Question

A state highway department is trying to decide whether it should “hot-patch” a short section of an existing country road or resurface it. If the hot-patch method is used, approximately 300 cubic meters of material would be required at a cost of $700 per cubic meter (in place). Additionally, the shoulders will have to be improved at the same time at a cost of $24,000. These improvements will last 2 years, at which time they will have to be redone. The annual cost of routine maintenance on the patched up road would be $5000. Alternatively, the state can resurface the road at a cost of $850,000. The surface will last 10 years if the road is maintained at a cost of $2000 per year beginning 4 years from now. No matter which alternative is selected, the road will be completely rebuilt in 10 years. At an interest rate of 9% per year,

a. make a recommendation on the best alternative based on the annual worth analysis.

b. make a recommendation based on the present worth method

Answer 1

The option1 requires material for 300 cubic meters and the rate is $700 per cubic meter and also needs shoulder improvement.
( 300 * 700 ) + 24000 = 234000
This is valid for 2 years and the end of 2nd year, the same cost will be again incurred.
The annual maintenance is $5000 per year.

Option B has an initial cost of $850000 and annual maintenance cost of $2000 per year.

PV = PMT / (1+Discount Rate)Duration

a) Option 1
PV of year 0 = 234000
PV of year 1 = 5000 / ( 1.09 ) = 4587.16

So we can create a table of this cash flow.

Year	Option 1	PW @ 9%	Option 2	PW @ 9%
0	234000	234000	850000	850000
1	5000	4587.16	2000	1834.86
2	239000	201161.52	2000	1683.36
3	5000	3860.92	2000	1544.37
4	239000	169313.63	2000	1416.85
5	5000	3249.66	2000	1299.86
6	239000	142507.89	2000	1192.53
7	5000	2735.17	2000	1094.07
8	239000	119946.04	2000	1003.73
9	5000	2302.14	2000	920.86
10	5000	2112.05	2000	844.82
NPW		885776.17		862835.32
PW Annuity	Factor	6.4177		6.4177
Annual Worth		138021.7		134446.2

The annual worth can be calculated by dividing the NPW by the PW annuity factor of 9% interest rate and 10 years.
Annual Worth = NPW / PW Annuity Factor

Option 1
885776.17 / 6.4177 = 138021.7

Option 2
862835.32 / 6.4177 = 134446.2

The annual cost of option 2 is lower so it should be selected.

b) The PW of the cash flow can be calculated by discounting the cash flow by the given interest rate.
Option 2 ,
PW of year 0 = 850000
year 1 = ( 2000 / 1.09 ) = 1834.86

NPW of the option 1 = 885776.17

NPW of the option 2 = 862835.32

The NPW of the option 2 is lower so it should be selected.