Ambiguity in portfolio...

IntroductionSolution techniques

Statistical estimation and confidence setsExample

Conclusions

Ambiguity in portfolio optimization

Georg Ch. Pflug

May/June 2006

Georg Ch. Pflug Ambiguity in portfolio optimization



Conclusions

Introduction: Risk and Ambiguity

Frank Knight ”Risk, Uncertainty and Profit” (1920)Risk: the decision-maker can assign mathematical probabilities torandom phenomenaUncertainty: randomness cannot be expressed in terms of specificmathematical models.

Ellsberg (1961)Uncertainty: the probabilistic model is known, but the realizationsof the random variables are unknown (”aleatoric uncertainty”)Ambiguity: the probability model itself is unknown (”epistemicuncertainty”).




Conclusions

It has frequently been observed that the solutions of portfoliooptimization problems are much sensitive w.r.t. the parameters ofthe distributions (Klein and Bawa (1976), Chopra and Ziemba(1993)).However, the distributions are estimated from data and thereforecontain some estimation error and model ambiguity.Therefore we need to combineStatistics (”how to estimate the probability model and itsparameters and to assess confidence”) andOptimization (”how to find the best decision”) in one approach.




Conclusions

Problem formulation

(ξ1, . . . , ξM) random returns for M asset categories(x1, . . . , xM) portfolio weights

Yx =∑M

m=1 xmξm portfolio returnA(Yx) acceptability functional

Let P be a baseline probability measure for (ξ1, . . . , ξM). If wefully trust in the validity of this model, we solve thenon-ambiguous portfolio optimization problem

∥∥∥∥∥∥∥∥∥∥

Maximize (in x) : EP(Yx)subject toAP(Yx) ≥ qx>1l = 1x ≥ 0




Conclusions

Acceptability functionals

Required properties:

(i) translation equivariance

A(Y + c) = A(Y ) + c

(ii) concavity

A(λY1 + (1− λ)Y2) ≥ λA(Y1) + (1− λ)A(Y2).

(iii) continuity Y 7→ A(Y ) continuous

(iv) version independence: If Y1 and Y2 have the samedistribution, then A(Y1) = A(Y2)




Conclusions

Examples for such acceptability functionals A are

I the expectationA(Y ) = E(Y )

I the average-value-at risk (conditional value-at-risk, expectedshortfall)

A(Y ) = AV@Rα(Y ) =1

α

∫ α

0G−1

Y (p) dp,

where G−1Y (p) = infv : PY ≤ v ≥ p

I the (lower) standard deviation corrected expectation(0 < λ < 1)

A(Y ) = E(Y )− λStd(Y ) or E(Y )− λStd−(Y ),

Std−(Y ) =√E[[Y − EY ]−)2




Conclusions

Stress testing

We solve the portfolio problem for the baseline probability model,but check its performance under new probability models.Example: 500 historic weakly returns from NYSEmodel composition optimal for weakly return acceptabilty [email protected]

(P) (P) 0.8% 0.92

(P−) (P) -0.13% 0.90

(P+) (P) 1.17% 0.93

(P−) (P−) -0.07% 0.92(P+) (P+) 1.9% 0.92

(P) : historic data(P−) : 5% higher prob. for bad scenarios(P+) : 5% lower prob. for bad scenarios




Conclusions

The ambiguity problem - maximin formulation

Let P be an ambiguity set, i.e. the set of probability models, towhich the modeler in indifferent. The portfolio selection modelunder P-ambiguity is of maximin type and reads

∥∥∥∥∥∥∥∥∥∥

Maximize (in x) : minEP(Yx) : P ∈ Psubject toAP(Yx) ≥ q for all P ∈ Px>1l = 1x ≥ 0




Conclusions

Dealing with uncertainty:

I Deterministic optimization:

maxF (x) : Fi (x) ≥ 0F is the objective,Fi are the constraint functions.

I Robust optimization (maximin approach):A set Ξ of possible parameters ξ is given.

maxminF (x , ξ) : ξ ∈ Ξ : Fi (x , ξ) ≥ 0; ξ ∈ Ξ

Robust optimization produces very conservative portfolios.




Conclusions

I Stochastic optimization:(Ξ,A, P) is a probability space, typical element is called ξ.

maxEP [F (x , ξ)];EP [Fi (x , ξ)] ≥ 0

or more generally

maxAP [F (x , ξ)];A(i)P [Fi (x , ξ)] ≥ 0

where A(·)P are aggregating probability functionals, mapping

the distribution of F (x , ξ) under P to the (extended) real line.Stochastic optimization produces well diversified portfolios.




Conclusions

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Conclusions

The ambiguity problem - Bayesian formulation

Let λ be a probability on the ambiguity set P. λ is called theBayesian prior on the set of probability models. The Bayesianformulation is

∥∥∥∥∥∥∥∥∥∥

Maximize (in x) :∫P AP [F (x , ξ)] dλ(P)

subject to∫P A

(i)P [Fi (x , ξ)] dλ(P) ≥ 0

x>1l = 1x ≥ 0




Conclusions

The ambiguity problem as robust stochastic optimization

We repeat the maximin formulation of the problem:

∥∥∥∥∥∥∥∥∥∥

Maximize (in x) : minEP(Yx) : P ∈ Psubject toAP(Yx) ≥ q for all P ∈ Px>1l = 1x ≥ 0




Conclusions

Solution techniques for the maximin portfolio problemLet

X = x : x>1l = 1, x ≥ 0,AP(Yx) ≥ q for all P ∈ P,then the ambiguity problem reads

maxx∈X

minP∈P

EP [Yx ].

By continuity and concavity of A, X is a compact convex set.Moreover, (P , x) 7→ EP [Yx ] is bilinear in P and x and henceconvex-concave. Therefore x∗ ∈ X is a solution of if and only ifthere is a P∗ ∈ P such that (P∗, x∗) is a saddle point, i.e.

EP∗ [Yx ] ≤ EP∗ [Yx∗ ] ≤ EP [Yx∗ ]

for all (P, x) ∈ P × X.Our problem is semi-infinite. Direct saddle point methods wereused by Rockafellar (1976), Nemirovskii and Yudin (1978).




Conclusions

Successive convex programming (SCP)1. Set n = 0 and P0 = P with P ∈ P.2. Solve the outer problem∥∥∥∥∥∥∥∥∥∥

Maximize (in x , t) : tsubject tot ≤ EP(Yx) for all P ∈ Pn

AP(Yx) ≥ q for all P ∈ Pn

x>1l = 1; x ≥ 0

and call the solution (xn, tn).3. Solve the first inner problem∥∥∥∥∥∥

Minimize (in P) : EP(Yxn)subject toP ∈ P

and call the solution P(1)n .




Conclusions

4. Solve the second inner problem

∥∥∥∥∥∥

Minimize (in P) : AP(Yxn)subject toP ∈ P

call the solution P(2)n and let Pn+1 = Pn ∪ P(1)

n ∪ P(2)n .

5. If Pn+1 = Pn then stop. Otherwise set n := n + 1 and goto 2.




Conclusions

Convergence

Proposition. Assume that P is compact and convex and that(P, x) 7→ EP [Yx ] as well as (P, x) 7→ AP [Yx ] are jointlycontinuous. Then every cluster point of (xn) is a solution of theminimax problem. If the saddle point is unique, then the algorithmconverges to the optimal solution.




Conclusions

P




Conclusions

P

P1




Conclusions

P

P1

P2




Conclusions

P

P1

P2

P3




Conclusions

Ambiguity sets as balls -distances of probability measures

We want to use an ambiguity sets of the formP = P : d(P ,P∗) ≤ ε as ambiguity sets. But which distance dto choose?

I Transportation distance, Wasserstein distance

dT (P1, P2) = sup∫

h(u) dP1(u)−∫

h(u) dP2(u) :

|h(u)− h(v)| ≤ |u − v |




Conclusions

I Kolmogorov-Smirnov distance

dKS(P1,P2) = sup|P1((−∞, a])− P2((−∞, a])| : a ∈ R

I Variational distance

dV (P1, P2) = sup|P1(A)− P2(A)| : A mesurable If densities exist, this is equivalent to the L1 distance:

dL1(P1, P2) =

∫ ∣∣∣∣dP1

dµ− dP2

dµ

∣∣∣∣ dµ.




Conclusions

Statistical estimation and confidence sets

I The ambiguity set must reflect our current information aboutP

I If our information is based on statistical estimation, theambiguity set must coincide with a confidence set

I If the distance is too fine, no statistical confidence set can beconstructed without further assumptions (e.g. for thevariational distance)

If we base our estimate for P on some historical data ξ1, ξ2, . . . , ξn,we use the empirical distribution

Pnξi =1

n




Conclusions

Confidence sets

IPdKS(P, Pn) ≥ M/

√n ≤ 58 exp(−2M2)

(Dvoretzky, Kiefer, Wolfowitz inequality)

IEP [dT (P, Pn)] ≤ Cn−1/M

for some constant C (Dudley (1969)). By Markov’s inequality

PdT (P , Pn) ≥ ε ≤ E[d(P, Pn)]/ε ≤ n−1/MC/ε.

Under smoothness conditions on P, the confidence sets mayimproved (Kersting (1978)).




Conclusions

Example6 assets

I IBM - International Business Machines CorporationI PRG - Procter & Gamble CorporationI ATT - AT&T CorporationI VER - Verizon Communications IncI INT - Intel CorporationI EXX - Exxon Mobil Corporation

Risk constraint: AV@R0,1 ≥ 0.9Ambiguity set: P = P : dT (P, P) ≤ ε.In order to make the ambiguity sets more interpretable, we define arobustness parameter γ as the maximal relative change of theexpected returns and relate this to ε by

ε = maxη : supdT (P,P)≤η

EP(ξ(i)) ≤ (1 + γ)EP(ξ(i)) : for all i.

γ is used in the figures.Georg Ch. Pflug Ambiguity in portfolio optimization



Conclusions

Working with Transportation distancesBy the well known theorem of Kantorovich-Rubinstein, theTransportation ambiguity set can be represented as

P : dT (P , P) ≤ ε = P : there is a bivariate probability K (·, ·) s.t.∫

vK (u, dv) = P(u);

∫

uK (du, v) = P(v);

∫

u

∫

v‖u − v‖1 K (du, dv) ≤ ε.

If the probability space Ω is finite, Ω = x (1), . . . , x (n), theambiguity set is a polyhedral set

P : dT (P , P) ≤ ε = P = (P1, . . . ,PS) : Pj =∑

i

Kij ;∑

j

Kij = Pi ;

Kij ≥ 0;∑

i ,j

‖x (i) − x (j)‖1Kij ≤ ε.




Conclusions

Illustration of the Transportation distance

P1

P2

dT (P1,P2) = minimal E ( transported mass × distance ).Georg Ch. Pflug Ambiguity in portfolio optimization



Conclusions

The bivariate probability K has the interpretation as the solutionof Monge’s mass transportation problem. The Transportationdistance describes the minimal effort (in terms of expectedtransportation distances), to change the mass distribution P intothe new mass distribution P (Rachev and Ruschendorf (1998)).In the case of a finite probability space Ω it is not difficult to find asolution for the inner problems, i.e. to determine

infEP(Yx) : dT (P , P) ≤ ε

andinfAP(Yx) : dT (P, P) ≤ ε

by a special algorithm.




Conclusions

The solution of the Example

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050.99

1

1.01

Robustness

Ret

urn

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050.06

0.07

0.08

0.09

0.1

Robustness

Ris

k

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

0.5

1

Robustness

Por

tfolio

Com

posi

tion

Basic Probability ModelWorst Case Probability Model

Basic Probability ModelWorst Case Probability Model

Expected returns, risks and portfolio composition. The assets fromtop to bottom are: EXX, VER, ATT, PRG, INT, IBM.




Conclusions

0

0.02

0.040.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

1.01

1.015

1.02

1.025

Risk

Robustness

Bas

ic R

etur

n

Efficient frontiers in dependence of the robustness parameter γ.Risk and return are calculated w.r.t. the basic model P.




Conclusions

0

0.01

0.02

0.03

0.04

0.05 0.050.1

0.150.2

0.25

0.98

0.99

1

1.01

1.02

1.03

Risk

Robustness

Wor

ts C

ase

Ret

urn

Efficient frontiers in dependence of the robustness parameter γ.Risk and return are calculated w.r.t. the worst case model.




Conclusions

Types of problems in Finance and Risk management

Modeling

Estimation

Pricing (Equation solving): F (x) = y

Optimization: maxF (x)

Game Theory (Equilibrium): max minF (x , y)

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Conclusions

Conclusions

I In order to capture scenario uncertainty (aleatoric uncertainty)and probability ambiguity (epistemic uncertainty) we use aprobabilistic maximin approach.

I The ambiguity neighborhood should be chosen in such a waythat it corresponds to a probabilistic confidence regions forwhich bounds for the covering probability are available.

I The result of model ambiguity is a further diversification of heportfolio

I It turns out that often the ”price” to be paid for includingambiguity in the optimization problem is very little.

I Minimax models are also relevant for the pricing of swingoptions in Electricity markets.


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