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COMINVEST · BL · Folie 1

Consistent Asset Return Estimatesvia Black-Litterman

-Theory and Application

Dr. Werner Koch, CEFAPM Balanced / Asset Allocation

- June 2003 -

COMINVEST · BL · Folie 2 / 2 /

Consistent Asset Return Estimates via Black-Litterman

Abstract

In portfolio management, the forecast of asset returns is essential within the asset allocationprocess. In this context, the Black-Litterman approach (1992) yields consistent asset returnforecasts as a weighted combination of (strategic) market equilibrium returns and (tactical)subjective forecasts (”views”). The Black-Litterman formalism allows to implement both absoluteviews (return levels) and relative views (outperforming vs. underperforming assets) for selectedassets for which a “strong” opinion exists. For any particular view, individual confidence levels forthe return estimates have to be specified. The formalism spreads these informations consistentlyacross all assets in the portfolio. The BL-revised returns can then serve as an input for mean-variance-optimisation procedures, thus allowing for the implementation of additional constraints.It turns out that BL-optimized portfolios overcome some well-known Markowitz insufficiencieslike unrealistic sensitivity to input factors or extreme portfolio weights. The BL-process will beintroduced both from its theoretical background and its implementation in practice.

Dr. Werner Koch Frankfurt, June 2003(COMINVEST)

[email protected]


Black-Litterman

§ Markowitz Theory Risk/Return Portfolio Sensitivity

§ Extending Markowitz Theory Concept of Equilibrium Returns

§ Black-Litterman Master Formula

Views & Degree of ConfidenceExample

Major Aspects Considered


On the Added Value of Theories...

Question:

What is the striking difference between option-pricing theory and portfolio theory ?

Answer:Black-Scholes-Theory is well established in practice

whereas

Markowitz-Theory is widely ignored in portfolio management - why ?

Quantitative approaches in practice


The Markowitz Approach - revisited

§ Starting point in a world of normally distributed returns:Any (risky) asset can be described by the first twomoments of return - its mean and variance.§ In an efficient portfolio the assets are weighted such that

for a given level of risk a maximum return is achieved(equivalently: for given return the risk is minimized).§ All efficient portfolios form the efficiency line 1. It starts in

the minimum variance portfolio and ends in the maximumreturn portfolio (which is the asset of maximum return).§ If a risk-free asset exists, all efficient portfolios lie on the

efficiency line 2, starting at the risk-free asset and tangentiallytouching efficiency line 1 of risky portfolios. Efficient portfolios on efficiency line 2 are acombination of the risky market portfolio and the risk-free asset (risk-free long or short ).

Markowitz - Optimal Portfolios

Efficient portfolios in the mean-variance-approach

Return

Risk

MinVar

MaxRet

MarketPortfolio

EffLine 2

EffLine 1

Risk-Free

Assets


The Markowitz Approach

Deficit Solution

High sensitivity on inputs (return estimates!) leads Black-Litterman

to large weight fluctuations in the optimal portfolio.

„Corner solutions“: Extreme portfolio weights (also in the case Black-Litterman

of optimization algorithms using constraints!)

Aggregation of information: Consistent aggregation of huge number Black-Litterman

of estimated returns overburdens the investment process

No quantification of confidence in estimated returns Black-Litterman

One-periodical approach Scenario via VAR

„Variance“= restriction of risk to (symmetrical) return volatility, Scenario via VAR

no alternative risk concept (VaR, SaR).

Requires ex-ante-estimates of covariance matrix GARCH, ...


Deficits of the mean/variance concept, solutions



§ Let return expectations be shifted by +2%pts for BANK, CNYL and MEDA.(here, for illustration, using the historical returns of STOXX sectors)

§ Only slight variations of estimated returns lead to huge unrealistic shifts in portfolio weights!§ Problems: Communication of results, re-allocation in real portfolios, acceptance of method ?!


Sensitivity of weights on changes in return estimates

Changes in portfolio weights due to changes of expected returns(18 historical returns (ave=13.3%) with 3 returns changed by +2%pts)

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY

wei

gh

t ch

ang

es in

%p

ts.


§ Markowitz Theory relates risk & return

§ Optimization problem:

§ Solution: Optimal portfolio weights (no constraints):

§ Markowitz provides a mechanism towards optimal portfolios - what about the input factors ???



The formal approach for risky assets, basic outline.

max 2

TT

w

www →

Ω⋅−Π

γ

( ) ΠΩ⋅= −∗ 1 γw

Π = vector of returnsΩ = covariance matrixγ = risk aversion parameterw = vector of weights


Extending the Markowitz Approach

Supply & demand§ Traditional approach of maximum return & minimum risk is demand-side perspective.§ Need to balance with supply-side...

Concept of equilibrium returns:§ The market portfolio exists in market equilibrium, i.e. supply & demand are in equilibrium.§ Therefore, equilibrium returns reflect neutral „fair“ reference returns.

§ Inverse optimization yields:

Conclusion§ Use of equilibrium returns as a long term strategic reference for any return estimate

(„market neutral starting point“).

Extending Markowitz

Equilibrium returns

( ) MCap wΩ⋅=Π γ


Extending the Markowitz Approach

Extending Markowitz

Equilibrium (or implicit) returns for the STOXX sectors

§ These implicit returns serve as reference returns for the following investigations(„market neutral starting point “)§ Note that equilibrium returns are just calculated; they do not require any estimation procedure.

Historical and equilibrium returns

0%

5%

10%

15%

20%

25%

30%


retu

rn in

%p

.a.

historical return equilibrium return


Black-Litterman Approach - Basic Steps

Black-Litterman Process

BL-Prozess provides a mechanism to merge long term equilibrium returns with short term return estimates, i.e. with tactical deviations from equilibrium returns.

Market Weights

Strategic Weights

Equilibrium Returns

Implicit Returns

Subjective Return Estimates

Specification of Confidence

BL-Revised Consistent Expected Returns

BL-Revised Portfolio Weights


Black-Litterman Approach

BL Optimization Problem

Optimization s.t. contraints

§ Determine the optimal estimate for E(R), which minimizes the variance of E(R) around

equilibrium returns Π:

s.t.

where: P· E(R) ~ N(V,Σ ) , Σ ii = ei

[ ] ( ) [ ] min )( )( 1T

E(R)RERE →Π−⋅Ω⋅Π− −τ

+=⋅

Viewsuncertain Viewscertain

)(eV

VREP




Equations for optimal BL-return estimates

§ Solution in the case of certain estimates (Σ ≡ zero matrix):

§ Solution in the case of uncertain estimates (Σ = diagonal matrix):

§ The constraints are implicitly fulfilled.

( ) ( )Π−⋅Ω⋅Ω+Π=−

) ( ) ()(1TT PVPPPRE ττ

( )[ ] ( )[ ]VPPPRE 1T111T1 )( −−−−− Σ+ΠΩ⋅Σ+Ω= ττ

VREP =⋅ )(




Formal proof for the case „certain estimates“

Proposition: The optimization problem s.t.

yields variance-minimum returns

Proof:

[ ] ( ) [ ] min )( )()(

1T

RERERE →Π−⋅Ω⋅Π− −τ VREP =⋅ )(

[ ] ( ) [ ]

◊

=−=∂∂

=−ΠΩ−Ω=∂∂

⇒

Ω⋅=Ω+Ω=

Ω

∂∂

=∂∂

⇒

=∂∂

=∂∂

⋅Π−⋅Ω⋅Π−=

−−

−−−−−−−−

−

∑∑∑∑

(1). Eq. with followsresult ,for expression yields Eq.(2)in insert , for (1) Eq. Solve

0 )2(

0 ) (2 ) (2 )1(

2 .).(

0 and 0 :sf.o.c.'

)( - : :function Lagrange

11

111111

,

11

1T

λ

λττ

ττττ

λτ

E

VPE?L

PEEL

EEEEEEE

Lge

?L

EL

PE-VEEL

kk

ikjj

jikk

ikkj

kjkj

ii

( ) ( )Π−⋅Ω⋅Ω+Π= − ) ( ) ()( 1TT PVPPPRE ττ


Black-Littermann - Master formula


§ First impression: Formula looks rather complex...§ Very strong interdependencies between market and subjective return expectations§ First factor: Normalisation („Denominator“),

§ Second factor („Numerator“): Balance between returns Π (= equilibrium returns) and V (= estimates), where the prefactors (τ Ω)-1 and P T Σ-1, resp., serve as weighting factors.

§ Furthermore: Scaled covariance of returns, τ Ω Matrix P, relating Views to expected returns

Matrix Σ = covariance of estimated Views

§ Limiting case 1: no estimates ⇔ P=0: E(R) = Π i.e. BL-returns = equil. returns.

§ Limiting case 2: no estimation errors ⇔ Σ−1→ ∞: E(R) = P -1 V i.e. BL-returns = View returns.

( )[ ] ( )[ ]VPPPRE 1T111T1 )( −−−−− Σ+ΠΩ⋅Σ+Ω= ττ


Black-Litterman Approach - Remarks


- Use of CAPM to determine equilibrium returns- BL-returns as a Bayesian a-posteriori estimator

CAPM

§ Instead of inverse optimization: derive equilibrium returns ΠΕq from CAPM§ Additional input for CAPM: Risk-free rate rf , Risk premium vs market (M), Beta coefficients§ Evaluation:

(fair return for asset i )

Bayes§ Bayes Theorem (“Law“ or “Rule“) states how to determine conditional expectations.§ Given an a-priori known distribution of a random variable. Adding new information leads to a revised conditional distribution, the so-called a-posteriori distribution („learning“).

BL-return estimates are a-posteriori (multivariate) normally distributed return expectations.

( )2, withM

iMifMifEqi rrr

σσ

ββ =−⋅+=Π



BL-Views

Views

§ Individual Views, i.e. return estimates differing from the strategic equilibrium returns arean essential input to the BL estimation process.

Specification of Views§ ... as absolute return expectations for individual assets

and / or§ ... as relative return expectations relating a number of assets or aggregates of assets.

Formal constraint: #Views ≤ #Assets.

Confidence§ Each View has to be assigned a level of confidence.

Selective Views§ Views are restricted to those assets for which in-depth analysis is available.



BL-Views

Relative Views

§ A relative View can be formulated as follows: „The sectors Pharmacy and Industry willoutperform Telecom and Technology by 3% with a confidence of 90% for ±1%“:

§ Thus: A long-portfolio with outperformers, a short-portfolio with underperformers.

[ ][ ] 2%)61.0(%3)()(

)()(

+=⋅+⋅−

⋅+⋅

TECHTECHTELETELE

INDUINDUPHRMPHRM

REwREw

REwREw

Views: relative changes in return estimates

-4,0%

-3,0%

-2,0%

-1,0%

0,0%

1,0%

2,0%

3,0%

4,0%


un

der

- /

ove

rwei

gh

t



BL-Views

Absolute Views

§ An absolute View can be formulated as follows: „The sektor of non-cyclical goods willperform better than stated by the equilibrium return of 6.66%. Our target return is 7.5% with

90% of confidence for an interval of ±1.5%“:.

2%)91.0(%5.7)(1 +=⋅ CNYLRE

Views: absolute return estimates

0,0%

1,0%

2,0%

3,0%

4,0%

5,0%

6,0%

7,0%

8,0%


un

der

- /

ove

rwei

gh

t



BL-Views

Aggregation of Views

§ Relative and absolute views form a system of linear equations as a constraint to theoptimization problem:

where (k = #Views and n = #Assets, with k ≤ n):

E(R) = n×1 vector of expected asset returns, unknownP = k×n matrix, representing the ViewsV = k×1 vector, absolute / relative return expectations (i.e., levels or over-/underperforming)e = k×1 vector of squared StDev‘sΣ = k×k diagonal matrix of confidence (assuming independent estimation errors), Σ ii = ei

eVREP +=⋅ )(



BL-Views

Combined Views

§ Combining the aforementioned Views using P⋅ E = V + e , we get:

§ with

+

=

⋅ 2

2

%)91.0(%)61.0(

%5.7%3

)(

)(

UTLY

AUTO

RE

REP M

=

=

0 0 1 0 00 0.49- 0.51- 0 0.66 00 0.34 00

.,2 . ,1

LLLLLL

absViewrelView

P

long positions short positions

asset selection for absolute estimate



BL-Views

Remark: Technical note on „confidence“

§ Comment on e (outlined for the relative return estimate): The fact that the amount ofrelative outperformance of 3% ±1% is assigned a 90% probability is interpreted within a

normal distribution, with mean = µ = 3% and variance = VAR = σ2 = (0.61%)2 ≡ e.

σ = 0.61%

2% 3% 4%

± 1.664 σ



Parameters in BL

Remark: Risk aversion parameter γ

§ Related to underlying investment universe

Satchell & Scowcroft and Best & Grauer:

Let γ = (rM-rf) / σM2

where (a) σM = StDev(Market Portfolio) or (b) σM

2 = wT Ω w , w = market cap.

Zimmermann et al.: (a) σM=16.95%p.a. from STOXX-Data (own calculation) Let γ = 3, which corresponds to a risk premium of 8.6%.

(b) Using wT Ω w yields σM=16.85%p.a. ⇒ risk premium=8.5%.

Idzorek: (DJIA, USA): Risk premium=7.5%: (a) γ = 2.05 (b) γ = 2.25.



Parameters in BL

Remark: Skaling parameter τ for the covariance matrix

Results/Setting

§ Covariances of expected returns proportional to historical covariances, via τ .

§ τ measures confidence in benchmark, i.e. balances between BM and Views.

§ τ small: VAR[E(R)] << VAR[historical returns]

§ τ = 0.3 “plausible“ (used for numerical evaluations throughout).

Sensitivity of BL-optimal portfolio weights on choice of τ

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

0,005 0,010 0,025 0,050 0,075 0,100 0,200 0,300 0,400 0,500 0,600 0,700 0,800 0,900 1,000

τ

po

rtfo

lio w

eig

ht



Parameters in BL

Additional remarks on the recent remarks

§ Calibration problems with parameter τ (“plausible“, “adjusted to IR=1“, ... )

§ Calibration problems with parameter γ (“world wide risk aversion“, ... )

§ Calibration problems with expressing the degree of confidence in various

publications (“1..3“, “0..100%“)



Example: DJ STOXX

„Sector allocation, Dow Jones STOXX“

§ Presentation based on: Chapter 10 from „Global Asset Allocation“, H.Zimmermann, W.Drobetz, P.Oertmann, 2002 „Integrating tactical and equilibrium portfolio management -

Putting the Black-Litterman model at work“.§ Notation, scenarios and data have been used, some data are missing.

§ Missing data - volatilities and covariances - had to be calculated, causing some deviations inthe numerical results between this presentation and Chapter 10 .Nevertheless, all relevant results are reproduced.

§ All calculations can be / have been implemented and performed in Excel (TM).



Example: DJ STOXX

Sectors in the Dow Jones STOXX index

§ Monthly returns in Sfr (Swiss francs), period: 06/1993 - 11/2000, annualized data.

Historical Data - Risk/Return Characteristics

UTLY

TELE

RETL

PHRM

MEDA

INSU

INDS

FBEV

FISV

ENGYCNYL

CYCLCONS

CHEM

BRES

BANK

AUTO

TECH

0%

5%

10%

15%

20%

25%

30%

14% 16% 18% 20% 22% 24% 26% 28% 30%

Risk

Ret

urn

Asset hist.Return hist.Volatility MarketCap

total: average: total:18 16,22% 100,01%

AUTO 8,32% 25,09% 1,65%BANK 15,14% 21,21% 15,04%BRES 7,31% 23,56% 1,22%CHEM 12,25% 20,81% 1,80%CONS 6,56% 18,92% 1,26%CYCL 5,24% 19,94% 2,85%CNYL 11,80% 16,66% 2,90%ENGY 14,92% 20,72% 10,30%FISV 13,01% 20,91% 4,12%FBEV 10,47% 15,72% 4,59%INDS 13,45% 19,35% 5,19%INSU 17,43% 19,68% 6,89%

MEDA 14,63% 25,17% 3,27%PHRM 22,83% 16,20% 10,24%RETL 9,49% 16,16% 2,27%TECH 25,95% 28,60% 11,03%TELE 18,99% 25,69% 10,56%UTLY 11,77% 17,25% 4,83%



Example: DJ STOXX

Correlation matrix of Dow Jones STOXX sectors


AUTO 100% 74% 73% 83% 78% 75% 73% 55% 73% 71% 79% 72% 46% 43% 68% 69% 65% 64%BANK 74% 100% 63% 73% 74% 75% 71% 59% 92% 75% 75% 87% 39% 63% 62% 67% 59% 64%BRES 73% 63% 100% 83% 81% 78% 60% 66% 69% 56% 78% 56% 44% 31% 59% 62% 52% 41%CHEM 83% 73% 83% 100% 85% 82% 72% 67% 72% 74% 82% 70% 51% 45% 69% 65% 57% 57%CONS 78% 74% 81% 85% 100% 90% 72% 66% 75% 76% 89% 64% 54% 39% 67% 66% 63% 64%CYCL 75% 75% 78% 82% 90% 100% 67% 64% 79% 70% 87% 63% 58% 43% 67% 70% 63% 56%CNYL 73% 71% 60% 72% 72% 67% 100% 55% 69% 75% 69% 74% 41% 58% 73% 53% 59% 71%ENGY 55% 59% 66% 67% 66% 64% 55% 100% 59% 59% 59% 54% 28% 43% 55% 43% 32% 46%FISV 73% 92% 69% 72% 75% 79% 69% 59% 100% 75% 73% 85% 39% 61% 58% 64% 57% 58%FBEV 71% 75% 56% 74% 76% 70% 75% 59% 75% 100% 62% 74% 27% 63% 61% 40% 41% 66%INDS 79% 75% 78% 82% 89% 87% 69% 59% 73% 62% 100% 65% 72% 38% 68% 82% 77% 67%INSU 72% 87% 56% 70% 64% 63% 74% 54% 85% 74% 65% 100% 36% 67% 61% 60% 56% 68%MEDA 46% 39% 44% 51% 54% 58% 41% 28% 39% 27% 72% 36% 100% 21% 42% 75% 77% 57%PHRM 43% 63% 31% 45% 39% 43% 58% 43% 61% 63% 38% 67% 21% 100% 43% 35% 37% 58%RETL 68% 62% 59% 69% 67% 67% 73% 55% 58% 61% 68% 61% 42% 43% 100% 52% 53% 57%TECH 69% 67% 62% 65% 66% 70% 53% 43% 64% 40% 82% 60% 75% 35% 52% 100% 81% 55%TELE 65% 59% 52% 57% 63% 63% 59% 32% 57% 41% 77% 56% 77% 37% 53% 81% 100% 70%UTLY 64% 64% 41% 57% 64% 56% 71% 46% 58% 66% 67% 68% 57% 58% 57% 55% 70% 100%

§ Calculation based on monthly returns

§ Covariance matrix via σij = σi σj ρij .



Example: DJ STOXX

A (very) first approach: Average return for all assets

§ Use the average historical return Raverage=16.22% as expected return across all assets.§ Calculate the optimal portfolio weights (no constraints) via

( ) averageaverage Rw ⋅Ω⋅= −1τ

Unconstrained weights for equal average return of 16,22%

-40%-30%

-20%-10%

0%10%

20%30%

40%50%

60%


asse

t w

eig

ht

in %



Example: DJ STOXX

BL-starting point: Equilibrium - (implicit) - returns of Market Portfolio

§ Calculation of equilibrium returns on the basis of the Market Portfolio (“inverse optimization“):

( ) PortfolioMarketmEquilibriumEquilibriumEquilibriu wwwR where =⋅Ω⋅= τ

Historical and equilibrium returns

0%

5%

10%

15%

20%

25%

30%


retu

rn in

%p

.a.

historical return equilibrium return



Example: DJ STOXX

From equilibrium returns to Black-Litterman returns (Views given on p.18-19)

Returns s.t. Views:§ Short portfolio Return expectations significantly lowered for TECH und TELE.§ Long portfolio Return expectation higher in PHRM but lower in INDS (ok, because the

relative View „INDS better than TELE und TECH“ remains intact !)§ For CNYL, expected return shifts from 6.66% to 7.48% (90% confidence in estimate of 7,5%).§ Via correlations (e.g.) MEDA (75% to TECH, 77% to TELE) has significantly lower return.

Equilibrium and Black-Litterman returns

0%

2%

4%

6%

8%

10%

12%

14%


retu

rn in

%p

.a.

equilibrium return Black-Litterman return



Example: DJ STOXX

Comparing weights based on equilibrium returns and Black-Litterman returns

§ Short portfolio: Significant weight reduction for TELE and TECH§ Long portfolio: Significant weight increase for INDS and PHRM§ Furthermore: Higher return expectation (+0.84%pts) leads to higher weight for CNYL.

Equilibrium and Black-Litterman weights

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%


wei

gh

ts in

%

equilibrium weights Black-Litterman weights



Example: DJ STOXX

Influence of degree of confidence on BL-returns and BL-weights, I

§ Focus on asset „CNYL“§ Implicit return: 6.66%§ View (Return estimate): 7.5% ± 1.5%

Observations:§ Low confidence: BL lowers the return

estimate, down to 5.1%.§ Moderate/higher confidence: Asymptotic

approach to the estimated value of 7.5%with increasing confidence.§ Limit: At a confidence level of 100%, BL fully accepts the strong view of 7.5%.

§ Weights: from 2.9% (=market cap, due to confidence of 0% de facto no view),

up to 25% (overweighting due to the strong view confidence of 100%).

CNYL: Return estimates vs. return target s.t. various confidence levels

5,0%

5,5%

6,0%

6,5%

7,0%

7,5%

8,0%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

confidence level

exp

ecte

d r

etu

rn

0%

5%

10%

15%

20%

25%

30%

po

rtfo

lio w

eig

ht

expected return target: 7,50% implicit: 6,66% weight (r.h.)



Example: DJ STOXX

Influence of degree of confidence on BL-returns and BL-weights, II

Observations:§ For low degrees of confidence the

BL-weights converge to the weightsin equilibrium (=market cap‘s).§ Weights approach equilibrium

values from either underweighting(short) or overweighting (long) path.§ Significant weight changes for the

assets with View.

Sensitivity of weights on degree of confidence

-30%

-20%

-10%

0%

10%

20%

30%

40%

high mid low

confidence

wei

gh

t dif

fere

nti

al to

eq

uili

bri

um


Black-LittermanApproach

Example: DJ STOXX

§ Large changes inweights due to the“strong views“(change up to 35%pt)

Strong confidence


0%

2%

4%

6%

8%

10%

12%

14%


retu

rn in

%p.

a.equilibrium return Black-Litterman return


-30%

-20%

-10%

0%

10%

20%

30%

40%

50%


wei

ghts

in %




Example: DJ STOXX

§ Moderate changes inweights(change up to 14%pt)

Mid confidence


0%

2%

4%

6%

8%

10%

12%

14%


retu

rn in

%p.



0%

5%

10%

15%

20%

25%

30%


wei

ghts

in %




Example: DJ STOXX

§ Weights close toequilibrium weights(change up to 3 %pt)

Poor confidence


0%

2%

4%

6%

8%

10%

12%

14%


retu

rn in

%p.



0%

2%

4%

6%

8%

10%

12%

14%

16%


wei

ghts

in %




Example: DJ STOXX

Only one View for CNYL:Return: from 6.6% to 7.5%(with strong confidence)

Result§ Realistic weight changes

in BL§ Volatile weight scenario

in “naiv“ approach

- compared to“naiv“ mean/variance


0%

2%

4%

6%

8%

10%

12%

14%

16%


wei

gh

ts in

%


Equilibrium and naiv weights

-20%

-10%

0%

10%

20%

30%

40%

50%


wei

gh

ts in

%equilibrium weights naiv weights



Example: DJ STOXX

Only one View for CNYL:Return: from 6.6% to 5.5%(with strong confidence)

Result§ Realistic weight changes

in BL§ Volatile weight scenario

in “naiv“ approach

- compared to“naiv“ mean/variance


-15%

-10%

-5%

0%

5%

10%

15%

20%


wei

gh

ts in

%


Equilibrium and naiv weights

-60%-50%-40%

-30%-20%-10%

0%

10%20%30%


wei

gh

ts in

%equilibrium weights naiv weights


Black-Litterman Approach - Conclusion I

Key Features

Traditional vs BL approach

simple approach Black-Litterman

§ Return estimates: for each asset for selected assetsassumed as certain degree of confidenceabsolute returns relative or absolute estimatessubjective consistent

§ Reference return none equilibrium returns


Black-Litterman Approach - Conclusion II

Key Features

BL within an improved portfolio construction process

Research Opinion

Views & Confidence

BL-Returns

BL-Weights(no Constraints)

(BL-) Weights viaMean-Variance-Optimizer

Subject to Constraints

Market

Key elements of the traditional process

Black-Litterman sub-process

BL-tactical level

BL-strategic level


Black-Litterman ApproachSuggestions for further reading

§ Black F. und Litterman R.: Global Portfolio Optimization, Fin.Analysts Journal, Sep.1992, p.28-43§ Black F. und Litterman R.: Asset Allocation: Combining investor views with market equilibrium,

Goldman-Sachs, Fixed Income Research, Sep.1990§ Zimmermann H., Drobetz W. und Oertmann P.: Global Asset Allocation: New methods and

applications, publ. by. Wiley & Sons, Nov.2002§ Drobetz T.: Einsatz des BL-Verfahrens in der Asset Allocation, Working paper, Mar.2002

www.unibas.ch/wwz/finance/publications/researchpapers/3-02%20Black-Litterman.pdf

§ Christodoulakis G.A.: Bayesian optimal portfolio selection: The BL approach, Nov.2002www.staff.city.ac.uk/~gchrist/Teaching/ QAP/optimalportfoliobl.pdf

§ He Q. und Litterman R.: The intuition behind BL-model portfolios, Dec.1999faculty.fuqua.duke.edu/~charvey/Teaching/BA453_2003/GS_The_intuition_behind.pdf

§ Idzorek T.: Step-by-step guide to the BL-model, Feb.2002faculty.fuqua.duke.edu/~charvey/Teaching/BA453_2002/How_to_do_Black_Litterman.doc

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