Question

X = 04

Q3. [20 marks] For this question, X is the first two digits (from the right) of your student ID. A Chinese official is considering construction of a lift system that transports visitors from the tops of the mountain peaks to the valley at the National Forest Parks in China. The lift system is expected to last for 30 years. Annual maintenance costs are expected to be $260,000. Moreover, inspection and repair costs of $40,000 every five years (including year 30) are expected. Annual benefits are estimated to be $425,000. If an interest rate of 15% per year is used and a conventional B-C ratio of at least 1.X (using the PW method) is targeted, what is the maximum investment cost allowed for the lift system?

Answer 1

Life = 30 years

Annual cost = $260,000

Annual benefits = $425,000

Repair cost every 5 years = $40,000

Interest rate = 15%

B-C artio = 1.04

Net annual benefit = Annual benefit - Annual cost = $400,000 - $260,000 = $140,000

Present value is calculated as: [Net annual benefit / (1 + Interest rate)^^Year]

Present value of net annual benefit in year 1 = [140,000/ (1 + 0.15)^¹]

Present value of net annual benefit in year 2 = [140,000 / (1 + 0.15)^²]

Present value of net annual benefit in year 3 = [140,000 / (1 + 0.15)^³]

and so on till year 30

Sum of present value of net annual benefit in 30 years = [140,000 / (1 + 0.15)^¹] + [140,000 / (1 + 0.15)^²] + [140,000 / (1 + 0.15)^³] + ............................ + [140,000 / (1 + 0.15)^³⁰]

This forms a G.P. whose sum can be calculated as: [a * (1 - r)^ⁿ / (1 - r)]

a = [140,000 / (1 + 0.15)^¹]

n = 30

r (ratio of two consecutive terms) = 0.87

Sum of G.P. = [140,000 / (1 + 0.15)^¹] * (1 - 0.87^³⁰) / (1 - 0.87) = 922,097.82

Present value of net annual benefits = 922,097.82

Present value of repair cost is calculated in the table below:

Year	Payment	Present value
5	40,000.00	19,887.07
10	40,000.00	9,887.39
15	40,000.00	4,915.78
20	40,000.00	2,444.01
25	40,000.00	1,215.11
30	40,000.00	604.12
		38,953.48

Total Cost = Initial Cost (X) + Present value of repair cost = X + 38,953.48

Benefit - cost ratio = (Present value of net annual benefit / Total Cost) = [922,097.82 / (X + 38,953.48)] = 1.04

X = 847,679.04

Maximum investment allowed for the lift system is 847,679.04