1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 12. nance notions as self-contained as possible.This thesis is organized as follows. In the
13. 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. 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. 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. 16. Introduction 5Bayes rule, the Black-Litterman model brings the
17. 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. 18. fth section and proposedirections for further research.1.1 Quantitative portfolio allocationLet us de
19. 19. ne si;t, the price at time t of security i (typically i can be a stock representing partialownership of a
20. 20. rm). Holding this security from time t