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 Outline Introduction Problem formulation Index Tracking Enhanced Indexation Computational results Conclusion

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

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

7 Problem Formulation (5) (6) (7) (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 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 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 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 Problem Formulation The modulus objective is nonlinear and can be linearised in a standard way (13) (14) (15) (16) (17)

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

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

19 Computational Results IndexNumber of stocks NNumber of selected stocks K Hang Seng3110 DAX FTSE S&P Nikkei S&P Russell Russell

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

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 Index Tracking Systematic Revision Results

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

24 Enhanced Indexation Systematic Revision Results

25 Conclusion Good computational results Reasonable computational times in all cases

26 Thank you for listening!