black litterman 91 global asset allocation with equities bonds and currencies

8/10/2019 Black Litterman 91 Global Asset Allocation With Equities Bonds and Currencies

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Fixed

Income

Research

October 1991

Global AssetAllocation WithEquities, Bonds,and Currencies

Fischer Black

Robert Litterman


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Acknowledgements Although we dont ha ve space to tha nk them each individu-ally, we would like to acknowledge the thoughtful comments

and suggest ions that we have received from our many col-

leagues at Goldman Sachs and the dozens of port folio man-agers w ith w hom we ha ve discussed our w ork this past year.

In a ddit ion, w e wa nt to offer special t ha nks to Alex Bergier,

Tom Iben, P iotr Ka ra sinski, Scott P inkus, Scott R icha rd, an d

Ken S ingleton for t heir car eful reading of an earlier dra ft of

t hi s p a p er. We a r e a l s o g r a t e f ul t o G r e g C u m b e r, G a ne s h

R a m ch a n d r a n , a n d S u r es h Wa d h w a n i f or t h e ir v a l ua b l e

analyt ical support .

Fischer Black

(212) 902-8859

Robert Litterman

(212) 902-1677

Fischer Bla ck is a P ar tner in G oldman S achs Asset Mana ge-

m e nt .

R o b e r t L i t t e r m a n i s a V i c e P r e s i d e nt i n t he F i x e d I nc o m eResearch Department at Goldman, Sachs & Co.

Editor: Ronald A. Krieger

C o py r i gh t 1 99 1 b y G o ld m a n , S a ch s & C o . S i xt h Pr i n t i n g , O ct o be r 1 99 2

This mat erial is for your private information, and we a re not solicit ing a ny a ction

based upon it. Certain transactions, including those involving futures, options, and

h i gh y i el d s e cur it i es , g i v e r i s e t o s ub st a n t i a l r i sk a n d a r e n ot s uit a b l e f o r a l l

investors. Opinions expressed a re our present opinions only. The ma teria l is ba sed

upon i n for m a t i on t h a t w e con s id e r r e l ia b l e, b ut w e d o n o t r e pr es en t t h a t i t i saccurate or complete, a nd i t should not be rel ied upon a s such. We, or persons

involved in the preparation or issuance of this material , may from time to t ime

have long or short posit ions in, a nd buy or sell, securit ies, futures, or options

identical with or related to those mentioned herein. Further information on any of

the securities, futures, or options mentioned in this material may be obtained upon

request . This material has been issued by Goldman, Sachs & Co. and has been

approved by Goldman Sachs International Limited, a member of The Securities and

Futures Authority, in connection with its distribution in the United Kingdom, and

by Goldman Sachs Canada in connection with i ts distr ibution in Canada.


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Global Asset Allocation WithEquities, Bonds, and Currencies

Contents

Executive Summary

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

II . Neutral Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

N a i v e Ap pr oa c h es . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

H istori cal Aver ages . . . . . . . . . . . . . . . . . . . . . . . . 7E q u a l M e a n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

R i sk-Ad j u sted E q u a l M ea n s . . . . . . . . . . . . . . . . . 10

The Eq uilibrium Approach . . . . . . . . . . . . . . . . . . . . . . 11

II I. Expressing Views . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3

IV. Combining Investor Views With MarketEquilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

Th re e-Asset E x a m p le . . . . . . . . . . . . . . . . . . . . . . . . . . 16

A S e ve n -Co u n t ry E x a m p le . . . . . . . . . . . . . . . . . . . . . . 21

V. Controlling the Balance of a Portfolio . . . . . . . . . . 23

VI. Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6

VII . Implied Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8

VII I. Quantifying the Benefits of Global

Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9

IX. Historical Simulations . . . . . . . . . . . . . . . . . . . . . . . 3 1

X. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 6

Appendix. A Mathematical Description of theBlack-Litterman Approach . . . . . . . . . . . . . . . . . . . 37

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 9

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0


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Executive Summary A ye a r a g o , G o ld m a n S a ch s in t rod u ce d a qu a n t i t a t ive m od e l t h a toffered a n innovat ive approach t o the ma na gement of fixed income

portfolios.* It provided a mecha nism for investors t o ma ke global

asset allocat ion decisions by combining t heir views on expectedreturns w ith Fischer B lacks universal hedging equilibrium. Giv-

e n a n in ve st o rs vie w s a b o u t in t e re st ra t e s a n d e x ch a n g e ra t e s,

this init ial version of the B lack-Litterma n G lobal Asset Allocat ion

Model has been used to generate portfolios with optimal weights

in bonds in different countr ies an d t he optimal d egree of currency

exposure.

I n t h is p a per, w e d es cr ib e a n u pd a t ed v er s ion of t h e B l a ck -

L it t e rm a n M od e l t h a t in corp ora t e s e qu it ie s a s w e ll a s b on d s a n d

currencies. The new version of the model will be especially useful

to portfolio managers who make global asset allocation decisions

across equity a nd fi xed income mar kets, but it will a lso have a dvan-

ta ges for pure fixed income ma nagers.

The add it ion of the equity a sset class t o the model allows us to use

an equilibrium based on both bonds and equities. This equilibrium

is more desirable from a theoretical st an dpoint because it incorpo-

ra tes a larger fra ction of the universe of investment a ssets. In our

m o d e l (a s in a n y Ca p it a l A sse t Pricin g M o d e l e qu il ib riu m ), t h e

equilibrium expected excess return on an asset is proportional to

the covariance of the asset s return with the return of the market

portfolio. E ven for pure fi xed income ma na gers, it is useful to use

as broad a measure of the market portfolio as is practical.

As w e described in our ea rlier paper, th e equilibrium is importa nt

in t h e m od e l b e ca u se i t p rovid es a n e u t ra l re fe ren ce p oin t f or

expected returns. This allows the investor to express views only

f o r t h e a sse t s t h a t h e d e sire s; vie w s f o r t h e o t h e r a sse t s a re d e -

rive d f ro m t h e e qu il ib riu m . B y p ro vid in g a ce n t e r o f g ra vit y f o r

expected returns, the equilibrium ma kes th e m odels portfolios

m o re b a la n ced t h a n t h o se fro m st a n d a rd qu a n t i t a t ive a sset a l loca -

tion models. Standard models tend to choose unbalanced portfolios

unless a rt ifi cial constr aint s ar e imposed on portfolio composit ion.

* See Bla ck and Litterma n (1990). [Note: a complete l isting of references ap-

pears at the end of this report on page 40.]


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Global Asset Allocation WithEquities, Bonds and Currencies

I. Introduction nvestors w ith globa l portfolios of equities a nd bonds a re

I generally aware that their asset a llocat ion decisions t he p r o p o r t i o ns o f f u nd s t ha t t he y i nv e s t i n t he a s s e tclasses of different countries and the degrees of currency

hedging a re the most importa nt investm ent decisions th ey

ma ke. In a t tempting t o decide on the appropriate a llocat ion,

they are usually comfortable with the simplifying assump-

tion t ha t their object ive is to ma ximize expected return for

a ny given level of risk, subject in most cases t o various ty pes

of constra ints.

Given the straightforward mathematics of this optimizat ion

problem, the many correlat ions among global asset c lasses

required in mea suring risk, and t he large a mounts of money

i nv ol ve d, one m i g ht e xp ect t ha t i n t o da y s com p ut e r iz ed

w o r l d , q u a nt i t a t i v e m o d e l s w o u l d p l a y a d o m i na nt r o l e i n

this global allocation process.

U nfortuna tely, when investors have tried to use quant itat ive

models to help optimize this critical allocation decision, the

unreasonable nature of the results has often thwarted their

efforts.2 Wh en i n ve st or s i m pos e n o con s t r a i n t s, a s s et

w e i g ht s i n t he op t im i ze d p or t f ol ios a l m o st a l w a y s or d a i n

large short posit ions in ma ny assets. When constra ints rule

out short positions, the models often prescribe corner solu-

t ions with zero weights in ma ny a ssets, as well as unreason-

ably large weights in the assets of markets with small capi-

talizat ions.

The s e u nr e a s ona b l e r e su l t s ha v e s t e m m ed f r om t w o w e l l-

recognized problems. First, expected returns are very diffi-

cult to est imate. Investors typically have views about abso-

l u t e o r r e l a t i v e r e t u r ns i n o nl y a f e w m a r k e t s . I n o r d e r t ouse a st an dar d optimizat ion model, however, they must sta te

a set of expected returns for all assets and currencies. Thus,

t he y m u s t a u g m e nt t he i r v i e w s w i t h a s e t o f a u x i l i a r y a s -

2For s ome a ca demic discu ss ions of t his is s u e, s ee G reen a nd Hollifi eld

(1990) and Best and Grauer (1991).

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sumptions, and the historical returns that port folio manag-

ers often use for this purpose provide poor guides to future

returns.

S e cond , t he op t im a l p or t f ol io a s s e t w e i ght s a nd cu r r ency

posit ions in sta nda rd a sset a llocat ion models a re extremely

sensitive to th e expected ret urn a ssumptions. The t w o prob-

lems compound ea ch other because t he sta nda rd m odel has

no w a y t o d i s t i ng u i s h s t r o ng l y he l d v i e w s f r o m a u x i l i a r y

a s s u m p t i o ns , a nd g i v e n i t s s e ns i t i v i t y t o t he e x p e c t e d r e -

t u r ns , t he op t im a l p or t f ol io i t g ene r a t e s of t en a p p ea r s t o

ha ve li t t le or no relat ion to the views th at the investor w ish-

es to express.

Confronting th ese problems, investors a re often disappoint ed

wh en they at t empt to use a st an dard a sset a llocat ion model.

Our experience has been t ha t in practice, despite the obvious

conceptua l a t tr act ions of a qua ntita t ive approach, few global

investment ma na gers regularly a llow q uan tita t ive models to

play a major role in their asset allocation decisions.

I n t h i s p a p er w e d es cr i be a n a p pr oa c h t h a t p rov id es a n

intuit ive solut ion to these two problems that have plagued

the use of quantitat ive asset a llocat ion models. The key to

our approach is the combining of two established tenets of

modern port folio t heory: the mean-varia nce optimizat ion

fra mew ork of Mar kowitz (1952) a nd t he capita l ass et pricingmodel (CAP M) of Sha rpe (1964) an d Lintner (1965). We

a llow th e investor to combine his views a bout th e outlook for

global equit ies, bonds, and currencies with the risk premi-

ums genera ted by B lacks (1989) global version of t he C AP M

equilibrium. These equilibrium risk premiums are the excess

returns tha t equa te the supply an d demand for global assets

and currencies.3

As w e w i ll i ll us t r a t e w i t h e xa m p le s, t h e m ea n -v a r ia n c e

op t im i z a t i on u s ed i n s t a nd a r d a s s e t a l l oca t i o n m od el s i s

extremely sensitive to the expected return assumptions that

the investor must provide. I n our model, th e equilibriumrisk premiums provide a neutral reference point for expected

returns. This, in t urn, a llows the m odel to genera te optima l

port folios that are much better behaved than the unreason-

a ble port folios tha t st a nda rd models typica lly produce, wh ich

3For a gu ide t o t erms u s ed in t h is p a p er, s ee t he G los s a ry, p a ge 39.

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often include large long an d short positions unless otherwise

cons t r a i ned . I ns t ea d , ou r m od el g r a v it a t e s t ow a r d a b a l -

anced i.e., market-capitalization-weighted portfolio but

tilts in the direction of assets favored by the investors views.

O u r m od el d oe s not a s s u m e t ha t t he w o r ld i s a l w a y s a t t he

CAPM equilibrium, but rather that when expected returns

m ov e a w a y f r om t he ir e q u il ib r iu m v a l u es , i m b a l a nc es i n

m a r k e t s w i ll t e nd t o p u s h t he m b a ck . Thu s , w e t hi nk i t i s

reasonable to assume t ha t expected returns a re not l ikely to

deviat e t oo fa r from equilibrium values. This intuit ive idea

suggests t ha t t he investor ma y profi t by combining his views

a b ou t r et u r n s i n d if fer en t m a r k et s w i t h t h e i n for m a t i on

contained in the equilibrium.

In our approach, we distinguish between views of the investor

an d the expected returns th at drive the optimiza tion a na lysis.

The equilibrium risk premium s provide a center of gra vity for

expected returns. The expected returns that we use in the

optimization deviate from equilibrium risk premiums when

the investor explicitly states views. The extent of the devia-

tions from equilibrium depends on the degree of confidence

the investor expresses in each view. Our model makes adjust-

ments in a manner as consistent as possible with historical

covar iances of returns of different a ssets a nd currencies.

Our use of equilibrium allows investors to specify views in am u ch m or e fl ex ib le a n d pow e r fu l w a y t h a n i s o t h er w i s e

possible. For example, rather than requiring a view about

t he a b s o l u t e r e t u r ns o n e v e r y a s s e t a nd c u r r e nc y , o u r a p -

proach allows investors to specify as many or as few views

as they wish. In addit ion, investors can specify views about

relat ive returns, a nd they can specify a degree of confidence

about ea ch of th e views.

In this paper, through a set of examples, we il lustrate how

t he i ncor p or a t i on of e q u il ib r iu m i nt o t he s t a nd a r d a s s e t

allocat ion model makes it bet ter behaved and enables i t to

genera te insight s for th e global investm ent ma na ger. To th a tend, we sta rt w ith a discussion of how equilibrium can help

an investor tra nslat e his views into a set of expected returns

for a ll assets an d currencies. We then follow with a set of

applications of the model that illustrate how the equilibrium

solves t he problems t ha t ha ve tra dit iona lly led t o unreason-

able results in standard mean-variance models.

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II. Neutral Views

Exhibit 1

Historical Excess Returns(January 1975 through August 1991)

Total Historical Excess Returns

Germany France J apan U.K. U.S. Canada Australia

C ur ren cies -20.8 3.2 23.3 13.4 12.6 3.0

B onds C H 15.3 -2.3 42.3 21.4 -4.9 -22.8 -13.1

E q u it ies C H 112.9 117.0 223.0 291.3 130.1 16.7 107.8

Annualized Historical Excess Returns

Germany France J apan U.K. U.S. Canada AustraliaC ur ren cies -1.4 0.2 1.3 0.8 0.7 0.2

B onds C H 0.9 -0.1 2.1 1.2 -0.3 -1.5 -0.8

E q u it ies C H 4.7 4.8 7.3 8.6 5.2 0.9 4.5

Annualized Volatility of Historical Excess Returns


C ur ren cies 12.1 11.7 12.3 11.9 4.7 10.3

B onds C H 4.5 4.5 6.5 9.9 6.8 7.8 5.5

E q u it ies C H 18.3 22.2 17.8 24.7 16.1 18.3 21.9

________________________________

Note: Bond and equity excess returns are in U.S. dollars currency hedged (CH). Excess returns on bonds and

equities are in excess of the London interbank offered rate (LIBOR), and those on currencies are in excess of

the one-month forward rat es. Volati l i t ies are expressed as annua lized sta ndard deviat ions.

hy s hou l d a n i nve st o r u s e a g l ob a l e q u il ib r iu m

W m od el t o he lp m a k e hi s g l ob a l a s s e t a l l oca t i ondecision? A neutra l reference is a critica lly impor-t a nt i npu t i n m a k ing u s e o f a m e a n-v a r i a nc e op t im i za t i on

model, an d a n equilibrium provides t he a ppropriat e neutra l

reference. Most of the time investors do have views feel-

ings that some assets or currencies are overvalued or under-

valued at current market prices. An asset a llocat ion modelshould help them to leverage those views to their greatest

a d v a nt a g e . B u t i t i s u nr ea l i s t ic t o e xp ect a n i nve s t or t o b e

able to state exact expected excess returns for every asset

and currency. The purpose of the equilibrium is to give the

i nv e s t o r a n a p p r o p r i a t e p o i nt o f r e f e r e nc e s o t ha t he c a n

express his views in a realist ic manner.

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We begin our discussion of how to combine views with the

e q u il ib r iu m b y s u p pos i ng t ha t a n i nve st o r ha s no v i ew s .

What then is the optimal portfolio? Answering this question

is a sensible point of departure beca use it demonstrat es theusefulness of the equilibrium risk premium as the appropri-

ate point of reference.

We consider th is q uest ion, an d exam ples t hroughout this

paper, using historical data on global equit ies, bonds, and

currencies. We use a seven-country model with monthly

r e t u r ns f or t he U ni t e d S t a t e s , J a p a n, G e r m a ny, F r a nce , t he

U nited Kingdom, Ca na da, a nd Austra lia from J an uary 1975

th rough August 1991.4

E x h ib it 1 p r es en t s t h e m ea n s a n d s t a n d a r d d ev ia t i on s of

excess returns. The correlations are in Exhibit 2 (pages 6-7).

All results in t his paper ha ve a U .S. dollar perspective, but

other currency points of view would give similar results. 5

4I n a ct u a l a p p lica t ions of t he model, w e t yp ica lly inclu de more a s s et

c l a s s e s a n d u s e d a i l y d a t a t o m o r e a c c u r a t e l y m e a s u r e t h e c u r r e n t

s t a t e of t he t ime-va rying r is k s t ru ct u re. We int end t o a ddres s is s u es

concerning u ncert a int y of t he cova ria nces in a not her p a p er. For t he

purposes of this paper, however, we t reat the t rue covar ian ces of excess

ret u rns a s know n.

5 We define exces s ret u rn on cu rrency hedged a s s et s t o be t ot a l ret u rn

less t h e s hort ra t e a nd exces s ret u rn on cu rrency p os it ions t o be t ot a l

return less the forward premium (see Glossary, page 39). In Exhibit 2,

a ll exces s ret u rns a nd vola t i l i t y a re in p ercent . The cu rrency hedged

exces s ret u rn on a bond or a n equ it y a t t ime t is given by:

Et

Pt 1 /Xt 1

Pt/Xt 1 100 (1 Rt)F Xt Rt

w h e r e P t is t he p rice of t he a s s et in foreign cu rrency, X t i s t h e e x -

cha nge ra t e in u nit s of foreign cu rrency p er U . S. dolla r , R t is t he do-

m e st i c s h or t r a t e , a n d F Xt i s t h e r e t u r n o n a f or w a r d c on t r a c t , a l l a t

t ime t . The ret u rn on a forw a rd cont ra ct , or equ iva lent ly t he exces sreturn on a foreign currency, is given by:

F Xt

F t 1t

Xt 1

Xt 1

1 00

w h e re i s t h e on e-p er i od f or w a r d e xch a n g e r a t e a t t i me t .F t 1t

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Exhibit 2

Historical Correlations of E xcess Returns(J a n u a r y 1 9 7 5 th r ou g h A u g u st 1 99 1)

Germany France J apanE q uit ies B on ds C ur ren cy E q uit ies B on ds C ur ren cy E q uit ies B on ds C ur ren cy

C H C H C H C H C H C H

GermanyE q u it ies C H 1.00

B on ds C H 0.28 1.00

C ur rency 0.02 0.36 1.00

FranceE q uit ies C H 0.52 0.17 0.03 1.00

B onds C H 0.23 0.46 0.15 0.36 1.00C ur ren cy 0.03 0.33 0.92 0.08 0.15 1.00

J apanE q uit ies C H 0.37 0.15 0.05 0.42 0.23 0.04 1.00

B onds C H 0.10 0.48 0.27 0.11 0.31 0.21 0.35 1.00

C ur ren cy 0.01 0.21 0.62 0.10 0.19 0.62 0.18 0.45 1.00

U.K.E q uit ies C H 0.42 0.20 0.01 0.50 0.21 0.04 0.37 0.09 0.04

B onds C H 0.14 0.36 0.09 0.20 0.31 0.09 0.20 0.33 0.19

C ur ren cy 0.02 0.22 0.66 0.05 0.05 0.66 0.06 0.24 0.54

U.S.

E q uit ies C H 0.43 0.23 0.03 0.52 0.21 0.06 0.41 0.12 0.02

B onds C H 0.17 0.50 0.26 0.10 0.33 0.22 0.11 0.28 0.18Canada

E q uit ies C H 0 .33 0.16 0.05 0.48 0.04 0.09 0.33 0.02 0.04

B onds C H 0.13 0.49 0.24 0.10 0.35 0.21 0.14 0.33 0.22

C ur ren cy 0.05 0.14 0.11 0.10 0.04 0.10 0.12 0.05 0.06

Australia

E q uit ies C H 0.34 0.07 0.00 0.39 0.07 0.05 0.25 0.02 0.12

B onds C H 0.24 0.19 0.09 0.04 0.16 0.08 0.12 0.16 0.09

C ur ren cy 0.01 0.05 0.25 0.07 0.03 0.29 0.05 0.10 0.27

Naive Approaches To motivat e the use of the equilibrium risk premiums as aneutral reference point, we first consider several other naive

a p p r oa c hes i nve st o r s m i g ht u s e t o cons t r u ct a n op t im a l

p or t f ol io w he n t he y ha v e no v i ew s a b o ut a s s e t s or cu r r en-

cies. We w ill call these th e historical avera ge a pproach, the

e q u a l m e a n a p p r oa c h, a nd t he r i s k-a d j us t e d e q u a l m e a n

approach.

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H istor ical Averages The historical a verag e a pproa ch defi nes a neutr a l position to

Exhibit 2 (Continued)

Historical Correlations of E xcess Returns(J a n u a r y 1 9 7 5 th r ou g h A u g u st 1 99 1)

United Kingdom United States Canada AustraliaE q uit ies B on ds C ur ren cy E q uit ies B on ds E q uit ies B on ds C ur ren cy E q uit ies B on ds

C H C H C H C H C H C H C H C H

U.K.E q u it ies C H 1. 00

B on ds C H 0.47 1.00

C ur rency 0.06 0.27 1.00

U.S.E q uit ies C H 0.58 0.23 0.02 1.00

B onds C H 0.12 0.28 0.18 0.32 1.00

CanadaE q uit ies C H 0.56 0.27 0.11 0.74 0.18 1.00

B onds C H 0.18 0.40 0.25 0.31 0.82 0.23 1.00

C ur ren cy 0.14 0.13 0.09 0.24 0.15 0.32 0.24 1.00

AustraliaE q uit ies C H 0.50 0.20 0.15 0.48 0.05 0.61 0.02 0.18 1.00

B onds C H 0.17 0.17 0.09 0.24 0.20 0.21 0.18 0.13 0.37 1.00

C ur ren cy 0.06 0.05 0.27 0.07 0.00 0.19 0.04 0.28 0.27 0.20

b e t he a s s u m pt i on t ha t e xce ss r e t u r ns w i l l e q u a l t he i r hi s -

t o r i c a l a v e r a g e s . T he p r o b l e m w i t h t hi s a p p r o a c h i s t ha t

historical means are very poor forecasts of future returns.

F or e xa m p le, i n E x h ib it 1 w e s ee m a n y n eg a t i ve v a l ues .

Let s see wh at ha ppens w hen w e use these historical excess

returns as our expected excess return assumptions. We may

optimize expected r eturn s for ea ch level of risk to get a fron-

t ier of optima l port folios. E xhibit 3 (page 8) i llustrat es t he

frontiers with the port folios that have 10.7%risk, with and

without short ing constraints.6

We m a y m a k e a nu m b er of p oi nt s a b o ut t he s e op t im a l port folios. First , they i l lustrat e wha t w e mean by unrea son-

6We choose to normalize on 10.7%risk here and throughout this paper

because it ha ppens to be the risk of the ma rket-capita lization-weight ed,

80%currency hedged portfolio that will be held in equilibrium in our

model.

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able when we claim that standard mean-variance optimiza-

t i on m od el s of t en g ene r a t e u nr ea s o na b l e p or t f ol ios . The

portfolio that does not constrain against short ing has many

large long and short posit ions with no obvious relat ionshipto the expected excess return assumptions. When we con-

s t r a i n s hor t i ng w e ha v e p os i t iv e w e i ght s i n o nly t w o of t he

14 potent ial a ssets. These portfolios a re ty pica l of those tha t

t he s t a nd a r d m od el g e ne r a t e s .

Exhibit 3

Optimal Portfolios Based onHistorical Average Approach

(percent of portfolio value)

Unconstrained

Germany F rance J apan U.K. U.S. Canada Australia

C ur ren cy exposur e 78.7 46.5 15.5 28.6 65.0 5.2

B onds 30.4 40.7 40.4 1.4 54.5 95.7 52.5

E q u it ies 4.4 4.4 15.5 13.3 44.0 44.2 9.0

With constraints against shorting assets



B onds 7.6 0.0 88.8 0.0 0.0 0.0 0.0

E q u it ies 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Given how we have set up the optimizat ion problem, there

i s no r e a s o n t o e x p e c t t ha t w e w o u l d g e t a b a l a nc e d s e t o f

weights. The use of past excess returns to represent a neu-

tra l set of views is equivalent t o assuming t ha t th e consta nt

port folio w eights t ha t w ould have performed best h istorica lly

ar e in some sense neutral . I n reality, of course, they a re not

ne u t r a l a t a l l , b u t r a t he r a r e a v e r y s p e c i a l s e t o f w e i g ht s

tha t go short a ssets tha t ha ve done poorly an d go long assets

that have done well in the part icular historical period.

E q u a l Me a n s Recognizing th e problem of using past return s, th e investor

might hope tha t a ssuming equa l means for returns a cross a ll

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countries for each a sset c lass w ould be a bet ter representa-

Exhibit 4

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

Unconstrained



B onds 11.6 4.2 1.8 10.8 13.9 18.9 32.7

E q u it ies 21.4 4.8 23.0 4.6 32.2 9.6 10.5




B onds 0.0 0.0 0.0 0.0 0.0 0.0 0.0

E q u it ies 17.5 0.0 22.1 0.0 27.0 8.2 7.3

t ion of a neutral reference. We show an example of the opti-

ma l port folio for this t ype of an a lysis in E xhibit 4. Aga in, w e

get an unreasonable portfolio.7

O f c ou r s e, one p r ob le m w i t h t he e q u a l m e a ns a p p r oa c h i s

th a t equa l expected excess return s do not compensa te inves-

tors appropriately for the different levels of risk in assets of

different countries. Investors diversify globally to reduce

risk. Everything else being equal, they prefer assets whose

r e t u r ns a r e l e s s v o l a t i l e a nd l e s s c o r r e l a t e d w i t h t ho s e o f

other assets.

Although such preferences a re obvious, it is perha ps surpris-

ing how unbala nced t he optima l port folio weights can be, a s

E x hi bi t 4 i l lu s t r a t e s , w he n w e t a k e e ve r y t hing e ls e b e ing

e q u a l t o s u c h a l i t e r a l e x t r e m e . W i t h no c o ns t r a i nt s , t helargest posit ion is short Austra lian bonds.

7For the purposes of this exercise, we arbitrarily assigned to each coun-

try the a verage hist orical excess return a cross countr ies, as follows: 0.2

for currencies, 0.4 for bonds, an d 5.1 for equities.

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Risk-Adjusted Equal Means O u r t hir d na i v e a p p r oa c h t o d e fi ni ng a ne ut r a l r ef er e nce

p o i nt i s t o a s s u m e t ha t b o nd s a nd e q u i t i e s ha v e t he s a m e

expected excess return per unit of risk, where the risk mea-

sure is simply the volat il i ty of asset returns. Currencies inth is ca se are a ssumed to ha ve no excess return . We show th e

op t im a l p or t f ol io f or t hi s ca s e i n E x hi bi t 5. Now w e ha v e

incorporated volat il i t ies, but the port folio behavior is no

b et t e r. O n e p r ob lem w i t h t h i s a p pr oa c h i s t h a t i t h a s n t

ta ken t he correlat ions of the a sset returns int o account . B ut

th ere is another problem a s well perhaps more subtle, but

also more serious.

Exhibit 5

Optimal Portfolios Based on

Equal Risk-Adjusted Means(percent of portfolio value)

Unconstrained



B onds 23.9 12.6 54.0 20.8 23.1 37.8 15.6

E q u it ies 9.9 8.5 12.4 0.3 14.1 13.2 20.1




B onds 0.0 0.0 0.0 7.8 0.0 19.3 0.0

E q u it ies 11.1 9.4 19.2 6.0 0.0 7.6 19.5

S o f a r , t h e a p p r o a c h e s w e h a v e u s e d t o t r y t o d e fi n e t h e

a p p r op r ia t e ne u t r a l m e a ns w he n t h e i nv es t or ha s no v i ew s

ha v e b e e n b a s e d o n w ha t m i g ht b e c a l l e d t he d e m a nd f o r

assets side of the equat ion tha t is , historical return s a nd

risk mea sures. The problem w ith such a pproa ches is obvious

when we bring in the supply side of the market .

Suppose the ma rket port folio comprises 80% of one asset

an d 20% of t he other. In a simple w orld, w ith identical in-

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v e s t o r s a l l ho l d i ng t he s a m e v i e w s a nd b o t h a s s e t s ha v i ng

e q u a l v ol a t i l it i es , e ve r y one ca nno t hol d e q u a l w e i g ht s of

e a c h a s s e t . P r i c e s a nd e x p e c t e d e x c e s s r e t u r ns i n s u c h a

w o r l d w o u l d ha v e t o a d j u s t a s t he e x c e s s d e m a nd f o r o neasset an d excess supply of the other a ffect the ma rket .

The Equilibrium Approach To us, the only sensible defi nit ion of neutral mea ns is th eset of expected returns that would clear the market i f a ll

investors h a d ident ica l views. The concept of equilibrium in

the context of a global portfolio of equities, bonds, and cur-

rencies is similar, but currencies do ra ise a complicat ing

question. H ow much currency hedging ta kes pla ce in equilib-

rium? The answer, as described in Black (1989), is that in a

global equilibrium investors worldwide will a ll w an t t o take

a small amount of currency risk.

This result arises because of a curiosity known in the cur-

r e ncy w o r l d a s S i eg el s p a r a d o x. The b a s i c i d ea i s t ha t

because investors in different countries measure returns in

different units, each w ill gain some expected return by ta k-

ing some currency risk. In vestors will accept currency risk

u p t o t he p o i nt w he r e t he a d d i t i o na l r i s k b a l a nc e s t he e x -

pected return. Under certain simplifying assumptions, the

percentage of foreign currency risk hedged will be the same

for investors of different count ries giving rise to the na me

universa l h edging for this equilibrium.

The equilibrium degree of hedging the universa l hedging

constant depends on three averages: the average across

countries of the mean return on the market port folio of as-

s e t s , t he a v e r a g e a c r o s s c o u nt r i e s o f t he v o l a t i l i t y o f t he

w o r ld m a r k e t p or t f ol io, a nd t he a v e r a g e a cr os s a l l p a i r s o f

countries of exchange rate volatility.

I t i s d i f f i c u l t t o p i n d o w n e x a c t l y t he r i g ht v a l u e f o r t he

universal hedging consta nt , primarily because the risk pre-

mium on the market port folio is a diff icult number to est i-

m a t e . Ne v er t he le ss , w e f ee l t ha t u ni ve r sa l he d gi ng v a l u esbetween 75%and 85%are reasonable. In our monthly data

set, t he former va lue corresponds t o a risk premium of 5.9%

on U.S . equities wh ile th e latt er corresponds t o a risk premi-

um of 9.8%. F or t he purposes of t his paper, we will use an

equilibrium value of 80%currency hedging.

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Exhibit 6

Equilibrium Risk Premiums(per cent ann ual i zed excess r etu r n)


C u rr encies 1.01 1.10 1.40 0.91 0.60 0.63

B onds 2.29 2.23 2.88 3.28 1.87 2.54 1.74

E q u it ies 6.27 8.48 8.72 10.27 7.32 7.28 6.45

E x hi bi t 6 s ho w s t he e q u il ib r iu m r i s k p r em i u m s f or a l l a s -

sets, given this va lue of th e universal hedging consta nt . 8

Let us consider wha t h appens w hen w e adopt t hese equilib-

rium risk premiums as our neutral means when we have no

views. Exhibit 7 shows the optimal portfolio. It is simply the

ma rket capita lizat ion portfolio w ith 80%of the currency risk

hedged. Other portfolios on th e frontier w ith different levels

of risk would correspond to combinations of risk-free borrow-

ing or lending plus more or less of this portfolio.

B y itself , the equilibrium is interest ing but not pa rt icularlyu s ef u l. The r e a l v a l u e o f t he e q u il ib r iu m i s t o p r ov id e a

ne u t r a l f r a m e w o r k t o w hi c h t he i nv e s t o r c a n a d d hi s o w n

perspective in terms of views, optimizat ion object ives, a nd

constraints. These are the issues to which we now turn.

8The universa l hedging eq uilibrium is, of course, ba sed on a set simpli-

f y in g a s s u m pt i on s , s u ch a s a w o r ld w i t h n o t a x es , n o c a p i t a l c on -

s t ra int s , no infl a t ion, et c. E xcha nge ra t es in t his w orld a re t he ra t es of

exchange between the different consumption bundles of individuals of

dif ferent cou nt ries . While s ome m a y fi nd t he a s s u mp t ions t ha t jus t ifyu nivers a l hedging overly res t r ict ive, t his equ ilibriu m does ha ve t he

virtue of being simpler than other global CAPM equilibriums that have

been described elsewhere, such as in Solnik (1974) or Grauer, Litzen-

berger , a nd St ehle ( 1976) . While t hes e s imp lifying a s s u mp t ions a re

necessary to justify the universal hedging equilibrium, we could easily

a p p ly t he ba s ic idea of t his p a p er t o combine a globa l equ ilibriu m

w it h inves t or view s t o a not her globa l equ ilibriu m derived from a

different, less restrictive, set of assumptions.

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II I. Expressing Views

Exhibit 7

Equilibrium Optimal Portfolio(percent of portfolio value)



B onds 2.9 1.9 6.0 1.8 16.3 1.4 0.3

E q u it ies 2.6 2.4 23.7 8.3 29.7 1.6 1.1

he b a s i c p r o b l e m t ha t c o nf r o nt s i nv e s t o r s t r y i ng t o

Tu s e q u a nt i t a t i v e a s s e t a l l oca t i on m od el s i s how t o

tra nslat e th eir views into a complete set of expected

e x c e s s r e t u r n s o n a s s e t s t h a t c a n b e u s e d a s a b a s i s f o r

portfolio optimization. As we will show here, the problem is

tha t optima l port folio weights from a mean-varia nce model

a re incredibly sensitive t o minor cha nges in expected excess

retur ns. The adva nt a ge of incorpora ting a globa l equilibrium

will become a ppar ent w hen w e show how to combine it w ith

a n invest ors views t o genera te w ell-behaved portfolios, w ith-

out requiring the investor to express a complete set of ex-

pected excess r eturn s.

We should emphasize tha t t he dist inct ion we a re ma king between investor views on t he one han d a nd a complete set

of expected excess returns for a ll assets on the other is

not usua lly recognized. In our a pproa ch, views r epresent t he

subject ive feelings of th e investor a bout relat ive va lues of-

fered in different markets.9 I f a n i nv e s t o r d o e s no t ha v e a

view about a given ma rket , he should not h ave to sta te one.

If some of his views ar e more strongly held th an others, the

i nv es t or s hou l d b e a b l e t o e xp r es s t ha t d i ff er e nce . M o s t

views a re relat ive for exa mple, w hen a n investor feels one

ma rket w ill outperform a nother or even w hen h e feels bull-

ish (above neutra l) or bearish (below n eutral) about a ma r-

ket . As w e w ill show, the equilibrium is th e key to a llow ingthe investor to express his views this w a y instead of as a set

of expected excess retur ns.

9As we will show in Section IX, views can also represent feelings about

t h e r e la t i o n sh i ps b et w e e n ob s er v a b l e con d i t ion s a n d s u ch r e la t i v e

values.

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bonds. This lack of apparent connection between the views

Exhibit 8

Optimal Portfolios Based on a Moderate View(percent of portfolio value)

Unconstrained



B onds 13.6 6.4 15.0 3.3 112.9 42.4 0.7

E q u it ies 3.7 6.3 27.2 14.5 30.6 24.8 6.0




B onds 0.0 0.0 0.0 0.0 35.7 0.0 0.0

E q u it ies 2.6 5.3 28.3 13.6 0.0 13.1 1.5

t he i nve s t or a t t e m pt s t o e x pr e ss a nd t he op t im a l p or t f ol io

t he m od el g e ner a t e s i s a p er v a s i ve p r ob le m w i t h s t a nd a r d

mean-varia nce optimizat ion. I t ar ises because, as we sa w intrying to generate a port folio representing no views, in the

optimiza tion th ere is a complex int eraction betw een expected

excess returns and the volat il i t ies and correlat ions used in

measuring risk.

IV. CombiningInvestor ViewsWith MarketEquilibrium

ow our a pproach t ra nslat es a few views into expect-

H e d e x c e s s r e t u r ns f o r a l l a s s e t s i s o ne o f i t s m o r ecomplex fea tur es, but a lso one of its most inn ovative.Here is t he intuit ion behind our a pproach:

(1) We believe there ar e tw o distinct sources of informa -tion about future excess returns: investor views and

ma rket equilibrium.

(2) We assume tha t both sources of informa tion are un-

certain and are best expressed as probability distri-

butions.

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(3) We choose expected excess returns t ha t a re as consis-

tent as possible w ith both sources of informa tion.

The a bove description captures t he ba sic idea , but th e imple-menta t ion of this a pproach can lead to some novel insights.

F o r e x a m p l e , w e w i l l no w s ho w ho w a r e l a t i v e v i e w a b o u t

t w o a s s e t s c a n i nfl u e nc e t he e x p e c t e d e x c e s s r e t u r n o n a

t hi r d a s s e t .11

Three-Asset Example Let us fi rst work through a very simple exam ple of our ap-proach. After th is illustr a tion, we w ill apply it in the cont ext

of our seven-country model. Suppose we know the true struc-

t u r e of a w o r ld t h a t ha s ju s t t h r e e a s s et s : A, B , a nd C . The

excess return for ea ch of t hese assets is known to be gener-

ated by an equilibrium risk premium plus four sources of

risk: a common fa ctor a nd independent sh ocks to each of the

t hr e e a s s et s .

We can wr ite t his model as follows:

R A = A + A Z + AR B = B + B Z + BR C = C + C Z + C

where:

R i is the excess return on the it h a s s e t ,

i is th e equilibrium risk premium on th e it h a s s e t ,

i i s t h e i m p a c t o n t h e it h a s s e t of Z , t he com m on

factor, an d

i is t he independent shock t o the it h a s s e t .

I n t hi s w or l d , t he c ov a r i a nc e m a t r i x , , of asset excess re-turns is determined by the relat ive impacts of the common

factor an d the independent shocks. The expected excess

returns of the assets 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.

11I n t his s ect ion, w e t ry t o develop t he int u it ion behind ou r a p p roa ch

using some basic concepts of sta tistics an d ma trix a lgebra. We provide

a more forma l ma t hema t ica l des cript ion in t he A p pendix.

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For example, the expected excess return of asset A, which

w e w r i t e a s E [ RA], is given by:

E [RA] = A + A E[Z] + E[A]

We a re not a ssuming th at the w orld is in equilibrium, i .e. ,

t h a t E [ Z ] a n d t h e E [i]s a r e e q u a l t o z e r o. We d o a s s u m et ha t t he m ea n, E [R A], is itself an unobservable random vari-

able whose distribution is centered at the equilibrium risk

premiums. Our uncertainty about E [R A] is due to our uncer-

t a i nt y a b o u t E [ Z ] a nd t he E [i]s. Furthermore, we assumet he d eg r ee o f u nc er t a i nt y a b o ut E [Z] a nd t he E [i]s is pro-port ional t o the volat il it ies of Z and the is themselves.

Thi s i m pl ie s t ha t E [RA

] i s d i s t r ib u t ed w i t h a cov a r i a nc e

structure proport ional to . We will refer to this covariancem a t r i x o f t he e x p e c t e d e x c e s s r e t u r ns a s . B e c a u s e t h euncerta inty in the mean is much sma ller tha n the uncerta in-

ty in t he return itself , w ill be close t o zero. The eq uilibriumrisk premiums together with determine t he equilibriumdistribution for expected excess r eturns. We assume this

i nf o r m a t i o n i s k no w n t o a l l ; i t i s no t a f u nc t i o n o f t he c i r -

cumsta nces of a ny individual investor.

In addit ion, we a ssume tha t ea ch investor provides a ddit ion-

al information about expected excess returns in the form of

views. For example, one type of view is a statement of theform: I expect Asset A to outperform Asset B by Q, where

Q is a given value.

We i n t e r pr et s u ch a v iew t o m e a n t h a t t h e i n ves t or h a s

subjective information about the future returns of A relative

to B. One wa y w e think about representing th at informa tion

i s t o a c t a s i f w e ha d a s u m m a r y s t a t i s t i c f r om a s a m p l e of

dat a dra wn from the distribution of future returns dat a in

wh ich all we were a ble to observe is the difference betw een

t he r e t u r ns o f A a n d B . A lt e r na t i v el y, w e c a n e xp r es s t hi s

view directly as a probability distribution for the difference

between the mea ns of the excess returns of A and B . I t does-n t m a t t e r w h i c h o f t h e s e a p p r o a c h e s w e w a n t t o u s e t o

think about our views; in the end we get the same result .

In both approaches, though, we need a measure of the inves-

tors confi dence in his views. We use t his mea sure to deter-

mine how much weight to give to the view when combining

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it with the equilibrium. We can think of this degree of confi-

dence in the first case as determining the number of obser-

vat ions t ha t w e have from the distribution of future returns,

i n t h e s e cond a s d et e r m ini ng t he s t a nd a r d d e v ia t i on of t h eprobability distribution.

In our example, consider the l imit ing case: the investor is

100% s u r e o f hi s one v ie w. We m i g ht t hi nk of t ha t a s t he

case where we have an unbounded number of observat ions

from th e distribution of future returns, an d tha t th e average

value of R A R B from these data is Q. In this special case,

w e ca n r ep res en t t h e v i ew a s a l in ea r r es t r ict i on on t h e

expected excess returns, i.e. E[R A] E[R B ] = Q .

Indeed, in this special, case we can compute the distribution

of E[R] = {E[RA], E[R B ], E[R C]} conditional on the equilibri-

um a nd th is informa tion. This is a relat ively stra ightforw a rd

p r ob le m f r om m u l t iv a r i a t e s t a t i s t i cs . To s i m pl if y, l et u s

a s s u m e a nor m a l d i st r i b ut i on f or t he m e a ns of t he r a nd o m

components.

We ha v e t he e q u il ib r iu m d i st r i b ut i on f or E [R ], w hi ch i s

given by Normal (, ), w here = {A, B , C }. We wish tocalculate a conditional distribution for the expected returns,

subject to the restrict ion that the expected returns sat isfy

the l inear restrict ion: E[RA] E[RB ] = Q .

L e t u s w r i t e t hi s r e st r i ct i on a s a l ine a r e q u a t i on i n t he e x-

pected returns:12

P E [ R ] = Q w here P is t he vector [1, 1, 0].

The condit iona l normal distribution ha s t he mea n

+ P [P P ]-1 [ Q P ],

wh ich is th e solut ion to th e problem of minimizing

(E [R] ) ()-1

(E [R] ) subject t o P E[R] = Q.

For the special case of 100% confi dence in a view, w e use

this condit ional mean as our vector of expected excess re-

t u r n s . I n t h e m o r e g e n e r a l c a s e w h e r e w e a r e n o t 1 0 0 %

12A prime sym bol (e.g. , P ) indicates a transposed vector or matrix.

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con fi d e n t , w e c a n t h i n k of a v iew a s r ep res en t i n g a fi x ed

number of observat ions drawn from the distribution of fu-

ture returns in which case we follow the mixed est ima-

tion strategy described in Theil (1971), or alternatively asdirectly reflecting a subjective distribution for the expected

excess returns in w hich case we use the t echnique given

in t he Appendix. The formula for th e expected excess ret urn s

vector is the same from both perspectives.

In either approach, we assume th at the view can be summa -

rized by a sta tement of the form P E[R] = Q + , w h e r e Pa nd Q a r e g iv en a nd is a n unobservable, norma lly distrib-u t e d r a nd om v a r i a b le w i t h m e a n 0 a nd v a r i a nce. repre-s ent s t he u nc er t a i nt y i n t he v i ew . I n t he l im i t a s goes tozero, t he result ing mean converges to th e condit iona l mea n

described above.

When there is more than one view, the vector of views can

be represented by P E[R]= Q + , wh ere we now interpretP a s a m a t r i x, a n d is a normally distributed random vectorw i t h m e a n 0 a nd d i a g ona l cov a r i a nc e m a t r i x. A diagonal corresponds to the assumption that the views representindependent draws from the future distribution of returns,

or tha t the deviat ions of expected returns from t he mean s of

t he d i st r i b ut i on r e pr e se nt ing e a ch v ie w a r e i nd ep end e nt ,

depending on which approach is used to think about subjec-

t i v e v ie w s . I n t he Ap pe nd ix , w e g i v e t he f or m u l a f or t heexpected excess return s th a t combine views w ith equ ilibrium

in the general case.

Now cons i d er ou r e xa m p l e, i n w hi ch cor r e la t i o ns a m o ng

assets result from the impact of one common factor. In gen-

e r a l , w e w i l l no t k no w t he i m p a c t s o f t he f a c t o r o n t he a s -

s e t s , t h a t i s t h e v a l u e s o f A, B , a n d C . B u t s u p p o s e t heunknown values are [3, 1, 2]. Suppose further that the inde-

pendent shocks ar e small so tha t t he a ssets are h ighly corre-

lated wit h volat il i t ies a pproximat ely in t he ra t ios 3:1:2.

For example, suppose t he covaria nce mat rix is as follows:

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9.13.0

6.0

3.01.1

2.0

6.02.0

4.1

Also, for s implicity, let th e equilibrium risk premiums (in

percent) be equa l for exa mple, [1, 1, 1]. There is a set of

market capitalizat ions for which that is the case.

Now consider what happens when we specify a view that A

w ill outperform B by 2. In th is example, since virtua lly all of

the volat il i ty of the assets is associated with movements in

t he com m on f a ct o r, w e cl ea r l y ou g ht t o i m pu t e f r om t he

higher return of A tha n of B (relat ive to equilibrium) that a

shock to the common factor is the most likely reason A will

outperform B . If so, C ought to perform better th an equilibri-

um as well .

The conditional mean in this case is [3.9, 1.9, 2.9], and in-

d ee d, t he v i ew of A r e l a t i v e t o B ha s r a i s e d t he e xp ect e d

returns on C by 1.9.

Now suppose the independent shocks have a much larger im-

pact than the common factor. Let the matrix be as follows:

Suppose the equilibrium risk premiums are again given by

19.03.06.0

3.011.02.0

6.02.0

14.0

[1, 1, 1], and again we specify a view that A will outperform

B by 2.

This t ime, more th an ha lf of the volat il ity of A is a ssociated

with its own independent shock not related to movements in

the common factor. Now, although we ought to impute some

change in the factor from the higher return of A relat ive to

B , the impact on C should be less th an in th e previous case.

In th is case, t he conditiona l mea n is [2.3, 0.3, 1.3]. H ere the

implied effect of the common factor shock on asset C is lower

t h a n i n t h e pr ev iou s ca s e . We m a y a t t r i b ut e m os t of t h e

outperforma nce of A rela tive to B to th e independent shocks;

indeed, the implica tion for E [R B ] is nega tive relat ive to equi-

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librium. The impa ct of the in dependent shock to B is expect-

ed to domina te, even though th e contribution of the common

factor t o asset B is posit ive.

Notice that we can identify the impact of the common factor

o n l y i f w e a s s u m e t h a t w e k n o w t h e t r u e s t r u c t u r e t h a t

g ene r a t e d t he c ov a r i a nc e m a t r i x o f r e t u r ns . Tha t i s t r u e

here, but i t will not be true in general. The computat ion of

th e conditiona l mea n, h owever, does not depend on t his spe-

cia l knowledge, but only on the covar ian ce ma tr ix of return s.

Fina lly, let s look a t the case wh ere we ha ve less confi dence

i n o u r v i e w , s o t h a t w e m i g h t s a y ( E [ R A] E [R B ] ) h a s a

m e a n o f 2 a nd a v a r i a nc e of 1 .

Consider the original case, where the covariance matrix of

returns is:

The difference is tha t in t his example the condit iona l mea n

9.13.06.0

3.01.12.0

6.02.04.1

i s b a s e d o n a n u nce r t a i n v ie w. U s i ng t he f or m u l a g i ve n i n

the Appendix, we fi nd t ha t t he condit iona l mean is given by:

[3.3, 1.7, 2.5].

B ecause w e ha ve stat ed less confi dence in our view, we ha ve

allowed the equilibrium to pull us away from our expected

r e la t i ve r e t u r ns o f 2 f or A B t o a v a l u e of 1 .6, w hi ch i s

closer t o the equilibrium value of 0. We a lso fin d a sma ller

effect of the common factor on the third asset because of the

uncerta inty expressed in the view.

A Seven-Country Example Now we apply this approach to our actual dat a . We will t ryto represent the view described previously on page 14 tha t

b a d ne w s a b ou t t he U . S . e conom y w i l l ca u s e U . S . b ond s t o

outperform U.S. stocks.

The crit ical difference between our approach here and our

earlier experiment that generated Exhibit 8 is that here we

say something a bout expected returns on U .S. bonds versus

U .S. equit ies and we a llow all other expected excess returns

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to adjust accordingly. Above we adjusted only the returns to

Exhibit 9

Expected Excess Returns

Combining Investor Views With Equilibrium(annualized percent)


C u rr encies 1.32 1.28 1.73 1.22 0.44 0.47

B onds 2.69 2.39 3.29 3.40 2.39 2.70 1.35

E q u it ies 5.28 6.42 7.71 7.83 4.39 4.58 3.86

U .S. bonds an d U .S. equit ies, holding fi xed a ll other expect-

ed excess retur ns. Anoth er difference is th a t here w e specify

a d i ff er e nt ia l of m e a ns , l et t i ng t he e q u il ib r iu m d e t er m i ne

th e actua l levels of mean s; above we ha d to specify the levels

directly.

In Exhibit 9 we show the complete set of expected excess

r e t u r ns w he n w e p u t 100% confi d ence i n a v ie w t ha t t he

differential of expected excess returns of U.S. equities over

bonds is 2.0 percenta ge points, below th e equilibrium differ-

entia l of 5.5 percenta ge points. E xhibit 10 shows th e optima l

portfolio associated with this view.

Exhibit 10

Optimal PortfolioCombining Investor Views With Equilibrium




B onds 3.6 2.4 7.5 2.3 67.0 1.7 0.3

E q u it ies 3.3 2.9 29.5 10.3 3.3 2.0 1.4

This is in contrast to the inexplicable results we saw earlier.

We see here a bala nced port folio in w hich the weights ha ve

t i l t e d a w a y f r o m m a r k e t c a p i t a l i z a t i o ns t o w a r d U . S . b o nd s

an d aw ay from U .S. equit ies. Given our view, we now obta in

a port folio tha t we consider reasona ble.

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V. Controlling theBalance ofa Portfolio

n the previous section, we illustrated how our approach

Iallows us to express a view tha t U .S. bonds w ill outper-

form U .S. equit ies, in a w ay tha t leads to a w ell-behaved

optimal port folio that expresses that view. In this sect ion,we focus more specifically on t he concept of a balan ced

p or t f ol io a nd s how a n a d d i t i ona l f ea t u r e o f o u r a p p r oa c h:

Cha nges in t he confi dence in views can be used to control

the balance of the optimal portfolio.

Exhibit 11

Goldman Sachs Economists Views

Currencies


J uly 31, 1991

C ur rent S pot R a t es 1.743 5.928 137.3 1.688 1.151 1.285

Three-Month Horizon

E xpect ed F ut ur e S pot 1.790 6.050 141.0 1.640 1.000 1.156 1.324

Annualized Expected

E xcess R et ur ns 7.48 4.61 8.85 6.16 0.77 8.14

Interest Rates


J uly 31, 1991B en ch m a r k B on d Yields 8.7 9.3 6.6 10.2 8.2 9.9 11.0

Three-Month Horizon

E xpect ed F ut ur e Yields 8.8 9.5 6.5 10.1 8.4 10.1 10.8

Annualized Expected

E xcess R et ur ns 3.31 5.31 1.78 1.66 3.03 3.48 5.68

We start by i llustrat ing wha t ha ppens when w e put a set of

stronger views, shown in Exhibit 11, into our model. These

ha p p e n t o ha v e b e e n t he s ho r t - t e r m i nt e r e s t r a t e a nd e x -

change rate views expressed by our Goldman Sachs econo-

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mists on J uly 31, 1991.12 We put 100% confi dence in t hese

Exhibit 12

Optimal Portfolio Based on Economists Views(percent of portfolio value)

Unconstrained



B onds 34.5 65.4 79.2 16.9 3.3 22.7 108.3

E q u it ies 2.2 0.6 6.6 0.7 3.6 5.2 0.5

views, solve for the expected excess returns on all assets,

a nd fi nd t he o p t i m a l p o r t f o l i o s ho w n i n E x hi b i t 1 2 . G i v e n

s u ch s t r ong v ie w s o n s o m a ny a s s e t s , a nd op t im i z ing w i t h-

out constra ints, w e genera te a ra ther extreme port folio.

Ana l y s t s ha v e t r i ed a nu m b er of a p p r oa c hes t o a m e li or a t e

t hi s p r o b l e m . S o m e p u t c o ns t r a i nt s o n m a ny o f t he a s s e t

weights. H owever, we resist using such a rt ifi cial constra ints.

When a sset w eights run up a gainst constraint s, the port folio

op t im i z a t i on no l ong e r b a l a nce s r e t u r n a nd r i s k a cr os s a l l

assets. Others specify a benchmark port folio and limit therisk relat ive to the benchmark until a reasonably balanced

port folio is obtain ed. This ma kes sense if th e objective of t he

optimizat ion is to m an a ge the port folio relat ive to a bench-

m a r k ,13 b ut a g a i n w e a r e u n com f or t a b le w h en i t i s u s ed

simply to ma ke the m odel bet ter beha ved.

An alt ernat e response w hen the optima l port folio seems t oo

extreme is to consider reducing the confidence expressed in

some or a ll of the views. E xhibit 13 shows a n optima l port fo-

lio where we have lowered the confidence in all of our views.

By putt ing less confidence in our views, we have generated

12For det a ils of t hes e view s , s ee t he follow ing G oldma n Sa chs p u blica -

tions: Th e I n t er n a t i on a l Fi x ed I n c om e A n a l y s t , August 2, 1991, for

int eres t ra t es a nd The International Economics Analyst, J uly /Augu st

1991, for exchange rates.

13 We discuss this situation in Section VI.

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a set of expected excess returns that more strongly reflect

t h e e q ui li br i um a n d h a v e pu ll ed t h e op t im a l p or t f ol io

weights toward a more balanced posit ion.

Exhibit 13

Optimal Portfolio With Less Confidencein the Economists Views




B onds 3.9 21.0 19.6 2.6 7.3 13.6 42.4

E q u it ies 0.8 2.2 24.7 7.1 26.6 4.2 1.2

Let us n ow expla in precisely a propert y of a portfolio th a t w e

call ba lance. We define ba lance as a measure of how simi-

lar a portfolio is to the global equilibrium portfolio th a t is,

the market capitalization portfolio with 80%of the currency

risk hedged. The dista nce meas ure th a t w e use is the volatil-

ity of t he difference in returns of t he t wo portfolios.

We fi nd this property of balan ce t o be a useful supplement

to the standard measures of portfolio optimization, expected

return a nd risk. In our a pproach, for an y given level of r isk

there will a continuum of portfolios that maximize expected

return depending on the relat ive levels of confidence that

are expressed in the views. The less confidence the investor

ha s, th e more bala nced will be h is port folio.

Suppose that an investor does not have equal confidence in

a l l hi s v i e w s . I f t he i nv e s t o r i s w i l l i ng t o r a nk t he r e l a t i v e

confi dence levels of his different views, th en he can genera te

a n e ve n m o r e p ow e r f u l r e s ul t . I n t hi s ca s e , t h e m o d el w i l l

move away from his less strongly held views more quickly

tha n from those in wh ich he h as more confi dence. For exam-

ple, we have specified higher confidence in our view of yield

d ecl in es i n t h e U n i t ed K i n gd om a n d y iel d i n cr ea s e s i nFra nce and G ermany. These are not t he biggest y ield cha ng-

e s t ha t w e e x p e c t , b u t t he y a r e t he f o r e c a s t s t ha t w e m o s t

strongly w a nt to represent in our port folio. In part icular, w e

put less confi dence in our view s of int erest ra te moves in the

U ni t e d S t a t e s a nd A u s t r a l i a .

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Before, when we put equal confidence in our views, we ob-

ta ined th e optima l port folio show n in Exhibit 13. The view

tha t dominat ed tha t port folio wa s the interest ra te decline in

Australia . Now, when we put less than 100%confidence inour views, we keep relat ively more confidence in the views

a b o ut s om e c ou nt r i es t ha n ot he r s . E x hi b it 14 s ho w s t he

optimal port folio for this case, in which the weights repre-

senting more strongly held view s a re larger.

Exhibit 14

Optimal Portfolio With Less Confidencein Certain Views




B onds 10.3 34.3 25.5 1.6 22.9 2.4 28.1

E q u it ies 0.1 2.3 25.9 7.0 26.3 6.0 1.3

VI. Benchmarks ne of th e most importa nt , but often overlooked, infl u-

O ences on the asset allocation decision is the choice oft he b e nc hm a r k b y w hi c h t o m e a s u r e r i s k . I n m e a n-variance optimizat ion, the object ive is to maximize return

per unit of portfolio risk. The investors benchmark defines

the point of origin for measuring this risk in other words,

the minimum risk portfolio.

In m an y investment problems, risk is measured, as we ha ve

done so far in this paper, as the volat il i ty of the port folios

excess returns. This can be interpreted as having no bench-

mark, or as defining the benchmark to be a port folio 100%

invested in the domestic short-term interest rate. In many

cases, however, an alternat ive benchmark is called for. For

e xa m p l e, m a ny p or t f ol io m a na g er s a r e g i ve n a n e xp li ci t

performa nce benchmark, such as a ma rket-capita lizat ion-weighted index. If such an explicit performance benchmark

exists, then invest ing in bills is c learly a r isky stra tegy, a nd

the appropriate measure of risk for the purpose of portfolio

o p t i m i z a t i o n i s t he v o l a t i l i t y o f t he t r a c k i ng e r r o r o f t he

port folio vis- -vis th e benchma rk.

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Another example of a situat ion with an obvious alternat ive

b e n c h m a r k i s t h a t o f a m a n a g e r f u n d i n g a k n o w n s e t o f

liabili t ies. I n such a case, the benchmark port folio repre-

sents the l iabili t ies.

Unfortunately, for many port folio managers, the object ives

f a l l i nt o a l e s s w e l l - d e fi ne d m i d d l e g r o u nd , a nd t he a s s e t

a l l o c a t i o n d e c i s i o n i s m a d e d i f f i c u l t b y t he f a c t t ha t t he i r

object ive is implicit ra ther tha n explicit . For exam ple, a

global equity port folio manager may feel his object ive is to

perform a mong the top rankings of a ll global equity ma na g-

e r s. Al t hou g h he d oe s not ha v e a n e xp li ci t p er f or m a nc e

b e nc hm a r k , hi s r i s k i s c l e a r l y r e l a t e d t o t he s t a nc e o f hi s

port folio relat ive to th e portfolios of his competition.

Other examples are an overfunded pension plan or a univer-

sity endowment where matching the measurable l iabili ty is

only a small part of the total investment object ive. In these

types of situat ions, a t tempts t o use qua ntita t ive approaches

a r e of t en f r us t r a t e d b y t h e a m b ig u it y of t h e i n ves t m en t

objective.

When a n explicit benchmark does not exist , tw o alterna t ive

approaches can be used. The first is to use the volat il i ty of

excess retu rns a s t he mea sure of risk. The second is to speci-

f y a nor m a l p or t f ol io o ne t ha t r e pr e se nt s t he d es i r ed

allocation of assets in the absence of views. Such a portfolio

might , for exam ple, be designed w ith a higher-tha n-ma rket

weight for domestic assets, to represent t he domestic nat ure

of liabilities without attempting to specify an explicit liabili-

ty benchmark.

A n e q u i l i b r i u m m o d e l c a n he l p i n t he d e s i g n o f a no r m a l

port folio by qua nt ifying some of th e risk and retu rn t ra deoffs

between different port folio weights in the absence of views.

For example, the optimal portfolio in equilibrium is market-

capitalization-weighted with 80%of the currency hedged. It

ha s a n expected return (using equilibrium risk premiums) of5.7%w ith a n a nnua lized volat ility of 10.7%.

A pension fund wishing to increase the domestic weight to

85%from t he current m a rket capita lizat ion of 45%, an d not

wishing to hedge the currency risk of th e remaining 15%in

interna t ional ma rkets, might consider a n a lternat e port folio

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such as the one shown in Exhibit 15. The higher domestic

Exhibit 15

Alternative Domestic Weighted Benchmark Portfolio(percent of portfolio value)



B onds 0.5 0.5 2.0 1.0 30.0 1.0 0.0

E q u it ies 1.0 1.0 5.0 2.0 55.0 1.0 0.0

weights lead t o 0.4 percenta ge points h igher a nnua lized vola-

tility and an expected excess return 30 bp below that of the

optima l portfolio. The pension fund ma y or ma y n ot feel th atits preference for domestic concentration is worth those costs.

VII. Implied Views nce a n i nv es t or ha s e st a b l i she d hi s ob je ct i v es , a n

O asset a llocat ion model defines a correspondence be-t w e en v iew s a n d opt i m a l por t f ol ios . R a t h e r t h a ntreating a quantitative model as a black box, successful port-

folio man agers use a model to invest igate t he na ture of this

relationship. In par ticular, it is often useful to start an an a ly-

sis by using a model to find the implied investor views for

which an existing portfolio is optimal relative to a benchmark.

To illustra te this t ype of an a lysis, we assu me tha t a portfolio

m a na g e r ha s a p o r t f o l i o w i t h w e i g ht s a s s ho w n i n E x hi b i t

16. The weights, relat ive to those of his benchmark, define

th e directions of the investors views. B y a ssuming t he inves-

tor s degree of risk a version, w e ca n fi nd t he expected excess

r e t u r ns f o r w hi c h t he p o r t f o l i o i s o p t i m a l . I n t hi s t y p e o f

a na l y si s , d i f fe r ent b enchm a r k s m a y i m pl y v er y d i ff er e nt

views for a given portfolio. In Exh ibit 17 (page 30) we show

the implied views of the portfolio shown in Exhibit 16, with

t he b enchm a r k a l t e r na t i v el y (1) t he m a r k e t ca p i t a l i za t i onweights, 80% hedged, or (2) the domestic weighted a lterna-

t i v e s ho w n i n E x hi b i t 1 5 . U nl e s s a p o r t f o l i o m a na g e r ha s

t ho u g ht c a r e f u l l y a b o u t w ha t hi s b e nc hm a r k i s a nd w he r e

his a llocat ions a re relat ive to i t , and ha s conducted the t ype

o f a na l y s i s s ho w n he r e , he m a y no t ha v e a c l e a r i d e a w ha t

views are being represented in his portfolio.

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Exhibit 16

Current Portfolio Weights for Implied View Analysis(percent of portfolio value)



B onds 1.0 0.5 4.7 2.5 13.0 0.3 3.5

E q u it ies 3.4 2.9 22.3 10.2 32.0 1.7 2.0

VII I. Quantifying theBenefits of GlobalDiversification

hile i t has long been recognized that most inves-

Wtors demonstrat e a substant ial bias t ow ar d domes-

tic assets, many recent studies have documented a

rapid growth in the internat ional component in port folios

worldwide. I t is perhaps not surprising, then, tha t t here has

been a react ion a mong many investment a dvisers, wh o have

s t a r t e d t o q u e s t ion t he t r a d i t iona l a r g u m e nt s t h a t s u pp or t

global diversificat ion. This ha s been part icularly t rue in t he

U nited St at es, where global port folios ha ve tended to under-

perform domestic portfolios in recent years.

O f c ou r s e, w ha t m a t t e r s f or i nve st o r s i s t he p r os pe ct i v e

returns from internat ional assets, and as noted in our earli-er discussion of neutral views, the historical returns are of

virtually no value in project ing future expected excess re-

turns. Historical analyses continue to be used in this context

simply because investment advisors argue there is nothing

better to m easure th e value of global diversificat ion.

We w ould suggest t ha t there is something bet ter. A rea son-

a b l e m e a s u r e o f t he v a l u e o f g l o b a l d i v e r s i fi c a t i o n i s t he

degree to w hich allow ing foreign a ssets into a port folio rais-

es the optimal portfolio frontier. A natural starting point for

quantifying this value is to compute it based on the neutral

v ie w s i m pl ie d b y a g l ob a l C AP M e q u il ib r iu m . The r e a r es om e l im i t a t i on s t o u s i n g t h i s m e a s u r e. I t a s s u me s t h a t

t he r e a r e no e x t r a cos t s t o i nt e r na t i ona l i nve st m e nt ; t h u s ,

r el a xi n g t h e con s t r a i n t a g a i n s t i n t er n a t i on a l i n ves t m en t

ca nno t m a k e t he i nv es t or w o r s e of f. O n t he o t he r ha nd , i n

m e a s u r i ng t he v a l u e o f g l o b a l d i v e r s i fi c a t i o n t hi s w a y , w e

a r e a l s o a s s u m i ng t ha t m a r k e t s a r e e ff ici ent a nd t he r ef or e

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we are neglect ing to capture any value that an internat ional

port folio mana ger might a dd th rough h aving informed views

about these ma rkets. We suspect t ha t a n importa nt benefi t

of i nt e r na t i ona l i nve st m e nt t ha t w e a r e m i s si ng he r e is t h efreedom it gives th e port folio mana ger to ta ke adva nta ge of

a l a r g er nu m b er of o pp or t u ni t i es t o a d d v a l u e t ha n a r e a f -

forded in domestic markets.

Exhibit 17

Expected Excess Returns Implied by a Given Portfolio

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


C ur ren cies 1.55 1.82 0.27 1.22 0.63 2.45

B onds 0.30 0.30 0.58 1.03 0.13 0.01 1.22

E q u it ies 2.82 3.97 0.30 6.73 4.15 5.01 5.88

Views Relative to the Domestic Weighted Benchmark(Annualized Expected Excess Returns)


C u rr encies 0.05 0.20 0.50 0.54 0.01 0.90

B onds 0.01 0.21 0.72 0.85 1.45 1.01 0.18

E q u it ies 2.24 2.83 5.24 4.83 1.49 0.28 2.38

In an y case, as a n i l lustrat ion of the value of the equilibrium

concept, w e use it h ere to calcula te t he va lue of global diver-

s i fi c a t i on f or a b ond p or t f ol io, a n e q u it y p or t f ol io, a nd a

port folio containing both bonds and equit ies, in each case

both with an d w ithout a llowing currency h edging. We nor-

ma lize the port folio volat il i t ies a t 10.7% the volat il i ty of

th e ma rket-capita lizat ion-w eighted portfolio, 80%hedged. In

E x hi bi t 18, w e s ho w t he a d d i t i ona l r e t u r n a v a i la b l e f r om

including int ernat iona l a ssets relat ive to the optimal domes-

t ic port folio with the sam e degree of r isk.

W ha t i s c l e a r f r o m t hi s t a b l e i s t ha t g l o b a l d i v e r s i fi c a t i o n

p r ov id es a s u bs t a nt i a l p ick u p i n e xp ect e d r e t u r n f or t he

domestic bond port folio ma na ger, both in absolute a nd per-

centa ge terms. The ga ins for a n equity ma na ger, or a port fo-

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l io manager with both bonds and equit ies, are a lso substan-

Exhibit 18

The Value of Global Diversification

Expected Excess Returns in Equilibriumat a Constant 10.7% Risk

Without Currency Risk Hedging

Basis Point PercentDomestic Global Difference Gain

B on ds On ly 2.14 2.63 49 22.9

E q u it ies Only 4.72 5.48 76 16.1

B on ds a nd E q u it ies 4.76 5.50 74 15.5

Allowing Currency Hedging

Basis Point PercentDomestic Global Difference Gain

B on ds On ly 2.14 3.20 106 49.5

E q u it ies Only 4.72 5.56 84 17.8

B on ds a nd E q u it ies 4.76 5.61 85 17.9

t i a l , t h o ug h m u ch s m a l le r a s a p er ce nt a g e of t h e ex ces s

returns of the domestic portfolio. These results also appear

to provide a just ifi cat ion for the common pract ice of bond

portfolio managers to currency hedge and of equity portfolio

managers not to hedge. In the absence of currency views, the

ga ins to currency hedging a re clearly more import a nt in both

absolute and relat ive terms for fi xed income investors.

IX. Historical

Simulations

t is na tura l to ask how a model such a s ours would ha ve

Iperformed in simulations. However, our approach does

not , in i tself , produce investment stra tegies. I t requiresa s e t o f v i e w s , a nd a ny s i m u l a t i o n i s a t e s t no t o nl y o f t he

model but a lso of the stra tegy producing th e views.

F o r e x a m p l e , o ne s t r a t e g y t ha t i s f a i r l y w e l l k no w n i n t he

i nv e s t m e nt w o r l d , a nd w hi c h ha s p e r f o r m e d q u i t e w e l l i n

recent years, is to invest funds in high yielding currencies.

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In this sect ion we il lustrate how a quantitat ive model such

as ours can be used to optimize such a strategy, and also to

compare the relat ive performances of different investment

s t r a t e gi es . I n p a r t icu la r , w e w i l l com pa r e t h e h i st or i ca lperforma nce of a stra tegy of invest ing in h igh yielding cur-

r e nci es v er s u s t w o ot he r s t r a t e g i es : (1) i nv es t i ng i n t he

bonds of countries w ith high bond yields a nd (2) investing in

the equities of countries with high ratios of dividend yield to

bond yield. Our purpose is to i l lustrate how a quantitat ive

approach can be used to make a useful comparison between

alterna t ive investment stra tegies. We a re not t rying t o pro-

mote or just ify th ese part icular stra tegies. We ha ve chosen

to focus on these three primarily because they are simple,

relatively comparable, and representative of standard invest-

ment approaches.

O u r s i m u l a t i o ns o f a l l t hr e e s t r a t e g i e s u s e t he s a m e b a s i c

methodology, the sa me dat a , an d t he sam e underlying secu-

r i t i e s . T he d i f f e r e nc e s i n t he t hr e e s i m u l a t i o ns a r e i n t he

sources of views about excess returns and in the assets to-

ward which those views are applied. All of the simulat ions

use our approach of adjusting expected excess returns away

from the global equilibrium a s a function of investor views.

In ea ch of the simulat ions, we test a stra tegy by performing

t he f ol low i ng s t e ps . S t a r t i ng i n J u l y 1981 a nd cont i nu ing

e a ch m on t h f or t h e n e xt 10 y e a r s , w e u s e d a t a u p t o t h a tpoint in t ime to est imate a covariance matrix of returns on

equities, bonds, a nd currencies. We compute t he equilibrium

risk premiums, add views according to t he part icular stra te-

gy, and calculate the set of expected excess returns for a ll

securities based on combining views with equilibrium.

We th en optimize the equit y, bond, an d currency w eights for

a g i ve n l e v el of r i s k w i t h no c ons t r a i nt s on t he p or t f ol io

w e i g ht s . We ca l cu l a t e t he e xce s s r e t u r ns t ha t w o u ld ha v e

accrued in tha t month. At the end of each month we updat e

the data and repeat the calculat ion. At the end of ten years

we compute the cumulat ive excess returns for each of thet h r e e s t r a t e g i e s a n d c o m p a r e t h e m w i t h o n e a n o t h e r a n d

with several passive investments.

The views for the three strategies represent very different

information but are generated using similar approaches. In

simulations of the high yielding currency strategy, our views

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a r e b a s e d o n t he a s s u m p t i o n t ha t t he e x p e c t e d e x c e s s r e -

turns from h olding a foreign currency a re a bove their equi-

librium value by a n a mount equal t o the forw ar d discount on

that currency.

For example, i f the equilibrium risk premium on yen, from

a U . S . d ol la r p er s pe ct i v e, i s 1 % a nd t he f or w a r d d i scou nt

(w hich, beca use of covered int erest ra te pa rity, approxima te-

ly equals the difference between the short rate on yen-de-

nominated deposits a nd t he short ra te on dollar-denomina t-

ed deposits) is 2%, then we assume the expected excess

return on yen currency exposures to be 3%. We compute

expected excess returns on bonds and equities by adjusting

their returns a wa y from equilibrium in a ma nner consistent

with 100%confidence in the currency views.

I n s i m ul a t i ons of a s t r a t e g y of i nv e st i ng i n fi x ed i ncom e

m a r k e t s w i t h hi g h y i e l d s , w e g e ne r a t e v i e w s b y a s s u m i ng

tha t expected excess returns on bonds ar e a bove their equi-

librium values by an amount equal to the difference between

t he b ond -e q u iv a l ent y i el d i n t ha t cou nt r y a nd t he g l ob a l

ma rket-c

black litterman 91 global asset allocation with equities bonds and currencies

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