The Black-Litterman Approach:
Original Model and Extensions1
Attilio Meucci2
attilio_meucci@symmys.com
This version: October 12 2010
Last version available at http://ssrn.com/abstract=1117574
Abstract
We walk the reader through the Black-Litterman approach, providing all the
proofs. We show how minor modifications of the original model greatly improve
its range of applications. We discuss full generalizations of this and related
models. Code is available at MATLAB Central File Exchange.
JEL Classification: C1, G11
1A shorter version of this article appears as Meucci A., The Black-Litterman Approach:
Original Model and Extensions, The Encyclopedia of Quantitative Finance, Wiley (2010).
Please cite as such.
2The author gratefully acknowledges the very helpful feedback from Bob Litterman and
Jay Walters
1
1 Introduction
At a time when portfolio optimization used to take as inputs only the expec-
tations and the covariances of a set of assets computed from a given reference
econometric model, the pathbreaking technique by Black and Litterman (1990)
(BL in the sequel) provided a framework in which more satisfactory results could
be obtained from a larger set of inputs: view portfolios, the expected returns
on those portfolios, the confidence in the view portfolios and the uncertainty
on the reference model. Using BL, a portfolio manager could process those in-
puts, blend them into the reference return distribution, and obtain an optimal
allocation that reflected the views in a consistent way without corner solutions.
In Section 2 we review the original BL methodology: the reference model is
normal, centered around the CAPM equilibrium; the views are normal; and the
distribution that blends these two inputs is obtained analytically using Bayes’
formula.
In Section 3 we rephrase BL in terms of views on the market, instead of
the market parameters: this market-based version is more parsimonious and it
allows for the inclusion of scenario analysis as a special case.
In Section 4 we review the literature related to BL and its extensions: ranking
views, stress-test of correlations and volatilities, views on generalized risk factors
rather than returns, views on external factors that influence the p&l indirectly
through correlation, non-normal markets, multiple users, markets of complex
derivatives.
2 The original model
Here we follow He and Litterman (2002), see also Satchell and Scowcroft (2000),
Idzorek (2004) and Walters (2008).
The market model
We consider a market ofN securities or asset classes, whose returns are normally
distributed:
X ∼ N(µ,Σ) . (1)
The covariance Σ is estimated by exponential smoothing of the past return
realizations. To specify µ, BL acknowledge and address the issue estimation
risk: since µ cannot be known with certainty, it is modeled as a random variable
whose dispersion represents the possible estimation error. In particular, BL state
that µ is normally distributed
µ ∼ N(π, τΣ) , (2)
where π represents the best guess for µ and τΣ the uncertainty on this guess.
To set π, BL invoke an equilibrium argument. Assuming there is no estima-
tion error, i.e. τ ≡ 0 in (2), the reference model (1) becomes
X ∼ N(π,Σ) . (3)
2
Assume that, consistently with this normal market, all investors maximize a
mean-variance trade-off and that the optimization is unconstrained:
wλ ≡ argmax
w
{w0π − λw0Σw} . (4)
Setting to zero the derivative with respect to w of the term in curly brackets
we can solve explicitly this problem, obtaining the relationship between the
equilibrium portfolio ew which stems from an average risk-aversion level λ and
the reference expected returns:
π ≡ 2λΣew. (5)
Therefore, π can be set in terms of ew, where BL set exogenously λ ≈ 1.2.
Giacometti, Bertocchi, Rachev, and Fabozzi (2007) generalize this argument to
stable-distributed markets.
Notice that historical information does not play a direct role in the deter-
mination of π: this is an instance of the shrinkage approach to estimation risk.
Indeed, portfolio optimization is extremely sensitive to the input parameters, in
particular the expected values. In BL the risk drivers X represent the returns
of a broad market that are independently and identically normally distributed
across time as in (3). The standard estimator of the expectations in this context
is the sample mean
bµ ≡ 1
T
TX
t=1
Xt ∼ N
µ
π,
Σ
T
¶
, (6)
where T is the length of the available time series. The sample mean is a very
inefficient estimator: therefore, the "optimal" allocations based on (6) vary
wildly when different time series are fed into the allocation process, see Jobson
and Korkie (1980), Best and Grauer (1991), Green and Hollifield (1992), Chopra
and Ziemba (1993), Britten-Jones (1999) and Meucci (2005) for a review. One
way to cope with this issue is to use as in Stein (1955) more efficient "shrinkage"
estimators:
µ(s) ≡ (1− s) bµ+ sπ0, (7)
where π0 is a generic shrinkage target and 0 ≤ s ≤ 1 is the amount of shrinkage.
Therefore, the BL prior can be seen as an extreme case of shrinkage toward the
theoretical expectations (5) implied by equilibrium. Modifications of BL with
different targets were considered early on by the authors, see Section 4.
To calibrate the overall uncertainty level τ in the reference model, we can
compare the specification (2) with the uncertainty on the sample estimator (6)
and set
τ ≈ 1
T
. (8)
Satchell and Scowcroft (2000) propose an ingenious model where τ is stochastic,
but extra parameters need to be calibrated. In practice, a tailor-made calibra-
tion that spans the interval (0, 1) is called for in most applications, see also the
discussion in Walters (2008).
3
To illustrate, we consider the oversimplified case of an international stock
fund that invests in the following six stock market national indexes: Italy, Spain,
Switzerland, Canada, US and Germany. The covariance matrix of daily returns
on the above classes Σ is estimated as follows in terms of the (annualized)
volatilities σ ≈ (21%, 24%, 24%, 25%, 29%, 31%) and the correlation matrix
C ≈
⎛
⎜⎜⎜⎜⎜⎜⎝
1 54% 62% 25% 41% 59%
· 1 69% 29% 36% 83%
· · 1 15% 46% 65%
· · · 1 47% 39%
· · · · 1 38%
· · · · · 1
⎞
⎟⎟⎟⎟⎟⎟⎠
(9)
To determine the prior expectation π we start from the market-weighted portfo-
lio ew ≈ (4%, 4%, 5%, 8%, 71%, 8%)0 and obtain from (5) the annualized expected
returns π ≈ (6%, 7%, 9%, 8%, 17%, 10%)0. Finally, we set τ ≈ 0.4 in (2).
The views
A view is a statement on the market that can potentially clash with the reference
market model (1). For instance, the portfolio manager might say that the third
asset class will outperform the second, in which case the view is X3 −X2 ≥ 0.
Another view could be that the fourth asset class experiences twice to three
times the volatility predicted by the model: 4Σ44 ≤ Var {X4} ≤ 9Σ44.
BL consider views on expectations. In the normal market (1), this corre-
sponds to statements on the parameter µ. Furthermore, BL focus on linear
views: K views are represented by a K × N "pick" matrix P, whose generic
k-th row determines the relative weight of each expected return in the respective
view. In order to associate uncertainty with the views, BL use a normal model:
Pµ ∼ N(v,Ω) , (10)
where the meta-parameters v and Ω quantify views and uncertainty thereof
respectively.
If the user has only qualitative views, it is convenient to set the entries of v
in terms of the volatility induced by the market:
vk ≡ (Pπ)k + ηk
q¡
PΣP0
¢
k,k
, k = 1, . . . ,K, (11)
where ηk ∈ {−β,−α,+α,+β} defines "very bearish", "bearish", "bullish" and
"very bullish" views respectively. Typical choices for these parameters are α ≡ 1
and β ≡ 2. Also, it is convenient to set as in Meucci (2005)
Ω ≡ 1
c
PΣP0, (12)
where the scatter structure of uncertainty is inherited from the market volatil-
ities and correlations and c ∈ (0,∞) represents an overall level of confidence
4
in the views. Also, in order to assign a scale-independent, relative uncertainty
level to the different views, it is convenient to modify (12) as follows:
Ω ≡ 1
c
diag (u)PΣP0 diag (u) , (13)
where u ∈ (0,∞)K .
To continue with our example, the manager might assess two views: the
Spanish index will rise by 12% on an annualized basis, and the spread US-
Germany will experience a negative annualized change of 10%. Therefore the
pick matrix reads
P ≡
µ
0 1 0 0 0 0
0 0 0 0 1 −1
¶
; (14)
and the annualized views vector becomes v ≡ (12%,−10%)0. We set the uncer-
tainty in the views of the same order of magnitude as the market, i.e. c ≡ 1 in
(12).
The posterior
In Appendix 5.1 we show how to obtain the distribution of µ given the views
using Bayes’ formula:
µ|v;Ω ∼ N(µBL,ΣµBL) , (15)
where
µBL ≡
³
(τΣ)−1 +P0Ω−1P
´−1 ³
(τΣ)−1 π +P0Ω−1v
´
(16)
and
ΣµBL ≡
³
(τΣ)−1 +P0Ω−1P
´−1
. (17)
However, we are interested in the distribution of the risk factors X. To
compute this distribution we rewrite the reference model (1) asX d= µ+Z, where
Z ∼ N(0,Σ). Therefore the posterior market model is X|v;Ω d= µ|v;Ω+Z or
X|v;Ω ∼ N(µBL,ΣBL) , (18)
where µBL is defined in (16) and ΣBL follows from (17) by assuming that µ
and Z are independent:
ΣBL ≡ Σ+ΣµBL. (19)
As we prove in Appendix 5.2, an equivalent, computationally more stable rep-
resentation of the posterior parameters (16) and (19) reads:
µBL = π + τΣP
0 ¡τPΣP0 +Ω¢−1 (v−Pπ) (20)
ΣBL = (1 + τ)Σ− τ2ΣP0
¡
τPΣP0 +Ω
¢−1
PΣ. (21)
The normal posterior distribution (18) with (20) and (21) represents the modi-
fication of the reference model (1) that incorporates the views (10).
5
The allocation
With the posterior distribution it is now possible to set and solve a mean-
variance optimization, possibly under a set of linear constraints, such as bound-
aries on securities/asset classes, or a budget constraint. This quadratic program-
ming problem can be easily solved numerically. The ensuing efficient frontier
represents a gentle twist to equilibrium that reflects the views without extreme
corner solutions.
0 . 0 1 1 0 . 0 1 2 0 . 0 1 3 0 . 0 1 4 0 . 0 1 5 0 . 0 1 6 0 . 0 1 7 0 . 0 1 8
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1
r e f e r e n c e
0 . 0 1 3 0 . 0 1 4 0 . 0 1 5 0 . 0 1 6 0 . 0 1 7 0 . 0 1 8 0 . 0 1 9 0 . 0 2
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1
B L v i e w s
co
m
po
si
tio
n
co
m
po
si
tio
n
volatility
volatility
REFERENCE MODEL
POSTERIOR MODEL
Italy
Spain
Switz.
Canada
US
Germany
Italy
Spain
Switz.
Canada US
Germany
Figure 1: BL: efficient frontier twisted according to the views
In our example we assume the standard long-only and budget constraints, i.e.
w ≥ 0 and w01 ≡ 1. In Figure 1 we plot the efficient frontier from the reference
model (3) and from the posterior model (18). Consistently with the views, the
exposure to the Spanish market increases for lower risk values; the exposure to
Germany increases across all levels of risk aversion; and the exposure to the US
market decreases.
3 The market formulation
The BL posterior distribution (18) presents two puzzles.
On one extreme, when the views are uninformative, i.e. Ω→∞ in (10), one
would expect the posterior to equal the reference model (3), which we report
here:
X ∼ N(π,Σ) . (22)
6
This condition is important to ensure that, when the user does not have strong
views, the model does not depart from the reference prior. In BL, in this limit
the posterior becomes
X ∼ N(π, (1 + τ)Σ) . (23)
In other words, the covariance of the uninformative posterior model appears
distorted, unless τ ≡ 0. This result is actually fully consistent. Indeed, the
reference model (22) was derived by assuming no estimation risk, i.e. τ ≡ 0.
Without that assumption, the reference model would coincide exactly with the
uninformative model (23). However, the covariance of (23) is not as easy to inter-
pret because it contains, in addition to the pure volatility-correlation structure
of the market Σ, the additional term τΣ, which represents the estimation error
on µ.
On the other extreme, when the confidence in the views v is full, i.e. Ω→ 0,
one would expect the posterior to become the reference model (22) conditioned
on the specific views, which, as proved in Meucci (2005), is also normal:
X|v ∼ N(µ|v,Σ|v) , (24)
where
µ|v ≡ π +ΣP0 ¡PΣP0¢−1 (v −Pπ) (25)
Σ|v ≡ Σ−ΣP0 ¡PΣP0¢−1PΣ. (26)
The conditional distribution (24) is the core of scenario analysis: the user selects
a set of deterministic scenarios v ≡ (v1, . . . , vK)0 for the combinations of factor
realizations that are assumed to take place and analyzes their effect on the
reference model. Indeed, (25)-(26) generalize the classical regression-like result
utilized e.g. in Mina and Xiao (2001).
In BL, the full-confidence posterior becomes
X ∼ N
¡
µΩ=0BL ,Σ
Ω=0
BL
¢
, (27)
where
µΩ=0BL = π +ΣP
0 ¡PΣP0¢−1 (v−Pπ) (28)
ΣΩ=0BL = (1 + τ)Σ− τΣP0
¡
PΣP0
¢−1
PΣ. (29)
Therefore, the expectation of the full-confidence posterior (28) equals the con-
ditional expectation of scenario analysis (25), but the full-confidence posterior
covariance (29) never yields the conditional covariance of scenario analysis (26):
the volatility-correlation portion Σ of the covariance in (19) prevents the BL
full-confidence posterior covariance from ever becoming degenerate. Again, this
apparent paradox can be explained: in BL, the views (10) are expressed on the
parameter µ, not on the market X: therefore, the confidence in the views is
only supposed to affect the estimation risk part of the covariance in (23), not
its volatility-correlation component Σ.
7
To summarize, although the null- and full-confidence limits of the BL pos-
terior distribution are fully consistent, they might clash with the practitioner’s
intuition. To fix this problem we can proceed as in Meucci (2005) and rephrase
BL in terms of views directly on the market X, instead of the parameter µ.
We start from the reference model (1). However, we do not consider µ as
a random variable. Therefore we set µ ≡ π as in (3) to obtain the reference
model without estimation risk (22). The user has views on linear functions of
the market V ≡ PX, where P is a pick matrix as in (10). The distribution of
the views in general clashes with the distribution implied by the reference model
(22): as a random variable, the realization of V according to the views could
turn out larger or smaller than the realization of PX according to the reference
model. Therefore, the views V as a random variable are a perturbation of the
outcome implied by the reference model and as such they are modeled as a
conditional distribution
V|x ∼ N(Px,Ω) , (30)
where Ω represents the uncertainty on the views as in (10).
Once the model is set up, the user quantifies his views, choosing a specific
value v for the random variable V. As we show in Appendix 5.3, applying
Bayes’ rule as in the original BL we obtain the posterior distribution of the
market given the views, which is also normal
X|v;Ω ∼ N(µmBL,ΣmBL) , (31)
where
µmBL ≡ π +ΣP0
¡
PΣP0 +Ω
¢−1
(v−Pπ) (32)
ΣmBL ≡ Σ−ΣP0
¡
PΣP0 +Ω
¢−1
PΣ. (33)
These market-posterior formulas are very similar to their counterparts (20)-(21)
in the original BL model. However, in the first place the parameter τ in (2)
never appears here. Second, as it turns out, in the market based-specification
there is no need to add the original covariance to the posterior as in (19).
As a result, the market specification is consistent with both the reference
model and with scenario analysis. Indeed, it is immediate to check that if the
confidence is null, i.e. Ω→∞, then the posterior (31) equals the prior (22). At
the other extreme, if the confidence in the views is full, i.e. Ω → 0, then the
posterior (31) equals the conditional distribution (24).
X ∼ N(π,Σ) (no confidence: Ω→∞)
% prior
X ∼ N(µmBL,ΣmBL)
mkt-posterior &
X ∼ N(µ|v,Σ|v) (full confidence: Ω→ 0)
conditional
8
4 Related literature
Scenario analysis (24) allows the practitioner to explore the implications on a
given portfolio of a set of subjective views on possible market realizations, see
e.g. Mina and Xiao (2001).
BL adds uncertainty on the views by means of Bayesian formulas, although
this is not the standard Bayesian framework of decision theory, see Bawa, Brown,
and Klein (1979) and Goel and Zellner (1986). In particular, in BL there is no
historical updating of the estimate of µ, which does not depend on the time
series of the past returns.
Under the normality assumption, Qian and Gorman (2001) process views
on expectations of a set of portfolios and covariances of the same portfolios;
Pezier (2007) processes full and partial views on expectations and covariances
by least discrimination; Almgren and Chriss (2006) provide a framework to
express ranking, "lax" views on expectations.
The equilibrium-based prior expectations (5) in the original BL and in
the above literature appears to restrict the potential applications of these ap-
proaches to the tactical management of a global diversified fund. However, the
posterior formulas (20)-(21) and their modified versions (32)-(33), as well as
the formulas in the above literature can be applied to any normal distribution,
not necessarily equilibrium. Indeed, active management was among the first
applications of BL by their authors, where the prior expectation was assumed
null.
Accordingly, Meucci (2009) applies the above approaches to fully generic risk
factors that map non-linearly into the final p&l, instead of securities returns,
thereby handling views and stress-testing in derivatives markets.
Further generalizations of BL should handle non-normal reference risk mod-
els; non-linear views; views on generalized features such as medians, ranges,
tail behaviors, etc.; equality and inequality (ranking) views; and simultaneous
inputs from multiple users with different hierarchical confidence levels.
The Bayesian formalism of BL cannot achieve this: aside from the insur-
mountable computational problems when leaving the normal assumption, the
Bayesian approach in BL acts on the parameter µ of a normal distribution: this
is correct, because the practitioner wishes to express views on the expectations
of the market and, under the normal assumption, these are represented by µ.
However, in non-normal markets with views on features other than expectations
one should develop a different approach for each different possible market para-
metrization and for each possible feature on which the practitioner wishes to
express views.
Instead, it is more natural to input views directly on the market, instead of
the combinations of parameters that correspond to those features. The COP
approach in Meucci (2006) proceeds in this direction, although it does not cover
all the above desired applications. More importantly, the COP relies on ad-hoc
manipulations.
The entropy pooling approach in Meucci (2008) solves the above issues:
a posterior consistent with the most general views is obtained naturally by
9
entropy minimization, whereas opinion pooling accounts for different confidence
levels and multiple users. Furthermore, the posterior is represented in terms
of the same Monte Carlo scenarios as the reference model, but with different
probabilities: therefore, even the most complex derivatives can be handled, as
no costly repricing is ever necessary.
We summarize in the table below the capabilities of the original BL and its
market formulation in Section 3, Almgren and Chriss (2006), Qian and Gor-
man (2001), Pezier (2007), Meucci (2009), the COP in Meucci (2006) and the
entropy-pooling approach in Meucci (2008).
BL AC QG P M COP EP
normal market & linear views X · X X X X X
scenario analysis X · X X X X X
correlation stress-test · · X X X · X
trading desk: non-linear pricing · · · · X X X
external factors: macro, etc. · · · · X X X
partial specifications · · · X X · X
non-normal market · · · · · X X
multiple users · · · · · X X
non-linear views · · · · · · X
trading desk: costly pricing
本文档为【The Black-Litterman Approach Original Model and Extensions】,请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑,
图片更改请在作品中右键图片并更换,文字修改请直接点击文字进行修改,也可以新增和删除文档中的内容。
该文档来自用户分享,如有侵权行为请发邮件ishare@vip.sina.com联系网站客服,我们会及时删除。
[版权声明] 本站所有资料为用户分享产生,若发现您的权利被侵害,请联系客服邮件isharekefu@iask.cn,我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。