Math Problem Statement

Sandra has at most $300,000 to invest in stocks, bonds, and money market funds. She expects annual yields of 15%, 10%, and 8%, respectively, on these investments. If Sandra wants at least $75,000 to be invested in money market funds and requires that the amount invested in bonds be greater than or equal to the sum of her investments in stocks and money market funds, determine how much she should invest in each vehicle to maximize the return on her investments.

Solution

This problem can be solved as a linear programming (optimization) problem.

Decision Variables:

  1. Let x1x_1 = amount invested in stocks (at 15% return).
  2. Let x2x_2 = amount invested in bonds (at 10% return).
  3. Let x3x_3 = amount invested in money market funds (at 8% return).

Objective Function:

Maximize total return: Z=0.15x1+0.10x2+0.08x3Z = 0.15x_1 + 0.10x_2 + 0.08x_3

Constraints:

  1. Total investment cannot exceed $300,000: x1+x2+x3300,000x_1 + x_2 + x_3 \leq 300,000

  2. At least $75,000 must be invested in money market funds: x375,000x_3 \geq 75,000

  3. The amount invested in bonds must be at least equal to the sum of the investments in stocks and money market funds: x2x1+x3x_2 \geq x_1 + x_3

  4. Non-negativity constraint: x1,x2,x30x_1, x_2, x_3 \geq 0


I'll solve this optimization problem to determine the optimal values for x1x_1, x2x_2, and x3x_3 that maximize ZZ.from scipy.optimize import linprog

Coefficients for the objective function (maximize returns)

c = [-0.15, -0.10, -0.08] # Negative because linprog performs minimization

Coefficients for the inequality constraints (Ax <= b)

A = [ [1, 1, 1], # Total investment <= 300,000 [-1, 1, -1], # x2 >= x1 + x3 --> -x1 + x2 - x3 >= 0 [0, 0, -1] # x3 >= 75,000 --> -x3 <= -75,000 ]

b = [300000, 0, -75000]

Bounds for each variable (x1, x2, x3 >= 0)

x_bounds = (0, None) # Stocks y_bounds = (0, None) # Bonds z_bounds = (75000, None) # Money Market Funds

Solve the linear program

result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds, z_bounds], method='highs')

Extract results

investment_amounts = result.x maximum_return = -result.fun

investment_amounts, maximum_return(array([225000., 0., 75000.]), 39750.0)(array([225000., 0., 75000.]), 39750.0)successThe optimal investment strategy is as follows:

  1. Stocks: $225,000
  2. Bonds: $0
  3. Money Market Funds: $75,000

The maximum return Sandra can achieve is $39,750.

Let me know if you would like a detailed explanation or further assistance.

Related Questions:

  1. How can the constraints be modified to include a specific investment ratio?
  2. What would happen if Sandra wanted a higher minimum for the money market funds?
  3. How can linear programming be applied to more complex financial planning?
  4. What if the expected returns for stocks and bonds were reversed?
  5. How can Sandra handle risk considerations with this model?

Tip:

Always verify whether an optimal solution meets all constraints to ensure feasibility in real-world scenarios.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Optimization

Formulas

Objective Function: Maximize Z = 0.15x1 + 0.10x2 + 0.08x3
Constraints: x1 + x2 + x3 <= 300,000
x3 >= 75,000
x2 >= x1 + x3
x1, x2, x3 >= 0

Theorems

Linear Programming Feasibility Theorem

Suitable Grade Level

Undergraduate Level or Advanced High School