A new SouthStar rail line is proposed. This project is expected to reduce travel time for 2,000 commuters by 30 minutes per day, in its completion year. The line only operates on weekdays (Monday-Friday). It will take 2 years to build and cost $320,000,000 total (Net Present Value). If the interest rate is 3 percent, above what value of time must average value of time for SouthStar passengers be in order for the benefit/cost ratio to exceed 1.
Assume a 30 year lifespan. The interest rate is annual*
Benefits considered are only Travel Time Savings.
Travel Time Savings = 2,000 Commuters x 0.5 Hours/day (30 minutes/day)
Travel Time Savings = 1000 Commuters-Hours/day x 5 days/week 52 weeks/year
Travel Time Savings = 260,000 Commuters-Hours/year
Travel Time Savings start after two years (given in the problem statement). Other assumptions are: Present Year is 0, no growth (constant commuters ev- ery year, and thus constant savings), and constant Value of Travel Time (VOT). Therefore, Benefits must be discounted to Present value for each year and added for a total during the lifespan of the project considered (30 years).
Adding up all Present Value of Travel Time
Present Value of Travel Time = (VOT )(260,000)(1/[1+0.03]^2 + 1/[1+0.03]^3 + ... + 1/[1+0.03]^29 + 1/[1+0.03]^30)
You can sum it up in an excel spreadsheet or recognize that this is a geometric series.
The sum is 18.63 inside the parentheses.
Total Present Value of Travel Time = (4,843,687.57)(VOT)
Costs are given by Total Present Value of $320,000,000.