Question

An industrial grade Farm wants to permanently expand its production and increase profits by an estimated 15,000 $/yr for the next 20 yrs. The expansion would need additional irrigation over the expected 20 years life of the expansion, therefore additional water consumption. The Farm has two options for increasing its water consumption: bring the needed water from a nearby reservoir, or dig a well on its property; each alternative has its own characteristics:

1. Bring water from a nearby reservoir: the farm would have to install a pipeline from the reservoir to the field, however the reservoir is at an higher elevation so the water would flow to the field by natural gravity without the need of a pump. The installation would be simple and would not require extensive maintenance. The farm will have to pay the Reservoir’s Consortium for the required water.
2. Drill a well: the well will need a pump to deliver water to the field, the system will require regular maintenance, and the pump will have to be replaced every ten years. The well would be on Farm’s property and the extracted water would be at no cost.

The Farm will have to borrow the necessary capital from a Lending Institution; however, farm’s management does not know what interest the Institution will charge. In order to prepare for negotiating the interest rate, the Farm assumes the following forecast of parameters for the two alternatives:

                        Installation Cost           O&M/yr           Pump Replacement      Water Cost/yr Life

Alternative A                25,000              1,000                N/A                              4,000              20       

Alternative B                35,000              3,000                5,000/10 yrs                 N/A                 20

Identify the interest ranges where the farm should go for Alternative A, for Alternative B, which interest rate would make the two alternatives equivalent, and at which interest rate none of the two alternatives are economically feasible

Answer 1

Alternative A;

Installation cost = 25000

O&M Per year = 1,000

Water Cost /yr = 4000

Increase profit estimate = 15000

So monthly inflow = 15000- 1000 - 4000 = 10,000 for the next 20 years.

Let the rate of interest be x

So PV of Alternative A = -25000 + 10000/(1+x/100)^1 + 10000/(1+x/100)^2 + ...  + 10000/(1+x/100)^20

= -25000 + 10000/(1+x/100) (1 + 1/(1+x/100)^1 + 1/(1+x/100)^2 + ... + 1/(1+x/100)^19)

= -25000 + 10000/(1+x/100) * (1/(1+x/100)^20 - 1) / (1/(1+x/100) - 1)

Alternative B:

Installation cost = 35000

O&M Per year = 3,000

Increase profit estimate = 15000

So monthly inflow = 15000- 3000 = 12,000 for the next 20 years.

Pump replacement = 5000 / 10 years

Let the rate of interest be x

So PV of Alternative B = -35000 + 12000/(1+x/100)^1 + 12000/(1+x/100)^2 + ...  + 12000/(1+x/100)^20 - 5000*(1 /(1+x/100)^10)

= -35000 + 12000/(1+x/100) (1 + 1/(1+x/100)^1 + 1/(1+x/100)^2 + ... + 1/(1+x/100)^19) - 5000*(1 /(1+x/100)^10)

= -35000 + 12000/(1+x/100) * (1/(1+x/100)^20 - 1) / (1/(1+x/100) - 1) - 5000*(1 /(1+x/100)^10)

Let 1/(1+x/100) = y

If Alternative A is to be selected, then

-25000 + 10000y * (y^20 - 1) / (y - 1) > -35000 + 12000y * (y^20 - 1) / (y - 1) - 5000*(y^10)

or 2000y(y^20-1) / (y-1) - 5000 * y^10 < 10000

or 2y(y^20 - 1) / (y - 1) - 5y^10 < 10

Solving, we get y<0.85

or 1/(1+x/100)<0.85

or 1+x/100 > 1.176

or x>17.6%

If Alternative B is to be selected,

-25000 + 10000y * (y^20 - 1) / (y - 1) < -35000 + 12000y * (y^20 - 1) / (y - 1) - 5000*(y^10)

or 2000y(y^20-1) / (y-1) - 5000 * y^10 > 10000

or 2y(y^20 - 1) / (y - 1) - 5y^10 > 10

Solving, we get y>0.85

or 1/(1+x/100)>0.85

or 1+x/100 < 1.176

or x<17.6%

If neither of the option has to be feasible,

-25000 + 10000y * (y^20 - 1) / (y - 1) <0 and -35000 + 12000y * (y^20 - 1) / (y - 1) - 5000*(y^10)<0

solving, we get y<0.714 and y<0.7466

So at y<0.714, the projects are not feasible

or 1/(1+x/100) < 0.714

or 1+x/100 > 1.4

or x>40%

So Alternative A can be selected when the interest rate is between 17.6% and 40%

Alternative B can be selected when interest rate is less than 17.6%

Both the alternatives are equal at interest rate of 17.6%

Neither of the projects will be feasible at interest rates above 40%.

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