I'm having trouble understanding the last bit of the solution to this problem:
An industrial juicer costs \$45 000. It will be used for five years and then sold to a remarketer for $25 000. If interest is 15 percent, what net yearly savings are needed to justify its purchase?
Solution:
Using the capital recovery formula: $$ A = (P − S)(A | P, i, N) + Si $$ $$= (45 000 − 25 000)(A|P, 0.15, 5) + 25 000(0.5)$$ $$= 20 000(0.29832) + 12 500$$ $$= 18 466.40 $$
I understand the (P-S) part because it represents the total money we have lost and multiplying it by the capital recovery factor will incorporate the effect of the interest rate in our yearly money we need to put aside. But what justifies adding Si ? and why is i=0.5 in that specific case when we are told that i=0.15? I'm really stuck on that.
Thanks.
There is a typo. It has to be $25,000\cdot 0.\color{red}15$. Therefore $Si=3750$.
It follows that $A=20,000\cdot 0.29832+3750\approx 9716.4$
This is the right result. In general an equation can be set up. The future value of the costs has to be equal to the future value of the savings plus the selling price of the machine.
$C_0:$ Present value of the purchase price of the machine
$A:$ Constant savings per year
$S:$ Selling price
$i:$ Interest rate
$q=1+i$
$C_0\cdot q^n=A\cdot \frac{q^n-1}{q-1}+S$
Solving for A
$A=\left( C_0\cdot q^n-S\right) \cdot \frac{q-1}{q^n-1}$
$A=\left( 45,000\cdot 1.15^5-25,000\right) \cdot \frac{1.15-1}{1.15^5-1}$
$A\approx 9716.4$