robust portfolio selection via utility optimisation with smaller...
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Problem ModellingExprimental results
Conclusion
Robust Portfolio Selection via Utility Optimisation
with smaller uncertainty sets
Denis Zuevdenis.zuev@maths.ox.ac.uk
OCIAM,
University of Oxford
May 18, 2007
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Main idea of the talk
Idea
Having got 1000 pounds we want to find an optimal way ofinvesting this money into a set of given shares in an optimal way asto maximise our expected returns and minimise risks after a fixedinvestment period.
This investment should be:
robust to estimation errors,
stable over time.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Main idea of the talk
Idea
Having got 1000 pounds we want to find an optimal way ofinvesting this money into a set of given shares in an optimal way asto maximise our expected returns and minimise risks after a fixedinvestment period.
This investment should be:
robust to estimation errors,
stable over time.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Main idea of the talk
Idea
Having got 1000 pounds we want to find an optimal way ofinvesting this money into a set of given shares in an optimal way asto maximise our expected returns and minimise risks after a fixedinvestment period.
This investment should be:
robust to estimation errors,
stable over time.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Definitions
Let Si be the closing price of n stocks on day i .
Let ri =Si−Si−1
Si−1be daily returns of stocks.
We assume that the period return r ∼ N(µ,Σ).
Let φ denote a share of our wealth in stocks.
Let R ∈ Rm×n - a matrix of observations, where m is the
number of stocks and n is the number of observations.
Let µ = 1n
∑
ri = R1n
and Σ = 1nR
(
I − 1n11T
)
RT .
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Definitions
Let Si be the closing price of n stocks on day i .
Let ri =Si−Si−1
Si−1be daily returns of stocks.
We assume that the period return r ∼ N(µ,Σ).
Let φ denote a share of our wealth in stocks.
Let R ∈ Rm×n - a matrix of observations, where m is the
number of stocks and n is the number of observations.
Let µ = 1n
∑
ri = R1n
and Σ = 1nR
(
I − 1n11T
)
RT .
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Definitions
Let Si be the closing price of n stocks on day i .
Let ri =Si−Si−1
Si−1be daily returns of stocks.
We assume that the period return r ∼ N(µ,Σ).
Let φ denote a share of our wealth in stocks.
Let R ∈ Rm×n - a matrix of observations, where m is the
number of stocks and n is the number of observations.
Let µ = 1n
∑
ri = R1n
and Σ = 1nR
(
I − 1n11T
)
RT .
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Definitions
Let Si be the closing price of n stocks on day i .
Let ri =Si−Si−1
Si−1be daily returns of stocks.
We assume that the period return r ∼ N(µ,Σ).
Let φ denote a share of our wealth in stocks.
Let R ∈ Rm×n - a matrix of observations, where m is the
number of stocks and n is the number of observations.
Let µ = 1n
∑
ri = R1n
and Σ = 1nR
(
I − 1n11T
)
RT .
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Definitions
Let Si be the closing price of n stocks on day i .
Let ri =Si−Si−1
Si−1be daily returns of stocks.
We assume that the period return r ∼ N(µ,Σ).
Let φ denote a share of our wealth in stocks.
Let R ∈ Rm×n - a matrix of observations, where m is the
number of stocks and n is the number of observations.
Let µ = 1n
∑
ri = R1n
and Σ = 1nR
(
I − 1n11T
)
RT .
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Classical models
The investor will choose a portfolio as to maximise the expectedutility of the portfolio return.
maxφ
µTφ − γφTΣφ
s.t. φ ∈ C,
where C is some convex set.
maxφ
µTφ
s.t. φT Σφ ≤ Rrisk
φ ∈ C,
and
minφ
φTΣφ
s.t. µTφ ≥ Rreturn
φ ∈ C,
These are all equivalent formulations. Pioneering work [Mar52].
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Classical models
The investor will choose a portfolio as to maximise the expectedutility of the portfolio return.
maxφ
µTφ − γφTΣφ
s.t. φ ∈ C,
where C is some convex set.
maxφ
µTφ
s.t. φT Σφ ≤ Rrisk
φ ∈ C,
and
minφ
φTΣφ
s.t. µTφ ≥ Rreturn
φ ∈ C,
These are all equivalent formulations. Pioneering work [Mar52].
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Drawbacks of classical models
Optimal portfolio is very sensitive to input parameters.
Bad portfolio diversification.
Hardly intuitive portfolios.
Unstable over time. Large transaction costs.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Drawbacks of classical models
Optimal portfolio is very sensitive to input parameters.
Bad portfolio diversification.
Hardly intuitive portfolios.
Unstable over time. Large transaction costs.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Drawbacks of classical models
Optimal portfolio is very sensitive to input parameters.
Bad portfolio diversification.
Hardly intuitive portfolios.
Unstable over time. Large transaction costs.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Drawbacks of classical models
Optimal portfolio is very sensitive to input parameters.
Bad portfolio diversification.
Hardly intuitive portfolios.
Unstable over time. Large transaction costs.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Some solutions to drawbacks
Change model
Imposing extra constraintsAssuming different model for returns, e.g. CAPM
Account for uncertainty in parameter estimation
Loss function approaches. Shrinkage estimators.
Robust optimisation methodology
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Some solutions to drawbacks
Change model
Imposing extra constraintsAssuming different model for returns, e.g. CAPM
Account for uncertainty in parameter estimation
Loss function approaches. Shrinkage estimators.
Robust optimisation methodology
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Some solutions to drawbacks
Change model
Imposing extra constraintsAssuming different model for returns, e.g. CAPM
Account for uncertainty in parameter estimation
Loss function approaches. Shrinkage estimators.
Robust optimisation methodology
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Robust optimisation in finance
An example of the robust portfolio selection models would be
minφ
maxΣ∈UΣ
φTΣφ
s.t. minµ∈Uµ
µTφ ≥ Rreturn
φ ∈ C,
whereUµ = {µ : µ ≤ µ ≤ µ}
andUΣ = {Σ : Σ ≤ Σ ≤ Σ, Σ � 0}.
This model was studied by B.V. Halldorsson, M. Koenig and R. H.Tutuncu [TK04].
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Drawbacks of robust models
There is no dependence between returns and risks in theuncertainty sets.
Cautious investments. Sometimes performance suffers.
Uncertainty sets considered in the literature are sometimestoo large than what they should be.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Drawbacks of robust models
There is no dependence between returns and risks in theuncertainty sets.
Cautious investments. Sometimes performance suffers.
Uncertainty sets considered in the literature are sometimestoo large than what they should be.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Drawbacks of robust models
There is no dependence between returns and risks in theuncertainty sets.
Cautious investments. Sometimes performance suffers.
Uncertainty sets considered in the literature are sometimestoo large than what they should be.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Modelling dependence between risks and returns
Given that r ∼ N(µ,Σ) the uncertainty set (1 − α confidenceinterval) for µ is an elliptic set
Uµ ={
µ = µ + u : ‖u‖2S−1 ≤ 1
}
,
where S−1 = n
ρ Σ−1 and ρ is a 1 − α p-value of the Hotelling Tdistribution. We model risks as a maximum likelihood estimatorgiven µ:
Σ(µ) =1
nRRT − µµT − µµT + µµT .
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Modelling dependence between risks and returns
Given that r ∼ N(µ,Σ) the uncertainty set (1 − α confidenceinterval) for µ is an elliptic set
Uµ ={
µ = µ + u : ‖u‖2S−1 ≤ 1
}
,
where S−1 = n
ρ Σ−1 and ρ is a 1 − α p-value of the Hotelling Tdistribution. We model risks as a maximum likelihood estimatorgiven µ:
Σ(µ) =1
nRRT − µµT − µµT + µµT .
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Model formulation: Problem to solve
The problem to solve is
minφ
maxµ∈Uµ
φT Σ(µ)φ − ΓµTφ
s.t. φ ∈ C,
where
Σ(µ) =1
nRRT − µµT − µµT + µµT
andUµ =
{
µ = µ + u : ‖u‖2S−1 ≤ 1
}
.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Reformulating the objective function
Proof is related to [DG02]. Consider the objective function.
φT Σ(µ)φ − Γ(µTφ) =[
1n
∥
∥RTφ∥
∥
2
2− (µTφ)2 − Γ(µT φ)
]
− Γ(uT φ) + (uT φ)2 =
A − Γx + x2,
where A =∥
∥Σ1/2φ∥
∥
2
2− Γ(µTφ) and x := uTφ.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Important lemma
Lemma
The optimal values for the following two problems are in fact the
same.
maxu
A − Γ(uTφ) + (uTφ)2
s.t. ‖u‖2G≤ 1,
maxu
A − Γ(uTφ) + (uTφ)2
s.t. (uTφ)2 ≤ ‖φ‖2G−1,
where G ≻ 0.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Solution outline
Proof.
1 Using S-lemma, transform the original problem into anonlinear matrix inequality problem.
2 Find an equivalent tractable convex formulation of thenonlinear matrix inequality problem.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Solution outline
Proof.
1 Using S-lemma, transform the original problem into anonlinear matrix inequality problem.
2 Find an equivalent tractable convex formulation of thenonlinear matrix inequality problem.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Problem Solution
minφ,ν,τ,α,β,δ,ζ
ν
s.t. φ ∈ C,ν ≥ α + β + δ
0 ≤ τ ≤ 0
4αΓ2 ≥ ζ − 1∥
∥
∥
∥
2S1/2φ
τ − δ
∥
∥
∥
∥
≤ τ + δ
∥
∥
∥
∥
2Σ1/2φ
1 − ΓµTφ − β
∥
∥
∥
∥
≤ 1 + ΓµTφ + β
∥
∥
∥
∥
2ζ + τ − 1
∥
∥
∥
∥
≤ ζ − τ + 1
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
IntroductionInvestment ModelsUncertainty ModellingSolution
Optimisation Software
1 SeDuMi, http://sedumi.mcmaster.ca/, [Stu99].
2 SDPT3, http://www.math.nus.edu.sg/mattohkc/sdpt3.html,[RHTT01].
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
Experiments on NASDAQ data from 1999 to 2006
0 2 4 6 8 10 12500
1000
1500
2000
2500
3000
3500
TIme periods n*130 days
Pou
nds
Real portfolio performance over NASDAQ data (1999−2006)
Equal weighted portfolioRobust−risk adjusted modelGoldfarb & Iyengar robust model
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
Final remarks
Developed a new robust portfolio selection model and showedthat it can be reformulated as a conic programming problem.
Uncertainty is described more accurately, therefore making therobust model more accurate.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
References I
A. Nemirovski A. Ben-Tal.Robust optimization - methodology and applications.Math. Program., 92:453–480, 2002.
G. Iyengar D. Goldfarb.Robust portfolio selection problems.Technical report, 2002.
H. M. Markowitz.Portfolio selection.Journal of Finance, 1(7):77–91, 1952.
K. C. Toh R. H. Tutuncu and M. J. Todd.SDPT3 - a Matlab software package for
semidefinite-quadratic-linear programming, version 3.0, 2001.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
Problem ModellingExprimental results
Conclusion
References II
J. Sturm.Using sedumi 1.02, a matlab toolbox for optimization oversymmetric cones.Optimization Methods and Software, 11–12:625–653, 1999.
R. H. Tutuncu and M. Koenig.Robust asset allocation.Annals of Operations Research, 132(157-187), 2004.
Denis Zuev denis.zuev@maths.ox.ac.uk Worst Case Utility in Portfolio Selection
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