Question

Suppose the City purchases five sidewalk plows worth $20,000 each to be used over 10 years. Determine the annual payments the City is responsible for if the interest rate is constant at 3%. Suppose the benefits of having cleared sidewalks in the neighbourhood these snowplows will be used are estimated at $20,000 each winter, and the annual operating costs are $500 per plow, should the City make this investment? Show your work.

Answer 1

Cost of 1 sideplow = 20,000

Cost of 5 sideplow = 100,000

Annual benefit (winter comes once in an year) = 20,000 of clear sideplow

Annual Interest rate = 3%

Annual operating cost = 500

If present value of net benefit = Cost of 5 sideplow + Present value of annual operating cost is less than present value of annual benefit, City should make this investment, otherwise not.

Present value of annual benefit:

Year	Annual benefit	Present value of annual benefit	Rough work to calculate annual benefit
1	20,000	19,417.48	[20,000 / 1.03^1]
2	20,000	18,851.92	[20,000 / 1.03^2]
3	20,000	18,302.83	[20,000 / 1.03^3]
4	20,000	17,769.74	[20,000 / 1.03^4]
5	20,000	17,252.18	[20,000 / 1.03^5]
6	20,000	16,749.69	[20,000 / 1.03^6]
7	20,000	16,261.83	[20,000 / 1.03^7]
8	20,000	15,788.18	[20,000 / 1.03^8]
9	20,000	15,328.33	[20,000 / 1.03^9]
10	20,000	14,881.88	[20,000 / 1.03^10]
		170,604.06

Present value of annual operating cost:

Year	Annual benefit	Present value of annual benefit	Rough work to calculate annual benefit
1	500	485.44	[500 / 1.03^1]
2	500	471.30	[500 / 1.03^2]
3	500	457.57	[500 / 1.03^3]
4	500	444.24	[500 / 1.03^4]
5	500	431.30	[500 / 1.03^5]
6	500	418.74	[500 / 1.03^6]
7	500	406.55	[500 / 1.03^7]
8	500	394.70	[500 / 1.03^8]
9	500	383.21	[500 / 1.03^9]
10	500	372.05	[500 / 1.03^10]
		4,265.10

Total cost = 100,000 + 4,265.10 = 104,265.10

Net benefit: 170,604.06 - 104,265.10 = 66,338.96

City should invest in sideplows due to positive net benefit.