CHAPTER 5 Optimization Modeling T his chapter introduces optimization—a methodology for selecting an op- timal strategy given an objective and a set of constraints. Optimiza- tion appears in a variety of financial applications, including portfolio allocation, trading strategies, identifying arbitrage opportunities, and pric- ing financial derivatives. We will encounter it in Chapters 7, 8, 9, 14, and 18, among others. In this chapter, we motivate the discussion by a simple example and describe how optimization problems are formulated and solved. Let us recall the retirement example from section 4.2.2. We showed how to compute the realized return on the portfolio of stocks and bonds if we allocate 50% of our capital in each of the two investments. Can we obtain a “better” portfolio return with a different allocation? (As discussed in section 4.2.3 of the previous chapter, a “better” return is not well-defined in the context of uncertainty, so for the sake of argument, let us assume that “better” means higher expected return.) We found that if the allocation is (100%, 0%) instead of (50%, 50%), we end up with a higher portfolio expected return, but also higher portfolio standard deviation. What about an allocation of (30%, 70%)? It turned out that the portfolio expected return is lower, and so is the standard deviation. What about an allocation of (20%, 80%)? In this example, we are dealing with only two investments, and we have no additional requirements on the portfolio structure. It is, however, still difficult to enumerate all the possibilities and find those that provide the optimal trade-off of return and risk. In practice, portfolio managers are handling thousands of investments and need to worry about transaction costs, requirements on the portfolio composition, and trading constraints, which makes it impossible to find the “best” portfolio allocation by trial- and-error. The optimization methodology provides a disciplined way to approach the problem of optimal asset allocation. 143 Simulation and Optimization in Finance: Modeling with MATLAB, @RISK, or VBA by Dessislava A. Pachamanova and Frank J. Fabozzi Copyright © 2010 John Wiley & Sons, Inc.