The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 666 Wednesday, February 4, 2004 1:12 PM

666 The Mathematics of Financial Modeling and Investment Management

The next step is to consider trading constraints, such as the need to

purchase “even” lots of assets. Under these constraints, assets can be

purchased only in multiples of some minimal quantity, the even lots. For

a large organization, purchasing smaller amounts, “odd” lots, might be

suboptimal and might result in substantial costs and illiquidity.

The optimization problem that results from the purchase of assets in

multiples of a minimal quantity is much more difficult. It is no longer a rel-

atively simple LP problem but it becomes a much harder mixed-integer pro-

gramming (MIP) problem. A MIP problem is conceptually more difficult

and computationally much more expensive to solve than an LP problem.

The next step involves allowing for transaction costs. The objective

of including transaction costs is to avoid portfolios made up of many

assets held in small quantities. Including transaction costs, which must

be divided between fixed and variable costs, will again result in a MIP

problem which will, in general, be quite difficult to solve.

In the formulation of the CFM problem discussed thus far, it was

implicitly assumed that the dates of positive cash flows and liabilities are

the same. This might not be the case. There might be small misalignment

due to the practical availability of funds or positive cash flows might be

missing when liabilities are due. To cope with these problems, one could

simply generate a bigger model with more dates so that all the dates cor-

responding to inflows and outflows are properly considered. In a number

of cases, this will be the only possible solution. A simpler solution, when

feasible, consists in adjusting the dates so that they match, considering the

positive interest earnings or negative costs incurred to match dates.

In the above formulation of the CFM problem, the initial investment

cost is the only variable to optimize: The eventual residual cash at the end of

the last period is considered lost. However, it is possible to design a different

model under the following scenario. One might try to maximize the final

cash position, subject to the constraint of meeting all the liabilities and

within the constraint of an investment budget. In other words, one starts

with an investment budget which should be at least sufficient to cover all the

liabilities. The optimization problem is to maximize the final cash position.

We have just described the CFM problem in a deterministic setting.

This is more than an academic exercise as many practical dedication

problems can be approximately cast into this framework. Generally

speaking, however, a dedication problem would require a stochastic for-

mulation, which in turn requires multistage stochastic optimization.

Dahl, Meeraus, and Zenios^12 discuss the stochastic case. Later in this

(^12) H. Dahl, A. Meeraus, and S.A. Zenios, “Some Financial Optimization Models,”
in S.A. Zenios (ed.), Financial Optimization (Cambridge: Cambridge University
Press, 1993).