To produce excellent results, many investments require deliberate ongoing management For example, the course of a project within a firm might be guided by a series of operational decisions Likewise, a portfolio of financial instruments might (and should be) modified systematically over time. The selection of an appropriate sequence of actions that affect an investment's cash flow stream is the problem of dynamic management.

Imagine, for example, that you have purchased an oil well. This is an investment project, and to obtain good results from it, it must be carefully managed, In this case you must decide, each month, whether to pump oil from your well or not If you do pump oil, you will incur operational costs and receive revenue from the sale of oil, leading to a profit; but you will also reduce the oil reserves, Your current pumping decision clearly influences the future possibilities of production If you believe that current oil prices are low, you may wisely choose not to pump now, but rather to save the oil fot a time of higher prices,

Discussion of this type of problem within the context of deterministic cash flow streams is especially useful—both because it is an important class of problems, and because the method used to solve these problems, dynamic programming, is used also in Part 3 of the book This simpler setting provides a good foundation for that later work,

A deterministic investment is defined by its cash flow stream, say, x = (.v(), .v¡, „vt, ,, -v,,), but the magnitudes of the cash flows in this stream often depend on management choices in a complex fashion. In order to solve dynamic management problems, we need a way to represent the possible choices at each period, and the effect that those choices have on future cash flows In short, we need a dynamic model There are several mathematical structures that can be used to construct such a model, but the simplest is a graph. In this structure, the time points at which cash flows occur are represented by points along the horizontal direction, as usual In the vertical direction above each such time point is laid out a set of nodes, which represent the different possible states or conditions of the process at that time Nodes from one time to the next are connected by branches or arcs. A branch represents a possible path from a node at one time to another node at the next time Different branches correspond to different management actions, which guide the course of the process. Simple examples of such graphs are that of a binomial tree and a binomial lattice, illustrated in Figure 53(a) and (b). In such a tree there are exactly two branches leaving each node. The leftmost node corresponds to the situation at the initial time, the next vertical pair of nodes represent the two possibilities at time 1, and so forth. (In the figure only four time points are shown. )

The best way to describe the meaning of the tree is to walk through an example, Let us again consider the management of the oil well you recently purchased, At any time you can either pump oil or not. A node in the tree represents the condition of the well, defined by the size of its reserves, the state of repair, and so forth, To model your choices as a tree, you should start at the leftmost node of the tree, which represents the initial condition of the well, You have only two choices at that time: pump or don't pump. Assign one of' these choices to an upward movement and the other to a downward movement; suppose that pumping corresponds to moving upward and nonpumping corresponds to moving downward. At the next time point your well is at one of the two nodes for that time. Again you make a choice and move either up or down. As you make your decisions, you move through the tree, from left to right, from node to node, along a particular path of branches, The path is uniquely determined by your choices; that is, the condition of the well through time and the magnitude of your overall profit are determined by your choices and represented by this unique path through the tree.

Suppose, specifically, that the well has initial reserves of 10 million barrels of oil. Each year it is possible to pump out 10% of the current reserves, but to do so a crew must be hired and paid. However, if a crew is already on hand, because it was used in the previous year, the hiring expenses are avoided Therefore, to calculate the profit that can be obtained in any year, it is necessary to know the level of oil reserves and whether a crew is already on hand. Hence we label each node of the tree showing the reserve level and the status of a crew For example, the label (9, YES) means that the reserves are 9,000,000 barrels and there is a crew on hand A complete tree for the two periods is shown in Figure 5 3(c/)

If crews can be assembled with no hiring cost, it is not necessary to keep track of the crew status. We can therefore drop one component from the node labels and keep only the reserve level. If we do that, some nodes that had distinct labels in the original tree will now have identical labels. In the example illustrated in Figure 5.4, two of the nodes at the final time both have a reserve level of 9 (meaning 9 million barrels) Since the labels are identical, we can combine these nodes into a single node, as shown in Figure 5.4(/>). If the tree were extended for additional time periods, this combining effect would happen frequently, and as a

(b) Binomial lattice

(a) Binomial tree

(b) Binomial lattice

FIGURE 5A Trees showing oil well states Pumping corresponds to an upward movement; no pumping corresponds to a downward movement The tree in (¿0 accounts for both the level of reserves and the status of a crew if only the reserve levels affect the profit, some nodes combine, forming a binomial lattice, as shown in {b)

result the tree could be collapsed to a binomial lattice A typical binomial lattice is shown in Figure 53(b). In such a graph, moving up and then down leads to the same node as moving down and then up There are fewer nodes in a binomial lattice than in a binomial tree

In terms of the oil well, if the only relevant factor for determining profit is the reserve level, it is clear that starting at any node, an upward movement in the tree (corresponding to pumping) followed by a downward movement (corresponding to not pumping) is identical in its influence on reserves to a downward movement followed by an upward movement. Both combinations deplete the reserves by the same amount. Hence a binomial lattice can be used to represent the management choices, as in Figure 5.4(b)

We used a binomial tree or a binomial lattice for the oil well example, which is appropriate when there are only two possible choices at each time. If there were three choices, we could form a trinomial tree or a trinomial lattice, having three branches emanating from each node. Clearly, any finite number of choices can be accommodated (It is only reasonable to draw small trees on paper, but a computer can handle larger trees quite effectively, up to a point,)

The description of the nodes of a graph as states of a process is only an intermediate step in the representation of a dynamic investment situation The essential part of the final representation is an assignment of cash flows to the various branches of the graph These cash flows are used to evaluate management alternatives

In the first oil well example, where crew hiring costs are not zero, suppose that the cost of hiring a crew is $100,000 (This represents just the initial hiring cost, not the wages paid.) Suppose the profit from oil production is $5.00 per barrel, Finally, suppose that at the beginning of a year the level of reserves in the well is .v Then the net profit for a year of production is $5 x . 10 x ,.v — $100,000 if a crew must be hired, and $5 x 10 x v if a crew is already on hand, We can enter these values on the branches of the tree, indicating that much profit is attained if that branch is selected These values are shown in Figure 5,5 in units of millions of dollars.

FIGURE 5.5 Oil well cash flow tree. The cash flow corresponding to a decision is listed on (he branch corresponding to that decision These cash flow values are determined by the node state and the decision

Since only the cash flow values on the branches are important for analysis, it would be possible (conceptually) to bypass the step of describing the nodes as states of the process However, in practice the node description is important because the cash flow values are determined from these descriptions by an accounting formula. If someone gave us the tree with cash flow values specified on all branches, that would be sufficient; we would not need the node descriptions. In practice, someone must first characterize the nodes, as we did earlier, so that the cash flows can be determined.

In representations of this land it must also be stated whether the cash flow of a branch occurs at the beginning or at the end of the corresponding time period, In reality, a branch cash flow is often spread out over the entire period, but the model assigns a lump value at one end or the other (or sometimes a part at the beginning and another part at the end). The choice may vary with the situation being represented.

In some cases there is cash flow associated with the termination of the process, whose value varies with the final node achieved This is a final reward or salvage value. These values are placed on the graph at the corresponding final nodes, in the oil well example, the final value might be the value for which the well could be sold.

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