(ebook) Goldman Sachs-Fixed Income Research

8/7/2019 (ebook) Goldman Sachs-Fixed Income Research

1/44

FixedIncomeResearch. . .

October 1991 Global AssetAllocation WithEquities, Bonds,and Currencies

Fischer BlackRdbert Litterman

43


2/44

FixedIncomeResearchAcknowledgements Although we don't have space to thank them each individu-

ally, we would like to acknowledge the thoughtful commentsand suggestions. that we have received from our many col-leagues at Goldman Sachs and the dozens of portfolio man-agers with whom we have discussed our.work this past year.In addition, we want to offer speeial thanks to Alex Bergier,''Ibm Iben,Piotr Karasinski, Scott Pinkus, Scott Richard, andKen Singleton for their careful reading of an earlier draft ofthis paper. We are also grateful to Greg Cumber, GaneshRam chandran, and Suresh Wadhwani for their valuableanalytical support.Fischer Black(212) 902-8859R.obert Iit term.an(212) 902-1677

Fischer Black is a Partner in Goldman Sachs Asset Manage-ment.Robert Litterman is a Vice President in the Fixed IncomeResearch Department at Goldman, Sachs & Co.

Editor: Ronald A.Krieger

Cop)'rigl&t" 1991 by Goldman, Sacha &: Co. Sixth PriDtiDg, October 1992' l "Jd. material is for your private iDformatioD, and _ are DOt aolicit.iDgany actionbaedupon it.CertaiD tnuISaCtioDa. mduding thOM in-tviDg futurea, optio-. andhigh yield securities. giw riM to I I11batantia l risk and are not auitabJe for aDin_ton. 0pini0D8 ~ are our praent opWcma only. The material is baaedupon inform~ that ,.. coD8ider teHable, but ,.. do not :rep:eeem that it isaccurate or complete, and it ahoWcl not be relied upon as .ueh. We, 01' pertIODainvohed in the preparation or iaauaDce of this material, may from time to timehave long or abort poaitioD8 in. and buy or . e n . MCUritiea, &turea, or optiollllideDtic:al with or related to thee. mentioned he:reiD. Further information on any oftheaeeurities, futwea, or opticma mentioned in this material may be obtained uponrequest. Thia material baa heen i B I I 1 1 e d by GoIdmaD. Sacha & Co . I IDI l has beenappzovecl by Goldman Sacha Ia tenaat iODal Limited, a member 0!The Sec:aziti_ and~ Authority, in connection with ita cUatribution inthe UDitecl Xiqdom, andby Goldman Sachs C._da in connecWm with itadistributiOD in: CaDad.a.

44


3/44

FixedIncomeResearchGlobal Asset Allocation WithEquities, Bonds, and Currencies

ContentsExecutive SummaryL Introduction. . . . . . . . . . ,. . . . .. 1

n. Neutral 'Views ...... 4Naiw Approaches .................... 6Kudorical Averages ....... 7Equal Means . . . . . . . . . . . .. 8Ris.,Adjusted EqU4l Means ..... 10The Equilibrium Approach . . . . . . . . . . 11IlL Esp~g~ews ..................... 13Iv. Combining Investor Views With Market

:Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15T'bree-.Asset Example . : .................. 16A Seven-Country Example ............... 21V. Controlling the Balance of a Portfolio 23VI. Benchmarks ................. 26VII. Implied Views .. . . . . . . . . . . . . . . . . . . . . . . . . ..... 28VIn. Quantifying the Benefits of Global

Diversification .. 29IX. mstorical Simulations 31X. Conclusion ........................ 36Appendi::lt. AMathematical Description of the

Black-Litterman Approach' .......... 37Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39References . . . . . . . . . . . . . . . . . . . . . . . . . . ..... '.' .... 40Selected Goldman Sachs Publications ............. 41

45


4/44

FixedIncomeResearchExecutive Summary A year ago, Goldman Sachs introduced a quantitative model thatoffered an innovative approach to the management of fixed incomeportfolios. It provided a mecbanism for investors to make global

asset allocation decisions by combining their views on expectedreturns with Fischer Black's "universal hedging" equilibrium. Giv-.en an investor's views about P i t e r E s t rates and exchange rates;this initial version of the B1ack-Litterman Global Asset AllocationModel has been used to generate portfolios with optimal weightsin bonds in different countries and the optimal degree of currencyexposure.In this paper, we describe an updated version of the Black-Litterman Model that incorporates equities as well asbonds andcurrencies. The new version of the model will be especially usefulto portfolio managers who make global asset allocation decisionsacross equity and fixed income markets, but it will also have advan-tages for pure fixed inCOlIlemanagers.The addition of the equity asset class to the model allows us to usean equibOrium based on both bonds and equities. This equilibriumismore desirable from a theoretical standpoint because it incorpo-rates a larger fraction of the universe of investment assets. In ourmodel (as in any Capital Asset Pricing Model equilibrium). theequilibrium expected excess return on an asset is proportional tothe covariance of the asset's return with the return of the marketportfolio. Even for pure fixed income managers, it is useful to use' ! lS broad a measure of the market portfolio as is practical.As we described in our earlier paper. the equilibrium is importantin the model because it provides a neutral reference point forexpected returns. This allows the investor to express views onlyfor the assets that he desires; views for the other assets are de-rived from the equilibrium. By providing a center of gravity forexpected returns, the equihbrium makes the model's portfoliosmore balanced than those from standard quantitative asset alloca-tion models. Standard models tend to choose unbalanced portfoliosunless artificial constraints are imposed on portfolio composition.

See Black aM Litterman (1990). [Note: a complete listing of references ap-. pears at the end of this report onpap 40.J

46


5/44

FixedIncomeResearchGlobal Asset Allocation WithEquities, Bonds and Currencies

I. Introduction Investors with global portfolioa of equities a n d bonds aregen!7ral1ya:vare that their asset all~cation .decisions -the proportions of funds that they Invest lD the asset.classes of different countries and the degrees of currencyhedging - are the most important investment decisions theymake. Inattempting to decide on the appropriate allocation,they are usually comfortable with the simplifying assump-tion that their objective is to maximize expected return forany given level ofrisk, subject inmost cases tovarious typesof constraints.Given the straightforward mathematics of this optimizationproblem, the many correlations among global asset classesrequired in measuring risk, and the large amounts ofmoneyinvolved, one might expect that in todays computerizedworld, quantitative models would playa dominant role inthis global allocation process.Unfortunately, when investors have tried to use quantitativemodels to help optimize this critical allocation decision, theunreasonable nature of the results has often thwarted theirefforts.I When investors impose no constraints, assetweights in the optimized portfolios almost always ordainlarge short positions in many assets. When constraints ruleout short positions, the models often prescribe "corner" solu-tions with zero weights in many assets, as well as unreason-ably large weights in the assets of markets with small capi-talizations.These unreasonable results have stemmed from two well-recognized problems. First, expected returns are very diffi-cult to estimate. Investors typieally have views about abso-lute or relative returns in only a few markets. In order touse a standard optimization model, however, they must statea set of expected returns for all assets and currencies. Thus,they must augment their views with a set of auxiliary as-

For some academic discussions of this issue, see Green and Hollifield(1990) and Best and Grauer (1991).

147


6/44

FixedIncomeResearchsumptions, and the historical returns that portfolio manag-ers often use for this purpose provide poor guides to futurereturns.Second, the optimal portfolio asset weights and. currencypositions in standard asset allocation models are extremely .sensitive to the expected return assumptions. rIle two prob-lems compound each other because the standard model hasno way to distinguish strongly held views from auxiliaryassumptions, and given its sensitivity to the expected re-turns, the optimal portfolio it generates often appears tohave little or no relation to the views that the investor wish-es to express. .Confronting these problems, investors are often disappointedwhen they attempt to use a standard asset allocation model.Our experience has been that in practice, despite the obviousconceptual attractions of a quantitative approach, few globalinvestment managers regularly allow quantitative models toplaya major role in their asset allocation decisions.In this paper we describe an approach that provides anintuitive solution to these two problems that have plaguedthe use of quantitative asset allocation models. The key to.our approach is the combining of two established tenets ofmodern portfolio theory: the mean-variance optimizationframework ofMarkowitz (1952) and the capital asset pricingmodel (CAPM) of Sharpe (1964) and Lintner (1965). Weallow the investor to combine his views about the outlook forglobal equities, bonds, and currencies with the risk premi-ums generated by Black's (1989) global version of the CAPMequilibrium. Tbese equilibrium risk premiums are the excessreturns that equate the supply and demand for global assetsand currencies.2As we will illustrate with examples, the mean-varianceoptimization used in standard asset allocation models isextremely sensitive to the expected return assumptions thatthe investor must provide. In our model, the equilibriumrisk premiums provide a neutral reference point for expectedreturns. This, intum, allows the model to generate optimalportfolios that are much better behaved than the unreason-able portfolios that standard models typically produce, which

2 For a guide to terms used inthis paper, see the G)~, page 39.2

48


7/44

FixedIncomeResearchoften include large long and short positions unless otherwiseconstrained. Instead, our model gravitates toward a' bal-anced - i.e., market-capitalization-weighted - portfolio but;tilts in the direction of assets favored by the investor's views.Our model-does not as~e that the world is always 'at the -CAPM equilibrium, but rather that when expected returnsmove away from their equilibrium values, imbalances in .markets will tend to push them back. Thus, we think it isreasonable to assume that expected returns are not likely todeviate too far from equilibrium values. This intuitive ideasuggests that the investor may profit by combining his viewsabout returns in different markets with the informationcontained in the equilibrium.In our approach, we distinguish between views of the investorand the expected returns that drive the optimization analysis,The equilibrium risk premiums provide a center of gravity forexpected returns. The expected returns that we use in theoptimization deviate from equilibrium risk premiums whenthe investor explicitly states views. The extent of the devia-tions from equihD~um depends on the degree of confidencethe investor expresses in each view. Our model makes adjust-ments in a manner as consistent as.pcssible with historicalcovariances ot returns of different assets and currencies.Our use of equilibrium allows investors to specify views in amuch more flexible and powerful way than is otherwisepossible. For example, rather than requiring a view aboutthe absolute returns' on every asset and currency, our ap-proach allows investors to specify as many or as few viewsas they wish. In addition, investors can specify views aboutrelative returns, and they can specify a degree of confidenceabout each of the views.In this paper, through a set of examples, we illustrate howthe incorporation of equilibrium into the standard assetallocation model makes it better behaved and enables it togenerate insights for the global investment manager. 'lb thatend, we start with a discussion of how equilibrium can helpan investor translate his views into a set of expected returnsfor all assets and currencies. We then follow with a set ofapplications of the model that illustrate how the equilibriumsolves the problems that have traditionally led to unreason-able results in standard mean-variance models.

349


8/44

Fixed, . ,comeResearchExhibit 1

mstoricaI Excess Returns(January 1975 through" August 1991)Total mstorical Excess BeturDs

Germany France Japan UJL u.s. Canada AustraliaCurrencies -20.8 3.2 23.3 13.4 12.6 3.0BondsCH 15.3 -2.3 42.3 21.4 -4.9 -22.8 -13.1Equities CH 112.9 117.0 223.0 291.3 130.1 16.7 107.8

Annualized Historical Excess ReturDsGermany France Japan UJL U.s. Cauada AustraliaCurrencies -1.4 0.2 1.3 0.8 0.7 0.2BondsCH 0.9 -0.1 2.1 1.2 -0.3 -1.5 -0.8EquitiesCH 4.7 4.8 7.3 8.6 5 . 2 0.9 4.5

Amaualized Volatility ofHistorical Excess ReturDsGermany France Japan UJL us, Cauada Australia

Currencies 12.1 11.7 12.3 11.9 4.7 10.3Bonds CH 4.5 4.5 6.5 9 . 9 6.8 7.8 5.5EquitiesCH 18.3 22.2 17.8 24.7 16.1 18.3 21.9Note: Bond and equity ezceae return. are inU.S. dollan c:uzremy hedged (CB). Excess returDa on bonds andequities are in exa:eaa of the London interbank o1l'eredrate (LIBOR). and those on aurencie. are in e:zceaaofthe one-month forward :rata. Volatilities are expreaaed as aJUlualiud .taDdard deviations.

II. Neutral Views Why should an investor use a global equilibriummodel to help make his global asset allocationdecision?A neutral reference is a critically impor-tant input in making use of a mean-variance optimizationmodel, and an equilibrium provides the appropriate neutralreference. Most of the time investors do have views - feel-ings that some assets or currencies are overvalued or under-valued at current market prices. An asset allocation modelshould help them to leverage those views to their greatestadvantage. But it is unrealistic to expect an investor to beable to state exact expected excess returns for every assetand currency; The purpose of the equilibrium is to give theinvestor an appropriate point of reference so that he canexpress his views in a realistic manner.

450


9/44

FixedIncomeResearchWe begin our discussion of how to combine views with theequilibrium by supposing that an investor has no views.What then is the optimal portfolio? Answering this questionis a sensible point of departure because it demonstrates theusefulness of the equilibrium risk premium as the appropri-ate point of.reference.We consider this question, and examples throughout thispaper; using historical data on global equities, bonds, andcurrencies. We use a seven-country model with monthlyreturns for the United States, Japan, Germany, France, theUnited Kingdom, Canada, and Australia from January 1975through August 1991.3Exhibit 1 presents the means and standard deviations ofexcess returns. The correlations are inExhibit 2 (pages 6-7).All results in this paper have a U.S. dollar perspective, butother currency points of view would give similar results;'

3 In actual applications or the model, we typically include more assetclasses and use daily data to more accurate1y measure the currentstate of the time-varying risk structure. We intend to address issuesconcerning uncertainty or the covariances. in another paper. For thepurposes of this paper, however. we treat the true cavariances ofexcessreturns as known.

.. We define excess return on eulrency hedged assets to be total returnless the short rate and excess return on currency positions to be totalreturn less the forward premium (see Glossary. page 39). In Exhibit 2,all ucess returns and volatility are in perc:ent. The currency hedgeducess return on a bond or an equity at time tisgiven by:

where P,is the price or the asset in foreign currency, ~ is the u-change rate in units of foreign currency per U.s. dollar, ~ is the d0-mestic short rate, and ~ is the return on a forwazd contract, all attime t.The return on a forward contract, or equivalently the excessreturn on a foreign currency. is given by:

F X , _ ( F : l;, X ' . l ) )C 100where F:+l is the one-period forward ezehange rate at time t.

551


10/44

FixedIncomeResearchExhibit 2

ffistorical Correlations of Excess Returns(January 1975 through August 1991)

Germany France JapanEquities Bonds Currency Equities Bonds Currency Equities Bonds CurrencyCH .C H CH CH CH CHGel'DlBDYEquities CH 1.00BondsCH 0.28 1.00Currency 0.02 - 0.36 LOOFnmcp.Equities CH 0.52 0.17 0.03 1.00Bonds CH 0.23 0.46 0.15 0.36 1.00Currency 0.03 0.33 0.92 0.08 0.15 1.00JapanEquities CH 0.37 0.15 0.05 0.42 0.23 0.04 1.00Bonds CH 0.10 0.43 0.27 0.11 0.31 0.21 0.35 1.00Currency 0.01 0.21 0.62 0.10 0.19 0.62 0.18 0.45 1.00U.K.Equities CH 0.42 0.20 -0.01 0.50 0.21 0.04 0.37 0.09 0.04Bonds CH 0.14 0.36 0.09 0.20 0.31 0.09 0.20 0.33 0.19Currency 0.02 0.22 0.66 0.05 0.05 0.66 0.06 0.24 0.54u.s.Equities CH 0.43 0.23 0.03 0.52 0.21 0.06 0.41 0.12 -0.02Bonds CH 0.17 0.50 0.26 0.10 0.33 0.22 0.11 0.28 0.18CanadaEquities CH 0.33 0.16 0.05 0.48 0.04 0.09 0.33 0.02 0.04BondsCH 0.13 0.49 0.24 0.10 0.35 0.21 0.14 0.33 0.22Currency 0.05 0.14 0.11 0.10 0.04 0.10 0.12 0.05 0.0&AustraliaEquities CH 0.34 0.07 -0.00 0.39 0.07 0.05 0.25 -0.02 0.12Bonds CH 0.24 0.19 0.09 0.04 0.16 0.08 0.12 0.16 0.09Currency -0.01 0.05 0.25 0.07 -0.03 0.29 0.05 0.10 0.27

Naive Approaches 'lb motivate the use of the eqUilibrium risk premiums as aneutral reference point,we first consider several other naiveapproaches investors might use to construct an optimalportfolio when they have no views about assets or curren-cies. Wewill call these the historical average approach, theequal mean approach, and the risk-adjusted equal meanapproach.

652


11/44

~FixedIncome

\,-j ResearchExhibit 2 (Continued)mstorica1 Correlations of Excess Returns

(January 1975through August 1991)

UnitedKingdom umted States C a D " " . AustranaEquities Bonds Currency Equities Bonds Equities Bonds Currency Equities BondsCH CH CH CH CH CH CH CHUXEquities CH 1.00BondsCH 0.47 1.00Currency 0.06 0.27 1.00us,Equities CH 0.58 0.23 -0.02 1.00Bonds CH 0.12 0.28 0.18 0.32 1.00CanadaEquities CH 0.56 0.27 0.11 0.74 0.18 1.00Bonds CH 0.18 0.40 0.25 0.31 0.82 0.23 1.00Currency 0.14 0.13 0.09 0.24 .0.15 0.32 0.24 1.00AustraliaEquities CH 0.50 0.20 0.15 0.48 -0.05 0.61 0.02 0.18 1.00Bonds CH 0.17 0.17 0.09 0.24 0.20 0.21 0.18 0.13 0.37 1.00'-. Currency 0.06 0.05 0.27 0.07 -0.00 0.19 0.04 0.28 0.27 0.20

Historical Averages The historical average approach defines a neutral position tobe the assumption that excess returns will equal their his-torical averages. The problem with this approach is thathistorical means are very poor forecasts of future returns.For example, in Exhibit 1 we see many negative values.Let's see what happens when we use these historical excess'returns as our expected excess return assumptions. Wemayoptimize expected returns for each level of risk to get a fron-tier of optimal portfolios..Exhibit 3 (page 8) illustrates thefrontiers with the portfolios that have 10.7% risk, with andwithout shorting constraints.5 .We may make a number of points about these "optimal"portfolios. First, they illustrate what we mean by "unreason-

6 We choose to normalize on 10.7% risk here and throughout this paperbecause it happens tobe the risk ofthemarket-alpitalization-weighted.80% currency hedged portf'olio that will be held inequilibrium inourmodel.7

53


12/44

FixedIncomeResearchable- when we claim that standard mean-variance optimiza-tion mOdels often generate unreasonable portfolios. Theportfolio that does not constrain against shorting has manylarge long and short positions with no obvious relationshipto the expected excess return assumptions. When we con-strain shorting we have positive weights in only two of the14 potential assets. These portfolios are typical of those thatthe standard model generates. .Given how we have set up the optimization problem, thereis no reason to expect that we would get a balanced set ofweights. The use of past excess returns to represent a "neu-tral- set of views is equivalent to assuming that the constantportfolio weights that would have performed. best historicallyare in some sense neutral. In reality, of course, they are notneutral at all, but rather are a very special set of weightsthat go short assets that have done poorly and go long assetsthat have done well in the particular historical period.

Equal Means Recognizing the problem of using past returns, the investormight hope that assuming equal means for returns across allcountries for each asset class would be a better representa-Exhibit 3

Optimal Portfolios Based onHistorical Average Approach(percent of portfolio value)

UnconstrainedGermany France Japan UA U.s. Canada Australia

Currency exposw-e -78.7 46.5 15.5 28.6 65.0 -5.2Bonds 30.4 -40.1 40.4 -1.4 54.5 -95.7 -52.5Equities 4.4 -4.4 15.5 IS.! 44.0 -44.2 9.0

With constraints against shorting assetsGermany France Japan UA U.s. Canada Australia

Currency exposure -160.0 115.2 18.0 23.7 77.8 -13.8Bonds 7. 6 0.0 88.8 0.0 0.0 0.0 0.0Equities 0.0 0.0 0.0 0.0 0 . 0 0 . 0 0.0

854


13/44


Optimal Portfolios Based on Equal Means(percent of portfoiia value)

UnconstrainedGermaDy France Japan u.x. u.s. Canada Australia

Currency exposure 14.5 -12.6 -0.9 4 . 4 -18.7 -2.1Bonds -11.6 4 . 2 -1.8 -10.8 13.9 -18.9 -32.7Equities 21.4 -4.8 23.0 -4.6 32.2 9.S. 10.5

With constraints against shorting assetsGermany France Japan U.x. U.s. cimada Australia

Currency exposure 14.3 - U . 2 -4.5 0 . 2 -25.9 -2.0Bonds 0.0 0 . 0 0 . 0 0 . 0 0 . 0 0.0 0 . 0Equities 17.5 0 . 0 22.1 0.0 2 7 . 0 8.2 7.3

tion of a neutral reference. We show an example Q f the opti-mal portfolio for this type of analysis inExhibit 4. Again, weget an unreasonable portfolio.6 --Of course. one problem with the equal means approach isthat equal expected excess returns donot compensate inves-tors appropriately for the different levels of risk in assets ofdifferent countries. Investors diversify globally to reducerisk. Everything else being equal, they prefer assets whosereturns are less volatile and less correlated with those ofother assets.Although such preferences are obvious, it is perhaps surpris-ing howunbalanced the optimal portfolio weights can be, asExhibit 4 illustrates. when we take -everything else beingequal- to such a literal extreme. With no constraints, thelargest position is short Australian bonds.

II For the purposes of this ezercise, we arbitrarily assigned to each coun-try the average historical ezcess return ac:roes countries, as follows: 0.2for currencies, 0.4 01' bonds, and 5.1 Orequities.

955


14/44

FixedIncomeResearchRisk-Adjusted Equal Means Our third naive approach to defining a neutral referencepoint is 'to assume that bonds and equities have the sameexpected excess return per unit of risk, where the risk mea-sure is simply the volatility of asset returns. Currencies in

this case are assumed to have no excess return, We show theoptimal portfolio for this ease inExhibjt 5. Nowwe haveincorporated volatilities, but the portfolio behavior is' no. better. One problem with this approach is. that it hasn'ttaken the correlations of the asset returns into account. Butthere is another problem as well - perhaps more subtle, butalso more serious.So far, the approaches we have used to try to define theappropriate neutral means when the investor has no viewshave been based on what might be called the "demand forassets- side of the equation - that is, historical returns andrisk measures. The problem with such approaches is obviouswhen we bring in the supply side o C the market.Suppose the market portfolio comprises 80% of one assetand 20% of the other. In a simple world, with identical in-vestors all bolding the same views and both assets having

Exhibit 5Optimal Portfolios Based onEqual Risk-Adjusted Means

(percent of portfolio value)

UDCODStraiDeclGermany France Japan U.K. u.s. Canada Australia

C~ncy exposure 5 . 6 11.3 -28.6 -20.3 -50.9 -4.9Bonds -23.9 12.6 54.0 20.8 23.1 3 7 . 3 15.6Equities 9.9 3 . 5 12.4 -0.3 -14.1 13.2 20.1

With constraints against shorting assetsGermany France Japan U.K. U.s. Canada Australia

Currency exposure 21.7 -8.9 -14.0 -12.2 -47.9 -6.7Bonds 0.0 0 . 0 0.0 7 . 3 0.0 19.3 0 . 0Equities u.i 9 . 4 19.2 6 . 0 0.0 7 . 6 19.5

1056


15/44

FixedIncomeResearchequal volatilities, everyone cannot hold equal weights ofeach asset. Prices and expected excess returns in such aworld would have to adjust as the excess demand for oneasset and exce~ supply of the o~. affect the market.

The Equih"brium Approach . 'lb us, the only sensible definition of neutral means is theset of expected returns that would "clear the market- if allinvestors had identical views. The concept of equilibrium inthe context of a global portfolio of equities, bonds, and cur-rencies is similar, but curreneies do raise a complicatingquestion. How much currency hedging takes place in equilib-rium? The answer, as described in Black (1989), is that in aglobal equilibrium investors worldwide will all want to takea small amount of currency risk.This result arises because of a curiosity known in the cur-rency world as Siegel's paradox. - The basic idea is thatbecause investors in different countries measure returns indifferent units, each will gain some expected return by tak-ing some currency risk. Investors will accept currency riskup to the point where the additional risk balances the ex-pected return. Under certain simplifying assumptions, thepercentage of foreign currency risk hedged will be the samefor investors of different countries - giving rise to the name"unrversal hedging- for this equilibrium.The equilibrium degree of hedging - the "universal hedgingconstant- - depends on three averages: the average acrosscountries of the mean return on"the market portfolio of as-sets, the average across countries of the volatility of theworld market portfolio, and the average across all pairs ofcountries of exchange rate volatility.It is difficult to pin down exactly the right value for theuniversal hedging constant, primarily because the risk pre-mium on the market portfolio is a difficult number to esti-mate. Nevertheless, we feel that universal hedging valuesbetween 75% and 85% are reasonable. In our monthly dataset, the former value corresponds to a risk premium of 5.9%on U.S. equities while the latter corresponds to a risk premi-um of 9.8%. For the purposes of this paper, we will use anequilibrium value of 80% currency hedging.

57 11


16/44


Equilibrium Risk Premiums(percent annualized excess return)

GermanyCurrencies 1.01Bonds 2.29Equities 6.27

FranceLIO2.238.48

Japan1.402.888.72

U.K.0.913.2810.27 1.877.32

u.s. Cauada Australia0.60 0.632.54 1.747.28 6.45

Exhibit 6 shows the equilibrium risk premiums for all as-sets, given this value of the universal hedging constant. 7Let us consider what happens when we adopt these equilib-rium risk premiums as our neutral means when we have noviews. Exhibit 7 shows the optimal portfolio. Itis simply themarket capitalization portfolio with 80%of the currency riskhedged. Other portfolios on the frontier with different levels/ofrisk would correspond to combinations ofrisk-free borrow-ing or lending plus more or less of this portfolio.

By itself, the equilibrium is interesting but not particularlyuseful. The real value of the equilibrium is to provide aneutral framework to which the investor can add his ownperspective in terms of views, optimization objectives, andconstraints. These are the issues towhich we now turn.

'1 The "universal hedging" equilibrium is, ot course, based on a set simpli-fying assumptions, such as a world with no taxes, no capital con-straints, no inftation, etc:.Exchange rates in this world are tberates otexchange between the dift"erent consumption bundles of individuals ofdifferent countries. While some may find the assumptions that justifyuniversal hedging overly restrictive, this equilibrium does have thevirtue ofbeing simpler than other global CAPM equilibriums that havebeen described elsewhere, such as inSolnik (1974) or Grauer, Litzen-berger, and Stehle (1976). While these simplifying assumptions arenecessary to justifY the universal hedging equib'brlum, we could easilyapply the basic idea of this peper - to combine a global equilibriumwith investor views - to another global equilibrium derived from adifferent, less restrictive, set of assumptions.

1258


17/44

FixedIncomeResearchExhibit 7EqullibriumOptimal Portfolio(percent of portfolio value).

G e r m a n y France JiapaD UJL U.s. Canada Australia .Currency exposure 1.1. 0.9 5.9 2.0 O . S 0.3Bonds 2 . 9 1.9 6.0 1.8 16.3 1.4 0.3Equities 2 . 6 2.4 23.7 8.3 29.7 1 . S 1 . 1

m. ~ressingViews The basic problem that confronts investors trying touse quantitative asset allocation models is how totranslate their views into a complete set of expectedexcess returns on assets that can be used as a basis forportfolio optimization. As wewill show here, the 'problem isthat optimal portfolio weights from a mean-variance modelare incredibly sensitive tominor changes in expected excessreturns. The advantage of incorporating a global equilibriumwill become apparent when we show how to combine it withan investor's views togenerate well-behaved portfolios, with-out requiring the investor to express a complete 'set of ex-pected excess returns.We should emphasize that the distinction we are making -between investor views on the one hand and a complete setof expected excess returns)"or all assets on the other - isnot usually recognized. Irrour approach, views represent thesubjective feelings of the investor about relative values of-fered in different markets.8 If an investor does not have' aview about a given market, he should not have to state one.Ifsome of his views are more strongly held than others, theinvestor should be able to express that difference. Mostviews are relative - for example, when an investor feels onemarket will outperform another or even when he feels bull-ish (above neutral) or bearish (below neutral) about a mar-ket. As we will show, the equilibrium isthe key to allowingthe investor ~ express his views this way instead ofas a setof expected excess returns. .

As we will show in Section IX,views can also represent feelings aboutthe relationships between observable conditions and such relatiwvalues.1359


18/44

FixedIncomeResearch'Ib see why this is so important, we start by illustrating theextreme sensitivity of portfolio weights to the expected ex-cess returns and the inability of investors to express viewsdirectly as a complete set of expected returns. We have al-ready seen how 'difficult it can be simply to translate noviews into.a. set of expected excess returns thabvill not lead'to an unreas.onable portfolio in an asset allocation model.But let us suppose that the investor has' already solved that'problem and happens to start with the equilibrium riskpremiums as his neutral means. He is comfortable with aportfolio that bas the market capitalization weights, 80%hedged. Consider what can happen when this investor now'tries to express one simple, extremely modest view.In this example, we suppose the investor's view is that overthe next three months it will become increasingly apparentthat the economicrecovery in the United States will beweak .and bonds will perform relatively well and equities poorly.We suppose that the investor's view isnot very strong,9 andhe quantifies this view by assuming that over the next threemonths the U.S. benchmark bond yieldwill drop 1basis point(bp) rather than rise 1 bp, as is consistent with the equilibri-um risk premium. Similarly, the investor expects U.S. shareprices to rise only 2.7% over the next three months ratherthan to rise 3.3%, as is consistent with the equilibrium.'lb implement the asset allocation optimization, the investorstarts with the expected excess returns equal to the equilib-rium risk premiums and adjusts them as follows: he movesthe annualized expected ex~ returns on U.S. bonds up by0.8 percentage points and expected excess returns on U.S.equities down by 2.5 percentage points. AU other expectedexcess returns remain unchanged. In Exhibit 8 we show theoptimal portfolio given this view.Note the remarkable effect of this very modest change in theexpected excess returns. The portfolio weights change indramatic and largely inexplicable ways. The optimal portfo-lio weights do shift out of U.S. equity into U.S. bonds, butthe model also suggests shorting Canadian and Germanbonds. This lack of apparent connection between the views

t Inthis paper. we use the term -strength- ofa ,new to refer- to itsmag-nitude. w. reeerve the term -confidence- to refer to the degree of cer-tainty with wbich it isheld.

1460


19/44


20/44

FixedIncomeResearch(3) We choose expected excess returns that are as consis-

tent as possible with both sources of information.The above description captures the basic idea, but the imple-mentation of this approach can lead to some novel insights.For example, we will now-show how Iir_elative View about-two assets can influence the expected excess return on athird asset. 10 -. .

Three-Asset Example Let us first work through a very simple example of our ap-proach. After this illustration, we will apply it in the contextof our seven-country model. Suppose we know the true struc-ture of a world that has just three assets: A, B, and C. Theexcess return for each of these assets is known to be gener-ated by an equilibrium risk premium plus four sources ofrisk: a common factor and independent shocks to each of thethree assets.We can write this model as follows:

RA=XA+TAxZ+'UA~ = " 1 3 + TBxZ +.~Rc :: Xc + TcxZ + Vcwhere:

~ is the excess return on the jthasset,Xi is the equilibrium risk g;emium on the jth asset,'Y i is the impact on the i asset of Z, the commonfactor, andVi is the independent shock to the ith ~set.

Inthis world, the covariance matrix, 1:, of asset excess re-turns is determined by the relative impacts of the commonfactor and the independent shocks. The expected excessreturns of the assets are a function of the equilibrium riskpremiums, the expected value of the common factor, and theexpected values of the independent shocks to each asset.

18 In this section, we try to develop the intuition behind our approachusingsome basic concepts of statistics and matrix algebra. We providea more formal mathematical description inthe Appendix.16

62


21/44

FixedIncomeResearchFor example, the expected excess retum of asset A, whichwe writ~ as E[RA), is given by: .

E[R,J = xA + lAxE[Z] + E[vA)We are not assuming that the world i s in equilibrium, i.e.,that E[Z] and the E["i]s are equal to zero..We do assumethat the mean, E[RA]' is itself an unobservable random vari-able whose distribution is centered at the equilibrium riskpremiums. Our uncertainty about E[RA] is due to our uncer-tainty about E[Z] and the E["~. Furthermore, we assumethe degree of uncertainty about E[Z] and the E[Vi]S is pro-portional to the volatilities of Z and the "is themselves.This implies that E[RA) is distributed with a covariancestructure proportional to ~. We will refer to this covariancematrix of the expected excess returns as 'tl:. Because theuncertainty in the mean is much s~aller than the uncertain-ty in the return itself, 'twill be clo~ to zero. The equilibriumrisk premiums together with'tl: determine the equilibriumdistribution for expected excess returns. We assume thisinformation is known to all; it is not a function of the cir-cumstances of a n y individual investor.In addition, we assume that each investor provides addition-al information aboutexpected excess returns in the form ofviews. For example, one type of view is a statement of theform: -I expect Asset A to outperform Asset B by Q,.whereQ is a given value.We interpret such a view to mean that the investor hassubjective information about the future returns ofA relativetoB. One way we think about representing that informationis to act as ifwe had a summary statistic from a sample ofdata drawn from the distribution of future returns - data inwhich all we were able to observe is the difference betweenthe returns of A and B. Alternatively, we can express thisview directly as a probability distribution for the differencebetween the means of the excess returns ofA and B. It does-n't matter which of these approaches we want to use tothink about our views; in the end we get the same result.In both approaches, though, we need a measure of the inves-tor's confidence in his views. We use this measure to deter-mine how much weight to give to the view when combining

1763


22/44

FIXedIncomeResearchit with the equilibrium. We can think of this degree of confi-dence inthe first case as determining the number of obser-vations that we have from the distribution of future returns,in the second as determining the standard deviation of theprobability distribution.In our example, consider the limiting case: the investor is100% sure of his one view. We might think of that as thecase where we have an unbounded number of observationsfrom the distribution of future returns, and that the averagevalue of RA - ~ from these data is Q. In this special case,we can represent the view as a linear restriction on theexpected excess returns, Le. E[RA] - E[R:sl = Q..Indeed, in this special, case we can compute the distributionof E[R] = {E[RA],E[R:sl, ERcD conditional on the equilibri-um and this information. This is a relatively straightforwardproblem from multivariate statistics. 'Ib simplify, let usassume a normal distribution f~ the means of the randomcomponents. . \We have the equilibrium distribution for E[R], which isgiven by Normal (li, '(1:), where" = {"A' ~ "ol. We wish tocalculate a conditional distribution for the expected returns,subject to the restriction that the expected returns satisfythe linear restriction: ElRA].- E[R:sl = Q.Let us write this restriction as a linear equation in. the ex-pected returns:l1

P x E[R)' = Q where P is the vector [1, -1. 0).The conditional normal distribution has the mean

x' + 'tl:.x P' x [P x -r1: x PT1x [Q - P x "1.which is the solution to the problem of minimizing

(E[R] - ,,) ('(1:)"1 (E[R) - x)' subject to P x E[R]' = Q.For the special case of 100% confidence in a view, we usethis conditional mean as our vector of expected excess re-turns. In the more general case where we are not 100%

11 A" . boI< P')' di --~ pnme sym e.g., m cates a _ ........~ vector ormatrix.18

64


23/44

FixedIncomeResearchconfident, we can think of a view as representing a fixednumber of observations drawn from the distribution of fu-ture returns - in which case we follow the "mixed estima-tion- strategy described in Theil (1971), or alternatively asdirectly refiecting a subjective distribution for the expectedexcess returns - in which case we use the technique givenin the Appendix. The "formula for the expected exceSs returnsvector is the same from both perspectives.Ineither approach, we assume that the view can be summa-rized by a statement of the form P x E[Rl' =Q + t,where Pand Q are given and e is an unobservable, normally distrib-uted random variable with m~ 0 and variance Q. Q repre-sents the uncertainty inthe ~ew. In the limit as Q goes tozero, the resulting mean converges to the conditional meandescribed above.When there is more than one view, the vector of views canbe represented by P x E[R]' = Q + e, where we now interpretP as a matrix, and e is a normally distributed random vectorwith mean 0 and diagonal covariance matrix Q. A diagonalQ corresponds to the assumption that the views representindependent draws from the future distribution of returns,or that the deviations of expected returns from the means ofthe distribution representing each view are independent,depending on which approach is used to think about subjec-tive views. In the Appendix, we give the formula for theexpected excess returns that combine views with equilibriumin the general case. .Now consider our example, in which correlations amongassets result from the impact of one common factor. In gen-eral, we will not know the impacts of the factor on the as-sets, that is the values of lA' lB, and Tc. But suppose theunknown Values are [3, 1, 2]. Suppose further that the inde-pendent shocks are small so that the assets are highly corre-lated with volatilities approximately in the ratios 3:1:2 -.For example, suppose the covariance matrix is as follows:

19.1 3.03.0 1.16.0 2.0

6.0 12 . 04 . 1

1965


24/44

FixedIncomeResearchAlso, fQr simplicity, let the equilibrium risk premiums (inpercent) be equal - for example, [1, 1, 1]. There is a set ofmarket capitalizations for which that is the case.Now consider what happens when we specify a view that Awill o~tperform B by 2.ln this exauiple; since virtually all ofthe volatility of the assets is associated with movements in~e common factor, we clearly. ought to impute from thehigher retuni ofA than ofB (relative to equilibrium) that ashock to the common factor is the most likely reason A willoutperform B.Ifso, C ought to perform better than equilibri-umaswell. ~The conditionai mean in this case is [3.9, 1.9, 2.9J,.and in-deed, the view of A relative to B has raised the expectedreturns on C by 1.9.Naw suppose the independent shocks have a much larger im-pact than the common factDr. Let the 1:matrix be as follows:

[19.0 . 3.03.0 11.06.0 2.0

6.0 1 .2.014.0 .Suppose the equilibrium risk premiums are again given by[1, 1, 1], and again we specify a view that A will outperformBby2. .This time, more than half of the volatility of A is associatedwith its own independent shock not related to movements inthe common factor. Now, althoughwe ought to impute somechange inthe factor from the higher return of A relative toB, the impact on C should be less than in the previous case.Inthis case, the condItional mean is [2.3, 0.3, 1.3]. Here theimplied effect of the common factor shock on asset C is lowerthan in the previous ease. We may attribute most of theoutperformance of A relative to B to the independent shocks;indeed, the implication for E(R~ isnegative relative to equi-librium. The impact of the independent shock to B is expect-. ed todominate, even though the contribution of the commonfactor to asset B is positive.Notice that we ~ identify the impact of the common factorOnly if we assume that we know -the true' structure that, .

66


25/44

FixedIncomeResearchgenerated the covariance matrix of returns. That is truehere, but it will not be true in general. The computation ofthe conditional mean, however, does not depend on this spe-cial knowledge, but only on the covariance matrix of returns.. .. . .Finally, let's look at the case ~ere we have less cOIm~enc~'in our view, so that we might say (E[RA] - E[RJJ> has .amean of 2and a variance of 1.Consjder the original case, where the covariance matrix ofreturns is:

[ 9.13.06.03.0 6.0 11.1 2.02.0 4.1

The difference is that inthis example the conditional meanis based on an uncertain view. Using the formula given inthe Appendix, we find that the conditional mean is given by:[3.3, 1.7, 2.51.

Because we have stated less confidence inour view, we haveallowed the equilibrium to pull us away from our expectedrelative returns of 2 for A - B to a value of 1.6, which iscloser to the equilibrium value of O . We also find a smallereffect of the common factor on the third asset because of theuncertainty expressed in the view.

A Seven-Country Example Now we apply this approach to our actual data. We will tryto represent the view described previously on page 14thatbad news about the U.S. economy will cause U.S. bonds tooutperform U.S. stocks.The critical- difference between our approach here and ourearlier experiment that generated Exhibit 8 is that here wesay something about expected returns on U.S. bonds versusU.S. equities and we allow all other expected excess returnsto adjust accordingly. Above we adjusted only the returns toU.S. bonds and U.S. equities, holding fixed all other expect-ed excess returns. Another difference is that here we specifya differential of means, letting the equilibrium determinethe actual levels ofmeans; above we had to specify the levelsdirectly.

2167


26/44


Expected Excess ReturnsCombining Investor ViewsWith Equilibri~(annualized pe~nt)

Germany France Japan UK U.s. Canada AustraliaCurrencies 1.32 1.28 1.73 1.22 0. 0.47Bonds" 2.69 2.39 3.29 3.40 2.39 2.70 1.35Equities " 5.28 6.42 7.71 7.83 4.39 4.58 3 . 8 6

InExhibit 9 we show the complete set of expected excessreturns when we put 100% confidence in a view that thedifferential of expected excess retums of U.S. equities overbonds is 2.0 pereentage pointe, below the equilibrium differ-ential of 5.5 percentage points. Exhibit 10 shows the optimalportfolio associated with this view.This is in contrast to the inexplicable results we saw earlier.We see here a balanced portfolio inwhich the weights havetilted away from market capitalizations toward U.S. bondsand away from U.s. equities. Given our view, we now obtaina portfolio that we consider reasonable.

Exhibit 10Optimal PortfolioCombining Investor ViewsWith Equilibrium(percent of portfolio value)

GenDaDy Franee Japan UA u.s. Canada AustraliaCurrency exposure 1 . 4 1 . 1 7 . 4 2 . 5 0 . 8 0 . 3Bonds 3 . 6 2 . 4 7 . 5 2 . 3 6 7 . 0 1 . 7 0 . 3Equities 3 . 3 2 . 9 29.5 1 0 . 3 3 . 3 " 2 . 0 1 . 4

'-_/22

68


27/44

FixedIncomeResearchV. Controlling theBalance of

aPortfolio I"n the previous section, we illustrated how our approachallows us to express a view that U.S. bonds will outper-form U.S. equities, ina way that leads to a well-behavedoptimal portfolio that expresses that view. In this section,we focus more specifically on the concept of ~ "balanced" .portfolio" and show an ~aditiOnaI "feature of our approach:Changes in the "confidence" inviews can be used to C O D t I ' c : > lthe balance of the optimal portfolio.We start by illustrating what happens when we put a set ofstronger views, shown inExhibit 11, into our model. Thesehappen to have been the short-term interest rate and ex-change rate views expressed by our Goldman Sachs econo-

Exhibit 11Goldman Sachs Economists' Views

CurreDciesGer.many Prance Japan v.x. v.s. CaDada AustraliaJuly 31. 1991Current Spot Rates 1.743 5.928 137.3 1.688 L151 1.235

Three-M011th HorizonExpected Future Spot 1.790 6.050 141.0 1.640 1.000 1.156 1.324Annualized ExpectedExcess Returns -7.48 -4.61 -8.35 -6.16 0.77 -8.14

IDterest RatesGer.many France Japan VA V.s. CeDad. AustraliaJuly 31, 1991Benchmark Bcod Yields 8.7 9.3 6.6 10.2 8.2 9.9 11.0

Tbree-Month Horizon&peeted Future Yields 8.8 9.5 6.5 10.1 8 . 4 10.1 10.8Annualized ExpectedExcess Returns -3.31 -5.31 1.78 1.66 -3.03 -3.48 5.68

2369


28/44


Optimal Portfolio Based on Economists' Views(percent of portfoliC value)

UDCODStra iDed /Germany France Japan UK U.s. CSDBda Australia

Currency exposUre 16.3 68.8 -35.2 -12.7 29.7 -51.4Bonds 34.5 - 6 5 . 4 79.2 16.9 3.3 -22.7 108.3Equities -2.2 0.6 6.6 0.7 3.6 5.2. 0.5

mists on July 31, 1991.12We put 100% confidence in theseviews, solve for the expected excess returns on all assets,and find the optimal portfolio shown inExhibit 12. Givensuch strong views on so many assets, and optimizing without constraints, we generate a rather extreme portfolio.Analysts have tried a number of approaches to amelioratethis problem. Some put constraints .on many of the assetweights. However, we resist using such artificial constraints.When asset weights run up against constraints, the portfoliooptimization no longer balances return and risk across allassets. Others specify a benchmark portfolio and limit therisk relative to the benchmark until a reasonably balancedportfolio is obtained. This makes sense if the objective of theoptimization is to manage the portfolio relative to a bench-mark,13 but again we are uncomfortable when it is usedsimply to make the model better behaved.An alternate response when the optimal portfolio seems tooextreme is to consider reducing the confidence expressed insome or all o(tbe views. Exhibit 13shows an optimal portfo-liowhere we have lowered the confidence in all of our views.By putting less confidence in our views, we have generated

12 For details of these views, see the following Goldman Sachs publica-tions: The I~ Fixed Income ~st. August 2, 1991, forinterest rates and TM Int e rnD .t io r uJ Econom i cs Analy., July/August1991, for exchange rates.13 We discuss this situation in Section VI.

2470


29/44

FixedIncomeResearcha set of.expected excess returns that more strongly reftectthe equilibrium and have pulled the optimal portfolioweights toward a m~ balanced ~sition~Let us now explain precisely a property of a portfolio thatwecall "balanee," We define balance as a measure of how simi-lar a portfolio is to'theglobal equilibrium portfolio"'::" that is,themarket capitalization portfolio with 80% of the currencyrisk liedged. The distance measure that we use is the volatil-ity of the difference in returns of the two portfolios.We\find this property of balance to be a useful 'supplementto the standard measures of portfolio optimization, expectedreturn and risk. Inour approach, for any given level of nskthere will a continuum of portfolios that maximize expectedreturn depending on the relative levels of confidence thatare expressed in the views. The less confidence ~e investorhas, the more balanced will be bis portfolio. 'Suppose that an investor does not have equal confidence inall 'his views. Ifthe investor is willing to rank the relativeconfidence levels of his dift'erentviews, then he can generatean even more powerful result. In ,this case, the model willmove away from bis less strongly held views mOre quicklythan from those in which he has more confidence. For exam-ple, we have specified higher confidence in our view of yielddeclines in the United Kingdom and yield increases inFrance and Germany. These are not the biggest yield chang-es that we expect, but they are the forecasts that we moststrongly want to represent in our portfolio. In particular,. we .put less confidence in our views of interest rate moves in theUnited States and Australia. .

Exhibit 13Optimal Portfolio With Less Confidence

in the Economists' Views(percent of portfolio value)

Germany France Japan 11.K.. 1 1 . 8 . Canad. AuatreliaCurrency exposure -12.9 -3.5 -10.0 -6.9 -0.4 -17.9Bonds -3.9 -21.0 19.6 2.6 7.3 -13.6 42.4Equities 0.8 2.2 24.7 7.1 26.6 4.2 1.2

'-...._.

2571


30/44

FixedIncomeResearchBefore, when we put equal confidence in our views, we ob-tained the optimal portfolio shown in Exhibit 13. The viewthat dominated that portfolio was the interest rate declin-einAustralia. Now, when~e put less than 100% confidence Inour views, we keep rela'tively more confidence in. the viewsabout some countries than others. -Exhibit 14- shows theoptimal- portfolio for this case, in which the weights- repre-senting more strongly held views- are larger. -

VI. Benchmarks

One of the most important, but often overlooked, influ-ences on the asset allocation decision is the choice ofthe benchmark by which to measure risk. In mean-variance optimization, the objective is to maximize returnper unit of portfolio risk. The investor's benchmark definesthe point of origin for measuring this risk - in other words,the minimum risk portfolio.

In many investment problems,-risk is measured, as we havedone so far in this paper, as the volatility of the portfolio'sexcess returns. This can be interpreted as having no bench-mark, or as defining the benchmark to be a portfolio 100$invested in the domestic short-term interest rate. In manycases, however, an alternative benchmark is called for.Forexample, many portfolio managers are given an explicitperformance benchmark, such as a market-capitalization-weighted index. If such an explicit performance benchmarkexists, then investing in bills is clearly a risky strategy, andthe appropriate measure of risk for the purpose of portfoliooptimization is the volatility of the tracking error of theportfolio vis-a-vis the benchmark.Exhibit 14

Optimal Portfolio With Less ConfidenceinCertain Views(percent of portfolio value)

Germany France Japan U.K. u.s. Canada AustraliaCurrency exposure -10.0 -0.4 -4_8 -2.8 -6.2 . -7.8Bonds -10.3 -34.3 25.5 1.6 22.9 -2.4 28.1Equities 0.1 2.3 25.9 7 . 0 26.3 6 . 0 1.3

2672


31/44

FixedIncomeResearchAnother example of a situation with an obvious alternativebenchmark is that of a manager funding a known set ofliabilities. In such a ~eJ the benchmark portfolio repre-sents the liabilities.Unfortunately, for many portfolio m~rs,tbe objectivesfall into a less well-defined middle ground, and the assetallocation decision is made diffiCUltby the "fact that theirobjective is implicit rather than explicit. For example, aglobal equity portfolio manager may feel his objective is toperform among the top rankings of all global e~ity manag-ers. Although he does not have an explicit performancebenchmark, his risk is clearly related to the stance of hisportfolio relative to the portfolios of his competition.Other examples are an overfunded pension plan or a univer-sity endowment where matching the measurable liability isonly a small part of the total investment objective. In thesetypes of situations, attempts to use quantitative approachesare often frustrated by the ambiguity of the investmentobjective.When an explicit benchmark does not exist, two alternativeapproaches "canbe used. The first is to use the volatility ofexcess returns as the measure of risk. The second is to speci-fy a "normal" portfolio - one that represents the desiredallocation of assets in the absence of views. Such a portfoliomight, for example, be designed with a higher-than-marketweight for domestic assets, to represent the domestie natureof liabilities without attempting to specify an explicit liabili-ty benchmark.An equilibrium model can help in the design of a normalportfolioby quantifying some of the risk and return tradeoffsbetween different portfolio weights in the absence of views.For example, the optimal portfolio in equilibrium is market-capitalization-weighted with 80% of the currency hedged. Ithas an expected return (using equilibrium risk premiums) of5.7%with an annualized volatility of 10.7%.A pension fund wishing to increase the domestic weight to85% from the current market capitalization of 45S, and notwishing to hedge the currency risk of the remaining 15% ininternational markets, might consider an alternate portfolio

2773


32/44

FixedIncomeResearchsuch as the one shown in E~bit 15. The higher domesticweights lead to 0.4 percen~ points higher annualized vola-tility and an expected excess return 30 bp below that of theoptimal portfolio. The pension fund mayor may nQtfeel thatits preference fordomestic concentration isworth those costs.. '. . ~

VII. Implied Views Once an investor has established his objectives, anasset allocation model defines a correspondence be-tween views and optimal portfolios. Rather thantreating a quantitative model as a black box, successful port-foliomanagers use a model to investigate the nature of thisrelationship. In particular, it isoftenuseful to start an analy-sis by using a model to find the implied investor views forwhich an existing portfoliois optimal relative toa benchmark.'Ib illustrate this type of analysis, we assume that a portfoliomanager has a portfoliowith weights as shown in Exhibit16. The weights, relative to those of his benchmark, definethe directions oftbe investor's views.By assumingthe inves-tor's degree of risk aversion, we can find the expected excessreturns for which the portfolio is optimal. In this type of.analysis, different benchmarks may imply very differentviews for a given portfolio. In Exhibit 17 (page 30)we showthe implied views of the portfolio shown in Exhibit 16, withthe benchmark alternatively (1) the market capitalizationweights, 80%hedged, or (2) the domestic weighted alterna-tive shown in Exhibit 15. Unless a portfolio manager basthought carefully about what his benchmark is and wherehis allocations are relative to it, and has conducted the typeof analysis shown here, he may not have a clear idea whatviews are being represented in his portfolio. .

Exhibit 15Alternative Domestic Weighted Benchmark Portfolio

(percent of portfolio value)

Germany France Japan U.K. U.s. Canada AustraliaCurrency exposure 1 . 5 1 . 5 7 . 0 3 . 0 2 . 0 0 . 0Bonds 0 . 5 0 . 5 2 . 0 1 . 0 3 0 . 0 1 . 0 0 . 0Equities 1 . 0 1 . 0 5 . 0 2 . 0 5 5 . 0 1 . 0 0 . 0

2874


33/44

FixedIncomeResearch\

Exhibit 16Current Portfolio Weights for Implied ViewAnalysis.(percent of portfolio value)

France Japan3.4 2.00.5 4.72.9 22.3

u.s. Canada Australia2.0 5.50.3 3.51.7 2.0

~yCurrency exposure 4.4Bonds 1.0Equities 3.4

u.x.2. 22.510.2 13.032.0

VIII. Quantifying theBenefits. of GlobalDiversification While it has long been reCOgDl.zedthat most inves-tors demonstrate a substantial bias toward domes-. tic assets, many recent studies have documented arapid growth in the international component in portfoliosworldwide. It is perhaps not surprising, then, that there has .been a reaction among many investment advisers, who havestarted to question the traditional arguments that supportglobal diversification. This has been particularly true in theUnited States, where global portfolios have tended tounder-perform domestic portfolios in recent years.Of course, what matters for investors is the prospectivereturns from international assets, and as noted in our earli-er discussion of neutral views, the historical returns are ofvirtually no value in projecting future expected excess re-turns. Historical analyses continue to be used in this contextsimply because investment advisors argue there is nothingbetter to measure the value of global diversification.Wewould suggest that there is something better. A reason-able measure of the value of global diversification is thedegree to which allowing foreign assets into a portfolio rais-es the optimal portfolio frontier. Anatural starting point forquantifying this value is to compute it based on the neutralviews implied by a global CAPM equilibrium. There aresome limitations to using this measure. It assumes thatthere are no extra costs to international investment; thus,relaxing the constraint against international investmentcannot make the investor worse off. On the other hand, inmeasuring the value of global diversification this way, weare also assuming that markets are efficient and therefore

2975


34/44

FixedIncomeResearchwe are neglecting to capture any value that an internationalportfolio manager might add through having informed viewsabout these markets. We suspect that an important benefitof international investment that we are missing here is thefreedom it gives the portfolio manager to take adv.antage of .a larger number of opportunities to add-value than are at- .forded iii domestic markets. .In any case,' a s an illustration of the value of the equilibriumconcept, we use it here to calculate the value of global diver-sification' for a bond portfolio, an equity portfolio, and aportfolio containing both bonds and equities, in each caseboth with and without allowing currency hedging, We nor-malize the portfolio volatilities at 10.7% - the volatility ofthe market-capitalization-weighted portfolio, 80% hedged. InExhibit 18, we show 'the additional return available fromincluding international assets relative to the optimal domes-tic portfolio with the same degree of risk.What-is clear from this tableis that global diversificationprovides a substantial pickup in expected return for thedomestic bond portfolio manager, both in absolute and per-centage terms. The gains for an equity manager, or a portfo-lio manager with both bonds and equities, are also substan- '

Exhibit 17,Expected Excess Returns Implied by a Given Portfolio

Views Relative to the Market Capitalization Benchmark(Annualized bpected Excess Returns)

Germany France Japan U.K. U.s. Canada AustraliaCurrencies 1.55 1.82 - -0.27 1.22 0.63 2.45Bonds 0.30 -0.30 -0.58 1.03 -0.13 -0.01 1.22Equities 2.82 3.97 -0.30 6.73 4.15 5.01 5.88

Views Relative to the Domestic Weighted Benchmark


35/44


The Value of Global Diversification

Expected Excess Returns inEquilibrium _, at 'a CoDStant 10.'7~RiskWithout Currency Risk BedgiDg

BasisPoiDt PercentDomestic Global Differepce Gain

Bonds Only 2.14 2.63 49 22.9Equities Only 4.72 5.43 76 16.1Bonds and Equities 4.76 5.50 74 15.5Allowing Currency Hedging

BasisPoiDt PercentDomestic Global Difference GainBonds Only 2.14 3.20 106 49.5Equities Only 4.72 5.56 34 17.8Bonds and Equities 4.76 5.61 85 17.9

tial, though much smaller as a percentage of the excessreturns of the domestic portfolio. These results also appearto provide a justification for the common practice of bondportfolio managers to currency hedge and of equity portfoliomanagers not to bedge. In the absence of currency views, thegains to currency hedging are clearly more important in bothabsolute and relative terms for fixed income investors.

IX.mstoricalSimulations Itis natural to ask how a model such as ours would haveperformed in simulations. However, our approach doesnot, in itself, produce investment strategies. It requiresa set of views, and any simulation is a test not only of themodel but also of the strategy producing the views.For example, one strategy that is fairly well known in theinvestment world, and which bas performed quite well inrecent years, is to invest funds in high yielding eurrencies.

3177


36/44

FixedIncomeResearchIn this section we illustrate how a quantitative model suchas ours can be used to optimize such a strategy, and also tocompare the relative performances of different investmentstrategies. In particular, we will 'compare the historicalperformance of a strategy of investing in high yielding cur-rencies -versus two other- strategies:' (1) investing in thebonds of countries with high bond yields and (2) investing inthe equities of countries with high-ratios of dividend yield tobond yield. Our purpose is to illustrate how a quantitativeapproach can be used to make a useful comparison betweenalternative investment strategies. We are not trying to pro--mote or justify these particular strategies. We have chosento focus on these three primarily because they are simple,relatively comparable, and representative of standard invest-ment approaches.Our simulations of all three strategies use the same basicmethodology, the same data, and the same underlying secu-rities. The differences in the three simulations are in thesources of views about excess returns and in the assets to-ward which those views are applied. All of the simulationsuse our approach of adjusting expected excess returns awayfrom the global equilibrium as a function of investor views.In each of the simulations, we test a strategy by performingthe following steps. Starting in July 1981 and continuingeach month for the next 10 years, we use data up to thatpoint in time to estimate a covariance matrix of returns onequities, bonds, and currencies. We compute the equilibriumrisk premiums, add views according to the particular strate-gy , and calculate the set of expected excess returns for. allsecurities based on combining views with equilibrium.We then optimize the equity, bond, and currency weights fora given level of risk with no constraints on the portfolioweights. We calculate the excess returns that would haveaccrued in that month. At the end of each month we updatethe data and repeat the calculation. At the end of ten yearswe compute the cumulative excess returns for each of thethree strategies and compare them with one another andwith several passive investments.The views for the three strategies represent very differentinformation but are generated using similar approaches. Insimulations of the high yielding currency strategy, our views

827 8


37/44

FixedIncomeResearch IIiare based on the assumption that the expected excess re-turns from holding a foreign currency are above their equi-librium value by an amount equal to the forward discount onthat currency.For example, if the equilibrium risk premium on yen,. froma U.S. dollar perspective, is 1% and the forward discount(which, because of covered interest rate parity, approximate-ly equals the difference between the short rate on yen-de-nominated deposits and the short rate on dollar-denominat-ed deposits) is 2%, then we assume the expected excessreturn on yen currency exposures to be 3%. We computeexpected excess returns on bonds and equities by adjustingtheir returns away from equilibrium ina manner consistentwith 100% confidence in the currency views.In simulations of a strategy of investing in fixed incomemarkets with bigh yields, we generate views by assumingthat expected excess returns on bonds are above their equi-librium values by an amount equal to the difference betweenthe bond-equivalent yield in that country and the globalmarket-capitalization-weighted average bond-equivalentyield.For example, if the equilibrium risk premium on bonds in agiven country is 1.0% and the yield on the 10-year bench-mark bond is 2.0 percentage points above the world averageyield, then we assume the expected excess return for bondsin that country to be 3.0%. We compute expected excessreturns on currencies and equities by assuming 100% confi-dence in these views for bonds and adjusting them awayfrom equilibrium in the appropriate manner.In simulations of a strategy of investing in equity marketswith high ratios of dividend yield to bond yield. we generateviews by assuming that expected excess returns on equitiesare above their equilibrium values by an amount equal to50times the difference between the ratio of dividend to bondyield in that country and the global market-capitalization-weighted average ratio of dividend to bond yield.For example, if the equilibrium risk premium on equities ina given country is 6.0% and the dividend to bond yield ratiois0.5 with a world average ratio of 0.4, then we assume theexpected excess return for equities in that country to be

33 79


38/44


39/44


40/44

FixedIncomeResearch /remarkably well over the past 10years. On the other hand, ,a strategy of investing in high yielding bond markets hasnot improved returns. Although past performance is certain-lyno guarantee of future performance, we believe that theseresults, and those of similar experiments with other strate-gies, suggest some interesting lines of inquiry. .,.

X. Conclusion uantitative asset allocation models have not playedthe important role that they should in global portfoliomanagement. We suspect that a good part of the

p em has been that users of such models have foundthem difficult to use and badly behaved. ..We have learned that the inclusion of a global CAPM equi-librium with equities, bonds, and currencies cansignificantlyimprove the behavior ofthese models. In particular, it allowsus to distinguish between the views of the investor and theset of expected excess returns used to drive the portfoliooptimization. This distinction in our approach allows us togenerate optimal portfolios that start at a set of neutralweights and then tilt in the direction of the investor's views.By adjusting the confidence in his views, the investor cancontrol how strongly the views influence the portfolioweights, Similarly, by spec;ifyinga ranking of confidence indifferent views, the investor can control which views areexpressed most strongly in the portfolio. The investor canexpress views about the relative performance of assets aswell as their absolute performance.Wehope that our series ofexamples - designed to illustratethe insights that quantitative modeling can provide - willstimulate investment managers to consider, or perhaps toreconsider, the application of such modeling to their ownportfolios.

3682


41/44

FixedIncomeResearchAppendix.AMathematicalDescription of theBlackLitterman.Approach

1. . n assets - bonds, equities, and currencies - areindexed by i = 1 , ... , n.2. For bonds and equities; the market capital~tioD is

given byMi' .8. Market weights of the n assets are given by the vec-

tor W= {Wl , ... , Wn}.Wedefine the Wisas follows:If asset i is a bond or equity:

Ifasset i is a currency of the jth country:

W. = AW.cI J

Where W_/is the country weight (the sum ofmarket.weights for bonds and equities in thejth country) andAis the universal hedging constant.

4. Assets' excess returns are given by a vector R ={Rl J' ,RJ.

5. Assets' excess returns are normally distributed witha covariance matrix :I.

6. The equilibrium risk premiums vector nis given byn = 6l:W,where 6 is a proportionality constant basedon the formulas in Black (1989).7. The expected excess retum E{Rl is unobservable. Itis assumed to have a probability distribution that is

proportional to a product oftwo normal distributions.The first distribution represents equilibrium; it iscentered at nwith a covariance matrix U:, where 1: isa constant.

83 87


42/44

FIXedIncomeResearch /The second distribution represents our views about klinear combinations of the elements of E(R]. Theseviews are expressed in the following form:

PE[R] = Q + ..Here P is a known k x nmatrix, Q is a k-dimenSionalveetor, and is an unobservable normally distributedrandom vector with zero mean and a diagonal covari-ance matrix C.

8. The resulting distribution for E [R] is normal with amean ElR]:

Inportfolio optimization, we use E [ R l as the vectorof expected excess returns.

3884


43/44

FixedIncomeResearch.Glossary (Note: Definitions on this page explain terms as used in thispoper.) .

Asset Excess Returns: Returns on assets less the domestic short rate (see formulBS~..in footnote 4 on p a g e 5). . .

Benchmark Portfolw:

A measure of how close a portfolio is to the equilibriumportfolio.The standard used to define the risk of other portfolios. If abenchmark is defined, the risk of a portfolio ismeasured asthe volatility of the tracking error - the difference betweenthe portfolio's returns and those of the benchmark.

. Balance:

Currency Excess Returns: Returns on forward contracts (see formulas in footnote 4 onpage 5).Ezpected Excess Returns: Expected values of the distribution of future excess returns.

Equilibrium. Portfolw:

The condition in which means (see below) equilibrate thedemand for assets with the outstanding supply.The portfolio held in equilibrium: in this paper. marketcapitalization weights, 8 0 C l > currency hedged.Expected excess returns.

Equilibrium:

Means:Neutral Portfolio: An optimal portfolio given neutral views.Neutral Views: Means when the investor has no views.Normal Portfolio:

Risk Premiums:

The portfolio that an investor feels comfortable with whenhe bas no views. He can use the normal portfolio to infer abenchmark when no explicit benclnnark exists.Means implied by the equilibrium model.

3985


44/44

FixedIncomeResearch/

References Best, Michael J., and Robert R. Grauer, "On the Sensitivity ofMean-Variance-Efficient Portfolios to Changes in Asset Means:Some Analytical and Computational Results; The Review of Fi-nancial Studies. Vol 4, No. 2, 1991, pp. 315-342.Black, Fischer, "Universal -Hedging: How to Optimize. CurrencyRisk and Reward in InternatioDal Equity Portfolios, F i1 uz.n cia l Analysts Journal, July/August 1989, Vol 45, No. 4 . pp. 16-22.'Black, Fischer, and Robert Littennan,Asset Allocanon: CombiningInvestor Views mth Market Equilibrium, Goldman, Sachs & Co.,September 1990.Grauer, Frederick L.A., Robert H.Litzenberger, and Richard E.Stehle, Sharing Rules and Equilibrium in an International Capi-tal Market Under Uncertainty; Journal of Financial Economics.Vol. 3, 1976, pp.233-256;Green, Richard C., and Burton Hollifield, -when Will MeanVariance Efficient Portfolios BeWell Diversified"- Working Paper.Graduate School of Industrial Administration, Carnegie-MellonUniversity, February 1990 (revised).Lintner, John, " 'The Valuation of Risk Assets and the Selection ofRisky Investments in Stock Portfolios and Capital Budgets, Re-view of Economics and SUWstics, Vol 47, No.1, February 1965.pp.I3-37.Markowitz, Harry, "Portfolic Selection," Journal of Finance, Vol. 7,March 1952, pp. 7791.Sharpe, William F., -Capital .Asset Prices: A Theory of MarketEquilibrium Under Conditions of Risk,. JoU17U1lo f ~nam:e, Vol.19, No.3, September 1964, pp. 425-442.Solnik, Bruno H., -An Equiborium Model of the IriternationalCapital Market; JournaJ. of Economic Theory. Vol. 8, August 1974,pp. 500-524.Theil, Henri, Principles of EcoTWmetrics , Wlley and So~ 1971.

40

(ebook) Goldman Sachs-Fixed Income Research

Documents