Question

3(b) A City government is planning to build a Light Rail Tunnel in the Central Business District (CBD). The funds have to be borrowed at an interest rate of 12%. For the tunnel to serve its function forever, the following funds will be required. .The construction contractor will require $500 million at beginning of year 1 and another 200 million at the end of year 2 when the construction will end. For routine maintenance, $100,000/year (end of year payment, forever) will be required after the tunnel will be open for service in year 3. . S1 million every 10 years for repairs. These repair bills will be paid as end of year payments. What is the total present worth of the costs of building, maintaining, and repairing the tunnel?

Answer 1

Consider the given problem here the rate of return is given by, “r=12%=0.12”. Now, the building cot is given by “$500 million” at the beginning and “$200 million” at the end of the year 2. So, the present value of the building cost is given by.

=> PV(building) = $500 + $200/1.12^2 = $500 + $200/1.12^2 = 659.438776 million = $659,438,776.

Now, the maintenance cost is given by “$100,000 per year” will be required after the tunnel will be open for the service that is after the year 3. Now, if the cost will require after the year one, => the present value will be “$100,000/0.12”. But here it will require after the “3^rd year”, => the present value of maintenance cost is given by.

=> PV(maintenance) = $100,000 /0.12 - $100,000/1.12 - $100,000/1.12^2 = $833,333.3333 - $89,285.7143 - $79,719.3877 = $664,328.23.

Now, the repairing cost is “$1 million every 10 years”. So, the present value will be.

=> PV(Repair) = ($1/1.12^10 + $1/1.12^20 + $1/1.12^30 + $1/1.12^40 + ……..), infinite geometric series.

=> PV(Repair) = $1/1.12^10*(1 + $1/1.12^10 + $1/1.12^20 + ……..).

=> PV(Repair) = $1/1.12^10*[1 /1- ( $1/1.12^10)] = $1/1.12^10*[1.12^10 /1.12^10 - 1].

=> PV(Repair) = $1*[1 /1.12^10 - 1] = $0.474868034 million = $474,868.034.

So, the present worth of the cost of building, maintaining and repairing is given by.

=> Present Worth = $659,438,776 + $664,328 + $474,868 = $660,577,972.