June 1999 In Pursuit of the Perfect Portfolio By Jenny Summerour Sometimes the best implementations in life come to those who wait. Or so it seems in the case of Grantham, Mayo, Van Otterloo and Company LLC (GMO), an investment-management firm in Boston. In 1996 GMO found that its value investing strategy, the principal strategy used in its quantitative stock division, was performing below expectations in many markets around the world. As a result, the investment returns of some of its largest and most important clients were trending down. The group's performance was hindered by an unusually large number of stocks (names) and transactions (trading tickets) in their clients' portfolios, the result of the classical quadratic-programming method GMO used. The additional custodial fees and transaction costs associated with the excessive names and tickets forced up the costs of trading, which reduced clients' net returns. GMO asked Dimitris Bertsimas, a professor at the MIT Sloan School of Management and Operations Research, to address the problem of reducing the number of names and trading tickets in clients' portfolios. Bertsimas decided to design a mixed-integer programming approach. Although he did not know of other cases where mixed-integer programming had been used in portfolio construction, he believed this method would work since the number of names and trading tickets are discrete numbers, and since he had been successful in applying mixed-integer programming in other areas. Testing the formula After Bertsimas outlined and formulated the problem in mathematical terms, the people in the firm began the actual testing and implementation. They went through a confidence testing first and then continued testing on a trial-and-error basis, seeking to mimic the large-name portfolio with the same liquidity, turnover and expected return as the target portfolio, but with fewer names and fewer securities. Robert Soucy, a member at GMO, says the implementation was not necessarily difficult, but it did present some challenges. "We were not familiar with integer programming," Soucy says. "There's a lot of trial-and-error to solve specific problems once you get the process in place. There are a whole host of parameters associated with the integer programming problem, and altering those parameters one at a time or in combination can make very big differences in how rapidly you can get a solution to the problem." The initial setup took only a couple of months, but the people at GMO, as well as Bertsimas, found that continual testing was a very time-consuming process. "It required quite a bit of effort," Bertsimas says. "In the beginning, it took one hour per one monthly optimization. You back-simulate, so for 20 years (240 months), you have to solve 240 integer programming problems. "To judge portfolios, there are many issues," Bertsimas says. "After you have finished the version of the algorithm, you are testing it historically. If there is something you don't like, you change the algorithm again. It is a very iterative process." "Simulations can be very time-consuming if each individual optimization takes a long time," Soucy agrees. "Once we were able to get to the point where a single optimization was done relatively quickly, then we could run these long historical tests and develop a lot more confidence in the process." Results GMO applied the mixed-integer programming methodology to portfolios with a total market value of $8.158 billion from October 1996 to January 1997. They achieved an average reduction of 48.7 percent in the number of names and an average reduction of 79.3 percent in the number of trading tickets. In addition, they realized annual savings of about $4 million from the sharp drop in trading tickets in the international and small stock U.S. funds. Bertsimas estimates that this figure is now closer to $10 million. "This has been a very beneficial implementation," Soucy says. "Now we can easily manage the problem of reducing the number of names in a portfolio. "It has reduced the amount of trading that we do because we have a hand on the number of securities each month we're getting ready to trade," Soucy continues. "And we can minimize that when we're developing our portfolios." Bertsimas is particularly pleased that now integer programming is a mainstream of sorts among the partners at GMO. "When we wrote the problem, the approach was being implemented on half the funds of the firm," he says. "Now it's implemented in pretty much all the funds. "Integer programming is something that everybody knows and uses in their everyday lives at the firm," Bertsimas adds. "One of the larger clients in the firm actually ordered a book I had written on integer programming. That is sort of unusual." Lessons learned Bertsimas breaks down the lessons he learned from this experience into four points: "I learned that solving an integer programming problem is a multi-faceted activity. It involves having a good idea and working well with data. Modeling is an iterative process — you start with a model and change it to accommodate the data and to make the algorithm faster." "To be successful in solving large-scale integer programming problems requires very good implementation and extensive experimentation." "In comparing this experience with other experiences where OR has been used, finance is an area where it is perhaps easier to convince people that OR is useful. The reason, I think, is that you can immediately see results." "On a personal side, having an environment where people trust you — where there is teamwork — is very important to success. This process required some of my knowledge as well as some of the people who implemented it and some of the people who run it." Soucy offers his advice to people who plan to implement a similar program: "They'll have to plan on spending some time playing with the problems they're trying to solve," he says. "It's likely they could get an initial solution fairly quickly; but I think there's a learning curve and that they need to spend some time — perhaps a fair amount of time — just really understanding exactly what the optimization is doing and what the different parameters mean." Jenny Summerour is managing editor of OR/MS Today. She can be reached via e-mail at jsummerour@lionhrtpub.com Table of Contents OR/MS Today Home Page OR/MS Today copyright © 1999 by the Institute for Operations Research and the Management Sciences. All rights reserved. Lionheart Publishing, Inc. 2555 Cumberland Parkway, Suite 299, Atlanta, GA 30339 USA Phone: 770-431-0867 | Fax: 770-432-6969 E-mail: lpi@lionhrtpub.com URL: http://www.lionhrtpub.com Web Site © Copyright 1999 by Lionheart Publishing, Inc. All rights reserved.