Question

1. (20 pts) The dome of the state capital building needs to be replaced and two options are being considered which use different materials: • Option 1: Initial cost of replacement is $1,200,000. Every 10 years the dome will need to have a thorough cleaning and sandblasting that will cost $400,000, and that cost will continue every 10 years. • Option 2: Initial cost of replacement is $1,000,000. Every 7 years the dome will need to have a thorough cleaning and sandblasting that will cost $300,000, and that cost will continue every 7 years. Additionally, there will be an annual maintenance cost of $10,000 for minor repairs. The state’s MARR for their economic evaluations is 4%, and they base decisions on a net present worth analysis. The dome of the capital will be there forever. Including all costs for each option, which option will you recommend to the state and why?

Answer 1

Answer:

The most economic option is: Option 1. because the present value of the costs is lower than Option 2.

Explanation:

Hi, first we need to introduce the equation to find the present value of a perpetuity, that is:

$PV=\frac{Annuity}{r}$

Annuity= consist in the amount to be paid periodically (not necessarily every year)

r = The discount rate that coincides with the periodicity of the annuity.

So, there are different periodities in both options. In option 1, every 10 years and forever the dome needs to be sandblast, therfore we need to find the effective equivalent rate to 4% annual (MARR=4%). That is:

$r(e.10-yrs)=(1+r(annual))^{10} -1$

therefore

$r(e.10-yrs)=(1+0.04)^{10} -1=0.4802$

So, the equivalent effective 10 years rate to MARR=4% is 48.02%

Now, the cost of this option is:

$PV=1,200,000+\frac{400,000}{0.4802} =2,032,909.44$

On the other hand, Option 2. has 2 periodicities we need to take into consideration, first is the sandblasting process that need to be performed every 7 years and the annual maintenance cost, so we need to find the effective 7 year rate to bring to present value this anuity (sandblasting cost)

$r(e.7-yrs)=(1+0.04)^{7} -1=0.3159$

o, the equivalent effective 7 years rate to MARR=4% is 31.59%

Now, the cost of Option 2 is:

$PV=1,000,000+\frac{300,000}{0.3159} +\frac{10,000}{0.04} =2,199,572.09$

For all of the above, we can conclude that the best choice is Option 1, because it is cheaper that Option 2.

Best of luck.