Question

Suppose that a city has to justify the net benefits of a proposed water treatment plant.

(A) Name and discuss the importance of the four things that the City must consider in terms of estimating the Present Value of Net Benefits (PVNB).

(B) Calculate the PVNB’s from the following information:

Year​ 0 ​1​ 2​ 3​ 4​ 5​ 6​

Cost​ 100 ​ 50​ 30​ 20​ 20​ 10​ 10​​

Benefit​ 10​ 20 ​20​ 40​ 100​ 120​ 150​

(i) PVNB (@ 4%) = _______________

(ii) PVNB (@ 8%) = _______________

(C) Given the benefit and cost schedules and PVNB calculations, explain why the City would prefer a 4% compared to an 8% rate.

Answer 1

We have the following information

Year	Cost ($)	Benefit ($)
0	100	10
1	50	20
2	30	20
3	20	40
4	20	100
5	10	120
6	10	150

When interest rate = 4%

Net Present Worth (4%) = –100 – 50(P/F,4%,1) – 30(P/F,4%,2) – 20(P/F,4%,3) – 20(P/F,4%,4) – 10(P/F,4%,5) – 10(P/F,4%,6) + 10 + 20(P/F,4%,1) + 20(P/F,4%,2) + 40(P/F,4%,3) + 100(P/F,4%,4) + 120(P/F,4%,5) + 150(P/F,4%,6)

Net Present Worth (4%) = –100 – 50/(1 + 0.04)¹ – 30/(1 + 0.04)² – 20/(1 + 0.04)³ – 20/(1 + 0.04)⁴ – 10/(1 + 0.04)⁵ – 10/(1 + 0.04)⁶ + 10 + 20/(1 + 0.04)¹ + 20/(1 + 0.04)² + 40/(1 + 0.04)³ + 100/(1 + 0.04)⁴ + 120/(1 + 0.04)⁵ + 150/(1 + 0.04)⁶

Net Present Worth (4%) = –226.81 + 385.94

Net Present Worth (4%) = $159.12

When interest rate = 8%

Net Present Worth (8%) = –100 – 50(P/F,8%,1) – 30(P/F,8%,2) – 20(P/F,8%,3) – 20(P/F,8%,4) – 10(P/F,8%,5) – 10(P/F,8%,6) + 10 + 20(P/F,8%,1) + 20(P/F,8%,2) + 40(P/F,8%,3) + 100(P/F,8%,4) + 120(P/F,8%,5) + 150(P/F,8%,6)

Net Present Worth (8%) = –100 – 50/(1 + 0.08)¹ – 30/(1 + 0.08)² – 20/(1 + 0.08)³ – 20/(1 + 0.08)⁴ – 10/(1 + 0.08)⁵ – 10/(1 + 0.08)⁶ + 10 + 20/(1 + 0.08)¹ + 20/(1 + 0.08)² + 40/(1 + 0.08)³ + 100/(1 + 0.08)⁴ + 120/(1 + 0.08)⁵ + 150/(1 + 0.08)⁶

Net Present Worth (8%) = –215.7 + 327.1

Net Present Worth (8%) = $111.4

Since, the net present worth is comparatively higher with 4% as compared to 8%, so the City would prefer a 4% compared to an 8% rate.