A tax exempt municipality is considering the construction of a new municipal was

Question

A tax exempt municipality is considering the construction of a new municipal waste water treatment facility. Two different sites have been selected as technically, politically, socially, and financially feasible. The city council uses 6% interest rate for all analyses for public projects. The expected cash flow for the two alternatives are as follow:

What is the incremental benefit/cost ratio?

Enter your answer as follow: 12.34

Year Alt. A Alt. B 0 - $13,003,049 - $25,542,331 1 - 75 $1,986,539/year $2,997,339/year

Explanation / Answer

In the initial year cost(C) is incurred on both the projects and from next year the benefit(B) will be recieved till the end of 75 years.

In order to do benefit/cost analysis over 76 years we need the present values of future benefits and costs dicounted at a interest rate of 6%

Present Value of Benefits, PVB = B1/(1+i)1 + B2/(1+i)2 + ..................+ B74/(1+i)74 + B75/(1+i)75

As B1 = B2 = ...................= B74 = B75 = B

=> PVB = B/(1+i)1 + B/(1+i)2 + ..................+ B/(1+i)74 + B/(1+i)75

=> PVB = B [ 1/(1+i)1 + 1/(1+i)2 + ..................+ 1/(1+i)74 + 1/(1+i)75]

The term in bracket is GP with common ratio,r = 1/(1+i)

=> sum of GP = [1/(1+i)] / [1 - 1/(1+i)] = 1/i

=> PVB = B * 1/i

=> PVB = B * 1/0.06

=> PVB = B * 16.67

For alternative (a):

PVBa = 2107656 * 16.67 = 35134625.52

PVCa = 13771609/(1+0.06)0 =13771609/1 = 13771609

B/C = 35134625.52 / 13771609 = 2.5512

=> (B/C)a = 2.55

For alternative (b):

PVBb = 3167602 * 16.67 = 52803925.34

PVCb = 26315860/(1+0.06)0 =26315860/1 = 26315860

B/C = 52803925.34 / 26315860 = 2.0065

=> (B/C)b = 2.01

Both alternatives have (B/C) ratio >1 => B> C & therefore both are feasible & acceptable projects.

Therefore we must do incemental analysis to find which project can be choosen.

Alternative (a) has lower cost and (B/C)a > 1 & hence it must be choosen.

Because (B/C)b is also greater than one therefore we must do an icremental analysis in which we find the increment in B/C ratio if we choose alternative (b) instead of alternative (a). If the incremental (B/C)b-a ratio > 1 then we accept alternative (b) even with higher cost and reject alternative (a). If the incremental (B/C)b-a ratio< 1 then we reject alternative (b) and accept alternative (a).

Incremental (B/C)b-a = [PVBb - PVBa] / [PVCb - PVCa]

= [52803925.34 - 35134625.52] / [26315860 - 13771609]

= 17669299.82 / 12544251

= 1.40855 > 1

Hence, opt alternative (b) and reject alternative (a).

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