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Math Help - try and error semi annual bond

  1. #1
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    try and error semi annual bond

    hi how do i work out trial and error method , the ytm of this 4.5 year bond
    for a semi annual coupon bond
    price is 104.5 the face value is 100

    year---discount factor d(t) --------semi an coupon rate z(t) --------ytm
    .6 ----------09999 ----------------------.001700 ----------------- .0017
    1 -----------.9884 ----------------------.002400 -------------------.0027
    1.5 ---------.9766 ----------------------.004200 ------------------.0042
    2 ----------.9696 -------------------- .005400 --------------------. 0053
    2.5 ----------.9500 --------------------.006896 -------------------.0069
    3 ----------- .9456 ---------------------- .008900--------------------.0090
    3.5 ----------.8907 ---------------------- .0104 --------------------.0105
    4 ------------ .8806 ------------------- .0113 -----------------------.0110
    Last edited by matlondon; February 18th 2011 at 03:47 AM.
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  2. #2
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    What's a try (trial?) and error bond?
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  3. #3
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    Quote Originally Posted by Wilmer View Post
    What's a try (trial?) and error bond?
    yep
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  4. #4
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    Please, write in English. What is ytm - yield to maturity?
    Are you trying to work out the implied interest rate from the table of various maturity bonds?
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  5. #5
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    Missing Coupon Rate

    Quote Originally Posted by matlondon View Post
    hi how do i work out trial and error method , the ytm of this 4.5 year bond
    for a semi annual coupon bond
    price is 104.5 the face value is 100

    year---discount factor d(t) --------semi an coupon rate z(t) --------ytm
    .6 ----------09999 ----------------------.001700 ----------------- .0017
    1 -----------.9884 ----------------------.002400 -------------------.0027
    1.5 ---------.9766 ----------------------.004200 ------------------.0042
    2 ----------.9696 -------------------- .005400 --------------------. 0053
    2.5 ----------.9500 --------------------.006896 -------------------.0069
    3 ----------- .9456 ---------------------- .008900--------------------.0090
    3.5 ----------.8907 ---------------------- .0104 --------------------.0105
    4 ------------ .8806 ------------------- .0113 -----------------------.0110
    For me to find YTM with trial and error you need to provide the coupon rate, I used a 5% coupon rate and plugged the values in the online YTM Calculation tool at YTM|Yield till Maturity calculation - ThinkandDone.com that provided the following yield calculations

    Bond Price at Lower YTM of 1.41%

    Compounding = semi-annual
    Par Value = 100
    Market Price = 104.50
    Coupon Rate = 5%
    YTM Low = 1.41%
    N = 4.5

    Non Zero Bond Price Formula
    Coupon Rate x Par Value x PVIFA(ytml%, n) + Par Value x PVIF(ytml%, n)

    PVIFA Formula
    PVIFA(ytml%, n) = [1 - v] / ytml%
    v = 1 / (1 + ytml%)^n
    PVIFA(ytml%, n) = [1 - { 1 / (1 + ytml%)^n }] / ytml%

    PVIFA Calculation
    v = 1 / (1+0.014)^4.5
    v = 0.93935381283725
    PVIFA(0.014, 4.5) = [1 - 0.93935381283725] / 0.014
    PVIFA(0.014, 4.5) = 0.060646187162752 / 0.014
    PVIFA(0.014, 4.5) = 4.3318705116251

    PVIF Formula
    PVIF(ytml%, n) = 1 / (1 + ytml%)^n

    PVIF Calculation
    PVIF(0.014, 4.5) = 1 / (1+0.014)^4.5
    PVIF(0.014, 4.5) = 1 / 1.0645616021716
    PVIF(0.014, 4.5) = 0.93935381283725

    Non Zero Bond Price Calculation
    Price = 0.05 x 100 x 4.3318705116251 + 100 x 0.93935381283725
    Price = 21.659352558126 + 93.935381283725
    Price @ 0.014 = 115.59

    Bond Price at Upper YTM of 6.41%

    Compounding = semi-annual
    Par Value = 100
    Market Price = 104.50
    Coupon Rate = 5%
    YTM High = 6.41%
    N = 4.5

    Non Zero Bond Price Formula
    Coupon Rate x Par Value x PVIFA(ytmh%, n) + Par Value x PVIF(ytmh%, n)

    PVIFA Formula
    PVIFA(ytmh%, n) = [1 - v] / ytmh%
    v = 1 / (1 + ytmh%)^n
    PVIFA(ytmh%, n) = [1 - { 1 / (1 + ytmh%)^n }] / ytmh%

    PVIFA Calculation
    v = 1 / (1+0.064)^4.5
    v = 0.75641942691628
    PVIFA(0.064, 4.5) = [1 - 0.75641942691628] / 0.064
    PVIFA(0.064, 4.5) = 0.24358057308372 / 0.064
    PVIFA(0.064, 4.5) = 3.8059464544331

    PVIF Formula
    PVIF(ytmh%, n) = 1 / (1 + ytmh%)^n

    PVIF Calculation
    PVIF(0.064, 4.5) = 1 / (1+0.064)^4.5
    PVIF(0.064, 4.5) = 1 / 1.3220178705308
    PVIF(0.064, 4.5) = 0.75641942691628

    Non Zero Bond Price Calculation
    Price = 0.05 x 100 x 3.8059464544331 + 100 x 0.75641942691628
    Price = 19.029732272166 + 75.641942691628
    Price @ 0.064 = 94.67

    YTM Approximation with Linear Interpolation
    ytmL = 1.41%
    ytmU = 6.41%
    pvL = 115.59
    pvU = 94.67

    ytm = ytmL + [(ytmH-ytmL)(pvL-pvYTM)] / [pvL-pvU]
    ytm = 0.014 + [(0.064-0.014)(115.59 - 104.50)] / [115.59-94.67]
    ytm = 0.014 + [(0.05)(11.09)] / [20.92]
    ytm = 0.014 + 0.5545 / 20.92
    ytm = 0.014 + 0.0265
    ytm = 0.0405
    ytm = 4.05%

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  6. #6
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    Quote Originally Posted by Wilmer View Post
    What's a try (trial?) and error bond?
    @Wilmer

    YTM on bonds is akin to internal rate of return at which present value of interest payments and terminal value of bond equals the market price of the bond. We use the Bond Price equation for calculating YTM with Trial and Error method, We select two rates one at which present value of bond is higher than market price of the bond and the second rate at which present value of bond is less than market price of the bond. We then use linear interpolation to arrive at an approximate value of YTM. See the following links

    What is YTM, Yield till maturity it's Definition, Formula, an Example Calculation, Linear Interpolation
    YTM|Yield till Maturity calculation - ThinkandDone.com
    YTM|Yield till Maturity calculator - ThinkandDone.com

    We can also use the RATE formula in Excel to find YTM as follows

    =RATE(NPER, PMT, PV, FV, type)

    here NPER will be number of periods in this case 9 (4.5 x 2)
    PMT is the interest payment if coupon rate was 5% then PMT will be 5
    PV is the market price -104.50
    FV is the terminal value 100
    Type is 0 for ordinary annuity (end of period payments)
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