Chapter 3.8, Problem 9P

Explanation of Solution

 Formulation of a Linear Programming (LP) to maximize the expected return on its investment:

 From the given information, let consider “x₁” bet the amount of money invested in bond 1, “x₂” bet the amount of money invested in bond 2, “x₃” bet the amount of money invested in bond 3, “x₄” bet the amount of money invested in bond 4.

 The objective is to maximize the expected return on its investments.

 $\begin{array}{c}z=\left(\begin{array}{l}\left(\text{percentage of expected annual return on bond 1}\right)\left(\text{the amount of money invested in bond 1}\right)+\\ \left(\text{percentage of expected annual return on bond 2}\right)\left(\text{the amount of money invested in bond 2}\right)+\\ \left(\text{percentage of expected annual return on bond 3}\right)\left(\text{the amount of money invested in bond 3}\right)+\\ \left(\text{percentage of expected annual return on bond 4}\right)\left(\text{the amount of money invested in bond 4}\right)\end{array}\right)\\ =0.13x_1+0.08x_2+0.12x_3+0.14x_4\end{array}$

 Therefore, the objective function is,

 Maximize $z=0.13x_1+0.08x_2+0.12x_3+0.14x_4$

 Constraint 1:

 At least, 8% must be the worst-case return of the bond portfolio.

 $\begin{array}{c}\frac{\left(\begin{array}{l}\left(\text{percentage of the worst-case annual return on each bond}\right)\times \\ \left(\text{amount of money invested in each bond}\right)\end{array}\right)}{1000000}\ge 0.8\%\\ 0.06x_1+0.08x_2+0.10x_3+0.09x_4\ge 0.08\left(1000000\right)\\ 0.06x_1+0.08x_2+0.10x_3+0.09x_4\ge 80000\end{array}$

 Constraint 2:

 At most, the average duration of the bond portfolio must be 6