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TRANSCRIPT
Asset Allocation under a
Conditional Diversification
Measure
d-fine GmbH
Kellogg College
University of Oxford
A thesis submitted in partial fulfillment of the requirements for the M.Sc. in
Mathematical Finance
September 30, 2011
Contents
1 Introduction 2
2 Principal Component Techniques for Asset Allocation 6
2.1 A Basic Framework for Portfolio Optimisation . . . . . . . . . . . . . 7
2.1.1 The Portfolio Model . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 The Optimization Framework . . . . . . . . . . . . . . . . . . 9
2.1.3 Example: Classical Mean-Variance Selection . . . . . . . . . . 10
2.2 Principal Component Techniques . . . . . . . . . . . . . . . . . . . . 11
2.2.1 The Classical Approach . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Portfolio Analysis of Principal Portfolios . . . . . . . . . . . . 12
2.2.2.1 Transformation Scheme . . . . . . . . . . . . . . . . 12
2.2.2.2 Characteristics of the Efficient Points . . . . . . . . . 13
2.2.3 Conditional Principal Component Decomposition . . . . . . . 15
2.2.4 Portfolio Analysis of Conditional Principal Portfolios . . . . . 17
2.2.4.1 Transformations . . . . . . . . . . . . . . . . . . . . 17
2.2.4.2 Characteristics of the Efficient Points . . . . . . . . . 19
2.2.4.3 Consideration of Inequality Constraints . . . . . . . 20
2.2.5 Theoretical Framework for Conditional Principal Component
Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 20
i
2.2.5.1 Existence of a Conditional Principal Component De-
composition . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.5.2 Computation of Conditional Principal Component De-
compositions . . . . . . . . . . . . . . . . . . . . . . 22
3 An Entropy Diversification Risk Measure 27
3.1 Detour: Entropy Measures and Diversification . . . . . . . . . . . . . 28
3.2 Diversification Risk Measure . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Portfolio Analysis of Diversified Portfolios . . . . . . . . . . . . . . . 32
4 Numerical Results 35
4.1 Allocation Results of Diversified Selections . . . . . . . . . . . . . . . 35
4.1.1 The Allocation Strategy . . . . . . . . . . . . . . . . . . . . . 36
4.1.2 The Investment Universe . . . . . . . . . . . . . . . . . . . . . 36
4.1.3 Portfolio Selection using PCA . . . . . . . . . . . . . . . . . . 39
4.1.3.1 Principal Portfolios . . . . . . . . . . . . . . . . . . . 39
4.1.3.2 Statistical Measures for Assets and Principal Portfolios 40
4.1.3.3 Efficient Portfolios . . . . . . . . . . . . . . . . . . . 41
4.1.4 Portfolio Selection using cPCA . . . . . . . . . . . . . . . . . 45
4.1.4.1 Principal Portfolios . . . . . . . . . . . . . . . . . . . 45
4.1.4.2 Statistical Measures for Assets and Conditional Prin-
cipal Portfolios . . . . . . . . . . . . . . . . . . . . . 47
4.1.4.3 Efficient Portfolios . . . . . . . . . . . . . . . . . . . 47
4.2 Proof of Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.1 Back-Testing Strategies . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Strategy I: Yearly Rebalancing, 2002 - 2010 . . . . . . . . . . 51
4.2.3 Strategy II: Monthly Rebalancing, 2002 . . . . . . . . . . . . . 53
4.2.4 Strategy II: Monthly Rebalancing, 2009 . . . . . . . . . . . . . 53
4.2.5 Back-Testing Results . . . . . . . . . . . . . . . . . . . . . . . 54
ii
5 Conclusions and Outlook 55
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
A Mathematical Proofs 59
A.1 Proof of Theorem 2.2.3 (Alternative II): . . . . . . . . . . . . . . . . 59
A.2 Proof of Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
B Numerical Details 64
B.1 Asset Allocation using PCA . . . . . . . . . . . . . . . . . . . . . . . 64
B.1.1 Mean-Variance Weights on Original Assets . . . . . . . . . . . 64
B.1.2 Mean-Diversification Weights on Original Assets . . . . . . . . 64
B.1.3 Mean-Variance Weights on Principal Portfolios . . . . . . . . . 65
B.1.4 Mean-Diversification Weights on Principal Portfolios . . . . . 66
B.1.5 Principal Portfolio Weights . . . . . . . . . . . . . . . . . . . . 66
B.2 Asset Allocation using cPCA . . . . . . . . . . . . . . . . . . . . . . . 68
B.2.1 Mean-Variance Weights on Original Portfolios . . . . . . . . . 68
B.2.2 Mean-Diversification Weights on Original Portfolios . . . . . . 69
B.2.3 Mean-Variance Weights on Conditional Principal Portfolios . . 70
B.2.4 Mean-Diversification Weights on Conditional Principal Portfolios 71
B.2.5 Conditional Principal Portfolio Weights . . . . . . . . . . . . . 71
B.3 Statistical Examinations . . . . . . . . . . . . . . . . . . . . . . . . . 73
B.3.1 Distribution Parameter . . . . . . . . . . . . . . . . . . . . . . 73
B.3.2 Proof of Identical Distributions . . . . . . . . . . . . . . . . . 75
B.3.2.1 European Market . . . . . . . . . . . . . . . . . . . . 76
B.3.2.2 Asian Market . . . . . . . . . . . . . . . . . . . . . . 78
B.3.2.3 American Market . . . . . . . . . . . . . . . . . . . . 80
B.3.3 Proof of Independent Random Variables . . . . . . . . . . . . 81
iii
List of Figures
4.1 Time series of European indices Source: yahoo.finance.com . . . . . . . . . . . 37
4.2 Time series of Asian indices Source: yahoo.finance.com . . . . . . . . . . . . . 38
4.3 Time series of American indices Source: yahoo.finance.com . . . . . . . . . . . 38
4.4 Principal portfolios using PCA . . . . . . . . . . . . . . . . . . . . . . 39
4.5 Principal sector portfolios using PCA . . . . . . . . . . . . . . . . . . 40
4.6 Multivariate statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.7 Efficient portfolios using PCA . . . . . . . . . . . . . . . . . . . . . . 42
4.8 Allocation weights using cPCA (target return 0.018 %) . . . . . . . . 44
4.9 Diversification distributions using PCA . . . . . . . . . . . . . . . . . 45
4.10 Conditional principal portfolios using cPCA . . . . . . . . . . . . . . 46
4.11 Conditional principal sector portfolios using PCA . . . . . . . . . . . 46
4.12 Multivariate statistics cPCA . . . . . . . . . . . . . . . . . . . . . . . 47
4.13 Efficient points of the risk return diagramm (cPCA approach) . . . . 48
4.14 Allocation weights using cPCA (target return 0.025 %) . . . . . . . . 49
4.15 Diversification distributions using cPCA . . . . . . . . . . . . . . . . 50
4.16 Considered index time series over 10 years . . . . . . . . . . . . . . . 51
4.17 Time series of the principal portfolios over 10 years . . . . . . . . . . 52
4.18 Back-test of Strategy II, 2002 . . . . . . . . . . . . . . . . . . . . . . 53
iv
4.19 Back-test of Strategy II, 2009 . . . . . . . . . . . . . . . . . . . . . . 54
A.1 Entropy plot for one dimension . . . . . . . . . . . . . . . . . . . . . 61
A.2 Diversification measure plots . . . . . . . . . . . . . . . . . . . . . . . 63
B.1 Distribution plots, European indices . . . . . . . . . . . . . . . . . . . 77
B.2 QQ-Plots plots, European indices . . . . . . . . . . . . . . . . . . . . 77
B.3 Distribution plots, Asian indices . . . . . . . . . . . . . . . . . . . . . 79
B.4 QQ-Plots plots, Asian indices . . . . . . . . . . . . . . . . . . . . . . 79
B.5 Distribution plots, American indices . . . . . . . . . . . . . . . . . . . 80
B.6 QQ-Plots plots, American indices . . . . . . . . . . . . . . . . . . . . 81
B.7 Scatter plots, European indices . . . . . . . . . . . . . . . . . . . . . 82
B.8 Scatter plots, Asian indices . . . . . . . . . . . . . . . . . . . . . . . . 82
B.9 Scatter plots, American indices . . . . . . . . . . . . . . . . . . . . . 83
v
Abstract
In this thesis we consider the problem of diversifying investments in com-
mon market securities under certain restrictions, such as budget con-
straints, etc. Therefore we adapt the entropy diversification measure as
well as the conditional principal component decomposition, both proposed
by Meucci (2009). We derive a rigorous and powerful theoretical frame-
work describing the geometry of the conditional principal components,
which particularly allows us to prove the existence of such decomposi-
tions. Furthermore, we apply numerical tests to selected index securities,
compare two approaches of portfolio selection (mean-variance vs. mean-
diversification) and illustrate the differences that arise between the effi-
cient frontiers. A propagated aim in this thesis is the asset allocation of
mutually uncorrelated portfolios, which are naturally given by the princi-
pal components. Thus, finally, we back-test several rebalancing strategies
based on a principal component decomposition and verify whether the
resulting portfolios are closely uncorrelated.
Chapter 1
Introduction
For several reasons, mutually uncorrelated portfolios are usually attractive investment
targets for asset allocation. This is primarily because such portfolios can be diversified
quite simply, as the allocation with the maximum diversification is just represented
by equal volatility-adjusted weights. A diversified allocation for correlated portfolios,
however, is normally not adaptable in such a way. From a mathematical point of
view, we simply use the fact that usually uncorrelated structures are represented by
diagonal matrices, so that complexity is perceptibly reduced whenever Multivariate
Stochastic or Linear Algebra is considered in that context. Against this background
it is worth the effort, for some cases, to project considered correlated markets on
uncorrelated structures.
The well-known principal component analysis (PCA), indeed, allows us a wholly gen-
eral and intuitive access to decorrelated structures since the resulting principal port-
folios are, per construction, uncorrelated (see Pearson(1901)). Previously Partovi
and Caputo (2004) proposed a (strategic) asset allocation based on the uncorrelated
principal portfolios, which are realizable portfolios whenever short-selling is allowed.
Substituting the principal portfolios into the Lagrange analysis, they provide a com-
pact view of the characteristics of efficient portfolios which are restricted by budget
constraints. However, the intuitive interpretation of this analysis framework becomes
complex if more general constraints are involved. This is because according to the
rank dimension of the constraints, the same amount of dimensions regarding an op-
timal solution is fixed before an optimization has even been started. As constraints
are not considered by the PCA, these fixed solution parts tend to be disregarded, so
that a distinction between determined and optimizable components is not possible,
at least not directly.
2
Meucci (2009) proposed a conditional principal component analysis (cPCA), that
considers any linearized equality constraints for the construction of the, so-called,
conditional principal portfolios. This technique allows the elimination of linear equa-
tions and maintains the advantages of a PCA-like framework. Applying cPCA, the
Lagrange analysis of optimization problems can be decomposed nicely, so that even
under complex equality constraints an intuitive interpretation is possible.
Even for the case that an investor has the complete knowledge of the uncorrelated
portfolios describing a considered market; the investor is still facing the problem
of finding a suitable diversification measure. Several diversification measures are
proposed in the literature but, similar to the quest of choosing a suitable risk measure,
there is no common understanding of a unique quantitative description. However, the
qualitative definition is accepted as a portfolio is diversified if its concentration risk
is minimized. Whereas concentration risk is understood more or less for credit risk,
in terms of market risk, there is no clear statement.
A valuable approach to describing the risk concentrations is also proposed by Meucci
(2009). He uses an active risk contribution to express the volatility concentrations
component-wise. The volatility concentrations are stated as the weighted first deriva-
tives of the portfolio volatility, differentiated by the principal portfolios. Normalizing
the volatility concentrations leads to a diversification distribution, which can be in-
terpreted as probability-like masses. The probability character, moreover, allows the
utilization of the entropy measure. The entropy function, furthermore, proves to be
a powerful measure, still accounting for the concentration by components, even af-
ter adding up to the total value of the scalar measure. That latter property is an
essential characteristic of the entropy function, so that the maximization of these
forces the diversification distributions to uniformity, which is the maximal achievable
diversification grade.
In the following chapters we examine the practicability of tracing uncorrelated mar-
ket structures and their diversification. One aim is the formulation of a quantitative
road map that provides the construction of uncorrelated portfolios under given con-
straints. For this, we follow the methodology proposed in Meucci (2009). His paper
contains the derivation of the entropy diversification measure as well as of the con-
nected diversification distribution. Both are constructed via principal components.
In addition, the conditional principal component decomposition is presented as an
important extension for restricted portfolio selections1. For applications a set of
1As far as known by the author, cPCA was proposed for the first time by Meucci.
3
practical mathematical programs is formulated that allow the conditional principal
component decompositions to be generated iteratively. Finally, a portfolio selection is
defined using the mean-diversification approach. Numerical results are also included.
Meucci’s publication is, thus, mainly focused on presenting the idea behind, the prac-
tical methodology as well as numerical results. The theoretical background is rather
barely described.
In this document we develop a complete theoretical background explaining the cPCA.
The analysis used leads to a compact notation and eventually to a clear formulation
of projection rules between the security space and the space of the principal port-
folios. By means of an alternative scalar product, defined by the covariance matrix
considered, we reveal the geometrical interpretation of these transformations. The
existence of a conditional principal component decomposition especially can thus be
proved easily. We present an alternative and more compact proof of the workability
with respect to the decomposition-generating mathematical programs proposed by
Meucci. Beside this we compare the results of mean-variance vs. mean-diversification
and investigate the characteristics of the corresponding efficient points using Lagrange
analysis. Within this theoretical framework, the kernel space of the constraint matrix
plays an important role because it allows the elimination of linear(ized) constraints.
Due to the elimination we are able to formulate an analytical and intuitive view of
efficient points.
In a second step we use the (conditional) principal portfolios to find a diversified
allocation based on the entropy diversification measure. We extensively analyze the
benefit of diversification as we compare numerical results of a portfolio selection using
the classical mean-variance (MV) approach from Markowitz (1952) and a portfolio
selection using Meucci’s mean-diversification (MD) approach. Finally, we will back-
test the stability of allocated uncorrelated portfolios over time and interpret the test
results.
The structure of this document is as follows:
Chapter 2 contains the main theory of principal component techniques, which is used
in the following chapters and sections. In the first section a compact survey to com-
mon portfolio theory and especially to the corresponding optimization framework is
given. The classical MV approach is introduced as example and will be hencefor-
ward the benchmarking case study for further contemplations. The second section
4
is dedicated to introducing the principal component theory starting with the classi-
cal PCA. Following the principle of Partovi and Caputo (2004), in Section 2.2.2 we
derive the MV-analysis of an efficient point characterized by the Lagrange criteria,
which are additionally substituted with the principal components. Similar analysis
views are made for the MV- as well as for the MD-approach based on the cPCA (see
Section 2.2.4 and Section 3.3). After the classical PCA the conditional PCA will be
discussed. It turns out that by means of a linear equality constraint matrix A, the
considered domain space can be separated into the kernel space K of A as well as a
complementary space R = IRn\K. This fact is used to obtain a powerful theoreti-
cal framework explaining the geometry of the cPCA. In the final section we discuss
an algorithm for the practical computation of the conditional principal portfolios,
which is proposed by Meucci (2009). However, due to the theoretical framework, we
are able to find more compact arguments explaining and proving that the algorithm
is well-defined. In addition we provide an theorem guaranteeing the existence of a
conditional principal decomposition.
The aim of Chapter 3 is to give a brief summary of Meucci’s entropy based diversifi-
cation measure, which is used for numerical results in Chapter 4. Following Meucci
(2009) we introduced the corresponding definitions, notation and terms very close to
his paper. A detour to the entropy function in general is given for a better under-
standing.
The numerical results are presented in Chapter 4. The first numerical experiments
compare the allocation results of the MV selection with the allocation results of a
MD selection. Thereby we use both techniques, the PCA as well as the cPCA. The
considered market is represented by selected main indices of the world. The second
numerical experiment examine the stability of market correlations respectively the
impact on the corresponding principal portfolios.
The appendix follows after some conclusions and outlooks. It contains the proof
of (non-)concavity regarding Meucci’s diversification measure and further numerical
results such as value tables and distribution checks for the indices used.
5
Chapter 2
Principal Component Techniquesfor Asset Allocation
The Principal Component Analysis (PCA) is a well-known and established theory,
repeatedly applied in statistics and other disciplines, whenever covariances, correla-
tions or similar data need to be decomposed into their eigenvalue components (see
[19]). The decomposition is quite useful to discover the main correlation drivers. Fur-
thermore - and this will be a fundamental result - we are going to use, the eigenvalues
themselves representing mutually uncorrelated portfolio allocations of the original
correlated portfolios. In fact, the eigenvectors are artificial but realizable portfolio
instances, namely the principal portfolios (see [18]). Depending on the traceability
of correlations, principal portfolios may allow powerful strategies for asset allocation,
whenever uncorrelated assets/funds are demanded and need to be provided, e.g., in
form of ETFs or as a translation console between a strategic and a tactical asset
allocation (SAA and TAA respectively).
As we shall recognize, the fact is that constraints to be considered are reducing the
freedom of allocation since the range space of the constraint derivative determines a
certain part of assets to be chosen. For the sake of separating these range effects,
we introduce a conditional principal component analysis (cPCA) proposed by Meucci
(2009). This technique allows us to differentiate between range and kernel effects.
A second important range of application of PCA and cPCA respectively shall be the
diversification theory presented in Chapter 3.
The aim of this chapter is to introduce the principal component techniques as well
as the mathematical notation we will need for the discussions below. In Section 2.1
6
we first give a short introduction to the basic formulation of optimizable portfolio
selection in a bi-criterial context. The classical Markowitz approach as an example
will furthermore be the comparison benchmark for the following diversification con-
templations. Section 2.2 is dedicated to introducing the notations of the PCA and
the cPCA.
For the MV approach and the MD approach as well as for the PCA and the cPCA ap-
proach each we will present the essential analysis of an efficient portfolio characterized
by its Lagrange functions.
2.1 A Basic Framework for Portfolio Optimisation
This section is about a general framework, which is valid for a wide range of portfolio
optimization approaches where the aim is to maximize the expected mean and to
minimize the risk simultaneously, which both are considered as objective functions
µ(w) and ρ(w). The framework is fully general with respect to the chosen objective
functions. Portfolio theory is widely used by banks and especially by asset managers.
The origin can be traced back to the famous Markowitz approach (see example in Sec-
tion 2.1.3). Important extensions are, e.g., the Capital Asset Pricing Model (CAPM),
proposed by Sharpe (1964), the Arbitrage Pricing Theory by Ross (1976) as well as
special views like Merton’s Mutual Fund Theory (1972). Today, portfolio theory is
usually enriched, considering further assumptions such as non-normal (joint) distri-
butions, copulas, fat tails, alternative risk measures, regime switching, robustness
etc.
2.1.1 The Portfolio Model
The basic notation for an investment framework shall be as follows:
A market is driven by n risk factors S1, . . . , Sn, which, in our context, are simple
securities or indices respectively. At time t an investor has to decide over unknown
absolute returns that comes out at time t + dt
dSit+td := Si
t+dt − Sit , i = 1, . . . , n (2.1)
7
where dt represents the investment horizon. The term Sit denotes the security level
of Si at time point t. The returns are assumed to be describable by a stochastic
distribution.
In most cases the distribution of logarithmic returns
rit+dt := ln
(
Sit+dt
Sit
)
(2.2)
are considered instead. Both return types (absolute and logarithmic) are equivalent
and can be transformed into each other. A common assumption is that the returns
are driven by a stochastic process and are independent of past events, such as
ridt =
(
µi −1
2σ2
i
)
dt + σi dW it . (2.3)
Since the investment interval dt is fixed and the returns are independent, the starting
time point is irrelevant, so we just write ri ≡ ridt and analogously dSi ≡ dSi
t+td.
The stochastic term dW it can be described in manifold ways. If dW i
t = W it+dt − W i
t
can be described as a Wiener process, ri is normal distributed and ri ∼ N (µi12σ2
i , σ2i )
holds. Then the relative returns
Ri ≡ dSi
Si:= eri − 1 (2.4)
where dSi
Si ∼ dSit+dt
Sit
, are log-normal distributed and, according to Ito’s Lemma, we
know thatdSi
Si= µidt + σidW i
t (2.5)
holds. Finally, the evolution of the security levels can be inferred as
Sit+dt = Si
teri
= Site
(µi−12σ2
i )dt+σidW it . (2.6)
A portfolio P is described by the weighted combination of securities, since wi repre-
sents the investment weight with respect to security Si. In the following sections, the
weights are interpreted as fractions, which are determined by the absolute investment
amount, normalized by the total investment budget, so thatn∑
i
wi = 1.
Knowing the expected return µi of the single securities Si, the expected return of
the portfolio can be calculated easily. Assuming, e.g., normal distributed security
logarithmic returns, the portfolio’s expected return is given by
µP =n
∑
i
wiµi. (2.7)
8
In terms of a vector notation, the equivalent term µP = wT µ will be used. The
portfolio’s volatility then is
σP =√
wT Σw, (2.8)
where Σ ∈ IRn×n denotes the corresponding covariance matrix of the considered re-
turns.
2.1.2 The Optimization Framework
Portfolio selection is mostly stated as a bi-criteria mathematical program, as long as
the expected return has to be maximized and the risk has to be minimized. Con-
sidering a general return function µ(w) as well as a general risk function ρ(w), the
bi-criteria portfolio optimization framework can be formulated as
minw∈C
F (w) = minw∈C
(
−µ(w)ρ(w)
)
(2.9)
The constraint set C shall be used in linear standard form
C = {x ∈ IRn |Ax = b, x ≥ l} (2.10)
for matrices A ∈ IRm×n, and vectors b ∈ IR
m, l ∈ IRn. Without loss of generality every
type of linear equality and linear inequality constraint can be transformed to this
standard form. Nonlinear inequality functions might also be allowed, but avoided
here, to maintain simplicity. Nonlinear equality functions, however, are not easily
adaptable. The reason for that will become clear when we consider the conditional
principal component analysis. Though, we may linearize the constraints by their
derivatives to utilize the local information. Nevertheless, a global approach with
nonlinear constraints is not provided absolutely.
For calculation reasons, usually, there are claimed further conditions to the risk func-
tions, which might be required as being convex and differentiable or even coherent
(see [2]).
The solutions for program (2.9) is the set of efficient points1, which is called the
efficient frontier2. The solutions can be approximated by several methods and tech-
niques. Common practices in a numerical manner are the following two:
1According to program (2.9), a vector F ∗ ∈ F = {F (w) |w ∈ C } ⊂ IRk is called an efficient point
or Pareto point of F if there exist no other vector F ∈ F with F ≤ F ∗ and Fi < F ∗i for at least one
index i ∈ 1, . . . , k2The (Pareto) efficient frontier of a set F is the set of Pareto points in F
9
Every solution of a bi-criteria optimization problem corresponds to a scalarized, single
criteria program
minw∈C
α µ(w) + (1 − α) ρ(w) (2.11)
combined with a certain scalarization factor α ∈ (0, 1).
The solution set can be discretized equivalently on a grid of target returns
minw∈C
ρ(w)
s.t. µ(w) = µ(2.12)
with respect to an open interval µ ∈ [µmin, µmax].
2.1.3 Example: Classical Mean-Variance Selection
The classical portfolio selection from Markowitz [17] is a mean-variance approach
(MV). Hence, the expected return of the portfolio is determined as
µ(w) = wT µ, (2.13)
and the portfolio risk is interpreted as the variance ρVΣ due to a covariance matrix Σ.
Set
ρ(w) = ρVΣ(w) ≡ wT Σw. (2.14)
Considering budget constraints and allowing unlimited short selling, the Markowitz
approach is completely represented by optimizing a bi-criteria optimization problem.
In terms of the discretized program (2.12) every point of the efficient set has its target
return µ, and solves
min ρVΣ(w)
s.t. Aw = bµT w = µ
(2.15)
Setting A = I1Tn and b = 1 represents especially the budget constraints. 3
3The vector I1n is a n-dimensional vector containing exclusively the value one as entry, i.e.,I1n := (1, . . . , 1). If the dimension is known we also write I1.
10
More extensions and variations are possible: The matrix A might also contain other
constraints. Furthermore, we can limit short selling by box constraints, e.g., −0.3 ≤w ≤ 1.3. Whenever short selling is not allowed 0 ≤ w ≤ 1 has to be set. Other
inequality constraints cl ≤ Cw ≤ cu can be added, too. In terms of the standard
constraints (2.10) inequality constraints can be substituted to equality constraints by
inserting slack variables s, so that, e.g., Cw + s = cu, s ≥ 0.
2.2 Principal Component Techniques
In this section we motivate the PCA and the cPCA and derive a consistent notation
for later usage.
2.2.1 The Classical Approach
A symmetric and positive definite covariance matrix Σ ∈ IRn×n can be factored as
ET ΣE = Λ ⇔ EΛET = Σ (2.16)
where Λ is a resulting diagonal matrix whose diagonal entries are the eigenvalues
(usually ordered as λ1 ≥ λ2 ≥ . . . λn), which are real numbers on the diagonal,
and E is an orthogonal matrix whose columns e1, e2, . . . , en are the corresponding
eigenvectors. This fact is a basic result of the Linear Algebra known as eigenvalue
decomposition or diagonalization of a square matrix.
Whenever unlimited short selling is allowed, the normalized eigenvectors themselves
are n realizable and furthermore uncorrelated portfolios. The eigenvalues indicate
the corresponding variances. In that context, the eigenvector ei is called a principal
portfolio.
Any portfolio w can be replicated as a linear combination of principal portfolios, so
that
Ew = w ⇔ w = E−1w (2.17)
holds and w stands for the vector of combination coefficients. Since E is orthogonal,
i.e., E−1 = ET , the matrix E−1 transforms a generic portfolio w to an equivalent
principal portfolio substitute w, which is an element of the vector space, where the
principal portfolios themselves are the orthogonal coordinates.
11
2.2.2 Portfolio Analysis of Principal Portfolios
The principal component decomposition implicitly defines a linear and bijective trans-
formation function
φ : IRN → R(E) φ(z) := Ez = z. (2.18)
where R(E) denotes the column range space of matrix E (see (2.35)). As a convention
we introduce the following notation. Each function f meets its transformed function
f as a composition of f as well as φ and so we write
f(z) := f(φ(z)) or f(φ−1(z)) = f(z)
respectively.
2.2.2.1 Transformation Scheme
According to the notation above, we get a transformation scheme in terms of the
general scheme, which is presented in Table 2.1.
Original TransformedExpected returns µ(w) µ(w)Risk measure ρ(w) ρ(w)
Constraints C C
Table 2.1: General transformation scheme
In case of the standard constraint set C as defined in (2.10), the transformed constraint
set C is determined as
C ={
x ∈ IRn∣
∣
∣Ax = b, Ex ≥ l
}
(2.19)
where A := AE.
Substituting these formulae into (2.12) leads to the equivalent program
minw∈C
ρ(w)
s.t. µ(w) = µ(2.20)
having an optimal solution w∗. The corresponding solution w∗ of program (2.12) is
simply given by the transformation φ−1(w∗).
12
As an application, consider the mean-variance program (2.15) which is transformed
as follows
Original TransformedExpected returns µT w µT wRisk measure ρV
Σ(w) ρVΣ(w)
Constraints Aw − b Aw − b
Table 2.2: Mean-variance transformations for PCA
with µ := ET µ and ρVΣ = ρV
Λ = wT Λw. Note, that the transformation functions
maintain the function results, i.e., ρVΣ(w) = ρV
Λ (w), µT w = µT w and Aw = Aw.
2.2.2.2 Characteristics of the Efficient Points
The aim of this section is to derive the analytical characteristic for an efficient so-
lutions of program (2.15) within the transformed space of principal components ac-
cording to the scheme in Table 2.2. The results will be compared with further results
of the cPCA which will be discussed Section 2.2.4.2.
The transformed program arises as
minw∈ IR
nwT Λw
s.t. Aw = bµT w = µ
(2.21)
The corresponding Lagrange function reads
L = wT Λw − (Aw − b)T ν − γ(µT w − µ) (2.22)
where ν ∈ IRm and γ ∈ IR are the Lagrange factors.
With ∂L∂w
= 2Λw − AT ν − γµ = 0 follows
w =1
2
(
Λ−1AT ν + γΛ−1µ)
(2.23)
Then, setting ∂L∂ν
= Aw − b = 0 leads to
b = Aw (2.24)
=1
2
(
AΛ−1AT ν + γAΛ−1µ)
(2.25)
13
Define D := AΛ−1AT and we obtain
ν = 2D−1b − γD−1AΛ−1µ (2.26)
Substituting ν back into Equation 2.23 reads
w =1
2
(
Λ−1AT[
2D−1b − γD−1AΛ−1µ]
+ γΛ−1µ)
(2.27)
= Λ−1AT D−1b +1
2γ
(
Λ−1 − Λ−1AT D−1AΛ−1)
µ (2.28)
and it remains only γ as unknown variable.
Finally, set ∂L∂γ
= µT w − µ = 0 and substitute w. It follows
µ = µT w (2.29)
= µT
[
Λ−1AT D−1b +1
2γ
(
Λ−1 − Λ−1AT D−1AΛ−1)
µ
]
(2.30)
= µT Λ−1AT D−1b +1
2γ µT
(
Λ−1 − Λ−1AT D−1AΛ−1)
µ (2.31)
Thus, we come to the result that
γ = 2 · µ − µT Λ−1AT D−1b
µT(
Λ−1 − Λ−1AT D−1AΛ−1)
µ. (2.32)
All together, the following terms were derived:
Variable Value
w 12
(
Λ−1AT ν + γΛ−1µ)
ν 2 · D−1b − γD−1AΛ−1µ
γ 2 · µ−µT Λ−1AT D−1b
µT (Λ−1−Λ−1AT D−1AΛ−1)µ
D AΛ−1AT
Table 2.3: Solutions of an efficient point using PCA
This result is not quite satisfactory for programs with general constraint matrix A,
although for simpler constraints, e.g. plain budget constraints a proper analysis was
made: Under unlimited short selling Partovi (2004) extracted two principal portfolios
representing the solution, which can be interpreted as analogue to Merton’s mutual
fund theory. However, even though one characterizes the solution by principal port-
folios, it still proves to be difficult to find a straight interpretation of the results.
14
2.2.3 Conditional Principal Component Decomposition
In this section we discuss the conditional principal component analysis (cPCA).
Meucci (2009) proposed this decomposition technique as an extension of the PCA
also considering linear equality constraints4, represented by the matrix A ∈ IRm×n
At the end of this section, an intuitive approach will result, providing powerful and
less complicated terms, which can be used for a natural interpretation of the solution
analysis.
The cPCA considers the constraint matrix A by separating the nullspace (or kernel
space) of A, namely
K ∼ K(A) = {x ∈ IRn |Ax = 0} dim (K) = n − m. (2.33)
and a complementary space
R ∼ IRn\K dim (R) = m. (2.34)
Initially, one can choose the row range space of A, which is the same as the column
range space of AT , reads 5
R(AT ) ={
AT y | y ∈ IRm
}
. (2.35)
Similar to the PCA the cPCA diagonalizes the covariance matrix S. Hence, both
subspaces need to be projected adequately. That can be achieved by applying a
Gram-Schmidt orthogonalization using the inner product, which is a scalar product
induced by the covariance matrix. The results are two bases of the spaces
K and R, (2.36)
which are mutually orthogonal regarding the alternative scalar product.
The space separation allows us to construct a conditional principal component anal-
ysis:
4The constraint matrix is assumed to have full row rank, otherwise the dimensions correspondto the actual rank
5From the Fundamental Theorem of Linear Algebra we know that K(A)⊥R(AT ) and K(A) ⊕R(AT ) = IR
n, where ⊕ denotes the orthogonal direct sum of two vector spaces. Hence, if a vectorspace is equipped with the Euclidean scalar product, an orthogonal space decomposition alwaysexists, which is induced by a matrix A.
15
Definition 2.2.1 Consider a positive definite matrix Σ ∈ IRn×n and a matrix A ∈
IRm×n having full row rank. A conditional principal component decomposi-
tion is given with a nonsingular matrix E ∈ IRn×n, which can be separated into two
submatrices E = (EK ER), so that,
• the columns of EK stand for a base of K(A)
• the columns of ER stand for a base of R
• E diagonalizes Σ, i.e., the equation
ET ΣE = Λ = diag(λ1, . . . , λn) (2.37)
and equivalently
E−T ΛE−1 = Σ (2.38)
holds.
The columns of EK are called the principal kernel portfolios. The columns of ER
are called the principal range portfolios.
The unique vectors w =(
wTK wT
R
)
satisfying
w = EKwK + ERwR (2.39)
are the kernel portfolio weights and range portfolio weights of w.
Note that the separation of w is well-defined because the conditional principal port-
folio weights can be separated as follows
w = Ew (2.40)
= (EK, ER)
(
wK
wR
)
(2.41)
= EKwK + ERwR (2.42)
Finally, Λ can be also separated analogously:
Λ =
(
ΛK 00 ΛR
)
A proof of existence of a conditional principal component decomposition will be given
in Section 2.2.5.
16
2.2.4 Portfolio Analysis of Conditional Principal Portfolios
Isolating the null space of A allows us to eliminate linear equality constraints from
a restricted optimization problem. This fact will play a special role for conditional
asset allocation, subject to budget constraints and other ones. The cPCA as defined in
Definition 2.2.1 leads to a deeper understanding of possible allocations. Furthermore,
the cPCA will be used to allocate feasible diversification capacities (see Chapter 3).
In this section we derive an elegant view of efficient principal portfolios subject to
linear equality constraints, namely Ax = b, whereby A ∈ IRm×n, m ≤ n. Nonlinear
constraints need to be locally linearized by its Jacobian matrix.
2.2.4.1 Transformations
Driven by Definition 2.2.1, the transformation function φ from Equation 2.18 can be
equivalently expressed as the sum of the kernel transformation
φK : IRn−m → K ⊆ IR
n, φK(zK) := EKzK (2.43)
as well as the range transformation
φR : IRm → R ⊆ IR
n, φR(zR) := ERzR, (2.44)
so that finally φ(z) ≡ φ(zK, zR) = φK(zK) + φR(zR).
For the linear equality constraints Aw = b, we get the following transformation:
b = Aw (2.45)
= Aφ(w) (2.46)
= A (EKwK + ERwR) (2.47)
= AERwR (2.48)
Define A := AER, then A ∈ IRm×m is non-singular since A has full row rank. Hence,
wR ≡ A−1b (2.49)
and so we can define the range part of w, which is fixed as
wR := φR(wR) = ERwR. (2.50)
17
The range part of w is thus well-known and determined by the equality constraints.
This fact allows us to redefine the transformation function to
φ(zK) = φ(zK, wR) = φK(zK) + wR (2.51)
Definition 2.2.1 reminds us that w has a particular structure, w =(
wTK wT
R
)
, which
shall be utilized now. Analogously to Section 2.2.2 we look again at the corresponding
transformation of program (2.15). According to this, the risk function ρVΣ will be
transformed as follows
ρVΛ (w) = wT Λw (2.52)
=(
wTK wT
R
)
(
ΛK 00 ΛR
)(
wK
wR
)
(2.53)
= ρVΛK
(wK) + ρVΛR
(wR) (2.54)
We define
ρVΛR
:= ρVΛR
(wR). (2.55)
Similarly, we derive the expected principal returns
µ(w) = µT Ew (2.56)
= µT(
EK ER
)
(
wK
wR
)
(2.57)
= µTKwK + µT
RwR (2.58)
whereby µR := ETRµ and µK := ET
Kµ. Again, the range part
µR := µTR (wR) (2.59)
is fixed due to the determined wR.
In summary, we get the terms of Table 2.4
Original TransformedExpected returns µT w µT
KwK + µR
Risk measure ρVΣ(w) ρV
ΛK(wK) + ρV
ΛR
Constraints Aw − b no constraints
Table 2.4: Mean-variance transformations for cPCA
18
2.2.4.2 Characteristics of the Efficient Points
Similar to Section 2.2.2, we examine the characteristics of efficient points. Therewith
the transformation of program (2.15) arises in the format
min ρVΛK
(wK) + ρVΛR
s.t. µTKwK = µ − µR
(2.60)
The Lagrange function appears as
L(µ) = wTKΛKwK + ρV
ΛR+ γ
(
µ − µR − µTKwK
)
(2.61)
Setting ∂L∂wK
= 2ΛKwK − γµK = 0 we read the optimal solution, which is
wK =γ
2Λ−1
K µK =γ
2ζK (2.62)
with ζK := Λ−1K µK.
Setting ∂L∂γ
= µ − µR − µTKwK = 0 gives us
µ − µR = µTKwK =
γ
2µTKζK (2.63)
so that
γ = 2µ − µR
µTKζK
(2.64)
and
w∗K =
µ − µR
µTKζK
ζK (2.65)
Summarized again, we are facing the terms in Table 2.5
Variable ValuewK
γ2ζK
γ 2 µ−µR
µTK
ζK
ζK Λ−1K µK
Table 2.5: Solutions of an efficient point using cPCA
The optimal variance reads
ρVΛ (w∗
K) =(µ − µR)2
(µTKζK)2
· ζTKΛKζK + ρV
ΛR(2.66)
=(µ − µR)2
(µTKζK)2
· µTKΛ−1
K µK + ρVΛR
(2.67)
=(µ − µR)2
(µTKζK)2
·n−m∑
i=1
(µK)2i (λK)−1
i + ρVΛR
(2.68)
19
The entries of ζK represent the inverted principal volatilities adjusted by the prin-
cipal expected returns, i.e., low volatilities and high returns will be preferred, high
volatilities and low returns will be avoided.
2.2.4.3 Consideration of Inequality Constraints
If inequality constraints w ≥ l are considered, after transformation they remain as
EKw ≥ l, so that the program reads
min ρVΛK
(wK) + ρVΛR
s.t. µTKwK = µ − µR
EKwK ≥ 0(2.69)
A stationary point can be characterized as Karush-Kuhn-Tucker conditions6, which
are stated as
−∇ρVΛK
(wK) + ETKs + νµK = 0 (2.75)
EKx ≥ l (2.76)
µTKwK = µ − µR (2.77)
sT EKx = 0 (2.78)
The stationary points need to be computed numerically, for instance with interior
point methods, SQP methods etc.
2.2.5 Theoretical Framework for Conditional Principal Com-ponent Decompositions
In this chapter we first show the existence of a conditional principal component de-
composition. The second section is dedicated to deriving an algorithm. As a result
of the decomposition algorithm we will obtain auxiliary mathematical programs on
an equivalent projection space.
6Under suitable regularity assumptions the local optimum of the program
min f(x) s.t. g(x) > 0, h(x) = 0 (2.70)
fulfills the Karush-Kuhn-Tucker conditions
−∇f(x) + ∇h(x)T ν + ∇g(x)T s = 0 (2.71)
h(x) = 0, g(x) ≥ 0 (2.72)
sT g(x) = 0 (2.73)
s ≥ 0 (2.74)
20
2.2.5.1 Existence of a Conditional Principal Component Decomposition
In this section we show that, under certain assumptions, a conditional principal com-
ponent decomposition exists. The following Theorem gives us the framework.
Theorem 2.2.2 (Conditional principal component decomposition)
Let S ∈ IRn×n be a symmetric and positive definite matrix. Furthermore, let A ∈
IRm×n be a matrix having full row rank, then a conditional principal component de-
composition exists as described in Definition 2.2.1.
Proof of Theorem 2.2.2: The matrix A represents a surjective linear map V →W , where V ∼ IR
n and W ∼ IRm. Choose the null space K(A) and the range space
R(AT ), which both are subspaces of V . Let us fix bases v1, . . . , vn−m of K(A) and
vn−m+1, . . . , vn of R(AT ). The bases together are spanning the vector space V such
that V = K(A) ⊕R(AT ).
The symmetric positive definite matrix S induces a scalar product
〈x, y〉S := xT Sy.
Two vectors are called S-conjugated if they are orthogonal with respect to the scalar
product 〈·, ·〉S.
In a vector space provided by a scalar product one can apply the Gram-Schmidt
orthogonalization and its conclusions 7. Doing so on the kernel base v1, . . . , vn−m
with respect to the scalar product 〈·, ·〉S results in a S-conjugated base e1, . . . , en−m
which is still a base of K(A).
7Consider the projection function projs(v) := 〈v,s〉〈s,s〉 s. Given a base v1, v2, . . . , vn, the Gram-
Schmidt orthogonalization transforms an arbitrary base recursively in an orthogonal base:
u1 = v1 (2.79)
u2 = v2 − projv1(v2) (2.80)
u3 = v3 − projv1(v3) − projv2
(v3) (2.81)
. . . (2.82)
uN = vN −N
∑
i=1
projvi(vn) (2.83)
The orthogonal base vectors ui can be also normalised by setting ei = ui
‖ui‖.
21
Afterwards, continue the S-orthogonalization by inserting the base vectors of the
range base vn−m+1, . . . , vn, starting with un−m+1 = vn−m+1 −n−m∑
i=1
projvi(vn−m+1) then
en−m+1 = un−m+1
‖un−m+1‖and so on. Note, that en−m+i /∈ K(A) still holds. After the
projection, the base vectors are also orthogonal with respect to the inner product
defined by S. However, en−m+i is no longer an element of R(AT ) but an element of
R ∼ IRn\K, what has to be shown.
In fact, the arbitrary chosen bases v1, . . . , vn−m and vn−m+1, . . . , vn can be S-orthogonalized
and normalized to vectors e1, . . . , en−m and en−m+1, . . . , en respectively. All pro-
duced base vectors are mutually orthogonal regarding the scalar product 〈·, ·〉S .
Hence, Equation (2.37) is fulfilled if the matrix E is set as (EK ER), where ER :=
(en−m+1 . . . en) and EK := (e1 . . . en−m) .
Due to the S-orthogonal property of the base vectors, they are linear independent as
well, and therefore E is a nonsingular matrix. ¤
The factorization (2.37) differs from the standard diagonalization as described in
Equation (2.16). The matrix S needs to be refactored by the inverse matrix of E
instead of its transposition, which is only possible for the case if E is an orthogonal
matrix. This, however, is not necessarily a given characteristic when the conditional
decomposition is applied. Furthermore, the columns of E are not necessarily eigen-
vectors of S. At least, they turns out to be generalized eigenvectors, i.e., Se = λBe,
where B := E−T E−1. This statement can be easily derived as we have
SE = E−T ΛE−1E (2.84)
= E−T Λ (2.85)
= E−T E−1EΛ (2.86)
= BEΛ. (2.87)
2.2.5.2 Computation of Conditional Principal Component Decomposi-tions
The aim of this section is to propose a practical algorithm to compute the conditional
principal decomposition. We follow the approach which is described in Meucci (2009).
However, the derivation is partly varied and enhanced with the theoretical background
we discussed above.
22
Before the actual algorithm is derived, a short detour is considered to motivate some
essential steps in the algorithm. Then the algorithm framework itself will be de-
scribed, followed by an auxiliary program, which, eventually, can be used for compu-
tation.
Detour: Eigenvalues results from quadratic optimization
The quadratic optimization problem
maxeT e=1
e⊤Se (2.88)
is a valid strategy to determine the maximum eigenvalue λmax of a square matrix
S. Although, quadratic optimization is not a recommendable approach because the
maximization of a convex function has local stationary points, which are not neces-
sarily the global maximums. However, optimization theory shall allow us to add and
to characterize further constraints intuitively regarding the eigenvalue problem.
A deeper understanding to this alternative eigenvalue approach can be achieved by
considering the Rayleigh quotient
τ(x) =xT Sx
xT x(2.89)
whose stationary values are the eigenvalues λmax = λ1 ≥ . . . ≥ λn = λmin of the
matrix S, so that λmax ≥ τ(x) ≥ λmin and the stationary vectors are the corresponding
eigenvectors.
On the other side, a stationary point of the quadratic program (2.88) is characterised
by fulfilling the first order condition
∂L∂x
= 0
of its Lagrange function
L(x, ν) = xT Sx − ν(
xT x − 1)
(2.90)
where∂L∂x
= 2Sx − 2νx (2.91)
Fulfilling the first order condition, thus, we are facing the original eigenvalue problem
Sx = νx, so that, similar to the Rayleigh quotient, each eigenvector together with its
eigenvalue is a stationary point of (2.88).
23
Indeed, both approaches are related since the Rayleigh quotient represents the fixed
points of Lx(ν) as L(x, τ(x)) = τ(x) holds.
The following reflections use conditional quadratic optimization problem to construct
the principal component decomposition in Theorem 2.2.2.
The algorithm
Given a matrix A ∈ IRm×n having full row rank. Regarding a matrix S ∈ IR
n×n we
recursively find the columns eK1 , . . . , eKn−m of EK (see the proof of Theorem (2.2.2))
by solving the eigenvalue-related programs
eKk := arg maxeT e=1
e⊤Se
s.t. Bk−1 e = 0(2.92)
for each k = 1, . . . , n − m, where BK0 := A
BKk :=
(
BKk−1
(
eKk)T
S
)
∈ IR(m+k)×n. (2.93)
The results are mutually S-conjugated and lie in the kernel space of A.
The remaining columns eR1 , . . . , eRm of matrix ER can be analogously determined by
setting BR0 = ET
KS and solving program (2.92) subject to the constraint matrices
BRj :=
(
BRj−1
(
eRj)T
S
)
∈ IR(n−m+j)×n, (2.94)
where j = 1, . . . ,m.
The mutually S-conjugated vectors eK1 , . . . , eKn−m of EK are linear independent and
they fully span the kernel space K. Forced by the constraint matrices BRj the results
eR1 , . . . , eRm of (2.94) are S-conjugates, too. For dimension reasons, they cannot be
elements of the kernel space K, so that they have to be elements of R.
According to Theorem 2.2.2 at least one feasible solution exists, thus, the program
is feasible. The algebraic multiplicity as well as the full row rank of A has impact to
the uniqueness.
Theorem 2.2.3 Suppose the quadratic optimization problem (2.92), subject to an
equality constraint matrix B, having full row rank and the objective function matrix S
24
is positive definite. Then the mathematical program’s set of stationary points equals
the set of stationary points of the auxiliary optimization program
maxeT e=1
eT P T SPe (2.95)
where
P := I − BT (BBT )−1B.
The following Lemma shall be helpful for the proof Theorem 2.2.3.
Lemma 2.2.4 Let A ∈ IRm×n be a matrix having full row rank m. Then
P := I − AT (AAT )−1A (2.96)
is a projection matrix onto the kernel space of A having the following properties:
i) P = P T (symmetry)
ii) PP = P and P T P T = P T (idempotence)
iii) Py = y ∈ K(A) (projection)
iv) Py = y for all y ∈ K(A) and Py = 0 for all y ∈ R(AT )
v) Each vector y ∈ K(A) is an eigenvector of P with eigenvalue 1. Each vector
y ∈ R(AT ) is an eigenvector of P with eigenvalue 0.
Proof of Lemma 2.2.4:
ad i) Trivial computation of P T
ad ii) For each y holds PPy :=(
I − AT (AAT )−1A)
Py = Py − AT (AAT )−1APy(iii)=
Py and therefore PP must be P.
ad iii) Read Ay = APy = Ay − AAT (AAT )−1Ay = 0
ad iv) First, suppose e ∈ K(A) then Pe = e − AT (AAT )−1Ae = e. Now suppose
e ∈ R(AT ) then there exist a coordinate vector x such that AT x = e and
Pe = e − AT (AAT )−1AAT x = 0.
ad v) Follows directly from item (iv).
¤
Now we show an proof for Theorem 2.2.3 An alternative proof is listed in Appendix
A.1
25
Proof of Theorem 2.2.3 (Alternative I):
Due to the fact that feasible points eK of (2.92) have to be in the kernel space K(B)
and knowing that K(B) = {Px | x ∈ V } , we can eliminate the equality constraint
and consider instead the equivalent program
maxeT e=1
eT P T SPe (2.97)
what had to be shown. The argmax of (2.92) can be calculated easily. If e∗ is an
optimal solution for (2.95) then eK = Pe∗ is an optimal solution for (2.92). ¤
An alternative but more constructive proof can be found in Section A.1.
26
Chapter 3
An Entropy Diversification RiskMeasure
The benefit and drawbacks of the classical Markowitz portfolio selection, the mean-
variance (MV) approach, are well discussed in the literature. Particularly the declared
assumptions are not necessarily fulfilled in praxis. Especially with regard to diver-
sification aspects, MV results may usually be too concentrated on some few assets1.
But what does that mean and how might diversification be understood? In fact, the
meaning of diversification is often not clearly quantified in the literature. Meucci
(2009) asserts that there is no common quantitative understanding about how port-
folio diversification can be defined. However, for the qualitative definition there is a
broad consensus, since a portfolio is diversified if its concentration risk is minimized.
Since concentration risk - more or less - is well understood for credit risk, in terms of
market risk, there is no clear statement.
Several popular diversification measures handle different aspects and approaches to
spread the investments adequately2. Bera and Park (2004), in particular, proposed
an entropy measure diversifying the portfolio weights, which is applicable if short-
selling will not be allowed. However, the diversification measure focuses solely on
the uniform distribution of the weights. Correlations, deviation or sensitivities can
merely handled within suitable constraints.
Meucci (2009) proposed a diversification measure, which is, similar to Bera and Park,
also based on the entropy measure. Instead of using a diversification number, how-
ever, he embedded a diversification distribution settled in the space of the principal
1See Litterman and Scheinkman (1991), Bera and Park (2004)2See Herfindahl, Hannah and Kay, Differential diversification index, idiosyncratic diversification
index
27
portfolios. Consequently, the approach is fully general, so that neither short-selling
nor asset correlations need to be excluded. Furthermore, provided by the derived
diversification analysis, an established quantitative definition of diversification can
eventually be formulated. Meucci’s diversification approach leads to uncorrelated
bets on principal portfolios, which can be retransformed to the original investment
universe.
3.1 Detour: Entropy Measures and Diversification
The entropy measure
H(p) = −N
∑
i=1
pi ln(pi) (3.1)
had been primarily applied in physics as the thermodynamic entropy (see Boltzmann
(1896), von Neumann (1927)). Due to the similarity of the formula used, Claude E,
Shannon (1948) adopted the name ”entropy” in informational theory, as well. He
operated with the entropy measure, analyzing the quality of reproducing a received
message, which includes transmitted signal mistakes, influence of statistics and white
noise respectively. Shannon quantified the expected value of the information con-
tained in a message. However, Jaynes (1957a, 1957b) was the first to close the missing
theoretical gap between informational and thermodynamic entropy. He showed that
thermodynamic entropy can rather be applied to applications of information theory,
and so, both theories can be transformed as equivalent views. He also defined the
principle of maximum entropy.
The principle of maximum entropy shall essentially used for measuring diversification
as derived below. Wikipedia initialized the principle as follows:
”In Bayesian probability, the principle of maximum entropy is a postulate which
states that, subject to known constraints (called testable information), the probability
distribution which best represents the current state of knowledge is the one with
largest entropy.” (See [27])
Furthermore, Keith Conrad writes in his working paper [7]:
28
”[The] principle [of maximum entropy] leads to the selection of a probability density
function which is consistent with our knowledge and introduces no unwarranted infor-
mation. Any probability density function satisfying the constraints which has smaller
entropy will contain more information (less uncertainty), and thus says something
stronger than what we are assuming. The probability density function with maxi-
mum entropy, satisfying whatever constraints we impose, is the one which should be
least surprising in terms of the predictions it makes.”
A main property of the entropy measure is the following:
Consider a probability density function p = (p1, . . . , pN) on a discrete set {w1, . . . , wN}where
∑
pi = 1, then the following holds
0 ≤ −N
∑
i=1
pi ln(pi) ≤ −N
∑
i=1
1
Nln(
1
N) = ln (N) (3.2)
with equality if and only if pi = 1/N , i.e., uniformly distributed, for all i. (see the
proof in [7]).
A compact survey and more references can be find on the Wikipedia pages [25], [26],
[27].
3.2 Diversification Risk Measure
This section discusses Meucci’s derivation of an diversification risk measure based on
the entropy measure (see [15]). In order to this, we start with the basic terminology
which is defined in the space of the (conditional) principal portfolios.
First define a variance concentration curve of a set of n uncorrelated principal port-
folios. The variance of an weighted principal portfolio k = 1, . . . , n is given by the
product of the squared principal portfolio weight wk and the portfolio variance λk
vk = w2kλk, k = 1, . . . , n (3.3)
so that the total portfolio variance can be calculated as ρVΛ (w) =
n∑
k=1
vk.
Second, marking the standard deviation as σΛ(w) :=√
ρVΛ (w), the term
sk =vk
σΛ(w)k = 1, . . . , n (3.4)
29
denotes the volatility concentration curve, which formally is the weighted contribution
to risk, since
sk = w∂σΛ(w)
∂wk
k = 1, . . . , n. (3.5)
Herewith, we obtain a sensitivity measure of how marginal changes in the princi-
pal weights have an impact on the total portfolio risk (see also Litterman (1996)).
Stochastic deviations from the expected return of the portfolio’s single investment
exposures precipitate the uncertainty of gains and losses. Hence, for diversification
reason, each investment should reveal the same impact. Since the principal portfolios
are uncorrelated, an investor should divide his investments for diversification into
uniform exposure parts (represented by sk) to make uncorrelated bets.
Indeed, the maximized entropy gives us an appropriate approach for that challenge
because it is related to providing the best uniform distribution especially under con-
sideration of constraints. However, probability-like masses are required, which, in
fact, is the motivation for the third step.
The normalized variance concentrations are denoted as diversification distributions
pk =w2
kλkn∑
i=1
w2i λi
=w2
kλk
ρVΛ (w)
(3.6)
Normalizing the variance contributions vk gives us the required property of a proba-
bility measure:
0 ≤ pk ≤ 1,n
∑
i=1
pi = 1 (3.7)
Finally, the so constructed diversification distributions can be interpreted as proba-
bility masses. With respect to diversification a concentration of these masses should
be avoided. Moreover, concentration aversion can be controlled by simply using the
maximized entropy. 3
3Beside the probability character Meucci proved that the term pn equals the r-square from aregression of the total portfolio return
µ =
n∑
i=1
wiµi
on the k-th principal portfolio (see [15] for more details).
30
Note, that considering m equality constraints, a same number of principal range
portfolio will be already fixed independently and not diversifiable. However, an asset
manager is still able to allocate the n−m free degrees of the principal kernel portfolios.
The conditional diversification distribution, then, is defined as
pk|A =wK
2kλk
ρVΛK
(wK)(3.8)
Based on the derivations above Meucci proposed the following entropy measures:
ρM(p) = exp
(
−n
∑
i=1
pi ln pi
)
(3.9)
and
ρM|A(p) = exp
(
−n−m∑
i=1
pi ln pi
)
(3.10)
respectively.
The entropy function is therefore utilized on the diversification distributions. The
exponential composition, in addition, is an equivalent transformation supporting the
human view on the risk results. Accordingly, an investor is fully concentrated if
ρM(p) = 1 and fully diversified if ρM(p) = n. Hence, a better readability is given
compared to pure entropy results as ln(1) or ln(n) respectively.
Maximizing the entropy without constraints pushes p to a uniformed distribution
(1/n, . . . , 1/n), so that each share of a principal portfolio is eventually exposed with
the same volatility adjusted proportion on the total volatility.
Now we use the entropy risk measure for portfolio selection. The efficient set of a
mean-diversification selection can be described as
max ρM(w)
s.t. Aw = bµT w = µ
(3.11)
when a plain principal portfolio decomposition is used and
max ρM|A(wK)
s.t. µTKwK = µ − µR
(3.12)
when a conditional principal portfolio decomposition is used.
31
The objective risk function ρM , however, is not strictly concave (see Section A.2). We
are facing an at most locally concave function to be maximized, thus. Consequently
we cannot simply use local optimization methods seeking for stationary points that
have to be a global maximum. Instead, one should use algorithms that are able to
handle general nonlinear mathematical programs. However, solving global optimiza-
tion programs without strict convexity can be complex and time-consuming. Genetic
algorithms (e.g. DEOptim) perform well for low dimensions and simple constraints.
Complex and high dimensional programs on the other hand are rather problematic.
However, depending on the aim, even local improvements can be useful, e.g., whenever
an asset manager wants to apply broader diversification to efficient mean-variance
portfolios. We will see later that the usage of efficient mean-variance points turned
out to be adequate starting points.
3.3 Portfolio Analysis of Diversified Portfolios
As done in (2.2.2) and (2.2.4), this section is dedicated to understanding the charac-
teristics of efficient points. Unlike in the case of the mean-variance framework, we do
not have a quadratic objective function anymore. Consequently, the Hessian matrix
is changing and has not even to be nonsingular, necessarily. Hence, instead of the
analytical solution we show the difference to the efficient solution gotten from the
corresponding mean-variance selection.
As above, we first present the transformation rules to the space of the principal
portfolios. Then we analyze the distance of a mean-diversification solution to a mean-
variance solution.
Transformations
Table 3.1 shows the transformation rules for a mean-diversification program using the
PCA:
Original TransformationExpected returns µT w µT
Kw + µR
Risk measure ρM(φ−1(w)) ρM(w)
Constraints Aw − b Aw − b
Table 3.1: Mean-diversification transformations
Considering the equality constraints via cPCA, reads
32
Original TransformationExpected returns – µT
Kw + µR
Risk measure – ρM|A(wK)
Constraints – no constraints
Table 3.2: Mean-diversification transformations
Characteristics of the efficient points
The Lagrange function for program (3.12) appears as
L(wK) = ρM|A(wK) + γ
(
µ − µR − µTKwK
)
(3.13)
A first order condition is given by equating the following derivatives to zero
∂L∂wK
=∂ρM
|A
∂wK
(wK) − γµK (3.14)
This gives us a non linear equation, which is not solvable analytically. However, we
can do some sensitivity analysis to a given feasible point v∗ = E
(
v∗K
v∗R
)
, chosen here
as the optimal mean-variance solution.
The gap to the optimal mean-diversification solution w∗K, then, is ∆wK := w∗
K − v∗K.
Define d := ∇ρM|A(v∗
K) for the gradient and H := ∇2ρM|A(v∗
K) for the Hessian matrix of
ρM|A at v∗
K.
Using the first order condition, then, in terms of the Taylor series we have
γµK =∂ρM
|A
∂wK
(wK) = d + H∆wK + o(∆wK), (3.15)
which implies
H∆wK = γµK − d + o(∆wK). (3.16)
If H is nonsingular ∆wK can be determined uniquely, such that
∆wK = H−1 (γµK − d) + o(∆wK) (3.17)
Given an efficient mean-variance solution v∗K of problem (2.60) the Lagrange condi-
tions
µ − µR = µTKv∗
K (3.18)
33
as well as
ΛKv∗K = γv∗
KµK (3.19)
of the mean-variance optimization are fulfilled, where γv∗
Kare the corresponding La-
grange factors of the mean-variance optimization.
To avoid clutter, we write µ instead of µ − µR and neglect the term o(∆wK).
Setting ∂L∂γ
= 0 as well as ∆γ := γ − γv∗
Kand substituting the derived results above
gives
µ = µTKw∗
K (3.20)
≈ µTK(v∗
K + ∆wK) (3.21)
= µ + (γv∗
K+ ∆γ) · µT
KH−1µK − µTKH−1d (3.22)
= µ + µTKH−1ΛKv∗
K + ∆γµTKH−1µK − µT
KH−1d (3.23)
so that γ can be approximated as
∆γ ≈ µTKH−1 (d − ΛKv∗
K)
µTKH−1µK
(3.24)
The results give us an impression of the differences between solutions of mean-variance
and mean-diversification. The natural view on such calculations is a Newton step. In-
deed, Newton iterations can be used for approximating maximal diversified solutions.
Interior-Point and other optimization algorithms are based certain variations of New-
ton steps. This fact is a another clue that mean-variance solutions are predestinated
as proper starting points for mean-diversification.
34
Chapter 4
Numerical Results
The aim of this chapter is the description and presentation of the numerical experi-
ments that were made.
4.1 Allocation Results of Diversified Selections
The aim of this section is the documentation of numerical results obtained by a
comparison of two allocation strategies:
• Mean-variance selection
• Mean-diversification selection are based on
– principal portfolios using a PCA
– conditional principal portfolios using a cPCA.
In the following sections, we first describe the applied allocation strategy in detail.
Then, in Section 4.1.2, the investment universe will be presented, which is a num-
ber of the world’s main indices like the Dow Jones, the Nikkei, the FTSE 100 etc.
Section 4.1.3 contains the results of an portfolio selection using a PCA. The presen-
tation structure is as follows: First the principal portfolios will be presented (Section
4.1.3.1). Then, the corresponding mean returns, the variances as well as the covari-
ances (Section 4.1.3.2) of the original space and the transformed space of the principal
portfolios will be depicted. Finally, Section 4.1.3.3 will bring out the efficient points of
both approaches (MV and MD). Analogously, Section 4.1.4 is dedicated to illustrate
the results of a portfolio selection, which was calculated with a cPCA.
35
4.1.1 The Allocation Strategy
As already mentioned in the introduction above, two approaches, MV and MD, will be
considered for asset allocation. The principal portfolios as well as the diversification
risk measure ρM based on a PCA as well as a cPCA. We always recycled the MV
solution as a starting point for the MD optimization.
The efficient set was approximated by ten equidistant target returns, starting with
the minimum mean return subject to the minimal allowed variance or a maximal
allowed diversification measure respectively. The end of the return interval is reached
by the maximum mean return of the original assets - the investment universe.
The drawn time series had been synchronized and transformed to daily compounded
returns rt = ln(
St+1
St
)
, where St denotes the prices of an asset at time t. Furthermore,
budget constraints were applied and long-short-selling were allowed tracing a 130/30
strategy.
A strategic asset allocation may use one of the portfolio selection strategies above.
The optimization results in a set of efficient points. Depending on what the investor’s
risk preference is, the asset allocation provides an optimal solution being a portfolio
that contains a weighted mix of assets in the considered investment universe. The use
of indices allows one to identify interesting market segments, which are represented
by the index ingredients, e.g., 10 % mid-cap German stock market if 10 % of MDAX
is chosen. A strategic asset allocation might use the optimal portfolios to dictate the
tactical asset allocation, a certain investment vector or a benchmark the portfolio
managers have to consider for their investments.
4.1.2 The Investment Universe
For testing, a selection of 27 indices of the main world markets were considered (see
Table 4.1). The time horizon was chosen over ten years from 11-Jan-2001 until 07-
Jan-2011, which are 2607 business days in total.
36
Nbr. Yahoo ID Index Region Country Sector
1 AEX AEX Europe Netherlands European Stocks2 ATX ATX Europe Austria European Stocks3 FCHI CAC 40 Europe France European Stocks4 GDAXI DAX Europe Germany European Stocks5 GREXP REXP Europe Germany European Yields6 FTSE FTSE 100 Europe UK European Stocks7 OMXSPI Stockholm General Europe Sweden European Stocks8 SSMI Swiss Market Europe Switzerland European Stocks9 OMXC20.CO OMX COPENHAGEN Europe Denmark European Stocks10 AORD ASX All Ordinaries Asia/Pacific Australia Asian (Australian) Stocks11 BSESN Bombay Stock Exchange Sensex Asia India Asian Stocks12 HSI Hang Seng Asia China Asian Stocks13 JKSE Jakarta Composite Asia Indonesia Asian Stocks14 KLSE KLSE Composite Asia Malaysia Asian Stocks15 KS11 Seoul Composite Asia South Korea Asian Stocks16 N225 Nikkei Asia Japan Asian Stocks17 STI Straits Times Asia Singapore Asian Stocks18 TWII Taiwan Weighted Asia Taiwan Asian Stocks19 DJC Dow Jones AIG Commodity n.v. America USA American Stocks20 FVX Treasury Yield 5Y America USA American Yields21 GSPC S&P 500 America USA American Stocks22 IXIC Nasdaq Composite America USA American Stocks23 NDX Nasdaq 100 America USA American Stocks24 RUA Russell 3000 America USA American Stocks25 TNX Treasury Yield 10Y America USA American Yields26 TYX Treasury Yield 30Y America USA American Yields27 TA100 TA-100 Africa/Middle East Israel Asian (Middle East) Stocks
Table 4.1: Investment universe (indices of main world markets)
As shown in Table 4.1, the indices are divided into sectors of the regions Europe,Asia and America (USA). The sector Asia, here, includes also the Middle East andAustralia. Europe and America are additionally divided into Stock and Yield markets.
The time series, representing the European market (9 indices), are shown in Figure4.1 (logarithmic scaling).
14−Dec−2000 08−Jul−2002 30−Jan−2004 24−Aug−2005 18−Mar−2007 10−Oct−2008 04−May−2010 27−Nov−201110
2
103
104
Time
Val
ues
(log
scal
e)
European Time Series
01−AEX02−ATX03−FCHI04−GDAXI05−GREXP06−FTSE07−OMXSPI08−SSMI09−OMXC20.CO
FinancialCrisis
REXP
ATX
Figure 4.1: Time series of European indices Source: yahoo.finance.com
The REXP (Deutscher Renten-Performanceindex) is a German yield index appearingwith a relative low volatility. The index includes information from 30 German govern-ment bonds and its yield incomes. Even during the financial crisis, when all indicesdropped down, the REXP stayed quite stable. Compared to other European stock
37
indices, the Austrian index ATX performed very stable before the financial crisis, butdecreased relatively severely during the crisis.
The time series, representing the Asian market (10 indices), are shown in Figure 4.2:
06−Jan−2001 31−Jul−2002 22−Feb−2004 16−Sep−2005 10−Apr−2007 02−Nov−2008 27−May−2010 20−Dec−201110
2
103
104
105
Time
Val
ues
(log
scal
e)
Asian Time Series
10−AORD11−BSESN12−HSI13−JKSE14−KLSE15−KS1116−N22517−STI18−TWII27−TA100
FinancialCrisis
TA100
AORD
N225HSI
Figure 4.2: Time series of Asian indices Source: yahoo.finance.com
The American market is represented by main indices of the USA (7 indices). Thetime series are plotted in Figure 4.3
14−Dec−2000 11−Oct−2002 08−Aug−2004 06−Jun−2006 02−Apr−2008 29−Jan−2010 27−Nov−201110
0
101
102
103
104
Time
Val
ues
(log
scal
e)
American Time Series
19−DJC20−FVX21−GSPC22−IXIC23−NDX24−RUA25−TNX26−TYX
FinancialCrisis
TNXTYX
FVX
DJC
RUA
Figure 4.3: Time series of American indices Source: yahoo.finance.com
Three indices, namely FVX, TNX and TVX, are yield indices mapping governmentbonds with maturities in 5, 10 and 30 years respectively. It is interesting to observethat their net values were very close before the financial crisis, and spread out afterthe crisis. In contrast with the German REXP, the crisis impacted severely on to theAmerican yield indices.
Appendix B.3 provides statistical examinations to prove the existence of market in-variants (for definition, see B.3).
38
4.1.3 Portfolio Selection using PCA
In this section we present the result of a strategic asset allocation based on a portfoliooptimization using the variance risk measure ρV
Σ (mean-variance) as well as the di-versification risk measure ρM (mean-diversification). For diversification we comparethe principal component and the conditional principal component method.
4.1.3.1 Principal Portfolios
The principal portfolios generated by a principal component decomposition are pre-sented in Figure 4.4 (see Tables B.5, B.6 and B.7 for exact numerical weight infor-mation).
0 5 10 15 20 25 30−300
−200
−100
0
100
200
300
400
500Principal Portfolio Structure
Principal Portfolio
Wei
ghts
01−AEXc
02−ATXc
03−FCHIc
04−GDAXIc
05−GREXPc
06−FTSEc
07−OMXSPIc
08−SSMIc
09−OMXC20.COc
10−AORDc
11−BSESNc
12−HSIc
13−JKSEc
14−KLSEc
15−KS11c
16−N225c
17−STIc
18−TWIIc
19−DJCc
20−FVXc
21−GSPCc
22−IXICc
23−NDXc
24−RUAc
25−TNXc
26−TYXc
27−TA100c
Figure 4.4: Principal portfolios using PCA
Figure 4.4 presents the percentage weights of original assets building a principal port-folio. The weights are simply the entries of the eigenvector matrix E. Since principalportfolios can also be realized portfolios, we refer to the 480% long investment neededby the first principal portfolio. The remaining portfolios particularly appear with along-short ratio more balanced, but the result shows that high overbuying/oversellingeffects are usually an issue for a single principal portfolio. However, the leverage effectcan be easily avoided by introducing a factor α, so that αl · I1 ≤ α I1
T E ≤ αu · I1 holdsfor limiting lower and upper bounds αl and αu respectively. The principal componentdecomposition is therewith still fulfilled as
α2ET ΣE = α2Λ (4.1)
39
is a valid solution as well.
An aggregated view on sector information allows a better understanding of the in-vestment structure for the principal portfolios:
0 5 10 15 20 25 30−3
−2
−1
0
1
2
3
4
5Principal Sector Portfolio Structure
Principal Sector Portfolio
Wei
ghts
EuropeYieldsEuropeStocksAsianStocksAmericanYieldsAmericanStocks
Figure 4.5: Principal sector portfolios using PCA
4.1.3.2 Statistical Measures for Assets and Principal Portfolios
The aim of this section is the presentation of the statistical measures, i.e., meanreturn, variance and correlation for the original assets as well as for the principalportfolios.
Figure 4.6 sketches the statistical information row-wise. The figure’s left columncontains corresponding bar and matrix plots for the original assets, the right columncontains bar and matrix plots for the corresponding principal portfolios.
40
0 10 20−10
−5
0
5
x 10−4
Log−
Ret
urns
µ
Original Assets
Original Investment Space
0 10 20−10
−5
0
5
x 10−4
Prin
c. R
etur
ns
Principal Portfolios
Principal Investment Space
0 10 200
0.5
1
1.5
2
2.5x 10
−3
Var
ianc
e ρV
Original Assets0 10 20
0
0.5
1
1.5
2
2.5x 10
−3
Prin
c. V
aria
nce
λ2 n
Principal Portfolios
Original Correlation Matrix
5 10 15 20 25
5
10
15
20
25
Principal Correlation Matrix
5 10 15 20 25
5
10
15
20
25
EuropeanMarket
AsianMarket
AmericanMarket
Figure 4.6: Multivariate statistics
The principal returns are given with µ, the variances are the diagonal entries of Λ(see Table B.3.1).
The correlation matrix of the uncorrelated principal portfolios (right lower plot) is thediagonal matrix Λ. On the other side we have the correlation matrix for the originalassets (left lower plot). Red areas reveal highly positively correlated indices, greenareas highly negatively correlated and blue areas uncorrelated ones. It is noticeablethat the REXP (index 5) is uncorrelated to each other index of the investment uni-verse. Furthermore we observe that the sectors Europe, Asia and America are eitheruncorrelated or negatively correlated to each other. Within the American market theyield indices (indices 20, 25, 26) are positively correlated to each other but negativelycorrelated to the American stock indices.
4.1.3.3 Efficient Portfolios
In this section, two efficient sets are considered. The first set is the result of a mean-variance selection (MV) using the risk measure ρV
Σ . The second set is the result of amean-diversification selection (MD) using the risk measure ρM .
The results are plotted in Figure 4.7 containing 3 × 2 plots. The left plot columnshows the MV results and the right column the MD results respectively.
41
The upper plot row is dedicated to present the risk-return diagrams. The middle rowsketches the optimal selections of principal portfolios with respect to certain targetreturns. Analogously, the lower row sketches the optimal selections of the originalassets.
0 5 10 15 20 25−10
−5
0
5
10
Variance ρV
Ret
urns
µ in
%
Mean−Variance−Plot
Pareto Set MVPareto Set MDOriginal AssetsPrincipal Portfolios
0 2 4 6 8 10 12 14−10
−5
0
5
10
Diversification measure ρM
Ret
urns
µ in
%
Mean−Diversification−Plot
Pareto Set MDPareto Set MVOriginal AssetsPrincipal Portfolios
1.82 2.54 3.26 3.98 4.70 5.42 6.14 6.86 7.58 8.30−3
−2
−1
0
1Principal Portfolio Weights (Mean−Variance)
Target Return in %
Wei
ghts
1.82 2.54 3.26 3.98 4.70 5.42 6.14 6.86 7.58 8.30−3
−2
−1
0
1Principal Portfolio Weights (Mean−Diversification)
Target Returns in %
Wei
ghts
pca1
pca2
pca3
pca4
pca5
pca6
pca7
pca8
pca9
pca10
pca11
pca12
pca13
pca14
pca15
pca16
pca17
pca18
pca19
pca20
pca21
pca22
pca23
pca24
pca25
pca26
pca27
1.82 2.54 3.26 3.98 4.70 5.42 6.14 6.86 7.58 8.30−0.5
0
0.5
1
1.5Original Portfolio Weights (Mean−Variance)
Target Return in %
Wei
ghts
01−AEXc
02−ATXc
03−FCHIc
04−GDAXIc
05−GREXPc
06−FTSEc
07−OMXSPIc
08−SSMIc
09−OMXC20.COc
10−AORDc
11−BSESNc
12−HSIc
13−JKSEc
14−KLSEc
15−KS11c
16−N225c
17−STIc
18−TWIIc
19−DJCc
20−FVXc
21−GSPCc
22−IXICc
23−NDXc
24−RUAc
25−TNXc
26−TYXc
27−TA100c
1.82 2.54 3.26 3.98 4.70 5.42 6.14 6.86 7.58 8.30−0.5
0
0.5
1
1.5Original Portfolio Weights (Mean−Diversification)
Target Return in %
Wei
ghts
Figure 4.7: Efficient portfolios using PCA
Starting with the left upper plot, which is the MV diagram, visualizing primarily theMV-efficient set itself (green line). Additionally, for comparative purposes, also thetransformed MD-efficient set is plotted (red line). Within the diagram, both sets lierather close to each other. Finally, the single MV-coordinates of the various originalindices (magenta dots) and the principal portfolios (blue dots) are inserted.
The same sets and values are analogously presented in the MD diagram (upper rightplot). In contradiction to the MV diagram, the MV and MD sets differs the morethe lower target returns are chosen. For the largest target return, both, MV and MDresults are equal, what is due to the fact that the asset with the highest expectedreturn has to be chosen. The spike that can be observed for target return 0.025362 iscaused by the non-concave objective function (see Section A.2). A finer grid of targetreturns would show a continuous zigzag-curve similar to the effects plotted in FigureA.2. Such outlier points are locally efficient but never globally efficient. A rationaldecision maker would not buy a portfolio like this because even the portfolios withhigher expected returns are connected to less diversification risk.
Note, that, compared to the MV diagram, MD-efficient points are mirror inverted,i.e., an efficient point with arbitrary coordinates (µ, ρ) never allows further pointslying in the affine cone (µ, ρ) + IR
2≥0. Whereas efficient points in the MV diagram
never allows further points lying in (µ, ρ) + {(x, y) |x ∈ IR≤0, y ∈ IR≥0} .
42
Table 4.2 provides a list of risk-returns coordinates (µ, ρ) for the plotted efficientpoints in Figure 4.7.
Iter targetExp µ MV ρV MV ρM MD ρV MD ρM
1 0.018160 0.018160 0.034452 1.640899 0.128900 11.2890892 0.025362 0.025362 0.044269 3.469579 0.109393 8.1061503 0.032563 0.032563 0.094835 8.661413 0.141562 10.6106614 0.039764 0.039764 0.200163 9.484739 0.233803 10.3445225 0.046965 0.046965 0.361091 9.159118 0.385754 9.7923386 0.054166 0.054166 0.577861 8.862393 0.610785 9.3704497 0.061368 0.061368 0.850476 8.675784 0.897432 9.1022418 0.068569 0.068569 1.178934 8.574808 1.247287 8.9425519 0.075770 0.075770 1.563236 8.529751 1.662129 8.85577010 0.082971 0.082971 2.054029 8.733935 2.054029 8.733935
Table 4.2: Efficient points of the risk return diagrams (PCA approach)
In the middle left bar plot, the MV-optimal allocation weights w for the principalportfolios are presented (see Table B.3 for numerical weight information). The middleright bar plot contains the corresponding MD-optimal allocation weights (see TableB.4). Again, we encounter conformity for the largest target return. For the lower tar-get returns, however, one can see the difference since, compared to the MV allocationin the left plot, other principal portfolios are weighted more significantly.
More meaningful are the upper bar plots presenting the optimal weights for the orig-inal asset allocation. The left plot contains again the MV weights; the right onecontains the MD weights. Numerical information are listed in Tables B.1 and B.2respectively.
For a better understanding of the diversification effect, we zoom into the lowest targetreturn, which is 0.01816%. The diversification effect can be retraced by consideringthe entropy based diversification measure ρM and its diversification distributions pi.Targeting a return of 0.01816% leads to an optimal allocation as plotted in Figure4.8
43
0 5 10 15 20 25
−0.8
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−0.4
−0.2
0
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MV
wei
ghts
Original Assets
Target return =0.01816%
0 5 10 15 20 25
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MV
wei
ghts
Principal Portfolios
Target return =0.01816%
0 5 10 15 20 25
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MD
wei
ghts
Original Assets
Target return =0.01816%
0 5 10 15 20 25
−0.6
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−0.2
0
0.2
0.4
0.6
MD
wei
ghts
Principal Portfolios
Target return =0.01816%
REXP
REXP
HSI
DJC
Figure 4.8: Allocation weights using cPCA (target return 0.018 %)
The upper bar plots present the optimal MV weights to the original assets (left side)as well as the optimal MV weights to the principal portfolios (right side). The lowerones present the corresponding optimal MD weights.
It is interesting to see that the MV allocation bets mainly 83% on the REXP, 12%on various Asian indices (AORD, BSESN, JKSE, KLSE, KS11, N225, TWII, TA100)and 5% on American indices (DJC, GSPC, TYX). The MD approach on the otherhand bets on 62% REXP, 12.5% HSI, 22% DJC and the rest 3.5% on JKSE, KLSEas well as TA100. Although, the MV approach bets on a higher number of indices,which may be naively assumed to be well-diversified, the entropy theory says thatunder missing distribution assumptions the better and targeted diversification will bereached with the MD approach.
A consideration of the diversification distributions pi gives a reason for this. Figure4.9 shows the distributions information of each principal portfolio. The distributionsin the left plot are based on the MV approach, the distributions in the right plot onthe MD approach.
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0 5 10 15 20 250
0.1
0.2
0.3
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1
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pn
Principal portfolio
Target return =0.01816%
0 5 10 15 20 250
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0.4
0.5
0.6
0.7
0.8
0.9
1
MD
pn
Principal portfolio
Target return =0.01816%
pca26
pca26
Figure 4.9: Diversification distributions using PCA
What is noticeable is the principal portfolio pca26, which is highly concentrated forthe MV solution. Portfolio pca26 contains mainly American indices and a few Asianindices. Optimizing via diversification measure reduces the concentration on pca26and put more weight into further principal portfolios. Eventually, these facts mightexplain the spread but not substantial allocation in the corresponding American andAsian indices when MV is considered. The MD approach, on the other hand, seemsto be able to achieve a clear investment structure by weighting the HSI and DJC.
4.1.4 Portfolio Selection using cPCA
This section presents the same structure as Section 4.1.3. Therefore, only the differ-ences and the particularities compared to the PCA will be mentioned. For declara-tions to the plot construction the reader to be referred to Section 4.1.3.
4.1.4.1 Principal Portfolios
In contrast to the PCA, the cPCA consider a range portfolios (here principal portfolioscpca1), which are determined by the equality constraint Aw = b. The number of thedetermined portfolios depends on the rank of A.
Analogous to in Section 4.1.3.1 we can plot the weight distribution of the one deter-mined range portfolio, which is cpca1, and the remaining 26 kernel portfolios.
45
0 5 10 15 20 25 30−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5Principal Portfolio Structure
Principal Portfolio
Wei
ghts
01−AEXc
02−ATXc
03−FCHIc
04−GDAXIc
05−GREXPc
06−FTSEc
07−OMXSPIc
08−SSMIc
09−OMXC20.COc
10−AORDc
11−BSESNc
12−HSIc
13−JKSEc
14−KLSEc
15−KS11c
16−N225c
17−STIc
18−TWIIc
19−DJCc
20−FVXc
21−GSPCc
22−IXICc
23−NDXc
24−RUAc
25−TNXc
26−TYXc
27−TA100c
Range portfolio
Figure 4.10: Conditional principal portfolios using cPCA
Note that, due to the budget constraints, the weight sum of the kernel portfoliosis always zero, which is absolutely clear, since each portfolio must a solution of thecorresponding, homogeneous linear equation.
The corresponding numerical weights are listed in Tables B.5, B.6 and B.7. Theaggregated sector distribution looks as follows:
0 5 10 15 20 25 30−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5Principal Sector Portfolio Structure
Principal Sector Portfolio
Wei
ghts
EuropeYieldsEuropeStocksAsianStocksAmericanYieldsAmericanStocks
RangePortfolio
Figure 4.11: Conditional principal sector portfolios using PCA
46
4.1.4.2 Statistical Measures for Assets and Conditional Principal Portfo-lios
Like in Section 4.1.3.2 we find the characteristic plots to the expected returns, thevariances and the covariances of the original assets and their conditional portfolios.The range portfolio, which is composed by almost 95% of the REXP index, is high-lighted in blue.
0 10 20−10
−5
0
5
x 10−4
Log−
Ret
urns
µ
Original Assets
Original Investment Space
0 10 20−10
−5
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5
x 10−4
Prin
c. R
etur
ns
Principal Portfolios
Principal Investment Space
0 10 200
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1
x 10−3
Var
ianc
e ρV
Original Assets0 10 20
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x 10−3
Prin
c. V
aria
nce
λ2 n
Principal Portfolios
Original Correlation Matrix
5
10
15
20
25
5 10 15 20 25
Principal Correlation Matrix
5 10 15 20 25
5
10
15
20
25
AmericanMarket
AsianMarket
EuropeanMarket
Figure 4.12: Multivariate statistics cPCA
The corresponding numerical results are listed in Tables B.12, B.13, B.1 and B.11.
The values of the principal portfolio’s mean returns and variances are listed in TableB.3.1
4.1.4.3 Efficient Portfolios
The efficient set due to cPCA presents itself as follows
47
0 1 2 3 4 5 6 7 8 9 10−10
−5
0
5
10
Variance ρV
Ret
urns
µ in
%
Mean−Variance−Plot
Pareto Set MVPareto Set MDOriginal AssetsPrincipal Portfolios
0 5 10 15 20 25−10
−5
0
5
10
Diversification measure ρM
Ret
urns
µ in
%
Mean−Diversification−Plot
Pareto Set MDPareto Set MVOriginal AssetsPrincipal Portfolios
1.81 2.53 3.26 3.98 4.70 5.42 6.14 6.86 7.58 8.30−4
−2
0
2
4Principal Portfolio Weights (Mean−Variance)
Target Return in %
Wei
ghts
1.81 2.53 3.26 3.98 4.70 5.42 6.14 6.86 7.58 8.30−4
−2
0
2
4Principal Portfolio Weights (Mean−Diversification)
Target Returns in %
Wei
ghts
cpcaFixed1
cpca2
cpca3
cpca4
cpca5
cpca6
cpca7
cpca8
cpca9
cpca10
cpca11
cpca12
cpca13
cpca14
cpca15
cpca16
cpca17
cpca18
cpca19
cpca20
cpca21
cpca22
cpca23
cpca24
cpca25
cpca26
cpca27
1.81 2.53 3.26 3.98 4.70 5.42 6.14 6.86 7.58 8.30−0.2
0
0.2
0.4
0.6
0.8
1Original Portfolio Weights (Mean−Variance)
Target Return in %
Wei
ghts
01−AEXc
02−ATXc
03−FCHIc
04−GDAXIc
05−GREXPc
06−FTSEc
07−OMXSPIc
08−SSMIc
09−OMXC20.COc
10−AORDc
11−BSESNc
12−HSIc
13−JKSEc
14−KLSEc
15−KS11c
16−N225c
17−STIc
18−TWIIc
19−DJCc
20−FVXc
21−GSPCc
22−IXICc
23−NDXc
24−RUAc
25−TNXc
26−TYXc
27−TA100c
1.81 2.53 3.26 3.98 4.70 5.42 6.14 6.86 7.58 8.30−0.2
0
0.2
0.4
0.6
0.8
1Original Portfolio Weights (Mean−Diversification)
Target Return in %
Wei
ghts
Fixed range portfolio
Figure 4.13: Efficient points of the risk return diagramm (cPCA approach)
In contrast to the PCA approach, the diversification measure ρM|A is a risk measure
in a projected space where the range information are separated. As a consequence,diversification can be made more efficient using the remaining conditional principalportfolios. Compared to the efficient set, when PCA was used (4.1.3.3), the pointsgotten by mean-diversification and mean-variance are no longer as close to each otheras in Figure 4.7. Furthermore, higher diversification units can be achieved, namely21.8 vs. 11.2 (compare Tables 4.2 and the following 4.3).
Altogether the results prove to be more stable in the numerical sense. The com-parison to the PCA approach shows that a broader diversification can be achieved(compare the lower right plots of Figures 4.7 and 4.13). The same look to the resultsin the transformed principal space (middle right plots) and particularly the higherdiversification level ρM verifies the impression even supported by the entropy theory.
Nbr. Target return µ MV ρV MV ρM MD ρV MD ρM
1 0.018145 0.018145 0.034452 9.456645 0.037174 21.8028512 0.025348 0.025348 0.044224 9.786076 0.382161 15.6196613 0.032551 0.032551 0.094708 7.028525 0.378285 16.2287814 0.039754 0.039754 0.199977 7.011108 0.697145 15.9271575 0.046957 0.046957 0.360870 7.232127 0.801070 14.4779286 0.054160 0.054160 0.577633 7.373139 0.950181 12.7573577 0.061362 0.061362 0.850265 7.467745 1.047944 11.7180228 0.068565 0.068565 1.178767 7.534540 1.267703 10.1125369 0.075768 0.075768 1.563140 7.584178 1.572288 8.25381310 0.082971 0.082971 2.054029 6.585184 2.054029 6.585184
Table 4.3: Pareto set points with cPCA
The allocation weights (Figure 4.14) are clearly dominated by the principal rangeportfolio (86% of the allocation). However, a higher diversification can be achieved in
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the remaining 14%. Subject to a target return of 0.025% the diversification effect hasa significantly impact on the allocation. Compared to the MV allocation that investsin the REXP with more than 80% of the budget, the MD allocation reduced the staketo less than 25% and considered instead TA100, DJC and other stock markets. Dueto the MD optimisation, the diversification level increases from a degree about circa4 to closely 15.
0 5 10 15 20 25
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MV
wei
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Target return = 0.025348 %
Range portfolioOptimal allocation
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Target return = 0.025348 %
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ghts
Target return = 0.025348 %
Range portfolioOptimal allocation
0 5 10 15 20 25
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MD
wei
ghts
Principal Portfolios
Target return = 0.025348 %
Kernel weightsRange weights
Kernel weightsRange weights
Figure 4.14: Allocation weights using cPCA (target return 0.025 %)
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0 5 10 15 20 250
0.1
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1
MV
pn
Principal Portfolios
Target return = 0.025348 %
0 5 10 15 20 250
0.1
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0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MD
pn
Principal Portfolios
Target return = 0.025348 %
Figure 4.15: Diversification distributions using cPCA
4.2 Proof of Concept
This section presents the numerical results we received from back-testing the con-sistence of the correlation structure regarding the principal portfolios 1. Here, weconsidered correlations instead of covariances because they appears more stable. Fur-thermore, a volatility change can be easily compensated by adjusted portfolio weights.
We tested two strategies, which are described in the following section.
4.2.1 Back-Testing Strategies
The first strategy (Strategy I) plans yearly rebalancing of the principal portfolios.A smoothing method with the aim of avoiding jumping reallocations due to jumpscaused from changing eigenvectors is not applied. Instead of this, we clear these jumpsout of the return data. The principal components will be determined by a principalcomponent decomposition of a correlation matrix, measured in a certain considerationperiod. The correlation matrix will be calculated with the daily log-return time series,which are considered one year back from the rebalancing time point. The principalportfolios represent the allocation for a holding period of one year. After the end ofthe year, the correlation of the returns of the principal portfolios will be measured.With this strategy, the consideration periods follow consecutively.
1A performance back-test was not the aim of this examinations
50
The second strategy (Strategy II) plans monthly rebalancing, the consideration periodis again one year back from the rebalancing time point. This time, the holding periodis one month. After one month, we restart the principal component decompositionand receive the allocation vector for the next period. With this strategy the one-yearconsideration periods overlap due the one-month shift.
The back-testing is completed successfully, if the resulting correlation matrix approxi-mates sufficiently to the unit matrix, i.e., the allocated principal portfolios are closelyuncorrelated over the holding period.
The following sections include the results of three back-tests: The first test appliesStrategy I from 2002 to 2010. The second test applies Strategy II for 2002, and so dothe third test for 2009.
Test objectives are the indices DAX, Dow Jones, FTSE and Nikkei. The index timeseries are plotted in Figure 4.16.
08−Dec−1999 10−Sep−2001 15−Jun−2003 19−Mar−2005 22−Dec−2006 25−Sep−2008 30−Jun−2010 03−Apr−20122.000
4.000
6.000
8.000
10.000
12.000
14.000
16.000
18.000
20.000
Time
Val
ues
Index Time Series
DJIFTSEGDAXIN225
Figure 4.16: Considered index time series over 10 years
4.2.2 Strategy I: Yearly Rebalancing, 2002 - 2010
We tested Strategy I and started with a consideration period from 23-Jan-2001 to23-Jan-2002. The consecutive holding period started from 23-Jan-2002 and expiredin 23-Jan-2003. After eight iterations, we stopped with the last consideration periodfrom 23-Jan-2008 to 23-Jan-2009 and the consecutive holding period from 24-Jan-2009to 23-Jan-2010.
The resulting investment time series are plotted in Figure 4.17. The y-axis representsthe respective levels of the principal portfolios. One can clearly see the jumps comingfrom the eigenvector changes, which again is due to different correlation matricesconsidered by the rebalancing. The jumps can be easily avoided by leveraging theinvestment into the principal portfolio.
51
14−Dec−2000 28−Apr−2002 10−Sep−2003 22−Jan−2005 06−Jun−2006 19−Oct−2007 02−Mar−2009 15−Jul−2010−15.000
−10.000
−5.000
0
5.000
10.000
15.000
20.000
Date
Val
ues
Time Series of the principal portfolios
pca1pca2pca3pca4
Figure 4.17: Time series of the principal portfolios over 10 years
The eight resulting correlations are displayed in the following table:
Principal portfolios pca1 pca2 pca3 pca423/01/2002 - 23/01/2003pca1 1pca2 -0.1404 1pca3 0.2430 -0.0044 1pca4 -0.5648 0.2150 -0.0811 124/01/2003 - 23/01/2004pca1 1pca2 -0.0190 1pca3 0.3183 -0.0356 1pca4 -0.6335 0.0681 -0.1192 124/01/2004 - 23/01/2005pca1 1pca2 -0.3067 1pca3 -0.0115 0.3092 1pca4 0.5318 -0.1453 0.4146 124/01/2005 - 23/01/2006pca1 1pca2 -0.2296 1pca3 0.0056 0.0312 1pca4 0.4093 0.0714 -0.1831 124/01/2006 - 23/01/2007pca1 1pca2 -0.2189 1pca3 0.0035 -0.6434 1pca4 0.3985 0.0007 -0.0780 124/01/2007 - 23/01/2008pca1 1pca2 -0.1935 1pca3 -0.1747 -0.0118 1pca4 -0.1157 -0.0966 -0.1358 124/01/2008 - 23/01/2009pca1 1pca2 -0.1679 1pca3 -0.2556 0.0999 1pca4 -0.0351 -0.3127 0.2345 124/01/2009’, ’23/01/2010’pca1 1pca2 0.2037 1pca3 -0.1043 -0.4532 1pca4 -0.3450 -0.1179 0.2925 1
Table 4.4: Correlations of Strategy II over the periods
We see that the principal portfolios are not really uncorrelated over the investmenttime. Rather, some correlations fluctuate strongly. For instance, the correlation ofthe first principal portfolio to the forth wiggles over the periods between -0.6 and 0.5(see Table 4.4). The results are not satisfying and so, the back-testing is doubtful.
52
4.2.3 Strategy II: Monthly Rebalancing, 2002
In this section the results of back-testing Strategy II are presented. The test startedwith a consideration period from 23-Jan-2001 to 23-Jan-2002. The consecutive hold-ing period was from 24-Jan-2002 to 23-Feb-2002. After twelve iterations, we stoppedwith the last consideration period from 21-Dec-2001 to 21-Dec-2002 and the consec-utive holding period from 20-Dec-2002 to 19-Jan-2003.
The corresponding time series of the invested principal portfolios are plotted in Figure4.18. Even though, we did not apply a smoothing method, the junctions are smoothand do not show huge reallocation jumps. The correlations (see Table 4.5) are mostlysmall but for pca3. Thus, the back-testing with Strategy II in 2002 is still doubtful.In total, however, it seems to produce better results than Strategy I. Nevertheless,the year 2001 and 2002 provide decreasing but quite stable markets, which also mightbe a explanation for the stability.
18−Jan−2002 16−Mar−2002 12−May−2002 08−Jul−2002 03−Sep−2002 30−Oct−2002 26−Dec−2002 22−Feb−2003−10.000
−5.000
0
5.000
10.000
15.000
Date
Val
ues
pca1pca2pca3pca4
Figure 4.18: Back-test of Strategy II, 2002
Principal portfolios pca1 pca2 pca3 pca423/01/2002 - 23/01/2003pca1 1pca2 0.0919 1pca3 0.2848 -0.3546 1pca4 -0.1390 0.0142 0.0121 1
Table 4.5: Correlations of Strategy II over the periods, 2002
4.2.4 Strategy II: Monthly Rebalancing, 2009
In this section the back-testing results of Strategy II are presented. The test startedwith an consideration period from 23-Jan-2008 to 23-Jan-2009. The consecutive hold-ing period was from 24-Jan-2009 to 23-Feb-2009. After twelve iterations, we stoppedwith the last consideration period from 21-Dec-2008 to 21-Dec-2009 and the consec-utive holding period from 20-Dec-2009 to 19-Jan-2010.
53
The corresponding time series of the invested principal portfolios are plotted in Figure4.19. Like in Strategy I we observe huge reallocation jumps, which are due to thechanging eigenvectors. The correlations (see Table 4.6) are mostly small but for pca3.Thus, the back-testing with Strategy II in 2009 is also doubtful. Again, it seems toproduce better results than Strategy I. The years 2008 and 2009 represent the criticalperiod during the financial crisis. Lehman defaulted September 2008. The marketswere changing and so were the correlation structures.
11−Jan−2009 09−Mar−2009 05−May−2009 01−Jul−2009 27−Aug−2009 23−Oct−2009 19−Dec−2009 15−Feb−2010−15.000
−10.000
−5.000
0
5.000
10.000
15.000
Date
Val
ues
pca1pca2pca3pca4
Figure 4.19: Back-test of Strategy II, 2009
Principal portfolios pca1 pca2 pca3 pca423/01/2002 - 23/01/2003pca1 1pca2 -0.0973 1pca3 0.5532 0.2520 1pca4 0.1185 -0.1956 -0.0776 1
Table 4.6: Correlations of Strategy II over the periods, 2009
4.2.5 Back-Testing Results
Both back-testing strategies I and II did not lead to satisfactory results. Strategy IIseems to be slightly better than Strategy I, but it is difficult to say what the actualdrivers of these effects are. The tests pointed out that the correlation structuresare changing constantly and perceptibly. Due to the moving average character ofcorrelation measure we are also facing the problem of finding a suitable considerationperiod as well as a suitable measure for determining the active correlation structure.
The test results identified the potentials but also the difficulties of finding uncorrelatedportfolios. More research on that issue is needed. There are several starting pointsas listed in the outlook in Chapter 5.
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Chapter 5
Conclusions and Outlook
The aim of this final chapter is to give a brief summary of the examination that hasbeen undertaken in this thesis and what, finally, are the conclusions drawn from theresults. Finally, an overview of further topics and extensions will be given.
5.1 Conclusions
The initial motivation for this thesis was the examination of the entropy-based di-versification measure as well as the conditional PCA, both are proposed in Meucci(2009). The paper describes the idea and the realization in an illustrative and concisefashion but also quite compactly and sometimes sketchily. Therefore, we were drivenby the quest to find a deeper theoretical understanding as well as a numerical reviewand the comparison with the classical Markowitz approach. Our investigations ledus among others to the question of the stability of the correlation structures we usefor diversification. In the course of this, the idea arises that given sufficiently stablestructures, a society may provide funds or ETFs, tracking the uncorrelated principalcomponents of a certain market. The benefit of such portfolios would be manifold forinvestors, since they could adapt the structures for their own investment aiming fordecorrelation and simultaneously diversification.
In the first chapter, we derived a comprehensive theory describing the geometry ofthe cPCA for linearized equality constraints represented by a matrix A. The theory’smain principle is the separation of the domain vector space in two subspaces: Thekernel space A and the transformed range space of R. From Linear Algebra we knowthat these subspaces are complements spanning the whole domain space. For the sakeof the principal component decomposition we did not use the Euclidean scalar productbut the scalar product induced by the covariance matrix Σ, so that we find a kernelbase and a transformed range base of Σ−conjugates. The decomposition could bepowerfully used for the reformulation of optimization problems, since we were able to
55
eliminate the linearized equality constraints. The corresponding transformations ledto equivalent but simple structured programs, whose Lagrange analysis became moresatisfactory. For the computation of the conditional principal component decomposi-tions we referred to Meucci (2009) again. However, the herein derived theory and thecorresponding notation allow us a very compact presentation of the algorithm and itsauxiliary formulation. Furthermore we proved the existence of conditional principalcomponent decompositions.
Since Chapter 3 solely summarized Meucci’s diversification theory expressed in thenotation we used, Chapter 4 presented the numerical results we found by optimizing amean-variance and a mean-diversification selection on selected leading world indices.For the two approaches we considered both the decomposition using the PCA and acPCA. Because of numerical difficulties that arose due to the missing strict concavityof the diversification measure, we decided to choose an efficient mean-variance alloca-tion as a starting point for the mean-diversification. This strategy was even a suitableheuristic, since an investor is seeking for better diversification close to a known so-lution. Otherwise we were facing local optima, so that the resulting efficient frontierdid not behave smoothly enough. The adoption of an evolutionary algorithm, such asDEOptim, did not perform, because of the relative high number of dimensions. Theeffort of considering other global optimizer was not made. To draw a conclusion, welocated diversification potential with both methods the PCA and the cPCA in themiddle and lower regions of target returns, what is pursuable and was even expected.Since the degree of freedom decreases for optimization all the more, the closer a max-imum expected return of an investment universe is required. It is interesting to notethat the portfolios’ variances did not change too much but their diversification gradedid. We observed a (relative) higher increase in diversification grade for the cPCA.However, diversification for the PCA is made on a different component space thanthe PCA diversification, because of the excluded range portfolio, and we disclaimedthe accomplishment of a bridge calculation. Nevertheless, the cPCA should be pre-ferred as an important and powerful view on the limited degrees of freedom, which isrestricted by the equality constraints. The diversification effects, then, are exclusivelyfocused on the relevant components and exclude the determined range portfolios. Theinvestor obtains a better knowledge of the component effects and simultaneously thenumeric becomes more stable.
The second part of Chapter 4 contains the result of the proof for stability. There-fore two back-testing strategies were designed, producing a multi-period allocation ofprincipal portfolios. Through the holding periods, the resulting correlation structureswere determined. A suitable strategy should produce nearly uncorrelated portfolios.However, the results of the applied strategies were not satisfactory. Nevertheless, theresults showed an optimistic tendency, so that further research is recommend. It omitstill a potential in diverse strategies.
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5.2 Outlook
The obvious shortcoming on the measuring of correlations is the determination ofactive and future-straighten correlation structures. Better results may be achievedby considering, e.g., the usage of:
• Exponential weighted time series - the weights allows a better consideration ofactive effects
• Weekly returns, instead of daily ones - the correlation behaviour is more stableand not affected by time shift effects on the exchanges in the main time zones(America, Europe, Asia)
• Filtering the main principal components - the first principal components aremostly stable and reveal a high percentage of the market drivers (see Potters,Bouchaud, Laloux (2005)) the remaining ones can be neglected
• Optimized rebalancing frequencies and consideration periods
• Smoothing techniques regarding the reallocation in conjunction with the chang-ing eigenvectors of the PCA
• Robust and random correlation measures - the aim is to filter out random noise
• Consideration of regime switching triggers - allows a future-straighten estima-tion, correlation switch usually in high and low volatile markets
• Bayes theory and/or other analytical approaches - the consideration of (condi-tional) probabilities, i.e., the determination of correlations distribution mightbe a valuable extension, autocorrelations are also significant drivers, which hasto be considered.
Regarding the cPCA and the diversification measure, global optimization is still anopen issue. Furthermore, numerical results for more complex constraints and partic-ularly the impact of inequality constraints are still unknown, e.g., constraints like theVaR considered as linearized nonlinear constraints (see Hallerbach (2003) and Gouri-erieux, Laurent, Scaillet (2000)). Finally, the allocation sensitivity on the change ofstatistical data (mean, variance, correlation) has not yet been examined.
In sum we can say: The cPCA described here as well as the entropy diversificationapproach are two powerful and applicable techniques for an advanced asset alloca-tion. It is less a stand-alone solution than a valuable add-on view on investmentstructures that supports the understanding of potential diversification effects in themarkets. The cPCA allows the separation of fixed investments, which are determinedby the consideration of equality constraints. This view is allowed by the theoreticalframework we obtain with the cPCA.
57
The challenging treatment of changing or unstable correlations is mostly still neglectedin practice. In particular, the regulator’s calls for sophisticated stress tests as well asthe volatile markets make the need to investigate the correlation structures and theirprincipal components more urgent than it was the case in the past.
58
Appendix A
Mathematical Proofs
A.1 Proof of Theorem 2.2.3 (Alternative II):
Finding stationary points of (2.92) is equivalent to fulfilling the Lagrange criteria ofthe Lagrange function
L = eT Se − ν(eT e − 1) − eT BT γ. (A.1)
A stationary point (e, ν, γ) of L solves
0 =∂L∂e
= 2Se − 2νe − BT γ (A.2)
Multiplying with B leads to
0 = 2BSe − 2νBe − BBT γ (A.3)Be=0= 2BSe − BBT γ (A.4)
Matrix B has full row rank. Therefore the matrix product BBT is non-singular andthe vector of Lagrange multipliers γ is determined as
γ = 2 · (BBT )−1BSe (A.5)
Hence,
0 = Se − νe − BT (BBT )−1BSe (A.6)
= −νe + (I − BT (BBT )−1B)Se (A.7)
= −νe + PSe (A.8)
where P := I − BT (BBT )−1B. It remains the eigenvalue problem
PSe = νe, (A.9)
59
so that a stationary point (e, ν) represents an eigenvector e of the matrix product PSand its eigenvalue ν. By construction, we additionally know, that all eigenvectors eof PS are normalised.
Hence, finding a stationary points of (2.92) is equivalent to finding a stationary pointof program
maxeT e=1
eT PSe, (A.10)
Furthermore, each eigenvector e ∈ V = K(B) ⊕ R(BT ) can be represented by thesum of the two vectors eK ∈ K(B) and eR ∈ R(BT ), so that e = eK + eR. Therewith,we infer
PSe = PPSe (A.11)
= νPe (A.12)
= νP (eK + eR) (A.13)
= νeK (A.14)
Thus, it does not matter if we consider an eigenvector e or its kernel component eK.Using the fact that eK is entirely determined by the projection of e, i.e., Pe = eK, wefound an equivalent formulation to (A.10), reads
maxeT e=1
eT PSPe, (A.15)
since the matrix PSP has the same eigenvalues and eigenvectors as PS. ¤
A.2 Proof of Concavity
This section is dedicated to inspecting the convexity/concavity property of risk mea-sure ρM . We will see that concavity is not given globally.
Before starting the contemplations with respect to the concavity structure of ρM wefirst introduce two important characteristics of convex and concave functions, whichwill be helpful in the discussion below 1 2:
If f is (twice) differentiable, one can show that convex functions fulfil the followingconditions:
f(y) ≥ f(x) + ∇f(x)T (x − y) (First-order condition) (A.16)
1A set C is convex if for each x, y ∈ C and any θ with 0 ≤ θ ≤ 1 the combined elementsθx + (1 − θ)y are elements in C again.
2A function f : V → W ⊆ IR is convex if V is a convex set and if for all x, y ∈ V, and θ with0 ≤ θ ≤ 1, the inequality f(θx + (1 − θ)y) ≤ θf(x) + (1 − θ)f(y) holds. A function f is concave if−f is convex.
60
and
∇2f(x) is positive semi-definite. (Second-order condition) (A.17)
The composition of two differentiable functions h and g, f := h◦g preserves convexityif and only if
f ′′(x) = h′′(g(x))g′(x)2 + h′(g(x))g′′(x) (A.18)
is positive semi-definite3.
The following discussion is dedicated to examining the structure of the diversificationmeasure ρM , which is a composition of the functions:
f1 : IRn → IR
n+, f1(x) = [x2
i ]i=1,...,n
f2 : IRn+ → [0, 1]n, f2(x) = [xiλ
2i ]i=1,...,n /
∑ni=1 xiλ
2i
f3 : (0, 1]n → [0, 1/e] , f3(x) = −xT ln(x)f4 : [0, 1/e] →
(
0, ne1/e]
, f4(x) = exp(x)
(A.19)
so that eventually ρM = f4 ◦ f3 ◦ f2 ◦ f1 : IRn → IR.
First, consider the partially composition F2 := f4 ◦ f3. The exponential function f4 isconvex and strictly increasing. Whereas the entropy measure f3 is concave, strictlyincreasing for x ∈ (0, 1/e], strictly decreasing for for x ∈ (1/e, 1], and so its maximumis reached in x = 1/e. Criteria (A.18) cannot be derived because the terms of theright hand side have opposite signs, so that further calculations are needed.
The first derivative of composition F2 := f4 ◦ f3 : (0, 1] → [0, e1/e] is
F ′2 = e−x ln(x)(− ln(x) − 1)
which is positive for x ∈ (0, 1/e), zero for x = 1/e and negative for x ∈ (1/e, 1].Hence, the slope behaviour of f3 has been preserved and so the concavity property (seeFigure A.1). Composition F2 has been considered on IR. Without loss of generality,the result is still valid for IR
n.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Entropy function
p
−p
* ln
(p)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45Exponent of the entropy function
p
exp(
−p
* ln
(p))
p=1/e
p=1/e
Figure A.1: Entropy plot for one dimension
3For more details we refer to basic literature for convex analysis and optimisation, e.g., [4].
61
Second, consider the composition F1 := f2 ◦ f1. The quadratic function f1 is convexand its image is the IR
n+. The following Lemma4 gives us the deciding argument for
proving the convexity property of the composition:
Lemma A.2.1 (Linear-fractional function) Consider an affine function
l(x) : IRn → IR
m+1, l(x) =
(
AcT
)
x +
(
bd
)
where A ∈ IRm×n, b ∈ IR
m, c ∈ IRn and d ∈ IR. Then the linear-fractional function
f(x) : V → IRm, f(x) =
(Ax + b)
(cT x + d), V ⊆
{
x ∈ IRn∣
∣cT x + d > 0}
is convex.
Set c = (λ21 . . . λ2
n)T , A = diag(c) and b = 0, d = 0. Substituting this variablesinto F1 gives us a linear-fractional function F1 = Ax
cT x. Hence, according to the Lemma,
F1 is convex since x = f1(y) = y2 ∈ V .
Finally, consider the whole composition ρM(x) = F2 ◦ F1. Again we take a look onthe second derivative
(ρM)′′(x) = F ′′2 (F1(x))F ′
1(x)2 + F ′2(F1(x))F ′′
1 (x)
Because of the concavity of F2, the first term F ′′2 (F1(x))F ′
1(x)2 is always negative. Thesecond derivative of F1 is always positive since F1 is convex. The first derivative of F2,however, is changing its slope, i.e., the function is decreasing on the interval [1/e, 1].and increasing on the interval (0, 1/e). Consequently the second term F ′
2(F1(x))F ′′1 (x)
changes its sign and on interval (0, 1/e). we might observe indefiniteness, whenever−F ′′
2 (F1(x))F ′1(x)2 < F ′
2(F1(x))F ′′1 (x) holds. Indeed, such effects can be observed in
graphic plots (see Figure A.2).
4See [4] for a comprehensive introduction to linear-fractional functions
62
Figure A.2: Diversification measure plots
As a conclusion we find that the diversification measure ρM is not necessarily concave.
63
Appendix B
Numerical Details
B.1 Asset Allocation using PCA
B.1.1 Mean-Variance Weights on Original Assets
Target mean 1.8% 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3%AEXATX 1.0%FCHIGDAXIGREXP 83.6% 83.8% 74.2% 62.9% 51.4% 39.9% 28.3% 16.8% 5.2%FTSEOMXSPISSMI 1.4%OMXC20.COAORD 3.0%BSESN 0.2% 2.2% 4.1% 5.9% 7.4% 8.8% 10.3% 11.8% 13.2%HSIJKSE 0.5% 6.2% 17.1% 27.7% 38.1% 48.5% 58.8% 69.2% 79.5% 100.0%KLSE 1.8% 1.9%KS11 1.8% 1.2% 0.1%N225 0.6%STI 2.4%TWII 0.5%DJC 0.9%FVXGSPC 1.1%IXICNDXRUATNXTYX 2.7%TA100 1.3% 3.0% 3.4% 3.4% 3.1% 2.9% 2.6% 2.3% 2.0%Total 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 1000%
Table B.1: Optimal weights of the original portfolios due to mean-variance
B.1.2 Mean-Diversification Weights on Original Assets
For optimal mean-variance weights, see Table B.1.
64
Target returns 1.8% 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3%AEXATXFCHIGDAXIGREXP 62.0% 73.8% 68.8% 62.1% 52.9% 41.6% 30.0% 18.1% 5.9%FTSEOMXSPISSMIOMXC20.COAORDBSESN 1.9% 3.0% 3.6% 3.8% 4.0% 4.2% 4.6%HSI 12.4%JKSE 1.4% 17.0% 23.4% 32.3% 42.4% 53.5% 64.8% 76.1% 87.5% 100.0%KLSE 2.0%KS11N225 4.0% 1.3%STITWIIDJC 22.2%FVX 7.7% 1.7% 1.3% 1.1% 1.1% 1.3% 1.6% 2.0%GSPCIXICNDX 1.4% 0.2%RUATNXTYXTA100
Total 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%
Table B.2: Optimal weights of the original portfolios due to mean-diversification
B.1.3 Mean-Variance Weights on Principal Portfolios
Target Return 1.8% 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3% 8.3%pca1 0.8% 0.8% 2.1% 3.6% 5.2% 6.7% 8.3% 9.9% 11.5% 11.6% 11.6%pca2 -0.2% -2.1% -4.3% -6.7% -9.2% -11.8% -14.3% -16.8% -19.3% -20.6% -20.6%pca3 -0.4% -0.9% -2.7% -4.6% -6.5% -8.5% -10.5% -12.5% -14.5% -16.4% -16.4%pca4 -0.2% -0.7% -1.9% -3.0% -4.2% -5.5% -6.8% -8.1% -9.3% -10.0% -10.0%pca5 -0.4% -3.3% -7.8% -12.3% -16.3% -20.2% -24.1% -28.0% -32.0% -27.8% -27.8%pca6 -1.4% -3.6% -7.8% -12.8% -17.6% -22.3% -27.0% -31.8% -36.5% -54.1% -54.1%pca7 -0.5% 2.6% 7.3% 11.8% 16.3% 20.8% 25.2% 29.7% 34.2% 42.3% 42.3%pca8 0.7% 0.4% -1.8% -5.0% -8.2% -11.3% -14.5% -17.6% -20.7% -28.9% -28.9%pca9 0.1% -0.6% -0.3% 0.0% 0.6% 1.2% 1.8% 2.5% 3.1% 4.7% 4.7%pca10 2.1% 0.0% -4.2% -7.0% -9.7% -12.5% -15.3% -18.1% -20.9% -26.9% -26.9%pca11 -0.2% -2.2% -7.7% -12.0% -16.1% -20.3% -24.5% -28.7% -32.8% -42.3% -42.3%pca12 0.3% 2.7% 3.6% 4.7% 6.1% 7.4% 8.8% 10.1% 11.5% 10.5% 10.5%pca13 -0.4% 1.0% -0.5% -2.1% -3.0% -3.9% -4.7% -5.6% -6.4% -6.0% -6.0%pca14 -0.5% 1.4% 1.3% 1.6% 2.0% 2.4% 2.8% 3.2% 3.6% 3.9% 3.9%pca15 1.0% -0.4% -0.2% -0.1% 0.1% 0.3% 0.5% 0.6% 0.8% 1.6% 1.6%pca16 -0.8% 2.1% 2.3% 2.6% 2.9% 3.2% 3.5% 3.7% 4.0% 4.2% 4.2%pca17 -1.6% -2.0% -1.8% -1.7% -1.6% -1.5% -1.4% -1.3% -1.2% -0.9% -0.9%pca18 1.5% -1.1% -1.4% -1.8% -2.2% -2.6% -3.0% -3.4% -3.8% -4.6% -4.6%pca19 1.0% 0.1% 0.5% 0.8% 1.0% 1.2% 1.4% 1.7% 1.9% 2.7% 2.7%pca20 -0.9% -0.2% -0.2% -0.3% -0.3% -0.2% -0.2% -0.1% -0.1% -0.2% -0.2%pca21 0.6% 1.9% 1.6% 1.3% 1.1% 0.8% 0.6% 0.3% 0.1% -0.2% -0.2%pca22 -1.3% -1.8% -1.7% -1.5% -1.4% -1.2% -1.1% -1.0% -0.8% -0.9% -0.9%pca23 0.3% 0.3% 0.3% 0.3% 0.3% 0.4% 0.4% 0.4% 0.4% 0.4% 0.4%pca24 -5.6% -4.2% -3.8% -3.2% -2.7% -2.1% -1.5% -1.0% -0.4% -0.1% -0.1%pca25 -83.4% -83.6% -74.1% -62.9% -51.4% -39.9% -28.5% -17.0% -5.5% -0.3% -0.3%pca26 2.0% 1.7% 1.6% 1.5% 1.3% 1.2% 1.0% 0.9% 0.7% 0.7% 0.7%pca27 -1.9% -1.1% -1.0% -0.9% -0.8% -0.6% -0.5% -0.4% -0.2% -0.2% -0.2%Total -89.5% -92.7% -102.6% -109.7% -114.5% -119.0% -123.6% -128.2% -132.7% -157.7% -157.7%
Table B.3: Transformed weights of the principal portfolios due to mean-variance
65
B.1.4 Mean-Diversification Weights on Principal Portfolios
Target Return 1.8% 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3% 8.3%pca1 3.4% 3.5% 3.1% 3.9% 5.0% 6.5% 8.1% 9.7% 11.4% 11.6% 11.6%pca2 -3.8% 1.6% -4.7% -6.3% -8.4% -10.8% -13.2% -15.5% -17.8% -20.6% -20.6%pca3 -2.1% -5.0% -4.6% -5.7% -7.2% -9.2% -11.3% -13.4% -15.7% -16.4% -16.4%pca4 -1.8% -2.0% -2.9% -3.4% -4.4% -5.5% -6.8% -8.0% -9.3% -10.0% -10.0%pca5 -4.0% -4.9% -6.7% -10.9% -14.6% -17.8% -21.0% -24.3% -27.7% -27.8% -27.8%pca6 -5.9% -9.4% -11.5% -16.0% -21.3% -27.2% -33.1% -39.0% -44.9% -54.1% -54.1%pca7 -9.9% 7.5% 8.4% 13.4% 18.3% 23.0% 27.7% 32.5% 37.3% 42.3% 42.3%pca8 6.1% -4.6% -7.1% -8.9% -11.6% -14.9% -18.2% -21.6% -24.9% -28.9% -28.9%pca9 11.6% 1.4% 2.1% 2.4% 2.7% 3.1% 3.5% 3.9% 4.4% 4.7% 4.7%pca10 9.2% -4.7% -6.8% -8.7% -11.2% -14.2% -17.3% -20.4% -23.5% -26.9% -26.9%pca11 -9.2% -7.2% -10.2% -13.6% -17.7% -22.3% -27.0% -31.8% -36.5% -42.3% -42.3%pca12 -9.3% 1.7% 4.3% 4.5% 5.3% 6.5% 7.7% 8.9% 10.1% 10.5% 10.5%pca13 1.9% -0.3% -3.1% -2.5% -2.6% -3.4% -4.1% -4.9% -5.7% -6.0% -6.0%pca14 -1.6% 0.7% 0.7% 1.1% 1.5% 2.0% 2.5% 3.0% 3.6% 3.9% 3.9%pca15 -0.4% -4.0% -1.1% -0.7% -0.3% -0.1% 0.1% 0.2% 0.2% 1.6% 1.6%pca16 6.7% 3.6% 2.9% 3.0% 3.2% 3.4% 3.7% 4.0% 4.3% 4.2% 4.2%pca17 -4.2% -0.3% -1.0% -1.3% -1.4% -1.3% -1.1% -0.9% -0.7% -0.9% -0.9%pca18 -1.3% -0.6% -2.3% -2.1% -2.2% -2.6% -3.0% -3.5% -3.9% -4.6% -4.6%pca19 -1.1% 0.0% 0.5% 0.9% 1.1% 1.4% 1.7% 2.0% 2.2% 2.7% 2.7%pca20 0.3% 0.1% -0.4% -0.3% -0.3% -0.3% -0.2% -0.2% -0.1% -0.2% -0.2%pca21 -0.4% 1.3% 1.6% 1.3% 1.0% 0.8% 0.5% 0.2% -0.1% -0.2% -0.2%pca22 0.6% -1.7% -1.6% -1.5% -1.4% -1.3% -1.2% -1.0% -0.9% -0.9% -0.9%pca23 0.1% 0.5% 0.3% 0.4% 0.4% 0.4% 0.4% 0.4% 0.4% 0.4% 0.4%pca24 -3.0% -5.8% -4.0% -3.6% -3.0% -2.5% -1.9% -1.4% -0.9% -0.1% -0.1%pca25 -61.7% -73.6% -68.7% -62.1% -52.9% -41.6% -30.1% -18.3% -6.2% -0.3% -0.3%pca26 1.2% 2.3% 1.6% 1.4% 1.3% 1.2% 1.1% 0.9% 0.8% 0.7% 0.7%pca27 -1.0% -0.8% -0.9% -0.9% -0.8% -0.7% -0.5% -0.4% -0.3% -0.2% -0.2%Total -77.8% -98.3% -109.0% -112.3% -116.6% -121.9% -127.0% -132.1% -137.0% -149.4% -149.4%
Table B.4: Transformed weights of the principal portfolios due to mean-diversification
B.1.5 Principal Portfolio Weights
Index / pPortfolio pca1 pca2 pca3 pca4 pca5 pca6 pca7 pca8 pca9 pca10AEX 28.4% -4.6% 11.2% 29.4% 7.6% 9.8% 12.4% -6.1% 0.7% 8.7%ATX 21.5% -10.6% -0.3% 16.2% -17.2% -36.4% -28.1% 1.4% 13.5% -30.1%FCHI 27.5% -4.3% 11.8% 29.2% 7.5% 8.0% 8.2% -3.5% 1.5% 5.5%GDAXI 27.8% -0.8% 15.1% 20.6% 9.9% 16.1% 15.3% -5.6% 2.4% 6.7%GREXP -1.5% 0.9% 1.3% 0.6% -0.3% -0.5% 0.2% 0.8% 1.1% 0.2%FTSE 22.3% -4.5% 8.3% 23.5% 2.3% 4.5% 0.1% -3.0% 5.6% 4.1%OMXSPI 24.3% -6.3% 9.1% 21.3% 3.1% -1.0% 0.7% 2.1% -4.7% -5.6%SSMI 21.1% -5.2% 8.2% 22.0% 6.5% 8.3% 7.8% -6.5% -0.1% 1.0%OMXC20.CO 20.1% -9.9% 2.2% 17.2% -4.7% -16.2% -9.8% 5.4% -8.6% -11.4%AORD 9.3% -16.5% -12.5% -5.9% 6.5% -4.1% -13.7% -5.6% -6.4% -3.3%BSESN 15.7% -21.1% -11.1% -11.1% -73.8% 52.2% 6.1% 8.0% 5.0% 3.5%HSI 17.6% -27.4% -17.9% -20.4% -1.3% 9.0% -20.9% -13.8% -15.4% 0.6%JKSE 11.6% -20.6% -16.4% -10.0% -27.8% -54.1% 42.3% -28.9% 4.7% -26.9%KLSE 6.1% -11.2% -9.4% -5.9% -1.3% -25.3% 5.1% -32.0% -15.9% 79.8%KS11 16.0% -25.1% -20.2% -23.0% 32.0% 18.0% 11.6% 14.0% 17.0% -3.9%N225 15.4% -25.4% -18.0% -15.2% 33.7% 11.8% -44.8% -23.6% 4.0% -15.4%STI 14.3% -19.7% -12.9% -11.2% -3.9% 2.3% -2.6% -9.0% -9.5% 9.9%TWII 12.4% -20.2% -17.5% -17.4% 25.8% -5.8% 49.1% 44.1% 21.2% 3.5%DJC 8.3% -1.8% -1.3% 2.9% -14.6% -24.7% -36.5% 37.9% 59.1% 35.1%FVX 30.0% 54.0% -47.2% -1.9% -0.4% 1.7% 1.3% -2.5% -1.8% -3.6%GSPC 20.0% 10.6% 23.6% -21.3% -4.8% -3.6% -5.3% -0.8% 5.5% -2.3%IXIC 23.3% 13.7% 31.1% -34.7% 0.2% -4.5% 0.8% 0.1% -3.6% -0.3%NDX 24.4% 16.3% 35.8% -41.8% 3.5% -2.6% 2.7% -1.7% -7.2% 0.7%RUA 20.3% 10.6% 23.9% -21.9% -5.2% -4.9% -5.4% -0.1% 5.6% -2.1%TNX 19.4% 36.2% -31.9% 0.1% -0.5% 2.3% -0.6% -1.0% -1.8% 0.7%TYX 13.1% 24.2% -21.9% 0.8% -2.0% -1.5% -3.0% 1.2% 0.2% 3.8%TA100 11.5% -9.5% -4.2% 3.0% -4.9% -18.2% -12.3% 58.8% -68.1% 4.5%Total positive 481.6% 166.4% 181.6% 186.8% 138.6% 143.8% 163.8% 173.8% 147.1% 168.3%Total negative -1.5% -244.5% -242.6% -241.7% -162.7% -203.3% -182.8% -143.7% -143.1% -104.9%Total 480.1% -78.1% -61.0% -55.0% -24.1% -59.5% -19.0% 30.1% 4.0% 63.4%
Table B.5: Weights of principal portfolios 1 - 10
66
Index / pPortfolio pca11 pca12 pca13 pca14 pca15 pca16 pca17 pca18 pca19 pca20AEX -10.9% -5.3% -5.8% 7.3% -5.4% -10.3% -5.8% 5.5% 0.1% 26.0%ATX 58.4% -1.3% 9.9% 40.4% 10.5% 3.4% 7.5% -0.9% -15.7% 8.7%FCHI -8.5% -3.4% -7.3% 5.8% 0.5% 0.7% -2.5% 2.0% 1.1% 7.5%GDAXI -14.2% -4.7% 0.6% 23.6% 13.5% 16.0% 15.6% 1.5% -38.9% -66.4%GREXP -0.1% 0.2% 0.6% -0.4% -0.5% 2.1% -2.0% -0.5% 0.1% -0.5%FTSE -7.6% -6.4% -3.8% 3.7% -5.7% -10.6% -6.8% 3.1% 10.8% 30.0%OMXSPI 9.9% 9.4% 9.1% -29.2% 16.0% 51.8% 23.8% 1.3% 61.0% -5.3%SSMI -4.2% 0.1% -5.6% 5.1% -9.3% -26.6% -18.1% -7.7% -3.5% 25.6%OMXC20.CO 13.2% 14.7% 19.6% -72.8% -18.7% -13.6% -20.4% -7.6% -33.9% -16.8%AORD 1.8% 1.0% -11.7% 0.7% -4.6% -13.5% -19.9% 88.1% 14.8% -19.3%BSESN 9.5% 21.8% -11.6% 0.7% -2.8% 4.4% -3.5% 0.2% -2.0% 0.8%HSI -8.8% -69.5% 14.8% -1.5% 2.4% 28.4% -32.7% -14.6% -3.0% 2.3%JKSE -42.3% 10.5% -6.0% 3.9% 1.6% 4.2% -0.9% -4.6% 2.7% -0.2%KLSE 26.7% 20.7% 2.5% 6.7% -5.2% 6.9% -8.2% -7.2% 0.8% -2.5%KS11 -7.4% 34.3% 67.2% 16.3% 5.3% -3.0% -6.0% 0.6% 0.4% 9.4%N225 -10.5% 34.4% -49.0% -0.2% -5.3% 3.5% 8.4% -25.1% -2.1% -2.5%STI 1.1% -27.9% 9.5% -17.2% 11.8% -46.0% 73.1% 3.6% 0.8% 3.9%TWII 35.0% -17.8% -39.5% -9.3% -4.4% 1.1% -4.9% -6.6% 0.8% -4.4%DJC -36.1% -5.7% -1.4% -5.9% -1.4% 7.4% 5.9% 4.6% -4.0% 1.4%FVX 1.2% -3.8% 5.2% 5.8% -51.8% 13.0% 16.1% 4.3% -1.4% 1.3%GSPC 2.6% -3.5% 3.8% 8.2% -12.5% -28.1% -13.4% -17.1% 32.0% -26.9%IXIC 1.2% 3.8% -3.0% -4.4% 3.9% 6.8% 3.1% 6.9% -8.2% 12.3%NDX -0.9% 5.4% -6.3% -9.3% 10.8% 22.8% 9.3% 19.1% -29.9% 29.7%RUA 2.6% -2.7% 3.8% 7.4% -11.9% -27.2% -12.7% -16.3% 31.5% -25.2%TNX -1.3% 2.5% -4.5% -5.5% 32.2% -4.6% -10.1% -2.9% 0.0% 0.2%TYX 0.1% 4.5% -6.2% -9.1% 67.2% -19.6% -21.2% -4.8% 3.1% -0.8%TA100 -21.4% 12.3% -5.4% 20.4% -1.6% -2.5% 2.1% -7.6% 1.2% 0.1%Total positive 163.5% 175.5% 146.6% 156.0% 175.6% 172.3% 164.9% 141.0% 161.1% 159.2%Total negative -174.2% -152.0% -167.1% -164.7% -141.1% -205.6% -189.0% -123.5% -142.5% -170.7%Total -10.7% 23.5% -20.5% -8.6% 34.5% -33.3% -24.1% 17.5% 18.6% -11.5%
Table B.6: Weights of principal portfolios 11 - 20
Index / pPortfolio pca21 pca22 pca23 pca24 pca25 pca26 pca27AEX 49.9% 48.0% -43.1% -3.1% -0.4% 1.1% -0.1%ATX 1.6% 3.5% -0.6% 1.8% -0.6% 0.8% -0.2%FCHI 19.6% 4.6% 85.8% 1.7% 0.3% -1.0% -0.1%GDAXI -10.9% -11.2% -16.1% -1.6% -0.1% -1.3% 0.6%GREXP 2.1% -2.0% 0.4% -5.1% -99.7% 1.9% -1.3%FTSE 9.2% -83.4% -19.8% 2.5% 1.3% -1.5% 0.4%OMXSPI -16.2% 6.5% -9.0% -1.7% 0.2% 2.1% -0.1%SSMI -79.9% 20.8% -4.4% 0.1% -2.6% 0.6% 0.0%OMXC20.CO 8.0% -2.8% -0.7% 0.6% 0.6% 0.2% -0.2%AORD -4.7% 1.7% 0.3% 1.4% -1.1% -0.4% -0.2%BSESN 1.0% 0.0% 0.6% -0.6% -0.4% 0.3% 0.1%HSI -1.3% 2.5% 0.3% -1.2% 0.0% -1.0% 0.2%JKSE -0.2% -0.9% 0.4% -0.1% -0.3% 0.7% -0.2%KLSE -0.6% -2.6% 0.2% 0.8% -0.1% 0.4% -0.1%KS11 2.8% -1.2% 2.8% 0.5% -0.3% 0.2% 0.0%N225 4.4% -0.1% -2.1% -1.6% -1.1% -0.2% 0.1%STI -2.2% 1.2% 2.9% 1.9% -3.4% 0.6% 0.0%TWII 0.8% -2.9% -1.5% 0.1% -0.1% 0.9% 0.1%DJC -7.0% 7.3% -0.7% 1.1% 0.7% 0.3% -0.7%FVX -2.7% -0.6% 1.8% -26.6% 0.9% -0.1% 0.0%GSPC 5.8% 0.5% 0.4% 2.3% 1.5% 29.4% -64.8%IXIC -1.6% 1.3% -0.2% -0.7% -1.4% -79.1% -20.2%NDX -4.3% -2.1% 1.4% -1.3% 0.7% 53.2% 11.9%RUA 5.5% 1.7% 0.4% 1.7% -1.0% -4.5% 72.4%TNX 1.2% 2.2% -2.6% 77.6% -4.4% -0.1% 0.1%TYX 1.9% -2.2% 0.5% -56.5% 2.4% 0.4% -0.2%TA100 -0.2% -1.1% -1.1% 0.2% -0.6% 1.2% -0.1%Total positive 114.0% 101.7% 98.0% 94.4% 8.6% 94.4% 85.9%Total negative -131.8% -112.9% -101.8% -100.0% -117.5% -89.2% -88.6%Total -17.9% -11.1% -3.8% -5.6% -108.9% 5.2% -2.7%
Table B.7: Weights of principal portfolios 21 - 27
67
B.2 Asset Allocation using cPCA
B.2.1 Mean-Variance Weights on Original Portfolios
The following two tables contain the weight results due to the mean-variance opti-misation for the target return 0.025%, shown in Figure 4.14. The first Table B.8includes the optimal weights cumulating the range and the kernel portfolio. In thesecond Table B.9 range and kernel portfolio weights are separated.
wR optimal 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3%Target Return 1.8% 1.8% 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3%AEX -1.0 %ATX -0.1 % 1.0 %FCHI -1.6 %GDAXIGREXP 81.8 % 83.5 % 83.8 % 74.2 % 63.0 % 51.4 % 39.9 % 28.3 % 16.8 % 5.2 %FTSE 0.3 %OMXSPI 0.3 %SSMI 3.6 % 1.4 %OMXC20.CO -0.4 %AORD 3.3 % 3.0 %BSESN 0.3 % 0.3 % 2.2 % 4.2 % 5.9 % 7.4 % 8.8 % 10.3 % 11.8 % 13.2 %HSI -0.9 %JKSE 0.7 % 0.5 % 6.2 % 17.0 % 27.7 % 38.1 % 48.4 % 58.8 % 69.2 % 79.5 % 100.0 %KLSE 1.9 % 1.8 % 2.0 %KS11 1.8 % 1.1 % 0.1 %N225 0.6 % 0.5 %STI 2.9 % 2.4 %TWII 0.6 % 0.5 %DJC 1.3 % 0.9 %FVX -1.6 %GSPC 19.1 % 1.2 %IXIC 1.8 %NDX -1.0 %RUA -18.1 %TNX 2.6 %TYX 2.1 % 2.7 %TA100 1.6 % 1.3 % 3.0 % 3.4 % 3.4 % 3.1 % 2.9 % 2.6 % 2.3 % 2.0 %Total 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 %
Table B.8: Cumulated optimal weights of the original portfolios due to conditionalmean-variance
68
Target Return 1.8 % 1.8 % 2.5 % 3.3 % 4.0 % 4.7 % 5.4 % 6.1 % 6.9 % 7.6 % 8.3 %AEX -1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 %ATX -0.1 % 0.1 % 1.1 % 0.1 % 0.1 % 0.1 % 0.1 % 0.1 % 0.1 % 0.1 % 0.1 %FCHI -1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 %GDAXI 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 %GREXP 81.8 % 1.8 % 2.0 % -7.6 % -18.8 % -30.4 % -41.9 % -53.5 % -65.0 % -76.6 % -81.8 %FTSE 0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 %OMXSPI 0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 % -0.3 %SSMI 3.6 % -2.2 % -3.6 % -3.6 % -3.6 % -3.6 % -3.6 % -3.6 % -3.6 % -3.6 % -3.6 %OMXC20.CO -0.4 % 0.4 % 0.4 % 0.4 % 0.4 % 0.4 % 0.4 % 0.4 % 0.4 % 0.4 % 0.4 %AORD 3.3 % -0.3 % -3.3 % -3.3 % -3.3 % -3.3 % -3.3 % -3.3 % -3.3 % -3.3 % -3.3 %BSESN 0.3 % -0.1 % 1.9 % 3.9 % 5.6 % 7.1 % 8.5 % 10.0 % 11.4 % 12.9 % -0.3 %HSI -0.9 % 0.9 % 0.9 % 0.9 % 0.9 % 0.9 % 0.9 % 0.9 % 0.9 % 0.9 % 0.9 %JKSE 0.7 % -0.2 % 5.5 % 16.3 % 27.0 % 37.4 % 47.8 % 58.1 % 68.5 % 78.8 % 99.3 %KLSE 1.9 % -0.1 % 0.1 % -1.9 % -1.9 % -1.9 % -1.9 % -1.9 % -1.9 % -1.9 % -1.9 %KS11 0.0 % 0.0 % 1.8 % 1.1 % 0.1 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 %N225 0.6 % -0.1 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 %STI 2.9 % -0.5 % -2.9 % -2.9 % -2.9 % -2.9 % -2.9 % -2.9 % -2.9 % -2.9 % -2.9 %TWII 0.6 % -0.1 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 % -0.6 %DJC 1.3 % -0.4 % -1.3 % -1.3 % -1.3 % -1.3 % -1.3 % -1.3 % -1.3 % -1.3 % -1.3 %FVX -1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 % 1.6 %GSPC 19.1 % -18.0 % -19.1 % -19.1 % -19.1 % -19.1 % -19.1 % -19.1 % -19.1 % -19.1 % -19.1 %IXIC 1.8 % -1.8 % -1.8 % -1.8 % -1.8 % -1.8 % -1.8 % -1.8 % -1.8 % -1.8 % -1.8 %NDX -1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 % 1.0 %RUA -18.1 % 18.1 % 18.1 % 18.1 % 18.1 % 18.1 % 18.1 % 18.1 % 18.1 % 18.1 % 18.1 %TNX 2.6 % -2.6 % -2.6 % -2.6 % -2.6 % -2.6 % -2.6 % -2.6 % -2.6 % -2.6 % -2.6 %TYX 2.1 % 0.6 % -2.1 % -2.1 % -2.1 % -2.1 % -2.1 % -2.1 % -2.1 % -2.1 % -2.1 %TA100 1.6 % -0.3 % 1.4 % 1.8 % 1.8 % 1.5 % 1.2 % 1.0 % 0.7 % 0.4 % -1.6 %Total 100 % 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 %
Table B.9: Optimal weights of the original portfolios due to conditional mean-variance
B.2.2 Mean-Diversification Weights on Original Portfolios
The following two tables contain the weight results due to the mean-diversificationoptimisation for the target return 0.025%, shown in Figure 4.14. The first Table B.10includes the optimal weights cumulating the range and the kernel portfolio. In thesecond Table B.11 range and kernel portfolio weights are separated.
Target Return 1.8 % 1.8 % 2.5 % 3.3 % 4.0 % 4.7 % 5.4 % 6.1 % 6.9 % 7.6 % 8.3 %AEX -1.0 % 0.4 %ATX -0.1 % 1.3 % 2.6 % 4.2 %FCHI -1.6 %GDAXIGREXP 81.8 % 80.2 % 24.9 % 23.3 %FTSE 0.3 % 2.5 % 2.6 %OMXSPI 0.3 %SSMI 3.6 %OMXC20.CO -0.4 % 0.5 % 0.2 %AORD 3.3 % 1.2 %BSESN 0.3 % 0.2 % 8.0 % 4.0 % 6.3 % 7.4 % 10.0 % 12.1 %HSI -0.9 %JKSE 0.7 % 0.6 % 18.1 % 26.8 % 39.4 % 49.8 % 52.2 % 62.9 % 77.9 % 100.0 %KLSE 1.9 % 3.2 % 3.8 % 4.2 % 13.7 % 6.6 %KS11 2.9 % 5.9 % 6.2 % 2.7 %N225 0.6 % 0.6 %STI 2.9 % 3.9 % 11.6 % 11.3 % 13.6 %TWII 0.6 % 0.5 %DJC 1.3 % 2.5 % 26.8 % 21.4 % 28.8 % 36.9 % 31.0 % 8.3 %FVX -1.6 %GSPC 19.1 %IXIC 1.8 % 0.6 %NDX -1.0 % 0.3 %RUA -18.1 %TNX 2.6 %TYX 2.1 % 4.7 %TA100 1.6 % 23.3 % 16.1 % 22.1 % 29.5 % 27.1 % 10.1 %Total 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 %
Table B.10: Cumulated optimal weights of the original portfolios due to conditionalmean-diversification
69
Target Return 1.8% 1.8% 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3%AEX -1.0% 1.4% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0%ATX -0.1% 1.4% 0.1% 2.7% 4.4% 0.1% 0.1% 0.1% 0.1% 0.1% 0.1%FCHI -1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6%GDAXIGREXP 81.8% -1.6% -56.9% -58.5% -81.8% -81.8% -81.8% -81.8% -81.8% -81.8% -81.8%FTSE 0.3% -0.3% 2.2% 2.3% -0.3% -0.3% -0.3% -0.3% -0.3% -0.3% -0.3%OMXSPI 0.3% -0.3% -0.3% -0.3% -0.3% -0.3% -0.3% -0.3% -0.3% -0.3% -0.3%SSMI 3.6% -3.6% -3.6% -3.6% -3.6% -3.6% -3.6% -3.6% -3.6% -3.6% -3.6%OMXC20.CO -0.4% 0.4% 0.4% 0.9% 0.6% 0.4% 0.4% 0.4% 0.4% 0.4% 0.4%AORD 3.3% -2.1% -3.3% -3.3% -3.3% -3.3% -3.3% -3.3% -3.3% -3.3% -3.3%BSESN 0.3% -0.1% 7.7% -0.3% -0.3% 3.7% 6.0% 7.1% 9.6% 11.7% -0.3%HSI -0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9% 0.9%JKSE 0.7% -0.1% -0.7% 17.4% 26.1% 38.7% 49.2% 51.5% 62.2% 77.2% 99.3%KLSE 1.9% 1.3% -1.9% 1.9% 2.3% 11.8% 4.7% -1.9% -1.9% -1.9% -1.9%KS11 2.9% 5.9% 6.2% 2.7%N225 0.6% -0.1% -0.6% -0.6% -0.6% -0.6% -0.6% -0.6% -0.6% -0.6% -0.6%STI 2.9% 1.0% 8.7% 8.3% 10.7% -2.9% -2.9% -2.9% -2.9% -2.9% -2.9%TWII 0.6% -0.1% -0.6% -0.6% -0.6% -0.6% -0.6% -0.6% -0.6% -0.6% -0.6%DJC 1.3% 1.2% 25.5% 20.1% 27.5% 35.6% 29.7% 7.0% -1.3% -1.3% -1.3%FVX -1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6%GSPC 19.1% -19.1% -19.1% -19.1% -19.1% -19.1% -19.1% -19.1% -19.1% -19.1% -19.1%IXIC 1.8% -1.2% -1.8% -1.8% -1.8% -1.8% -1.8% -1.8% -1.8% -1.8% -1.8%NDX -1.0% 1.0% 1.0% 1.4% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0%RUA -18.1% 18.1% 18.1% 18.1% 18.1% 18.1% 18.1% 18.1% 18.1% 18.1% 18.1%TNX 2.6% -2.6% -2.6% -2.6% -2.6% -2.6% -2.6% -2.6% -2.6% -2.6% -2.6%TYX 2.1% 2.6% -2.1% -2.1% -2.1% -2.1% -2.1% -2.1% -2.1% -2.1% -2.1%TA100 1.6% -1.6% 21.7% 14.5% 20.5% -1.6% -1.6% 27.9% 25.5% 8.5% -1.6%
Total 100 % 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 %
Table B.11: Optimal weights of the original portfolios due to conditional mean-diversification
B.2.3 Mean-Variance Weights on Conditional Principal Port-folios
Target Return 1.8% 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3%Range prtf. 86.3% 86.3% 86.3% 86.3% 86.3% 86.3% 86.3% 86.3% 86.3% 86.3%cpca2 0.1% -1.6% -3.2% -4.8% -6.4% -8.1% -9.8% -11.4% -13.1% -14.4%cpca3 0.0% -0.1% -0.6% -1.0% -1.5% -2.0% -2.5% -3.0% -3.5% -5.0%cpca4 0.4% 0.4% 1.2% 2.2% 3.3% 4.3% 5.4% 6.4% 7.4% 7.6%cpca5 -0.3% -0.8% -4.5% -8.3% -12.6% -16.9% -21.2% -25.5% -29.9% -30.4%cpca6 0.1% 3.1% 8.4% 13.7% 18.5% 23.4% 28.2% 33.0% 37.8% 32.2%cpca7 -0.2% -4.3% -12.4% -21.2% -29.8% -38.3% -46.9% -55.4% -64.0% -84.7%cpca8 -0.2% 0.7% 0.4% -0.7% -1.8% -2.8% -3.9% -4.9% -5.9% -10.9%cpca9 -0.1% -1.5% -2.3% -2.7% -3.0% -3.1% -3.3% -3.5% -3.7% -3.3%cpca10 0.1% 0.8% 2.8% 4.2% 5.8% 7.5% 9.1% 10.8% 12.4% 15.4%cpca11 -0.2% 2.1% 8.2% 12.8% 17.3% 21.7% 26.2% 30.7% 35.2% 45.3%cpca12 -0.6% 1.5% 2.5% 3.3% 4.2% 5.1% 6.0% 6.9% 7.8% 7.0%cpca13 -0.3% -2.5% -1.6% -1.2% -1.4% -1.8% -2.1% -2.4% -2.8% -2.9%cpca14 -0.2% -2.7% -4.8% -8.0% -11.1% -14.2% -17.3% -20.4% -23.5% -24.3%cpca15 0.5% 0.9% 3.2% 6.0% 8.7% 11.3% 14.0% 16.6% 19.3% 19.7%cpca16 -0.1% -1.0% -1.7% -2.5% -3.4% -4.2% -5.0% -5.9% -6.7% -7.6%cpca17 0.8% 2.5% 1.8% 1.1% 0.3% -0.5% -1.2% -2.0% -2.8% -3.5%cpca18 -0.6% 0.8% 2.5% 4.6% 6.8% 9.0% 11.2% 13.4% 15.5% 16.8%cpca19 0.2% -0.8% -0.3% 0.5% 1.3% 2.2% 3.0% 3.8% 4.6% 4.1%cpca20 -0.2% -0.5% 0.8% 2.4% 4.2% 5.9% 7.6% 9.4% 11.1% 11.8%cpca21 0.2% -1.2% 3.4% 8.9% 14.4% 20.0% 25.6% 31.1% 36.7% 39.3%cpca22 3.0% 5.2% 0.4% -5.4% -11.3% -17.2% -23.1% -29.0% -34.9% -37.8%cpca23 -1.0% -1.0% 1.1% 3.6% 6.1% 8.6% 11.2% 13.7% 16.2% 17.6%cpca24 -1.4% -1.4% -1.0% -0.4% 0.1% 0.6% 1.2% 1.7% 2.3% 2.5%cpca25 -2.9% -1.4% -1.5% -1.6% -1.7% -1.7% -1.8% -1.9% -1.9% -1.9%cpca26 4.2% 4.5% 4.2% 3.8% 3.4% 3.0% 2.7% 2.3% 1.9% 1.7%cpca27 25.3% 26.0% 25.9% 25.8% 25.7% 25.6% 25.5% 25.3% 25.2% 25.1%Total 112.9% 114.0% 119.2% 121.4% 122.6% 123.7% 124.9% 126.0% 127.2% 105.9%
Table B.12: Optimal weights of the principal portfolios due to conditional mean-variance
70
B.2.4 Mean-Diversification Weights on Conditional Princi-pal Portfolios
Target Return 1.8% 2.5% 3.3% 4.0% 4.7% 5.4% 6.1% 6.9% 7.6% 8.3%Range prtf. 86.3% 86.3% 86.3% 86.3% 86.3% 86.3% 86.3% 86.3% 86.3% 86.3%cpca2 0.5% -2.7% -3.9% -5.8% -7.5% -8.8% -9.6% -11.1% -13.0% -14.4%cpca3 -0.5% 2.3% 1.3% 0.8% -1.7% -2.0% -1.1% -1.8% -3.2% -5.0%cpca4 0.7% 7.2% 7.4% 9.5% 6.4% 6.8% 9.1% 9.1% 8.3% 7.6%cpca5 -0.9% -12.3% -12.9% -17.7% -17.2% -21.3% -24.6% -27.2% -30.2% -30.4%cpca6 0.7% 11.7% 10.4% 14.7% 18.0% 23.2% 25.1% 30.6% 36.6% 32.2%cpca7 -0.8% 8.6% -12.0% -18.1% -28.0% -35.8% -38.2% -48.6% -62.2% -84.7%cpca8 -1.2% 1.1% -5.0% -7.0% -8.5% -7.5% -0.8% -1.4% -4.9% -10.9%cpca9 1.2% -15.6% -11.5% -16.5% 8.9% 6.4% -26.0% -25.9% -11.0% -3.3%cpca10 1.1% 17.1% 15.6% 21.9% 24.6% 25.8% 13.4% 9.3% 12.0% 15.4%cpca11 -1.3% 10.0% 13.0% 18.5% 21.5% 27.6% 30.6% 32.5% 36.0% 45.3%cpca12 -1.7% -11.8% -10.6% -13.3% -9.5% -5.5% 1.8% 5.4% 7.2% 7.0%cpca13 -1.3% -12.5% -10.2% -13.4% -18.6% -15.4% -9.5% -5.7% -3.8% -2.9%cpca14 -1.1% -23.8% -23.1% -32.1% -33.3% -30.4% -32.4% -30.9% -26.9% -24.3%cpca15 1.9% 21.6% 20.9% 27.7% 32.1% 28.8% 21.7% 20.1% 20.3% 19.7%cpca16 -2.1% -2.4% -3.2% -5.0% -8.0% -7.9% -5.9% -5.9% -6.7% -7.6%cpca17 -1.4% -14.5% -13.8% -18.6% -4.7% -4.7% -5.5% -4.8% -3.7% -3.5%cpca18 1.5% 10.6% 11.4% 15.8% 16.6% 16.2% 18.0% 18.0% 17.0% 16.8%cpca19 0.9% 7.1% 6.6% 9.2% 9.5% 8.7% 6.1% 5.1% 5.0% 4.1%cpca20 0.7% 10.5% 10.3% 13.4% 13.6% 13.4% 12.7% 12.1% 12.0% 11.8%cpca21 2.1% 28.1% 28.6% 40.4% 41.6% 40.8% 40.4% 40.1% 39.5% 39.3%cpca22 2.4% -28.0% -28.2% -41.8% -42.8% -41.6% -39.2% -38.2% -37.8% -37.8%cpca23 -0.4% 12.3% 13.1% 15.7% 15.9% 16.0% 17.3% 17.7% 17.5% 17.6%cpca24 -1.1% 2.1% 2.2% 2.9% 2.9% 2.7% 3.0% 2.9% 2.6% 2.5%cpca25 -4.0% -1.1% -1.1% -1.1% -1.3% -1.5% -1.7% -1.9% -1.9% -1.9%cpca26 4.9% 2.7% 2.4% 1.8% 1.9% 1.9% 1.6% 1.6% 1.7% 1.7%cpca27 25.9% 25.4% 25.4% 25.1% 25.0% 25.0% 25.2% 25.2% 25.2% 25.1%Total 112.8% 140.1% 119.4% 113.3% 143.8% 147.3% 117.5% 112.8% 122.0% 105.9%
Table B.13: Optimal weights of the principal portfolios due to conditional mean-diversification
B.2.5 Conditional Principal Portfolio Weights
Index / cpPortfolio cpca1 cpca2 cpca3 cpca4 cpca5 cpca6 cpca7 cpca8 cpca9 cpca10AEX -1.2% 6.2% 20.6% 32.8% -6.8% -3.7% -2.0% 9.6% 9.6% -10.5%ATX -0.1% -4.4% 6.6% 19.0% 6.8% 12.7% -16.3% -38.3% -17.1% 43.7%FCHI -1.9% 6.1% 20.5% 31.6% -3.9% -4.7% 0.2% 7.3% 7.5% -6.7%GDAXI 0.0% 9.6% 22.5% 23.3% -9.9% -4.7% 0.7% 13.9% 12.4% -11.0%GREXP 94.8% -6.4% -7.6% -14.2% 42.0% -11.4% 8.5% 6.6% 8.7% -9.5%FTSE 0.3% 2.6% 14.1% 23.2% 5.1% -2.7% 3.8% 1.2% 9.0% -2.2%OMXSPI 0.3% 2.1% 16.3% 23.2% -1.2% -2.2% -1.9% 0.1% -6.9% 1.2%SSMI 4.2% 1.3% 13.5% 21.4% 4.1% -6.1% 1.3% 6.6% 9.0% -10.9%OMXC20.CO -0.5% -4.2% 8.2% 18.3% 7.5% 1.2% -5.5% -11.1% -17.9% 8.3%AORD 3.8% -17.9% -10.8% -4.6% 6.1% -10.1% 6.4% -12.6% -2.6% -6.9%BSESN 0.4% -18.5% -5.3% -3.4% -6.2% 81.9% 31.8% 23.6% 12.4% -8.0%HSI -1.1% -24.0% -9.3% -5.8% -32.0% 10.2% 7.4% -25.6% -13.3% -6.5%JKSE 0.8% -20.9% -12.6% -5.3% 6.6% 22.0% -77.1% -5.5% 4.0% 6.1%KLSE 2.2% -14.9% -11.5% -10.5% 28.2% -11.6% -15.8% -9.7% 23.1% -54.7%KS11 0.0% -22.9% -12.8% -8.7% -36.6% -22.7% 9.0% 21.4% 10.3% 13.4%N225 0.7% -23.2% -10.7% -3.3% -28.1% -27.5% 27.3% -42.9% 9.4% -3.5%STI 3.4% -18.2% -8.0% -4.1% -10.0% 6.0% -0.4% -6.7% -2.2% -11.9%TWII 0.7% -20.2% -13.4% -9.5% -14.5% -25.0% -17.2% 60.7% 2.0% 28.2%DJC 1.5% -4.4% -4.9% -6.2% 42.6% -3.2% 27.9% -4.8% 28.0% 53.9%FVX -1.8% 54.6% -47.9% 10.8% -23.7% 7.0% -5.9% -3.9% -5.5% 4.6%GSPC 22.2% 15.0% 20.1% -24.7% 1.9% 3.1% 0.7% -5.2% 2.8% 5.3%IXIC 2.1% 19.8% 27.0% -36.1% -9.2% 0.6% -4.4% -1.3% -4.3% -0.3%NDX -1.2% 23.1% 30.8% -42.9% -14.1% -1.4% -5.0% -0.5% -5.4% -4.6%RUA -21.0% 15.1% 20.4% -25.1% 1.6% 3.3% 0.0% -5.4% 2.1% 6.3%TNX 3.0% 33.9% -35.3% 3.0% 2.4% -0.7% 3.5% 1.2% 1.9% -5.4%TYX 2.4% 20.4% -26.8% -2.0% 18.4% -4.4% 6.9% 2.3% 5.3% -7.0%TA100 1.9% -9.6% -3.6% -0.1% 23.1% -6.0% 16.1% 19.0% -82.4% -11.5%Total positive 144.7% 209.9% 220.5% 206.5% 196.5% 148.0% 151.7% 173.5% 157.7% 170.9%Total negative -28.8% -209.9% -220.5% -206.5% -196.5% -148.0% -151.7% -173.5% -157.7% -170.9%Total 115.9% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
Table B.14: Weights of conditional principal portfolios 1 - 10
71
Index / cpPortfolio cpca11 cpca12 cpca13 cpca14 cpca15 cpca16 cpca17 cpca18 cpca19 cpca20AEX 9.0% -8.3% 3.8% -9.6% -1.1% 10.4% 0.2% 6.0% 9.4% 31.4%ATX -53.2% 11.2% -3.6% -21.0% -34.2% -10.5% -7.5% 4.2% 12.1% 2.8%FCHI 7.4% -4.9% 5.7% -7.8% -0.7% -1.0% 1.6% 1.7% 2.5% 10.2%GDAXI 12.7% -7.5% -1.4% -17.3% -15.6% -21.9% -9.3% -2.9% 23.2% -74.9%GREXP 1.7% 6.3% 8.0% 22.0% -22.6% 3.5% 9.0% -16.5% -10.9% -14.9%FTSE 6.7% -6.6% 5.2% 0.2% -6.1% 12.5% 3.3% -1.3% -4.4% 30.4%OMXSPI -8.4% 10.3% -9.1% 18.1% 25.3% -47.2% 1.5% -40.0% -57.7% 4.1%SSMI 4.9% 2.8% 8.5% 5.5% -14.7% 27.3% 7.0% 14.1% 1.3% 17.8%OMXC20.CO -10.6% 16.2% -17.5% 54.2% 49.9% 23.4% 13.2% 23.3% 25.8% -22.3%AORD -0.1% 6.5% 15.7% 11.6% -15.8% 18.2% 19.3% -68.0% 34.0% 2.0%BSESN -8.1% 22.5% 6.0% -7.0% 5.1% -0.8% 5.2% 0.4% 1.7% 0.3%HSI 4.2% -66.5% 2.9% 13.9% -8.2% -16.3% 39.9% 13.0% -5.4% -2.1%JKSE 46.8% 12.6% 4.7% -5.2% -0.4% -4.1% 1.8% 2.8% -4.4% 0.1%KLSE -40.1% -7.0% -26.9% -33.8% 22.4% -2.3% 7.9% 10.9% -1.4% 0.4%KS11 8.6% 24.0% -69.1% 4.5% -24.7% -0.4% 3.5% 2.2% 1.5% 9.3%N225 16.1% 40.7% 38.0% -20.5% 18.4% -1.0% -7.1% 17.6% -12.0% -6.9%STI -4.4% -28.6% -2.0% 21.5% 2.3% 13.3% -85.2% -10.2% -0.6% 2.8%TWII -35.5% -9.6% 40.4% -3.4% 14.8% 2.3% 4.6% 6.4% -3.5% -3.8%DJC 26.8% -27.8% -14.3% -20.6% 27.9% -6.8% -4.4% -0.2% 7.3% 3.2%FVX -1.8% -6.4% -8.8% -21.9% 19.8% 27.6% 4.4% -23.2% -5.4% -6.9%GSPC -2.2% -1.6% -0.4% 3.3% -15.9% 30.3% 3.7% 8.7% -33.9% -15.7%IXIC -1.1% 3.2% 0.7% -1.4% 7.7% -8.3% -1.4% -2.3% 11.3% 10.3%NDX 0.9% 4.1% 2.5% -2.7% 16.2% -25.1% -2.1% -8.0% 34.7% 20.2%RUA -2.3% -1.3% -1.0% 2.7% -14.1% 29.0% 3.1% 8.6% -32.8% -13.8%TNX 1.9% 4.5% 6.6% 13.7% -9.5% -19.6% -1.4% 14.3% 3.8% 4.7%TYX 0.7% 8.2% 11.3% 30.1% -25.3% -34.0% -6.6% 30.0% 6.9% 10.4%TA100 19.6% 2.8% -5.9% -29.1% -0.9% 1.3% -4.4% 8.4% -3.2% 1.1%Total positive 167.9% 176.1% 160.0% 201.3% 209.8% 199.1% 129.4% 172.6% 175.7% 161.4%Total negative -167.9% -176.1% -160.0% -201.3% -209.8% -199.1% -129.4% -172.6% -175.7% -161.4%Total 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
Table B.15: Weights of conditional principal portfolios 11 - 20
Index / cpPortfolio cpca21 cpca22 cpca23 cpca24 cpca25 cpca26 cpca27AEX 28.1% 32.0% -51.2% 42.5% -3.0% -1.1% -0.1%ATX -9.3% 5.5% -4.8% 0.5% 1.9% -0.8% -0.2%FCHI 12.9% 12.2% -4.5% -85.7% 1.8% 0.9% -0.1%GDAXI -2.2% -4.9% 10.8% 16.1% -1.6% 1.3% 0.6%GREXP -48.0% 50.6% -21.0% -5.1% 0.4% 2.6% 0.9%FTSE -8.4% 24.4% 80.0% 20.2% 2.4% 1.5% 0.4%OMXSPI -1.2% -15.4% -5.0% 9.0% -1.7% -2.1% -0.1%SSMI -50.2% -65.4% -14.6% 4.5% 0.1% -0.5% 0.0%OMXC20.CO 3.9% 6.8% 2.3% 0.7% 0.6% -0.2% -0.2%AORD 34.0% -21.1% 3.1% 0.1% 1.3% 0.5% -0.1%BSESN -0.4% 1.2% -0.2% -0.6% -0.6% -0.3% 0.1%HSI -8.8% 3.2% -3.6% -0.4% -1.2% 1.0% 0.1%JKSE 0.5% -0.7% 1.1% -0.4% -0.1% -0.7% -0.2%KLSE 8.7% -6.7% 4.8% 0.0% 0.7% -0.4% 0.0%KS11 -1.5% 3.7% 0.6% -2.8% 0.6% -0.2% 0.0%N225 -4.0% 6.9% -1.2% 1.9% -1.5% 0.3% 0.1%STI -3.8% -0.4% -1.6% -3.1% 2.0% -0.4% 0.0%TWII 3.0% -0.5% 3.2% 1.5% 0.1% -0.9% 0.1%DJC 4.2% -13.1% -4.4% 0.9% 1.0% -0.3% -0.7%FVX -19.0% 8.3% -2.3% -2.1% -26.5% 0.1% -0.1%GSPC 23.5% -3.8% 1.1% -0.3% 2.2% -29.5% -64.8%IXIC -4.7% -1.0% -1.0% 0.3% -0.7% 79.1% -20.3%NDX -20.6% 4.0% 0.6% -1.5% -1.2% -53.2% 12.0%RUA 23.9% -5.1% 0.3% -0.2% 1.7% 4.6% 72.4%TNX 10.7% -5.6% -0.4% 2.8% 77.7% 0.3% 0.2%TYX 24.8% -12.2% 6.0% -0.2% -56.6% -0.4% -0.1%TA100 3.7% -3.1% 2.1% 1.2% 0.2% -1.1% 0.0%Total positive 182.0% 158.8% 115.9% 102.4% 94.8% 92.1% 87.1%Total negative -182.0% -158.8% -115.9% -102.4% -94.8% -92.1% -87.1%Total 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
Table B.16: Weights of conditional principal portfolios 21 - 27
72
B.3 Statistical Examinations
The aim of this section is the proof for market invariants.
Market invariants are independent and identically distributed random variables Xt,t ∈ Dt,τ , where Dt,τ ≡
{
t, t + τ , t + 2τ , . . .}
are equally-spaced dates for a startingpoint t and an estimation interval τ. A time-homogenous invariant does not dependon the reference time t. See [14] for details.
For an arbitrary time series
xt, t = t, t + τ , . . . , T
to be applied, the qualities of a time-homogenous market invariants have to be provedfor the quest of portfolio selection. We apply two tests to verify time series to bemarket invariants (see Section B.3.2 and Section B.3.3).
Mean-diversification relies simply on a linear correlation structure between the in-dices. Both approaches (MV and MD) have more quality when stable and robustmean returns, variances as well as correlations were put in. In basic approaches,independent random variables xt are additionally required.
The statistics below will show that not all indices are endowed with proper qualities.At least for the chosen return type and the chosen time horizon switching distribu-tions, fat tails and non-normal distributions lower the quality of a portfolio selection,so that further and/or alternative estimation techniques should be applied. However,because of the missing distribution assumption MD approach reacts more robust withinformation uncertainty as it is the case for the MV approach.
B.3.1 Distribution Parameter
In this section, a summary of parameters is given. Among others, mean, variance,skewness and kurtosis are discussed for the original indices and their principal port-folios.
73
Iteration TS Name Min Max Mean Variance Skewness Kurtosis
1 AEX -9.590334 10.028274 -0.021879 2.635519 -0.034130 8.9763882 ATX -10.252637 12.021037 0.037770 2.213167 -0.361171 11.9809963 FCHI -9.471537 10.594590 -0.014581 2.418631 0.072736 8.6652034 GDAXI -7.433464 10.797465 0.003633 2.594755 0.066979 7.6458575 GREXP -1.895914 1.935395 0.018855 0.053280 -0.173563 9.1943886 FTSE -9.264555 9.384244 -0.000486 1.700715 -0.111866 9.8321067 OMXSPI -7.991479 8.628890 0.010788 2.123749 0.028129 7.0630378 SSMI -8.107794 10.787642 -0.007302 1.643215 0.055472 9.3060999 OMXC20.CO -11.723193 9.496355 0.014300 1.800101 -0.255434 9.66782610 AORD -8.553591 5.360118 0.015835 1.019719 -0.659359 10.53597811 BSESN -11.809177 15.989984 0.060685 2.603264 -0.167372 10.88629512 HSI -13.582024 13.406809 0.016426 2.484593 0.019429 12.72692813 JKSE -10.953868 7.623376 0.082971 2.054029 -0.613347 9.20324814 KLSE -19.246395 19.860493 0.032162 1.263242 0.668443 103.76489615 KS11 -12.804697 11.284352 0.050392 2.488646 -0.577163 8.84042316 N225 -12.111026 10.086498 -0.009299 2.378883 -0.519539 9.01677517 STI -8.695982 7.530528 0.019338 1.522671 -0.242417 8.45211718 TWII -6.912347 6.524620 0.018397 2.067516 -0.182622 5.32490219 DJC -6.402840 5.647702 0.011968 1.288064 -0.266587 5.50984220 FVX -26.592389 17.499063 -0.034436 6.446026 -0.094081 11.15882221 GSPC -9.469514 10.957196 -0.001240 1.777266 -0.111651 11.89520422 IXIC -9.587690 11.159442 0.002628 2.648849 0.028051 7.21096723 NDX -11.114930 11.849331 -0.002242 3.400909 0.081197 7.25584124 RUA -9.740323 10.862791 0.001987 1.812652 -0.169020 11.43554725 TNX -17.039299 9.160353 -0.016200 2.868469 -0.235153 9.18032626 TYX -7.598591 7.512063 -0.007643 1.589392 -0.020955 6.96307827 TA100 -8.005506 9.714340 0.038074 1.519194 -0.133455 7.746726
Table B.17: Statistical data to daily compounded returns in %
Normal distribution had skewness zero and a kurtosis of 3. The statistic results showthat at least the kurtosis criterion does not need to be completely satisfied. Al-though, weekly or monthly compounded returns are more stable and it may worth toexamining these variants as well.
The next two tables list mean returns and variances of the principal portfolios, shownin Figure 4.6 and Figure 4.12.
74
Principal portfolio Mean Variancepca1 0.0308 23.6739pca2 -0.0924 9.1746pca3 -0.0238 6.5498pca4 -0.0355 4.2677pca5 -0.0636 1.8754pca6 -0.0436 1.3714pca7 0.0283 1.2529pca8 0.0148 1.2006pca9 -0.0061 1.1423pca10 -0.0024 0.9833pca11 -0.0021 0.936pca12 0.0314 0.8814pca13 0.03 0.8095pca14 0.0154 0.6916pca15 0.0139 0.5496pca16 0.0076 0.5309pca17 0.0001 0.5025pca18 0.0021 0.437pca19 -0.0008 0.4163pca20 -0.0099 0.3785pca21 -0.0081 0.2942pca22 -0.0126 0.2169pca23 -0.0025 0.1332pca24 0.0017 0.0702pca25 -0.02 0.0449pca26 -0.0014 0.0286pca27 0.0009 0.0033
Table B.18: Mean returns and variances of the principal portfolios
Conditional principal portfolio Mean Variancecpca1 (range portfolio) 0.021 0.0451
cpca2 -0.092 9.6911cpca3 -0.0114 6.8882cpca4 -0.0261 4.6166cpca5 0.0192 2.5777cpca6 0.0518 1.8275cpca7 -0.0436 1.2833cpca8 0.0147 1.2356cpca9 -0.0184 1.1467cpca10 0.0121 1.1357cpca11 0.0028 0.9375cpca12 0.0225 0.8944cpca13 -0.0381 0.8236cpca14 -0.01 0.7077cpca15 -0.0119 0.6745cpca16 -0.015 0.5405cpca17 0.0004 0.51cpca18 -0.0024 0.451cpca19 0.0005 0.4244cpca20 -0.0102 0.3841cpca21 -0.0048 0.3339cpca22 -0.001 0.2714cpca23 0.0105 0.2129cpca24 0.0019 0.133cpca25 0.0027 0.0701cpca26 0.0023 0.0286cpca27 0.0014 0.0033
Table B.19: Mean returns and variances of the conditional principal portfolios
B.3.2 Proof of Identical Distributions
For a proof of identically distributed time series [14] proposes to split the time seriesxt into two halves getting
xt, t = t, . . . , t +
[
T − t
2τ
]
τ
and
xt, t = t +
([
T − t
2τ
]
+ 1
)
, . . . , T
Doing so, we obtain the following parameters and distribution
75
B.3.2.1 European Market
European market, proof of market invariants
Sample tsName Min Max Mean Var Skewness Kurtosis1st Half 01-AEX -7.530973 9.516867 -0.026249 2.647732 0.091102 6.9594472nd Half 01-AEX -9.590334 10.028274 -0.016841 2.626728 -0.161920 11.025628rel. Diff. 01-AEX -27% -5% 36% 1% 278% -58%1st Half 02-ATX -4.483706 3.236725 0.096683 0.692837 -0.485351 5.3400882nd Half 02-ATX -10.252637 12.021037 -0.020577 3.729598 -0.217945 8.235059rel. Diff. 02-ATX -129% -271% 121% -438% 55% -54%1st Half 03-FCHI -7.678085 7.002286 -0.011367 2.212846 -0.027079 6.3543382nd Half 03-FCHI -9.471537 10.594590 -0.017600 2.628062 0.150060 10.191427rel. Diff. 03-FCHI -23% -51% -55% -19% 654% -60%1st Half 04-GDAXI -6.499882 7.552676 -0.010150 2.858072 -0.004290 5.3481362nd Half 04-GDAXI -7.433464 10.797465 0.018008 2.334568 0.167932 10.888629rel. Diff. 04-GDAXI -14% -43% 277% 18% 4014% -104%1st Half 05-GREXP -0.892898 1.036703 0.021778 0.038787 -0.340350 4.6945002nd Half 05-GREXP -1.895914 1.935395 0.016134 0.067785 -0.081932 9.838039rel. Diff. 05-GREXP -112% -87% 26% -75% 76% -110%1st Half 06-FTSE -5.588781 5.903779 -0.004285 1.329178 -0.121494 6.7890402nd Half 06-FTSE -9.264555 9.384244 0.003886 2.074402 -0.107863 10.446055rel. Diff. 06-FTSE -66% -59% 191% -56% 11% -54%1st Half 07-OMXSPI -7.991479 7.257002 0.007565 1.797736 -0.012928 6.3178822nd Half 07-OMXSPI -7.382235 8.628890 0.014515 2.452669 0.050928 7.208630rel. Diff. 07-OMXSPI 8% -19% -92% -37% 494% -14%1st Half 08-SSMI -5.780385 6.487236 -0.001423 1.574476 -0.001766 7.2235702nd Half 08-SSMI -8.107794 10.787642 -0.012668 1.714073 0.105816 11.031523rel. Diff. 08-SSMI -40% -66% -790% -9% 6091% -53%1st Half 09-OMXC20.CO -5.592228 4.969850 0.015241 1.237844 -0.209626 5.7506762nd Half 09-OMXC20.CO -11.723193 9.496355 0.013842 2.364817 -0.260233 9.640734rel. Diff. 09-OMXC20.CO -110% -91% 9% -91% -24% -68%
Table B.20: European market, statistical data to daily compounded returns in %
According to the parameters above and the historical returns, the following resultscan be plotted. Figure B.3.2.1 sketches the histograms as well as the correspondingnormal distributions based on the two split sample halves:
76
−8 −6 −4 −2 0 2 4 6 8
0.050.1
0.150.2
01−AEXc 1st sample half
−8 −6 −4 −2 0 2 4 6 8
0.050.1
0.150.2
01−AEXc 2nd sample half
−10 −8 −6 −4 −2 0 2 4 6 8 10
0.2
0.402−ATX
c 1st sample half
−10 −8 −6 −4 −2 0 2 4 6 8 10
0.2
0.402−ATX
c 2nd sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.050.1
0.150.2
0.2503−FCHI
c 1st sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.050.1
0.150.2
0.2503−FCHI
c 2nd sample half
−6 −4 −2 0 2 4 6 8 10
0.050.1
0.150.2
0.2504−GDAXI
c 1st sample half
−6 −4 −2 0 2 4 6 8 10
0.050.1
0.150.2
0.2504−GDAXI
c 2nd sample half
−1.5 −1 −0.5 0 0.5 1 1.5
0.51
1.52
05−GREXPc 1st sample half
−1.5 −1 −0.5 0 0.5 1 1.5
0.51
1.52
05−GREXPc 2nd sample half
−8 −6 −4 −2 0 2 4 6 8
0.10.20.3
06−FTSEc 1st sample half
−8 −6 −4 −2 0 2 4 6 8
0.10.20.3
06−FTSEc 2nd sample half
−6 −4 −2 0 2 4 6 8
0.050.1
0.150.2
0.2507−OMXSPI
c 1st sample half
−6 −4 −2 0 2 4 6 8
0.050.1
0.150.2
0.2507−OMXSPI
c 2nd sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.10.20.3
08−SSMIc 1st sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.10.20.3
08−SSMIc 2nd sample half
−10 −8 −6 −4 −2 0 2 4 6 8
0.10.20.3
09−OMXC20.COc 1st sample half
−10 −8 −6 −4 −2 0 2 4 6 8
0.10.20.3
09−OMXC20.COc 2nd sample half
Figure B.1: Distribution plots, European indices
The following plots sketch the overlaid qqplots confronting the quantiles of normaldistribution to the quantiles of the two halves:
−5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
01−AEXc
1st normal distr. 2nd normal distribution first half second half
−10 −5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
02−ATXc
−5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
03−FCHIc
−5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
04−GDAXIc
−1 0 1
0.050.1
0.250.5
0.750.9
0.950.99
0.999
Data
Pro
babi
lity
05−GREXPc
−5 0 5
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
06−FTSEc
−5 0 5
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
07−OMXSPIc
−5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
08−SSMIc
−10 −5 0 5
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
09−OMXC20.COc
Figure B.2: QQ-Plots plots, European indices
77
B.3.2.2 Asian Market
For explanations, see Section B.3.2.2.
Sample tsName Min Max Mean Var Skewness Kurtosis1st Half 10-AORD -4.899737 3.387204 0.031103 0.383803 -0.659646 8.5968942nd Half 10-AORD -8.553591 5.360118 0.000806 1.656667 -0.537074 7.508705rel. Diff. 10-AORD -75% -58% 97% -332% 19% 3%1st Half 11-BSESN -11.809177 7.931097 0.066148 1.747134 -0.833949 10.5600602nd Half 11-BSESN -11.604445 15.989984 0.056384 3.461587 0.083413 9.641895rel. Diff. 11-BSESN 2% -102% 15% -98% 110% 9%1st Half 12-HSI -9.285365 4.345420 0.000553 1.318364 -0.388249 7.3832772nd Half 12-HSI -13.582024 13.406809 0.032200 3.654125 0.090170 10.814022rel. Diff. 12-HSI -46% -209% -5722% -177% 123% -46%1st Half 13-JKSE -10.933211 4.850479 0.083853 1.591694 -0.699214 9.4299702nd Half 13-JKSE -10.953868 7.623376 0.081655 2.519270 -0.551417 8.484640rel. Diff. 13-JKSE -0% -57% 3% -58% 21% 10%1st Half 14-KLSE -6.342201 4.170897 0.022704 0.634627 -0.761154 11.0525982nd Half 14-KLSE -19.246395 19.860493 0.041821 1.893562 0.862693 91.198591rel. Diff. 14-KLSE -203% -376% -84% -198% 213% -725%1st Half 15-KS11 -12.804697 5.742081 0.070666 2.611487 -0.599728 7.0900942nd Half 15-KS11 -11.172001 11.284352 0.030815 2.368198 -0.555480 10.932043rel. Diff. 15-KS11 13% -97% 56% 9% 7% -54%1st Half 16-N225 -6.864457 5.735232 0.014378 1.839434 -0.217452 4.5930722nd Half 16-N225 -12.111026 10.086498 -0.032621 2.920717 -0.641054 10.124417rel. Diff. 16-N225 -76% -76% 327% -59% -195% -120%1st Half 17-STI -7.713594 4.905236 0.015982 1.061416 -0.424856 7.6675412nd Half 17-STI -8.695982 7.530528 0.022825 1.986219 -0.163380 7.756456rel. Diff. 17-STI -13% -54% -43% -87% 62% -1%1st Half 18-TWII -6.912347 5.612609 0.016518 2.099201 0.042188 4.8268532nd Half 18-TWII -6.735079 6.524620 0.020689 2.038775 -0.418140 5.850469rel. Diff. 18-TWII 3% -16% -25% 3% 1091% -21%1st Half 27-TA100 -4.009637 5.144642 0.045855 1.043139 0.111170 4.6111762nd Half 27-TA100 -8.005506 9.714340 0.030836 1.997085 -0.212388 7.712328rel. Diff. 27-TA100 -100% -89% 33% -91% 291% -67%
Table B.21: Asian market, statistical data to daily compounded returns in %
78
−8 −6 −4 −2 0 2 4
0.20.40.6
10−AORDc 1st sample half
−8 −6 −4 −2 0 2 4
0.20.40.6
10−AORDc 2nd sample half
−10 −5 0 5 10 15
0.10.20.3
11−BSESNc 1st sample half
−10 −5 0 5 10 15
0.10.20.3
11−BSESNc 2nd sample half
−10 −5 0 5 10
0.10.20.3
12−HSIc 1st sample half
−10 −5 0 5 10
0.10.20.3
12−HSIc 2nd sample half
−10 −8 −6 −4 −2 0 2 4 6
0.10.20.3
13−JKSEc 1st sample half
−10 −8 −6 −4 −2 0 2 4 6
0.10.20.3
13−JKSEc 2nd sample half
−15 −10 −5 0 5 10 15
0.10.20.30.40.5
14−KLSEc 1st sample half
−15 −10 −5 0 5 10 15
0.10.20.30.40.5
14−KLSEc 2nd sample half
−10 −5 0 5 10
0.050.1
0.150.2
0.2515−KS11
c 1st sample half
−10 −5 0 5 10
0.050.1
0.150.2
0.2515−KS11
c 2nd sample half
−12 −10 −8 −6 −4 −2 0 2 4 6 80.050.10.150.20.25
16−N225c 1st sample half
−12 −10 −8 −6 −4 −2 0 2 4 6 80.050.10.150.20.25
16−N225c 2nd sample half
−8 −6 −4 −2 0 2 4 6
0.10.20.3 17−STI
c 1st sample half
−8 −6 −4 −2 0 2 4 6
0.10.20.3 17−STI
c 2nd sample half
−6 −4 −2 0 2 4 60.05
0.10.15
0.20.25
18−TWIIc 1st sample half
−6 −4 −2 0 2 4 60.05
0.10.15
0.20.25
18−TWIIc 2nd sample half
−6 −4 −2 0 2 4 6 8
0.10.20.3 27−TA100
c 1st sample half
−6 −4 −2 0 2 4 6 8
0.10.20.3 27−TA100
c 2nd sample half
Figure B.3: Distribution plots, Asian indices
−5 0 5
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
10−AORDc
1st normal distr.2nd normal distributionfirst halfsecond half
−10 −5 0 5 10 15
0.050.10.250.50.750.90.950.990.999
Data
Pro
babi
lity
11−BSESNc
−10 −5 0 5 10
0.050.10.250.50.750.90.950.99
0.999
Data
Pro
babi
lity
12−HSIc
−10 −5 0 5
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
13−JKSEc
−15 −10 −5 0 5 10 15
0.050.10.250.50.750.90.950.990.999
Data
Pro
babi
lity
14−KLSEc
−10 −5 0 5 10
0.050.10.250.5
0.750.90.950.99
0.999
Data
Pro
babi
lity
15−KS11c
−10 −5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
16−N225c
−5 0 5
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
17−STIc
−5 0 5
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
18−TWIIc
−5 0 5
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
27−TA100c
Figure B.4: QQ-Plots plots, Asian indices
79
B.3.2.3 American Market
For explanations, see Section B.3.2.3.
Sample tsName Min Max Mean Var Skewness Kurtosis1st Half 19-DJC -4.274953 4.824465 0.029237 0.839101 -0.001349 4.4996572nd Half 19-DJC -6.402840 5.647702 -0.005042 1.738330 -0.320193 4.990863rel. Diff. 19-DJC -50% -17% 117% -107% -23631% -11%1st Half 20-FVX -7.616136 9.371401 -0.008424 3.640905 0.335060 4.8909032nd Half 20-FVX -26.592389 17.499063 -0.061361 9.258564 -0.175620 10.064374rel. Diff. 20-FVX -249% -87% -628% -154% 152% -106%1st Half 21-GSPC -4.414078 5.574432 -0.001363 1.197875 0.188564 5.2760532nd Half 21-GSPC -9.469514 10.957196 -0.001090 2.359386 -0.214515 12.154902rel. Diff. 21-GSPC -115% -97% 20% -97% 214% -130%1st Half 22-IXIC -6.511092 8.545415 -0.006517 2.739188 0.176943 5.0859652nd Half 22-IXIC -9.587690 11.159442 0.011721 2.562410 -0.135265 9.624335rel. Diff. 22-IXIC -47% -31% 280% 6% 176% -89%1st Half 23-NDX -8.066436 10.272710 -0.025034 4.182243 0.169122 5.4738082nd Half 23-NDX -11.114930 11.849331 0.020455 2.623751 -0.075212 10.522079rel. Diff. 23-NDX -38% -15% 182% 37% 144% -92%1st Half 24-RUA -4.361893 5.367126 0.003121 1.190903 0.162309 4.9800882nd Half 24-RUA -9.740323 10.862791 0.000802 2.437180 -0.271260 11.475967rel. Diff. 24-RUA -123% -102% 74% -105% 267% -130%1st Half 25-TNX -5.090157 5.971923 -0.011379 1.877718 0.390196 4.2163632nd Half 25-TNX -17.039299 9.160353 -0.021905 3.862553 -0.428551 9.134837rel. Diff. 25-TNX -235% -53% -93% -106% 210% -117%1st Half 26-TYX -6.748592 3.522869 -0.013936 0.971400 0.051298 5.1701122nd Half 26-TYX -7.598591 7.512063 -0.002024 2.209160 -0.046474 6.217797rel. Diff. 26-TYX -13% -113% 85% -127% 191% -20%
Table B.22: American market, statistical data to daily compounded returns in %
−6 −4 −2 0 2 4
0.2
0.419−DJC
c 1st sample half
−6 −4 −2 0 2 4
0.2
0.419−DJC
c 2nd sample half
−25 −20 −15 −10 −5 0 5 10 15
0.050.1
0.150.2
20−FVXc 1st sample half
−25 −20 −15 −10 −5 0 5 10 15
0.050.1
0.150.2
20−FVXc 2nd sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.10.20.3
21−GSPCc 1st sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.10.20.3
21−GSPCc 2nd sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.050.1
0.150.2
22−IXICc 1st sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.050.1
0.150.2
22−IXICc 2nd sample half
−10 −8 −6 −4 −2 0 2 4 6 8 10
0.050.1
0.150.2
23−NDXc 1st sample half
−10 −8 −6 −4 −2 0 2 4 6 8 10
0.050.1
0.150.2
23−NDXc 2nd sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.10.20.3
24−RUAc 1st sample half
−8 −6 −4 −2 0 2 4 6 8 10
0.10.20.3
24−RUAc 2nd sample half
−15 −10 −5 0 5
0.050.1
0.150.2
0.2525−TNX
c 1st sample half
−15 −10 −5 0 5
0.050.1
0.150.2
0.2525−TNX
c 2nd sample half
−6 −4 −2 0 2 4 6
0.2
0.4
26−TYXc 1st sample half
−6 −4 −2 0 2 4 6
0.2
0.4
26−TYXc 2nd sample half
Figure B.5: Distribution plots, American indices
80
−6 −4 −2 0 2 4
0.050.1
0.250.5
0.750.9
0.950.99
0.999
Data
Pro
babi
lity
19−DJCc
1st normal distr.2nd normal distributionfirst halfsecond half
−20 −10 0 10
0.050.10.250.50.750.90.950.990.999
Data
Pro
babi
lity
20−FVXc
−5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
21−GSPCc
−5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
22−IXICc
−10 −5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
23−NDXc
−5 0 5 10
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
24−RUAc
−15 −10 −5 0 5
0.050.10.250.5
0.750.90.950.99
0.999
Data
Pro
babi
lity
25−TNXc
−6 −4 −2 0 2 4 6
0.050.10.25
0.50.75
0.90.950.99
0.999
Data
Pro
babi
lity
26−TYXc
Figure B.6: QQ-Plots plots, American indices
B.3.3 Proof of Independent Random Variables
This test compares the time series xt and the lagged time series xt−τ . If the two seriesare independent of each other, the scatter plot is symmetrical. If they are identicallydistributed the scatter plot must lie in a circular cloud. (See [14])
81
−5 0 5 10
−5
0
5
10
01−AEXc
−10 0 10−10
−5
0
5
10
02−ATXc
−5 0 5 10
−5
0
5
10
03−FCHIc
−5 0 5 10
−5
0
5
10
04−GDAXIc
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
05−GREXPc
−5 0 5
−5
0
5
06−FTSEc
−5 0 5
−5
0
5
07−OMXSPIc
−5 0 5 10
−5
0
5
10
08−SSMIc
−10 −5 0 5
−10
−5
0
5
09−OMXC20.COc
Figure B.7: Scatter plots, European indices
−5 0 5
−5
0
5
10−AORDc
−10 0 10−10
0
10
11−BSESNc
−10 0 10
−10
−5
0
5
10
12−HSIc
−10 −5 0 5−10
−5
0
5
13−JKSEc
−10 0 10
−10
0
10
14−KLSEc
−10 0 10
−10
−5
0
5
10
15−KS11c
−10 0 10
−10
−5
0
5
10
16−N225c
−5 0 5
−5
0
5
17−STIc
−5 0 5
−5
0
5
18−TWIIc
−5 0 5
−5
0
5
27−TA100c
Figure B.8: Scatter plots, Asian indices
82
−5 0 5−6
−4
−2
0
2
4
19−DJCc
−20 −10 0 10
−20
−10
0
10
20−FVXc
−5 0 5 10
−5
0
5
10
21−GSPCc
−5 0 5 10
−5
0
5
10
22−IXICc
−10 0 10
−10
−5
0
5
10
23−NDXc
−5 0 5 10
−5
0
5
10
24−RUAc
−15 −10 −5 0 5
−15
−10
−5
0
5
25−TNXc
−5 0 5
−6
−4
−2
0
2
4
6
26−TYXc
Figure B.9: Scatter plots, American indices
83
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