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Math Help - Annual Yield of bond

  1. #1
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    Annual Yield of bond

    A US government bond matures in 10 years. Its quoted price is now 97.8, which means the buyer will pay $97.80 per $100 of the bond's face value. The bond pays 5% interest on its face value each year. If $10,000 (the face value) worth of these bonds is purchased now, what is the yield to the investor who holds the bonds for 10 years?

    This is from an engineering economics course.

    Here's the equation I came up with:

    9780 = 10000 (P/F, i'%, 10) + 10000(.05)(P/A,i'%, 10)

    I'm really not sure what to do, so I went with the guess and check method. I eventually figured out that i'% is going to be in between 5% and 6%. But after that, I don't know what to do.

    5%: 10000(.6139) + 10000(.05)(7.7217) = 9999.85
    6%: 10000(.5584) + 10000(.05)(7.3601) = 9264.05
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  2. #2
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    Did your book mention that Coupons normally are ever half-year? This would make them, $250 every six months, rather than $500 every year.

    I think you're close. Try:

    j' = (1+i')^{1/2} - 1

    9780 = 10000 (P/F, i'%, 10) + 10000(.05)(P/A,j'%, 20)

    Or, you can simplify your life a little doing it this way.

    9780 = 10000 (P/F, j'%, 20) + 10000(.05)(P/A,j'%, 20)

    Make sure that (P/A...) calculation is set to "End of Period".

    I get,

    i' = 0.05 ==> 10047.67 > 9780
    i' = 0.06 ==> 9318.39 < 9780

    This is the tricky part. Solving for an internal rate cannot be done simply. It is an interative process. Unless your calculator has an IRR button, you must now guess at a better value. A typical way is linear interpolation.

    10047.67-9780 = 267.67 <== Distance from 5% to where we want to be.
    10047.67-9318.39 = 729.28 <== Distance from 5% to 6%
    267.67/729.28 = 0.367033238 <== Fraction to next guess. Use as many decimal places as you like. The process should correct itself.

    i' = 0.05 + 0.367*(0.06-0.05) = 0.05367 ==> 9771.82 < 9780

    See, we improved it quite a bit. Do it again. This time, throw out the 6% and use the 5.367%. Keep the desired answer between the two values.

    10047.67-9780 = 267.67 <== Distance from 5% to where we want to be.
    10047.67-9771.82 = 275.85 <== Distance from 5% to 5.367%
    267.67/275.85 = 0.970346203 <== Fraction to next guess.

    i' = 0.05 + 0.97*(0.05367-0.05) = 0.05356 ==> 9779.95 < 9780

    That may be close enough, or you can just do it all day long. What fun!

    If your calcualtor will do it for you, you should notice that it takes a while. It is not an immediate response like most calculator functions.

    Anyway, I get 5.535593% doing it the way I proposed. If the coupons are REALLY annual, then I get 5.28892%. It does make a little difference - those nasty coupons.
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  3. #3
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    The quoted price is less than the bond’s face value. Thus, the bond was bought at a discount and the investor’s yield rate is greater than the bond rate of 5%. An approximation to this yield rate is

    <br />
j = \frac{{\tfrac{{10 \times 10,000 \times .05 + \left( {10,000 - 9,780} \right)}}<br />
{{10}}}}<br />
{{\tfrac{{10,000 + 9,780}}<br />
{2}}}<br />

    If you want a closer approximation of this yield rate, either use a CAS like MAPLE or Mathematica, or your calculator’s built in solve function to solve for j in the following “general method” for computing the value of a bond on an interest date:
    <br />
9,780 = 10,000\left( {1 + \tfrac{j}<br />
{1}} \right)^{ - \left( {10 \times 1} \right)}  + \left( {10,000 \times \tfrac{{.05}}<br />
{1}} \right) \cdot \frac{{1 - \left( {1 + \tfrac{j}<br />
{1}} \right)^{ - \left( {10 \times 1} \right)} }}<br />
{{\tfrac{j}<br />
{1}}}<br />
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  4. #4
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    Quote Originally Posted by TKHunny View Post
    Did your book mention that Coupons normally are ever half-year? This would make them, $250 every six months, rather than $500 every year.
    No, my book didn't mention them being every half-year. In fact, the book spent only about three paragraphs and two examples on bonds. Should this be implied, or will they include it in the question?

    But thank you showing me the interpolation process. I understand it now.
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  5. #5
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    Quote Originally Posted by tn2k7 View Post
    Should this be implied, or will they include it in the question?
    Tough call. U.S. Bonds are almost always semi-annual. Other places? Not so much. If it doesn't specify, I'd read the problem very carefully and indicate very clearly what I had construed from it. Personally, I'd do it a couple of ways to see how much difference it makes, but not many can do that under exam pressure.
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