Overview Optimization Robust Portfolio Optimization

Overview

1. Estimation Error

2. RobustOptimization

3. ReverseOptimization

D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 1 / 37

Robust Portfolio Optimization

November 2007

Prof. Dr. Heinz ZimmermannWirtschaftswissenschaftliches Zentrum (WWZ), Universitat [email protected]

Daniel NierdermayerWirtschaftswissenschaftliches Zentrum (WWZ), Universitat [email protected]

Overview

. Overview

Overview

1. Estimation Error




Overview

Overview

. Overview

1. Estimation Error




1. Estimation Error

2. Robust Optimization

3. Reverse Optimization

1. Estimation Error

Overview

.1. EstimationError

a. Monte Carlo

b. Experiment

c. Results




Estimation Error

Overview

1. Estimation Error

a. Monte Carlo

b. Experiment

c. Results




Traditional Mean-Variance optimization (Markowitz) assumesknown expected returns µ and covariance matrix Σ.

In practice, µ and Σ must be estimated and thereforecontain estimation error.

This section presents a simulation that highlights the impactof estimation error on optimal asset allocation.

As the simulation uses multivariate normal returns, we firstdiscuss a return generating Monte Carlo method.

Estimation Error

Overview

1. Estimation Error

a. Monte Carlo

b. Experiment

c. Results




a. Generating multivariate normal returns (Monte CarloSimulation)

b. Experiment

a. Monte Carlo - Generating multivariate normal returns

Overview

1. Estimation Error

. a. Monte Carlo

b. Experiment

c. Results




Let x be an N × 1 vector with

x ∼ N (0, IN×N) ,

where IN×N is an N ×N identity matrix.

Then E(x) = 0 and E(xx′) = I.

Let y be defined asy = Ax , (1)

where A is an N ×N matrix.

From Eq. (1) follows E(y) = E(Ax) = AE(x) = 0 andΣ ≡ E(yy′) = E(Axx′A′) = AE(xx′)A′ = AA′

For a known positive definite N ×N matrix Σ an N ×N matrixA must be found such that Σ = AA′.

a. Monte Carlo - Cholesky Decomposition

Overview

1. Estimation Error

. a. Monte Carlo

b. Experiment

c. Results




Use the Cholesky decomposition of Σ (in Matlabchol(Sigma)’)

A = chol(Σ) check that A is a lower triangular matrix (as Matlab returns

an upper triangular matrix, it has to be transposed) A has the property of Σ = AA′

The multivariate return series can be generated by

r = y + µ ,

where µ is a vector of constants (here expected returns).

The random vector r satisfies the properties E(r) = µ andCov(r) = E(rr′) − E(r)E(r′) = E(y + µ)(y + µ)′ − µµ′ =E(yy′) + 2µE(y′) + µµ′ − µµ′ = E(yy′) = Σ.

b. Experiment

Overview

1. Estimation Error

a. Monte Carlo

. b. Experiment

c. Results




The following experiment is based on DeMiguel/Nogales (2007).

Given are the (theoretical) moments of monthly returns

µ =

0.010.010.010.01

, Σ = 0.04 × IN×N .

The Mean-Variance optimal weights for any risk aversioncoefficient are 0.25 for each asset!1

1To verify this, set µ = a1 and Σ = bI in Eq. (3).

b. Experiment

Overview

1. Estimation Error

a. Monte Carlo

. b. Experiment

c. Results




Generate 140 returns ri ∼ N (µ,Σ), i = 1 . . . 140.

For different values of t calculate sample moments from

µ =1

120

120+t−1∑

i=t

ri , Σ =1

120 − 1

120+t−1∑

i=t

(ri − µ)(ri − µ)′

Rolling window estimation:

Rebalancing date t = 1: Calculate µ and Σ using r1 . . . r120

Rebalancing date t = 2: Calculate µ and Σ using r2 . . . r121

...

..until t = 20.

In the following, for each µ, Σ optimal weights are calculated.

c. Results

Overview

1. Estimation Error

a. Monte Carlo

b. Experiment

. c. Results




µ and Σ are unknown:

2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

1.5

2

wei

ghts

rebalancing dates

T=120

µ and Σ are estimated using 120 monthly returns ri (rolling window).The dashed line shows optimal portfolio weights without estimationerror (= 0.25) and γ = 1.

c. Results

Overview

1. Estimation Error

a. Monte Carlo

b. Experiment

. c. Results




µ is estimated and Σ is known:

2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

1.5

2

wei

ghts

rebalancing dates

T=120

c. Results

Overview

1. Estimation Error

a. Monte Carlo

b. Experiment

. c. Results




µ is known and Σ is estimated:

2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

1.5

2

wei

ghts

rebalancing dates

T=120

c. Results

Overview

1. Estimation Error

a. Monte Carlo

b. Experiment

. c. Results




Remarks:

Estimation error can cause strong deviation of estimatedportfolio weights from theoretically optimal weights.

– In the above example where µ and Σ are unknown themaximal estimated weight is 0.78 and the minimal weightis −0.09 while (theoretically) optimal weights are all 0.25.

Estimation error in expected returns has larger impact onasset allocation than estimation error in the covariancematrix.

Higher risk aversion leads to a portfolio allocation closer tothe global minimum variance portfolio (GMVP) (see nextsection). As the GMVP does not rely on expected returns,higher risk aversion reduces estimation error (besidesreducing portfolio risk)!

2. Robust Optimization

Overview

1. Estimation Error

.2. RobustOptimization

TraditionalOptimization

Robust Optimization

a. Worst Case µ

b. Derivation

c. Example



Robust Optimization

Overview

1. Estimation Error



Robust Optimization

a. Worst Case µ

b. Derivation

c. Example



The previous section has shown the effects of estimationerror in expected returns and the covariance matrix onoptimal asset allocation.

Optimal weights using sample estimates (esp. of expectedreturns) can strongly deviate from their theoretical values.

This section shows a robust portfolio optimization methodbased on Scherer (2007) (and Tutuncu and Konig (2004))that mitigates the problem of estimation error.

Traditional Optimization

Overview

1. Estimation Error


.TraditionalOptimization

Robust Optimization

a. Worst Case µ

b. Derivation

c. Example



In traditional mean variance optimization of a portfolio with Nassets, expected utility is maximized using the followingLagrangian function

L = µ′w − 1

2γw′

Σw − λ(w′1 − 1) (2)

where

µ: is an N × 1 vector of expected returns,Σ: is the N ×N covariance matrix of assets’ returns,w: is an N × 1 vector of portfolio weights,γ: is the parameter of relative risk aversion,λ: is a Lagrange multiplier.

Note that (w′1 − 1) is the constraint requiring that the optimal

weight vector’s elements sum up to 1.

Traditional Optimization

Overview

1. Estimation Error


.TraditionalOptimization

Robust Optimization

a. Worst Case µ

b. Derivation

c. Example



The weight vector that maximizes Eq. (2) can be found by

∂L

∂w= µ − γΣw − λ1 = 0

∂L

∂λ= w′

1 − 1 = 0

The solution is given by

w∗

mv =1

γΣ

−1

(

µ − 1′Σ

−1µ

1′Σ

−111

)

︸︷︷︸

w∗spec

+Σ

−11

1′Σ

−11

︸︷︷︸

w∗

min

(3)

where w∗

min is the portfolio weight of the global minimumvariance portfolio and w∗

spec is a speculative demand.

Robust Optimization, Scherer(2007)

Overview

1. Estimation Error



.RobustOptimization

a. Worst Case µ

b. Derivation

c. Example



Overview: Scherer(2007) (based on Tutuncu/Konig (2004))

Estimation error in sample means µ has larger effect onportfolio allocation than estimation error in the covariancematrix.

Instead of plugging the mean vector into the Mean-Varianceoptimizer, we will use a ‘worst case’ vector µ∗.

The worst case expected portfolio return w′µ∗ is lower than(or equal to) w′µ; this makes it generally less attractive toinvest into the speculative weight w∗

spec which depends on µ

(or µ∗).

The robust optimization method presented here leads tohigher investment into the global minimum variance portfoliow∗

min than in traditional Mean-Variance optimization.

Robust Optimization – a. Worst Case Expected Returns

Overview

1. Estimation Error



Robust Optimization

. a. Worst Case µ

b. Derivation

c. Example



The vector µ is drawn from a multivariate normal distribution

µ ∼ N (µ,Ω) ,

where Ω is an N ×N positive definite covariance matrix.

We consider a realization of µ as an extreme event if

(µ − µ)′Ω−1(µ − µ) ≥ κ21−α,N .

Note that (µ − µ)′Ω−1(µ − µ) is χ2N distributed (chi-square

distributed with N degrees of freedom – N is the number ofassets).


Overview

1. Estimation Error



Robust Optimization

. a. Worst Case µ

b. Derivation

c. Example



0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

chi s

quar

e

κ1−α,N2

N = 4α = 5%

5%

where κ21−α,N = Inv-χ2

N (1 − α) and Inv-χ2N is the inverse

chi-square cumulative density function with N degrees offreedom.Example: κ2

0.95,4 = 9.4877. (use chi2inv(0.95,4) in Matlab).


Overview

1. Estimation Error



Robust Optimization

. a. Worst Case µ

b. Derivation

c. Example



Now consider the following set of expected return vectors

(µ − µ)′Ω−1(µ − µ) = κ21−α,N . (4)

Geometrically, Eq. (4) describes an N -dimensional ellipsoid. Wewill restrict our analysis to expected return vectors µ that lie onthis ellipsoid.

On this ellipsoid the value of µ with the lowest expected portfolioreturn is obtained by maximizing the Lagrangian function

L(µ, λ) = w′µ − w′µ − λ

2

((µ − µ)′Ω−1(µ − µ) − κ2

1−α,N

)

where w is any portfolio weight vector.


Overview

1. Estimation Error



Robust Optimization

. a. Worst Case µ

b. Derivation

c. Example



Setting ∂L/∂µ = 0 and ∂L/∂λ = 0 and solving this system oflinear equations with respect to µ yields

µ∗ = µ − κ1−α,N√w′

ΩwΩw

andw′µ∗ = w′µ − κ1−α,N

√

w′Ωw . (5)

From Eq. (5) it is obvious that w′µ∗ ≤ w′µ.

The value w′µ∗ is particularly low for high values of κ1−α,N .

For Ω = 1

TΣ Eq. (5) can be rewritten as

w′µ∗ = w′µ − κ1−α,NT−1/2

√

w′Σw , (6)

where Σ is the covariance matrix of assets’ returns.

Robust Optimization – b. Deriving Optimal Weights

Overview

1. Estimation Error



Robust Optimization

a. Worst Case µ

. b. Derivation

c. Example



The traditional Mean-Variance objective is to maximize theLagrangian

L = µ′w − 1

2γw′

Σw − λ(w′1 − 1) .

Lrob = µ′

∗w − 1

2γw′

Σw − λ(w′1 − 1)

= µ′w − κ1−α,NT−1/2

√

w′Σw − 1

2γw′

Σw − λ(w′1 − 1) .


Overview

1. Estimation Error



Robust Optimization

a. Worst Case µ

. b. Derivation

c. Example



Solving ∂Lrob/∂w = 0 and ∂Lrob/∂λ = 0 gives

w∗

rob =

(

1 − T−1/2κ1−α,N

γσ∗p + T−1/2κ1−α,N

)

1

γΣ

−1

(

µ − 1′Σ

−1µ

1′Σ

−111

)

+Σ

−11

1′Σ

−11

w∗

rob =

(

1 − T−1/2κ1−α,N

γσ∗p + T−1/2κ1−α,N

)

w∗

spec + w∗

min (7)

Note that σ∗p =√

w∗′

robΣw∗

rob which makes Eq. (7) not directly

solvable. However, Eq. (7) can be solved numerically.


Overview

1. Estimation Error



Robust Optimization

a. Worst Case µ

. b. Derivation

c. Example



Moreover, note that

If κ1−α,N → 0 or T → ∞,

w∗

rob =1

γΣ

−1

(

µ − 1′Σ

−1µ

1′Σ

−111

)

+Σ

−11

1′Σ

−11

= w∗

mv

If κ1−α,N → ∞ or T → 0,

w∗

rob =Σ

−11

1′Σ

−11

= w∗

min

Robust Optimization – c. Example

Overview

1. Estimation Error



Robust Optimization

a. Worst Case µ

b. Derivation

. c. Example



Impact of estimation error in traditional M-V Optimization

2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

1.5

2

wei

ghts

rebalancing dates

T=120

(γ = 1)


Overview

1. Estimation Error



Robust Optimization

a. Worst Case µ

b. Derivation

. c. Example



Impact of estimation error in robust M-V Optimization

2 4 6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1

1.5

2

wei

ghts

rebalancing dates

T=120

(γ = 1)


Overview

1. Estimation Error



Robust Optimization

a. Worst Case µ

b. Derivation

. c. Example



Investment into w∗

spec increases with higher sample size T .(Note again that σ∗

p is not constant in Eq. (7) - the curve below wassimulated numerically.)

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

T

ψ

Shrinkage Intensity

ψ =

„

1 −T−1/2κ1−α,N

γσ∗

p + T−1/2κ1−α,N

«

≤ 1 (8)


Overview

1. Estimation Error



Robust Optimization

a. Worst Case µ

b. Derivation

. c. Example



Remarks:

as w′µ∗ ≤ w′µ, robust optimization using µ∗ leads to ‘lessattractive’ expected returns which reduces speculativedemand w∗

spec. This is shown by Eq. (7), where ψ ≤ 1(compare also Eq. (8)).

Thus, the fraction of wealth invested into the globalminimum variance portfolio w∗

min increases with robustoptimization.

As w∗

min does not depend on µ, the resulting portfolioallocation is more robust than in traditional Mean-Varianceoptimization.

This type of robust optimization is equivalent to traditionaloptimization where the risk aversion coefficient is increasedto 1

ψγ. Note that ψ depends on the ‘uncertainty aversion’κ1−α,N and T .

3. Reverse Optimization

Overview

1. Estimation Error


.3. ReverseOptimization


Reverse Optimization

Overview

1. Estimation Error




Reverse optimization (applied in e.g. Black and Litterman(1992)) avoids the problem arising from estimation error inexpected returns.

Traditional Mean-Variance optimization uses expectedreturns (and the covariance matrix) as input and providesportfolio weights as output.

In contrast, reverse optimization uses portfolio weights (andthe covariance matrix) as input and provides expectedreturns as output.

The rationale behind using portfolio weights as input is thatin some cases weights are easier to estimate than expectedreturns.


Overview

1. Estimation Error




As in Black/Litterman (1992) we confine our analysis toexpected excess returns µe (expected returns minus the riskfreerate).Then in Eq. (2) the constraint λ(w′

1 − 1) can be dropped asweights do not have to sum up to 1 (because implicitly,remaining wealth is invested into the riskfree asset)

L = µ′

ew − 1

2γw′

Σw

This simplifies the solution of the optimal weight. Setting∂L/∂w = 0 gives

µe − γΣw = 0 .

Rearranging yields

w∗ =1

γΣ

−1µe . (9)


Overview

1. Estimation Error




From Eq. (9) follows the main equation for reverse optimization:

µ∗

e= γΣw∗ .

In e.g. the Black/Litterman model value weighted marketportfolio weights (or strategic asset allocation weights) are usedfor w∗.


Overview

1. Estimation Error




Remarks:

By definition, plugging µ∗

einto Eq. (9) yields w∗.

However, when changing the value of the risk aversioncoefficient γ, optimal weights obtained in Eq. (9) differ fromthe original reference weights w∗.

A ‘sensible’ value should be chosen for the reference weightsw∗. This improves the economic intuition of the results.


Overview

1. Estimation Error




Advantages:

The advantage of reverse optimization is that it does not relyon the estimation of expected returns. This leads to portfolioallocations more robust than in traditional Mean-Varianceoptimization.

Resulting weights are often economically easier to interpretthan weights from traditional Mean-Variance optimization(for reasonable reference weights w∗).

Disadvantage:

Reverse optimization may throw away useful informationcontained in (estimated) expected returns.

Literature

Overview

1. Estimation Error




Black, F., and R. Litterman, 1992 (September-October),Global Portfolio Optimization, Financial Analysts Journal, p.28-43.

DeMiguel, V. and F. J. Nogales (2007), Portfolio Selectionwith Robust Estimation, Working Paper Series, SSRN eLibrary,http://ssrn.com/paper=911596

Scherer, B. (2007): Portfolio Construction and RiskBudgeting, Risk Books, 3rd edition.

Tutuncu, R. H. and M. Konig (2004), Robust AssetAllocation, Annals of Operations Research, 132(1), p. 157-187.

Documents

Overview Optimization Robust Portfolio Optimization