[Anonymous]

Pole Co. is investing in a machine with a 3-year life. The machine is expected to reduce annual cash operating costs by $30,000 in each of the first two years and by $20,000 in Year 3. Present values of an annuity of $1 at 14% are:

Period (1) 0.8772

Period (2) 1.6467

Period (3) 2.3216

Using a 14% cost of capital, what is the present value of these future savings?

A. $59,600

B. $60,800

C. $62,900

D. $69,500

This is the explanation: (my actual question is below)

Present value of Year 1

and 2 savings of $30,000 = Savings x Annuity factor for 2 years

= $30,000 x 1.6467

= $49,401

Present value of Year 3

savings of $20,000 = Savings x Difference between second- and

third-year annuity factor

= $20,000 x (2.3216 -1.6467)

= $20,000 x 0.6749

= $13,498

Present value of

Years 1-3 savings = Present value of Year 1 and 2 savings +

Present value of Year 3 savings

= $49,401 + $13,498

= $62,899, or $62,900 rounded

This may be dumb but I do not understand why the factors get substracted. Can anyone explain the logic behind subtracting the 2nd and 3rd year factors? I want to make sure that I know when I have to subtract them when I see other scenarios.

[Biff-1955-Tannen]

That's a weird way of doing it… I would have just done the PV of an annuity of $10,000 for 2 years + PV of an annuity of $20,000 for 3 years

What question is this from? Becker? Wiley?

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[Anonymous]

The question is from Ninja.

[NebCPA]

The reason for the subtraction is that an annuity is an annual payment, so a three year annuity is 3 payments, not one

If you simply took the PV of the 3 year annuity amount (2.3216) and multiplied it by the $20,000 savings in year 3, then you are saying that you have (3) $20,000 savings amounts – not one. You are using an annuity amount – a 3 year annuity is saving you are getting an amount each year for 3 years.

Where people will get tripped up is multiplying the given annuity rates by the one-time savings. The actual savings are (below) so you will want to use three PV of a lump sum numbers, not three PV of annuity numbers:

Year 1 – $30,000
Year 2 – $30,000
Year 3 – $20,000

If you do not change the PV annuity numbers to represent the PV of a single time period (ie, you simply multiply the given PV of the annuity numbers by the one-time savings given), then this is what you are saying the savings are :

Year 1 – $30,000 $30,000 $20,000
Year 2 – $30,000 $20,000
Year 3 – $20,000

But this is incorrect.

What the question is really trying to teach is that you can manipulate PV calculations pretty easily. As Bill stated above, you can manipulate the dollar amounts too. This is how his cash flow looks (he is assuming (1) 3-year annuity and (1) 2-year annuity:

Year 1 – $20,000 $10,000
Year 2 – $20,000 $10,000
Year 3 – $20,000

The take away here is not that you need to learn when to subtract or not subtract, add or not add PV numbers. The take away from this question is you need to understand what the annuity numbers you are given mean (ie, and annuity is an annual payment, and lump sum is a one-time payment) and how they can be manipulated (ie, you can add/subtract PV amounts, dollar amounts, etc.).

[Anonymous]

Thanks NebCPA. It totally makes sense.

[Author]

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