High School

Al Ferris has $60,000 to invest for purchasing a retirement annuity in 5 years. He is considering four fixed-income investments, labeled as A, B, C, and D.

- Investments A and B are available at the beginning of each of the next 5 years (years 1 to 5).
- Each dollar invested in A returns $1.5 two years later.
- Each dollar invested in B returns $1.8 three years later.
- Investments C and D will be available only once in the future.
- Each dollar invested in C at the beginning of year 2 returns $2 at the end of year 5.
- Each dollar invested in D at the beginning of year 5 returns $1.4 at the end of year 5.

Al wishes to know which investment plan maximizes the amount of money accumulated by the beginning of year 6. Identify the complete linear programming model for this problem:

a) Maximize \( P = 1.5A_4 + 1.8B_3 + 2C_2 + 1.4D_5 + R_5 \)

Subject to:
1. \( A_1 + B_1 + R_1 = 60000 \)
2. \( A_2 + B_2 + C_2 - R_1 + R_2 = 0 \)
3. \( -1.5A_1 + A_3 + B_3 - R_2 + R_3 = 0 \)
4. \( -1.5A_2 + A_4 - 1.8B_1 - R_3 + R_4 = 0 \)
5. \( -1.5A_3 - 1.8B_2 + D_5 - R_4 + R_5 = 0 \)

b) Maximize \( P = 1.5A_1 + 1.8B_2 + 2C_3 + 1.4D_4 + R_5 \)

Subject to:
1. \( A_1 + B_1 + R_1 = 60000 \)
2. \( A_2 + B_2 + C_2 - R_1 + R_2 = 0 \)
3. \( -1.5A_1 + 2A_3 + B_3 - R_2 + R_3 = 0 \)
4. \( -1.5A_2 + 2A_4 - 1.8B_1 - R_3 + R_4 = 0 \)
5. \( -1.5A_3 - 1.8B_2 + D_5 - R_4 + R_5 = 0 \)

c) Minimize \( P = 1.5A_4 + 1.8B_3 + 2C_2 + 1.4D_5 + R_5 \)

Subject to:
1. \( A_1 + B_1 + R_1 = 60000 \)
2. \( A_2 + B_2 + C_2 - R_1 + R_2 = 0 \)
3. \( -1.5A_1 + A_3 + 1.4B_3 - R_2 + R_3 = 0 \)
4. \( -1.8A_2 + 1.4A_4 - 1.8B_1 - R_3 + R_4 = 0 \)
5. \( -2A_3 - 1.8B_2 + D_5 - R_4 + R_5 = 0 \)

d) Maximize \( P = 1.4A_4 + 1.8B_3 + 1.5C_2 + 2D_5 + R_5 \)

Subject to:
1. \( A_1 + B_1 + R_1 = 60000 \)
2. \( A_2 + B_2 + C_2 - R_1 + R_2 = 0 \)
3. \( -1.5A_1 + A_3 + B_3 - R_2 + R_3 = 0 \)
4. \( -1.5A_2 + A_4 - 1.8B_1 - R_3 + R_4 = 0 \)
5. \( -1.5A_3 - 1.8B_2 + D_5 - R_4 + R_5 = 0 \)

Answer :

Final answer:

The problem discusses how to maximize returns from different investments using linear programming. The examples provided in the reference information illustrate the concept of intertemporal choice in financial planning, highlighting the impact of rate of return and the power of compound interest.

Explanation:

In the given problem, Al Ferris is faced with a decision on how to best invest his $60,000 for optimal returns at retirement in 5 years. He is considering four investments A, B, C and D, each with different rates of return and time of returns. It reminds us of the financial planning in real life, like our examples of Yelberton and Quentin, who saved during their working years for future consumption after retirement.

The linear programming models given in the question are supposed to maximize the total profit, P, subject to several constraints based on the money invested and returned by each types of investment over the five years. Keep in mind the power of compound interest over substantial periods of time as we saw in Yelberton's case. As we can see from Quentin's example, decisions on investment and consumption can also be modeled by 'budget constraints' and 'indifference curves'.

Interestingly, changes in rate of return as seen in Yelberton's case, alter the slope of the intertemporal budget constraint, affecting decisions on present and future consumption. Unfortunately, without further information, we cannot definitively say which of the given models correctly represents Al Ferris' problem. With a better understanding of the problem parameters and constraints, the correct model could be more accurately identified.

Learn more about Financial Planning Using Linear Programming here:

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