Problem 33 - College Algebra Essentials

Question

Suppose that you inherit 10,000 dollar. The will states how you must invest the money. Some (or all) of the money must be invested in stocks and bonds. The requirements are that at least 3000 dollar be invested in bonds, with expected returns of 0.08 dollar per dollar, and at least 2000 dollar be invested in stocks, with expected returns of 0.12 dollar per dollar. Because the stocks are medium risk, the final stipulation requires that the investment in bonds should never be less than the investment in stocks. How should the money be invested so as to maximize your expected returns?

Step-by-Step Solution

Answer

You should invest 5000 dollars each in stocks and bonds to maximise your expected returns.

Step 1: Identify and interpret the constraints

Let \( X \) and \( Y \) represent the amount of money invested in stocks and bonds respectively. From the problem, the constraints can be formulated thus: \n1) \( X \leq Y \) (investment in bonds should never be less than the investment in stocks) \n2) \( Y \geq 3000 \) (at least 3000 dollar must be invested in bonds) \n3) \( X \geq 2000 \) (at least 2000 dollar must be invested in stocks)\n4) \( X + Y = 10000 \) (the total investment must sum to 10,000 dollar).

Step 2: Formulate the objective function

The expected returns from the investments are determined by the rates of return and the amount invested in each category. Therefore the objective function to maximise is: \( 0.12X+0.08Y \).

Step 3: Solve the linear program

The problem becomes that of solving the following linear program:\nMaximise \( Z = 0.12X+0.08Y \) \nsubject to: \( X \leq Y; Y \geq 3000; X \geq 2000; X + Y = 10000 \).\nSolution to this linear program will give the exact amounts to be invested in stocks and bonds for maximum returns.

Step 4: Determine optimal values for X and Y

The constraints \( X \) and \( Y \) essentially means that \( X = Y \). Therefore, from the constraint \( X + Y = 10000 \), we get \( 2X = 10000 \) or \( X = 5000 \) dollars. Likewise, \( Y = 5000 \) dollars. Thus, equal amounts of 5000 dollars should be invested in stocks and bonds.

Key Concepts

Investment Optimization

Investment optimization is all about making your money work as hard as it can for you, especially when you have certain rules or constraints to follow.
In the context of the exercise, we are trying to figure out the best way to invest $10,000 to get the highest returns while adhering to specific requirements.
In linear programming, investment optimization often involves maximizing returns while considering constraints like minimum and maximum investment levels or risk factors.
For instance, in the provided problem, we know exactly how much needs to be allocated to stocks and bonds, the expected returns, and the need to balance investments between the two to adhere to given stipulations.

• Invest the minimum of $3,000 in bonds to ensure a stable return of 0.08 dollars per dollar.
• Invest at least $2,000 in stocks for a higher expected return of 0.12 dollars per dollar.
• Ensure that the stocks are medium risk by maintaining a balance where bonds aren't less than stocks.

This strategic allocation and balancing act ensure that your inherited money not only respects the conditions set forth but also positions you for maximum returns.

Constraint Satisfaction

Constraints in linear programming are like the rules of a game.
They are non-negotiable conditions that an optimal solution must satisfy. The investment optimization problem comes with several constraints derived from the rules given in the will.
Our primary goal here is to manage these constraints effectively while still trying to maximize return.

• The first rule is that the investment in bonds must never be less than that in stocks. This is expressed as the constraint \( X \leq Y \).
• Next, you need to invest at least \(3,000 in bonds, which is represented mathematically as \( Y \geq 3000 \).
• Similarly, a minimum of \)2,000 must go into stocks, conveyed as \( X \geq 2000 \).
• Lastly, the sum of investments in stocks and bonds must total up exactly to $10,000, denoted by \( X + Y = 10000 \).

These constraints define the boundaries of how you can split up your money, and only by adhering to them can we find a valid solution.

Objective Function

Creating an objective function is like setting your goal post.
For the investment problem at hand, the objective function is a linear equation representing the total expected return from both stocks and bonds.
This function is what you aim to maximize, given:\[ Z = 0.12X + 0.08Y \]
Here, \( Z \) is the total return, \( X \) represents the amount invested in stocks, and \( Y \) represents the amount in bonds.
The coefficients 0.12 and 0.08 represent the expected returns for each dollar invested in stocks and bonds, respectively.
Thus, for every dollar put into stocks, you expect to gain 12 cents in return, whereas each dollar in bonds yields 8 cents.

• To find the optimal investment strategy, you need to maximize \( Z \) while respecting all the constraints.
• This process often involves testing the limits of the constraints to pinpoint the values of \( X \) and \( Y \) that will yield the highest possible \( Z \).

By successfully maximizing this objective function, you ensure that your investment strategy is the best possible within the given rules, thereby securing the highest financial return from your inheritance.