A city plans a pipeline to transport water from a distant watershed area to the city. The pipeline will cost $8 million and will have an expected life of 70 years. The city expects to keep the water line in service indefinitely. Compute the capitalized cost, assuming 7% interest.

Question

Accepted Answer

The capitalized cost equation

[latex]P=\frac{A}{i}[/latex]

is simple to apply when there are end-of-period disbursements A. Here we have renewals of the pipeline every 70 years. To compute the capitalized cost, it is necessary to first compute an end-of-period disbursement A that is equivalent to $8 million every 70 years.

The $8 million disbursement at the end of each 70-year period may be resolved into an equivalent A.

[latex]A=F(A/F,i,n)=\$8 million (A/F,7\%,70)[/latex]

[latex]=\$8 million (0.00062)[/latex]

[latex]=\$4960[/latex]

Each 70-year period is identical to this one, and the infinite series is shown in Figure 5–2.

[latex]Capitalized cost P=8 million+\frac{A}{i}=8 million +\frac{4960}{0.07}[/latex]

[latex]=\$8,071,000[/latex]

1)

Instead of solving for an equivalent end-of-period payment A based on a future $8 million disbursement, we could find A, given a present $8 million disbursement.

[latex]A=P(A/P,i,n)=8 million (A/P,7\%,70)[/latex]

[latex]=8 million(0.0706)=\$565,000[/latex]

On this basis, the infinite series is shown in Figure 5–3. Carefully note the difference between this and Figure 5–2. Now:

[latex]Capitalized cost P=\frac{A}{i}=\frac{565,000}{0.07}=\$8,071,000[/latex]

2)

Another way of solving the problem is to assume the interest period is 70 years long and compute an equivalent interest rate for the 70-year period.
Then the capitalized cost may be computed by using Equation 3–8 for m = 70

[latex] ext{Effective annual interest rate } \quad i_a = (1+i)^m -1 [/latex] (3-8)

[latex]i_{70 yr}=(1+i_{1 yr})^{70}-1=(1+0.07)^{70}-1=112.989[/latex]

[latex]Capitalized cost =8 million+\frac{8 million}{112.989}=\$8,070,803[/latex]

Question 5.7: A city plans a pipeline to transport water from a distant wa...

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