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Chapter 7: Problem 46

An investor has up to \(450,000\) to invest in two types of investments. Type \(A\) pays \(6 \%\) annually and type B pays \(10 \%\) annually. To have a well- balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth of the portfolio is to be allocated to type \(B\) investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return?

Short Answer

Expert verified
Optimal investments are $225,000 in A and $225,000 in B, giving a return of $36,000.

Step by step solution

01

Representation of the Problem

Let \(x\) represent the amount to be invested in type A and \(y\) the amount to be invested in type B. This implies \(x + y = 450,000\). Now, according to the conditions, \(x \geq 0.5 * 450,000\) and \(y \geq 0.25 * 450,000\). The return from these investments would be \(6%\) from type A and \(10%\) from type B, so in terms of the variables, the total return is \(0.06x + 0.10y\).
02

Setup the Inequality Constraints

From previous step, we know: \(x \geq 0.5 * 450,000\) and \(y \geq 0.25 * 450,000\), which simplifies to \(x \geq 225,000\) and \(y \geq 112,500\). We also have the constraint that the total amount invested is \(450,000\), so \(x + y \leq 450,000\).
03

Use Constraints to find Optimal Investment

We have three constraints to work with: \(x \geq 225,000\), \(y \geq 112,500\), and \(x + y \leq 450,000\). Since both investments give some return and we are trying to maximize that, we intuitively try to put as much as possible in the investment with the higher return. Here, type B gives a higher return. So we put maximum in type B while respecting the constraints. So, initially try if we can put \(0.75 * 450,000\) in B, but that鈥檚 over. So we can only put \(450,000 - 225,000 = 225,000\) in B and the rest, which is \(450,000 - 225,000 = 225,000\) in A. Check that this does not violate the constraint for B which gives \(y \geq 112,500\), and it doesn't so this is the optimal distribution.
04

Calculate the Optimal Return

Using the optimal investment amounts, calculate the total return. This is \(0.06*225,000 + 0.10*225,000 = 13,500 + 22,500 = 36,000\). So the optimal return is \(36,000\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Constraints
Inequality constraints are an essential concept in linear programming. These constraints are used to define the boundaries or limits within which a solution must be found. In the context of the given problem, inequality constraints are applied to determine how the investor's money can be divided into two types of investments.

For the given problem, we have the inequality constraints:
  • \( x \geq 225,000 \): At least half of the total investment must be allocated to type A.
  • \( y \geq 112,500 \): At least one-fourth of the total investment must be allocated to type B.
  • \( x + y \leq 450,000 \): The total investment must not exceed the available \(450,000\).
These constraints help define a feasible region, where solutions are possible. By solving these inequalities along with the objective function, the maximum return can be identified.
Investment Portfolio Optimization
Investment portfolio optimization is about finding the best way to allocate funds among different investment options to achieve the highest possible return. In linear programming, this typically involves maximizing or minimizing an objective function while respecting set constraints.

In this problem, the objective function to maximize is the return of the portfolio, given by:
  • Return from type A: \(0.06x\)
  • Return from type B: \(0.10y\)

Thus, the overall return is calculated as \(0.06x + 0.10y\). The challenge is to allocate the \(450,000\) such that the constraints are met in a way that this return is maximized. The process involves increasing the investment in type B, as it offers a higher return, while ensuring that neither constraint for type A nor type B is violated. This strategic allocation allows the highest possible return within the provided limits.
Mathematical Modeling
Mathematical modeling involves the creation of a mathematical representation of a real-world scenario. This model can be used to analyze and make predictions about such situations. For the investment problem posed, mathematical modeling helped reformulate the investor's requirements and constraints into a linear programming problem that could be solved systematically.

Here, the problem is modeled by defining variables \(x\) and \(y\), representing the amounts invested in types A and B respectively. The model defines constraints based on the minimum investment conditions and the total amount available. The goal, indicated by an objective function, is to maximize the total return.
  • Variables: Describe specific unknown quantities in the problem.
  • Constraints: Govern the relationships between variables, dictated by real-world limits.
  • Objective Function: Measures what needs to be optimized, such as return.

This systematic representation allows not just solution finding, but also adaptability to changes in variables or constraints, showcasing the powerful applications of mathematical modeling in portfolio management.

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Most popular questions from this chapter

A humanitarian agency can use two models of vehicles for a refugee rescue mission. Each model A vehicle costs \(1000\) and each model B vehicle costs \(1500 .\) Mission strategies and objectives indicate the following constraints. " The agency must use a total of at least 20 vehicles. \(\cdot \mathrm{A}\) model \(\mathrm{A}\) vehicle can hold 45 boxes of supplies. \(\mathrm{A}\) model B vehicle can hold 30 boxes of supplies. The agency must deliver at least 690 boxes of supplies to the refugee camp. \(\cdot \mathrm{A}\) model \(\mathrm{A}\) vehicle can hold 20 refugees. A model \(\mathrm{B}\) vehicle can hold 32 refugees. The agency must rescue at least 520 refugees. What is the optimal number of vehicles of each model that should be used? What is the optimal cost?

Write the partial fraction decomposition of the rational expression. Then assign a value to the constant \(a\) to check the result algebraically and graphically. $$\frac{1}{a^{2}-x^{2}}$$

Acid Mixture A chemist needs 10 liters of a \(25 \%\) acid solution. The solution is to be mixed from three solutions whose concentrations are \(10 \%, 20 \%,\) and \(50\%\). How many liters of each solution will satisfy each condition? (a) Use 2 liters of the \(50 \%\) solution. (b) Use as little as possible of the \(50 \%\) solution. (c) Use as much as possible of the \(50 \%\) solution.

The sums have been evaluated. Solve the given system for \(a, b,\) and \(c\) to find the least squares regression parabola for the points. Use a graphing utility to confirm the result. $$\left\\{\begin{array}{c} 4 c+9 b+29 a=20 \\ 9 c+29 b+99 a=70 \\ 29 c+99 b+353 a=254 \end{array}\right.$$

\(\mathrm{A}\) store sells two models of laptop computers. Because of the demand, the store stocks at least twice as many units of model \(\mathrm{A}\) as of model \(\mathrm{B}\). The costs to the store for the two models are \(\$ 800\) and \(\$ 1200\), respectively. The management does not want more than \(\$ 20,000\) in computer inventory at any one time, and it wants at least four model A laptop computers and two model B laptop computers in inventory at all times. Find and graph a system of inequalities describing all possible inventory levels.
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