Portfolio Diversity and Robustness

TOC

Markowitz Model Diversification Robustness

Random returns Random covariance

Extensions Conclusion

Introduction & Background

The classic model

S - Covariance matrix (deterministic) r – Return vector (deterministic) Solution via KKT conditions

min

min

. . 0

1 1

T

T

T

x Sx

s t x

x

r x r

Introduction & Background

The efficient frontier

Problems and Concerns

Number of assets vs. time period Empirical estimate of Covariance matrix is

noisy Slight changes in Covariance matrix can

significantly change the optimal allocations

Sparse solution vectors Without diversity constraints the optimal

solution allows for large idiosyncratic exposure

Outline

Diversity Constraints L1/L2-norms Robust optimization via variation in

returns vector Variation in Covariance Estimators

via Random Matrix theory Results Further developments

Original problem : extension of Markowitz

portfolio optimization

min

{0.1* }

[ ]1

min

. . 0

1 1

T

T

T

n

ii

x Sx

s t x

x

r x r

x

Diversity Extension

Adding The L-2 norm constraint

min

{0.1* }

[ ]1

2

min

. . 0

1 1

T

T

T

n

ii

x Sx

s t x

x

r x r

x

x u

L-1 norm constraint:

min

{0.1* }

[ ]1

1

min

. . 1 1

T

T

n

ii

x Sx

s t x

r x r

x

x u

Robust optimization

The classic model

Robust: letting r vary i.e. adding infinitely many constraints

min

min

. . 0

1 1

T

T

T

x Sx

s t x

x

r x r

Robust Model

The robust model

E is an ellipsoid min 2

min

. . 0

1 1

, { ||| || 1}

T

T

T

x Sx

s t x

x

r x r r r Pu u

Robust Model (cont’d)

Family of constraints: it can be shown that

The new Robust Model:

min 2, { ||| || 1}Tr x r r r Pu u

min 2 min{ | , } { | || || }Tn T n Tx R r x r r x R r x P x r

2 min

min

. . 0

1 1

|| ||

T

T

T T

x Sx

s t x

x

r x P x r

Robust Optimization (cont’d)

min 2 min{ | , } { | || || }Tn T n Tx R r x r r x R r x P x r

22

2 2

2

min min

| | 1,| | 1

| | 1 | | 1

2| | 1

, inf

inf inf inf ( )

inf sup( )

sup( ) | |

n T T

r E

T T T

r E ur r Pu u

T TT T T T

u u

T TT T T

u

x R satisfy r x r r iff r x r

r x r x r Pu x

r x u P x r x u P x

r x u P x r x P x

Robust Optimization Ellipsoids

Ellipsoids

Fact iff

1 2

12

{ | ,|| || 1}

{ |( ) ( ) }

n

n T

E x R x r Pu u

E x R x r Q x r

1 2E E 1/2P Q

Random Matrix Theory

Covariance Matrix is estimated rather than deterministic

The Eigenvalue/Eigenvector combinations represent the effect of factors on the variation of the matrix

The largest eigenvalue is interpreted as the broad market effect on the estimated Covariance Matrix

Random Matrix Implementation

compute the covariance and eigenvalues of the

empirical covariance matrices

Estimate the eigenvalue series for the decomposed

historical covariance matrices

Calculate the parameters of the eigenvalue

distribution

Perturb the eigenvalue estimate according to the

variability of the estimator

Random Matrix Confidence Interval

Confidence interval

max max0.95 max 0.95[ ] 0.95P t tn n

Random Matrix Formulation

Problem to solve

max (1 )%CImin max T T

xx E DEx

min. . Ts t x r r1

1

{( ) ( ) ( ) }

0

1

T T

n

ii

r E r r E DE r r F

x

x

Markowitz and Robust Portfolio

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.8 1 1.2

stdev of returns

me

an

re

turn

s

Markowitz Efficient Frontier

Markowitz Optimal Portfolio

Robust Optimal Portfolio

Return is assumed to be random r~N(m,S)Robust portfolio also lies on efficient frontier

Efficient Frontier Perturbed Covariance

0

0.005

0.01

0.015

0.02

0.025

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

stdev of returns

mean r

etu

rns

Markowitz Efficient Frontier

Perturbed Cov EfficientFrontierAssets

The worst case perturbed Covariance matrix shifts the entire efficient frontier

Further Extensions

Contribution to variance constraints Multi-Moment Models Extreme Tail Loss (ETL) Shortfall Optimization

Contribution to Variance Model

1 1 2 2 ... ...i

i i i ii n niT

x x x xx

x Sx

2

min

( )

_ min

1

T

T

T

x Sx

st

Diag x Sx x Sxe

x r r

x

QQP Formulation Add artificial : 0x

0 0 0

0 0

min

( ) 0

( _ min) 0

1 0

1 0

T

T

T T

T

T

x Sx

st

Diag x Sx x Sxe

x r x x rx

x x

x x

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