Question

Suppose that you are again working for your state government but that instead of working on health and human services issues, you are running the highway department. Your state turnpike is in poor shape, with large potholes and crumbling shoulders that slow down traffic and pose an accident risk. You have been charged by the governor with the task of considering whether the state should invest in repairing this road assuming it will last for 100 years.

The data for the project is indicated in the Table below. Note that all values are in nominal dollar terms (actual market values).

Benefit and Cost:                        

Making improvements will require the following inputs:

1. Initial cost: 1 million bags of asphalt, $100/bag;
2. Initial cost: 1 million hours of construction labor (500 workers for 2,000 hours each), at $20/hour
3. $10 million per year in the future for maintenance costs

There are two main benefits to these road improvements:

1. Driving time for producers (trucks) and consumers will be reduced by 500,000 hours per year. Hourly value is $19/hour.
2. The road will be safer, resulting in five fewer fatalities per year. The estimated life value (using wages) is $8.7 million/life.

Question 1: What is the net present value of the investment at 5%, 7%, and 10%? Should the project be undertaken?

Question 2: Now assume that the maintenance cost is increasing at the rate of 5% per year in nominal terms. Hourly value of the reduced driving time is rising at 2% per year in nominal terms. Life values increase by 1% per year in nominal terms. The inflation rate is 2% per annum. All annual values accrue at the end of each year. Assume the nominal discount rate is 7%.

1. What is the real discount rate?
2. What is the Net Annual Worth (annualized net benefit) of this project?
3. What is the net present value? Should the project be undertaken?

Answer 1

1. Initial cost = 1 million bags* $100/bag+ 1 million hours * $20/hour = $120 million

Annual benfit = 500000 hours* $19/hour + $8.7 million*5 - $10 million = $43 million

NPV (5%) = -120+43/0.05*(1-1/1.05^100) = $733.46 million

NPV (7%) = -120+43/0.07*(1-1/1.07^100) = $493.58 million

NPV (5%) = -120+43/0.05*(1-1/1.05^100) = $309.97 million

As the NPV is positive at all discount rates, the project should be undertaken

2.

a) Real discount rate = (1+nominal rate)/(1+inflation rate) -1 = 1.07/1.02-1 = 0.04902 or 4.902%

b) Present value of maintenance costs = 10/1.07+10*1.05/1.07^2+... +10*1.05^99/1.07^100

= 10/1.07 * (1-(1.05/1.07)^100)/(1-1.05/1.07) = $424.2257 million

Driving time benefit in 1st year = 500000 hours* $19/hour = $9.5 million

Present values of benefit due to driving time = 9.5/1.07+9.5*1.02/1.07^2+....+9.5*1.02^99/1.07^100

=9.5/1.07*(1-(1.02/1.07)^100)/(1-1.02/1.07)

= $188.4137 million

Life benefit in 1st year = $8.7 million/life * 5 lives = $43.5 million

Present values of benefit due to driving time = 43.5/1.07+43.5*1.01/1.07^2+....+43.5*1.01^99/1.07^100

=43.5/1.07*(1-(1.01/1.07)^100)/(1-1.01/1.07)

= $722.7401 million

So, NPV = -120-424.2257+188.4137+722.7401 = $366.9281 million

Net Annual Worth (A) is given by

A/0.07*(1-1/1.07^100) = 366.9281

=> 14.26925*A = 366.9281

=> A =25.7146

So Annual Worth is $25.7146 Million

c) NPV of the project is $366.9281 million as calculated above.

As NPV is positive , the project should be undertaken