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Chapter 6: Problem 35

Madison Finance has a total of $$\$ 20$$ million earmarked for homeowner and auto loans. On the average, homeowner loans have a \(10 \%\) annual rate of return, whereas auto loans yield a \(12 \%\) annual rate of return. Management has also stipulated that the total amount of homeowner loans should be greater than or equal to 4 times the total amount of automobile loans. Determine the total amount of loans of each type that Madison should extend to each category in order to maximize its returns. What are the optimal returns?

Short Answer

Expert verified
To maximize its returns, Madison Finance should extend \( \$ 4 \) million in homeowner loans and \( \$ 16 \) million in auto loans, resulting in an optimal annual return of \( \$ 2.24 \) million.

Step by step solution

01

Define the variables and constraints

Let: - H = total amount of homeowner loans (in millions) - A = total amount of auto loans (in millions) Constraints: 1. The total amount of homeowner loans should be greater than or equal to 4 times the total amount of automobile loans: \(H \ge 4A\) 2. Madison Finance has a total of \( \$20 \) million earmarked for homeowner and auto loans: \(H + A = 20\)
02

Define the objective function

The objective function represents Madison Finance's total returns from both types of loans. For homeowner loans: \(0.10H\) For auto loans: \(0.12A\) Therefore, the objective function to maximize is: \(R(H, A) = 0.10H + 0.12A\)
03

Formulate the problem as a linear program

Maximize \(R(H, A) = 0.10H + 0.12A\) subject to: 1. \(H \ge 4A\) 2. \(H + A = 20\)
04

Solve the problem using graphical method

We can solve this problem graphically by plotting the constraints and identifying the feasible region: 1. Plot the constraint \(H + A = 20\). This line has a slope of -1 and passes through H = 20 and A = 20. 2. Plot the constraint \(H \ge 4A\). This line represents \(H = 4A\) with a slope of 4. The region above this line represents \(H \ge 4A\). 3. Identify the feasible region, which is the set of points on the graph that satisfy both constraints. In this case, it's the area above the line \(H = 4A\) and to the left of the line \(H + A = 20\). 4. Identify the vertices of the feasible region: the points where the constraints lines intersect. In this case, there are 2 vertices: (4, 16) and (15, 5).
05

Find the optimal solution

Evaluate the objective function, R(H, A), at each vertex of the feasible region to find the maximum return: 1. At vertex (4, 16): \(R(4, 16) = 0.10(4) + 0.12(16) = 2.24\) 2. At vertex (15, 5): \(R(15, 5) = 0.10(15) + 0.12(5) = 1.80\) The maximum return is achieved at vertex (4, 16), where the annual return is \( \$ 2.24 \) million. In order to maximize its returns, Madison Finance should extend: - Homeowner loans: \( \$ 4 \) million - Auto loans: \( \$ 16 \) million The optimal returns will be \( \$ 2.24 \) million per year.

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Key Concepts

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

Optimization Problems in Finance
Optimization problems in finance are crucial for making decisions that aim to achieve the best outcome within a given set of constraints. For instance, companies like Madison Finance need to determine the optimal allocation of resources鈥攊n this case, funds鈥攖o maximize returns. An optimization problem typically involves an objective, which in a financial context could be maximizing profit or minimizing costs, and is subject to certain limitations known as constraints.

These problems require identifying the best possible solution from a range of feasible options. In the case of Madison Finance, the optimization was used to allocate funds between homeowner and auto loans to achieve maximum returns. Through mathematical modeling, as demonstrated in the textbook solution, finance professionals can use optimization to make informed strategic decisions that contribute positively to a company's financial objectives.
Objective Function in Financial Decision-Making
The objective function is the heartbeat of any optimization problem. It is a formula that needs to be maximized or minimized. In finance, the objective function represents the goal of an investment or allocation decision鈥攖his could be to maximize returns, as with Madison Finance, or to minimize risks or costs depending on the scenario.

In our exercise, the objective function was about maximizing returns from homeowner and auto loans. It was expressed as a mathematical equation \( R(H, A) = 0.10H + 0.12A \), indicating the total return based on the amounts allocated to homeowner loans \( H \) and auto loans \( A \). The solution process focused on finding the values of \( H \) and \( A \) that would yield the maximum return, demonstrating how pivotal the objective function is in steering the course towards the optimal financial decision.
Feasible Region in Linear Programming
The feasible region is a cornerstone concept in linear programming and represents all possible solutions that satisfy the problem's constraints. In the context of finance and our problem, it is the set of all possible combinations of homeowner and auto loans that Madison Finance could extend within its limits.

To visualize this, imagine graphing the constraints on a two-dimensional plane, with each point representing a potential strategy for loan allocation. The area that encompasses all the strategies that meet Madison Finance's criteria constitutes the feasible region. For example, in our solved exercise, the feasible region was bounded by the constraint that homeowner loans must be at least four times greater than auto loans, and the total amount for loans was fixed at $20 million. The feasible region guided us to the point that maximized the returns, providing a vivid demonstration of how constraints define the boundaries of financial strategy and decision-making.

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Most popular questions from this chapter

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Solve each linear programming problem by the method of corners. $$ \begin{array}{l} \text { Maximize } P=4 x+2 y \\ \text { subject to } \quad x+y \leq 8 \\ \quad 2 x+y \leq 10 \\ x \geq 0, y \geq 0 \end{array} $$

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