Proceedings of World Business Research Conference
21 - 23 April 2014, Novotel World Trade Centre, Dubai, UAE, ISBN: 978-1-922069-48-1
Optimization of Black-Litterman Model for Portfolio Assets
Allocation
A. Hidalgo, A. Desportes, E. Bonin+, A. Kadaoui and T. Bouaricha
Present paper is concerning portfolio management with Black-Litterman (B-L) model.
Considered stocks are exclusively limited to large companies stocks on US market.
Results obtained by application of the model are presented. From analysis of
collected Dow Jones stock data, remarkable explicit analytical expression of optimal
B-L parameter
which scales dispersion of normal distribution of assets mean
return, is proposed in terms of standard deviation of covariance matrix.
Implementation has been developed in Matlab environment to split optimization in
Markovitz sense from specific elements related to B-L representation.
Keywords: Black-Litterman, Markowitz, Market Data, Portfolio Manager Opinion
Introduction
The seventies have withstood very large financial uncertainties related to inflation, interest
rates, volatility and change rate instability, later followed by strong financial markets growth very
favorable for investments. “Modern” portfolio theory was limited until the nineties to mechanical
construction of the “efficient boundary” representing the optimum balance between risk and
return on investment. To ease the work of portfolio managers, various “rational” methods have
been designed for return optimization [3]. Typically since the future dividends of most securities
are unknown, it is claimed that the value of a security should be the net present value of future
expected returns (ER). Another important element is that investors should take into account the
co-movements represented by co-variances of assets. The idea is that if investors take covariances into consideration, then they can construct portfolios that generate higher ER at the
same (or lower) level of risk than portfolios ignoring the co-movements of asset returns. The
risk here is as usual assessed as the portfolio variance, which depends both on assets variance
in the portfolio and on co-variances between assets. This mean-variance portfolio model is the
base on which much researches about portfolio theory are performed.
It rapidly came out however from observation of application cases that if mathematically correct,
the parameters implied in formulae were still left to specific evaluations outside the model. In
consequence, according to [2,3], optimizers in [1] are in fact maximizing errors. Since there are
no correct and exact estimates of either ER or variances and co-variances in the models, these
estimates are subject to estimation errors. Typically, Markowitz optimizers overweight securities
with high ER and negative correlation and underweight those with low ER and positive
correlation. These securities are most prone to be subject to large estimation errors [2]. The
argument appears however somewhat contradictory. The reason for investors to estimate high
ER on an asset is their belief that it will indeed generate high return. It then seems reasonable
that a manager would appreciate that the model over-weights this asset in the portfolio (taking
co-variances into consideration).
_________________________________________________________________
Undergraduate Students, ECE Paris School of Engineering, France.
+Corresponding Author: ebonin@ece.fr
Proceedings of World Business Research Conference
21 - 23 April 2014, Novotel World Trade Centre, Dubai, UAE, ISBN: 978-1-922069-48-1
Furthermore, to balanced portfolio in previous Markowitz model, B-L model has been bringing a
new component containing investor anticipations. Main advantage of this association is to be a
natural place where investor can input his experience and personal vision, but still keep the
control of assets weighting in the portfolio.
Portfolio Models
1-Markovitz Mean-Variance Model : In a seminal paper on quantitative portfolio selection [1,4], H. Markowitz
identified the forecasted return with expected return and risk with variance of return (VR). Postulating that this
objective is the one to strive for, he developed a mathematical model for portfolio selection. In terms of variance,
the objective is to minimize VR for a certain level of expected return. As expected value is often called the mean
value, this kind of optimization is called mean-variance (M-V) optimization. So a portfolio can be defined as
efficient when it minimizes variance for a given level of ER, or equivalently it maximizes ER for a given level of
variance. After weights selection, portfolio variance and ER can be computed. Risk and portfolio ER are then
drawn in a risk-return diagram, see Figure 1 where three assets (equity, bonds and cash for instance) have been
considered. The limiting curve of efficient portfolios (the efficient frontier) is a tipped parabola [2]. No portfolio
can exist beyond the efficient frontier with smaller variance for the ER level. The top of the parabola corresponds
to the portfolio having the minimal variance over all possible combinations in considered investment universe.
Portfolios corresponding to risk-averse investor preferences are located on top half of the parabola. These
portfolios all share the characteristic that they have maximal expected return for a given level of variance.
Figure 1 : Efficient Frontier of a Three-asset Portfolio in VR-Expected Return Plane
2-Black and Litterman (B-L) Model : The MV model was a groundbreaking mathematically rigorous [3] model in
portfolio selection by maximizing ER and minimizing risks [7]. It allows investors to quantitatively select a
portfolio on the basis of their views. However the model is almost unusable as results are often extreme.
Practitioners who did use this model were working around the difficulties by adding many constraints. In this
context B-L did set out another approach to accomplish more intuitive portfolios by computing a better estimate
for the ER vector. This vector could then be directly used to compute the portfolio weights, or ER vector could be
fed to a mean-variance optimizer for providing a solution to a constrained optimization problem.
BL identified two sources of information about the expected returns and they combined these two sources in a
single ER formula. The first source of information is a quantitative one represented by the ER following from the
CAPM (Capital Asset Pricing Model [5]), and which should hold if the market is in equilibrium [6]. The CAPM
returns form a backbone to the process, and are used to dampen the possibly extreme views of the second source of
information. This is the base of B-L model [11].
The second source of information consists in the “views” held by investment manager. He has access to different
information and could therefore have different opinions about the ER of each asset than those that would hold in
equilibrium. Investor “views” are used to tilt equilibrium “views”: they provide information to invest less or more
in some assets than it would follow from equilibrium “views”. Combination of these two sources of information
results in a new ER vector. This improved ER vector can then be used in portfolio optimization process.
Proceedings of World Business Research Conference
21 - 23 April 2014, Novotel World Trade Centre, Dubai, UAE, ISBN: 978-1-922069-48-1
Mathematical Aspects of B-L model
Mathematically, the main challenge is to combine the two separate sources of information into one ER vector. This
should be done in such a way that mathematics remains tractable and parameters are intuitive to the user [8]. One
could combine heuristically the ER of neutral reference points with investor “views”. If positive about an asset,
then one should simply increase the weight of the asset, and conversely for assets with negative “view”. The
question then becomes how much to increase it. Furthermore, assets are correlated: if one asset is expected to
perform well and therefore its weight is increased, then the weights of other positively correlated assets should also
be increased. It would be very cumbersome to do this all by hand. A more constructive approach is needed. B-L
combined these two separate sources of information in a constructive manner and suggested two methods to
accomplish this. First, the mixed estimation method of Theil (1971), related to generalized least square method,
will be used to estimate dependent parameters. Secondly, B-L suggests that the new ER vector should be assumed
to have a probability distribution product of two normal distributions [18]. A Bayesian approach is proposed in
[13] to accomplish this blending of probability distributions.
1)-A portfolio consisting of n assets, is represented mathematically by a vector
with ∑
Let the
* ( )+
return of an asset be denoted
and ER of asset is noted ( ) ER vector is then ( )
(
), The assets covariance in the portfolio are stored in the covariance matrix
Denote by
the covariance between asset and asset [9]. Finally, the return of portfolio is determined by assets return in
∑
the portfolio itself,
, and ER of the portfolio can be expressed as
(∑
( )
)
∑
(
)
∑
( )
( )
(1)
Besides, the variance of return of the portfolio can also be computed as
(∑
( )
∑
)
∑
∑
∑
(
(
)
)
∑
∑
(2)
The ER of equilibrium portfolio is then evaluated
(3)
where
is the market portfolio and is risk aversion coefficient [10].
2)-An improvement to B-L model is in the possibility to integrate directly investor “views” in the model in a really
intuitive way [1,8]. Those “views” can be relative – for instance asset X will outperform asset Y by 5% - or
absolute – for instance asset X will perform by 3%. Let P denote the vector where are specified the assets
concerned by the “views” and Q the vector of their performances. The mathematical aspect of those “views” has to
match with some constraints. The “views” are relative to ER vector. Therefore, the user should to be able to fix a
level of confidence in his “views” [8]. Such a specification may read
( )
(4)
, n the number of assets and k the number of “views”,
the performance vector [9].
is the random normal vector of error terms, ie
(
) with diagonal variance matrix . ER vector
( )
needs to be estimated.
The assumption is that, as the market is always moving around an equilibrium point, so does the investor portfolio,
regarding the CAPM hypothesis [4,5]. Assume that the mean of ( )
, the ER of equilibrium portfolio
Therefore, its variance is assumed to be proportional to , with factor of uncertainty [12]. Mathematically,
( )
(
). Taking into account the previous consideration, the estimator of ( ) is finally given by
where
̂
( )
,( )
When there is no “views” from the investor, P = Q = 0, and ̂
( )
-
,( )
-
, the equilibrium portfolio ER
(5)
Proceedings of World Business Research Conference
21 - 23 April 2014, Novotel World Trade Centre, Dubai, UAE, ISBN: 978-1-922069-48-1
Despite solid mathematical aspect, there remain some grey zones in the model, especially around the specification
of parameter
Rare (and contradictory) information is given on its calibration, even in B-L analysis. As a
reminder, is a constant introduced to scale ER variance. B-L did advise to fix it close to 0, whereas, according to
[13], it should be fixed to 1, both values merely justified. This excessive liberty of interpretation is a great
weakness to be cured [17]. Most alternative attempts are based on predictive analysis from time series analysis [9].
Empirical B-L Analysis and Evaluation
In this view, implementation of Portfolio Asset Allocation based on B-L model has been developed in Matlab
environment. Stable assets have been chosen amongst largest U.S. companies. Five sectors from U.S. companies
are covered. Portfolio equilibrium is formed from capitalization market over 10 years from 1995 to 2005. The
following chosen “views” are accurate, in the sense that they follow directly from equity result of these companies:
Absolute views
Boeing
Coca-Cola and Exxon
Relative views
Wal-Mart compared to Microsoft
The purpose is to reduce price volatility. Application of B-L model produces an expected return ( ) and a new
portfolio allocation [15]. These two final results are changing when is modified. Present study will be based on
both variables. As the assets are supposed to be quite stable and correspond to large stock values, the objective is to
obtain the best ER regarding the variance of new allocation, in order not to fall in extreme weighting. In the
analysis, twenty values of between 0.05 and 1 have been used, with a step of 0.05, and the analysis is conducted
over a period of 10 years, from 1995 to 2005, see Figure 2.
Figure 2 : Dow Jones Stock Chart 1995-2005
Simulations have been performed with the 20 values of τ for each year giving
-The expected returns
-The market weight
-The optimal weight for each asset
-The variation between the new allocation and old one.
As indicated earlier, best ER is researched regarding the variance of new allocation. So the ratio
( ) (
) has been calculated for the 20 values of τ (from 0.05 to 1, with steps of 0.05) for each
year. To get a general view of the evolution of  with respect to τ, the curves for three values of τ : 0.05, 0.5 and 1
have been selected, see Figure 3.
Proceedings of World Business Research Conference
21 - 23 April 2014, Novotel World Trade Centre, Dubai, UAE, ISBN: 978-1-922069-48-1
1.20
Expected Return - Δ Allocation ratio regarding Tau
1.00
0.80
0.60
Tau = 0,05
0.40
Tau = 0,5
0.20
Tau = 1
0.00
1994
1996
1998
2000
2002
2004
2006
Figure 3 :  vs time for different τ
On the graph, it is observed that the maximum of
is not always reached for the value τ = 1, so let τ(max,i) the
value of τ corresponding to the maximum values of . As well known [2,11], main difficulty is to determine the
origin of these τ values, so it is interesting to relate τ (max,i) to a “global” system parameter. The idea is to
investigate the possibility of finding a “state” equation as for mechanics of continuous media [16], and attention
will be focused on covariance matrix which, in some sense, gives a global metrics of market “coherence”. Standard
deviation of covariance matrix (SDCM) has been evaluated and normalized in order to compare it to τ(max,i).
Plotting both τ(max,i) and normalized SDCM curves vs time on the same graph, it appears that there are two
situations. Before 1996, in time interval [1998,2000] and after 2003, τ(max,i) is within error equal to 1, and is
almost constant (unless before 1996), whereas in other intervals where  varies more rapidly with time, τ(max,i)
exhibit an almost oppositely correlated variation, see Figure 4. Two important results can be already drawn from
this observation.
Normalized Standard Development of the Covariance and Tau
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1994
1996
1998
2000
2002
Normalized Std of the Covariance Matrix
2004
2006
Tau(max,i)
Figure 4 : Plot of τ(max,i) and  vs time
First, “natural” value of τ is 1 in agreement with [13], contrary to [1] where it is stated at 0. Second, τ changes in
time only if covariance matrix is itself fast enough time varying, and in a strictly opposite way. So it is interesting
to determine a relation between τ and  in the form
(reg,i)
,
( 
̇ )-
(6)
where ̇ is the time derivative of First elementary evaluation by linear regression[14] would suggest simple
expression ( 
̇ )  .5t is not really interesting because comparison of regressed τ with original one
indicates a higher localized discrepancy of this representation around year 2001 which is not removed with more
accurate second order regression, see Figure 5.
Proceedings of World Business Research Conference
21 - 23 April 2014, Novotel World Trade Centre, Dubai, UAE, ISBN: 978-1-922069-48-1
More elaborate form rests upon global representation of curve (t) which is typically of the form (t) = <(t)> +
jBj(ttj) with slowly varying curve <(t)> and a set of displaced “bumps” Bj(t) always of same functional
dependence Bj(t) = ki(tti). There are two such “bumps” around year t1 = 1997 and around year t2 = 2001.
Interesting point is that τ response curve can be represented in the same form τ = 1 jf(ki) (ttj). On the other
hand the “bumps” are not overlapping ((tti)(ttj)dt = N2 ij), and can thus be considered independently, so
elimination of time between (t) and τ(t) is elementary and gives
( 
̇ ) = f()

(7)

with (t) = (t)  <(t)> the “bumpy” part, and normalizing function (t) to 1.
Comparison of original and regressed Tau
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1994
1996
Tau Reg
1998
Tau
2000
2002
2004
2006
Normalized STD of the covariance matrix
Figure 5 : Plots of τ(max,i), τ(reg,i) and  vs time
Function f(.) can now be very simply determined from the table linking ki and f(ki) for different i from observed
“bumps”. Taking here the two biggest ones at t1 and t2 and considering the limit of small ones at k0 = 0 for which
f(k0) = 0, it is possible from these three sets of data to represent function f(.) in (7) by the parabola through the
origin
f(k) =k( k + ), where 
(
)
and 
(8)
with
( ) measured on data curve in Figure 4 after removal of <> from (t). The results, reported on
Figure 6, are showing the excellent agreement of “state” equation (6) where f(.) is given by (8) with observed data,
see Figure 6. Little discrepancies on both sides of time interval are related to imprecise evaluations of (t)
amplitude. Error with this model is reduced to 2.5% as compared to
Figure 6 : Observed τ (blue) and Obtained One from “State” Equations (6,8)
13.2% with usual regression analysis.
Proceedings of World Business Research Conference
21 - 23 April 2014, Novotel World Trade Centre, Dubai, UAE, ISBN: 978-1-922069-48-1
Conclusion
Analysis of highly questionable parameter τ in B-L model has been performed to determine its value from
(hopefully) accurate functional dependence in terms of global parameters extracted from market data observation.
Attention has been focusing on Standard deviation of covariance matrix (SDCM) which contains interesting
information on market “coherence”. It appears that τ is basically equal to 1 as long as (observed) SDCM  is itself
slowly varying, and only varies antagonistically to  when variation frequency is above a critical threshold
value. An explicit functional law relating τ to  can be proposed which is much better than linear or quadratic
regression. However, owing to system uncertainties, further analysis on the real microscopic financial phenomena
at investor level is required for backing up the proposed “state” equation between τ and . Generalization of
present study to other assets is under way.
Aknowledgments
The authors are very much indebted to Pr M. Cotsaftis for constant help in conducting the analysis and in writing
the manuscript.
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