1. 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

2. 2 Outline Introduction Problem formulation Index Tracking Enhanced Indexation Computational results Conclusion

3. 3 Introduction Passive fund management Index tracking Full replication Fewer stocks Tracking portfolio (TP)

4. 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

5. 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

6. 6 Problem Formulation Constraints (1) (2) (3) (4)

7. 7 Problem Formulation (5) (6) (7) (8)

8. 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

9. 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

10. 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)

11. 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

12. 12 Problem Formulation The modulus objective is nonlinear and can be linearised in a standard way (13) (14) (15) (16) (17)

13. 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

14. 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)

15. 15 Problem Formulation Enhanced indexation One-stage approach to enhanced indexation is: subject to (1)-(11),(13)-(17) and

16. 16 Problem Formulation Two-stage approach is precisely the same as seen before, namely minimise (19) subject to (1)-(11), (13)-(17), (20), (21)

17. 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)

18. 18 Computational Results The initial TP composed of the first K stocks in equal proportions, i.e.

19. 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

20. 20 Index Tracking In-Sample vs. Out-of-Sample Results

21. 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)

22. 22 Index Tracking Systematic Revision Results

23. 23 Enhanced Indexation In-Sample vs. Out-of-Sample Results

24. 24 Enhanced Indexation Systematic Revision Results

25. 25 Conclusion Good computational results Reasonable computational times in all cases

26. 26 Thank you for listening!