portfolio diversity and robustness
DESCRIPTION
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 PresentationTRANSCRIPT
![Page 1: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/1.jpg)
Portfolio Diversity and Robustness
![Page 2: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/2.jpg)
TOC
Markowitz Model Diversification Robustness
Random returns Random covariance
Extensions Conclusion
![Page 3: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/3.jpg)
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
![Page 4: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/4.jpg)
Introduction & Background
The efficient frontier
![Page 5: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/5.jpg)
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
![Page 6: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/6.jpg)
Outline
Diversity Constraints L1/L2-norms Robust optimization via variation in
returns vector Variation in Covariance Estimators
via Random Matrix theory Results Further developments
![Page 7: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/7.jpg)
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
![Page 8: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/8.jpg)
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
![Page 9: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/9.jpg)
![Page 10: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/10.jpg)
![Page 11: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/11.jpg)
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
![Page 12: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/12.jpg)
![Page 13: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/13.jpg)
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
![Page 14: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/14.jpg)
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
![Page 15: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/15.jpg)
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
![Page 16: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/16.jpg)
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
![Page 17: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/17.jpg)
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
![Page 18: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/18.jpg)
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
![Page 19: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/19.jpg)
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
![Page 20: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/20.jpg)
Random Matrix Confidence Interval
Confidence interval
max max0.95 max 0.95[ ] 0.95P t tn n
![Page 21: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/21.jpg)
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
![Page 22: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/22.jpg)
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
![Page 23: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/23.jpg)
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
![Page 24: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/24.jpg)
Further Extensions
Contribution to variance constraints Multi-Moment Models Extreme Tail Loss (ETL) Shortfall Optimization
![Page 25: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/25.jpg)
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
![Page 26: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/26.jpg)
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
![Page 27: Portfolio Diversity and Robustness](https://reader035.vdocuments.us/reader035/viewer/2022062304/56813d50550346895da70b74/html5/thumbnails/27.jpg)
We’d Like To Thank