Present Value Analysis For my exmaple, we have machine A with an initial cost of 15K and O&amp;M of 2k per year for the next 4 years. After 4 years, the O&amp;M rise to $2.7k per year. No salvage value and life span of 20 years. 10% interest. What's the present worth of machine A?

It is easier for me to understand the problem by first stating that all cost out of pocket is negative. Since we do not have any salvage value in this problem, it will all be negative.

Here are my thoughts:

The cost of purchase the machine is out of pocket. 2k per year of O&amp;M for first 4 year, at an uniform rate which is annual. 2.7k after the 4 years. 

My equation came out to : 

Present (A) = -15k - 2k (P/A, 10%, 4) - 2.7k (P/A, 10%, 16)

However at the solution, they added one more variable that I do not understand why.

Present (A) = 15k + 2k (P/A, 10%, 4) + 2.7k (P/A, 10%, 16) * (P/F,10%,4)

Please explain to me why do we need the last part (P/F,10%,4)? to get the present value of machine A. 

thanks
You need the last part because you want the present value. If you were to only use the 16 you‘d get to 4 years in the future, or you‘d be calculating the PV for 4 year operating cost of 2000 and concurrent 16 year cost of 2700. without the last part you‘d basically have two operating costs (one of 2000 and one of 2700) for the next four years and then 1  operating cost of 2700 for the next twelve years.

This is the way you can think of the cost in more granular terms

Purchasing cost: $15k$ no discount because purchased today.

1st four years overhead:
$
\sum^4_t{2'000\over(1+r)^t}
$ 

2nd 16 years of overhead: $ {\sum^{16}_t{2'000\over(1+r)^t}}\over(1+r)^4 $ 

The first part gives you the PV of the 16 periods, but because this would only start 4 years in the future you'd need to take the PV of the PV in a way.

The total of the cash flow is then the following:
$15'000 + \sum^4_t{2'000\over(1+r)^t} +{{\sum^{16}_t{2'000\over(1+r)^t}}\over(1+r)^4}$

Alternatively you could write
$15'000 + \sum^4_t{2'000\over(1+r)^t} +{{\sum^{20}_{t=4}{2'000\over(1+r)^t}}}$

Hope this helps.