Robust Portfolio Selection via Utility Optimisation with smaller...

Problem ModellingExprimental results

Conclusion

Robust Portfolio Selection via Utility Optimisation

with smaller uncertainty sets

Denis [email protected]

OCIAM,

University of Oxford

May 18, 2007

Denis Zuev [email protected] Worst Case Utility in Portfolio Selection

Conclusion

IntroductionInvestment ModelsUncertainty ModellingSolution

Main idea of the talk

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.

Conclusion

stable over time.

Conclusion

stable over time.

Conclusion

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

Conclusion

Definitions

Let ri =Si−Si−1

Let µ = 1n

ri = R1n

and Σ = 1nR

I − 1n11T

Conclusion

Definitions

Let ri =Si−Si−1

Let µ = 1n

ri = R1n

and Σ = 1nR

I − 1n11T

Conclusion

Definitions

Let ri =Si−Si−1

Let µ = 1n

ri = R1n

and Σ = 1nR

I − 1n11T

Conclusion

Definitions

Let ri =Si−Si−1

Let µ = 1n

ri = R1n

and Σ = 1nR

I − 1n11T

Conclusion

Classical models

The investor will choose a portfolio as to maximise the expectedutility of the portfolio return.

µTφ − γφTΣφ

s.t. φ ∈ C,

where C is some convex set.

s.t. φT Σφ ≤ Rrisk

φ ∈ C,

φTΣφ

s.t. µTφ ≥ Rreturn

φ ∈ C,

These are all equivalent formulations. Pioneering work [Mar52].

Conclusion

Classical models

The investor will choose a portfolio as to maximise the expectedutility of the portfolio return.

µTφ − γφTΣφ

s.t. φ ∈ C,

where C is some convex set.

s.t. φT Σφ ≤ Rrisk

φ ∈ C,

φTΣφ

s.t. µTφ ≥ Rreturn

φ ∈ C,

These are all equivalent formulations. Pioneering work [Mar52].

Conclusion

Drawbacks of classical models

Optimal portfolio is very sensitive to input parameters.

Bad portfolio diversification.

Hardly intuitive portfolios.

Unstable over time. Large transaction costs.

Conclusion

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

Conclusion

Change model

Conclusion

Change model

Conclusion

Robust optimisation in finance

An example of the robust portfolio selection models would be

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].

Conclusion

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.

Conclusion

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 .

Conclusion

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 .

Conclusion

Model formulation: Problem to solve

The problem to solve is

maxµ∈Uµ

φT Σ(µ)φ − ΓµTφ

s.t. φ ∈ C,

Σ(µ) =1

nRRT − µµT − µµT + µµT

andUµ =

µ = µ + u : ‖u‖2S−1 ≤ 1

Conclusion

Reformulating the objective function

Proof is related to [DG02]. Consider the objective function.

φT Σ(µ)φ − Γ(µTφ) =[

∥RTφ∥

2− (µTφ)2 − Γ(µT φ)

− Γ(uT φ) + (uT φ)2 =

A − Γx + x2,

where A =∥

∥Σ1/2φ∥

2− Γ(µTφ) and x := uTφ.

Conclusion

Important lemma

The optimal values for the following two problems are in fact the

A − Γ(uTφ) + (uTφ)2

s.t. ‖u‖2G≤ 1,

A − Γ(uTφ) + (uTφ)2

s.t. (uTφ)2 ≤ ‖φ‖2G−1,

where G ≻ 0.

Conclusion

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.

Conclusion

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.

Conclusion

Problem Solution

minφ,ν,τ,α,β,δ,ζ

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

Conclusion

Optimisation Software

1 SeDuMi, http://sedumi.mcmaster.ca/, [Stu99].

2 SDPT3, http://www.math.nus.edu.sg/mattohkc/sdpt3.html,[RHTT01].

Conclusion

Experiments on NASDAQ data from 1999 to 2006

0 2 4 6 8 10 12500

TIme periods n*130 days

Real portfolio performance over NASDAQ data (1999−2006)

Equal weighted portfolioRobust−risk adjusted modelGoldfarb & Iyengar robust model

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.

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.

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.

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