1
ABSTRACT
In the classical mean-variance approach, an investor is required to provide estimates of the
expected returns and covariances of all the securities in the investment universe considered.
Obviously, this is a humongous task: typically, portfolio managers focus on a small segment of
the investment universe and they pick those stocks in which they believe to achieve superior
performances. The unsatisfactory use of mean-variance approach among practitioners as well as
the need to consider the peculiarities of investment industry motivated Black and Litterman
(1990-2) to formalize a model for combining subjective views with market equilibrium returns.
In particular, Black and Litterman focus on portfolios that behave badly in the sense that
unrealistic and no-intuitive weights may occur when a mean-variance model is utilized in the
asset allocation. The Black and Litterman’s approach allows investors to combine market
information with subjective views in a consistent way. The model relies on the use of implied
equilibrium returns obtained from reverse optimization as reference model that is “tilted away”
in the direction of the assets most favoured by investor’s views. An average investor without
particular information on specific assets should invest according to the equilibrium weighting
scheme. Portfolio manager may have different expectations about future returns; the model
allows investor to incorporate subjective (tactical) views and to combine them with neutral
(equilibrium, strategic) views in a consistent way. Moreover, investor indicates the degree of
confidence in his views in such a way that higher confidence produces revised expected returns
more tilted towards investor’s views. Then, the vector of revised expected returns is utilized as
input of an optimizer to obtain optimal portfolio weights. The work reflects upon the analysis on
the intuition behind the model presenting details of Bayesian portfolio selection. Furthermore,
each input is described in detail explaining some of the most important approaches used. Finally,
we analyze how to use the model in practice introducing a factor model to compute views: an
example on European sectors will help us to describe the growing importance of the model in the
asset allocation.
2
RESUMEN
En el enfoque clásico de media-varianza, el inversor está obligado a proporcionar las
estimaciones de los retornos esperados y de las covarianzas de todos los valores en el universo
de inversión considerados. Obviamente, esta es una tarea gigantesca: en general, los gestores se
centran en un pequeño segmento del universo de inversión y eligen los activos que ellos creen
para lograr resultados superiores. Puesto que hay un uso insatisfactorio del enfoque de media
varianza entre los profesionales, así como hay la necesidad de considerar las peculiaridades de la
industria de inversión, Black y Litterman (1990-2) han formalizado un modelo para la
combinación de puntos de vista subjetivos con retornos de equilibrio del mercado. En particular,
Black y Litterman enfocan en portfolios que se comportan mal en el sentido de que los pesos no
reales y no intuitivos pueden ocurrir cuando el modelo de media-varianza se utiliza en la
asignación de los activos. El enfoque de Black y Litterman permite a los inversores combinar la
información de mercado con puntos de vista subjetivos de una manera consistente. El modelo se
basa en el uso de los rendimientos implícitos de equilibrio, obtenidos mediante optimización
inversa, como modelo de referencia que es “inclinado lejos” en la dirección de los activos más
favoritos del inversor. El inversor medio sin información particular sobre activos específicos
debe invertir de acuerdo con el esquema de ponderación de equilibrio. Inversor puede tener
diferentes expectativas sobre los rendimientos futuros; el modelo permite a los inversores de
incorporar subjetivos (tácticos) puntos de vista y de combinarlos con (de equilibrio, estratégicos)
puntos de vista neutrales de una manera consistente. Por otra parte, el inversor indica el grado de
confianza en sus puntos de vista de tal manera que el aumento de la confianza produce revisados
retornos más inclinados en la directiòn de los puntos de vista favoritos del inversor. Entonces, el
vector de los revisados rendimientos esperados se utiliza como entrada de un optimizador para
obtener óptimos pesos de cartera financiera. La composixiòn reflexiona sobre el análisis de la
intuición detrás del modelo y presenta en detalles la selección Bayesiana de la cartera. Además,
cada entrada se describe en detalle explicando algunos de los planteamientos más importantes
que se utilizan. Finalmente, se analiza cómo utilizar el modelo en la práctica introduciendo un
modelo factoriales para calcular los puntos de vista subjetivos: un ejemplo en los sectores
europeos nos ayudará a describir la creciente importancia del modelo en la asignación de activos.
6
Introduction
Portfolio selection is one of the most significant problems in practical investment
management. The Mean-Variance framework of Markowitz (1952) analytically formalizes the
risk/return trade-off in selecting optimal portfolios. Unfortunately, mean-variance approach often
leads to portfolios that behave badly: scarcity of diversification (extreme portfolios or corner
solutions), sensitivity of the solution to inputs and estimation arguments explain why its use is
not widespread [see Merton (1980), Michaud (1989) and Best and Grauer (1991)].
The unsatisfactory use of mean-variance approach among practitioners as well as the
need to consider the peculiarities of investment industry motivated Black and Litterman (1990-2)
to formalize a model for combining subjective views with market equilibrium returns. The model
influences asset allocation in two directions: first, it defines an equilibrium distribution which
summarizes neutral information and it is related to market equilibrium portfolio; second, it
allows investors to consider their subjective views on specific assets for which “core
competence” is available. Therefore, assets returns’ forecasts will be weighted combinations of
market equilibrium returns and subjective views in which the weights depend on (1) the
volatility of each asset and its correlation with the other assets and (2) the degree of confidence
in each view. Finally, the Black and Litterman’s revised returns serve as a consistent input for
mean-variance portfolio optimization procedures which provide less unstable and more
diversified and intuitive portfolios [Black and Litterman (1992), He and Litterman (1999) and
Litterman et al. (2003)].
In particular, the classical approach to estimating future expected returns considers the
“true” means and covariances of returns as unknown and fixed. Forecasting models of observed
market data or proprietary data are used to form a point estimate. Unfortunately, it is difficult to
produce accurate estimates and, thus, mean-variance portfolio allocation is heavily influenced by
estimation errors. Conversely, the Bayesian approach assumes that the “true” expected returns
are unknown and random. Furthermore, investor’s belief on the probability that a specific event
will actually occur might change after more information is provided. Bayes’ rule is the formula
applied to combine prior distribution plus new data in order to obtain the posterior distribution.
7
In the Black and Litterman’s model, such framework is represented by the combination
of market equilibrium (e.g. the CAPM equilibrium) with investor’s subjective views. Then, the
mean of the predictive distribution is used as input of the optimization process; portfolio
weighting schemes computed by using this method tend to be more intuitive and less sensible to
changes in the inputs.
Extreme and not intuitive weights returned by mean-variance model appear inappropriate
for being implemented in a client’s portfolio. In addition, mean-variance approach requires to
provide estimates of expected returns and covariances of all securities considered. Typically,
asset managers focus on a small segment of the investment universe and they pick those stocks
in which they believe to achieve superior performances. It is unrealistic for portfolio manager to
generate estimates of all the inputs and, therefore, it represented one of the motivations of Black
and Litterman’s model development. Thus, the model can be interpreted as a method to lead
asset allocation in which only the assets for which specific views are produced receive revised
weights.
In particular, the field in which practitioners apply the model is represented by the asset
allocation. Usually, we denote by class a homogeneous set of assets with certain peculiarities
and a specific risk/return profile. A more detailed taxonomy focuses on the investment horizon;
we can briefly distinguish the following approaches: buy-and-hold, a passive strategy that
consists in the acceptance of a position that is maintained until the end of the investment;
strategic asset allocation, a strategy that minimizes the level of risk given a target return and the
tactical asset allocation (TAA) which represents the approach used by investor to vary portfolio
weights periodically according to the information available at a given time. In particular, TAA is
an active strategy which seeks to improve the performances by varying portfolio weights when
market scenarios change; from a mean-variance point of view, TAA is an approach used by
investor to make bets against the benchmark in order to achieve better performance in terms of
risk/return. Therefore, the judgement on the TAA’s performance is expressed relative to the
benchmark; common used measures are alpha, which represents the extra-returns over the
benchmark, and the tracking error volatility which indicates the difference between portfolio’s
risk and benchmark’s risk. Moreover, the information ratio computed by using the ratio of the
previous measures provides a risk/return judgement of portfolio manager’s performance. The
starting point of the TAA is the public information which allows investor to form portfolio
weights according to observed measures of means and covariances. In addition, investor may
8
have personal information based on his knowledge in core segments. The Black and Litterman’s
model is a method to combine both sources of information in a consistent way. Therefore, in a
TAA investor makes a bet on the strategic mix, that represents long-term forecasts (i.e. the
benchmark), and he makes also a tactical bet that depends on his personal information. If
investor’s forecast on a certain asset is different from equilibrium return, then the respective
portfolio weight will deviate away from the strategic weight. Optimal weights will depend on the
degree of confidence in subjective views and, therefore, personal information will be considered
in the model in order to achieve better performances: equilibrium weights, i.e. strategic
weighting schemes, will be modified in the direction of the assets most favoured by investor’s
views.
The Black and Litterman’s model was formalized by the Quantitative Resources Group at
Goldman Sachs Asset Management in the early 1990s and it represents the development of
Black’s (1989) study on global hedged portfolios. Moreover, Bevan and Winkelmann (1998), He
and Litterman (1999), Satchell and Scowcroft (2000), Drobetz (2001), Litterman et alt (2003)
and Idzorek (2004) clarified and extended it further. In particular, some future directions were
exposed by Qian and Gorman (2001), Blamont and Firoozy (2003), Fusai and Meucci (2003),
Meucci (2005-8), Almgren and Chriss (2006) and Pezier (2007).
The thesis is divided into three chapters. In the first part, a description of the mean-
variance approach is shown. In particular, we focus on the framework of the model and on its
derivation; then, a brief explanation of the CAPM is analyzed. The second part reflects upon the
Black and Litterman’s model: paragraph 2.1 explains the intuition behind the model while
paragraph 2.2 focuses on the Bayesian portfolio selection. Paragraph 2.3 describes the model and
analyzes each input in detail explaining some of the most important approaches used.
The third Chapter presents a case study. We will show how the model can be used in
practice: an example on European sectors will help us to describe the growing importance of the
model in the asset allocation. Moreover, an original way to produce views based on either a
factor model or the use of the Composite Leading Indicator will be analyzed and, in particular,
we will explain how the Black and Litterman’s strategy based on this procedure achieves
remarkable risk/return performances.
9
Chapter I
Mean Variance Analysis and Modern
Portfolio Theory
1.1 Mean-Variance Analysis
Portfolio theory received a significant impact by Harry Markowitz’s article Portfolio
Selection
1
published in 1952 in the Journal of Finance. His intuition attracted interest in a field
previously poorly investigated
2
. Markowitz’s approach represents the foundations of what is now
popularly referred to as mean-variance analysis. In this context, key roles are played by the net
present value of expected future returns and the measure of risk. He focuses on the contribution
of each security within portfolios; given a set of securities, investors should consider the
movements of each security in relationship with the other assets forming the portfolio. In its
simplest form, mean-variance analysis provides a framework to construct and select portfolios
based on the expected performance of the assets and the risk appetite of the investor. Taking
covariance into consideration, investors can select portfolios that generate higher expected return
at the same level of risk or lower level of risk at the same level of expected return. This is a well
known benefit of diversification and it follows Markowitz’s acceptance of variance as the
measure of risk. The risk of a portfolio in turn depends on the variance of the assets in the
portfolio and on the covariance between its assets.
The theory of portfolio selection is a normative theory. It describes how investors should
behave in constructing a portfolio; under certain assumptions, Markowitz shows that investors
can select a combination of assets that minimizes the risk given a level of expected return or
maximizes expected return given a specified level of risk.
1
Markowitz, H., (1952), “Portfolio Selection”. The Journal of Finance, Vol. 7, No. 1 (March): pp 77-91.
2
See e.g. Williams James Burr, (1938), “Theory of investment value”. Amsterdam: North-Holland Publishing
Company for an analysis of pre-Markowitz way of analyzing portfolio theory.
10
Markowitz’s main contribution to portfolio theory is represented by the analysis of
investors’ behaviour in a context of mean-variance. Even if Markowitz’s approach does not rely
on joint normality of securities returns, it can be shown that his model is consistent either with
the assumption that securities returns are jointly normally distributed or with expected utility
maximization under certain hypothesis
3
.
1.1.1 Markowitz’s Model Framework
Diversification represents an important feature of Markowitz’s model. When investors
select assets and form portfolios, the statistical contribution of each asset within a portfolio
influences not only its expected return but also it shows significant implications in the measure
of risk, i.e. in the computation of the variance of the portfolio. Diversification is related to the
Central Limit Theorem, which states that the sum of identical and independent random variables
with bounded variance is asymptotically Gaussian. It can be shown that the variance of a
portfolio of N identically and independently distributed assets is inversely related to the number
N of assets. In particular, given this relation we can notice that increasing the number of assets
the variance of the portfolio decreases towards zero. Despite this optimistic view, for real-world
portfolios we cannot expect a portfolio variance of zero due to no-vanishing correlations, even if
a large number of assets are involved.
Markowitz’s approach considers several assumptions under which mean-variance
analysis can hold. The starting point is represented by a ration investor who plans his investment
in a one-period horizon; taxes, transaction costs and any other market imperfections are omitted.
Leit motive is the trade-off between risk and expected return. For any given level of expected
return, a rational investor would choose the portfolio with minimum variance from amongst the
set of all possible portfolios, i.e. within the feasible set. Moreover, the set of all mean-variance
efficient portfolios is called the efficient frontier which represents the set of portfolios with the
lowest level of risk given a certain level of expected return or, equivalently, the portfolios that
maximize expected return for a given level of risk.
3
See Markowitz, H., and Usmen, N., (1996), “The likelihood of various stock market return distributions, part I:
Principles of inference”. Journal of Risk and Uncertainty, 13: pp 207-219. See also Rachev, S., Ortobelli, S., and
Schwartz, E., (2004), “The problem of optimal asset allocation with stable distributed returns” in Krinik, A., and
Swift, R. (eds.), “Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M.
Rao”. Lecture Notes in Pure and Applied Mathematics: pp 295-347. New York: Marcel Dekker.
11
In this section, a step-by-step explanation of the model is shown trying to focus on the
fundamental contributions of the mean-variance approach. In mathematical terms, an investor’s
investment in a portfolio composed by N assets is represented by a vector w = (w
1
, w
2
, …, w
n
)‟
of weights, where each w
i
is the percentage of the i-th asset held in the portfolio and
Moreover, the expected returns vector is μ = (μ
1
, μ
2
, …, μ
n
)‟ and the N x N covariance matrix Σ
is:
where σ
ij
denotes the covariance between asset i and asset j. If we suppose that the assets’ returns
vector is R = (R
1
, R
2
,…, R
n
)‟ then the return on a portfolio with weights w = (w
1
, w
2
, …, w
n
)‟
will be a random variable R
p
= w’R with expected return and variance respectively:
Mean-variance analysis
4
requires the specification of a given level of risk or equivalently
a given level of expected return. In the first case, the aim is to find the highest level of expect
return given a certain risk so that investor solves a maximization problem; in the second case, the
investor’s purpose is to reach the lowest risk given a certain level of expected return so that he
solves a minimization problem. In this work, we adopt the second method. This constrained
minimization problem assumes that investor must seek:
4
The paragraph is based on Fabozzi, Frank J., Focardi, Sergio M., and Kolm, Petter M., (2006), “Financial
Modeling of the Equity Market from CAPM to Cointegration”. Hoboken, New Jersey: John Wiley & Sons, Inc.
For more details, see Bodie, Z., Kane, A., and Marcus, Alan j., (2003), “Investments”. McGraw-Hill, V Edition.
12
subject to constraints
where μ
0
is a certain targeted level of expected return. In particular, this is a quadratic
optimization problem with equality constraints and the solution is given by w = g + hμ
0
where
the vectors g and h are respectively:
and
The efficient frontier is obtained by solving the optimization problem above for different choices
of μ
0
In its simplest form, the optimization problem can be solved analytically. Numerical
optimization methods must be used in more complex cases.
Alternative formulations of mean-variance model are possible. One of these depends on
the use of a risk aversion parameter λ. The following formulation is called the risk aversion
formulation of the classical mean-variance optimization problem:
subject to
13
The risk aversion parameter λ, also known as the Arrow-Pratt risk aversion index, has a
significant influence in the definition of the weights. It is a measure for the rate at which the
investor is willing to accept additional risk for a one unit increase in expected return. Therefore,
a small λ determines a small penalty leading to a more risky portfolio; conversely, if λ is large
then the penalty will increase. Each portfolio along the efficient frontier can be calculated
considering λ from zero to higher levels. Often, back testing methods with historical data are
utilized for the calibration of λ. This formulation determines optimal portfolio’s weights:
w* = (λΣ)
-1
μ
The risk aversion parameter λ represents the relation between risk and expected returns. In
particular, many practitioners consider it as proportional to the Sharpe Ratio of the market
portfolio divided by its standard deviation. Usually a range between 2 and 4 is used.
1.1.2 Markowitz’s Model Limits
Several problems arise when the model is utilized in practice and, although mean-
variance model seems reasonable, it often leads to unrealistic optimal portfolios. Michaud
5
(1989) reflects upon the practical problems and he argues that one of the reasons of mean-
variance model use among practitioners is its quantitative background that has a strong influence
on top management. In particular, Michaud analyzes the limits that affect the model.
First of all, Markowitz’s optimizers maximize errors. The impossibility to use exact
estimates of expected returns and variances leads to estimation errors. In particular, the model
tends to overweight securities with high expected return and negative correlation and to
underweight those with low expected returns and positive correlation. According to Michaud,
large estimation errors affect these securities.
The model does not consider securities’ market capitalizations. For instance, the model
can suggest high portfolio weights for assets with low level of capitalization and high expected
returns if they are negatively correlated with the other assets in the portfolio. Unfortunately, very
high weights for small capitalized assets are unreasonably possible.
5
Michaud, Richard O., (1989), “The Markowitz Optimization Enigma: Is „Optimized‟ Optimal?” Financial Analyst
Journal, vol. 45, no. 1 (January/February): pp 31-42.
14
Mean-variance approach does not take into account the levels of confidence in the inputs.
The uncertainty associated with them should influence differently the optimizer results. Even if it
makes sense for the uncertainty of inputs to be part of the optimization process, thus making the
model more realistic, unfortunately, optimizers completely ignore this problem.
One of the most important limits is that the model is often unstable and small changes in
one of the inputs may change the entire composition of the portfolio. Empirical evidences show
that minimally varying the expected returns, the consequences may be great and the weights may
change dramatically. Michaud argues that it depends mainly on the covariance matrix estimated
by using insufficient historical data.
In particular, the model suggests portfolio with large negative weights in many assets
when the optimizer runs without constraints. This result could be a problem for many investors
due to their constraints in short positions. Therefore, if a limit in short positions is added, the
model will recommend zero weights in many securities and very large positions in few
securities. Investors find such portfolios unreasonable and it represents one of the main limits in
the use of mean-variance model.
Investors wish to invest in portfolios that show an intuitive link to both the reality and the
estimated variables. Many authors suggest that the model is not able to define a vector of weights
that represents a reasonable combination of assets; thus, without short selling constraints, the
optimizer gives too negative weights in some assets and, conversely, adding short selling
constraints, the model shows few positive weights and some of them are really high. These
evidences suggest to consider with prudence Markowitz’s model portfolios and to focus on the
problems of the estimation of risks and expected returns. Michaud suggests that a better
estimation of the variables could lead to more intuitive portfolios.
Although the model has theoretical foundations and shows a realistic trade-off between
risk and expected return, empirical evidences are in contrast with optimizer’s results. Markowitz
argues that, especially in large portfolios, most of the problems are linked to the estimation of the
covariance matrix. For instance, a portfolio with 4 securities needs 6 estimated covariances and 4
estimated variances while a portfolio with 10 securities has 45 covariances and 10 variances to
be estimated. In particular, given n assets the estimated covariances will be (n
2
– n)/2. It is
evident that as the number of assets increases, the number of estimates grows quickly and the
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