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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)
COMINVEST · BL · Folie 3 / 3 /
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
COMINVEST · BL · Folie 4 / 4 /
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
COMINVEST · BL · Folie 5 / 5 /
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
COMINVEST · BL · Folie 6 / 6 /
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, ...
Markowitz - Optimal Portfolios
Deficits of the mean/variance concept, solutions
COMINVEST · BL · Folie 7 / 7 /
The Markowitz Approach
§ 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 ?!
Markowitz - Optimal Portfolios
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.
COMINVEST · BL · Folie 8 / 8 /
§ 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 Markowitz Approach
Markowitz - Optimal Portfolios
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
COMINVEST · BL · Folie 9 / 9 /
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
COMINVEST · BL · Folie 10 / 10 /
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%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
retu
rn in
%p
.a.
historical return equilibrium return
COMINVEST · BL · Folie 11 / 11 /
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
COMINVEST · BL · Folie 12 / 12 /
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
COMINVEST · BL · Folie 13 / 13 /
Black-Litterman Approach
BL Optimization Problem
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 =⋅ )(
COMINVEST · BL · Folie 14 / 14 /
Black-Litterman Approach
BL Optimization Problem
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
COMINVEST · BL · Folie 15 / 15 /
Black-Littermann - Master formula
BL Optimization Problem
§ 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
COMINVEST · BL · Folie 16 / 16 /
Black-Litterman Approach - Remarks
BL Optimization Problem
- 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
COMINVEST · BL · Folie 17 / 17 /
Black-Litterman Approach
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.
COMINVEST · BL · Folie 18 / 18 /
Black-Litterman Approach
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%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
un
der
- /
ove
rwei
gh
t
COMINVEST · BL · Folie 19 / 19 /
Black-Litterman Approach
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%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
un
der
- /
ove
rwei
gh
t
COMINVEST · BL · Folie 20 / 20 /
Black-Litterman Approach
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 +=⋅ )(
COMINVEST · BL · Folie 21 / 21 /
Black-Litterman Approach
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
COMINVEST · BL · Folie 22 / 22 /
Black-Litterman Approach
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
COMINVEST · BL · Folie 23 / 23 /
Black-Litterman Approach
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.
COMINVEST · BL · Folie 24 / 24 /
Black-Litterman Approach
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
COMINVEST · BL · Folie 25 / 25 /
Black-Litterman Approach
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%“)
COMINVEST · BL · Folie 26 / 26 /
Black-Litterman Approach
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).
COMINVEST · BL · Folie 27 / 27 /
Black-Litterman Approach
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%
COMINVEST · BL · Folie 28 / 28 /
Black-Litterman Approach
Example: DJ STOXX
Correlation matrix of Dow Jones STOXX sectors
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
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 .
COMINVEST · BL · Folie 29 / 29 /
Black-Litterman Approach
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%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
asse
t w
eig
ht
in %
COMINVEST · BL · Folie 30 / 30 /
Black-Litterman Approach
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%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
retu
rn in
%p
.a.
historical return equilibrium return
COMINVEST · BL · Folie 31 / 31 /
Black-Litterman Approach
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%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
retu
rn in
%p
.a.
equilibrium return Black-Litterman return
COMINVEST · BL · Folie 32 / 32 /
Black-Litterman Approach
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%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
wei
gh
ts in
%
equilibrium weights Black-Litterman weights
COMINVEST · BL · Folie 33 / 33 /
Black-Litterman Approach
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.)
COMINVEST · BL · Folie 34 / 34 /
Black-Litterman Approach
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
COMINVEST · BL · Folie 35 / 35 /
Black-LittermanApproach
Example: DJ STOXX
§ Large changes inweights due to the“strong views“(change up to 35%pt)
Strong confidence
Equilibrium and Black-Litterman returns
0%
2%
4%
6%
8%
10%
12%
14%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
retu
rn in
%p.
a.equilibrium return Black-Litterman return
Equilibrium and Black-Litterman weights
-30%
-20%
-10%
0%
10%
20%
30%
40%
50%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
wei
ghts
in %
equilibrium weights Black-Litterman weights
COMINVEST · BL · Folie 36 / 36 /
Black-LittermanApproach
Example: DJ STOXX
§ Moderate changes inweights(change up to 14%pt)
Mid confidence
Equilibrium and Black-Litterman returns
0%
2%
4%
6%
8%
10%
12%
14%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
retu
rn in
%p.
a.equilibrium return Black-Litterman return
Equilibrium and Black-Litterman weights
0%
5%
10%
15%
20%
25%
30%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
wei
ghts
in %
equilibrium weights Black-Litterman weights
COMINVEST · BL · Folie 37 / 37 /
Black-LittermanApproach
Example: DJ STOXX
§ Weights close toequilibrium weights(change up to 3 %pt)
Poor confidence
Equilibrium and Black-Litterman returns
0%
2%
4%
6%
8%
10%
12%
14%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
retu
rn in
%p.
a.equilibrium return Black-Litterman return
Equilibrium and Black-Litterman weights
0%
2%
4%
6%
8%
10%
12%
14%
16%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
wei
ghts
in %
equilibrium weights Black-Litterman weights
COMINVEST · BL · Folie 38 / 38 /
Black-LittermanApproach
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
Equilibrium and Black-Litterman weights
0%
2%
4%
6%
8%
10%
12%
14%
16%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
wei
gh
ts in
%
equilibrium weights Black-Litterman weights
Equilibrium and naiv weights
-20%
-10%
0%
10%
20%
30%
40%
50%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
wei
gh
ts in
%equilibrium weights naiv weights
COMINVEST · BL · Folie 39 / 39 /
Black-LittermanApproach
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
Equilibrium and Black-Litterman weights
-15%
-10%
-5%
0%
5%
10%
15%
20%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
wei
gh
ts in
%
equilibrium weights Black-Litterman weights
Equilibrium and naiv weights
-60%-50%-40%
-30%-20%-10%
0%
10%20%30%
AUTO BANK BRES CHEM CONS CYCL CNYL ENGY FISV FBEV INDS INSU MEDA PHRM RETL TECH TELE UTLY
wei
gh
ts in
%equilibrium weights naiv weights
COMINVEST · BL · Folie 40 / 40 /
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
COMINVEST · BL · Folie 41 / 41 /
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
COMINVEST · BL · Folie 42 / 42 /
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
Some literature