Question

"To decrease the cost of operating a lock in a large river, a new system of operation is proposed. The system will cost $840,000 to design and build. It is estimated that it will have to be reworked every 7 years at a cost of $130,000. In addition, a one-time expenditure of $72,000 will have to be made at the end of the fifth year for a new type of gear that will not be available until then. Annual operating costs are expected to be $72,000 for the first 12 years and $102,000 thereafter. Compute the capitalized cost of perpetual service at i = 6.3%. Enter your answer as a POSITIVE number."

Answer 1

Solution:

Capitalized cost of perpetual service is Present value of cash flows for all type of outflows discounted at 6.3%

Initial System Cost = $840,000

Reworked cost = $130,000 every 7 years

Present value of reworked cost = [$130,000/(1+0.063)^7] + [$130,000/(1+0.063)^14] + [$130,000/(1+0.063)^21].....up to infinity

This is sum of infinity terms of GP with first term(a) = $130,000/(1.063)^7, common ratio = 1/(1.063)^7

Sum of infinity terms of GP = a / (1-r)

= [$130,000/(1.063)^7] / [1 - 1/(1.063)^7]

= $84,764 / 0.347970 =$243,596

Present value of one time expenditure = $72,000 * pv factor at 5 years at 6.3%

=$72,000*0.736773 = $$53,048

Annual operating cost = $72000 for first 12 years and 102000 thereafter.

Present value of annual operating cost = Present value of operating cost of 12 years + PV of annual operating cost after 12 years

Present value of operating cost of first 12 years = $72,000 * cumulative pv factor for 12 years at 6.30%

= $72,000*8.247657 = $593,831

PV of annual operating cost after 12 years = $102,000/(1.063)^13 + $102,000/(1.063)^14 + $102,000/(1.063)^15......upto infinity

This is sum of infinity terms of GP with first term(a) = $102,000/(1.063)^13, common ratio = 1/1.063

Sum of infinity terms of GP = a / (1-r)

= [$102,000/(1.063)^13] / [1 - 1/(1.063)]

= $46,096.48 / 0.059266 =$777,790

Present value of annual operating cost = $593,831 + $777,790 = $1,371,621

Capitalized cost of perpetual service = Initial cost + PV of rework cost + PV of one time expenditure + PV of annual operating cost

= $840,000 + $243,596 + $53,048 + $1,371,621 = $2,508,265