(Transcript of audio with descriptions. Transcript includes narrator headings and description headings of the visual content)
(Speaker)
When making investment decisions today, it is often the case that the benefits of these decisions are not realized until some date in the future. The appendix reviewed the concepts of present value and how to make an investment decision when the costs are incurred today while the benefits are realized in the future. This problem will analyze the decision-making process when the costs for a highway project are incurred today with the benefits realized over future years.
(Description)
Suppose that a major city's main thoroughfare, which is also an interstate highway, will be completely closed to traffic for two years, from January 2014 to December 2015, for reconstruction at a cost of 535 million dollars. If the construction company were to keep the highway open for traffic during construction, the highway reconstruction project would take much longer and be more expensive. Suppose that construction would take four years if the highway were kept open, at a total cost of 800 million dollars. The state department of transportation had to make its decision in 2014, one year before the start of construction (so that the first payment was one year away). So the department of transportation had the following choices:
(i) Close the highway during construction, at an annual cost of 267.5 million dollars per year for two years.
(ii) Keep the highway open during construction, at annual cost of 200 million dollars per year for four years.
(Speaker)
For this particular project, the city must decide between closing a major highway for two years at a cost of 535 million dollars, 267.5 million dollars each year, or keep the highway partially open, which reduces the annual costs to 200 million dollars, but the project will take four years to complete. To determine the cost of each scenario, we must calculate the present value of the costs incurred under each plan. In the first scenario, the city incurs a cost of 267.5 million dollars at an interest rate of 10 percent. To calculate the present value, we must take the discounted value of 267.5 million dollars for both years. We will start with the general equation for calculating the present value. Next we want to insert the correct values for scenario A.
(Description)
Suppose the interest rate is 10 percent. Calculate the present value of the costs incurred under each plan. (i) Close the highway during construction, at an annual cost of 267.5 million dollars per year for two years. The present value of plan (i) is present value equals Costs 1 divided by (1 plus i) to the first power plus Costs 2 divided by (1 plus i) to the second power. Present value equals 267.5 million dollars divided by (1 plus 0.10) to the first power plus 267.5 million dollars divided by (1 plus 0.10) to the second power. Present value equals 243.18 million dollars plus 221.07 milion dollars. Present value equals 464.25 million dollars.
(Speaker)
Closing the highway for two years comes at a cost of 267.5 million dollars each year with an interest rate of 10 percent. Calculating each term, we find the present value of 267.5 million dollars at the end of the first year is 243.18 million dollars, and at the end of the second year is 221.07 million dollars. Adding up the present values for each amount, we arrive at a final cost of 464.25 million dollars. This means that today's dollars, the cost of closing the highway is 464.25 million dollars. The second option involves partially closing the highway, which delays the time to complete the project. Under this scenario, the costs are 200 million dollars per year for four years. You can see that we have extended our general present value equation to account for the two additional years. Next, we want to use the correct values for scenario B, which costs 200 million dollars each year at an interest rate of 10 percent.
(Description)
(ii) Keep the highway open during construction, at annual cost of 200 million dollars per year for four years. The present value of plan (ii) is present value equals Costs 1 divided by (1 plus i) to the first power plus Costs 2 divided by (1 plus i) to the second power plus Costs 3 divided by (1 plus i) to the third power plus Costs 4 divided by (1 plus i) to the fourth power. Present value equals 200 million dollars divided by (1 plus 0.10) to the first power plus 200 million dollars divided by (1 plus 0.10) to the second power plus 200 million dollars divided by (1 plus 0.10) to the third power plus 200 million dollars divided by (1 plus 0.10) to the fourth power. Present value equals 181.82 million dollars plus 165.29 milion dollars plus 150.26 million dollars plus 136.60 milion dollars. Present value equals 633.97 million dollars.
(Speaker)
Finally, solving, you will find the present value of partially closing the highway will come at a cost of 633.97 million dollars. Under scenario A, it costs 464.25 million dollars, which is much cheaper.