Table Of ContentA thesis submitted for the degree of
Master of Technology
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Financial Portfolio Optimization:
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Computationally guided agents to investigate, analyse and invest!?
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v Ankit Dangi
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CENTRE FOR MODELING AND SIMULATION
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1. UNIVERSITY OF PUNE
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June, 2012
CERTIFICATE
This is to certify that the work incorporated in this thesis entitled "Financial Portfolio Optimization:
Computationally guided agents to investigate, analyse and invest!?" is a bonafide work carried out
by Ankit Dangi in partial fulfillment of the award of the degree of Master of Technology in Modeling
and Simulation of University of Pune, during the second half of Academic year 2011-12 under our
supervision. Material as has been obtained from other sources has been duly acknowledged in the thesis
and that this work has not been submitted elsewhere for a degree.
Dr. Abhijit Kulkarni Dr. Sukratu Barve
SAS Solutions OnDemand (SSO) Assistant Professor
Advanced Analytics Laboratory Centre for Modeling and Simulation
SAS Research and Development India University of Pune
Prof. Anjali Kshirsagar
Director
Date: June 15, 2012 Centre for Modeling and Simulation
Place: Pune, India University of Pune
DECLARATION
I hereby declare that the thesis “Financial Portfolio Optimization: Computationally guided
agents to investigate, analyse and invest!?” submitted for the degree of Master of Technology in
Modeling and Simulation to the University of Pune has not been submitted by me for a degree to
any other University.
Date: June 15, 2012 Ankit Dangi
Place: Pune, India M. Tech. (M&S), Batch 2010-12
Acknowledgements
Wordswouldn’tsufficeforthedeepsenseofgratitudewithwhichIwouldliketosincerely
thankmyguideDr. AbhijitKulkarniforhisguidancethroughoutthecourseofthiswork.
I would like to whole-heartedly acknowledge Prof. Uttara V. Naik-Nimbalkar and Dr.
T. V. Ramanathan from Department of Statistics, University of Pune.
I am grateful to Dr. Mihir Arjunwadkar, Dr. Sukratu Barve and Prof. Anjali Kshir-
sagar from Centre for Modeling and Simulation, University of Pune.
I am thankful to Dr. Lokesh Nagar, Mr. Viswanath Pothinindi, Dr. Sourish Das,
Mr. Prasad Paranjpe, Mr. Gaurav Singh, Dr. Swarup De, Mr. Mousum Dutta and Mr.
Ashwin Deokar from SAS Solutions OnDemand, SAS Research and Development India.
I am thankful to my parents, family and friends for their support and encouragement.
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Contents
1 Portfolio Optimization 2
1.1 Introduction, Background & Motivation . . . . . . . . . . . . . . . . . . 2
1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Problem Formulations 10
2.1 Markowitz Modern Portfolio Theory (MMPT). . . . . . . . . . . . . . . 10
2.2 Robust Formulation addressing Parameter Uncertainty . . . . . . . . . . 11
2.3 Risk-based Asset Allocation Strategies . . . . . . . . . . . . . . . . . . . 13
3 Computationally Guided Agents Approach 17
3.1 Conceptual Representation . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Classes of Solver Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Super-agent Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Empirical Validation and Results 25
4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Nature of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Scenario 1: Bearish Market . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Scenario 2: Bullish Market. . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Conclusion and Future Work 35
A Return and Risk Measures 37
A.1 Return Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.1.1 Absolute Return Measures. . . . . . . . . . . . . . . . . . . . . . 37
A.1.2 Relative Return Measures . . . . . . . . . . . . . . . . . . . . . . 37
A.1.3 Absolute Risk-Adjusted Return Measures . . . . . . . . . . . . . 38
A.1.4 Relative Risk-Adjusted Return Measures . . . . . . . . . . . . . 38
A.2 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A.2.1 Absolute Risk Measures . . . . . . . . . . . . . . . . . . . . . . . 39
A.2.2 Relative Risk Measures . . . . . . . . . . . . . . . . . . . . . . . 39
A.2.3 Tail Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 39
B Business & Technical Constraints 40
C Characteristics of Solver Agents 42
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Abstract
Financial portfolio optimization is a widely studied problem in mathematics, statistics,
financial and computational literature. It adheres to determining an optimal combina-
tion of weights that are associated with financial assets held in a portfolio. In practice,
portfolio optimization faces challenges by virtue of varying mathematical formulations,
parameters, business constraints and complex financial instruments. Empirical nature
of data is no longer one-sided; thereby reflecting upside and downside trends with re-
peated yet unidentifiable cyclic behaviours potentially caused due to high frequency
volatile movements in asset trades. Portfolio optimization under such circumstances is
theoretically and computationally challenging. This work presents a novel mechanism
to reach to an optimal solution by encoding a variety of optimal solutions in a solution
bank to guide the search process with regard to the global investment objective for-
mulation. It conceptualizes the role of individual solver agents that contribute optimal
solutions to a bank of solutions, and a super-agent solver that learns from the solution
bank, and, thus reflects a knowledge-based computationally guided agents approach to
investigate, analyse and reach to optimal solution for informed investment decisions.
Conceptual understanding of classes of solver agents that represent varying problem
formulations and, mathematically oriented deterministic solvers along with stochastic-
search driven evolutionary and swarm-intelligence based techniques for optimal weights
are discussed in this work. Algorithmic implementation of the computational guidance
approach from a bank of optimal solutions is presented by an enhanced neighbourhood
generationmechanismintheSimulatedAnnealingalgorithm. Aframeworkforinclusion
of heuristic knowledge and human expertise from financial literature related to invest-
mentdecisionmakingprocessisreflectedviatheintroductionofcontrolledperturbation
strategies using a decision matrix for neighbourhood generation. Empirical validation of
theproposedmethodologyhasbeencarriedoutforBearishandBullishmarketscenarios.
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Contributions
Academic/Computational Contributions
• Computationally guided agents approach to financial portfolio optimization that
incorporates varying problem formulations, variety of parameters, and complex
business constraints.
• Conceptualizationofacomputationalmodelcomprisingofvariousclassesofsolver
agentsthatcontributeoptimalsolutionstoabankofsolutionsthatguidesasuper-
agent solver towards obtaining optimal solutions for a global investment objective.
• Development of a novel mechanism in Simulated Annealing algorithm for neigh-
bourhood generation using solution bank approach.
• Conceptualization of a perturbation strategy using decision matrix for controlled
disturbances to decisions variables during neighbourhood generation.
Contributions to SAS Research and Development
• Portfolio Optimization Component of Asset Performance Management Solution
- Development of Robust Portfolio Optimization Formulation
- Development of Risk-based Asset Allocation Strategies
• PlatformforInnovation(SASFoundry)andKnowledgeManagement(SASToolPool)
- Simulated Annealing for Global Optimization (Algorithm Development)
- Ant Colony Optimization for Continuous Domains (Algorithm Development)
- A Suite of Benchmark Functions for Global Optimization
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Notations
n/N number of assets
x n x 1 column-vector of portfolio weights
xT transpose of x ; 1 x n row-vector of portfolio weights
x ith element of x, denoting the weight of asset i in the portfolio
i
r expected return of the portfolio
p
σ2 variance of the portfolio
p
r n x 1 vector of expected returns for n assets
s n x 1 vector of expected risks for n assets
r ith element of r, denoting the expected return of ith asset
i
s ith element of s, denoting the expected risk of ith asset
i
Σ n x n covariance matrix for the returns on the assets in the portfolio
ρ correlation coefficient between returns on asset i and j ∈ [−1,1]
ij
Γ budget of uncertainty of constraints for all assets in the portfolio ∈ [0,n]
p uncertainty optimization parameter for budget of uncertainty Γ parameter
q uncertainty optimization parameter for expected risk of asset i in the portfolio
i
(1)
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List of Figures
3.1 Computationally Guided Agents Framework: Basic Model . . . . . . . . 18
3.2 Classes of Solver Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Neighbourhood Generation: Solution Bank Approach . . . . . . . . . . . 22
3.4 Solution Bank Approach with Perturbation Strategy (Decision Matrix) . 23
4.1 Computationally Guided Agents Framework: Extended Model . . . . . 26
4.2 BSE Sensex Index (2000-2012) . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Bearish & Bullish Market Scenarios. . . . . . . . . . . . . . . . . . . . . 28
4.4 Portfolio Constituents: 29 BSE Sensex Equities . . . . . . . . . . . . . . 29
4.5 Bearish Market Scenario: Exploratory Analysis . . . . . . . . . . . . . . 30
4.6 Bearish Market Scenario: Solution Bank Inclinations . . . . . . . . . . . 30
4.7 Bearish Market Scenario: Experimental Results . . . . . . . . . . . . . . 31
4.8 Bullish Market Scenario: Exploratory Analysis . . . . . . . . . . . . . . 32
4.9 Bullish Market Scenario: Solution Bank Inclinations . . . . . . . . . . . 32
4.10 Bullish Market Scenario: Experimental Results . . . . . . . . . . . . . . 33
4.11 Experimental Set-up: Solver Parameters . . . . . . . . . . . . . . . . . . 34
4.12 Experimental Set-up: Super-Agent Configuration . . . . . . . . . . . . . 34
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