9781118041581

Summary 739

The portfolio manager must determine the proportions of the individual’s total dol-

lar investment to invest in the securities. These proportions are denoted by B, T, C, M, and

J for the respective securities. (For instance, if the manager divided the portfolio equally

among the five assets, the values would be B T C M J  .2.) The actual size of

the manager’s investment fund does not enter into the formulation. The optimal pro-

portions will be the same whether the manager is investing $20,000 or $20 million.

The objective function lists the average (or expected) return of the portfolio. The

first constraint indicates that the portfolio’s average risk rating must be at least 3.5. The

second and third constraints list the bounds on the portfolio’s average maturity. The final

constraint ensures that the portfolio proportions sum exactly to 1.

What portfolio will maximize the investor’s expected return, subject to the risk and

maturity constraints? From Table 17.3, we see that the optimal portfolio puts 26 percent

of the individual’s total dollar investment in treasury bills, 36.5 percent in treasury bonds,

and the remainder in junk bonds. This portfolio has a risk rating of exactly 3.5 (which just

meets this constraint), has a maturity of exactly 2.5 years (which just meets the upper

maturity constraint), and delivers a maximum portfolio return of 6.23 percent.

The spreadsheet also lists the relevant shadow prices as calculated by the LP program.

The shadow prices associated with the risk and maturity constraints are of some interest. The

former shadow price shows that allowing a unit reduction in the portfolio’s risk index

(reflecting a tolerance for greater risk) would raise the portfolio’s expected return by .708

percent. According to the latter shadow price, increasing the average maturity of the port-

folio by a year would increase the expected return by .555 percent. (In either case, the port-

folio would shift toward a greater share of junk bonds and a smaller share of treasury bills.)

SUMMARY

Decision-Making Principles

1. Linear programming is a method of formulating and solving decision
   problems that involve explicit resource constraints. The range of LP
   problems includes product mix, cost minimization, transportation,
   scheduling, inventory, and financial and budgeting decisions.


2. To qualify as a linear program, all decision variables must enter linearly
   into the objective function and into all constraints. As long as the LP
   problem is feasible, an optimal solution always exists at one of the
   corners of the feasible region. The optimal corner can be found by
   graphical means or by computer algorithms.


3. The shadow price of a resource shows the change in the value of the
   objective function associated with a unit change in the resource. Thus, the
   shadow price measures the improvement in the objective from relaxing a
   constraint or, conversely, the decline in the objective from tightening a
   constraint. A nonbinding constraint has a shadow price of zero.

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