. The backtested results
are consistent with the trends illustrated in Figure 3—assetrisk parity in large universes is characterized by overweighting small-cap stocks and a substantial portion of return
arising from low-beta positioning. These two factor bets are
largely responsible for increased tracking error, which in
turn results in inferior risk-adjusted active returns.
Risk Parity Over Groupings Of Stocks
In the previous example of asset-risk parity on the
Euro Stoxx 50, risk was not being diversified across 50
independent sources; in fact, as Figures 5A and 5B show,
there is still considerable concentration along sector and
country lines. More generally, in any long-only equityonly portfolio, the broad market (beta) accounts for a
large portion of risk, and stocks in the same sector or
country remain highly correlated.
An alternative to asset-risk parity may be to equalize over systematic sources of risk, such as sectors or
countries. For global investment universes, the presence of some very small countries renders country risk
parity somewhat impractical, and sectors may not be
homogeneous across geographies; hence, one could consider region-sector groupings, such as Asian technology,
European banks, etc. In general, risk parity over groupings of stocks can be expressed mathematically by summing up Equation 3 over all stocks in a subset S:
Here, m represents the number of groupings S. Despite
the similarity of Equation 4 to the asset-risk parity case of
Figure 4
Performance Of Asset Risk Parity Strategies In Select Markets,
Quarterly Backtest, 2005-2013
Eurozone
EUR.STX.50 EUR.STX.
France
CAC
CAC-All
Germany
DAX
H+SDAX
Spain
IBEX
Switzerland
IGBM
SMI
SMI Expd
Total Return
6.88% 6.82% 6.39% 7.52%10.38%
11.04%6.52%1.82% 8.56%8.79%
Realized Risk
16.29% 16.61% 18.32% 16.83% 18.00%18.36%19.11%17.63% 13.97%15.47%
Sharpe Ratio
Active Return
1.77% 1.21% 1.94% 1.86% 1.50%1.76%1.07%-2.89% 1.49%1.61%
Specific
0.87% 2.34% 1.51% 2.93% 0.87%4.59%2.48%-0.21% 1.17%2.83%
0.42 0.41 0.35 0.45 0.580.600.340.10 0.610.57
Factor
0.91% -1.13% 0.43% -1.06% 0.62%-2.82%-1.41%-2.68% 0.32%-1.22%
Tracking Error
2.17% 3.96% 3.39% 7.25% 4.98%7.91%5.08%9.80% 5.10%6.89%
Information Ratio 0.82 0.31 0.57 0.26 0.300.220.21-0.29 0.290.23
Hit Rate
62.96% 62.04% 57.41% 64.81% 67.59%66.67%56.48%53.70% 61.11%65.74%
Average Beta
Average Names
0.95 0.87 0.97 0.70 0.910.810.910.74 0.950.90
50 311
40 260 30160 35118 20 50
Average Turnover 36.43% 39.83% 28.05% 52.84% 29.98%48.26%35.37%50.90% 29.50%35.89%
Sources: Axioma, Stoxx, Thomson Reuters
Note: Using monthly gross returns in EUR (CHF for SMI). Risk and return figures are annualized. *The CAC All-Tradable replaced the SBF 250 in early 2011.
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May / June 2014
21
Figures 5a and 5b
Risk Breakdown Of Euro Stoxx 50 Asset Risk Parity Portfolio, September 2013.
a) Risk Contribution
By Sector
■ Financials
■ Industrials
■ Consumer Goods
■ Utilities
■ Consumer Services
■ Oil & Gas
■ Telecommunications
■ Basic Materials
■ Health Care
b) Risk Contribution
By Country
■ France
■ Germany
■ Spain
■ Italy
■ Netherlands
■ Ireland
■ Belgium
■ Technology
Sources: Axioma, Stoxx
Equation 3, the problem is continuous nonconvex, therefore
significantly more difficult to solve. Optimality cannot be
guaranteed and neither can the existence of a unique solution. Solution quality and the mere ability to find feasible
solutions will depend heavily on the numerical techniques
employed by one’s solver. As such, the advantages of enforcing parity via optimization constraints rather than the objective function become apparent. (Note also that the number
of names is free to vary, no longer dictated by the breadth
of the universe.) Given such complexities in implementing
the optimization, some practitioners may opt to construct
portfolios by trial and error, perhaps starting with the capitalization-weighted benchmark and manually adjusting
allocations to the groupings in an ad hoc fashion until all
their risk contributions fall into the same “ballpark.” Chaves
et al. (2012) proposes numerical algorithms for computing
risk-parity portfolios, but it remains unclear how or whether
such methods can be adapted to groupings of stocks where
the number of names held can vary.
In the following examples, we solve the problem heuristically and are generally able to find solutions that satisfy, or
come very close to, risk parity. Occasionally, the optimizer
fails to find a solution, usually in small universes where some
groupings are sparsely populated, thereby not providing sufficient choices for diversification. Equally problematic are small
sectors or countries entering and leaving the universe, inducing additional turnover as well as instability in the relative marginal risk contribution share of the remaining groupings present. In many cases, reducing the right-hand side of Equation 4
by a tiny e is sufficient to yield a solution, though one or more
groupings may be marginally out of parity. Having an ample
pool of stocks to choose from is thus helpful in ensuring a solution. In contrast to asset risk parity, which is best suited to concentrated universes with few names, risk parity of groupings of
stocks is more appropriate for broad universes.
Multiple Solutions And The Choice Of Objective Function
By modeling and enforcing risk parity via optimization
constraints, we “free up” the objective function to pursue
other investment goals such as minimizing risk, turnover/
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May / June 2014
trading or deviation from a benchmark index. This is especially important in risk parity over groupings where solution
uniqueness cannot be established; in these cases, minimizing violations of risk parity between groupings alone is not
sufficient for producing an attractive investment portfolio.
Consider the trivial example where we select one stock at
random from each sector and apply asset risk parity to the
portfolio—this solution satisfies sector risk parity but is
unlikely to have any real-world appeal.
Figure 6 contains several examples of combining risk
parity of groupings with other portfolio construction rules,
again using the Euro Stoxx universe. By changing the optimization objective and/or applying additional constraints,
we are able to obtain significantly different performance. In
particular, note the differences in risk, beta, name count and
turnover. While other risk-based strategies are often accompanied by large deviations from the capitalization-weighted
benchmark, here it appears that sector risk parity can be
achieved with a surprisingly small tracking error budget
(about 2 percent). We leave it to readers to experiment with
additional permutations; for example, constraining risk factor exposures, imposing holdings constraints or even incorporating an expected returns (alpha) objective.
In Bai et al. (2013), risk parity over sector groupings is solved
the conventional way—by minimizing an objective function
encapsulating total violations of risk parity—but it is unclear
what properties the optimal portfolio satisfies, apart from
exhibiting sector risk parity. For example, all three portfolios in
Figure 6 are optimal under their formulation, but the user has
no control over which variant is returned as the solution.
In short, risk parity over groupings of stocks is a complex
problem and requires careful research and calibration of
optimization parameters to the particular investment goal
and universe. Figure 7 shows two different solutions to sector risk parity; both allocate risk equally across sectors but
pursue radically different investment objectives—Figure
7a uses sector risk parity in conjunction with a minimum
variance strategy, and Figure 7b applies sector risk parity
to a passive indexing strategy. We reiterate and summarize
thusly: Asset risk parity has a unique solution and fully
Figure 6
Performance Of Sector Risk Parity Strategies On The Euro
Stoxx Universe, Quarterly Backtest, 2002-2013
Benchmark Sector RP
Min Risk
Sector RP
Min Risk
Target 30%
Turnover
Sector RP
Min Tracking
Error
Target 30%
Turnover
Total Return 3.57% 7.77% 8.39%3.74%
Realized Risk 18.69% 10.83% 10.93%18.70%
Sharpe Ratio 0.19 0.72 0.770.20
X is a matrix of factor exposures, Ω is the factor covariance matrix and Δ is the diagonal matrix of stock-specific
variances. Let f = XTw be the factor exposures of a portfolio
w. Suppose the exposure matrix X is made up of a set of
country, industry and style risk factors:
The factor covariance matrix can be divided into blocks
corresponding to factor types/groups:
Active Return 4.20% 4.82%0.17%
Specific 1.04% 2.11%-0.22%
Factor 3.17% 2.71%0.39%
Tracking Error12.43% 11.13% 2.05%
Information Ratio 0.34 0.430.08
Hit Rate 69.44% 68.06%59.72%
Average Beta 0.51 0.561.00
We can enforce risk parity across the set of country risk
factors by:
Average Names 41 52172
Average Turnover222.08% 42.98% 32.82%
Sources: Axioma, Stoxx
Note: Using monthly gross returns in EUR. Risk and return figures are annualized.
defines a portfolio strategy; risk parity over groupings of
stocks, on the other hand, is a risk control component of a
larger overall strategy. Previously, researchers may not have
had adequately flexible tools to separate risk-parity requirements from the optimization objective; hence, this notion of
using risk parity as an “overlay” has largely gone unexplored.
Factor Risk Parity
Let us begin with a factor risk model that decomposes risk
(the asset-asset covariance matrix Q) into common factor
and stock-specific components:
Like Equation 3, this can be formulated as a set of
convex constraints on relative marginal contribution
to risk. A similar condition can be used to implement
industry-factor risk parity. Other factors, such as those
of Fama-French-Carhart (Roncalli and Weisang (2012))
or principal components (Lohre et al. (2012)) are also
possible, but researchers have noted difficulties with
the optimization arising from the fact that exposures to
these factors can take on negative values.
Parity across country factors is different from parity
across country groupings, described previously, and
likewise for industries/sectors. Equation 6 should be
interpreted as “ensure the contribution to country risk
from each country factor is equal.” Measuring country
risk (the denominator in Equation 6) requires under-
Figures 7a and 7b
Time Series Sector Contributions To Total Risk For Two Sector Risk Parity Strategies
40%
a) Minimize Total Risk
35%
40%
b) Minimize Tracking Error
35%
30%
30%
25%
25%
20%
20%
15%
15%
10%
10%
5%
5%
0%
0%
2001
2003
2005
2007
2009
2011
2013
2001
2003
2005
2007
2009
2011
2013
■ Basic Materials ■ Consumer Goods ■ Consumer Services ■ Financials ■ Health Care ■ Industrials ■ Oil & Gas
■ Technology ■ Telecommunications ■ Utilities
Euro Stoxx Risk
Sources: Axioma, Stoxx
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May / June 2014
23
Figure 8
Country Weights And Risk Contributions Of Country Factor Risk Parity Strategies On Euro Stoxx Universe
Benchmark
Weight
Factor RP (Pure)
Weight
Factor RP (w/ Mkt)
% of Risk
Weight
Country RP (groupings)
% of Risk
Weight
% of Risk
Austria
0.86%
7.78%7.30% 8.80%8.19% 9.89%9.09%
Belgium
3.64%
10.27%8.49% 9.31%7.63%11.17%9.09%
Finland
France
Germany
Greece
0.27%
2.82%2.70% 8.21%9.97% 9.12%9.09%
Ireland
1.71%
8.39%6.48%10.73%8.05% 9.60%9.09%
Italy
7.71%
6.31%8.88%8.97%
11.78%8.55%9.09%
3.24%
11.20%12.22% 8.47% 9.36% 10.73% 9.09%
33.93%
13.98%
15.07%8.85%9.10%8.08%9.09%
28.84%
17.38%17.59% 10.31%10.71% 9.28% 9.09%
Netherlands
8.72%
12.43%
11.53%9.76%8.96%9.55%9.09%
Portugal
0.67%
3.98%3.54% 8.39%7.71% 7.60%9.09%
Spain
10.41%
5.47%6.20% 8.21%8.54% 6.42%9.09%
Tracking Error
1.96%3.60% 9.25%
Sources: Axioma, Stoxx
standing how the risk model attributes aggregate portfolio risk to individual risk factors.
Importance Of Risk Model Specification
Although a comprehensive discussion of empirical
asset pricing and factor risk modeling is beyond the
scope of this analysis, we will provide a simple example
to illustrate how the portfolio construction implications
of Equation 6 depend on nuances pertaining to factor
risk model construction. Figure 8 shows country weights
and risk contributions from three strategies: The first two
use factor risk parity with country factors, and the last
one uses risk parity with country groupings.
Furthermore, the first factor risk parity example uses a factor risk model with “pure” country factors, while the second
takes the same risk model but embeds broad-market effects
into the country factors by means of a linear transformation.
There are large differences in country allocations
between the two factor risk-parity strategies even though
the risk models used differ only in how they decompose
risk by factors. Figure 9 explains this result. France has
the lowest “pure” factor volatility; much of its country
risk coincides with that of the broad European market.
In contrast, Greece’s high risk is largely country-specific,
distinct from other core European markets. As a result, if
one were to only consider “pure” country factors, Greece
would appear almost seven times as risky as France, thus
receiving significantly less weight in the first factor riskparity portfolio. In the second factor risk-parity variant,
correlations with the broad market are taken into account,
and Greece becomes only twice as risky as France, resulting in less extreme weight differences. The interpretation
of risk model factors, therefore, is of vital importance to
structure of the factor risk-parity portfolio.
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May / June 2014
The first factor risk-parity portfolio in Figure 8 may
appear attractive because of its similarity to benchmark
country allocations, but it is questionable whether risk
is being diversified in an intuitive or desired manner.
Mathematically, Equation 6 is being satisfied, but only
in the context of “pure” factors. The second factor riskparity strategy provides a more egalitarian risk breakdown, but still leaves some puzzling results unresolved.
For example, Belgium has one of the highest weights
but the lowest risk contribution. Per Figure 9, Belgium
has the lowest average levels of stock-specific risk.
Figure 9
Characteristics Of Country Factors
From The Two Risk Models Used In Figure 8
Factor Volatility
Pure
w/Mkt
Corr. With
Mkt Factor
Avg
Specific
Risk
Austria
6.39%15.07% 0.16 21.64%
Belgium
4.73%13.96% 0.10 15.80%
Finland
6.78%16.02% 0.29 19.90%
France
3.38%14.32% 0.38 17.44%
Germany
3.80%12.63% -0.17 19.04%
Greece
25.91%27.88% -0.08 30.85%
Ireland
10.01%14.79% -0.17 23.94%
Italy
8.35%15.43% 0.03 22.24%
Netherlands 4.44%13.38% -0.01 17.56%
Portugal
12.35%17.93% 0.03 21.86%
Spain
9.26%16.80% 0.15 19.41%
Market
12.69%12.69% 1.00 19.14%
Source: Axioma
Equation 6 will only equalize contributions to country
factor risk and does not take stock-specific risk into
account. Compared with Belgium, other countries claim
a larger share of risk owing to their higher average levels
of stock-specific risk, despite country factor risk parity
having been met. Finally, the third risk-parity portfolio
has equal sharing of portfolio risk across countries but
involves some very aggressive active country bets.
Additional Interpretations
Recall that Equation 6 seeks to diversify country risk
. It is not controlling, contrary to what many
may expect, the contribution of country factors to total portfolio risk. Such an interpretation of factor risk parity can be
accomplished by:
(7)
Risk parity at the factor level allows for a very granular
level of risk control but may be too flexible for many users.
Without a full understanding of the factor risk model construction process, factor risk parity may yield portfolios that
are ultimately inconsistent with the user’s investment needs.
Positioning Versus Other Risk-Based Strategies
Risk parity is often compared with risk-based strategies
such as minimum variance or maximum diversification.
The first-order condition of a minimum-variance problem
requires that the marginal contribution to risk (MCTR) of
each stock be equal:
Risk parity, on the other hand, equalizes relative MCTR.
Maximum diversification and risk parity are both driven by
the concept of diversification. Risk parity spreads out risk
across its various sources; maximum diversification aims to
reduce the share of portfolio risk coming from correlations:
The right-hand size of Equation 9 is defined as the “diversification ratio,” and its numerator is equivalent to the risk of a portfolio where all stocks have a correlation of 1. Mathematically,
Equation 9 is also analogous to maximizing the Sharpe ratio,
where each stock’s expected return is equal to its volatility.
Real-world application of risk-based strategies is often
accompanied by investment constraints such as trading
limits and restrictions on stock weights or risk exposures,
because the unconstrained solution may not be implementable. For example, low-volatility themed strategies
typically load heavily on components with low risk and/or
low correlation with the market, resulting in concentrated
weights, high turnover and sometimes poor liquidity. The
performance of these strategies will be very sensitive to
choice of constraints and their associated parameters.
Conclusion
We have presented an optimization-based approach to
finding risk-parity portfolios that models risk parity as a
set of optimization constraints. Compared with the various numerical or analytical methods sometimes discussed
in the literature, this approach is scalable, efficient and
allows us to analyze different applications of risk parity.
This discussion began by showcasing asset risk parity
because, for a given universe of stocks, there is a unique longonly, fully invested portfolio that exhibits asset risk parity. It
can be used as an “off the shelf” strategy with no fine-tuning.
Unlike many other risk-based strategies, risk parity, by construction, embraces diversification and avoids concentration,
offering a high degree of built-in risk and turnover control.
Simulated backtests in various markets suggest that asset
risk-parity strategies can outperform their corresponding
capitalization-weighted benchmarks with reasonable levels
of tracking error and trading. The diversification benefits and
superior returns make asset risk parity ideal for concentrated
benchmark universes comprising relatively few stocks.
Other forms of risk parity can loosely be labeled as principles of portfolio construction and do not alone define a set
of portfolio weights. Their applicability, therefore, depends
on the other components of the investment strategy. Rather
than serving as a risk-based strategy, risk parity over groupings of stocks and factor risk parity are best applied as a form
of risk control to augment an existing investment process,
preferably on a sufficiently broad universe of stocks.
References
Anderson, Robert M., Stephen W. Bianchi and Lisa R. Goldberg, “Will My Risk Parity Strategy Outperform?” Financial Analysts Journal, 68(6):75-93, 2012.
Asness, Clifford, Andrea Frazzini and Lasse Heje Pedersen, “Leverage Aversion and Risk Parity,” Financial Analysts Journal, 68(1):47-59, 2012.
Bai, Xi, Katya Scheinberg and Reha Tutuncu, “Least-squares approach to risk parity in portfolio selection,” http://www.optimization-online.org/DB_FILE/2013/10/4089.pdf, 2013.
Chaves, Denis, Jason Hsu, Feifei Li and Omid Shakernia, “Efficient Algorithms for Computing Risk Parity Portfolio Weights,” Journal of Investing, pages 21(3): 150-163, 2012.
Clarke, Roger, Harindra de Silva and Steven Thorley, “Risk Parity, Maximum Diversification, and Minimum Variance: An Analytic Perspective. Journal of Portfolio
Management, 39(3):39-53, 2013.
Kaya, Hakan and Wai Lee, “Demystifying Risk Parity,” Neuberger Berman white paper, 2012.
Kazemi, Hossein, “An Introduction to Risk Parity,” Alternative Investment Analyst Review, 1, 2012.
Lohre, Harald, Ulrich Neugebauer and Carsten Zimmer, “Diversified Risk Parity Strategies for Equity Portfolio Selection,” Journal of Investing, 21(3): 111-128, 2012.
Maillard, Sébastien, Thierry Roncalli and Jérôme Teiletche, “On the properties of equally-weighted risk contribution portfolios,” Journal of Portfolio Management, 36(4):
60-70, 2010.
Roncalli, Thierry and Guillaume Weisang, “Risk Parity Portfolios with Risk Factors,” working paper, 2012.
Stubbs, Robert A., “Risk Parity: Applying the Concept to Equity Strategies,” http://updatefrom.com/axioma/2012_q1/newsletter.asp, March 2012.
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