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- 1. Views Integration in a Quantitative Portfolio
AllocationMaster Thesis of Thibault Vatter1;2Supervized by David
Morton de Lachapelle2 and Paolo De Los Rios1Abstract Modi
- 2. cations of statistical forecasts by investors having a
particular percep-tionof future market conditions prove to be of
utmost importance in practice. In thisthesis, we investigate the
eects of market views and review dierent possibilities
toincorporate them in a quantitative scheme. In the
- 3. rst section, we start by recallingthe optimal allocation
problem and set the notations. In the second section, we re-viewthe
concepts of information sets and ecient market hypothesis and
formalize theincorporation of views in a quantitative framework. In
the third section, we presentthe path-breaking approach of Black
and Litterman, capitalizing on Gaussian markets,the CAPM and Bayes
rule. In the fourth section, we oer new insights into
Meucci'sapproach, translating views into information gain using
f-divergences as a measure ofdistortion between distributions. In
the
- 4. fth section, we conclude on the project andgive directions
of interest for the future.1LBS - Institute of Theoretical Physics,
EPFL, thibault.vatter@epfl.ch2QAM Department, Swissquote Bank SA,
thibault.vatter@swissquote.ch
- 5. 2 Views Integration in Quantitative Portfolio
AllocationContents1 Introduction 41.1 Quantitative portfolio
allocation . . . . . . . . . . . . . . . . . . . . . . . . . . .
51.2 The mean-variance allocation scheme . . . . . . . . . . . . .
. . . . . . . . . . . . 61.3 A general formalism . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Dimension
reduction and linear factor models . . . . . . . . . . . . . . . .
. . . . 121.5 The Fama{French three-factor model and market
benchmarks . . . . . . . . . . . 142 Quantitative integration of
market views 172.1 Information sets, ecient market hypothesis and
market views . . . . . . . . . . 172.2 Problem formalization . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1
Views focus . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 222.2.2 Views integration . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 233 The Black-Litterman model
and extensions 233.1 First pillar: a Gaussian market . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 243.2 Second pillar: CAPM
reverse optimization . . . . . . . . . . . . . . . . . . . . . .
253.3 Third pillar: Bayesian views integration . . . . . . . . . .
. . . . . . . . . . . . . 283.4 The Augmented Black-Litterman model
. . . . . . . . . . . . . . . . . . . . . . . 32
- 6. CONTENTS 34 The scenario-based approach 364.1 Learning from
disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 374.2 Analytical example . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 404.3 Fullyexible probabilities
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4
Relative entropy minimization . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 444.5 Copulas andexible market models . . . . .
. . . . . . . . . . . . . . . . . . . . . 464.5.1 Time-varying
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
474.5.2 Some useful copulas . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 504.5.3 Dependence structure estimation . . . .
. . . . . . . . . . . . . . . . . . . 534.5.4 Closing the model . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.6
Numerical example : portfolio stress testing . . . . . . . . . . .
. . . . . . . . . . 574.6.1 Measures of market risk . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 574.6.2 Stress testing . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
594.7 Numerical example: mean-variance, market equilibrium and
relative entropy . . . 605 Conclusion 616 Acknowledgments 63A The
Fama and French factors 68B Jensen-Shannon's divergence
minimization 69C Mean-CV aR optimization 70
- 7. 4 Views Integration in Quantitative Portfolio Allocation1
IntroductionTo use the [expected returns-variance] rule in the
selection of securities we must have proce-dures for
- 8. nding reasonable i and ij . These procedures, I believe,
should combine statisticaltechniques and the judgment of practical
men.Harry Markowitz, 1952Since its begining with Markowitz in 1952,
modern portfolio theory mixes art and science: judg-ment of
practical men and powerful statistical techniques. The two
approaches, althoughcomplementary, are sometimes dicult to
conciliate. While practitioners frequently discardquantitative
strategies as obscures mathematical complications,
- 9. nancial engineers forget thatportfolio management is often
about common sense. We attempt to reconcile both in a
soundtheoretical and practical framework. In this scope, we review
methods allowing the alterationof statistical forecasts by an
investor having a particular perception of future market
conditions;these modi
- 10. ed forecasts contain the investor's views in suitable form
for a quantitative portfolioallocation.The idea is to set in a
single frame current approaches and related concepts to formalize
thisproblem and bring sound theoretical and practical answers. We
choose to adopt a general for-malismwithout elaborating much on the
underlying technical concepts. In order to keep thereader a oat, we
try to give as many heuristic justi
- 11. cations, intuitions and numerical exampleas possible rather
than hard proofs. We start with the basics and progress towards
increasinglycomplex methods while keeping practical applicability
in mind, at the cost of simplifying some-timesdrastically
real-world situations and behaviors. We assume only that the reader
is familiarwith the basics of probability and statistics (from an
introductory university level course), andwe try to make the
- 12. nance notions as self-contained as possible.This thesis is
organized as follows. In the
- 13. rst section, we expose the problem of optimal
assetallocation along with useful notations and practical examples.
We also introduce the issue ofdimension reduction and linear
models, as exogenous factors are often the focus of
practitionersviews. In the second section, we de
- 14. ne what kind of information is relevant for portfolio
opti-mizationin the context of the ecient market hypothesis of
Fama. Then, we formalize theproblem of incorporating this
information into an actual allocation. In the third section, we
startas often in
- 15. nance with the Gaussian description of markets. Using the
CAPM1 equilibrium and1Introduced by William Sharpe and John
Lintner, the Capital Asset Pricing Model describes the
relationshipbetween a security risk and its associated
premium.
- 16. Introduction 5Bayes rule, the Black-Litterman model brings
the
- 17. rst answer to the problem. In the fourthsection, we leave
Gaussian markets for more advanced statistical modelling and use
the conceptof distortion between distributions to translate market
views into information gain. To generatethe required Monte-Carlo
simulations and test this technique in various situations, we
presentan advanced andexible market model. Finally, we conclude in
the
- 18. fth section and proposedirections for further research.1.1
Quantitative portfolio allocationLet us de
- 19. ne si;t, the price at time t of security i (typically i can
be a stock representing partialownership of a
- 20. rm). Holding this security from time t