black litterman satchelland scowcroft.pdf

colleagues as idiot savants. Seniormanagement is rarely prepared tointervene when managers are successfuland profitable, however they made theirdecisions. These disharmonies can makecompany-wide risk-management andportfolio analysis non-operational and canhave deleterious effects on companyprofitability and staff morale.

One model which has the potential tobe used to integrate these diverse

IntroductionOne of the major difficulties in financialmanagement is trying to integratequantitative and traditional managementinto a joint framework. Typically,traditional fund managers are resistant toquantitative management, as they feelthat techniques of mean-variance analysisand related procedures do not captureeffectively their value added. Quantitativemanagers often regard their judgmental

138 Journal of Asset Management Vol. 1, 2, 138–150 � Henry Stewart Publications 1470-8272 (2000)

A demystification of theBlack–Litterman model: Managingquantitative and traditionalportfolio constructionReceived: 20th January, 2000

Stephen Satchell*is a fellow of Trinity College, Cambridge, a lecturer in Economics at Cambridge University and a Visiting Professor at CityUniversity Business School, London. His specialities are finance and econometrics, on which he has written at least 80papers. An academic advisor and consultant to a wide range of financial institutions covering areas such as actuarialvaluation, asset management, risk management and strategy design, Dr Satchell has links with research organisations suchas the Department of Applied Economics (Cambridge) and INQUIRE, and has doctorates from the Universities of Cambridgeand London.

Alan Scowcroftwas educated at Ruskin College, Oxford and Wolfson College, Cambridge where he was awarded the Jennings prize foracademic achievement. He taught econometrics at Clare College, Cambridge before joining Phillips and Drew as aneconometrician in 1984. There he worked with the leading macro research group of the time building macro-econometricforecasting models and developing innovative software for solving large scale dynamic input-output models. In 1991 heestablished the equities quantitative research group at UBS with Alun Jones and since that time has worked on every aspectof quantitative modelling from stock valuation to TAA. He has been closely associated with the pioneering work on equitystyle and portfolio analysis developed by UBS Warburg. Mr Scowcroft’s research interests include practical applications ofBayesian econometrics and portfolio optimisation. He is currently the head of equities quantitative research at UBS Warburg.

*Trinity College, Cambridge University, Trinity Lane, Cambridge CB2 1TQ, UKTel: �44(0) 1223 338409; Fax: �44(0) 1223 335475; e-mail: [email protected]

Abstract The purpose of this paper is to present details of Bayesian portfolioconstruction procedures which have become known in the asset management industryas Black–Litterman models. We explain their construction, present some extensions andargue that these models are valuable tools for financial management.

Keywords: Bayesian portfolio construction; Black–Litterman model; optimisation; assetallocation

have a well-defined n-dimensionalcovariance matrix �; in particular, theircovariance matrix is non-singular. If thereturns for period t are denoted by rt,we shall write E(r) to mean expectedforecasted returns. This is shorthand forE(rt � 1 ��t), where �t refers to allinformation up to and including period t.A second related concept is the (n � 1)vector � representing equilibrium excessreturns, either in terms of a theory suchas the capital asset pricing model(CAPM) or in the sense of the prevailingsupply of value-weighted assets. Thelatter interpretation corresponds to aglobal market portfolio demonetised indomestic currency.

Algebraically, assuming the validity ofthe CAPM,

� � �(�m � rf)

where �m is the return on the globalmarket in domestic currency, rf is theriskless (cash) domestic rate of return, �is an (n � 1) vector of asset betas, where

� � Cov(r, r�wm)/2m

where r�wm is the return on the globalmarket, wm are the weights on theglobal market, determined by marketvalues, and 2

m is the variance of the rateof return on the world market.

If we let � � Cov(r, r�) be thecovariance matrix of the n asset classes,then

� � �wm

where � (�m � rf)/2m is a positive

constant. If returns were arithmetic withno reinvestment, would be invariant totime, since both numerator anddenominator would be linear in time.However, if compounding is present,there may be some time effect.

In this paper, we shall only consider

approaches is the Black–Litterman (BL)model (Black and Litterman, 1991,1992). This is based on a Bayesianmethodology which effectively updatescurrently held opinions with data toform new opinions. This frameworkallows the judgmental managers to givetheir views/forecasts, these views areadded to the quantitative model and thefinal forecasts reflect a blend of bothviewpoints. A lucid discussion of themodel appears in Lee (1999).

Given the importance of this model,however, there appears to be no readabledescription of the mathematicsunderlying it. The purpose of this paperis to present such a description. In thesecond and third sections we describe theworkings of the model and present someexamples. In the fourth section wepresent an alternative formulation whichtakes into account prior beliefs onvolatility. In the second and thirdsections, particular attention is paid tothe interesting issue of how to connectthe subjective views of our managersinto information usable in the model.This is not a trivial matter and lies at theheart of Bayesian analysis. Indeed Rev.Bayes had such misgivings aboutapplying Bayes theorem to real-worldphenomena that he did not publish hispaper (Bayes, 1763): it was presented tothe Royal Society by his literaryexecutor (Bernstein, 1996: 129–34).

Workings of the modelBefore we present Bayes’ theorem and itsapplication by BL to asset managementproblems, we shall present our notationand basic concepts. We assume that thereis an (n � 1) vector of asset returns r;these are, typically, excess returnsmeasured in the domestic currency andsubtracting the domestic cash returnwhich is not included in the vectors.With this convention the asset returns

� Henry Stewart Publications 1470-8272 (2000) Vol. 1, 2, 138–150 Journal of Asset Management 139

Black–Litterman model

the data equilibrium return, given theforecasts held by the investor.

The result of the theorem pdf(E(r) ��) isthe ‘combined’ return or posteriorforecast given the equilibriuminformation. It represents the forecasts ofthe manager/investor after updating forthe information from the quantitativemodel.

The contribution of BL was to placethis problem into a tractable form with aprior distribution that was both sensibleand communicable to investors. Bayesiananalysis has, historically, been weakenedby difficulties in matching tractablemathematical distributions to individual’sviews.

We now review and extend BL’sresults. We make the followingassumptions:

A1 pdf(E(r)) is represented in thefollowing way. The investor has a setof k beliefs represented as linearrelationships. More formally, weknow the (k � n) matrix P and aknown (k � 1) vector q. Lety � PE(r) be a (k � 1) vector. It isassumed that y � N(q,�), where �is a (k � k) diagonal matrix withdiagonal elements �ii. A larger �ii

represents a larger degree of disbeliefin the relationship represented by yi,�ii � 0 represents absolute certainty,and, as a practical matter, we bound�ii above zero. The parameters qand � are called by Bayesianshyperparameters; they parameterise theprior pdf and are known to theinvestor.

A2 pdf(� �E(r)) is assumed to beN(E(r), �) where � is thecovariance matrix of excess returnsand is a (known) scaling factoroften set to 1. This assumptionmeans that the equilibrium excessreturns conditional upon the

(global) equity in our n assets. Extendingthe model to domestic and foreignequities and bonds presents fewdifficulties.

Considering Foreign Exchange (FX)as an additional asset class does presentdifficulties as we need to ‘convert’currencies into a domestic value, whichrequires making assumptions abouthedge ratios. Black (1990) proves that,in an international CAPM (ICAPM)1

under very stringent assumptions, allinvestors hedge the same proportion ofoverseas investment, and uses this resultto justify a global or universal hedgingfactor which is the same for allinvestors facing all currencies. Adlerand Prasad (1992) discuss Black’s resultand show how restrictive the resultactually is.

It is natural to think of � as being theimplied returns from the equilibriummodel and, as the above discussionshows, these would depend upon ourdata and represent the input of thequantitative manager. How can werepresent the views of the fundmanagers? To answer this question,consider Bayes’ theorem. In the notationwe have defined above, Bayes’ theoremstates that

pdf(E(r) ��) �pdf(� �E(r))pdf(E(r))

pdf(�)

where pdf(.) means probability densityfunction. The above terms have thefollowing interpretations:

— pdf(E(r)) is the prior pdf thatexpresses the (prior) views of the fundmanager/investor

— pdf(�) represents the marginal pdf ofequilibrium returns. In the treatmentthat follows, it is not modelled. Aswe will demonstrate, it disappears inthe constant of integration.

— pdf(� �E(r)) is the conditional pdf of


Satchell and Scowcroft

volatilities as well. The natural equilibriumvalue for volatility is the Black–Scholes(BS) model, so that if option data wereavailable, the prior on volatility could beupdated by the observed implied volatility.Unfortunately, the pdf of impliedvolatility would depend on the nature ofthe stochastic volatility ignored by theBS formula, and there appears to be nosimple way forward. An alternativewould be to formulate a prior on .Although we have no obvious data toupdate our beliefs, a solution similar toProposition 12.3 in Hamilton (1994)could be attempted. We present details inthe fourth section.

ExamplesIn this section, we consider variousexamples which illustrate themethodology.

Example 1

In this example, we consider the casewhere a sterling-based investor believesthat the Swiss equity market willoutperform the German by 0.5 per centper annum. All returns are measured insterling and are unhedged. This is amodest target and is intended toemphasise that the forecast represents anew equilibrium and not short-termoutperformance. In the notation of thesecond section, we have one belief,k � 1 in A1. Using the universe of 11European equity markets listed in Table2, P is a (1 � 11) vector of the form

P � [l, �1, 0, 0, 0, 0, 0, 0, 0, 0, 0]

and q � 0.5 per cent. Table 1 lists theparameters used to compute theconditional forecast.

The computed values E(r��) are shownin Table 2. In addition to the prior viewof the relative performance of Swiss and

individual’s forecasts equals theindividual’s forecast on average. Thismay not hold in practice as theauthors have met many practitionerswho have exhibited the mostalarming biases relative to the marketview. The conditioning needs to beunderstood in the sense that, if allindividuals hold this view and investin a CAPM-type world, then �represents the equilibrium returnsconditional upon the individuals’common beliefs.2

Given A1 and A2, it is a straightforwardresult to show the following theorem.

Theorem 1. The pdf of E(r) given � isgiven by

pdf(E(r) ��) � N([( �)�1 � P��1P]�1

[( �)�1� � P��1q],[( �)�1 � P��1P]�1]

Proof: See Appendix.We emphasise that Theorem 1 is a resultknown to Bayesian econometricians andto BL, although they did not report thevariance formula in the papers. Also, ourinterpretation of what is prior and whatis sample information may differ fromBL.

It should be clear from the previousanalysis that neither A1 nor A2 areessential for the model to be used. Mostpriors used in finance, however, tend toconvey little information about theinvestors’ beliefs. Various alternatives suchas a diffuse prior (see Harvey and Zhou(HZ), 1990: Equation 6; or Klein andBawa (KB) 1976: Equation 3) or themore detailed priors presented inHamilton (1994: Chap. 12) cannot beeasily understood in behavioural terms.In Bayesian terms, the prior chosen byBL is called the natural conjugate prior.

Extensions could be considered for



weight (ie from unhedged to fullyhedged). The assumed benchmark holdingof currency is zero for all markets. Thesolution weights for currencies, which arenot shown in the table, are all negligible.

The Sharpe ratio for the solution is0.16 with a tracking error3 of 0.39, theportfolio is beta constrained to 1.0.

Example 2

In this second example, we consider thecase where a US dollar-based investorbelieves that six hard currency markets willoutperform nine other European markets,on average, by 1.5 per cent per annum.This could be interpreted as a possibleEMU scenario. As in Example 1, thisstill represents one view and P is now a(1 � 15) vector equal to

[1/6. . .1/6 � 1/9. . .�1/9]

The values for the other parameters areas shown in Table 5. Note that Delta ()

German markets, larger changes from theimplied view for other markets areassociated with low covariance with theSwiss market. In Table 3, we reportcertain key parameters associated with ourportfolio construction.

We now consider the impact of theconditional forecast in an optimisationproblem, where the objective is a simplemean-variance utility function. The risk-aversion parameter has been set withreference to delta. The beta of theportfolio and the sum of the weights areconstrained to unity. The resultspresented in Table 4, show, as expected,a switch from the German to the Swissmarket. Some large differences inforecasts, Italy for example, are translatedinto small changes in the portfolioweights as the optimiser takes intoaccount the benchmark weight, the assetbeta and the impact of covariances.

In both this and the following example,currency holdings were free to varybetween zero and minus the market



Table 1 Bayesian parameters

Parameter Value Symbol

DeltaTauViewConfidence

5.001.000.050.05

q�

Table 2 Forecast results

MarketBenchweight

SwissCov � 100 � E(r��) Difference

SwitzerlandGermanyDenmarkSpainFinlandFranceItalyNetherlandsNorwaySwedenUK

0.09820.15110.01360.04090.01250.12340.05680.08700.01030.04740.3588

0.18840.07240.07660.06660.06660.10160.00610.08260.09790.07760.0784

5.346.465.318.07

10.697.938.065.648.437.716.33

5.536.315.297.99

10.557.897.885.628.407.676.33

�0.19�0.15�0.02�0.08�0.14�0.03�0.18�0.03�0.03�0.04�0.00

Table 3 Optimisation parameters

Parameter Value

Risk aversion �Tracking error limitPortfolio beta

2.52.51.0

broadly in line with the imposed view,with the exception that the forecast forIreland actually goes down whileSwitzerland increases slightly.

To consider the impact of theconditional forecast, we solve a simpleoptimisation problem, where, as inExample 1, the asset weights areconstrained to be positive and sum tounity. The asset beta is constrained tounity and currency weights are free tovary between unhedged and fully hedgedfor each market. The tracking error isbounded at 2.5 (see Table 7).

The optimisation results are shown

has now been set at 3 to ensure that thelevel of the conditional forecast accordswith historical experience.

The conditional forecast is shown inTable 6. The difference between theimplied and conditional forecasts is



Table 4 Optimisation results

Market BetaBenchmarkweight (%)

Solutionweight (%) Difference

SwitzerlandGermanyDenmarkSpainFinlandFranceItalyNetherlandsNorwaySwedenUK

0.800.970.801.201.571.181.200.851.251.150.95

9.8215.111.364.091.25

12.345.688.701.034.74

35.88

12.1912.811.224.271.37

12.615.638.041.104.77

36.00

�2.37�2.30�0.14�0.18�0.11�0.27�0.05�0.66�0.07�0.02�0.12

Table 5 Bayesian parameters

Parameter Value Symbol

DeltaTauViewConfidence

3.0001.0000.0150.025

q�

Table 6 Forecast results

Benchweight � E(r��) Difference

‘Hard’ marketsAustriaBelgiumFranceGermanyNetherlandsIreland

0.00600.02440.11810.14460.08320.0076

14.8413.7514.8613.5712.3311.18

15.0513.8314.9813.6012.3811.03

�0.21�0.08�0.12�0.04�0.06�0.15

‘Soft’ marketsDenmarkFinlandItalyNorwayPortugalSpainSwedenSwitzerlandUK

0.01300.01200.05430.00980.00490.03920.04540.09400.3433

12.2418.8316.6215.5511.8413.6313.0413.2712.74

11.9217.5815.4215.0411.6713.0312.2813.2912.68

�0.32�1.24�1.20�0.51�0.17�0.60�0.76�0.02�0.06

properly the views held by theinvestor. When these excess returns aresubsequently fed back into theoptimisation process, the investor’soptimal weights will reflect the priorview.

It is this usage of implied excessreturns in the data model which alsohelps to address one of the principalreservations many practitioners havewith respect to the use of optimisersin portfolio construction, namely theirextreme sensitivity to changes inforecasts. Raw forecast alphas areinevitably volatile and, if used asoptimiser inputs, give rise tocompletely unacceptable revisions toportfolio weights. By combining neutralmodel forecasts with the investor’sviews, the Bayesian formulationproduces robust inputs for theoptimiser.

Alternative formulationsIn this section, we present twoalternative formulations of the BL model,the first of which takes into accountprior beliefs about overall volatility. Todo this, we make the followingadjustment. We shall assume that isnow unknown and stochastic so that

in Table 8 and not surprisingly show apositive tilt in favour of the strongcurrency markets. Interestingly, eventhough the optimiser was free to holdcurrency up to a fully hedged position,all the solution weights for currenciesare zero. The Sharpe ratio for thesolution is 0.18 with a tracking errorof 1.8. The portfolio beta isconstrained to unity.

Overall, we feel that the examplesjustify our confidence in the approach.Care needs to be taken interpreting theconditional forecast, however, since it isthe product of the prior view and thedata model. In these examples, the datamodel has been taken to be theimplied excess returns generated by amean-variance optimisation problem.Even though such excess returns canbe counter-intuitive, as in the case ofIreland in Example 2, they may beunderstood as the extent to which theneutral forecast has to change to reflect



Table 7 Optimisation parameters

Parameter Value

Risk aversion �Tracking error limitPortfolio beta

1.52.51.0

Table 8 Optimisation results

Market BetaBenchmarkweight (%)

Solutionweight (%) Difference

AustriaBelgiumFranceGermanyNetherlandsIreland

DenmarkFinlandItalyNorwayPortugalSpainSwedenSwitzerlandUK

1.091.021.101.000.920.84

0.911.371.221.140.881.010.970.980.95

0.602.44

11.8114.468.320.76

1.301.205.430.980.493.924.549.40

34.33

3.274.82

15.9018.639.283.39

0.000.002.660.000.000.981.826.79

32.46

�2.67�2.38�4.09�4.17�0.96�2.63

�1.30�1.20�2.77�0.98�0.49�2.94�2.72�2.61�1.88

Theorem 2. If we assume A3 and A4,then

pdf(E(r) ��) �

[m � (E(r) � �)��*V (E(r) � �)]�(m�n)/2

which is a multivariate t distribution.The vector � is the term E(r ��) givenin Theorem 1, the matrix V is theVar(r ��) given in Theorem 1 while

�* �m

� � A � C�H�1C

where A, C, H are defined in the proofof Theorem 1.

Proof: See Appendix.

An immediate corollary of Theorem 2 isthe following.

Corollary 2.1: pdf(� �E(r), �) is ascale gamma with ‘degrees of freedom’m � n and scale factor G � �, whereG � (� � E(r))��1 (� � E(r)) �(PE(r) � q)��1 (PE(r) � q).

Proof: See Appendix.The consequence of Corollary 2.1 is

that we can now compute

E(��E(r),�) � (m � n)(G � �)

and

Var(� �E(r),�) � 2(m � n)(G � �)2

The increase in precision can becomputed as

E(��E(r),�) � E(w)� (m � n) (G � �) � m�� mG � n(G � �)

It is interesting to note that, althoughour expected returns now have amultivariate t distribution, such a returnsdistribution is consistent withmean-variance analysis and the CAPM.

A3

pdf(� �E(r), ) � N(E(r), �).

Furthermore,

pdf(E(r) � ) � N(q, �),

A4 The marginal (prior) pdf of� � 1/ is given by the following,4

pdf(�) �

(�/2)m/2 �(m/2)�1exp��

2 ��(m/2)

,

0 � � ��

This pdf has two hyperparameters mand �, and we assume it isindependent of pdf(�).

Remark 1. Here we treat as afundamental parameter that measuresthe overall dispersion of � about E(r).Considering pdf(E(r) � ), we define theelements of � relative to so that�ii � 1 reflects a degree of disbeliefequal in scale to the dispersion measureof � about E(r), a value �ii �1implies greater disbelief than before andan increase in not only moves thedispersion of equilibrium expectedreturns about the forecasts but alsoincreases the overall degree of disbeliefin the forecasts.

Remark 2. The prior pdf of � � 1/ isa scale gamma, where � is often calledthe precision. It follows that E(�) � m�and Var(�) � 2m�2. This means that forfixed E(�) as m → �, Var(�) → 0 andhence is a more reliable prior.

We are now in a position to state ournew result which is, again, a standardresult in the Bayesian literature. For asimilar result, see Hamilton (1994,Proposition 12.3).



pdf(� �PE(r) � qi) � �i (1)

� � 12��

k/2 1det(P�P�)

exp ��(P� � qi)� (P�P�)�1 (P� � qi)2 �

Combining the above, we deduce thatpi* the posterior probability of scenario ibecomes

pi* �pi �i

m�i�1

pi �i

(2)

Equation (2) gives us an updating ruleon the prior probabilities which allows usto rescale our pi by value of thelikelihood function with expected returnsevaluated at qi normalised so that thesum of the weights is one. Thus, if theequilibrium return p satisfied the �condition P� � qi, �i would reach itsmaximum value. We note that since theterm in front of the exponential in (1) iscommon for all �i, we can simplify pi*to be

pi exp� �(P� � qi)� (P�P�)�1 (P� � qi)2 �

m�i�1

pi exp��(P� � qi)� (P�P�)�1 (P� � qi)2 �

(3)

Our new weights take a maximum valueof 1 and a minimum value of 0. Table 9provides an illustration of the calculationsbased on ten scenarios for the EMUexample given in the third section.Column one shows the assumedoutperformance of the strong currencymarkets for each scenario. For simplicity,we have assumed that the managerbelieves each scenario to be equallylikely; pi � 0.1. The calculated posteriorprobabilities pi* show clearly thatsubstantial outperformance is much less

[This is proved in Klein and Bawa(1976)]. Thus, our extended analysisleaves us with a mean vector and acovariance matrix which, up to a scalefactor, are the same as before. What wegain is that probability computations willnow involve the use of the t distribution.This will give the same probabilities asthe normal for large m, but for small mwill put more weights in the tails of ourforecast distribution. Thus, we canmanipulate this feature to give extradiagnostics to capture uncertainties aboutour forecasts.

We do not present numericalcalculations for this model, as thenature of the prior is too complex tocapture the beliefs of a typicallynon-mathematical fund manager.However, in our experience, fundmanagers are able to provide a rangeof scenarios for expected returns andassociate probabilities with thesescenarios. We shall explore such amodel next, this being the second‘extension’ of the BL model referred toearlier.

A5 The prior pdf for E(r) is of theform PE(r) � qi, i � 1, . . .,m. Each(vector value qi has priorprobability pi, where �m

i�1pi � 1. P

and E(r) have the same definitionas before.

If we combine A5 with A3, it isstraightforward to compute

pi* � prob(PE(r) � qi ��);m�

i�1

pi* � 1

pi* �pdf(� �PE(r ) � qi)prob(E(r ) � qi)

pdf(�)

and

pdf(�) �m�

i�1

pdf(� �PE(r) � qi)prob(E(r) � qi)



modern financial institutions where bothviewpoints are represented. We presentan exposition of these models so thatreaders should be able to apply thesemethods themselves. We also presentseveral extensions.

Notes1 Here we use the acronym ICAPM to mean

international CAPM. The standard usage for ICAPMis for intertemporal CAPM. Since the internationalCAPM is a particular application of Merton’sintertemporal CAPM, this should cause noconfusion.

2 This rather loose interpretation can be tightened; seeHiemstra (1997) for a construction of a CAPMmodel based on heterogeneous expectations byinvestors.

3 The tracking error or active risk of a portfolio isconventionally defined as the annualised standarddeviation of portfolio active return (ie the excessreturn attributable to holding portfolio weightsdifferent from the benchmark weights).

4 �(.) is the gamma function, �(n) � �0�xn � 1exp(�x)dx.

ReferencesAdler, M. and Prasad, B. (1992) ‘On Universal

Currency Hedges’, Journal of Financial andQuantitative Analysis, 27(1), 19–39.

Bayes, T. (1763) ‘An Essay Toward Solving a Problemin the Doctrine of Chances’, Philosophical Transactions,Essay LII, 370–418.

Bernstein, P. L. (1996) Against the Gods, John Wiley,Chichester.

Black, F. (1989) ‘Universal Hedging: Optimal CurrencyRisk and Reward in International Equity Portfolios’,Financial Analysts Journal, July–Aug., 16–22, reprintedin Financial Analysts Journal, Jan– Feb., 161–7.

Black, F. (1990) ‘Equilibrium Exchange Rate Hedging’,Journal of Finance, July, 45(3), 899–907

likely given the historic covariancesbetween these markets. In this example,each scenario is associated with only oneview. If the scenario contained manyviews, the posterior probability wouldstill relate to the entire scenario and notan individual view.

In practical terms, the judgmental fundmanager can use the posterior probabilitypi* as a consistency check of the priorbelief associated with scenario i expressedas probability pi. If the scenario seemsunlikely when tested against the datausing (3) the confidence numbers �ii

defined in A1 can be revised upwardsaccordingly. Equation (3) can thereforebe regarded as a useful adjunct toTheorem 1 by helping the rationalmanager formulate the inputs required ina Bayesian manner. As observed by noless an authority than Harry Markowitz,‘the rational investor is a Bayesian’(Markowitz, 1987: 57, italics in original).

ConclusionWe have presented several examples ofBayesian asset allocation portfolioconstruction models and showed howthey combine judgmental andquantitative views. It is our belief thatthese models are potentially ofconsiderable importance in themanagement of the investment process in



Table 9 Posterior probabilities

Scenariooutperformance (%)

Priorprobability (%)

Posteriorprobability (%)

0.51.01.52.02.53.03.54.04.55.0

10.0010.0010.0010.0010.0010.0010.0010.0010.0010.00

10.0710.0610.0510.0410.0210.009.989.969.939.90

University of Strathclyde.Klein, R. W. and Bawa, V. S. (1976) ‘The Effect of

Estimation Risk on Optimal Portfolio Choice’,Journal of Financial Economics, 3, 215–31.

Lee, W. (1999) ‘Advanced Theory and Methodology ofTactical Asset Allocation’, unpublished manuscript,available athttp://faculty.fuqua.duke.edu/~charvey/Teaching/BA453 2000/lee.pdf

Markowitz, H. (1987) ‘Mean-Variance Analysis inPortfolio Choice and Capital Markets, BasilBlackwell, Oxford.

Zellner, A. (1971) An Introduction to Bayesian Inference inEconometrics, John Wiley, New York.

Black, F. and Litterman, R. (1991) ‘Global AssetAllocation with Equities, Bonds and Currencies’,Goldman Sachs and Co., October.

Black, F. and Litterman, R. (1992) ‘Global PortfolioOptimization’, Financial Analysts Journal, Sept–Oct,28–43.

Hamilton, J. (1994) Time Series Analysis, BreedonUniversity Press, Princeton, NJ.

Harvey, C. R. and Zhou, G. (1990) ‘Bayesian Inferencein Asset Pricing Tests’, Journal of Financial Economics,26, 221–54.

Hiemstra, C. (1997) Addressing Nonlinear Dynamics inAsset Prices with a Behavioural Capital Asset PricingModel, Accounting and Finance Department,



AppendixProof of Theorem 1Using Bayes’ theorem and Assumptions A1 and A2, we see that

pdf(E(r) ��) �k exp(� 1—

2 (� � E(r))��1(� � E(r)) � 1–2(PE(r) � q)��1(PE(r) � q))

pdf(�)

where k is an appropriate constant.We next simplify the quadratic term in the exponent.

E(r)�( �)�1E(r) � 2�( �)�1E(r) � ��( �)�1� � E(r)�P��1PE(r)�2q��1PE(r)� q��1q

� E(r)�(( �)�1 � P��P)(( �)�1 � P��1P)�1(( �)�1 � P��1P)E(r)�2(��( �)�1

� q��1P)(( �)�1 � P��1P)�1(( �)�1 � P��1E(r))�q��1q � ��( �)�1�

LetC � ( �)�1� � P��1qH � ( �)�1 � P��1P, we note that H is symmetrical so H � H�A � q��1q � ��( �)�1�

We can rewrite the exponent as equal to

E(r)�H�H�1HE(r) � 2C�H�1HE(r) � A� (HE(r) � C)�H�1(HE(r) � C) � A � C�H�1C� (E(r) � H�1C)�H( E(r) � H�1C) � A � C�H�1C

In terms of E(r), terms such as A � C�HC disappear into the constant of integration.Thus,

pdf(E(r) ��) � exp(�1–2(E(r) � H�1C)�H(E(r) � H�1C)) (A1)

so that E(r) �� has mean � H�1 C (A2)

� [( �)�1 � P��1P]�1 [( �)�1� � P��1q] (A3)

and Var(r ��) � [( �)�1 � P��1 P]�1 (A4)



Proof of Theorem 2First

pdf(E(r),w ��) �pdf(� �E(r),w)pdf(E(r) �w)pdf(w)

pdf(�)(A5)

From Assumption A3, we can write

pdf(� �E(r),w)pdf(E(r) �w) � kwn/2 exp � �wG2 � (A6)

where G � (� � E(r))��1 (� � E(r)) � (PE(r) � q)��1(PE(r) � q)

If we now use Assumption 4 and Equation (A6), we see that

pdf(E(r), w ��) �

k exp ��w2

(G � �)��

2 �m/2

w(m�n)/2�1

� �m2 � pdf(�)

(A7)

To compute pdf(E(r) ��), we integrate out w.Let

v �w2

(G � �), w �2v

(G � �), dw �

2(G � �)

dv

pdf(E(r) ��) � k��

2�m/2 ��

0

exp(�v) � 2vG � ��

(m�n)/2�1

� 2G � �� dv (A8)

then

�

k��

2 �m/2

2(m�n)/2 � �m � n2 �

� �m2 � (G � �)(m�n)/2

(A9)

The multivariate t is defined (see Zellner, 1971: 383, B20) for matrices �(l � 1) andV(l � l) and positive constant v as

pdf(x ��, V, v, l) �vv/2 �((v � l )/2 �V � 1/2 [v � (x � �)�V(x � �)](l�v)/2

�l/2 �(v/2)

If we re-write G � � in terms of A, C and H as defined in the proof of Theorem1, we see that

G � � � (E(r) � H�1C)�H(E(r) � H�1C) � A � C�H�1C � �

� (E(r) � H�1C)�mH

� � A � C�H�1C(E(r) � H�1C) � m (A10)



This shows that pdf(E(r) ��) is multivariate t,

� � H�1C (as before)

V �m

� � A � C�H�1CH, l � n and v � m.

Proof of Corollary 2.1Factorising pdf(E(r),w ��) � pdf(w �E(r), �)pdf(E(r) ��) gives us

pdf(w �E(r),�) �pdf(E(r),w ��)pdf(E(r) ��)

thus

pdf(w �E(r), �) �

k exp��w2

(G � �)��

2 �m/2

w(m�n)/2�1

� �m2 �pdf(�)

(A11)

k��

2 �m/2

2(m�n)/2 � �m � n2 �

(G � �)(m�n)/2 � �m2 � pdf(�)

using Equations (A6) and A7).Simplifying,

pdf(w �E(r), �) �

k� exp ��w2

(G � �)�w(m�n)/2�1G(m�n)/2

� �m � n2 �

(A12)

black litterman satchelland scowcroft.pdf

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