Introduction

Decisions must often be made in a sequential manner over time. Earlier decisions may affect the feasibility and performance of later decisions. In such environments, myopic decisions that optimize only the immediate impact are usually suboptimal for the overall process. To find optimal strategies one must consider current and future decisions simultaneously. These types of multi-stage decision problems are the typical settings where one employs dynamic programming, or DP. Dynamic programming is a term used both for the modeling methodology and the solution approaches developed to solve sequential decision problems. In some cases the sequential nature of the decision process is obvious and natural, in other cases one reinterprets the original problem as a sequential decision problem. We will consider examples of both types below.

Dynamic programming models and methods are based on Bellman's Principle of Optimality, namely that for overall optimality in a sequential decision process, all the remaining decisions after reaching a particular state must be optimal with respect to that state. In other words, if a strategy for a sequential decision problem makes a sub-optimal decision in any one of the intermediate stages, it cannot be optimal for the overall problem. This principle allows one to formulate recursive relationships between the optimal strategies of successive decision stages and these relationships form the backbone of DP algorithms.

The structuring of collateralized mortgage obligations will give us an opportunity to apply the dynamic programming approach studied in Chapter 13.

Mortgages represent the largest single sector of the US debt market, surpassing even the federal government. In 2000, there were over $5 trillion in outstanding mortgages. Because of the enormous volume of mortgages and the importance of housing in the US economy, numerous mechanisms have been developed to facilitate the provision of credit to this sector. The predominant method by which this has been accomplished since 1970 is securitization, the bundling of individual mortgage loans into capital market instruments. In 2000, $2.3 trillion of mortgage-backed securities were outstanding, an amount comparable to the $2.1 trillion corporate bond market and $3.4 trillion market in federal government securities. A mortgage-backed security (MBS) is a bond backed by a pool of mortgage loans. Principal and interest payments received from the underlying loans are passed through to the bondholders. These securities contain at least one type of embedded option due to the right of the home buyer to prepay the mortgage loan before maturity. Mortgage payers may prepay for a variety of reasons. By far the most important factor is the level of interest rates. As interest rates fall, those who have fixed rate mortgages tend to repay their mortgages faster.

Introduction

A linear programming model for constructing a portfolio of assets might produce a solution with 3205.7 shares of stock XYZ and similarly complicated figures for the other assets. Most portfolio managers would have no trouble rounding the value 3205.7 to 3205 shares or even 3200 shares. In this case, a linear programming model would be appropriate. Its optimal solution can be used effectively by the decision maker, with minor modifications. On the other hand, suppose that the problem is to find the best among many alternatives (for example, a traveling salesman wants to find a shortest route going through ten specified cities). A model that suggests taking fractions of the roads between the various cities would be of little value. A 0,1 decision has to be made (a road between a pair of cities is either on the shortest route or it is not), and we would like the model to reflect this.

This integrality restriction on the variables is the central aspect of integer programming. From a modeling standpoint, integer programming has turned out to be useful in a wide variety of applications. With integer variables, one can model logical requirements, fixed costs, and many other problem aspects. Many software products can change a linear programming problem into an integer program with a single command.

The downside of this power, however, is that problems with more than a thousand variables are often not possible to solve unless they show a specific exploitable structure.

As we discussed in Chapter 2, in each iteration of the simplex method, we first choose an entering variable looking at the objective row of the current tableau, and then identify a leaving variable by comparing the ratios of the numbers on the right-hand side and the column for the entering variable. Once these two variables are identified we update the tableau. Clearly, the most time-consuming job among these steps of the method is the tableau update. If we can save some time on this bottleneck step then we can make the simplex method much faster. The revised simplex method is a variant of the simplex method developed with precisely that intention.

The crucial question here is whether it is necessary to update the whole tableau in every iteration. To answer this question, let us try to identify what parts of the tableau are absolutely necessary to run the simplex algorithm. As we mentioned before, the first task in each iteration is to find an entering variable. Let us recall how we do that. In a maximization problem, we look for a nonbasic variable with a positive rate of improvement. In terms of the tableau notation, this translates into having a negative coefficient in the objective row, where Z is the basic variable.

To facilitate the discussion below let us represent a simplex tableau in an algebraic form, using the notation from Section 2.4.1.