Question

8-15 consider four mutually exclusive and a do - nothing alternatives, each having an 10 - year useful life:
first cost uniform annual benefit salvage value
a $1000 125 750
b $800 120 500
c $600 100 250
d $500 125 0
(a) construct a choice table for interest rates from 0% to 100%.
(b) if the minimum attractive rate of return is 8%, which alternative should be selected?

Explanation:

Step1: Calculate the Net - Present - Value (NPV) formula for each alternative

The NPV formula for a project with first cost $P$, uniform annual benefit $A$, salvage value $S$, and interest rate $i$ over $n$ years is $NPV=-P + A(P/A,i,n)+S(P/F,i,n)$, where $(P/A,i,n)=\frac{(1 + i)^n-1}{i(1 + i)^n}$ and $(P/F,i,n)=\frac{1}{(1 + i)^n}$, and $n = 10$ years.

Step2: Calculate NPV for alternative A

For alternative A: $P = 1000$, $A=125$, $S = 750$.
$(P/A,i,10)=\frac{(1 + i)^{10}-1}{i(1 + i)^{10}}$, $(P/F,i,10)=\frac{1}{(1 + i)^{10}}$
$NPV_A=-1000 + 125(P/A,i,10)+750(P/F,i,10)$

Step3: Calculate NPV for alternative B

For alternative B: $P = 800$, $A = 120$, $S=500$
$NPV_B=-800+120(P/A,i,10)+500(P/F,i,10)$

Step4: Calculate NPV for alternative C

For alternative C: $P = 600$, $A = 100$, $S = 250$
$NPV_C=-600+100(P/A,i,10)+250(P/F,i,10)$

Step5: Calculate NPV for alternative D

For alternative D: $P = 500$, $A = 125$, $S = 0$
$NPV_D=-500+125(P/A,i,10)+0(P/F,i,10)=-500 + 125(P/A,i,10)$

Step6: When $i = 8\%$

$(P/A,8\%,10)=\frac{(1 + 0.08)^{10}-1}{0.08(1 + 0.08)^{10}}=\frac{2.158925 - 1}{0.08\times2.158925}=\frac{1.158925}{0.172714}\approx6.7101$
$(P/F,8\%,10)=\frac{1}{(1 + 0.08)^{10}}=\frac{1}{2.158925}\approx0.4632$

$NPV_A=-1000+125\times6.7101 + 750\times0.4632=-1000 + 838.7625+347.4=186.1625$
$NPV_B=-800+120\times6.7101+500\times0.4632=-800 + 805.212+231.6=236.812$
$NPV_C=-600+100\times6.7101+250\times0.4632=-600 + 671.01+115.8=186.81$
$NPV_D=-500+125\times6.7101=-500 + 838.7625 = 338.7625$

Answer:

Alternative D should be selected as it has the highest NPV when the minimum attractive rate of return is $8\%$.