Question

Three sites (A, B & C) are being considered for flood control dams. The three construction costs are $10 million, $12 million and $20 million; and the annual maintenance costs are expected to be $15,000, $20,000 and $23,000 respectively for sites A, B and C. In addition, a $75,000 refurbishing expenditure will be required every 10 years at each site. Every year without the dams, $2 million in flood damage occurs. It is projected that dam A, by itself, would reduce the annual damage to $1.6 million. Similarly, Dam B would reduce the damage to $1.2 million, and Dam C to $0.77 million. Since the dams would be built on different branches of a large river, any one or all of the dams could be constructed, and the flood damage reductions would be additive. The planning interest rate is 5% per year, and the dams would be considered permanent.

a. Are the dam options Mutually Exclusive or Independent alternatives?
b. Determine which dams, if any, should be built on the basis of their Standard
Benefit Cost Ratios.

Answer 1

A.

Two or more events are considered to be mutually exclusive, when their occurrence cannot happen together i.e.Occurrence of one event will result in the non-occurrence of the other event/s. Similarily two or more events are considered as independent, when the occurrence of one event does not affect happening of other event/s.  

In the given case study since dams would be built on different branches of a large river and any one or all of the dams could be constructed, construction of dams would be considered as independent event. That is construction of one or any dam is not affecting construction of rest of the dams, so they are independent.

B.

Standard benefit cost ratio formula = Discounted value of Benefits / Discounted value of costs

Projects with a benefit-cost ratio greater than 1 have greater benefits than costs; hence they have positive net benefits. So we will calculate standard cost benefit ratio of each dam sites individually and collectively and accordingly will take the decision.

Taking the values from the question:

Sites		A	B	C	Total
Initial cost		$ 1,000,000.00	$ 1,200,000.00	$      2,000,000.00	$ 4,200,000.00
Annual recurring cost		$        15,000.00	$        20,000.00	$            23,000.00	$        58,000.00
Cost every 10th Yr		$        75,000.00	$        75,000.00	$            75,000.00	$      225,000.00
Benefits Every year
Expected Damage without Dams	I	$ 2,000,000.00	$ 2,000,000.00	$      2,000,000.00	$ 2,000,000.00
Expected Damage after Dam Construction	II	$ 1,600,000.00	$ 1,200,000.00	$          770,000.00
Annual Benefit of Dam (Max ristricted to $ 2mn	I-II	$      400,000.00	$      800,000.00	$      1,230,000.00	$ 2,000,000.00

Now we will calculate discounted value of cost and benefits for site A
		n
Formula of PV of annuity is	P-P(1/1+r)
	(1+r)-1
Where
P= Periodic value
r=rate per period
n= no of periods

PV of annual maintenance cost =[15000-15000(1/1.05) to power 10]/[1.05-1] = $115820

and PV of 10th year cost =P/(1+r)power n i.e. 75000/1.05 to power10 = $46044

Similarily PV of annual benfits would be [400000-400000(1/1.05) to power10]/[1.05-1] = $3088800

Similarily we will compute for site B and C and values arrived would be as under:

Sites		A	B	C	Total
Initial cost PV	1	$ 1,000,000.00	$ 1,200,000.00	$      2,000,000.00	$    4,200,000.00
Annual recurring cost PV	2	$      115,820.00	$      154,440.00	$          177,606.00	$        447,866.00
Cost every 10th Yr PV	3	$        46,044.00	$        46,044.00	$            46,044.00	$        138,132.00
Total Discounted Cost	X= 1+2+3	$ 1,161,864.00	$ 1,400,484.00	$      2,223,650.00	$    4,785,998.00
Total Discounted Benefits of Benefits Every year PV Max restricted to PV of $2mn	y	$ 3,088,800.00	$ 6,177,600.00	$      9,498,060.00	$ 15,444,000.00
Cost Benefit Ratio	Y/X	2.658486708	4.411046467	4.271382637	3.226913175

In view of the above, since cost benefit ratio is greater than one for all sites,Dams can be constructed on all sites i.e. A, B & C.

Assumption: Period of 10 year has been considered for cost benefit ratio appraisal.