Goldman

Andrew BevanKurt Winkelmann

Fixed IncomeResearchGLOBAL FIXED INCOME PORTFOLIO STRATEGY

June 1998 Using theBlack-LittermanGlobal AssetAllocation Model:Three Years ofPractical Experience

Goldman, Sachs & Co. Global Fixed Income Portfolio Strategy Fixed Income Research

Acknowledgments

We wish to acknowledge the comments and suggestions of Joanne Hill, Robert Litterman, Robert Litzenberger,Tom Macirowski, Scott McDermott, Victor Ng, and Scott Pinkus. Over the past three years, we have also re-ceived valuable feedback on our portfolio process from the Global Economics Group.

Andrew BevanLondon44-171-774-1168

Kurt WinkelmannLondon44-171-774-5545

Andrew Bevan is an Executive Director in the Economic Research Group at Goldman Sachs International inLondon.

Kurt Winkelmann is an Executive Director in the Fixed Income Research Department at Goldman Sachs Interna-tional in London.

Editor: Ronald A. Krieger

Copyright © 1998 by Goldman, Sachs & Co.

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Using the Black-Litterman Global Asset Allocation Model:Three Years of Practical Experience

Contents

Executive Summary

I. Introduction.................................................................................................................................................... 1

II. An Overview of Our Process ........................................................................................................................ 2

III. Calculating Equilibrium Returns............................................................................................................... 2

IV. Setting the Weight-on-Views and Confidence Levels................................................................................. 4

V. Setting Target Risk Levels............................................................................................................................ 5

VI. Optimization and Risk Decomposition ....................................................................................................... 7

VII. How Have We Done? ................................................................................................................................. 8

VIII. Conclusions............................................................................................................................................. 10

Appendix A: Interpreting the Published Portfolios ....................................................................................... 12

Appendix B: Practical Issues in Developing a Process................................................................................... 13

Bibliography .................................................................................................................................................... 15


Executive Summary

The Black-Litterman model, developed at Goldman Sachs in the early 1990s, provides a framework for combin-ing investor views with a global capital market equilibrium. Its purpose is to help investment managers determinean optimal portfolio allocation for specific classes of assets in a manner consistent with their market views. Withthis model, we can calculate optimal portfolio weights by using volatilities and correlations across asset classes.

For the past three years, we have been publishing optimal global fixed income portfolios using the Black-Litterman framework — both to offer portfolio advice to our clients consistent with the views of our economistsand to illustrate how the model can be used to solve practical investment management problems. This paper pro-vides a summary of our experiences in using the model for investment strategy.

Following an overview of our “investment process,” we first explain how we set the key parameters in the Black-Litterman framework. This involves a discussion of how we use the model to observe the equilibrium returns inglobal capital markets and then blend the equilibrium returns with our own views to provide a set of expectedreturns. We explain how we determine the weight and confidence levels on our own views relative to the equilib-rium. Next we discuss risk control and optimization. We describe the process we follow to set tracking error riskand Market Exposure (a statistical measure of a portfolio’s sensitivity to market moves). Finally, we discuss ourperformance over the three-year period and consider how the same framework can be applied to other fund man-agement issues.

It turns out that in the aggregate, our published portfolio has outperformed its benchmark over the last threeyears. However, we focus on our performance in this paper only to illustrate how the Black-Litterman frameworkcan be used for designing investment strategies. Clearly, the performance relative to the benchmark will in largepart reflect the accuracy of our views. At the same time, however, it is at least as important to consider the impactof the risk control mechanisms used in the model. We have constructed our own portfolio against a global gov-ernment bond index, but we can easily apply the approach to other asset classes and other benchmarks. Designingrisk-controlled portfolios is likely to become increasingly important in global fixed income markets with the ad-vent of European economic and monetary union and with the growth of local currency emerging governmentbond markets.


June 1998 1

Using the Black-Litterman Global Asset Allocation Model:Three Years of Practical Experience

I. Introduction

The basic investment management problem is tosimultaneously maximize performance and managerisk: Investors determine risk-controlled allocationsto specific asset classes that make best use of theinformation at their disposal. Quantitative invest-ment theory holds out the potential of providing asystematic framework for solving this problem. Atthe heart of this framework is an asset allocationmodel, which practitioners can use to find portfolioallocations that best reflect their investment views.They can determine optimal portfolio weights by usingthe volatilities and correlations across asset classes.

However, many practitioners have not fully incorpo-rated this framework into their investment manage-ment processes, for two reasons: First, this approachcan lead to dramatic swings in the optimal portfolioweights for small changes in investment views. Sec-ond, the optimal portfolio weights in the traditionalmean/variance framework often seem to be takingrisk positions that appear to be at odds with the strong-est investment views. The Black-Litterman model(see Black and Litterman, 1990, and Black and Lit-terman, 1991) was developed to provide a system-atic resolution to these problems.1

A central feature of the Black-Litterman frameworkis the notion that investors should take risk wherethey have views, and correspondingly, they shouldtake the most risk where they have the strongestviews. In the Black-Litterman framework, all ex-pected returns are viewed as a blend of a set ofequilibrium returns (reflecting a neutral referencepoint) and an actual set of investor views (whichshould differ from the equilibrium returns). Conse-quently, the problem facing the practitioner is todetermine the weight given to the actual views.

In February 1995, we started publishing optimalglobal fixed income portfolios using the Black-Litterman framework. These portfolios, which re-flect both our economists’ forecasts and the equilib-

1 A list of references appears at the end of this report, on page 15.

rium returns, appear in our monthly publication,Global Fixed Income Asset Allocation. We had twopurposes in starting this publication: First, wewanted to provide portfolio advice to our clientsconsistent with the views of our Global EconomicsGroup. Second, we wanted to illustrate how theBlack-Litterman model could be used to solve prac-tical investment management problems.

In addition to using the equilibrium concept to findoptimal portfolio weights, our published portfolioshave incorporated risk control as an explicit compo-nent of the investment management process. In par-ticular, we have used the concepts of Hot Spots(see Litterman, 1996) and Market Exposure (see Lit-terman and Winkelmann, 1996) to identify the riskdistribution and directional bias in our portfolios.2

This paper provides a summary of our experiencesin using the Black-Litterman model for investmentstrategy. In the aggregate, over a three-year period,our published portfolio has outperformed its bench-mark. However, the primary purpose of this paper isnot to discuss performance. Rather, our publishedportfolio (see Appendix A) provides an illustrationof how the Black-Litterman framework can be usedfor designing investment strategies. While we haveconstructed our portfolio against a global govern-ment bond index, the approach can be easily appliedto other asset classes and other benchmarks. Indeed,designing risk-controlled portfolios is likely to be-come increasingly important in global fixed incomemarkets as the effects of European economic andmonetary union (EMU) begin to take hold.

The paper is organized as follows: In the next sec-tion we give an overview of our “investment proc-ess.” In Sections III and IV we explain how we setthe key parameters in the Black-Litterman frame-work. In Sections V and VI we discuss risk controland optimization, and in Section VII we review ouractual performance over the three-year period. Sec-tion VIII presents our conclusions and considerssome open issues.

2 Hot Spots is a trademark of Goldman, Sachs & Co.

Goldman, Sachs & Co . Global Fixed Income Portfolio Strategy Fixed Income Research

June 19982

II. An Overview of Our Process

As discussed in the Introduction, in the Black-Litterman framework, expected returns are viewedas a combination of a specific set of investor viewswith a neutral reference point. By extension, theframework has implications for risk control. As aresult, successful implementation of the Black-Litterman model requires careful consideration ofthe key parameters influencing both expected returnsand risk control.

Exhibit 1Optimization Procedure

Step Action Purpose

1 Calculate equilibriumreturns.

Set neutral referencepoint.

2 Determine weightingsfor views.

Dampen impact ofaggressive views.

3 Set target tracking error. Control risk relative tobenchmark.

4 Set target MarketExposure.

Control directionaleffects.

5 Determine optimalportfolio weights.

Find allocations thatmaximize performance.

6 Examine riskdistribution.

Determine whether riskis diversified.

The process that we have adopted is explicit in itsdetermination of these parameters. It has six princi-pal steps, which we summarize in Exhibit 1. First,we find the neutral reference point as the returnsassociated with a global capital market equilibrium.Considered from a different perspective, the assetreturns from our neutral reference point are thoserequired for a representative investor to hold theglobal capitalization-weighted portfolio.

Our second step is to determine how much weight toput on the neutral reference point returns relative toour actual views on asset returns. In this step, wealso set a relative weighting for each of our individ-ual views. The effect of this step is to dampen theimpact on the portfolio composition of particularlyaggressive views.

In our third and fourth steps, we set target risk lev-els. We choose our portfolio weights relative to thecharacteristics of a benchmark portfolio. Conse-quently, the two dimensions of risk for which targetsare set are the tracking error (i.e., projected volatilityof performance differences) and Market Exposure

(i.e., the directional bias of the portfolio relative tothe benchmark).

Our fifth step is to find an optimal portfolio thatmaximizes expected return subject to the risk con-straints. Finally, we consider the risk decompositionof the optimal portfolio to determine whether it sat-isfies our diversification requirements and whetherthe sources of risk are consistent with our moststrongly held views. In the event the portfolio doesnot satisfy these requirements, we return to the sec-ond step and recalibrate both the weight assigned tothe equilibrium returns and the relative weightsgiven to individual views. We will discuss each ofthese six steps in more detail in the sections thatfollow. Also, in Appendix B, we consider somepractical issues involved in distinguishing betweentactical and strategic portfolio positioning.

III. Calculating Equilibrium Returns

This section describes in more detail how we findequilibrium returns. In the Black-Litterman framework,expected returns are viewed as a blend of equilibriumreturns and an actual set of investor views. The equi-librium returns can be interpreted as the long-run re-turns provided by the global capital markets. Under thisinterpretation, the equilibrium returns represent the in-formation that is available through the capital markets.Investor views, by contrast, correspond to the interpre-tation of information that is unique to the individualinvestor. Thus, expected returns are a blend of the in-formation available through the capital markets andinformation unique to a specific investor. As the mix-ture of sources of information changes, expected re-turns will also change.

While equilibrium returns represent a useful neutralstarting point, one immediate problem is that theyare unobservable. However, with a few straightfor-ward assumptions, we can easily infer equilibriumreturns from other data that are readily available.

A natural way to proceed is as follows: We can startby assuming that if asset markets are in equilibrium,a representative investor would hold some propor-tion of the global capitalization-weighted portfolio.This assumption provides us with one readily ob-servable piece of information. We can work fromthe observable capitalization weights to equilibrium


June 1998 3

returns by calculating this portfolio’s volatility andthen finding the asset returns that are consistent witha target Sharpe Ratio (the ratio of projected portfolioexcess returns over cash to portfolio volatility).

Since our problem is a global fixed income assetallocation problem, our capitalization-weighted port-folio consists of the sector allocations for the Gold-man Sachs government bond indexes. These indexesgive the capitalization weights for the one- to three-year, three- to seven-year, seven- to 10- (or 11-)year, and greater-than-10- (or 11-) year maturitysectors in 13 government bond markets. For marketswhere index data are not available, we use a short-

and a long-maturity bond. Representing the globalfixed income markets in this way allows us to uscapture the effects of country allocation, overallMarket Exposure, duration within each country, andsteepness of the curve in each market. Exhibit 2shows an illustration of the capitalization weights.

After finding the capitalization weights, we need toestimate a covariance matrix. At Goldman Sachs, weestimate covariance matrices using daily data.Volatilities and correlations are determined withweighted averages of daily squared returns. In thiscalculation, the weight depends upon the rebalanc-ing horizon (see Litterman and Winkelmann, 1998,for more detail on our covariance matrix estimationprocedures). For the purposes of our Global FixedIncome Asset Allocation portfolio, we assume aweight that is consistent with a two- to three-monthrebalancing horizon. We combine the covariancematrix with the capitalization weights to find thevolatility of the capitalization-weighted portfolio.

Once we have determined the cap-weighted portfo-lio’s volatility, we calibrate asset returns to producea Sharpe Ratio of 1.0. We calibrate the model in thisway for two reasons: first, a Sharpe Ratio of 1.0 canbe interpreted equivalently as a one-standard-deviation event. When returns are distributed nor-mally, a one-standard-deviation event occurs ap-proximately 66% of the time. Since one-standard-deviation events occur with such a high frequency,we can be reasonably sure that they will have beenobserved historically, with the corresponding impli-cation that the equilibrium returns are not dependent

on events that have fewhistorical precedents.

The second reason wecalibrate the model toproduce a Sharpe Ratioof 1.0 is because thatfigure is roughly consis-tent with historical ex-perience. Exhibit 3shows the historicalSharpe Ratios for 17portfolios from January1988 through October1997. The portfoliosconsist of the Group ofSeven equity and bond

Exhibit 2Benchmark Characteristics

Market Weight DurationAustralia 0.74 4.43

Austria 0.98 4.13

Belgium 2.32 4.80

Canada 3.73 5.17

Denmark 1.56 4.27

France 7.19 5.12

Germany 8.06 4.21

Italy 6.51 3.82

Japan 18.34 5.80

Netherlands 2.96 4.84

Spain 2.83 3.75

Sweden 1.58 3.87

U.K. 6.64 5.93

U.S. 36.58 5.02

Exhibit 3Historical Sharpe RatiosJanuary 1988 – October 1997

Hedged excess returnsG-7 Sharpe Ratios are for cap-weighted portfoliosSharpe Ratio over all G-7 assets is 1.05

Canada France Germany Italy Japan U.K. U.S. G-7-0.5

0.0

0.5

1.0

1.5

2.0

Country

Sha

rpe

Rat

io

Bonds Equity


June 19984

markets individually, a capitalization-weighted port-folio of the G-7 equity markets, a capitalization-weighted portfolio of the G-7 bond markets, and acapitalization-weighted portfolio of the combined G-7 equity and bond markets. As the chart illustrates,historical Sharpe Ratios have ranged between -0.2and 1.5, with an average over all 17 portfolios of0.8. The last figure is roughly consistent with ourassumption of an equilibrium Sharpe Ratio of 1.0.

Exhibit 4 shows two sets of equilibrium returns. Thetwo sets of returns correspond to projected SharpeRatios of 1.0 and 0.50. It is important to note thatthe equilibrium returns are not the result of aneconometric forecasting exercise. Instead, the equi-librium returns represent the idea of a set of long-runreturns that are consistent with market clearing. Oneclear role for econometric forecasting models is toprovide indications about short-term movementsaround the long-run equilibrium.

As previously discussed, the Black-Litterman ap-proach combines equilibrium returns with an ex-plicit set of views. Expected returns can, in somesense, be interpreted as a complicated weighted av-erage of the neutral (or equilibrium) returns and aninvestor’s views. In the next section, we will discussthe procedure that we follow for determining howmuch weight to put on the equilibrium returns andhow to set the relative weights for each specificview.

IV. Setting the Weight-on-Views andConfidence Levels

In the next stage of the optimization process, wedetermine how much weight to put on the neutralreference point (equilibrium) returns relative to anexplicit set of investor views. For the latter, we willtake our own views as an example, using the interestrate and currency forecasts provided by the GoldmanSachs Economic Research Group in London (wewill not discuss the forecasting procedure itself).

The blending of individual market views with theequilibrium returns is an important step in ourmethodology. There are two major reasons for fol-lowing this procedure. The first reason is that byreferring to the covariance matrix of historical re-turns (implicit in calculating the equilibrium), weensure greater consistency across our own views. Inthis sense, the equilibrium returns help to serve as amacro constraint on our forecasts. The second rea-son is that, in damping extreme views relative to theimplied equilibrium, the methodology producesmore balanced portfolios than are typically producedfrom an unconstrained mean-variance optimization.

In determining the “weight-on-views” (WOV), weconsider the projected excess return on the portfoliorelative to the benchmark. In particular, we look atthe portfolio’s projected “information ratio” (IR).We define the IR as the forecast excess return on theportfolio over the benchmark, divided by its trackingerror, where the latter is one standard deviation ofexcess returns relative to the benchmark (see thenext section for a discussion of tracking error risk).

As such, the IR can beconsidered as the fore-cast excess returns on theportfolio expressed interms of standard devia-tions, based on the co-variance matrix ofhistorical returns. Wecan interpret the IR as ameasure of the informa-tion contained in the in-vestor’s views. In ourprocess, we select theWOV to produce an IRof no greater than 2.0, ongrounds that returns of

Exhibit 4Equilibrium Expected Returns

Returns are excess over local cash

ATS AUS BGM CDA DKK FRA GER ITA JPN NDL SPN SEK U.K. U.S.0

1

2

3

4

5

Market

Ret

urn

(%)

SR = 1.0 SR = 0.50


June 1998 5

greater than two standard deviations in excess of thebenchmark would be statistically unlikely to occur.

From this we can see how the WOV serves, in ef-fect, partly as a constraint at the macro level on ourown views. An optimization based on the uncon-strained views often will lead to a projected IR sig-nificantly in excess of 2.0. This would usually bethe result of one or more extreme views relative tothe others. Controlling the WOV usually leads tomore-balanced portfolios. If left unconstrained, theportfolio will overweight the views with the largestrisk-adjusted returns. By combining our forecastswith the implied equilibrium returns, we dampen theinfluence of extreme views. Typically, in order tokeep the IR at 2.0 or lower, we have attached aWOV of 50–70%.

Having determined the WOV and the set of expectedreturns to be used in the optimization, we then con-sider constraints at the micro level on the individualmarket views, which we refer to as “confidenceweightings.” We use the model to determine theconditional probability of the projected excess re-turns on each asset and currency. In other words,using the covariance matrix of historical returns forN assets, we determine the probability of observingthe return on the Nth asset, conditional on the returnforecast for the N-1 assets. In the same way that weimpose the constraint at the macro level that wewould not want the projected IR of the portfolio tobe in excess of 2.0, we attempt also to ensure thatthe individual market views are not greater-than-two-standard-deviation events.

We constrain the views on projected returns on theindividual exposures by allocating confidenceweightings. We attach either a Low, Medium, orHigh confidence weighting to each of the separateviews. If, for example, the forecast excess return onAustralian bonds is a three-standard-deviation eventrelative to the other views, then we would give aLow confidence weighting to the Australian bondforecast. This has the effect of using less of the in-formation in the actual view, and more of the infor-mation in the equilibrium returns.

Why would we dampen the influence of a particularview? The principal reason is because our statisticalanalysis is telling us that this view is unlikely toprove correct, based on the historical behavior of

asset returns. Thus, putting less emphasis on a par-ticular view lets us incorporate the information thatgave rise to that view, and at the same time producea portfolio whose performance is not overly depend-ent on an asset return that is statistically unlikely tooccur.

An alternative to using the WOV and confidenceweightings to constrain the projected excess returnswould be to revisit the currency and interest rateforecasts. Indeed, this occurs in the process of pro-ducing our own optimal portfolio; the model is used,to a certain extent, as part of an iterative process inhelping to build the forecasts of the Economic Re-search Group. The intention is not to suppress thedeliberate expression of strongly held views, how-ever. After all, as we will discuss in Section VI, it isthe identification of these key views that we use tomake sure that the portfolio is taking risk where weare comfortable. Rather, we use WOV and confi-dence intervals to ensure greater statistical consis-tency across the bond and currency forecasts.

V. Setting Target Risk Levels

After finding expected returns, we then set targetrisk levels. Since we construct our optimal portfoliorelative to a benchmark, we consider all of our riskmeasures as risks relative to the benchmark. The tworisks that we care most about are the tracking errorand the Market Exposure. (See Litterman andWinkelmann, 1996, for a detailed discussion of bothof these measures.)

Tracking error measures the volatility of the portfo-lio and benchmark performance differences. Sincetracking error is a standard deviation, we can give ita natural probabilistic interpretation. For example, atracking error of 100 basis points tells us that(roughly) 66% of the time, the actual performance ofthe portfolio should be within 100 bp of the bench-mark’s performance, irrespective of whether thebenchmark return is positive or negative.

To measure the directional bias of the portfolio, welook at the Market Exposure. A portfolio’s MarketExposure tells us how responsive the portfolio’s per-formance is for given levels of benchmark perform-ance, with the caveat that all other influences areassumed to be constant. For example, a Market Ex-


June 19986

posure of 1.2 tells us that if the benchmark has a1.0% return, then the portfolio will have a 1.2% re-turn, other variables held constant. When the MarketExposure is greater than 1.0 the portfolio has abullish bias, and when the Market Exposure is lessthan 1.0 the portfolio has a bearish bias.3

3 Practitioners sometimes look at other types of risk measures.

For example, portfolio managers might look at the Value-at-Risk (VaR) of a portfolios. The VaR shows the size of thepotential loss for specific probability choice. For example,suppose that we set the probability level at 1.7% and we findthat our portfolio will lose 200 bp or more relative to itsbenchmark at this probability level. In this case, our VaR is200 bp.

When assets have a symmetric payoff, VaR has a natural in-terpretation in terms of tracking error. Continuing with theexample, when asset returns follow a Normal distribution,the probability level of 1.7% corresponds to two standarddeviations. Thus, the portfolio has a VaR that corresponds toa two-standard-deviation event. Since tracking error is ex-pressed in terms of one standard deviation, we find that theVaR of 200 bp corresponds to a tracking error of 100 bp. Wecan do the same kind of calculation for other choices ofprobability and VaR whenever the assets have symmetricpayoffs.

A second popular risk measure is called “downside risk.”The basic idea behind downside risk is to find the expectedvalue of portfolio returns less than some critical level. Forexample, if the critical value is set at zero, we can calculateboth the expected value of all portfolio returns less than zeroand the probability that the portfolio will have negative port-folio returns. As with VaR, when the distribution of assetreturns is symmetric, downside risk has equivalent interpre-tations in terms of tracking error. Also in common with VaR,when payoffs are not symmetric (i.e., when options are pres-ent), downside risk does not have an equivalent interpreta-tion in terms of standard deviations.

How do we set our tworisk parameters? Let’slook first at the MarketExposure. When ourviews on market direc-tion are relatively neu-tral, we constrain theoptimal portfolio to havea Market Exposure of1.0. Imposing this con-straint means that ouroptimal portfolios reflectour curve views andcountry views but areneutral in any overalldirectional sense. Bycontrast, when we have

strong directional views, we let the optimizer findthe Market Exposure number. For example, whenour views are strongly bullish, the optimizer willselect a portfolio whose Market Exposure is greaterthan one.

Our tracking error figures have been set to produce aportfolio with a tracking error of roughly 100 bp.While we also analyze portfolios with 200 bp oftracking error, our focus is on the lower risk portfo-lio. The reason for this is that in our experience, thehigher risk optimal portfolios have tended to con-centrate risk in a smaller number of positions. Inother words, we have found that a tracking error of100 bp has been more appropriate to give adequateexpression to our views.

We also consider actual industry practice in choos-ing our tracking error figures. Recent experienceshows that a tracking error of 100 bp has been con-sistent with the practice of global fixed income fundmanagers. Exhibit 5 illustrates this point. The chartplots the tracking error and Market Exposure forparticipants in a survey of global fixed income port-folio managers. As shown in the chart, our trackingerror of 100 bp is approximately the median trackingerror for the survey participants.

Exhibit 5Survey Risk Characteristics

Source: GS Global Fixed Income Survey, August 1997

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80

50

100

150

200

250

Market Exposure

Trac

king

Err

or (

bp)


June 1998 7

VI. Optimization and Risk Decomposition

Once our target risk levels have been set, we findportfolios that maximize expected return. We thentake the optimal portfolio weights and identify themajor sources of risk (see Litterman, 1996), on thepremise that the major sources of risk should also beconsistent with the most strongly held views. In ourprocess, we typically look for optimal portfolioweights that satisfy three criteria. First (as discussedabove), we want to make sure that the risk exposuresin the optimal portfolio correspond to positionswhere we want to take risk. Second, we typicallyimpose an upper bound of 20% of the risk from anyparticular position: That is, any currency or bondmarket position should contribute no more than 20%of the tracking error. Finally, we try to ensure thatthe tracking error is evenly balanced between ourbond and currency positions.

We look for a balanced distribution of risk for oneprincipal reason: Spreading the risk across severalpositions means that we can exploit the power ofdiversification. When the risk in a multi-asset port-folio is concentrated in one or two positions, theimplication is that all other portfolio managementdecisions are not likely to have a significant impacton the portfolio’s performance. (In this case, the in-vestment manager could be well advised to considerimplementing the optimal portfolio through optionsstrategies, thereby limiting the potential losses). Inpractice, of course, we are likely to have more than

one or two strongly heldviews that we would liketo see expressed in theportfolio.Why would the optimalportfolio concentrate riskin one particular posi-tion? Simply put, theoptimal portfolio willallocate its risk to themost aggressively heldviews. Thus, when oneor two views are muchmore aggressive than theremaining views, theoptimal portfolio willconcentrate its risk inthose positions.

It is important to keep in mind that views can beaggressive in two senses: relative to their own his-tory (as captured by the volatility) and relative to allother views. In the first case, the portfolio concen-trates its risk in positions that are driven by viewsthat are not likely to prove correct in the context oftheir own history. In the second case, the portfolioconcentrates its risk in positions based on views thatare not likely to prove correct if all other views areborne out, assuming that the structure of the covari-ance matrix remains unchanged. An implication ofthe second point is that views also have implicationsfor correlations. We illustrate this point in Exhibit 6.

Exhibit 6 plots the correlation between German andItalian 10-year bond returns against the conditionalZ-Score for Italian bonds.4 The assumption underly-ing the chart is that German bonds will sell off andItalian bonds will rally. The size of the respectivesell-off and rally is set equal to one standard devia-tion. As the chart illustrates, given a correlation

4 We calculate the conditional Z-scores as follows: Suppose

that we have N assets and (for illustrative purposes) we haveviews on all N. For each of the N assets, we can compare theactual view with the view that is most likely given the otherN-1 views. We find the most likely views by using the co-variance matrix to project the Nth asset on the N-1 assetsand then substitute in the views on the N-1 assets to find theconditional return on the Nth. We then find the differencebetween the actual view and the most likely view. To makecomparisons across assets, we take the ratio of the differenceto the conditional volatility.

Exhibit 6Correlation and Performance

Constant one-standard-deviation rally in Italian 10-yearConstant one-standard-deviation sell-off in German 10-year

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.500.8

1.0

1.2

1.4

1.6

1.8

Correlation

Con

ditio

nal P

erfo

rman

ce Standardized Cond. Performance


June 19988

between German and Italian bond returns of 0.50, ifa one-standard-deviation sell-off in German bondreturns occurs, then a one-standard-deviation rally inItalian bonds is a very unlikely event (as measuredby the conditional Z-score). Indeed, the correlationbetween Italian and German bond returns must be-0.05 in order for the conditional Z-score to be zero.

One natural way to correct for the implicit view onthe correlation matrix is to dampen the views — inother words, take less-extreme views. In fact, this isexactly the role of the weight-on-views and confi-dence parameters. Decreasing the weight-on-viewsor assigning a Low confidence to a particular viewhas the effect of pulling the views back toward theequilibrium views. From a correlation perspective,decreasing the weight-on-views (or assigning a Lowconfidence) has the effect of pulling the correlationthat makes the expected returns statistically likelymore consistent with the actual covariance matrix.The net result is that the ultimate portfolio providesa better diversification of the portfolio’s risk — i.e.,less-aggressive risk positions are taken.

The risk distribution has important implications forcomparing the optimal portfolio with alternativeportfolio weights. For example, it is often heard thatportfolios with the same risk level as an optimalportfolio, but interior to the efficient frontier, mayhave desirable properties. These properties can besummarized as lower exposure to adverse moves ina particular set of asset prices.5

Let’s look at these desirable properties in more de-tail. The idea that a point that is interior to the fron-tier is less exposed to potentially adverse moves insome asset prices can be interpreted as saying thatrisk is better diversified than in the optimal portfo-lio. As discussed above, the risk distribution in theoptimal portfolio depends on the relative strength ofthe investment views: The most risk is taken whereviews are the most aggressive. Thus, the desirableproperties of suboptimal portfolios are a result of theoptimal portfolio seeking the maximum leverage

5 The observation that points interior to the efficient frontier

have potentially desirable properties is sometimes used tojustify so-called “scenario-dependent” optimization.

from very aggressive investment views. Using theequilibrium in a systematic way to dampen the im-pact of aggressive views has the effect of providinga portfolio that is well diversified in a risk sense andexpressing views that are internally consistent.

VII. How Have We Done?

In this section, we consider the performance of ourportfolio strategy recommendations. Clearly, theperformance relative to the benchmark will in largepart reflect the accuracy of our views. At the sametime, however, it is at least as important to considerthe impact of the various control mechanisms usedin the Black-Litterman model. That is because theprocedures that we have outlined above are designedto help in the process of risk management. Inde-pendently of the accuracy of our views, it is of inter-est to know whether these control procedures haveworked effectively.

We have been publishing regular updates to ourportfolio recommendations since the beginning of1995. We calculate the relative performance of theportfolio on a monthly basis by adjusting the over-or underweighting in each market for the recom-mended duration relative to the index. Similarly, wecalculate the relative foreign exchange performanceas the product of the actual performance of the cur-rency and the relative overweighting (or under-weighting) in that currency.

For two reasons, these results should be regarded asonly an approximate measure of portfolio perform-ance. First, the calculation of returns is based on theportfolios at the beginning of each month, thoughthe actual portfolio revisions were often made atdifferent times. Second, no allowance is made forshort-term trading views that have been expressed inother regular commentaries.


June 1998 9

Last year, our base case portfolio underperformedthe benchmark by 26 bp (through end-December1997). In 1996, the portfolio outperformed thebenchmark by 83 bp, and in 1995, the portfolio’sreturn exceeded that of the benchmark by 103 bp.The average over the three-year period has been 52bp. Given that we aim for a projected tracking errorof 100 bp, the ex-post information ratio (i.e., per-formance relative to risk) has been 0.52.

Monthly Returns: Exhibit 7 shows the portfolio’smonthly excess returns over the benchmark. Thechart further decomposes total relative performanceinto its fixed income and currency components. Ex-hibit 8 shows a history of standardized monthly per-formance for close to three years. As we discussbelow, this history provides some evidence of theefficacy of our risk control procedures

In spite of a bearish outlook for global bond marketsat the outset of 1995, the portfolio outperformed thebenchmark over the year as a whole, for three rea-sons. The first reason was that, although bearishviews predominated in the major markets at the outsetof 1995, we switched to a bullish near-term outlook forboth the Japanese and European bond markets and ex-tended the duration of the portfolio to longer than thatof the index for April–August. The second reason isthat we remained appropriately underweighted in theUnited States relative to Europe and were mostly closeto index weight in Japan. The third reason is that weadopted a bullish outlook on the U.S. dollar

against both the yen and European currencies fromFebruary, and the portfolio then remained long thedollar, mostly in terms of European currencies, for therest of the year.

In 1996, the dominant source of outperformancerelative to the benchmark was our currency position-ing. Cumulatively, our bullish stance on the U.S.dollar for the bulk of 1996 provided 106 bp of theportfolio’s relative outperformance. In only four ofthe 12 months were our foreign exchange positionsat odds with actual events in the currency market.The figures also indicate that the portfolio’s bondpositioning acted as a drag on performance: Thebond component of the portfolio underperformed thefixed income component of the benchmark byroughly 23 bp. This is because the portfolio was notpositioned for the global bond rally in the latter partof the year. In the period up until the end of July, thefixed income portion of the portfolio outperformedthe benchmark by 42 bp on the benchmark, whilefrom August through December, relative perform-ance was -65 bp.

Over the first eight months of 1997, the portfolio’sperformance was dominated by the incorrect bearishstance that we adopted, for the most part, on overallmarket direction. The duration of the portfolio wasshort relative to the benchmark for the four monthsending in July. In August, we set a constraint thatthe bond exposures should have a Market Exposure(or Beta) coefficient of 1.0, and we retained thisuntil December, when we lengthened the Market

Exposure of the hedgedbond holdings to 1.08.For the year as a whole,the bond component ofthe portfolio outperformedthe benchmark by 12 bp,while the currency com-ponent underperformed by39 bp.

Exhibit 7Monthly PerformanceFebruary 1995 – December 1997

GS Global Fixed Income Asset AllocationLow Risk Portfolio/Three-Month HorizonPredicted Tracking Error = 100 bp

2/95 8/95 2/96 8/96 2/97 8/97-80

-60

-40

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June 199810

Risk Analysis: Clearly, even after taking into ac-count the control mechanisms detailed above and theapproach that we take to blending expected excessreturns with the equilibrium, the ex-post perform-ance relative to the benchmark will depend heavilyon the accuracy of our views. We would make twoobservations, however. First, we should compare theperformance with an unconstrained optimization.This would be one way of measuring the efficacy ofthe model. Second, it is at least as important to con-sider whether the risk control parameters applied inthe model work efficiently. We can explore the latterissue by referring to Exhibit 8.

Recall that Exhibit 8 shows the entire history of theportfolio’s performance on a standardized basis. Wecalculated the figures in the chart by taking the ac-tual monthly performance and dividing by the pre-dicted monthly tracking error. Standardizedperformance provides a method for evaluating therisk control elements of the portfolio constructionprocess; our objective is to eliminate very largefluctuations in monthly performance.

To illustrate, for an annualized tracking error of 100bp, we would anticipate a monthly tracking error ofroughly 29 bp. That is, if our risk measurement andcontrol procedures are accurate, then 66% of thetime we would expect the actual monthly perform-ance of the portfolio to be within 29 bp of the per-formance of the benchmark. Similarly, 34% of thetime we would expect monthly relative performanceto exceed 29 bp.

As Exhibit 8 shows, in 22of the 33 months (theportfolio started in Febru-ary 1995) we had actualperformance within 29 bpof the benchmark. Bycontrast, in only one ofthe 33 months was ouractual performance out-side the two-standard-deviation band. On bal-ance, then, we can con-clude that our portfolioprocess is succeeding inits approach to risk con-trol.

VIII. Conclusions

This paper has focused on our use of the Black-Litterman model to develop portfolios that best re-flect our investment views. We have discussed theprocedures that we use to set all of the Black-Litterman parameters and shown how the choice ofthese parameters affects the structure of risk-controlled optimal portfolios. Finally, we have dis-cussed the performance of our particular portfolios.Two features of our actual performance stand out:First, we have outperformed the benchmark over thethree-year period. Second, and in our view of equalimportance, our portfolio’s performance has beenenhanced by using the risk-control characteristicsinherent in the Black-Litterman model.

We have not discussed procedures for actual imple-mentation of the optimal portfolios. The output fromour exercise is allocations, sector exposures (e.g.,the one- to three-year sector in Italy), and durationsin each of the markets in our optimization problem.Investors clearly cannot purchase the individualsectors and must implement the portfolio throughthe purchase of actual securities. The natural solu-tion to this problem is to use a fitted curve to iden-tify cheap securities in each of the relevant sectors.Investors would choose weights in the individualsecurities to match the allocations and durationsproduced by the optimization process.

Exhibit 8Standardized Monthly Performance

GS Global Fixed Income Asset AllocationLow Risk Portfolio/Three-Month HorizonPredicted Tracking Error = 100 bp

2/95 8/95 2/96 8/96 2/97 8/97-2

-1

0

1

2

3

Date

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June 1998 11

The discussion in this paper has been oriented to-ward selecting an optimal asset allocation for a port-folio whose performance is measured against aglobal bond index. However, the approach can beextended to other applications. For instance, onepopular investment strategy is to allow internationalinvesting versus a domestic benchmark, an invest-ment style that has been labeled “opportunistic in-ternational investing.” For example, Frenchinvestors may purchase Japanese government bonds,even though their benchmark is a French govern-ment bond benchmark. Alternatively, investors witha global bond benchmark may take exposure inemerging markets on an opportunistic basis (wehave included Poland and South Africa in our rec-ommended portfolio since January 1998). Two keyaspects of our framework can provide insight intothis process:

• First, since investors are taking risk in interna-tional positions because they expect interna-tional markets to outperform domestic markets,the risk control dimensions of our frameworkprovide a means for ensuring that risk is wellbalanced.

• Second, the identification of aggressive views

(both absolute and relative) helps investors de-termine the probability of international outper-formance contingent on their views of domesticperformance.

Active managers add value by using information totake risk positions that deviate from their bench-marks. We have shown through a practical examplehow the Black-Litterman model can be used in thisprocess. The model has provided an effectivemeans to combine a particular set of investor views(i.e., investor information) with the returns givenby a capital market equilibrium (i.e., the market’sinformation). As the effects of risk positions be-come more pronounced — e.g., through the influ-ence of EMU — the ability to efficiently useinformation to produce risk-controlled portfoliosbecomes more acute. Our framework provides onemeans to this end. P


June 199812

Appendix A:Interpreting the Published Portfolios

In Exhibit 9, we provide an example of a recentlypublished portfolio taken from Global Fixed IncomeAsset Allocation. Drawing on the discussion in thevarious sections above, we will illustrate how tointerpret the portfolio recommendations and the riskanalysis.

In the “Bonds” section of the table, the first columnshows the market-capitalization weights of thebenchmark, using Goldman Sachs government bondindexes. Alongside this, in the second column, weshow the recommended bond weightings derivedfrom our optimization.

In the “Duration” section, the first column is theduration of the benchmark sectors. The second col-umn shows the recommended duration. At the footof the table, we can see both the aggregate durationof the benchmark and the aggregate duration of the

portfolio. We would caution against using durationas a measure of the portfolio’s risk. Our preferredmeasure is the Market Exposure of the portfolio, asdiscussed in Section V of this report but not shownhere.

The “Forex Overlay” column shows the additionalunits of foreign currency to be purchased or sold,which — combined with the bond exposures — willproduce the total currency exposure recommendedin the optimal portfolio. The total currency positionis shown in the “Net Allocation” column. The shortor long currency exposures of the portfolio can befound by comparing this column with the bench-mark in the first column.

Finally, the “Risk” section shows the marginal con-tribution to the tracking error risk of the portfolioarising from the individual bond and currency expo-sures relative to the benchmark. (For a detailed dis-cussion of the contribution to risk, see Litterman,1996.)

Exhibit 9Recommended Bond and Currency Weightings

GS Global Bond Portfolio - One Month Horizon (Dollar Based)

NetBonds (%) Duration (years) Allocation Risk*(%)

(%)

GS GS Forex GS Index Recomm- Index Recomm- Overlay Recomm- Bonds Currency

ended ended (%) ended

US 35.9 11 5.3 10.5 23 34 27.1 -Canada 3.4 0 5.6 0.0 3 3 3.6 0.0Australia 0.7 2 4.3 6.5 -1 1 2.3 0.0Japan 17.7 12 5.8 9.1 6 18 1.0 0.0Euroland 32.5 39 4.7 5.0 -2 37 16.7 12.0UK 6.9 4 6.3 10.2 0 4 0.2 0.1Denmark 1.5 18 4.9 3.6 -16 2 17.0 0.0Sweden 1.4 8 4.3 4.1 -7 1 12.1 0.0Poland - 5 - 1.9 -5 0 7.4 0.0South Africa - 1 - 5.0 -1 0 0.5 0.0

TOTAL 100 100 5.3 5.9 - 100 87.9 12.1

*Percentage contributions to tracking error risk of the portfolio. Errors due to rounding.

Source: Global Fixed Income Asset Allocation, Goldman Sachs International, June 1, 1998


June 1998 13

Appendix B:Practical Issues in Developing a Process

In the past three years, we have attempted differentapproaches to distinguish between strategic and tac-tical portfolio positioning. We constructed a strate-gic portfolio to draw a distinction between the bondholdings that reflect our longer-term views (basedon our interest rate and currency forecasts) and port-folio positions that represent shorter-term tacticalviews. Strategic portfolio positions should, in prin-ciple, exhibit little change on a month-to-month ba-sis (until, of course, our fundamental views change).By contrast, tactical trading positions can changefrequently.

Our procedure was relatively straightforward. Usingour Low Risk (100 bp of tracking error) portfolioderived from our three-month horizon views, weidentified combinations of deviations from thebenchmark portfolio that gave rise to an informationratio (projected performance divided by portfoliorisk) that was about 80% of the optimal portfolio’sinformation ratio. Among the set of portfolios thatsatisfied this condition, we then selected the portfo-lio that satisfied two additional conditions. First, welooked to minimize the number of deviations fromthe benchmark, and second, we looked for the port-folio whose tracking error (risk relative to thebenchmark) was closest to the risk of the Low Riskoptimal portfolio. We then assumed that all otherbond positions in the portfolio were held at indexweight and at index duration.

Using this framework, we then introduced tacticaltrades to overlay the strategic portfolio. The ideawas to retain a portfolio that best represented the keyelements of our three-month horizon interest rateand currency forecasts but that at the same timewould allow for flexibility to express near-termtrading views. In this way, we intended to moreclosely replicate actual practice in the fund man-agement industry.

However, we encountered a number of difficulties inour attempt to distinguish between tactical and stra-tegic portfolio positioning. An obvious problemarose when our near-term trading views differedfrom the views expressed in our three-month hori-zon forecasts. For example, although the

forecasts might look for a rise in U.S. interest ratesover a three-month horizon, that should not precludeexpressing a trading view that is bullish for the verynear term.

A second related difficulty was one of presentation.Readers failed to understand that portfolio alloca-tions are not always intuitive based on any given setof expected returns. The problem is compoundedwhen we distinguish between trading views andthree-month horizon forecasts. Trading views aredeveloped in isolation and take no account of therelative volatility of returns across asset classes orthe correlation structure. Thus, it would be perfectlypossible — and not necessarily inconsistent — toexpress a bullish view on an individual market andthen to be underweighted in the portfolio. Neverthe-less, such positioning served to generate confusion.

Partly intending to resolve the difficulties that arosein attempting to distinguish tactical and strategicportfolios, we decided to explicitly develop bothportfolios that would be more consistent with ourtrading views and portfolios based on our strategicviews. In the case of the former, we developed amethodology that would ensure consistency with ourtrading views. In addition, we continued to publish astrategic portfolio using our three-month horizonforecasts, though not employing the approach tolimiting the portfolio to a small number of expo-sures, as set forth above.

In the first step of developing our portfolios, weidentified the small number of trading views in thebond markets that we held most strongly. We thengenerated a set of one-month horizon excess returnson two-year and 10-year bonds consistent with theseviews. Using the Black-Litterman model to producean optimal bond portfolio, we then continued withthe procedure described in Section IV to use weight-on-views and confidence levels.

In addition, as a further control procedure, we usu-ally looked for the aggregate currency positions tocontribute between 40% and 60% of the overall risk(measured as tracking error), with bond positionscontributing the remaining risk. Moreover, we typi-cally looked for any individual position to contributeno more than 20% of the aggregate risk. Finally,again to ensure consistency with trading views, we


June 199814

used the risk analysis to ensure that we were takingrisk where we had the strongest views.

At the same time as making this distinction betweentactical and strategic views, we also separated thebond and currency allocation decisions. The moti-vation was twofold. First, it allowed for better trac-tability; in other words, it avoided problems that hadarisen where it was not possible to be sure whether itwas the bond view or the currency view that wasdriving the allocation. Second, it more closely re-sembled industry practice.

In making the distinction between the bond and cur-rency allocation decisions, we first optimized thebond holdings relative to a fully hedged benchmark.We next considered the currency exposure sepa-rately as an overlay. We assume that currency posi-tions are taken through outright positions in theforward market.


June 1998 15

Bibliography

Black, Fischer, and Robert Litterman, Asset Allocation: Combining Investor Views With Market Equilibrium,Goldman, Sachs & Co., Fixed Income Research, September 1990.

Black, Fischer, and Robert Litterman, Global Asset Allocation With Equities, Bonds, and Currencies, Goldman,Sachs & Co., Fixed Income Research, October 1991.

Litterman, Robert, Hot Spots and Hedges, Goldman, Sachs & Co., Risk Management Series, October 1996

Litterman, Robert, and Kurt Winkelmann, Managing Market Exposure, Goldman, Sachs & Co., Risk Manage-ment Series, January 1996.

Litterman, Robert, and Kurt Winkelmann, Estimating Covariance Matrices, Goldman, Sachs & Co., Risk Man-agement Series, January 1998.

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