Give 5 examples of Capital Recovery problems with a solution.
The Correct Answer and Explanation is :
Capital Recovery Problems and Solutions
Capital recovery is the process of recovering the initial investment in a project or asset over time. It is commonly used in finance, engineering, and business to determine the annual equivalent amount needed to recover a capital investment.
Here are five examples of capital recovery problems with solutions:
1. Problem:
A company invests $50,000 in new equipment. If the company expects the equipment to last for 10 years with an interest rate of 8%, what is the annual amount required to recover the investment?
Solution:
Using the capital recovery factor formula:
[
A = P \times \frac{i(1+i)^n}{(1+i)^n – 1}
]
Where:
- ( P = 50,000 ) (initial investment)
- ( i = 0.08 ) (interest rate)
- ( n = 10 ) (number of years)
[
A = 50,000 \times \frac{0.08(1+0.08)^{10}}{(1+0.08)^{10} – 1}
]
[
A = 50,000 \times \frac{0.08(2.1589)}{2.1589 – 1}
]
[
A = 50,000 \times \frac{0.17271}{1.1589}
]
[
A = 50,000 \times 0.14889
]
[
A = 7,444.49
]
Answer: The annual amount required is $7,444.49.
2. Problem:
A municipal bond worth $100,000 is issued for a 20-year period with an interest rate of 5%. What is the annual payment the city must make to cover the principal and interest?
Solution:
Using the same formula:
[
A = 100,000 \times \frac{0.05(1+0.05)^{20}}{(1+0.05)^{20} – 1}
]
[
A = 100,000 \times \frac{0.05(2.6533)}{2.6533 – 1}
]
[
A = 100,000 \times \frac{0.132665}{1.6533}
]
[
A = 100,000 \times 0.0802
]
[
A = 8,020.16
]
Answer: The annual payment is $8,020.16.
3. Problem:
A business owner invests $120,000 in a new facility. The investment has a life span of 15 years, and the interest rate is 6%. What is the annual recovery amount?
Solution:
Using the formula:
[
A = 120,000 \times \frac{0.06(1+0.06)^{15}}{(1+0.06)^{15} – 1}
]
[
A = 120,000 \times \frac{0.06(2.3966)}{2.3966 – 1}
]
[
A = 120,000 \times \frac{0.1438}{1.3966}
]
[
A = 120,000 \times 0.1029
]
[
A = 12,348.44
]
Answer: The annual recovery amount is $12,348.44.
4. Problem:
A car dealership purchases new cars worth $200,000. The dealership expects the cars to last for 5 years with an interest rate of 7%. What is the annual recovery amount?
Solution:
Using the formula:
[
A = 200,000 \times \frac{0.07(1+0.07)^5}{(1+0.07)^5 – 1}
]
[
A = 200,000 \times \frac{0.07(1.40255)}{1.40255 – 1}
]
[
A = 200,000 \times \frac{0.0981785}{0.40255}
]
[
A = 200,000 \times 0.2434
]
[
A = 48,687.80
]
Answer: The annual recovery amount is $48,687.80.
5. Problem:
A college builds a new dormitory for $1,000,000. The dormitory will be used for 25 years, and the interest rate is 4%. How much will the college need to set aside annually to recover the investment?
Solution:
Using the formula:
[
A = 1,000,000 \times \frac{0.04(1+0.04)^{25}}{(1+0.04)^{25} – 1}
]
[
A = 1,000,000 \times \frac{0.04(2.6666)}{2.6666 – 1}
]
[
A = 1,000,000 \times \frac{0.106664}{1.6666}
]
[
A = 1,000,000 \times 0.064
]
[
A = 64,000
]
Answer: The annual recovery amount is $64,000.
Explanation:
The capital recovery factor formula calculates the equal annual payment required to recover the initial investment (P) over a specified period with a fixed interest rate. This payment includes both principal repayment and interest costs.
- ( P ): Initial investment.
- ( A ): Annual recovery amount.
- ( i ): Interest rate per period (as a decimal).
- ( n ): Number of periods.
The formula ensures that the sum of annual payments (A) over the given time period covers the initial investment, including interest costs. The method is used in various fields, such as finance and engineering, for evaluating investments, calculating loan repayments, and determining required revenue for capital-intensive projects.