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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. - PowerPoint PPT Presentation
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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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