Math Problem Statement

A pension fund manager decides to invest a total of at most ​$40 million in U.S. Treasury bonds paying 6​% annual interest and in mutual funds paying 9​% annual interest. He plans to invest at least ​$5 million in bonds and at least ​$15 million in mutual funds. Bonds have an initial fee of​ $100 per million​ dollars, while the fee for mutual funds is​ $200 per million. The fund manager is allowed to spend no more than ​$7000 on fees. How much should be invested in each to maximize annual​ interest? What is the maximum annual​ interest?

Solution

To solve this problem, we can set it up as a linear programming optimization problem where the objective is to maximize the annual interest, subject to constraints on the investments and fees.

Let:

  • xx = Amount invested in U.S. Treasury bonds (in million dollars)
  • yy = Amount invested in mutual funds (in million dollars)

Objective function:

The goal is to maximize the total annual interest, which is: Maximize Z=0.06x+0.09y\text{Maximize} \ Z = 0.06x + 0.09y Where:

  • 6% interest from bonds gives 0.06x0.06x
  • 9% interest from mutual funds gives 0.09y0.09y

Constraints:

Total investment:

The total investment cannot exceed $40 million, so: x+y40x + y \leq 40

Minimum investments:

The manager must invest at least $5 million in bonds and at least $15 million in mutual funds, so: x5x \geq 5 y15y \geq 15

Fee constraint:

Bonds have a fee of $100 per million, and mutual funds have a fee of $200 per million. The total fee cannot exceed $7000, so: 100x+200y7000100x + 200y \leq 7000 Simplifying: x+2y70x + 2y \leq 70

Summary of constraints:

x+y40x + y \leq 40 x+2y70x + 2y \leq 70 x5x \geq 5 y15y \geq 15

Solution:

We now solve this as a linear programming problem. The next step is to find the feasible region defined by the constraints and identify the corner points, where the maximum interest can occur.

Let me calculate the optimal values of xx and yy at these points.The pension fund manager should invest:

  • x=10x = 10 million dollars in U.S. Treasury bonds, and
  • y=30y = 30 million dollars in mutual funds.

The maximum annual interest that can be earned is 3.33.3 million dollars.

Would you like more details or have any questions?

Here are 5 related questions to further explore:

  1. How would changes in the interest rates affect the optimal investment strategy?
  2. What if the total fee allowed was reduced to $5000 instead of $7000?
  3. How would the solution change if the maximum total investment increased to $50 million?
  4. What happens to the solution if the minimum mutual fund investment requirement is removed?
  5. How do you calculate the break-even point if interest rates for bonds and mutual funds fluctuate?

Tip: In linear programming, constraints define the feasible region, and the optimal solution typically occurs at one of the corner points of this region.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Optimization
Inequalities
Interest Calculation

Formulas

Objective function: Maximize Z = 0.06x + 0.09y
Total investment constraint: x + y ≤ 40
Fee constraint: x + 2y ≤ 70
Minimum investments: x ≥ 5, y ≥ 15

Theorems

Corner Point Theorem for Linear Programming

Suitable Grade Level

Grades 11-12, College