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Math Help - Modified Duration vs. Effective Duration

  1. #1
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    Modified Duration vs. Effective Duration

    I was always told that Modified Duration = Effective Duration when there are no embedded options, although have not been able to prove this numerically as I get quite different answers using the two formulas. I guess it is as this formula is really just an approximation of the modified duration?

    Effective Duration = (P- - P+ ) / [(2)*(P0)*(Y+ - Y-)]

    How would you explain the difference? Partial Convexity? or just that this is only meant to be an approximation?

    Thanks for any thoughts!


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    Re: Modified Duration vs. Effective Duration

    Aidie

    I shall TRY to answer, but I have not worked with duration for many years.

    But before I even try, please give the formulas for both modified duration and effective duration as well as the definitions of the variables involved in BOTH formulas.

    Abraham may be more help than I am because he seems to have all the formulas down by heart, and I certainly do not.
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    Re: Modified Duration vs. Effective Duration

    Quote Originally Posted by JeffM View Post
    Aidie

    I shall TRY to answer, but I have not worked with duration for many years.

    But before I even try, please give the formulas for both modified duration and effective duration as well as the definitions of the variables involved in BOTH formulas.

    Abraham may be more help than I am because he seems to have all the formulas down by heart, and I certainly do not.
    Modified duration is the ratio of first order derivative of Bond price and bond price

    Modified Duration = -\frac{\frac{dB}{di}}{B}

    It gives the approximate change in bond price if the interest rate were to increase/decrease by 100 BPS or 1%
    Last edited by AbrahamA; June 1st 2014 at 07:30 PM.
    Thanks from JeffM
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    Re: Modified Duration vs. Effective Duration

    Quote Originally Posted by Aidie View Post
    I was always told that Modified Duration = Effective Duration when there are no embedded options, although have not been able to prove this numerically as I get quite different answers using the two formulas. I guess it is as this formula is really just an approximation of the modified duration?

    Effective Duration = (P- - P+ ) / [(2)*(P0)*(Y+ - Y-)]

    How would you explain the difference? Partial Convexity? or just that this is only meant to be an approximation?

    Thanks for any thoughts!
    Duration is measured in years and is the time period at which the discounted cash flows balance the scale.

    It also indicates a roughly estimated change in bond price if the interest rate were to increase/decrease by 100 BPS or 1%

    Code:
    Semi-annual compounding of interest
    Par value = 1000
    YTM = 6%
    Coupon = 4%
    Years to maturity = 10
    
    Modified duration = 8.169
    Thus the price of the bond is expected to decrease by approximately 8.169% percent (actual price at YTM=5% is slighly different)


    t Ct (1+i)^-t Ct(1+i)^-t tCt(1+i)^-t
    1 20 0.97087378640777 19.417475728155 19.417475728155
    2 20 0.94259590913375 18.851918182675 37.70383636535
    3 20 0.91514165935316 18.302833187063 54.90849956119
    4 20 0.88848704791569 17.769740958314 71.078963833255
    5 20 0.86260878438416 17.252175687683 86.260878438416
    6 20 0.83748425668365 16.749685133673 100.49811080204
    7 20 0.81309151134335 16.261830226867 113.83281158807
    8 20 0.78940923431394 15.788184686279 126.30547749023
    9 20 0.76641673234363 15.328334646873 137.95501182185
    10 20 0.74409391489673 14.881878297935 148.81878297935
    11 20 0.72242127659876 14.448425531975 158.93268085173
    12 20 0.70137988019297 14.027597603859 168.33117124631
    13 20 0.68095133999318 13.619026799864 177.04734839823
    14 20 0.66111780581862 13.222356116372 185.11298562921
    15 20 0.64186194739672 12.837238947934 192.55858421902
    16 20 0.62316693922011 12.463338784402 199.41342055044
    17 20 0.60501644584477 12.100328916895 205.70559158722
    18 20 0.58739460761628 11.747892152326 211.46205874186
    19 20 0.57028602681192 11.405720536238 216.70869018853
    20 1020 0.55367575418633 564.74926927006 11294.985385401
    851.22525139544 13907.037765422
    D=Σ(t)Ct(1+i)^-t/P 16.338
    Duration=D/2 8.169

    Effective duration as given by the formula in your post gives the relative change in bond price by a spread of 1%

    Code:
    Semi-annual compounding of interest
    Par value = 1000
    YTM = 6%
    Coupon = 4%
    Years to maturity = 10
    
    Y- = 5.5%/2
    Y+ = 6.5%/2
    
    P0 (6%/2) = 851.23
    P- (Y-) = 885.8
    P+ (Y+) = 818.26
    
    Effective duration = [P- - P+] / [ 2 * P0 (Y+ - Y-)]
    Effective duration = 885.80 - 818.26 / [ 2 * 851.23 (0.0325 - 0.0275)]
    Effective duration = 67.54 / [ 1702.46 (0.005)]
    Effective duration = 67.54 / 8.5123
    Effective duration = 7.9344
    This effective duration of 7.9344 is quite different from modified duration of 8.169

    Data output
    Interest compounded
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    YTM on bond is 0.03
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.03, 20) = 14.877474860456
    PVIF(0.03, 20) = 0.55367575418633
    Price = 0.02 x 1000 x 14.877474860456 + 1000 x 0.55367575418633
    Price = 297.54949720911 + 553.67575418633
    Price = 851.23

    Data output
    Interest compounded
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    YTM on bond is 0.0275
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.0275, 20) = 15.227252133776
    PVIF(0.0275, 20) = 0.58125056632117
    Price = 0.02 x 1000 x 15.227252133776 + 1000 x 0.58125056632117
    Price = 304.54504267551 + 581.25056632117
    Price = 885.8

    Data output
    Interest compounded
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    YTM on bond is 0.0325
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.0325, 20) = 14.539346146934
    PVIF(0.0325, 20) = 0.52747125022464
    Price = 0.02 x 1000 x 14.539346146934 + 1000 x 0.52747125022464
    Price = 290.78692293868 + 527.47125022464
    Price = 818.26

    The toolkit

    Modified Duration
    Calculate bond duration
    http://finance.thinkanddone.com/tadBDuration.exe

    Bond price
    Non zero bond price calculation
    http://finance.thinkanddone.com/tadBPrice.exe
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  5. #5
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    Re: Modified Duration vs. Effective Duration

    @mods

    I need a correction made to my last post, and if possible to delete this one when the correction is made

    The following text in my last reply contains error

    Thus the price of the bond is expected to decrease by approximately 8.169% percent (actual price at YTM=5% is slighly different)
    Thus it needs to be replaced with this text

    Thus the price of the bond is expected to decrease by approximately 8.169% percent (actual price at YTM=7% is slightly different)
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  6. #6
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    Re: Modified Duration vs. Effective Duration

    Quote Originally Posted by Aidie View Post
    I was always told that Modified Duration = Effective Duration when there are no embedded options, although have not been able to prove this numerically as I get quite different answers using the two formulas. I guess it is as this formula is really just an approximation of the modified duration?

    Effective Duration = (P- - P+ ) / [(2)*(P0)*(Y+ - Y-)]

    How would you explain the difference? Partial Convexity? or just that this is only meant to be an approximation?

    Thanks for any thoughts!
    Duration is a very useful tool that helps a bond analyst to help immunize the bond from sudden changes in interest rates.

    The yield from bond may change from its initial value thus the bond holder would want to find her actual rate of return which happens to be the YTH - yield to horizon.

    YTH - yield to horizon is a better indicator of a return as compared to YTM - yield to maturity. But finding YTH is no easy task as this will entail forecasting interest rates at future dates.

    Having said that let me show you how using duration, the bond owner can immunize her investment in fixed income security by making use of duration as the horizon.

    But first let us calculate the YTH - yield to horizon for the same bond that was issued at 4% semi-annual coupon at par of 1000 dollars with 10 years to maturity. The initial yield on the bond was 6% but after the purchase of the bond the very next day, the interest rate jumped to 7%.

    Now if she wanted to hold on to the investment for 5 years, she would like to know her YTH - yield to horizon

    Her YTH turns out to be 5.38% if she decided to hold on to the investment for 5 years

    Code:
    Semi-annual compounding of interest
    Par value = 1000
    YTM0 = 6%
    YTM1 = 7%
    Coupon = 4%
    Years to maturity = 10
    Horizon = 5
    
    Bond price before change in YTM
    
    Interest compounded semi annually
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    Initial YTM on bond is 0.03
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.03, 20) = 14.877474860456
    PVIF(0.03, 20) = 0.55367575418633
    Price = 0.02 x 1000 x 14.877474860456 + 1000 x 0.55367575418633
    Price = 297.54949720911 + 553.67575418633
    Price = 851.23
    
    Bond price after change in YTM
    
    Interest compounded semi annually
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    Initial YTM on bond is 0.035
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.035, 20) = 14.212403301952
    PVIF(0.035, 20) = 0.50256588443167
    Price = 0.02 x 1000 x 14.212403301952 + 1000 x 0.50256588443167
    Price = 284.24806603905 + 502.56588443167
    Price = 786.81
    
    Horizon rate of return
    
    Bond price using initial YTM = 851.23
    Bond price after change in YTM = 786.81
    Change in YTM = 0.035
    Bond horizon = 10
    horizon rate = (786.81/851.23)^(1/10) * (1+0.035) - 1
    horizon rate = (0.9243310735682)^(0.1) * 1.035 - 1
    horizon rate = 0.99216237890656 * 1.035 - 1
    horizon rate = 1.0268880621683 - 1
    
    horizon rate = 2.69%
    Annual horizon rate = 5.38%
    Now she remembers from her college financial management course, that her YTH - yield to horizon will turn out to be the same as initial YTM if she were to use Duration as her Horizon.

    Thus she would be able to immunize the bond from any change in interest rates. It turns out that indeed her YTH - yield to horizon is 6%, the same as initial YTM when her horizon is the same as Duration of the bond

    The rates have jumped to 7% from 6% but YTH has remained at 6%

    Code:
    Semi-annual compounding of interest
    Par value = 1000
    YTM0 = 6%
    YTM1 = 7%
    Coupon = 4%
    Years to maturity = 10
    Horizon = Duration = 8.1688353010109609266845229091446
    
    Bond price before change in YTM
    
    Interest compounded semi annually
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    Initial YTM on bond is 0.03
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.03, 20) = 14.877474860456
    PVIF(0.03, 20) = 0.55367575418633
    Price = 0.02 x 1000 x 14.877474860456 + 1000 x 0.55367575418633
    Price = 297.54949720911 + 553.67575418633
    Price = 851.23
    
    Bond price after change in YTM
    
    Interest compounded semi annually
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    Initial YTM on bond is 0.035
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.035, 20) = 14.212403301952
    PVIF(0.035, 20) = 0.50256588443167
    Price = 0.02 x 1000 x 14.212403301952 + 1000 x 0.50256588443167
    Price = 284.24806603905 + 502.56588443167
    Price = 786.81
    
    Horizon rate of return
    
    Bond price using initial YTM = 851.23
    Bond price after change in YTM = 786.81
    Change in YTM = 0.035
    Bond horizon = 16.337670602022
    horizon rate = (786.81/851.23)^(1/16.337670602022) * (1+0.035) - 1
    horizon rate = (0.9243310735682)^(0.06120823612861) * 1.035 - 1
    horizon rate = 0.99519541111377 * 1.035 - 1
    horizon rate = 1.0300272505027 - 1
    
    horizon rate = 3%
    Annual horizon rate = 6.01%
    Now the rates have dropped to 5% from 6% but here YTH is the still 6% or the same as initial YTM

    Code:
    Semi-annual compounding of interest
    Par value = 1000
    YTM0 = 6%
    YTM1 = 5%
    Coupon = 4%
    Years to maturity = 10
    Horizon = Duration = 8.1688353010109609266845229091446
    
    Bond price before change in YTM
    
    Interest compounded semi annually
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    Initial YTM on bond is 0.03
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.03, 20) = 14.877474860456
    PVIF(0.03, 20) = 0.55367575418633
    Price = 0.02 x 1000 x 14.877474860456 + 1000 x 0.55367575418633
    Price = 297.54949720911 + 553.67575418633
    Price = 851.23
    
    Bond price after change in YTM
    
    Interest compounded semi annually
    Par value of bond is 1000
    Coupon rate on bond is 0.02
    Initial YTM on bond is 0.025
    Years till maturity are 20
    Price = coupon rate x par value x PVIFA(ytm%, n) + par value x PVIF(ytm%, n)
    PVIFA(0.025, 20) = 15.589162285647
    PVIF(0.025, 20) = 0.61027094285883
    Price = 0.02 x 1000 x 15.589162285647 + 1000 x 0.61027094285883
    Price = 311.78324571294 + 610.27094285883
    Price = 922.05
    
    Horizon rate of return
    
    Bond price using initial YTM = 851.23
    Bond price after change in YTM = 922.05
    Change in YTM = 0.025
    Bond horizon = 16.337670602022
    horizon rate = (922.05/851.23)^(1/16.337670602022) * (1+0.025) - 1
    horizon rate = (1.0832082190468)^(0.06120823612861) * 1.025 - 1
    horizon rate = 1.0049041899641 * 1.025 - 1
    horizon rate = 1.0300267947132 - 1
    
    horizon rate = 3%
    Annual horizon rate = 6.01%
    The toolkit

    Yield to Horizon - YTH
    Bond horizon rate of return calculation

    YTH calculator
    http://finance.thinkanddone.com/tadBHRR.exe
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