Question

An area on the Colorado River is subject to periodic flood damage that​ occurs, on the​ average, every two years​ (i.e., probability of flooding in any year is​ 0.5) and results in a $2,200,000 loss. It has been proposed that the river channel be improved by straightening and deepening it at a cost of$2,400,000​, to reduce the annual probability of flooding to 0.364  during a period of 18 years before it would have to be deepened again. This procedure would also involve annual expenditures of $90,000 for minimal maintenance.

            One legislator in the area has proposed that a better solution would be to construct a​ flood-control dam at a cost of $8,000,000​, which would last​ indefinitely, with annual maintenance costs of not over $50,000.

She estimates that this project would reduce the annual probability of flooding to 0.205. In​ addition, this solution would provide a substantial amount of irrigation water that would produce annual revenue of ​$150,000

and recreational​ facilities, which she estimates would be worth at least $45,000 per year to the adjacent populace.

            If the​ state's capital is worth 11​%, perform​ B-C analysis and recommend the proposal to be adopted. Note that one of the two proposals must be​ adopted, as the​ do-nothing alternative will not be accepted by the​ legislature, no matter what the analysis shows. This means that only the incremental analysis between the two proposals need be done.

The​ B-C ratio for the​ "Improve channel" proposal is ______ ​(Round to 2 decimal places​)

The​ B-C ratio for the​ "Construct dam" proposal is ________ ​(Round to 2 decimal places​)

The incremental​ B-C ratio of​ "Construct dam" vs.​ "Improve channel" is _____ ​(Round to 2 decimal places​)

​Therefore, the recommended proposal is to

A.Improve the channel

B.Construct Dam

Answer 1

General Definition of BCR

A benefit-cost ratio (BCR) is an indicator showing the relationship between the relative costs and benefits of a proposed project, expressed in monetary or qualitative terms. If a project has a BCR greater than 1.0, the project is expected to deliver a positive net present value to a firm and its investors.

Calculation of Benefit - Cost Ratio or B-C Analysis

= Present Value of Projects Benefits/ Present Value of Project Cost

Discount rate = 11%

Yearly Loss on account of periodic Floods = 2,200,000*0.5 = $ 1,100,000

Option A - Improve the channel

Life of Channel = 18 Years

Annual Loss Reduction if channel is improved (Cash Inflow) = (0.5 - 0.364) * $1,100,000 = $149,600

Present Value of Annual Loss Reduction at 11% discount rate for 18 years

= 149,600 * Present Value Annuity Factor (11%, 18 Years) = 149,600 * 7.7016 = $1,152,162

Annual Maintenance Cost (Cash Outflow) = $90,000

Present Value of Annual Maintenance at 11% discount rate for 18 years

= 90,000 * Present Value Annuity Factor (11%, 18 Years) = 90,000 * 7.7016 = $693,144

Cost of Project (Cash Outflow) = $2,400,000

B-C Ratio = Benefits from Project/Cost of Project

= 1,152,162 / (693,144 + 2,400,000)

= 1,152,162 / 3,093,144 = 0.37

Option B - Construct Dam

Life of Dam = Indefinite (Perpetual)

Annual Loss Reduction if dam is constructed (Cash Inflow) = (0.5 - 0.205) * $1,100,000 = $324,500

Present Value of Annual Loss Reduction at 11% discount rate for perpetual period

= 324,500 / Discount Rate = 324,500 / 0.11 = $2,950,000

Annual Maintenance Cost (Cash Outflow) = $50,000

Present Value of Annual Maintenance at 11% discount rate for perpetual period

= 50,000 / Discount Rate = 50,000 / 0.11 = $454,545

Annual Revenue from water for Irrigation and Recreational Facilities (Cash Inflow) = (150,000 + 45,000) = $195,000

Present Value of Annual Additional Revenue at 11% discount rate for perpetual period

= 195,000 / Discount Rate = 195,000 / 0.11 = $1,772,727

Cost of Project (Cash Outflow) = $8,000,000

B-C Ratio = Benefits from Project/Cost of Project

= (2,950,000 + 1,772,727) / (8,000,000 + 454,545)

= 4,722,727 / 8,454,545 = 0.56

Decision = Under both the options, BCR is coming out to be less than one which implies that neither of the projects will deliver a positive Net Present Value to legislature.

However, since one of the options has to be adopted, Option B of Constructing the Dam is better compared to improving the channel since the BCR of option B is 0.56 which is greater than 0.37 of Option A of improving the channel.

Incremental BCR of the two projects

Increasing Order

Option B = Benefit = 4,722,727

Cost = 8,454,545

Option A = Benefit = 1,152,162

Cost = 3,093,144

Incremental BCR = Incremental Benefits / Incremental Costs

= (4,722,727 - 1,152,162) / (8,454,545 - 3,093,144)

= 3,570,565 / 5,361,401 = 0.67

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