Demystifying Risk ParityHakan Kaya, Ph.DVice President
Quantitative Investment Group Neuberger Berman
INTRODUCTION
Wai Lee, Ph.D.Managing Director Chief Investment Officer and
Director of Research Quantitative Investment Group Neuberger
Berman
The majority of risk parity analysis is treated as a heuristic
process and compares the backtests of different allocation methods
with less of an emphasis on investment rationale. We investigate
risk parity under different settings, highlight its potential
utility, and provide insight into when this method may be expected
to outperform by conducting path-independent controlled simulation
experiments. Following an extended period of market turbulence,
macroeconomic dislocations and increased cross-asset class
correlations, the investment community is increasingly looking
beyond traditional one size fits all asset allocation strategies to
find solutions that may be effective in an increasingly variable
environment. The past few years have undoubtedly been challenging
for investors. From increased market volatility to historical trend
deviations to myriad macro-level events that have impacted market
and asset price behavior in often extreme ways, investors have been
left to question long-held assumptions underlying various asset
allocation methodologies as well as their own approaches. The
environment has shifted to one in which constructing multi-asset
class portfolios that can deliver on investment objectives over a
period of several years seems infinitely more complicated than just
a few years ago. In fact, the third quarter of 2011 punctuated this
perhaps most distinctly, as many markets posted their worst quarter
since early in the financial crisis, only to be followed by a month
that was one of the best for certain equity markets since the
1970s. How should investors handle these extremes and how can they
effectively build portfolios to weather such storms and changing
conditions? These questions highlight many of the assumptions that
led investors to more traditional asset allocation methodologies in
the first place. For institutions, for example, typical plan
restrictions might prohibit the use of leverage or shorting of
securitiesyet, at the same time, have a required return, which has
led long-term allocations to relatively risky assets, such as
equities. Over time, the portfolio mix of 60% equities and 40%
fixed income, or slight variations thereof, emerged as typical
because it was thought to have a good chance of meeting the
required return. This was accepted despite the known concentration
in equity risk resulting from the mismatch between equity and fixed
income risks in such a portfolio. More recently, the investment
communitys focus has shifted to developing alternative approaches
to asset allocation and multi-asset class portfolios. As part of
this trend, both risk budgeting and diversification are being
re-evaluated. On risk budgeting, Bender, Briand, Nielsen, and
Stefek (2010), and more recently Page and Taborsky (2011), promoted
the so-called risk class, or factor-based approach instead of risk
modeling and budgeting based on assets. Others, including Kaya,
Lee, and Wan (2011) and Kowara
March 2012
NOT FOR RETAIL CLIENT USE IN EUROPE
1
DEMYSTIFYING RISK PARITY
and Idzorek (2011), however, pointed out that while the risk
class approach may offer complementary insights to the asset class
approach, no magic really exists behind such a new approach.
Regardless of approach, risk budgeting remains just as challenging
as it was before. Diversification is one of the most used terms in
investing, but its meaning and application is not always clear (see
discussions in Meucci (2009), Levell (2010), and Lee (2011) on the
ill-defined nature of diversification). We agree with Markowitzs
(1952) Modern Portfolio Theory (MPT) that investors should
diversify to the degree they are uncertain. In other words, if
investors had perfect forecasting ability, diversification would be
unnecessary. Once an investment objective such as a preference of
return with aversion to risk as proxied by volatility is clearly
stated, Lee (2011) argues that the efficient portfolio (as
constructed according to MPT) is the best portfolio in that it is
expected to maximize return and diversify to the appropriate degree
given the investors perceived degree of uncertainty of the future.
To be clear, diversification does not necessarily mean that one
should hold a portfolio that is expected to have the minimum
volatility. Scherer (2010, p.3) questions whether the minimization
of risk on its own in the spirit of minimum volatility, and also
conceptually in relation to maximizing diversification, is a
meaningful objective. A number of studies have examined the
empirical properties of some alternative asset allocation
approaches, collectively categorized as risk-based asset allocation
portfolios (see Lee (2011)), that do not require explicit forecasts
of expected returns. Instead, these portfolios can be constructed
solely based on forecasts of risks, which are typically represented
by a covariance matrix. In the literature, these risk-based
approaches have been applied to stock selection in different
regions during different sample periods, with encouraging simulated
outperformance of such portfolios relative to the market
capitalization-weighted portfolios. Historical simulations,
however, have to be interpreted with caution, particularly for the
portfolios requiring ad hoc constraints in order to keep the
portfolios well-behaved with acceptable turnovers and less extreme
position concentration (see Demey, Maillard, and Roncalli (2010)
and Lee (2011) for more discussions). By revealing the lack of a
clearly defined investment objective behind these portfolios, Lee
(2011) advocates shifting the focus of studies on these portfolios
from being largely heuristic, intuitive, and empirical, to more
formal, conceptual, and theoretical. Among these risk-based
approaches, the Risk Parity (RP) portfolio plays a unique role and
has moved into the spotlight. While risk parity is most often
implemented at the asset class level to construct a global risk
balanced portfolio, applications for individual securities within a
particular asset class have also gained momentum. Demey, Maillard,
and Roncalli (2010) examine the application of the risk parity
approach to individual stocks in different regions and find that,
compared with other alternative risk-based approaches, turnover
requirements of the RP portfolio are far more feasible, with well
balanced portfolio positions and risks. Therefore, ad hoc
constraints on positions and risks are not required, unlike with
other approaches. Risk parity also seems to have generated the most
interest in recent literature (see for example, Thiagarajan and
Schachter (2011), Inker (2011), Ruban and Melas (2011), Chaves,
Hsu, Li, and Shakernia (2011), Qian (2011), Peters (2011), and
Bhansali (2011), among others).
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DEMYSTIFYING RISK PARITY
To date, justification of risk parity remains conceptual and
intuitive, rather than theoretical, and therefore, risk parity is
largely considered a heuristic asset allocation approach with
intuitive appeal (Thiagarajan and Schachter, 2011, p.80). Recently,
Asness, Frazzini, and Pedersen (2012) explored the empirical
performance of risk parity as driven by leverage aversion. The goal
of this paper was to further advance our understanding and
appreciation of the RP portfolio through conceptual and theoretical
developments. We structured the paper as follows: Building on Lees
(2011) point on the importance of a clearly defined investment
objective, we put risk parity in the context of Mean-Variance
Optimality as a natural starting point. We give conditions of
efficiency and study the properties of the RP portfolio in a one
factor world in which the covariances are modeled by a market
factor and idiosyncratic risks. We compare the input sensitivity
and, hence, the turnover characteristics of risk parity, and next,
we analyze its potential utility function and interpret it in a
Bayesian sense to shed some light on the investment rationale.
Finally, we report path-independent simulation results, and
conclude that risk parity may be a preferable method in a regime in
which input parameters are very noisy and returns are
fat-tailed.Risk Parity as a Special Case of Mean-Variance
Optimization
In the absence of a clearly defined objective, we follow Lee
(2011) in using Mean-Variance Optimality as our metric in an
attempt to provide additional context in understanding the risk
parity approach. Using simple portfolio mathematics, we derive the
underlying conditional expectations of the investment opportunity
set that are expected to ensure the optimality of the RP portfolio.
In the appendix, we demonstrate that a RP portfolio is
mean-variance efficient when: 1. Sharpe ratios of all assets are
identical, and 2. Correlations among assets are the same.1 A number
of studies have examined and discussed the historical performance
of a risk parity approach to asset allocation among some asset
classes, mostly relying on backtests in the last two decades.
Examples of these studies include Allen (2010), Levell (2010), and
Chaves, Hsu, Li, and Shakernia (2011). Below, we take a different
approach. Having established the conditions for the optimality of
risk parity, we may interpret the performance of risk parity
relative to other alternatives in historical samples as an
indication that the identical Sharpe ratios and same correlations
among all assets might have been a better or worse proxy of the
realized investment opportunity set than those alternatives during
the sample periods of interests. To further illustrate the point,
we proceed to examine the realized performance characteristics of
some widely followed asset classes in the U.S., including
large-capitalization stocks, small-capitalization stocks, long-term
corporate bonds, and long-term government bonds. We use monthly
returns from the Ibbotson dataset that goes back to 1926. Table 1
summarizes the historical returns, volatilities, and
correlations.
1
An informal proof of the same results can be found in Maillard,
Thierry and Teiletche (2010).
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DEMYSTIFYING RISK PARITY
TA B L E 1 : L O N G -T E R M R I S K / R E T U R N P R O F I L
E O F U N D E R LY I N G A SS E T SLONG-TERM CORPORATE BONDS 2.65
7.35 0.21 0.16 LONG-TERM GOVERNMENT BONDS 2.32 8.04 0.12 0.06
0.84
Annualized Mean (%) Annualized Standard Deviation (%)
Correlations S&P 500 Index Small-Cap Long-Term Corporate Bonds
Long-Term Government Bonds
S&P 500 Index 7.67 19.23
SMALL-CAP 11.79 29.14 0.83
Source: Ibbotson. Note: Estimates are based on monthly total
return data from 19262010.
In our empirical exercise, we attempt to understand the
performance of risk parity by gauging how closely its optimality
conditions might have proxied the realized return and risk
characteristics of assets. To start, Figure 1 plots rolling 10-year
Sharpe ratios. We see that the Sharpe ratios of these assets were
indeed close to each other at times, such as in the mid 70s and
late 90s. However, before the 70s, they were quite different. While
the condition of identical Sharpe ratios was clearly violated for
shorter periods of time over this long sample time period, the
median of 10-year Sharpe ratios of these assets as plotted in
Figure 2 were remarkably similar. Similarly, Figure 3 and Figure 4,
which plot the 10-year rolling correlations among these assets and
their medians in the whole sample, respectively, suggest that the
condition of same correlation was violated, especially between
asset class pairs. Evidently, observed Sharpe ratios and
correlations have not been the same all the time and therefore,
theoretically, other more efficient portfolios had to exist. While
seeking efficient portfolios ex-ante is generally the objective, it
is still interesting to examine the extent to which such violations
of the conditions of RP portfolio optimality may lead to the loss
of portfolio efficiency. The following analysis focuses on the
violation of same correlation conditions only. Similar analysis can
be done on the violation of identical Sharpe ratios.F IG U R E 1 :
ROLLIN G 10-Y EA R SH A RPE RA TIOS2.5 2.0 1.5 1.0 0.5 0.0 -0.5
-1.0 Dec-35
Dec-43
Dec-51
Dec-59
Dec-67 LT Gov
Dec-75
Dec-83
Dec-91
Dec-99
Dec-07
S&P 500 Index Source: Ibbotson.
LT Corp
Small-Cap
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DEMYSTIFYING RISK PARITY
F IG U R E 2 : MED IA N OF ROLLIN G 10-Y EA R SH A RPE RA TIOS
(1936 2011)0.500 0.450 0.400 0.350 0.300 0.250 0.200 0.150 0.100
0.050 0.000 S&P 500 Index Source: Ibbotson. Small-Cap Long-Term
Corporate Long-Term Government 0.41 0.43 0.39 0.37
To quantify the potential loss of efficiency, we follow the cash
equivalent comparison methodology as described in Chopra and Ziemba
(1993). First, a mean variance efficient portfolio of the four
assets is constructed based on the whole sample statistics as
reported in Table 1 and a risk tolerance of 50 as suggested by
these authors. The cash equivalent value of this portfolio is
estimated to be 0.26% per month. Next, we construct two other
efficient portfolios based on the same inputs of returns,
volatilities, and risk tolerance, but one portfolio with the same
correlation of all assets equal to the average correlation during
the whole sample (0.37 to be specific), and one portfolio with
correlations of all assets at zero. The cash equivalent measures of
these portfolios were estimated to be 0.24% and 0.19% per month,
respectively, corresponding to a 6% and 25% loss in cash equivalent
terms versus the first efficient portfolio constructed with the
true ex-post correlations. Figure 5 demonstrates how these losses
change when the risk tolerance parameter varies. In the case of
non-zero average correlations, the loss in cash equivalent due to
the violation of the same correlation assumption is found to
increase as the investor becomes more and more risk seeking. In the
case of the zero correlation assumption, however, the loss in cash
equivalent increases with risk aversion instead. With a zero
correlation assumption, small-cap stocks become more attractive and
their weight in the efficient portfolio is higher. As the investor
becomes more risk averse, the higher weight in small-cap stocks
causes more loss in utility as measured by the cash equivalent.
Therefore, averaging correlations is close to efficiency when the
investor is becoming more risk averse. However, the assumption of
zero correlation among assets can result in unintended risk
concentration in riskier assets and hence reduce efficiency. To
summarize, identical Sharpe ratios and constant correlations when
measured with shorter sample periods were certainly violated.
However, judging from the relatively small loss of cash equivalents
relative to the true efficient portfolio with perfect hindsight,
the optimality conditions of the RP portfolio appears to be a
reasonable starting point in achieving portfolio efficiency
ex-ante.5
DEMYSTIFYING RISK PARITY
F IG U R E 3 : ROLLIN G 10-Y EA R CORRELA TION S1.20 1.00 0.80
0.60 0.40 0.20 0.00 -0.20 -0.40 Dec-35 Dec-43 Dec-51 Dec-59 Dec-67
Dec-75 Dec-83 Dec-91 Dec-99 Dec-07 S&P 500-Small
Small-CorpSource: Ibbotson.
S&P 500-Corp Small-Govt
S&P 500-Govt Corp-Govt
F IG U R E 4 : M ED IA N OF ROLLIN G 10-Y EA R CORRELA TION
S0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 S&P
500-Small S&P 500-Corp S&P 500-Govt Source: Ibbotson.
Small-Corp Small-Govt Corp-Govt 0.28 0.22 0.17 0.12 0.82 0.76
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DEMYSTIFYING RISK PARITY
F IG U R E 5 : P ERCEN TA G E CA SH EQU IV A LEN T LOSS A S A FU
N CTION O F R IS K T OL E R A N CE FOR A CON STA N T CORRELA TION
PORTFOLIO40% 35% 30% 25% 20% 15% 10% 5% 0% 25 30 35 40 45 50 55 60
65 70 75 Risk Tolerance () Average Correlation Zero Correlation
Source: Neuberger Berman Quantitative Investment Group. Note:
The two lines shown correspond to cases when the constant
correlation is equal to the average of all pairwise correlations or
zero.
Risk Parity in a Single-Factor World
In an attempt to shed light on the properties and apparent
outperformance of the Minimum Variance (MV) portfolio relative to
the market capitalization-weighted portfolio, Scherer (2010) and
Clarke, de Silva, and Thorley (2011) report that the MV portfolio
implicitly loads up on low beta and low idiosyncratic risk assets,
and therefore captures these pricing anomalies, as documented in
Jensen, Black, Scholes (1972) and Ang, Hodrick, Xing, and Zhang
(2009), that might have existed in the sample period. Our goal in
this section is to take a similar path to further understand the
portfolio compositions of a RP portfolio. In the Appendix, we
demonstrate that a RP portfolio, similar to an MV portfolio, is by
construction, biased towards low beta, low idiosyncratic assets
and, therefore, is able to capture these anomalies as well. Here we
use a simple example to demonstrate the beta and idiosyncratic risk
properties of RP portfolios and use minimum variance for comparison
(see Figure 6 for our results). Due to its additional dependence on
expected return parameters, the mean variance method is skipped in
these comparisons. In the case of idiosyncratic risks, we consider
a universe of 50 assets, each with a beta of 0.25, but with the
idiosyncratic risk of assets monotonically increasing from 1% to
30%. Without loss of generality, we also assume a 12% market risk.
Using these statistics, we create a covariance matrix from the
factor model to be used as an input to the MV and RP portfolios.
Figure 6a exhibits the weight and risk allocations to these assets
as a
Cash Equivalent Loss
7
DEMYSTIFYING RISK PARITY
function of their idiosyncratic risks. Needless to say, both
risk-driven models prefer low idiosyncratic risk assets when beta
is kept constant. Another eye-catching pattern is the fact that
minimum variance loads on a few low idiosyncratic assets while risk
parity is less concentrated. (Also noted by Clarke, de Silva and
Thorley (2011)). Second, in the case of varying beta, we again
consider a universe of 50 assets, each with an idiosyncratic risk
of 25%, but now with the betas of assets monotonically increasing
from -2 to 5. Again, assuming a 12% market risk and using these
inputs, we create a covariance matrix from the factor model used in
optimizing MV and RP portfolios. From Figure 6b, we observe that
both portfolios like low beta assets. Although the betas range in
between extreme thresholds from -2 to 5, the difference in weights
is not as dramatic as the case in varying idiosyncratic risks. One
interesting fact is that as beta increases, MV is likely to short
high beta assets while RP still stays long-only.F IG U R E 6 A : IM
PA CT OF ID IOSY N CRA TIC RISK ON RISK -D RIV EN P OR T F OL
IOSWeights of Risk Parity vs Minimum Variance Impact of
Idiosyncratic Risk
50 40 Weight 30 20 Weight 10 0 0 10 20 AssetPCTR of Risk Parity
vs Minimum Variance Impact of Idiosyncratic Risk
50
Weights as a Function of Idiosyncratic Risk
40
30
30
40
50
20
50 40
Weight
30 20 10 0 0 10 20 Asset Risk Parity Minimum Variance 30 40
50
10
0
0
5
10
15
20
25
30
Asset Idiosyncratic Risk Risk Parity Minimum Variance
Source: Neuberger Berman Quantitative Investment Group. Note:
PCTR: Percentage contribution to portfolio risk.
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DEMYSTIFYING RISK PARITY
F IG U R E 6 B : IM PA CT OF M A RK ET BETA ON RISK -D RIV EN
PORTFO LI O SWeights of Risk Parity vs Minimum Variance Impact of
Weights as a Function of
6 5Weight
6 5 4 3 0 5 10 15 20 25Asset PCTR of Risk Parity vs Minimum
Variance Impact of
4 3 2 1 0 -1 30 35 40 45 50Weight
2 1 0 -1
5 4Weight
3 2 1 0 -1 0 5 10 15 20 25 30 35 40 45 50Asset
-2
-1
0
1 2 Asset
3
4
5
Risk Parity
Minimum Variance
Risk Parity
Minimum Variance
Source: Neuberger Berman Quantitative Investment Group. Note:
PCTR: Percentage contribution to portfolio risk.
Sensitivities and Robustness of Portfolio Weights
Having explored the efficiency and factor properties of RP
portfolios, we switch our attention to the topic of portfolio
sensitivity to changes in input parameters, an important subject
for a variety of reasons. To cite a few, first, the majority of
applications of risk parity, especially a RP portfolio of global
asset classes, typically require leverage in order to deliver
reasonable required returns. In some cases, over-the-counter
derivatives with lower liquidity are included. Therefore, it would
be an undesirable property if small changes in risk estimates could
potentially result in high turnover statistics. Second, we are
supplying the optimizers only with estimates, with errors, of the
true parameters. While our estimates are at best in the
neighborhood of the unknown true parameters, it would be a positive
if the allocation method remained consistent around these
estimates. Mitigating this type of portfolio sensitivity has been
studied extensively in the literature. Remedies ranging from simply
imposing constraints to a complex approach employing second order
cone programming, have been offered as potential solutions to the
robust asset allocation problem. Each potential remedy, however,
brings varying degrees of challenges including, for example,
becoming ad hoc with the introduction of discretionary elements in
the case of constraints, loss of transparency, and high cost of
applicability (see Ledoit and Wolf (2003), Jagannathan and Ma
(2003), Tutuncu and Koenig (2004), Garlappi et al. (2007)).
Therefore, if we can show that risk parity is robust by nature, or
at least more robust than alternative methods, we can then hope to
deal less with these intricacies of quantitative portfolio
management.
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DEMYSTIFYING RISK PARITY
Given the notoriety of mean-variance optimization with respect
to its sensitivity to input parameters (as discussed in Chopra and
Ziemba (1993), among many others), we again exclude mean variance
and our analyses will focus on risk parity and its similarly
spirited counterpart, minimum variance. On this quest, we focus on
two measures of interest. First, we investigate the concentration
in dollar weights and risk contributions. Second, we quantify the
extent that optimal solutions change when a particular parameter is
perturbed by a small amount and what the resulting turnover will
be. To assess these impacts, we use sum of squared weights S2 := xT
x, and the change in sum of square weights with respect toT a
change in parameter a, S2(a) := x x
respectively, where x is a vector of weights. To illustrate,
when the weights sum to unity, S2 is minimized when all the
components are the same, and it is maximized when all the weights
are concentrated in one asset. Similarly, if there is no change in
the portfolio weights after a perturbation, S2 will be zero.
Moreover, any deviations from zero in both signs will proxy the
level of sensitivity and, therefore, the level of turnover.
Although the analytical computation of this derivative is available
in the case of the unconstrained MV portfolio, the same is not true
for the RP portfolio given its endogenously determined optimal
weights. Therefore, we start by studying a simplified example to
demonstrate the impact of sensitivity by considering three assets
with the following volatility and correlation structure.TA B L E 2
: A S S U M E D R I S K PA R A M E T E R SA 10% 1 B 12% 0.1 1 C 14%
0.2 0.3 1
a
as measures of concentration and turnover
Volatilities Correlations A B C Souce: Neuberger Berman
Quantitative Investment Group.
To explore the sensitivity of these portfolio weights with
respect to changes in volatility, we vary the volatility of Asset A
from 10% to 20% in 1% increments and calculate S2 and S2 (A). In
Figure 7, we compare snapshots of weights and risk contributions
when the volatility of Asset A is 10% and 20%, respectively. The MV
portfolio at first loads heavily on Asset A at about 90%, then cuts
the weight significantly to a little less than 50%, when volatility
is increased from 10% to 20%. Its risk contribution profile is
equally concentrated if not more alarmingly so, with Asset A
accounting for most of the portfolios risk when its volatility is
low at 10% (indeed, the percentage contributions to risk of an
unconstrained MV portfolio can be shown to equal its weights).
While the RP portfolio also adjusts its weight when the volatility
of Asset A is changed, its sensitivity is clearly much lower than
that of the MV portfolio.
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DEMYSTIFYING RISK PARITY
F IG U R E 7 : S N A PSH OT OF PORTFOLIO WEIG H TS WH EN AND
WHEN
A = 20% (RIG H T)% 100 90 80 70 60 50 40 30 20 10 0
A = 10% ( LEFT)
% 100 90 80 70 60 50 40 30 20 10 0
Weights of Risk Parity vs Minimum Variance Case: A = 10.0
Weights of Risk Parity vs Minimum Variance Case: A = 20.0
A
B
C
A
B
C
% 100 90 80 70 60 50 40 30 20 10 0
PCTR of Risk Parity vs Minimum Variance Case: A = 10.0
A
B
C Risk Parity
% 100 90 80 70 60 50 40 30 20 10 0
PCTR of Risk Parity vs Minimum Variance Case: A = 20.0
A Minimum Variance
B
C
Source: Neuberger Berman Quantitative Investment Group. Note:
PCTR: Percentage contribution to portfolio risk.
To demonstrate the impact of a volatility change in continuum,
Figure 8a shows that the MV portfolio is always more concentrated
than the RP portfolio, regardless of the volatility of Asset A, as
it increases from 10% to 20%. Next, Figure 8b illustrates the
extent to which the MV portfolio can be more sensitive with respect
to changes in the volatility of Asset A when compared to the RP
portfolio. As an example, when A is increased to 11% from 10%, the
resulting change in the sum of squared weights in the MV portfolio
is around -0.07, more than double the value of -0.03 of the RP
portfolio. In other words, both portfolios cut the weight of Asset
A as its volatility increases, but the corresponding response from
the RP portfolio is much less than the MV portfolio, which had very
dramatic changes in response. Moreover, the absolute delta is
always greater in the case of minimum variance, meaning that
regardless of the volatility of Asset A, any change in volatility
will result in greater turnover than risk parity.
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DEMYSTIFYING RISK PARITY
F IG U R E 8 A : IM PA CT TO PORTFOLIO CON CEN TRA TION WH EN V
O LATI LI TY OF A S S E T A V A RIES FROM 10% TO 20%Sum of Squared
Weights vs. Volatility of Asset A 1.0
0.8 Sum of Squared Weights
0.6
0.4
0.2
0.0 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% Volatility Risk
Parity Minimum Variance
Source: Neuberger Berman Quantitative Investment Group.
F IG U R E 8 B : IM PA CT TO PORTFOLIO TU RN OV ER WH EN V OLA
TILI TY O F A S S E T A VA RIES FROM 10% TO 20%Delta in Sum of
Squared Weights vs. Volatility of Asset A 0.00
Delta in Sum of Squared Weights
-0.02
-0.04
-0.06
-0.08 11% 12% 13% 14% 15% Volatility Risk Parity Source:
Neuberger Berman Quantitative Investment Group. Minimum Variance
16% 17% 18% 19% 20%
12
DEMYSTIFYING RISK PARITY
In a similar fashion, we analyze the impact of changes in
correlations to portfolio robustness. In doing so, we keep the
volatilities constant and only vary the correlation between Assets
A and B, denoted as AB. The parameters of the base case scenario
are summarized in Table 3.TA B L E 3 : A S S U M E D PA R A M E T E
R S W H E N A N A LY Z I N G T H E I M PA C T O F C O R R E L AT I
O N C H A N G E S O N P O R T F O L I O W E I G H T SA 10% 1 B 10%
0.1 1 C 10% 0.1 0.1 1
Volatilities Correlations A B C Souce: Neuberger Berman
Quantitative Investment Group.
F IG U R E 9 A : IM PA CT TO PORTFOLIO CON CEN TRA TION WH EN
THE C OR R E L A T IO N BETWEEN A SSET A A N D A SSET B V A RIES
FROM - 0. 5 TO 1Sum of Squared Weights vs. AB 0.40
0.38 Sum of Squared Weights
0.36
0.34
0.32
0.30 -0.5 0.0 0.5 1.0
Correlation Risk Parity Minimum Variance
Source: Neuberger Berman Quantitative Investment Group.
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DEMYSTIFYING RISK PARITY
F IG U R E 9 B : IM PA CT TO PORTFOLIO TU RN OV ER WH EN TH E
CORRELATI O N B E T W E E N A SSET A A N D A SSET B V A RIES FROM
-0.5 TO 1Delta in Sum of Squared Weights vs. AB 0.04
Delta in Sum of Squared Weights
0.02
0.00
-0.02
-0.04
-0.06
-0.08 -0.5 0.0 Correlation Risk Parity Source: Neuberger Berman
Quantitative Investment Group. Minimum Variance 0.5 1.0
Figures 9a and 9b again show that, except at a tangential point,
the MV portfolio always takes highly concentrated weights relative
to the RP portfolio (upper chart). Furthermore, as correlation
varies, the MV portfolios turnover is always greater than the RP
portfolio as changes in squared weights deviates from zero
significantly more than the risk paritys deviations.Potential
Utility that Leads to Risk Parity
Earlier, we studied the RP portfolio in a one-factor world and
also as a specific efficient portfolio within the mean-variance
paradigm. However, none of these sections provided insight into
what general utility function risk parity is built to maximize. As
Lee (2011) points out, a challenge to many of the risk-based asset
allocation approaches, including risk parity, is the lack of a
clearly defined investment objective. As a result, it is not
immediately clear what investment problems these portfolios are
built to solve. In this section, we further expand the analysis to
focus on a more generalized utility maximization problem in which
risk parity is shown to be a special case. With this generalized
utility function, we allow different portfolio constituents to
contribute different amounts of risks to the portfolio volatility.
Risk parity, then, is a special case in which the risk
contributions from all constituents are identical.
14
DEMYSTIFYING RISK PARITY
We impose a long-only constraint to make our analysis more
relevant to a typical investor. Next, we also assume that the
investor has an overall risk budget as measured by a limit on the
portfolio volatility; exceeding this limit would be beyond the
investors appetite and tolerance for volatility. This leads us to
analyze the generalized risk budgeting problem specified as
follows: Maximize ln(x) Subject to (x) T x0N
(1)
where > 1 is an N x 1 vector, x is the vector of weights, and
(x) is the volatility of the portfolio x, which is constrained to
be less than a target T. In the Appendix, we demonstrate that the
vector can be interpreted as the vector of risk budgets, defined as
contributions to the portfolio volatility of the assets. The RP
portfolio, in the specific case when i for i= 1, 2, , N, is set to
be identical, is shown to be a solution to the utility maximization
problem above. In addition, existence and uniqueness of the RP
portfolio are also established.Likelihood Maximization
Interpretation in a Bayesian Setting
To shed some light on the economic meaning of this utility
function, we form an objective prior on the portfolio weights (see
objective-based priors Avramov, Zhou (2006)). To get an idea of the
form of this prior, we first express the objective function in the
following equivalent form. L( x ; ) = x i ii =1 N
(2)
Maximization objective expressed this way resembles the
likelihood function of a multivariate Dirichlet distribution
(Frigyik 2010) when asset weights sum to unity. Indeed, given the
parameters , the likelihood of observing portfolio weights x (under
an uninformative prior) is given by (2). In the case with no total
risk constraint, L(x; ) is maximized at x =
weights, and as such in the absence of a risk target, has the
interpretation of portfolio weights. However, when the total risk
constraint is imposed, the maximum likelihood is achieved at a
generalized risk parity solution, and thus has a risk budget
interpretation. As an example, if an investor starts with a prior
belief that a portfolio should consist of 60% stocks and 40% bonds
and is not constrained by the total risk taken, then the investor
will invest according to the prior beliefs. However, if the
investor is risk averse and has a limited total risk budget, the
investor will most likely hold a portfolio with a risk allocation
identical to the prior belief on portfolio weights. In other words,
the generalized risk parity approach attempts to find a set of
portfolio weights that is closest to the prior belief without
exceeding the total risk budget constraint. Needless to say, when
the elements of are the same, the resulting RP portfolio is the
portfolio that is closest to an equal dollar weighted allocation,
but with risk being constrained. The above analysis helps justify
the rationale for generalized risk parity investments. The problem
maximizes allocation to those assets that the investor likes and at
the same time, strives to control the risk of the resulting
portfolio. Although expected returns do not appear directly as in
the case of mean variance, the relative preference relations
defined by s provide us with the necessary link to justify the
relative allocation of funds.15
to yield the maximum likelihood estimates of portfolio 1T
DEMYSTIFYING RISK PARITY
Those assets expected to yield higher returns can be assigned
relatively higher s and the resulting dollar weights will be tilted
towards these assets accordingly in a risk diversified manner.When
Should Risk Parity Be a Preferred Method?
Having identified some important properties of RP portfolios, we
now address the intricate issue of what may make risk parity a
better approach than the readily available, widely understood and
adopted portfolio construction methodologies such as mean variance
and minimum variance. In essence, our goal is to examine the
scenarios or regimes where risk parity is likely to be more
promising with respect to some efficiency criteria. To achieve this
task, we devise simulation-based controlled experiments where the
aim is to create a statistically static environment, and then run
horse races between asset allocation methodologies to see which one
dominates in terms of portfolio efficiency as measured by the
reward-to-risk ratio. The stochastic investment opportunity set is
simulated by first setting expected returns, the covariance matrix
of assets, and the degree of freedom in a multivariate
t-distribution that introduces different degrees of tail thickness.
We pretend that investors are told that the true underlying returns
generating function is static, but unknown. As a result, the
investors have to use different lengths of rolling windows of the
observed returns data in an attempt to estimate the true underlying
return generating function. Thus, in this experiment, the size of
the rolling estimation window becomes a proxy for the level of
noise in estimation. Investors then construct their portfolios
according to their estimates as well as their preferred portfolio
construction approaches, and out-ofsample performances of these
portfolios are recorded and compared. For each degree of freedom
parameter and window length choice, we repeat the above steps
numerous times and record the corresponding performance of these
portfolios for subsequent analysis of their portfolio
efficiency.
16
DEMYSTIFYING RISK PARITY
F IG U R E 1 0 : RISK PA RITY M A Y BETTER N A V IG A TE A WORLD
WIT H FAT T A IL S A N D N OISE800
400
Less Noise
MEAN / VARIANCE MORE EFFICIENT
200 Estimation Window Length 150
100
More Noise
50
RISK PARITY MORE EFFICIENT
40
30
20
10
2
3
4
5
6
7
8
9
10 12
15
20
25 30
50
80 100 120
200
500
1000
Degrees of Freedom (DoF) for the Multivariate Tdistribution
(Tail Thickness: as DoF Decreases Tail Becomes Thicker)
More Fat Tails
Less Fat Tails/More Normal
Source: Neuberger Berman Quantitative Investment Group.
For the sake of illustration, we take the point (7,40) on Figure
10 as an example. We first create 20 replicas of history, each
consisting of 1,200 months of simulated returns with a tail index
corresponding to a degree of freedom equal to 7, using the average
returns and covariance matrix from Table 1 on four asset classes.
Next, at any point in time, we use the previous 40 months of
observed returns data to estimate the expected returns and
covariance matrix assuming returns follow a normal distribution.
Given these empirically estimated parameters, mean-variance, MV,
and RP portfolios are then constructed and held for one month, with
performance in the following month recorded. We repeat this
estimation, optimization, rebalancing, and performance recording
cycle until we exhaust the 1,200 months of simulated returns. We
then repeat the above steps for each of the 20 replicas of history,
and count in how many of these 20 runs each method wins as the most
efficient portfolio as measured by the highest reward-to-risk
ratio. In the case of (7,40), among these 20 runs, the risk parity
methodology dominates, being the most efficient in 17 out of 20
times, than the other portfolios. As such, we conclude that the RP
portfolio is the most dominant portfolio given the tail index as 7
and estimation window length of 40 months under the aforementioned
returns and covariance matrix environment. Therefore, the (7,40)
point is contained in the Risk Parity More Efficient part of the
plane. We highlight three important observations from Figure 10.
First, when the tail parameter is fixed, risk parity is most
efficient when estimation noise, as represented by shorter window
lengths, is high. Here, we note that in our experiment, all
return-generating parameters are fixed. On the contrary, in the
real world, the true return-generating function is time-varying,
changing as we move in and out of different regimes, making
parameters estimation even more noisy and imprecise given the same
window length.
17
DEMYSTIFYING RISK PARITY
Moreover, real world investment problems are much more complex
and involve many more assets. In other words, the error
maximization problem in the real world is many times more
significant than what we have built into our experiment and, as
such, the true boundary in Figure 10 is pushed much further north
in the real world, making risk parity even more dominating in a
noisy environment. Furthermore, as the asset management industry
continues to evolve with new ideas and strategies as investment
solutions, insufficient track records become more a norm than an
exception, presenting challenges to strategic asset allocation
analysis that normally requires sufficient historical data. In
light of our findings above, and together with Mertons (1980)
findings that precision of estimating risks is higher than the
precision of estimating expected returns, risk parity appears to be
a reasonable solution in allocating to new strategies. Second,
given a certain degree of noise associated with a fixed window
length, risk parity is the most efficient portfolio as returns
become more fat tailed, deviating from the normal distribution. By
now, few would disagree that the real investment world does not
follow a normal distribution. For instance, the Ibbotson dataset
used in the simulation from 1926 through 2010, the fitted
multivariate t-distribution, indicates that the tail parameter is
estimated at 3.11, a level far from the value 30 of normality. At
such a value of the tail parameter, Figure 10 suggests that one
requires many decades of data for the mean-variance approach to be
more efficient than the RP portfolio, and that is based on an
unrealistic environment of static returns generation. Lastly, when
returns are well-proxied by a normal distribution with negligible
fat tailsthat is, when the tail parameter is greater than 30the
decision boundary plateaus, making tail thickness less of a concern
than estimation noise.Conclusion
Financial crises remind investors of the necessity of risk
management, and financial institutions of the need to innovate new
products that are better suited for increasingly uncertain and risk
averse environments. The aftermath of the 2008 financial meltdown
lead to the (re)emergence of risk parity as a panacea for what was
missing before the crises in asset allocation practices before the
crises. Thenceforth, risk parity has enjoyed constant fund flows
from institutional investors and popularity from index providers.
In this paper, our aim was to help investors enhance their
understanding of risk parity in a number of ways to assess whether
this new remedy is really a panacea or a temporal placebo. First,
for those who still look at the world in a mean variance setting,
we provided conditions that make risk parity efficient and argued
that although these conditions are not expected to be met over
short time horizons, they may be good proxies for longer-term
allocations. Second, to relate historical risk parity
outperformance to known price anomalies, we studied risk parity in
a one-factor setting and showed that risk parity favors low beta
and low idiosyncratic risk assets. As such, we argued that the
performance of risk parity may very well be related to the
underperformance of high beta investments in the last two decades.
Third, we analyzed risk parity sensitivity to input parameters and
found it more stable than other risk-driven allocation methods. We
argued that this insensitivity is important for risk parity
managers as turnover resulting from covariance instabilities may
significantly reduce any value added. In particular, we showed that
the risk parity method is influenced more by changes in
volatilities than correlations, but the impact of these changes is
much lower when compared to minimum variance.18
DEMYSTIFYING RISK PARITY
Fourth, we studied the risk parity problem and related its
objective function to a constrained maximum likelihood estimation.
In this setting, we argued that a generalized risk parity investor
has prior beliefs about what portfolio weights should be, and is
trying to be statistically as close as possible to those weights
without exceeding a risk budget. In the special case, we showed
that if the investor does not have any beliefs, or treats each
investment equally, then pure risk parity is the portfolio held
under a binding risk constraint. Finally, acknowledging the path,
asset domain, and macroeconomic dependencies of the risk parity
backtests, we conducted extensive controlled simulation experiments
in order to produce a fair judgment of risk-based asset allocation
methods. In these runs, we exposed mean variance, minimum variance
and risk parity to environments that we statistically set, and
found that risk parity statistically dominates other methods when
there is high uncertainty around input parameters (i.e., high
noise) and/or there are fat tails in the asset return series. In
summary, although risk parity may not be truly efficient, we have a
number of reasons to believe that it is efficient enough, ex-ante,
when compared to other allocation methods. As such, even if it is
not a cure-all solution to long-term strategic allocation, it may
be a good starting point.APPENDIX Notation
N Number of assets x : N x 1 vector of portfolio weights : N x 1
vector of expected returns : N x 1 positive vector of risk
multiples associated with each asset 0 : N x 1 vector of zeros (A)i
: ith row of a matrix A : N x N covariance matrix : N x N
correlation matrix V : N x N matrix of zeros except nonzero asset
volatilities on the diagonal X : N x N matrix of zeros except asset
weights x on the diagonal : N x 1 vector of volatilities T :
Volatility target
( x ) := x T x the portfolio volatility : N x 1 vector of
Lagrange multipliers for the constraint x 0 : Lagrange multiplier
for the constraint (x) T ( x )i := x i ( x ) : Contribution to risk
of asset i =1,2,..., N x i
> 0 : Coefficient of risk aversion
19
DEMYSTIFYING RISK PARITY
Proof of Risk Parity portfolios preference of low beta, low
idiosyncratic risk assets
To show this formally, let us consider a one-factor world in
which asset excess returns follow, i = 1, 2,..., N i = ai + bi f +
iE[i j ] = E[i ] E[ j ] E[i f ]= E[i ] E[ f ] Var[ f ] = 2 f
=0 =0
(3)
Var[i ] = 2i
where f is the excess return of the single factor, ai and bi are
constants and i is a mean zero error term assumed to be
uncorrelated with the factor and with other error terms. In this
case, we can represent variance and covariances of each asset i,j
=1,2,..., N as i2 = ij = Then, the risk parity condition, i , j {1,
2,..., N }, bi2 f2 + i2 bi bj f2 (4)
( Xx )i (x)
=
( Xx ) j (x)
,
or x ' x NN
(5)
( Xx )i = ( Xx ) j =leads toN
x i2 2i + x i 2 bi bk x k = x 2 2j + x j 2 b j bk x k f j fk =1
k =1
2 N f bk x k ( x i bi x j b j ) = ( x j j x i i )( x j j + x i i
) k =1 x j j x i i x i bi x j b j = ( x j j + x i i ) 2 N f bk x k
k =1
(6)
Assuming that the beta of the portfolio x is positive, which is
typically the case for most long-only portfolios, (6) has the
following consequences for those assets which have nonnegative
betas: Ifidiosyncraticrisksarethesame,i.e., i = j , then bi > bj
0 xi < xj, and Ifbetasarethesame,i.e.,bi =bj 0, then i >
j
xi < x j
Briefly, the first bullet point follows from the fact that if by
assuming by contradiction xi xj , then xi bi xjbi > xjbj
implying that xi bi xjbi > 0. Therefore, the left-hand side of
the last row in (1.4) becomes positive implying that in the
right-hand side, with the common positive idiosyncratic risk
factored out, is also positive, i.e., xj xi > 0. Hence, we
arrive at a contradiction. The second bullet can be proven
similarly.
20
DEMYSTIFYING RISK PARITY
Therefore, without loss of generality, we can conclude that a
typical long-only RP portfolio tends to prefer low beta assets with
low idiosyncratic risks and as such it is also capturing these
pricing anomalies.Proof of mean-variance efficiency of the RP
portfolio if Sharpe ratios and correlations among assets are
identical
We first state the proven result that the mean-variance
efficient portfolio can be represented by: 1 (7) x = 1
where > 0 represents the coefficient of relative risk
aversion. The contribution to risk of each asset can then be
written as
( x ) =
Xx (x) 1 X 1 = (x) 1 X = (x)
(8)
Therefore, the ith row of (x) is ( x )i
i 1 1 ( )i ( x ) = 2 i ( 1 )i (x)=
(9)
Here, the inverse of the covariance matrix can be expressed in
terms of volatilities and correlations as: 1 = [VV]1 = V1 1 V1
(10)
If we denote by bij the entries of the inverse of the
correlation matrix 1, then we can b further write the elements of 1
as ij 1 = ij i j r Putting (10) in (9), and letting s i := i denote
the reward to risk ratios, we get i ( x )i =
i 1b1i 2b2i i bii N bNi + + ... + 2 + ... + (x) 1 i 2 i i N i
2
i2 2 b b b = 2 i 1 i 1i + 2 i 2i + ... + bii + ... + N i Ni ( x
) 1 i 2 i N i = s1 s i2 sN s2 2 s b1i + s b2i + ... + bii + ... + s
bNi (x) i i i
(11)
21
DEMYSTIFYING RISK PARITY
Then, from (11) it follows that
s s1 s 2 b + 2 b + ... + bii + ... + N bNi s i s i 1i s i 2i ( x
)i si = s s1 s2 ( x ) j s j b1 j + b2 j + ... + b jj + ... + N bNj
sj sj sj
(12)
In the special case, when all assets have the same Sharpe ratios
such that si = s, i = 1, 2 ,..., N , and the off-diagonal terms of
the correlation matrix, , equal to the same constant, we get2 ( x
)i =1 (13) ( x ) j which shows each asset contributes equal risk to
the portfolio. In other words, a RP portfolio is mean-variance
efficient if Sharpe ratios and correlations among assets are
identical.Risk Parity as a Solution to Utility Maximization
We claim that RP portfolios, in the general case, are the
solution of the following problem: Maximize T ln(x) Subject to (x)
T x0 (14)
Here, Karush-Kuhn-Tucker (KKT) conditions provide us with
necessary conditions for optimality. Because the constraints and
the objective function are convex in this case, these necessary
conditions are also sufficient. Let L(x, , ) denote the Lagrangian
function: L(x, , )= T ln(x) + ((x) T) T x) Then, we have the
following KKT conditions: (15)
i ( x ) + i = 0 , i = 1, 2 ,..., N x i xi(16)
, 0 ( ( x ) T ) = 0 i x i = 0 , i = 1, 2 ,..., N
Because i = 1, 2 ,..., N , xi Dom(ln), and i xi = 0, we get i =
0. Then, the first row in (1.14) becomes ( x ) (17) i + = 0 , i =
1, 2 ,..., N xi x i Furthermore, from the last row in (16) and from
(17), we see that > 0, because otherwise xi = , i = 1, 2 ,..., N
. Therefore, at optimality, we have (x) = T.
2
Here we use the fact that if is a constant correlation matrix,
then both the diagonal and the off-diagonal elements of constant
(not necessarily equal to each other).
1
are
22
DEMYSTIFYING RISK PARITY
Rearranging the terms in (17), the contribution to risk of asset
I can be given as ( x ) i (18) ( x )i := x i , i = 1, 2 ,..., N = x
i Therefore,
( x )i i = , i , j = 1, 2 ,..., N ( x ) j j
(19)
If i = j, i , j = 1, 2 ,..., N , then (19) implies ( x )i = ( x
) j , i , j = 1, 2 ,..., N which is the original RP
portfolio.Existence of RP Portfolios
(20)
In order to make sure that a RP portfolio can always be formed,
we need to show that the feasible set for the problem in (14) is
nonempty. However, this is not a difficult task because for any
given volatility target, T > 0, one can find small enough
nonnegative weights to satisfy (x) T, e.g., a portfolio consisting
of only the first asset scaled properly to have less than T risk is
feasible. Given that a local solution of the problem (14)
necessarily satisfies risk parity conditions, and such a solution
can always be found, we conclude that RP portfolios can always be
formed.Uniqueness of RP Portfolios
Given that any RP portfolio with nonnegative weights and less
than or equal to T volatility target solves problem (14), the set
of all RP portfolios and the set of all solutions of problem (14)
coincide. Therefore, in order to show that the RP portfolio is
unique, we need to show that the solution of problem (14) is
unique. At first look, problem (14) is a maximization of a concave
function subject to non-convex constraints, i.e., (x) T. Therefore,
we are not guaranteed a unique solution. However, the following
result helps us transform it to a convex problem. Proposition 1 A
local solution of problem (14) is also a solution of Maximize T
ln(x) Subject to (x)2 (T)2 x0 Proof Let x be a local solution of
problem (14). Then, by KKT we have the conditions in (16) hold
true, ( x ) i = 0 , i = 1, 2 ,..., N i + xi x i (21)
, 0
( ( x ) T ) = 0
i x i = 0 , i = 1, 2 ,..., N
(22)
23
DEMYSTIFYING RISK PARITY
Remember, all risk budget is strictly used at optimality (i.e.,
T = (x)), let us define := 2 T = 2 ( x ) 0 . Then we have,
i ( x )2 + i = 0 , i = 1, 2 ,..., N xi x i(23)
, 0
i x i = 0 , i = 1, 2 ,..., N
( ( x )2 ( T )2 ) = 0
However, these are the KKT conditions for problem (21) at point
x. Therefore, x is a solution of the problem (1.19). Given that the
maximization problem (21) is convex (with convex constraints and
strictly concave objective function); its solution is guaranteed to
be unique. Because each solution of (14) is a solution of (21), and
the solution of (21) is unique, the solution of problem (14) has to
be unique.
24
DEMYSTIFYING RISK PARITY
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DEMYSTIFYING RISK PARITY
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