(Industrial Engineering Help) A city is offered the ownership of a bus company now operating under franchise.?

Question

The proposal requires no immediate cash from the city. The buses will make a profit of $400,000 per year for the first 2 years of city operation. During year 3 the city will purchase a new fleet of buses and pay off the present owners at a cost of $1,900,000. The new buses will be operated for 4 years at a net profit of $300,000 per year. No salvage value is expected on either the present buses or the new buses. In other words, the costs of hauling them away will be exactly balanced by the revenue received from their sale. The life of the new buses is 4 years. At the end of year 7, therefore, the buses will be retired. At this time, the matter will be reinvestigated before any new investment is made. The opportunity cost of capital to the city is considered to be 10 percent. What is the true rate of return on this investment? Should the city accept the deal?

If anyone can give me any pointers or help at all... I'd be very grateful.

Answer 1

year . profit interest . . . balance . . . . . . loan
0 . . . . . . . . . . . . . . . . . . . . . . . . 0
1 400,000 . . . . . . .. . . . 400,000
2 400,000 . . . . . .. . . -1,100,000 1,900,000
3 300,000 -110,000 . . -690,000
4 300,000 . -69,000 . . -321,000
5 300,000 . -32,100 . . . .11,100
6 300,000 . . . . .. . 0 . . . 311,100
7 300,000 . . . . .. . 0 . . . 611,100

Net ROI = 611,100/1,900,000 = 32.163%
Comparable compound interest rate = 5.7358%
This is NOT the IRR interest. To calculate that we need
V = (((P(1 + i)^2 - p(1 + i) - p)(1 + i) - p)(1 + i) - p)(1 + i) - p
V = ((P(1 + i)^3 - p(1 + i)^2 - p(1 + i) - p)(1 + i) - p)(1 + i) - p
V = (P(1 + i)^4 - p(1 + i)^3 - p(1 + i)^2 - p(1 + i) - p)(1 + i) - p
V = P(1 + i)^5 - p(1 + i)^4 - p(1 + i)^3 - p(1 + i)^2 - p(1 + i) - p
V = P(1 + i)^5 - p((1 + i)^4 + (1 + i)^3 + (1 + i)^2 + (1 + i) + 1)
V = P(1 + i)^5 - (p/i)(1 + i)^5
V = (1 + i)^5(P - (p/i))
V/(1 + i)^5 - P = -p/i
i = -p/(V/(1 + i)^5 - P), which must be iterated to find IRR
i = 300,000/(611,100/(1 + i)^5 + 1,900,000),
i = 1.533246%

Note that this IRR incorporates the 10% interest paid out on the loan.

At the end of 2 years, the city would have a balance of $800,000. After the 7 year program, the city would have a balance of $611,100. Each successive iteration would require borrowing an additional $188,900. If year 3 income is $400,000 instead of $300,000, the payout and IRR are much better.

Based on the increasing cost of supporting the program, I would recommend against it.