Question

(4 pts) A city is trying to attract a new business to the area. It has offered to install and operate a water pumping plant to provide service to the proposed plant site. This would cost $50,000 now, plus $5,000 per year in operating costs for the next ten years. To reimburse the city, the new business must pay a fixed uniform annual fee, A, at the end of each year for ten years. In addition, they are to pay the city $50,000 at the end of ten years. As the assistant city engineer, it is your job to determine the amount of the uniform annual fee, A, which would make the total amount paid by the company equivalent to the amount paid by the city. Assume an interest rate of 10% per year

Answer 1

Answer: A = $10,000

Step-by-step explanation:

This question requires that we find the amount of A that will equate the present value of costs to the present value of the benefits.

So we are looking for the amount of A that will bring a Break-Even.

Starting with the Present Value of the Costs.

It will cost $50,000 now and $5,000 per year for 10 years at an interest rate of 10%.

The $5,000 is a constant cash flow so we can treat it as an annuity.

We can use the Present Value of an Annuity Interest factor. I have attached a table showing the different factors.

$5,000 each year for 10 years at a 10% rate.

Looking at the table where 10 years intercept 10% we have, 6.1446.

Present value of $5,000 every year for 10 years is

= 5,000 * 6.1446

= $30,723

Adding that to the original cost of $50,000 we have,

= 50,000 + 30,723

= $80,723

Present value of Benefits.

A is an annuity as it is a fixed amount that will be paid every year.

They are also to pay $50,000 at the end of the 10th year so we would have to discount the $50,000 to the present.

The formula will look like this

= A * (6.1446) + 50,000/ (1 + 10% ) ^ 10

= 6.1446A + 19,277.1644715

= 6.1446A + 19,277.16

The above is the formula for the present value of the benefits.

Equating both figures we have,

6.1446A + 19,277.16 = 80,723

6.1446A = 80,723 - 19,277.16

6.1446A = 61,445.84

A = 61,445.84/6.1446

= 9999.97396088

= $10,000 ( nearest dollar. )

A = $10,000

$10,000 is the amount the company should pay yearly for 10 years to reimburse the city in addition to the $50,000 at the end of the year.

Answer 2

Present value of cost = 50000 + 5000 * (P/A,10%,10) = 50000 + 5000 * 6.144567 = 80722.83

Let annual fee be A, then

Present value of benefit = A *(P/A,10%,10) + 50000 * (P/F,10%,10)

= A *6.144567 + 50000 * 0.385543

As per given condition

A *6.144567 + 50000 * 0.385543 = 80722.83

A = (80722.83 - 50000 * 0.385543) / 6.144567 = 10000.00163 ~ 10000 (Nearest Dollar)