The Theory and Application of the B-L Model by Andrei Costea

The Theory and Application of the Black‐Litterman Model Bachelorarbeit

eingereicht bei Prof. Dr. Raimond Maurer Lehrstuhl für Betriebswirtschaftslehre, insbesondere Investment, Portfolio Management und Alterssicherung Fachbereich Wirtschaftswissenschaften Johann Wolfgang Goethe‐Universität Frankfurt am Main von cand. rer. pol. Andrei Costea Calea Dorobanților 10 010572, Bukarest, Rumänien +40‐(0)‐726‐282829 andrei.costea [at] inbox.com Studienrichtung: Wirtschaftswissenschaften 6. Fachsemester Matrikelnummer: 3152560

Initial draft: 8th September 2008 This draft: 28th March 2009

‐I‐

Table of Contents List of Figures .................................................................................................................................... III List of Tables ..................................................................................................................................... IV List of Abbrevations ........................................................................................................................... V List of Symbols .................................................................................................................................. VI 1 Introduction .......................................................................................................................... 1 2 Laying the Foundation for the Black‐Litterman Model ........................................................... 1 2.1 The Criticism of Classical Portfolio Optimization ................................................................. 1 2.1.1 Sampling .................................................................................................................... 2 2.1.2 Concentration and Sensitivity .................................................................................... 2 2.1.3 Estimation Risk ........................................................................................................... 3 2.2 Introducing the Black‐Litterman Approach .......................................................................... 5 2.2.1 Model Assumptions ................................................................................................... 5 2.2.2 The Capital Asset Pricing Model – the economic intuition ........................................ 5 2.2.3 The Bayesian Framework – the formal intuition ....................................................... 7 3 The Derivation of the Model .................................................................................................. 8 3.1 The Prior and the Reference Model ..................................................................................... 8 3.2 The Conditional .................................................................................................................... 9 3.3 The Posterior ...................................................................................................................... 11 3.4 Reformulations of the Posterior ........................................................................................ 13 3.5 Interpretations of the Posterior ......................................................................................... 14 3.6 The Posterior Portfolio and View Weights ......................................................................... 16 4 Theoretical and Practical Aspects ........................................................................................ 20 4.1 Model Inputs ...................................................................................................................... 20 4.1.1 The Parameter ...................................................................................................... 20 4.1.2 The Matrix ............................................................................................................ 21 4.2 Further Developments ....................................................................................................... 22 4.2.1 Implied Views ........................................................................................................... 22 4.2.2 A Consistency Measure ............................................................................................ 23 4.2.3 The UBS Warburg Approach .................................................................................... 24 4.3 Aspects of Practical Implementation ................................................................................. 26 4.3.1 Strategic and Tactical Portfolio Positioning ............................................................. 26 4.3.2 Behavioral Finance ................................................................................................... 28 4.3.3 The Experience at Goldman Sachs ........................................................................... 29

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5 An Exemplifying Application ................................................................................................ 30 5.1 Introducing the Data .......................................................................................................... 30 5.2 Comparison of Classical M‐V Portfolios with B‐L Portfolios ............................................... 32 5.4 Comparison of Different B‐L Portfolios .............................................................................. 34 6 Conclusion ........................................................................................................................... 37 Appendix ................................................................................................................................. 39 Bibliography ............................................................................................................................ 53 Ehrenwörtliche Erklärung ........................................................................................................ 57

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List of Figures Figure 1:

Evolution of Time Series (Aug. 1988 – Aug. 2008). ................................................ 48

Figure 2:

Classical Unconstrained Stacked Weights for

Figure 3:

B‐L Unconstrained Stacked Weights for

Figure 4:

Classical Constrained Stacked Weights for

Figure 5:

B‐L Constrained Stacked Weights for

Figure 6:

The Unconstrained Efficient Frontiers and Relevant Portfolios. ........................... 52

Figure 7:

The Constrained Efficient Frontiers and Relevant Portfolios ................................ 52

0.625, ∞ . ............................. 50

0.625, ∞ . ...................................... 50 0.625, ∞ ................................... 51

0.625, ∞ . .......................................... 51

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List of Tables Table 1:

Sample Statistical Properties. ................................................................................ 31

Table 2:

Prior Information. .................................................................................................. 32

Table 3:

Expected Excess Returns with Zero Views. ............................................................ 32

Table 4:

Optimal Portfolio Weights with Zero Views. ......................................................... 33

Table 5:

Expected Excess Returns with 2 Views. ................................................................. 33

Table 6:

Optimal Portfolio Weights with 2 Views. ............................................................... 34

Table 7:

Changes of Optimal Portfolio Weights from Zero Views to Two Views. ............... 34

Table 8:

Expected Excess Returns with Two Relative Views. .............................................. 35

Table 9:

Optimal Portfolio Weights with Two Relative Views. ............................................ 35

Table 10:

Expected Excess Returns with Two Modified Relative Views. ............................... 35

Table 11:

Optimal Portfolio Weights with Two Modified Relative Views. ............................ 36

Table 12:

Expected Excess Returns with Four Views. ............................................................ 36

Table 13:

Optimal Portfolio Weights with Two Modified Relative Views. ............................ 37

Table 14:

Company Specific Information. .............................................................................. 48

Table 15:

The Covariance Matrix. .......................................................................................... 49

Table 16:

The Correlation Matrix. .......................................................................................... 49

‐V‐

List of Abbrevations APT

arbitrage pricing theory

B‐L

Black‐Litterman

CAPM

capital asset pricing model

CML

capital market line

eds.

editors

e.g.

exempli gratia

et al.

et alii

IAPT

international abitrage pricing theory

i.i.d.

independently and identically distributed

information ratio

M‐V

mean‐variance

MVP

minimum variance portfolio

OLS

ordinary least squares

Sharpe ratio

tracking error

tangency portfolio

‐ VI ‐

List of Symbols ·

Expected value ·

Variance ·

pdf ·

Covariance Probability density function

Likelihood function

Number of assets

Number of views

Return

Returns vector

Expected return

Expected returns vector

Standard deviation

Covariance matrix of the returns

Expected equilibrium risk‐premium

Expected equilibrium risk‐premiums vector

Error terms vector

Risk‐aversion parameter

Scalar / Weight‐on‐views

Weight

Weights vector

View portfolios matrix

View portfolio weights vector

Expected view returns vector

Covariance matrix of the views

Covariance matrix of the posterior expected returns

Weights vector of the view portfolios

Vector of Lagrangian multipliers

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Introduction

Classical portfolio optimization, one of the most important advances in modern financial theory introduced in 1952 by MARKOWITZ (1952), has not had the hoped for impact in prac‐ tice. BLACK and LITTERMAN (1990, 1992) have expanded the setting of classical portfolio op‐ timization towards an intuitive and feasible framework which has gained wide application in the field of portfolio management. The first part of this thesis includes a motivational outline for considering the Black‐ Litterman model and offers a formal derivation of the model after presenting its theoreti‐ cal underpinnings. In subsection 2.1 the shortcomings of classical portfolio optimization, that have stimulated practitioners to disregard its application, are considered. Once the model is introduced intuitively in subsection 2.2 by means of the CAPM and Bayes’ theo‐ rem, subsequent to a discussion of its assumptions, section 3 will be concerned with its mathematical derivation and the interpretation of the formal results. The second part of the thesis is intended to touch on the strengths and weaknesses of the Black‐Litterman model by conferring about the model inputs, by presenting further devel‐ opments to the model and by providing an insight to its practical implementation. To that purpose, an exemplifying application of the model is also considered. Subsection 4.1 spe‐ cifically deals with inputs that are essential to expressing confidence in the equilibrium and the views. Broadening developments including implied views, a consistency measure and the UBS Warburg approach are referred to in subsection 4.2. Subsection 4.3 will then ex‐ amine the benefits for using the Black‐Litterman model both in a strategic as well as a tac‐ tical portfolio positioning landscape, discuss the implications stemming from behavioral finance and offer a grasp of how the model was employed within Goldman Sachs. Section 5 presents an application of the model and is intended to highlight the properties and benefits of the model more concretely than in the first part. Finally, section 6 concludes the thesis.

Laying the Foundation for the Black‐Litterman Model

2.1

The Criticism of Classical Portfolio Optimization

Before discussing the pitfalls of classical portfolio optimization it is important to make a distinction between classical portfolio optimization and the mean‐variance methodology

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of optimizing a portfolio in itself. The mean‐variance methodology is a model onto which the Black‐Litterman approach is built and which can be altered to consider other classes of risk‐to‐return tradeoffs, utility functions or constraints. Theoretically, there is nothing wrong with this methodology as it presents a coherent optimization procedure given a set of future expectations about its inputs1 and considering that the returns belong to a sym‐ metric distribution with two parameters2. Classical portfolio optimization is the process of optimizing portfolios where the set of future expectations is replaced by a set of historical performances and it is this approach that bears the critique. Like the Black‐Litterman ap‐ proach it is also based on the mean‐variance portfolio selection model, but since it over‐ looked to accommodate such concepts like the market equilibrium or Bayesian inference it has ended in being unfeasible in practice. 2.1.1

Sampling

A major shortcoming of classical portfolio optimization lies in the specification of the mod‐ el inputs. Particularly in the setting of global portfolio selection investors possess limited information about the targeted assets or asset classes. Furthermore, when returns are stated they must not only be based on absolute expectations but also on relative expecta‐ tions between assets. The specification of correlations hence becomes important. Pro‐ vided that the series of expected returns, variances and correlations is large and that in‐ vestors must provide a complete set of aggregated information about these inputs, the sampling problem becomes a real impediment to a consistent investment process. The imminent alternative to a comprehensive analysis utilized in classical portfolio optimiza‐ tion is the use of historical data as a proxy for future expected returns, but as subsection 2.1.2 will indicate, this method provides a poor guide for a reliable estimation. 2.1.2

Concentration and Sensitivity

Another important deficiency of classical portfolio optimization is that it leads to portfolios that have either extreme weights in several assets when the short‐selling constraint is omitted or to highly concentrated allocations in few assets when this constraint is consi‐ dered. In the former case, the optimizer tends to concentrate large positive weights to assets with high expected returns and low variances and covariances, and large negative weights to assets with low expected returns and high variances and covariances, whereas 1

MARKOWITZ (1991, p. 206) is clear about the fact that the role of the portfolio analyst begins where the role of the security analyst (i.e. who forms judgments about the inputs) ends. 2 See TOBIN (1958).

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the assets with characteristics that fall in‐between are almost left out. In the latter case, it is usually the assets with low market capitalizations that are included in the portfolio, since market weights are not accounted for in the optimization and since these assets usually exhibit better risk‐to‐return profiles. When faced with the problem of concentration, the investment practice generally calls for the use of heuristic approaches. These approaches indeed result in less concentrated port‐ folios, but they do not possess any insightful grounds and skip to provide an explicit solu‐ tion to the actual problem. When used in the framework of classical portfolio optimization, the mean‐variance model delivers weights that prove to be highly sensible to changes in the inputs. The highest sen‐ sitivity stems from changes in the vector of expected returns. Moreover, a small alteration of the expected return of a single asset can lead to large changes in the weights of all the other assets, but, as BEST and GRAUER (1991) point out, the return and variance of the port‐ folio remain unchanged. The consistency problem of the sampling procedure on the one hand and the lack of diver‐ sification and stability on the other hand are the shortcomings which lead BLACK et al. (1992, p. 28) to consider an approach that should drive portfolio optimization towards an intuitive framework with well behaved results. 2.1.3

Estimation Risk

As suggested above, the use of historical data as a substitute for future expected returns, variances and covariances does not provide a reliable estimate for these inputs; rather it increases the total risk of the assets beyond its intrinsic value. When the sample return vector and covariance matrix are utilized in the mean‐variance model, BLACK et al. (1992) and MICHAUD (1989) speak of estimation‐error maximization. A further inadequacy of clas‐ sical portfolio optimization is that it does not account for different uncertainty levels of the estimations. HERLOD and MAURER (2002) ascertain that even if the true sample moments where known with certainty the mean‐variance model does not guarantee a consistent outperformance of every benchmark portfolio in each period.3

Note that this characteristic is a weakness of the mean‐variance model, but as it was stated above the model should be theoretically correct. Per se this is still true, as the model was constructed to con‐ sider an optimal risk‐to‐return tradeoff and not a consistent outperformance.

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CHOPRA and ZIEMBA (1993) approximate that the estimation errors in the returns are ten times as important as the estimation errors in the variances and that the estimation errors in the variances are twice as important as the estimation errors in the covariances. How‐ ever, BENGTSSON (2004) concludes that the results of CHOPRA et al. (1993) are flawed due to the fact that constraints have significant effects on the impact of estimation errors and arrives at some contrary conclusions. Conservative investors are more exposed to estima‐ tion risk than aggressive investors, but they encounter nearly the same loss from the esti‐ mation error in the returns as from the estimation error in the covariance matrix. He also shows that short‐selling restrictions reduce estimation risk, but when eliminated they lead to the loss from the estimation error in the covariance matrix being greater that the one from the estimation error in the mean vector, even for relatively aggressive investors. BRITTEN‐JONES (1999) realizes that the sampling error in estimates of the weights of a global efficient portfolio is large when using OLS regression like statistics to test the weights of mean‐variance efficient portfolios. JORION (1985) provides evidence for a weak out‐of‐sample performance of classical mean‐ variance optimized portfolios when compared to trivially optimized portfolios or portfolios based on heuristic approaches. The literature also considers several notable alternatives to classical portfolio optimization. Not including the Black‐Litterman model, these alterna‐ tives are the resampling methodology of MICHAUD (1989), the Bayes‐Stein estimation pro‐ cedure of JORION (1985, 1986), which uses Bayesian inference and the MVP as the prior towards which the sample means are shrunk, and the approach of PÁSTOR (2000), PÁSTOR and STAMBAUGH (1999, 2000) and WANG (2001), which is similar to the latter approach in that it also implements a conditional methodology. The Pástor approach is also similar to the Black‐Litterman model due to the use of an economically motivated prior, but the shrinkage is performed starting from a conditional based on historical estimates and not on the proprietary views of the investors.4 Altogether, uncertainty can never be eliminated from a portfolio selection model, but the Black‐Litterman approach does manage to re‐ duce this risk.

HEROLD et al. (2002) document an economically significant out‐of‐sample outperformance when as‐ suming predictability and implementing instrumental variables, instead of historical returns direct‐ ly, to estimate the conditional means.

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2.2

Introducing the Black‐Litterman Approach

2.2.1

Model Assumptions

The keystone of the B‐L model is the integration of a general market equilibrium, which reflects the investor’s investment universe, and of a set of views, that the investor pos‐ sesses about certain assets in that universe. The merged result of this integration is then used as input in the mean‐variance model. Starting from this approach to asset allocation, a series of implicit and overriding explicit assumptions where made by BLACK et al. (1990, 1992). The implicit assumptions include those made by modern financial theory: normally distributed returns; rational risk‐averse investors who are price‐takers, trade off risk for return and maximize a quadratic utility function; the existence of a risk‐free rate; no incurrence of transaction costs; efficient and arbitrage‐free markets. The B‐L model overrides the last implicit assumption, i.e. it as‐ sumes that markets are not efficient and that there exist arbitrage opportunities of which investors can take advantage, and adds several others: investors hold the equilibrium port‐ folio on the long‐term and can express views about the market that can deviate from the equilibrium in the short‐term; the consideration of different levels of uncertainty in the expressed views and respectively in the market equilibrium; the use of a benchmark as a standard to evaluate the risk of different portfolios. The last explicit assumption is implied by the inclusion of the equilibrium which acts as an inherent reference. The detailed tech‐ nical assumptions will be referred to separately in subsection 3.1 for more intelligibility. 2.2.2

The Capital Asset Pricing Model – the economic intuition

As BLACK et al. (1992, p. 28) indicate, the basis of an intuitive quantitative portfolio selec‐ tion model is the combination “of two tenets of modern financial theory”, namely the pre‐ viously mentioned mean‐variance model and the capital asset pricing model5 of SHARPE (1964) and LINTNER (1965), which is used as a surrogate for the concept of market equili‐ brium. When all investors hold the same views about the market and deviations from the views towards riskier assets are rewarded with risk premiums the market is in equilibrium and consequently the demand for assets equals the outstanding supply for these assets.6 The long‐term returns are a result of market clearing. In equilibrium, the investors hold the 5

When referring to an equilibrium model, BLACK et al. (1992) also mention the use of a global CAPM, introduced by BLACK (1989), due to the consideration of a global investment universe for which a universal optimal hedge ratio is computed. 6 In classical portfolio optimization only the demand side of the market is taken into account.

‐ 6 ‐

market portfolio.7 The CAPM assumes that the non‐diversifiable risk is priced in a linear fashion, so that the expected equilibrium risk‐premiums of the assets are given by ·

where

is the vector of expected equilibrium risk‐premiums,

of expected returns,

the expected market return and

(2.1) the vector

the vector of systematic risk factors, the risk‐free rate. BLACK et al. (1992)

and HE and LITTERMAN (1999a) suggest that the input of the expected equilibrium risk‐ premiums in the B‐L model should be performed through reverse optimization8, which entails that

(2.2)

and consequently ,

where

(2.3)

is the covariance matrix of the expected excess returns and

the equilibrium weights vector of the assets.

is a risk‐aversion factor representing

the average risk‐tolerance and given that (2.1) and (2.3) are equivalent it is derived as ,

where

(2.4)

is the variance of the equilibrium portfolio. This process follows the logic of

an unconstrained portfolio due to the fact that the unity restriction is implicit for the mar‐ ket portfolio. The B‐L model is designed to cope with any prevailing market equilibrium, so that the CAPM equilibrium is only a specific type of market equilibrium. Due to the critique of the CAPM, both owing to its empirical invalidity, see BLACK (1993), as well as, recently, its theo‐ retical invalidity, see MARKOWITZ (2008), it should be worth considering alternative models like the arbitrage pricing theory of ROSS (1976). Moreover, since it may often be difficult to 7

Note that in the context of the two‐fund theorem of TOBIN (1958) investors are allowed to devote a portion of their wealth to the risk‐free asset. Nevertheless, the risky portion is still invested in the market portfolio. 8 See SHARPE (1974).

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define the market spectrum, especially in an international context, usually a benchmark is employed, which would suffice to represent the investment universe. 2.2.3

The Bayesian Framework – the formal intuition

The B‐L model differentiates itself from classical portfolio optimization by entailing the idea of Bayesian inference and is distinct amongst models that identify the informative prior as a market equilibrium, i.e. the Pástor approach, by incorporating subjective inves‐ tor views. The adjunct of this unique feature is the consideration of view uncertainty. These two elements confer the B‐L model the formal intuition required in a quantitative portfolio choice model which assumes that markets are predictable. From Bayes’ theorem, see BAYES (1763), the posterior distribution of the revised prior conditional on is

pdf

| pdf pdf

pdf | pdf pdf | pdf

pdf

(2.5)

Where pdf · is the probability density function and · the likelihood function. In the B‐L model, the prior is assumed to be conjugate and belonging to the Gaussian family. Consis‐ tent with Bayesian statistics, is a random variable. The literature considers several speci‐ fications of the prior and conditional distributions. SATCHELL and SCOWCROFT (2000) and CHRISTODOUALKIS (2002) associate these with the expected returns and the equilibrium risk‐ premiums respectively, but this specification is incorrect as it leaves out the views.9 A second formulation belongs to MEUCCI (2008)10 and SALOMONS (2007) who correctly relate the prior to the expected returns conditional on the views. The latter specification is intui‐ tive because the investors are assumed to arrive at the expected returns by updating their initial set of expected equilibrium returns with the views. Alternatively to Bayesian infe‐ rence, MANKERT (2006) uses traditional statistics to identify the posterior mean and arrives at an interesting conclusion mentioned in 4.1.1. In sampling theory, is regarded as an unknown constant which is estimated using the maximum‐likelihood method.

In the derivations they replace the equilibrium returns with the views to arrive at the correct result. See SALOMONS (2007, p. 45). 10 Note that this author uses a modified relation where the true mean of the views varies around its realizations, which cannot be correct. This will become clearer in subsection 3.2. However, this modification does not impact the later course of the derivation as the parameters can be reshuffled to the correct form.

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Ultimately, HEROLD et al. (2002) conclude that it seems to be advantageous to employ Bayesian models in portfolio selection, when the i.i.d. assumption is violated, as this re‐ duces volatility and turnover and leads to a slight outperformance.

The Derivation of the Model

3.1

The Prior and the Reference Model 1, … ,

The market consists of assets , i.e.

. The vector of returns11

for

these assets is given by: ,

(3.1)

is invertible and positive semi‐definite. In the B‐L model it

The covariance matrix

is assumed to be a known constant12. BLACK et al. (1992) argue that the market is not nec‐ essarily at equilibrium, but continuously moving towards the equilibrium. Hence, the ex‐ pected returns

are state variables and, as defined in HE et al. (1999a), they are

centered at the equilibrium values, so that ,

where variance

(3.2)

is a multivariate normally distributed random vector, with mean 0 and , representing the estimation error: ~

BLACK et al. (1992) claim that the covariance matrix of the expected returns should be pro‐ portional to the covariance matrix of the returns and thus define indicates the uncertainty in the CAPM prior. and

as a scalar13 which

are considered to be independent.

The CAPM prior has a multivariate normal distribution and is defined as

(3.3)

From this, the reference model is defined as 11

From this point on, the term “return” is taken to represent “excess return” throughout the thesis. The term “excess return” will be used only for emphasis. 12 This is similar to the Black‐Scholes formula for European options pricing. QIAN and GORMAN (2001) extend the B‐L model by specifying a conditional covariance matrix onto which views can be ex‐ pressed. LITTERMAN and WINKELMAN (1998) and RISKMETRICS GROUP (2001) propose a series of estima‐ tion techniques for the covariance matrix of returns. 13 A thorough discussion about this parameter is provided in subsection 4.1.1.

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(3.4)

Verifying the reference model in relation to results in the following conclusions. If the investor is certain about her estimate, then the reference model reduces to ~

(3.5)

This is intuitive, as the more certainty there is in the estimate the less it will fluctuate around its true value . If the investor asserts the estimate of the equilibrium returns to entirely inherit the uncertainty of the returns, then

1 and the reference model be‐

comes ~

, 2 .

(3.6)

The literature also specifies an alternative reference model which identifies the variance of the posterior expected returns with the variance of the prior returns14. Finally, the follow‐ ing assumptions can be denoted: Assumption. A1 The returns vector is multivariate normally distributed with mean and variance

.15

Assumption. A2 The covariance matrix is deterministic.

3.2

The Conditional

In addition to the estimate of the expected returns, the investor also has the choice to express her particular views about the individual assets. Furthermore, she can also express her uncertainty in the views. The views are expressed as expected returns and they can be of absolute nature, e.g. “Asset A will have a performance of X%.”, or of relative nature16, e.g. “Asset A will outperform Asset B by Y%.”, respectively “Assets A and B will underperform assets C and D by Z%.”. This indirectly implies that 1. See WALTERS (2008). GIACOMETTI, BERTOCCHI, RACHEV and FABOZZI (2005) propose an extension of the model to ‐stable distri‐ butions and the t‐student distribution. 16 Note that in this case the return per se is still an absolute value. 14 15

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The intuitiveness behind this representation lies in the conceptualization of the market portfolio as the initial portfolio of the investor and of the views as smaller portfolios that are added to the initial portfolio to form a new portfolio, which better reflects the expec‐ tations of the investor. The portfolio of an absolute view contains a weight of 1, i.e. 100%, in the concerned asset and weights of 0 in all the other assets. A portfolio of a relative view contains positive weights in the outperforming assets that sum up to 1 and negative weights in the underperforming assets that sum up to 1. This can be interpreted as a long position in a portfolio of outperforming assets financed by a short position in a port‐ folio of underperforming assets. The weights within these two portfolios are set either proportionally to the market weights, as in IDZOREK (2004), or by using an equal‐weighting scheme, as in SATCHELL et al. (2000). 1, … ,

The investor has the option to express views , i.e. that

. MANKERT (2006) argues

, but DROBETZ (2002) leaves unconstrained. The weights of the view portfo‐

lios are represented by the matrix

, where each row

represents the portfolio

weights for a specific view, i.e.

, where

The expected returns on the view portfolios are denoted by the vector

, such that,

according to HE et al. (1999a), ,

where variance

(3.7)

is a multivariate normally distributed random vector, with mean 0 and , representing the estimation error: ~

Consequently

(3.8)

‐ 11 ‐

expresses the uncertainty in the views and, as in BLACK et al. (1992), it is treated to be diagonal. This presupposes that the investor forms uncorrelated views.17 and

of the view portfolio

is the variance

is called its confidence. In Bayesian statistics

denoted as the precision, but when the matrix is singular, i.e. the views are certain, the value of the precision cannot be identified as it tends towards infinity. If

, the matrix

of the view portfolios loses its full rank and becomes non‐invertible as well. When the investor is certain about his views, goes to zero and the estimate of the views reduces to the true mean.

and

are independent: ~

Finally, the following assumption can be denoted: Assumption. A3 The returns vector

of the view portfolios is multivariate normally

distributed with mean and variance , where is a diagonal covariance matrix.18

3.3

The Posterior

Given that the prior and conditional distributions are specified, the posterior distribution of the expected returns will be derived considering the general case of uncertain views.19 The steps of this derivation are adapted from CHRISTODOULAKIS (2002) and SALOMONS (2007). Concisely, Bayes’ theorem is first applied to the problem, the probability density functions are then substituted and finally the intermediary result is manipulated to arrive at a form which identifies the distribution of the posterior expected returns. Theorem 3.1. When the views are uncertain, i.e. returns vector

, then the posterior expected

is multivariate normally distributed with mean |

(3.9)

and variance

(3.10)

From Bayes’ theorem the following relation is obtained 17

When , redundancies can occur and the views may become correlated. Subsection 4.1.2 dis‐ cusses the matrix further. 18 MEUCCI (2006) proposes an extension of the model to non‐normal markets and views. 19 The derivation of the mean posterior for the case with certain views is presented in Appendix B.

‐ 12 ‐

where pdf pdf

pdf

| pdf

pdf

is the probability density function of the posterior expected returns, |

the probability density function of the prior, pdf

ty function of

given and pdf

the joint probability densi‐

the normalizing constant. The prior information

contained in the market equilibrium is combined with the sample represented by the views to form the posterior expected returns. (3.3) and (3.8) conclude that the Gaussians of 1

pdf

det|

det| |

are stated respectively as:

1 2

exp

and

1 2

exp

Substituting into the Bayesian formulation yields pdf

det| | det|

exp

1 2

1 2 . 20

The exponential can be simplified as follows: exp

1 2

exp

1 2

· exp

where . 21 Hence, the posterior probability density function of the expected returns is given by

pdf disappears into the constant of integration. The details of this mathematical manipulation are presented in Appendix A.

‐ 13 ‐

pdf

det| | det|

1 2

· exp In terms of , the factor

exp

| |

and the exponential exp

disappear into the constant of integration, resulting in pdf

exp

1 2

It follows that the posterior expected returns vector

. |

is multivariate normally dis‐

tributed with mean |

and variance |

3.4

Reformulations of the Posterior

The mean and the variance of the posterior expected returns can be rearranged to yield a more intuitive representation. There seem to be two possible ways to restate the mean and one way to restate the variance. The first reformulation of the mean belongs to MANKERT (2006). She shows that |

. 22

(3.11)

The posterior expected returns are equal to a sum consisting of the expected equilibrium returns and a second expression. The second, less intuitive, expression is interpreted as a revision of the implied expected returns. The second reformulation of the mean belongs to CHRISTODOULAKIS (2002). He indicates that |

can be rearranged from (3.9) as |

From 22

The mathematical transformations that lead to this formulation are presented in Appendix C.

‐ 14 ‐

is interpreted to be the regressor of , and the vector of the coefficients to be esti‐ .23 The mean of

mated. The least squares estimate of is then given by the posterior expected returns is then formulated as |

(3.12)

and can be interpreted as a normalized weighted average of the expected equilibrium re‐ turns and the least squares estimate of the expected returns. The normalized weights sa‐ tisfy the unity condition24. This representation is akin to the representations of shrinkage approaches, such as Bayes‐Stein or Pástor. If the investor is confident about his views, i.e. is large, then there is more weight given to the view estimate of the expected returns. Alternatively, when the investor feels more confident about the CAPM prior, i.e. is small and

is, respectively, large, then there is more weight given to the expected equili‐

brium returns. The rearrangement of the variance is provided by WALTERS (2008). The reformulated va‐ riance of the posterior expected returns is given by |

. 25

(3.13)

WALTERS (2008) claims that this formula is computationally more stable than the initial one because it does not require the inversion of the matrix , but it does require the number of the views to equal the number of the assets, i.e.

. Interpreting equation (3.13)

leads to the conclusion that any specification of a view reduces the variance of the post‐ erior, i.e. the revised estimate of the expected returns becomes more accurate when new information is accounted for.

3.5

Interpretations of the Posterior

A boundary analysis of the mean and variance of the posterior utilizing their standard and reformulated representations proves to be useful for testing the theoretical consistency of the B‐L model and for understanding its behavior when the fundamental inputs change.26

Note that a necessary condition for the least squares estimate is . This is because 1. Note that formula (3.9) can also be interpreted as a weighted average of and but without satisfying the unity restriction. 25 The mathematical transformations that lead to this formulation are presented in Appendix D. 26 Corollaries 3.1 to 3.4 follow from equations (3.11) and (3.13). The naïve corollary 3.5 follows from equations (3.9) and (3.10). For corollaries 3.2 and 3.5 see Appendix E for more details. 24

‐ 15 ‐

Corollary 3.1. When the views are absent, i.e.

0 , the mean of the posterior expected

returns reduces to the equilibrium risk‐premiums and the variance of the posterior ex‐ pected returns reduces to the variance of the prior expected returns

This corollary shows that the CAPM prior is considered to be a neutral reference point in the B‐L model and that the statement of opinions is regarded as an imbedded option. Corollary 3.2. When the views are certain, i.e.

0, the mean of the posterior expected

returns reduces to |

lim

and the variance of the posterior expected returns reduces to zero. This corollary is consistent with the result of Appendix B. If is of full rank, it implies that |

, which is a restatement of equation (3.7) under the circumstance of

the corollary. It entails that the views under the posterior equal the actual forecasted views. The new posterior expected returns are thus given by

which shows that they are a function solely dependent on the expressed views. This means that the information stemming from the equilibrium, and hence also from , is irre‐ levant. The second implication of this corollary is that the variance of the posterior returns shrinks towards the value of

. Together with the first implication a new vector

of posterior returns with higher precision is defined as |

Corollary 3.3. When the views are completely uncertain, i.e.

∞, the mean of the

posterior expected returns reduces to the equilibrium risk‐premiums and the variance of the posterior expected returns reduces to the variance of the prior expected returns

Corollary 3.3 leads to the observation that when investors are completely uncertain about their opinions, their views become irrelevant and the model only considers the informa‐ tion coming from the, less uncertain, prior. Corollary 3.4. When the equilibrium is certain, i.e.

0, the mean of the posterior ex‐

pected returns reduces to the equilibrium expected returns and the variance of the post‐ erior expected returns reduces to zero. 27

Note that this new mean is equivalent to the least squares estimate of equation (3.12).

‐ 16 ‐

The intuitiveness behind this corollary is that when the equilibrium is specified with cer‐ tainty any view becomes inappropriate and is ignored by the model. Due to a certain long‐ run equilibrium only the risk of the short‐term dynamics are inherited and the variance of |

the posterior returns decreases to

. ∞, the mean of the

Corollary 3.5. When the equilibrium is completely uncertain, i.e. posterior expected returns reduces to lim

and the variance of the posterior expected returns reduces to lim

The mean of corollary 3.5 is higher than the initial mean due to a lower decrease in the numerator than in the denominator, when that

1. If is of full rank, this corollary implies

and, in regarding the mean, is equivalent to corollary 3.2. A

further implication is that the complete uncertainty of the equilibrium risk‐premiums re‐ duces the precision of the posterior. Per se, a completely uncertain equilibrium does not entirely correspond to certain views, as in corollary 3.2. The mean from corollary 3.5 is indeed equal to that of corollary 3.2, but the variance of corollary 3.5 reduces to the va‐ riance implied by the views, whereas in corollary 3.2 it reduces to that implied by the equi‐ librium. The new vector of posterior returns is defined as

. Only when the views are explicitly defined as certain will the corollary 3.5 be a complement of corollary 3.2.

3.6

The Posterior Portfolio and View Weights

This subsection provides the optimal portfolio weights when the unconstrained mean‐ variance optimization is implemented. This optimization form is motivated by the fact that the intuitiveness of the B‐L model only becomes apparent in the absence of constraints.28 After a mathematical rearrangement of the portfolio weights, the weight vector of the view portfolio weights will be identified and interpreted. The reference model from equation (3.4), combined with the posterior result from equa‐ tions (3.9) and (3.10), states, as in HE et al. (1999a), that 28

See IDZOREK (2004, p. 19). For an inclusion of constraints see HE and LITTERMAN (1999b, p. 18).

‐ 17 ‐

(3.14)

where |

The unconstrained mean‐variance optimization problem entails max where δ

0, and the optimal portfolio weights are given by 1

(3.15)

From HE et al. (1999a)29, the optimal portfolio weights can intuitively be rewritten as 1

where

and τ

where

(3.16)

1 δ

1 , δ

(3.17)

such that

It should be observed that the unconstrained efficient frontier resulting from equations (3.15) and (3.16), respectively, is linear. If

is a constant and not a function of δ, as in

equation (2.2), then the resulting frontier of equation (3.16) would be hyperbolic, tangent to the formerly described linear efficient frontier and suboptimal. The tangency point would be given by the posterior portfolio at the initial average risk‐aversion. The reformulation of the optimal portfolio weights shows that the investor’s optimal port‐ folio is composed of the market equilibrium portfolio and of a weighted sum of the portfo‐ lios forming the views, which are then normalized by the factor 1 29

Note that HE et al. (1999a) do not restrict the matrix to be diagonal.

. is called the

‐ 18 ‐

weight vector of the view portfolio weights, or, more concisely, the views weights vector. When there are no views specified, the optimal portfolio weights are reduced to ⁄ 1

. This is due to the fact that, as HE et al. (1999a) claim, when the inves‐

tor is unsure about the CAPM prior she will not be entirely invested in the equilibrium portfolio.30 can also be interpreted, at this point, as the magnitude of the confidence in the expressed views and

⁄ as the level of uncertainty in these views. The vector that

weights the view portfolios can be interpreted as follows: The first term is an increasing function of the strength of the views. The strength can ei‐ ther be expressed through a higher expected return of a view or with a lower level of uncertainty in the views. The higher the term is the more weight it carries in the optimal portfolio. The second and third terms represent penalizations of the weight of a view, due to the correlation between the view portfolio and the market equilibrium portfolio and between the view portfolio and other view portfolios respectively. These penalizations seem right as the market portfolio and the other view portfolios already contain a part of the infor‐ mation that is brought new by that view. HE et al. (1999a) articulate two properties of the view weights . To that end, an additional view

1 denoted as

is considered. The new views weights vec‐

tor is then given by and the weight on the additional view by where

and

. Ensuing from the uncon‐

strained portfolio weights in equation (3.15), the implied expected returns of the first views are given by

.31 From these specifications the following properties are

articulated: 30

The remainder is invested in a risk‐free asset, which has a certain expected return and zero variance. Note that the implied expected returns of the view portfolios can also be written as .

Property 3.1. Since

‐ 19 ‐

0, the weight on the additional view portfolio

will

have the same sign as the difference between the expected return of the view and its im‐ plied expected return. This property denotes that when a view, with an expected return lower than the implied expected return, is specified, then the model will tilt the optimal portfolio away from the bearish asset(s). The contrary is true when the expected return of the view is higher than its equilibrium value, whereas when the expected return of the view is identical to its equi‐ librium value, the model remains unresponsive to the view. Property 3.2. The weight of a view is an increasing function of its expected return . The absolute value of is an increasing function of its precision

From this property it follows that a view has a stronger influence on the optimal portfolio weights either when the absolute value of its expected return is high or when it enters the model with low uncertainty. From DROBETZ (2002) a third property is inferred: Property 3.3. An absolute or relative view j modifies the view weight and leaves the view weights of the uncorrelated views unaltered. This property implies that an uncorrelated view only has an impact on the asset(s) to which it is related. If a view is correlated with other views, it will also impact the weights of the assets to which the other views are related. An uncorrelated view will also influence the posterior returns of all the assets to the degree to which the returns of these assets are correlated. WALTERS (2008, p. 14) claims that if

, then the weights are tilted towards assets with

lower variance and away from assets with higher variance.32 KOCH (2005, p. 37) and IDZOREK (2004, p. 19) caution that, when the optimization is unconstrained, relative views do not affect the unity33 of the optimal weights, whereas absolute views do.

The tilts of the weights from their equilibrium values are represented by . This vector will hence be denoted as the weights tilts vector. Please note that the tilts are an effect of the view weights. 33 Unity in this case is understood as 100%⁄ 1 . The reason for this was explained above.

‐ 20 ‐

Theoretical and Practical Aspects

4.1

Model Inputs

In the search for an intuitive quantitative model which should be easily applied into prac‐ tice, Black and Litterman have also produced some less intuitive facets of the model which can act as an obstacle for the practical implementation. 4.1.1

The Parameter

In defining the variance of the prior expected returns, BLACK et al. (1992) used a heuristic manner. They have assumed that the expected value of the stochastic returns is in itself a state variable. They have further assumed that the long‐run dynamics of the market inhe‐ rit the volatility of the short‐run dynamics, which move the market towards equilibrium in a linear fashion. The scale to which this attainment takes place is comprised in the para‐ meter .

is therefore the covariance matrix of the prior expected returns. Black and

Litterman claim the scalar to be in the vicinity of zero, because the uncertainty regarding the estimate of the return must be lower than the uncertainty of the return itself. None‐ theless, they have omitted to provide a method for deriving this parameter. The confusion about is amplified by the fact that other authors, such as SATCHELL et al. (2000), claim that this parameter is often set to 1.34 From the discussion in section 3, is either a scalar of uncertainty in the equilibrium prior or a level of certainty in the conditional, called the weight‐on‐views. It thus represents a trade‐off between the confidence in the prior expected returns and the confidence in the views. A straightforward way to define the scalar is setting it equal to 1⁄ , where is the num‐ ber of sampled periods. This intuitively follows from the law of large numbers and is equivalent to saying that the uncertainty in the prior is inversely proportional to the num‐ ber of the sampled periods.35 In this case, is close to zero and not greater than 1.36 Using the sampling theory approach to derive the mean of the posterior expected returns, MANKERT (2006) has found that and

⁄

, where

is the sample size of the prior

the sample size of the conditional. She explains that the scalar is a ratio between

Moreover, SATCHELL et al. (2000) adjust the B‐L model to cope with a stochastic scalar and hence a stochastic volatility. 35 is also interpreted as the standard error of the expected equilibrium returns, such that 1⁄ . See IDZOREK (2004, p. 14). 36 When the latter is true, corollaries 3.5 and 3.10 become unrealistic.

‐ 21 ‐

the samples observed by the investor and the samples observed by the market. But WALTERS (2008) indicates that this approach is only valid when the alternative reference model is used. HE et al. (1999a) claim that the level of uncertainty

⁄ should be equal to

and

. As HE et al. (1999a) claim and SALOMONS (2007) and

assume the relation

WALTERS (2008) prove, this leads the scalar to be eliminated from equation (3.9), as only the ratio ⁄ enters the model, and thus there is no need for the specification of any more when considering the alternative reference model. This assumption is however another heuristic approach which excludes an important parameter of the model. WALTERS (2008) cites a less stringent relation, i.e. as

1 enters into the model

, where

1⁄ 1

and hence represents a substitute for .

Conclusively, it is not the meaning of which is condemned, as the intuition behind it is obvious, but the deficiency in specifying an intuitive value for it. The main advantage be‐ hind this parameter is the flexibility it confers to the optimization process. The same ratio‐ nale applies to the matrix , which is discussed next. 4.1.2

The Matrix

The definition of the certainty of the views presents another impediment to the practical implementation of the B‐L model. As for the scalar, BLACK et al. (1992) do not describe a method for specifying the covariance matrix . One alternative of computing the covariance matrix of the view portfolios is described above. In accordance with assumption A2, this relation is modified as

diag

But as already pointed out, this is a poor assumption. A second alternative, proposed by DROBETZ (2002), uses the normality assumption and in‐ volves the specification of a confidence interval

·Φ

;

·Φ

where α is the significance level and Φ · the cumulative distribution function of the nor‐ mal distribution. The investor is required to define a confidence level and the boundaries of the interval in order to arrive at the variance of the view portfolio. This method may seem to be more intuitive to practitioners, as analyst forecasts usually span over an inter‐ val. A third alternative suggested by WALTERS (2008) is based on modeling the views via a multi‐ factor model of the form

∑

, where is the return of asset ,

are

‐ 22 ‐

the factor loadings,

the return due to the ‐th factor loading and an independent

normally distributed residual. This method suggests that the variance of the residuals can be used to obtain the variance of the view portfolios. The variance of the residuals is ob‐ tained from

, where is the vector of the factor loadings and

the vector of the factorial returns. Given this relation, the variances of the residuals can be obtained. A fourth alternative belongs to IDZOREK (2004) and is similar to the second alternative, in that it requires the specification of a confidence level for each view portfolio, but adopts a different approach. It starts from the idea that the confidence level of a ‐th view portfolio %

is defined as in %

tor,

, where

is the equilibrium weights vec‐

the weights vector under view certainty and the vector including the confi‐

dence level of a single view . The confidence level is included only in the vector entries which are related to the view.

is obtained from the optimization by computing the

mean of the posterior expected returns for each view separately37 as and the variance as Thus, a series of estimated weights vectors

is obtained for each view. The variances

result iteratively from minimizing the sum of squared differences between the estimated weights

and the actual weights

function of

, i.e. min ∑

. .

0, where

is a

. An advantage of this method is that it is insensitive to the scalar , since it

is held constant during the process.

4.2

Further Developments

The B‐L model has proved itself to be a solid platform for an enhanced theoretical as well as practical framework. Several noteworthy developments are presented next. 4.2.1

Implied Views

By modeling subjectivity into the B‐L model, BLACK et al. (1992) have reasoned that this characteristic can be used to follow a reverse procedure in the investment process. Rather than trying to find the optimal weights for a portfolio, an investor who already has an ex‐ isting portfolio with implicit views39 can identify these views with certainty by relating to a 37

The reason for this is to account for assumption A3.

Where

This does not necessarily mean that the investor does not know the views of the portfolio.

‐ 23 ‐

particular benchmark. The reasoning of this procedure is to verify whether the benchmark is a good reference for the investor to adhere to, by examining if the views implied by the benchmark correspond with the views of the investor. This reasoning is based on the as‐ sumption that the tracking error of the portfolio is regarded as a risk. Following the steps from Appendix F, the optimal unconstrained portfolio weights of the benchmark are 1

The implied views are obtained by replacing the weights of the benchmark optimal weights of the investor’s portfolio

B with the

and then solving for the views. The latter

step can be performed in two ways. If the investor wishes to obtain the expected returns of the views for a set of specific view portfolios the following equation is employed ,

(4.1)

where represents the view portfolio weights of the specific views40 and

the implied

expected returns of these views. If the naturally implied views are to be obtained then the following equation is considered ,

where

(4.2)

represents the implied view portfolio weights and has a full rank. Due to re‐ 41

verse optimization the following relation holds

and hence

. The latter relation also follows from corollary 3.2. Given that the number of unknowns is equal to the number of equations,

and

can be ob‐

tained. 4.2.2

A Consistency Measure

In the investment practice, an asset management firm uses a firm‐wide forecasting model to set up a uniform basis for its portfolios. Because a portfolio manager acting on a certain market niche is allowed to specify particular views on the market, the resulting portfolio can depart inconsistently from the forecasting model and damage the brand name of the 40

The specific views can also be the initial views specified in the portfolio.

This relation holds if and . Also note that the equilibrium values of the market in which the investor is invested must correspond to the equilibrium values of the (correct) benchmark; if this necessary condition is broken then the benchmark is unsuitable. Corresponding views between the investor and the benchmark are a sufficient condition for the benchmark to be suitable.

‐ 24 ‐

asset management firm. This has motivated FUSAI and MEUCCI (2003) to define a consisten‐ cy measure in accordance with which a portfolio manager should actively allocate. They propose to specify the measure as a probability of the consistency, presuming that the forecasting model is true. In the B‐L model, the forecasting model is identified with the equilibrium. The z‐score of the probability is given by the Mahalanobis distance |

which measures the standardized distance of the posterior from the prior and is distri‐ buted as a chi‐square with degrees of freedom.42 The consistency measure is defined as ,

where

(4.3)

· is the chi‐square cumulative distribution function with degrees of freedom.

The probability

is a decreasing function in , meaning that if the manager has speci‐

fied extreme views then she will likely be below the firm‐wide threshold. To be consistent with the equilibrium the manager has to identify the extreme views and modify these in order to reach the threshold. For this reason, FUSAI et al. (2003) advise that the sensitivity of the probability to the views should be computed as | |

The views with the highest absolute values of marginal probabilities are prone to be changed. If the marginal probability is positive (negative) the expected return of the view should be increased (decreased). FUSAI et al. (2003) claim that this consistency measure is advantageous because it cannot be misled by the allocation, since it accounts for the cor‐ relations between the assets, and because the distribution of the Mahalanobis distance is independent of the investor’s utility. 4.2.3

The UBS Warburg Approach

SCOWCROFT and SEFTON (2003) have proposed an alternative to the B‐L model which should relax the difficulty of specifying difficult inputs. Their alternative undertakes a different manner in identifying the returns. The two models are nonetheless conceptually equiva‐ lent as both integrate the Bayesian framework. The UBS Warburg approach presupposes 42

Note that FUSAI et. el (2003) use the alternative reference model. WALTERS (2008) provides the mod‐ ified equations when the reference model is used.

‐ 25 ‐

that investors specify returns as absolute next period values relative to the equilibrium, instead of expected returns as in the B‐L model. The returns vector for a market consisting of assets is given by

and defined as ,

where

is a multivariate normally distributed random vector, with mean 0 and

variance , representing the next‐period excess return over the long‐run equilibrium: ~

The returns are multivariate normally distributed as ~ where

. SCOWCROFT et al. (2003) conversely assume that the long‐run equilibrium

is known with certainty. Similarly to the B‐L model the investor has the option to express absolute or relative views, however in the form of an excess return of the next period over the expected equilibrium return. The views for the next period are given by

and

defined as , where

is the matrix of the view portfolios and

is a multivariate normally

distributed random vector, with mean 0 and variance , representing the error terms: ~ where

and

are independent of each other. Conversely to the B‐L mod‐

el, the views have a conditional dimension and a higher variance than that intrinsically implied by their error terms. The conditional views vector is multivariate normally distri‐ buted as |

From Bayes’ theorem it follows that the combination of the prior returns with the views yields a multivariate normally distributed posterior returns vector with mean |

and variance |

‐ 26 ‐

The expected excess returns for the next period relative to the equilibrium are denoted by . Compared with the B‐L model, the UBS Warburg approach presents several analogous characteristics. When the views are expressed with certainty, i.e.

0 as in corollary

3.2, the views under the posterior equal the actual forecasted views, i.e. . If the views are absent, i.e.

0 as in corollary 3.1, or uncertain, i.e.

∞ as in

corollary 3.3, the posterior returns are reduced to the equilibrium returns and the ex‐ pected excess returns are equal to zero. The advantage of the UBS Warburg approach arises from the disposal of confusing para‐ meters such as the scalar in the B‐L model. But the simplistic approach comes at a cost, as it does not integrate the reality of an uncertain equilibrium and trades‐off the benefits from using the scalar. A further shortcoming comes from the less intuitive specification of the returns.

4.3

Aspects of Practical Implementation

Without the practical usefulness, the theoretical consistency is not sufficient for a model to attest its value. This subsection is intended to shed light on the practical implementa‐ tion of the B‐L model. 4.3.1

Strategic and Tactical Portfolio Positioning

The recent controversy about the efficient market hypothesis has signaled the possibility for significant market outperformance and encouraged the implementation of tactical asset allocation decisions.43 The encapsulation of a long‐run equilibrium, serving the stra‐ tegic asset allocation, and of conditioning subjective short‐term views, serving the tactical asset allocation, recommends the B‐L model as an appropriate instrument for tactical port‐ folio positioning. In assessing the advantages of the B‐L model against a given benchmark or portfolios stemming from other portfolio selection models, the literature generally founds the out‐of‐sample performance analysis on the systematic fact based methodology rather than the discretionary alternative. The variables commonly substantiated to deliver an increased potential of predictability include macroeconomic variables or market indica‐ tors which better reflect business cycles44. 43

See FERSON and HARVEY (1991). E.g. the term spread, the change in industrial production or the credit spread.

‐ 27 ‐

The empirical results of SALOMONS (2007) confirm the suitability of the B‐L model as a tac‐ tical asset allocation tool against the classical approach of portfolio optimization and points towards an outperformance relative to a benchmark. The criterions she uses in the comparison with the classical M‐V approach include the Sharpe ratio, the hit ratio, higher moments and portfolio turnover. She finds out that the B‐L portfolio features a better risk‐ adjusted performance with relatively low portfolio turnover and a higher hit ratio and that the returns of this portfolio seem to be normally distributed, whereas the returns of the classical M‐V portfolio exhibit positive skewness and significant platikurtosis. Salomons also finds out that both portfolios outperform the benchmark, which does not necessarily speak in favor of the B‐L model, and argues that this outperformance strongly depends on the choice of the benchmark. She finally claims that the B‐L approach possesses the quali‐ ties of an enhanced indexing model. MERTENS and ZIMMERMANN (2002) implement a conditional version of the IAPT and instru‐ mental variables based on OLS regression to define the time‐varying expected returns. They indicate that when the B‐L model is used in an international investment context the decisions concerning currency hedging matter more than the conditioning macroeconomic tactical asset allocation strategy, but once the latter conditions are specified tactical asset allocation can add value. This conclusion is natural because currency hedging is part of strategic asset allocation. The authors also reveal that the classical M‐V portfolio behaves badly in the tactical context since it does not account for estimation risk. MARTELLINI and ZIEMANN (2007) investigate the consequences of implementing tactical port‐ folio positioning in an alternative investment universe through the B‐L model. Due to the non‐normal characteristics of hedge fund returns, they extend the B‐L model to a setting where higher moments are accounted for both in the CAPM prior as well as in the risk measure, which is identified with the value‐at‐risk. The conditioning views are obtained from an active style allocation procedure based on macroeconomic and market variables. When the minimum value‐at‐risk portfolio is used as a benchmark, the B‐L portfolios, spe‐ cified with different levels of the scalar, are outperforming and display higher Sharpe ra‐ tios and lower value‐at‐risk measures relative to the benchmark. The scalar also proves to determine the level of departure from the benchmark and, in this case, of the outperfor‐ mance, as increases in are associated with higher tracking errors and information ratios. But transaction costs are excluded from this investigation and are suspected to have signif‐ icant effects on the outperformance.

‐ 28 ‐

4.3.2

Behavioral Finance

In treating the usability of the B‐L model, MANKERT (2006) argues that the benefits and de‐ triments of the model should also be identified from a behavioral standpoint and shifts the perspective from quantitative finance to behavioral finance. She identifies two features of the model that are related to behavioral finance: i) the equilibrium and ii) the views uncer‐ tainty. Traditional finance, onto which the B‐L model is constructed, assumes an entirely concave utility function whereas in reality investors guide their preferences according to a utility function which is convex in the domain of losses. The implication of this is that gains and losses are judged relative to a point of reference or a benchmark45, which in the B‐L model is identified with the equilibrium, and that investors act as risk‐seekers on the negative domain of the utility. This only speaks partly in favor of the B‐L model since it still pre‐ sumes a traditional utility function and, consequently, risk‐seeking, i.e. loss‐averse, inves‐ tors may not find the posterior return of the model to completely reflect their intuitive feelings. The former is due to the fact that biases, such as regret, the status‐quo bias, the endowment effect or the herd behavior, are not represented on an entirely risk‐averse utility function. A loss‐averse investor will not assess the expected return, delivered by the traditional utility, to be, in relation to risk, high enough as to compensate her to risk leav‐ ing the status‐quo and the herd, falling behind the benchmark and feeling regrets. The second feature of the B‐L model is associated with the problem of overconfidence. This problem is exacerbated when confidence intervals have to be estimated and entails that overconfident investors have a lower expected utility than rational investors. Mankert argues that an overconfident investor who assigns a matrix with relatively low variances will also assign a high weight‐on‐views. Assuming that the confidence levels of each view are specified with similar overconfidence, it is possible to counteract this problem by set‐ ting a lower weight‐on‐views. Consequently, this problem can be solved in an agency con‐ text where the portfolio manager or security analyst, who are identified as the agent, spe‐ cify and the investment supervisory committee, which is identified as the principal, spe‐ cifies . The use of a consistency measure, as in subsection 4.2.1, may be an alternative solution. The home bias is another consequence of overconfidence that enters the B‐L

The relative point is located in the kink of the utility function.

‐ 29 ‐

model and which disappears in classical portfolio optimization due to the use of historical data. Ultimately, the incorporation of the equilibrium and the flexible structure of the model make provide the B‐L approach with an advantage over classically optimized portfolios, but its consideration of entirely risk‐averse investors is not compliant with reality. 4.3.3

The Experience at Goldman Sachs

BEVAN and WINKELMANN (1997) of Goldman Sachs offer an insight on how the B‐L model was implemented in the company which has pioneered it. The investment universe they use includes fixed income and currency markets of industrialized countries whereas the benchmark is associated with a global government bond index. The asset allocation process they implemented over a time span of three years is mainly focused on estimating the equilibrium inputs and determining the views confidence, setting target risk levels and calibrating the results. The equilibrium excess returns over the local risk‐free rate are reversely determined from the Sharpe ratio by using a proprietary estimated covariance matrix from historical daily returns. A Sharpe ratio with unitary value is used and motivated empirically with historical data and according to a statistical reasoning, which assumes normally distributed excess returns46. The views confidence is interpreted as having a macroeconomic dimension, through the weight‐on‐views, and a microeconomic dimension, through the confidence levels of the individual views. The weight‐on‐views is set consistently with an information ratio47 not higher than 2, which is interpreted as an unlikely two‐standard‐deviation event. This is motivated by the fact that a weight‐on‐views that would shift the portfolio towards a higher value of the IR would imply returns that are statistically improbable to occur. The scalar is adjusted by computing an anticipated IR based on the initial posterior returns and then recursively attuning it until the desired IR is attained. The result is a weight‐on‐views of between 0.5 and 0.7. Similarly, the confidence levels are determined in accordance with a SR of 2. A view resulting in a SR higher (lower) than 2, i.e. 3 (1), is associated with a low (high) confidence level, whereas the views that fall in between are considered to be of medium confidence. The inclusion of equilibrium information in the two methods produc‐ es results that avoid extreme views. Bevan and Winkelmann however note that “the inten‐ 46 47

A SR of 1 is interpreted as a one‐standard‐deviation event occurring two thirds of the time. The IR is defined as the expected excess return over the benchmark divided by the TE.

‐ 30 ‐

tion is not to suppress the deliberate expression of strongly held views” but to, conversely, bear risks where these are well compensated and in accordance with the strongest fore‐ casts.48 The target risk levels that are input in the optimization process are based on two measures typically used for indexed funds: the tracking error and the market exposure relative to the benchmark. The tracking error is set, commonly with industry practice, at 1, i.e. 100 basis points, whereas the market exposure is determined according to the views on general market direction. Neutral views with respect to market direction involve an exogenously set market exposure of 1, whereas in the case of directional views the market exposure is left undefined and the portfolio unconstrained to this respect. Finally, subsequent to the optimization, the risk is decomposed in order to make sure that chances are taken mostly in the positions with strongly held views. The intention is also to ascertain that risk is well diversified across the views such that the risk stemming from one view, relative to total TE, is restricted to an upper bound of 20%. The requirement of an even balance between the TE of the bond positions and that of the currency positions adds further diversification. A recalibration of the inputs is necessitated when these criteria are not satisfied. The experience of Bevan and Winkelmann of using the B‐L model in a global fixed income context has confirmed the benefits of risk‐control innate to the model and the prospect of an outperformance, which in this case amounted to 0.52% and hence an IR of 0.52.

An Exemplifying Application

5.1

Introducing the Data

This section is intended to exemplify the characteristics and the properties of the B‐L model discussed earlier. However, its role is not to replicate an enhanced investment process but to concentrate on the workings of the B‐L model in a simple and intelligible manner. For this reason, the simplifying assumption of reducing the investment universe to twelve risky assets and the risk‐free asset, respectively, is made. The twelve risky assets were selected as to provide a good potential for diversification and represent the major industrial sectors of the broader investment universe. From these, a market weighted benchmark was constructed. The time series involved belong to the U.S. 48

They also denote the iterative character of the process in relation to the exogenous forecasts.

‐ 31 ‐

market and span over 241 months, from August 1988 to August 2008.49 The risky assets are highly capitalized corporations commonly included in major stock market indices, such as the Dow Jones Industrial Average or the S&P 500. The risk‐free asset is assumed to be the three‐month U.S. Treasury bill. End of month prices are used for the risky assets whe‐ reas the risk‐free rate represents the monthly average. Table 14 from Appendix G contains general company specific information and Figure 1 illustrates the evolution of the time series. The equilibrium weights are computed as average historical market weights over the re‐ spective time span. The returns of the risky assets represent excess returns over the risk‐ free rate of the corresponding month. The prior expected returns and the covariance ma‐ trix stem from logarithmic historical data. The values represented next and used subse‐ quently in the optimization are of discrete nature. Table 1 includes various statistical prop‐ erties of the sample. All twelve risky assets are, according to the Jarque‐Bera test, normally distributed at a significance level of 13.29%, whereas the null‐hypothesis regarding the benchmark is only accepted at a significance level of 1.39% or below. The results for the twelve risky assets endorse the normality assumption A1. All the expected returns exhibit positive values. Table 1: Sample Statistical Properties. Table 15 and Table 16 from Appendix H present the covariance and correlation matrices, respectively, for the annual returns used in the optimization. The relatively low correla‐ tions of the assets support the potential of a well diversified portfolio. Table 2 provides the annual expected equilibrium returns and the equilibrium weights, the latter pointing to‐ wards a fairly well allocated market portfolio with no extreme weights. The average risk‐aversion parameter, given the annualized excess return and volatility of the market, is also included in the above table alongside with the scalar. Because BLACK et 49

The sources of the time series are the U.S. Federal Reserve and Yahoo! Finance.

‐ 32 ‐

al. (1992) argue that should be close to zero it is hence reasonably set at 0.05. Using the sample to determine the scalar would yield a value of 0.0042, which indicates a relatively high confidence in the equilibrium. Table 2: Prior Information.

5.2

Comparison of Classical M‐V Portfolios with B‐L Portfolios

This subsection touches on the differences between the classical approach and the B‐L approach to optimizing portfolios. The case of absent views is treated first. Conversely to classical portfolio optimization, the investor is not obliged to specify any views about the assets if she does not have any. Her expected returns vector will then be represented by the expected equilibrium returns vector. Curtailing the sampling problem represents an advantage of the B‐L portfolios over classically M‐V optimized portfolios. The expected returns used in the two optimization approaches as well as the expected equilibrium returns are depicted in Table 3. The expected returns of the classical M‐V op‐ timization equal the historical returns, whereas those of the B‐L optimization are equal to the expected equilibrium returns due to absent views. Table 3: Expected Excess Returns with Zero Views. Table 4 presents the optimal portfolio weights resulting from the two optimization ap‐ proaches. Both optimizations were performed including either no constraints or a budget constraint. The classical portfolios exhibit extreme weights relative to the B‐L portfolios, especially in XOM and PG on the long‐buy side and GM on the short‐sell side. The B‐L port‐ folios differ slightly from their equilibrium values. The unconstrained B‐L portfolio is not fully invested in the risky assets due to an uncertain CAPM prior. If would have been set at 0 its weights would have equaled the equilibrium weights.

‐ 33 ‐

Table 4: Optimal Portfolio Weights with Zero Views. The second case considered includes a relative and an absolute view. The investor first expects XOM to outperform GM by 2% and secondly WMT to have a performance of 14.20%. When using the classical approach she redefines the vector of historical returns to match these views by holding the spreads between each asset not involved in the views and the other assets, as well as the historical market return unchanged.50 The view portfo‐ lios and the expected returns on the views used in the B‐L optimization are defined as 1

and, respectively,

2% . 14.20%

In expressing the views the investor has also assumed the following confidence intervals 1%; 5%

√

10.20%; 18.20%

·Φ

; 2%

√ ·Φ

14.20%

·Φ .

and, respectively,

; 14.20%

·Φ

From these assumptions the diagonal covariance matrix of the views is given by 0.00033 0 . 0 0.00059 Table 5 depicts the redefined classical expected returns and the means of the posterior expected returns. The methodology the investor has used reflects her views indeed, but since she did not account for the correlations between the assets these expected returns are deficient. In contrast, the B‐L vector of expected returns is similarly suited to cope with the views, but it also accounts for the correlations between the assets as well as the level of uncertainty in the prior. Table 5: Expected Excess Returns with 2 Views. The resulting optimal portfolio weights are shown in Table 6. The classical portfolios are concentrated mainly in five assets, i.e. XOM, PG, MSFT, WMT and JNJ. The allocations of these portfolios are unintuitive as, relatively to the previous case, the weight given to the 50

In computing the spreads and the market return the equilibrium weights were used.

‐ 34 ‐

outperforming asset XOM is reduced whereas the one given to the underperforming asset GM is increased. Additionally, the unconstrained weights of the assets not included in the views have also changed. Conversely, the B‐L portfolios are well behaved and intuitive.51 The intuition is obvious when considering the unconstrained B‐L portfolio, as the weights of the assets affected by the views change correspondingly and those of the other assets remain unaltered. This result enforces property 3.3. Table 6: Optimal Portfolio Weights with 2 Views. Table 7 reveals that the changes in the optimal weights from the case with zero views to the case with 2 views are larger for the classical portfolios than for the B‐L portfolios. Table 7: Changes of Optimal Portfolio Weights from Zero Views to Two Views.

5.3

Comparison of Different B‐L Portfolios

Subsequently to the differences of the B‐L model to classical portfolio optimization, the properties of the optimal weights and further aspects discussed in subsection 3.6 are ex‐ emplified next. From these properties, property 3.3 was already highlighted above. The first considered case includes two relative views. The first view expresses an outper‐ formance of XOM over GM of 4%. The second view involves an outperformance of MSFT and IBM relative to BAC and JPM of 1.44%. The weightings of the view portfolios are pro‐ portional to the equilibrium weights. The variances of the view portfolio returns are com‐ puted using the confidence intervals 2%; 6%

√

0.56%; 3.44%

·Φ

1.44%

; 4%

√ ·Φ

·Φ .

and, respectively,

; 1.44%

·Φ

From these assumptions the diagonal covariance matrix of the views is given by 51

Appendix I illustrates the optimal allocations of the four portfolios over an interval of risk classes. This interval is constant for all figures and defined as 0.625, ∞ . The figures support the relatively higher diversification of the B‐L portfolios over the classical M‐V portfolios.

‐ 35 ‐

0.00088 0 . 0 0.00015 The resulting means of the posterior expected returns are shown in Table 8. All the ex‐ pected returns change correspondingly, the most influenced being GM. Table 8: Expected Excess Returns with Two Relative Views. Table 9 presents the resulting optimal weights of the two B‐L portfolios. The weights of the assets affected by the first view change noticeably, but the weights of the assets affected by the second view remain unchanged. The latter outcome is due to the fact that the ex‐ pected return of the second view was specified as to correspond to its implied expected return. Hence, only the weights tilts of MSFT, IBM, BAC and JPM and the weight of the first view are non‐zero.52 This supports property 3.1. Table 9: Optimal Portfolio Weights with Two Relative Views. When the expected return of the second view is increased to 5% and the significance level of the first view is increased to 0.495 the covariance matrix of the view portfolio returns is given by 2.54635 0 . 0 0.00009 These modifications entail that the strength of the first view decreases with a higher va‐ riance of the view portfolio return, whereas the strength of the second view increases with a higher expected return of the view. Table 10 shows that the reaction of the B‐L model is to leave the mean of the posterior expected returns of the assets involved in the first view relatively unchanged from their equilibrium values and to alter those of the assets in‐ volved in the second view in the corresponding direction. Table 10: Expected Excess Returns with Two Modified Relative Views. 52

Note that the weights tilts and the view weights are related to the unconstrained optimal portfolio weights.

‐ 36 ‐

According to Table 11 the first view looses strength due to weights tilts and a view weight close to zero while the second view gains strength by receiving weights tilts and a view weight that are noticeably different from zero. These results support property 3.2. Table 11: Optimal Portfolio Weights with Two Modified Relative Views. The second case considers the inclusion of two additional absolute views to the first mod‐ ified case. The third view expresses a performance of JPM of 10% whereas the fourth view involves a performance of GE of 7%. The variances of the view portfolio returns are com‐ puted using the confidence intervals 7%; 13%

10% 4%; 10%

·Φ 7%

; 10% ·Φ

·Φ .

; 7%

and, respectively, ·Φ

From these assumptions the variances of the absolute views are given by and

. 0.00033

0.00033, respectively. The means of the posterior expected returns are de‐

picted in Table 12 and are intuitive with respect to the expressed views. The expected re‐ turn of the unconstrained portfolio with four views amounts to 6.97% and is lower than the expected return of the unconstrained portfolio with two modified views by 5.89%. This lower expected return is due to the two pessimistic absolute views. Table 12: Expected Excess Returns with Four Views. Table 13 entails that any specification of absolute views will force the weights of the un‐ constrained portfolio to depart from unity. The third view also leads to a modification of all the view weights implied by the second view. On the one hand, the weight tilt of BAC, which has a lower volatility than JPM, increases by 1.42%, whereas the weight tilt of JPM decreases by 4.05%. On the other hand, the weight tilt of IBM, which has a lower volatility than MSFT, decreases by 1.07%, which is lower than the decrease in the weight tilt of

‐ 37 ‐

MSFT of 1.52%. These results indicate that the B‐L model tilts the weights towards assets with relatively lower volatilities.53 Table 13: Optimal Portfolio Weights with Four Views.

Conclusion

The classical mean‐variance methodology for portfolio selection did not have the hoped for impact in the investment practice. The sampling difficulties, the highly concentrated and sensitive portfolio allocations and the estimation risk are the causes for this. These shortcomings have led Black and Litterman to create an intuitive and feasible model which should circumvent these shortcomings. The B‐L model solves the first two problems of classical portfolio optimization and reduces the estimation risk. The assumptions underlying the B‐L model are to an extent inherited from the mean‐ variance model and to another extent particularly defined. The existence of arbitrage op‐ portunities is an overriding assumption of the B‐L model with respect to modern financial theory. The keystone of the B‐L model is to blend the information stemming from the market equilibrium with conditioning proprietary views in order to redefine the expected returns. The equilibrium is treated as an initial starting point defined by the market, whe‐ reas the expression of views represents an option to the individual investor. The market is not assumed to be at equilibrium on the short‐run, but constantly moving towards it. Another inherent feature of the model is the specification of levels of confidence both in the prior as well as in the conditional. The parameter which scales the equilibrium uncer‐ tainty is also defined as a weight‐on‐views, as it trades off the certainty in the prior versus the confidence in the conditional. The boundary analysis of the posterior expected returns relative to the views and the equilibrium reveals that the workings of the B‐L model are theoretically consistent. The posterior weights, resulting from the mean‐variance optimi‐ zation, possess intuitive properties. These properties are typically observable in an uncon‐ strained optimization. 53

Appendix J contains the unconstrained and constrained efficient frontiers, together with the relevant portfolios, for the case with two modified views and the case with four views when δ 0.625, ∞ .

‐ 38 ‐

The B‐L model also has two unattractive facets, namely the missing methods to specify intuitive values for the parameter and the matrix . A series of techniques are consi‐ dered in the literature to evade this problem. The framework onto which the model is build has also allowed several developments to be made, which should enhance the usa‐ bility of the B‐L model and the concept underlying it by means of extensions or alternative model specifications, respectively. The incorporation of an uncertain equilibrium and of a Bayesian inference enables the B‐L model to be an attractive instrument for strategic as well as tactical portfolio positioning. The practical experience indicates that the model can deliver consistent results and that it has an attractive innate risk‐control mechanism. From the standpoint of behavioral finance, the implementation of a benchmark and a flexible structure in defining the confidence of the prior and the conditional recommends the B‐L model to be more realistic than the classical mean‐variance approach. Its assumption of entirely risk‐averse investors is however not compliant with reality. The exemplifying application has evidenced the sampling, concentration and sensitivity shortcomings of classically optimized portfolios and demonstrated that the Black‐ Litterman portfolios, regarding both the mean of the posterior expected returns as well as the optimal posterior weights, are well behaved and intuitive.

‐ 39 ‐

Appendix Appendix A In line with CHRISTODOULAKIS (2002) and SALOMONS (2007), the exponential can be simplified as follows: exp

1 2

exp

1 2 , ,

exp

1 2

exp

1 2 2 .

Let ,

, , .

‐ 40 ‐

The exponential can thus be rewritten as: exp

1 2

exp

1 2

exp

1 2

exp

· exp exp

1 2

· exp 1 2

exp

because and are both

Note that

1 2

· exp

1 vectors.

Given the fact that , the exponential can be rearranged as exp

1 2

· exp

1 2

‐ 41 ‐

Appendix B ~

The solution to the optimization problem with certain views, i.e.

, 0 , follows the

lines of non‐linear constrained optimization, which is methodically different from the technique using Bayesian statistics, but conceptually the same, as the constraints have a conditioning character. The reasoning is to minimize the Mahalanobis distance, i.e. deter‐ mining the optimal estimate of the expected returns which minimizes the variance of around the expected equilibrium returns and satisfies the linear constraints. The deri‐ vation of this problem is inspired from CHRISTODOULAKIS (2002). The objective function and the constraints are stated as min

. .

The Lagrangian is given by , where

is the vector of Lagrangian multipliers. Hence, the first order conditions are 0

0 0 2

2 2

0 ,

. 0

‐ 42 ‐

0 Solving equation (3.9) with respect to leads to 2

and substituting into equation (3.10) results in 2

Substituting back into equation (3.9) to obtain the optimal estimator for entails 2

0 .

‐ 43 ‐

Appendix C Starting from the mean of the posterior expected returns, as in equation (3.9), the following matrix manipulations, according to MANKERT (2006, p. 40), are performed to yield an alternartive formulation: |

, , , ,

(C.1)

, , , . Note that this manipulation requires that is of full rank. This condition can be avoided starting with equation (C.1) as follows:

‐ 44 ‐

‐ 45 ‐

Appendix D Starting from the variance of the posterior expected returns, as in equation (3.10), the following matrix manipulations are performed, according to WALETRS (2008, p. 37), to result in an alternative formulation: |

‐ 46 ‐

Appendix E 0, starting from equations 3.11 and, respectively,

When the views are certain, i.e.

3.13, the following results can be obtained for the mean and, respectively, the variance of the posterior expected returns: |

and |

∞, starting from equation (3.9) the following

When the equilibrium is uncertain, i.e.

result can be obtained for the mean of the posterior expected returns: lim

lim

lim lim

1 ,

‐ 47 ‐

Appendix F From corollary 3.2, the expected posterior returns and, respectively, the variance of the posterior returns of the benchmark, under view certainty, are given by | |

From equation (3.15), the unconstrained optimization process delivers the optimal portfo‐ lio weights of the benchmark as 1

and from this it also results that .

‐ 48 ‐

Appendix G

Table 14: Company Specific Information.

Figure 1: Evolution of Time Series (Aug. 1988 – Aug. 2008).

‐ 49 ‐

Appendix H

Table 15: The Covariance Matrix.

Table 16: The Correlation Matrix.

‐ 50 ‐

Appendix I

Figure 2: Classical Unconstrained Stacked Weights for

0.625, ∞ .

Figure 3: B‐L Unconstrained Stacked Weights for

0.625, ∞ .

‐ 51 ‐

Figure 4: Classical Constrained Stacked Weights for

0.625, ∞ .

Figure 5: B‐L Constrained Stacked Weights for

0.625, ∞ .

‐ 52 ‐

Appendix J

Figure 6: The Unconstrained Efficient Frontiers and Relevent Portfolios.

Figure 7: The Constrained Efficient Frontiers and Relevant Portfolios.

‐ 53 ‐

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Ehrenwörtliche Erklärung Versicherung U

Ich versichere hiermit, daß ich die vorliegende Arbeit selbständig und nur unter Benutzung der angegebenen Literatur und Hilfsmittel angefertigt habe. Wörtlich übernommene Sätze und Satzteile sind als Zitate belegt, andere Anlehnungen hinsichtlich Aussage und Umfang unter den Quellenangaben kenntlich gemacht. Die Arbeit hat in gleicher oder ähnlicher Form noch keiner Prüfungsbehörde vorgelegen und ist nicht veröffentlicht. Ort, Datum: ________________________ Unterschrift: ________________________