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Global Portfolio Optimization Author(s): Fischer Black and Robert Litterman Source: Financial Analysts Journal, Vol. 48, No. 5 (Sep. - Oct., 1992), pp. 28-43 Published by: CFA Institute Stable URL: http://www.jstor.org/stable/4479577 . Accessed: 24/02/2011 03:26 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=cfa. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Global Portfolio Optimization

Fischer Black and Robert Litterman

Quantitativeasset allocation models have not played the importantrole theyshould in global portfolio management.A good part of theproblem is that such models are difficultto use and tend to resultin portfoliosthat are badly behaved.

cn

Considerationof theglobal CAPMequilibriumcan significantlyimprovethe usefulness of thesemodels.In particular,equilibriumreturnsfor equities,bonds and currenciesprovide neutral startingpointsfor estimatingthe set of expected excessreturns needed to drive theportfolio optimizationprocess. Thisset of neutral weights can then be tilted in accordance with the investor's views.

LU

LU

LU

z D

-J

z U

z Z

28

If the investorhas no particular viewsabout asset returns,he can use the neutral valuesgiven by the equilibriummodel. If the investordoes have one or more viewsabout the relativeperformancesof assets, or theirabsoluteperformances, he can adjust equilibriumvalues in accordance with those views. Furthernore,the investor can control how strongly a

particular view influences portfolio weights,in accordance with the degree of confidence with whichhe holds the view. Investorswithglobalportfoliosof equities and bonds are generally aware that their asset allocation decisions-the proportions of funds they invest in the asset classes of differentcountriesand the degrees of currency hedging-are the most importantinvestmentdecisions they make.In deciding on the appropriateallocation, they are usuallycomfortable making the simplifyingassumptionthattheirobjectiveis to maximize expected return for a given level of risk (subject, in most cases, to various types of constraints).

These unreasonableresults stem from two well recognized problems. First,expected returnsare very difficultto estimate. Investorstypicallyhaveknowledgeable views about absolute or relative returnsin only a few markets.A standard optimization model, however, requires them to provide expected returnsfor all assets and currencies.Thus investors must augment their views with a set of auxiliaryassumptions, and the historicalreturns they often use for this purpose providepoor guides to futurereturns.

Second, the optimalportfolioasset weights and currency positions of standardasset allocation models are extremelysensitiveto the returnassumptionsused. The two problems compound each other;the standardmodel has no way to distinguishstronglyheld views fromauxiliaryassumptions, mathe- and the optimalportfolioit genGiventhe straightforward maticsof this optimizationprob- erates,given its sensitivityto the lem, the many correlations expected returns, often appears among global asset classes re- to bear little or no relationto the quiredin measuringrisk,and the views the investorwishes to exlargeamountsof moneyinvolved, press. In practice,therefore,deone might expect that,in today's spite the obvious conceptual computerizedworld, quantitative attractionsof a quantitativeapmodels would play a dominant proach, few global investment role in the global allocationpro- managersregularlyallow quanticess. Unfortunately,when inves- tativemodels to playa majorrole tors have tried to use quantitative in theirassetallocationdecisions. models to help optimizethe critical allocation decision, the un- This article describes an apreasonablenature of the results proach that providesan intuitive has often thwartedtheir efforts.' solutionto the two problemsthat When investors impose no con- have plagued quantitativeasset straints,the models almostalways allocation models. The key is ordain large short positions in combiningtwo establishedtenets many assets. When constraints of modern portfoliotheory-the rule out shortpositions,the mod- mean-variance optimization els often prescribe"corner"solu- frameworkof Markowitzand the tions with zero weights in many capital asset pricing model assets, as well as unreasonably (CAPM)of Sharpe and Lintner.2 largeweightsin the assetsof markets with small capitalizations. Copyright 1991 by Goldman Sachs.

Glossary *"AssetExcess Returns:

*'Risk Premiums: Means implied by the equilibrium model.

In this article, returns on assets less the domestic short rate (see formulas in footnote 5).

loBalance:

A measure of how close a portfolio is to the equilibrium portfolio.

*Bencbmark Portfolio: The standard used to define the risk of other portfolios. If a benchmark is defined, the risk of a portfolio is measured as the volatility of the tracking error-the difference between the portfolio's rerturns and those of the benchmark.

* Currency Excess Returns: Returns on forward contracts (see formulas in footnote 5).

l-Expected Excess Returns: Expected values of the distribution of future excess returns.

*Equilibrium: The condition in which means (see below) equilibrate the demand for assets with the outstanding supply.

loEquilibrium Portfolio:

The portfolio held in equilibrium; in this article, market capitalization weights, 80% currency hedged.

*Means: Expected excess returns.

*Neutral Portfolio: An optimal portfolio given neutral views.

NoNeutral Views: Means when the investor has no views.

*Normal Portfolio: The portfolio that an investor feels comfortable with when he has no views. He can use the normal portfolio to infer a benchmark when no explicit benchmark exists.

Our approach allows the investor to combine his views about the outlook for global equities, bonds and currencies with the risk premiums generated by Black'sglobal version of CAPM equilibrium.3 These equilibrium risk premiums are the excess returns that equate the supply and demand for global assets and currencies. As we have noted, and will illustrate,the mean-varianceoptimization used in standardasset allocation models is extremely sensitive to the expected return assumptions the investor must provide. In our model, equilibrium risk premiums provide a neutral reference point for expected returns. This, in turn, allows the model to generate optimal portfolios that are much better behaved than the unreasonable portfolios that standard models typically produce, which often include large long and short positions unless otherwise constrained. Instead, our model gravitates toward a balanced-i.e., market-capitalization-weightedportfolio that tilts in the direction of assets favored by the investor. Our model does not assume that the world is always at CAPMequilibrium, but rather that when expected returns move away from their equilibrium values, imbalances in marketswill tend to push them back. We thus think it is reasonable to assume that expected returns are not likely to deviate too far from equilibrium values. This suggests that the investor may profit by combining his views about returns in different markets with the information contained in equilibrium prices and returns.

drive optimization analysis. Equilibrium risk premiums provide a center of gravity for expected returns. The expected returns used in our optimization will deviate from equilibrium risk premiums in accordance with the investor's explicitly stated views. The extent of the deviations from equilibrium will depend on the degree of confidence the investor has in each view. Our model makes adjustments in a manner as consistent as possible with historical covariances of returns of different assets and currencies. Our use of equilibrium allows investors to specify views in a much more flexible and powerful way than is otherwise possible. For example, rather than requiring the investor to have a view about the absolute return on every asset and currency, our approach allows the investor to specify as many or as few views as he wishes. In addition, the investor can specify views about relative returns and can specify a degree of confidence about each view.

A set of examples illustrates how the incorporation of equilibrium into the standard asset allocation model makes it better behaved and enables it to generate insights for the global investment manager. To that end, we start with a discussion of how equilibrium 0 can help an investor translate his rn views into a set of expected re- 0 turns for all assets and currencies. co We then follow with a set of ap- uJ plications of the model that illus- H trate how the equilibrium solves 2VIlU the problems that have tradition- z ally led to unreasonable results in D 0 standard mean-variance models. w

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La

Neutral Views

-

Why should an investor use a global equilibrium model to help -j make his global asset allocation decision? A neutral reference is a Our approach distinguishes be- critically important input in mak- z twveenthe views of the investor ing use of a mean-variance optiand the expected returns that mization model, and an equilib- 29 z

u z

Table I Historical Excess Returns, January 1975-August 1991* Germany

France

Japan

UK

US.

Canada

Australia

Currencies Bonds Equities

-20.8 15.3 112.9

Total Mean Excess Return 3.2 23.3 13.4 -2.3 42.3 21.4 -4.9 117.0 223.0 291.3 130.1

12.6 -22.8 16.7

3.0 -13.1 107.8

Currencies Bonds Equities

-1.4 0.9 4.7

Annualized Mean Excess Return 0.2 0.8 1.3 2.1 -0.1 1.2 -0.3 4.8 8.6 7.3 5.2

0.7 -1.5 0.9

0.2 -0.8 4.5

tions that bear no obvious relation to the expected excess return assumptions. When we constrain shorting,we have positive weights in only two of the 14 potential assets. These portfolios are typical of those generated by standard optimization models.

The use of past excess returns to represent a "neutral"set of views is equivalent to assuming that the Annualized Standard Deviation constant portfolio weights that 12.1 Currencies 11.7 12.3 11.9 4.7 10.3 would have performed best hisBonds 6.8 7.8 4.5 4.5 6.5 9.9 5.5 torically are in some sense neu22.2 Equities 18.3 17.8 24.7 16.1 18.3 21.9 tral. In reality, of course, they are * Bondandequityreturnsin U.S.dollars,currencyhedgedandin excessof the Londoninterbank not neutral at all, but rather are a very special set of weights that go offeredrate(LIBOR); returnson currenciesare in excess of the one-monthforwardrates. short assets that have done poorly and go long assets that have done in the particular historical well rium provides the appropriate Of course, besides equilibrium period. neutral reference. Most of the risk premiums, there are several time investors have viewsother naive approaches investors Equal Means feelings that some assets or cur- might use to construct an optimal The investor might hope that asrencies are overvalued or under- portfolio when they have no means for returns suming equal valued at current market prices. views about assets or currencies. all across countries for each asset An asset allocation model can We examine some of these-the in class would result an approprihelp them to apply those views to historical average approach, the their advantage. But it is unrealis- equal mean approach and the ate neutral reference. Table IV tic to expect an investor to be risk-adjusted equal mean ap- gives an example of the optimal portfolio for this type of analysis. able to state exact expected ex- proach-below. we get an unreasonable Again, cess returns for every asset and portfolio.7 Historical Averages currency. The equilibrium, however, can provide the investor an The historical average approach Of course, one problem with this appropriate point of reference. assumes, as a neutral reference, approach is that equal expected that excess returns will equal excess returns do not compenSuppose, for example, that an in- their historical averages. The sate investors appropriately for vestor has no views. How then, problem with this approach is the different levels of risk in ascan he define his optimal portfo- that historical means provide very sets of different countries. Inveslio? Answering this question dem- poor forecasts of future returns. tors diversify globally to reduce on onstrates the usefulness of the For example, Table I shows many risk. Everythingelse being equal, negative values. Table III shows they prefer assets whose returns equilibrium risk premium. what happens when we use such are less volatile and less corre0 U In considering this question, and returns as expected excess return lated with those of other assets. others throughout this article, we assumptions. We may optimize use historical data on global eq- expected returns for each level of Although such preferences are uities, bonds and currencies. We risk to get a frontier of optimal obvious, it is perhaps surprising use a seven-country model with portfolios. The table illustrates how unbalanced the optimal cca monthly returns for the United the frontiers with the portfolios portfolio weights can be, as Table States, Japan, Germany, France, that have 10.7% risk, with and IV illustrates, when we take "everything else being equal" to the United Kingdom, Canada and without shorting constraints.6 such a literal extreme. With no Australia from January 1975 We can make a number of points constraints, the largest position is through August 1991.4 about these "optimal" portfolios. short Australianbonds. z Table I presents the means and First,they illustratewhat we mean Risk-Adjusted Equal Means standard deviations of excess re- when we claim that standard 30 turns and Table II the correla- mean-varianceoptimization mod- Our third naive approach to detions. All the results in this article els often generate unreasonable fining a neutral reference point is are given from a U.S. dollar per- portfolios. The portfolio that does to assume that bonds and equities spective; use of other currencies not constrain against shorting has have the same expected excess would give similar results.5 many large long and short posi- return per unit of risk, where the cn

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Table II Historical Correlations of Excess Returns, January 1975-August 1991 Germany Equities

France

Bonds

Currency

Equities

Bonds

Japan Currency

Equities

Bonds

Currency

Germany Equities Bonds Currency

1.00 0.28 0.02

1.00 0.36

1.00

France Equities Bonds Currency

0.52 0.23 0.03

0.17 0.46 0.33

0.03 0.15 0.92

1.00 0.36 0.08

1.00 0.15

1.00

Japan Equities Bonds Currency

0.37 0.10 0.01

0.15 0.48 0.21

0.05 0.27 0.62

0.42 0.11 0.10

0.23 0.31 0.19

0.04 0.21 0.62

1.00 0.35 0.18

1.00 0.45

1.00

UK Equities Bonds Currency

0.42 0.14 0.02

0.20 0.36 0.22

-0.01 0.09 0.66

0.50 0.20 0.05

0.21 0.31 0.05

0.04 0.09 0.66

0.37 0.20 0.06

0.09 0.33 0.24

0.04 0.19 0.54

Equities Bonds

0.43 0.17

0.23 0.50

0.03 0.26

0.52 0.10

0.21 0.33

0.06 0.22

0.41 0.11

0.12 0.28

-0.02 0.18

Canada Equities Bonds Currency

0.33 0.13 0.05

0.16 0.49 0.14

0.05 0.24 0.11

0.48 0.10 0.10

0.04 0.35 0.04

0.09 0.21 0.10

0.33 0.14 0.12

0.02 0.33 0.05

0.04 0.22 0.06

Australia Equities Bonds Currency

0.34 0.24 -0.01

0.07 0.19 0.05

-0.00 0.09 0.25

0.39 0.04 0.07

0.07 0.16 -0.03

0.05 0.08 0.29

0.25 0.12 0.05

-0.02 0.16 0.10

0.12 0.09 0.27

Us

United States

United Kingdom

Bonds

Canada Bonds

Australia

Equities

Bonds

1.00 0.47 0.06

1.00 0.27

1.00

Equities Bonds

0.58 0.12

0.23 0.28

-0.02 0.18

1.00 0.32

1.00

Canada Equities Bonds Currency

0.56 0.18 0.14

0.27 0.40 0.13

0.11 0.25 0.09

0.74 0.31 0.24

0.18 0.82 0.15

1.00 0.23 0.32

1.00 0.24

1.00

Australia Equities Bonds Currency

0.50 0.17 0.06

0.20 0.17 0.05

0.15 0.09 0.27

0.48 0.24 0.07

-0.05 0.20 -0.00

0.61 0.21 0.19

0.02 0.18 0.04

0.18 0.13 0.28

UK Equities Bonds Currency

Currency

Equities

Equities

Currency

Equities

Bonds

US. an

LU

0

U LU

1.00 0.37 0.27

1.00 0.20

LU

4:

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risk measure is simply the volatility of asset returns. Currencies in this case are assumed to have no excess return. Table V shows the optimal portfolio for this case.

is no better. One problem with this approach is that it hasn't taken the correlations of the asset returns into account. But there is another problem as well-perNow we have incorporated vola- haps more subtle, but also more tilities, but the portfolio behavior serious.

This approach, and the others we z have so far used, are based on what might be called the "demand for assets" side of the equa- 4: z tion-that is, historical returns and risk measures. The problem with such approaches is obvious 31

Table III Optimal Portfolios Based on Historical Average Approach The equilibrium degree of hedging-the "universalhedging conGermany France Japan UK US. Canada Australia stant"-depends on three averages-the averageacross countries Unconstrained of the mean return on the market Currency -78.7 46.5 28.6 15.5 65.0 -5.2 portfolio of assets, the average Exposure (%) across countries of the volatility Bonds (%) 30.4 -40.7 40.4 -1.4 54.5 -95.7 -52.5 of the world market portfolio, Equities (%) 4.4 -4.4 44.0 -44.2 15.5 13.3 9.0 and the average across all pairs of countries of exchange rate volatilWith Constraints Against Shorting Assets ity. -160.0 Currency 18.0 115.2 23.7 77.8 -13.8 Exposure (%) Bonds (%) Equities (%)

7.6 0.0

0.0 0.0

88.8 0.0

0.0 0.0

0.0 0.0

0.0 0.0

0.0 0.0

when we bring in the supply side of the market.

It is difficult to pin down exactly the right value for the universal hedging constant, primarily because the risk premium on the market portfolio is a difficult number to estimate. Nevertheless, we feel that universal hedging values between 75% and 85% are reasonable. In our monthly data set, the former value corresponds to a risk premium of 5.9% on U.S. equities, while the latter corresponds to a risk premium of 9.8%. For this article, we will use

rency risk up to the point where the additional risk balances the expected return. Under certain Suppose the market portfolio simplifying assumptions, the percomprises two assets, with centage of foreign currency risk weights 80% and 20%.In a simple hedged will be the same for inworld, with identical investors all vestors of different countriesholding the same views and both giving rise to the name "universal assets having equal volatilities, ev- hedging" for this equilibrium. eryone cannot hold equal weights of each asset. Prices and expected excess returns in such a world would have to adjustas the excess Table IV Optimal Portfolios Based on Equal Means demand for one asset and excess supply of the other affectthe marGermany France Japan UK US. Canada Australia ket.

The Equilibrium Approach

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32

To us, the only sensible definition of neutral means is the set of expected returns that would "clear the market" if all investors had identical views. The concept of equilibrium in the context of a global portfolio of equities, bonds and currencies is similar, although currencies do raise a complicating question. How much currency hedging takes place in equilibrium? The answer is that, in a global equilibrium, investors worldwide will all want to take a small amount of currency risk.8

Currency Exposure (%) Bonds (%) Equities (%)

Currency Exposure (%) Bonds (%) Equities (%)

14.5 -11.6 21.4

Unconstrained -12.6 -0.9 4.2 -4.8

-1.8 23.0

4.4 -10.8 -4.6

13.9 32.2

-18.7

-2.1

-18.9 9.6

-32.7 10.5

With Constraints Against Shorting Assets -11.2 14.3 0.2 -4.5 -25.9 0.0 17.5

0.0 0.0

0.0 22.1

0.0 0.0 0.0 27.0

0.0 8.2

-2.0 0.0 7.3

Table V Optimal Portfolios Based on Equal Risk-AdjustedMeans Germany France Japan Currency Exposure (%) Bonds (%) Equities (%)

5.6

UK

Unconstrained 11.3 -28.6 -20.3

US.

Canada Australia -50.9

This result arises because of a 12.6 54.0 -23.9 20.8 23.1 37.8 curiosity known in the currency 12.4 -0.3 -14.1 9.9 8.5 13.2 world as "Siegel's paradox." The basic idea is that, because invesWith Constraints Against Shorting Assets tors in different countries mea21.7 -14.0 -12.2 Currency -8.9 -47.9 sure returns in different units, Exposure (%) each will gain some expected re- Bonds (%) 0.0 0.0 0.0 7.8 0.0 19.3 turn by taking some currency Equities (%) 11.1 9.4 6.0 0.0 7.6 19.2 risk. Investors will accept cur-

-4.9 15.6 20.1

-6.7 0.0 19.5

Table VI Equilibrium Risk Premiums (% annualized excess returns) Germany France Japan Currencies Bonds Equities

1.01 2.29 6.27

1.10 2.23 8.48

1.40 2.88 8.72

UK 0.91 3.28 10.27

US.

Canada

Australia

1.87 7.32

0.60 2.54 7.28

0.63 1.74 6.45

an equilibriumvaluefor currency hedging of 80%. Table VI gives the equilibrium risk premiums for all assets, given this value of the universalhedging constant.9

between investor views on the one hand and a complete set of expected excess returns for all assets on the other-is not usually recognized.In our approach, views represent the subjective Considerwhathappenswhen we feelings of the investoraboutreladopt these equilibriumrisk pre- ative values offered in different miums as our neutral means markets.10 If an investordoes not when we haveno views.TableVII havea view abouta given market, shows the optimalportfolio.It is he should not have to state one. simply the market-capitalizationAndif some of his views are more portfolio with 80% of the cur- strongly held than others, he rency risk hedged. Other portfo- should be able to express the lios on the frontierwith different differences. levels of risk would correspond to combinationsof risk-freebor- Mostviews are relative.Forexamrowing or lending plus more or ple, the investor may feel one less of this portfolio. marketwill outperformanother. Or he may feel bullish (above By itself,the equilibriumconcept neutral) or bearish (below neuis interestingbut not particularly tral) about a market.As we will useful.Its real value is to provide show, the equilibriumallows the a neutralframeworkthe investor investorto express his views this can adjustaccordingto his own way, instead of as a set of exviews, optimization objectives pected excess returns. and constraints.

To see why this is so important, we start by illustratingthe extreme sensitivity of portfolio weights to the expected excess returnsand the inabilityof investors to expressviews directlyas a completeset of expectedreturns. We have alreadyseen how difficult it can be simply to translate no views into a set of expected excess returnsthatwill not lead an asset allocationmodel to produce an unreasonableportfolio. But suppose thatthe investorhas alreadysolved that problem, using equilibriumriskpremiumsas the neutralmeans.He is comfortable with a portfoliothathas market capitalizationweights, 80% hedged. Considerwhat can happen when this investornow tries to express one simple, extremely modest view. Suppose the investor's view is that,over the next three months, the economic recovery in the United States will be weak and bondswill performrelativelywell and equities poorly. The investor'sview is not very strong,and he quantifiesit by assumingthat, over the next three months, the U.S. benchmarkbond yield will drop 1 basispointratherthanrise

Table VII Equilibrium Optimal Portfolio Expressing Views Investorstrying to use quantitaGermany France Japan UK US. Canada tive asset allocationmodels must 1.1 2.0 0.9 5.9 0.6 translatetheir views into a com- Currency Exposure (%) plete set of expected excess re- Bonds (%) 2.9 1.9 6.0 1.8 16.3 1.4 turnson assetsthatcanbe used as Equities (%) 2.4 2.6 23.7 8.3 29.7 1.6 a basis for portfoliooptimization. As we will show here, the problem is that optimal portfolio weights from a mean-variance Table VIII Optimal Portfolios Based on a Moderate View model are incrediblysensitiveto minor changes in expected exGermany France Japan UK US. Canada cess returns. The advantageof incorporating a global equilibUnconstrained riumwill become apparentwhen Currency -6.4 -1.3 8.3 -3.3 8.5 Exposure (%) we show how to combine it with 6.4 -13.6 15.0 -3.3 112.9 -42.4 an investor's views to generate Bonds (%) (%) Equities 3.7 6.3 27.2 14.5 24.8 -30.6 well-behavedportfolios,without requiringthe investorto express With Constraints Against Shorting Assets a completeset of expectedexcess 9.2 2.3 4.3 5.0 -3.0 Currency returns. Exposure (%)

We should emphasize that the distinction we are making

Bonds (%) Equities (%)

0.0 2.6

0.0 5.3

0.0 28.3

0.0 13.6

35.7 0.0

0.0 13.1

Australia 0.3 0.3 1.1

r'i 0) 0a

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m LU L1J

Australia

D z

-1.9 0.7 6.0

-0.6

-J

U

z z

0.0 1.5

33

1 basis point, as is consistent with the equilibrium risk premium.11 Similarly, the investor expects U.S. share prices to rise only 2.7% over the next three months, rather than to rise the 3.3% consistent with the equilibrium view. To implement the asset allocation optimization, the investor starts with expected excess returns equal to the equilibrium risk premiums and adjusts them as follows. He moves the annualized expected excess returns on U.S. bonds up by 0.8 percentage points and the expected excess returns on U.S. equities down by 2.5 percentage points. All other expected excess returns remain unchanged. Table VIII shows the optimal portfolio, given this view.

2. We assume that both sources of informationare uncertainand are best expressedas probabilitydistributions. 3. We choose expected excess returnsthatare as consistent as possible with both sources of information. The above description captures the basic idea, but the implementationof the approachcan lead to some novel insights.We will now show how a relativeview about two assets can influence the expected excess return on a third asset.12

Three-Asset Example Let us first work through a very

simple exampleof our approach. Afterthis illustration,we will apply it in the contextof our sevencountry model. Suppose we know the true structure of a world thathas justthree assets,A, B and C. The excess return for each of these assets is known to be generated by an equilibrium risk premium plus four sources of risk-a common factor and independent shocks to each of the three assets.We can express this model as follows:

Note the remarkable effect of this very modest change in expected excess returns. The portfolio weights change in dramatic and largely inexplicable ways. The optimal portfolio weights do shift out of U.S. equity into U.S. bonds, as might be expected, but the model also suggests shorting Canadian and German bonds. The lack of apparent connection between the view the investor is attempting to express and the optimal portfolio the model gener- RA = 'TA + YAZ + VA) ates is a pervasive problem with standardmean-varianceoptimization. It arises because there is a RB = 7rB + YBZ + VB, complex interaction between expected excess returns and the vol- RC= 'TC+ YCZ+ VC, atilities and correlations used in where: measuring risk. UJ 0 R, = the excess return on the Combining Investor LU

are a function of the equilibrium risk premiums, the expected value of the common factor, and the expected values of the independent shocks to each asset. For example, the expected excess return of asset A, which we write as E[RA],is given by: E[RA]=

7TA +

yAE[Z]+ E[vj.

We are not assuming that the world is in equilibrium (i.e., that E[Z] and the E[vi]s are equal to zero). We do assume that the mean, E[RA],is itself an unobservable random variable whose distribution is centered at the equilibrium risk premium. Our uncertainty about E[RA]is due to our uncertainty about E[Z] and the E[vi]s. Furthermore, we assume the degree of uncertainty about E[Z] and the E[vi]s is proportional to the volatilities of Z and the vis themselves. This implies that E[RA]is distributed with a covariance structure proportional to E. We will refer to this covariance matrix of the expected excess returns as rY. Because the uncertainty in the mean is much smaller than the uncertainty in the return itself, r will be close to zero. The equilibrium risk premiums together with rX determine the equilibrium distribution for expected excess returns. We assume this information is known to all; it is not a function of the circumstances of any individual investor.

In addition, we assume that each investor provides additional inViews with Market ,wi = the equilibrium risk pre- formation about expected excess Equilibrium returns in the form of views. For mium on the ith asset, example, one type of view is a How our approach translates a yi = the impact on the ith as- statement of the form: "I expect few views into expected excess z set of Z, asset A to outperform asset B by returns for all assets is one of its cc = Z the common factor, and Q,"where Q is a given value. more complex features, but also vi = the independent shock one of its most innovative. Here is to the ith asset. We interpret such a view to mean the intuition behind our apthat the investor has subjective z proach. In this world, the covariancema- information about the future re1. We believe there are two trix, E, of asset excess returnsis turns of A relative to B. One way U z distinct sources of informa- determined by the relative im- we think about representing that z tion about future excess re- pacts of the common factorand information is to act as if we had a turns-investor views and the independentshocks.The ex- summary statistic from a sample market equilibrium. pectedexcess returnsof the assets of data drawn from the distribu34 LU

-J

ith asset,

TableIX ExpectedExcessAnnualizedPercentageReturnsCombin- In the more general case where ing InvestorViews With Equilibrium we are not 100%confident,we can thinkof a view as representUS Germany France UK ing a fixed number of observaJapan Canada Australia tions drawnfrom the distribution Currencies 1.28 1.22 0.44 1.32 1.73 0.47 of futurereturns.In this case, we Bonds 2.69 3.40 2.39 3.29 2.70 2.39 1.35 Equities 6.42 5.28 7.71 7.83 4.39 3.86 4.58 follow the "mixed estimation" strategydescribed in Theil.14Alternatively,we can think of the view as directlyreflectinga subjective distribution for the extion of future returns, data in the means of the random compo- pected excess returns. In this whichallwe were able to observe nents. case, we use the Black-Litterman approach, given in the appenis the differencebetween the returnsof A and B. Alternatively, we We have the equilibrium distribu- dix.15 The formula for the excan express the view directlyas a tion for E[R],which is given by pected excess returns vector is probability distribution for the Normal (w, rY,),where w = {WTA, the same fromeitherperspective. differencebetween the means of 7B' 7r} We wish to calculate a the excess returnsof A and B. It conditional distribution for the In either approach,we assume doesn'tmatterwhich of these ap- expected returns, subject to the thatthe view can be summarized proaches we use to think about restriction that the expected re- by a statement of the form our views;in the end we get the turns satisfy the linear restriction P*E[RI'= Q + s,where PandQ same result. E[RA1- E[RB]= Q. We can write are given and e is an unobservthis restriction as a linear equa- able, normally distributed ranIn both approaches,though, we tion in the expected returns:13 dom variablewith mean 0 and need a measureof the investor's variancefl. fl representsthe unconfidence in his views. We use P *E[R]' = Q, certaintyin the view. In the limit, this measure to determine how as Q goes to zero, the resulting much weight to give to the view where P is the vector [1, -1, 0O. mean converges to the condiwhen combiningit with the equitionalmean describedabove. librium.We can thinkof this de- The conditional normal distribugree of confidence as determin- tion has the following mean: When there is more than one ing, in the first case, the number view, the vector of views can be of observationsthatwe havefrom 77" + Ty,.p, .[P - T .pt]- I representedby P *E[R]'= Q + , the distributionof futurereturns where we now interpretP as a or as determining,in the second, matrix,and e is a normallydis-[ I P -1 V, +Q the standard deviation of the tributed random vector with probabilitydistribution. mean 0 and diagonalcovariance which is the solution to the prob- matrix fQ. A diagonal fl correIn our example,considerthe lim- lem of minimizing spondsto the assumptionthatthe iting case: The investor is 100% views represent independent sure of his view. We might think (E[R]- )T draws from the future distribu1(E[R]- 7) of this as the case where we have tion of returns,or thatthe deviaan unboundednumberof obser- subject to P *E[R]I= Q. tions of expected returns from ,0 vations from the distributionof the meansof the distributionrepfuturereturns,and where the av- For the special case of 100%con- resentingeach view are indepen- UJ erage value of RA- RBfrom these fidence in a view, we use this dent, depending on which ap- UJ datais Q. In this special case, we conditional mean as our vector of proach is used to think about LU can representthe view as a linear expected excess returns. subjective views. The appendix H restriction on the expected excess returns-i.e., E[RA]- E[RBI

= Q.

In this special case, we can compute the distributionof E[R] =

Table X

Optimal Portfolio Combining Investor Views With Equilibrium

-J

{E[R,j, E[RB], E[RJ} conditional

on the equilibriumand this information. This is a relatively straightforward problem from multivariatestatistics.To simplify, assume a normaldistributionfor

z

Currency Exposure (%) Bonds (%) Equities (%)

Germany 1.4

France 1.1

Japan 7.4

UK 2.5

US.

Canada 0.8

Australia 0.3

3.6 3.3

2.4 2.9

7.5 29.5

2.3 10.3

67.0 3.3

1.7 2.0

0.3 1.4

z z

35

Table XI

Economists' Views France

Japan

UK

1.743

5.928

137.3

1.688

1.790

6.050

141.0

1.640

Germany Currencies July 31, 1991 Current Spot Rates Three-Month Horizon Expected Future Spot Annualized Expected Excess Returns Interest Rates July 31, 1991 Benchmark Bond Yields Three-Month Horizon Expected Future Yields Annualized Expected Excess Returns

-7.48

-4.61

-8.85

-6.16

8.7

9.3

6.6

10.2

8.8

9.5

6.5

10.1

-3.31

-5.31

1.78

1.66

gives the formula for the expected excess returns that combine views with equilibrium in the general case.

exceeds that of B by more than it does in equilibrium. From this, we clearly ought to impute that a shock to the common factor is the most likely reason A will outperNow consider our example, in form B. If so, C ought to perform which asset correlations result better than equilibrium as well. from the impact of one common The conditional mean in this case factor. In general, we will not is [3.9, 1.9, 2.9]. Indeed, the invesknow the impacts of the factor on tor's view of A relative to B has the assets-that is, the values of raised the expected return on C yA, yB and yYcBut suppose the by 1.9 percentage points. unknown values are [3, 1, 2]. Suppose further that the independent But now suppose the indepenshocks are small, so that the assets dent shocks have a much larger are highly correlated with volatil- impact than the common factor. ities approximately in the ratios Let the I matrix be as follows: 3:1:2.

(r

0) LU

Suppose, for example, the covariance matrix is as follows:

[9.13.0

19.0 3.0

3.0 11.0

6.0 2.0

6.0

2.0

14.0]

US

1.000

Canada

Australia

1.151

1.285

1.156

1.324

0.77

-8.14

8.2

9.9

11.0

8.4

10.1

10.8

-3.03

-3.48

5.68

own independent shock. Although we should impute some change in the factor from the higher return of A relative to B, the impact on C should be less than in the previous case. In this case, the conditional mean is [2.3, 0.3, 1.3]. Here the implied effect of the common-factor shock on asset C is lower than in the previous case. We may attribute most of the outperformance of A relative to B to the independent shocks; indeed, the implication for E[RB]is negative relative to equilibrium. The impact of the independent shock to B is expected to dominate, even though the contribution of the common factor to asset B is positive.

Note that we can identify the impact of the common factor only if 0 3.0 1.1 2.0 we assume that we know the true 6.0 2.0 4.1 structure that generated the cova0~ riance matrix of returns. That is true here, but it will not be true in Assume also, for simplicity, that the percentage equilibrium risk This time, more than half of the general. The computation of the premiums are equal-for exam- volatilityof A is associated with its conditional mean, however, does z ple, [1, 1, 1]. There is a set of VI D :: market capitalizations for which -J that is the case. r-J Table XII Optimal (Unconstrained) Portfolio Based on Economists' Views U z Now consider what happens when the investor expects A to Germany France Japan UK US Canada Australia : outperform B by 2%. In this ex68.8 -12.7 Currency 16.3 -35.2 29.7 -51.4 Z ample, virtuallyall of the volatility Exposure (%) z of the assets is associated with Bonds (%) 34.5 -65.4 79.2 16.9 3.3 -22.7 108.3 movements in the common fac- Equities (%) -2.2 6.6 3.6 0.6 0.7 5.2 0.5 36 tor, and the expected return of A

LU m

U

LU

6.0

Suppose the equilibrium risk premiums are again given by [1, 1, 1]. Now assume the investor expects that A will outperform B by 2%.

Table XIII Optimal Portfolio With Less Confidence in the Economists' Views Germany Currency Exposure (%) Bonds (%) Equities (%)

-12.9 -3.9 0.8

France

Japan

UK

-3.5

-10.0

-6.9

-21.0 2.2

19.6 24.7

2.6 7.1

US.

7.3 26.6

Canada

Australia

-0.4

-17.9

-13.6 4.2

42.4 1.2

not depend on this special knowl- only the returnsto U.S.bonds and edge, but only on the covariance U.S. equities, holding fixed all other expected excess returns. matrix of returns. Finally, let's look at the case where the investor has less confidence in his view. We might say (E[RA]- E[RB])has a mean of 2 and a variance of 1, and the covariance matrix of returns is, as it was originally:

Anotherdifferenceis thathere we specify a differentialof means, lettingthe equilibriumdetermine the actuallevels of means;above we had to specify the levels directly.

TableIX shows the complete set of expected excess returnswhen we put 100%confidencein a view [9.1 3.0 6.0 that the differentialof expected 1.1 2.0 3.0 excess returns of U.S. equities 4.1] 2.0 6.0 over bondswill be 2.0 percentage pointsbelow the equilibriumdifIn this example, however, the ferentialof 5.5 percentagepoints. conditional mean is based on an Table X shows the optimalportuncertain view. Using the formula folio associatedwith this view. given in the appendix, we find that the conditional mean is given by: [3.3, 1.7, 2.5]. Because the investor has less confidence in his view, the expected relative return of 2%for A - B is reduced to a value of 1.6, which is closer to the equilibrium value of 0. There will also be a smaller effect of the common factor on the third asset because of the uncertainty of the view.

Seven-Country Example

tion we focus more specifically on the concept of a "balanced" portfolioand show an additional featureof our approach:Changes in the "confidence"in views can be used to controlthe balanceof the optimalportfolio. We startby illustratingwhat happens when we put a set of stronger views, shown in TableXI,into our model. These happento have been the short-terminterestrate and exchange rate views expressed by GoldmanSachseconomistson July31, 1991.16 We put 100%confidence in these views, solve for the expected excess returns on all assets, and find the optimalportfolio,shown in Table XII. Given such strong views on so many assets, and optimizing without constraints,we generate a ratherextreme portfolio.

Analystshave tried a number of approaches to ameliorate this problem. Some put constraints on manyof the assetweights.We resist using such artificialconstraints.When asset weights run up againstconstraints,the portfoThese results contrast with the lio optimizationno longer balinexplicableresults we saw ear- ances return and risk across all lier.We see here a balancedport- assets. folio in which the weights have tilted away from marketcapitali- Others specify a benchmark zations toward U.S. bonds and portfolio and limit the risk relaawayfrom U.S.equities.We now tive to the benchmarkuntila reaobtain a portfolio that we con- sonablybalancedportfoliois obsider reasonable,given our view. tained. This makes sense if the objectiveof the optimizationis to Controlling the Balance managethe portfoliorelativeto a of a Portfolio benchmark. We are uncomfortIn the previoussection, we illus- able when it is used simply to tratedhow our approachallows make the model betterbehaved. us to express a view that U.S. bonds will outperformU.S.equi- An alternateresponse when the ties, in a way thatleads to a well- optimal portfolio seems too exbehaved optimal portfolio that tremeis to considerreducingthe expresses that view. In this sec- confidenceexpressedin some or

a')

co LU

ax

H Now we will attempt to apply our LU view that bad news about the U.S. z economy will cause U.S. bonds to outperform U.S. stocks to the actual data. The critical difference between our approach here and Table XIV Optimal Portfolio With Less Confidence in Certain Views z our earlier experiment that generated Table VIII is that here we US. Canada Australia -J UK Germany France Japan say something about expected re-7.8 -4.8 -2.8 -6.2 -10.0 -0.4 Currency turns on U.S. bonds versus U.S. 0L Exposure (%) equities and we allow all other Bonds (%) -2.4 28.1 1.6 22.9 -10.3 -34.3 25.5 expected excess returns to adjust Equities (%) 6.0 0.1 7.0 26.3 1.3 2.3 25.9 accordingly. Before we adjusted

Table XV Alternative Domestic-Weighted Benchmark Portfolio Currency Exposure (%) Bonds (%) Equities (%)

Germany 1.5

France 1.5

Japan 7.0

0.5 1.0

0.5 1.0

2.0 5.0

all of the views. Table XIII shows the optimal portfolio that results when we lower the confidence in all of our views. By putting less confidence in our views, we generate a set of expected excess returns that more strongly reflect equilibrium. This pulls the optimal portfolio weights toward a more balanced position. We define balance as a measure of how similar a portfolio is to the

global equilibrium

portfo-

lio-that is, the market-capitalization portfolio with 80% of the currency risk hedged. The distance measure we use is the volatility of the difference between the returns on the two portfolios.

on

LU

Lii

cL/

H-

-J

We find this property of balance to be a useful supplement to the standard measures of portfolio optimization, expected return and risk. In our approach, for any given level of risk there will be a continuum of portfolios that maximize expected return depending on the relative levels of confidence that are expressed in the views. The less confidence the investor has, the more balanced his portfolio will be. Suppose an investor does not have equal confidence in all his views. If the investor is willing to rank the relative confidence levels of his different views, then he can generate an even more powerful result. In this case, the model will move away from his less strongly held views more quickly than from his more strongly held ones.

0 z

38

We have specified higher confidence in our view of yield declines in the United Kingdom and yield increases in France and Ger-

UK 3.0

US.

Canada 2.0

Australia 0.0

invested in the domestic shortterm interest rate. In many cases, however, an alternative benchmark is called for.

Many portfolio managers are given an explicit performance benchmark, such as a marketcapitalization-weighted index. If an explicit performance benchmany. These are not the biggest mark exists, then the appropriate yield changes that we expect, but measure of risk for the purpose they are the forecasts that we of portfolio optimization is the most strongly want to represent volatility of the tracking error of in our portfolio. We put less con- the portfolio vis-a-vis the benchfidence in our views of interest mark.And for a manager funding rate moves in the United States a known set of liabilities, the appropriate benchmark portfolio and Australia. represents the liabilities. When we put equal confidence in For many portfolio managers, the our views, we obtained the opti- performance objective is less exmal portfolio shown in Table XIII. plicit, and the asset allocation deThe view that dominated that cision is therefore more difficult. portfolio was the interest rate de- For example, a global equity portcline in Australia.Now, when we folio manager may feel his objecput less than 100% confidence in tive is to perform in the top rankour views, we have relatively ings of all global equity managers. more confidence in some views Although he does not have an than in others. Table XIV shows explicit performance benchmark, the optimal portfolio for this case. his risk is clearly related to the stance of his portfolio relative to Benchmarks the portfolios of his competitors. One of the most important, but often overlooked, influences on Other examples are an overthe asset allocation decision is the funded pension plan or a univerchoice of the benchmark by sity endowment where matching which to measure risk. In mean- the measurable liability is only a variance optimization, the objec- small part of the total investment tive is to maximize return per objective. In these types of situaunit of portfolio risk. The inves- tions, attempts to use quantitative tor's benchmark defines the point approaches are often frustrated of origin for measuring this risk. by the ambiguity of the investIn other words, it represents the ment objective. minimum-risk portfolio. When an explicit benchmark In many investment problems, does not exist, two alternativeaprisk is measured as the volatility proaches can be used. The first is of the portfolio's excess returns. to use the volatility of excess reThis is equivalent to having no turns as the measure of risk. The benchmark, or to defining the second is to specify a "normal" benchmark as a portfolio 100% portfolio, one that represents 1.0 2.0

30.0 55.0

1.0 1.0

0.0 0.0

Table XVI Current Portfolio Weights for Implied-View Analysis Currency Exposure (%) Bonds (%) Equities (%)

Germany

France

4.4

3.4

2.0

2.2

1.0 3.4

0.5 2.9

4.7 22.3

2.5 10.2

Japan

UK

US.

13.0 32.0

Canada

Australia

2.0

5.5

0.3 1.7

3.5 2.0

TableXVII Annualized Expected Excess Returns Implied by a views of the portfolio shown in Table XVI,given that the benchGiven Portfolio markis, alternatively,(1) a marportUS. UK Canada Australia ket-capitalization-weighted Germany France Japan folio, 80% hedged, or (2) the Views Relative to the Market-Capitalization Benchmark domestic-weighted alternative 2.45 1.22 0.63 1.82 -0.27 1.55 Currencies shown in TableXV.Unlessa port1.22 -0.01 1.03 -0.13 -0.58 0.30 -0.30 Bonds folio managerhas thought care5.88 5.01 6.73 4.15 3.97 -0.30 2.82 Equities fullyaboutwhathis benchmarkis andwhere his allocationsare relViews Relative to the Domestic-Weighted Benchmark ativeto it, and has conductedthe 0.01 0.90 0.20 0.50 0.54 Currencies 0.05 type of analysisshown here, he -1.01 0.18 0.21 -1.45 0.72 0.85 Bonds -0.01 maynot have a clearidea of what 0.28 2.38 -1.49 2.83 5.24 4.83 2.24 Equities views his portfoliorepresents.

Quantifying the Benefits the desired allocation of assets in the absence of views. Such a portfolio might, for example, be designed with a higher-than-market weight for domestic assets in order to represent the domestic nature of liabilities without attempting to specify an explicit liability benchmark. An equilibrium model can help in the design of a normal portfolio by quantifying some of the risk and return tradeoffs in the absence of views. The optimal portfolio in equilibrium is marketcapitalization-weighted and is 80% currency hedged. It has an expected excess return (using equilibrium risk premiums) of 5.7%and an annualized volatility of 10.7%. A pension fund wishing to increase the domestic weight of its portfolio to 85%from the current market capitalization of 45%,and not wishing to hedge the currency risk of the remaining 15% in international markets, might consider an alternative portfolio such as the one shown in Table XV. The higher domestic weights lead to an annualized volatility0.4 percentage points higher than and an expected excess return 30 basis points below those of the optimal portfolio. The pension fund may or may not feel that its preference for domestic concentration is worth those costs.

Implied Views Once an investor has established his objectives, an asset allocation

model establishes a correspon- of Global Diversification dence betweenviews andoptimal While most investors demonportfolios.Ratherthan treatinga strate a substantialbias toward quantitativemodel as a blackbox, domestic assets, many recent successfulportfoliomanagersuse studieshave documenteda rapid a model to investigatethe nature growth in the internationalcomof this relationship.In particular, ponents of portfoliosworldwide. it is often useful to startan analy- It is perhapsnot surprising,then, sis by using a model to find the that investment advisers have implied investorviews for which startedto questionthe traditional an existing portfolio is optimal argumentsthatsupportglobal direlativeto a benchmark. versification.Thishasbeen particularly true in the United States, For example,we assume a port- where global portfolios have folio managerhasa portfoliowith tended to underperformdomesweights as shown in Table XVI. tic portfoliosin recentyears. The weights, relativeto those of his benchmark,define the direc- Of course,whatmattersfor investions of the investor'sviews. By tors is the prospective returns assumingthe investor'sdegree of from internationalassets, and as risk aversion, we can find the noted in our earlierdiscussionof expected excess returns for neutral views, the historicalrewhich the portfoliois optimal. turns are of virtuallyno value in projectingfutureexpectedexcess In this type of analysis,different returns.Historicalanalyses conbenchmarksmay imply very dif- tinue to be used in this context ferentviews for a given portfolio. simplybecause investmentadvisTable XVII shows the implied ers argue there is nothingbetter

CN

a' Lr 0a

U

m

0

LU

Table XVIII The Value of Global Diversification (expected excess returns in equilibrium at a constant 10.7% risk)

LU LU

~: Domestic Wlthout

Global Currency

Basis-Point Difference

Percentage Gain

Hedging

Bonds Only Equities Only Bonds and Equities

2.14 4.72 4.76

Bonds Only Equities Only Bonds and Equities

With Currency Hedging 2.14 3.20 5.56 4.72 5.61 4.76

2.63 5.48 5.50

z D

0

-J

49 76 74

22.9 16.1 15.5

z -j u z

106 84 85

49.5 17.8 17.9

z

39

to measure the value of global diversification. We would suggest that there is something better. A reasonable measure of the value of global diversification is the degree to which allowing foreign assets into a portfolio raises the optimal portfolio frontier. A natural starting point for quantifying this value is to compute it based on the neutral views implied by a global CAPMequilibrium. There are some limitations to using this measure. It assumes that there are no extra costs to international investment; thus relaxing the constraint against international investment cannot make the investor worse off. On the other hand, in measuring the value of global diversification this way, we are also assuming that markets are efficient and therefore we are neglecting to capture any value that an international portfolio manager might add through having informed views about these markets. We suspect that an important benefit of international investment that we are missing here is the freedom it gives the portfolio manager to take advantageof a larger number of opportunities to add value than are afforded by domestic markets alone. We use the equilibrium concept here to calculate the value of global diversification for a bond portfolio, an equity portfolio and 0 U a portfolio containing both bonds 0 and equities (in each case both with and without allowing currency hedging). We normalize .u the portfolio volatilities at marof the 10.7%-the volatility H ket-capitalization-weighted portfolio, 80% hedged. Table XVIII w shows the additional return available from including international Z: assets relative to the optimal domestic portfolio with the same degree of risk. LU

LU m

LLI

z L#) -J

U

z

What is clear from this table is that global diversification provides a substantial increase in ex40 pected return for the domestic z-

bond portfolio manager, both in absolute and percentage terms. The gains for an equity manager, or a portfolio manager with both bonds and equities, are also substantial,although much smaller as a percentage of the excess returns of the domestic portfolio. These results also appear to provide a justificationfor the common practice of bond portfolio managers to hedge currency risk and of equity portfolio managers not to hedge. In the absence of currency views, the gains to currency hedging are clearly more important in both absolute and relative terms for fixed income investors.

Historical Simulations

Our simulations of all three strategies use the same basic methodology, the same data and the same underlying securities. The strategies differ in the sources of views about excess returns and in the assets to which those views are applied. All the simulations use our approach of adjusting expected excess returns away from the global equilibrium as a function of investor views. In each of the simulations, we test a strategy by performing the following steps. Startingin July 1981 and continuing each month for the next 10 years, we use data up to that point in time to estimate a covariance matrix of returns on equities, bonds and currencies. We compute the equilibrium risk premiums, add views according to the particularstrategy, and calculate the set of expected excess returns for all securities based on combining views with equilibrium.

It is natural to ask how a model such as ours would have performed in simulations. However, our approach does not, in itself, produce investment strategies. It requires a set of views, and any simulation is a test not only of the model but also of the strategy We then optimize the equity, producing the views. bond and currency weights for a One strategy that is fairly well given level of risk with no conknown in the investment world, straints on the portfolio weights. and that has performed quite well We calculate the excess returns in recent years, is to invest funds that would have accrued in that in high-yielding currencies. Be- month. At the end of each month, low, we show how a quantitative we update the data and repeat the model such as ours can be used calculation.At the end of 10 years, to optimize such a strategy. In we compute the cumulative exparticular, we will compare the cess returns for each of the three historical performance of a strat- strategies and compare them with egy of investing in high-yielding one another and with several pascurrencies with two other strate- sive investments. gies-(1) investing in the bonds The views for the three strategies of countries with high bond represent very different informayields and (2) investing in the tion but are generated using simequities of countries with high ilar approaches. In simulations of ratios of dividend yield to bond the high-yielding currency stratyield. egy, our views are based on the assumption that the expected exOur purpose is to illustrate how a cess returns from holding a forquantitative approach can be eign currency are above their used to make a useful compari- equilibrium value by an amount son of alternative investment equal to the forward discount on strategies. We are not trying to that currency. promote or justify a particular strategy.We have chosen to focus For example, if the equilibrium on these three primarily because risk premium on yen, from a U.S. they are simple, relatively compa- dollar perspective,is 1% and the rable, and representative of stan- forward discount (which, because of covered interest rate parity, dard investment approaches.

Historical Cumulative Monthly Returns, U.S.-dollar-based perspective

Figure A

$ 600

For example, if the equilibrium risk premium on equities in a given country is 6.0% and the dividendto bond yield ratiois 0.5 with a world averageratioof 0.4, then we assumethe expected excess return for equities in that countryto be 11%.We compute expected excess returnson currencies and bonds by assuming 100%confidence in these views for equities and adjustingthe returns away from equilibriumin the appropriatemanner.

Currency Strategy

r

500

Equilibrium 1

400

300-

200 -

Bond Strategy

00

X

t

~~~~~~~~~~Equity

Strategy|

0u June-'81

Sept-'82

Dec.-'83

Mar.-'85

approximatelyequals the difference between the short rate on yen-denominated deposits and the short rate on dollar-denominated deposits) is 2%, then we assume the expected excess return on yen currencyexposures to be 3%.We compute expected excess returnson bonds and equities by adjustingtheir returns awayfrom equilibriumin a manner consistent with 100%confidence in the currencyviews.

Sept-'87

June-'86

Mar.-'90 June-'91

Dec.-'88

global market-capitalizationweighted average ratio of dividend to bond yield.

views and adjustingreturnsaway fromequilibriumin the appropriate manner. In simulations of a strategy of investingin equity marketswith high ratios of dividend yield to bond yield, we generateviews by assumingthatexpectedexcess returnson equities are above their equilibriumvaluesby an amount equal to 50 times the difference between the ratio of dividendto bond yield in thatcountryandthe

FiguresA and B show the results graphically.Figure A compares the cumulativevalue of $100 invested in each of the three strategies as well as in the equilibrium portfolio,which is a global market portfolio of equities and bonds, with 80% currencyhedging. The strategies were structuredto have riskequal to thatof the equilibriumportfolio.While the graphgives a clear pictureof the relativeperformancesof the differentstrategies,it cannot easily convey the tradeoffbetween risk and return that can be obtained by takinga more or less

In simulations of a strategy of investingin fixed income markets with high yields, we generate Figure B HistoricalRisk/ReturnTradeoffs,July 1981 - August 1991 views by assumingthat expected excess returns on bonds are 10abovetheirequilibriumvaluesby an amountequalto the difference 9 Currency Strategy between the bond-equivalent 8 yield in thatcountryand the global market-capitalizationEquity Strategy weighted average bond-equivaU.S. Equities U 6 lent yield. S;

For example, if the equilibrium riskpremiumon bonds in a given countryis 1%andthe yield on the 10-yearbenchmarkbond is 2 percentage points above the world averageyield,thenwe assumethe expected excess returnfor bonds in that country to be 3%. We computeexpected excess returns on currenciesand equities by assuming 100%confidencein these

5-

P4

/ UnhedgedE wGlobal /

V

GlobalHedged

4-

4

/ 2-

LU

cn

z

^Equilibrium

0

Bond Strategy

/

/

2U.S. lIBOR

(N

z

S. Bonds

-J

U 0-

0

2

4

6

8

Risk (%)

10

12

14

16

z

41

aggressive position for any given strategy. Figure B makes such a comparison. Because the simulations have no constraints on asset weights, the risk/return tradeoffs obtained by combining the simulation portfolios with cash are linear and define the appropriate frontier for each strategy. We show each frontier, together with the risk/return positions of several benchmark portfolios-domestic bond and equity portfolios, the equilibrium portfolio and global market-capitalization-weighted bond and equity portfolios with and without currency hedging. What we find is that strategies of investing in high-yielding currencies and in the equity markets of countries with high ratios of dividend yields to bond yields have performed remarkably well over the past 10 years. By contrast, a strategy of investing in highyielding bond markets has not added value. Although past performance is certainly no guarantee of future performance, we believe that these results, and those of similar experiments with other strategies, suggest some interesting lines of inquiry.

Conclusion

r54

0) 0) cl:

Quantitativeasset allocation models have not played the important role that they should in global portfolio management. We suspect that a good part of the problem has been that users of such models have found them difficult to use and badly behaved.

LLJ

views. By adjusting the confi- 6. The equilibrium-risk-premidence in his views, the investor ums vector HIis given by Hl = can control how strongly the 8K:W,where 5 is a proportionviews influence the portfolio ality constant based on the forweights. Similarly,by specifying a mulas in Black."8 ranking of confidence in different views, the investor can control 7. The expected excess return, E[R],is unobservable. It is aswhich views are expressed most sumed to have a probability strongly in the portfolio. The indistribution that is proporvestor can express views about tional to a product of two northe relative performance of assets mal distributions. The first as well as their absolute perfordistribution represents equimance. librium; it is centered at 1I with a covariance matrix r-, We hope that our series of examples-designed to illustrate the where r is a constant. The secinsights that quantitative modelond distribution represents ing can provide-will stimulate the investor's views about k linear combinations of the elinvestment managers to consider, or perhaps to reconsider, the apements of E(R]. These views plication of such modeling to are expressed in the following form: their own portfolios.

Appendix

PE[R] =

1. n assets-bonds, equities and currencies-are indexed by i = n,...,n.

Here P is a known k *n matrix, Q is a k-dimensional vector, and ? is an unobservable normally distributed random vector with zero mean and a diagonal covariance

2. For bonds and equities, the market capitalization is given by Mi. 3. Marketweights of the n assets are given by the vector W = {W1, ... ., W,J. We define

We have learned that the inclusion of a global CAPMequilib4 z rium with equities, bonds and currencies can significantly improve the behavior of these models. In particular, it allows us to Ul distinguish between the views of 4:D the investor and the set of ex4 pected excess returns used to 4 drive the portfolio optimization. This distinction in our approach z allows us to generate optimal 4portfolios that start at a set of neutral weights and then tilt in 42 the direction of the investor's

If asset i is a bond or equity: Mi

Wi=

-J Fz

8. The resulting distribution for E[R is normal with a mean E[RI: E[R]= [(72)-1 + Pf-V1P]-1 lrH+ P'0 - 1Q]. [(,Ty, In portfolio optimization, we use E[R] as the vector of expected excess returns.

If asset i is a currency of the jth Footnotes country: 1. For some academic discussions of Wi = AWjcl

LL

L/,I LU r-

matrixQ.

the

Wis as follows:

0~

LU

Q+

where Wjc is the country weight (the sum of market weights for bonds and equities in the jth country) and A is the universal hedging constant. 4. Assets' excess returns are iven by a vector R= R1, . . .,

5. Assets excess returns are normally distributed with a covariance matrix L:.

this issue, see R. C Green and B. Hollifeld, "WhenWill Mean-Variance Efficient Portfolios Be Well Diversified?"Journal of Finance, forthcoming, and M j Best and R. R. Grauer, "On the Sensitivityof Mean-Variance Efficient Portfolios to Changes in AssetMeans: Some Analytical and Computational Results," Review of Financial Studies 4 (1991), pp. 16-22. 2. H Markowitz, "PortfolioSelection," Journal of Finance, March 1952; j Lintner, "TheValuation of RiskAssets and the Selection of RiskyInvestments in Stock Portfolios and Capital Budgets," Review of Eco-

nomics and Statistics,February 1965; and W F Sbarpe, "Capital Asset Prices:A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance, September 1964 3. F Black, "UniversalHedging: How to Optimize Currency Risk and Reward in International Equity Portfolios," Financial AnalystsJoumaL July/August 1989. 4 In actual applications of the model, we typically include more asset classes and use daily data to measure more accurately the current state of the time-varying risk structure. We intend to address issues concerning uncertainty of the covariances in another paper. For the purposes of this article, we treat the true covariances of excess returns as known. 5. We define excess return on currency-hedged assets to be total return less the short rate and excess return on currency positions to be total return less theforward premium. In Table II, all excess returns and volatilities are percentages. The currency-hedged excess return on a bond or equity at time t is given by:

Et =

Pt + l/Xt + 1

* 100

Pt/xt -

(1 + Rt)FXt-Rt,

where P, is the price of the asset in foreign currency, Xt the exchange rate in units offoreign currency per US dollar, R, the domestic short rate and FX, the return on a forward contract, all at time t. The return on a forward contract or, equivalently, the excess return on a foreign currency, is given by: Ftt +1 -

FXt=

xt

t1 1

' 100,

where F't+' is the one-periodforward exchange rate at time t. 6 We choose to normalize on 10. 7% risk here and throughout the article because it happens to be the risk of the market-capitalization-weighted 80% currency-hedgedportfolio that will be held in equilibrium in our model. 7. For the purposes of this exercise, we arbitrarily assigned to each country the average historical excess return for across countries, as follows-.2 currencies, 0.4 for bonds and 5.1 for equities.

8. See Black, "UniversalHedging," op. cit 9. The "universal hedging" equilibrium is, of course, based on a set of simplifying assumptions, such as a world with no taxes, no capital constraints and no inflation. Exchange rates in this world are the rates of exchange between the different consumption bundles of individuals of diferent countries. While some may find the assumptions that justify universal hedging overly restrictive, this equilibrium does have the virtue of being simpler than other global CAPMequilibriums that have been described elsewhere. (See B. H. Solnik, "AnEquilibrium Model of the International Capital Market,"Journal of Economic Theory, August 1974, or F. L. A. Grauer, R. H Litzenbergerand R. E Stehle, "SharingRules and Equilibrium in an International Capital Market Under Uncertainty," Journal of Financial Economics 3 (1976), pp. 233-56) While these simplifying assumptions are necessary to justify the universal hedging equilibrium, we could easily apply the basic idea of this article-combining a global equilibrium witb investors' views-to another global equilibrium derivedfrom a different, less restrictive,set of assumptions. 10. Views can representfeelings about the relationships between observable conditions and such relative values. 11. In this article we use the term "strength"of a view to refer to its magnitude. We reservethe term "confidence"to refer to the degree of certainty with which a view is held. 12. We try here to develop the intuition behind our approach using some basic concepts of statistics and matrix algebra. A more formal mathematical description is given in the appendix. 13. A 'prime" symbol (e.g., P') indicates a transposed vector or matrix. 14. H Theil, Principles of Econometrics (New York:Wiley and Sons, 1971). 15. F Black and R. Litterman, "Asset Allocation: Combining Investor Views with Market Equilibrium" (Goldman, Sachs & Co., September 1990). 16 For details of these views, see the following Goldman Sachs publications: The International Fixed Income Analyst August 2, 1991, for interest rates and The International Economics Analyst,July/August 1991, for exchange rates. 17. We discuss this situation later.

18. Black, "Universal Hedging,"Op. Cit

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