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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    FixedIncomeResearch

    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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    FixedIncomeResearch

    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

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