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Risk Budgeting with Asset Classand Risk Class ApproachesHAKAN
KAYA, WAI LEE, AND YI WAN
HAKAN KAYAis a vice presidentof the QuantitativeInvestment Group
atNcuberger Berniaii111 New York, NY.
WAI LEEIS a managing director,CIO. and director ofresearch of
the Quantita-tive Investment Groupat Neuberger Dermanin New York,
NY.
Yi W A Nis a vice presidentof the QuantitativeInvestment Group
atNeuberger Bermanm New York. NY.
In describing a portfolio, a set of portfolioweights that
reflect capital allocation isthe natural starting point because of
itsclear definition, timeliness, and the factthat it can be
precisely measured. A descriptionof a fund in Bloomberg, for
example, typicallyincludes asset class and regional and
sectorallocation, followed by the top ten and otherportfolio
holdings, which are expressed as per-centage weights of the total
investment value.
By now, however, most investors realizethat more analysis is
required in order to under-stand the risks of the portfolio behind
the setof measurable portfolio weights. This is whereinvestors may
have differing opinions. How willthis portfolio perform if economic
growth inthe next quarter disappoints? What if inflationaccelerates
more than expected, or oil reaches$125 a barrel? Unlike capital
allocation, whichis accounting driven, risk allocation is based
onone's estimation of multiple parameters overpotentially different
time frames with differenttechniques, and may even begin with
differentdefinitions and interpretations of risks.
According to modern portfolio theory(MPT), investors seek
balance between returnand risk. While over time many
alternativerisk measures have been developed and betterunderstood,
the one proxy for risk that allinvestors continue to estimate and
consider isthe volatility of a portfolio, which takes intoaccount
the volatilities and correlations of allassets. Meanvariance
optimization (MVO),
whose required inputs include expectedreturns and a covariance
matrix, is often usedtogether with MPT. The theory and practicego
hand in hand, so that an investor will seekto maximize portfolio
return subject to his/her degree of risk aversion, which
determinesthe level of portfolio volatility one may
findappropriate. Although the theories provide apowerful and
intuitive framework, they donot dictate how the required inputs
shouldbe determined. For instance, the investmenthorizon, estimates
of expected returns, vola-tilities, and correlations among assets
are allsubject to an investor's approach.
Practitioners have offered some "nextgeneration solutions" to
what "went wrong"as their response to the most recent
financialcrisis. One of the most common refrains hasbeen that
MPT/MVO tailed investors inproviding diversification when it was
mostneeded.' In particular, the failure of risk man-agement points
to the interpretation and mod-eling of assets and portfolio risks.
Some critics,for example, consider the use of a covariancematrix of
assets as traditional and flawed. Theypoint out that the
traditional approach promotesseeing the world as defined by assets,
suchas stocks, bonds, commodities, and the like.Their arguments go
even further, emphasizingthat the ways in which these assets
respondedduring the financial crisis clearly indicated thatasset
classes, while different, could be exposedto the same risks. As
such, some of these critics
SPRING 2012 THEJOURNAL OF INVESTING 1 0 9
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propose to displace "asset classes" by "risk classes" for
thepurpose of asset allocation. In this new risk class approach,the
investor will determine the optimal mix of assetsthrough which
target exposures to different risks areachieved.
We think that this risk class approach makes a lot ofsense
conceptually, which should make the discussions andthought process
of asset allocation more interesting andunderstandable. However, we
do not believe that puttingit into practice is any easier than the
asset class approach.In the following sections, we discuss the
motivation oftherisk class approach through the most basic concept
of assetpricing. We argue that the risk class approach is a
morestructural approach to modeling a covariance matrix ofassets.
However, our numerical examples illustrate that itcomes with a
pricemore model misspecification risks andparameter uncertainties.
We emphasize that when the spec-ification of risks is complete and
idiosyncratic elements arealmost negligible and uncorrelated, the
risk class approachconverges with the asset class approach.
ASSET PRICES AND RISKS
Understanding the risks embedded in a portfolio isone ofthe most
important steps in putting MPT to work.To establish a comparison
ofthe asset class and the risk classperspectives, we went back to
the drawing board usingone ofthe most fundamental concepts in
financial theory,namely, the present value of assets.
Financial assets are valued based on the stream ofuncertain cash
flows discounted by a risk-adjusted discountrate. In equation form,
current price can be expressed as
Price,,
+
(1)where C is the uncertain cash flow at the end of periodf, k^
is the discount rate for the period (, and Eg[.] denotesthe
expected value as ofthe current time. This equationis a
mathematical identity that defines current price. Inthe simplified
version, one may assume that the discountrate stays constant in all
periods so that k^ = k. In the caseof stocks, C^ are future
dividends, while in the case ofbonds, C are interests to be earned
and the face valueofthe bond to be returned in the last time
period. Riskmodeling of an asset is thus rooted in otir
understanding
of how prices move in accordance with this present
valuedetermination.
First, consider the volatility of a single asset. Thecauses
behind an asset's price moving from one period toanother must
correspond to the changes in expectationsof future cash flows and
the perceived risk of the assetthat is reflected by revisions to
the discount rates used incalculating the present value. For
instance, a "growth"factor can impact asset prices when business
conditionsare expected to deteriorate. In such a scenario, one
mayrevise the expected future dividends of stocks to reflectslower
growth and revise the dividend discount rate up toreflect the
higher expected risk premium given the higherperceived uncertainty.
Both revisions will put a down-ward pressure on the stock price. In
the case of bonds,if coupons are fixed, then the price change must
be theresult of changes in discount rates in response to a changein
the growth factor. An "inflation" factor can work ina similar way.
Suppose future inflation is expected toaccelerate. Expected future
dividends may be revised upif one believes that companies with
pricing power willpass the nominal increase in goods price to their
earningsand, therefore, dividends. Of course, the revision can
alsobe downward in case one expects economic conditions tostart to
deteriorate in response to accelerating inflation,which ultimately
has a negative impact on growth andthe propensity to consume.
Inflation also tends to leadto upward revisions of future dividend
discount rates notonly as a nominal effect, but also as higher
perceivedrisk as related to unstable prices. If coupons of bonds
arefixed, then higher inflation has an unambiguous negativeimpact
on bond prices through a higher discount rate.
Second, consider how assets are correlated with eachother.
Correlation measures how one asset moves withanother. As a result,
the correlation between two assetsmust be determined by how their
prices change over time,which is in turn driven by how their
respective expectedcash flows and discount rates are revised in
response tochanges in conditions. In the example above, the
correla-tion between stocks and bonds will be driven by how
theirprices move in response to changes in the "growth"
and"inflation" factors. Needless to say, response to changes
infactors ofthe same assets can be different at different
times.Forecasting correlations precisely, therefore, requires a
fullunderstanding of how assets react to changes in
differentfactors over time. In addition, at times, one asset may
reactto conditions that are unique to that particular asset,
whileother assets show no reaction whatsoever.
110 RISK BUDGETING WITH ASSET CLASS AND RISK CLASS APPROACHES
SPRING 2012
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IFROM ASSET CLASS TO RISK CLASS
The practice of asset allocation requires modelingrisks of
assets. For example, to determine the optimalallocation between
stocks and bonds in a portfolio, weneed a covariance matrix that
includes the volatility ofstocks, volatility of bonds, and
correlation between thetwo assets. The risk class approach differs
from the assetclass approach by inserting an additional step in the
pro-cess of modeling the covariance matrix. Instead of mod-eling
the volatilities and correlations of assets directly asin the asset
class approach, the risk class approach firstimposes a factor
structure on all assets so that the volatilityof an asset is driven
by its exposures to these factors, thefactor volatilities, as well
as the idiosyncratic volatilityof an asset that is unrelated to
these factors. Since idio-syncratic volatilities are asset specific
and do not overlapwith other assets, by definition, the correlation
of assetsis entirely through their relative exposures to the
sameset of factors.
In the earlier example with growth and inflation asthe only two
factors, we may specify the structure of stocksand bonds as
follows:
Stock Return = Constant.. + Beta^ ^^ . x Crowth + ^^,X Ination +
Stock Specific (2)
Bond Return = Constant,, + , X Growth +X Inflation + Bond
Specific (3)
where Beta denotes the exposure of an asset with respectto a
factor and subscripts S, B, C, and I denote stock, bond,growth, and
inflation factors, respectively. Assuming thatthe factor structure
above is correct, the volatilities ofstocks and bonds can then be
determined by their respec-tive exposures to these factors, the
factor volatilities, andthe idiosyncratic volatilities of the
assets. The correlationbetween stocks and bonds, given tbis factor
structure, isentirely driven by how the stocks and bonds are
exposedto these two factors. If both assets have positive
exposuresto the growth factor, for the purpose of illustration,
thena shock to the growth factor will lead to a positive
cor-relation, everything else being equal.
Obviously, the risk class approach is intuitivelyappealing it
the factor structure assumed is an accuratereflection of reality.
With a factor structure such as theabove, one can interpret, and
even forecast, tbe correla-tion of stocks and bonds based on our
understanding ofhow each asset responds to factor shocks, such as
through
the lens of the present value model discussed earlier, inan
attempt to understand how the expected cash flowsand discount rates
react.^
It is a big "if," however. First, what if our estimatesof asset
exposures to factors are off? If we get the signs ofexposures
correct but not the magnitudes, our forecasto correlation likely
has the correct sign, but is either toohigh or too low. In that
case, if the signs of exposures areincorrect, our correlation
forecast can be in the wrongdirection as well, which will clearly
affect our asset allo-cation decisions.
Second, what if there are missing factors? In thisscenario, the
assumed factor structure will not capture thecorrelation
completely, and therefore, diversification ben-efits will not be
fully captured and forecasts of portfoliovolatilities will be
inaccurate. One simple diagnostic onthe completeness of the risk
class approach is to check theidiosyncratic components as defined
by the factor struc-ture. If the tactor structure is complete so
that the cor-relation of assets is fully captured, then by
construction,the idiosyncratic components must be uncorrelated
andhave a relatively small contribution to asset variance
whencompared with factor-related volatility. If the idiosyn-cratic
components are found to be correlated, there mustbe some other
factors that were missed by the currentstructure that still have an
impact on asset correlations.Lastly, if the extent of asset
volatility explained by thefactors is insignificant, it may also
indicate the possibilityot missing factors.
Furthermore, details of model specifications, suchas the
definitions of factors, linear versus nonlinear, timehorizons
(e.g., monthly, quarterly, annual, or longer), andother
considerations, are all subject to debate. As a preview,the
disappointing results of using the risk class approachin describing
the realized risk characteristics of stocks andbonds in the
following example could have been the resultof model specifications
to an extent.
EXAMPLE
Recently, the risk class approach for asset allocationhas caught
the attention of the investment community.'However, there is
generally no agreement on the meth-odology behind determining tbe
factor structure. Forillustrative purposes, we have estimated a
factor struc-ture for the S&P 500 Index and U.S. 10-year
govern-ment bonds witb a set of variables that can be related
togrowth and inflation factors. Given that our primary goal
SPRINC; 2012 THEJOURNAL OF INVESTING 111
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1is to shed some light on the merits and challenges of therisk
class approach rather than estimating the best factorstructure, we
simply followed a classical study by Chen,Roll, and Ross [1986] in
selecting and defining the set offactors. For simplicity, we have
grouped the factors intotwo broad categoriesgrowth and
inflation:
Growth: Monthly percentage change of the Industrial Pro-
duction Index Credit spread, defined as the difference between
the
Moody's Baa yield and Aaa yield Slope of the yield curve,
defined as the difference
between the 10-year and 1-year U.S. Treasury yields
Inflation'': Monthly change in expected inflation rate
Unexpected inflation
The factor structure in the risk class approach is esti-mated
using a sample period of monthly data from April1953-December
2010.
It should be noted that if the factor structure is per-fect,
meaning that the growth and inflation factors (asdefined)
completely capture the risk characteristics of theassets as well as
their correlation, then the volatilities andcorrelation as
determined by the factors should exactlymatch the realized
volatilities and correlation duringthis sample period. Recognizing
the possibility that thefactors alone may still miss some of the
unaccountedvariations of stock returns and bond returns, one mayadd
back the stock- and bond-specific volatilities on topof the
factor-driven components. However, there is noguarantee that the
second set of estimates, even takinginto account risks specific to
the assets that are unre-lated to the factors, will match the
realized risk statisticsexactly. The reason is that if there are
additional factorsbeyond growth and inflation that drove the
correlationand volatilities of the assets, then what we consider as
thespecific risks of the assets will not be uncorrelated, as
themissing factors are embedded there. Comparing thesethree
cases(I) Factors Only (II) Factors + Uncorre-lated Specific Risks
(III) Factors + Correlated SpecificRiskswhere the last case should
be identical to therealized sample statistics, will give us the
extent to whichhow well the factors captured the volatilities and
co-movements of the assets and how important the missing
factors are, if there are any, in determining the risk
char-acteristics of the assets.
Note that the empirical exercise here is not to fore-cast risks
but merely to try to assess how successful arisk class approach,
subject to the specification of factorstructure, might have
captured or described the realizedrisk characteristics of the
assets. Exhibit 1 reports theresults, which are very interesting,
if not alarming.
According to the estimated factor structure labeledFactors Only
in Exhibit 1, the volatilities of the S&P500 and government
bonds during this period, as drivenby the growth and inflation
factors, should have been2.04% and 2.60%, respectively. Compared
with theirrealized volatilities of 14.77% and 9.40%, these
surpris-ingly low factor-driven volatilities suggest that
duringthis sample period, most of the volatilities of stocks
andbonds were not driven by the growth and inflation fac-tors
together. Besides, the correlation between stocksand bonds, in
accordance with the factor structure alone,should have been 0.41,
compared with the realized cor-relation of 0.16.
Next, we add the specific risks of stocks and bondsto the
Factors Only, and the results are grouped underColumn 2 in Exhibit
1. Note that the volatilities of stocksand bonds and their
correlation, after taking into accounttheir specific risks, are now
estimated to be 14.82%, 9.43%,and 0.02, respectively. These values
are still different from,but much closer to, the realized sample
values.
What could have been the missing factors beyondgrowth and
inflation accounting for the gaps in assetvolatilities and
correlation during this period? Of course,the poor results could
have been because of the way wespecified the factors. As discussed
earlier, a challenge ofthe risk class approach is that the true
factor structure isunobservable. However, recent history may
suggest thatone such factor could have been related to risk
aver-sion, or what some investors interpret as flight to
safety.During the last decade or so, we have often observed
that
E X H I B I T 1Risk Characteristics of the S&P 500 Index and
U.S.10-Year Government Bonds (April 1953-December 2010)
Volatility of S&P 500Volatility of Gov. BondCorrelation
(I) FactorsOnly2.04%2.60%0.41
(II) Factors +UncorrelatedSpecinc Risks
14.82%9.43%0.02
(HI) Realized Sample =Factors + Correlated
Specific Risks14.77%9.40%0.16
1 1 2 RISK BUDGETING WITH ASSET CLASS AND RISK CLASS APPROACHES
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stocks and bonds moved in opposite directions duringand around
crises, such as the 2008 global financial crisis.It may be helpful
to refer back to the present value defini-tion in Equation (1).
Presumably, if investors attribute therisk-aversion impact entirely
to the assets through futuregrowth and/or inflation, then the
factor structure withgrowth and inflation factors should have
captured the riskcharacteristics. However, if investors raise the
discountrate for stocks and lower the discount rate for bonds
inanticipation of higher risk aversion (having nothing to dowith
the future growth and inflation perspectives), thenthe
growth/inflation factor structure is clearly misspeci-fied, and its
degree of misestimation of assets' risk char-acteristics will
depend on the relative importance of themissing factors versus the
included factors. In addition,even if risk aversion is a factor
that can be quantified,there is no guarantee that including it as
the third factorcould have perfectly captured the assets' risk
characteris-tics. Other unknown factors may have been at work.
To further compare the asset and risk class approaches,we apply
the results above to a 60/40 portfolio of the S&P500 Index and
U.S. long-term government bonds. The60/40 portfolio's volatility is
calculated as 10.16%) duringthe sample period. Exhibits 2, 3, and 4
represent threedifferent descriptions of the same 60/40
portfolio.
While Exhibit 2 is just the standard representationof a
portfolio by portfolio weights. Exhibit 3 representsthe percentage
of risks of the 60/40 portfolio that canbe attributed to each asset
class.^ Based on the historical,realized sample risk statistics of
stocks and bonds duringtlie sample period of April 1953 to December
2010,stocks accounted for over 80% of the 10.16% volatility of
E X H I B I T 2Portfolio Weights of a 60/40 Portfolio
40%
60%
the 60/40 portfolio. This reaffirms the observation
thatportfolio weights of assets do not fully reflect risk
con-tributions from the assets.
Exhibit 4 provides risk contributions by risk factorsor
classes.'' Given the results discussed earlier, it shouldnot be a
surprise to see that the volatility of the 60/40portfolio that can
be attributed to the risk factors is verylow. Both the growth and
inflation factors are shown tohave accounted for about 2% of the
total volatility of the60/40 portfolio. In other words, about 96%
of the vola-tility of the portfolio is the result of either missing
factorsand/or risks that are specific to the asset classes
stocksand bonds.^
E X H I B I T 3Risk Contribution of a 60/40 Portfolio byAsset
Classes (April 1953-December 2010)
81.20%
I S&P 500 U.S. LT Govt. Bond
E X H I B I T 4Risk Contribution of a 60/40 Portfolio byRisk
Classes (April 1953-December 2010)
-1.72%2.07%
96.22%I S&P 500 U.S. LT Govt. Bond I Growth Inflation D
Specific
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DISCUSSIONS AND SUMMARY ENDNOTES
From a conceptual standpoint, the risk class approachis
superior; it recognizes that investable and tradable assetsin a
portfolio are merely a vehicle for investors to gainexposure to a
set of risks that are believed to be rewarded.Therefore, this
approach provides a deeper understandingof what drives the risks
and returns of a portfolio.
In practice, however, we believe that there is littlemagic
behind the risk class approach, and it doesn't nec-essarily offer a
superior investment framework to thetraditional asset class
approach. As a matter of fact, whenrisk is measured by volatilities
and correlations, we arguethat technically, the risk class approach
is a structural,or factor, approach of modeling the covariance
matrixof the assets, using similar practices that can be datedback
to the 1970s.'' In addition, the success of such anapproach depends
on whether the investors can come upwith a set of risk factors that
not only are of interest andrelevant to their strategic and/or
tactical concerns, butalso successful in capturing the exposures of
the assetswith respect to all of these factors. Besides, a
majorityof the risk characteristics of the assets that make up
theportfolio should be driven by these factors rather thanthe
idiosyncrasies. If the factors account for only a smallportion of
assets' risks, then the risk class approach pro-vides insignificant
insights beyond the historical riskcharacteristics of the
assets.
Finally, many, if not all, of the challenges in applyingthe
asset class approach are equally, if not more, relevant tothe risk
class approach. For instance, critics point out that theasset class
approach relies on the assumption that the cor-relation and
volatilities of assets remain stable over a pre-determined
investment horizon. As discussed earlier, therisk class approach is
a structural modeling of the cova-riance matrix of the assets.
Therefore, criticism of theasset class approach implies that
implementation of therisk class approach would require stable
factor structures,exposures of assets with respect to the risk
factors, as wellas stable correlations and volatilities of all the
factors.
In short, we appreciate the deeper insights behindthe risk class
approach beyond the observable and invest-able assets in the
portfolio. Its implementation, however,presents a new set of
challenges.
'See Kritznian [2011] for different perspectives.A^ relevant
analogy is the structural macro-econometric
modeling of the economy, particularly popular in the 1970s,such
as the work hy the Wharton Econometric ForecastingAssociates
(WEFA). Hundreds of structural equations weremodeled and linked in
order to understand and forecast eco-nomic variables. In contrast,
vector autoregression (VAR),introduced in the 1980s, is often
considered as the reduced-form approach to estimate economic
relationships.
'See Rue [2009] and Meketa Investment Group [2010]on risk
budgeting for examples.
*We follow the methodology of Fama and Gibbons[1984] in
estimating expected and unexpected inflation.
'See Lee [2011] for details of risk decompositions
bypositions.
"See Grinold and Kahn [1999, Chap. 3] for details of
riskdecompositions by risk factors.
^This result may appear to be in sharp contrast tostudies that
report high R-squared of using factors or styles ine.xplaining
portfolios' performance. Note that that style analysesof Sharpe
[1992], for example, as well as recent literature ofhedge funds
replication, typically use asset returns as factors orstyles in
capturing return and risk characteristics of portfolios.Factors
used in these studies, such as small size premium, valuepremium,
credit premium, and the like, are all constructedusing combinations
of different asset returns; therefore, theybypass the focus of our
example, which is to use fimdamentalfactors such as growth and
inflation to capture the risk char-acteristics of asset returns. As
such, we believe that the styleanalysis used in the industry can be
grouped into the asset classapproach instead of the risk class
approach.
"For examples, in equity, there are many specializedfirms that
offer their factor models of the covariance matrixof individual
stocks. These factors can include industry, style(e.g., value or
growth), characteristics (e.g., liquidity), andso on, in the
fundamental approach. Others may use a sta-tistical approach, such
as principal component analysis, tocome up with a set of
statistical factors, or combinations ofboth fundamental and
statistical approaches. Similarly, in thefixed-income world, factor
models of covariance matrices areequally popular, using interest
rate level and steepness andconvexity of the yield curve as
factors, among others.
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DisclosureThis article reflects the views ofthe authors and does
not reflect the officialviews ofthe authors' employer, Neuberger
Berman.
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