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Portfolios and Optimization. Andrew Mullhaupt. Maximize profit with risk bound:. In ‘unit risk’ coordinates:. Mean-variance portfolio. Portfolio Selection. THE END. Transaction Costs. Commissions and Fees. Taxes. Slippage -. Slippage. Induced Costs. Expected Costs. - PowerPoint PPT Presentation
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Portfolios and Optimization
Andrew Mullhaupt
Portfolio Selection
2 1maxT
T
u C ur u
Maximize profit with risk bound:
1
1maxT
T
Cu CuC r Cu
In ‘unit risk’ coordinates:
2
* 1/ 22T
C rur C r
Mean-variance portfolio
THE END
Transaction Costs
Commissions and Fees
Taxes
Slippage -
Slippage
0Initial portfolio: u
1Final portfolio: u
0Fair market price at time of decision: p
1 :price d transacteActual p
1 0 1 0Slippage: Tu u p p
Trade size
Expe
cted
Co
sts
Proportional Costs
Induced Costs
‘Eating the Book’
Portfolio
Loss
Initial Portfolio
Mean-variance portfolio
Cost relative to Initial Portfolio
Total LossRisk relative to optimal mean-variance portfolio
Optimal Portfolio
Portfolio Selection With Deterministic Costs
Portfolio
Loss
*0 uu
*0 and risk, cost,by determined is Trade uu
Original Portfolio
Mean-variance Portfolio
:enough small is When *0 uu
Original Portfolio
Trading cannot reduce the loss
Mean-variance Portfolio
tradeno of tradeThe
No Trade Regions
* 0Set of portfolio differences where the slopeof the risk is less than slopes of cost at zero trade
u u
Proportional Costs:Always trades to the no-trade region
Independent of induced costs
No Trade Region = Optimality for Proportional Costs
Optimality for Superproportional Costs Contains The No Trade Region
Take the gradient with respect to wOPTSolve for the optimal trade wUse duality to exchange the order of optimization 1
Trick substitution: maxT T
zT w e T w z
OPT
2OPT * 0
T
w ww u u C T w z
2
* 0 0TC w u u T w z
TT w e TT w z1
maxz
minw
The no trade region is a Parallelopiped
2* 0 * 0
12
Tw u u C w u u
linear image of the cube 1z
OPTNo Trade Region: 0 :w
2* 0 0
T
wu u C T w z
Proportional Costs are Incredible!
as
T ww
w
Proportional costs are toooptimistic for large trades,so reasonable costs are :superlinear
Who Says Say’s Law?
• Say’s Law: Supply Creates Demand• In the large? (Supply Side Economics).• In the small? Look for sublinear transaction
costs (‘volume attracts volume’).• Not frequent enough to explain the
expectation but it could be a variance component.
Special case quadratic costs Tw diagwc diagw2d
convexity requires d 0
being on the buy side requires c 0
wzTTw z c 2diagz dw
w u u0 C 2 wzTTw
I 2C 2 diagz d 1u u0 C 2z c
d 0 w u u0 C 2z caka "proportional costs"
"proportional costs" w u u0 C 2z c
max z 112 z cTC 2z c z cTu u0 C 2z c
max z 1 12 z cTC 2z c z cTu u0
min z 112 z cTC 2z c z cTu u0
min |qk | ck12 q
TC 2q qTu u0
w u u0 C 2q
Bound constrained quadratic program:
Quadratic loss with bounded trades:
min |w | 12 w u u0 TC2w u u0
equivalent to
min |w | 12 w
TC2w wTC2u u0
also a bound constrained quadratic program.
minAx b 12 x
TGx dTx
KKT Conditions
Gx AT d 0Ax y b 0
y 0yT 0
# # # #
G AT 0A 0 I0 Y
x y
Gx AT dAx y by
Central path 1n yT
G AT 0A 0 I0 Y
x y
Gx AT dAx y by
00e
Newton direction
Choose step size t 0 such that
Ax t x b t 0
# #
update
Interior point iteration
x, x t x, t
y Ax b # #
...what about ?
G AT 0A 0 I0 Y
x y
Gx AT dAx y by
00e
eliminate y A x
G AT
A Y x
Gx AT d
Y
0e
add ATY 1 times bottom part to the top part:
I ATY 1
0 IG AT
A Y
G ATY 1 A 0 A Y
The top part is the ‘reduced system’
G ATY 1 A x Gx d ATY 1e
G ATY 1 A x Gx d ATY 1e
Y Y e A x y A x
Structure of A and G can provide great computational advantage.
Bound Constraints:
I I
x l
u
A I I
ATY 1 A is diagonal
Any structure for G that accomodates addition of a diagonal matrix
(banded, sparse, low grade, factor, etc.)
I XXT I XI XTX 1XT I XXT XI XTX 1XT XXTXI XTX 1XT
I XXT XI XTXI XTX 1XT
I XXT XXT
II XXT 1 I XI XTX 1XT
Solve D VVT x yD D1/2XD1/2XT x y
D1/2I XXT D1/2x y
x D 1/2 I XI XTX 1XT D 1/2y
Iterated Diagonal Box QP
x zD V
V Ixz
z V x
x D VV x
D VV I
I 0
VD 1 ID 00 I VD 1V
I D 1V0 I
min l x uz V x
f 12 Vz
x 12 x
TDx
Iterated Diagonal Box QP
xk 1 arg minl x u
f 12 Vzk
x 1
2 xDx
zk 1 V xk 1
#
#
stationarity: 0 f 12 Vz j
Djjx j
x j min uj ,max l j , Djj 1 f 12 Vz j
min l x uz V x
f 12 Vz
x 12 x
TDx
Modified Steepest Descent
Alternate between:
Move as far as feasible
1) toward the vertex
2) Toward the minimum along the gradient direction
Postprocessing
Gx d l u 0x l y l 0u x yu 0y l l 0yu u 0
Once we have u and d we can solve for x via x G 1 l u d
Accuracy Comparison
Time Comparison – 5 instances3000x150
Unstructured (ip) 90.8 sec
Matlab Factor (qp) 0.690
Homegrown factor (bq) 0.995
Diagonal Iterate (di) 0.059
Mod. Steepest Descent (ms) 0.128
The unstructured method is too slow to compare for enough instances
Time and Accuracy 150 instances3000x15
Method Time Max Inaccuracy
QP 17.16 0.3525
BQ 8.5 0.1075
MS 0.91 0.0360
DI 0.37 0.1075
Time and Accuracy 150 instances3000x50
Method Time Max Inaccuracy
QP 19.43 0.2356
BQ 18.55 0.0744
MS 2.04 0.0057
DI 0.722 0.0745
Time and Accuracy 150 instances3000x150
Method Time Max Inaccuracy
QP 17.6 0.1347
BQ 33.2 0.0401
MS 3.8 0.0355
DI 1.72 0.0399
Equity Curve
Covariance Distortion Hedgeminw 1
2 u1 u C2u1 u eT|Tu1 u0 | 2 u1
ThhTu1
u min hTu 0uTC2u 1
rTu
u minuT C2 hhT u 1 rTu
minw 12 u1 u C2 hhT u1 u eT|Tu1 u0 |
Question TimeYes, you have questions.