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Why does this economical method work? Net annual cost comparison for two alternatives

  1. Apr 23, 2012 #1
    I've been searching everywhere for an answer to this question but haven't been able to get one.

    Let's say we're coming two things at 12% compounded annually.

    A's investment, salvage, life, and expense/year are given as:

    $50,000 ; $10,000 ; 11 ; $5000

    While B's are:

    $40,000 ; $0 ; 10 ; $2000

    If I do 50000(A/P, 0.12, 11) - 10000(A/F, 0.12, 11) + 5000 for A and 40000(A/P, 0.12, 10) + 2000 for B, can someone explain why this comparison "works?" For this problem in particular, we end up with the values $12000.94/year for A and $9000.08/year for B, so obviously B is better, but exactly what method is being used? I've tried researching "net annual cost" for a similar method, but haven't been able to find anything useful.

    Any help?
     
  2. jcsd
  3. Apr 23, 2012 #2

    HallsofIvy

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    Re: Why does this economical method work? Net annual cost comparison for two alternat

    Not if you don't tell us what "50000(A/P, 0.12, 11)" means! I'm not even sure what "A/P" , means. It clearly does not have to do with "investment A" because you use it in the calculation for B also.

    What I would do is this- A has a total cost of $50000, but we can recover $10000 so it really costs $40000, and lasts 11 years so we can prorate it at 40000/11= $3636.36 per year. It also has a "cost per year" of 5000 so it costs us $8636.36 per year.

    B has a total cost of $40000 and we can recover nothing and lasts 10 years so we can prorate it at 40000/10= $4000 per year. It also has a "cost per year" of $2000 per year so it costs $6000 per year. It may be that you are doing a projected cost of not banking the money but that would require an assumed interest rate which is not given.
     
  4. Apr 23, 2012 #3
    Re: Why does this economical method work? Net annual cost comparison for two alternat

    He is using conversion factors that convert certain types of money to other types. The A/P converts a present value of money (i.e. $200 dollars today) to an equivalent annual rate at 12% per unit time over 11 units of time. The A/F does the same thing, except it does the conversion, using a future sum of money (you salvage the product at $10,000 in the future. Since there is an interest rate of 12%, you are effectively losing money by salvaging it then as opposed to now).
     
  5. Apr 23, 2012 #4

    Ray Vickson

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    Re: Why does this economical method work? Net annual cost comparison for two alternat

    Why do you add and subtract the way you do? Measuring in $000s, the cash outflow stream for A is (50,10,10,10,10,10,10,10,10,10,10,10-5) [we invest 50, then also spend 10 per year for 11 years, but get back 5 at the end of year 11]. The cash outflow stream for B is (40,0,0,0,0,0,0,0,0,0,0-2). You can compute the net present value (NPV) of each stream and compare them (assuming that in A the first payment, 50, occurs at time t = 0, the next payment, 10, at t = 1 year, etc.) I get NPVs of
    [tex] \text{NPV}(A) = 50 + \sum_{n=1}^{11} \frac{10}{1.12^n} - \frac{5}{1.12^{11}} = 107.940\; (\$000),[/tex]
    and
    [tex] \text{NPV}(B) = 40 - \frac{2}{1.12^{10}} = 39.356 \; (\$000). [/tex]

    You could, of course, convert these to equivalent annual amounts, but why bother?

    RGV
     
  6. Apr 23, 2012 #5
    Re: Why does this economical method work? Net annual cost comparison for two alternat

    It works by converting the present money and future money into an equivalent annual rate, given i = 12% per term and there are 11 terms. You then sum up all of the annuities to find the total annual loss (which is positive here) for each option.
     
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