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Math Help - Deferred annuity problem

  1. #1
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    Deferred annuity problem

    Hi,

    The problem i'm trying to figure out says that a debt of $60,000 will be repaid over 6 years by making payments of $4000 quarterly for the first 3 years, followed by a year where no payments are made, then quarterly payments again in the final two years. I have to find the size of the payments for the final two years at an annual rate of 12%.

    Here's what i tried that gave me the wrong answer(s):

    S = 4000 ( (1 + 0.03)^12 - 1 / 0.03)
    S = 56,768.12

    56,768.12 = R ( 1 - (1 + 0.03)^-12 - 1 / 0.03) * (1.03) * (1.03)^-4


    If anyone can spot it from that let me know - thanks in advance
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  2. #2
    Member jonah's Avatar
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    You need to get used to equations of value. Make a time diagram in the form of line graph from 0 to 24 (6 years times 4 quarters). Mark the point zero as the beginning of the 1st year and use that as the comparison date (you could of course use other points from zero to 24 as your comparison date). On one side of an equation write 60,000 as is. One the other side of that equation, write an expression for the present value of the 12 quarterly payments of $4,000 plus an expression for the present value of the 8 unknown R quarterly payments at the beginning of the 5th year (which is the point 16 on your time diagram) which is to be discounted for 4 years (at point 0).

    <br />
60,000 = 4000\frac{{1 - \left( {1 + {\textstyle{{0.12} \over 4}}} \right)^{ - \left( {3 \times 4} \right)} }}{{{\textstyle{{0.12} \over 4}}}} + R\frac{{1 - \left( {1 + {\textstyle{{0.12} \over 4}}} \right)^{ - \left( {2 \times 4} \right)} }}{{{\textstyle{{0.12} \over 4}}}}\left( {1 + {\textstyle{{0.12} \over 4}}} \right)^{ - \left( {4 \times 4} \right)} <br />
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  3. #3
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    OK, if i could just ask a quick question, and sorry if it's beyond stupid but - when do you know to multiply by that last factor (where you discount 4 years) and why 4 years?
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  4. #4
    Member jonah's Avatar
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    To paraphrase from two excellent authors, Dr. Zima and Dr. Brown, the most efficient way to solve an annuity problem is to make a time diagram, determine the type of annuity, and then apply the proper formula(s). If you had bothered to read my post carefully and if you had bothered to draw that time diagram like I suggested, you’d have a clearer picture of why I did what I had to do. In one of your earlier posts, I saw that TKHunny gave you pretty much the same solution that I did. Like I said earlier, you need first to get used to (and work out) equations of value before you tackle annuity problems. Since the point zero was used as the comparison date, surely it should be clear to you that the value of the debt at that point of time is $60,000. This debt is then equated with the value of the first 12 $4,000 quarterly payments at the point zero by use of the standard end of period amortization formula; the same goes for the next 8 unknown quarterly payments – find an expression for the value of the next 8 unknown quarterly payments (also) at the point zero. As you can see from my post, this was first accomplished by use of the standard end of period amortization formula at the point 16 (which is the end of the 4th year or the beginning of the 5th year depending on how you look at it). Since this valuation expression is for the point 16 on the line graph (or the end of year 4) and since the point or date of comparison is at the point zero, you need to discount this valuation expression for 4 years, back to the point zero.
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