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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 (5)

(6)

(7)

(8)

NiGx ii ,...,10,

Nizi ,...,11,0

N

ii

N

iiiT GCxV

11

N

ii CG

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

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Conclusion Good computational results Reasonable computational times in

all cases

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Thank you for listening!