Integer Programming Review Paper (Fall 2001) Muthiah Prabhakar Ponnambalam (University of Texas Austin)
Portfolio Construction Through Mixed Integer Programming at Grantham, Mayo, Van Otterloo and Company (Dimitris Bertsimas, Christopher Darnell, Robert Soucy) Goal: The main purpose of this paper was to develop a technology that could improve the firm s capabilities in two critical areas: diversification of investment styles and control of the process of constructing portfolios and trading. Diversification is a concept used by most investment firms to reduce the risk of investing in the stock market by investing in both conservative and risky stocks. While diversification reduces the risk, it significantly increases the number of securities in order to maintain a desired rate of return. So they needed a technology that could support a large scale and comprehensive multiple investment style process. The number of positions and trades in the composite portfolio is an inherently integer quantity. These constraints lead to the use of mixed integer - programming methods to globally optimize the portfolio. Mixed Integer Programming Approach: Here we consider a single portfolio and compare it with a target portfolio (portfolio that is desirable to own after rebalancing). The objective function is to minimize the difference between the actual portfolio and the target portfolio. The different factors like exposure to different factors, minimum number of securities and transactions, high return, high liquidity and minimum total transaction cost are in built into the objective function.
User specified penalty lambda is used to keep the number of securities and transactions to a minimum value. The model uses various penalties to capture the relative importance of the various objectives. Penalties are chosen heuristically after extensive experimentation. Once the formulation was complete, it was solved in FORTRAN using CPLEX 4.0 as the underlying mixed integer programming solver. Once the input file is parsed into a dynamic data structure the problem is solved by CPLEX. The single portfolio problem with 1,500 securities in the portfolio took about 15 hours to solve. This is undesirable for simulation purposes. So they followed a branch and bound strategy and by adjusting the stopping criteria they could improve the solution time. Also by strengthening the formulation they could improve the LP relaxation bounds. They used the structure of the problem to set node selection and branching priorities. It is unlikely that the largest positions in the target portfolio will be eliminated, so branching is done on the variables that correspond to the largest positions in the target portfolio. The above-mentioned approach was tried out on a portfolio and the optimized portfolio has approximately 55% that of the target portfolio and the average number of transactions was reduced by 60% with only a marginal decrease in liquidity. Comments: I have always wanted to work in the area of Finance and wondered about the application of integer programming in Finance. I was really surprised to see a direct application of the branch and bound procedure in the construction of a portfolio.
A Note on Management Decision and Integer Programming (David L. Currin and W. Allen Spicey) Objective: This paper addresses the question whether solving a corresponding linear programming model without an integer restriction and rounding the components of an optimal vector to integer values is an appropriate line of attack. It is one thing to study about optimization in theory, but when it comes to solving such problems in real life various issues complicate the situation. Many real world optimization problems have the property that optimal decisions must be stated in terms of integers or whole numbers. A model for many such problems is that of integer programming. This makes matters worse as none of the solution algorithms currently available can handle large or even medium sized problem of this kind. There are numerous examples in the literature in which the use of a linear programming model to come up with a solution for the I.P problem is neither feasible nor optimal. Since this is not the case, we could set things straight by considering a simple example. A manager seeks the cheapest way to invest in four selected projects by buying one or more of four available options shown in Table I. Options Projects Option 1 2 3 4 cost 1 30 15 52 53 40 2 31 56 50 10 40 3 40 36 40 30 40 4 20 85 30 40 40
Solving this model gives the following optimal solution. figure 1 figure 2 Thus by buying one half of each option 1,3 and 4, the manager could attain minimum costs of 180.5. However since the manager can only purchase or not purchase options, the L.P model provides an approximation only. When the above L.P model is solved with integer restrictions, we get the feasible (Optimal) solution obtained by rounding the vectors x and y, which is shown in figure 2. Conclusion: This paper addresses the basic problem involved in solving an IP Problem. It also clearly states that an IP solution could be obtained by rounding off the solution to the corresponding LP problem.