1 Mixed Integer Programming Approaches for Index Tracking and Enhanced Indexation Nilgun Canakgoz,...
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Transcript of 1 Mixed Integer Programming Approaches for Index Tracking and Enhanced Indexation Nilgun Canakgoz,...
1
Mixed Integer Programming Approaches for Index Tracking and Enhanced Indexation
Nilgun Canakgoz, John BeasleyDepartment of Mathematical Sciences, Brunel University
CARISMA: Centre for the Analysis of Risk and Optimisation Modelling Applications
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Outline Introduction Problem formulation
Index Tracking Enhanced Indexation
Computational results Conclusion
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Introduction Passive fund management Index tracking
Full replication Fewer stocks
Tracking portfolio (TP)
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Problem Formulation Notation
N : number of stocks K : number of stocks in the TP εi : min proportion of TP held in stock i
δi : max proportion of TP held in stock i
Xi : number of units of stock i in the current TP
Vit : value of one unit of stock i at time t
It : value of index at time t
Rt : single period cont. return given by index
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Problem Formulation C : total value of TP :be the fractional cost of selling one unit of
stock i at time T :be the fractional cost of buying one unit of
stock i at time T : limit on the proportion of C consumed by TC xi : number of units of stock i in the new TP Gi : TC incurred in selling/buying stock i zi = 1 if any stock i is held in the new TP
= 0 otherwise rt : single period cont. return by the new TP
N
i iit
N
i iitt xVxVr1 11
ln
sif
bif
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Problem Formulation Constraints
(1)
(2)
(3)
(4)
KzN
ii
1
NizCxVz iiiitii ,...,1
NiVxXfG iTiisii ,...,1)(
NiVXxfG iTiibii ,...,1)(
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Problem Formulation Index Tracking Objective
Single period continuous time return for the TP (in period t)
is a nonlinear function of the decision variables
To linearise, we shall assume Linear weighted sum of individual returns Weights summing to 1
N
i iti
N
i iti VxVx1 11
ln
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Problem Formulation Hence the return on the TP at time t
Approximate Wit by a constant term which is independent of time
Hence the return on the TP at time t
N
j jtjitiitit
N
i it VxVxWwhererW11
tWandN
i it 1
1
N
j jTjiTii VxVxw1
N
i itirw1
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Problem Formulation Our expression for wi is nonlinear, to linearise
it we first use equation (6) and then equation (5) to get
(9)
Finally we have a linear expression (approximation) for the return of the TP
If we regress these TP returns against the index returns
(10), (11)
NiCCVxw iTii ,...,1)(
NtrwN
i iti ,...,11
N
i iiw1 ˆˆ
N
i iiw1ˆˆ
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Problem Formulation Ideally, we would like, for index-
tracking, to choose K stocks and their quantities (xi) such that we achieve
We adopt the single weighted objective
, user defined weights
1ˆ0ˆmin
1ˆ0ˆ and
0,
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Problem Formulation The modulus objective is nonlinear and
can be linearised in a standard way (13)
(14)
(15)
(16)
(17)
DD
1ˆ E
1ˆ E
0, ED
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Problem Formulation Our full MIP formulation for solving
index-tracking problem is
subject to (1)-(11) and (13)-(17)
This formulation has 3N+4 continuous variables, N zero-one variables and 4N+9 constraints
ED min
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Problem Formulation Two-stage approach
Let and be numeric values forand when we use our formulation above
Then the second stage is
(19) subject to (1)-(11) and (13)-(17) and
(20)(21)
opt opt
opt ˆopt ˆ
N
iiG
1
min
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Problem Formulation Enhanced indexation
One-stage approach to enhanced indexation is:
subject to (1)-(11),(13)-(17) and
Emin
*ˆ
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Problem Formulation Two-stage approach is precisely the
same as seen before, namely
minimise (19) subject to (1)-(11), (13)-(17), (20), (21)
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Computational Results Data sets
Hang Seng, DAX, FTSE, S&P 100, Nikkei, S&P 500, Russell 2000 and Russell 3000
Weekly closing prices between March 1992 and September 1997 (T=291)
Model coded in C++ and solved by the solver Cplex 9.0 (Intel Pentium 4, 3.00Ghz, 4GB RAM)
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Computational Results The initial TP composed of the first
K stocks in equal proportions, i.e.
6
0
100
;,...,2,1/)/(
CwithKiX
KiVKCX
i
ii
iii 1&01.0 01.0 b
isi ff
1
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Computational Results
Index Number of stocks N Number of selected stocks K
Hang Seng 31 10
DAX 100 85 10
FTSE 100 89 10
S&P 100 98 10
Nikkei 225 225 10
S&P 500 457 40
Russell 2000 1318 90
Russell 3000 2151 70
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Alpha Beta Alpha Beta Time Alpha Beta TimeHang Seng 0.00028 1.00000 -0.00084 0.96435 0.1 -0.00052 0.99680 0.1DAX -0.00118 0.88476 0.00114 0.80548 0.1 0.00097 0.84596 0.2FTSE 0.00019 1.00000 0.00184 0.75941 0.1 0.00161 0.72776 0.1S&P 0.00000 1.00000 -0.00036 0.88089 0.2 -0.00040 0.94016 0.3Nikkei 0.00000 1.00000 0.00009 1.01419 0.3 -0.00023 0.94745 1.2S&P 500 0.00073 1.01354 0.00213 1.19389 2.8 0.00250 1.05111 5.2Russell 2000 0.00000 1.00000 0.00174 1.15996 4.6 0.00201 1.31635 10.6Russell 3000 0.00000 1.00000 0.00357 1.16305 8.6 0.00355 1.10387 32.1Average 0.00000 0.98688 0.00103 0.98725 2.0 0.00116 0.98069 6.1
IndexOne Stage Two Stage
Average In-Sample Average Out-Of-Sample Average Out-Of-Sample
Index Tracking In-Sample vs. Out-of-Sample Results
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Systematic Revision To investigate the performance of
our approach over time we systematically revise our TP
a) Set T=150b) Use our two-stage approach to
decide the new TP [xi]
c) Set [Xi]=[xi] (replace the current TP by the new TP)
d) Set T=T+20 and if T 270 go to (b)
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Index Tracking Systematic Revision Results
Alpha Beta Alpha Beta
Hang Seng 0.00031 1.00000 -0.00128 0.96525 0.5DAX -0.00101 0.88127 0.00080 0.86855 0.8FTSE 0.00021 1.00000 0.00071 0.75515 0.4S&P 0.00000 1.00000 -0.00104 0.95216 0.4Nikkei 0.00000 1.00000 -0.00005 0.98554 5.5S&P 500 0.00076 1.01479 0.00366 1.02045 21.0Russell 2000 0.00000 1.00000 0.00749 0.96127 104.4Russell 3000 0.00000 1.00000 0.00868 1.05301 178.9
Average In-Sample
IndexAverage
Out-of-Sample Average
Computation time
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Enhanced Indexation In-Sample vs. Out-of-Sample Results
Average In-Sample
Beta Alpha Beta
Hang Seng 0.99935 -0.00050 0.99169 -2.77 0.1 -DAX 0.87413 0.00089 0.99579 4.74 0.2 0.00FTSE 1.00000 0.00150 0.74962 3.18 0.2 0.00S&P 1.00000 0.00003 0.94572 -0.71 0.3 -Nikkei 0.99621 0.00009 0.98116 0.59 1.4 13.56S&P 500 1.00973 0.00237 1.03358 12.82 3.1 0.00Russell 2000 1.00000 0.00184 1.31621 11.49 10.5 0.00Russell 3000 1.00000 0.00364 1.12532 19.85 19.0 0.00
Sign. Level (%)
IndexAverage
Out-of-Sample Average
AER (yearly)
Average Computation
time
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Enhanced Indexation Systematic Revision Results
Average In-Sample
Beta Alpha Beta
Hang Seng 1.00000 -0.00068 0.96553 -4.30 0.4 -DAX 0.89813 0.00041 0.96246 1.41 0.7 5.49FTSE 1.00000 0.00092 0.76325 0.40 1.1 10.97S&P 1.00000 -0.00031 0.96525 -2.76 1.6 -Nikkei 0.99631 0.00026 0.99164 1.46 3.8 3.52S&P 500 1.00840 0.00317 1.06737 17.33 55.9 0.00Russell 2000 1.00000 0.00496 1.03162 30.13 143.1 0.00Russell 3000 1.00000 0.00782 0.97111 51.41 393.8 0.00
Sign. Level (%)
IndexAverage
Out-of-Sample Average
AER (yearly)
Average Computation
time