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1 Mixed Integer Programming Approaches for Index Tracking and Enhanced Indexation Nilgun Canakgoz, John Beasley Department of Mathematical Sciences, Brunel University CARISMA: Centre for the Analysis of Risk and Optimisation Modelling Applications
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2 Outline Introduction Problem formulation Index Tracking Enhanced Indexation Computational results Conclusion
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3 Introduction Passive fund management Index tracking Full replication Fewer stocks Tracking portfolio (TP)
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4 Problem Formulation Notation N : number of stocks K : number of stocks in the TP ε i : min proportion of TP held in stock i δ i : max proportion of TP held in stock i X i : number of units of stock i in the current TP V it : value of one unit of stock i at time t I t : value of index at time t R t : single period cont. return given by index
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5 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 x i : number of units of stock i in the new TP G i : TC incurred in selling/buying stock i z i = 1 if any stock i is held in the new TP = 0 otherwise r t : single period cont. return by the new TP
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6 Problem Formulation Constraints (1) (2) (3) (4)
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7 Problem Formulation (5) (6) (7) (8)
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8 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
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9 Problem Formulation Hence the return on the TP at time t Approximate W it by a constant term which is independent of time Hence the return on the TP at time t
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10 Problem Formulation Our expression for w i 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)
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11 Problem Formulation Ideally, we would like, for index-tracking, to choose K stocks and their quantities (x i ) such that we achieve We adopt the single weighted objective, user defined weights
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12 Problem Formulation The modulus objective is nonlinear and can be linearised in a standard way (13) (14) (15) (16) (17)
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13 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
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14 Problem Formulation Two-stage approach Let and be numeric values for and when we use our formulation above Then the second stage is (19) subject to (1)-(11) and (13)-(17) and (20) (21)
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15 Problem Formulation Enhanced indexation One-stage approach to enhanced indexation is: subject to (1)-(11),(13)-(17) and
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16 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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17 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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18 Computational Results The initial TP composed of the first K stocks in equal proportions, i.e.
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19 Computational Results IndexNumber of stocks NNumber of selected stocks K Hang Seng3110 DAX 1008510 FTSE 1008910 S&P 1009810 Nikkei 22522510 S&P 50045740 Russell 2000131890 Russell 3000215170
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20 Index Tracking In-Sample vs. Out-of-Sample Results
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21 Systematic Revision To investigate the performance of our approach over time we systematically revise our TP a)Set T=150 b)Use our two-stage approach to decide the new TP [x i ] c)Set [X i ]=[x i ] (replace the current TP by the new TP) d)Set T=T+20 and if T 270 go to (b)
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22 Index Tracking Systematic Revision Results
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23 Enhanced Indexation In-Sample vs. Out-of-Sample Results
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24 Enhanced Indexation Systematic Revision Results
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25 Conclusion Good computational results Reasonable computational times in all cases
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26 Thank you for listening!
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