Question

5-71 consider the following three alternatives. there is also a \do nothing\ alternative.

at the end of the 5 - year useful life of a, a replacement is made. if a 10 - year analysis period and a 8% interest rate are selected, which is the preferred alternative?

Explanation:

Step1: Calculate the present - worth of Alternative A

For Alternative A, the cost is $P_{A1}=50$ and after 5 years, another cost of $P_{A2}=50$ is incurred. The net - annual benefit is $A = 12$ for 10 years.
The present - worth formula for a single payment is $P = F(P/F,i,n)$ and for an annuity is $P=A(P/A,i,n)$.
$P_{A}=50 + 50(P/F,8\%,5)-12(P/A,8\%,10)$
Using the compound - interest factor tables: $(P/F,8\%,5)=\frac{1}{(1 + 0.08)^{5}}\approx0.6806$ and $(P/A,8\%,10)=\frac{(1 + 0.08)^{10}-1}{0.08(1 + 0.08)^{10}}\approx6.7101$
$P_{A}=50+50\times0.6806 - 12\times6.7101$
$P_{A}=50 + 34.03-80.5212$
$P_{A}=3.5088$

Step2: Calculate the present - worth of Alternative B

The cost of Alternative B is $P_{B}=30$, and the net - annual benefit $A = 4.5$ for 10 years.
$P_{B}=30-4.5(P/A,8\%,10)$
$P_{B}=30 - 4.5\times6.7101$
$P_{B}=30-30.19545=- 0.19545$

Step3: Calculate the present - worth of Alternative C

The cost of Alternative C is $P_{C}=40$, and the net - annual benefit $A = 6$ for 10 years.
$P_{C}=40-6(P/A,8\%,10)$
$P_{C}=40-6\times6.7101$
$P_{C}=40 - 40.2606=-0.2606$

Step4: Analyze the results

The "do nothing" alternative has a present - worth of $P = 0$.
Since $P_{A}=3.5088>0$, $P_{B}=-0.19545<0$, $P_{C}=-0.2606<0$, the preferred alternative is the one with the highest present - worth.

Answer:

Alternative A