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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FixedIncomeResearch

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