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

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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

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

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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 SPRING 2012

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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Fama, E.F., and M.R. Gibbons. "A Comparison of Infla-tion Forecasts." JoMWij/ of Monetary Economics, Vol. 13, No. 3(1984), pp. 327-348.

Grinold, R., and R. Kahn. Active Portfolio Management: AQuantitative Approach for Producing Superior Returns and Control-ling Risk, 2nd ed. New York, NY: McGraw-Hill, 1999.

Kritzman, M. "The Graceful Aging of Mean-Variance Opti-mization." TheJournal of Portfolio Management, Vol. 37, No. 2(Winter 2011), pp. 3-5.

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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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