Question

1. A highway department is considering building a temporary bridge to cut travel time during the three years it will take to build a permanent bridge. The temporary bridge can be put up in a few weeks at a cost of $740,000. At the end of three years, it would be removed, and the steel would be sold for scrap. The real net costs of this would be $81,000. Based on estimated time savings and wage rates, fuel savings, and reductions in risks of accidents, department analysts predict that the benefits in real dollars would be $275,000 during the first year, $295,000 during the second year, and $315,000 during the third year. Departmental regulations require use of a real discount rate of 4 percent.

   1. Calculate the present value of net benefits assuming that the benefits are realized at the end of each of the three years.

   2. Calculate the present value of net benefits assuming that the benefits are realized at the beginning of each of the three years.

   3. Calculate the present value of net benefits assuming that the benefits are realized in the middle of each of the three years.

   4. Calculate the present value of net benefits assuming that half of each year’s benefits are realized at the beginning of the year and the other half at the end of the year.

   5. Does the temporary bridge pass the net benefits test?

Answer 1

Begin by calculating the present value of the costs.This includes the construction cost of the temporary bridge, which occurs at the beginning of year 1, and the net cost of decommissioning the bridge at the end of year 3:

PV(C) = $740,000+$81,000/(1+.04)3

= $740,000+$72,009

= $812,009.

a.

Benefits accrue at the end of the year:

PV(B) = $275,000/(1+.04)1+$295,000/(1+.04)2+$315,000/(1+.04)3

  = $264,423 + $272,744 + $280,034

= $817,201

NPV = $817,201-$812,009 = $5,192

b.

Benefits accrue at the beginning of the year:

PV(B) = $275,000+$295,000/(1+.04)1+$315,000/(1+.04)2

= $275,000 + $283,654 + $291,235

=$849,889

NPV = $849,889-$812,009 = $37,880

c.

Benefits accrue at mid-year:

PV(B) = $275,000/(1+.05).5+$295,000/(1+.05)1.5+$315,000/(1+.05)2.5

= $269,660 + $278,145 + $285,580

= $833,385

NPV = $833,385-$812,009 = $21,376

d.

Benefits split equally between beginning and end of year:

PV(B) = ($817,201 + $849,889)/2 = $833,545

NPV = $833,545-$812,009 = $21,536

Alternatively, average the answers to parts a and b: ($5,192 + $37,880)/2 = $21,536.

e.

Although the NPVs vary depending on when the benefits actually arise, they are all positive, implying that the department should construct a temporary bridge, assuming that is the only alternative to the status quo.